Properties

Label 3800.2.d.k.3649.6
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.6
Root \(1.32001 - 1.32001i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.k.3649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.12489i q^{3} -1.51514i q^{7} -6.76491 q^{9} +4.24977 q^{11} -4.15516i q^{13} -3.51514i q^{17} +1.00000 q^{19} +4.73463 q^{21} +8.73463i q^{23} -11.7649i q^{27} -1.45459 q^{29} -4.96972 q^{31} +13.2800i q^{33} +7.60975i q^{37} +12.9844 q^{39} -9.21949 q^{41} +8.31032i q^{43} +5.28005i q^{47} +4.70436 q^{49} +10.9844 q^{51} -0.155162i q^{53} +3.12489i q^{57} +2.48486 q^{59} -4.49954 q^{61} +10.2498i q^{63} +7.43521i q^{67} -27.2947 q^{69} +8.49954 q^{71} +15.0450i q^{73} -6.43899i q^{77} +0.310323 q^{79} +16.4693 q^{81} +8.96972i q^{83} -4.54541i q^{87} -0.719953 q^{89} -6.29564 q^{91} -15.5298i q^{93} +17.3893i q^{97} -28.7493 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.12489i 1.80415i 0.431576 + 0.902077i \(0.357958\pi\)
−0.431576 + 0.902077i \(0.642042\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.51514i − 0.572668i −0.958130 0.286334i \(-0.907563\pi\)
0.958130 0.286334i \(-0.0924368\pi\)
\(8\) 0 0
\(9\) −6.76491 −2.25497
\(10\) 0 0
\(11\) 4.24977 1.28135 0.640677 0.767810i \(-0.278654\pi\)
0.640677 + 0.767810i \(0.278654\pi\)
\(12\) 0 0
\(13\) − 4.15516i − 1.15243i −0.817297 0.576217i \(-0.804528\pi\)
0.817297 0.576217i \(-0.195472\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.51514i − 0.852546i −0.904595 0.426273i \(-0.859826\pi\)
0.904595 0.426273i \(-0.140174\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 4.73463 1.03318
\(22\) 0 0
\(23\) 8.73463i 1.82130i 0.413182 + 0.910648i \(0.364417\pi\)
−0.413182 + 0.910648i \(0.635583\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 11.7649i − 2.26416i
\(28\) 0 0
\(29\) −1.45459 −0.270110 −0.135055 0.990838i \(-0.543121\pi\)
−0.135055 + 0.990838i \(0.543121\pi\)
\(30\) 0 0
\(31\) −4.96972 −0.892589 −0.446294 0.894886i \(-0.647257\pi\)
−0.446294 + 0.894886i \(0.647257\pi\)
\(32\) 0 0
\(33\) 13.2800i 2.31176i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.60975i 1.25103i 0.780210 + 0.625517i \(0.215112\pi\)
−0.780210 + 0.625517i \(0.784888\pi\)
\(38\) 0 0
\(39\) 12.9844 2.07917
\(40\) 0 0
\(41\) −9.21949 −1.43984 −0.719922 0.694055i \(-0.755822\pi\)
−0.719922 + 0.694055i \(0.755822\pi\)
\(42\) 0 0
\(43\) 8.31032i 1.26731i 0.773615 + 0.633656i \(0.218447\pi\)
−0.773615 + 0.633656i \(0.781553\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.28005i 0.770174i 0.922880 + 0.385087i \(0.125829\pi\)
−0.922880 + 0.385087i \(0.874171\pi\)
\(48\) 0 0
\(49\) 4.70436 0.672051
\(50\) 0 0
\(51\) 10.9844 1.53812
\(52\) 0 0
\(53\) − 0.155162i − 0.0213131i −0.999943 0.0106565i \(-0.996608\pi\)
0.999943 0.0106565i \(-0.00339215\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.12489i 0.413901i
\(58\) 0 0
\(59\) 2.48486 0.323501 0.161751 0.986832i \(-0.448286\pi\)
0.161751 + 0.986832i \(0.448286\pi\)
\(60\) 0 0
\(61\) −4.49954 −0.576107 −0.288054 0.957614i \(-0.593008\pi\)
−0.288054 + 0.957614i \(0.593008\pi\)
\(62\) 0 0
\(63\) 10.2498i 1.29135i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.43521i 0.908355i 0.890911 + 0.454178i \(0.150067\pi\)
−0.890911 + 0.454178i \(0.849933\pi\)
\(68\) 0 0
\(69\) −27.2947 −3.28590
\(70\) 0 0
\(71\) 8.49954 1.00871 0.504355 0.863496i \(-0.331730\pi\)
0.504355 + 0.863496i \(0.331730\pi\)
\(72\) 0 0
\(73\) 15.0450i 1.76088i 0.474159 + 0.880439i \(0.342752\pi\)
−0.474159 + 0.880439i \(0.657248\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.43899i − 0.733791i
\(78\) 0 0
\(79\) 0.310323 0.0349141 0.0174570 0.999848i \(-0.494443\pi\)
0.0174570 + 0.999848i \(0.494443\pi\)
\(80\) 0 0
\(81\) 16.4693 1.82992
\(82\) 0 0
\(83\) 8.96972i 0.984555i 0.870438 + 0.492278i \(0.163835\pi\)
−0.870438 + 0.492278i \(0.836165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.54541i − 0.487320i
\(88\) 0 0
\(89\) −0.719953 −0.0763149 −0.0381574 0.999272i \(-0.512149\pi\)
−0.0381574 + 0.999272i \(0.512149\pi\)
\(90\) 0 0
\(91\) −6.29564 −0.659963
\(92\) 0 0
\(93\) − 15.5298i − 1.61037i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 17.3893i 1.76562i 0.469731 + 0.882810i \(0.344351\pi\)
−0.469731 + 0.882810i \(0.655649\pi\)
\(98\) 0 0
\(99\) −28.7493 −2.88941
\(100\) 0 0
\(101\) −12.4995 −1.24375 −0.621875 0.783116i \(-0.713629\pi\)
−0.621875 + 0.783116i \(0.713629\pi\)
\(102\) 0 0
\(103\) 10.8898i 1.07300i 0.843899 + 0.536502i \(0.180254\pi\)
−0.843899 + 0.536502i \(0.819746\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.43521i − 0.718789i −0.933186 0.359394i \(-0.882983\pi\)
0.933186 0.359394i \(-0.117017\pi\)
\(108\) 0 0
\(109\) 19.3553 1.85390 0.926950 0.375186i \(-0.122421\pi\)
0.926950 + 0.375186i \(0.122421\pi\)
\(110\) 0 0
\(111\) −23.7796 −2.25706
\(112\) 0 0
\(113\) 17.8595i 1.68008i 0.542523 + 0.840041i \(0.317469\pi\)
−0.542523 + 0.840041i \(0.682531\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 28.1093i 2.59870i
\(118\) 0 0
\(119\) −5.32592 −0.488226
\(120\) 0 0
\(121\) 7.06055 0.641868
\(122\) 0 0
\(123\) − 28.8099i − 2.59770i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.70058i 0.239637i 0.992796 + 0.119819i \(0.0382313\pi\)
−0.992796 + 0.119819i \(0.961769\pi\)
\(128\) 0 0
\(129\) −25.9688 −2.28643
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) − 1.51514i − 0.131379i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 7.51514i − 0.642062i −0.947069 0.321031i \(-0.895971\pi\)
0.947069 0.321031i \(-0.104029\pi\)
\(138\) 0 0
\(139\) −16.7493 −1.42066 −0.710329 0.703870i \(-0.751454\pi\)
−0.710329 + 0.703870i \(0.751454\pi\)
\(140\) 0 0
\(141\) −16.4995 −1.38951
\(142\) 0 0
\(143\) − 17.6585i − 1.47668i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 14.7006i 1.21248i
\(148\) 0 0
\(149\) 12.4995 1.02400 0.512001 0.858985i \(-0.328904\pi\)
0.512001 + 0.858985i \(0.328904\pi\)
\(150\) 0 0
\(151\) 14.2498 1.15963 0.579815 0.814748i \(-0.303125\pi\)
0.579815 + 0.814748i \(0.303125\pi\)
\(152\) 0 0
\(153\) 23.7796i 1.92247i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 4.71995i 0.376693i 0.982103 + 0.188347i \(0.0603128\pi\)
−0.982103 + 0.188347i \(0.939687\pi\)
\(158\) 0 0
\(159\) 0.484862 0.0384521
\(160\) 0 0
\(161\) 13.2342 1.04300
\(162\) 0 0
\(163\) − 10.7493i − 0.841951i −0.907072 0.420976i \(-0.861688\pi\)
0.907072 0.420976i \(-0.138312\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 10.7006i − 0.828035i −0.910269 0.414018i \(-0.864125\pi\)
0.910269 0.414018i \(-0.135875\pi\)
\(168\) 0 0
\(169\) −4.26537 −0.328105
\(170\) 0 0
\(171\) −6.76491 −0.517326
\(172\) 0 0
\(173\) 4.39025i 0.333785i 0.985975 + 0.166892i \(0.0533732\pi\)
−0.985975 + 0.166892i \(0.946627\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 7.76491i 0.583646i
\(178\) 0 0
\(179\) 3.21949 0.240636 0.120318 0.992735i \(-0.461609\pi\)
0.120318 + 0.992735i \(0.461609\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) − 14.0606i − 1.03939i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 14.9385i − 1.09241i
\(188\) 0 0
\(189\) −17.8255 −1.29661
\(190\) 0 0
\(191\) 13.2342 0.957591 0.478796 0.877926i \(-0.341073\pi\)
0.478796 + 0.877926i \(0.341073\pi\)
\(192\) 0 0
\(193\) − 17.3893i − 1.25171i −0.779939 0.625856i \(-0.784750\pi\)
0.779939 0.625856i \(-0.215250\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 0 0
\(199\) 14.3250 1.01547 0.507736 0.861513i \(-0.330482\pi\)
0.507736 + 0.861513i \(0.330482\pi\)
\(200\) 0 0
\(201\) −23.2342 −1.63881
\(202\) 0 0
\(203\) 2.20390i 0.154683i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 59.0890i − 4.10697i
\(208\) 0 0
\(209\) 4.24977 0.293963
\(210\) 0 0
\(211\) −6.98440 −0.480826 −0.240413 0.970671i \(-0.577283\pi\)
−0.240413 + 0.970671i \(0.577283\pi\)
\(212\) 0 0
\(213\) 26.5601i 1.81987i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.52982i 0.511157i
\(218\) 0 0
\(219\) −47.0138 −3.17690
\(220\) 0 0
\(221\) −14.6060 −0.982504
\(222\) 0 0
\(223\) − 24.4802i − 1.63931i −0.572855 0.819657i \(-0.694164\pi\)
0.572855 0.819657i \(-0.305836\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 24.9045i − 1.65297i −0.562960 0.826484i \(-0.690338\pi\)
0.562960 0.826484i \(-0.309662\pi\)
\(228\) 0 0
\(229\) −4.47018 −0.295398 −0.147699 0.989032i \(-0.547187\pi\)
−0.147699 + 0.989032i \(0.547187\pi\)
\(230\) 0 0
\(231\) 20.1211 1.32387
\(232\) 0 0
\(233\) 3.93945i 0.258082i 0.991639 + 0.129041i \(0.0411899\pi\)
−0.991639 + 0.129041i \(0.958810\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.969724i 0.0629903i
\(238\) 0 0
\(239\) 17.8255 1.15303 0.576517 0.817085i \(-0.304412\pi\)
0.576517 + 0.817085i \(0.304412\pi\)
\(240\) 0 0
\(241\) −27.3094 −1.75915 −0.879577 0.475757i \(-0.842174\pi\)
−0.879577 + 0.475757i \(0.842174\pi\)
\(242\) 0 0
\(243\) 16.1698i 1.03730i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.15516i − 0.264387i
\(248\) 0 0
\(249\) −28.0294 −1.77629
\(250\) 0 0
\(251\) −18.4390 −1.16386 −0.581929 0.813239i \(-0.697702\pi\)
−0.581929 + 0.813239i \(0.697702\pi\)
\(252\) 0 0
\(253\) 37.1202i 2.33373i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 0.889794i − 0.0555038i −0.999615 0.0277519i \(-0.991165\pi\)
0.999615 0.0277519i \(-0.00883485\pi\)
\(258\) 0 0
\(259\) 11.5298 0.716428
\(260\) 0 0
\(261\) 9.84014 0.609089
\(262\) 0 0
\(263\) 12.6206i 0.778222i 0.921191 + 0.389111i \(0.127218\pi\)
−0.921191 + 0.389111i \(0.872782\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.24977i − 0.137684i
\(268\) 0 0
\(269\) 16.4390 1.00230 0.501151 0.865360i \(-0.332910\pi\)
0.501151 + 0.865360i \(0.332910\pi\)
\(270\) 0 0
\(271\) 25.6050 1.55540 0.777698 0.628638i \(-0.216387\pi\)
0.777698 + 0.628638i \(0.216387\pi\)
\(272\) 0 0
\(273\) − 19.6732i − 1.19067i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.749313i 0.0450218i 0.999747 + 0.0225109i \(0.00716605\pi\)
−0.999747 + 0.0225109i \(0.992834\pi\)
\(278\) 0 0
\(279\) 33.6197 2.01276
\(280\) 0 0
\(281\) 3.77959 0.225471 0.112736 0.993625i \(-0.464039\pi\)
0.112736 + 0.993625i \(0.464039\pi\)
\(282\) 0 0
\(283\) 9.46927i 0.562889i 0.959577 + 0.281445i \(0.0908136\pi\)
−0.959577 + 0.281445i \(0.909186\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.9688i 0.824553i
\(288\) 0 0
\(289\) 4.64380 0.273165
\(290\) 0 0
\(291\) −54.3397 −3.18545
\(292\) 0 0
\(293\) − 24.6841i − 1.44206i −0.692905 0.721029i \(-0.743669\pi\)
0.692905 0.721029i \(-0.256331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 49.9982i − 2.90119i
\(298\) 0 0
\(299\) 36.2938 2.09893
\(300\) 0 0
\(301\) 12.5913 0.725750
\(302\) 0 0
\(303\) − 39.0596i − 2.24392i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.3288i 0.760714i 0.924840 + 0.380357i \(0.124199\pi\)
−0.924840 + 0.380357i \(0.875801\pi\)
\(308\) 0 0
\(309\) −34.0294 −1.93586
\(310\) 0 0
\(311\) −13.4546 −0.762940 −0.381470 0.924381i \(-0.624582\pi\)
−0.381470 + 0.924381i \(0.624582\pi\)
\(312\) 0 0
\(313\) 21.7649i 1.23023i 0.788439 + 0.615113i \(0.210889\pi\)
−0.788439 + 0.615113i \(0.789111\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.5639i 1.32348i 0.749734 + 0.661740i \(0.230182\pi\)
−0.749734 + 0.661740i \(0.769818\pi\)
\(318\) 0 0
\(319\) −6.18166 −0.346106
\(320\) 0 0
\(321\) 23.2342 1.29681
\(322\) 0 0
\(323\) − 3.51514i − 0.195588i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 60.4830i 3.34472i
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −11.9541 −0.657058 −0.328529 0.944494i \(-0.606553\pi\)
−0.328529 + 0.944494i \(0.606553\pi\)
\(332\) 0 0
\(333\) − 51.4792i − 2.82105i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 22.8292i − 1.24359i −0.783181 0.621794i \(-0.786404\pi\)
0.783181 0.621794i \(-0.213596\pi\)
\(338\) 0 0
\(339\) −55.8089 −3.03113
\(340\) 0 0
\(341\) −21.1202 −1.14372
\(342\) 0 0
\(343\) − 17.7337i − 0.957531i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.4693i 1.58199i 0.611821 + 0.790997i \(0.290437\pi\)
−0.611821 + 0.790997i \(0.709563\pi\)
\(348\) 0 0
\(349\) −4.56009 −0.244096 −0.122048 0.992524i \(-0.538946\pi\)
−0.122048 + 0.992524i \(0.538946\pi\)
\(350\) 0 0
\(351\) −48.8851 −2.60929
\(352\) 0 0
\(353\) 24.4849i 1.30320i 0.758564 + 0.651599i \(0.225901\pi\)
−0.758564 + 0.651599i \(0.774099\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 16.6429i − 0.880835i
\(358\) 0 0
\(359\) −12.0147 −0.634111 −0.317055 0.948407i \(-0.602694\pi\)
−0.317055 + 0.948407i \(0.602694\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 22.0634i 1.15803i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 25.1589i − 1.31329i −0.754201 0.656643i \(-0.771976\pi\)
0.754201 0.656643i \(-0.228024\pi\)
\(368\) 0 0
\(369\) 62.3690 3.24680
\(370\) 0 0
\(371\) −0.235091 −0.0122053
\(372\) 0 0
\(373\) − 9.62443i − 0.498334i −0.968461 0.249167i \(-0.919843\pi\)
0.968461 0.249167i \(-0.0801568\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.04404i 0.311284i
\(378\) 0 0
\(379\) −19.7943 −1.01676 −0.508382 0.861132i \(-0.669756\pi\)
−0.508382 + 0.861132i \(0.669756\pi\)
\(380\) 0 0
\(381\) −8.43899 −0.432343
\(382\) 0 0
\(383\) − 33.3288i − 1.70302i −0.524337 0.851511i \(-0.675687\pi\)
0.524337 0.851511i \(-0.324313\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 56.2186i − 2.85775i
\(388\) 0 0
\(389\) −11.5904 −0.587655 −0.293828 0.955858i \(-0.594929\pi\)
−0.293828 + 0.955858i \(0.594929\pi\)
\(390\) 0 0
\(391\) 30.7034 1.55274
\(392\) 0 0
\(393\) − 12.4995i − 0.630518i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8.74931i − 0.439115i −0.975600 0.219558i \(-0.929539\pi\)
0.975600 0.219558i \(-0.0704614\pi\)
\(398\) 0 0
\(399\) 4.73463 0.237028
\(400\) 0 0
\(401\) −18.6206 −0.929871 −0.464935 0.885345i \(-0.653922\pi\)
−0.464935 + 0.885345i \(0.653922\pi\)
\(402\) 0 0
\(403\) 20.6500i 1.02865i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.3397i 1.60302i
\(408\) 0 0
\(409\) −7.93945 −0.392580 −0.196290 0.980546i \(-0.562889\pi\)
−0.196290 + 0.980546i \(0.562889\pi\)
\(410\) 0 0
\(411\) 23.4839 1.15838
\(412\) 0 0
\(413\) − 3.76491i − 0.185259i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 52.3397i − 2.56309i
\(418\) 0 0
\(419\) 13.0908 0.639529 0.319764 0.947497i \(-0.396396\pi\)
0.319764 + 0.947497i \(0.396396\pi\)
\(420\) 0 0
\(421\) 19.8936 0.969554 0.484777 0.874638i \(-0.338901\pi\)
0.484777 + 0.874638i \(0.338901\pi\)
\(422\) 0 0
\(423\) − 35.7190i − 1.73672i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 6.81743i 0.329918i
\(428\) 0 0
\(429\) 55.1807 2.66415
\(430\) 0 0
\(431\) −33.7796 −1.62711 −0.813553 0.581491i \(-0.802470\pi\)
−0.813553 + 0.581491i \(0.802470\pi\)
\(432\) 0 0
\(433\) − 9.07901i − 0.436310i −0.975914 0.218155i \(-0.929996\pi\)
0.975914 0.218155i \(-0.0700038\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.73463i 0.417834i
\(438\) 0 0
\(439\) 12.3103 0.587540 0.293770 0.955876i \(-0.405090\pi\)
0.293770 + 0.955876i \(0.405090\pi\)
\(440\) 0 0
\(441\) −31.8245 −1.51545
\(442\) 0 0
\(443\) − 2.56009i − 0.121634i −0.998149 0.0608169i \(-0.980629\pi\)
0.998149 0.0608169i \(-0.0193706\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 39.0596i 1.84746i
\(448\) 0 0
\(449\) −22.9991 −1.08539 −0.542697 0.839929i \(-0.682597\pi\)
−0.542697 + 0.839929i \(0.682597\pi\)
\(450\) 0 0
\(451\) −39.1807 −1.84495
\(452\) 0 0
\(453\) 44.5289i 2.09215i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6741i 0.967093i 0.875319 + 0.483546i \(0.160652\pi\)
−0.875319 + 0.483546i \(0.839348\pi\)
\(458\) 0 0
\(459\) −41.3553 −1.93030
\(460\) 0 0
\(461\) −5.87890 −0.273807 −0.136904 0.990584i \(-0.543715\pi\)
−0.136904 + 0.990584i \(0.543715\pi\)
\(462\) 0 0
\(463\) − 35.5592i − 1.65258i −0.563249 0.826288i \(-0.690449\pi\)
0.563249 0.826288i \(-0.309551\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 41.7408i 1.93154i 0.259408 + 0.965768i \(0.416472\pi\)
−0.259408 + 0.965768i \(0.583528\pi\)
\(468\) 0 0
\(469\) 11.2654 0.520186
\(470\) 0 0
\(471\) −14.7493 −0.679612
\(472\) 0 0
\(473\) 35.3170i 1.62388i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.04965i 0.0480603i
\(478\) 0 0
\(479\) −16.8704 −0.770829 −0.385415 0.922744i \(-0.625942\pi\)
−0.385415 + 0.922744i \(0.625942\pi\)
\(480\) 0 0
\(481\) 31.6197 1.44174
\(482\) 0 0
\(483\) 41.3553i 1.88173i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.64094i 0.164987i 0.996592 + 0.0824934i \(0.0262883\pi\)
−0.996592 + 0.0824934i \(0.973712\pi\)
\(488\) 0 0
\(489\) 33.5904 1.51901
\(490\) 0 0
\(491\) −1.96881 −0.0888510 −0.0444255 0.999013i \(-0.514146\pi\)
−0.0444255 + 0.999013i \(0.514146\pi\)
\(492\) 0 0
\(493\) 5.11307i 0.230281i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 12.8780i − 0.577656i
\(498\) 0 0
\(499\) 36.7787 1.64644 0.823220 0.567723i \(-0.192175\pi\)
0.823220 + 0.567723i \(0.192175\pi\)
\(500\) 0 0
\(501\) 33.4381 1.49390
\(502\) 0 0
\(503\) − 15.8860i − 0.708322i −0.935184 0.354161i \(-0.884766\pi\)
0.935184 0.354161i \(-0.115234\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 13.3288i − 0.591952i
\(508\) 0 0
\(509\) −20.2791 −0.898857 −0.449428 0.893316i \(-0.648372\pi\)
−0.449428 + 0.893316i \(0.648372\pi\)
\(510\) 0 0
\(511\) 22.7952 1.00840
\(512\) 0 0
\(513\) − 11.7649i − 0.519433i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.4390i 0.986866i
\(518\) 0 0
\(519\) −13.7190 −0.602199
\(520\) 0 0
\(521\) −23.4693 −1.02821 −0.514104 0.857728i \(-0.671875\pi\)
−0.514104 + 0.857728i \(0.671875\pi\)
\(522\) 0 0
\(523\) 26.3737i 1.15324i 0.817011 + 0.576622i \(0.195629\pi\)
−0.817011 + 0.576622i \(0.804371\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.4693i 0.760973i
\(528\) 0 0
\(529\) −53.2938 −2.31712
\(530\) 0 0
\(531\) −16.8099 −0.729486
\(532\) 0 0
\(533\) 38.3085i 1.65932i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 10.0606i 0.434145i
\(538\) 0 0
\(539\) 19.9924 0.861135
\(540\) 0 0
\(541\) 22.5601 0.969934 0.484967 0.874532i \(-0.338832\pi\)
0.484967 + 0.874532i \(0.338832\pi\)
\(542\) 0 0
\(543\) 68.7475i 2.95024i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.6694i 0.883759i 0.897074 + 0.441879i \(0.145688\pi\)
−0.897074 + 0.441879i \(0.854312\pi\)
\(548\) 0 0
\(549\) 30.4390 1.29910
\(550\) 0 0
\(551\) −1.45459 −0.0619674
\(552\) 0 0
\(553\) − 0.470182i − 0.0199942i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 37.7484i − 1.59945i −0.600366 0.799725i \(-0.704978\pi\)
0.600366 0.799725i \(-0.295022\pi\)
\(558\) 0 0
\(559\) 34.5307 1.46049
\(560\) 0 0
\(561\) 46.6812 1.97088
\(562\) 0 0
\(563\) 36.6694i 1.54543i 0.634753 + 0.772715i \(0.281102\pi\)
−0.634753 + 0.772715i \(0.718898\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 24.9532i − 1.04794i
\(568\) 0 0
\(569\) −2.84862 −0.119420 −0.0597102 0.998216i \(-0.519018\pi\)
−0.0597102 + 0.998216i \(0.519018\pi\)
\(570\) 0 0
\(571\) 28.4002 1.18851 0.594256 0.804276i \(-0.297446\pi\)
0.594256 + 0.804276i \(0.297446\pi\)
\(572\) 0 0
\(573\) 41.3553i 1.72764i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 29.2947i − 1.21956i −0.792572 0.609778i \(-0.791259\pi\)
0.792572 0.609778i \(-0.208741\pi\)
\(578\) 0 0
\(579\) 54.3397 2.25828
\(580\) 0 0
\(581\) 13.5904 0.563824
\(582\) 0 0
\(583\) − 0.659401i − 0.0273096i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4683i 0.927368i 0.886001 + 0.463684i \(0.153473\pi\)
−0.886001 + 0.463684i \(0.846527\pi\)
\(588\) 0 0
\(589\) −4.96972 −0.204774
\(590\) 0 0
\(591\) 6.24977 0.257081
\(592\) 0 0
\(593\) − 16.0606i − 0.659528i −0.944063 0.329764i \(-0.893031\pi\)
0.944063 0.329764i \(-0.106969\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 44.7640i 1.83207i
\(598\) 0 0
\(599\) 23.2489 0.949922 0.474961 0.880007i \(-0.342462\pi\)
0.474961 + 0.880007i \(0.342462\pi\)
\(600\) 0 0
\(601\) −0.841057 −0.0343074 −0.0171537 0.999853i \(-0.505460\pi\)
−0.0171537 + 0.999853i \(0.505460\pi\)
\(602\) 0 0
\(603\) − 50.2985i − 2.04831i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.4802i − 1.31833i −0.751999 0.659165i \(-0.770910\pi\)
0.751999 0.659165i \(-0.229090\pi\)
\(608\) 0 0
\(609\) −6.88693 −0.279072
\(610\) 0 0
\(611\) 21.9394 0.887575
\(612\) 0 0
\(613\) 19.4305i 0.784791i 0.919797 + 0.392395i \(0.128354\pi\)
−0.919797 + 0.392395i \(0.871646\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.7787i 0.836518i 0.908328 + 0.418259i \(0.137360\pi\)
−0.908328 + 0.418259i \(0.862640\pi\)
\(618\) 0 0
\(619\) −18.9310 −0.760900 −0.380450 0.924802i \(-0.624231\pi\)
−0.380450 + 0.924802i \(0.624231\pi\)
\(620\) 0 0
\(621\) 102.762 4.12370
\(622\) 0 0
\(623\) 1.09083i 0.0437031i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 13.2800i 0.530354i
\(628\) 0 0
\(629\) 26.7493 1.06656
\(630\) 0 0
\(631\) −0.370875 −0.0147643 −0.00738215 0.999973i \(-0.502350\pi\)
−0.00738215 + 0.999973i \(0.502350\pi\)
\(632\) 0 0
\(633\) − 21.8255i − 0.867484i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 19.5474i − 0.774495i
\(638\) 0 0
\(639\) −57.4986 −2.27461
\(640\) 0 0
\(641\) 26.8099 1.05893 0.529463 0.848333i \(-0.322393\pi\)
0.529463 + 0.848333i \(0.322393\pi\)
\(642\) 0 0
\(643\) 0.969724i 0.0382422i 0.999817 + 0.0191211i \(0.00608680\pi\)
−0.999817 + 0.0191211i \(0.993913\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.20482i 0.0473662i 0.999720 + 0.0236831i \(0.00753926\pi\)
−0.999720 + 0.0236831i \(0.992461\pi\)
\(648\) 0 0
\(649\) 10.5601 0.414520
\(650\) 0 0
\(651\) −23.5298 −0.922206
\(652\) 0 0
\(653\) 22.9310i 0.897358i 0.893693 + 0.448679i \(0.148105\pi\)
−0.893693 + 0.448679i \(0.851895\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 101.778i − 3.97073i
\(658\) 0 0
\(659\) −15.1055 −0.588427 −0.294214 0.955740i \(-0.595058\pi\)
−0.294214 + 0.955740i \(0.595058\pi\)
\(660\) 0 0
\(661\) 19.0450 0.740763 0.370381 0.928880i \(-0.379227\pi\)
0.370381 + 0.928880i \(0.379227\pi\)
\(662\) 0 0
\(663\) − 45.6420i − 1.77259i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.7053i − 0.491950i
\(668\) 0 0
\(669\) 76.4977 2.95757
\(670\) 0 0
\(671\) −19.1220 −0.738197
\(672\) 0 0
\(673\) − 15.4205i − 0.594418i −0.954812 0.297209i \(-0.903944\pi\)
0.954812 0.297209i \(-0.0960558\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.62534i 0.331499i 0.986168 + 0.165749i \(0.0530043\pi\)
−0.986168 + 0.165749i \(0.946996\pi\)
\(678\) 0 0
\(679\) 26.3472 1.01111
\(680\) 0 0
\(681\) 77.8236 2.98221
\(682\) 0 0
\(683\) − 38.8898i − 1.48808i −0.668137 0.744038i \(-0.732908\pi\)
0.668137 0.744038i \(-0.267092\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 13.9688i − 0.532943i
\(688\) 0 0
\(689\) −0.644721 −0.0245619
\(690\) 0 0
\(691\) −21.5979 −0.821624 −0.410812 0.911720i \(-0.634755\pi\)
−0.410812 + 0.911720i \(0.634755\pi\)
\(692\) 0 0
\(693\) 43.5592i 1.65468i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.4078i 1.22753i
\(698\) 0 0
\(699\) −12.3103 −0.465619
\(700\) 0 0
\(701\) 17.8183 0.672990 0.336495 0.941685i \(-0.390759\pi\)
0.336495 + 0.941685i \(0.390759\pi\)
\(702\) 0 0
\(703\) 7.60975i 0.287007i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.9385i 0.712257i
\(708\) 0 0
\(709\) −47.4986 −1.78385 −0.891924 0.452185i \(-0.850645\pi\)
−0.891924 + 0.452185i \(0.850645\pi\)
\(710\) 0 0
\(711\) −2.09931 −0.0787302
\(712\) 0 0
\(713\) − 43.4087i − 1.62567i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 55.7025i 2.08025i
\(718\) 0 0
\(719\) 25.4546 0.949296 0.474648 0.880176i \(-0.342575\pi\)
0.474648 + 0.880176i \(0.342575\pi\)
\(720\) 0 0
\(721\) 16.4995 0.614475
\(722\) 0 0
\(723\) − 85.3388i − 3.17378i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 21.5445i 0.799041i 0.916724 + 0.399521i \(0.130823\pi\)
−0.916724 + 0.399521i \(0.869177\pi\)
\(728\) 0 0
\(729\) −1.12110 −0.0415224
\(730\) 0 0
\(731\) 29.2119 1.08044
\(732\) 0 0
\(733\) − 24.2110i − 0.894254i −0.894470 0.447127i \(-0.852447\pi\)
0.894470 0.447127i \(-0.147553\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.5979i 1.16392i
\(738\) 0 0
\(739\) 37.4693 1.37833 0.689165 0.724605i \(-0.257978\pi\)
0.689165 + 0.724605i \(0.257978\pi\)
\(740\) 0 0
\(741\) 12.9844 0.476994
\(742\) 0 0
\(743\) 15.8595i 0.581829i 0.956749 + 0.290915i \(0.0939596\pi\)
−0.956749 + 0.290915i \(0.906040\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 60.6794i − 2.22014i
\(748\) 0 0
\(749\) −11.2654 −0.411628
\(750\) 0 0
\(751\) 5.12958 0.187181 0.0935906 0.995611i \(-0.470166\pi\)
0.0935906 + 0.995611i \(0.470166\pi\)
\(752\) 0 0
\(753\) − 57.6197i − 2.09978i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 37.4381i 1.36071i 0.732883 + 0.680355i \(0.238174\pi\)
−0.732883 + 0.680355i \(0.761826\pi\)
\(758\) 0 0
\(759\) −115.996 −4.21040
\(760\) 0 0
\(761\) 21.5833 0.782392 0.391196 0.920307i \(-0.372061\pi\)
0.391196 + 0.920307i \(0.372061\pi\)
\(762\) 0 0
\(763\) − 29.3259i − 1.06167i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 10.3250i − 0.372814i
\(768\) 0 0
\(769\) 40.7640 1.46999 0.734994 0.678074i \(-0.237185\pi\)
0.734994 + 0.678074i \(0.237185\pi\)
\(770\) 0 0
\(771\) 2.78051 0.100137
\(772\) 0 0
\(773\) − 48.1845i − 1.73308i −0.499110 0.866538i \(-0.666340\pi\)
0.499110 0.866538i \(-0.333660\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 36.0294i 1.29255i
\(778\) 0 0
\(779\) −9.21949 −0.330323
\(780\) 0 0
\(781\) 36.1211 1.29251
\(782\) 0 0
\(783\) 17.1131i 0.611571i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.6925i 0.987132i 0.869708 + 0.493566i \(0.164307\pi\)
−0.869708 + 0.493566i \(0.835693\pi\)
\(788\) 0 0
\(789\) −39.4381 −1.40403
\(790\) 0 0
\(791\) 27.0596 0.962130
\(792\) 0 0
\(793\) 18.6963i 0.663926i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.3368i 0.366149i 0.983099 + 0.183074i \(0.0586049\pi\)
−0.983099 + 0.183074i \(0.941395\pi\)
\(798\) 0 0
\(799\) 18.5601 0.656609
\(800\) 0 0
\(801\) 4.87042 0.172088
\(802\) 0 0
\(803\) 63.9376i 2.25631i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 51.3700i 1.80831i
\(808\) 0 0
\(809\) 23.2342 0.816870 0.408435 0.912787i \(-0.366075\pi\)
0.408435 + 0.912787i \(0.366075\pi\)
\(810\) 0 0
\(811\) 32.1433 1.12871 0.564353 0.825534i \(-0.309126\pi\)
0.564353 + 0.825534i \(0.309126\pi\)
\(812\) 0 0
\(813\) 80.0128i 2.80617i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.31032i 0.290741i
\(818\) 0 0
\(819\) 42.5895 1.48820
\(820\) 0 0
\(821\) −27.0908 −0.945476 −0.472738 0.881203i \(-0.656734\pi\)
−0.472738 + 0.881203i \(0.656734\pi\)
\(822\) 0 0
\(823\) 14.4849i 0.504911i 0.967609 + 0.252455i \(0.0812381\pi\)
−0.967609 + 0.252455i \(0.918762\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 54.0247i − 1.87862i −0.343067 0.939311i \(-0.611466\pi\)
0.343067 0.939311i \(-0.388534\pi\)
\(828\) 0 0
\(829\) −18.7640 −0.651700 −0.325850 0.945421i \(-0.605651\pi\)
−0.325850 + 0.945421i \(0.605651\pi\)
\(830\) 0 0
\(831\) −2.34152 −0.0812263
\(832\) 0 0
\(833\) − 16.5365i − 0.572954i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 58.4683i 2.02096i
\(838\) 0 0
\(839\) 7.84014 0.270672 0.135336 0.990800i \(-0.456789\pi\)
0.135336 + 0.990800i \(0.456789\pi\)
\(840\) 0 0
\(841\) −26.8842 −0.927041
\(842\) 0 0
\(843\) 11.8108i 0.406785i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.6977i − 0.367578i
\(848\) 0 0
\(849\) −29.5904 −1.01554
\(850\) 0 0
\(851\) −66.4683 −2.27851
\(852\) 0 0
\(853\) − 36.8174i − 1.26060i −0.776350 0.630302i \(-0.782931\pi\)
0.776350 0.630302i \(-0.217069\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 5.23887i − 0.178956i −0.995989 0.0894782i \(-0.971480\pi\)
0.995989 0.0894782i \(-0.0285199\pi\)
\(858\) 0 0
\(859\) 29.6197 1.01061 0.505306 0.862940i \(-0.331380\pi\)
0.505306 + 0.862940i \(0.331380\pi\)
\(860\) 0 0
\(861\) −43.6509 −1.48762
\(862\) 0 0
\(863\) 35.8889i 1.22167i 0.791757 + 0.610836i \(0.209166\pi\)
−0.791757 + 0.610836i \(0.790834\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.5114i 0.492832i
\(868\) 0 0
\(869\) 1.31880 0.0447373
\(870\) 0 0
\(871\) 30.8945 1.04682
\(872\) 0 0
\(873\) − 117.637i − 3.98142i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 29.8136i − 1.00674i −0.864072 0.503368i \(-0.832094\pi\)
0.864072 0.503368i \(-0.167906\pi\)
\(878\) 0 0
\(879\) 77.1349 2.60169
\(880\) 0 0
\(881\) −38.9679 −1.31286 −0.656431 0.754386i \(-0.727935\pi\)
−0.656431 + 0.754386i \(0.727935\pi\)
\(882\) 0 0
\(883\) 28.5677i 0.961378i 0.876891 + 0.480689i \(0.159613\pi\)
−0.876891 + 0.480689i \(0.840387\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.5104i 0.923710i 0.886955 + 0.461855i \(0.152816\pi\)
−0.886955 + 0.461855i \(0.847184\pi\)
\(888\) 0 0
\(889\) 4.09174 0.137233
\(890\) 0 0
\(891\) 69.9906 2.34477
\(892\) 0 0
\(893\) 5.28005i 0.176690i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 113.414i 3.78678i
\(898\) 0 0
\(899\) 7.22889 0.241097
\(900\) 0 0
\(901\) −0.545414 −0.0181704
\(902\) 0 0
\(903\) 39.3463i 1.30936i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 55.4939i − 1.84265i −0.388799 0.921323i \(-0.627110\pi\)
0.388799 0.921323i \(-0.372890\pi\)
\(908\) 0 0
\(909\) 84.5583 2.80462
\(910\) 0 0
\(911\) 6.84106 0.226654 0.113327 0.993558i \(-0.463849\pi\)
0.113327 + 0.993558i \(0.463849\pi\)
\(912\) 0 0
\(913\) 38.1193i 1.26156i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.06055i 0.200137i
\(918\) 0 0
\(919\) 6.17454 0.203679 0.101840 0.994801i \(-0.467527\pi\)
0.101840 + 0.994801i \(0.467527\pi\)
\(920\) 0 0
\(921\) −41.6509 −1.37244
\(922\) 0 0
\(923\) − 35.3170i − 1.16247i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 73.6685i − 2.41959i
\(928\) 0 0
\(929\) −44.7034 −1.46667 −0.733336 0.679866i \(-0.762038\pi\)
−0.733336 + 0.679866i \(0.762038\pi\)
\(930\) 0 0
\(931\) 4.70436 0.154179
\(932\) 0 0
\(933\) − 42.0440i − 1.37646i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 12.1064i − 0.395500i −0.980253 0.197750i \(-0.936637\pi\)
0.980253 0.197750i \(-0.0633634\pi\)
\(938\) 0 0
\(939\) −68.0128 −2.21952
\(940\) 0 0
\(941\) −26.2645 −0.856197 −0.428098 0.903732i \(-0.640816\pi\)
−0.428098 + 0.903732i \(0.640816\pi\)
\(942\) 0 0
\(943\) − 80.5289i − 2.62238i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.37935i 0.239797i 0.992786 + 0.119898i \(0.0382569\pi\)
−0.992786 + 0.119898i \(0.961743\pi\)
\(948\) 0 0
\(949\) 62.5142 2.02930
\(950\) 0 0
\(951\) −73.6344 −2.38776
\(952\) 0 0
\(953\) 21.5398i 0.697743i 0.937171 + 0.348871i \(0.113435\pi\)
−0.937171 + 0.348871i \(0.886565\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 19.3170i − 0.624429i
\(958\) 0 0
\(959\) −11.3865 −0.367688
\(960\) 0 0
\(961\) −6.30184 −0.203285
\(962\) 0 0
\(963\) 50.2985i 1.62085i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 39.8789i − 1.28242i −0.767366 0.641209i \(-0.778433\pi\)
0.767366 0.641209i \(-0.221567\pi\)
\(968\) 0 0
\(969\) 10.9844 0.352870
\(970\) 0 0
\(971\) −29.7408 −0.954429 −0.477214 0.878787i \(-0.658353\pi\)
−0.477214 + 0.878787i \(0.658353\pi\)
\(972\) 0 0
\(973\) 25.3775i 0.813566i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 58.6694i − 1.87700i −0.345279 0.938500i \(-0.612216\pi\)
0.345279 0.938500i \(-0.387784\pi\)
\(978\) 0 0
\(979\) −3.05964 −0.0977864
\(980\) 0 0
\(981\) −130.937 −4.18049
\(982\) 0 0
\(983\) − 11.7384i − 0.374397i −0.982322 0.187199i \(-0.940059\pi\)
0.982322 0.187199i \(-0.0599408\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 24.9991i 0.795730i
\(988\) 0 0
\(989\) −72.5876 −2.30815
\(990\) 0 0
\(991\) −26.7493 −0.849720 −0.424860 0.905259i \(-0.639677\pi\)
−0.424860 + 0.905259i \(0.639677\pi\)
\(992\) 0 0
\(993\) − 37.3553i − 1.18543i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 45.8401i 1.45177i 0.687815 + 0.725886i \(0.258570\pi\)
−0.687815 + 0.725886i \(0.741430\pi\)
\(998\) 0 0
\(999\) 89.5280 2.83254
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 3800.2.d.k.3649.6 6
5.2 odd 4 3800.2.a.y.1.3 3
5.3 odd 4 760.2.a.h.1.1 3
5.4 even 2 inner 3800.2.d.k.3649.1 6
15.8 even 4 6840.2.a.bj.1.2 3
20.3 even 4 1520.2.a.r.1.3 3
20.7 even 4 7600.2.a.bo.1.1 3
40.3 even 4 6080.2.a.bs.1.1 3
40.13 odd 4 6080.2.a.bw.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.1 3 5.3 odd 4
1520.2.a.r.1.3 3 20.3 even 4
3800.2.a.y.1.3 3 5.2 odd 4
3800.2.d.k.3649.1 6 5.4 even 2 inner
3800.2.d.k.3649.6 6 1.1 even 1 trivial
6080.2.a.bs.1.1 3 40.3 even 4
6080.2.a.bw.1.3 3 40.13 odd 4
6840.2.a.bj.1.2 3 15.8 even 4
7600.2.a.bo.1.1 3 20.7 even 4