Properties

Label 1336.2.a.c.1.1
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 1.1
Root \(2.70492\) 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)
 
\(f(q)\) \(=\) \(q-2.70492 q^{3} -4.24392 q^{5} +1.27743 q^{7} +4.31658 q^{9} +O(q^{10})\) \(q-2.70492 q^{3} -4.24392 q^{5} +1.27743 q^{7} +4.31658 q^{9} -0.403841 q^{11} +1.48621 q^{13} +11.4794 q^{15} +3.32654 q^{17} +6.12675 q^{19} -3.45535 q^{21} -4.20028 q^{23} +13.0108 q^{25} -3.56123 q^{27} -9.76774 q^{29} +3.32050 q^{31} +1.09236 q^{33} -5.42131 q^{35} -5.09030 q^{37} -4.02007 q^{39} -4.16778 q^{41} +9.20912 q^{43} -18.3192 q^{45} +1.54157 q^{47} -5.36817 q^{49} -8.99803 q^{51} -0.932016 q^{53} +1.71387 q^{55} -16.5723 q^{57} -1.70002 q^{59} -9.98119 q^{61} +5.51413 q^{63} -6.30735 q^{65} -4.92232 q^{67} +11.3614 q^{69} +6.72094 q^{71} +7.05470 q^{73} -35.1932 q^{75} -0.515879 q^{77} -0.843450 q^{79} -3.31690 q^{81} +4.04828 q^{83} -14.1176 q^{85} +26.4209 q^{87} +1.42451 q^{89} +1.89853 q^{91} -8.98167 q^{93} -26.0014 q^{95} -14.5450 q^{97} -1.74321 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) 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
\(3\) −2.70492 −1.56168 −0.780842 0.624728i \(-0.785210\pi\)
−0.780842 + 0.624728i \(0.785210\pi\)
\(4\) 0 0
\(5\) −4.24392 −1.89794 −0.948969 0.315370i \(-0.897871\pi\)
−0.948969 + 0.315370i \(0.897871\pi\)
\(6\) 0 0
\(7\) 1.27743 0.482824 0.241412 0.970423i \(-0.422390\pi\)
0.241412 + 0.970423i \(0.422390\pi\)
\(8\) 0 0
\(9\) 4.31658 1.43886
\(10\) 0 0
\(11\) −0.403841 −0.121763 −0.0608813 0.998145i \(-0.519391\pi\)
−0.0608813 + 0.998145i \(0.519391\pi\)
\(12\) 0 0
\(13\) 1.48621 0.412200 0.206100 0.978531i \(-0.433923\pi\)
0.206100 + 0.978531i \(0.433923\pi\)
\(14\) 0 0
\(15\) 11.4794 2.96398
\(16\) 0 0
\(17\) 3.32654 0.806805 0.403403 0.915023i \(-0.367827\pi\)
0.403403 + 0.915023i \(0.367827\pi\)
\(18\) 0 0
\(19\) 6.12675 1.40557 0.702786 0.711401i \(-0.251939\pi\)
0.702786 + 0.711401i \(0.251939\pi\)
\(20\) 0 0
\(21\) −3.45535 −0.754019
\(22\) 0 0
\(23\) −4.20028 −0.875819 −0.437910 0.899019i \(-0.644281\pi\)
−0.437910 + 0.899019i \(0.644281\pi\)
\(24\) 0 0
\(25\) 13.0108 2.60217
\(26\) 0 0
\(27\) −3.56123 −0.685359
\(28\) 0 0
\(29\) −9.76774 −1.81382 −0.906912 0.421320i \(-0.861567\pi\)
−0.906912 + 0.421320i \(0.861567\pi\)
\(30\) 0 0
\(31\) 3.32050 0.596379 0.298189 0.954507i \(-0.403617\pi\)
0.298189 + 0.954507i \(0.403617\pi\)
\(32\) 0 0
\(33\) 1.09236 0.190155
\(34\) 0 0
\(35\) −5.42131 −0.916369
\(36\) 0 0
\(37\) −5.09030 −0.836839 −0.418420 0.908254i \(-0.637416\pi\)
−0.418420 + 0.908254i \(0.637416\pi\)
\(38\) 0 0
\(39\) −4.02007 −0.643727
\(40\) 0 0
\(41\) −4.16778 −0.650898 −0.325449 0.945560i \(-0.605515\pi\)
−0.325449 + 0.945560i \(0.605515\pi\)
\(42\) 0 0
\(43\) 9.20912 1.40438 0.702189 0.711991i \(-0.252206\pi\)
0.702189 + 0.711991i \(0.252206\pi\)
\(44\) 0 0
\(45\) −18.3192 −2.73086
\(46\) 0 0
\(47\) 1.54157 0.224860 0.112430 0.993660i \(-0.464137\pi\)
0.112430 + 0.993660i \(0.464137\pi\)
\(48\) 0 0
\(49\) −5.36817 −0.766881
\(50\) 0 0
\(51\) −8.99803 −1.25998
\(52\) 0 0
\(53\) −0.932016 −0.128022 −0.0640111 0.997949i \(-0.520389\pi\)
−0.0640111 + 0.997949i \(0.520389\pi\)
\(54\) 0 0
\(55\) 1.71387 0.231098
\(56\) 0 0
\(57\) −16.5723 −2.19506
\(58\) 0 0
\(59\) −1.70002 −0.221324 −0.110662 0.993858i \(-0.535297\pi\)
−0.110662 + 0.993858i \(0.535297\pi\)
\(60\) 0 0
\(61\) −9.98119 −1.27796 −0.638980 0.769223i \(-0.720643\pi\)
−0.638980 + 0.769223i \(0.720643\pi\)
\(62\) 0 0
\(63\) 5.51413 0.694715
\(64\) 0 0
\(65\) −6.30735 −0.782330
\(66\) 0 0
\(67\) −4.92232 −0.601357 −0.300679 0.953726i \(-0.597213\pi\)
−0.300679 + 0.953726i \(0.597213\pi\)
\(68\) 0 0
\(69\) 11.3614 1.36775
\(70\) 0 0
\(71\) 6.72094 0.797629 0.398814 0.917032i \(-0.369422\pi\)
0.398814 + 0.917032i \(0.369422\pi\)
\(72\) 0 0
\(73\) 7.05470 0.825690 0.412845 0.910801i \(-0.364535\pi\)
0.412845 + 0.910801i \(0.364535\pi\)
\(74\) 0 0
\(75\) −35.1932 −4.06376
\(76\) 0 0
\(77\) −0.515879 −0.0587899
\(78\) 0 0
\(79\) −0.843450 −0.0948956 −0.0474478 0.998874i \(-0.515109\pi\)
−0.0474478 + 0.998874i \(0.515109\pi\)
\(80\) 0 0
\(81\) −3.31690 −0.368545
\(82\) 0 0
\(83\) 4.04828 0.444357 0.222178 0.975006i \(-0.428683\pi\)
0.222178 + 0.975006i \(0.428683\pi\)
\(84\) 0 0
\(85\) −14.1176 −1.53127
\(86\) 0 0
\(87\) 26.4209 2.83262
\(88\) 0 0
\(89\) 1.42451 0.150997 0.0754986 0.997146i \(-0.475945\pi\)
0.0754986 + 0.997146i \(0.475945\pi\)
\(90\) 0 0
\(91\) 1.89853 0.199020
\(92\) 0 0
\(93\) −8.98167 −0.931356
\(94\) 0 0
\(95\) −26.0014 −2.66769
\(96\) 0 0
\(97\) −14.5450 −1.47682 −0.738409 0.674353i \(-0.764423\pi\)
−0.738409 + 0.674353i \(0.764423\pi\)
\(98\) 0 0
\(99\) −1.74321 −0.175199
\(100\) 0 0
\(101\) 15.7304 1.56524 0.782618 0.622503i \(-0.213884\pi\)
0.782618 + 0.622503i \(0.213884\pi\)
\(102\) 0 0
\(103\) −7.86506 −0.774967 −0.387484 0.921877i \(-0.626656\pi\)
−0.387484 + 0.921877i \(0.626656\pi\)
\(104\) 0 0
\(105\) 14.6642 1.43108
\(106\) 0 0
\(107\) −12.9508 −1.25200 −0.625999 0.779823i \(-0.715309\pi\)
−0.625999 + 0.779823i \(0.715309\pi\)
\(108\) 0 0
\(109\) −12.2339 −1.17180 −0.585898 0.810385i \(-0.699258\pi\)
−0.585898 + 0.810385i \(0.699258\pi\)
\(110\) 0 0
\(111\) 13.7688 1.30688
\(112\) 0 0
\(113\) −3.89621 −0.366525 −0.183262 0.983064i \(-0.558666\pi\)
−0.183262 + 0.983064i \(0.558666\pi\)
\(114\) 0 0
\(115\) 17.8256 1.66225
\(116\) 0 0
\(117\) 6.41533 0.593098
\(118\) 0 0
\(119\) 4.24943 0.389545
\(120\) 0 0
\(121\) −10.8369 −0.985174
\(122\) 0 0
\(123\) 11.2735 1.01650
\(124\) 0 0
\(125\) −33.9973 −3.04081
\(126\) 0 0
\(127\) −3.72218 −0.330290 −0.165145 0.986269i \(-0.552809\pi\)
−0.165145 + 0.986269i \(0.552809\pi\)
\(128\) 0 0
\(129\) −24.9099 −2.19320
\(130\) 0 0
\(131\) 17.5303 1.53163 0.765815 0.643061i \(-0.222336\pi\)
0.765815 + 0.643061i \(0.222336\pi\)
\(132\) 0 0
\(133\) 7.82650 0.678644
\(134\) 0 0
\(135\) 15.1136 1.30077
\(136\) 0 0
\(137\) 9.75708 0.833604 0.416802 0.908997i \(-0.363151\pi\)
0.416802 + 0.908997i \(0.363151\pi\)
\(138\) 0 0
\(139\) −5.10285 −0.432818 −0.216409 0.976303i \(-0.569435\pi\)
−0.216409 + 0.976303i \(0.569435\pi\)
\(140\) 0 0
\(141\) −4.16981 −0.351161
\(142\) 0 0
\(143\) −0.600192 −0.0501905
\(144\) 0 0
\(145\) 41.4535 3.44252
\(146\) 0 0
\(147\) 14.5204 1.19763
\(148\) 0 0
\(149\) −22.9309 −1.87857 −0.939287 0.343134i \(-0.888512\pi\)
−0.939287 + 0.343134i \(0.888512\pi\)
\(150\) 0 0
\(151\) −8.96454 −0.729524 −0.364762 0.931101i \(-0.618850\pi\)
−0.364762 + 0.931101i \(0.618850\pi\)
\(152\) 0 0
\(153\) 14.3593 1.16088
\(154\) 0 0
\(155\) −14.0919 −1.13189
\(156\) 0 0
\(157\) 3.77912 0.301607 0.150803 0.988564i \(-0.451814\pi\)
0.150803 + 0.988564i \(0.451814\pi\)
\(158\) 0 0
\(159\) 2.52103 0.199930
\(160\) 0 0
\(161\) −5.36557 −0.422866
\(162\) 0 0
\(163\) 14.8808 1.16555 0.582776 0.812633i \(-0.301966\pi\)
0.582776 + 0.812633i \(0.301966\pi\)
\(164\) 0 0
\(165\) −4.63587 −0.360902
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −10.7912 −0.830091
\(170\) 0 0
\(171\) 26.4466 2.02242
\(172\) 0 0
\(173\) 23.4547 1.78323 0.891615 0.452794i \(-0.149573\pi\)
0.891615 + 0.452794i \(0.149573\pi\)
\(174\) 0 0
\(175\) 16.6204 1.25639
\(176\) 0 0
\(177\) 4.59841 0.345638
\(178\) 0 0
\(179\) −9.88961 −0.739184 −0.369592 0.929194i \(-0.620503\pi\)
−0.369592 + 0.929194i \(0.620503\pi\)
\(180\) 0 0
\(181\) −8.80485 −0.654460 −0.327230 0.944945i \(-0.606115\pi\)
−0.327230 + 0.944945i \(0.606115\pi\)
\(182\) 0 0
\(183\) 26.9983 1.99577
\(184\) 0 0
\(185\) 21.6028 1.58827
\(186\) 0 0
\(187\) −1.34339 −0.0982387
\(188\) 0 0
\(189\) −4.54923 −0.330907
\(190\) 0 0
\(191\) 11.9396 0.863916 0.431958 0.901894i \(-0.357823\pi\)
0.431958 + 0.901894i \(0.357823\pi\)
\(192\) 0 0
\(193\) 10.9297 0.786738 0.393369 0.919381i \(-0.371310\pi\)
0.393369 + 0.919381i \(0.371310\pi\)
\(194\) 0 0
\(195\) 17.0609 1.22175
\(196\) 0 0
\(197\) −9.63480 −0.686451 −0.343226 0.939253i \(-0.611520\pi\)
−0.343226 + 0.939253i \(0.611520\pi\)
\(198\) 0 0
\(199\) 18.1255 1.28489 0.642443 0.766334i \(-0.277921\pi\)
0.642443 + 0.766334i \(0.277921\pi\)
\(200\) 0 0
\(201\) 13.3145 0.939130
\(202\) 0 0
\(203\) −12.4776 −0.875757
\(204\) 0 0
\(205\) 17.6877 1.23536
\(206\) 0 0
\(207\) −18.1308 −1.26018
\(208\) 0 0
\(209\) −2.47423 −0.171146
\(210\) 0 0
\(211\) −17.4655 −1.20237 −0.601187 0.799108i \(-0.705305\pi\)
−0.601187 + 0.799108i \(0.705305\pi\)
\(212\) 0 0
\(213\) −18.1796 −1.24564
\(214\) 0 0
\(215\) −39.0827 −2.66542
\(216\) 0 0
\(217\) 4.24171 0.287946
\(218\) 0 0
\(219\) −19.0824 −1.28947
\(220\) 0 0
\(221\) 4.94394 0.332565
\(222\) 0 0
\(223\) −0.166086 −0.0111219 −0.00556096 0.999985i \(-0.501770\pi\)
−0.00556096 + 0.999985i \(0.501770\pi\)
\(224\) 0 0
\(225\) 56.1622 3.74415
\(226\) 0 0
\(227\) −26.7551 −1.77580 −0.887898 0.460041i \(-0.847835\pi\)
−0.887898 + 0.460041i \(0.847835\pi\)
\(228\) 0 0
\(229\) 26.5428 1.75400 0.877000 0.480491i \(-0.159542\pi\)
0.877000 + 0.480491i \(0.159542\pi\)
\(230\) 0 0
\(231\) 1.39541 0.0918112
\(232\) 0 0
\(233\) 27.6384 1.81065 0.905325 0.424719i \(-0.139627\pi\)
0.905325 + 0.424719i \(0.139627\pi\)
\(234\) 0 0
\(235\) −6.54227 −0.426771
\(236\) 0 0
\(237\) 2.28146 0.148197
\(238\) 0 0
\(239\) −30.3260 −1.96163 −0.980813 0.194951i \(-0.937545\pi\)
−0.980813 + 0.194951i \(0.937545\pi\)
\(240\) 0 0
\(241\) −24.6851 −1.59011 −0.795054 0.606539i \(-0.792558\pi\)
−0.795054 + 0.606539i \(0.792558\pi\)
\(242\) 0 0
\(243\) 19.6556 1.26091
\(244\) 0 0
\(245\) 22.7821 1.45549
\(246\) 0 0
\(247\) 9.10563 0.579377
\(248\) 0 0
\(249\) −10.9503 −0.693945
\(250\) 0 0
\(251\) −26.1189 −1.64861 −0.824304 0.566147i \(-0.808434\pi\)
−0.824304 + 0.566147i \(0.808434\pi\)
\(252\) 0 0
\(253\) 1.69624 0.106642
\(254\) 0 0
\(255\) 38.1869 2.39135
\(256\) 0 0
\(257\) −3.37604 −0.210592 −0.105296 0.994441i \(-0.533579\pi\)
−0.105296 + 0.994441i \(0.533579\pi\)
\(258\) 0 0
\(259\) −6.50251 −0.404046
\(260\) 0 0
\(261\) −42.1632 −2.60984
\(262\) 0 0
\(263\) −25.2221 −1.55526 −0.777631 0.628721i \(-0.783579\pi\)
−0.777631 + 0.628721i \(0.783579\pi\)
\(264\) 0 0
\(265\) 3.95540 0.242978
\(266\) 0 0
\(267\) −3.85317 −0.235810
\(268\) 0 0
\(269\) −18.0869 −1.10278 −0.551390 0.834248i \(-0.685902\pi\)
−0.551390 + 0.834248i \(0.685902\pi\)
\(270\) 0 0
\(271\) −23.4229 −1.42284 −0.711421 0.702766i \(-0.751948\pi\)
−0.711421 + 0.702766i \(0.751948\pi\)
\(272\) 0 0
\(273\) −5.13537 −0.310807
\(274\) 0 0
\(275\) −5.25430 −0.316846
\(276\) 0 0
\(277\) 5.90502 0.354798 0.177399 0.984139i \(-0.443232\pi\)
0.177399 + 0.984139i \(0.443232\pi\)
\(278\) 0 0
\(279\) 14.3332 0.858105
\(280\) 0 0
\(281\) −30.9175 −1.84439 −0.922193 0.386730i \(-0.873604\pi\)
−0.922193 + 0.386730i \(0.873604\pi\)
\(282\) 0 0
\(283\) −3.55188 −0.211137 −0.105569 0.994412i \(-0.533666\pi\)
−0.105569 + 0.994412i \(0.533666\pi\)
\(284\) 0 0
\(285\) 70.3317 4.16609
\(286\) 0 0
\(287\) −5.32405 −0.314269
\(288\) 0 0
\(289\) −5.93410 −0.349065
\(290\) 0 0
\(291\) 39.3429 2.30632
\(292\) 0 0
\(293\) 19.2122 1.12239 0.561195 0.827683i \(-0.310342\pi\)
0.561195 + 0.827683i \(0.310342\pi\)
\(294\) 0 0
\(295\) 7.21475 0.420059
\(296\) 0 0
\(297\) 1.43817 0.0834510
\(298\) 0 0
\(299\) −6.24250 −0.361013
\(300\) 0 0
\(301\) 11.7640 0.678067
\(302\) 0 0
\(303\) −42.5495 −2.44440
\(304\) 0 0
\(305\) 42.3593 2.42549
\(306\) 0 0
\(307\) −1.07774 −0.0615096 −0.0307548 0.999527i \(-0.509791\pi\)
−0.0307548 + 0.999527i \(0.509791\pi\)
\(308\) 0 0
\(309\) 21.2743 1.21025
\(310\) 0 0
\(311\) 5.71693 0.324177 0.162089 0.986776i \(-0.448177\pi\)
0.162089 + 0.986776i \(0.448177\pi\)
\(312\) 0 0
\(313\) −11.6515 −0.658582 −0.329291 0.944228i \(-0.606810\pi\)
−0.329291 + 0.944228i \(0.606810\pi\)
\(314\) 0 0
\(315\) −23.4015 −1.31853
\(316\) 0 0
\(317\) −3.79584 −0.213196 −0.106598 0.994302i \(-0.533996\pi\)
−0.106598 + 0.994302i \(0.533996\pi\)
\(318\) 0 0
\(319\) 3.94461 0.220856
\(320\) 0 0
\(321\) 35.0308 1.95523
\(322\) 0 0
\(323\) 20.3809 1.13402
\(324\) 0 0
\(325\) 19.3368 1.07261
\(326\) 0 0
\(327\) 33.0917 1.82998
\(328\) 0 0
\(329\) 1.96924 0.108568
\(330\) 0 0
\(331\) 34.3425 1.88764 0.943818 0.330466i \(-0.107206\pi\)
0.943818 + 0.330466i \(0.107206\pi\)
\(332\) 0 0
\(333\) −21.9726 −1.20409
\(334\) 0 0
\(335\) 20.8899 1.14134
\(336\) 0 0
\(337\) −5.31445 −0.289497 −0.144748 0.989469i \(-0.546237\pi\)
−0.144748 + 0.989469i \(0.546237\pi\)
\(338\) 0 0
\(339\) 10.5389 0.572396
\(340\) 0 0
\(341\) −1.34095 −0.0726166
\(342\) 0 0
\(343\) −15.7995 −0.853092
\(344\) 0 0
\(345\) −48.2169 −2.59591
\(346\) 0 0
\(347\) 14.5429 0.780704 0.390352 0.920666i \(-0.372353\pi\)
0.390352 + 0.920666i \(0.372353\pi\)
\(348\) 0 0
\(349\) −11.9375 −0.638999 −0.319499 0.947587i \(-0.603515\pi\)
−0.319499 + 0.947587i \(0.603515\pi\)
\(350\) 0 0
\(351\) −5.29273 −0.282505
\(352\) 0 0
\(353\) 31.5607 1.67980 0.839902 0.542738i \(-0.182612\pi\)
0.839902 + 0.542738i \(0.182612\pi\)
\(354\) 0 0
\(355\) −28.5231 −1.51385
\(356\) 0 0
\(357\) −11.4944 −0.608346
\(358\) 0 0
\(359\) 16.8047 0.886916 0.443458 0.896295i \(-0.353752\pi\)
0.443458 + 0.896295i \(0.353752\pi\)
\(360\) 0 0
\(361\) 18.5371 0.975634
\(362\) 0 0
\(363\) 29.3129 1.53853
\(364\) 0 0
\(365\) −29.9396 −1.56711
\(366\) 0 0
\(367\) −9.86034 −0.514705 −0.257353 0.966318i \(-0.582850\pi\)
−0.257353 + 0.966318i \(0.582850\pi\)
\(368\) 0 0
\(369\) −17.9905 −0.936550
\(370\) 0 0
\(371\) −1.19059 −0.0618122
\(372\) 0 0
\(373\) −4.70042 −0.243379 −0.121689 0.992568i \(-0.538831\pi\)
−0.121689 + 0.992568i \(0.538831\pi\)
\(374\) 0 0
\(375\) 91.9599 4.74879
\(376\) 0 0
\(377\) −14.5169 −0.747659
\(378\) 0 0
\(379\) 14.2996 0.734523 0.367261 0.930118i \(-0.380295\pi\)
0.367261 + 0.930118i \(0.380295\pi\)
\(380\) 0 0
\(381\) 10.0682 0.515808
\(382\) 0 0
\(383\) −13.0374 −0.666179 −0.333089 0.942895i \(-0.608091\pi\)
−0.333089 + 0.942895i \(0.608091\pi\)
\(384\) 0 0
\(385\) 2.18935 0.111579
\(386\) 0 0
\(387\) 39.7519 2.02070
\(388\) 0 0
\(389\) 10.2817 0.521305 0.260652 0.965433i \(-0.416062\pi\)
0.260652 + 0.965433i \(0.416062\pi\)
\(390\) 0 0
\(391\) −13.9724 −0.706616
\(392\) 0 0
\(393\) −47.4180 −2.39192
\(394\) 0 0
\(395\) 3.57953 0.180106
\(396\) 0 0
\(397\) −18.1577 −0.911311 −0.455656 0.890156i \(-0.650595\pi\)
−0.455656 + 0.890156i \(0.650595\pi\)
\(398\) 0 0
\(399\) −21.1700 −1.05983
\(400\) 0 0
\(401\) −8.93888 −0.446386 −0.223193 0.974774i \(-0.571648\pi\)
−0.223193 + 0.974774i \(0.571648\pi\)
\(402\) 0 0
\(403\) 4.93495 0.245828
\(404\) 0 0
\(405\) 14.0767 0.699475
\(406\) 0 0
\(407\) 2.05567 0.101896
\(408\) 0 0
\(409\) −12.3707 −0.611694 −0.305847 0.952081i \(-0.598940\pi\)
−0.305847 + 0.952081i \(0.598940\pi\)
\(410\) 0 0
\(411\) −26.3921 −1.30183
\(412\) 0 0
\(413\) −2.17166 −0.106860
\(414\) 0 0
\(415\) −17.1806 −0.843362
\(416\) 0 0
\(417\) 13.8028 0.675926
\(418\) 0 0
\(419\) −29.2192 −1.42745 −0.713726 0.700425i \(-0.752994\pi\)
−0.713726 + 0.700425i \(0.752994\pi\)
\(420\) 0 0
\(421\) 4.49591 0.219117 0.109559 0.993980i \(-0.465056\pi\)
0.109559 + 0.993980i \(0.465056\pi\)
\(422\) 0 0
\(423\) 6.65428 0.323542
\(424\) 0 0
\(425\) 43.2811 2.09944
\(426\) 0 0
\(427\) −12.7503 −0.617030
\(428\) 0 0
\(429\) 1.62347 0.0783818
\(430\) 0 0
\(431\) 1.46354 0.0704962 0.0352481 0.999379i \(-0.488778\pi\)
0.0352481 + 0.999379i \(0.488778\pi\)
\(432\) 0 0
\(433\) −17.4259 −0.837434 −0.418717 0.908117i \(-0.637520\pi\)
−0.418717 + 0.908117i \(0.637520\pi\)
\(434\) 0 0
\(435\) −112.128 −5.37614
\(436\) 0 0
\(437\) −25.7341 −1.23103
\(438\) 0 0
\(439\) 3.55586 0.169712 0.0848559 0.996393i \(-0.472957\pi\)
0.0848559 + 0.996393i \(0.472957\pi\)
\(440\) 0 0
\(441\) −23.1721 −1.10343
\(442\) 0 0
\(443\) −28.4618 −1.35226 −0.676131 0.736781i \(-0.736345\pi\)
−0.676131 + 0.736781i \(0.736345\pi\)
\(444\) 0 0
\(445\) −6.04548 −0.286583
\(446\) 0 0
\(447\) 62.0262 2.93374
\(448\) 0 0
\(449\) −9.59661 −0.452892 −0.226446 0.974024i \(-0.572711\pi\)
−0.226446 + 0.974024i \(0.572711\pi\)
\(450\) 0 0
\(451\) 1.68312 0.0792550
\(452\) 0 0
\(453\) 24.2483 1.13929
\(454\) 0 0
\(455\) −8.05721 −0.377728
\(456\) 0 0
\(457\) −21.7289 −1.01644 −0.508218 0.861228i \(-0.669696\pi\)
−0.508218 + 0.861228i \(0.669696\pi\)
\(458\) 0 0
\(459\) −11.8466 −0.552951
\(460\) 0 0
\(461\) 28.8932 1.34569 0.672845 0.739783i \(-0.265072\pi\)
0.672845 + 0.739783i \(0.265072\pi\)
\(462\) 0 0
\(463\) −23.4238 −1.08860 −0.544299 0.838891i \(-0.683204\pi\)
−0.544299 + 0.838891i \(0.683204\pi\)
\(464\) 0 0
\(465\) 38.1175 1.76765
\(466\) 0 0
\(467\) 31.5852 1.46159 0.730793 0.682599i \(-0.239150\pi\)
0.730793 + 0.682599i \(0.239150\pi\)
\(468\) 0 0
\(469\) −6.28793 −0.290350
\(470\) 0 0
\(471\) −10.2222 −0.471015
\(472\) 0 0
\(473\) −3.71902 −0.171001
\(474\) 0 0
\(475\) 79.7141 3.65753
\(476\) 0 0
\(477\) −4.02312 −0.184206
\(478\) 0 0
\(479\) −33.7955 −1.54416 −0.772078 0.635527i \(-0.780783\pi\)
−0.772078 + 0.635527i \(0.780783\pi\)
\(480\) 0 0
\(481\) −7.56524 −0.344945
\(482\) 0 0
\(483\) 14.5134 0.660384
\(484\) 0 0
\(485\) 61.7277 2.80291
\(486\) 0 0
\(487\) 38.2331 1.73251 0.866253 0.499605i \(-0.166521\pi\)
0.866253 + 0.499605i \(0.166521\pi\)
\(488\) 0 0
\(489\) −40.2513 −1.82023
\(490\) 0 0
\(491\) 2.32648 0.104992 0.0524962 0.998621i \(-0.483282\pi\)
0.0524962 + 0.998621i \(0.483282\pi\)
\(492\) 0 0
\(493\) −32.4928 −1.46340
\(494\) 0 0
\(495\) 7.39803 0.332517
\(496\) 0 0
\(497\) 8.58554 0.385114
\(498\) 0 0
\(499\) −31.8384 −1.42528 −0.712642 0.701528i \(-0.752502\pi\)
−0.712642 + 0.701528i \(0.752502\pi\)
\(500\) 0 0
\(501\) −2.70492 −0.120847
\(502\) 0 0
\(503\) 4.97511 0.221829 0.110915 0.993830i \(-0.464622\pi\)
0.110915 + 0.993830i \(0.464622\pi\)
\(504\) 0 0
\(505\) −66.7586 −2.97072
\(506\) 0 0
\(507\) 29.1893 1.29634
\(508\) 0 0
\(509\) −34.2648 −1.51876 −0.759381 0.650647i \(-0.774498\pi\)
−0.759381 + 0.650647i \(0.774498\pi\)
\(510\) 0 0
\(511\) 9.01190 0.398663
\(512\) 0 0
\(513\) −21.8187 −0.963321
\(514\) 0 0
\(515\) 33.3787 1.47084
\(516\) 0 0
\(517\) −0.622547 −0.0273796
\(518\) 0 0
\(519\) −63.4431 −2.78484
\(520\) 0 0
\(521\) 10.8100 0.473596 0.236798 0.971559i \(-0.423902\pi\)
0.236798 + 0.971559i \(0.423902\pi\)
\(522\) 0 0
\(523\) −15.1590 −0.662856 −0.331428 0.943481i \(-0.607530\pi\)
−0.331428 + 0.943481i \(0.607530\pi\)
\(524\) 0 0
\(525\) −44.9569 −1.96208
\(526\) 0 0
\(527\) 11.0458 0.481162
\(528\) 0 0
\(529\) −5.35764 −0.232941
\(530\) 0 0
\(531\) −7.33827 −0.318454
\(532\) 0 0
\(533\) −6.19419 −0.268300
\(534\) 0 0
\(535\) 54.9620 2.37622
\(536\) 0 0
\(537\) 26.7506 1.15437
\(538\) 0 0
\(539\) 2.16788 0.0933774
\(540\) 0 0
\(541\) −7.68358 −0.330343 −0.165171 0.986265i \(-0.552818\pi\)
−0.165171 + 0.986265i \(0.552818\pi\)
\(542\) 0 0
\(543\) 23.8164 1.02206
\(544\) 0 0
\(545\) 51.9197 2.22400
\(546\) 0 0
\(547\) −5.79317 −0.247698 −0.123849 0.992301i \(-0.539524\pi\)
−0.123849 + 0.992301i \(0.539524\pi\)
\(548\) 0 0
\(549\) −43.0846 −1.83880
\(550\) 0 0
\(551\) −59.8445 −2.54946
\(552\) 0 0
\(553\) −1.07745 −0.0458178
\(554\) 0 0
\(555\) −58.4338 −2.48037
\(556\) 0 0
\(557\) −15.2484 −0.646094 −0.323047 0.946383i \(-0.604707\pi\)
−0.323047 + 0.946383i \(0.604707\pi\)
\(558\) 0 0
\(559\) 13.6867 0.578885
\(560\) 0 0
\(561\) 3.63377 0.153418
\(562\) 0 0
\(563\) 13.5679 0.571818 0.285909 0.958257i \(-0.407704\pi\)
0.285909 + 0.958257i \(0.407704\pi\)
\(564\) 0 0
\(565\) 16.5352 0.695641
\(566\) 0 0
\(567\) −4.23712 −0.177942
\(568\) 0 0
\(569\) 3.43479 0.143994 0.0719970 0.997405i \(-0.477063\pi\)
0.0719970 + 0.997405i \(0.477063\pi\)
\(570\) 0 0
\(571\) 43.1389 1.80531 0.902654 0.430368i \(-0.141616\pi\)
0.902654 + 0.430368i \(0.141616\pi\)
\(572\) 0 0
\(573\) −32.2955 −1.34916
\(574\) 0 0
\(575\) −54.6491 −2.27903
\(576\) 0 0
\(577\) 17.6883 0.736372 0.368186 0.929752i \(-0.379979\pi\)
0.368186 + 0.929752i \(0.379979\pi\)
\(578\) 0 0
\(579\) −29.5640 −1.22864
\(580\) 0 0
\(581\) 5.17141 0.214546
\(582\) 0 0
\(583\) 0.376386 0.0155883
\(584\) 0 0
\(585\) −27.2261 −1.12566
\(586\) 0 0
\(587\) 32.5290 1.34261 0.671307 0.741180i \(-0.265733\pi\)
0.671307 + 0.741180i \(0.265733\pi\)
\(588\) 0 0
\(589\) 20.3439 0.838254
\(590\) 0 0
\(591\) 26.0613 1.07202
\(592\) 0 0
\(593\) −1.04886 −0.0430716 −0.0215358 0.999768i \(-0.506856\pi\)
−0.0215358 + 0.999768i \(0.506856\pi\)
\(594\) 0 0
\(595\) −18.0342 −0.739332
\(596\) 0 0
\(597\) −49.0281 −2.00659
\(598\) 0 0
\(599\) −42.5697 −1.73935 −0.869675 0.493624i \(-0.835672\pi\)
−0.869675 + 0.493624i \(0.835672\pi\)
\(600\) 0 0
\(601\) −9.46471 −0.386073 −0.193037 0.981192i \(-0.561834\pi\)
−0.193037 + 0.981192i \(0.561834\pi\)
\(602\) 0 0
\(603\) −21.2476 −0.865268
\(604\) 0 0
\(605\) 45.9910 1.86980
\(606\) 0 0
\(607\) −34.1161 −1.38473 −0.692365 0.721548i \(-0.743431\pi\)
−0.692365 + 0.721548i \(0.743431\pi\)
\(608\) 0 0
\(609\) 33.7509 1.36766
\(610\) 0 0
\(611\) 2.29109 0.0926875
\(612\) 0 0
\(613\) 17.8616 0.721425 0.360712 0.932677i \(-0.382534\pi\)
0.360712 + 0.932677i \(0.382534\pi\)
\(614\) 0 0
\(615\) −47.8438 −1.92925
\(616\) 0 0
\(617\) −36.4070 −1.46569 −0.732846 0.680395i \(-0.761808\pi\)
−0.732846 + 0.680395i \(0.761808\pi\)
\(618\) 0 0
\(619\) 15.8973 0.638968 0.319484 0.947592i \(-0.396490\pi\)
0.319484 + 0.947592i \(0.396490\pi\)
\(620\) 0 0
\(621\) 14.9582 0.600250
\(622\) 0 0
\(623\) 1.81971 0.0729051
\(624\) 0 0
\(625\) 79.2276 3.16910
\(626\) 0 0
\(627\) 6.69259 0.267276
\(628\) 0 0
\(629\) −16.9331 −0.675167
\(630\) 0 0
\(631\) 17.2964 0.688557 0.344279 0.938868i \(-0.388124\pi\)
0.344279 + 0.938868i \(0.388124\pi\)
\(632\) 0 0
\(633\) 47.2427 1.87773
\(634\) 0 0
\(635\) 15.7966 0.626869
\(636\) 0 0
\(637\) −7.97822 −0.316109
\(638\) 0 0
\(639\) 29.0114 1.14767
\(640\) 0 0
\(641\) 19.0444 0.752207 0.376103 0.926578i \(-0.377264\pi\)
0.376103 + 0.926578i \(0.377264\pi\)
\(642\) 0 0
\(643\) −17.0174 −0.671100 −0.335550 0.942022i \(-0.608922\pi\)
−0.335550 + 0.942022i \(0.608922\pi\)
\(644\) 0 0
\(645\) 105.716 4.16255
\(646\) 0 0
\(647\) 11.0099 0.432844 0.216422 0.976300i \(-0.430561\pi\)
0.216422 + 0.976300i \(0.430561\pi\)
\(648\) 0 0
\(649\) 0.686537 0.0269489
\(650\) 0 0
\(651\) −11.4735 −0.449681
\(652\) 0 0
\(653\) −19.1819 −0.750644 −0.375322 0.926894i \(-0.622468\pi\)
−0.375322 + 0.926894i \(0.622468\pi\)
\(654\) 0 0
\(655\) −74.3972 −2.90694
\(656\) 0 0
\(657\) 30.4521 1.18805
\(658\) 0 0
\(659\) 30.3453 1.18209 0.591043 0.806640i \(-0.298717\pi\)
0.591043 + 0.806640i \(0.298717\pi\)
\(660\) 0 0
\(661\) −14.9037 −0.579685 −0.289842 0.957074i \(-0.593603\pi\)
−0.289842 + 0.957074i \(0.593603\pi\)
\(662\) 0 0
\(663\) −13.3729 −0.519362
\(664\) 0 0
\(665\) −33.2150 −1.28802
\(666\) 0 0
\(667\) 41.0273 1.58858
\(668\) 0 0
\(669\) 0.449248 0.0173689
\(670\) 0 0
\(671\) 4.03081 0.155608
\(672\) 0 0
\(673\) 12.6160 0.486313 0.243156 0.969987i \(-0.421817\pi\)
0.243156 + 0.969987i \(0.421817\pi\)
\(674\) 0 0
\(675\) −46.3345 −1.78342
\(676\) 0 0
\(677\) −11.9550 −0.459468 −0.229734 0.973253i \(-0.573786\pi\)
−0.229734 + 0.973253i \(0.573786\pi\)
\(678\) 0 0
\(679\) −18.5802 −0.713043
\(680\) 0 0
\(681\) 72.3702 2.77323
\(682\) 0 0
\(683\) −35.8260 −1.37084 −0.685421 0.728147i \(-0.740382\pi\)
−0.685421 + 0.728147i \(0.740382\pi\)
\(684\) 0 0
\(685\) −41.4082 −1.58213
\(686\) 0 0
\(687\) −71.7961 −2.73919
\(688\) 0 0
\(689\) −1.38517 −0.0527708
\(690\) 0 0
\(691\) 24.0214 0.913816 0.456908 0.889514i \(-0.348957\pi\)
0.456908 + 0.889514i \(0.348957\pi\)
\(692\) 0 0
\(693\) −2.22683 −0.0845903
\(694\) 0 0
\(695\) 21.6561 0.821462
\(696\) 0 0
\(697\) −13.8643 −0.525148
\(698\) 0 0
\(699\) −74.7595 −2.82767
\(700\) 0 0
\(701\) −25.0403 −0.945759 −0.472879 0.881127i \(-0.656785\pi\)
−0.472879 + 0.881127i \(0.656785\pi\)
\(702\) 0 0
\(703\) −31.1870 −1.17624
\(704\) 0 0
\(705\) 17.6963 0.666482
\(706\) 0 0
\(707\) 20.0945 0.755733
\(708\) 0 0
\(709\) 27.7976 1.04396 0.521980 0.852957i \(-0.325193\pi\)
0.521980 + 0.852957i \(0.325193\pi\)
\(710\) 0 0
\(711\) −3.64082 −0.136541
\(712\) 0 0
\(713\) −13.9470 −0.522320
\(714\) 0 0
\(715\) 2.54716 0.0952585
\(716\) 0 0
\(717\) 82.0293 3.06344
\(718\) 0 0
\(719\) 36.4516 1.35942 0.679708 0.733483i \(-0.262106\pi\)
0.679708 + 0.733483i \(0.262106\pi\)
\(720\) 0 0
\(721\) −10.0471 −0.374173
\(722\) 0 0
\(723\) 66.7712 2.48325
\(724\) 0 0
\(725\) −127.086 −4.71987
\(726\) 0 0
\(727\) −21.0330 −0.780070 −0.390035 0.920800i \(-0.627537\pi\)
−0.390035 + 0.920800i \(0.627537\pi\)
\(728\) 0 0
\(729\) −43.2161 −1.60060
\(730\) 0 0
\(731\) 30.6345 1.13306
\(732\) 0 0
\(733\) 11.5226 0.425596 0.212798 0.977096i \(-0.431742\pi\)
0.212798 + 0.977096i \(0.431742\pi\)
\(734\) 0 0
\(735\) −61.6236 −2.27302
\(736\) 0 0
\(737\) 1.98783 0.0732228
\(738\) 0 0
\(739\) −10.4455 −0.384245 −0.192123 0.981371i \(-0.561537\pi\)
−0.192123 + 0.981371i \(0.561537\pi\)
\(740\) 0 0
\(741\) −24.6300 −0.904805
\(742\) 0 0
\(743\) −29.0541 −1.06589 −0.532946 0.846149i \(-0.678915\pi\)
−0.532946 + 0.846149i \(0.678915\pi\)
\(744\) 0 0
\(745\) 97.3169 3.56541
\(746\) 0 0
\(747\) 17.4747 0.639367
\(748\) 0 0
\(749\) −16.5437 −0.604495
\(750\) 0 0
\(751\) −36.3643 −1.32695 −0.663476 0.748198i \(-0.730920\pi\)
−0.663476 + 0.748198i \(0.730920\pi\)
\(752\) 0 0
\(753\) 70.6494 2.57461
\(754\) 0 0
\(755\) 38.0448 1.38459
\(756\) 0 0
\(757\) −36.7704 −1.33644 −0.668221 0.743963i \(-0.732944\pi\)
−0.668221 + 0.743963i \(0.732944\pi\)
\(758\) 0 0
\(759\) −4.58820 −0.166541
\(760\) 0 0
\(761\) −28.6610 −1.03896 −0.519481 0.854482i \(-0.673875\pi\)
−0.519481 + 0.854482i \(0.673875\pi\)
\(762\) 0 0
\(763\) −15.6280 −0.565771
\(764\) 0 0
\(765\) −60.9396 −2.20328
\(766\) 0 0
\(767\) −2.52659 −0.0912297
\(768\) 0 0
\(769\) −40.6341 −1.46530 −0.732652 0.680604i \(-0.761718\pi\)
−0.732652 + 0.680604i \(0.761718\pi\)
\(770\) 0 0
\(771\) 9.13191 0.328878
\(772\) 0 0
\(773\) −12.1122 −0.435646 −0.217823 0.975988i \(-0.569895\pi\)
−0.217823 + 0.975988i \(0.569895\pi\)
\(774\) 0 0
\(775\) 43.2024 1.55188
\(776\) 0 0
\(777\) 17.5887 0.630992
\(778\) 0 0
\(779\) −25.5349 −0.914884
\(780\) 0 0
\(781\) −2.71419 −0.0971213
\(782\) 0 0
\(783\) 34.7852 1.24312
\(784\) 0 0
\(785\) −16.0383 −0.572431
\(786\) 0 0
\(787\) 43.2132 1.54038 0.770191 0.637813i \(-0.220161\pi\)
0.770191 + 0.637813i \(0.220161\pi\)
\(788\) 0 0
\(789\) 68.2237 2.42883
\(790\) 0 0
\(791\) −4.97714 −0.176967
\(792\) 0 0
\(793\) −14.8341 −0.526776
\(794\) 0 0
\(795\) −10.6990 −0.379455
\(796\) 0 0
\(797\) 32.6861 1.15780 0.578901 0.815398i \(-0.303482\pi\)
0.578901 + 0.815398i \(0.303482\pi\)
\(798\) 0 0
\(799\) 5.12808 0.181419
\(800\) 0 0
\(801\) 6.14899 0.217264
\(802\) 0 0
\(803\) −2.84897 −0.100538
\(804\) 0 0
\(805\) 22.7710 0.802574
\(806\) 0 0
\(807\) 48.9237 1.72219
\(808\) 0 0
\(809\) −10.4760 −0.368316 −0.184158 0.982897i \(-0.558956\pi\)
−0.184158 + 0.982897i \(0.558956\pi\)
\(810\) 0 0
\(811\) −12.9589 −0.455049 −0.227525 0.973772i \(-0.573063\pi\)
−0.227525 + 0.973772i \(0.573063\pi\)
\(812\) 0 0
\(813\) 63.3571 2.22203
\(814\) 0 0
\(815\) −63.1528 −2.21215
\(816\) 0 0
\(817\) 56.4220 1.97395
\(818\) 0 0
\(819\) 8.19515 0.286362
\(820\) 0 0
\(821\) −16.4621 −0.574532 −0.287266 0.957851i \(-0.592746\pi\)
−0.287266 + 0.957851i \(0.592746\pi\)
\(822\) 0 0
\(823\) 42.7238 1.48926 0.744630 0.667478i \(-0.232626\pi\)
0.744630 + 0.667478i \(0.232626\pi\)
\(824\) 0 0
\(825\) 14.2124 0.494814
\(826\) 0 0
\(827\) 33.0454 1.14910 0.574550 0.818469i \(-0.305177\pi\)
0.574550 + 0.818469i \(0.305177\pi\)
\(828\) 0 0
\(829\) 18.0587 0.627206 0.313603 0.949554i \(-0.398464\pi\)
0.313603 + 0.949554i \(0.398464\pi\)
\(830\) 0 0
\(831\) −15.9726 −0.554083
\(832\) 0 0
\(833\) −17.8574 −0.618724
\(834\) 0 0
\(835\) −4.24392 −0.146867
\(836\) 0 0
\(837\) −11.8250 −0.408733
\(838\) 0 0
\(839\) −28.1830 −0.972986 −0.486493 0.873685i \(-0.661724\pi\)
−0.486493 + 0.873685i \(0.661724\pi\)
\(840\) 0 0
\(841\) 66.4088 2.28996
\(842\) 0 0
\(843\) 83.6294 2.88035
\(844\) 0 0
\(845\) 45.7969 1.57546
\(846\) 0 0
\(847\) −13.8434 −0.475665
\(848\) 0 0
\(849\) 9.60755 0.329730
\(850\) 0 0
\(851\) 21.3807 0.732920
\(852\) 0 0
\(853\) 42.4867 1.45472 0.727359 0.686257i \(-0.240747\pi\)
0.727359 + 0.686257i \(0.240747\pi\)
\(854\) 0 0
\(855\) −112.237 −3.83843
\(856\) 0 0
\(857\) 49.9048 1.70471 0.852357 0.522960i \(-0.175172\pi\)
0.852357 + 0.522960i \(0.175172\pi\)
\(858\) 0 0
\(859\) 8.47277 0.289087 0.144544 0.989498i \(-0.453829\pi\)
0.144544 + 0.989498i \(0.453829\pi\)
\(860\) 0 0
\(861\) 14.4011 0.490789
\(862\) 0 0
\(863\) 1.46856 0.0499902 0.0249951 0.999688i \(-0.492043\pi\)
0.0249951 + 0.999688i \(0.492043\pi\)
\(864\) 0 0
\(865\) −99.5399 −3.38446
\(866\) 0 0
\(867\) 16.0513 0.545129
\(868\) 0 0
\(869\) 0.340619 0.0115547
\(870\) 0 0
\(871\) −7.31560 −0.247880
\(872\) 0 0
\(873\) −62.7845 −2.12493
\(874\) 0 0
\(875\) −43.4292 −1.46818
\(876\) 0 0
\(877\) −39.0735 −1.31942 −0.659709 0.751521i \(-0.729321\pi\)
−0.659709 + 0.751521i \(0.729321\pi\)
\(878\) 0 0
\(879\) −51.9675 −1.75282
\(880\) 0 0
\(881\) −23.4707 −0.790747 −0.395374 0.918520i \(-0.629385\pi\)
−0.395374 + 0.918520i \(0.629385\pi\)
\(882\) 0 0
\(883\) 8.49116 0.285750 0.142875 0.989741i \(-0.454365\pi\)
0.142875 + 0.989741i \(0.454365\pi\)
\(884\) 0 0
\(885\) −19.5153 −0.655999
\(886\) 0 0
\(887\) −8.79252 −0.295224 −0.147612 0.989045i \(-0.547159\pi\)
−0.147612 + 0.989045i \(0.547159\pi\)
\(888\) 0 0
\(889\) −4.75483 −0.159472
\(890\) 0 0
\(891\) 1.33950 0.0448749
\(892\) 0 0
\(893\) 9.44478 0.316058
\(894\) 0 0
\(895\) 41.9707 1.40293
\(896\) 0 0
\(897\) 16.8854 0.563788
\(898\) 0 0
\(899\) −32.4338 −1.08173
\(900\) 0 0
\(901\) −3.10039 −0.103289
\(902\) 0 0
\(903\) −31.8207 −1.05893
\(904\) 0 0
\(905\) 37.3671 1.24212
\(906\) 0 0
\(907\) 19.8502 0.659116 0.329558 0.944135i \(-0.393100\pi\)
0.329558 + 0.944135i \(0.393100\pi\)
\(908\) 0 0
\(909\) 67.9016 2.25215
\(910\) 0 0
\(911\) 28.6417 0.948942 0.474471 0.880271i \(-0.342639\pi\)
0.474471 + 0.880271i \(0.342639\pi\)
\(912\) 0 0
\(913\) −1.63486 −0.0541060
\(914\) 0 0
\(915\) −114.579 −3.78785
\(916\) 0 0
\(917\) 22.3938 0.739508
\(918\) 0 0
\(919\) 2.99821 0.0989018 0.0494509 0.998777i \(-0.484253\pi\)
0.0494509 + 0.998777i \(0.484253\pi\)
\(920\) 0 0
\(921\) 2.91519 0.0960586
\(922\) 0 0
\(923\) 9.98872 0.328783
\(924\) 0 0
\(925\) −66.2290 −2.17760
\(926\) 0 0
\(927\) −33.9501 −1.11507
\(928\) 0 0
\(929\) 9.34190 0.306498 0.153249 0.988188i \(-0.451026\pi\)
0.153249 + 0.988188i \(0.451026\pi\)
\(930\) 0 0
\(931\) −32.8894 −1.07791
\(932\) 0 0
\(933\) −15.4638 −0.506262
\(934\) 0 0
\(935\) 5.70125 0.186451
\(936\) 0 0
\(937\) −37.3671 −1.22073 −0.610365 0.792121i \(-0.708977\pi\)
−0.610365 + 0.792121i \(0.708977\pi\)
\(938\) 0 0
\(939\) 31.5164 1.02850
\(940\) 0 0
\(941\) −11.7055 −0.381587 −0.190794 0.981630i \(-0.561106\pi\)
−0.190794 + 0.981630i \(0.561106\pi\)
\(942\) 0 0
\(943\) 17.5058 0.570069
\(944\) 0 0
\(945\) 19.3065 0.628042
\(946\) 0 0
\(947\) −33.8526 −1.10006 −0.550030 0.835145i \(-0.685384\pi\)
−0.550030 + 0.835145i \(0.685384\pi\)
\(948\) 0 0
\(949\) 10.4848 0.340350
\(950\) 0 0
\(951\) 10.2674 0.332944
\(952\) 0 0
\(953\) 21.3850 0.692728 0.346364 0.938100i \(-0.387416\pi\)
0.346364 + 0.938100i \(0.387416\pi\)
\(954\) 0 0
\(955\) −50.6705 −1.63966
\(956\) 0 0
\(957\) −10.6698 −0.344907
\(958\) 0 0
\(959\) 12.4640 0.402484
\(960\) 0 0
\(961\) −19.9743 −0.644332
\(962\) 0 0
\(963\) −55.9030 −1.80145
\(964\) 0 0
\(965\) −46.3848 −1.49318
\(966\) 0 0
\(967\) −38.4901 −1.23776 −0.618879 0.785486i \(-0.712413\pi\)
−0.618879 + 0.785486i \(0.712413\pi\)
\(968\) 0 0
\(969\) −55.1286 −1.77099
\(970\) 0 0
\(971\) −11.1981 −0.359365 −0.179682 0.983725i \(-0.557507\pi\)
−0.179682 + 0.983725i \(0.557507\pi\)
\(972\) 0 0
\(973\) −6.51855 −0.208975
\(974\) 0 0
\(975\) −52.3045 −1.67508
\(976\) 0 0
\(977\) 45.5724 1.45799 0.728996 0.684518i \(-0.239987\pi\)
0.728996 + 0.684518i \(0.239987\pi\)
\(978\) 0 0
\(979\) −0.575273 −0.0183858
\(980\) 0 0
\(981\) −52.8086 −1.68605
\(982\) 0 0
\(983\) 53.6365 1.71074 0.855370 0.518018i \(-0.173330\pi\)
0.855370 + 0.518018i \(0.173330\pi\)
\(984\) 0 0
\(985\) 40.8893 1.30284
\(986\) 0 0
\(987\) −5.32664 −0.169549
\(988\) 0 0
\(989\) −38.6809 −1.22998
\(990\) 0 0
\(991\) 27.2527 0.865710 0.432855 0.901464i \(-0.357506\pi\)
0.432855 + 0.901464i \(0.357506\pi\)
\(992\) 0 0
\(993\) −92.8937 −2.94789
\(994\) 0 0
\(995\) −76.9233 −2.43863
\(996\) 0 0
\(997\) 33.3821 1.05722 0.528610 0.848865i \(-0.322713\pi\)
0.528610 + 0.848865i \(0.322713\pi\)
\(998\) 0 0
\(999\) 18.1277 0.573535
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.1.1 9
4.3 odd 2 2672.2.a.m.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.c.1.1 9 1.1 even 1 trivial
2672.2.a.m.1.9 9 4.3 odd 2