Properties

Label 768.4.c.o.767.1
Level $768$
Weight $4$
Character 768.767
Analytic conductor $45.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [768,4,Mod(767,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.767");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{-14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 767.1
Root \(-0.646084i\) of defining polynomial
Character \(\chi\) \(=\) 768.767
Dual form 768.4.c.o.767.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.58258 - 2.44949i) q^{3} -2.31464i q^{5} -29.6636i q^{7} +(15.0000 + 22.4499i) q^{9} +O(q^{10})\) \(q+(-4.58258 - 2.44949i) q^{3} -2.31464i q^{5} -29.6636i q^{7} +(15.0000 + 22.4499i) q^{9} +22.8348 q^{11} +8.66061 q^{13} +(-5.66970 + 10.6070i) q^{15} +93.3503i q^{17} +85.4402i q^{19} +(-72.6606 + 135.936i) q^{21} +116.661 q^{23} +119.642 q^{25} +(-13.7477 - 139.621i) q^{27} +103.531i q^{29} -10.6070i q^{31} +(-104.642 - 55.9337i) q^{33} -68.6606 q^{35} -380.624 q^{37} +(-39.6879 - 21.2141i) q^{39} -257.983i q^{41} +359.783i q^{43} +(51.9636 - 34.7197i) q^{45} +293.285 q^{47} -536.927 q^{49} +(228.661 - 427.785i) q^{51} +679.678i q^{53} -52.8545i q^{55} +(209.285 - 391.536i) q^{57} +45.8258 q^{59} +595.230 q^{61} +(665.945 - 444.954i) q^{63} -20.0462i q^{65} +206.341i q^{67} +(-534.606 - 285.759i) q^{69} +996.515 q^{71} +593.927 q^{73} +(-548.270 - 293.063i) q^{75} -677.363i q^{77} -910.940i q^{79} +(-279.000 + 673.498i) q^{81} -332.889 q^{83} +216.073 q^{85} +(253.597 - 474.436i) q^{87} +821.636i q^{89} -256.904i q^{91} +(-25.9818 + 48.6075i) q^{93} +197.764 q^{95} +420.715 q^{97} +(342.523 + 512.641i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 60 q^{9} + 128 q^{11} - 112 q^{13} - 96 q^{15} - 144 q^{21} + 320 q^{23} - 108 q^{25} + 168 q^{33} - 128 q^{35} - 496 q^{37} - 672 q^{39} - 672 q^{45} - 388 q^{49} + 768 q^{51} - 336 q^{57} - 112 q^{61} + 1344 q^{63} - 672 q^{69} + 320 q^{71} + 616 q^{73} - 2688 q^{75} - 1116 q^{81} - 2688 q^{83} + 2624 q^{85} - 672 q^{87} + 336 q^{93} - 4928 q^{95} + 2856 q^{97} + 1920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) −4.58258 2.44949i −0.881917 0.471405i
\(4\) 0 0
\(5\) 2.31464i 0.207028i −0.994628 0.103514i \(-0.966991\pi\)
0.994628 0.103514i \(-0.0330087\pi\)
\(6\) 0 0
\(7\) 29.6636i 1.60168i −0.598877 0.800841i \(-0.704386\pi\)
0.598877 0.800841i \(-0.295614\pi\)
\(8\) 0 0
\(9\) 15.0000 + 22.4499i 0.555556 + 0.831479i
\(10\) 0 0
\(11\) 22.8348 0.625906 0.312953 0.949769i \(-0.398682\pi\)
0.312953 + 0.949769i \(0.398682\pi\)
\(12\) 0 0
\(13\) 8.66061 0.184771 0.0923854 0.995723i \(-0.470551\pi\)
0.0923854 + 0.995723i \(0.470551\pi\)
\(14\) 0 0
\(15\) −5.66970 + 10.6070i −0.0975940 + 0.182582i
\(16\) 0 0
\(17\) 93.3503i 1.33181i 0.746036 + 0.665905i \(0.231954\pi\)
−0.746036 + 0.665905i \(0.768046\pi\)
\(18\) 0 0
\(19\) 85.4402i 1.03165i 0.856694 + 0.515824i \(0.172514\pi\)
−0.856694 + 0.515824i \(0.827486\pi\)
\(20\) 0 0
\(21\) −72.6606 + 135.936i −0.755040 + 1.41255i
\(22\) 0 0
\(23\) 116.661 1.05763 0.528813 0.848738i \(-0.322637\pi\)
0.528813 + 0.848738i \(0.322637\pi\)
\(24\) 0 0
\(25\) 119.642 0.957139
\(26\) 0 0
\(27\) −13.7477 139.621i −0.0979908 0.995187i
\(28\) 0 0
\(29\) 103.531i 0.662936i 0.943467 + 0.331468i \(0.107544\pi\)
−0.943467 + 0.331468i \(0.892456\pi\)
\(30\) 0 0
\(31\) 10.6070i 0.0614542i −0.999528 0.0307271i \(-0.990218\pi\)
0.999528 0.0307271i \(-0.00978227\pi\)
\(32\) 0 0
\(33\) −104.642 55.9337i −0.551997 0.295055i
\(34\) 0 0
\(35\) −68.6606 −0.331593
\(36\) 0 0
\(37\) −380.624 −1.69120 −0.845598 0.533820i \(-0.820756\pi\)
−0.845598 + 0.533820i \(0.820756\pi\)
\(38\) 0 0
\(39\) −39.6879 21.2141i −0.162952 0.0871018i
\(40\) 0 0
\(41\) 257.983i 0.982688i −0.870966 0.491344i \(-0.836506\pi\)
0.870966 0.491344i \(-0.163494\pi\)
\(42\) 0 0
\(43\) 359.783i 1.27596i 0.770052 + 0.637981i \(0.220230\pi\)
−0.770052 + 0.637981i \(0.779770\pi\)
\(44\) 0 0
\(45\) 51.9636 34.7197i 0.172140 0.115016i
\(46\) 0 0
\(47\) 293.285 0.910213 0.455106 0.890437i \(-0.349601\pi\)
0.455106 + 0.890437i \(0.349601\pi\)
\(48\) 0 0
\(49\) −536.927 −1.56539
\(50\) 0 0
\(51\) 228.661 427.785i 0.627821 1.17455i
\(52\) 0 0
\(53\) 679.678i 1.76153i 0.473557 + 0.880763i \(0.342970\pi\)
−0.473557 + 0.880763i \(0.657030\pi\)
\(54\) 0 0
\(55\) 52.8545i 0.129580i
\(56\) 0 0
\(57\) 209.285 391.536i 0.486324 0.909828i
\(58\) 0 0
\(59\) 45.8258 0.101119 0.0505594 0.998721i \(-0.483900\pi\)
0.0505594 + 0.998721i \(0.483900\pi\)
\(60\) 0 0
\(61\) 595.230 1.24937 0.624684 0.780878i \(-0.285228\pi\)
0.624684 + 0.780878i \(0.285228\pi\)
\(62\) 0 0
\(63\) 665.945 444.954i 1.33177 0.889823i
\(64\) 0 0
\(65\) 20.0462i 0.0382527i
\(66\) 0 0
\(67\) 206.341i 0.376247i 0.982145 + 0.188124i \(0.0602406\pi\)
−0.982145 + 0.188124i \(0.939759\pi\)
\(68\) 0 0
\(69\) −534.606 285.759i −0.932739 0.498570i
\(70\) 0 0
\(71\) 996.515 1.66570 0.832849 0.553500i \(-0.186708\pi\)
0.832849 + 0.553500i \(0.186708\pi\)
\(72\) 0 0
\(73\) 593.927 0.952246 0.476123 0.879379i \(-0.342042\pi\)
0.476123 + 0.879379i \(0.342042\pi\)
\(74\) 0 0
\(75\) −548.270 293.063i −0.844118 0.451200i
\(76\) 0 0
\(77\) 677.363i 1.00250i
\(78\) 0 0
\(79\) 910.940i 1.29733i −0.761075 0.648663i \(-0.775328\pi\)
0.761075 0.648663i \(-0.224672\pi\)
\(80\) 0 0
\(81\) −279.000 + 673.498i −0.382716 + 0.923866i
\(82\) 0 0
\(83\) −332.889 −0.440233 −0.220117 0.975474i \(-0.570644\pi\)
−0.220117 + 0.975474i \(0.570644\pi\)
\(84\) 0 0
\(85\) 216.073 0.275722
\(86\) 0 0
\(87\) 253.597 474.436i 0.312511 0.584654i
\(88\) 0 0
\(89\) 821.636i 0.978575i 0.872122 + 0.489288i \(0.162743\pi\)
−0.872122 + 0.489288i \(0.837257\pi\)
\(90\) 0 0
\(91\) 256.904i 0.295944i
\(92\) 0 0
\(93\) −25.9818 + 48.6075i −0.0289698 + 0.0541975i
\(94\) 0 0
\(95\) 197.764 0.213580
\(96\) 0 0
\(97\) 420.715 0.440383 0.220192 0.975457i \(-0.429332\pi\)
0.220192 + 0.975457i \(0.429332\pi\)
\(98\) 0 0
\(99\) 342.523 + 512.641i 0.347726 + 0.520428i
\(100\) 0 0
\(101\) 923.145i 0.909469i −0.890627 0.454734i \(-0.849734\pi\)
0.890627 0.454734i \(-0.150266\pi\)
\(102\) 0 0
\(103\) 1426.19i 1.36434i −0.731195 0.682168i \(-0.761037\pi\)
0.731195 0.682168i \(-0.238963\pi\)
\(104\) 0 0
\(105\) 314.642 + 168.183i 0.292438 + 0.156315i
\(106\) 0 0
\(107\) 1581.83 1.42917 0.714583 0.699550i \(-0.246616\pi\)
0.714583 + 0.699550i \(0.246616\pi\)
\(108\) 0 0
\(109\) 766.091 0.673195 0.336597 0.941649i \(-0.390724\pi\)
0.336597 + 0.941649i \(0.390724\pi\)
\(110\) 0 0
\(111\) 1744.24 + 932.335i 1.49149 + 0.797237i
\(112\) 0 0
\(113\) 1047.66i 0.872177i 0.899904 + 0.436088i \(0.143636\pi\)
−0.899904 + 0.436088i \(0.856364\pi\)
\(114\) 0 0
\(115\) 270.028i 0.218958i
\(116\) 0 0
\(117\) 129.909 + 194.430i 0.102650 + 0.153633i
\(118\) 0 0
\(119\) 2769.10 2.13314
\(120\) 0 0
\(121\) −809.570 −0.608242
\(122\) 0 0
\(123\) −631.927 + 1182.23i −0.463244 + 0.866649i
\(124\) 0 0
\(125\) 566.260i 0.405183i
\(126\) 0 0
\(127\) 553.181i 0.386511i 0.981148 + 0.193256i \(0.0619046\pi\)
−0.981148 + 0.193256i \(0.938095\pi\)
\(128\) 0 0
\(129\) 881.285 1648.73i 0.601495 1.12529i
\(130\) 0 0
\(131\) −2151.63 −1.43503 −0.717513 0.696545i \(-0.754720\pi\)
−0.717513 + 0.696545i \(0.754720\pi\)
\(132\) 0 0
\(133\) 2534.46 1.65237
\(134\) 0 0
\(135\) −323.173 + 31.8211i −0.206032 + 0.0202868i
\(136\) 0 0
\(137\) 1476.62i 0.920846i −0.887700 0.460423i \(-0.847698\pi\)
0.887700 0.460423i \(-0.152302\pi\)
\(138\) 0 0
\(139\) 1984.09i 1.21071i −0.795957 0.605353i \(-0.793032\pi\)
0.795957 0.605353i \(-0.206968\pi\)
\(140\) 0 0
\(141\) −1344.00 718.398i −0.802732 0.429078i
\(142\) 0 0
\(143\) 197.764 0.115649
\(144\) 0 0
\(145\) 239.636 0.137246
\(146\) 0 0
\(147\) 2460.51 + 1315.20i 1.38054 + 0.737930i
\(148\) 0 0
\(149\) 3026.56i 1.66406i −0.554729 0.832031i \(-0.687178\pi\)
0.554729 0.832031i \(-0.312822\pi\)
\(150\) 0 0
\(151\) 350.032i 0.188644i 0.995542 + 0.0943219i \(0.0300683\pi\)
−0.995542 + 0.0943219i \(0.969932\pi\)
\(152\) 0 0
\(153\) −2095.71 + 1400.25i −1.10737 + 0.739895i
\(154\) 0 0
\(155\) −24.5515 −0.0127227
\(156\) 0 0
\(157\) −372.406 −0.189307 −0.0946536 0.995510i \(-0.530174\pi\)
−0.0946536 + 0.995510i \(0.530174\pi\)
\(158\) 0 0
\(159\) 1664.86 3114.67i 0.830392 1.55352i
\(160\) 0 0
\(161\) 3460.57i 1.69398i
\(162\) 0 0
\(163\) 1353.69i 0.650487i −0.945630 0.325243i \(-0.894554\pi\)
0.945630 0.325243i \(-0.105446\pi\)
\(164\) 0 0
\(165\) −129.467 + 242.210i −0.0610846 + 0.114279i
\(166\) 0 0
\(167\) 2592.48 1.20127 0.600635 0.799523i \(-0.294915\pi\)
0.600635 + 0.799523i \(0.294915\pi\)
\(168\) 0 0
\(169\) −2121.99 −0.965860
\(170\) 0 0
\(171\) −1918.13 + 1281.60i −0.857794 + 0.573138i
\(172\) 0 0
\(173\) 3643.74i 1.60132i 0.599118 + 0.800660i \(0.295518\pi\)
−0.599118 + 0.800660i \(0.704482\pi\)
\(174\) 0 0
\(175\) 3549.02i 1.53303i
\(176\) 0 0
\(177\) −210.000 112.250i −0.0891783 0.0476678i
\(178\) 0 0
\(179\) 874.435 0.365130 0.182565 0.983194i \(-0.441560\pi\)
0.182565 + 0.983194i \(0.441560\pi\)
\(180\) 0 0
\(181\) 2699.01 1.10838 0.554188 0.832392i \(-0.313029\pi\)
0.554188 + 0.832392i \(0.313029\pi\)
\(182\) 0 0
\(183\) −2727.69 1458.01i −1.10184 0.588958i
\(184\) 0 0
\(185\) 881.010i 0.350125i
\(186\) 0 0
\(187\) 2131.64i 0.833588i
\(188\) 0 0
\(189\) −4141.65 + 407.807i −1.59397 + 0.156950i
\(190\) 0 0
\(191\) 708.036 0.268229 0.134114 0.990966i \(-0.457181\pi\)
0.134114 + 0.990966i \(0.457181\pi\)
\(192\) 0 0
\(193\) 3531.07 1.31695 0.658476 0.752602i \(-0.271201\pi\)
0.658476 + 0.752602i \(0.271201\pi\)
\(194\) 0 0
\(195\) −49.1030 + 91.8633i −0.0180325 + 0.0337357i
\(196\) 0 0
\(197\) 4919.77i 1.77928i 0.456658 + 0.889642i \(0.349046\pi\)
−0.456658 + 0.889642i \(0.650954\pi\)
\(198\) 0 0
\(199\) 264.092i 0.0940754i −0.998893 0.0470377i \(-0.985022\pi\)
0.998893 0.0470377i \(-0.0149781\pi\)
\(200\) 0 0
\(201\) 505.430 945.574i 0.177365 0.331819i
\(202\) 0 0
\(203\) 3071.08 1.06181
\(204\) 0 0
\(205\) −597.139 −0.203444
\(206\) 0 0
\(207\) 1749.91 + 2619.02i 0.587570 + 0.879395i
\(208\) 0 0
\(209\) 1951.01i 0.645715i
\(210\) 0 0
\(211\) 3422.86i 1.11678i −0.829580 0.558388i \(-0.811420\pi\)
0.829580 0.558388i \(-0.188580\pi\)
\(212\) 0 0
\(213\) −4566.61 2440.95i −1.46901 0.785218i
\(214\) 0 0
\(215\) 832.770 0.264160
\(216\) 0 0
\(217\) −314.642 −0.0984300
\(218\) 0 0
\(219\) −2721.72 1454.82i −0.839802 0.448893i
\(220\) 0 0
\(221\) 808.470i 0.246080i
\(222\) 0 0
\(223\) 4432.09i 1.33092i 0.746434 + 0.665460i \(0.231764\pi\)
−0.746434 + 0.665460i \(0.768236\pi\)
\(224\) 0 0
\(225\) 1794.64 + 2685.97i 0.531744 + 0.795842i
\(226\) 0 0
\(227\) −4674.99 −1.36692 −0.683458 0.729990i \(-0.739525\pi\)
−0.683458 + 0.729990i \(0.739525\pi\)
\(228\) 0 0
\(229\) 2317.95 0.668883 0.334441 0.942417i \(-0.391452\pi\)
0.334441 + 0.942417i \(0.391452\pi\)
\(230\) 0 0
\(231\) −1659.19 + 3104.07i −0.472584 + 0.884124i
\(232\) 0 0
\(233\) 5342.54i 1.50215i 0.660215 + 0.751076i \(0.270465\pi\)
−0.660215 + 0.751076i \(0.729535\pi\)
\(234\) 0 0
\(235\) 678.850i 0.188440i
\(236\) 0 0
\(237\) −2231.34 + 4174.45i −0.611566 + 1.14413i
\(238\) 0 0
\(239\) −5306.42 −1.43617 −0.718084 0.695957i \(-0.754981\pi\)
−0.718084 + 0.695957i \(0.754981\pi\)
\(240\) 0 0
\(241\) −1220.35 −0.326182 −0.163091 0.986611i \(-0.552146\pi\)
−0.163091 + 0.986611i \(0.552146\pi\)
\(242\) 0 0
\(243\) 2928.27 2402.95i 0.773038 0.634359i
\(244\) 0 0
\(245\) 1242.80i 0.324079i
\(246\) 0 0
\(247\) 739.964i 0.190618i
\(248\) 0 0
\(249\) 1525.49 + 815.409i 0.388249 + 0.207528i
\(250\) 0 0
\(251\) 385.695 0.0969916 0.0484958 0.998823i \(-0.484557\pi\)
0.0484958 + 0.998823i \(0.484557\pi\)
\(252\) 0 0
\(253\) 2663.93 0.661975
\(254\) 0 0
\(255\) −990.170 529.268i −0.243164 0.129977i
\(256\) 0 0
\(257\) 1158.77i 0.281253i −0.990063 0.140626i \(-0.955088\pi\)
0.990063 0.140626i \(-0.0449116\pi\)
\(258\) 0 0
\(259\) 11290.7i 2.70876i
\(260\) 0 0
\(261\) −2324.25 + 1552.96i −0.551217 + 0.368298i
\(262\) 0 0
\(263\) −1677.25 −0.393247 −0.196623 0.980479i \(-0.562998\pi\)
−0.196623 + 0.980479i \(0.562998\pi\)
\(264\) 0 0
\(265\) 1573.21 0.364685
\(266\) 0 0
\(267\) 2012.59 3765.21i 0.461305 0.863022i
\(268\) 0 0
\(269\) 4080.38i 0.924852i −0.886658 0.462426i \(-0.846979\pi\)
0.886658 0.462426i \(-0.153021\pi\)
\(270\) 0 0
\(271\) 550.301i 0.123352i 0.998096 + 0.0616761i \(0.0196446\pi\)
−0.998096 + 0.0616761i \(0.980355\pi\)
\(272\) 0 0
\(273\) −629.285 + 1177.28i −0.139509 + 0.260998i
\(274\) 0 0
\(275\) 2732.02 0.599079
\(276\) 0 0
\(277\) −468.115 −0.101539 −0.0507695 0.998710i \(-0.516167\pi\)
−0.0507695 + 0.998710i \(0.516167\pi\)
\(278\) 0 0
\(279\) 238.127 159.105i 0.0510979 0.0341412i
\(280\) 0 0
\(281\) 5322.22i 1.12988i −0.825131 0.564942i \(-0.808899\pi\)
0.825131 0.564942i \(-0.191101\pi\)
\(282\) 0 0
\(283\) 109.886i 0.0230814i −0.999933 0.0115407i \(-0.996326\pi\)
0.999933 0.0115407i \(-0.00367359\pi\)
\(284\) 0 0
\(285\) −906.267 484.420i −0.188360 0.100683i
\(286\) 0 0
\(287\) −7652.70 −1.57395
\(288\) 0 0
\(289\) −3801.28 −0.773718
\(290\) 0 0
\(291\) −1927.96 1030.54i −0.388381 0.207599i
\(292\) 0 0
\(293\) 2671.59i 0.532683i −0.963879 0.266341i \(-0.914185\pi\)
0.963879 0.266341i \(-0.0858148\pi\)
\(294\) 0 0
\(295\) 106.070i 0.0209344i
\(296\) 0 0
\(297\) −313.927 3188.22i −0.0613330 0.622894i
\(298\) 0 0
\(299\) 1010.35 0.195419
\(300\) 0 0
\(301\) 10672.4 2.04369
\(302\) 0 0
\(303\) −2261.23 + 4230.38i −0.428728 + 0.802076i
\(304\) 0 0
\(305\) 1377.75i 0.258654i
\(306\) 0 0
\(307\) 2441.80i 0.453944i −0.973901 0.226972i \(-0.927117\pi\)
0.973901 0.226972i \(-0.0728826\pi\)
\(308\) 0 0
\(309\) −3493.44 + 6535.62i −0.643154 + 1.20323i
\(310\) 0 0
\(311\) −6897.58 −1.25764 −0.628820 0.777551i \(-0.716462\pi\)
−0.628820 + 0.777551i \(0.716462\pi\)
\(312\) 0 0
\(313\) 5278.92 0.953298 0.476649 0.879094i \(-0.341851\pi\)
0.476649 + 0.879094i \(0.341851\pi\)
\(314\) 0 0
\(315\) −1029.91 1541.43i −0.184218 0.275713i
\(316\) 0 0
\(317\) 1537.58i 0.272427i −0.990680 0.136213i \(-0.956507\pi\)
0.990680 0.136213i \(-0.0434933\pi\)
\(318\) 0 0
\(319\) 2364.10i 0.414935i
\(320\) 0 0
\(321\) −7248.84 3874.67i −1.26041 0.673716i
\(322\) 0 0
\(323\) −7975.87 −1.37396
\(324\) 0 0
\(325\) 1036.18 0.176851
\(326\) 0 0
\(327\) −3510.67 1876.53i −0.593702 0.317347i
\(328\) 0 0
\(329\) 8699.87i 1.45787i
\(330\) 0 0
\(331\) 8894.54i 1.47700i 0.674252 + 0.738502i \(0.264466\pi\)
−0.674252 + 0.738502i \(0.735534\pi\)
\(332\) 0 0
\(333\) −5709.36 8544.99i −0.939553 1.40619i
\(334\) 0 0
\(335\) 477.606 0.0778938
\(336\) 0 0
\(337\) −6422.48 −1.03815 −0.519073 0.854730i \(-0.673723\pi\)
−0.519073 + 0.854730i \(0.673723\pi\)
\(338\) 0 0
\(339\) 2566.24 4801.00i 0.411148 0.769187i
\(340\) 0 0
\(341\) 242.210i 0.0384645i
\(342\) 0 0
\(343\) 5752.57i 0.905568i
\(344\) 0 0
\(345\) −661.430 + 1237.42i −0.103218 + 0.193103i
\(346\) 0 0
\(347\) −3357.43 −0.519413 −0.259707 0.965688i \(-0.583626\pi\)
−0.259707 + 0.965688i \(0.583626\pi\)
\(348\) 0 0
\(349\) 11693.7 1.79354 0.896772 0.442493i \(-0.145906\pi\)
0.896772 + 0.442493i \(0.145906\pi\)
\(350\) 0 0
\(351\) −119.064 1209.20i −0.0181058 0.183882i
\(352\) 0 0
\(353\) 9945.93i 1.49963i 0.661649 + 0.749813i \(0.269857\pi\)
−0.661649 + 0.749813i \(0.730143\pi\)
\(354\) 0 0
\(355\) 2306.58i 0.344846i
\(356\) 0 0
\(357\) −12689.6 6782.89i −1.88125 1.00557i
\(358\) 0 0
\(359\) −8527.75 −1.25370 −0.626849 0.779141i \(-0.715656\pi\)
−0.626849 + 0.779141i \(0.715656\pi\)
\(360\) 0 0
\(361\) −441.024 −0.0642986
\(362\) 0 0
\(363\) 3709.91 + 1983.03i 0.536419 + 0.286728i
\(364\) 0 0
\(365\) 1374.73i 0.197142i
\(366\) 0 0
\(367\) 891.162i 0.126753i 0.997990 + 0.0633764i \(0.0201869\pi\)
−0.997990 + 0.0633764i \(0.979813\pi\)
\(368\) 0 0
\(369\) 5791.71 3869.75i 0.817085 0.545938i
\(370\) 0 0
\(371\) 20161.7 2.82141
\(372\) 0 0
\(373\) 9107.23 1.26422 0.632111 0.774878i \(-0.282189\pi\)
0.632111 + 0.774878i \(0.282189\pi\)
\(374\) 0 0
\(375\) −1387.05 + 2594.93i −0.191005 + 0.357338i
\(376\) 0 0
\(377\) 896.637i 0.122491i
\(378\) 0 0
\(379\) 1451.32i 0.196699i −0.995152 0.0983497i \(-0.968644\pi\)
0.995152 0.0983497i \(-0.0313564\pi\)
\(380\) 0 0
\(381\) 1355.01 2535.00i 0.182203 0.340871i
\(382\) 0 0
\(383\) −3320.30 −0.442975 −0.221488 0.975163i \(-0.571091\pi\)
−0.221488 + 0.975163i \(0.571091\pi\)
\(384\) 0 0
\(385\) −1567.85 −0.207546
\(386\) 0 0
\(387\) −8077.11 + 5396.75i −1.06094 + 0.708868i
\(388\) 0 0
\(389\) 5726.58i 0.746399i −0.927751 0.373200i \(-0.878261\pi\)
0.927751 0.373200i \(-0.121739\pi\)
\(390\) 0 0
\(391\) 10890.3i 1.40856i
\(392\) 0 0
\(393\) 9859.99 + 5270.39i 1.26557 + 0.676478i
\(394\) 0 0
\(395\) −2108.50 −0.268583
\(396\) 0 0
\(397\) 5389.36 0.681321 0.340660 0.940186i \(-0.389349\pi\)
0.340660 + 0.940186i \(0.389349\pi\)
\(398\) 0 0
\(399\) −11614.4 6208.14i −1.45726 0.778936i
\(400\) 0 0
\(401\) 573.267i 0.0713905i 0.999363 + 0.0356953i \(0.0113646\pi\)
−0.999363 + 0.0356953i \(0.988635\pi\)
\(402\) 0 0
\(403\) 91.8633i 0.0113549i
\(404\) 0 0
\(405\) 1558.91 + 645.786i 0.191266 + 0.0792330i
\(406\) 0 0
\(407\) −8691.50 −1.05853
\(408\) 0 0
\(409\) 6576.42 0.795069 0.397535 0.917587i \(-0.369866\pi\)
0.397535 + 0.917587i \(0.369866\pi\)
\(410\) 0 0
\(411\) −3616.96 + 6766.71i −0.434091 + 0.812110i
\(412\) 0 0
\(413\) 1359.36i 0.161960i
\(414\) 0 0
\(415\) 770.521i 0.0911406i
\(416\) 0 0
\(417\) −4860.00 + 9092.23i −0.570732 + 1.06774i
\(418\) 0 0
\(419\) −8544.87 −0.996286 −0.498143 0.867095i \(-0.665985\pi\)
−0.498143 + 0.867095i \(0.665985\pi\)
\(420\) 0 0
\(421\) 1822.38 0.210968 0.105484 0.994421i \(-0.466361\pi\)
0.105484 + 0.994421i \(0.466361\pi\)
\(422\) 0 0
\(423\) 4399.27 + 6584.23i 0.505674 + 0.756823i
\(424\) 0 0
\(425\) 11168.7i 1.27473i
\(426\) 0 0
\(427\) 17656.7i 2.00109i
\(428\) 0 0
\(429\) −906.267 484.420i −0.101993 0.0545175i
\(430\) 0 0
\(431\) 2373.58 0.265269 0.132635 0.991165i \(-0.457656\pi\)
0.132635 + 0.991165i \(0.457656\pi\)
\(432\) 0 0
\(433\) 12307.8 1.36600 0.682999 0.730419i \(-0.260675\pi\)
0.682999 + 0.730419i \(0.260675\pi\)
\(434\) 0 0
\(435\) −1098.15 586.987i −0.121040 0.0646985i
\(436\) 0 0
\(437\) 9967.50i 1.09110i
\(438\) 0 0
\(439\) 5371.26i 0.583955i −0.956425 0.291978i \(-0.905687\pi\)
0.956425 0.291978i \(-0.0943133\pi\)
\(440\) 0 0
\(441\) −8053.91 12054.0i −0.869659 1.30159i
\(442\) 0 0
\(443\) −9066.41 −0.972366 −0.486183 0.873857i \(-0.661611\pi\)
−0.486183 + 0.873857i \(0.661611\pi\)
\(444\) 0 0
\(445\) 1901.79 0.202593
\(446\) 0 0
\(447\) −7413.52 + 13869.4i −0.784447 + 1.46757i
\(448\) 0 0
\(449\) 6049.97i 0.635892i 0.948109 + 0.317946i \(0.102993\pi\)
−0.948109 + 0.317946i \(0.897007\pi\)
\(450\) 0 0
\(451\) 5891.01i 0.615070i
\(452\) 0 0
\(453\) 857.400 1604.05i 0.0889275 0.166368i
\(454\) 0 0
\(455\) −594.642 −0.0612687
\(456\) 0 0
\(457\) −6664.98 −0.682220 −0.341110 0.940023i \(-0.610803\pi\)
−0.341110 + 0.940023i \(0.610803\pi\)
\(458\) 0 0
\(459\) 13033.7 1283.35i 1.32540 0.130505i
\(460\) 0 0
\(461\) 2588.66i 0.261531i −0.991413 0.130766i \(-0.958256\pi\)
0.991413 0.130766i \(-0.0417436\pi\)
\(462\) 0 0
\(463\) 12241.7i 1.22877i 0.789006 + 0.614385i \(0.210596\pi\)
−0.789006 + 0.614385i \(0.789404\pi\)
\(464\) 0 0
\(465\) 112.509 + 60.1387i 0.0112204 + 0.00599756i
\(466\) 0 0
\(467\) 6144.26 0.608828 0.304414 0.952540i \(-0.401539\pi\)
0.304414 + 0.952540i \(0.401539\pi\)
\(468\) 0 0
\(469\) 6120.81 0.602629
\(470\) 0 0
\(471\) 1706.58 + 912.205i 0.166953 + 0.0892403i
\(472\) 0 0
\(473\) 8215.59i 0.798633i
\(474\) 0 0
\(475\) 10222.3i 0.987431i
\(476\) 0 0
\(477\) −15258.7 + 10195.2i −1.46467 + 0.978626i
\(478\) 0 0
\(479\) 12210.4 1.16473 0.582365 0.812928i \(-0.302128\pi\)
0.582365 + 0.812928i \(0.302128\pi\)
\(480\) 0 0
\(481\) −3296.44 −0.312483
\(482\) 0 0
\(483\) −8476.63 + 15858.3i −0.798551 + 1.49395i
\(484\) 0 0
\(485\) 973.806i 0.0911716i
\(486\) 0 0
\(487\) 14354.5i 1.33565i −0.744317 0.667827i \(-0.767225\pi\)
0.744317 0.667827i \(-0.232775\pi\)
\(488\) 0 0
\(489\) −3315.85 + 6203.40i −0.306642 + 0.573675i
\(490\) 0 0
\(491\) 3894.57 0.357962 0.178981 0.983853i \(-0.442720\pi\)
0.178981 + 0.983853i \(0.442720\pi\)
\(492\) 0 0
\(493\) −9664.61 −0.882905
\(494\) 0 0
\(495\) 1186.58 792.818i 0.107743 0.0719889i
\(496\) 0 0
\(497\) 29560.2i 2.66792i
\(498\) 0 0
\(499\) 13373.6i 1.19977i −0.800086 0.599886i \(-0.795213\pi\)
0.800086 0.599886i \(-0.204787\pi\)
\(500\) 0 0
\(501\) −11880.2 6350.25i −1.05942 0.566284i
\(502\) 0 0
\(503\) 3757.55 0.333083 0.166541 0.986034i \(-0.446740\pi\)
0.166541 + 0.986034i \(0.446740\pi\)
\(504\) 0 0
\(505\) −2136.75 −0.188286
\(506\) 0 0
\(507\) 9724.20 + 5197.80i 0.851808 + 0.455311i
\(508\) 0 0
\(509\) 5968.10i 0.519708i 0.965648 + 0.259854i \(0.0836745\pi\)
−0.965648 + 0.259854i \(0.916326\pi\)
\(510\) 0 0
\(511\) 17618.0i 1.52519i
\(512\) 0 0
\(513\) 11929.2 1174.61i 1.02668 0.101092i
\(514\) 0 0
\(515\) −3301.12 −0.282456
\(516\) 0 0
\(517\) 6697.12 0.569708
\(518\) 0 0
\(519\) 8925.31 16697.7i 0.754870 1.41223i
\(520\) 0 0
\(521\) 10169.5i 0.855151i −0.903980 0.427575i \(-0.859368\pi\)
0.903980 0.427575i \(-0.140632\pi\)
\(522\) 0 0
\(523\) 13759.2i 1.15038i −0.818021 0.575188i \(-0.804929\pi\)
0.818021 0.575188i \(-0.195071\pi\)
\(524\) 0 0
\(525\) −8693.29 + 16263.7i −0.722679 + 1.35201i
\(526\) 0 0
\(527\) 990.170 0.0818453
\(528\) 0 0
\(529\) 1442.70 0.118575
\(530\) 0 0
\(531\) 687.386 + 1028.79i 0.0561771 + 0.0840781i
\(532\) 0 0
\(533\) 2234.29i 0.181572i
\(534\) 0 0
\(535\) 3661.36i 0.295878i
\(536\) 0 0
\(537\) −4007.16 2141.92i −0.322015 0.172124i
\(538\) 0 0
\(539\) −12260.7 −0.979784
\(540\) 0 0
\(541\) −10310.6 −0.819386 −0.409693 0.912223i \(-0.634364\pi\)
−0.409693 + 0.912223i \(0.634364\pi\)
\(542\) 0 0
\(543\) −12368.4 6611.20i −0.977495 0.522493i
\(544\) 0 0
\(545\) 1773.23i 0.139370i
\(546\) 0 0
\(547\) 19226.5i 1.50287i 0.659810 + 0.751433i \(0.270637\pi\)
−0.659810 + 0.751433i \(0.729363\pi\)
\(548\) 0 0
\(549\) 8928.45 + 13362.9i 0.694093 + 1.03882i
\(550\) 0 0
\(551\) −8845.67 −0.683917
\(552\) 0 0
\(553\) −27021.7 −2.07791
\(554\) 0 0
\(555\) 2158.02 4037.29i 0.165050 0.308781i
\(556\) 0 0
\(557\) 14675.8i 1.11640i 0.829707 + 0.558199i \(0.188508\pi\)
−0.829707 + 0.558199i \(0.811492\pi\)
\(558\) 0 0
\(559\) 3115.94i 0.235761i
\(560\) 0 0
\(561\) 5221.43 9768.40i 0.392957 0.735156i
\(562\) 0 0
\(563\) 21002.2 1.57218 0.786092 0.618110i \(-0.212101\pi\)
0.786092 + 0.618110i \(0.212101\pi\)
\(564\) 0 0
\(565\) 2424.97 0.180565
\(566\) 0 0
\(567\) 19978.4 + 8276.14i 1.47974 + 0.612989i
\(568\) 0 0
\(569\) 4389.21i 0.323384i −0.986841 0.161692i \(-0.948305\pi\)
0.986841 0.161692i \(-0.0516951\pi\)
\(570\) 0 0
\(571\) 3964.70i 0.290574i −0.989390 0.145287i \(-0.953590\pi\)
0.989390 0.145287i \(-0.0464105\pi\)
\(572\) 0 0
\(573\) −3244.63 1734.33i −0.236556 0.126444i
\(574\) 0 0
\(575\) 13957.6 1.01230
\(576\) 0 0
\(577\) 5848.93 0.422000 0.211000 0.977486i \(-0.432328\pi\)
0.211000 + 0.977486i \(0.432328\pi\)
\(578\) 0 0
\(579\) −16181.4 8649.31i −1.16144 0.620817i
\(580\) 0 0
\(581\) 9874.69i 0.705114i
\(582\) 0 0
\(583\) 15520.3i 1.10255i
\(584\) 0 0
\(585\) 450.037 300.693i 0.0318064 0.0212515i
\(586\) 0 0
\(587\) −7642.83 −0.537399 −0.268700 0.963224i \(-0.586594\pi\)
−0.268700 + 0.963224i \(0.586594\pi\)
\(588\) 0 0
\(589\) 906.267 0.0633991
\(590\) 0 0
\(591\) 12050.9 22545.2i 0.838763 1.56918i
\(592\) 0 0
\(593\) 16852.3i 1.16702i 0.812107 + 0.583509i \(0.198321\pi\)
−0.812107 + 0.583509i \(0.801679\pi\)
\(594\) 0 0
\(595\) 6409.49i 0.441619i
\(596\) 0 0
\(597\) −646.891 + 1210.22i −0.0443475 + 0.0829667i
\(598\) 0 0
\(599\) 7456.85 0.508646 0.254323 0.967119i \(-0.418147\pi\)
0.254323 + 0.967119i \(0.418147\pi\)
\(600\) 0 0
\(601\) 8982.48 0.609656 0.304828 0.952407i \(-0.401401\pi\)
0.304828 + 0.952407i \(0.401401\pi\)
\(602\) 0 0
\(603\) −4632.35 + 3095.12i −0.312842 + 0.209026i
\(604\) 0 0
\(605\) 1873.87i 0.125923i
\(606\) 0 0
\(607\) 6289.93i 0.420594i −0.977638 0.210297i \(-0.932557\pi\)
0.977638 0.210297i \(-0.0674431\pi\)
\(608\) 0 0
\(609\) −14073.5 7522.59i −0.936430 0.500543i
\(610\) 0 0
\(611\) 2540.02 0.168181
\(612\) 0 0
\(613\) 11497.6 0.757556 0.378778 0.925488i \(-0.376344\pi\)
0.378778 + 0.925488i \(0.376344\pi\)
\(614\) 0 0
\(615\) 2736.44 + 1462.69i 0.179421 + 0.0959044i
\(616\) 0 0
\(617\) 13336.4i 0.870181i 0.900387 + 0.435090i \(0.143284\pi\)
−0.900387 + 0.435090i \(0.856716\pi\)
\(618\) 0 0
\(619\) 27960.9i 1.81558i 0.419429 + 0.907788i \(0.362230\pi\)
−0.419429 + 0.907788i \(0.637770\pi\)
\(620\) 0 0
\(621\) −1603.82 16288.3i −0.103638 1.05254i
\(622\) 0 0
\(623\) 24372.6 1.56737
\(624\) 0 0
\(625\) 13644.6 0.873255
\(626\) 0 0
\(627\) 4778.99 8940.67i 0.304393 0.569467i
\(628\) 0 0
\(629\) 35531.4i 2.25235i
\(630\) 0 0
\(631\) 2435.82i 0.153675i −0.997044 0.0768373i \(-0.975518\pi\)
0.997044 0.0768373i \(-0.0244822\pi\)
\(632\) 0 0
\(633\) −8384.27 + 15685.5i −0.526453 + 0.984903i
\(634\) 0 0
\(635\) 1280.42 0.0800186
\(636\) 0 0
\(637\) −4650.12 −0.289237
\(638\) 0 0
\(639\) 14947.7 + 22371.7i 0.925388 + 1.38499i
\(640\) 0 0
\(641\) 882.215i 0.0543610i −0.999631 0.0271805i \(-0.991347\pi\)
0.999631 0.0271805i \(-0.00865288\pi\)
\(642\) 0 0
\(643\) 3761.20i 0.230680i 0.993326 + 0.115340i \(0.0367957\pi\)
−0.993326 + 0.115340i \(0.963204\pi\)
\(644\) 0 0
\(645\) −3816.23 2039.86i −0.232967 0.124526i
\(646\) 0 0
\(647\) 6601.59 0.401137 0.200568 0.979680i \(-0.435721\pi\)
0.200568 + 0.979680i \(0.435721\pi\)
\(648\) 0 0
\(649\) 1046.42 0.0632908
\(650\) 0 0
\(651\) 1441.87 + 770.713i 0.0868071 + 0.0464004i
\(652\) 0 0
\(653\) 13790.5i 0.826436i 0.910632 + 0.413218i \(0.135595\pi\)
−0.910632 + 0.413218i \(0.864405\pi\)
\(654\) 0 0
\(655\) 4980.25i 0.297091i
\(656\) 0 0
\(657\) 8908.91 + 13333.6i 0.529025 + 0.791773i
\(658\) 0 0
\(659\) 12482.7 0.737868 0.368934 0.929456i \(-0.379723\pi\)
0.368934 + 0.929456i \(0.379723\pi\)
\(660\) 0 0
\(661\) −6286.85 −0.369940 −0.184970 0.982744i \(-0.559219\pi\)
−0.184970 + 0.982744i \(0.559219\pi\)
\(662\) 0 0
\(663\) 1980.34 3704.88i 0.116003 0.217022i
\(664\) 0 0
\(665\) 5866.37i 0.342088i
\(666\) 0 0
\(667\) 12077.9i 0.701139i
\(668\) 0 0
\(669\) 10856.4 20310.4i 0.627401 1.17376i
\(670\) 0 0
\(671\) 13592.0 0.781987
\(672\) 0 0
\(673\) 13687.8 0.783992 0.391996 0.919967i \(-0.371785\pi\)
0.391996 + 0.919967i \(0.371785\pi\)
\(674\) 0 0
\(675\) −1644.81 16704.6i −0.0937908 0.952533i
\(676\) 0 0
\(677\) 20519.0i 1.16486i 0.812882 + 0.582428i \(0.197897\pi\)
−0.812882 + 0.582428i \(0.802103\pi\)
\(678\) 0 0
\(679\) 12479.9i 0.705354i
\(680\) 0 0
\(681\) 21423.5 + 11451.3i 1.20551 + 0.644370i
\(682\) 0 0
\(683\) 11851.0 0.663933 0.331967 0.943291i \(-0.392288\pi\)
0.331967 + 0.943291i \(0.392288\pi\)
\(684\) 0 0
\(685\) −3417.84 −0.190641
\(686\) 0 0
\(687\) −10622.2 5677.78i −0.589899 0.315314i
\(688\) 0 0
\(689\) 5886.42i 0.325479i
\(690\) 0 0
\(691\) 720.586i 0.0396706i 0.999803 + 0.0198353i \(0.00631418\pi\)
−0.999803 + 0.0198353i \(0.993686\pi\)
\(692\) 0 0
\(693\) 15206.8 10160.4i 0.833560 0.556946i
\(694\) 0 0
\(695\) −4592.45 −0.250650
\(696\) 0 0
\(697\) 24082.8 1.30875
\(698\) 0 0
\(699\) 13086.5 24482.6i 0.708122 1.32477i
\(700\) 0 0
\(701\) 3475.26i 0.187245i −0.995608 0.0936226i \(-0.970155\pi\)
0.995608 0.0936226i \(-0.0298447\pi\)
\(702\) 0 0
\(703\) 32520.6i 1.74472i
\(704\) 0 0
\(705\) −1662.84 + 3110.88i −0.0888313 + 0.166188i
\(706\) 0 0
\(707\) −27383.8 −1.45668
\(708\) 0 0
\(709\) −3963.46 −0.209945 −0.104972 0.994475i \(-0.533475\pi\)
−0.104972 + 0.994475i \(0.533475\pi\)
\(710\) 0 0
\(711\) 20450.6 13664.1i 1.07870 0.720737i
\(712\) 0 0
\(713\) 1237.42i 0.0649956i
\(714\) 0 0
\(715\) 457.752i 0.0239426i
\(716\) 0 0
\(717\) 24317.1 + 12998.0i 1.26658 + 0.677016i
\(718\) 0 0
\(719\) 3328.38 0.172639 0.0863195 0.996268i \(-0.472489\pi\)
0.0863195 + 0.996268i \(0.472489\pi\)
\(720\) 0 0
\(721\) −42305.9 −2.18523
\(722\) 0 0
\(723\) 5592.35 + 2989.24i 0.287665 + 0.153763i
\(724\) 0 0
\(725\) 12386.6i 0.634522i
\(726\) 0 0
\(727\) 19880.2i 1.01419i −0.861891 0.507094i \(-0.830720\pi\)
0.861891 0.507094i \(-0.169280\pi\)
\(728\) 0 0
\(729\) −19305.0 + 3838.94i −0.980796 + 0.195038i
\(730\) 0 0
\(731\) −33585.9 −1.69934
\(732\) 0 0
\(733\) 11000.8 0.554330 0.277165 0.960822i \(-0.410605\pi\)
0.277165 + 0.960822i \(0.410605\pi\)
\(734\) 0 0
\(735\) 3044.22 5695.20i 0.152772 0.285811i
\(736\) 0 0
\(737\) 4711.77i 0.235495i
\(738\) 0 0
\(739\) 24841.3i 1.23654i 0.785966 + 0.618269i \(0.212166\pi\)
−0.785966 + 0.618269i \(0.787834\pi\)
\(740\) 0 0
\(741\) 1812.53 3390.94i 0.0898584 0.168110i
\(742\) 0 0
\(743\) 40146.8 1.98229 0.991147 0.132769i \(-0.0423870\pi\)
0.991147 + 0.132769i \(0.0423870\pi\)
\(744\) 0 0
\(745\) −7005.41 −0.344508
\(746\) 0 0
\(747\) −4993.34 7473.35i −0.244574 0.366045i
\(748\) 0 0
\(749\) 46922.6i 2.28907i
\(750\) 0 0
\(751\) 14430.3i 0.701159i −0.936533 0.350580i \(-0.885985\pi\)
0.936533 0.350580i \(-0.114015\pi\)
\(752\) 0 0
\(753\) −1767.48 944.757i −0.0855385 0.0457223i
\(754\) 0 0
\(755\) 810.200 0.0390546
\(756\) 0 0
\(757\) −36743.8 −1.76417 −0.882085 0.471091i \(-0.843860\pi\)
−0.882085 + 0.471091i \(0.843860\pi\)
\(758\) 0 0
\(759\) −12207.6 6525.26i −0.583807 0.312058i
\(760\) 0 0
\(761\) 12211.3i 0.581681i −0.956772 0.290841i \(-0.906065\pi\)
0.956772 0.290841i \(-0.0939350\pi\)
\(762\) 0 0
\(763\) 22725.0i 1.07824i
\(764\) 0 0
\(765\) 3241.09 + 4850.82i 0.153179 + 0.229257i
\(766\) 0 0
\(767\) 396.879 0.0186838
\(768\) 0 0
\(769\) 547.115 0.0256560 0.0128280 0.999918i \(-0.495917\pi\)
0.0128280 + 0.999918i \(0.495917\pi\)
\(770\) 0 0
\(771\) −2838.39 + 5310.14i −0.132584 + 0.248041i
\(772\) 0 0
\(773\) 25467.3i 1.18499i −0.805575 0.592494i \(-0.798143\pi\)
0.805575 0.592494i \(-0.201857\pi\)
\(774\) 0 0
\(775\) 1269.05i 0.0588202i
\(776\) 0 0
\(777\) 27656.4 51740.4i 1.27692 2.38890i
\(778\) 0 0
\(779\) 22042.1 1.01379
\(780\) 0 0
\(781\) 22755.3 1.04257
\(782\) 0 0
\(783\) 14455.0 1423.31i 0.659745 0.0649616i
\(784\) 0 0
\(785\) 861.987i 0.0391919i
\(786\) 0 0
\(787\) 7196.98i 0.325978i 0.986628 + 0.162989i \(0.0521135\pi\)
−0.986628 + 0.162989i \(0.947887\pi\)
\(788\) 0 0
\(789\) 7686.15 + 4108.42i 0.346811 + 0.185378i
\(790\) 0 0
\(791\) 31077.5 1.39695
\(792\) 0 0
\(793\) 5155.05 0.230847
\(794\) 0 0
\(795\) −7209.36 3853.57i −0.321622 0.171914i
\(796\) 0 0
\(797\) 30718.6i 1.36526i −0.730766 0.682628i \(-0.760837\pi\)
0.730766 0.682628i \(-0.239163\pi\)
\(798\) 0 0
\(799\) 27378.2i 1.21223i
\(800\) 0 0
\(801\) −18445.7 + 12324.5i −0.813665 + 0.543653i
\(802\) 0 0
\(803\) 13562.2 0.596016
\(804\) 0 0
\(805\) −8009.99 −0.350702
\(806\) 0 0
\(807\) −9994.85 + 18698.7i −0.435979 + 0.815643i
\(808\) 0 0
\(809\) 10427.2i 0.453153i −0.973993 0.226577i \(-0.927247\pi\)
0.973993 0.226577i \(-0.0727534\pi\)
\(810\) 0 0
\(811\) 6840.19i 0.296167i −0.988975 0.148084i \(-0.952690\pi\)
0.988975 0.148084i \(-0.0473105\pi\)
\(812\) 0 0
\(813\) 1347.96 2521.80i 0.0581488 0.108786i
\(814\) 0 0
\(815\) −3133.32 −0.134669
\(816\) 0 0
\(817\) −30739.9 −1.31635
\(818\) 0 0
\(819\) 5767.49 3853.57i 0.246071 0.164413i
\(820\) 0 0
\(821\) 19729.3i 0.838683i 0.907829 + 0.419341i \(0.137739\pi\)
−0.907829 + 0.419341i \(0.862261\pi\)
\(822\) 0 0
\(823\) 13924.4i 0.589763i −0.955534 0.294882i \(-0.904720\pi\)
0.955534 0.294882i \(-0.0952803\pi\)
\(824\) 0 0
\(825\) −12519.7 6692.05i −0.528338 0.282409i
\(826\) 0 0
\(827\) 40307.7 1.69484 0.847421 0.530921i \(-0.178154\pi\)
0.847421 + 0.530921i \(0.178154\pi\)
\(828\) 0 0
\(829\) 9091.40 0.380889 0.190445 0.981698i \(-0.439007\pi\)
0.190445 + 0.981698i \(0.439007\pi\)
\(830\) 0 0
\(831\) 2145.17 + 1146.64i 0.0895490 + 0.0478660i
\(832\) 0 0
\(833\) 50122.3i 2.08480i
\(834\) 0 0
\(835\) 6000.67i 0.248697i
\(836\) 0 0
\(837\) −1480.96 + 145.823i −0.0611584 + 0.00602194i
\(838\) 0 0
\(839\) −22517.0 −0.926549 −0.463274 0.886215i \(-0.653326\pi\)
−0.463274 + 0.886215i \(0.653326\pi\)
\(840\) 0 0
\(841\) 13670.4 0.560516
\(842\) 0 0
\(843\) −13036.7 + 24389.5i −0.532632 + 0.996464i
\(844\) 0 0
\(845\) 4911.66i 0.199960i
\(846\) 0 0
\(847\) 24014.7i 0.974210i
\(848\) 0 0
\(849\) −269.164 + 503.559i −0.0108807 + 0.0203558i
\(850\) 0 0
\(851\) −44403.9 −1.78865
\(852\) 0 0
\(853\) 8230.04 0.330353 0.165177 0.986264i \(-0.447181\pi\)
0.165177 + 0.986264i \(0.447181\pi\)
\(854\) 0 0
\(855\) 2966.45 + 4439.78i 0.118656 + 0.177588i
\(856\) 0 0
\(857\) 12042.6i 0.480009i 0.970772 + 0.240005i \(0.0771490\pi\)
−0.970772 + 0.240005i \(0.922851\pi\)
\(858\) 0 0
\(859\) 26661.8i 1.05901i 0.848307 + 0.529505i \(0.177622\pi\)
−0.848307 + 0.529505i \(0.822378\pi\)
\(860\) 0 0
\(861\) 35069.1 + 18745.2i 1.38810 + 0.741969i
\(862\) 0 0
\(863\) −18444.3 −0.727521 −0.363760 0.931493i \(-0.618507\pi\)
−0.363760 + 0.931493i \(0.618507\pi\)
\(864\) 0 0
\(865\) 8433.96 0.331518
\(866\) 0 0
\(867\) 17419.6 + 9311.19i 0.682356 + 0.364734i
\(868\) 0 0
\(869\) 20801.2i 0.812005i
\(870\) 0 0
\(871\) 1787.04i 0.0695195i
\(872\) 0 0
\(873\) 6310.73 + 9445.03i 0.244657 + 0.366169i
\(874\) 0 0
\(875\) −16797.3 −0.648974
\(876\) 0 0
\(877\) −17206.7 −0.662520 −0.331260 0.943539i \(-0.607474\pi\)
−0.331260 + 0.943539i \(0.607474\pi\)
\(878\) 0 0
\(879\) −6544.03 + 12242.8i −0.251109 + 0.469782i
\(880\) 0 0
\(881\) 34837.3i 1.33223i 0.745847 + 0.666117i \(0.232045\pi\)
−0.745847 + 0.666117i \(0.767955\pi\)
\(882\) 0 0
\(883\) 27943.9i 1.06499i −0.846432 0.532496i \(-0.821254\pi\)
0.846432 0.532496i \(-0.178746\pi\)
\(884\) 0 0
\(885\) −259.818 + 486.075i −0.00986858 + 0.0184624i
\(886\) 0 0
\(887\) 3273.25 0.123907 0.0619533 0.998079i \(-0.480267\pi\)
0.0619533 + 0.998079i \(0.480267\pi\)
\(888\) 0 0
\(889\) 16409.3 0.619068
\(890\) 0 0
\(891\) −6370.92 + 15379.2i −0.239544 + 0.578253i
\(892\) 0 0
\(893\) 25058.3i 0.939019i
\(894\) 0 0
\(895\) 2024.01i 0.0755922i
\(896\) 0 0
\(897\) −4630.01 2474.85i −0.172343 0.0921212i
\(898\) 0 0
\(899\) 1098.15 0.0407402
\(900\) 0 0
\(901\) −63448.1 −2.34602
\(902\) 0 0
\(903\) −48907.3 26142.1i −1.80236 0.963403i
\(904\) 0 0
\(905\) 6247.25i 0.229465i
\(906\) 0 0
\(907\) 21401.7i 0.783499i −0.920072 0.391749i \(-0.871870\pi\)
0.920072 0.391749i \(-0.128130\pi\)
\(908\) 0 0
\(909\) 20724.5 13847.2i 0.756204 0.505260i
\(910\) 0 0
\(911\) 3638.18 0.132314 0.0661572 0.997809i \(-0.478926\pi\)
0.0661572 + 0.997809i \(0.478926\pi\)
\(912\) 0 0
\(913\) −7601.48 −0.275545
\(914\) 0 0
\(915\) −3374.78 + 6313.63i −0.121931 + 0.228112i
\(916\) 0 0
\(917\) 63824.9i 2.29846i
\(918\) 0 0
\(919\) 53008.1i 1.90269i 0.308124 + 0.951346i \(0.400299\pi\)
−0.308124 + 0.951346i \(0.599701\pi\)
\(920\) 0 0
\(921\) −5981.16 + 11189.7i −0.213991 + 0.400341i
\(922\) 0 0
\(923\) 8630.42 0.307772
\(924\) 0 0
\(925\) −45538.8 −1.61871
\(926\) 0 0
\(927\) 32017.9 21392.8i 1.13442 0.757964i
\(928\) 0 0
\(929\) 24985.8i 0.882409i 0.897407 + 0.441205i \(0.145449\pi\)
−0.897407 + 0.441205i \(0.854551\pi\)
\(930\) 0 0
\(931\) 45875.2i 1.61493i
\(932\) 0 0
\(933\) 31608.7 + 16895.6i 1.10913 + 0.592857i
\(934\) 0 0
\(935\) 4933.99 0.172576
\(936\) 0 0
\(937\) −14614.3 −0.509528 −0.254764 0.967003i \(-0.581998\pi\)
−0.254764 + 0.967003i \(0.581998\pi\)
\(938\) 0 0
\(939\) −24191.1 12930.7i −0.840730 0.449389i
\(940\) 0 0
\(941\) 1389.68i 0.0481425i 0.999710 + 0.0240713i \(0.00766286\pi\)
−0.999710 + 0.0240713i \(0.992337\pi\)
\(942\) 0 0
\(943\) 30096.5i 1.03932i
\(944\) 0 0
\(945\) 943.927 + 9586.46i 0.0324931 + 0.329997i
\(946\) 0 0
\(947\) 28674.0 0.983928 0.491964 0.870616i \(-0.336279\pi\)
0.491964 + 0.870616i \(0.336279\pi\)
\(948\) 0 0
\(949\) 5143.77 0.175947
\(950\) 0 0
\(951\) −3766.29 + 7046.09i −0.128423 + 0.240258i
\(952\) 0 0
\(953\) 13984.4i 0.475339i 0.971346 + 0.237669i \(0.0763835\pi\)
−0.971346 + 0.237669i \(0.923616\pi\)
\(954\) 0 0
\(955\) 1638.85i 0.0555309i
\(956\) 0 0
\(957\) 5790.85 10833.7i 0.195602 0.365939i
\(958\) 0 0
\(959\) −43801.7 −1.47490
\(960\) 0 0
\(961\) 29678.5 0.996223
\(962\) 0 0
\(963\) 23727.4 + 35511.9i 0.793982 + 1.18832i
\(964\) 0 0
\(965\) 8173.16i 0.272646i
\(966\) 0 0
\(967\) 27834.7i 0.925649i 0.886450 + 0.462825i \(0.153164\pi\)
−0.886450 + 0.462825i \(0.846836\pi\)
\(968\) 0 0
\(969\) 36550.0 + 19536.8i 1.21172 + 0.647691i
\(970\) 0 0
\(971\) 12546.9 0.414676 0.207338 0.978269i \(-0.433520\pi\)
0.207338 + 0.978269i \(0.433520\pi\)
\(972\) 0 0
\(973\) −58855.1 −1.93917
\(974\) 0 0
\(975\) −4748.35 2538.10i −0.155968 0.0833685i
\(976\) 0 0
\(977\) 47934.6i 1.56967i 0.619708 + 0.784833i \(0.287251\pi\)
−0.619708 + 0.784833i \(0.712749\pi\)
\(978\) 0 0
\(979\) 18761.9i 0.612496i
\(980\) 0 0
\(981\) 11491.4 + 17198.7i 0.373997 + 0.559747i
\(982\) 0 0
\(983\) −44432.7 −1.44169 −0.720846 0.693095i \(-0.756247\pi\)
−0.720846 + 0.693095i \(0.756247\pi\)
\(984\) 0 0
\(985\) 11387.5 0.368362
\(986\) 0 0
\(987\) −21310.3 + 39867.8i −0.687247 + 1.28572i
\(988\) 0 0
\(989\) 41972.5i 1.34949i
\(990\) 0 0
\(991\) 24714.1i 0.792199i 0.918208 + 0.396100i \(0.129637\pi\)
−0.918208 + 0.396100i \(0.870363\pi\)
\(992\) 0 0
\(993\) 21787.1 40759.9i 0.696266 1.30259i
\(994\) 0 0
\(995\) −611.279 −0.0194762
\(996\) 0 0
\(997\) −49387.0 −1.56881 −0.784405 0.620250i \(-0.787031\pi\)
−0.784405 + 0.620250i \(0.787031\pi\)
\(998\) 0 0
\(999\) 5232.72 + 53143.1i 0.165722 + 1.68306i
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 768.4.c.o.767.1 4
3.2 odd 2 768.4.c.m.767.3 4
4.3 odd 2 768.4.c.m.767.4 4
8.3 odd 2 768.4.c.p.767.1 4
8.5 even 2 768.4.c.n.767.4 4
12.11 even 2 inner 768.4.c.o.767.2 4
16.3 odd 4 384.4.f.h.191.4 yes 8
16.5 even 4 384.4.f.g.191.3 yes 8
16.11 odd 4 384.4.f.h.191.5 yes 8
16.13 even 4 384.4.f.g.191.6 yes 8
24.5 odd 2 768.4.c.p.767.2 4
24.11 even 2 768.4.c.n.767.3 4
48.5 odd 4 384.4.f.h.191.2 yes 8
48.11 even 4 384.4.f.g.191.8 yes 8
48.29 odd 4 384.4.f.h.191.7 yes 8
48.35 even 4 384.4.f.g.191.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.f.g.191.1 8 48.35 even 4
384.4.f.g.191.3 yes 8 16.5 even 4
384.4.f.g.191.6 yes 8 16.13 even 4
384.4.f.g.191.8 yes 8 48.11 even 4
384.4.f.h.191.2 yes 8 48.5 odd 4
384.4.f.h.191.4 yes 8 16.3 odd 4
384.4.f.h.191.5 yes 8 16.11 odd 4
384.4.f.h.191.7 yes 8 48.29 odd 4
768.4.c.m.767.3 4 3.2 odd 2
768.4.c.m.767.4 4 4.3 odd 2
768.4.c.n.767.3 4 24.11 even 2
768.4.c.n.767.4 4 8.5 even 2
768.4.c.o.767.1 4 1.1 even 1 trivial
768.4.c.o.767.2 4 12.11 even 2 inner
768.4.c.p.767.1 4 8.3 odd 2
768.4.c.p.767.2 4 24.5 odd 2