Properties

Label 2704.2.f.o.337.5
Level $2704$
Weight $2$
Character 2704.337
Analytic conductor $21.592$
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] = [2704,2,Mod(337,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.5
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 2704.337
Dual form 2704.2.f.o.337.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.24698 q^{3} -0.246980i q^{5} +2.35690i q^{7} +2.04892 q^{9} +O(q^{10})\) \(q+2.24698 q^{3} -0.246980i q^{5} +2.35690i q^{7} +2.04892 q^{9} +4.24698i q^{11} -0.554958i q^{15} -2.15883 q^{17} -0.0881460i q^{19} +5.29590i q^{21} +1.49396 q^{23} +4.93900 q^{25} -2.13706 q^{27} +4.63102 q^{29} -6.63102i q^{31} +9.54288i q^{33} +0.582105 q^{35} +5.69202i q^{37} +11.5918i q^{41} -0.295897 q^{43} -0.506041i q^{45} +7.35690i q^{47} +1.44504 q^{49} -4.85086 q^{51} -10.3937 q^{53} +1.04892 q^{55} -0.198062i q^{57} +6.78017i q^{59} +3.47219 q^{61} +4.82908i q^{63} +7.67994i q^{67} +3.35690 q^{69} -8.66487i q^{71} +6.73556i q^{73} +11.0978 q^{75} -10.0097 q^{77} -9.97046 q^{79} -10.9487 q^{81} +1.60925i q^{83} +0.533188i q^{85} +10.4058 q^{87} -2.88471i q^{89} -14.8998i q^{93} -0.0217703 q^{95} +8.05861i q^{97} +8.70171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 6 q^{9} + 4 q^{17} - 10 q^{23} + 10 q^{25} - 2 q^{27} - 2 q^{29} - 8 q^{35} + 26 q^{43} + 8 q^{49} - 2 q^{51} + 2 q^{53} - 12 q^{55} + 8 q^{61} + 12 q^{69} + 30 q^{75} - 16 q^{77} + 10 q^{79} - 2 q^{81} + 36 q^{87} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 2.24698 1.29729 0.648647 0.761089i \(-0.275335\pi\)
0.648647 + 0.761089i \(0.275335\pi\)
\(4\) 0 0
\(5\) − 0.246980i − 0.110453i −0.998474 0.0552263i \(-0.982412\pi\)
0.998474 0.0552263i \(-0.0175880\pi\)
\(6\) 0 0
\(7\) 2.35690i 0.890823i 0.895326 + 0.445411i \(0.146943\pi\)
−0.895326 + 0.445411i \(0.853057\pi\)
\(8\) 0 0
\(9\) 2.04892 0.682972
\(10\) 0 0
\(11\) 4.24698i 1.28051i 0.768161 + 0.640256i \(0.221172\pi\)
−0.768161 + 0.640256i \(0.778828\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) − 0.554958i − 0.143290i
\(16\) 0 0
\(17\) −2.15883 −0.523594 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) 0 0
\(19\) − 0.0881460i − 0.0202221i −0.999949 0.0101110i \(-0.996782\pi\)
0.999949 0.0101110i \(-0.00321850\pi\)
\(20\) 0 0
\(21\) 5.29590i 1.15566i
\(22\) 0 0
\(23\) 1.49396 0.311512 0.155756 0.987796i \(-0.450219\pi\)
0.155756 + 0.987796i \(0.450219\pi\)
\(24\) 0 0
\(25\) 4.93900 0.987800
\(26\) 0 0
\(27\) −2.13706 −0.411278
\(28\) 0 0
\(29\) 4.63102 0.859959 0.429980 0.902839i \(-0.358521\pi\)
0.429980 + 0.902839i \(0.358521\pi\)
\(30\) 0 0
\(31\) − 6.63102i − 1.19097i −0.803368 0.595483i \(-0.796961\pi\)
0.803368 0.595483i \(-0.203039\pi\)
\(32\) 0 0
\(33\) 9.54288i 1.66120i
\(34\) 0 0
\(35\) 0.582105 0.0983937
\(36\) 0 0
\(37\) 5.69202i 0.935763i 0.883791 + 0.467881i \(0.154983\pi\)
−0.883791 + 0.467881i \(0.845017\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.5918i 1.81033i 0.425056 + 0.905167i \(0.360254\pi\)
−0.425056 + 0.905167i \(0.639746\pi\)
\(42\) 0 0
\(43\) −0.295897 −0.0451239 −0.0225619 0.999745i \(-0.507182\pi\)
−0.0225619 + 0.999745i \(0.507182\pi\)
\(44\) 0 0
\(45\) − 0.506041i − 0.0754361i
\(46\) 0 0
\(47\) 7.35690i 1.07311i 0.843864 + 0.536557i \(0.180275\pi\)
−0.843864 + 0.536557i \(0.819725\pi\)
\(48\) 0 0
\(49\) 1.44504 0.206435
\(50\) 0 0
\(51\) −4.85086 −0.679256
\(52\) 0 0
\(53\) −10.3937 −1.42769 −0.713844 0.700304i \(-0.753048\pi\)
−0.713844 + 0.700304i \(0.753048\pi\)
\(54\) 0 0
\(55\) 1.04892 0.141436
\(56\) 0 0
\(57\) − 0.198062i − 0.0262340i
\(58\) 0 0
\(59\) 6.78017i 0.882703i 0.897334 + 0.441351i \(0.145501\pi\)
−0.897334 + 0.441351i \(0.854499\pi\)
\(60\) 0 0
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 0 0
\(63\) 4.82908i 0.608407i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.67994i 0.938254i 0.883131 + 0.469127i \(0.155431\pi\)
−0.883131 + 0.469127i \(0.844569\pi\)
\(68\) 0 0
\(69\) 3.35690 0.404123
\(70\) 0 0
\(71\) − 8.66487i − 1.02833i −0.857691 0.514166i \(-0.828102\pi\)
0.857691 0.514166i \(-0.171898\pi\)
\(72\) 0 0
\(73\) 6.73556i 0.788338i 0.919038 + 0.394169i \(0.128968\pi\)
−0.919038 + 0.394169i \(0.871032\pi\)
\(74\) 0 0
\(75\) 11.0978 1.28147
\(76\) 0 0
\(77\) −10.0097 −1.14071
\(78\) 0 0
\(79\) −9.97046 −1.12176 −0.560882 0.827896i \(-0.689538\pi\)
−0.560882 + 0.827896i \(0.689538\pi\)
\(80\) 0 0
\(81\) −10.9487 −1.21652
\(82\) 0 0
\(83\) 1.60925i 0.176638i 0.996092 + 0.0883192i \(0.0281495\pi\)
−0.996092 + 0.0883192i \(0.971850\pi\)
\(84\) 0 0
\(85\) 0.533188i 0.0578323i
\(86\) 0 0
\(87\) 10.4058 1.11562
\(88\) 0 0
\(89\) − 2.88471i − 0.305778i −0.988243 0.152889i \(-0.951142\pi\)
0.988243 0.152889i \(-0.0488577\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 14.8998i − 1.54503i
\(94\) 0 0
\(95\) −0.0217703 −0.00223358
\(96\) 0 0
\(97\) 8.05861i 0.818227i 0.912483 + 0.409114i \(0.134162\pi\)
−0.912483 + 0.409114i \(0.865838\pi\)
\(98\) 0 0
\(99\) 8.70171i 0.874555i
\(100\) 0 0
\(101\) 13.3545 1.32882 0.664411 0.747367i \(-0.268682\pi\)
0.664411 + 0.747367i \(0.268682\pi\)
\(102\) 0 0
\(103\) 1.36227 0.134229 0.0671144 0.997745i \(-0.478621\pi\)
0.0671144 + 0.997745i \(0.478621\pi\)
\(104\) 0 0
\(105\) 1.30798 0.127646
\(106\) 0 0
\(107\) −3.26875 −0.316002 −0.158001 0.987439i \(-0.550505\pi\)
−0.158001 + 0.987439i \(0.550505\pi\)
\(108\) 0 0
\(109\) − 15.7017i − 1.50395i −0.659191 0.751976i \(-0.729101\pi\)
0.659191 0.751976i \(-0.270899\pi\)
\(110\) 0 0
\(111\) 12.7899i 1.21396i
\(112\) 0 0
\(113\) 12.0489 1.13347 0.566733 0.823901i \(-0.308207\pi\)
0.566733 + 0.823901i \(0.308207\pi\)
\(114\) 0 0
\(115\) − 0.368977i − 0.0344073i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 5.08815i − 0.466430i
\(120\) 0 0
\(121\) −7.03684 −0.639712
\(122\) 0 0
\(123\) 26.0465i 2.34854i
\(124\) 0 0
\(125\) − 2.45473i − 0.219558i
\(126\) 0 0
\(127\) −9.80731 −0.870258 −0.435129 0.900368i \(-0.643297\pi\)
−0.435129 + 0.900368i \(0.643297\pi\)
\(128\) 0 0
\(129\) −0.664874 −0.0585389
\(130\) 0 0
\(131\) 6.57673 0.574611 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(132\) 0 0
\(133\) 0.207751 0.0180143
\(134\) 0 0
\(135\) 0.527811i 0.0454267i
\(136\) 0 0
\(137\) 6.21983i 0.531396i 0.964056 + 0.265698i \(0.0856024\pi\)
−0.964056 + 0.265698i \(0.914398\pi\)
\(138\) 0 0
\(139\) 14.7071 1.24744 0.623719 0.781648i \(-0.285621\pi\)
0.623719 + 0.781648i \(0.285621\pi\)
\(140\) 0 0
\(141\) 16.5308i 1.39214i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 1.14377i − 0.0949848i
\(146\) 0 0
\(147\) 3.24698 0.267806
\(148\) 0 0
\(149\) − 4.33513i − 0.355147i −0.984108 0.177574i \(-0.943175\pi\)
0.984108 0.177574i \(-0.0568248\pi\)
\(150\) 0 0
\(151\) − 3.94438i − 0.320989i −0.987037 0.160494i \(-0.948691\pi\)
0.987037 0.160494i \(-0.0513089\pi\)
\(152\) 0 0
\(153\) −4.42327 −0.357600
\(154\) 0 0
\(155\) −1.63773 −0.131545
\(156\) 0 0
\(157\) 4.45473 0.355526 0.177763 0.984073i \(-0.443114\pi\)
0.177763 + 0.984073i \(0.443114\pi\)
\(158\) 0 0
\(159\) −23.3545 −1.85213
\(160\) 0 0
\(161\) 3.52111i 0.277502i
\(162\) 0 0
\(163\) − 16.1588i − 1.26566i −0.774292 0.632829i \(-0.781894\pi\)
0.774292 0.632829i \(-0.218106\pi\)
\(164\) 0 0
\(165\) 2.35690 0.183484
\(166\) 0 0
\(167\) − 16.1172i − 1.24719i −0.781749 0.623594i \(-0.785672\pi\)
0.781749 0.623594i \(-0.214328\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) − 0.180604i − 0.0138111i
\(172\) 0 0
\(173\) 21.5362 1.63736 0.818682 0.574247i \(-0.194705\pi\)
0.818682 + 0.574247i \(0.194705\pi\)
\(174\) 0 0
\(175\) 11.6407i 0.879955i
\(176\) 0 0
\(177\) 15.2349i 1.14513i
\(178\) 0 0
\(179\) 11.4330 0.854540 0.427270 0.904124i \(-0.359475\pi\)
0.427270 + 0.904124i \(0.359475\pi\)
\(180\) 0 0
\(181\) −20.9705 −1.55872 −0.779361 0.626575i \(-0.784456\pi\)
−0.779361 + 0.626575i \(0.784456\pi\)
\(182\) 0 0
\(183\) 7.80194 0.576736
\(184\) 0 0
\(185\) 1.40581 0.103357
\(186\) 0 0
\(187\) − 9.16852i − 0.670469i
\(188\) 0 0
\(189\) − 5.03684i − 0.366376i
\(190\) 0 0
\(191\) 14.4373 1.04464 0.522322 0.852748i \(-0.325066\pi\)
0.522322 + 0.852748i \(0.325066\pi\)
\(192\) 0 0
\(193\) − 13.5797i − 0.977489i −0.872427 0.488745i \(-0.837455\pi\)
0.872427 0.488745i \(-0.162545\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.560335i 0.0399222i 0.999801 + 0.0199611i \(0.00635424\pi\)
−0.999801 + 0.0199611i \(0.993646\pi\)
\(198\) 0 0
\(199\) 11.4916 0.814616 0.407308 0.913291i \(-0.366468\pi\)
0.407308 + 0.913291i \(0.366468\pi\)
\(200\) 0 0
\(201\) 17.2567i 1.21719i
\(202\) 0 0
\(203\) 10.9148i 0.766071i
\(204\) 0 0
\(205\) 2.86294 0.199956
\(206\) 0 0
\(207\) 3.06100 0.212754
\(208\) 0 0
\(209\) 0.374354 0.0258946
\(210\) 0 0
\(211\) −8.78448 −0.604748 −0.302374 0.953189i \(-0.597779\pi\)
−0.302374 + 0.953189i \(0.597779\pi\)
\(212\) 0 0
\(213\) − 19.4698i − 1.33405i
\(214\) 0 0
\(215\) 0.0730805i 0.00498405i
\(216\) 0 0
\(217\) 15.6286 1.06094
\(218\) 0 0
\(219\) 15.1347i 1.02271i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 2.25906i − 0.151278i −0.997135 0.0756390i \(-0.975900\pi\)
0.997135 0.0756390i \(-0.0240996\pi\)
\(224\) 0 0
\(225\) 10.1196 0.674640
\(226\) 0 0
\(227\) − 6.96615i − 0.462359i −0.972911 0.231180i \(-0.925741\pi\)
0.972911 0.231180i \(-0.0742585\pi\)
\(228\) 0 0
\(229\) − 24.1739i − 1.59746i −0.601692 0.798728i \(-0.705507\pi\)
0.601692 0.798728i \(-0.294493\pi\)
\(230\) 0 0
\(231\) −22.4916 −1.47984
\(232\) 0 0
\(233\) 3.06100 0.200533 0.100266 0.994961i \(-0.468031\pi\)
0.100266 + 0.994961i \(0.468031\pi\)
\(234\) 0 0
\(235\) 1.81700 0.118528
\(236\) 0 0
\(237\) −22.4034 −1.45526
\(238\) 0 0
\(239\) − 25.1468i − 1.62661i −0.581839 0.813304i \(-0.697667\pi\)
0.581839 0.813304i \(-0.302333\pi\)
\(240\) 0 0
\(241\) − 20.2664i − 1.30547i −0.757586 0.652735i \(-0.773621\pi\)
0.757586 0.652735i \(-0.226379\pi\)
\(242\) 0 0
\(243\) −18.1903 −1.16691
\(244\) 0 0
\(245\) − 0.356896i − 0.0228012i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.61596i 0.229152i
\(250\) 0 0
\(251\) −23.7211 −1.49726 −0.748631 0.662987i \(-0.769288\pi\)
−0.748631 + 0.662987i \(0.769288\pi\)
\(252\) 0 0
\(253\) 6.34481i 0.398895i
\(254\) 0 0
\(255\) 1.19806i 0.0750256i
\(256\) 0 0
\(257\) −14.2241 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(258\) 0 0
\(259\) −13.4155 −0.833599
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) 0 0
\(263\) 17.0954 1.05415 0.527075 0.849819i \(-0.323289\pi\)
0.527075 + 0.849819i \(0.323289\pi\)
\(264\) 0 0
\(265\) 2.56704i 0.157692i
\(266\) 0 0
\(267\) − 6.48188i − 0.396684i
\(268\) 0 0
\(269\) −6.46681 −0.394288 −0.197144 0.980374i \(-0.563167\pi\)
−0.197144 + 0.980374i \(0.563167\pi\)
\(270\) 0 0
\(271\) − 6.44803i − 0.391690i −0.980635 0.195845i \(-0.937255\pi\)
0.980635 0.195845i \(-0.0627449\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.9758i 1.26489i
\(276\) 0 0
\(277\) −13.4601 −0.808739 −0.404370 0.914596i \(-0.632509\pi\)
−0.404370 + 0.914596i \(0.632509\pi\)
\(278\) 0 0
\(279\) − 13.5864i − 0.813398i
\(280\) 0 0
\(281\) 5.03684i 0.300472i 0.988650 + 0.150236i \(0.0480034\pi\)
−0.988650 + 0.150236i \(0.951997\pi\)
\(282\) 0 0
\(283\) 22.1280 1.31537 0.657686 0.753293i \(-0.271536\pi\)
0.657686 + 0.753293i \(0.271536\pi\)
\(284\) 0 0
\(285\) −0.0489173 −0.00289761
\(286\) 0 0
\(287\) −27.3207 −1.61269
\(288\) 0 0
\(289\) −12.3394 −0.725849
\(290\) 0 0
\(291\) 18.1075i 1.06148i
\(292\) 0 0
\(293\) 14.9463i 0.873172i 0.899663 + 0.436586i \(0.143813\pi\)
−0.899663 + 0.436586i \(0.856187\pi\)
\(294\) 0 0
\(295\) 1.67456 0.0974968
\(296\) 0 0
\(297\) − 9.07606i − 0.526647i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) − 0.697398i − 0.0401974i
\(302\) 0 0
\(303\) 30.0073 1.72387
\(304\) 0 0
\(305\) − 0.857560i − 0.0491037i
\(306\) 0 0
\(307\) − 19.1293i − 1.09177i −0.837861 0.545883i \(-0.816194\pi\)
0.837861 0.545883i \(-0.183806\pi\)
\(308\) 0 0
\(309\) 3.06100 0.174134
\(310\) 0 0
\(311\) −0.269815 −0.0152998 −0.00764990 0.999971i \(-0.502435\pi\)
−0.00764990 + 0.999971i \(0.502435\pi\)
\(312\) 0 0
\(313\) −23.3937 −1.32229 −0.661146 0.750257i \(-0.729930\pi\)
−0.661146 + 0.750257i \(0.729930\pi\)
\(314\) 0 0
\(315\) 1.19269 0.0672002
\(316\) 0 0
\(317\) 13.9952i 0.786050i 0.919528 + 0.393025i \(0.128571\pi\)
−0.919528 + 0.393025i \(0.871429\pi\)
\(318\) 0 0
\(319\) 19.6679i 1.10119i
\(320\) 0 0
\(321\) −7.34481 −0.409948
\(322\) 0 0
\(323\) 0.190293i 0.0105882i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 35.2814i − 1.95107i
\(328\) 0 0
\(329\) −17.3394 −0.955954
\(330\) 0 0
\(331\) 17.8213i 0.979548i 0.871849 + 0.489774i \(0.162921\pi\)
−0.871849 + 0.489774i \(0.837079\pi\)
\(332\) 0 0
\(333\) 11.6625i 0.639100i
\(334\) 0 0
\(335\) 1.89679 0.103633
\(336\) 0 0
\(337\) 27.8485 1.51700 0.758501 0.651672i \(-0.225932\pi\)
0.758501 + 0.651672i \(0.225932\pi\)
\(338\) 0 0
\(339\) 27.0737 1.47044
\(340\) 0 0
\(341\) 28.1618 1.52505
\(342\) 0 0
\(343\) 19.9041i 1.07472i
\(344\) 0 0
\(345\) − 0.829085i − 0.0446364i
\(346\) 0 0
\(347\) −1.50365 −0.0807200 −0.0403600 0.999185i \(-0.512850\pi\)
−0.0403600 + 0.999185i \(0.512850\pi\)
\(348\) 0 0
\(349\) − 14.1860i − 0.759358i −0.925118 0.379679i \(-0.876034\pi\)
0.925118 0.379679i \(-0.123966\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.16852i 0.381542i 0.981635 + 0.190771i \(0.0610988\pi\)
−0.981635 + 0.190771i \(0.938901\pi\)
\(354\) 0 0
\(355\) −2.14005 −0.113582
\(356\) 0 0
\(357\) − 11.4330i − 0.605096i
\(358\) 0 0
\(359\) − 19.8853i − 1.04951i −0.851255 0.524753i \(-0.824158\pi\)
0.851255 0.524753i \(-0.175842\pi\)
\(360\) 0 0
\(361\) 18.9922 0.999591
\(362\) 0 0
\(363\) −15.8116 −0.829895
\(364\) 0 0
\(365\) 1.66355 0.0870740
\(366\) 0 0
\(367\) −1.08383 −0.0565757 −0.0282878 0.999600i \(-0.509006\pi\)
−0.0282878 + 0.999600i \(0.509006\pi\)
\(368\) 0 0
\(369\) 23.7506i 1.23641i
\(370\) 0 0
\(371\) − 24.4969i − 1.27182i
\(372\) 0 0
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) 0 0
\(375\) − 5.51573i − 0.284831i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) − 2.40880i − 0.123732i −0.998084 0.0618658i \(-0.980295\pi\)
0.998084 0.0618658i \(-0.0197051\pi\)
\(380\) 0 0
\(381\) −22.0368 −1.12898
\(382\) 0 0
\(383\) − 30.3913i − 1.55292i −0.630164 0.776462i \(-0.717012\pi\)
0.630164 0.776462i \(-0.282988\pi\)
\(384\) 0 0
\(385\) 2.47219i 0.125994i
\(386\) 0 0
\(387\) −0.606268 −0.0308184
\(388\) 0 0
\(389\) 15.9409 0.808237 0.404118 0.914707i \(-0.367578\pi\)
0.404118 + 0.914707i \(0.367578\pi\)
\(390\) 0 0
\(391\) −3.22521 −0.163106
\(392\) 0 0
\(393\) 14.7778 0.745440
\(394\) 0 0
\(395\) 2.46250i 0.123902i
\(396\) 0 0
\(397\) 16.9148i 0.848931i 0.905444 + 0.424466i \(0.139538\pi\)
−0.905444 + 0.424466i \(0.860462\pi\)
\(398\) 0 0
\(399\) 0.466812 0.0233698
\(400\) 0 0
\(401\) 26.6625i 1.33146i 0.746192 + 0.665730i \(0.231880\pi\)
−0.746192 + 0.665730i \(0.768120\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.70410i 0.134368i
\(406\) 0 0
\(407\) −24.1739 −1.19826
\(408\) 0 0
\(409\) − 28.5163i − 1.41004i −0.709187 0.705021i \(-0.750938\pi\)
0.709187 0.705021i \(-0.249062\pi\)
\(410\) 0 0
\(411\) 13.9758i 0.689377i
\(412\) 0 0
\(413\) −15.9801 −0.786332
\(414\) 0 0
\(415\) 0.397452 0.0195102
\(416\) 0 0
\(417\) 33.0465 1.61830
\(418\) 0 0
\(419\) 29.6093 1.44651 0.723253 0.690583i \(-0.242646\pi\)
0.723253 + 0.690583i \(0.242646\pi\)
\(420\) 0 0
\(421\) 11.6606i 0.568301i 0.958780 + 0.284151i \(0.0917115\pi\)
−0.958780 + 0.284151i \(0.908288\pi\)
\(422\) 0 0
\(423\) 15.0737i 0.732907i
\(424\) 0 0
\(425\) −10.6625 −0.517206
\(426\) 0 0
\(427\) 8.18359i 0.396032i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.34913i 0.209490i 0.994499 + 0.104745i \(0.0334026\pi\)
−0.994499 + 0.104745i \(0.966597\pi\)
\(432\) 0 0
\(433\) 14.3884 0.691460 0.345730 0.938334i \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(434\) 0 0
\(435\) − 2.57002i − 0.123223i
\(436\) 0 0
\(437\) − 0.131687i − 0.00629942i
\(438\) 0 0
\(439\) −20.2325 −0.965645 −0.482822 0.875718i \(-0.660388\pi\)
−0.482822 + 0.875718i \(0.660388\pi\)
\(440\) 0 0
\(441\) 2.96077 0.140989
\(442\) 0 0
\(443\) −8.12200 −0.385888 −0.192944 0.981210i \(-0.561804\pi\)
−0.192944 + 0.981210i \(0.561804\pi\)
\(444\) 0 0
\(445\) −0.712464 −0.0337740
\(446\) 0 0
\(447\) − 9.74094i − 0.460731i
\(448\) 0 0
\(449\) 12.4916i 0.589513i 0.955572 + 0.294757i \(0.0952386\pi\)
−0.955572 + 0.294757i \(0.904761\pi\)
\(450\) 0 0
\(451\) −49.2301 −2.31816
\(452\) 0 0
\(453\) − 8.86294i − 0.416417i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 5.98121i − 0.279789i −0.990166 0.139895i \(-0.955324\pi\)
0.990166 0.139895i \(-0.0446764\pi\)
\(458\) 0 0
\(459\) 4.61356 0.215343
\(460\) 0 0
\(461\) 2.05669i 0.0957895i 0.998852 + 0.0478947i \(0.0152512\pi\)
−0.998852 + 0.0478947i \(0.984749\pi\)
\(462\) 0 0
\(463\) 8.44935i 0.392675i 0.980536 + 0.196337i \(0.0629048\pi\)
−0.980536 + 0.196337i \(0.937095\pi\)
\(464\) 0 0
\(465\) −3.67994 −0.170653
\(466\) 0 0
\(467\) 33.5139 1.55084 0.775420 0.631446i \(-0.217538\pi\)
0.775420 + 0.631446i \(0.217538\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) 0 0
\(471\) 10.0097 0.461222
\(472\) 0 0
\(473\) − 1.25667i − 0.0577817i
\(474\) 0 0
\(475\) − 0.435353i − 0.0199754i
\(476\) 0 0
\(477\) −21.2959 −0.975072
\(478\) 0 0
\(479\) 24.7313i 1.13000i 0.825091 + 0.565000i \(0.191124\pi\)
−0.825091 + 0.565000i \(0.808876\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 7.91185i 0.360002i
\(484\) 0 0
\(485\) 1.99031 0.0903754
\(486\) 0 0
\(487\) − 37.7555i − 1.71087i −0.517913 0.855433i \(-0.673291\pi\)
0.517913 0.855433i \(-0.326709\pi\)
\(488\) 0 0
\(489\) − 36.3086i − 1.64193i
\(490\) 0 0
\(491\) 31.3110 1.41304 0.706522 0.707691i \(-0.250263\pi\)
0.706522 + 0.707691i \(0.250263\pi\)
\(492\) 0 0
\(493\) −9.99761 −0.450270
\(494\) 0 0
\(495\) 2.14914 0.0965969
\(496\) 0 0
\(497\) 20.4222 0.916061
\(498\) 0 0
\(499\) 21.4873i 0.961902i 0.876748 + 0.480951i \(0.159708\pi\)
−0.876748 + 0.480951i \(0.840292\pi\)
\(500\) 0 0
\(501\) − 36.2150i − 1.61797i
\(502\) 0 0
\(503\) −37.5924 −1.67616 −0.838081 0.545546i \(-0.816322\pi\)
−0.838081 + 0.545546i \(0.816322\pi\)
\(504\) 0 0
\(505\) − 3.29829i − 0.146772i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.1075i 0.758278i 0.925340 + 0.379139i \(0.123780\pi\)
−0.925340 + 0.379139i \(0.876220\pi\)
\(510\) 0 0
\(511\) −15.8750 −0.702269
\(512\) 0 0
\(513\) 0.188374i 0.00831690i
\(514\) 0 0
\(515\) − 0.336454i − 0.0148259i
\(516\) 0 0
\(517\) −31.2446 −1.37414
\(518\) 0 0
\(519\) 48.3913 2.12414
\(520\) 0 0
\(521\) −19.8465 −0.869493 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(522\) 0 0
\(523\) 11.4300 0.499798 0.249899 0.968272i \(-0.419603\pi\)
0.249899 + 0.968272i \(0.419603\pi\)
\(524\) 0 0
\(525\) 26.1564i 1.14156i
\(526\) 0 0
\(527\) 14.3153i 0.623583i
\(528\) 0 0
\(529\) −20.7681 −0.902960
\(530\) 0 0
\(531\) 13.8920i 0.602862i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0.807315i 0.0349033i
\(536\) 0 0
\(537\) 25.6896 1.10859
\(538\) 0 0
\(539\) 6.13706i 0.264342i
\(540\) 0 0
\(541\) 16.1884i 0.695993i 0.937496 + 0.347996i \(0.113138\pi\)
−0.937496 + 0.347996i \(0.886862\pi\)
\(542\) 0 0
\(543\) −47.1202 −2.02212
\(544\) 0 0
\(545\) −3.87800 −0.166115
\(546\) 0 0
\(547\) −5.33081 −0.227929 −0.113965 0.993485i \(-0.536355\pi\)
−0.113965 + 0.993485i \(0.536355\pi\)
\(548\) 0 0
\(549\) 7.11423 0.303628
\(550\) 0 0
\(551\) − 0.408206i − 0.0173902i
\(552\) 0 0
\(553\) − 23.4993i − 0.999293i
\(554\) 0 0
\(555\) 3.15883 0.134085
\(556\) 0 0
\(557\) 7.39075i 0.313156i 0.987666 + 0.156578i \(0.0500463\pi\)
−0.987666 + 0.156578i \(0.949954\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) − 20.6015i − 0.869795i
\(562\) 0 0
\(563\) −9.47889 −0.399488 −0.199744 0.979848i \(-0.564011\pi\)
−0.199744 + 0.979848i \(0.564011\pi\)
\(564\) 0 0
\(565\) − 2.97584i − 0.125194i
\(566\) 0 0
\(567\) − 25.8049i − 1.08370i
\(568\) 0 0
\(569\) 10.1438 0.425249 0.212624 0.977134i \(-0.431799\pi\)
0.212624 + 0.977134i \(0.431799\pi\)
\(570\) 0 0
\(571\) −14.0925 −0.589751 −0.294876 0.955536i \(-0.595278\pi\)
−0.294876 + 0.955536i \(0.595278\pi\)
\(572\) 0 0
\(573\) 32.4403 1.35521
\(574\) 0 0
\(575\) 7.37867 0.307712
\(576\) 0 0
\(577\) − 25.1545i − 1.04720i −0.851965 0.523598i \(-0.824589\pi\)
0.851965 0.523598i \(-0.175411\pi\)
\(578\) 0 0
\(579\) − 30.5133i − 1.26809i
\(580\) 0 0
\(581\) −3.79284 −0.157354
\(582\) 0 0
\(583\) − 44.1420i − 1.82817i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 43.8353i 1.80928i 0.426180 + 0.904639i \(0.359859\pi\)
−0.426180 + 0.904639i \(0.640141\pi\)
\(588\) 0 0
\(589\) −0.584498 −0.0240838
\(590\) 0 0
\(591\) 1.25906i 0.0517909i
\(592\) 0 0
\(593\) − 24.9965i − 1.02648i −0.858244 0.513242i \(-0.828444\pi\)
0.858244 0.513242i \(-0.171556\pi\)
\(594\) 0 0
\(595\) −1.25667 −0.0515184
\(596\) 0 0
\(597\) 25.8213 1.05680
\(598\) 0 0
\(599\) 6.24027 0.254971 0.127485 0.991840i \(-0.459309\pi\)
0.127485 + 0.991840i \(0.459309\pi\)
\(600\) 0 0
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) 0 0
\(603\) 15.7356i 0.640802i
\(604\) 0 0
\(605\) 1.73795i 0.0706579i
\(606\) 0 0
\(607\) 43.6480 1.77162 0.885809 0.464050i \(-0.153604\pi\)
0.885809 + 0.464050i \(0.153604\pi\)
\(608\) 0 0
\(609\) 24.5254i 0.993820i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 25.9541i 1.04827i 0.851634 + 0.524137i \(0.175612\pi\)
−0.851634 + 0.524137i \(0.824388\pi\)
\(614\) 0 0
\(615\) 6.43296 0.259402
\(616\) 0 0
\(617\) − 45.9396i − 1.84946i −0.380626 0.924729i \(-0.624291\pi\)
0.380626 0.924729i \(-0.375709\pi\)
\(618\) 0 0
\(619\) − 6.73556i − 0.270725i −0.990796 0.135363i \(-0.956780\pi\)
0.990796 0.135363i \(-0.0432199\pi\)
\(620\) 0 0
\(621\) −3.19269 −0.128118
\(622\) 0 0
\(623\) 6.79895 0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) 0 0
\(627\) 0.841166 0.0335930
\(628\) 0 0
\(629\) − 12.2881i − 0.489960i
\(630\) 0 0
\(631\) − 45.0998i − 1.79539i −0.440614 0.897696i \(-0.645239\pi\)
0.440614 0.897696i \(-0.354761\pi\)
\(632\) 0 0
\(633\) −19.7385 −0.784537
\(634\) 0 0
\(635\) 2.42221i 0.0961223i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 17.7536i − 0.702322i
\(640\) 0 0
\(641\) −32.5821 −1.28692 −0.643458 0.765482i \(-0.722501\pi\)
−0.643458 + 0.765482i \(0.722501\pi\)
\(642\) 0 0
\(643\) 25.5754i 1.00860i 0.863530 + 0.504298i \(0.168249\pi\)
−0.863530 + 0.504298i \(0.831751\pi\)
\(644\) 0 0
\(645\) 0.164210i 0.00646578i
\(646\) 0 0
\(647\) −30.1715 −1.18616 −0.593082 0.805142i \(-0.702089\pi\)
−0.593082 + 0.805142i \(0.702089\pi\)
\(648\) 0 0
\(649\) −28.7952 −1.13031
\(650\) 0 0
\(651\) 35.1172 1.37635
\(652\) 0 0
\(653\) 36.9028 1.44412 0.722058 0.691832i \(-0.243196\pi\)
0.722058 + 0.691832i \(0.243196\pi\)
\(654\) 0 0
\(655\) − 1.62432i − 0.0634673i
\(656\) 0 0
\(657\) 13.8006i 0.538413i
\(658\) 0 0
\(659\) −23.6866 −0.922701 −0.461350 0.887218i \(-0.652635\pi\)
−0.461350 + 0.887218i \(0.652635\pi\)
\(660\) 0 0
\(661\) − 31.7590i − 1.23528i −0.786460 0.617641i \(-0.788089\pi\)
0.786460 0.617641i \(-0.211911\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 0.0513102i − 0.00198973i
\(666\) 0 0
\(667\) 6.91856 0.267888
\(668\) 0 0
\(669\) − 5.07606i − 0.196252i
\(670\) 0 0
\(671\) 14.7463i 0.569275i
\(672\) 0 0
\(673\) 7.50232 0.289193 0.144597 0.989491i \(-0.453812\pi\)
0.144597 + 0.989491i \(0.453812\pi\)
\(674\) 0 0
\(675\) −10.5550 −0.406261
\(676\) 0 0
\(677\) −35.0315 −1.34637 −0.673184 0.739475i \(-0.735074\pi\)
−0.673184 + 0.739475i \(0.735074\pi\)
\(678\) 0 0
\(679\) −18.9933 −0.728896
\(680\) 0 0
\(681\) − 15.6528i − 0.599816i
\(682\) 0 0
\(683\) 24.0834i 0.921524i 0.887524 + 0.460762i \(0.152424\pi\)
−0.887524 + 0.460762i \(0.847576\pi\)
\(684\) 0 0
\(685\) 1.53617 0.0586941
\(686\) 0 0
\(687\) − 54.3183i − 2.07237i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2.01447i 0.0766342i 0.999266 + 0.0383171i \(0.0121997\pi\)
−0.999266 + 0.0383171i \(0.987800\pi\)
\(692\) 0 0
\(693\) −20.5090 −0.779073
\(694\) 0 0
\(695\) − 3.63235i − 0.137783i
\(696\) 0 0
\(697\) − 25.0248i − 0.947880i
\(698\) 0 0
\(699\) 6.87800 0.260150
\(700\) 0 0
\(701\) 48.8189 1.84387 0.921933 0.387350i \(-0.126610\pi\)
0.921933 + 0.387350i \(0.126610\pi\)
\(702\) 0 0
\(703\) 0.501729 0.0189231
\(704\) 0 0
\(705\) 4.08277 0.153766
\(706\) 0 0
\(707\) 31.4752i 1.18375i
\(708\) 0 0
\(709\) − 20.8060i − 0.781385i −0.920521 0.390693i \(-0.872236\pi\)
0.920521 0.390693i \(-0.127764\pi\)
\(710\) 0 0
\(711\) −20.4286 −0.766134
\(712\) 0 0
\(713\) − 9.90648i − 0.371000i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 56.5042i − 2.11019i
\(718\) 0 0
\(719\) 21.4306 0.799225 0.399613 0.916684i \(-0.369145\pi\)
0.399613 + 0.916684i \(0.369145\pi\)
\(720\) 0 0
\(721\) 3.21073i 0.119574i
\(722\) 0 0
\(723\) − 45.5381i − 1.69358i
\(724\) 0 0
\(725\) 22.8726 0.849468
\(726\) 0 0
\(727\) 13.4862 0.500175 0.250088 0.968223i \(-0.419541\pi\)
0.250088 + 0.968223i \(0.419541\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) 0 0
\(731\) 0.638792 0.0236266
\(732\) 0 0
\(733\) 43.5424i 1.60828i 0.594443 + 0.804138i \(0.297373\pi\)
−0.594443 + 0.804138i \(0.702627\pi\)
\(734\) 0 0
\(735\) − 0.801938i − 0.0295799i
\(736\) 0 0
\(737\) −32.6165 −1.20145
\(738\) 0 0
\(739\) − 20.0543i − 0.737709i −0.929487 0.368855i \(-0.879750\pi\)
0.929487 0.368855i \(-0.120250\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 33.1685i − 1.21684i −0.793617 0.608418i \(-0.791805\pi\)
0.793617 0.608418i \(-0.208195\pi\)
\(744\) 0 0
\(745\) −1.07069 −0.0392270
\(746\) 0 0
\(747\) 3.29722i 0.120639i
\(748\) 0 0
\(749\) − 7.70410i − 0.281502i
\(750\) 0 0
\(751\) 39.2814 1.43340 0.716700 0.697382i \(-0.245652\pi\)
0.716700 + 0.697382i \(0.245652\pi\)
\(752\) 0 0
\(753\) −53.3008 −1.94239
\(754\) 0 0
\(755\) −0.974181 −0.0354541
\(756\) 0 0
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) 0 0
\(759\) 14.2567i 0.517484i
\(760\) 0 0
\(761\) − 21.8984i − 0.793818i −0.917858 0.396909i \(-0.870083\pi\)
0.917858 0.396909i \(-0.129917\pi\)
\(762\) 0 0
\(763\) 37.0073 1.33975
\(764\) 0 0
\(765\) 1.09246i 0.0394979i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) − 46.7096i − 1.68439i −0.539172 0.842196i \(-0.681263\pi\)
0.539172 0.842196i \(-0.318737\pi\)
\(770\) 0 0
\(771\) −31.9614 −1.15106
\(772\) 0 0
\(773\) 30.2416i 1.08771i 0.839178 + 0.543857i \(0.183037\pi\)
−0.839178 + 0.543857i \(0.816963\pi\)
\(774\) 0 0
\(775\) − 32.7506i − 1.17644i
\(776\) 0 0
\(777\) −30.1444 −1.08142
\(778\) 0 0
\(779\) 1.02177 0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 0 0
\(783\) −9.89679 −0.353682
\(784\) 0 0
\(785\) − 1.10023i − 0.0392688i
\(786\) 0 0
\(787\) − 28.7023i − 1.02313i −0.859246 0.511563i \(-0.829067\pi\)
0.859246 0.511563i \(-0.170933\pi\)
\(788\) 0 0
\(789\) 38.4131 1.36754
\(790\) 0 0
\(791\) 28.3980i 1.00972i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 5.76809i 0.204573i
\(796\) 0 0
\(797\) 18.5418 0.656785 0.328392 0.944541i \(-0.393493\pi\)
0.328392 + 0.944541i \(0.393493\pi\)
\(798\) 0 0
\(799\) − 15.8823i − 0.561876i
\(800\) 0 0
\(801\) − 5.91053i − 0.208838i
\(802\) 0 0
\(803\) −28.6058 −1.00948
\(804\) 0 0
\(805\) 0.869641 0.0306508
\(806\) 0 0
\(807\) −14.5308 −0.511508
\(808\) 0 0
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) 0 0
\(811\) − 10.0285i − 0.352147i −0.984377 0.176074i \(-0.943660\pi\)
0.984377 0.176074i \(-0.0563397\pi\)
\(812\) 0 0
\(813\) − 14.4886i − 0.508137i
\(814\) 0 0
\(815\) −3.99090 −0.139795
\(816\) 0 0
\(817\) 0.0260821i 0 0.000912498i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.1704i 0.913355i 0.889632 + 0.456677i \(0.150961\pi\)
−0.889632 + 0.456677i \(0.849039\pi\)
\(822\) 0 0
\(823\) 1.82238 0.0635242 0.0317621 0.999495i \(-0.489888\pi\)
0.0317621 + 0.999495i \(0.489888\pi\)
\(824\) 0 0
\(825\) 47.1323i 1.64094i
\(826\) 0 0
\(827\) 32.2941i 1.12298i 0.827485 + 0.561488i \(0.189771\pi\)
−0.827485 + 0.561488i \(0.810229\pi\)
\(828\) 0 0
\(829\) −15.1002 −0.524453 −0.262226 0.965006i \(-0.584457\pi\)
−0.262226 + 0.965006i \(0.584457\pi\)
\(830\) 0 0
\(831\) −30.2446 −1.04917
\(832\) 0 0
\(833\) −3.11960 −0.108088
\(834\) 0 0
\(835\) −3.98062 −0.137755
\(836\) 0 0
\(837\) 14.1709i 0.489818i
\(838\) 0 0
\(839\) 32.9965i 1.13917i 0.821933 + 0.569584i \(0.192896\pi\)
−0.821933 + 0.569584i \(0.807104\pi\)
\(840\) 0 0
\(841\) −7.55363 −0.260470
\(842\) 0 0
\(843\) 11.3177i 0.389801i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 16.5851i − 0.569870i
\(848\) 0 0
\(849\) 49.7211 1.70642
\(850\) 0 0
\(851\) 8.50365i 0.291501i
\(852\) 0 0
\(853\) − 37.7802i − 1.29357i −0.762673 0.646784i \(-0.776113\pi\)
0.762673 0.646784i \(-0.223887\pi\)
\(854\) 0 0
\(855\) −0.0446055 −0.00152547
\(856\) 0 0
\(857\) −27.3623 −0.934677 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(858\) 0 0
\(859\) 20.0629 0.684538 0.342269 0.939602i \(-0.388805\pi\)
0.342269 + 0.939602i \(0.388805\pi\)
\(860\) 0 0
\(861\) −61.3889 −2.09213
\(862\) 0 0
\(863\) − 6.14483i − 0.209173i −0.994516 0.104586i \(-0.966648\pi\)
0.994516 0.104586i \(-0.0333518\pi\)
\(864\) 0 0
\(865\) − 5.31900i − 0.180851i
\(866\) 0 0
\(867\) −27.7265 −0.941640
\(868\) 0 0
\(869\) − 42.3443i − 1.43643i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 16.5114i 0.558827i
\(874\) 0 0
\(875\) 5.78554 0.195587
\(876\) 0 0
\(877\) 13.5077i 0.456123i 0.973647 + 0.228061i \(0.0732386\pi\)
−0.973647 + 0.228061i \(0.926761\pi\)
\(878\) 0 0
\(879\) 33.5840i 1.13276i
\(880\) 0 0
\(881\) 5.23431 0.176348 0.0881741 0.996105i \(-0.471897\pi\)
0.0881741 + 0.996105i \(0.471897\pi\)
\(882\) 0 0
\(883\) −4.57301 −0.153894 −0.0769470 0.997035i \(-0.524517\pi\)
−0.0769470 + 0.997035i \(0.524517\pi\)
\(884\) 0 0
\(885\) 3.76271 0.126482
\(886\) 0 0
\(887\) 1.64071 0.0550897 0.0275448 0.999621i \(-0.491231\pi\)
0.0275448 + 0.999621i \(0.491231\pi\)
\(888\) 0 0
\(889\) − 23.1148i − 0.775246i
\(890\) 0 0
\(891\) − 46.4989i − 1.55777i
\(892\) 0 0
\(893\) 0.648481 0.0217006
\(894\) 0 0
\(895\) − 2.82371i − 0.0943861i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 30.7084i − 1.02418i
\(900\) 0 0
\(901\) 22.4383 0.747529
\(902\) 0 0
\(903\) − 1.56704i − 0.0521478i
\(904\) 0 0
\(905\) 5.17928i 0.172165i
\(906\) 0 0
\(907\) 8.10215 0.269027 0.134514 0.990912i \(-0.457053\pi\)
0.134514 + 0.990912i \(0.457053\pi\)
\(908\) 0 0
\(909\) 27.3623 0.907549
\(910\) 0 0
\(911\) 9.18119 0.304187 0.152093 0.988366i \(-0.451399\pi\)
0.152093 + 0.988366i \(0.451399\pi\)
\(912\) 0 0
\(913\) −6.83446 −0.226188
\(914\) 0 0
\(915\) − 1.92692i − 0.0637020i
\(916\) 0 0
\(917\) 15.5007i 0.511877i
\(918\) 0 0
\(919\) −27.5036 −0.907262 −0.453631 0.891190i \(-0.649872\pi\)
−0.453631 + 0.891190i \(0.649872\pi\)
\(920\) 0 0
\(921\) − 42.9831i − 1.41634i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 28.1129i 0.924346i
\(926\) 0 0
\(927\) 2.79118 0.0916745
\(928\) 0 0
\(929\) − 24.2131i − 0.794407i −0.917731 0.397203i \(-0.869981\pi\)
0.917731 0.397203i \(-0.130019\pi\)
\(930\) 0 0
\(931\) − 0.127375i − 0.00417454i
\(932\) 0 0
\(933\) −0.606268 −0.0198483
\(934\) 0 0
\(935\) −2.26444 −0.0740550
\(936\) 0 0
\(937\) 11.1830 0.365333 0.182666 0.983175i \(-0.441527\pi\)
0.182666 + 0.983175i \(0.441527\pi\)
\(938\) 0 0
\(939\) −52.5652 −1.71540
\(940\) 0 0
\(941\) 15.9638i 0.520404i 0.965554 + 0.260202i \(0.0837891\pi\)
−0.965554 + 0.260202i \(0.916211\pi\)
\(942\) 0 0
\(943\) 17.3177i 0.563941i
\(944\) 0 0
\(945\) −1.24400 −0.0404672
\(946\) 0 0
\(947\) − 6.51466i − 0.211698i −0.994382 0.105849i \(-0.966244\pi\)
0.994382 0.105849i \(-0.0337560\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 31.4470i 1.01974i
\(952\) 0 0
\(953\) −47.6469 −1.54344 −0.771718 0.635965i \(-0.780602\pi\)
−0.771718 + 0.635965i \(0.780602\pi\)
\(954\) 0 0
\(955\) − 3.56571i − 0.115384i
\(956\) 0 0
\(957\) 44.1933i 1.42857i
\(958\) 0 0
\(959\) −14.6595 −0.473380
\(960\) 0 0
\(961\) −12.9705 −0.418402
\(962\) 0 0
\(963\) −6.69740 −0.215821
\(964\) 0 0
\(965\) −3.35391 −0.107966
\(966\) 0 0
\(967\) − 43.8122i − 1.40891i −0.709751 0.704453i \(-0.751192\pi\)
0.709751 0.704453i \(-0.248808\pi\)
\(968\) 0 0
\(969\) 0.427583i 0.0137360i
\(970\) 0 0
\(971\) −4.29483 −0.137828 −0.0689139 0.997623i \(-0.521953\pi\)
−0.0689139 + 0.997623i \(0.521953\pi\)
\(972\) 0 0
\(973\) 34.6631i 1.11125i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 26.8019i − 0.857470i −0.903430 0.428735i \(-0.858959\pi\)
0.903430 0.428735i \(-0.141041\pi\)
\(978\) 0 0
\(979\) 12.2513 0.391553
\(980\) 0 0
\(981\) − 32.1715i − 1.02716i
\(982\) 0 0
\(983\) − 27.2495i − 0.869124i −0.900642 0.434562i \(-0.856903\pi\)
0.900642 0.434562i \(-0.143097\pi\)
\(984\) 0 0
\(985\) 0.138391 0.00440951
\(986\) 0 0
\(987\) −38.9614 −1.24015
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) 0 0
\(991\) −24.3889 −0.774740 −0.387370 0.921924i \(-0.626616\pi\)
−0.387370 + 0.921924i \(0.626616\pi\)
\(992\) 0 0
\(993\) 40.0441i 1.27076i
\(994\) 0 0
\(995\) − 2.83818i − 0.0899764i
\(996\) 0 0
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) 0 0
\(999\) − 12.1642i − 0.384859i
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 2704.2.f.o.337.5 6
4.3 odd 2 169.2.b.b.168.2 6
12.11 even 2 1521.2.b.l.1351.5 6
13.5 odd 4 2704.2.a.ba.1.3 3
13.8 odd 4 2704.2.a.z.1.3 3
13.12 even 2 inner 2704.2.f.o.337.6 6
52.3 odd 6 169.2.e.b.147.5 12
52.7 even 12 169.2.c.c.146.1 6
52.11 even 12 169.2.c.c.22.1 6
52.15 even 12 169.2.c.b.22.3 6
52.19 even 12 169.2.c.b.146.3 6
52.23 odd 6 169.2.e.b.147.2 12
52.31 even 4 169.2.a.c.1.1 yes 3
52.35 odd 6 169.2.e.b.23.2 12
52.43 odd 6 169.2.e.b.23.5 12
52.47 even 4 169.2.a.b.1.3 3
52.51 odd 2 169.2.b.b.168.5 6
156.47 odd 4 1521.2.a.r.1.1 3
156.83 odd 4 1521.2.a.o.1.3 3
156.155 even 2 1521.2.b.l.1351.2 6
260.99 even 4 4225.2.a.bg.1.1 3
260.239 even 4 4225.2.a.bb.1.3 3
364.83 odd 4 8281.2.a.bj.1.1 3
364.307 odd 4 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 52.47 even 4
169.2.a.c.1.1 yes 3 52.31 even 4
169.2.b.b.168.2 6 4.3 odd 2
169.2.b.b.168.5 6 52.51 odd 2
169.2.c.b.22.3 6 52.15 even 12
169.2.c.b.146.3 6 52.19 even 12
169.2.c.c.22.1 6 52.11 even 12
169.2.c.c.146.1 6 52.7 even 12
169.2.e.b.23.2 12 52.35 odd 6
169.2.e.b.23.5 12 52.43 odd 6
169.2.e.b.147.2 12 52.23 odd 6
169.2.e.b.147.5 12 52.3 odd 6
1521.2.a.o.1.3 3 156.83 odd 4
1521.2.a.r.1.1 3 156.47 odd 4
1521.2.b.l.1351.2 6 156.155 even 2
1521.2.b.l.1351.5 6 12.11 even 2
2704.2.a.z.1.3 3 13.8 odd 4
2704.2.a.ba.1.3 3 13.5 odd 4
2704.2.f.o.337.5 6 1.1 even 1 trivial
2704.2.f.o.337.6 6 13.12 even 2 inner
4225.2.a.bb.1.3 3 260.239 even 4
4225.2.a.bg.1.1 3 260.99 even 4
8281.2.a.bf.1.3 3 364.307 odd 4
8281.2.a.bj.1.1 3 364.83 odd 4