Properties

Label 7938.2.a.bg.1.2
Level $7938$
Weight $2$
Character 7938.1
Self dual yes
Analytic conductor $63.385$
Analytic rank $1$
Dimension $2$
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] = [7938,2,Mod(1,7938)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7938, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7938.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7938.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: \(63.3852491245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1134)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7938.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} -1.00000 q^{8} +0.267949 q^{10} +6.19615 q^{11} +6.46410 q^{13} +1.00000 q^{16} -7.00000 q^{17} -0.732051 q^{19} -0.267949 q^{20} -6.19615 q^{22} -4.19615 q^{23} -4.92820 q^{25} -6.46410 q^{26} +1.53590 q^{29} -8.19615 q^{31} -1.00000 q^{32} +7.00000 q^{34} +10.6603 q^{37} +0.732051 q^{38} +0.267949 q^{40} -2.53590 q^{41} -1.46410 q^{43} +6.19615 q^{44} +4.19615 q^{46} -4.73205 q^{47} +4.92820 q^{50} +6.46410 q^{52} -9.46410 q^{53} -1.66025 q^{55} -1.53590 q^{58} -4.19615 q^{59} -3.92820 q^{61} +8.19615 q^{62} +1.00000 q^{64} -1.73205 q^{65} -6.73205 q^{67} -7.00000 q^{68} +6.53590 q^{71} -8.26795 q^{73} -10.6603 q^{74} -0.732051 q^{76} -9.12436 q^{79} -0.267949 q^{80} +2.53590 q^{82} -16.5885 q^{83} +1.87564 q^{85} +1.46410 q^{86} -6.19615 q^{88} -9.92820 q^{89} -4.19615 q^{92} +4.73205 q^{94} +0.196152 q^{95} -10.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} + 4 q^{10} + 2 q^{11} + 6 q^{13} + 2 q^{16} - 14 q^{17} + 2 q^{19} - 4 q^{20} - 2 q^{22} + 2 q^{23} + 4 q^{25} - 6 q^{26} + 10 q^{29} - 6 q^{31} - 2 q^{32} + 14 q^{34} + 4 q^{37} - 2 q^{38} + 4 q^{40} - 12 q^{41} + 4 q^{43} + 2 q^{44} - 2 q^{46} - 6 q^{47} - 4 q^{50} + 6 q^{52} - 12 q^{53} + 14 q^{55} - 10 q^{58} + 2 q^{59} + 6 q^{61} + 6 q^{62} + 2 q^{64} - 10 q^{67} - 14 q^{68} + 20 q^{71} - 20 q^{73} - 4 q^{74} + 2 q^{76} + 6 q^{79} - 4 q^{80} + 12 q^{82} - 2 q^{83} + 28 q^{85} - 4 q^{86} - 2 q^{88} - 6 q^{89} + 2 q^{92} + 6 q^{94} - 10 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.267949 −0.119831 −0.0599153 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.267949 0.0847330
\(11\) 6.19615 1.86821 0.934105 0.356998i \(-0.116200\pi\)
0.934105 + 0.356998i \(0.116200\pi\)
\(12\) 0 0
\(13\) 6.46410 1.79282 0.896410 0.443227i \(-0.146166\pi\)
0.896410 + 0.443227i \(0.146166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −7.00000 −1.69775 −0.848875 0.528594i \(-0.822719\pi\)
−0.848875 + 0.528594i \(0.822719\pi\)
\(18\) 0 0
\(19\) −0.732051 −0.167944 −0.0839720 0.996468i \(-0.526761\pi\)
−0.0839720 + 0.996468i \(0.526761\pi\)
\(20\) −0.267949 −0.0599153
\(21\) 0 0
\(22\) −6.19615 −1.32102
\(23\) −4.19615 −0.874958 −0.437479 0.899229i \(-0.644129\pi\)
−0.437479 + 0.899229i \(0.644129\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) −6.46410 −1.26771
\(27\) 0 0
\(28\) 0 0
\(29\) 1.53590 0.285209 0.142605 0.989780i \(-0.454452\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) 0 0
\(31\) −8.19615 −1.47207 −0.736036 0.676942i \(-0.763305\pi\)
−0.736036 + 0.676942i \(0.763305\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6603 1.75253 0.876267 0.481825i \(-0.160026\pi\)
0.876267 + 0.481825i \(0.160026\pi\)
\(38\) 0.732051 0.118754
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) −2.53590 −0.396041 −0.198020 0.980198i \(-0.563451\pi\)
−0.198020 + 0.980198i \(0.563451\pi\)
\(42\) 0 0
\(43\) −1.46410 −0.223273 −0.111637 0.993749i \(-0.535609\pi\)
−0.111637 + 0.993749i \(0.535609\pi\)
\(44\) 6.19615 0.934105
\(45\) 0 0
\(46\) 4.19615 0.618689
\(47\) −4.73205 −0.690241 −0.345120 0.938558i \(-0.612162\pi\)
−0.345120 + 0.938558i \(0.612162\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.92820 0.696953
\(51\) 0 0
\(52\) 6.46410 0.896410
\(53\) −9.46410 −1.29999 −0.649997 0.759937i \(-0.725230\pi\)
−0.649997 + 0.759937i \(0.725230\pi\)
\(54\) 0 0
\(55\) −1.66025 −0.223869
\(56\) 0 0
\(57\) 0 0
\(58\) −1.53590 −0.201673
\(59\) −4.19615 −0.546293 −0.273146 0.961973i \(-0.588064\pi\)
−0.273146 + 0.961973i \(0.588064\pi\)
\(60\) 0 0
\(61\) −3.92820 −0.502955 −0.251477 0.967863i \(-0.580916\pi\)
−0.251477 + 0.967863i \(0.580916\pi\)
\(62\) 8.19615 1.04091
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.73205 −0.214834
\(66\) 0 0
\(67\) −6.73205 −0.822451 −0.411225 0.911534i \(-0.634899\pi\)
−0.411225 + 0.911534i \(0.634899\pi\)
\(68\) −7.00000 −0.848875
\(69\) 0 0
\(70\) 0 0
\(71\) 6.53590 0.775668 0.387834 0.921729i \(-0.373223\pi\)
0.387834 + 0.921729i \(0.373223\pi\)
\(72\) 0 0
\(73\) −8.26795 −0.967690 −0.483845 0.875154i \(-0.660760\pi\)
−0.483845 + 0.875154i \(0.660760\pi\)
\(74\) −10.6603 −1.23923
\(75\) 0 0
\(76\) −0.732051 −0.0839720
\(77\) 0 0
\(78\) 0 0
\(79\) −9.12436 −1.02657 −0.513285 0.858218i \(-0.671572\pi\)
−0.513285 + 0.858218i \(0.671572\pi\)
\(80\) −0.267949 −0.0299576
\(81\) 0 0
\(82\) 2.53590 0.280043
\(83\) −16.5885 −1.82082 −0.910410 0.413707i \(-0.864234\pi\)
−0.910410 + 0.413707i \(0.864234\pi\)
\(84\) 0 0
\(85\) 1.87564 0.203442
\(86\) 1.46410 0.157878
\(87\) 0 0
\(88\) −6.19615 −0.660512
\(89\) −9.92820 −1.05239 −0.526194 0.850365i \(-0.676381\pi\)
−0.526194 + 0.850365i \(0.676381\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.19615 −0.437479
\(93\) 0 0
\(94\) 4.73205 0.488074
\(95\) 0.196152 0.0201248
\(96\) 0 0
\(97\) −10.9282 −1.10959 −0.554795 0.831987i \(-0.687203\pi\)
−0.554795 + 0.831987i \(0.687203\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.92820 −0.492820
\(101\) −8.92820 −0.888389 −0.444195 0.895930i \(-0.646510\pi\)
−0.444195 + 0.895930i \(0.646510\pi\)
\(102\) 0 0
\(103\) 8.39230 0.826918 0.413459 0.910523i \(-0.364320\pi\)
0.413459 + 0.910523i \(0.364320\pi\)
\(104\) −6.46410 −0.633857
\(105\) 0 0
\(106\) 9.46410 0.919235
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 3.19615 0.306136 0.153068 0.988216i \(-0.451085\pi\)
0.153068 + 0.988216i \(0.451085\pi\)
\(110\) 1.66025 0.158299
\(111\) 0 0
\(112\) 0 0
\(113\) −5.73205 −0.539226 −0.269613 0.962969i \(-0.586896\pi\)
−0.269613 + 0.962969i \(0.586896\pi\)
\(114\) 0 0
\(115\) 1.12436 0.104847
\(116\) 1.53590 0.142605
\(117\) 0 0
\(118\) 4.19615 0.386287
\(119\) 0 0
\(120\) 0 0
\(121\) 27.3923 2.49021
\(122\) 3.92820 0.355643
\(123\) 0 0
\(124\) −8.19615 −0.736036
\(125\) 2.66025 0.237940
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.73205 0.151911
\(131\) −10.5359 −0.920526 −0.460263 0.887783i \(-0.652245\pi\)
−0.460263 + 0.887783i \(0.652245\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 6.73205 0.581561
\(135\) 0 0
\(136\) 7.00000 0.600245
\(137\) −8.26795 −0.706379 −0.353189 0.935552i \(-0.614903\pi\)
−0.353189 + 0.935552i \(0.614903\pi\)
\(138\) 0 0
\(139\) 3.26795 0.277184 0.138592 0.990350i \(-0.455742\pi\)
0.138592 + 0.990350i \(0.455742\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.53590 −0.548480
\(143\) 40.0526 3.34936
\(144\) 0 0
\(145\) −0.411543 −0.0341768
\(146\) 8.26795 0.684260
\(147\) 0 0
\(148\) 10.6603 0.876267
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) −5.80385 −0.472310 −0.236155 0.971715i \(-0.575887\pi\)
−0.236155 + 0.971715i \(0.575887\pi\)
\(152\) 0.732051 0.0593772
\(153\) 0 0
\(154\) 0 0
\(155\) 2.19615 0.176399
\(156\) 0 0
\(157\) 1.00000 0.0798087 0.0399043 0.999204i \(-0.487295\pi\)
0.0399043 + 0.999204i \(0.487295\pi\)
\(158\) 9.12436 0.725895
\(159\) 0 0
\(160\) 0.267949 0.0211832
\(161\) 0 0
\(162\) 0 0
\(163\) −13.4641 −1.05459 −0.527295 0.849682i \(-0.676794\pi\)
−0.527295 + 0.849682i \(0.676794\pi\)
\(164\) −2.53590 −0.198020
\(165\) 0 0
\(166\) 16.5885 1.28751
\(167\) 1.80385 0.139586 0.0697930 0.997561i \(-0.477766\pi\)
0.0697930 + 0.997561i \(0.477766\pi\)
\(168\) 0 0
\(169\) 28.7846 2.21420
\(170\) −1.87564 −0.143855
\(171\) 0 0
\(172\) −1.46410 −0.111637
\(173\) −6.26795 −0.476543 −0.238272 0.971199i \(-0.576581\pi\)
−0.238272 + 0.971199i \(0.576581\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.19615 0.467053
\(177\) 0 0
\(178\) 9.92820 0.744150
\(179\) −2.19615 −0.164148 −0.0820741 0.996626i \(-0.526154\pi\)
−0.0820741 + 0.996626i \(0.526154\pi\)
\(180\) 0 0
\(181\) 16.3923 1.21843 0.609215 0.793005i \(-0.291485\pi\)
0.609215 + 0.793005i \(0.291485\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 4.19615 0.309344
\(185\) −2.85641 −0.210007
\(186\) 0 0
\(187\) −43.3731 −3.17175
\(188\) −4.73205 −0.345120
\(189\) 0 0
\(190\) −0.196152 −0.0142304
\(191\) −5.66025 −0.409562 −0.204781 0.978808i \(-0.565648\pi\)
−0.204781 + 0.978808i \(0.565648\pi\)
\(192\) 0 0
\(193\) 18.8564 1.35731 0.678657 0.734455i \(-0.262562\pi\)
0.678657 + 0.734455i \(0.262562\pi\)
\(194\) 10.9282 0.784599
\(195\) 0 0
\(196\) 0 0
\(197\) 15.7846 1.12461 0.562303 0.826931i \(-0.309915\pi\)
0.562303 + 0.826931i \(0.309915\pi\)
\(198\) 0 0
\(199\) 19.1244 1.35569 0.677845 0.735205i \(-0.262914\pi\)
0.677845 + 0.735205i \(0.262914\pi\)
\(200\) 4.92820 0.348477
\(201\) 0 0
\(202\) 8.92820 0.628186
\(203\) 0 0
\(204\) 0 0
\(205\) 0.679492 0.0474578
\(206\) −8.39230 −0.584720
\(207\) 0 0
\(208\) 6.46410 0.448205
\(209\) −4.53590 −0.313755
\(210\) 0 0
\(211\) −17.2679 −1.18877 −0.594387 0.804179i \(-0.702605\pi\)
−0.594387 + 0.804179i \(0.702605\pi\)
\(212\) −9.46410 −0.649997
\(213\) 0 0
\(214\) 0 0
\(215\) 0.392305 0.0267550
\(216\) 0 0
\(217\) 0 0
\(218\) −3.19615 −0.216471
\(219\) 0 0
\(220\) −1.66025 −0.111934
\(221\) −45.2487 −3.04376
\(222\) 0 0
\(223\) 25.4641 1.70520 0.852601 0.522562i \(-0.175024\pi\)
0.852601 + 0.522562i \(0.175024\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 5.73205 0.381290
\(227\) 18.9282 1.25631 0.628154 0.778089i \(-0.283811\pi\)
0.628154 + 0.778089i \(0.283811\pi\)
\(228\) 0 0
\(229\) 2.46410 0.162832 0.0814162 0.996680i \(-0.474056\pi\)
0.0814162 + 0.996680i \(0.474056\pi\)
\(230\) −1.12436 −0.0741378
\(231\) 0 0
\(232\) −1.53590 −0.100837
\(233\) −2.80385 −0.183686 −0.0918431 0.995773i \(-0.529276\pi\)
−0.0918431 + 0.995773i \(0.529276\pi\)
\(234\) 0 0
\(235\) 1.26795 0.0827119
\(236\) −4.19615 −0.273146
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0526 −0.650246 −0.325123 0.945672i \(-0.605406\pi\)
−0.325123 + 0.945672i \(0.605406\pi\)
\(240\) 0 0
\(241\) 14.2679 0.919079 0.459540 0.888157i \(-0.348014\pi\)
0.459540 + 0.888157i \(0.348014\pi\)
\(242\) −27.3923 −1.76084
\(243\) 0 0
\(244\) −3.92820 −0.251477
\(245\) 0 0
\(246\) 0 0
\(247\) −4.73205 −0.301093
\(248\) 8.19615 0.520456
\(249\) 0 0
\(250\) −2.66025 −0.168249
\(251\) 22.0526 1.39195 0.695973 0.718068i \(-0.254973\pi\)
0.695973 + 0.718068i \(0.254973\pi\)
\(252\) 0 0
\(253\) −26.0000 −1.63461
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.46410 −0.403220 −0.201610 0.979466i \(-0.564617\pi\)
−0.201610 + 0.979466i \(0.564617\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.73205 −0.107417
\(261\) 0 0
\(262\) 10.5359 0.650910
\(263\) −6.33975 −0.390925 −0.195463 0.980711i \(-0.562621\pi\)
−0.195463 + 0.980711i \(0.562621\pi\)
\(264\) 0 0
\(265\) 2.53590 0.155779
\(266\) 0 0
\(267\) 0 0
\(268\) −6.73205 −0.411225
\(269\) −5.58846 −0.340734 −0.170367 0.985381i \(-0.554495\pi\)
−0.170367 + 0.985381i \(0.554495\pi\)
\(270\) 0 0
\(271\) 19.5167 1.18555 0.592776 0.805367i \(-0.298032\pi\)
0.592776 + 0.805367i \(0.298032\pi\)
\(272\) −7.00000 −0.424437
\(273\) 0 0
\(274\) 8.26795 0.499485
\(275\) −30.5359 −1.84138
\(276\) 0 0
\(277\) −18.7846 −1.12866 −0.564329 0.825550i \(-0.690865\pi\)
−0.564329 + 0.825550i \(0.690865\pi\)
\(278\) −3.26795 −0.195999
\(279\) 0 0
\(280\) 0 0
\(281\) −13.1962 −0.787216 −0.393608 0.919278i \(-0.628773\pi\)
−0.393608 + 0.919278i \(0.628773\pi\)
\(282\) 0 0
\(283\) −15.3205 −0.910710 −0.455355 0.890310i \(-0.650488\pi\)
−0.455355 + 0.890310i \(0.650488\pi\)
\(284\) 6.53590 0.387834
\(285\) 0 0
\(286\) −40.0526 −2.36836
\(287\) 0 0
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 0.411543 0.0241666
\(291\) 0 0
\(292\) −8.26795 −0.483845
\(293\) −3.33975 −0.195110 −0.0975550 0.995230i \(-0.531102\pi\)
−0.0975550 + 0.995230i \(0.531102\pi\)
\(294\) 0 0
\(295\) 1.12436 0.0654625
\(296\) −10.6603 −0.619615
\(297\) 0 0
\(298\) −9.00000 −0.521356
\(299\) −27.1244 −1.56864
\(300\) 0 0
\(301\) 0 0
\(302\) 5.80385 0.333974
\(303\) 0 0
\(304\) −0.732051 −0.0419860
\(305\) 1.05256 0.0602693
\(306\) 0 0
\(307\) 21.8564 1.24741 0.623706 0.781659i \(-0.285626\pi\)
0.623706 + 0.781659i \(0.285626\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.19615 −0.124733
\(311\) −10.1962 −0.578171 −0.289085 0.957303i \(-0.593351\pi\)
−0.289085 + 0.957303i \(0.593351\pi\)
\(312\) 0 0
\(313\) −25.5885 −1.44635 −0.723173 0.690667i \(-0.757317\pi\)
−0.723173 + 0.690667i \(0.757317\pi\)
\(314\) −1.00000 −0.0564333
\(315\) 0 0
\(316\) −9.12436 −0.513285
\(317\) −31.3923 −1.76317 −0.881584 0.472028i \(-0.843522\pi\)
−0.881584 + 0.472028i \(0.843522\pi\)
\(318\) 0 0
\(319\) 9.51666 0.532831
\(320\) −0.267949 −0.0149788
\(321\) 0 0
\(322\) 0 0
\(323\) 5.12436 0.285127
\(324\) 0 0
\(325\) −31.8564 −1.76708
\(326\) 13.4641 0.745708
\(327\) 0 0
\(328\) 2.53590 0.140022
\(329\) 0 0
\(330\) 0 0
\(331\) 12.3923 0.681143 0.340571 0.940219i \(-0.389380\pi\)
0.340571 + 0.940219i \(0.389380\pi\)
\(332\) −16.5885 −0.910410
\(333\) 0 0
\(334\) −1.80385 −0.0987021
\(335\) 1.80385 0.0985547
\(336\) 0 0
\(337\) −16.3923 −0.892946 −0.446473 0.894797i \(-0.647320\pi\)
−0.446473 + 0.894797i \(0.647320\pi\)
\(338\) −28.7846 −1.56568
\(339\) 0 0
\(340\) 1.87564 0.101721
\(341\) −50.7846 −2.75014
\(342\) 0 0
\(343\) 0 0
\(344\) 1.46410 0.0789391
\(345\) 0 0
\(346\) 6.26795 0.336967
\(347\) 21.4641 1.15225 0.576127 0.817360i \(-0.304563\pi\)
0.576127 + 0.817360i \(0.304563\pi\)
\(348\) 0 0
\(349\) −1.46410 −0.0783716 −0.0391858 0.999232i \(-0.512476\pi\)
−0.0391858 + 0.999232i \(0.512476\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6.19615 −0.330256
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) −1.75129 −0.0929488
\(356\) −9.92820 −0.526194
\(357\) 0 0
\(358\) 2.19615 0.116070
\(359\) −10.9282 −0.576769 −0.288384 0.957515i \(-0.593118\pi\)
−0.288384 + 0.957515i \(0.593118\pi\)
\(360\) 0 0
\(361\) −18.4641 −0.971795
\(362\) −16.3923 −0.861560
\(363\) 0 0
\(364\) 0 0
\(365\) 2.21539 0.115959
\(366\) 0 0
\(367\) −11.1244 −0.580687 −0.290343 0.956923i \(-0.593770\pi\)
−0.290343 + 0.956923i \(0.593770\pi\)
\(368\) −4.19615 −0.218740
\(369\) 0 0
\(370\) 2.85641 0.148498
\(371\) 0 0
\(372\) 0 0
\(373\) −6.14359 −0.318103 −0.159052 0.987270i \(-0.550844\pi\)
−0.159052 + 0.987270i \(0.550844\pi\)
\(374\) 43.3731 2.24277
\(375\) 0 0
\(376\) 4.73205 0.244037
\(377\) 9.92820 0.511328
\(378\) 0 0
\(379\) −27.5167 −1.41344 −0.706718 0.707495i \(-0.749825\pi\)
−0.706718 + 0.707495i \(0.749825\pi\)
\(380\) 0.196152 0.0100624
\(381\) 0 0
\(382\) 5.66025 0.289604
\(383\) −19.7128 −1.00728 −0.503639 0.863914i \(-0.668006\pi\)
−0.503639 + 0.863914i \(0.668006\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −18.8564 −0.959766
\(387\) 0 0
\(388\) −10.9282 −0.554795
\(389\) −13.4641 −0.682657 −0.341329 0.939944i \(-0.610877\pi\)
−0.341329 + 0.939944i \(0.610877\pi\)
\(390\) 0 0
\(391\) 29.3731 1.48546
\(392\) 0 0
\(393\) 0 0
\(394\) −15.7846 −0.795217
\(395\) 2.44486 0.123014
\(396\) 0 0
\(397\) 21.0000 1.05396 0.526980 0.849878i \(-0.323324\pi\)
0.526980 + 0.849878i \(0.323324\pi\)
\(398\) −19.1244 −0.958617
\(399\) 0 0
\(400\) −4.92820 −0.246410
\(401\) 10.5167 0.525177 0.262588 0.964908i \(-0.415424\pi\)
0.262588 + 0.964908i \(0.415424\pi\)
\(402\) 0 0
\(403\) −52.9808 −2.63916
\(404\) −8.92820 −0.444195
\(405\) 0 0
\(406\) 0 0
\(407\) 66.0526 3.27410
\(408\) 0 0
\(409\) −17.3397 −0.857395 −0.428698 0.903448i \(-0.641027\pi\)
−0.428698 + 0.903448i \(0.641027\pi\)
\(410\) −0.679492 −0.0335577
\(411\) 0 0
\(412\) 8.39230 0.413459
\(413\) 0 0
\(414\) 0 0
\(415\) 4.44486 0.218190
\(416\) −6.46410 −0.316929
\(417\) 0 0
\(418\) 4.53590 0.221858
\(419\) 9.46410 0.462352 0.231176 0.972912i \(-0.425743\pi\)
0.231176 + 0.972912i \(0.425743\pi\)
\(420\) 0 0
\(421\) 0.124356 0.00606072 0.00303036 0.999995i \(-0.499035\pi\)
0.00303036 + 0.999995i \(0.499035\pi\)
\(422\) 17.2679 0.840591
\(423\) 0 0
\(424\) 9.46410 0.459617
\(425\) 34.4974 1.67337
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −0.392305 −0.0189186
\(431\) −14.5359 −0.700170 −0.350085 0.936718i \(-0.613847\pi\)
−0.350085 + 0.936718i \(0.613847\pi\)
\(432\) 0 0
\(433\) −15.7321 −0.756034 −0.378017 0.925799i \(-0.623394\pi\)
−0.378017 + 0.925799i \(0.623394\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 3.19615 0.153068
\(437\) 3.07180 0.146944
\(438\) 0 0
\(439\) −23.3205 −1.11303 −0.556514 0.830839i \(-0.687861\pi\)
−0.556514 + 0.830839i \(0.687861\pi\)
\(440\) 1.66025 0.0791495
\(441\) 0 0
\(442\) 45.2487 2.15226
\(443\) −15.2679 −0.725402 −0.362701 0.931906i \(-0.618145\pi\)
−0.362701 + 0.931906i \(0.618145\pi\)
\(444\) 0 0
\(445\) 2.66025 0.126108
\(446\) −25.4641 −1.20576
\(447\) 0 0
\(448\) 0 0
\(449\) 15.8564 0.748310 0.374155 0.927366i \(-0.377933\pi\)
0.374155 + 0.927366i \(0.377933\pi\)
\(450\) 0 0
\(451\) −15.7128 −0.739887
\(452\) −5.73205 −0.269613
\(453\) 0 0
\(454\) −18.9282 −0.888345
\(455\) 0 0
\(456\) 0 0
\(457\) −6.85641 −0.320729 −0.160365 0.987058i \(-0.551267\pi\)
−0.160365 + 0.987058i \(0.551267\pi\)
\(458\) −2.46410 −0.115140
\(459\) 0 0
\(460\) 1.12436 0.0524234
\(461\) 6.78461 0.315991 0.157995 0.987440i \(-0.449497\pi\)
0.157995 + 0.987440i \(0.449497\pi\)
\(462\) 0 0
\(463\) −1.41154 −0.0656000 −0.0328000 0.999462i \(-0.510442\pi\)
−0.0328000 + 0.999462i \(0.510442\pi\)
\(464\) 1.53590 0.0713023
\(465\) 0 0
\(466\) 2.80385 0.129886
\(467\) 16.5885 0.767622 0.383811 0.923412i \(-0.374611\pi\)
0.383811 + 0.923412i \(0.374611\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.26795 −0.0584861
\(471\) 0 0
\(472\) 4.19615 0.193144
\(473\) −9.07180 −0.417122
\(474\) 0 0
\(475\) 3.60770 0.165532
\(476\) 0 0
\(477\) 0 0
\(478\) 10.0526 0.459793
\(479\) 21.5167 0.983121 0.491561 0.870843i \(-0.336427\pi\)
0.491561 + 0.870843i \(0.336427\pi\)
\(480\) 0 0
\(481\) 68.9090 3.14198
\(482\) −14.2679 −0.649887
\(483\) 0 0
\(484\) 27.3923 1.24510
\(485\) 2.92820 0.132963
\(486\) 0 0
\(487\) 2.58846 0.117294 0.0586471 0.998279i \(-0.481321\pi\)
0.0586471 + 0.998279i \(0.481321\pi\)
\(488\) 3.92820 0.177821
\(489\) 0 0
\(490\) 0 0
\(491\) 26.5359 1.19755 0.598774 0.800918i \(-0.295655\pi\)
0.598774 + 0.800918i \(0.295655\pi\)
\(492\) 0 0
\(493\) −10.7513 −0.484214
\(494\) 4.73205 0.212905
\(495\) 0 0
\(496\) −8.19615 −0.368018
\(497\) 0 0
\(498\) 0 0
\(499\) 19.8038 0.886542 0.443271 0.896388i \(-0.353818\pi\)
0.443271 + 0.896388i \(0.353818\pi\)
\(500\) 2.66025 0.118970
\(501\) 0 0
\(502\) −22.0526 −0.984254
\(503\) −40.0526 −1.78586 −0.892928 0.450200i \(-0.851353\pi\)
−0.892928 + 0.450200i \(0.851353\pi\)
\(504\) 0 0
\(505\) 2.39230 0.106456
\(506\) 26.0000 1.15584
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −31.8564 −1.41201 −0.706005 0.708207i \(-0.749504\pi\)
−0.706005 + 0.708207i \(0.749504\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.46410 0.285119
\(515\) −2.24871 −0.0990901
\(516\) 0 0
\(517\) −29.3205 −1.28951
\(518\) 0 0
\(519\) 0 0
\(520\) 1.73205 0.0759555
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) −33.1769 −1.45073 −0.725363 0.688367i \(-0.758328\pi\)
−0.725363 + 0.688367i \(0.758328\pi\)
\(524\) −10.5359 −0.460263
\(525\) 0 0
\(526\) 6.33975 0.276426
\(527\) 57.3731 2.49921
\(528\) 0 0
\(529\) −5.39230 −0.234448
\(530\) −2.53590 −0.110152
\(531\) 0 0
\(532\) 0 0
\(533\) −16.3923 −0.710030
\(534\) 0 0
\(535\) 0 0
\(536\) 6.73205 0.290780
\(537\) 0 0
\(538\) 5.58846 0.240936
\(539\) 0 0
\(540\) 0 0
\(541\) 3.33975 0.143587 0.0717934 0.997420i \(-0.477128\pi\)
0.0717934 + 0.997420i \(0.477128\pi\)
\(542\) −19.5167 −0.838312
\(543\) 0 0
\(544\) 7.00000 0.300123
\(545\) −0.856406 −0.0366844
\(546\) 0 0
\(547\) −22.7321 −0.971952 −0.485976 0.873972i \(-0.661536\pi\)
−0.485976 + 0.873972i \(0.661536\pi\)
\(548\) −8.26795 −0.353189
\(549\) 0 0
\(550\) 30.5359 1.30206
\(551\) −1.12436 −0.0478992
\(552\) 0 0
\(553\) 0 0
\(554\) 18.7846 0.798082
\(555\) 0 0
\(556\) 3.26795 0.138592
\(557\) 23.9282 1.01387 0.506935 0.861984i \(-0.330778\pi\)
0.506935 + 0.861984i \(0.330778\pi\)
\(558\) 0 0
\(559\) −9.46410 −0.400289
\(560\) 0 0
\(561\) 0 0
\(562\) 13.1962 0.556646
\(563\) −19.7128 −0.830796 −0.415398 0.909640i \(-0.636358\pi\)
−0.415398 + 0.909640i \(0.636358\pi\)
\(564\) 0 0
\(565\) 1.53590 0.0646157
\(566\) 15.3205 0.643969
\(567\) 0 0
\(568\) −6.53590 −0.274240
\(569\) 23.1962 0.972433 0.486217 0.873838i \(-0.338377\pi\)
0.486217 + 0.873838i \(0.338377\pi\)
\(570\) 0 0
\(571\) −22.7321 −0.951307 −0.475653 0.879633i \(-0.657788\pi\)
−0.475653 + 0.879633i \(0.657788\pi\)
\(572\) 40.0526 1.67468
\(573\) 0 0
\(574\) 0 0
\(575\) 20.6795 0.862394
\(576\) 0 0
\(577\) −24.6603 −1.02662 −0.513310 0.858203i \(-0.671581\pi\)
−0.513310 + 0.858203i \(0.671581\pi\)
\(578\) −32.0000 −1.33102
\(579\) 0 0
\(580\) −0.411543 −0.0170884
\(581\) 0 0
\(582\) 0 0
\(583\) −58.6410 −2.42866
\(584\) 8.26795 0.342130
\(585\) 0 0
\(586\) 3.33975 0.137964
\(587\) −14.7321 −0.608057 −0.304028 0.952663i \(-0.598332\pi\)
−0.304028 + 0.952663i \(0.598332\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) −1.12436 −0.0462890
\(591\) 0 0
\(592\) 10.6603 0.438134
\(593\) 22.1769 0.910697 0.455348 0.890313i \(-0.349515\pi\)
0.455348 + 0.890313i \(0.349515\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.00000 0.368654
\(597\) 0 0
\(598\) 27.1244 1.10920
\(599\) −15.1244 −0.617964 −0.308982 0.951068i \(-0.599988\pi\)
−0.308982 + 0.951068i \(0.599988\pi\)
\(600\) 0 0
\(601\) −19.1962 −0.783027 −0.391514 0.920172i \(-0.628048\pi\)
−0.391514 + 0.920172i \(0.628048\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −5.80385 −0.236155
\(605\) −7.33975 −0.298403
\(606\) 0 0
\(607\) 24.5885 0.998015 0.499007 0.866598i \(-0.333698\pi\)
0.499007 + 0.866598i \(0.333698\pi\)
\(608\) 0.732051 0.0296886
\(609\) 0 0
\(610\) −1.05256 −0.0426169
\(611\) −30.5885 −1.23748
\(612\) 0 0
\(613\) 26.7846 1.08182 0.540910 0.841080i \(-0.318080\pi\)
0.540910 + 0.841080i \(0.318080\pi\)
\(614\) −21.8564 −0.882053
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9808 0.482327 0.241164 0.970484i \(-0.422471\pi\)
0.241164 + 0.970484i \(0.422471\pi\)
\(618\) 0 0
\(619\) 23.7128 0.953098 0.476549 0.879148i \(-0.341887\pi\)
0.476549 + 0.879148i \(0.341887\pi\)
\(620\) 2.19615 0.0881996
\(621\) 0 0
\(622\) 10.1962 0.408828
\(623\) 0 0
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 25.5885 1.02272
\(627\) 0 0
\(628\) 1.00000 0.0399043
\(629\) −74.6218 −2.97537
\(630\) 0 0
\(631\) −3.66025 −0.145712 −0.0728562 0.997342i \(-0.523211\pi\)
−0.0728562 + 0.997342i \(0.523211\pi\)
\(632\) 9.12436 0.362947
\(633\) 0 0
\(634\) 31.3923 1.24675
\(635\) −3.21539 −0.127599
\(636\) 0 0
\(637\) 0 0
\(638\) −9.51666 −0.376768
\(639\) 0 0
\(640\) 0.267949 0.0105916
\(641\) 39.4449 1.55798 0.778989 0.627037i \(-0.215733\pi\)
0.778989 + 0.627037i \(0.215733\pi\)
\(642\) 0 0
\(643\) 9.41154 0.371155 0.185578 0.982630i \(-0.440584\pi\)
0.185578 + 0.982630i \(0.440584\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −5.12436 −0.201615
\(647\) 4.39230 0.172679 0.0863397 0.996266i \(-0.472483\pi\)
0.0863397 + 0.996266i \(0.472483\pi\)
\(648\) 0 0
\(649\) −26.0000 −1.02059
\(650\) 31.8564 1.24951
\(651\) 0 0
\(652\) −13.4641 −0.527295
\(653\) −30.2487 −1.18372 −0.591862 0.806039i \(-0.701607\pi\)
−0.591862 + 0.806039i \(0.701607\pi\)
\(654\) 0 0
\(655\) 2.82309 0.110307
\(656\) −2.53590 −0.0990102
\(657\) 0 0
\(658\) 0 0
\(659\) −36.3923 −1.41764 −0.708821 0.705388i \(-0.750773\pi\)
−0.708821 + 0.705388i \(0.750773\pi\)
\(660\) 0 0
\(661\) 12.8564 0.500056 0.250028 0.968239i \(-0.419560\pi\)
0.250028 + 0.968239i \(0.419560\pi\)
\(662\) −12.3923 −0.481641
\(663\) 0 0
\(664\) 16.5885 0.643757
\(665\) 0 0
\(666\) 0 0
\(667\) −6.44486 −0.249546
\(668\) 1.80385 0.0697930
\(669\) 0 0
\(670\) −1.80385 −0.0696887
\(671\) −24.3397 −0.939625
\(672\) 0 0
\(673\) −18.3205 −0.706204 −0.353102 0.935585i \(-0.614873\pi\)
−0.353102 + 0.935585i \(0.614873\pi\)
\(674\) 16.3923 0.631408
\(675\) 0 0
\(676\) 28.7846 1.10710
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.87564 −0.0719277
\(681\) 0 0
\(682\) 50.7846 1.94464
\(683\) −1.85641 −0.0710334 −0.0355167 0.999369i \(-0.511308\pi\)
−0.0355167 + 0.999369i \(0.511308\pi\)
\(684\) 0 0
\(685\) 2.21539 0.0846457
\(686\) 0 0
\(687\) 0 0
\(688\) −1.46410 −0.0558184
\(689\) −61.1769 −2.33065
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −6.26795 −0.238272
\(693\) 0 0
\(694\) −21.4641 −0.814766
\(695\) −0.875644 −0.0332151
\(696\) 0 0
\(697\) 17.7513 0.672378
\(698\) 1.46410 0.0554171
\(699\) 0 0
\(700\) 0 0
\(701\) 27.3923 1.03459 0.517297 0.855806i \(-0.326938\pi\)
0.517297 + 0.855806i \(0.326938\pi\)
\(702\) 0 0
\(703\) −7.80385 −0.294328
\(704\) 6.19615 0.233526
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) 0 0
\(709\) −3.87564 −0.145553 −0.0727764 0.997348i \(-0.523186\pi\)
−0.0727764 + 0.997348i \(0.523186\pi\)
\(710\) 1.75129 0.0657247
\(711\) 0 0
\(712\) 9.92820 0.372075
\(713\) 34.3923 1.28800
\(714\) 0 0
\(715\) −10.7321 −0.401356
\(716\) −2.19615 −0.0820741
\(717\) 0 0
\(718\) 10.9282 0.407837
\(719\) 9.46410 0.352951 0.176476 0.984305i \(-0.443530\pi\)
0.176476 + 0.984305i \(0.443530\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.4641 0.687163
\(723\) 0 0
\(724\) 16.3923 0.609215
\(725\) −7.56922 −0.281114
\(726\) 0 0
\(727\) −51.3205 −1.90337 −0.951686 0.307072i \(-0.900651\pi\)
−0.951686 + 0.307072i \(0.900651\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.21539 −0.0819953
\(731\) 10.2487 0.379062
\(732\) 0 0
\(733\) −47.3205 −1.74782 −0.873911 0.486085i \(-0.838424\pi\)
−0.873911 + 0.486085i \(0.838424\pi\)
\(734\) 11.1244 0.410607
\(735\) 0 0
\(736\) 4.19615 0.154672
\(737\) −41.7128 −1.53651
\(738\) 0 0
\(739\) −13.2679 −0.488069 −0.244035 0.969767i \(-0.578471\pi\)
−0.244035 + 0.969767i \(0.578471\pi\)
\(740\) −2.85641 −0.105004
\(741\) 0 0
\(742\) 0 0
\(743\) −40.3923 −1.48185 −0.740925 0.671588i \(-0.765613\pi\)
−0.740925 + 0.671588i \(0.765613\pi\)
\(744\) 0 0
\(745\) −2.41154 −0.0883521
\(746\) 6.14359 0.224933
\(747\) 0 0
\(748\) −43.3731 −1.58588
\(749\) 0 0
\(750\) 0 0
\(751\) −22.1436 −0.808031 −0.404016 0.914752i \(-0.632386\pi\)
−0.404016 + 0.914752i \(0.632386\pi\)
\(752\) −4.73205 −0.172560
\(753\) 0 0
\(754\) −9.92820 −0.361564
\(755\) 1.55514 0.0565972
\(756\) 0 0
\(757\) 20.7846 0.755429 0.377715 0.925922i \(-0.376710\pi\)
0.377715 + 0.925922i \(0.376710\pi\)
\(758\) 27.5167 0.999450
\(759\) 0 0
\(760\) −0.196152 −0.00711520
\(761\) −37.0000 −1.34125 −0.670624 0.741797i \(-0.733974\pi\)
−0.670624 + 0.741797i \(0.733974\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.66025 −0.204781
\(765\) 0 0
\(766\) 19.7128 0.712253
\(767\) −27.1244 −0.979404
\(768\) 0 0
\(769\) 4.41154 0.159084 0.0795421 0.996832i \(-0.474654\pi\)
0.0795421 + 0.996832i \(0.474654\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18.8564 0.678657
\(773\) 4.12436 0.148343 0.0741714 0.997246i \(-0.476369\pi\)
0.0741714 + 0.997246i \(0.476369\pi\)
\(774\) 0 0
\(775\) 40.3923 1.45093
\(776\) 10.9282 0.392300
\(777\) 0 0
\(778\) 13.4641 0.482711
\(779\) 1.85641 0.0665127
\(780\) 0 0
\(781\) 40.4974 1.44911
\(782\) −29.3731 −1.05038
\(783\) 0 0
\(784\) 0 0
\(785\) −0.267949 −0.00956352
\(786\) 0 0
\(787\) 28.3923 1.01208 0.506038 0.862511i \(-0.331109\pi\)
0.506038 + 0.862511i \(0.331109\pi\)
\(788\) 15.7846 0.562303
\(789\) 0 0
\(790\) −2.44486 −0.0869843
\(791\) 0 0
\(792\) 0 0
\(793\) −25.3923 −0.901707
\(794\) −21.0000 −0.745262
\(795\) 0 0
\(796\) 19.1244 0.677845
\(797\) 29.4449 1.04299 0.521495 0.853254i \(-0.325374\pi\)
0.521495 + 0.853254i \(0.325374\pi\)
\(798\) 0 0
\(799\) 33.1244 1.17186
\(800\) 4.92820 0.174238
\(801\) 0 0
\(802\) −10.5167 −0.371356
\(803\) −51.2295 −1.80785
\(804\) 0 0
\(805\) 0 0
\(806\) 52.9808 1.86617
\(807\) 0 0
\(808\) 8.92820 0.314093
\(809\) 32.1244 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(810\) 0 0
\(811\) −18.1962 −0.638953 −0.319477 0.947594i \(-0.603507\pi\)
−0.319477 + 0.947594i \(0.603507\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −66.0526 −2.31514
\(815\) 3.60770 0.126372
\(816\) 0 0
\(817\) 1.07180 0.0374974
\(818\) 17.3397 0.606270
\(819\) 0 0
\(820\) 0.679492 0.0237289
\(821\) 25.9282 0.904901 0.452450 0.891790i \(-0.350550\pi\)
0.452450 + 0.891790i \(0.350550\pi\)
\(822\) 0 0
\(823\) 0.784610 0.0273498 0.0136749 0.999906i \(-0.495647\pi\)
0.0136749 + 0.999906i \(0.495647\pi\)
\(824\) −8.39230 −0.292360
\(825\) 0 0
\(826\) 0 0
\(827\) −23.3205 −0.810934 −0.405467 0.914110i \(-0.632891\pi\)
−0.405467 + 0.914110i \(0.632891\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −4.44486 −0.154283
\(831\) 0 0
\(832\) 6.46410 0.224102
\(833\) 0 0
\(834\) 0 0
\(835\) −0.483340 −0.0167267
\(836\) −4.53590 −0.156877
\(837\) 0 0
\(838\) −9.46410 −0.326932
\(839\) −1.46410 −0.0505464 −0.0252732 0.999681i \(-0.508046\pi\)
−0.0252732 + 0.999681i \(0.508046\pi\)
\(840\) 0 0
\(841\) −26.6410 −0.918656
\(842\) −0.124356 −0.00428558
\(843\) 0 0
\(844\) −17.2679 −0.594387
\(845\) −7.71281 −0.265329
\(846\) 0 0
\(847\) 0 0
\(848\) −9.46410 −0.324999
\(849\) 0 0
\(850\) −34.4974 −1.18325
\(851\) −44.7321 −1.53339
\(852\) 0 0
\(853\) 49.7128 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.8564 0.575804 0.287902 0.957660i \(-0.407042\pi\)
0.287902 + 0.957660i \(0.407042\pi\)
\(858\) 0 0
\(859\) −24.3923 −0.832255 −0.416127 0.909306i \(-0.636613\pi\)
−0.416127 + 0.909306i \(0.636613\pi\)
\(860\) 0.392305 0.0133775
\(861\) 0 0
\(862\) 14.5359 0.495095
\(863\) −7.12436 −0.242516 −0.121258 0.992621i \(-0.538693\pi\)
−0.121258 + 0.992621i \(0.538693\pi\)
\(864\) 0 0
\(865\) 1.67949 0.0571044
\(866\) 15.7321 0.534597
\(867\) 0 0
\(868\) 0 0
\(869\) −56.5359 −1.91785
\(870\) 0 0
\(871\) −43.5167 −1.47451
\(872\) −3.19615 −0.108235
\(873\) 0 0
\(874\) −3.07180 −0.103905
\(875\) 0 0
\(876\) 0 0
\(877\) −36.5167 −1.23308 −0.616540 0.787324i \(-0.711466\pi\)
−0.616540 + 0.787324i \(0.711466\pi\)
\(878\) 23.3205 0.787029
\(879\) 0 0
\(880\) −1.66025 −0.0559672
\(881\) −30.2487 −1.01910 −0.509552 0.860440i \(-0.670189\pi\)
−0.509552 + 0.860440i \(0.670189\pi\)
\(882\) 0 0
\(883\) −9.66025 −0.325093 −0.162547 0.986701i \(-0.551971\pi\)
−0.162547 + 0.986701i \(0.551971\pi\)
\(884\) −45.2487 −1.52188
\(885\) 0 0
\(886\) 15.2679 0.512937
\(887\) 56.4449 1.89523 0.947617 0.319410i \(-0.103485\pi\)
0.947617 + 0.319410i \(0.103485\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −2.66025 −0.0891719
\(891\) 0 0
\(892\) 25.4641 0.852601
\(893\) 3.46410 0.115922
\(894\) 0 0
\(895\) 0.588457 0.0196700
\(896\) 0 0
\(897\) 0 0
\(898\) −15.8564 −0.529135
\(899\) −12.5885 −0.419849
\(900\) 0 0
\(901\) 66.2487 2.20706
\(902\) 15.7128 0.523179
\(903\) 0 0
\(904\) 5.73205 0.190645
\(905\) −4.39230 −0.146005
\(906\) 0 0
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 18.9282 0.628154
\(909\) 0 0
\(910\) 0 0
\(911\) 42.2487 1.39976 0.699881 0.714259i \(-0.253236\pi\)
0.699881 + 0.714259i \(0.253236\pi\)
\(912\) 0 0
\(913\) −102.785 −3.40167
\(914\) 6.85641 0.226790
\(915\) 0 0
\(916\) 2.46410 0.0814162
\(917\) 0 0
\(918\) 0 0
\(919\) 26.9808 0.890013 0.445007 0.895527i \(-0.353201\pi\)
0.445007 + 0.895527i \(0.353201\pi\)
\(920\) −1.12436 −0.0370689
\(921\) 0 0
\(922\) −6.78461 −0.223439
\(923\) 42.2487 1.39063
\(924\) 0 0
\(925\) −52.5359 −1.72737
\(926\) 1.41154 0.0463862
\(927\) 0 0
\(928\) −1.53590 −0.0504183
\(929\) 51.4974 1.68958 0.844788 0.535101i \(-0.179727\pi\)
0.844788 + 0.535101i \(0.179727\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.80385 −0.0918431
\(933\) 0 0
\(934\) −16.5885 −0.542791
\(935\) 11.6218 0.380073
\(936\) 0 0
\(937\) 25.8372 0.844064 0.422032 0.906581i \(-0.361317\pi\)
0.422032 + 0.906581i \(0.361317\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 1.26795 0.0413559
\(941\) 34.1244 1.11242 0.556211 0.831041i \(-0.312255\pi\)
0.556211 + 0.831041i \(0.312255\pi\)
\(942\) 0 0
\(943\) 10.6410 0.346519
\(944\) −4.19615 −0.136573
\(945\) 0 0
\(946\) 9.07180 0.294950
\(947\) −46.2487 −1.50288 −0.751441 0.659801i \(-0.770641\pi\)
−0.751441 + 0.659801i \(0.770641\pi\)
\(948\) 0 0
\(949\) −53.4449 −1.73489
\(950\) −3.60770 −0.117049
\(951\) 0 0
\(952\) 0 0
\(953\) −41.5885 −1.34718 −0.673591 0.739104i \(-0.735249\pi\)
−0.673591 + 0.739104i \(0.735249\pi\)
\(954\) 0 0
\(955\) 1.51666 0.0490780
\(956\) −10.0526 −0.325123
\(957\) 0 0
\(958\) −21.5167 −0.695172
\(959\) 0 0
\(960\) 0 0
\(961\) 36.1769 1.16700
\(962\) −68.9090 −2.22171
\(963\) 0 0
\(964\) 14.2679 0.459540
\(965\) −5.05256 −0.162648
\(966\) 0 0
\(967\) −3.66025 −0.117706 −0.0588529 0.998267i \(-0.518744\pi\)
−0.0588529 + 0.998267i \(0.518744\pi\)
\(968\) −27.3923 −0.880422
\(969\) 0 0
\(970\) −2.92820 −0.0940189
\(971\) −8.87564 −0.284833 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.58846 −0.0829395
\(975\) 0 0
\(976\) −3.92820 −0.125739
\(977\) 57.7128 1.84640 0.923198 0.384324i \(-0.125565\pi\)
0.923198 + 0.384324i \(0.125565\pi\)
\(978\) 0 0
\(979\) −61.5167 −1.96608
\(980\) 0 0
\(981\) 0 0
\(982\) −26.5359 −0.846795
\(983\) −58.6410 −1.87036 −0.935179 0.354176i \(-0.884762\pi\)
−0.935179 + 0.354176i \(0.884762\pi\)
\(984\) 0 0
\(985\) −4.22947 −0.134762
\(986\) 10.7513 0.342391
\(987\) 0 0
\(988\) −4.73205 −0.150547
\(989\) 6.14359 0.195355
\(990\) 0 0
\(991\) −27.6603 −0.878657 −0.439328 0.898327i \(-0.644784\pi\)
−0.439328 + 0.898327i \(0.644784\pi\)
\(992\) 8.19615 0.260228
\(993\) 0 0
\(994\) 0 0
\(995\) −5.12436 −0.162453
\(996\) 0 0
\(997\) −41.2487 −1.30636 −0.653180 0.757203i \(-0.726565\pi\)
−0.653180 + 0.757203i \(0.726565\pi\)
\(998\) −19.8038 −0.626880
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7938.2.a.bg.1.2 2
3.2 odd 2 7938.2.a.bt.1.1 2
7.6 odd 2 1134.2.a.l.1.1 2
21.20 even 2 1134.2.a.m.1.2 yes 2
28.27 even 2 9072.2.a.bp.1.1 2
63.13 odd 6 1134.2.f.s.379.2 4
63.20 even 6 1134.2.f.r.757.1 4
63.34 odd 6 1134.2.f.s.757.2 4
63.41 even 6 1134.2.f.r.379.1 4
84.83 odd 2 9072.2.a.y.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1134.2.a.l.1.1 2 7.6 odd 2
1134.2.a.m.1.2 yes 2 21.20 even 2
1134.2.f.r.379.1 4 63.41 even 6
1134.2.f.r.757.1 4 63.20 even 6
1134.2.f.s.379.2 4 63.13 odd 6
1134.2.f.s.757.2 4 63.34 odd 6
7938.2.a.bg.1.2 2 1.1 even 1 trivial
7938.2.a.bt.1.1 2 3.2 odd 2
9072.2.a.y.1.2 2 84.83 odd 2
9072.2.a.bp.1.1 2 28.27 even 2