Properties

Label 6032.2.a.bc.1.11
Level $6032$
Weight $2$
Character 6032.1
Self dual yes
Analytic conductor $48.166$
Analytic rank $0$
Dimension $11$
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] = [6032,2,Mod(1,6032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6032.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: \(48.1657624992\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 5x^{10} - 9x^{9} + 65x^{8} + 19x^{7} - 298x^{6} + 17x^{5} + 541x^{4} - 60x^{3} - 287x^{2} + 16x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 3016)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-2.12599\) of defining polynomial
Character \(\chi\) \(=\) 6032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.12599 q^{3} -3.76339 q^{5} +2.42274 q^{7} +6.77179 q^{9} +O(q^{10})\) \(q+3.12599 q^{3} -3.76339 q^{5} +2.42274 q^{7} +6.77179 q^{9} +6.57467 q^{11} +1.00000 q^{13} -11.7643 q^{15} -1.11681 q^{17} -3.43474 q^{19} +7.57346 q^{21} -1.69657 q^{23} +9.16312 q^{25} +11.7906 q^{27} -1.00000 q^{29} -0.408632 q^{31} +20.5523 q^{33} -9.11773 q^{35} +11.7164 q^{37} +3.12599 q^{39} +3.63876 q^{41} +8.08949 q^{43} -25.4849 q^{45} -5.68287 q^{47} -1.13032 q^{49} -3.49113 q^{51} +1.27099 q^{53} -24.7431 q^{55} -10.7370 q^{57} -7.03303 q^{59} -8.02110 q^{61} +16.4063 q^{63} -3.76339 q^{65} -0.218288 q^{67} -5.30347 q^{69} +14.8425 q^{71} +6.83100 q^{73} +28.6438 q^{75} +15.9287 q^{77} -10.2164 q^{79} +16.5418 q^{81} -9.23571 q^{83} +4.20299 q^{85} -3.12599 q^{87} +6.26275 q^{89} +2.42274 q^{91} -1.27738 q^{93} +12.9263 q^{95} +14.3964 q^{97} +44.5223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 11 q + 6 q^{3} - 2 q^{5} + 3 q^{7} + 11 q^{9} + 13 q^{11} + 11 q^{13} + 8 q^{15} - 6 q^{17} + 12 q^{19} + q^{21} + 13 q^{23} + 11 q^{25} + 24 q^{27} - 11 q^{29} + 11 q^{31} + 17 q^{33} + 4 q^{35} + 11 q^{37} + 6 q^{39} - 9 q^{41} + 30 q^{43} - 16 q^{45} + q^{47} - 4 q^{49} + 13 q^{51} - 9 q^{53} + q^{55} + 2 q^{57} + 9 q^{59} - 5 q^{61} + 6 q^{63} - 2 q^{65} + 25 q^{67} + 26 q^{71} + 10 q^{73} + 41 q^{75} - 8 q^{77} + 14 q^{79} + 3 q^{81} + 6 q^{83} + 19 q^{85} - 6 q^{87} - 11 q^{89} + 3 q^{91} - 3 q^{93} + 31 q^{95} + 12 q^{97} + 35 q^{99}+O(q^{100}) \) 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
\(3\) 3.12599 1.80479 0.902395 0.430911i \(-0.141808\pi\)
0.902395 + 0.430911i \(0.141808\pi\)
\(4\) 0 0
\(5\) −3.76339 −1.68304 −0.841520 0.540226i \(-0.818339\pi\)
−0.841520 + 0.540226i \(0.818339\pi\)
\(6\) 0 0
\(7\) 2.42274 0.915711 0.457855 0.889027i \(-0.348618\pi\)
0.457855 + 0.889027i \(0.348618\pi\)
\(8\) 0 0
\(9\) 6.77179 2.25726
\(10\) 0 0
\(11\) 6.57467 1.98234 0.991169 0.132606i \(-0.0423345\pi\)
0.991169 + 0.132606i \(0.0423345\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −11.7643 −3.03753
\(16\) 0 0
\(17\) −1.11681 −0.270866 −0.135433 0.990786i \(-0.543243\pi\)
−0.135433 + 0.990786i \(0.543243\pi\)
\(18\) 0 0
\(19\) −3.43474 −0.787984 −0.393992 0.919114i \(-0.628906\pi\)
−0.393992 + 0.919114i \(0.628906\pi\)
\(20\) 0 0
\(21\) 7.57346 1.65266
\(22\) 0 0
\(23\) −1.69657 −0.353760 −0.176880 0.984232i \(-0.556600\pi\)
−0.176880 + 0.984232i \(0.556600\pi\)
\(24\) 0 0
\(25\) 9.16312 1.83262
\(26\) 0 0
\(27\) 11.7906 2.26910
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −0.408632 −0.0733924 −0.0366962 0.999326i \(-0.511683\pi\)
−0.0366962 + 0.999326i \(0.511683\pi\)
\(32\) 0 0
\(33\) 20.5523 3.57770
\(34\) 0 0
\(35\) −9.11773 −1.54118
\(36\) 0 0
\(37\) 11.7164 1.92616 0.963080 0.269216i \(-0.0867645\pi\)
0.963080 + 0.269216i \(0.0867645\pi\)
\(38\) 0 0
\(39\) 3.12599 0.500558
\(40\) 0 0
\(41\) 3.63876 0.568278 0.284139 0.958783i \(-0.408292\pi\)
0.284139 + 0.958783i \(0.408292\pi\)
\(42\) 0 0
\(43\) 8.08949 1.23364 0.616818 0.787106i \(-0.288421\pi\)
0.616818 + 0.787106i \(0.288421\pi\)
\(44\) 0 0
\(45\) −25.4849 −3.79907
\(46\) 0 0
\(47\) −5.68287 −0.828931 −0.414466 0.910065i \(-0.636031\pi\)
−0.414466 + 0.910065i \(0.636031\pi\)
\(48\) 0 0
\(49\) −1.13032 −0.161474
\(50\) 0 0
\(51\) −3.49113 −0.488856
\(52\) 0 0
\(53\) 1.27099 0.174584 0.0872918 0.996183i \(-0.472179\pi\)
0.0872918 + 0.996183i \(0.472179\pi\)
\(54\) 0 0
\(55\) −24.7431 −3.33635
\(56\) 0 0
\(57\) −10.7370 −1.42215
\(58\) 0 0
\(59\) −7.03303 −0.915623 −0.457811 0.889049i \(-0.651366\pi\)
−0.457811 + 0.889049i \(0.651366\pi\)
\(60\) 0 0
\(61\) −8.02110 −1.02700 −0.513498 0.858091i \(-0.671651\pi\)
−0.513498 + 0.858091i \(0.671651\pi\)
\(62\) 0 0
\(63\) 16.4063 2.06700
\(64\) 0 0
\(65\) −3.76339 −0.466791
\(66\) 0 0
\(67\) −0.218288 −0.0266681 −0.0133340 0.999911i \(-0.504244\pi\)
−0.0133340 + 0.999911i \(0.504244\pi\)
\(68\) 0 0
\(69\) −5.30347 −0.638462
\(70\) 0 0
\(71\) 14.8425 1.76148 0.880738 0.473604i \(-0.157047\pi\)
0.880738 + 0.473604i \(0.157047\pi\)
\(72\) 0 0
\(73\) 6.83100 0.799508 0.399754 0.916623i \(-0.369096\pi\)
0.399754 + 0.916623i \(0.369096\pi\)
\(74\) 0 0
\(75\) 28.6438 3.30750
\(76\) 0 0
\(77\) 15.9287 1.81525
\(78\) 0 0
\(79\) −10.2164 −1.14944 −0.574718 0.818352i \(-0.694888\pi\)
−0.574718 + 0.818352i \(0.694888\pi\)
\(80\) 0 0
\(81\) 16.5418 1.83798
\(82\) 0 0
\(83\) −9.23571 −1.01375 −0.506876 0.862019i \(-0.669200\pi\)
−0.506876 + 0.862019i \(0.669200\pi\)
\(84\) 0 0
\(85\) 4.20299 0.455879
\(86\) 0 0
\(87\) −3.12599 −0.335141
\(88\) 0 0
\(89\) 6.26275 0.663850 0.331925 0.943306i \(-0.392302\pi\)
0.331925 + 0.943306i \(0.392302\pi\)
\(90\) 0 0
\(91\) 2.42274 0.253972
\(92\) 0 0
\(93\) −1.27738 −0.132458
\(94\) 0 0
\(95\) 12.9263 1.32621
\(96\) 0 0
\(97\) 14.3964 1.46173 0.730865 0.682522i \(-0.239117\pi\)
0.730865 + 0.682522i \(0.239117\pi\)
\(98\) 0 0
\(99\) 44.5223 4.47466
\(100\) 0 0
\(101\) −13.0169 −1.29523 −0.647613 0.761970i \(-0.724232\pi\)
−0.647613 + 0.761970i \(0.724232\pi\)
\(102\) 0 0
\(103\) −12.4214 −1.22392 −0.611959 0.790890i \(-0.709618\pi\)
−0.611959 + 0.790890i \(0.709618\pi\)
\(104\) 0 0
\(105\) −28.5019 −2.78150
\(106\) 0 0
\(107\) 6.23328 0.602594 0.301297 0.953530i \(-0.402580\pi\)
0.301297 + 0.953530i \(0.402580\pi\)
\(108\) 0 0
\(109\) −4.49199 −0.430255 −0.215127 0.976586i \(-0.569017\pi\)
−0.215127 + 0.976586i \(0.569017\pi\)
\(110\) 0 0
\(111\) 36.6252 3.47631
\(112\) 0 0
\(113\) −9.13873 −0.859699 −0.429850 0.902901i \(-0.641433\pi\)
−0.429850 + 0.902901i \(0.641433\pi\)
\(114\) 0 0
\(115\) 6.38487 0.595392
\(116\) 0 0
\(117\) 6.77179 0.626052
\(118\) 0 0
\(119\) −2.70574 −0.248035
\(120\) 0 0
\(121\) 32.2263 2.92966
\(122\) 0 0
\(123\) 11.3747 1.02562
\(124\) 0 0
\(125\) −15.6674 −1.40134
\(126\) 0 0
\(127\) 20.4929 1.81845 0.909226 0.416303i \(-0.136674\pi\)
0.909226 + 0.416303i \(0.136674\pi\)
\(128\) 0 0
\(129\) 25.2876 2.22645
\(130\) 0 0
\(131\) 11.7999 1.03096 0.515479 0.856902i \(-0.327614\pi\)
0.515479 + 0.856902i \(0.327614\pi\)
\(132\) 0 0
\(133\) −8.32150 −0.721566
\(134\) 0 0
\(135\) −44.3725 −3.81898
\(136\) 0 0
\(137\) −17.5849 −1.50238 −0.751191 0.660085i \(-0.770520\pi\)
−0.751191 + 0.660085i \(0.770520\pi\)
\(138\) 0 0
\(139\) −6.28759 −0.533307 −0.266653 0.963792i \(-0.585918\pi\)
−0.266653 + 0.963792i \(0.585918\pi\)
\(140\) 0 0
\(141\) −17.7646 −1.49605
\(142\) 0 0
\(143\) 6.57467 0.549802
\(144\) 0 0
\(145\) 3.76339 0.312533
\(146\) 0 0
\(147\) −3.53336 −0.291426
\(148\) 0 0
\(149\) 3.69125 0.302399 0.151199 0.988503i \(-0.451686\pi\)
0.151199 + 0.988503i \(0.451686\pi\)
\(150\) 0 0
\(151\) 9.22570 0.750777 0.375388 0.926868i \(-0.377509\pi\)
0.375388 + 0.926868i \(0.377509\pi\)
\(152\) 0 0
\(153\) −7.56280 −0.611416
\(154\) 0 0
\(155\) 1.53784 0.123522
\(156\) 0 0
\(157\) 9.45242 0.754385 0.377192 0.926135i \(-0.376890\pi\)
0.377192 + 0.926135i \(0.376890\pi\)
\(158\) 0 0
\(159\) 3.97309 0.315087
\(160\) 0 0
\(161\) −4.11036 −0.323942
\(162\) 0 0
\(163\) 7.77314 0.608840 0.304420 0.952538i \(-0.401537\pi\)
0.304420 + 0.952538i \(0.401537\pi\)
\(164\) 0 0
\(165\) −77.3465 −6.02141
\(166\) 0 0
\(167\) −15.3061 −1.18442 −0.592209 0.805784i \(-0.701744\pi\)
−0.592209 + 0.805784i \(0.701744\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −23.2594 −1.77869
\(172\) 0 0
\(173\) −8.65999 −0.658407 −0.329203 0.944259i \(-0.606780\pi\)
−0.329203 + 0.944259i \(0.606780\pi\)
\(174\) 0 0
\(175\) 22.1999 1.67815
\(176\) 0 0
\(177\) −21.9852 −1.65251
\(178\) 0 0
\(179\) −18.2826 −1.36650 −0.683251 0.730183i \(-0.739435\pi\)
−0.683251 + 0.730183i \(0.739435\pi\)
\(180\) 0 0
\(181\) −3.84457 −0.285765 −0.142882 0.989740i \(-0.545637\pi\)
−0.142882 + 0.989740i \(0.545637\pi\)
\(182\) 0 0
\(183\) −25.0738 −1.85351
\(184\) 0 0
\(185\) −44.0933 −3.24180
\(186\) 0 0
\(187\) −7.34266 −0.536948
\(188\) 0 0
\(189\) 28.5655 2.07784
\(190\) 0 0
\(191\) 20.9257 1.51413 0.757067 0.653338i \(-0.226632\pi\)
0.757067 + 0.653338i \(0.226632\pi\)
\(192\) 0 0
\(193\) 6.82112 0.490995 0.245497 0.969397i \(-0.421049\pi\)
0.245497 + 0.969397i \(0.421049\pi\)
\(194\) 0 0
\(195\) −11.7643 −0.842460
\(196\) 0 0
\(197\) 19.7650 1.40820 0.704100 0.710101i \(-0.251351\pi\)
0.704100 + 0.710101i \(0.251351\pi\)
\(198\) 0 0
\(199\) −5.66470 −0.401560 −0.200780 0.979636i \(-0.564348\pi\)
−0.200780 + 0.979636i \(0.564348\pi\)
\(200\) 0 0
\(201\) −0.682364 −0.0481303
\(202\) 0 0
\(203\) −2.42274 −0.170043
\(204\) 0 0
\(205\) −13.6941 −0.956435
\(206\) 0 0
\(207\) −11.4888 −0.798530
\(208\) 0 0
\(209\) −22.5823 −1.56205
\(210\) 0 0
\(211\) 17.2355 1.18654 0.593271 0.805002i \(-0.297836\pi\)
0.593271 + 0.805002i \(0.297836\pi\)
\(212\) 0 0
\(213\) 46.3973 3.17909
\(214\) 0 0
\(215\) −30.4439 −2.07626
\(216\) 0 0
\(217\) −0.990010 −0.0672062
\(218\) 0 0
\(219\) 21.3536 1.44294
\(220\) 0 0
\(221\) −1.11681 −0.0751248
\(222\) 0 0
\(223\) −21.0933 −1.41251 −0.706255 0.707957i \(-0.749617\pi\)
−0.706255 + 0.707957i \(0.749617\pi\)
\(224\) 0 0
\(225\) 62.0507 4.13672
\(226\) 0 0
\(227\) 22.6870 1.50579 0.752895 0.658141i \(-0.228657\pi\)
0.752895 + 0.658141i \(0.228657\pi\)
\(228\) 0 0
\(229\) 21.2248 1.40257 0.701286 0.712880i \(-0.252610\pi\)
0.701286 + 0.712880i \(0.252610\pi\)
\(230\) 0 0
\(231\) 49.7930 3.27614
\(232\) 0 0
\(233\) 5.05642 0.331257 0.165628 0.986188i \(-0.447035\pi\)
0.165628 + 0.986188i \(0.447035\pi\)
\(234\) 0 0
\(235\) 21.3869 1.39512
\(236\) 0 0
\(237\) −31.9363 −2.07449
\(238\) 0 0
\(239\) 6.04827 0.391230 0.195615 0.980681i \(-0.437330\pi\)
0.195615 + 0.980681i \(0.437330\pi\)
\(240\) 0 0
\(241\) −19.2437 −1.23959 −0.619797 0.784762i \(-0.712785\pi\)
−0.619797 + 0.784762i \(0.712785\pi\)
\(242\) 0 0
\(243\) 16.3377 1.04806
\(244\) 0 0
\(245\) 4.25383 0.271767
\(246\) 0 0
\(247\) −3.43474 −0.218547
\(248\) 0 0
\(249\) −28.8707 −1.82961
\(250\) 0 0
\(251\) 29.1177 1.83789 0.918946 0.394383i \(-0.129042\pi\)
0.918946 + 0.394383i \(0.129042\pi\)
\(252\) 0 0
\(253\) −11.1544 −0.701272
\(254\) 0 0
\(255\) 13.1385 0.822765
\(256\) 0 0
\(257\) −11.1233 −0.693851 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(258\) 0 0
\(259\) 28.3858 1.76381
\(260\) 0 0
\(261\) −6.77179 −0.419163
\(262\) 0 0
\(263\) −27.2917 −1.68288 −0.841439 0.540351i \(-0.818291\pi\)
−0.841439 + 0.540351i \(0.818291\pi\)
\(264\) 0 0
\(265\) −4.78323 −0.293831
\(266\) 0 0
\(267\) 19.5773 1.19811
\(268\) 0 0
\(269\) 4.42461 0.269773 0.134887 0.990861i \(-0.456933\pi\)
0.134887 + 0.990861i \(0.456933\pi\)
\(270\) 0 0
\(271\) 9.76700 0.593303 0.296651 0.954986i \(-0.404130\pi\)
0.296651 + 0.954986i \(0.404130\pi\)
\(272\) 0 0
\(273\) 7.57346 0.458367
\(274\) 0 0
\(275\) 60.2445 3.63288
\(276\) 0 0
\(277\) −13.4342 −0.807182 −0.403591 0.914939i \(-0.632238\pi\)
−0.403591 + 0.914939i \(0.632238\pi\)
\(278\) 0 0
\(279\) −2.76717 −0.165666
\(280\) 0 0
\(281\) 22.8153 1.36105 0.680523 0.732727i \(-0.261753\pi\)
0.680523 + 0.732727i \(0.261753\pi\)
\(282\) 0 0
\(283\) 2.18390 0.129819 0.0649097 0.997891i \(-0.479324\pi\)
0.0649097 + 0.997891i \(0.479324\pi\)
\(284\) 0 0
\(285\) 40.4074 2.39353
\(286\) 0 0
\(287\) 8.81577 0.520378
\(288\) 0 0
\(289\) −15.7527 −0.926632
\(290\) 0 0
\(291\) 45.0029 2.63812
\(292\) 0 0
\(293\) −4.55576 −0.266151 −0.133075 0.991106i \(-0.542485\pi\)
−0.133075 + 0.991106i \(0.542485\pi\)
\(294\) 0 0
\(295\) 26.4680 1.54103
\(296\) 0 0
\(297\) 77.5191 4.49811
\(298\) 0 0
\(299\) −1.69657 −0.0981154
\(300\) 0 0
\(301\) 19.5988 1.12965
\(302\) 0 0
\(303\) −40.6905 −2.33761
\(304\) 0 0
\(305\) 30.1865 1.72848
\(306\) 0 0
\(307\) −24.9372 −1.42324 −0.711620 0.702565i \(-0.752038\pi\)
−0.711620 + 0.702565i \(0.752038\pi\)
\(308\) 0 0
\(309\) −38.8291 −2.20891
\(310\) 0 0
\(311\) −26.9496 −1.52817 −0.764087 0.645113i \(-0.776810\pi\)
−0.764087 + 0.645113i \(0.776810\pi\)
\(312\) 0 0
\(313\) −20.3533 −1.15044 −0.575218 0.818000i \(-0.695083\pi\)
−0.575218 + 0.818000i \(0.695083\pi\)
\(314\) 0 0
\(315\) −61.7434 −3.47884
\(316\) 0 0
\(317\) 8.40162 0.471882 0.235941 0.971767i \(-0.424183\pi\)
0.235941 + 0.971767i \(0.424183\pi\)
\(318\) 0 0
\(319\) −6.57467 −0.368111
\(320\) 0 0
\(321\) 19.4851 1.08755
\(322\) 0 0
\(323\) 3.83596 0.213438
\(324\) 0 0
\(325\) 9.16312 0.508278
\(326\) 0 0
\(327\) −14.0419 −0.776519
\(328\) 0 0
\(329\) −13.7681 −0.759061
\(330\) 0 0
\(331\) −10.6995 −0.588100 −0.294050 0.955790i \(-0.595003\pi\)
−0.294050 + 0.955790i \(0.595003\pi\)
\(332\) 0 0
\(333\) 79.3408 4.34785
\(334\) 0 0
\(335\) 0.821502 0.0448835
\(336\) 0 0
\(337\) 16.1513 0.879816 0.439908 0.898043i \(-0.355011\pi\)
0.439908 + 0.898043i \(0.355011\pi\)
\(338\) 0 0
\(339\) −28.5675 −1.55158
\(340\) 0 0
\(341\) −2.68662 −0.145489
\(342\) 0 0
\(343\) −19.6977 −1.06357
\(344\) 0 0
\(345\) 19.9590 1.07456
\(346\) 0 0
\(347\) −18.4653 −0.991267 −0.495633 0.868532i \(-0.665064\pi\)
−0.495633 + 0.868532i \(0.665064\pi\)
\(348\) 0 0
\(349\) 20.2890 1.08605 0.543024 0.839717i \(-0.317279\pi\)
0.543024 + 0.839717i \(0.317279\pi\)
\(350\) 0 0
\(351\) 11.7906 0.629334
\(352\) 0 0
\(353\) 15.4547 0.822568 0.411284 0.911507i \(-0.365080\pi\)
0.411284 + 0.911507i \(0.365080\pi\)
\(354\) 0 0
\(355\) −55.8580 −2.96463
\(356\) 0 0
\(357\) −8.45812 −0.447651
\(358\) 0 0
\(359\) 32.0776 1.69299 0.846497 0.532394i \(-0.178708\pi\)
0.846497 + 0.532394i \(0.178708\pi\)
\(360\) 0 0
\(361\) −7.20254 −0.379081
\(362\) 0 0
\(363\) 100.739 5.28742
\(364\) 0 0
\(365\) −25.7077 −1.34560
\(366\) 0 0
\(367\) 13.7189 0.716119 0.358059 0.933699i \(-0.383439\pi\)
0.358059 + 0.933699i \(0.383439\pi\)
\(368\) 0 0
\(369\) 24.6409 1.28275
\(370\) 0 0
\(371\) 3.07928 0.159868
\(372\) 0 0
\(373\) 0.802214 0.0415371 0.0207685 0.999784i \(-0.493389\pi\)
0.0207685 + 0.999784i \(0.493389\pi\)
\(374\) 0 0
\(375\) −48.9762 −2.52912
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) 21.0194 1.07969 0.539846 0.841764i \(-0.318483\pi\)
0.539846 + 0.841764i \(0.318483\pi\)
\(380\) 0 0
\(381\) 64.0606 3.28192
\(382\) 0 0
\(383\) −34.1486 −1.74491 −0.872455 0.488694i \(-0.837473\pi\)
−0.872455 + 0.488694i \(0.837473\pi\)
\(384\) 0 0
\(385\) −59.9461 −3.05513
\(386\) 0 0
\(387\) 54.7803 2.78464
\(388\) 0 0
\(389\) 31.7963 1.61213 0.806067 0.591824i \(-0.201592\pi\)
0.806067 + 0.591824i \(0.201592\pi\)
\(390\) 0 0
\(391\) 1.89475 0.0958217
\(392\) 0 0
\(393\) 36.8862 1.86066
\(394\) 0 0
\(395\) 38.4483 1.93455
\(396\) 0 0
\(397\) 31.8909 1.60056 0.800280 0.599627i \(-0.204684\pi\)
0.800280 + 0.599627i \(0.204684\pi\)
\(398\) 0 0
\(399\) −26.0129 −1.30227
\(400\) 0 0
\(401\) −1.38173 −0.0690002 −0.0345001 0.999405i \(-0.510984\pi\)
−0.0345001 + 0.999405i \(0.510984\pi\)
\(402\) 0 0
\(403\) −0.408632 −0.0203554
\(404\) 0 0
\(405\) −62.2532 −3.09339
\(406\) 0 0
\(407\) 77.0313 3.81830
\(408\) 0 0
\(409\) 12.8107 0.633450 0.316725 0.948517i \(-0.397417\pi\)
0.316725 + 0.948517i \(0.397417\pi\)
\(410\) 0 0
\(411\) −54.9703 −2.71148
\(412\) 0 0
\(413\) −17.0392 −0.838445
\(414\) 0 0
\(415\) 34.7576 1.70618
\(416\) 0 0
\(417\) −19.6549 −0.962506
\(418\) 0 0
\(419\) 9.30321 0.454492 0.227246 0.973837i \(-0.427028\pi\)
0.227246 + 0.973837i \(0.427028\pi\)
\(420\) 0 0
\(421\) 5.60630 0.273234 0.136617 0.990624i \(-0.456377\pi\)
0.136617 + 0.990624i \(0.456377\pi\)
\(422\) 0 0
\(423\) −38.4832 −1.87112
\(424\) 0 0
\(425\) −10.2335 −0.496396
\(426\) 0 0
\(427\) −19.4331 −0.940431
\(428\) 0 0
\(429\) 20.5523 0.992276
\(430\) 0 0
\(431\) −29.2724 −1.41000 −0.705000 0.709207i \(-0.749053\pi\)
−0.705000 + 0.709207i \(0.749053\pi\)
\(432\) 0 0
\(433\) −31.3039 −1.50437 −0.752184 0.658954i \(-0.770999\pi\)
−0.752184 + 0.658954i \(0.770999\pi\)
\(434\) 0 0
\(435\) 11.7643 0.564056
\(436\) 0 0
\(437\) 5.82730 0.278757
\(438\) 0 0
\(439\) 12.1088 0.577922 0.288961 0.957341i \(-0.406690\pi\)
0.288961 + 0.957341i \(0.406690\pi\)
\(440\) 0 0
\(441\) −7.65428 −0.364489
\(442\) 0 0
\(443\) −19.5007 −0.926506 −0.463253 0.886226i \(-0.653318\pi\)
−0.463253 + 0.886226i \(0.653318\pi\)
\(444\) 0 0
\(445\) −23.5692 −1.11729
\(446\) 0 0
\(447\) 11.5388 0.545766
\(448\) 0 0
\(449\) −25.9909 −1.22659 −0.613293 0.789856i \(-0.710155\pi\)
−0.613293 + 0.789856i \(0.710155\pi\)
\(450\) 0 0
\(451\) 23.9236 1.12652
\(452\) 0 0
\(453\) 28.8394 1.35499
\(454\) 0 0
\(455\) −9.11773 −0.427446
\(456\) 0 0
\(457\) 0.863395 0.0403879 0.0201940 0.999796i \(-0.493572\pi\)
0.0201940 + 0.999796i \(0.493572\pi\)
\(458\) 0 0
\(459\) −13.1678 −0.614621
\(460\) 0 0
\(461\) −25.9598 −1.20907 −0.604533 0.796580i \(-0.706640\pi\)
−0.604533 + 0.796580i \(0.706640\pi\)
\(462\) 0 0
\(463\) 27.0249 1.25595 0.627977 0.778232i \(-0.283883\pi\)
0.627977 + 0.778232i \(0.283883\pi\)
\(464\) 0 0
\(465\) 4.80727 0.222932
\(466\) 0 0
\(467\) −21.2094 −0.981456 −0.490728 0.871313i \(-0.663269\pi\)
−0.490728 + 0.871313i \(0.663269\pi\)
\(468\) 0 0
\(469\) −0.528855 −0.0244202
\(470\) 0 0
\(471\) 29.5481 1.36151
\(472\) 0 0
\(473\) 53.1857 2.44548
\(474\) 0 0
\(475\) −31.4730 −1.44408
\(476\) 0 0
\(477\) 8.60687 0.394081
\(478\) 0 0
\(479\) 9.17046 0.419009 0.209504 0.977808i \(-0.432815\pi\)
0.209504 + 0.977808i \(0.432815\pi\)
\(480\) 0 0
\(481\) 11.7164 0.534221
\(482\) 0 0
\(483\) −12.8489 −0.584647
\(484\) 0 0
\(485\) −54.1792 −2.46015
\(486\) 0 0
\(487\) −28.0228 −1.26983 −0.634916 0.772581i \(-0.718965\pi\)
−0.634916 + 0.772581i \(0.718965\pi\)
\(488\) 0 0
\(489\) 24.2987 1.09883
\(490\) 0 0
\(491\) −36.5743 −1.65058 −0.825288 0.564712i \(-0.808987\pi\)
−0.825288 + 0.564712i \(0.808987\pi\)
\(492\) 0 0
\(493\) 1.11681 0.0502986
\(494\) 0 0
\(495\) −167.555 −7.53103
\(496\) 0 0
\(497\) 35.9595 1.61300
\(498\) 0 0
\(499\) −4.84413 −0.216853 −0.108427 0.994104i \(-0.534581\pi\)
−0.108427 + 0.994104i \(0.534581\pi\)
\(500\) 0 0
\(501\) −47.8465 −2.13762
\(502\) 0 0
\(503\) −6.90875 −0.308046 −0.154023 0.988067i \(-0.549223\pi\)
−0.154023 + 0.988067i \(0.549223\pi\)
\(504\) 0 0
\(505\) 48.9875 2.17992
\(506\) 0 0
\(507\) 3.12599 0.138830
\(508\) 0 0
\(509\) −4.83407 −0.214266 −0.107133 0.994245i \(-0.534167\pi\)
−0.107133 + 0.994245i \(0.534167\pi\)
\(510\) 0 0
\(511\) 16.5497 0.732118
\(512\) 0 0
\(513\) −40.4976 −1.78801
\(514\) 0 0
\(515\) 46.7466 2.05990
\(516\) 0 0
\(517\) −37.3630 −1.64322
\(518\) 0 0
\(519\) −27.0710 −1.18829
\(520\) 0 0
\(521\) −31.8930 −1.39726 −0.698629 0.715484i \(-0.746206\pi\)
−0.698629 + 0.715484i \(0.746206\pi\)
\(522\) 0 0
\(523\) 17.0320 0.744759 0.372379 0.928081i \(-0.378542\pi\)
0.372379 + 0.928081i \(0.378542\pi\)
\(524\) 0 0
\(525\) 69.3965 3.02871
\(526\) 0 0
\(527\) 0.456364 0.0198795
\(528\) 0 0
\(529\) −20.1216 −0.874854
\(530\) 0 0
\(531\) −47.6262 −2.06680
\(532\) 0 0
\(533\) 3.63876 0.157612
\(534\) 0 0
\(535\) −23.4583 −1.01419
\(536\) 0 0
\(537\) −57.1510 −2.46625
\(538\) 0 0
\(539\) −7.43147 −0.320096
\(540\) 0 0
\(541\) 4.71304 0.202630 0.101315 0.994854i \(-0.467695\pi\)
0.101315 + 0.994854i \(0.467695\pi\)
\(542\) 0 0
\(543\) −12.0181 −0.515745
\(544\) 0 0
\(545\) 16.9051 0.724136
\(546\) 0 0
\(547\) 26.4078 1.12911 0.564557 0.825394i \(-0.309047\pi\)
0.564557 + 0.825394i \(0.309047\pi\)
\(548\) 0 0
\(549\) −54.3172 −2.31820
\(550\) 0 0
\(551\) 3.43474 0.146325
\(552\) 0 0
\(553\) −24.7517 −1.05255
\(554\) 0 0
\(555\) −137.835 −5.85077
\(556\) 0 0
\(557\) 0.111362 0.00471856 0.00235928 0.999997i \(-0.499249\pi\)
0.00235928 + 0.999997i \(0.499249\pi\)
\(558\) 0 0
\(559\) 8.08949 0.342149
\(560\) 0 0
\(561\) −22.9530 −0.969078
\(562\) 0 0
\(563\) −23.6688 −0.997519 −0.498760 0.866740i \(-0.666211\pi\)
−0.498760 + 0.866740i \(0.666211\pi\)
\(564\) 0 0
\(565\) 34.3926 1.44691
\(566\) 0 0
\(567\) 40.0765 1.68305
\(568\) 0 0
\(569\) −19.1601 −0.803233 −0.401617 0.915808i \(-0.631552\pi\)
−0.401617 + 0.915808i \(0.631552\pi\)
\(570\) 0 0
\(571\) −27.9804 −1.17095 −0.585473 0.810692i \(-0.699091\pi\)
−0.585473 + 0.810692i \(0.699091\pi\)
\(572\) 0 0
\(573\) 65.4136 2.73269
\(574\) 0 0
\(575\) −15.5459 −0.648309
\(576\) 0 0
\(577\) −21.7366 −0.904906 −0.452453 0.891788i \(-0.649451\pi\)
−0.452453 + 0.891788i \(0.649451\pi\)
\(578\) 0 0
\(579\) 21.3227 0.886142
\(580\) 0 0
\(581\) −22.3758 −0.928303
\(582\) 0 0
\(583\) 8.35633 0.346084
\(584\) 0 0
\(585\) −25.4849 −1.05367
\(586\) 0 0
\(587\) 11.6844 0.482266 0.241133 0.970492i \(-0.422481\pi\)
0.241133 + 0.970492i \(0.422481\pi\)
\(588\) 0 0
\(589\) 1.40355 0.0578321
\(590\) 0 0
\(591\) 61.7852 2.54150
\(592\) 0 0
\(593\) −5.32509 −0.218675 −0.109338 0.994005i \(-0.534873\pi\)
−0.109338 + 0.994005i \(0.534873\pi\)
\(594\) 0 0
\(595\) 10.1828 0.417453
\(596\) 0 0
\(597\) −17.7078 −0.724731
\(598\) 0 0
\(599\) 26.4499 1.08071 0.540356 0.841437i \(-0.318290\pi\)
0.540356 + 0.841437i \(0.318290\pi\)
\(600\) 0 0
\(601\) 18.3389 0.748058 0.374029 0.927417i \(-0.377976\pi\)
0.374029 + 0.927417i \(0.377976\pi\)
\(602\) 0 0
\(603\) −1.47820 −0.0601969
\(604\) 0 0
\(605\) −121.280 −4.93074
\(606\) 0 0
\(607\) −13.9881 −0.567761 −0.283881 0.958860i \(-0.591622\pi\)
−0.283881 + 0.958860i \(0.591622\pi\)
\(608\) 0 0
\(609\) −7.57346 −0.306892
\(610\) 0 0
\(611\) −5.68287 −0.229904
\(612\) 0 0
\(613\) −10.7353 −0.433594 −0.216797 0.976217i \(-0.569561\pi\)
−0.216797 + 0.976217i \(0.569561\pi\)
\(614\) 0 0
\(615\) −42.8075 −1.72616
\(616\) 0 0
\(617\) −27.5898 −1.11072 −0.555361 0.831609i \(-0.687420\pi\)
−0.555361 + 0.831609i \(0.687420\pi\)
\(618\) 0 0
\(619\) −17.3912 −0.699012 −0.349506 0.936934i \(-0.613651\pi\)
−0.349506 + 0.936934i \(0.613651\pi\)
\(620\) 0 0
\(621\) −20.0036 −0.802716
\(622\) 0 0
\(623\) 15.1730 0.607895
\(624\) 0 0
\(625\) 13.1471 0.525886
\(626\) 0 0
\(627\) −70.5920 −2.81917
\(628\) 0 0
\(629\) −13.0850 −0.521732
\(630\) 0 0
\(631\) −1.93788 −0.0771458 −0.0385729 0.999256i \(-0.512281\pi\)
−0.0385729 + 0.999256i \(0.512281\pi\)
\(632\) 0 0
\(633\) 53.8780 2.14146
\(634\) 0 0
\(635\) −77.1229 −3.06053
\(636\) 0 0
\(637\) −1.13032 −0.0447848
\(638\) 0 0
\(639\) 100.510 3.97612
\(640\) 0 0
\(641\) −47.7710 −1.88684 −0.943421 0.331597i \(-0.892413\pi\)
−0.943421 + 0.331597i \(0.892413\pi\)
\(642\) 0 0
\(643\) −5.20337 −0.205201 −0.102600 0.994723i \(-0.532716\pi\)
−0.102600 + 0.994723i \(0.532716\pi\)
\(644\) 0 0
\(645\) −95.1673 −3.74721
\(646\) 0 0
\(647\) −16.6049 −0.652806 −0.326403 0.945231i \(-0.605837\pi\)
−0.326403 + 0.945231i \(0.605837\pi\)
\(648\) 0 0
\(649\) −46.2399 −1.81507
\(650\) 0 0
\(651\) −3.09476 −0.121293
\(652\) 0 0
\(653\) −15.4293 −0.603795 −0.301898 0.953340i \(-0.597620\pi\)
−0.301898 + 0.953340i \(0.597620\pi\)
\(654\) 0 0
\(655\) −44.4075 −1.73514
\(656\) 0 0
\(657\) 46.2581 1.80470
\(658\) 0 0
\(659\) −25.7062 −1.00137 −0.500687 0.865629i \(-0.666919\pi\)
−0.500687 + 0.865629i \(0.666919\pi\)
\(660\) 0 0
\(661\) 29.4703 1.14626 0.573130 0.819464i \(-0.305729\pi\)
0.573130 + 0.819464i \(0.305729\pi\)
\(662\) 0 0
\(663\) −3.49113 −0.135584
\(664\) 0 0
\(665\) 31.3171 1.21442
\(666\) 0 0
\(667\) 1.69657 0.0656916
\(668\) 0 0
\(669\) −65.9373 −2.54928
\(670\) 0 0
\(671\) −52.7361 −2.03585
\(672\) 0 0
\(673\) 8.76278 0.337780 0.168890 0.985635i \(-0.445982\pi\)
0.168890 + 0.985635i \(0.445982\pi\)
\(674\) 0 0
\(675\) 108.038 4.15840
\(676\) 0 0
\(677\) −0.174750 −0.00671620 −0.00335810 0.999994i \(-0.501069\pi\)
−0.00335810 + 0.999994i \(0.501069\pi\)
\(678\) 0 0
\(679\) 34.8787 1.33852
\(680\) 0 0
\(681\) 70.9193 2.71763
\(682\) 0 0
\(683\) 23.9721 0.917268 0.458634 0.888625i \(-0.348339\pi\)
0.458634 + 0.888625i \(0.348339\pi\)
\(684\) 0 0
\(685\) 66.1790 2.52857
\(686\) 0 0
\(687\) 66.3483 2.53135
\(688\) 0 0
\(689\) 1.27099 0.0484208
\(690\) 0 0
\(691\) 29.9400 1.13897 0.569485 0.822002i \(-0.307143\pi\)
0.569485 + 0.822002i \(0.307143\pi\)
\(692\) 0 0
\(693\) 107.866 4.09749
\(694\) 0 0
\(695\) 23.6627 0.897577
\(696\) 0 0
\(697\) −4.06380 −0.153927
\(698\) 0 0
\(699\) 15.8063 0.597849
\(700\) 0 0
\(701\) −34.7099 −1.31098 −0.655488 0.755206i \(-0.727537\pi\)
−0.655488 + 0.755206i \(0.727537\pi\)
\(702\) 0 0
\(703\) −40.2427 −1.51778
\(704\) 0 0
\(705\) 66.8550 2.51791
\(706\) 0 0
\(707\) −31.5365 −1.18605
\(708\) 0 0
\(709\) 29.3569 1.10252 0.551261 0.834333i \(-0.314147\pi\)
0.551261 + 0.834333i \(0.314147\pi\)
\(710\) 0 0
\(711\) −69.1834 −2.59458
\(712\) 0 0
\(713\) 0.693274 0.0259633
\(714\) 0 0
\(715\) −24.7431 −0.925338
\(716\) 0 0
\(717\) 18.9068 0.706088
\(718\) 0 0
\(719\) −26.6369 −0.993390 −0.496695 0.867925i \(-0.665453\pi\)
−0.496695 + 0.867925i \(0.665453\pi\)
\(720\) 0 0
\(721\) −30.0939 −1.12075
\(722\) 0 0
\(723\) −60.1555 −2.23721
\(724\) 0 0
\(725\) −9.16312 −0.340310
\(726\) 0 0
\(727\) 22.0278 0.816964 0.408482 0.912766i \(-0.366058\pi\)
0.408482 + 0.912766i \(0.366058\pi\)
\(728\) 0 0
\(729\) 1.44605 0.0535573
\(730\) 0 0
\(731\) −9.03442 −0.334150
\(732\) 0 0
\(733\) −1.27136 −0.0469586 −0.0234793 0.999724i \(-0.507474\pi\)
−0.0234793 + 0.999724i \(0.507474\pi\)
\(734\) 0 0
\(735\) 13.2974 0.490482
\(736\) 0 0
\(737\) −1.43517 −0.0528651
\(738\) 0 0
\(739\) −12.1865 −0.448287 −0.224144 0.974556i \(-0.571958\pi\)
−0.224144 + 0.974556i \(0.571958\pi\)
\(740\) 0 0
\(741\) −10.7370 −0.394432
\(742\) 0 0
\(743\) 30.7183 1.12695 0.563473 0.826134i \(-0.309465\pi\)
0.563473 + 0.826134i \(0.309465\pi\)
\(744\) 0 0
\(745\) −13.8916 −0.508949
\(746\) 0 0
\(747\) −62.5423 −2.28830
\(748\) 0 0
\(749\) 15.1016 0.551802
\(750\) 0 0
\(751\) −8.23116 −0.300359 −0.150180 0.988659i \(-0.547985\pi\)
−0.150180 + 0.988659i \(0.547985\pi\)
\(752\) 0 0
\(753\) 91.0215 3.31701
\(754\) 0 0
\(755\) −34.7199 −1.26359
\(756\) 0 0
\(757\) −14.6350 −0.531917 −0.265958 0.963985i \(-0.585688\pi\)
−0.265958 + 0.963985i \(0.585688\pi\)
\(758\) 0 0
\(759\) −34.8685 −1.26565
\(760\) 0 0
\(761\) −41.6624 −1.51026 −0.755131 0.655574i \(-0.772427\pi\)
−0.755131 + 0.655574i \(0.772427\pi\)
\(762\) 0 0
\(763\) −10.8829 −0.393989
\(764\) 0 0
\(765\) 28.4618 1.02904
\(766\) 0 0
\(767\) −7.03303 −0.253948
\(768\) 0 0
\(769\) −29.3784 −1.05941 −0.529706 0.848181i \(-0.677698\pi\)
−0.529706 + 0.848181i \(0.677698\pi\)
\(770\) 0 0
\(771\) −34.7712 −1.25225
\(772\) 0 0
\(773\) −14.7495 −0.530504 −0.265252 0.964179i \(-0.585455\pi\)
−0.265252 + 0.964179i \(0.585455\pi\)
\(774\) 0 0
\(775\) −3.74434 −0.134501
\(776\) 0 0
\(777\) 88.7335 3.18330
\(778\) 0 0
\(779\) −12.4982 −0.447794
\(780\) 0 0
\(781\) 97.5843 3.49184
\(782\) 0 0
\(783\) −11.7906 −0.421361
\(784\) 0 0
\(785\) −35.5731 −1.26966
\(786\) 0 0
\(787\) −13.6447 −0.486383 −0.243191 0.969978i \(-0.578194\pi\)
−0.243191 + 0.969978i \(0.578194\pi\)
\(788\) 0 0
\(789\) −85.3135 −3.03724
\(790\) 0 0
\(791\) −22.1408 −0.787236
\(792\) 0 0
\(793\) −8.02110 −0.284837
\(794\) 0 0
\(795\) −14.9523 −0.530304
\(796\) 0 0
\(797\) −24.5309 −0.868929 −0.434465 0.900689i \(-0.643062\pi\)
−0.434465 + 0.900689i \(0.643062\pi\)
\(798\) 0 0
\(799\) 6.34668 0.224529
\(800\) 0 0
\(801\) 42.4100 1.49849
\(802\) 0 0
\(803\) 44.9116 1.58489
\(804\) 0 0
\(805\) 15.4689 0.545207
\(806\) 0 0
\(807\) 13.8313 0.486884
\(808\) 0 0
\(809\) −17.8032 −0.625926 −0.312963 0.949765i \(-0.601322\pi\)
−0.312963 + 0.949765i \(0.601322\pi\)
\(810\) 0 0
\(811\) 24.7178 0.867958 0.433979 0.900923i \(-0.357109\pi\)
0.433979 + 0.900923i \(0.357109\pi\)
\(812\) 0 0
\(813\) 30.5315 1.07079
\(814\) 0 0
\(815\) −29.2534 −1.02470
\(816\) 0 0
\(817\) −27.7853 −0.972086
\(818\) 0 0
\(819\) 16.4063 0.573283
\(820\) 0 0
\(821\) −46.4017 −1.61943 −0.809715 0.586824i \(-0.800378\pi\)
−0.809715 + 0.586824i \(0.800378\pi\)
\(822\) 0 0
\(823\) −25.7389 −0.897202 −0.448601 0.893732i \(-0.648078\pi\)
−0.448601 + 0.893732i \(0.648078\pi\)
\(824\) 0 0
\(825\) 188.323 6.55658
\(826\) 0 0
\(827\) 37.2509 1.29534 0.647671 0.761920i \(-0.275743\pi\)
0.647671 + 0.761920i \(0.275743\pi\)
\(828\) 0 0
\(829\) −1.74066 −0.0604557 −0.0302278 0.999543i \(-0.509623\pi\)
−0.0302278 + 0.999543i \(0.509623\pi\)
\(830\) 0 0
\(831\) −41.9951 −1.45679
\(832\) 0 0
\(833\) 1.26235 0.0437378
\(834\) 0 0
\(835\) 57.6027 1.99342
\(836\) 0 0
\(837\) −4.81800 −0.166534
\(838\) 0 0
\(839\) 3.77361 0.130279 0.0651397 0.997876i \(-0.479251\pi\)
0.0651397 + 0.997876i \(0.479251\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 71.3202 2.45640
\(844\) 0 0
\(845\) −3.76339 −0.129465
\(846\) 0 0
\(847\) 78.0760 2.68272
\(848\) 0 0
\(849\) 6.82684 0.234297
\(850\) 0 0
\(851\) −19.8777 −0.681399
\(852\) 0 0
\(853\) 16.5330 0.566077 0.283039 0.959108i \(-0.408658\pi\)
0.283039 + 0.959108i \(0.408658\pi\)
\(854\) 0 0
\(855\) 87.5341 2.99360
\(856\) 0 0
\(857\) 43.0155 1.46938 0.734690 0.678403i \(-0.237327\pi\)
0.734690 + 0.678403i \(0.237327\pi\)
\(858\) 0 0
\(859\) 19.3081 0.658784 0.329392 0.944193i \(-0.393156\pi\)
0.329392 + 0.944193i \(0.393156\pi\)
\(860\) 0 0
\(861\) 27.5580 0.939173
\(862\) 0 0
\(863\) −24.4677 −0.832889 −0.416444 0.909161i \(-0.636724\pi\)
−0.416444 + 0.909161i \(0.636724\pi\)
\(864\) 0 0
\(865\) 32.5909 1.10813
\(866\) 0 0
\(867\) −49.2428 −1.67237
\(868\) 0 0
\(869\) −67.1695 −2.27857
\(870\) 0 0
\(871\) −0.218288 −0.00739640
\(872\) 0 0
\(873\) 97.4892 3.29951
\(874\) 0 0
\(875\) −37.9582 −1.28322
\(876\) 0 0
\(877\) −17.0083 −0.574328 −0.287164 0.957881i \(-0.592712\pi\)
−0.287164 + 0.957881i \(0.592712\pi\)
\(878\) 0 0
\(879\) −14.2413 −0.480346
\(880\) 0 0
\(881\) 21.4085 0.721269 0.360635 0.932707i \(-0.382560\pi\)
0.360635 + 0.932707i \(0.382560\pi\)
\(882\) 0 0
\(883\) −15.8993 −0.535053 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(884\) 0 0
\(885\) 82.7388 2.78123
\(886\) 0 0
\(887\) −42.7696 −1.43606 −0.718031 0.696011i \(-0.754956\pi\)
−0.718031 + 0.696011i \(0.754956\pi\)
\(888\) 0 0
\(889\) 49.6491 1.66518
\(890\) 0 0
\(891\) 108.757 3.64349
\(892\) 0 0
\(893\) 19.5192 0.653185
\(894\) 0 0
\(895\) 68.8044 2.29988
\(896\) 0 0
\(897\) −5.30347 −0.177078
\(898\) 0 0
\(899\) 0.408632 0.0136286
\(900\) 0 0
\(901\) −1.41945 −0.0472888
\(902\) 0 0
\(903\) 61.2654 2.03879
\(904\) 0 0
\(905\) 14.4686 0.480953
\(906\) 0 0
\(907\) 37.1644 1.23402 0.617012 0.786954i \(-0.288343\pi\)
0.617012 + 0.786954i \(0.288343\pi\)
\(908\) 0 0
\(909\) −88.1474 −2.92367
\(910\) 0 0
\(911\) 37.1149 1.22967 0.614836 0.788655i \(-0.289222\pi\)
0.614836 + 0.788655i \(0.289222\pi\)
\(912\) 0 0
\(913\) −60.7218 −2.00960
\(914\) 0 0
\(915\) 94.3627 3.11953
\(916\) 0 0
\(917\) 28.5880 0.944060
\(918\) 0 0
\(919\) −3.63662 −0.119961 −0.0599806 0.998200i \(-0.519104\pi\)
−0.0599806 + 0.998200i \(0.519104\pi\)
\(920\) 0 0
\(921\) −77.9533 −2.56865
\(922\) 0 0
\(923\) 14.8425 0.488546
\(924\) 0 0
\(925\) 107.359 3.52993
\(926\) 0 0
\(927\) −84.1151 −2.76270
\(928\) 0 0
\(929\) 4.75340 0.155954 0.0779770 0.996955i \(-0.475154\pi\)
0.0779770 + 0.996955i \(0.475154\pi\)
\(930\) 0 0
\(931\) 3.88235 0.127239
\(932\) 0 0
\(933\) −84.2442 −2.75803
\(934\) 0 0
\(935\) 27.6333 0.903705
\(936\) 0 0
\(937\) 41.3383 1.35046 0.675232 0.737605i \(-0.264043\pi\)
0.675232 + 0.737605i \(0.264043\pi\)
\(938\) 0 0
\(939\) −63.6241 −2.07629
\(940\) 0 0
\(941\) 11.7623 0.383439 0.191719 0.981450i \(-0.438594\pi\)
0.191719 + 0.981450i \(0.438594\pi\)
\(942\) 0 0
\(943\) −6.17342 −0.201034
\(944\) 0 0
\(945\) −107.503 −3.49708
\(946\) 0 0
\(947\) −29.7489 −0.966709 −0.483354 0.875425i \(-0.660582\pi\)
−0.483354 + 0.875425i \(0.660582\pi\)
\(948\) 0 0
\(949\) 6.83100 0.221744
\(950\) 0 0
\(951\) 26.2633 0.851647
\(952\) 0 0
\(953\) −29.7480 −0.963632 −0.481816 0.876272i \(-0.660023\pi\)
−0.481816 + 0.876272i \(0.660023\pi\)
\(954\) 0 0
\(955\) −78.7518 −2.54835
\(956\) 0 0
\(957\) −20.5523 −0.664362
\(958\) 0 0
\(959\) −42.6038 −1.37575
\(960\) 0 0
\(961\) −30.8330 −0.994614
\(962\) 0 0
\(963\) 42.2105 1.36021
\(964\) 0 0
\(965\) −25.6705 −0.826364
\(966\) 0 0
\(967\) 43.1335 1.38708 0.693541 0.720417i \(-0.256050\pi\)
0.693541 + 0.720417i \(0.256050\pi\)
\(968\) 0 0
\(969\) 11.9911 0.385211
\(970\) 0 0
\(971\) 14.8830 0.477618 0.238809 0.971067i \(-0.423243\pi\)
0.238809 + 0.971067i \(0.423243\pi\)
\(972\) 0 0
\(973\) −15.2332 −0.488355
\(974\) 0 0
\(975\) 28.6438 0.917335
\(976\) 0 0
\(977\) −35.4616 −1.13452 −0.567259 0.823540i \(-0.691996\pi\)
−0.567259 + 0.823540i \(0.691996\pi\)
\(978\) 0 0
\(979\) 41.1755 1.31598
\(980\) 0 0
\(981\) −30.4188 −0.971198
\(982\) 0 0
\(983\) 37.6307 1.20023 0.600116 0.799913i \(-0.295121\pi\)
0.600116 + 0.799913i \(0.295121\pi\)
\(984\) 0 0
\(985\) −74.3836 −2.37006
\(986\) 0 0
\(987\) −43.0390 −1.36995
\(988\) 0 0
\(989\) −13.7244 −0.436411
\(990\) 0 0
\(991\) −59.1447 −1.87879 −0.939396 0.342833i \(-0.888614\pi\)
−0.939396 + 0.342833i \(0.888614\pi\)
\(992\) 0 0
\(993\) −33.4466 −1.06140
\(994\) 0 0
\(995\) 21.3185 0.675841
\(996\) 0 0
\(997\) 19.2679 0.610220 0.305110 0.952317i \(-0.401307\pi\)
0.305110 + 0.952317i \(0.401307\pi\)
\(998\) 0 0
\(999\) 138.143 4.37064
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 6032.2.a.bc.1.11 11
4.3 odd 2 3016.2.a.i.1.1 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3016.2.a.i.1.1 11 4.3 odd 2
6032.2.a.bc.1.11 11 1.1 even 1 trivial