Properties

Label 1336.2.a.c.1.8
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $1$
Dimension $9$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, 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("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 13x^{7} + 8x^{6} + 56x^{5} - 15x^{4} - 81x^{3} + 2x^{2} + 13x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-1.76335\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.76335 q^{3} -2.14065 q^{5} -2.31472 q^{7} +0.109415 q^{9} +O(q^{10})\) \(q+1.76335 q^{3} -2.14065 q^{5} -2.31472 q^{7} +0.109415 q^{9} +0.602850 q^{11} +2.31508 q^{13} -3.77473 q^{15} -4.36219 q^{17} +3.77368 q^{19} -4.08166 q^{21} -7.45921 q^{23} -0.417611 q^{25} -5.09712 q^{27} +8.78791 q^{29} -5.26991 q^{31} +1.06304 q^{33} +4.95500 q^{35} -7.89738 q^{37} +4.08230 q^{39} -9.11012 q^{41} +2.06676 q^{43} -0.234220 q^{45} -9.53507 q^{47} -1.64209 q^{49} -7.69209 q^{51} -2.21424 q^{53} -1.29049 q^{55} +6.65433 q^{57} -5.79683 q^{59} -13.9341 q^{61} -0.253266 q^{63} -4.95578 q^{65} +12.8800 q^{67} -13.1532 q^{69} -7.93032 q^{71} +16.6124 q^{73} -0.736396 q^{75} -1.39543 q^{77} -2.75232 q^{79} -9.31627 q^{81} +7.18217 q^{83} +9.33793 q^{85} +15.4962 q^{87} +7.33737 q^{89} -5.35875 q^{91} -9.29272 q^{93} -8.07813 q^{95} +15.4665 q^{97} +0.0659611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} - 8 q^{5} + 2 q^{7} - 10 q^{11} - 13 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - 19 q^{21} - 3 q^{23} + 3 q^{25} - 10 q^{27} - 25 q^{29} - q^{31} - 12 q^{33} - 17 q^{35} - 35 q^{37} - 4 q^{39} - 16 q^{41} + 9 q^{43} - 24 q^{45} - q^{47} - q^{49} - 10 q^{51} - 29 q^{53} + 9 q^{55} - 17 q^{57} - 14 q^{59} - 28 q^{61} + 4 q^{63} - 31 q^{65} + 19 q^{67} - 19 q^{69} - 9 q^{71} - 7 q^{75} - 33 q^{77} - 18 q^{79} - 27 q^{81} - 13 q^{83} - 36 q^{85} + 18 q^{87} - 21 q^{89} + 20 q^{91} - 35 q^{93} - 12 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.76335 1.01807 0.509036 0.860745i \(-0.330002\pi\)
0.509036 + 0.860745i \(0.330002\pi\)
\(4\) 0 0
\(5\) −2.14065 −0.957328 −0.478664 0.877998i \(-0.658879\pi\)
−0.478664 + 0.877998i \(0.658879\pi\)
\(6\) 0 0
\(7\) −2.31472 −0.874881 −0.437440 0.899247i \(-0.644115\pi\)
−0.437440 + 0.899247i \(0.644115\pi\)
\(8\) 0 0
\(9\) 0.109415 0.0364718
\(10\) 0 0
\(11\) 0.602850 0.181766 0.0908831 0.995862i \(-0.471031\pi\)
0.0908831 + 0.995862i \(0.471031\pi\)
\(12\) 0 0
\(13\) 2.31508 0.642087 0.321044 0.947064i \(-0.395966\pi\)
0.321044 + 0.947064i \(0.395966\pi\)
\(14\) 0 0
\(15\) −3.77473 −0.974630
\(16\) 0 0
\(17\) −4.36219 −1.05799 −0.528993 0.848626i \(-0.677430\pi\)
−0.528993 + 0.848626i \(0.677430\pi\)
\(18\) 0 0
\(19\) 3.77368 0.865741 0.432871 0.901456i \(-0.357501\pi\)
0.432871 + 0.901456i \(0.357501\pi\)
\(20\) 0 0
\(21\) −4.08166 −0.890692
\(22\) 0 0
\(23\) −7.45921 −1.55535 −0.777677 0.628665i \(-0.783602\pi\)
−0.777677 + 0.628665i \(0.783602\pi\)
\(24\) 0 0
\(25\) −0.417611 −0.0835222
\(26\) 0 0
\(27\) −5.09712 −0.980942
\(28\) 0 0
\(29\) 8.78791 1.63187 0.815937 0.578141i \(-0.196222\pi\)
0.815937 + 0.578141i \(0.196222\pi\)
\(30\) 0 0
\(31\) −5.26991 −0.946505 −0.473252 0.880927i \(-0.656920\pi\)
−0.473252 + 0.880927i \(0.656920\pi\)
\(32\) 0 0
\(33\) 1.06304 0.185051
\(34\) 0 0
\(35\) 4.95500 0.837548
\(36\) 0 0
\(37\) −7.89738 −1.29832 −0.649161 0.760651i \(-0.724880\pi\)
−0.649161 + 0.760651i \(0.724880\pi\)
\(38\) 0 0
\(39\) 4.08230 0.653691
\(40\) 0 0
\(41\) −9.11012 −1.42276 −0.711381 0.702807i \(-0.751930\pi\)
−0.711381 + 0.702807i \(0.751930\pi\)
\(42\) 0 0
\(43\) 2.06676 0.315178 0.157589 0.987505i \(-0.449628\pi\)
0.157589 + 0.987505i \(0.449628\pi\)
\(44\) 0 0
\(45\) −0.234220 −0.0349155
\(46\) 0 0
\(47\) −9.53507 −1.39083 −0.695416 0.718607i \(-0.744780\pi\)
−0.695416 + 0.718607i \(0.744780\pi\)
\(48\) 0 0
\(49\) −1.64209 −0.234584
\(50\) 0 0
\(51\) −7.69209 −1.07711
\(52\) 0 0
\(53\) −2.21424 −0.304150 −0.152075 0.988369i \(-0.548595\pi\)
−0.152075 + 0.988369i \(0.548595\pi\)
\(54\) 0 0
\(55\) −1.29049 −0.174010
\(56\) 0 0
\(57\) 6.65433 0.881388
\(58\) 0 0
\(59\) −5.79683 −0.754682 −0.377341 0.926074i \(-0.623162\pi\)
−0.377341 + 0.926074i \(0.623162\pi\)
\(60\) 0 0
\(61\) −13.9341 −1.78407 −0.892037 0.451961i \(-0.850724\pi\)
−0.892037 + 0.451961i \(0.850724\pi\)
\(62\) 0 0
\(63\) −0.253266 −0.0319085
\(64\) 0 0
\(65\) −4.95578 −0.614688
\(66\) 0 0
\(67\) 12.8800 1.57354 0.786769 0.617247i \(-0.211752\pi\)
0.786769 + 0.617247i \(0.211752\pi\)
\(68\) 0 0
\(69\) −13.1532 −1.58346
\(70\) 0 0
\(71\) −7.93032 −0.941156 −0.470578 0.882358i \(-0.655955\pi\)
−0.470578 + 0.882358i \(0.655955\pi\)
\(72\) 0 0
\(73\) 16.6124 1.94433 0.972166 0.234295i \(-0.0752782\pi\)
0.972166 + 0.234295i \(0.0752782\pi\)
\(74\) 0 0
\(75\) −0.736396 −0.0850317
\(76\) 0 0
\(77\) −1.39543 −0.159024
\(78\) 0 0
\(79\) −2.75232 −0.309661 −0.154830 0.987941i \(-0.549483\pi\)
−0.154830 + 0.987941i \(0.549483\pi\)
\(80\) 0 0
\(81\) −9.31627 −1.03514
\(82\) 0 0
\(83\) 7.18217 0.788346 0.394173 0.919036i \(-0.371031\pi\)
0.394173 + 0.919036i \(0.371031\pi\)
\(84\) 0 0
\(85\) 9.33793 1.01284
\(86\) 0 0
\(87\) 15.4962 1.66137
\(88\) 0 0
\(89\) 7.33737 0.777759 0.388880 0.921289i \(-0.372862\pi\)
0.388880 + 0.921289i \(0.372862\pi\)
\(90\) 0 0
\(91\) −5.35875 −0.561750
\(92\) 0 0
\(93\) −9.29272 −0.963610
\(94\) 0 0
\(95\) −8.07813 −0.828799
\(96\) 0 0
\(97\) 15.4665 1.57039 0.785193 0.619251i \(-0.212564\pi\)
0.785193 + 0.619251i \(0.212564\pi\)
\(98\) 0 0
\(99\) 0.0659611 0.00662934
\(100\) 0 0
\(101\) 7.27579 0.723968 0.361984 0.932184i \(-0.382099\pi\)
0.361984 + 0.932184i \(0.382099\pi\)
\(102\) 0 0
\(103\) 15.1140 1.48923 0.744615 0.667494i \(-0.232633\pi\)
0.744615 + 0.667494i \(0.232633\pi\)
\(104\) 0 0
\(105\) 8.73742 0.852685
\(106\) 0 0
\(107\) 6.69621 0.647348 0.323674 0.946169i \(-0.395082\pi\)
0.323674 + 0.946169i \(0.395082\pi\)
\(108\) 0 0
\(109\) −18.7429 −1.79524 −0.897620 0.440770i \(-0.854705\pi\)
−0.897620 + 0.440770i \(0.854705\pi\)
\(110\) 0 0
\(111\) −13.9259 −1.32179
\(112\) 0 0
\(113\) −3.44430 −0.324012 −0.162006 0.986790i \(-0.551796\pi\)
−0.162006 + 0.986790i \(0.551796\pi\)
\(114\) 0 0
\(115\) 15.9676 1.48898
\(116\) 0 0
\(117\) 0.253305 0.0234181
\(118\) 0 0
\(119\) 10.0972 0.925612
\(120\) 0 0
\(121\) −10.6366 −0.966961
\(122\) 0 0
\(123\) −16.0644 −1.44848
\(124\) 0 0
\(125\) 11.5972 1.03729
\(126\) 0 0
\(127\) 11.9612 1.06139 0.530693 0.847564i \(-0.321932\pi\)
0.530693 + 0.847564i \(0.321932\pi\)
\(128\) 0 0
\(129\) 3.64443 0.320875
\(130\) 0 0
\(131\) 10.3596 0.905124 0.452562 0.891733i \(-0.350510\pi\)
0.452562 + 0.891733i \(0.350510\pi\)
\(132\) 0 0
\(133\) −8.73500 −0.757420
\(134\) 0 0
\(135\) 10.9112 0.939083
\(136\) 0 0
\(137\) −7.58902 −0.648373 −0.324187 0.945993i \(-0.605091\pi\)
−0.324187 + 0.945993i \(0.605091\pi\)
\(138\) 0 0
\(139\) 18.3794 1.55892 0.779459 0.626453i \(-0.215494\pi\)
0.779459 + 0.626453i \(0.215494\pi\)
\(140\) 0 0
\(141\) −16.8137 −1.41597
\(142\) 0 0
\(143\) 1.39565 0.116710
\(144\) 0 0
\(145\) −18.8118 −1.56224
\(146\) 0 0
\(147\) −2.89558 −0.238823
\(148\) 0 0
\(149\) −12.8525 −1.05292 −0.526460 0.850200i \(-0.676481\pi\)
−0.526460 + 0.850200i \(0.676481\pi\)
\(150\) 0 0
\(151\) 22.6877 1.84630 0.923148 0.384445i \(-0.125607\pi\)
0.923148 + 0.384445i \(0.125607\pi\)
\(152\) 0 0
\(153\) −0.477291 −0.0385867
\(154\) 0 0
\(155\) 11.2811 0.906116
\(156\) 0 0
\(157\) −4.90630 −0.391565 −0.195783 0.980647i \(-0.562725\pi\)
−0.195783 + 0.980647i \(0.562725\pi\)
\(158\) 0 0
\(159\) −3.90449 −0.309647
\(160\) 0 0
\(161\) 17.2660 1.36075
\(162\) 0 0
\(163\) 13.2900 1.04095 0.520476 0.853876i \(-0.325754\pi\)
0.520476 + 0.853876i \(0.325754\pi\)
\(164\) 0 0
\(165\) −2.27559 −0.177155
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −7.64041 −0.587724
\(170\) 0 0
\(171\) 0.412898 0.0315751
\(172\) 0 0
\(173\) 1.46160 0.111123 0.0555616 0.998455i \(-0.482305\pi\)
0.0555616 + 0.998455i \(0.482305\pi\)
\(174\) 0 0
\(175\) 0.966651 0.0730720
\(176\) 0 0
\(177\) −10.2219 −0.768321
\(178\) 0 0
\(179\) −5.13991 −0.384175 −0.192087 0.981378i \(-0.561526\pi\)
−0.192087 + 0.981378i \(0.561526\pi\)
\(180\) 0 0
\(181\) −22.6366 −1.68257 −0.841283 0.540596i \(-0.818199\pi\)
−0.841283 + 0.540596i \(0.818199\pi\)
\(182\) 0 0
\(183\) −24.5707 −1.81632
\(184\) 0 0
\(185\) 16.9055 1.24292
\(186\) 0 0
\(187\) −2.62975 −0.192306
\(188\) 0 0
\(189\) 11.7984 0.858207
\(190\) 0 0
\(191\) 13.6449 0.987309 0.493655 0.869658i \(-0.335661\pi\)
0.493655 + 0.869658i \(0.335661\pi\)
\(192\) 0 0
\(193\) −11.9607 −0.860951 −0.430476 0.902602i \(-0.641654\pi\)
−0.430476 + 0.902602i \(0.641654\pi\)
\(194\) 0 0
\(195\) −8.73878 −0.625797
\(196\) 0 0
\(197\) 3.35883 0.239307 0.119653 0.992816i \(-0.461822\pi\)
0.119653 + 0.992816i \(0.461822\pi\)
\(198\) 0 0
\(199\) −20.2433 −1.43501 −0.717506 0.696552i \(-0.754716\pi\)
−0.717506 + 0.696552i \(0.754716\pi\)
\(200\) 0 0
\(201\) 22.7119 1.60198
\(202\) 0 0
\(203\) −20.3415 −1.42769
\(204\) 0 0
\(205\) 19.5016 1.36205
\(206\) 0 0
\(207\) −0.816152 −0.0567265
\(208\) 0 0
\(209\) 2.27496 0.157362
\(210\) 0 0
\(211\) −4.00805 −0.275926 −0.137963 0.990437i \(-0.544055\pi\)
−0.137963 + 0.990437i \(0.544055\pi\)
\(212\) 0 0
\(213\) −13.9840 −0.958166
\(214\) 0 0
\(215\) −4.42422 −0.301729
\(216\) 0 0
\(217\) 12.1984 0.828079
\(218\) 0 0
\(219\) 29.2935 1.97947
\(220\) 0 0
\(221\) −10.0988 −0.679320
\(222\) 0 0
\(223\) −16.8926 −1.13122 −0.565608 0.824674i \(-0.691358\pi\)
−0.565608 + 0.824674i \(0.691358\pi\)
\(224\) 0 0
\(225\) −0.0456931 −0.00304620
\(226\) 0 0
\(227\) −21.8277 −1.44875 −0.724377 0.689404i \(-0.757873\pi\)
−0.724377 + 0.689404i \(0.757873\pi\)
\(228\) 0 0
\(229\) −9.80083 −0.647657 −0.323829 0.946116i \(-0.604970\pi\)
−0.323829 + 0.946116i \(0.604970\pi\)
\(230\) 0 0
\(231\) −2.46063 −0.161898
\(232\) 0 0
\(233\) −17.4242 −1.14150 −0.570749 0.821125i \(-0.693347\pi\)
−0.570749 + 0.821125i \(0.693347\pi\)
\(234\) 0 0
\(235\) 20.4113 1.33148
\(236\) 0 0
\(237\) −4.85332 −0.315257
\(238\) 0 0
\(239\) −0.238358 −0.0154181 −0.00770905 0.999970i \(-0.502454\pi\)
−0.00770905 + 0.999970i \(0.502454\pi\)
\(240\) 0 0
\(241\) 17.7271 1.14190 0.570951 0.820984i \(-0.306575\pi\)
0.570951 + 0.820984i \(0.306575\pi\)
\(242\) 0 0
\(243\) −1.13652 −0.0729076
\(244\) 0 0
\(245\) 3.51513 0.224574
\(246\) 0 0
\(247\) 8.73636 0.555881
\(248\) 0 0
\(249\) 12.6647 0.802593
\(250\) 0 0
\(251\) 4.15416 0.262208 0.131104 0.991369i \(-0.458148\pi\)
0.131104 + 0.991369i \(0.458148\pi\)
\(252\) 0 0
\(253\) −4.49679 −0.282711
\(254\) 0 0
\(255\) 16.4661 1.03115
\(256\) 0 0
\(257\) −8.11780 −0.506374 −0.253187 0.967417i \(-0.581479\pi\)
−0.253187 + 0.967417i \(0.581479\pi\)
\(258\) 0 0
\(259\) 18.2802 1.13588
\(260\) 0 0
\(261\) 0.961532 0.0595173
\(262\) 0 0
\(263\) −0.194729 −0.0120075 −0.00600376 0.999982i \(-0.501911\pi\)
−0.00600376 + 0.999982i \(0.501911\pi\)
\(264\) 0 0
\(265\) 4.73992 0.291171
\(266\) 0 0
\(267\) 12.9384 0.791816
\(268\) 0 0
\(269\) −15.6456 −0.953932 −0.476966 0.878922i \(-0.658264\pi\)
−0.476966 + 0.878922i \(0.658264\pi\)
\(270\) 0 0
\(271\) 7.69848 0.467650 0.233825 0.972279i \(-0.424876\pi\)
0.233825 + 0.972279i \(0.424876\pi\)
\(272\) 0 0
\(273\) −9.44937 −0.571902
\(274\) 0 0
\(275\) −0.251757 −0.0151815
\(276\) 0 0
\(277\) 20.0696 1.20586 0.602932 0.797793i \(-0.293999\pi\)
0.602932 + 0.797793i \(0.293999\pi\)
\(278\) 0 0
\(279\) −0.576610 −0.0345207
\(280\) 0 0
\(281\) −3.54093 −0.211234 −0.105617 0.994407i \(-0.533682\pi\)
−0.105617 + 0.994407i \(0.533682\pi\)
\(282\) 0 0
\(283\) −6.22156 −0.369833 −0.184917 0.982754i \(-0.559201\pi\)
−0.184917 + 0.982754i \(0.559201\pi\)
\(284\) 0 0
\(285\) −14.2446 −0.843777
\(286\) 0 0
\(287\) 21.0874 1.24475
\(288\) 0 0
\(289\) 2.02872 0.119336
\(290\) 0 0
\(291\) 27.2729 1.59877
\(292\) 0 0
\(293\) −13.1746 −0.769667 −0.384834 0.922986i \(-0.625741\pi\)
−0.384834 + 0.922986i \(0.625741\pi\)
\(294\) 0 0
\(295\) 12.4090 0.722479
\(296\) 0 0
\(297\) −3.07280 −0.178302
\(298\) 0 0
\(299\) −17.2687 −0.998672
\(300\) 0 0
\(301\) −4.78397 −0.275744
\(302\) 0 0
\(303\) 12.8298 0.737052
\(304\) 0 0
\(305\) 29.8280 1.70795
\(306\) 0 0
\(307\) 13.5356 0.772519 0.386260 0.922390i \(-0.373767\pi\)
0.386260 + 0.922390i \(0.373767\pi\)
\(308\) 0 0
\(309\) 26.6514 1.51614
\(310\) 0 0
\(311\) −23.9439 −1.35774 −0.678868 0.734261i \(-0.737529\pi\)
−0.678868 + 0.734261i \(0.737529\pi\)
\(312\) 0 0
\(313\) −9.22216 −0.521267 −0.260634 0.965438i \(-0.583931\pi\)
−0.260634 + 0.965438i \(0.583931\pi\)
\(314\) 0 0
\(315\) 0.542153 0.0305469
\(316\) 0 0
\(317\) 7.95458 0.446774 0.223387 0.974730i \(-0.428289\pi\)
0.223387 + 0.974730i \(0.428289\pi\)
\(318\) 0 0
\(319\) 5.29779 0.296619
\(320\) 0 0
\(321\) 11.8078 0.659047
\(322\) 0 0
\(323\) −16.4615 −0.915943
\(324\) 0 0
\(325\) −0.966802 −0.0536285
\(326\) 0 0
\(327\) −33.0503 −1.82768
\(328\) 0 0
\(329\) 22.0710 1.21681
\(330\) 0 0
\(331\) −27.4389 −1.50818 −0.754089 0.656772i \(-0.771921\pi\)
−0.754089 + 0.656772i \(0.771921\pi\)
\(332\) 0 0
\(333\) −0.864095 −0.0473521
\(334\) 0 0
\(335\) −27.5715 −1.50639
\(336\) 0 0
\(337\) 0.751936 0.0409606 0.0204803 0.999790i \(-0.493480\pi\)
0.0204803 + 0.999790i \(0.493480\pi\)
\(338\) 0 0
\(339\) −6.07352 −0.329868
\(340\) 0 0
\(341\) −3.17697 −0.172043
\(342\) 0 0
\(343\) 20.0040 1.08011
\(344\) 0 0
\(345\) 28.1565 1.51589
\(346\) 0 0
\(347\) −4.02151 −0.215886 −0.107943 0.994157i \(-0.534426\pi\)
−0.107943 + 0.994157i \(0.534426\pi\)
\(348\) 0 0
\(349\) 34.2514 1.83344 0.916719 0.399533i \(-0.130828\pi\)
0.916719 + 0.399533i \(0.130828\pi\)
\(350\) 0 0
\(351\) −11.8002 −0.629850
\(352\) 0 0
\(353\) −31.1247 −1.65660 −0.828299 0.560286i \(-0.810691\pi\)
−0.828299 + 0.560286i \(0.810691\pi\)
\(354\) 0 0
\(355\) 16.9761 0.900996
\(356\) 0 0
\(357\) 17.8050 0.942341
\(358\) 0 0
\(359\) −10.8992 −0.575237 −0.287618 0.957745i \(-0.592863\pi\)
−0.287618 + 0.957745i \(0.592863\pi\)
\(360\) 0 0
\(361\) −4.75935 −0.250492
\(362\) 0 0
\(363\) −18.7560 −0.984437
\(364\) 0 0
\(365\) −35.5613 −1.86136
\(366\) 0 0
\(367\) 9.47211 0.494440 0.247220 0.968959i \(-0.420483\pi\)
0.247220 + 0.968959i \(0.420483\pi\)
\(368\) 0 0
\(369\) −0.996787 −0.0518907
\(370\) 0 0
\(371\) 5.12535 0.266095
\(372\) 0 0
\(373\) 11.8698 0.614593 0.307297 0.951614i \(-0.400576\pi\)
0.307297 + 0.951614i \(0.400576\pi\)
\(374\) 0 0
\(375\) 20.4500 1.05603
\(376\) 0 0
\(377\) 20.3447 1.04780
\(378\) 0 0
\(379\) 6.91179 0.355035 0.177517 0.984118i \(-0.443193\pi\)
0.177517 + 0.984118i \(0.443193\pi\)
\(380\) 0 0
\(381\) 21.0918 1.08057
\(382\) 0 0
\(383\) 16.5500 0.845666 0.422833 0.906208i \(-0.361036\pi\)
0.422833 + 0.906208i \(0.361036\pi\)
\(384\) 0 0
\(385\) 2.98712 0.152238
\(386\) 0 0
\(387\) 0.226136 0.0114951
\(388\) 0 0
\(389\) 12.0288 0.609884 0.304942 0.952371i \(-0.401363\pi\)
0.304942 + 0.952371i \(0.401363\pi\)
\(390\) 0 0
\(391\) 32.5385 1.64554
\(392\) 0 0
\(393\) 18.2677 0.921482
\(394\) 0 0
\(395\) 5.89177 0.296447
\(396\) 0 0
\(397\) 31.2217 1.56697 0.783487 0.621409i \(-0.213439\pi\)
0.783487 + 0.621409i \(0.213439\pi\)
\(398\) 0 0
\(399\) −15.4029 −0.771109
\(400\) 0 0
\(401\) −25.4271 −1.26977 −0.634885 0.772607i \(-0.718953\pi\)
−0.634885 + 0.772607i \(0.718953\pi\)
\(402\) 0 0
\(403\) −12.2003 −0.607738
\(404\) 0 0
\(405\) 19.9429 0.990971
\(406\) 0 0
\(407\) −4.76094 −0.235991
\(408\) 0 0
\(409\) −1.95783 −0.0968086 −0.0484043 0.998828i \(-0.515414\pi\)
−0.0484043 + 0.998828i \(0.515414\pi\)
\(410\) 0 0
\(411\) −13.3821 −0.660091
\(412\) 0 0
\(413\) 13.4180 0.660257
\(414\) 0 0
\(415\) −15.3745 −0.754706
\(416\) 0 0
\(417\) 32.4093 1.58709
\(418\) 0 0
\(419\) −18.7504 −0.916016 −0.458008 0.888948i \(-0.651437\pi\)
−0.458008 + 0.888948i \(0.651437\pi\)
\(420\) 0 0
\(421\) 20.1681 0.982934 0.491467 0.870896i \(-0.336461\pi\)
0.491467 + 0.870896i \(0.336461\pi\)
\(422\) 0 0
\(423\) −1.04328 −0.0507261
\(424\) 0 0
\(425\) 1.82170 0.0883654
\(426\) 0 0
\(427\) 32.2534 1.56085
\(428\) 0 0
\(429\) 2.46102 0.118819
\(430\) 0 0
\(431\) −21.3780 −1.02974 −0.514872 0.857267i \(-0.672161\pi\)
−0.514872 + 0.857267i \(0.672161\pi\)
\(432\) 0 0
\(433\) 0.303474 0.0145841 0.00729203 0.999973i \(-0.497679\pi\)
0.00729203 + 0.999973i \(0.497679\pi\)
\(434\) 0 0
\(435\) −33.1719 −1.59047
\(436\) 0 0
\(437\) −28.1487 −1.34653
\(438\) 0 0
\(439\) 4.21788 0.201308 0.100654 0.994921i \(-0.467906\pi\)
0.100654 + 0.994921i \(0.467906\pi\)
\(440\) 0 0
\(441\) −0.179669 −0.00855569
\(442\) 0 0
\(443\) 32.4183 1.54024 0.770121 0.637898i \(-0.220196\pi\)
0.770121 + 0.637898i \(0.220196\pi\)
\(444\) 0 0
\(445\) −15.7067 −0.744571
\(446\) 0 0
\(447\) −22.6636 −1.07195
\(448\) 0 0
\(449\) 19.4412 0.917485 0.458742 0.888569i \(-0.348300\pi\)
0.458742 + 0.888569i \(0.348300\pi\)
\(450\) 0 0
\(451\) −5.49204 −0.258610
\(452\) 0 0
\(453\) 40.0064 1.87966
\(454\) 0 0
\(455\) 11.4712 0.537779
\(456\) 0 0
\(457\) −2.33537 −0.109244 −0.0546219 0.998507i \(-0.517395\pi\)
−0.0546219 + 0.998507i \(0.517395\pi\)
\(458\) 0 0
\(459\) 22.2346 1.03782
\(460\) 0 0
\(461\) 34.2317 1.59433 0.797165 0.603762i \(-0.206332\pi\)
0.797165 + 0.603762i \(0.206332\pi\)
\(462\) 0 0
\(463\) 23.3015 1.08291 0.541457 0.840729i \(-0.317873\pi\)
0.541457 + 0.840729i \(0.317873\pi\)
\(464\) 0 0
\(465\) 19.8925 0.922492
\(466\) 0 0
\(467\) −25.0070 −1.15718 −0.578592 0.815617i \(-0.696398\pi\)
−0.578592 + 0.815617i \(0.696398\pi\)
\(468\) 0 0
\(469\) −29.8135 −1.37666
\(470\) 0 0
\(471\) −8.65153 −0.398642
\(472\) 0 0
\(473\) 1.24595 0.0572888
\(474\) 0 0
\(475\) −1.57593 −0.0723086
\(476\) 0 0
\(477\) −0.242272 −0.0110929
\(478\) 0 0
\(479\) −15.8036 −0.722085 −0.361042 0.932549i \(-0.617579\pi\)
−0.361042 + 0.932549i \(0.617579\pi\)
\(480\) 0 0
\(481\) −18.2831 −0.833636
\(482\) 0 0
\(483\) 30.4460 1.38534
\(484\) 0 0
\(485\) −33.1084 −1.50337
\(486\) 0 0
\(487\) 14.4205 0.653456 0.326728 0.945118i \(-0.394054\pi\)
0.326728 + 0.945118i \(0.394054\pi\)
\(488\) 0 0
\(489\) 23.4349 1.05976
\(490\) 0 0
\(491\) 7.85762 0.354610 0.177305 0.984156i \(-0.443262\pi\)
0.177305 + 0.984156i \(0.443262\pi\)
\(492\) 0 0
\(493\) −38.3345 −1.72650
\(494\) 0 0
\(495\) −0.141200 −0.00634645
\(496\) 0 0
\(497\) 18.3565 0.823400
\(498\) 0 0
\(499\) −19.0143 −0.851196 −0.425598 0.904912i \(-0.639936\pi\)
−0.425598 + 0.904912i \(0.639936\pi\)
\(500\) 0 0
\(501\) 1.76335 0.0787808
\(502\) 0 0
\(503\) 15.6269 0.696768 0.348384 0.937352i \(-0.386731\pi\)
0.348384 + 0.937352i \(0.386731\pi\)
\(504\) 0 0
\(505\) −15.5749 −0.693076
\(506\) 0 0
\(507\) −13.4728 −0.598346
\(508\) 0 0
\(509\) −32.0033 −1.41852 −0.709261 0.704946i \(-0.750971\pi\)
−0.709261 + 0.704946i \(0.750971\pi\)
\(510\) 0 0
\(511\) −38.4529 −1.70106
\(512\) 0 0
\(513\) −19.2349 −0.849242
\(514\) 0 0
\(515\) −32.3539 −1.42568
\(516\) 0 0
\(517\) −5.74822 −0.252806
\(518\) 0 0
\(519\) 2.57731 0.113131
\(520\) 0 0
\(521\) −38.6637 −1.69389 −0.846943 0.531684i \(-0.821559\pi\)
−0.846943 + 0.531684i \(0.821559\pi\)
\(522\) 0 0
\(523\) −32.8532 −1.43657 −0.718285 0.695749i \(-0.755073\pi\)
−0.718285 + 0.695749i \(0.755073\pi\)
\(524\) 0 0
\(525\) 1.70455 0.0743926
\(526\) 0 0
\(527\) 22.9884 1.00139
\(528\) 0 0
\(529\) 32.6398 1.41912
\(530\) 0 0
\(531\) −0.634262 −0.0275246
\(532\) 0 0
\(533\) −21.0906 −0.913537
\(534\) 0 0
\(535\) −14.3343 −0.619724
\(536\) 0 0
\(537\) −9.06348 −0.391118
\(538\) 0 0
\(539\) −0.989932 −0.0426394
\(540\) 0 0
\(541\) −2.89302 −0.124381 −0.0621904 0.998064i \(-0.519809\pi\)
−0.0621904 + 0.998064i \(0.519809\pi\)
\(542\) 0 0
\(543\) −39.9163 −1.71297
\(544\) 0 0
\(545\) 40.1219 1.71863
\(546\) 0 0
\(547\) −32.4045 −1.38552 −0.692758 0.721170i \(-0.743605\pi\)
−0.692758 + 0.721170i \(0.743605\pi\)
\(548\) 0 0
\(549\) −1.52460 −0.0650684
\(550\) 0 0
\(551\) 33.1627 1.41278
\(552\) 0 0
\(553\) 6.37085 0.270916
\(554\) 0 0
\(555\) 29.8105 1.26538
\(556\) 0 0
\(557\) 17.0178 0.721068 0.360534 0.932746i \(-0.382595\pi\)
0.360534 + 0.932746i \(0.382595\pi\)
\(558\) 0 0
\(559\) 4.78472 0.202372
\(560\) 0 0
\(561\) −4.63718 −0.195782
\(562\) 0 0
\(563\) 4.04420 0.170443 0.0852214 0.996362i \(-0.472840\pi\)
0.0852214 + 0.996362i \(0.472840\pi\)
\(564\) 0 0
\(565\) 7.37304 0.310186
\(566\) 0 0
\(567\) 21.5645 0.905625
\(568\) 0 0
\(569\) −37.4572 −1.57029 −0.785143 0.619315i \(-0.787411\pi\)
−0.785143 + 0.619315i \(0.787411\pi\)
\(570\) 0 0
\(571\) −27.5962 −1.15486 −0.577432 0.816439i \(-0.695945\pi\)
−0.577432 + 0.816439i \(0.695945\pi\)
\(572\) 0 0
\(573\) 24.0608 1.00515
\(574\) 0 0
\(575\) 3.11505 0.129907
\(576\) 0 0
\(577\) 5.00542 0.208378 0.104189 0.994558i \(-0.466775\pi\)
0.104189 + 0.994558i \(0.466775\pi\)
\(578\) 0 0
\(579\) −21.0910 −0.876511
\(580\) 0 0
\(581\) −16.6247 −0.689708
\(582\) 0 0
\(583\) −1.33486 −0.0552841
\(584\) 0 0
\(585\) −0.542238 −0.0224188
\(586\) 0 0
\(587\) −21.4526 −0.885444 −0.442722 0.896659i \(-0.645987\pi\)
−0.442722 + 0.896659i \(0.645987\pi\)
\(588\) 0 0
\(589\) −19.8870 −0.819428
\(590\) 0 0
\(591\) 5.92281 0.243632
\(592\) 0 0
\(593\) −23.5505 −0.967105 −0.483552 0.875315i \(-0.660654\pi\)
−0.483552 + 0.875315i \(0.660654\pi\)
\(594\) 0 0
\(595\) −21.6147 −0.886115
\(596\) 0 0
\(597\) −35.6962 −1.46095
\(598\) 0 0
\(599\) 28.2263 1.15329 0.576647 0.816994i \(-0.304361\pi\)
0.576647 + 0.816994i \(0.304361\pi\)
\(600\) 0 0
\(601\) 18.4269 0.751649 0.375825 0.926691i \(-0.377359\pi\)
0.375825 + 0.926691i \(0.377359\pi\)
\(602\) 0 0
\(603\) 1.40927 0.0573898
\(604\) 0 0
\(605\) 22.7692 0.925699
\(606\) 0 0
\(607\) 7.58224 0.307753 0.153877 0.988090i \(-0.450824\pi\)
0.153877 + 0.988090i \(0.450824\pi\)
\(608\) 0 0
\(609\) −35.8693 −1.45350
\(610\) 0 0
\(611\) −22.0744 −0.893035
\(612\) 0 0
\(613\) −29.9490 −1.20963 −0.604813 0.796367i \(-0.706752\pi\)
−0.604813 + 0.796367i \(0.706752\pi\)
\(614\) 0 0
\(615\) 34.3882 1.38667
\(616\) 0 0
\(617\) 21.9020 0.881743 0.440871 0.897570i \(-0.354669\pi\)
0.440871 + 0.897570i \(0.354669\pi\)
\(618\) 0 0
\(619\) 42.7587 1.71862 0.859308 0.511459i \(-0.170895\pi\)
0.859308 + 0.511459i \(0.170895\pi\)
\(620\) 0 0
\(621\) 38.0205 1.52571
\(622\) 0 0
\(623\) −16.9839 −0.680447
\(624\) 0 0
\(625\) −22.7375 −0.909502
\(626\) 0 0
\(627\) 4.01156 0.160206
\(628\) 0 0
\(629\) 34.4499 1.37361
\(630\) 0 0
\(631\) −31.2171 −1.24273 −0.621366 0.783521i \(-0.713422\pi\)
−0.621366 + 0.783521i \(0.713422\pi\)
\(632\) 0 0
\(633\) −7.06761 −0.280912
\(634\) 0 0
\(635\) −25.6048 −1.01610
\(636\) 0 0
\(637\) −3.80156 −0.150623
\(638\) 0 0
\(639\) −0.867699 −0.0343257
\(640\) 0 0
\(641\) 0.411286 0.0162448 0.00812241 0.999967i \(-0.497415\pi\)
0.00812241 + 0.999967i \(0.497415\pi\)
\(642\) 0 0
\(643\) 22.8285 0.900267 0.450134 0.892961i \(-0.351376\pi\)
0.450134 + 0.892961i \(0.351376\pi\)
\(644\) 0 0
\(645\) −7.80146 −0.307182
\(646\) 0 0
\(647\) 48.5055 1.90695 0.953475 0.301473i \(-0.0974783\pi\)
0.953475 + 0.301473i \(0.0974783\pi\)
\(648\) 0 0
\(649\) −3.49462 −0.137176
\(650\) 0 0
\(651\) 21.5100 0.843044
\(652\) 0 0
\(653\) −31.3576 −1.22712 −0.613558 0.789649i \(-0.710262\pi\)
−0.613558 + 0.789649i \(0.710262\pi\)
\(654\) 0 0
\(655\) −22.1763 −0.866501
\(656\) 0 0
\(657\) 1.81765 0.0709132
\(658\) 0 0
\(659\) −10.5454 −0.410792 −0.205396 0.978679i \(-0.565848\pi\)
−0.205396 + 0.978679i \(0.565848\pi\)
\(660\) 0 0
\(661\) 24.0755 0.936429 0.468215 0.883615i \(-0.344897\pi\)
0.468215 + 0.883615i \(0.344897\pi\)
\(662\) 0 0
\(663\) −17.8078 −0.691597
\(664\) 0 0
\(665\) 18.6986 0.725100
\(666\) 0 0
\(667\) −65.5509 −2.53814
\(668\) 0 0
\(669\) −29.7877 −1.15166
\(670\) 0 0
\(671\) −8.40016 −0.324284
\(672\) 0 0
\(673\) −42.3524 −1.63257 −0.816283 0.577652i \(-0.803969\pi\)
−0.816283 + 0.577652i \(0.803969\pi\)
\(674\) 0 0
\(675\) 2.12861 0.0819304
\(676\) 0 0
\(677\) −33.0861 −1.27160 −0.635801 0.771853i \(-0.719330\pi\)
−0.635801 + 0.771853i \(0.719330\pi\)
\(678\) 0 0
\(679\) −35.8006 −1.37390
\(680\) 0 0
\(681\) −38.4899 −1.47494
\(682\) 0 0
\(683\) 31.9254 1.22159 0.610796 0.791788i \(-0.290850\pi\)
0.610796 + 0.791788i \(0.290850\pi\)
\(684\) 0 0
\(685\) 16.2454 0.620706
\(686\) 0 0
\(687\) −17.2823 −0.659362
\(688\) 0 0
\(689\) −5.12615 −0.195291
\(690\) 0 0
\(691\) −31.5093 −1.19867 −0.599336 0.800498i \(-0.704569\pi\)
−0.599336 + 0.800498i \(0.704569\pi\)
\(692\) 0 0
\(693\) −0.152681 −0.00579988
\(694\) 0 0
\(695\) −39.3438 −1.49240
\(696\) 0 0
\(697\) 39.7401 1.50526
\(698\) 0 0
\(699\) −30.7250 −1.16213
\(700\) 0 0
\(701\) 0.352719 0.0133220 0.00666100 0.999978i \(-0.497880\pi\)
0.00666100 + 0.999978i \(0.497880\pi\)
\(702\) 0 0
\(703\) −29.8022 −1.12401
\(704\) 0 0
\(705\) 35.9923 1.35555
\(706\) 0 0
\(707\) −16.8414 −0.633386
\(708\) 0 0
\(709\) −16.1858 −0.607870 −0.303935 0.952693i \(-0.598301\pi\)
−0.303935 + 0.952693i \(0.598301\pi\)
\(710\) 0 0
\(711\) −0.301147 −0.0112939
\(712\) 0 0
\(713\) 39.3094 1.47215
\(714\) 0 0
\(715\) −2.98759 −0.111730
\(716\) 0 0
\(717\) −0.420310 −0.0156968
\(718\) 0 0
\(719\) −35.9403 −1.34035 −0.670174 0.742204i \(-0.733780\pi\)
−0.670174 + 0.742204i \(0.733780\pi\)
\(720\) 0 0
\(721\) −34.9847 −1.30290
\(722\) 0 0
\(723\) 31.2591 1.16254
\(724\) 0 0
\(725\) −3.66993 −0.136298
\(726\) 0 0
\(727\) −6.13113 −0.227391 −0.113696 0.993516i \(-0.536269\pi\)
−0.113696 + 0.993516i \(0.536269\pi\)
\(728\) 0 0
\(729\) 25.9447 0.960916
\(730\) 0 0
\(731\) −9.01562 −0.333455
\(732\) 0 0
\(733\) 21.2168 0.783660 0.391830 0.920038i \(-0.371842\pi\)
0.391830 + 0.920038i \(0.371842\pi\)
\(734\) 0 0
\(735\) 6.19842 0.228632
\(736\) 0 0
\(737\) 7.76469 0.286016
\(738\) 0 0
\(739\) 6.35371 0.233725 0.116862 0.993148i \(-0.462716\pi\)
0.116862 + 0.993148i \(0.462716\pi\)
\(740\) 0 0
\(741\) 15.4053 0.565928
\(742\) 0 0
\(743\) 16.1390 0.592082 0.296041 0.955175i \(-0.404333\pi\)
0.296041 + 0.955175i \(0.404333\pi\)
\(744\) 0 0
\(745\) 27.5128 1.00799
\(746\) 0 0
\(747\) 0.785840 0.0287524
\(748\) 0 0
\(749\) −15.4998 −0.566352
\(750\) 0 0
\(751\) −18.0766 −0.659626 −0.329813 0.944046i \(-0.606986\pi\)
−0.329813 + 0.944046i \(0.606986\pi\)
\(752\) 0 0
\(753\) 7.32526 0.266947
\(754\) 0 0
\(755\) −48.5664 −1.76751
\(756\) 0 0
\(757\) −7.73240 −0.281039 −0.140519 0.990078i \(-0.544877\pi\)
−0.140519 + 0.990078i \(0.544877\pi\)
\(758\) 0 0
\(759\) −7.92943 −0.287820
\(760\) 0 0
\(761\) −14.3809 −0.521307 −0.260653 0.965432i \(-0.583938\pi\)
−0.260653 + 0.965432i \(0.583938\pi\)
\(762\) 0 0
\(763\) 43.3844 1.57062
\(764\) 0 0
\(765\) 1.02171 0.0369401
\(766\) 0 0
\(767\) −13.4201 −0.484572
\(768\) 0 0
\(769\) 27.8199 1.00321 0.501606 0.865096i \(-0.332743\pi\)
0.501606 + 0.865096i \(0.332743\pi\)
\(770\) 0 0
\(771\) −14.3145 −0.515526
\(772\) 0 0
\(773\) −50.1686 −1.80444 −0.902220 0.431276i \(-0.858063\pi\)
−0.902220 + 0.431276i \(0.858063\pi\)
\(774\) 0 0
\(775\) 2.20077 0.0790542
\(776\) 0 0
\(777\) 32.2345 1.15641
\(778\) 0 0
\(779\) −34.3787 −1.23174
\(780\) 0 0
\(781\) −4.78080 −0.171070
\(782\) 0 0
\(783\) −44.7930 −1.60077
\(784\) 0 0
\(785\) 10.5027 0.374856
\(786\) 0 0
\(787\) 3.18585 0.113563 0.0567816 0.998387i \(-0.481916\pi\)
0.0567816 + 0.998387i \(0.481916\pi\)
\(788\) 0 0
\(789\) −0.343376 −0.0122245
\(790\) 0 0
\(791\) 7.97258 0.283472
\(792\) 0 0
\(793\) −32.2585 −1.14553
\(794\) 0 0
\(795\) 8.35816 0.296433
\(796\) 0 0
\(797\) −42.9946 −1.52295 −0.761473 0.648197i \(-0.775523\pi\)
−0.761473 + 0.648197i \(0.775523\pi\)
\(798\) 0 0
\(799\) 41.5938 1.47148
\(800\) 0 0
\(801\) 0.802821 0.0283663
\(802\) 0 0
\(803\) 10.0148 0.353414
\(804\) 0 0
\(805\) −36.9604 −1.30268
\(806\) 0 0
\(807\) −27.5888 −0.971172
\(808\) 0 0
\(809\) 6.74852 0.237265 0.118633 0.992938i \(-0.462149\pi\)
0.118633 + 0.992938i \(0.462149\pi\)
\(810\) 0 0
\(811\) 47.5200 1.66865 0.834326 0.551271i \(-0.185857\pi\)
0.834326 + 0.551271i \(0.185857\pi\)
\(812\) 0 0
\(813\) 13.5751 0.476101
\(814\) 0 0
\(815\) −28.4492 −0.996533
\(816\) 0 0
\(817\) 7.79930 0.272863
\(818\) 0 0
\(819\) −0.586330 −0.0204880
\(820\) 0 0
\(821\) −8.64018 −0.301544 −0.150772 0.988569i \(-0.548176\pi\)
−0.150772 + 0.988569i \(0.548176\pi\)
\(822\) 0 0
\(823\) 49.5776 1.72817 0.864084 0.503348i \(-0.167899\pi\)
0.864084 + 0.503348i \(0.167899\pi\)
\(824\) 0 0
\(825\) −0.443936 −0.0154559
\(826\) 0 0
\(827\) −41.5098 −1.44344 −0.721718 0.692187i \(-0.756647\pi\)
−0.721718 + 0.692187i \(0.756647\pi\)
\(828\) 0 0
\(829\) −34.3918 −1.19448 −0.597238 0.802064i \(-0.703735\pi\)
−0.597238 + 0.802064i \(0.703735\pi\)
\(830\) 0 0
\(831\) 35.3897 1.22766
\(832\) 0 0
\(833\) 7.16309 0.248186
\(834\) 0 0
\(835\) −2.14065 −0.0740803
\(836\) 0 0
\(837\) 26.8614 0.928466
\(838\) 0 0
\(839\) −21.9413 −0.757499 −0.378749 0.925499i \(-0.623646\pi\)
−0.378749 + 0.925499i \(0.623646\pi\)
\(840\) 0 0
\(841\) 48.2273 1.66301
\(842\) 0 0
\(843\) −6.24391 −0.215052
\(844\) 0 0
\(845\) 16.3555 0.562645
\(846\) 0 0
\(847\) 24.6207 0.845976
\(848\) 0 0
\(849\) −10.9708 −0.376517
\(850\) 0 0
\(851\) 58.9083 2.01935
\(852\) 0 0
\(853\) −29.6049 −1.01365 −0.506826 0.862048i \(-0.669181\pi\)
−0.506826 + 0.862048i \(0.669181\pi\)
\(854\) 0 0
\(855\) −0.883872 −0.0302278
\(856\) 0 0
\(857\) 44.9376 1.53504 0.767519 0.641026i \(-0.221491\pi\)
0.767519 + 0.641026i \(0.221491\pi\)
\(858\) 0 0
\(859\) 32.8861 1.12206 0.561029 0.827796i \(-0.310406\pi\)
0.561029 + 0.827796i \(0.310406\pi\)
\(860\) 0 0
\(861\) 37.1845 1.26724
\(862\) 0 0
\(863\) 35.2095 1.19855 0.599273 0.800545i \(-0.295457\pi\)
0.599273 + 0.800545i \(0.295457\pi\)
\(864\) 0 0
\(865\) −3.12877 −0.106381
\(866\) 0 0
\(867\) 3.57734 0.121493
\(868\) 0 0
\(869\) −1.65924 −0.0562858
\(870\) 0 0
\(871\) 29.8181 1.01035
\(872\) 0 0
\(873\) 1.69227 0.0572748
\(874\) 0 0
\(875\) −26.8443 −0.907502
\(876\) 0 0
\(877\) 30.3650 1.02535 0.512676 0.858582i \(-0.328654\pi\)
0.512676 + 0.858582i \(0.328654\pi\)
\(878\) 0 0
\(879\) −23.2314 −0.783577
\(880\) 0 0
\(881\) −17.2880 −0.582448 −0.291224 0.956655i \(-0.594062\pi\)
−0.291224 + 0.956655i \(0.594062\pi\)
\(882\) 0 0
\(883\) 6.02538 0.202770 0.101385 0.994847i \(-0.467673\pi\)
0.101385 + 0.994847i \(0.467673\pi\)
\(884\) 0 0
\(885\) 21.8814 0.735536
\(886\) 0 0
\(887\) 5.18060 0.173948 0.0869738 0.996211i \(-0.472280\pi\)
0.0869738 + 0.996211i \(0.472280\pi\)
\(888\) 0 0
\(889\) −27.6868 −0.928586
\(890\) 0 0
\(891\) −5.61632 −0.188154
\(892\) 0 0
\(893\) −35.9823 −1.20410
\(894\) 0 0
\(895\) 11.0028 0.367782
\(896\) 0 0
\(897\) −30.4507 −1.01672
\(898\) 0 0
\(899\) −46.3115 −1.54458
\(900\) 0 0
\(901\) 9.65895 0.321786
\(902\) 0 0
\(903\) −8.43583 −0.280727
\(904\) 0 0
\(905\) 48.4571 1.61077
\(906\) 0 0
\(907\) 32.1119 1.06626 0.533129 0.846034i \(-0.321016\pi\)
0.533129 + 0.846034i \(0.321016\pi\)
\(908\) 0 0
\(909\) 0.796084 0.0264044
\(910\) 0 0
\(911\) −8.93962 −0.296183 −0.148091 0.988974i \(-0.547313\pi\)
−0.148091 + 0.988974i \(0.547313\pi\)
\(912\) 0 0
\(913\) 4.32977 0.143295
\(914\) 0 0
\(915\) 52.5973 1.73881
\(916\) 0 0
\(917\) −23.9796 −0.791876
\(918\) 0 0
\(919\) 10.0469 0.331415 0.165708 0.986175i \(-0.447009\pi\)
0.165708 + 0.986175i \(0.447009\pi\)
\(920\) 0 0
\(921\) 23.8681 0.786480
\(922\) 0 0
\(923\) −18.3593 −0.604304
\(924\) 0 0
\(925\) 3.29803 0.108439
\(926\) 0 0
\(927\) 1.65371 0.0543149
\(928\) 0 0
\(929\) −3.81527 −0.125175 −0.0625875 0.998039i \(-0.519935\pi\)
−0.0625875 + 0.998039i \(0.519935\pi\)
\(930\) 0 0
\(931\) −6.19670 −0.203089
\(932\) 0 0
\(933\) −42.2216 −1.38227
\(934\) 0 0
\(935\) 5.62937 0.184100
\(936\) 0 0
\(937\) −10.5607 −0.345004 −0.172502 0.985009i \(-0.555185\pi\)
−0.172502 + 0.985009i \(0.555185\pi\)
\(938\) 0 0
\(939\) −16.2619 −0.530688
\(940\) 0 0
\(941\) 36.0712 1.17589 0.587944 0.808902i \(-0.299938\pi\)
0.587944 + 0.808902i \(0.299938\pi\)
\(942\) 0 0
\(943\) 67.9543 2.21290
\(944\) 0 0
\(945\) −25.2563 −0.821586
\(946\) 0 0
\(947\) 2.80219 0.0910589 0.0455294 0.998963i \(-0.485503\pi\)
0.0455294 + 0.998963i \(0.485503\pi\)
\(948\) 0 0
\(949\) 38.4589 1.24843
\(950\) 0 0
\(951\) 14.0267 0.454848
\(952\) 0 0
\(953\) −5.17253 −0.167555 −0.0837773 0.996485i \(-0.526698\pi\)
−0.0837773 + 0.996485i \(0.526698\pi\)
\(954\) 0 0
\(955\) −29.2089 −0.945179
\(956\) 0 0
\(957\) 9.34188 0.301980
\(958\) 0 0
\(959\) 17.5664 0.567249
\(960\) 0 0
\(961\) −3.22800 −0.104129
\(962\) 0 0
\(963\) 0.732669 0.0236099
\(964\) 0 0
\(965\) 25.6037 0.824213
\(966\) 0 0
\(967\) −1.37994 −0.0443760 −0.0221880 0.999754i \(-0.507063\pi\)
−0.0221880 + 0.999754i \(0.507063\pi\)
\(968\) 0 0
\(969\) −29.0275 −0.932496
\(970\) 0 0
\(971\) −17.0946 −0.548591 −0.274295 0.961645i \(-0.588445\pi\)
−0.274295 + 0.961645i \(0.588445\pi\)
\(972\) 0 0
\(973\) −42.5431 −1.36387
\(974\) 0 0
\(975\) −1.70481 −0.0545977
\(976\) 0 0
\(977\) 36.2402 1.15943 0.579714 0.814820i \(-0.303164\pi\)
0.579714 + 0.814820i \(0.303164\pi\)
\(978\) 0 0
\(979\) 4.42333 0.141370
\(980\) 0 0
\(981\) −2.05076 −0.0654756
\(982\) 0 0
\(983\) 19.1807 0.611770 0.305885 0.952068i \(-0.401048\pi\)
0.305885 + 0.952068i \(0.401048\pi\)
\(984\) 0 0
\(985\) −7.19009 −0.229095
\(986\) 0 0
\(987\) 38.9189 1.23880
\(988\) 0 0
\(989\) −15.4164 −0.490214
\(990\) 0 0
\(991\) −39.9073 −1.26770 −0.633849 0.773457i \(-0.718526\pi\)
−0.633849 + 0.773457i \(0.718526\pi\)
\(992\) 0 0
\(993\) −48.3845 −1.53544
\(994\) 0 0
\(995\) 43.3339 1.37378
\(996\) 0 0
\(997\) 2.95261 0.0935101 0.0467551 0.998906i \(-0.485112\pi\)
0.0467551 + 0.998906i \(0.485112\pi\)
\(998\) 0 0
\(999\) 40.2539 1.27358
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 1336.2.a.c.1.8 9
4.3 odd 2 2672.2.a.m.1.2 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.c.1.8 9 1.1 even 1 trivial
2672.2.a.m.1.2 9 4.3 odd 2