Properties

Label 4096.2.a.s.1.7
Level $4096$
Weight $2$
Character 4096.1
Self dual yes
Analytic conductor $32.707$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4096,2,Mod(1,4096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4096, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4096 = 2^{12} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4096.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: \(32.7067246679\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{48})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1024)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.98289\) of defining polynomial
Character \(\chi\) \(=\) 4096.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.93185 q^{3} -0.396183 q^{5} +4.33496 q^{7} +5.59575 q^{9} +O(q^{10})\) \(q+2.93185 q^{3} -0.396183 q^{5} +4.33496 q^{7} +5.59575 q^{9} +3.96713 q^{11} -2.03886 q^{13} -1.16155 q^{15} +4.54587 q^{17} +0.578738 q^{19} +12.7095 q^{21} -6.24790 q^{23} -4.84304 q^{25} +7.61037 q^{27} +5.13851 q^{29} -0.0539984 q^{31} +11.6310 q^{33} -1.71744 q^{35} -0.864289 q^{37} -5.97764 q^{39} -0.878315 q^{41} +2.23972 q^{43} -2.21694 q^{45} -9.44387 q^{47} +11.7919 q^{49} +13.3278 q^{51} +10.8869 q^{53} -1.57171 q^{55} +1.69677 q^{57} -7.77025 q^{59} -10.5307 q^{61} +24.2574 q^{63} +0.807763 q^{65} +8.15988 q^{67} -18.3179 q^{69} -4.23065 q^{71} -4.12560 q^{73} -14.1991 q^{75} +17.1974 q^{77} -5.74836 q^{79} +5.52520 q^{81} -3.57749 q^{83} -1.80100 q^{85} +15.0654 q^{87} -3.36773 q^{89} -8.83839 q^{91} -0.158315 q^{93} -0.229286 q^{95} +13.2672 q^{97} +22.1991 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{11} + 8 q^{17} - 8 q^{25} + 8 q^{27} + 40 q^{33} - 8 q^{35} + 32 q^{41} + 56 q^{43} + 8 q^{49} + 48 q^{51} + 8 q^{57} + 32 q^{59} - 16 q^{65} + 24 q^{67} - 8 q^{73} + 16 q^{81} + 48 q^{83} - 8 q^{89} + 8 q^{91} + 8 q^{97} + 64 q^{99}+O(q^{100}) \) 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
\(3\) 2.93185 1.69271 0.846353 0.532623i \(-0.178794\pi\)
0.846353 + 0.532623i \(0.178794\pi\)
\(4\) 0 0
\(5\) −0.396183 −0.177178 −0.0885892 0.996068i \(-0.528236\pi\)
−0.0885892 + 0.996068i \(0.528236\pi\)
\(6\) 0 0
\(7\) 4.33496 1.63846 0.819231 0.573464i \(-0.194401\pi\)
0.819231 + 0.573464i \(0.194401\pi\)
\(8\) 0 0
\(9\) 5.59575 1.86525
\(10\) 0 0
\(11\) 3.96713 1.19613 0.598067 0.801446i \(-0.295936\pi\)
0.598067 + 0.801446i \(0.295936\pi\)
\(12\) 0 0
\(13\) −2.03886 −0.565479 −0.282739 0.959197i \(-0.591243\pi\)
−0.282739 + 0.959197i \(0.591243\pi\)
\(14\) 0 0
\(15\) −1.16155 −0.299911
\(16\) 0 0
\(17\) 4.54587 1.10253 0.551267 0.834329i \(-0.314144\pi\)
0.551267 + 0.834329i \(0.314144\pi\)
\(18\) 0 0
\(19\) 0.578738 0.132772 0.0663858 0.997794i \(-0.478853\pi\)
0.0663858 + 0.997794i \(0.478853\pi\)
\(20\) 0 0
\(21\) 12.7095 2.77343
\(22\) 0 0
\(23\) −6.24790 −1.30278 −0.651389 0.758744i \(-0.725813\pi\)
−0.651389 + 0.758744i \(0.725813\pi\)
\(24\) 0 0
\(25\) −4.84304 −0.968608
\(26\) 0 0
\(27\) 7.61037 1.46462
\(28\) 0 0
\(29\) 5.13851 0.954198 0.477099 0.878850i \(-0.341688\pi\)
0.477099 + 0.878850i \(0.341688\pi\)
\(30\) 0 0
\(31\) −0.0539984 −0.00969841 −0.00484920 0.999988i \(-0.501544\pi\)
−0.00484920 + 0.999988i \(0.501544\pi\)
\(32\) 0 0
\(33\) 11.6310 2.02470
\(34\) 0 0
\(35\) −1.71744 −0.290300
\(36\) 0 0
\(37\) −0.864289 −0.142088 −0.0710441 0.997473i \(-0.522633\pi\)
−0.0710441 + 0.997473i \(0.522633\pi\)
\(38\) 0 0
\(39\) −5.97764 −0.957189
\(40\) 0 0
\(41\) −0.878315 −0.137170 −0.0685849 0.997645i \(-0.521848\pi\)
−0.0685849 + 0.997645i \(0.521848\pi\)
\(42\) 0 0
\(43\) 2.23972 0.341554 0.170777 0.985310i \(-0.445372\pi\)
0.170777 + 0.985310i \(0.445372\pi\)
\(44\) 0 0
\(45\) −2.21694 −0.330482
\(46\) 0 0
\(47\) −9.44387 −1.37753 −0.688765 0.724984i \(-0.741847\pi\)
−0.688765 + 0.724984i \(0.741847\pi\)
\(48\) 0 0
\(49\) 11.7919 1.68456
\(50\) 0 0
\(51\) 13.3278 1.86627
\(52\) 0 0
\(53\) 10.8869 1.49543 0.747713 0.664022i \(-0.231152\pi\)
0.747713 + 0.664022i \(0.231152\pi\)
\(54\) 0 0
\(55\) −1.57171 −0.211929
\(56\) 0 0
\(57\) 1.69677 0.224743
\(58\) 0 0
\(59\) −7.77025 −1.01160 −0.505800 0.862651i \(-0.668803\pi\)
−0.505800 + 0.862651i \(0.668803\pi\)
\(60\) 0 0
\(61\) −10.5307 −1.34832 −0.674160 0.738586i \(-0.735494\pi\)
−0.674160 + 0.738586i \(0.735494\pi\)
\(62\) 0 0
\(63\) 24.2574 3.05614
\(64\) 0 0
\(65\) 0.807763 0.100191
\(66\) 0 0
\(67\) 8.15988 0.996888 0.498444 0.866922i \(-0.333905\pi\)
0.498444 + 0.866922i \(0.333905\pi\)
\(68\) 0 0
\(69\) −18.3179 −2.20522
\(70\) 0 0
\(71\) −4.23065 −0.502085 −0.251043 0.967976i \(-0.580773\pi\)
−0.251043 + 0.967976i \(0.580773\pi\)
\(72\) 0 0
\(73\) −4.12560 −0.482865 −0.241432 0.970418i \(-0.577617\pi\)
−0.241432 + 0.970418i \(0.577617\pi\)
\(74\) 0 0
\(75\) −14.1991 −1.63957
\(76\) 0 0
\(77\) 17.1974 1.95982
\(78\) 0 0
\(79\) −5.74836 −0.646741 −0.323370 0.946272i \(-0.604816\pi\)
−0.323370 + 0.946272i \(0.604816\pi\)
\(80\) 0 0
\(81\) 5.52520 0.613911
\(82\) 0 0
\(83\) −3.57749 −0.392681 −0.196340 0.980536i \(-0.562906\pi\)
−0.196340 + 0.980536i \(0.562906\pi\)
\(84\) 0 0
\(85\) −1.80100 −0.195345
\(86\) 0 0
\(87\) 15.0654 1.61518
\(88\) 0 0
\(89\) −3.36773 −0.356978 −0.178489 0.983942i \(-0.557121\pi\)
−0.178489 + 0.983942i \(0.557121\pi\)
\(90\) 0 0
\(91\) −8.83839 −0.926516
\(92\) 0 0
\(93\) −0.158315 −0.0164165
\(94\) 0 0
\(95\) −0.229286 −0.0235243
\(96\) 0 0
\(97\) 13.2672 1.34708 0.673541 0.739150i \(-0.264772\pi\)
0.673541 + 0.739150i \(0.264772\pi\)
\(98\) 0 0
\(99\) 22.1991 2.23109
\(100\) 0 0
\(101\) −14.7865 −1.47131 −0.735657 0.677355i \(-0.763126\pi\)
−0.735657 + 0.677355i \(0.763126\pi\)
\(102\) 0 0
\(103\) 6.60406 0.650718 0.325359 0.945591i \(-0.394515\pi\)
0.325359 + 0.945591i \(0.394515\pi\)
\(104\) 0 0
\(105\) −5.03528 −0.491393
\(106\) 0 0
\(107\) −8.25966 −0.798491 −0.399246 0.916844i \(-0.630728\pi\)
−0.399246 + 0.916844i \(0.630728\pi\)
\(108\) 0 0
\(109\) −2.66528 −0.255288 −0.127644 0.991820i \(-0.540741\pi\)
−0.127644 + 0.991820i \(0.540741\pi\)
\(110\) 0 0
\(111\) −2.53397 −0.240514
\(112\) 0 0
\(113\) 4.96472 0.467042 0.233521 0.972352i \(-0.424975\pi\)
0.233521 + 0.972352i \(0.424975\pi\)
\(114\) 0 0
\(115\) 2.47531 0.230824
\(116\) 0 0
\(117\) −11.4090 −1.05476
\(118\) 0 0
\(119\) 19.7062 1.80646
\(120\) 0 0
\(121\) 4.73810 0.430737
\(122\) 0 0
\(123\) −2.57509 −0.232188
\(124\) 0 0
\(125\) 3.89965 0.348795
\(126\) 0 0
\(127\) −15.4530 −1.37123 −0.685614 0.727965i \(-0.740466\pi\)
−0.685614 + 0.727965i \(0.740466\pi\)
\(128\) 0 0
\(129\) 6.56653 0.578151
\(130\) 0 0
\(131\) 10.4617 0.914043 0.457021 0.889456i \(-0.348916\pi\)
0.457021 + 0.889456i \(0.348916\pi\)
\(132\) 0 0
\(133\) 2.50881 0.217541
\(134\) 0 0
\(135\) −3.01510 −0.259498
\(136\) 0 0
\(137\) −16.2474 −1.38811 −0.694057 0.719920i \(-0.744178\pi\)
−0.694057 + 0.719920i \(0.744178\pi\)
\(138\) 0 0
\(139\) 7.49697 0.635885 0.317943 0.948110i \(-0.397008\pi\)
0.317943 + 0.948110i \(0.397008\pi\)
\(140\) 0 0
\(141\) −27.6880 −2.33175
\(142\) 0 0
\(143\) −8.08843 −0.676388
\(144\) 0 0
\(145\) −2.03579 −0.169063
\(146\) 0 0
\(147\) 34.5721 2.85146
\(148\) 0 0
\(149\) −6.33025 −0.518594 −0.259297 0.965798i \(-0.583491\pi\)
−0.259297 + 0.965798i \(0.583491\pi\)
\(150\) 0 0
\(151\) −10.7097 −0.871546 −0.435773 0.900057i \(-0.643525\pi\)
−0.435773 + 0.900057i \(0.643525\pi\)
\(152\) 0 0
\(153\) 25.4375 2.05650
\(154\) 0 0
\(155\) 0.0213933 0.00171835
\(156\) 0 0
\(157\) −8.15361 −0.650729 −0.325364 0.945589i \(-0.605487\pi\)
−0.325364 + 0.945589i \(0.605487\pi\)
\(158\) 0 0
\(159\) 31.9187 2.53132
\(160\) 0 0
\(161\) −27.0844 −2.13455
\(162\) 0 0
\(163\) −5.95983 −0.466810 −0.233405 0.972380i \(-0.574987\pi\)
−0.233405 + 0.972380i \(0.574987\pi\)
\(164\) 0 0
\(165\) −4.60802 −0.358734
\(166\) 0 0
\(167\) −20.3851 −1.57745 −0.788725 0.614747i \(-0.789258\pi\)
−0.788725 + 0.614747i \(0.789258\pi\)
\(168\) 0 0
\(169\) −8.84304 −0.680234
\(170\) 0 0
\(171\) 3.23848 0.247653
\(172\) 0 0
\(173\) 17.3354 1.31799 0.658993 0.752149i \(-0.270982\pi\)
0.658993 + 0.752149i \(0.270982\pi\)
\(174\) 0 0
\(175\) −20.9944 −1.58703
\(176\) 0 0
\(177\) −22.7812 −1.71234
\(178\) 0 0
\(179\) 9.40717 0.703125 0.351562 0.936164i \(-0.385651\pi\)
0.351562 + 0.936164i \(0.385651\pi\)
\(180\) 0 0
\(181\) 0.721790 0.0536502 0.0268251 0.999640i \(-0.491460\pi\)
0.0268251 + 0.999640i \(0.491460\pi\)
\(182\) 0 0
\(183\) −30.8745 −2.28231
\(184\) 0 0
\(185\) 0.342417 0.0251750
\(186\) 0 0
\(187\) 18.0340 1.31878
\(188\) 0 0
\(189\) 32.9907 2.39972
\(190\) 0 0
\(191\) 10.1746 0.736209 0.368105 0.929784i \(-0.380007\pi\)
0.368105 + 0.929784i \(0.380007\pi\)
\(192\) 0 0
\(193\) 11.7515 0.845889 0.422945 0.906156i \(-0.360996\pi\)
0.422945 + 0.906156i \(0.360996\pi\)
\(194\) 0 0
\(195\) 2.36824 0.169593
\(196\) 0 0
\(197\) 15.7016 1.11869 0.559347 0.828934i \(-0.311052\pi\)
0.559347 + 0.828934i \(0.311052\pi\)
\(198\) 0 0
\(199\) 3.29075 0.233275 0.116638 0.993175i \(-0.462788\pi\)
0.116638 + 0.993175i \(0.462788\pi\)
\(200\) 0 0
\(201\) 23.9236 1.68744
\(202\) 0 0
\(203\) 22.2753 1.56342
\(204\) 0 0
\(205\) 0.347974 0.0243035
\(206\) 0 0
\(207\) −34.9617 −2.43001
\(208\) 0 0
\(209\) 2.29593 0.158813
\(210\) 0 0
\(211\) 18.2988 1.25974 0.629872 0.776699i \(-0.283107\pi\)
0.629872 + 0.776699i \(0.283107\pi\)
\(212\) 0 0
\(213\) −12.4036 −0.849883
\(214\) 0 0
\(215\) −0.887340 −0.0605161
\(216\) 0 0
\(217\) −0.234081 −0.0158905
\(218\) 0 0
\(219\) −12.0956 −0.817348
\(220\) 0 0
\(221\) −9.26840 −0.623460
\(222\) 0 0
\(223\) 21.6471 1.44959 0.724797 0.688962i \(-0.241934\pi\)
0.724797 + 0.688962i \(0.241934\pi\)
\(224\) 0 0
\(225\) −27.1005 −1.80670
\(226\) 0 0
\(227\) 9.59476 0.636826 0.318413 0.947952i \(-0.396850\pi\)
0.318413 + 0.947952i \(0.396850\pi\)
\(228\) 0 0
\(229\) 28.7686 1.90108 0.950541 0.310599i \(-0.100530\pi\)
0.950541 + 0.310599i \(0.100530\pi\)
\(230\) 0 0
\(231\) 50.4201 3.31740
\(232\) 0 0
\(233\) 25.5580 1.67436 0.837180 0.546928i \(-0.184203\pi\)
0.837180 + 0.546928i \(0.184203\pi\)
\(234\) 0 0
\(235\) 3.74150 0.244069
\(236\) 0 0
\(237\) −16.8533 −1.09474
\(238\) 0 0
\(239\) 24.0765 1.55738 0.778690 0.627409i \(-0.215884\pi\)
0.778690 + 0.627409i \(0.215884\pi\)
\(240\) 0 0
\(241\) −10.3823 −0.668785 −0.334393 0.942434i \(-0.608531\pi\)
−0.334393 + 0.942434i \(0.608531\pi\)
\(242\) 0 0
\(243\) −6.63203 −0.425445
\(244\) 0 0
\(245\) −4.67175 −0.298467
\(246\) 0 0
\(247\) −1.17997 −0.0750796
\(248\) 0 0
\(249\) −10.4887 −0.664693
\(250\) 0 0
\(251\) 24.9345 1.57385 0.786927 0.617047i \(-0.211671\pi\)
0.786927 + 0.617047i \(0.211671\pi\)
\(252\) 0 0
\(253\) −24.7862 −1.55830
\(254\) 0 0
\(255\) −5.28025 −0.330662
\(256\) 0 0
\(257\) −18.7121 −1.16723 −0.583613 0.812032i \(-0.698362\pi\)
−0.583613 + 0.812032i \(0.698362\pi\)
\(258\) 0 0
\(259\) −3.74666 −0.232806
\(260\) 0 0
\(261\) 28.7539 1.77982
\(262\) 0 0
\(263\) 1.25113 0.0771479 0.0385740 0.999256i \(-0.487718\pi\)
0.0385740 + 0.999256i \(0.487718\pi\)
\(264\) 0 0
\(265\) −4.31319 −0.264957
\(266\) 0 0
\(267\) −9.87367 −0.604259
\(268\) 0 0
\(269\) 14.2249 0.867306 0.433653 0.901080i \(-0.357224\pi\)
0.433653 + 0.901080i \(0.357224\pi\)
\(270\) 0 0
\(271\) 17.5598 1.06668 0.533341 0.845900i \(-0.320936\pi\)
0.533341 + 0.845900i \(0.320936\pi\)
\(272\) 0 0
\(273\) −25.9129 −1.56832
\(274\) 0 0
\(275\) −19.2130 −1.15858
\(276\) 0 0
\(277\) −26.3830 −1.58520 −0.792602 0.609740i \(-0.791274\pi\)
−0.792602 + 0.609740i \(0.791274\pi\)
\(278\) 0 0
\(279\) −0.302162 −0.0180900
\(280\) 0 0
\(281\) −24.0082 −1.43221 −0.716105 0.697993i \(-0.754077\pi\)
−0.716105 + 0.697993i \(0.754077\pi\)
\(282\) 0 0
\(283\) 3.52104 0.209304 0.104652 0.994509i \(-0.466627\pi\)
0.104652 + 0.994509i \(0.466627\pi\)
\(284\) 0 0
\(285\) −0.672233 −0.0398197
\(286\) 0 0
\(287\) −3.80746 −0.224747
\(288\) 0 0
\(289\) 3.66490 0.215582
\(290\) 0 0
\(291\) 38.8975 2.28021
\(292\) 0 0
\(293\) −3.12386 −0.182498 −0.0912488 0.995828i \(-0.529086\pi\)
−0.0912488 + 0.995828i \(0.529086\pi\)
\(294\) 0 0
\(295\) 3.07844 0.179234
\(296\) 0 0
\(297\) 30.1913 1.75188
\(298\) 0 0
\(299\) 12.7386 0.736693
\(300\) 0 0
\(301\) 9.70911 0.559624
\(302\) 0 0
\(303\) −43.3519 −2.49050
\(304\) 0 0
\(305\) 4.17209 0.238893
\(306\) 0 0
\(307\) −8.52709 −0.486667 −0.243333 0.969943i \(-0.578241\pi\)
−0.243333 + 0.969943i \(0.578241\pi\)
\(308\) 0 0
\(309\) 19.3621 1.10147
\(310\) 0 0
\(311\) 4.78833 0.271521 0.135761 0.990742i \(-0.456652\pi\)
0.135761 + 0.990742i \(0.456652\pi\)
\(312\) 0 0
\(313\) −29.7972 −1.68424 −0.842120 0.539291i \(-0.818692\pi\)
−0.842120 + 0.539291i \(0.818692\pi\)
\(314\) 0 0
\(315\) −9.61037 −0.541483
\(316\) 0 0
\(317\) 28.1859 1.58308 0.791540 0.611117i \(-0.209280\pi\)
0.791540 + 0.611117i \(0.209280\pi\)
\(318\) 0 0
\(319\) 20.3851 1.14135
\(320\) 0 0
\(321\) −24.2161 −1.35161
\(322\) 0 0
\(323\) 2.63087 0.146385
\(324\) 0 0
\(325\) 9.87429 0.547727
\(326\) 0 0
\(327\) −7.81422 −0.432127
\(328\) 0 0
\(329\) −40.9389 −2.25703
\(330\) 0 0
\(331\) −31.3733 −1.72443 −0.862217 0.506540i \(-0.830924\pi\)
−0.862217 + 0.506540i \(0.830924\pi\)
\(332\) 0 0
\(333\) −4.83635 −0.265030
\(334\) 0 0
\(335\) −3.23281 −0.176627
\(336\) 0 0
\(337\) −35.6957 −1.94447 −0.972234 0.234010i \(-0.924815\pi\)
−0.972234 + 0.234010i \(0.924815\pi\)
\(338\) 0 0
\(339\) 14.5558 0.790564
\(340\) 0 0
\(341\) −0.214219 −0.0116006
\(342\) 0 0
\(343\) 20.7727 1.12162
\(344\) 0 0
\(345\) 7.25725 0.390717
\(346\) 0 0
\(347\) 28.2555 1.51684 0.758418 0.651768i \(-0.225973\pi\)
0.758418 + 0.651768i \(0.225973\pi\)
\(348\) 0 0
\(349\) 13.0593 0.699047 0.349524 0.936928i \(-0.386343\pi\)
0.349524 + 0.936928i \(0.386343\pi\)
\(350\) 0 0
\(351\) −15.5165 −0.828209
\(352\) 0 0
\(353\) −16.4488 −0.875479 −0.437740 0.899102i \(-0.644221\pi\)
−0.437740 + 0.899102i \(0.644221\pi\)
\(354\) 0 0
\(355\) 1.67611 0.0889587
\(356\) 0 0
\(357\) 57.7755 3.05781
\(358\) 0 0
\(359\) −5.54907 −0.292869 −0.146434 0.989220i \(-0.546780\pi\)
−0.146434 + 0.989220i \(0.546780\pi\)
\(360\) 0 0
\(361\) −18.6651 −0.982372
\(362\) 0 0
\(363\) 13.8914 0.729110
\(364\) 0 0
\(365\) 1.63449 0.0855533
\(366\) 0 0
\(367\) 18.9285 0.988061 0.494031 0.869444i \(-0.335523\pi\)
0.494031 + 0.869444i \(0.335523\pi\)
\(368\) 0 0
\(369\) −4.91484 −0.255856
\(370\) 0 0
\(371\) 47.1942 2.45020
\(372\) 0 0
\(373\) −15.4739 −0.801206 −0.400603 0.916252i \(-0.631199\pi\)
−0.400603 + 0.916252i \(0.631199\pi\)
\(374\) 0 0
\(375\) 11.4332 0.590407
\(376\) 0 0
\(377\) −10.4767 −0.539579
\(378\) 0 0
\(379\) −21.5269 −1.10576 −0.552881 0.833260i \(-0.686471\pi\)
−0.552881 + 0.833260i \(0.686471\pi\)
\(380\) 0 0
\(381\) −45.3058 −2.32109
\(382\) 0 0
\(383\) 14.8953 0.761113 0.380556 0.924758i \(-0.375732\pi\)
0.380556 + 0.924758i \(0.375732\pi\)
\(384\) 0 0
\(385\) −6.81330 −0.347238
\(386\) 0 0
\(387\) 12.5329 0.637085
\(388\) 0 0
\(389\) 19.9048 1.00921 0.504606 0.863350i \(-0.331638\pi\)
0.504606 + 0.863350i \(0.331638\pi\)
\(390\) 0 0
\(391\) −28.4021 −1.43636
\(392\) 0 0
\(393\) 30.6721 1.54721
\(394\) 0 0
\(395\) 2.27740 0.114589
\(396\) 0 0
\(397\) 16.2817 0.817153 0.408577 0.912724i \(-0.366025\pi\)
0.408577 + 0.912724i \(0.366025\pi\)
\(398\) 0 0
\(399\) 7.35546 0.368233
\(400\) 0 0
\(401\) 26.1297 1.30485 0.652427 0.757852i \(-0.273751\pi\)
0.652427 + 0.757852i \(0.273751\pi\)
\(402\) 0 0
\(403\) 0.110095 0.00548424
\(404\) 0 0
\(405\) −2.18899 −0.108772
\(406\) 0 0
\(407\) −3.42875 −0.169957
\(408\) 0 0
\(409\) −21.8489 −1.08036 −0.540179 0.841550i \(-0.681643\pi\)
−0.540179 + 0.841550i \(0.681643\pi\)
\(410\) 0 0
\(411\) −47.6351 −2.34967
\(412\) 0 0
\(413\) −33.6837 −1.65747
\(414\) 0 0
\(415\) 1.41734 0.0695746
\(416\) 0 0
\(417\) 21.9800 1.07637
\(418\) 0 0
\(419\) −11.1397 −0.544212 −0.272106 0.962267i \(-0.587720\pi\)
−0.272106 + 0.962267i \(0.587720\pi\)
\(420\) 0 0
\(421\) −26.2496 −1.27932 −0.639662 0.768656i \(-0.720926\pi\)
−0.639662 + 0.768656i \(0.720926\pi\)
\(422\) 0 0
\(423\) −52.8456 −2.56944
\(424\) 0 0
\(425\) −22.0158 −1.06792
\(426\) 0 0
\(427\) −45.6502 −2.20917
\(428\) 0 0
\(429\) −23.7141 −1.14493
\(430\) 0 0
\(431\) 11.4592 0.551972 0.275986 0.961162i \(-0.410996\pi\)
0.275986 + 0.961162i \(0.410996\pi\)
\(432\) 0 0
\(433\) −15.0019 −0.720946 −0.360473 0.932770i \(-0.617385\pi\)
−0.360473 + 0.932770i \(0.617385\pi\)
\(434\) 0 0
\(435\) −5.96864 −0.286174
\(436\) 0 0
\(437\) −3.61590 −0.172972
\(438\) 0 0
\(439\) −19.7287 −0.941600 −0.470800 0.882240i \(-0.656035\pi\)
−0.470800 + 0.882240i \(0.656035\pi\)
\(440\) 0 0
\(441\) 65.9846 3.14212
\(442\) 0 0
\(443\) −11.7035 −0.556049 −0.278024 0.960574i \(-0.589680\pi\)
−0.278024 + 0.960574i \(0.589680\pi\)
\(444\) 0 0
\(445\) 1.33424 0.0632488
\(446\) 0 0
\(447\) −18.5594 −0.877827
\(448\) 0 0
\(449\) 4.74061 0.223723 0.111862 0.993724i \(-0.464319\pi\)
0.111862 + 0.993724i \(0.464319\pi\)
\(450\) 0 0
\(451\) −3.48439 −0.164073
\(452\) 0 0
\(453\) −31.3994 −1.47527
\(454\) 0 0
\(455\) 3.50162 0.164159
\(456\) 0 0
\(457\) 21.4833 1.00495 0.502473 0.864593i \(-0.332423\pi\)
0.502473 + 0.864593i \(0.332423\pi\)
\(458\) 0 0
\(459\) 34.5957 1.61479
\(460\) 0 0
\(461\) 16.1181 0.750697 0.375348 0.926884i \(-0.377523\pi\)
0.375348 + 0.926884i \(0.377523\pi\)
\(462\) 0 0
\(463\) −11.4145 −0.530477 −0.265238 0.964183i \(-0.585451\pi\)
−0.265238 + 0.964183i \(0.585451\pi\)
\(464\) 0 0
\(465\) 0.0627219 0.00290866
\(466\) 0 0
\(467\) −6.91459 −0.319969 −0.159984 0.987120i \(-0.551144\pi\)
−0.159984 + 0.987120i \(0.551144\pi\)
\(468\) 0 0
\(469\) 35.3728 1.63336
\(470\) 0 0
\(471\) −23.9052 −1.10149
\(472\) 0 0
\(473\) 8.88526 0.408545
\(474\) 0 0
\(475\) −2.80285 −0.128604
\(476\) 0 0
\(477\) 60.9202 2.78935
\(478\) 0 0
\(479\) −16.2733 −0.743545 −0.371772 0.928324i \(-0.621250\pi\)
−0.371772 + 0.928324i \(0.621250\pi\)
\(480\) 0 0
\(481\) 1.76217 0.0803479
\(482\) 0 0
\(483\) −79.4075 −3.61317
\(484\) 0 0
\(485\) −5.25625 −0.238674
\(486\) 0 0
\(487\) −18.4658 −0.836766 −0.418383 0.908271i \(-0.637403\pi\)
−0.418383 + 0.908271i \(0.637403\pi\)
\(488\) 0 0
\(489\) −17.4733 −0.790172
\(490\) 0 0
\(491\) −14.3350 −0.646930 −0.323465 0.946240i \(-0.604848\pi\)
−0.323465 + 0.946240i \(0.604848\pi\)
\(492\) 0 0
\(493\) 23.3590 1.05204
\(494\) 0 0
\(495\) −8.79490 −0.395301
\(496\) 0 0
\(497\) −18.3397 −0.822648
\(498\) 0 0
\(499\) −24.6551 −1.10372 −0.551858 0.833938i \(-0.686081\pi\)
−0.551858 + 0.833938i \(0.686081\pi\)
\(500\) 0 0
\(501\) −59.7662 −2.67016
\(502\) 0 0
\(503\) 41.0356 1.82969 0.914844 0.403808i \(-0.132314\pi\)
0.914844 + 0.403808i \(0.132314\pi\)
\(504\) 0 0
\(505\) 5.85817 0.260685
\(506\) 0 0
\(507\) −25.9265 −1.15144
\(508\) 0 0
\(509\) −0.260792 −0.0115594 −0.00577971 0.999983i \(-0.501840\pi\)
−0.00577971 + 0.999983i \(0.501840\pi\)
\(510\) 0 0
\(511\) −17.8843 −0.791156
\(512\) 0 0
\(513\) 4.40441 0.194459
\(514\) 0 0
\(515\) −2.61642 −0.115293
\(516\) 0 0
\(517\) −37.4651 −1.64771
\(518\) 0 0
\(519\) 50.8249 2.23096
\(520\) 0 0
\(521\) 9.32605 0.408581 0.204291 0.978910i \(-0.434511\pi\)
0.204291 + 0.978910i \(0.434511\pi\)
\(522\) 0 0
\(523\) −32.6379 −1.42715 −0.713577 0.700577i \(-0.752926\pi\)
−0.713577 + 0.700577i \(0.752926\pi\)
\(524\) 0 0
\(525\) −61.5525 −2.68637
\(526\) 0 0
\(527\) −0.245470 −0.0106928
\(528\) 0 0
\(529\) 16.0363 0.697231
\(530\) 0 0
\(531\) −43.4804 −1.88689
\(532\) 0 0
\(533\) 1.79076 0.0775666
\(534\) 0 0
\(535\) 3.27234 0.141475
\(536\) 0 0
\(537\) 27.5804 1.19018
\(538\) 0 0
\(539\) 46.7800 2.01496
\(540\) 0 0
\(541\) −14.2713 −0.613573 −0.306786 0.951778i \(-0.599254\pi\)
−0.306786 + 0.951778i \(0.599254\pi\)
\(542\) 0 0
\(543\) 2.11618 0.0908140
\(544\) 0 0
\(545\) 1.05594 0.0452315
\(546\) 0 0
\(547\) 24.7376 1.05771 0.528853 0.848714i \(-0.322622\pi\)
0.528853 + 0.848714i \(0.322622\pi\)
\(548\) 0 0
\(549\) −58.9273 −2.51495
\(550\) 0 0
\(551\) 2.97385 0.126690
\(552\) 0 0
\(553\) −24.9189 −1.05966
\(554\) 0 0
\(555\) 1.00392 0.0426138
\(556\) 0 0
\(557\) 19.1124 0.809819 0.404909 0.914357i \(-0.367303\pi\)
0.404909 + 0.914357i \(0.367303\pi\)
\(558\) 0 0
\(559\) −4.56648 −0.193142
\(560\) 0 0
\(561\) 52.8731 2.23230
\(562\) 0 0
\(563\) −31.4787 −1.32667 −0.663336 0.748322i \(-0.730860\pi\)
−0.663336 + 0.748322i \(0.730860\pi\)
\(564\) 0 0
\(565\) −1.96694 −0.0827498
\(566\) 0 0
\(567\) 23.9515 1.00587
\(568\) 0 0
\(569\) −25.6420 −1.07497 −0.537485 0.843273i \(-0.680625\pi\)
−0.537485 + 0.843273i \(0.680625\pi\)
\(570\) 0 0
\(571\) −8.41040 −0.351964 −0.175982 0.984393i \(-0.556310\pi\)
−0.175982 + 0.984393i \(0.556310\pi\)
\(572\) 0 0
\(573\) 29.8305 1.24619
\(574\) 0 0
\(575\) 30.2588 1.26188
\(576\) 0 0
\(577\) −13.2275 −0.550669 −0.275335 0.961348i \(-0.588789\pi\)
−0.275335 + 0.961348i \(0.588789\pi\)
\(578\) 0 0
\(579\) 34.4536 1.43184
\(580\) 0 0
\(581\) −15.5083 −0.643393
\(582\) 0 0
\(583\) 43.1896 1.78873
\(584\) 0 0
\(585\) 4.52004 0.186881
\(586\) 0 0
\(587\) 34.6840 1.43156 0.715781 0.698325i \(-0.246071\pi\)
0.715781 + 0.698325i \(0.246071\pi\)
\(588\) 0 0
\(589\) −0.0312510 −0.00128767
\(590\) 0 0
\(591\) 46.0348 1.89362
\(592\) 0 0
\(593\) 37.4869 1.53940 0.769700 0.638405i \(-0.220406\pi\)
0.769700 + 0.638405i \(0.220406\pi\)
\(594\) 0 0
\(595\) −7.80725 −0.320066
\(596\) 0 0
\(597\) 9.64800 0.394866
\(598\) 0 0
\(599\) 5.73533 0.234339 0.117170 0.993112i \(-0.462618\pi\)
0.117170 + 0.993112i \(0.462618\pi\)
\(600\) 0 0
\(601\) −1.12581 −0.0459229 −0.0229614 0.999736i \(-0.507309\pi\)
−0.0229614 + 0.999736i \(0.507309\pi\)
\(602\) 0 0
\(603\) 45.6607 1.85945
\(604\) 0 0
\(605\) −1.87716 −0.0763173
\(606\) 0 0
\(607\) −13.8854 −0.563591 −0.281795 0.959475i \(-0.590930\pi\)
−0.281795 + 0.959475i \(0.590930\pi\)
\(608\) 0 0
\(609\) 65.3078 2.64640
\(610\) 0 0
\(611\) 19.2548 0.778964
\(612\) 0 0
\(613\) 10.2426 0.413694 0.206847 0.978373i \(-0.433680\pi\)
0.206847 + 0.978373i \(0.433680\pi\)
\(614\) 0 0
\(615\) 1.02021 0.0411387
\(616\) 0 0
\(617\) −7.63302 −0.307294 −0.153647 0.988126i \(-0.549102\pi\)
−0.153647 + 0.988126i \(0.549102\pi\)
\(618\) 0 0
\(619\) 33.7880 1.35806 0.679028 0.734112i \(-0.262401\pi\)
0.679028 + 0.734112i \(0.262401\pi\)
\(620\) 0 0
\(621\) −47.5488 −1.90807
\(622\) 0 0
\(623\) −14.5990 −0.584895
\(624\) 0 0
\(625\) 22.6702 0.906809
\(626\) 0 0
\(627\) 6.73132 0.268823
\(628\) 0 0
\(629\) −3.92894 −0.156657
\(630\) 0 0
\(631\) −30.9285 −1.23124 −0.615622 0.788042i \(-0.711095\pi\)
−0.615622 + 0.788042i \(0.711095\pi\)
\(632\) 0 0
\(633\) 53.6495 2.13238
\(634\) 0 0
\(635\) 6.12220 0.242952
\(636\) 0 0
\(637\) −24.0421 −0.952582
\(638\) 0 0
\(639\) −23.6737 −0.936515
\(640\) 0 0
\(641\) 34.0037 1.34307 0.671533 0.740975i \(-0.265636\pi\)
0.671533 + 0.740975i \(0.265636\pi\)
\(642\) 0 0
\(643\) −10.1833 −0.401589 −0.200794 0.979633i \(-0.564352\pi\)
−0.200794 + 0.979633i \(0.564352\pi\)
\(644\) 0 0
\(645\) −2.60155 −0.102436
\(646\) 0 0
\(647\) −16.9038 −0.664558 −0.332279 0.943181i \(-0.607818\pi\)
−0.332279 + 0.943181i \(0.607818\pi\)
\(648\) 0 0
\(649\) −30.8256 −1.21001
\(650\) 0 0
\(651\) −0.686292 −0.0268979
\(652\) 0 0
\(653\) 17.2863 0.676467 0.338234 0.941062i \(-0.390171\pi\)
0.338234 + 0.941062i \(0.390171\pi\)
\(654\) 0 0
\(655\) −4.14475 −0.161949
\(656\) 0 0
\(657\) −23.0858 −0.900665
\(658\) 0 0
\(659\) 3.37890 0.131623 0.0658117 0.997832i \(-0.479036\pi\)
0.0658117 + 0.997832i \(0.479036\pi\)
\(660\) 0 0
\(661\) 46.4429 1.80642 0.903210 0.429198i \(-0.141204\pi\)
0.903210 + 0.429198i \(0.141204\pi\)
\(662\) 0 0
\(663\) −27.1736 −1.05533
\(664\) 0 0
\(665\) −0.993948 −0.0385436
\(666\) 0 0
\(667\) −32.1049 −1.24311
\(668\) 0 0
\(669\) 63.4660 2.45374
\(670\) 0 0
\(671\) −41.7767 −1.61277
\(672\) 0 0
\(673\) 0.107425 0.00414091 0.00207046 0.999998i \(-0.499341\pi\)
0.00207046 + 0.999998i \(0.499341\pi\)
\(674\) 0 0
\(675\) −36.8573 −1.41864
\(676\) 0 0
\(677\) −11.0929 −0.426335 −0.213167 0.977016i \(-0.568378\pi\)
−0.213167 + 0.977016i \(0.568378\pi\)
\(678\) 0 0
\(679\) 57.5129 2.20714
\(680\) 0 0
\(681\) 28.1304 1.07796
\(682\) 0 0
\(683\) 27.3645 1.04707 0.523537 0.852003i \(-0.324612\pi\)
0.523537 + 0.852003i \(0.324612\pi\)
\(684\) 0 0
\(685\) 6.43696 0.245944
\(686\) 0 0
\(687\) 84.3452 3.21797
\(688\) 0 0
\(689\) −22.1968 −0.845632
\(690\) 0 0
\(691\) 30.3845 1.15588 0.577941 0.816079i \(-0.303856\pi\)
0.577941 + 0.816079i \(0.303856\pi\)
\(692\) 0 0
\(693\) 96.2322 3.65556
\(694\) 0 0
\(695\) −2.97017 −0.112665
\(696\) 0 0
\(697\) −3.99270 −0.151234
\(698\) 0 0
\(699\) 74.9322 2.83420
\(700\) 0 0
\(701\) −27.3877 −1.03442 −0.517210 0.855859i \(-0.673029\pi\)
−0.517210 + 0.855859i \(0.673029\pi\)
\(702\) 0 0
\(703\) −0.500197 −0.0188653
\(704\) 0 0
\(705\) 10.9695 0.413136
\(706\) 0 0
\(707\) −64.0990 −2.41069
\(708\) 0 0
\(709\) −47.6900 −1.79104 −0.895518 0.445025i \(-0.853195\pi\)
−0.895518 + 0.445025i \(0.853195\pi\)
\(710\) 0 0
\(711\) −32.1664 −1.20633
\(712\) 0 0
\(713\) 0.337377 0.0126349
\(714\) 0 0
\(715\) 3.20450 0.119841
\(716\) 0 0
\(717\) 70.5888 2.63619
\(718\) 0 0
\(719\) 20.5621 0.766835 0.383418 0.923575i \(-0.374747\pi\)
0.383418 + 0.923575i \(0.374747\pi\)
\(720\) 0 0
\(721\) 28.6284 1.06618
\(722\) 0 0
\(723\) −30.4395 −1.13206
\(724\) 0 0
\(725\) −24.8860 −0.924244
\(726\) 0 0
\(727\) −4.21618 −0.156369 −0.0781847 0.996939i \(-0.524912\pi\)
−0.0781847 + 0.996939i \(0.524912\pi\)
\(728\) 0 0
\(729\) −36.0197 −1.33406
\(730\) 0 0
\(731\) 10.1815 0.376575
\(732\) 0 0
\(733\) −47.9118 −1.76966 −0.884831 0.465912i \(-0.845726\pi\)
−0.884831 + 0.465912i \(0.845726\pi\)
\(734\) 0 0
\(735\) −13.6969 −0.505217
\(736\) 0 0
\(737\) 32.3713 1.19241
\(738\) 0 0
\(739\) −16.8399 −0.619466 −0.309733 0.950824i \(-0.600240\pi\)
−0.309733 + 0.950824i \(0.600240\pi\)
\(740\) 0 0
\(741\) −3.45949 −0.127088
\(742\) 0 0
\(743\) −21.5221 −0.789569 −0.394784 0.918774i \(-0.629181\pi\)
−0.394784 + 0.918774i \(0.629181\pi\)
\(744\) 0 0
\(745\) 2.50794 0.0918837
\(746\) 0 0
\(747\) −20.0188 −0.732449
\(748\) 0 0
\(749\) −35.8053 −1.30830
\(750\) 0 0
\(751\) 4.40389 0.160700 0.0803501 0.996767i \(-0.474396\pi\)
0.0803501 + 0.996767i \(0.474396\pi\)
\(752\) 0 0
\(753\) 73.1043 2.66407
\(754\) 0 0
\(755\) 4.24302 0.154419
\(756\) 0 0
\(757\) 24.2106 0.879950 0.439975 0.898010i \(-0.354987\pi\)
0.439975 + 0.898010i \(0.354987\pi\)
\(758\) 0 0
\(759\) −72.6696 −2.63774
\(760\) 0 0
\(761\) −9.63619 −0.349312 −0.174656 0.984630i \(-0.555881\pi\)
−0.174656 + 0.984630i \(0.555881\pi\)
\(762\) 0 0
\(763\) −11.5539 −0.418280
\(764\) 0 0
\(765\) −10.0779 −0.364368
\(766\) 0 0
\(767\) 15.8425 0.572038
\(768\) 0 0
\(769\) −16.7366 −0.603538 −0.301769 0.953381i \(-0.597577\pi\)
−0.301769 + 0.953381i \(0.597577\pi\)
\(770\) 0 0
\(771\) −54.8610 −1.97577
\(772\) 0 0
\(773\) 14.7678 0.531162 0.265581 0.964089i \(-0.414436\pi\)
0.265581 + 0.964089i \(0.414436\pi\)
\(774\) 0 0
\(775\) 0.261517 0.00939395
\(776\) 0 0
\(777\) −10.9847 −0.394072
\(778\) 0 0
\(779\) −0.508315 −0.0182123
\(780\) 0 0
\(781\) −16.7835 −0.600561
\(782\) 0 0
\(783\) 39.1060 1.39753
\(784\) 0 0
\(785\) 3.23032 0.115295
\(786\) 0 0
\(787\) −9.81125 −0.349733 −0.174867 0.984592i \(-0.555949\pi\)
−0.174867 + 0.984592i \(0.555949\pi\)
\(788\) 0 0
\(789\) 3.66812 0.130589
\(790\) 0 0
\(791\) 21.5219 0.765231
\(792\) 0 0
\(793\) 21.4707 0.762446
\(794\) 0 0
\(795\) −12.6456 −0.448495
\(796\) 0 0
\(797\) −37.6674 −1.33425 −0.667125 0.744946i \(-0.732475\pi\)
−0.667125 + 0.744946i \(0.732475\pi\)
\(798\) 0 0
\(799\) −42.9306 −1.51878
\(800\) 0 0
\(801\) −18.8450 −0.665854
\(802\) 0 0
\(803\) −16.3668 −0.577571
\(804\) 0 0
\(805\) 10.7304 0.378197
\(806\) 0 0
\(807\) 41.7052 1.46809
\(808\) 0 0
\(809\) 9.32508 0.327853 0.163926 0.986473i \(-0.447584\pi\)
0.163926 + 0.986473i \(0.447584\pi\)
\(810\) 0 0
\(811\) 31.3933 1.10237 0.551183 0.834384i \(-0.314177\pi\)
0.551183 + 0.834384i \(0.314177\pi\)
\(812\) 0 0
\(813\) 51.4827 1.80558
\(814\) 0 0
\(815\) 2.36118 0.0827087
\(816\) 0 0
\(817\) 1.29621 0.0453487
\(818\) 0 0
\(819\) −49.4575 −1.72818
\(820\) 0 0
\(821\) 29.2233 1.01990 0.509950 0.860204i \(-0.329664\pi\)
0.509950 + 0.860204i \(0.329664\pi\)
\(822\) 0 0
\(823\) 0.704233 0.0245480 0.0122740 0.999925i \(-0.496093\pi\)
0.0122740 + 0.999925i \(0.496093\pi\)
\(824\) 0 0
\(825\) −56.3295 −1.96114
\(826\) 0 0
\(827\) −47.4198 −1.64895 −0.824474 0.565900i \(-0.808529\pi\)
−0.824474 + 0.565900i \(0.808529\pi\)
\(828\) 0 0
\(829\) 31.4060 1.09078 0.545388 0.838184i \(-0.316382\pi\)
0.545388 + 0.838184i \(0.316382\pi\)
\(830\) 0 0
\(831\) −77.3512 −2.68328
\(832\) 0 0
\(833\) 53.6044 1.85728
\(834\) 0 0
\(835\) 8.07625 0.279490
\(836\) 0 0
\(837\) −0.410948 −0.0142044
\(838\) 0 0
\(839\) −39.8294 −1.37506 −0.687532 0.726154i \(-0.741306\pi\)
−0.687532 + 0.726154i \(0.741306\pi\)
\(840\) 0 0
\(841\) −2.59569 −0.0895064
\(842\) 0 0
\(843\) −70.3885 −2.42431
\(844\) 0 0
\(845\) 3.50346 0.120523
\(846\) 0 0
\(847\) 20.5395 0.705746
\(848\) 0 0
\(849\) 10.3232 0.354290
\(850\) 0 0
\(851\) 5.40000 0.185110
\(852\) 0 0
\(853\) −8.25003 −0.282476 −0.141238 0.989976i \(-0.545108\pi\)
−0.141238 + 0.989976i \(0.545108\pi\)
\(854\) 0 0
\(855\) −1.28303 −0.0438787
\(856\) 0 0
\(857\) −12.0746 −0.412462 −0.206231 0.978503i \(-0.566120\pi\)
−0.206231 + 0.978503i \(0.566120\pi\)
\(858\) 0 0
\(859\) −9.85490 −0.336245 −0.168122 0.985766i \(-0.553770\pi\)
−0.168122 + 0.985766i \(0.553770\pi\)
\(860\) 0 0
\(861\) −11.1629 −0.380431
\(862\) 0 0
\(863\) 17.7816 0.605294 0.302647 0.953103i \(-0.402130\pi\)
0.302647 + 0.953103i \(0.402130\pi\)
\(864\) 0 0
\(865\) −6.86800 −0.233519
\(866\) 0 0
\(867\) 10.7449 0.364917
\(868\) 0 0
\(869\) −22.8045 −0.773589
\(870\) 0 0
\(871\) −16.6369 −0.563719
\(872\) 0 0
\(873\) 74.2401 2.51265
\(874\) 0 0
\(875\) 16.9048 0.571487
\(876\) 0 0
\(877\) 23.1862 0.782942 0.391471 0.920191i \(-0.371966\pi\)
0.391471 + 0.920191i \(0.371966\pi\)
\(878\) 0 0
\(879\) −9.15869 −0.308915
\(880\) 0 0
\(881\) −21.5225 −0.725113 −0.362557 0.931962i \(-0.618096\pi\)
−0.362557 + 0.931962i \(0.618096\pi\)
\(882\) 0 0
\(883\) −30.1426 −1.01438 −0.507190 0.861834i \(-0.669316\pi\)
−0.507190 + 0.861834i \(0.669316\pi\)
\(884\) 0 0
\(885\) 9.02553 0.303390
\(886\) 0 0
\(887\) 35.7744 1.20119 0.600594 0.799554i \(-0.294931\pi\)
0.600594 + 0.799554i \(0.294931\pi\)
\(888\) 0 0
\(889\) −66.9880 −2.24671
\(890\) 0 0
\(891\) 21.9192 0.734320
\(892\) 0 0
\(893\) −5.46553 −0.182897
\(894\) 0 0
\(895\) −3.72696 −0.124579
\(896\) 0 0
\(897\) 37.3477 1.24700
\(898\) 0 0
\(899\) −0.277472 −0.00925420
\(900\) 0 0
\(901\) 49.4903 1.64876
\(902\) 0 0
\(903\) 28.4657 0.947278
\(904\) 0 0
\(905\) −0.285961 −0.00950566
\(906\) 0 0
\(907\) 20.1020 0.667475 0.333737 0.942666i \(-0.391690\pi\)
0.333737 + 0.942666i \(0.391690\pi\)
\(908\) 0 0
\(909\) −82.7417 −2.74437
\(910\) 0 0
\(911\) −57.0332 −1.88960 −0.944798 0.327655i \(-0.893742\pi\)
−0.944798 + 0.327655i \(0.893742\pi\)
\(912\) 0 0
\(913\) −14.1924 −0.469699
\(914\) 0 0
\(915\) 12.2319 0.404376
\(916\) 0 0
\(917\) 45.3511 1.49762
\(918\) 0 0
\(919\) 10.1609 0.335178 0.167589 0.985857i \(-0.446402\pi\)
0.167589 + 0.985857i \(0.446402\pi\)
\(920\) 0 0
\(921\) −25.0002 −0.823783
\(922\) 0 0
\(923\) 8.62571 0.283919
\(924\) 0 0
\(925\) 4.18579 0.137628
\(926\) 0 0
\(927\) 36.9547 1.21375
\(928\) 0 0
\(929\) 23.2751 0.763630 0.381815 0.924239i \(-0.375299\pi\)
0.381815 + 0.924239i \(0.375299\pi\)
\(930\) 0 0
\(931\) 6.82443 0.223662
\(932\) 0 0
\(933\) 14.0387 0.459606
\(934\) 0 0
\(935\) −7.14478 −0.233659
\(936\) 0 0
\(937\) −10.7545 −0.351333 −0.175667 0.984450i \(-0.556208\pi\)
−0.175667 + 0.984450i \(0.556208\pi\)
\(938\) 0 0
\(939\) −87.3611 −2.85092
\(940\) 0 0
\(941\) 2.97303 0.0969180 0.0484590 0.998825i \(-0.484569\pi\)
0.0484590 + 0.998825i \(0.484569\pi\)
\(942\) 0 0
\(943\) 5.48763 0.178702
\(944\) 0 0
\(945\) −13.0703 −0.425178
\(946\) 0 0
\(947\) −23.5098 −0.763964 −0.381982 0.924170i \(-0.624758\pi\)
−0.381982 + 0.924170i \(0.624758\pi\)
\(948\) 0 0
\(949\) 8.41153 0.273050
\(950\) 0 0
\(951\) 82.6370 2.67969
\(952\) 0 0
\(953\) −0.840764 −0.0272350 −0.0136175 0.999907i \(-0.504335\pi\)
−0.0136175 + 0.999907i \(0.504335\pi\)
\(954\) 0 0
\(955\) −4.03101 −0.130440
\(956\) 0 0
\(957\) 59.7662 1.93197
\(958\) 0 0
\(959\) −70.4321 −2.27437
\(960\) 0 0
\(961\) −30.9971 −0.999906
\(962\) 0 0
\(963\) −46.2190 −1.48939
\(964\) 0 0
\(965\) −4.65573 −0.149873
\(966\) 0 0
\(967\) −7.46721 −0.240129 −0.120065 0.992766i \(-0.538310\pi\)
−0.120065 + 0.992766i \(0.538310\pi\)
\(968\) 0 0
\(969\) 7.71331 0.247787
\(970\) 0 0
\(971\) 49.5807 1.59112 0.795560 0.605875i \(-0.207177\pi\)
0.795560 + 0.605875i \(0.207177\pi\)
\(972\) 0 0
\(973\) 32.4991 1.04187
\(974\) 0 0
\(975\) 28.9500 0.927141
\(976\) 0 0
\(977\) −27.5870 −0.882586 −0.441293 0.897363i \(-0.645480\pi\)
−0.441293 + 0.897363i \(0.645480\pi\)
\(978\) 0 0
\(979\) −13.3602 −0.426994
\(980\) 0 0
\(981\) −14.9143 −0.476176
\(982\) 0 0
\(983\) 45.9415 1.46531 0.732654 0.680601i \(-0.238281\pi\)
0.732654 + 0.680601i \(0.238281\pi\)
\(984\) 0 0
\(985\) −6.22071 −0.198208
\(986\) 0 0
\(987\) −120.027 −3.82049
\(988\) 0 0
\(989\) −13.9936 −0.444969
\(990\) 0 0
\(991\) 43.6148 1.38547 0.692735 0.721192i \(-0.256405\pi\)
0.692735 + 0.721192i \(0.256405\pi\)
\(992\) 0 0
\(993\) −91.9819 −2.91896
\(994\) 0 0
\(995\) −1.30374 −0.0413314
\(996\) 0 0
\(997\) −32.9880 −1.04474 −0.522370 0.852719i \(-0.674952\pi\)
−0.522370 + 0.852719i \(0.674952\pi\)
\(998\) 0 0
\(999\) −6.57756 −0.208105
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 4096.2.a.s.1.7 8
4.3 odd 2 4096.2.a.i.1.1 8
8.3 odd 2 inner 4096.2.a.s.1.8 8
8.5 even 2 4096.2.a.i.1.2 8
64.3 odd 16 1024.2.g.d.641.4 yes 16
64.5 even 16 1024.2.g.f.897.4 yes 16
64.11 odd 16 1024.2.g.g.385.1 yes 16
64.13 even 16 1024.2.g.f.129.4 yes 16
64.19 odd 16 1024.2.g.a.129.4 yes 16
64.21 even 16 1024.2.g.d.385.1 yes 16
64.27 odd 16 1024.2.g.a.897.4 yes 16
64.29 even 16 1024.2.g.g.641.4 yes 16
64.35 odd 16 1024.2.g.g.641.1 yes 16
64.37 even 16 1024.2.g.a.897.1 yes 16
64.43 odd 16 1024.2.g.d.385.4 yes 16
64.45 even 16 1024.2.g.a.129.1 16
64.51 odd 16 1024.2.g.f.129.1 yes 16
64.53 even 16 1024.2.g.g.385.4 yes 16
64.59 odd 16 1024.2.g.f.897.1 yes 16
64.61 even 16 1024.2.g.d.641.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.a.129.1 16 64.45 even 16
1024.2.g.a.129.4 yes 16 64.19 odd 16
1024.2.g.a.897.1 yes 16 64.37 even 16
1024.2.g.a.897.4 yes 16 64.27 odd 16
1024.2.g.d.385.1 yes 16 64.21 even 16
1024.2.g.d.385.4 yes 16 64.43 odd 16
1024.2.g.d.641.1 yes 16 64.61 even 16
1024.2.g.d.641.4 yes 16 64.3 odd 16
1024.2.g.f.129.1 yes 16 64.51 odd 16
1024.2.g.f.129.4 yes 16 64.13 even 16
1024.2.g.f.897.1 yes 16 64.59 odd 16
1024.2.g.f.897.4 yes 16 64.5 even 16
1024.2.g.g.385.1 yes 16 64.11 odd 16
1024.2.g.g.385.4 yes 16 64.53 even 16
1024.2.g.g.641.1 yes 16 64.35 odd 16
1024.2.g.g.641.4 yes 16 64.29 even 16
4096.2.a.i.1.1 8 4.3 odd 2
4096.2.a.i.1.2 8 8.5 even 2
4096.2.a.s.1.7 8 1.1 even 1 trivial
4096.2.a.s.1.8 8 8.3 odd 2 inner