Properties

Label 2368.2.g.h.961.2
Level $2368$
Weight $2$
Character 2368.961
Analytic conductor $18.909$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2368,2,Mod(961,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, 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("2368.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.g (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: \(18.9085751986\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 961.2
Root \(2.79129i\) of defining polynomial
Character \(\chi\) \(=\) 2368.961
Dual form 2368.2.g.h.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.79129 q^{3} +3.79129i q^{5} +2.00000 q^{7} +4.79129 q^{9} +O(q^{10})\) \(q-2.79129 q^{3} +3.79129i q^{5} +2.00000 q^{7} +4.79129 q^{9} +3.79129 q^{11} -0.791288i q^{13} -10.5826i q^{15} -1.58258i q^{17} -7.58258i q^{19} -5.58258 q^{21} -0.791288i q^{23} -9.37386 q^{25} -5.00000 q^{27} +0.791288i q^{29} -5.37386i q^{31} -10.5826 q^{33} +7.58258i q^{35} +(-4.00000 - 4.58258i) q^{37} +2.20871i q^{39} +5.20871 q^{41} -6.00000i q^{43} +18.1652i q^{45} -1.58258 q^{47} -3.00000 q^{49} +4.41742i q^{51} -7.58258 q^{53} +14.3739i q^{55} +21.1652i q^{57} -7.58258i q^{59} -8.20871i q^{61} +9.58258 q^{63} +3.00000 q^{65} -7.37386 q^{67} +2.20871i q^{69} -9.16515 q^{71} +9.37386 q^{73} +26.1652 q^{75} +7.58258 q^{77} -12.7913i q^{79} -0.417424 q^{81} -3.16515 q^{83} +6.00000 q^{85} -2.20871i q^{87} +6.00000i q^{89} -1.58258i q^{91} +15.0000i q^{93} +28.7477 q^{95} -4.41742i q^{97} +18.1652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 4 q^{21} - 10 q^{25} - 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 20 q^{63} + 12 q^{65} - 2 q^{67} + 10 q^{73} + 68 q^{75} + 12 q^{77} - 20 q^{81} + 24 q^{83} + 24 q^{85} + 60 q^{95} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(1407\) \(1925\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −2.79129 −1.61155 −0.805775 0.592221i \(-0.798251\pi\)
−0.805775 + 0.592221i \(0.798251\pi\)
\(4\) 0 0
\(5\) 3.79129i 1.69552i 0.530384 + 0.847758i \(0.322048\pi\)
−0.530384 + 0.847758i \(0.677952\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 3.79129 1.14312 0.571558 0.820562i \(-0.306339\pi\)
0.571558 + 0.820562i \(0.306339\pi\)
\(12\) 0 0
\(13\) 0.791288i 0.219464i −0.993961 0.109732i \(-0.965001\pi\)
0.993961 0.109732i \(-0.0349992\pi\)
\(14\) 0 0
\(15\) 10.5826i 2.73241i
\(16\) 0 0
\(17\) 1.58258i 0.383831i −0.981411 0.191915i \(-0.938530\pi\)
0.981411 0.191915i \(-0.0614700\pi\)
\(18\) 0 0
\(19\) 7.58258i 1.73956i −0.493438 0.869781i \(-0.664260\pi\)
0.493438 0.869781i \(-0.335740\pi\)
\(20\) 0 0
\(21\) −5.58258 −1.21822
\(22\) 0 0
\(23\) 0.791288i 0.164995i −0.996591 0.0824975i \(-0.973710\pi\)
0.996591 0.0824975i \(-0.0262896\pi\)
\(24\) 0 0
\(25\) −9.37386 −1.87477
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 0.791288i 0.146938i 0.997297 + 0.0734692i \(0.0234071\pi\)
−0.997297 + 0.0734692i \(0.976593\pi\)
\(30\) 0 0
\(31\) 5.37386i 0.965174i −0.875848 0.482587i \(-0.839697\pi\)
0.875848 0.482587i \(-0.160303\pi\)
\(32\) 0 0
\(33\) −10.5826 −1.84219
\(34\) 0 0
\(35\) 7.58258i 1.28169i
\(36\) 0 0
\(37\) −4.00000 4.58258i −0.657596 0.753371i
\(38\) 0 0
\(39\) 2.20871i 0.353677i
\(40\) 0 0
\(41\) 5.20871 0.813464 0.406732 0.913547i \(-0.366668\pi\)
0.406732 + 0.913547i \(0.366668\pi\)
\(42\) 0 0
\(43\) 6.00000i 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 18.1652i 2.70790i
\(46\) 0 0
\(47\) −1.58258 −0.230842 −0.115421 0.993317i \(-0.536822\pi\)
−0.115421 + 0.993317i \(0.536822\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 4.41742i 0.618563i
\(52\) 0 0
\(53\) −7.58258 −1.04155 −0.520773 0.853695i \(-0.674356\pi\)
−0.520773 + 0.853695i \(0.674356\pi\)
\(54\) 0 0
\(55\) 14.3739i 1.93817i
\(56\) 0 0
\(57\) 21.1652i 2.80339i
\(58\) 0 0
\(59\) 7.58258i 0.987167i −0.869698 0.493584i \(-0.835687\pi\)
0.869698 0.493584i \(-0.164313\pi\)
\(60\) 0 0
\(61\) 8.20871i 1.05102i −0.850788 0.525509i \(-0.823875\pi\)
0.850788 0.525509i \(-0.176125\pi\)
\(62\) 0 0
\(63\) 9.58258 1.20729
\(64\) 0 0
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) −7.37386 −0.900861 −0.450430 0.892812i \(-0.648729\pi\)
−0.450430 + 0.892812i \(0.648729\pi\)
\(68\) 0 0
\(69\) 2.20871i 0.265898i
\(70\) 0 0
\(71\) −9.16515 −1.08770 −0.543852 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(72\) 0 0
\(73\) 9.37386 1.09713 0.548564 0.836109i \(-0.315175\pi\)
0.548564 + 0.836109i \(0.315175\pi\)
\(74\) 0 0
\(75\) 26.1652 3.02129
\(76\) 0 0
\(77\) 7.58258 0.864115
\(78\) 0 0
\(79\) 12.7913i 1.43913i −0.694424 0.719566i \(-0.744341\pi\)
0.694424 0.719566i \(-0.255659\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) −3.16515 −0.347421 −0.173710 0.984797i \(-0.555576\pi\)
−0.173710 + 0.984797i \(0.555576\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 2.20871i 0.236799i
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 1.58258i 0.165899i
\(92\) 0 0
\(93\) 15.0000i 1.55543i
\(94\) 0 0
\(95\) 28.7477 2.94945
\(96\) 0 0
\(97\) 4.41742i 0.448521i −0.974529 0.224261i \(-0.928003\pi\)
0.974529 0.224261i \(-0.0719967\pi\)
\(98\) 0 0
\(99\) 18.1652 1.82567
\(100\) 0 0
\(101\) 1.58258 0.157472 0.0787361 0.996895i \(-0.474912\pi\)
0.0787361 + 0.996895i \(0.474912\pi\)
\(102\) 0 0
\(103\) 2.20871i 0.217631i −0.994062 0.108815i \(-0.965294\pi\)
0.994062 0.108815i \(-0.0347058\pi\)
\(104\) 0 0
\(105\) 21.1652i 2.06551i
\(106\) 0 0
\(107\) 8.37386 0.809532 0.404766 0.914420i \(-0.367353\pi\)
0.404766 + 0.914420i \(0.367353\pi\)
\(108\) 0 0
\(109\) 6.00000i 0.574696i −0.957826 0.287348i \(-0.907226\pi\)
0.957826 0.287348i \(-0.0927736\pi\)
\(110\) 0 0
\(111\) 11.1652 + 12.7913i 1.05975 + 1.21410i
\(112\) 0 0
\(113\) 19.5826i 1.84217i −0.389357 0.921087i \(-0.627303\pi\)
0.389357 0.921087i \(-0.372697\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 3.79129i 0.350505i
\(118\) 0 0
\(119\) 3.16515i 0.290149i
\(120\) 0 0
\(121\) 3.37386 0.306715
\(122\) 0 0
\(123\) −14.5390 −1.31094
\(124\) 0 0
\(125\) 16.5826i 1.48319i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 16.7477i 1.47456i
\(130\) 0 0
\(131\) 10.7477i 0.939033i −0.882924 0.469517i \(-0.844428\pi\)
0.882924 0.469517i \(-0.155572\pi\)
\(132\) 0 0
\(133\) 15.1652i 1.31499i
\(134\) 0 0
\(135\) 18.9564i 1.63151i
\(136\) 0 0
\(137\) −17.3739 −1.48435 −0.742175 0.670207i \(-0.766206\pi\)
−0.742175 + 0.670207i \(0.766206\pi\)
\(138\) 0 0
\(139\) 0.373864 0.0317107 0.0158553 0.999874i \(-0.494953\pi\)
0.0158553 + 0.999874i \(0.494953\pi\)
\(140\) 0 0
\(141\) 4.41742 0.372014
\(142\) 0 0
\(143\) 3.00000i 0.250873i
\(144\) 0 0
\(145\) −3.00000 −0.249136
\(146\) 0 0
\(147\) 8.37386 0.690665
\(148\) 0 0
\(149\) 13.5826 1.11273 0.556364 0.830939i \(-0.312196\pi\)
0.556364 + 0.830939i \(0.312196\pi\)
\(150\) 0 0
\(151\) −12.7477 −1.03740 −0.518698 0.854958i \(-0.673583\pi\)
−0.518698 + 0.854958i \(0.673583\pi\)
\(152\) 0 0
\(153\) 7.58258i 0.613015i
\(154\) 0 0
\(155\) 20.3739 1.63647
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 21.1652 1.67851
\(160\) 0 0
\(161\) 1.58258i 0.124724i
\(162\) 0 0
\(163\) 10.4174i 0.815956i 0.912992 + 0.407978i \(0.133766\pi\)
−0.912992 + 0.407978i \(0.866234\pi\)
\(164\) 0 0
\(165\) 40.1216i 3.12346i
\(166\) 0 0
\(167\) 0.956439i 0.0740115i −0.999315 0.0370057i \(-0.988218\pi\)
0.999315 0.0370057i \(-0.0117820\pi\)
\(168\) 0 0
\(169\) 12.3739 0.951836
\(170\) 0 0
\(171\) 36.3303i 2.77825i
\(172\) 0 0
\(173\) 3.16515 0.240642 0.120321 0.992735i \(-0.461608\pi\)
0.120321 + 0.992735i \(0.461608\pi\)
\(174\) 0 0
\(175\) −18.7477 −1.41719
\(176\) 0 0
\(177\) 21.1652i 1.59087i
\(178\) 0 0
\(179\) 7.58258i 0.566748i −0.959009 0.283374i \(-0.908546\pi\)
0.959009 0.283374i \(-0.0914538\pi\)
\(180\) 0 0
\(181\) 18.7477 1.39351 0.696754 0.717310i \(-0.254627\pi\)
0.696754 + 0.717310i \(0.254627\pi\)
\(182\) 0 0
\(183\) 22.9129i 1.69377i
\(184\) 0 0
\(185\) 17.3739 15.1652i 1.27735 1.11496i
\(186\) 0 0
\(187\) 6.00000i 0.438763i
\(188\) 0 0
\(189\) −10.0000 −0.727393
\(190\) 0 0
\(191\) 5.37386i 0.388839i 0.980918 + 0.194420i \(0.0622823\pi\)
−0.980918 + 0.194420i \(0.937718\pi\)
\(192\) 0 0
\(193\) 18.3303i 1.31944i 0.751510 + 0.659722i \(0.229326\pi\)
−0.751510 + 0.659722i \(0.770674\pi\)
\(194\) 0 0
\(195\) −8.37386 −0.599665
\(196\) 0 0
\(197\) −25.9129 −1.84622 −0.923108 0.384541i \(-0.874360\pi\)
−0.923108 + 0.384541i \(0.874360\pi\)
\(198\) 0 0
\(199\) 3.16515i 0.224372i −0.993687 0.112186i \(-0.964215\pi\)
0.993687 0.112186i \(-0.0357852\pi\)
\(200\) 0 0
\(201\) 20.5826 1.45178
\(202\) 0 0
\(203\) 1.58258i 0.111075i
\(204\) 0 0
\(205\) 19.7477i 1.37924i
\(206\) 0 0
\(207\) 3.79129i 0.263513i
\(208\) 0 0
\(209\) 28.7477i 1.98852i
\(210\) 0 0
\(211\) −3.37386 −0.232266 −0.116133 0.993234i \(-0.537050\pi\)
−0.116133 + 0.993234i \(0.537050\pi\)
\(212\) 0 0
\(213\) 25.5826 1.75289
\(214\) 0 0
\(215\) 22.7477 1.55138
\(216\) 0 0
\(217\) 10.7477i 0.729603i
\(218\) 0 0
\(219\) −26.1652 −1.76808
\(220\) 0 0
\(221\) −1.25227 −0.0842370
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 0 0
\(225\) −44.9129 −2.99419
\(226\) 0 0
\(227\) 19.9129i 1.32166i 0.750534 + 0.660832i \(0.229796\pi\)
−0.750534 + 0.660832i \(0.770204\pi\)
\(228\) 0 0
\(229\) 20.7477 1.37105 0.685524 0.728050i \(-0.259573\pi\)
0.685524 + 0.728050i \(0.259573\pi\)
\(230\) 0 0
\(231\) −21.1652 −1.39256
\(232\) 0 0
\(233\) 25.1216 1.64577 0.822885 0.568208i \(-0.192363\pi\)
0.822885 + 0.568208i \(0.192363\pi\)
\(234\) 0 0
\(235\) 6.00000i 0.391397i
\(236\) 0 0
\(237\) 35.7042i 2.31923i
\(238\) 0 0
\(239\) 24.9564i 1.61430i −0.590348 0.807149i \(-0.701009\pi\)
0.590348 0.807149i \(-0.298991\pi\)
\(240\) 0 0
\(241\) 13.5826i 0.874931i 0.899235 + 0.437465i \(0.144124\pi\)
−0.899235 + 0.437465i \(0.855876\pi\)
\(242\) 0 0
\(243\) 16.1652 1.03699
\(244\) 0 0
\(245\) 11.3739i 0.726649i
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 8.83485 0.559886
\(250\) 0 0
\(251\) 13.5826i 0.857325i 0.903465 + 0.428662i \(0.141015\pi\)
−0.903465 + 0.428662i \(0.858985\pi\)
\(252\) 0 0
\(253\) 3.00000i 0.188608i
\(254\) 0 0
\(255\) −16.7477 −1.04878
\(256\) 0 0
\(257\) 22.7477i 1.41896i 0.704723 + 0.709482i \(0.251071\pi\)
−0.704723 + 0.709482i \(0.748929\pi\)
\(258\) 0 0
\(259\) −8.00000 9.16515i −0.497096 0.569495i
\(260\) 0 0
\(261\) 3.79129i 0.234675i
\(262\) 0 0
\(263\) 8.83485 0.544780 0.272390 0.962187i \(-0.412186\pi\)
0.272390 + 0.962187i \(0.412186\pi\)
\(264\) 0 0
\(265\) 28.7477i 1.76596i
\(266\) 0 0
\(267\) 16.7477i 1.02494i
\(268\) 0 0
\(269\) 10.7477 0.655300 0.327650 0.944799i \(-0.393743\pi\)
0.327650 + 0.944799i \(0.393743\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 0 0
\(273\) 4.41742i 0.267355i
\(274\) 0 0
\(275\) −35.5390 −2.14308
\(276\) 0 0
\(277\) 23.3739i 1.40440i 0.711980 + 0.702200i \(0.247799\pi\)
−0.711980 + 0.702200i \(0.752201\pi\)
\(278\) 0 0
\(279\) 25.7477i 1.54148i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 24.0000i 1.42665i 0.700832 + 0.713326i \(0.252812\pi\)
−0.700832 + 0.713326i \(0.747188\pi\)
\(284\) 0 0
\(285\) −80.2432 −4.75320
\(286\) 0 0
\(287\) 10.4174 0.614921
\(288\) 0 0
\(289\) 14.4955 0.852674
\(290\) 0 0
\(291\) 12.3303i 0.722815i
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 28.7477 1.67376
\(296\) 0 0
\(297\) −18.9564 −1.09996
\(298\) 0 0
\(299\) −0.626136 −0.0362104
\(300\) 0 0
\(301\) 12.0000i 0.691669i
\(302\) 0 0
\(303\) −4.41742 −0.253774
\(304\) 0 0
\(305\) 31.1216 1.78202
\(306\) 0 0
\(307\) −12.3739 −0.706214 −0.353107 0.935583i \(-0.614875\pi\)
−0.353107 + 0.935583i \(0.614875\pi\)
\(308\) 0 0
\(309\) 6.16515i 0.350723i
\(310\) 0 0
\(311\) 9.62614i 0.545848i 0.962036 + 0.272924i \(0.0879908\pi\)
−0.962036 + 0.272924i \(0.912009\pi\)
\(312\) 0 0
\(313\) 33.1652i 1.87461i −0.348517 0.937303i \(-0.613315\pi\)
0.348517 0.937303i \(-0.386685\pi\)
\(314\) 0 0
\(315\) 36.3303i 2.04698i
\(316\) 0 0
\(317\) 1.25227 0.0703347 0.0351673 0.999381i \(-0.488804\pi\)
0.0351673 + 0.999381i \(0.488804\pi\)
\(318\) 0 0
\(319\) 3.00000i 0.167968i
\(320\) 0 0
\(321\) −23.3739 −1.30460
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 7.41742i 0.411445i
\(326\) 0 0
\(327\) 16.7477i 0.926151i
\(328\) 0 0
\(329\) −3.16515 −0.174500
\(330\) 0 0
\(331\) 16.4174i 0.902383i −0.892427 0.451192i \(-0.850999\pi\)
0.892427 0.451192i \(-0.149001\pi\)
\(332\) 0 0
\(333\) −19.1652 21.9564i −1.05024 1.20321i
\(334\) 0 0
\(335\) 27.9564i 1.52742i
\(336\) 0 0
\(337\) −8.12159 −0.442411 −0.221206 0.975227i \(-0.570999\pi\)
−0.221206 + 0.975227i \(0.570999\pi\)
\(338\) 0 0
\(339\) 54.6606i 2.96876i
\(340\) 0 0
\(341\) 20.3739i 1.10331i
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −8.37386 −0.450834
\(346\) 0 0
\(347\) 1.58258i 0.0849571i −0.999097 0.0424786i \(-0.986475\pi\)
0.999097 0.0424786i \(-0.0135254\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 3.95644i 0.211179i
\(352\) 0 0
\(353\) 7.58258i 0.403580i 0.979429 + 0.201790i \(0.0646758\pi\)
−0.979429 + 0.201790i \(0.935324\pi\)
\(354\) 0 0
\(355\) 34.7477i 1.84422i
\(356\) 0 0
\(357\) 8.83485i 0.467590i
\(358\) 0 0
\(359\) 27.1652 1.43372 0.716861 0.697216i \(-0.245578\pi\)
0.716861 + 0.697216i \(0.245578\pi\)
\(360\) 0 0
\(361\) −38.4955 −2.02608
\(362\) 0 0
\(363\) −9.41742 −0.494287
\(364\) 0 0
\(365\) 35.5390i 1.86020i
\(366\) 0 0
\(367\) 32.7477 1.70942 0.854709 0.519108i \(-0.173736\pi\)
0.854709 + 0.519108i \(0.173736\pi\)
\(368\) 0 0
\(369\) 24.9564 1.29918
\(370\) 0 0
\(371\) −15.1652 −0.787335
\(372\) 0 0
\(373\) −14.7477 −0.763608 −0.381804 0.924243i \(-0.624697\pi\)
−0.381804 + 0.924243i \(0.624697\pi\)
\(374\) 0 0
\(375\) 46.2867i 2.39024i
\(376\) 0 0
\(377\) 0.626136 0.0322477
\(378\) 0 0
\(379\) −21.1216 −1.08494 −0.542472 0.840074i \(-0.682511\pi\)
−0.542472 + 0.840074i \(0.682511\pi\)
\(380\) 0 0
\(381\) 22.3303 1.14402
\(382\) 0 0
\(383\) 18.3303i 0.936635i −0.883560 0.468317i \(-0.844860\pi\)
0.883560 0.468317i \(-0.155140\pi\)
\(384\) 0 0
\(385\) 28.7477i 1.46512i
\(386\) 0 0
\(387\) 28.7477i 1.46133i
\(388\) 0 0
\(389\) 35.8693i 1.81865i 0.416090 + 0.909323i \(0.363400\pi\)
−0.416090 + 0.909323i \(0.636600\pi\)
\(390\) 0 0
\(391\) −1.25227 −0.0633302
\(392\) 0 0
\(393\) 30.0000i 1.51330i
\(394\) 0 0
\(395\) 48.4955 2.44007
\(396\) 0 0
\(397\) 12.7477 0.639790 0.319895 0.947453i \(-0.396352\pi\)
0.319895 + 0.947453i \(0.396352\pi\)
\(398\) 0 0
\(399\) 42.3303i 2.11917i
\(400\) 0 0
\(401\) 10.7477i 0.536716i 0.963319 + 0.268358i \(0.0864810\pi\)
−0.963319 + 0.268358i \(0.913519\pi\)
\(402\) 0 0
\(403\) −4.25227 −0.211821
\(404\) 0 0
\(405\) 1.58258i 0.0786388i
\(406\) 0 0
\(407\) −15.1652 17.3739i −0.751709 0.861190i
\(408\) 0 0
\(409\) 8.83485i 0.436855i 0.975853 + 0.218428i \(0.0700927\pi\)
−0.975853 + 0.218428i \(0.929907\pi\)
\(410\) 0 0
\(411\) 48.4955 2.39210
\(412\) 0 0
\(413\) 15.1652i 0.746228i
\(414\) 0 0
\(415\) 12.0000i 0.589057i
\(416\) 0 0
\(417\) −1.04356 −0.0511034
\(418\) 0 0
\(419\) 6.79129 0.331776 0.165888 0.986145i \(-0.446951\pi\)
0.165888 + 0.986145i \(0.446951\pi\)
\(420\) 0 0
\(421\) 3.95644i 0.192825i −0.995341 0.0964125i \(-0.969263\pi\)
0.995341 0.0964125i \(-0.0307368\pi\)
\(422\) 0 0
\(423\) −7.58258 −0.368677
\(424\) 0 0
\(425\) 14.8348i 0.719596i
\(426\) 0 0
\(427\) 16.4174i 0.794495i
\(428\) 0 0
\(429\) 8.37386i 0.404294i
\(430\) 0 0
\(431\) 27.1652i 1.30850i 0.756279 + 0.654250i \(0.227015\pi\)
−0.756279 + 0.654250i \(0.772985\pi\)
\(432\) 0 0
\(433\) 24.3739 1.17133 0.585667 0.810552i \(-0.300833\pi\)
0.585667 + 0.810552i \(0.300833\pi\)
\(434\) 0 0
\(435\) 8.37386 0.401496
\(436\) 0 0
\(437\) −6.00000 −0.287019
\(438\) 0 0
\(439\) 26.3739i 1.25876i −0.777099 0.629378i \(-0.783310\pi\)
0.777099 0.629378i \(-0.216690\pi\)
\(440\) 0 0
\(441\) −14.3739 −0.684470
\(442\) 0 0
\(443\) 5.04356 0.239627 0.119813 0.992796i \(-0.461770\pi\)
0.119813 + 0.992796i \(0.461770\pi\)
\(444\) 0 0
\(445\) −22.7477 −1.07835
\(446\) 0 0
\(447\) −37.9129 −1.79322
\(448\) 0 0
\(449\) 21.1652i 0.998845i −0.866358 0.499423i \(-0.833546\pi\)
0.866358 0.499423i \(-0.166454\pi\)
\(450\) 0 0
\(451\) 19.7477 0.929884
\(452\) 0 0
\(453\) 35.5826 1.67182
\(454\) 0 0
\(455\) 6.00000 0.281284
\(456\) 0 0
\(457\) 33.4955i 1.56685i 0.621486 + 0.783426i \(0.286529\pi\)
−0.621486 + 0.783426i \(0.713471\pi\)
\(458\) 0 0
\(459\) 7.91288i 0.369342i
\(460\) 0 0
\(461\) 24.3303i 1.13318i 0.824002 + 0.566588i \(0.191737\pi\)
−0.824002 + 0.566588i \(0.808263\pi\)
\(462\) 0 0
\(463\) 20.7042i 0.962204i 0.876665 + 0.481102i \(0.159763\pi\)
−0.876665 + 0.481102i \(0.840237\pi\)
\(464\) 0 0
\(465\) −56.8693 −2.63725
\(466\) 0 0
\(467\) 25.5826i 1.18382i 0.806004 + 0.591910i \(0.201626\pi\)
−0.806004 + 0.591910i \(0.798374\pi\)
\(468\) 0 0
\(469\) −14.7477 −0.680987
\(470\) 0 0
\(471\) −5.58258 −0.257232
\(472\) 0 0
\(473\) 22.7477i 1.04594i
\(474\) 0 0
\(475\) 71.0780i 3.26128i
\(476\) 0 0
\(477\) −36.3303 −1.66345
\(478\) 0 0
\(479\) 0.791288i 0.0361549i 0.999837 + 0.0180774i \(0.00575454\pi\)
−0.999837 + 0.0180774i \(0.994245\pi\)
\(480\) 0 0
\(481\) −3.62614 + 3.16515i −0.165338 + 0.144318i
\(482\) 0 0
\(483\) 4.41742i 0.201000i
\(484\) 0 0
\(485\) 16.7477 0.760475
\(486\) 0 0
\(487\) 36.6606i 1.66125i 0.556832 + 0.830625i \(0.312017\pi\)
−0.556832 + 0.830625i \(0.687983\pi\)
\(488\) 0 0
\(489\) 29.0780i 1.31495i
\(490\) 0 0
\(491\) −8.37386 −0.377907 −0.188954 0.981986i \(-0.560510\pi\)
−0.188954 + 0.981986i \(0.560510\pi\)
\(492\) 0 0
\(493\) 1.25227 0.0563995
\(494\) 0 0
\(495\) 68.8693i 3.09545i
\(496\) 0 0
\(497\) −18.3303 −0.822226
\(498\) 0 0
\(499\) 16.7477i 0.749731i 0.927079 + 0.374866i \(0.122311\pi\)
−0.927079 + 0.374866i \(0.877689\pi\)
\(500\) 0 0
\(501\) 2.66970i 0.119273i
\(502\) 0 0
\(503\) 29.3739i 1.30972i −0.755752 0.654858i \(-0.772728\pi\)
0.755752 0.654858i \(-0.227272\pi\)
\(504\) 0 0
\(505\) 6.00000i 0.266996i
\(506\) 0 0
\(507\) −34.5390 −1.53393
\(508\) 0 0
\(509\) −13.5826 −0.602037 −0.301019 0.953618i \(-0.597327\pi\)
−0.301019 + 0.953618i \(0.597327\pi\)
\(510\) 0 0
\(511\) 18.7477 0.829351
\(512\) 0 0
\(513\) 37.9129i 1.67389i
\(514\) 0 0
\(515\) 8.37386 0.368997
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) −8.83485 −0.387807
\(520\) 0 0
\(521\) 9.16515 0.401533 0.200766 0.979639i \(-0.435657\pi\)
0.200766 + 0.979639i \(0.435657\pi\)
\(522\) 0 0
\(523\) 3.16515i 0.138402i −0.997603 0.0692012i \(-0.977955\pi\)
0.997603 0.0692012i \(-0.0220450\pi\)
\(524\) 0 0
\(525\) 52.3303 2.28388
\(526\) 0 0
\(527\) −8.50455 −0.370464
\(528\) 0 0
\(529\) 22.3739 0.972777
\(530\) 0 0
\(531\) 36.3303i 1.57660i
\(532\) 0 0
\(533\) 4.12159i 0.178526i
\(534\) 0 0
\(535\) 31.7477i 1.37257i
\(536\) 0 0
\(537\) 21.1652i 0.913344i
\(538\) 0 0
\(539\) −11.3739 −0.489907
\(540\) 0 0
\(541\) 8.20871i 0.352920i 0.984308 + 0.176460i \(0.0564646\pi\)
−0.984308 + 0.176460i \(0.943535\pi\)
\(542\) 0 0
\(543\) −52.3303 −2.24571
\(544\) 0 0
\(545\) 22.7477 0.974406
\(546\) 0 0
\(547\) 19.9129i 0.851413i 0.904861 + 0.425707i \(0.139974\pi\)
−0.904861 + 0.425707i \(0.860026\pi\)
\(548\) 0 0
\(549\) 39.3303i 1.67858i
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 25.5826i 1.08788i
\(554\) 0 0
\(555\) −48.4955 + 42.3303i −2.05852 + 1.79682i
\(556\) 0 0
\(557\) 41.7042i 1.76706i −0.468372 0.883531i \(-0.655159\pi\)
0.468372 0.883531i \(-0.344841\pi\)
\(558\) 0 0
\(559\) −4.74773 −0.200807
\(560\) 0 0
\(561\) 16.7477i 0.707090i
\(562\) 0 0
\(563\) 16.7477i 0.705833i −0.935655 0.352916i \(-0.885190\pi\)
0.935655 0.352916i \(-0.114810\pi\)
\(564\) 0 0
\(565\) 74.2432 3.12343
\(566\) 0 0
\(567\) −0.834849 −0.0350603
\(568\) 0 0
\(569\) 26.8348i 1.12498i −0.826806 0.562488i \(-0.809844\pi\)
0.826806 0.562488i \(-0.190156\pi\)
\(570\) 0 0
\(571\) −28.3739 −1.18741 −0.593705 0.804683i \(-0.702335\pi\)
−0.593705 + 0.804683i \(0.702335\pi\)
\(572\) 0 0
\(573\) 15.0000i 0.626634i
\(574\) 0 0
\(575\) 7.41742i 0.309328i
\(576\) 0 0
\(577\) 4.41742i 0.183900i −0.995764 0.0919499i \(-0.970690\pi\)
0.995764 0.0919499i \(-0.0293100\pi\)
\(578\) 0 0
\(579\) 51.1652i 2.12635i
\(580\) 0 0
\(581\) −6.33030 −0.262625
\(582\) 0 0
\(583\) −28.7477 −1.19061
\(584\) 0 0
\(585\) 14.3739 0.594286
\(586\) 0 0
\(587\) 25.9129i 1.06954i −0.844998 0.534769i \(-0.820398\pi\)
0.844998 0.534769i \(-0.179602\pi\)
\(588\) 0 0
\(589\) −40.7477 −1.67898
\(590\) 0 0
\(591\) 72.3303 2.97527
\(592\) 0 0
\(593\) −14.2087 −0.583482 −0.291741 0.956497i \(-0.594235\pi\)
−0.291741 + 0.956497i \(0.594235\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) 8.83485i 0.361586i
\(598\) 0 0
\(599\) −27.1652 −1.10994 −0.554969 0.831871i \(-0.687270\pi\)
−0.554969 + 0.831871i \(0.687270\pi\)
\(600\) 0 0
\(601\) 8.12159 0.331287 0.165643 0.986186i \(-0.447030\pi\)
0.165643 + 0.986186i \(0.447030\pi\)
\(602\) 0 0
\(603\) −35.3303 −1.43876
\(604\) 0 0
\(605\) 12.7913i 0.520040i
\(606\) 0 0
\(607\) 5.53901i 0.224822i 0.993662 + 0.112411i \(0.0358573\pi\)
−0.993662 + 0.112411i \(0.964143\pi\)
\(608\) 0 0
\(609\) 4.41742i 0.179003i
\(610\) 0 0
\(611\) 1.25227i 0.0506615i
\(612\) 0 0
\(613\) −5.49545 −0.221959 −0.110980 0.993823i \(-0.535399\pi\)
−0.110980 + 0.993823i \(0.535399\pi\)
\(614\) 0 0
\(615\) 55.1216i 2.22272i
\(616\) 0 0
\(617\) −45.9564 −1.85014 −0.925068 0.379801i \(-0.875993\pi\)
−0.925068 + 0.379801i \(0.875993\pi\)
\(618\) 0 0
\(619\) 6.12159 0.246048 0.123024 0.992404i \(-0.460741\pi\)
0.123024 + 0.992404i \(0.460741\pi\)
\(620\) 0 0
\(621\) 3.95644i 0.158766i
\(622\) 0 0
\(623\) 12.0000i 0.480770i
\(624\) 0 0
\(625\) 16.0000 0.640000
\(626\) 0 0
\(627\) 80.2432i 3.20460i
\(628\) 0 0
\(629\) −7.25227 + 6.33030i −0.289167 + 0.252406i
\(630\) 0 0
\(631\) 3.95644i 0.157503i 0.996894 + 0.0787517i \(0.0250934\pi\)
−0.996894 + 0.0787517i \(0.974907\pi\)
\(632\) 0 0
\(633\) 9.41742 0.374309
\(634\) 0 0
\(635\) 30.3303i 1.20362i
\(636\) 0 0
\(637\) 2.37386i 0.0940559i
\(638\) 0 0
\(639\) −43.9129 −1.73717
\(640\) 0 0
\(641\) −35.5390 −1.40371 −0.701853 0.712321i \(-0.747644\pi\)
−0.701853 + 0.712321i \(0.747644\pi\)
\(642\) 0 0
\(643\) 15.4955i 0.611081i 0.952179 + 0.305541i \(0.0988371\pi\)
−0.952179 + 0.305541i \(0.901163\pi\)
\(644\) 0 0
\(645\) −63.4955 −2.50013
\(646\) 0 0
\(647\) 32.7042i 1.28573i 0.765978 + 0.642867i \(0.222255\pi\)
−0.765978 + 0.642867i \(0.777745\pi\)
\(648\) 0 0
\(649\) 28.7477i 1.12845i
\(650\) 0 0
\(651\) 30.0000i 1.17579i
\(652\) 0 0
\(653\) 14.3739i 0.562493i −0.959636 0.281246i \(-0.909252\pi\)
0.959636 0.281246i \(-0.0907478\pi\)
\(654\) 0 0
\(655\) 40.7477 1.59215
\(656\) 0 0
\(657\) 44.9129 1.75222
\(658\) 0 0
\(659\) 33.9564 1.32276 0.661378 0.750053i \(-0.269972\pi\)
0.661378 + 0.750053i \(0.269972\pi\)
\(660\) 0 0
\(661\) 36.7913i 1.43102i −0.698605 0.715508i \(-0.746195\pi\)
0.698605 0.715508i \(-0.253805\pi\)
\(662\) 0 0
\(663\) 3.49545 0.135752
\(664\) 0 0
\(665\) 57.4955 2.22958
\(666\) 0 0
\(667\) 0.626136 0.0242441
\(668\) 0 0
\(669\) 39.0780 1.51084
\(670\) 0 0
\(671\) 31.1216i 1.20144i
\(672\) 0 0
\(673\) 5.12159 0.197423 0.0987114 0.995116i \(-0.468528\pi\)
0.0987114 + 0.995116i \(0.468528\pi\)
\(674\) 0 0
\(675\) 46.8693 1.80400
\(676\) 0 0
\(677\) 9.16515 0.352245 0.176123 0.984368i \(-0.443644\pi\)
0.176123 + 0.984368i \(0.443644\pi\)
\(678\) 0 0
\(679\) 8.83485i 0.339050i
\(680\) 0 0
\(681\) 55.5826i 2.12993i
\(682\) 0 0
\(683\) 22.4174i 0.857779i −0.903357 0.428889i \(-0.858905\pi\)
0.903357 0.428889i \(-0.141095\pi\)
\(684\) 0 0
\(685\) 65.8693i 2.51674i
\(686\) 0 0
\(687\) −57.9129 −2.20951
\(688\) 0 0
\(689\) 6.00000i 0.228582i
\(690\) 0 0
\(691\) −29.4955 −1.12206 −0.561030 0.827795i \(-0.689595\pi\)
−0.561030 + 0.827795i \(0.689595\pi\)
\(692\) 0 0
\(693\) 36.3303 1.38007
\(694\) 0 0
\(695\) 1.41742i 0.0537660i
\(696\) 0 0
\(697\) 8.24318i 0.312233i
\(698\) 0 0
\(699\) −70.1216 −2.65224
\(700\) 0 0
\(701\) 18.9564i 0.715975i 0.933726 + 0.357987i \(0.116537\pi\)
−0.933726 + 0.357987i \(0.883463\pi\)
\(702\) 0 0
\(703\) −34.7477 + 30.3303i −1.31054 + 1.14393i
\(704\) 0 0
\(705\) 16.7477i 0.630756i
\(706\) 0 0
\(707\) 3.16515 0.119038
\(708\) 0 0
\(709\) 27.7913i 1.04372i −0.853030 0.521862i \(-0.825238\pi\)
0.853030 0.521862i \(-0.174762\pi\)
\(710\) 0 0
\(711\) 61.2867i 2.29843i
\(712\) 0 0
\(713\) −4.25227 −0.159249
\(714\) 0 0
\(715\) 11.3739 0.425358
\(716\) 0 0
\(717\) 69.6606i 2.60152i
\(718\) 0 0
\(719\) 2.83485 0.105722 0.0528610 0.998602i \(-0.483166\pi\)
0.0528610 + 0.998602i \(0.483166\pi\)
\(720\) 0 0
\(721\) 4.41742i 0.164513i
\(722\) 0 0
\(723\) 37.9129i 1.41000i
\(724\) 0 0
\(725\) 7.41742i 0.275476i
\(726\) 0 0
\(727\) 28.1216i 1.04297i −0.853260 0.521486i \(-0.825378\pi\)
0.853260 0.521486i \(-0.174622\pi\)
\(728\) 0 0
\(729\) −43.8693 −1.62479
\(730\) 0 0
\(731\) −9.49545 −0.351202
\(732\) 0 0
\(733\) 7.49545 0.276851 0.138425 0.990373i \(-0.455796\pi\)
0.138425 + 0.990373i \(0.455796\pi\)
\(734\) 0 0
\(735\) 31.7477i 1.17103i
\(736\) 0 0
\(737\) −27.9564 −1.02979
\(738\) 0 0
\(739\) 6.12159 0.225186 0.112593 0.993641i \(-0.464084\pi\)
0.112593 + 0.993641i \(0.464084\pi\)
\(740\) 0 0
\(741\) 16.7477 0.615243
\(742\) 0 0
\(743\) 3.16515 0.116118 0.0580591 0.998313i \(-0.481509\pi\)
0.0580591 + 0.998313i \(0.481509\pi\)
\(744\) 0 0
\(745\) 51.4955i 1.88665i
\(746\) 0 0
\(747\) −15.1652 −0.554864
\(748\) 0 0
\(749\) 16.7477 0.611949
\(750\) 0 0
\(751\) −13.4955 −0.492456 −0.246228 0.969212i \(-0.579191\pi\)
−0.246228 + 0.969212i \(0.579191\pi\)
\(752\) 0 0
\(753\) 37.9129i 1.38162i
\(754\) 0 0
\(755\) 48.3303i 1.75892i
\(756\) 0 0
\(757\) 39.7913i 1.44624i 0.690723 + 0.723119i \(0.257292\pi\)
−0.690723 + 0.723119i \(0.742708\pi\)
\(758\) 0 0
\(759\) 8.37386i 0.303952i
\(760\) 0 0
\(761\) 18.7913 0.681184 0.340592 0.940211i \(-0.389373\pi\)
0.340592 + 0.940211i \(0.389373\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) 28.7477 1.03938
\(766\) 0 0
\(767\) −6.00000 −0.216647
\(768\) 0 0
\(769\) 48.3303i 1.74284i −0.490542 0.871418i \(-0.663201\pi\)
0.490542 0.871418i \(-0.336799\pi\)
\(770\) 0 0
\(771\) 63.4955i 2.28673i
\(772\) 0 0
\(773\) 13.9129 0.500411 0.250206 0.968193i \(-0.419502\pi\)
0.250206 + 0.968193i \(0.419502\pi\)
\(774\) 0 0
\(775\) 50.3739i 1.80948i
\(776\) 0 0
\(777\) 22.3303 + 25.5826i 0.801095 + 0.917770i
\(778\) 0 0
\(779\) 39.4955i 1.41507i
\(780\) 0 0
\(781\) −34.7477 −1.24337
\(782\) 0 0
\(783\) 3.95644i 0.141392i
\(784\) 0 0
\(785\) 7.58258i 0.270634i
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 0 0
\(789\) −24.6606 −0.877941
\(790\) 0 0
\(791\) 39.1652i 1.39255i
\(792\) 0 0
\(793\) −6.49545 −0.230660
\(794\) 0 0
\(795\) 80.2432i 2.84593i
\(796\) 0 0
\(797\) 21.6261i 0.766037i −0.923741 0.383019i \(-0.874885\pi\)
0.923741 0.383019i \(-0.125115\pi\)
\(798\) 0 0
\(799\) 2.50455i 0.0886045i
\(800\) 0 0
\(801\) 28.7477i 1.01575i
\(802\) 0 0
\(803\) 35.5390 1.25414
\(804\) 0 0
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) 11.0780i 0.389483i 0.980855 + 0.194741i \(0.0623868\pi\)
−0.980855 + 0.194741i \(0.937613\pi\)
\(810\) 0 0
\(811\) 48.8693 1.71603 0.858017 0.513621i \(-0.171696\pi\)
0.858017 + 0.513621i \(0.171696\pi\)
\(812\) 0 0
\(813\) 61.4083 2.15368
\(814\) 0 0
\(815\) −39.4955 −1.38347
\(816\) 0 0
\(817\) −45.4955 −1.59168
\(818\) 0 0
\(819\) 7.58258i 0.264957i
\(820\) 0 0
\(821\) 15.1652 0.529267 0.264634 0.964349i \(-0.414749\pi\)
0.264634 + 0.964349i \(0.414749\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 99.1996 3.45369
\(826\) 0 0
\(827\) 14.8348i 0.515858i 0.966164 + 0.257929i \(0.0830401\pi\)
−0.966164 + 0.257929i \(0.916960\pi\)
\(828\) 0 0
\(829\) 39.9564i 1.38774i −0.720098 0.693872i \(-0.755903\pi\)
0.720098 0.693872i \(-0.244097\pi\)
\(830\) 0 0
\(831\) 65.2432i 2.26326i
\(832\) 0 0
\(833\) 4.74773i 0.164499i
\(834\) 0 0
\(835\) 3.62614 0.125488
\(836\) 0 0
\(837\) 26.8693i 0.928739i
\(838\) 0 0
\(839\) 51.4955 1.77782 0.888910 0.458081i \(-0.151463\pi\)
0.888910 + 0.458081i \(0.151463\pi\)
\(840\) 0 0
\(841\) 28.3739 0.978409
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 46.9129i 1.61385i
\(846\) 0 0
\(847\) 6.74773 0.231855
\(848\) 0 0
\(849\) 66.9909i 2.29912i
\(850\) 0 0
\(851\) −3.62614 + 3.16515i −0.124302 + 0.108500i
\(852\) 0 0
\(853\) 5.70417i 0.195307i 0.995220 + 0.0976535i \(0.0311337\pi\)
−0.995220 + 0.0976535i \(0.968866\pi\)
\(854\) 0 0
\(855\) 137.739 4.71056
\(856\) 0 0
\(857\) 14.8348i 0.506749i 0.967368 + 0.253374i \(0.0815405\pi\)
−0.967368 + 0.253374i \(0.918460\pi\)
\(858\) 0 0
\(859\) 54.6606i 1.86500i 0.361175 + 0.932498i \(0.382376\pi\)
−0.361175 + 0.932498i \(0.617624\pi\)
\(860\) 0 0
\(861\) −29.0780 −0.990977
\(862\) 0 0
\(863\) 46.7477 1.59131 0.795656 0.605749i \(-0.207127\pi\)
0.795656 + 0.605749i \(0.207127\pi\)
\(864\) 0 0
\(865\) 12.0000i 0.408012i
\(866\) 0 0
\(867\) −40.4610 −1.37413
\(868\) 0 0
\(869\) 48.4955i 1.64510i
\(870\) 0 0
\(871\) 5.83485i 0.197706i
\(872\) 0 0
\(873\) 21.1652i 0.716332i
\(874\) 0 0
\(875\) 33.1652i 1.12119i
\(876\) 0 0
\(877\) −38.7477 −1.30842 −0.654209 0.756314i \(-0.726998\pi\)
−0.654209 + 0.756314i \(0.726998\pi\)
\(878\) 0 0
\(879\) −16.7477 −0.564887
\(880\) 0 0
\(881\) 28.1216 0.947440 0.473720 0.880675i \(-0.342911\pi\)
0.473720 + 0.880675i \(0.342911\pi\)
\(882\) 0 0
\(883\) 13.9129i 0.468206i −0.972212 0.234103i \(-0.924785\pi\)
0.972212 0.234103i \(-0.0752152\pi\)
\(884\) 0 0
\(885\) −80.2432 −2.69735
\(886\) 0 0
\(887\) −33.8258 −1.13576 −0.567879 0.823112i \(-0.692236\pi\)
−0.567879 + 0.823112i \(0.692236\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −1.58258 −0.0530183
\(892\) 0 0
\(893\) 12.0000i 0.401565i
\(894\) 0 0
\(895\) 28.7477 0.960931
\(896\) 0 0
\(897\) 1.74773 0.0583549
\(898\) 0 0
\(899\) 4.25227 0.141821
\(900\) 0 0
\(901\) 12.0000i 0.399778i
\(902\) 0 0
\(903\) 33.4955i 1.11466i
\(904\) 0 0
\(905\) 71.0780i 2.36271i
\(906\) 0 0
\(907\) 6.33030i 0.210194i 0.994462 + 0.105097i \(0.0335153\pi\)
−0.994462 + 0.105097i \(0.966485\pi\)
\(908\) 0 0
\(909\) 7.58258 0.251498
\(910\) 0 0
\(911\) 54.3303i 1.80004i −0.435845 0.900022i \(-0.643551\pi\)
0.435845 0.900022i \(-0.356449\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −86.8693 −2.87181
\(916\) 0 0
\(917\) 21.4955i 0.709842i
\(918\) 0 0
\(919\) 36.0000i 1.18753i −0.804638 0.593765i \(-0.797641\pi\)
0.804638 0.593765i \(-0.202359\pi\)
\(920\) 0 0
\(921\) 34.5390 1.13810
\(922\) 0 0
\(923\) 7.25227i 0.238711i
\(924\) 0 0
\(925\) 37.4955 + 42.9564i 1.23284 + 1.41240i
\(926\) 0 0
\(927\) 10.5826i 0.347577i
\(928\) 0 0
\(929\) −5.37386 −0.176311 −0.0881554 0.996107i \(-0.528097\pi\)
−0.0881554 + 0.996107i \(0.528097\pi\)
\(930\) 0 0
\(931\) 22.7477i 0.745527i
\(932\) 0 0
\(933\) 26.8693i 0.879662i
\(934\) 0 0
\(935\) 22.7477 0.743930
\(936\) 0 0
\(937\) −33.8693 −1.10646 −0.553231 0.833028i \(-0.686605\pi\)
−0.553231 + 0.833028i \(0.686605\pi\)
\(938\) 0 0
\(939\) 92.5735i 3.02102i
\(940\) 0 0
\(941\) −33.4955 −1.09192 −0.545960 0.837811i \(-0.683835\pi\)
−0.545960 + 0.837811i \(0.683835\pi\)
\(942\) 0 0
\(943\) 4.12159i 0.134217i
\(944\) 0 0
\(945\) 37.9129i 1.23331i
\(946\) 0 0
\(947\) 28.7477i 0.934176i −0.884211 0.467088i \(-0.845303\pi\)
0.884211 0.467088i \(-0.154697\pi\)
\(948\) 0 0
\(949\) 7.41742i 0.240780i
\(950\) 0 0
\(951\) −3.49545 −0.113348
\(952\) 0 0
\(953\) −46.4519 −1.50472 −0.752362 0.658750i \(-0.771086\pi\)
−0.752362 + 0.658750i \(0.771086\pi\)
\(954\) 0 0
\(955\) −20.3739 −0.659283
\(956\) 0 0
\(957\) 8.37386i 0.270689i
\(958\) 0 0
\(959\) −34.7477 −1.12206
\(960\) 0 0
\(961\) 2.12159 0.0684384
\(962\) 0 0
\(963\) 40.1216 1.29290
\(964\) 0 0
\(965\) −69.4955 −2.23714
\(966\) 0 0
\(967\) 0.956439i 0.0307570i −0.999882 0.0153785i \(-0.995105\pi\)
0.999882 0.0153785i \(-0.00489532\pi\)
\(968\) 0 0
\(969\) 33.4955 1.07603
\(970\) 0 0
\(971\) 36.6261 1.17539 0.587694 0.809083i \(-0.300036\pi\)
0.587694 + 0.809083i \(0.300036\pi\)
\(972\) 0 0
\(973\) 0.747727 0.0239710
\(974\) 0 0
\(975\) 20.7042i 0.663064i
\(976\) 0 0
\(977\) 23.0780i 0.738332i −0.929364 0.369166i \(-0.879643\pi\)
0.929364 0.369166i \(-0.120357\pi\)
\(978\) 0 0
\(979\) 22.7477i 0.727021i
\(980\) 0 0
\(981\) 28.7477i 0.917844i
\(982\) 0 0
\(983\) −18.3303 −0.584646 −0.292323 0.956320i \(-0.594428\pi\)
−0.292323 + 0.956320i \(0.594428\pi\)
\(984\) 0 0
\(985\) 98.2432i 3.13029i
\(986\) 0 0
\(987\) 8.83485 0.281216
\(988\) 0 0
\(989\) −4.74773 −0.150969
\(990\) 0 0
\(991\) 3.95644i 0.125680i 0.998024 + 0.0628402i \(0.0200159\pi\)
−0.998024 + 0.0628402i \(0.979984\pi\)
\(992\) 0 0
\(993\) 45.8258i 1.45424i
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 0 0
\(999\) 20.0000 + 22.9129i 0.632772 + 0.724931i
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 2368.2.g.h.961.2 4
4.3 odd 2 2368.2.g.j.961.4 4
8.3 odd 2 74.2.b.a.73.1 4
8.5 even 2 592.2.g.c.369.3 4
24.5 odd 2 5328.2.h.m.2737.4 4
24.11 even 2 666.2.c.b.73.4 4
37.36 even 2 inner 2368.2.g.h.961.1 4
40.3 even 4 1850.2.c.g.1849.1 4
40.19 odd 2 1850.2.d.e.1701.4 4
40.27 even 4 1850.2.c.h.1849.4 4
148.147 odd 2 2368.2.g.j.961.3 4
296.43 even 4 2738.2.a.h.1.2 2
296.147 odd 2 74.2.b.a.73.3 yes 4
296.179 even 4 2738.2.a.k.1.2 2
296.221 even 2 592.2.g.c.369.4 4
888.221 odd 2 5328.2.h.m.2737.1 4
888.443 even 2 666.2.c.b.73.1 4
1480.147 even 4 1850.2.c.g.1849.4 4
1480.443 even 4 1850.2.c.h.1849.1 4
1480.739 odd 2 1850.2.d.e.1701.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.b.a.73.1 4 8.3 odd 2
74.2.b.a.73.3 yes 4 296.147 odd 2
592.2.g.c.369.3 4 8.5 even 2
592.2.g.c.369.4 4 296.221 even 2
666.2.c.b.73.1 4 888.443 even 2
666.2.c.b.73.4 4 24.11 even 2
1850.2.c.g.1849.1 4 40.3 even 4
1850.2.c.g.1849.4 4 1480.147 even 4
1850.2.c.h.1849.1 4 1480.443 even 4
1850.2.c.h.1849.4 4 40.27 even 4
1850.2.d.e.1701.2 4 1480.739 odd 2
1850.2.d.e.1701.4 4 40.19 odd 2
2368.2.g.h.961.1 4 37.36 even 2 inner
2368.2.g.h.961.2 4 1.1 even 1 trivial
2368.2.g.j.961.3 4 148.147 odd 2
2368.2.g.j.961.4 4 4.3 odd 2
2738.2.a.h.1.2 2 296.43 even 4
2738.2.a.k.1.2 2 296.179 even 4
5328.2.h.m.2737.1 4 888.221 odd 2
5328.2.h.m.2737.4 4 24.5 odd 2