Properties

Label 4368.2.h.l.337.4
Level $4368$
Weight $2$
Character 4368.337
Analytic conductor $34.879$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4368,2,Mod(337,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("4368.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1092)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4368.337
Dual form 4368.2.h.l.337.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.56155i q^{5} +1.00000i q^{7} +1.00000 q^{9} +5.12311i q^{11} +(0.561553 - 3.56155i) q^{13} +3.56155i q^{15} -2.00000 q^{17} +1.56155i q^{19} +1.00000i q^{21} -8.68466 q^{23} -7.68466 q^{25} +1.00000 q^{27} +7.56155 q^{29} +1.56155i q^{31} +5.12311i q^{33} -3.56155 q^{35} +(0.561553 - 3.56155i) q^{39} +10.0000i q^{41} -5.56155 q^{43} +3.56155i q^{45} -0.438447i q^{47} -1.00000 q^{49} -2.00000 q^{51} +3.56155 q^{53} -18.2462 q^{55} +1.56155i q^{57} -6.87689i q^{59} -2.87689 q^{61} +1.00000i q^{63} +(12.6847 + 2.00000i) q^{65} -4.87689i q^{67} -8.68466 q^{69} +15.3693i q^{71} -6.43845i q^{73} -7.68466 q^{75} -5.12311 q^{77} -8.68466 q^{79} +1.00000 q^{81} +6.68466i q^{83} -7.12311i q^{85} +7.56155 q^{87} -6.68466i q^{89} +(3.56155 + 0.561553i) q^{91} +1.56155i q^{93} -5.56155 q^{95} -3.31534i q^{97} +5.12311i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} - 6 q^{13} - 8 q^{17} - 10 q^{23} - 6 q^{25} + 4 q^{27} + 22 q^{29} - 6 q^{35} - 6 q^{39} - 14 q^{43} - 4 q^{49} - 8 q^{51} + 6 q^{53} - 40 q^{55} - 28 q^{61} + 26 q^{65} - 10 q^{69} - 6 q^{75} - 4 q^{77} - 10 q^{79} + 4 q^{81} + 22 q^{87} + 6 q^{91} - 14 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.56155i 1.59277i 0.604787 + 0.796387i \(0.293258\pi\)
−0.604787 + 0.796387i \(0.706742\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.12311i 1.54467i 0.635213 + 0.772337i \(0.280912\pi\)
−0.635213 + 0.772337i \(0.719088\pi\)
\(12\) 0 0
\(13\) 0.561553 3.56155i 0.155747 0.987797i
\(14\) 0 0
\(15\) 3.56155i 0.919589i
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 1.56155i 0.358245i 0.983827 + 0.179122i \(0.0573258\pi\)
−0.983827 + 0.179122i \(0.942674\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) −8.68466 −1.81088 −0.905438 0.424478i \(-0.860458\pi\)
−0.905438 + 0.424478i \(0.860458\pi\)
\(24\) 0 0
\(25\) −7.68466 −1.53693
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.56155 1.40415 0.702073 0.712105i \(-0.252258\pi\)
0.702073 + 0.712105i \(0.252258\pi\)
\(30\) 0 0
\(31\) 1.56155i 0.280463i 0.990119 + 0.140232i \(0.0447847\pi\)
−0.990119 + 0.140232i \(0.955215\pi\)
\(32\) 0 0
\(33\) 5.12311i 0.891818i
\(34\) 0 0
\(35\) −3.56155 −0.602012
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0.561553 3.56155i 0.0899204 0.570305i
\(40\) 0 0
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −5.56155 −0.848129 −0.424064 0.905632i \(-0.639397\pi\)
−0.424064 + 0.905632i \(0.639397\pi\)
\(44\) 0 0
\(45\) 3.56155i 0.530925i
\(46\) 0 0
\(47\) 0.438447i 0.0639541i −0.999489 0.0319770i \(-0.989820\pi\)
0.999489 0.0319770i \(-0.0101803\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 3.56155 0.489217 0.244608 0.969622i \(-0.421341\pi\)
0.244608 + 0.969622i \(0.421341\pi\)
\(54\) 0 0
\(55\) −18.2462 −2.46032
\(56\) 0 0
\(57\) 1.56155i 0.206833i
\(58\) 0 0
\(59\) 6.87689i 0.895295i −0.894210 0.447648i \(-0.852262\pi\)
0.894210 0.447648i \(-0.147738\pi\)
\(60\) 0 0
\(61\) −2.87689 −0.368349 −0.184174 0.982894i \(-0.558961\pi\)
−0.184174 + 0.982894i \(0.558961\pi\)
\(62\) 0 0
\(63\) 1.00000i 0.125988i
\(64\) 0 0
\(65\) 12.6847 + 2.00000i 1.57334 + 0.248069i
\(66\) 0 0
\(67\) 4.87689i 0.595807i −0.954596 0.297904i \(-0.903713\pi\)
0.954596 0.297904i \(-0.0962874\pi\)
\(68\) 0 0
\(69\) −8.68466 −1.04551
\(70\) 0 0
\(71\) 15.3693i 1.82400i 0.410188 + 0.912001i \(0.365463\pi\)
−0.410188 + 0.912001i \(0.634537\pi\)
\(72\) 0 0
\(73\) 6.43845i 0.753563i −0.926302 0.376782i \(-0.877031\pi\)
0.926302 0.376782i \(-0.122969\pi\)
\(74\) 0 0
\(75\) −7.68466 −0.887348
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −8.68466 −0.977100 −0.488550 0.872536i \(-0.662474\pi\)
−0.488550 + 0.872536i \(0.662474\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.68466i 0.733737i 0.930273 + 0.366868i \(0.119570\pi\)
−0.930273 + 0.366868i \(0.880430\pi\)
\(84\) 0 0
\(85\) 7.12311i 0.772609i
\(86\) 0 0
\(87\) 7.56155 0.810684
\(88\) 0 0
\(89\) 6.68466i 0.708572i −0.935137 0.354286i \(-0.884724\pi\)
0.935137 0.354286i \(-0.115276\pi\)
\(90\) 0 0
\(91\) 3.56155 + 0.561553i 0.373352 + 0.0588667i
\(92\) 0 0
\(93\) 1.56155i 0.161925i
\(94\) 0 0
\(95\) −5.56155 −0.570603
\(96\) 0 0
\(97\) 3.31534i 0.336622i −0.985734 0.168311i \(-0.946169\pi\)
0.985734 0.168311i \(-0.0538313\pi\)
\(98\) 0 0
\(99\) 5.12311i 0.514891i
\(100\) 0 0
\(101\) 15.3693 1.52930 0.764652 0.644443i \(-0.222911\pi\)
0.764652 + 0.644443i \(0.222911\pi\)
\(102\) 0 0
\(103\) 18.2462 1.79785 0.898926 0.438100i \(-0.144348\pi\)
0.898926 + 0.438100i \(0.144348\pi\)
\(104\) 0 0
\(105\) −3.56155 −0.347572
\(106\) 0 0
\(107\) −16.4924 −1.59438 −0.797191 0.603727i \(-0.793682\pi\)
−0.797191 + 0.603727i \(0.793682\pi\)
\(108\) 0 0
\(109\) 7.12311i 0.682270i −0.940014 0.341135i \(-0.889189\pi\)
0.940014 0.341135i \(-0.110811\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −5.80776 −0.546348 −0.273174 0.961965i \(-0.588074\pi\)
−0.273174 + 0.961965i \(0.588074\pi\)
\(114\) 0 0
\(115\) 30.9309i 2.88432i
\(116\) 0 0
\(117\) 0.561553 3.56155i 0.0519156 0.329266i
\(118\) 0 0
\(119\) 2.00000i 0.183340i
\(120\) 0 0
\(121\) −15.2462 −1.38602
\(122\) 0 0
\(123\) 10.0000i 0.901670i
\(124\) 0 0
\(125\) 9.56155i 0.855211i
\(126\) 0 0
\(127\) 20.4924 1.81841 0.909204 0.416350i \(-0.136691\pi\)
0.909204 + 0.416350i \(0.136691\pi\)
\(128\) 0 0
\(129\) −5.56155 −0.489667
\(130\) 0 0
\(131\) −15.1231 −1.32131 −0.660656 0.750689i \(-0.729722\pi\)
−0.660656 + 0.750689i \(0.729722\pi\)
\(132\) 0 0
\(133\) −1.56155 −0.135404
\(134\) 0 0
\(135\) 3.56155i 0.306530i
\(136\) 0 0
\(137\) 19.3693i 1.65483i −0.561589 0.827416i \(-0.689810\pi\)
0.561589 0.827416i \(-0.310190\pi\)
\(138\) 0 0
\(139\) −22.2462 −1.88690 −0.943450 0.331516i \(-0.892440\pi\)
−0.943450 + 0.331516i \(0.892440\pi\)
\(140\) 0 0
\(141\) 0.438447i 0.0369239i
\(142\) 0 0
\(143\) 18.2462 + 2.87689i 1.52582 + 0.240578i
\(144\) 0 0
\(145\) 26.9309i 2.23649i
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 19.3693i 1.58680i 0.608703 + 0.793398i \(0.291690\pi\)
−0.608703 + 0.793398i \(0.708310\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i 0.872691 + 0.488273i \(0.162373\pi\)
−0.872691 + 0.488273i \(0.837627\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −5.56155 −0.446715
\(156\) 0 0
\(157\) 8.24621 0.658119 0.329060 0.944309i \(-0.393268\pi\)
0.329060 + 0.944309i \(0.393268\pi\)
\(158\) 0 0
\(159\) 3.56155 0.282450
\(160\) 0 0
\(161\) 8.68466i 0.684447i
\(162\) 0 0
\(163\) 15.1231i 1.18453i −0.805742 0.592267i \(-0.798233\pi\)
0.805742 0.592267i \(-0.201767\pi\)
\(164\) 0 0
\(165\) −18.2462 −1.42047
\(166\) 0 0
\(167\) 3.56155i 0.275601i −0.990460 0.137801i \(-0.955997\pi\)
0.990460 0.137801i \(-0.0440033\pi\)
\(168\) 0 0
\(169\) −12.3693 4.00000i −0.951486 0.307692i
\(170\) 0 0
\(171\) 1.56155i 0.119415i
\(172\) 0 0
\(173\) −13.1231 −0.997731 −0.498866 0.866679i \(-0.666250\pi\)
−0.498866 + 0.866679i \(0.666250\pi\)
\(174\) 0 0
\(175\) 7.68466i 0.580906i
\(176\) 0 0
\(177\) 6.87689i 0.516899i
\(178\) 0 0
\(179\) −6.43845 −0.481232 −0.240616 0.970620i \(-0.577349\pi\)
−0.240616 + 0.970620i \(0.577349\pi\)
\(180\) 0 0
\(181\) −7.36932 −0.547757 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(182\) 0 0
\(183\) −2.87689 −0.212666
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 10.2462i 0.749277i
\(188\) 0 0
\(189\) 1.00000i 0.0727393i
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) 9.36932i 0.674418i 0.941430 + 0.337209i \(0.109483\pi\)
−0.941430 + 0.337209i \(0.890517\pi\)
\(194\) 0 0
\(195\) 12.6847 + 2.00000i 0.908367 + 0.143223i
\(196\) 0 0
\(197\) 3.36932i 0.240054i −0.992771 0.120027i \(-0.961702\pi\)
0.992771 0.120027i \(-0.0382981\pi\)
\(198\) 0 0
\(199\) −15.6155 −1.10696 −0.553478 0.832864i \(-0.686700\pi\)
−0.553478 + 0.832864i \(0.686700\pi\)
\(200\) 0 0
\(201\) 4.87689i 0.343990i
\(202\) 0 0
\(203\) 7.56155i 0.530717i
\(204\) 0 0
\(205\) −35.6155 −2.48750
\(206\) 0 0
\(207\) −8.68466 −0.603625
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) 1.56155 0.107502 0.0537509 0.998554i \(-0.482882\pi\)
0.0537509 + 0.998554i \(0.482882\pi\)
\(212\) 0 0
\(213\) 15.3693i 1.05309i
\(214\) 0 0
\(215\) 19.8078i 1.35088i
\(216\) 0 0
\(217\) −1.56155 −0.106005
\(218\) 0 0
\(219\) 6.43845i 0.435070i
\(220\) 0 0
\(221\) −1.12311 + 7.12311i −0.0755483 + 0.479152i
\(222\) 0 0
\(223\) 6.43845i 0.431150i 0.976487 + 0.215575i \(0.0691626\pi\)
−0.976487 + 0.215575i \(0.930837\pi\)
\(224\) 0 0
\(225\) −7.68466 −0.512311
\(226\) 0 0
\(227\) 1.12311i 0.0745431i −0.999305 0.0372716i \(-0.988133\pi\)
0.999305 0.0372716i \(-0.0118667\pi\)
\(228\) 0 0
\(229\) 5.36932i 0.354814i 0.984138 + 0.177407i \(0.0567710\pi\)
−0.984138 + 0.177407i \(0.943229\pi\)
\(230\) 0 0
\(231\) −5.12311 −0.337076
\(232\) 0 0
\(233\) −5.31534 −0.348220 −0.174110 0.984726i \(-0.555705\pi\)
−0.174110 + 0.984726i \(0.555705\pi\)
\(234\) 0 0
\(235\) 1.56155 0.101864
\(236\) 0 0
\(237\) −8.68466 −0.564129
\(238\) 0 0
\(239\) 8.24621i 0.533403i −0.963779 0.266702i \(-0.914066\pi\)
0.963779 0.266702i \(-0.0859338\pi\)
\(240\) 0 0
\(241\) 14.0540i 0.905296i 0.891689 + 0.452648i \(0.149521\pi\)
−0.891689 + 0.452648i \(0.850479\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 3.56155i 0.227539i
\(246\) 0 0
\(247\) 5.56155 + 0.876894i 0.353873 + 0.0557955i
\(248\) 0 0
\(249\) 6.68466i 0.423623i
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 44.4924i 2.79721i
\(254\) 0 0
\(255\) 7.12311i 0.446066i
\(256\) 0 0
\(257\) 19.3693 1.20822 0.604112 0.796899i \(-0.293528\pi\)
0.604112 + 0.796899i \(0.293528\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 7.56155 0.468048
\(262\) 0 0
\(263\) 10.0540 0.619955 0.309977 0.950744i \(-0.399679\pi\)
0.309977 + 0.950744i \(0.399679\pi\)
\(264\) 0 0
\(265\) 12.6847i 0.779212i
\(266\) 0 0
\(267\) 6.68466i 0.409094i
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 26.7386i 1.62426i 0.583479 + 0.812128i \(0.301691\pi\)
−0.583479 + 0.812128i \(0.698309\pi\)
\(272\) 0 0
\(273\) 3.56155 + 0.561553i 0.215555 + 0.0339867i
\(274\) 0 0
\(275\) 39.3693i 2.37406i
\(276\) 0 0
\(277\) 8.43845 0.507017 0.253509 0.967333i \(-0.418415\pi\)
0.253509 + 0.967333i \(0.418415\pi\)
\(278\) 0 0
\(279\) 1.56155i 0.0934877i
\(280\) 0 0
\(281\) 18.0000i 1.07379i −0.843649 0.536895i \(-0.819597\pi\)
0.843649 0.536895i \(-0.180403\pi\)
\(282\) 0 0
\(283\) 26.2462 1.56018 0.780088 0.625670i \(-0.215174\pi\)
0.780088 + 0.625670i \(0.215174\pi\)
\(284\) 0 0
\(285\) −5.56155 −0.329438
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 3.31534i 0.194349i
\(292\) 0 0
\(293\) 20.9309i 1.22279i 0.791324 + 0.611397i \(0.209392\pi\)
−0.791324 + 0.611397i \(0.790608\pi\)
\(294\) 0 0
\(295\) 24.4924 1.42600
\(296\) 0 0
\(297\) 5.12311i 0.297273i
\(298\) 0 0
\(299\) −4.87689 + 30.9309i −0.282038 + 1.78878i
\(300\) 0 0
\(301\) 5.56155i 0.320563i
\(302\) 0 0
\(303\) 15.3693 0.882944
\(304\) 0 0
\(305\) 10.2462i 0.586696i
\(306\) 0 0
\(307\) 11.8078i 0.673905i 0.941522 + 0.336952i \(0.109396\pi\)
−0.941522 + 0.336952i \(0.890604\pi\)
\(308\) 0 0
\(309\) 18.2462 1.03799
\(310\) 0 0
\(311\) −15.6155 −0.885475 −0.442738 0.896651i \(-0.645993\pi\)
−0.442738 + 0.896651i \(0.645993\pi\)
\(312\) 0 0
\(313\) −25.1231 −1.42004 −0.710021 0.704181i \(-0.751315\pi\)
−0.710021 + 0.704181i \(0.751315\pi\)
\(314\) 0 0
\(315\) −3.56155 −0.200671
\(316\) 0 0
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) 0 0
\(319\) 38.7386i 2.16895i
\(320\) 0 0
\(321\) −16.4924 −0.920517
\(322\) 0 0
\(323\) 3.12311i 0.173774i
\(324\) 0 0
\(325\) −4.31534 + 27.3693i −0.239372 + 1.51818i
\(326\) 0 0
\(327\) 7.12311i 0.393909i
\(328\) 0 0
\(329\) 0.438447 0.0241724
\(330\) 0 0
\(331\) 9.75379i 0.536117i 0.963403 + 0.268058i \(0.0863820\pi\)
−0.963403 + 0.268058i \(0.913618\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.3693 0.948987
\(336\) 0 0
\(337\) 10.6847 0.582030 0.291015 0.956718i \(-0.406007\pi\)
0.291015 + 0.956718i \(0.406007\pi\)
\(338\) 0 0
\(339\) −5.80776 −0.315434
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 30.9309i 1.66526i
\(346\) 0 0
\(347\) 24.4924 1.31482 0.657411 0.753532i \(-0.271652\pi\)
0.657411 + 0.753532i \(0.271652\pi\)
\(348\) 0 0
\(349\) 35.4233i 1.89617i −0.318024 0.948083i \(-0.603019\pi\)
0.318024 0.948083i \(-0.396981\pi\)
\(350\) 0 0
\(351\) 0.561553 3.56155i 0.0299735 0.190102i
\(352\) 0 0
\(353\) 11.7538i 0.625591i 0.949820 + 0.312796i \(0.101265\pi\)
−0.949820 + 0.312796i \(0.898735\pi\)
\(354\) 0 0
\(355\) −54.7386 −2.90523
\(356\) 0 0
\(357\) 2.00000i 0.105851i
\(358\) 0 0
\(359\) 3.75379i 0.198117i 0.995082 + 0.0990587i \(0.0315832\pi\)
−0.995082 + 0.0990587i \(0.968417\pi\)
\(360\) 0 0
\(361\) 16.5616 0.871661
\(362\) 0 0
\(363\) −15.2462 −0.800219
\(364\) 0 0
\(365\) 22.9309 1.20026
\(366\) 0 0
\(367\) −13.3693 −0.697873 −0.348936 0.937146i \(-0.613457\pi\)
−0.348936 + 0.937146i \(0.613457\pi\)
\(368\) 0 0
\(369\) 10.0000i 0.520579i
\(370\) 0 0
\(371\) 3.56155i 0.184907i
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 9.56155i 0.493756i
\(376\) 0 0
\(377\) 4.24621 26.9309i 0.218691 1.38701i
\(378\) 0 0
\(379\) 28.9848i 1.48885i 0.667705 + 0.744426i \(0.267277\pi\)
−0.667705 + 0.744426i \(0.732723\pi\)
\(380\) 0 0
\(381\) 20.4924 1.04986
\(382\) 0 0
\(383\) 29.1231i 1.48812i 0.668112 + 0.744061i \(0.267103\pi\)
−0.668112 + 0.744061i \(0.732897\pi\)
\(384\) 0 0
\(385\) 18.2462i 0.929913i
\(386\) 0 0
\(387\) −5.56155 −0.282710
\(388\) 0 0
\(389\) −12.2462 −0.620908 −0.310454 0.950588i \(-0.600481\pi\)
−0.310454 + 0.950588i \(0.600481\pi\)
\(390\) 0 0
\(391\) 17.3693 0.878404
\(392\) 0 0
\(393\) −15.1231 −0.762860
\(394\) 0 0
\(395\) 30.9309i 1.55630i
\(396\) 0 0
\(397\) 26.0540i 1.30761i 0.756662 + 0.653806i \(0.226829\pi\)
−0.756662 + 0.653806i \(0.773171\pi\)
\(398\) 0 0
\(399\) −1.56155 −0.0781754
\(400\) 0 0
\(401\) 30.4924i 1.52272i 0.648330 + 0.761359i \(0.275468\pi\)
−0.648330 + 0.761359i \(0.724532\pi\)
\(402\) 0 0
\(403\) 5.56155 + 0.876894i 0.277041 + 0.0436812i
\(404\) 0 0
\(405\) 3.56155i 0.176975i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.56155i 0.0772138i −0.999254 0.0386069i \(-0.987708\pi\)
0.999254 0.0386069i \(-0.0122920\pi\)
\(410\) 0 0
\(411\) 19.3693i 0.955418i
\(412\) 0 0
\(413\) 6.87689 0.338390
\(414\) 0 0
\(415\) −23.8078 −1.16868
\(416\) 0 0
\(417\) −22.2462 −1.08940
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) 29.3693i 1.43137i 0.698422 + 0.715686i \(0.253886\pi\)
−0.698422 + 0.715686i \(0.746114\pi\)
\(422\) 0 0
\(423\) 0.438447i 0.0213180i
\(424\) 0 0
\(425\) 15.3693 0.745521
\(426\) 0 0
\(427\) 2.87689i 0.139223i
\(428\) 0 0
\(429\) 18.2462 + 2.87689i 0.880935 + 0.138898i
\(430\) 0 0
\(431\) 17.1231i 0.824791i −0.911005 0.412395i \(-0.864692\pi\)
0.911005 0.412395i \(-0.135308\pi\)
\(432\) 0 0
\(433\) −28.7386 −1.38109 −0.690545 0.723289i \(-0.742629\pi\)
−0.690545 + 0.723289i \(0.742629\pi\)
\(434\) 0 0
\(435\) 26.9309i 1.29124i
\(436\) 0 0
\(437\) 13.5616i 0.648737i
\(438\) 0 0
\(439\) 10.2462 0.489025 0.244512 0.969646i \(-0.421372\pi\)
0.244512 + 0.969646i \(0.421372\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) 36.6847 1.74294 0.871470 0.490448i \(-0.163167\pi\)
0.871470 + 0.490448i \(0.163167\pi\)
\(444\) 0 0
\(445\) 23.8078 1.12860
\(446\) 0 0
\(447\) 19.3693i 0.916137i
\(448\) 0 0
\(449\) 10.0000i 0.471929i 0.971762 + 0.235965i \(0.0758249\pi\)
−0.971762 + 0.235965i \(0.924175\pi\)
\(450\) 0 0
\(451\) −51.2311 −2.41238
\(452\) 0 0
\(453\) 12.0000i 0.563809i
\(454\) 0 0
\(455\) −2.00000 + 12.6847i −0.0937614 + 0.594666i
\(456\) 0 0
\(457\) 29.3693i 1.37384i 0.726734 + 0.686919i \(0.241037\pi\)
−0.726734 + 0.686919i \(0.758963\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 20.2462i 0.942960i 0.881877 + 0.471480i \(0.156280\pi\)
−0.881877 + 0.471480i \(0.843720\pi\)
\(462\) 0 0
\(463\) 23.6155i 1.09751i −0.835984 0.548753i \(-0.815103\pi\)
0.835984 0.548753i \(-0.184897\pi\)
\(464\) 0 0
\(465\) −5.56155 −0.257911
\(466\) 0 0
\(467\) −8.87689 −0.410774 −0.205387 0.978681i \(-0.565845\pi\)
−0.205387 + 0.978681i \(0.565845\pi\)
\(468\) 0 0
\(469\) 4.87689 0.225194
\(470\) 0 0
\(471\) 8.24621 0.379965
\(472\) 0 0
\(473\) 28.4924i 1.31008i
\(474\) 0 0
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) 3.56155 0.163072
\(478\) 0 0
\(479\) 18.3002i 0.836157i −0.908411 0.418078i \(-0.862704\pi\)
0.908411 0.418078i \(-0.137296\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 8.68466i 0.395166i
\(484\) 0 0
\(485\) 11.8078 0.536163
\(486\) 0 0
\(487\) 10.2462i 0.464300i −0.972680 0.232150i \(-0.925424\pi\)
0.972680 0.232150i \(-0.0745760\pi\)
\(488\) 0 0
\(489\) 15.1231i 0.683890i
\(490\) 0 0
\(491\) 10.2462 0.462405 0.231203 0.972906i \(-0.425734\pi\)
0.231203 + 0.972906i \(0.425734\pi\)
\(492\) 0 0
\(493\) −15.1231 −0.681110
\(494\) 0 0
\(495\) −18.2462 −0.820106
\(496\) 0 0
\(497\) −15.3693 −0.689408
\(498\) 0 0
\(499\) 0.384472i 0.0172113i −0.999963 0.00860566i \(-0.997261\pi\)
0.999963 0.00860566i \(-0.00273930\pi\)
\(500\) 0 0
\(501\) 3.56155i 0.159118i
\(502\) 0 0
\(503\) 4.49242 0.200307 0.100154 0.994972i \(-0.468067\pi\)
0.100154 + 0.994972i \(0.468067\pi\)
\(504\) 0 0
\(505\) 54.7386i 2.43584i
\(506\) 0 0
\(507\) −12.3693 4.00000i −0.549341 0.177646i
\(508\) 0 0
\(509\) 29.3153i 1.29938i 0.760199 + 0.649690i \(0.225101\pi\)
−0.760199 + 0.649690i \(0.774899\pi\)
\(510\) 0 0
\(511\) 6.43845 0.284820
\(512\) 0 0
\(513\) 1.56155i 0.0689442i
\(514\) 0 0
\(515\) 64.9848i 2.86357i
\(516\) 0 0
\(517\) 2.24621 0.0987883
\(518\) 0 0
\(519\) −13.1231 −0.576040
\(520\) 0 0
\(521\) −3.36932 −0.147612 −0.0738062 0.997273i \(-0.523515\pi\)
−0.0738062 + 0.997273i \(0.523515\pi\)
\(522\) 0 0
\(523\) −22.2462 −0.972759 −0.486379 0.873748i \(-0.661683\pi\)
−0.486379 + 0.873748i \(0.661683\pi\)
\(524\) 0 0
\(525\) 7.68466i 0.335386i
\(526\) 0 0
\(527\) 3.12311i 0.136045i
\(528\) 0 0
\(529\) 52.4233 2.27927
\(530\) 0 0
\(531\) 6.87689i 0.298432i
\(532\) 0 0
\(533\) 35.6155 + 5.61553i 1.54268 + 0.243236i
\(534\) 0 0
\(535\) 58.7386i 2.53949i
\(536\) 0 0
\(537\) −6.43845 −0.277840
\(538\) 0 0
\(539\) 5.12311i 0.220668i
\(540\) 0 0
\(541\) 44.9848i 1.93405i −0.254682 0.967025i \(-0.581971\pi\)
0.254682 0.967025i \(-0.418029\pi\)
\(542\) 0 0
\(543\) −7.36932 −0.316248
\(544\) 0 0
\(545\) 25.3693 1.08670
\(546\) 0 0
\(547\) 13.1771 0.563411 0.281706 0.959501i \(-0.409100\pi\)
0.281706 + 0.959501i \(0.409100\pi\)
\(548\) 0 0
\(549\) −2.87689 −0.122783
\(550\) 0 0
\(551\) 11.8078i 0.503028i
\(552\) 0 0
\(553\) 8.68466i 0.369309i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.12311i 0.217073i −0.994092 0.108536i \(-0.965384\pi\)
0.994092 0.108536i \(-0.0346164\pi\)
\(558\) 0 0
\(559\) −3.12311 + 19.8078i −0.132093 + 0.837779i
\(560\) 0 0
\(561\) 10.2462i 0.432595i
\(562\) 0 0
\(563\) 21.3693 0.900609 0.450305 0.892875i \(-0.351315\pi\)
0.450305 + 0.892875i \(0.351315\pi\)
\(564\) 0 0
\(565\) 20.6847i 0.870210i
\(566\) 0 0
\(567\) 1.00000i 0.0419961i
\(568\) 0 0
\(569\) 29.4233 1.23349 0.616744 0.787164i \(-0.288451\pi\)
0.616744 + 0.787164i \(0.288451\pi\)
\(570\) 0 0
\(571\) −7.80776 −0.326745 −0.163372 0.986564i \(-0.552237\pi\)
−0.163372 + 0.986564i \(0.552237\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 66.7386 2.78319
\(576\) 0 0
\(577\) 7.61553i 0.317039i −0.987356 0.158519i \(-0.949328\pi\)
0.987356 0.158519i \(-0.0506720\pi\)
\(578\) 0 0
\(579\) 9.36932i 0.389376i
\(580\) 0 0
\(581\) −6.68466 −0.277326
\(582\) 0 0
\(583\) 18.2462i 0.755681i
\(584\) 0 0
\(585\) 12.6847 + 2.00000i 0.524446 + 0.0826898i
\(586\) 0 0
\(587\) 15.5616i 0.642294i −0.947029 0.321147i \(-0.895932\pi\)
0.947029 0.321147i \(-0.104068\pi\)
\(588\) 0 0
\(589\) −2.43845 −0.100474
\(590\) 0 0
\(591\) 3.36932i 0.138595i
\(592\) 0 0
\(593\) 37.4233i 1.53679i 0.639976 + 0.768395i \(0.278944\pi\)
−0.639976 + 0.768395i \(0.721056\pi\)
\(594\) 0 0
\(595\) 7.12311 0.292019
\(596\) 0 0
\(597\) −15.6155 −0.639101
\(598\) 0 0
\(599\) 18.4384 0.753375 0.376687 0.926340i \(-0.377063\pi\)
0.376687 + 0.926340i \(0.377063\pi\)
\(600\) 0 0
\(601\) −10.4924 −0.427995 −0.213997 0.976834i \(-0.568648\pi\)
−0.213997 + 0.976834i \(0.568648\pi\)
\(602\) 0 0
\(603\) 4.87689i 0.198602i
\(604\) 0 0
\(605\) 54.3002i 2.20762i
\(606\) 0 0
\(607\) 32.4924 1.31883 0.659413 0.751781i \(-0.270805\pi\)
0.659413 + 0.751781i \(0.270805\pi\)
\(608\) 0 0
\(609\) 7.56155i 0.306410i
\(610\) 0 0
\(611\) −1.56155 0.246211i −0.0631737 0.00996064i
\(612\) 0 0
\(613\) 3.12311i 0.126141i −0.998009 0.0630705i \(-0.979911\pi\)
0.998009 0.0630705i \(-0.0200893\pi\)
\(614\) 0 0
\(615\) −35.6155 −1.43616
\(616\) 0 0
\(617\) 19.8617i 0.799604i −0.916602 0.399802i \(-0.869079\pi\)
0.916602 0.399802i \(-0.130921\pi\)
\(618\) 0 0
\(619\) 14.2462i 0.572604i 0.958139 + 0.286302i \(0.0924260\pi\)
−0.958139 + 0.286302i \(0.907574\pi\)
\(620\) 0 0
\(621\) −8.68466 −0.348503
\(622\) 0 0
\(623\) 6.68466 0.267815
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 22.7386i 0.905211i 0.891711 + 0.452605i \(0.149505\pi\)
−0.891711 + 0.452605i \(0.850495\pi\)
\(632\) 0 0
\(633\) 1.56155 0.0620662
\(634\) 0 0
\(635\) 72.9848i 2.89632i
\(636\) 0 0
\(637\) −0.561553 + 3.56155i −0.0222495 + 0.141114i
\(638\) 0 0
\(639\) 15.3693i 0.608001i
\(640\) 0 0
\(641\) 21.3153 0.841905 0.420953 0.907083i \(-0.361696\pi\)
0.420953 + 0.907083i \(0.361696\pi\)
\(642\) 0 0
\(643\) 38.2462i 1.50828i 0.656712 + 0.754142i \(0.271947\pi\)
−0.656712 + 0.754142i \(0.728053\pi\)
\(644\) 0 0
\(645\) 19.8078i 0.779930i
\(646\) 0 0
\(647\) −17.3693 −0.682858 −0.341429 0.939908i \(-0.610911\pi\)
−0.341429 + 0.939908i \(0.610911\pi\)
\(648\) 0 0
\(649\) 35.2311 1.38294
\(650\) 0 0
\(651\) −1.56155 −0.0612021
\(652\) 0 0
\(653\) −28.2462 −1.10536 −0.552680 0.833394i \(-0.686395\pi\)
−0.552680 + 0.833394i \(0.686395\pi\)
\(654\) 0 0
\(655\) 53.8617i 2.10455i
\(656\) 0 0
\(657\) 6.43845i 0.251188i
\(658\) 0 0
\(659\) −45.6695 −1.77903 −0.889516 0.456905i \(-0.848958\pi\)
−0.889516 + 0.456905i \(0.848958\pi\)
\(660\) 0 0
\(661\) 42.0540i 1.63571i 0.575424 + 0.817855i \(0.304837\pi\)
−0.575424 + 0.817855i \(0.695163\pi\)
\(662\) 0 0
\(663\) −1.12311 + 7.12311i −0.0436178 + 0.276638i
\(664\) 0 0
\(665\) 5.56155i 0.215668i
\(666\) 0 0
\(667\) −65.6695 −2.54273
\(668\) 0 0
\(669\) 6.43845i 0.248925i
\(670\) 0 0
\(671\) 14.7386i 0.568979i
\(672\) 0 0
\(673\) 35.1771 1.35598 0.677988 0.735073i \(-0.262852\pi\)
0.677988 + 0.735073i \(0.262852\pi\)
\(674\) 0 0
\(675\) −7.68466 −0.295783
\(676\) 0 0
\(677\) −44.7386 −1.71945 −0.859723 0.510761i \(-0.829364\pi\)
−0.859723 + 0.510761i \(0.829364\pi\)
\(678\) 0 0
\(679\) 3.31534 0.127231
\(680\) 0 0
\(681\) 1.12311i 0.0430375i
\(682\) 0 0
\(683\) 23.3693i 0.894202i −0.894483 0.447101i \(-0.852456\pi\)
0.894483 0.447101i \(-0.147544\pi\)
\(684\) 0 0
\(685\) 68.9848 2.63578
\(686\) 0 0
\(687\) 5.36932i 0.204852i
\(688\) 0 0
\(689\) 2.00000 12.6847i 0.0761939 0.483247i
\(690\) 0 0
\(691\) 22.9309i 0.872331i 0.899866 + 0.436166i \(0.143664\pi\)
−0.899866 + 0.436166i \(0.856336\pi\)
\(692\) 0 0
\(693\) −5.12311 −0.194611
\(694\) 0 0
\(695\) 79.2311i 3.00541i
\(696\) 0 0
\(697\) 20.0000i 0.757554i
\(698\) 0 0
\(699\) −5.31534 −0.201045
\(700\) 0 0
\(701\) −47.6695 −1.80045 −0.900226 0.435423i \(-0.856599\pi\)
−0.900226 + 0.435423i \(0.856599\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.56155 0.0588115
\(706\) 0 0
\(707\) 15.3693i 0.578023i
\(708\) 0 0
\(709\) 34.2462i 1.28614i −0.765806 0.643072i \(-0.777660\pi\)
0.765806 0.643072i \(-0.222340\pi\)
\(710\) 0 0
\(711\) −8.68466 −0.325700
\(712\) 0 0
\(713\) 13.5616i 0.507884i
\(714\) 0 0
\(715\) −10.2462 + 64.9848i −0.383187 + 2.43030i
\(716\) 0 0
\(717\) 8.24621i 0.307960i
\(718\) 0 0
\(719\) 50.3542 1.87789 0.938947 0.344063i \(-0.111803\pi\)
0.938947 + 0.344063i \(0.111803\pi\)
\(720\) 0 0
\(721\) 18.2462i 0.679524i
\(722\) 0 0
\(723\) 14.0540i 0.522673i
\(724\) 0 0
\(725\) −58.1080 −2.15808
\(726\) 0 0
\(727\) 23.6155 0.875851 0.437926 0.899011i \(-0.355713\pi\)
0.437926 + 0.899011i \(0.355713\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 11.1231 0.411403
\(732\) 0 0
\(733\) 38.4384i 1.41976i 0.704324 + 0.709878i \(0.251250\pi\)
−0.704324 + 0.709878i \(0.748750\pi\)
\(734\) 0 0
\(735\) 3.56155i 0.131370i
\(736\) 0 0
\(737\) 24.9848 0.920329
\(738\) 0 0
\(739\) 33.7538i 1.24165i −0.783948 0.620827i \(-0.786797\pi\)
0.783948 0.620827i \(-0.213203\pi\)
\(740\) 0 0
\(741\) 5.56155 + 0.876894i 0.204309 + 0.0322135i
\(742\) 0 0
\(743\) 20.7386i 0.760827i −0.924817 0.380413i \(-0.875782\pi\)
0.924817 0.380413i \(-0.124218\pi\)
\(744\) 0 0
\(745\) −68.9848 −2.52741
\(746\) 0 0
\(747\) 6.68466i 0.244579i
\(748\) 0 0
\(749\) 16.4924i 0.602620i
\(750\) 0 0
\(751\) −33.5616 −1.22468 −0.612339 0.790595i \(-0.709771\pi\)
−0.612339 + 0.790595i \(0.709771\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) −42.7386 −1.55542
\(756\) 0 0
\(757\) 28.0540 1.01964 0.509820 0.860281i \(-0.329712\pi\)
0.509820 + 0.860281i \(0.329712\pi\)
\(758\) 0 0
\(759\) 44.4924i 1.61497i
\(760\) 0 0
\(761\) 4.93087i 0.178744i 0.995998 + 0.0893719i \(0.0284860\pi\)
−0.995998 + 0.0893719i \(0.971514\pi\)
\(762\) 0 0
\(763\) 7.12311 0.257874
\(764\) 0 0
\(765\) 7.12311i 0.257536i
\(766\) 0 0
\(767\) −24.4924 3.86174i −0.884370 0.139439i
\(768\) 0 0
\(769\) 2.82292i 0.101797i 0.998704 + 0.0508985i \(0.0162085\pi\)
−0.998704 + 0.0508985i \(0.983791\pi\)
\(770\) 0 0
\(771\) 19.3693 0.697569
\(772\) 0 0
\(773\) 32.2462i 1.15982i 0.814682 + 0.579908i \(0.196911\pi\)
−0.814682 + 0.579908i \(0.803089\pi\)
\(774\) 0 0
\(775\) 12.0000i 0.431053i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −15.6155 −0.559484
\(780\) 0 0
\(781\) −78.7386 −2.81749
\(782\) 0 0
\(783\) 7.56155 0.270228
\(784\) 0 0
\(785\) 29.3693i 1.04824i
\(786\) 0 0
\(787\) 11.8078i 0.420901i −0.977604 0.210451i \(-0.932507\pi\)
0.977604 0.210451i \(-0.0674931\pi\)
\(788\) 0 0
\(789\) 10.0540 0.357931
\(790\) 0 0
\(791\) 5.80776i 0.206500i
\(792\) 0 0
\(793\) −1.61553 + 10.2462i −0.0573691 + 0.363854i
\(794\) 0 0
\(795\) 12.6847i 0.449878i
\(796\) 0 0
\(797\) 41.2311 1.46048 0.730239 0.683191i \(-0.239408\pi\)
0.730239 + 0.683191i \(0.239408\pi\)
\(798\) 0 0
\(799\) 0.876894i 0.0310223i
\(800\) 0 0
\(801\) 6.68466i 0.236191i
\(802\) 0 0
\(803\) 32.9848 1.16401
\(804\) 0 0
\(805\) 30.9309 1.09017
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) 54.6847 1.92261 0.961305 0.275486i \(-0.0888387\pi\)
0.961305 + 0.275486i \(0.0888387\pi\)
\(810\) 0 0
\(811\) 36.0000i 1.26413i −0.774915 0.632065i \(-0.782207\pi\)
0.774915 0.632065i \(-0.217793\pi\)
\(812\) 0 0
\(813\) 26.7386i 0.937765i
\(814\) 0 0
\(815\) 53.8617 1.88669
\(816\) 0 0
\(817\) 8.68466i 0.303838i
\(818\) 0 0
\(819\) 3.56155 + 0.561553i 0.124451 + 0.0196222i
\(820\) 0 0
\(821\) 38.4924i 1.34339i 0.740826 + 0.671697i \(0.234434\pi\)
−0.740826 + 0.671697i \(0.765566\pi\)
\(822\) 0 0
\(823\) 6.73863 0.234894 0.117447 0.993079i \(-0.462529\pi\)
0.117447 + 0.993079i \(0.462529\pi\)
\(824\) 0 0
\(825\) 39.3693i 1.37066i
\(826\) 0 0
\(827\) 32.7386i 1.13843i −0.822187 0.569217i \(-0.807246\pi\)
0.822187 0.569217i \(-0.192754\pi\)
\(828\) 0 0
\(829\) 28.7386 0.998134 0.499067 0.866563i \(-0.333676\pi\)
0.499067 + 0.866563i \(0.333676\pi\)
\(830\) 0 0
\(831\) 8.43845 0.292726
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 12.6847 0.438971
\(836\) 0 0
\(837\) 1.56155i 0.0539752i
\(838\) 0 0
\(839\) 5.12311i 0.176869i −0.996082 0.0884346i \(-0.971814\pi\)
0.996082 0.0884346i \(-0.0281864\pi\)
\(840\) 0 0
\(841\) 28.1771 0.971623
\(842\) 0 0
\(843\) 18.0000i 0.619953i
\(844\) 0 0
\(845\) 14.2462 44.0540i 0.490085 1.51550i
\(846\) 0 0
\(847\) 15.2462i 0.523866i
\(848\) 0 0
\(849\) 26.2462 0.900768
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 17.0691i 0.584436i −0.956352 0.292218i \(-0.905607\pi\)
0.956352 0.292218i \(-0.0943933\pi\)
\(854\) 0 0
\(855\) −5.56155 −0.190201
\(856\) 0 0
\(857\) 0.630683 0.0215437 0.0107719 0.999942i \(-0.496571\pi\)
0.0107719 + 0.999942i \(0.496571\pi\)
\(858\) 0 0
\(859\) −7.12311 −0.243037 −0.121519 0.992589i \(-0.538776\pi\)
−0.121519 + 0.992589i \(0.538776\pi\)
\(860\) 0 0
\(861\) −10.0000 −0.340799
\(862\) 0 0
\(863\) 34.1080i 1.16105i −0.814243 0.580524i \(-0.802848\pi\)
0.814243 0.580524i \(-0.197152\pi\)
\(864\) 0 0
\(865\) 46.7386i 1.58916i
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 44.4924i 1.50930i
\(870\) 0 0
\(871\) −17.3693 2.73863i −0.588537 0.0927951i
\(872\) 0 0
\(873\) 3.31534i 0.112207i
\(874\) 0 0
\(875\) 9.56155 0.323239
\(876\) 0 0
\(877\) 31.6155i 1.06758i −0.845617 0.533790i \(-0.820767\pi\)
0.845617 0.533790i \(-0.179233\pi\)
\(878\) 0 0
\(879\) 20.9309i 0.705981i
\(880\) 0 0
\(881\) −23.7538 −0.800285 −0.400143 0.916453i \(-0.631039\pi\)
−0.400143 + 0.916453i \(0.631039\pi\)
\(882\) 0 0
\(883\) −26.2462 −0.883255 −0.441628 0.897198i \(-0.645599\pi\)
−0.441628 + 0.897198i \(0.645599\pi\)
\(884\) 0 0
\(885\) 24.4924 0.823304
\(886\) 0 0
\(887\) 13.8617 0.465432 0.232716 0.972545i \(-0.425239\pi\)
0.232716 + 0.972545i \(0.425239\pi\)
\(888\) 0 0
\(889\) 20.4924i 0.687294i
\(890\) 0 0
\(891\) 5.12311i 0.171630i
\(892\) 0 0
\(893\) 0.684658 0.0229112
\(894\) 0 0
\(895\) 22.9309i 0.766494i
\(896\) 0 0
\(897\) −4.87689 + 30.9309i −0.162835 + 1.03275i
\(898\) 0 0
\(899\) 11.8078i 0.393811i
\(900\) 0 0
\(901\) −7.12311 −0.237305
\(902\) 0 0
\(903\) 5.56155i 0.185077i
\(904\) 0 0
\(905\) 26.2462i 0.872454i
\(906\) 0 0
\(907\) 46.4384 1.54196 0.770982 0.636857i \(-0.219766\pi\)
0.770982 + 0.636857i \(0.219766\pi\)
\(908\) 0 0
\(909\) 15.3693 0.509768
\(910\) 0 0
\(911\) −4.19224 −0.138895 −0.0694475 0.997586i \(-0.522124\pi\)
−0.0694475 + 0.997586i \(0.522124\pi\)
\(912\) 0 0
\(913\) −34.2462 −1.13338
\(914\) 0 0
\(915\) 10.2462i 0.338729i
\(916\) 0 0
\(917\) 15.1231i 0.499409i
\(918\) 0 0
\(919\) 20.4924 0.675983 0.337991 0.941149i \(-0.390253\pi\)
0.337991 + 0.941149i \(0.390253\pi\)
\(920\) 0 0
\(921\) 11.8078i 0.389079i
\(922\) 0 0
\(923\) 54.7386 + 8.63068i 1.80174 + 0.284082i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 18.2462 0.599284
\(928\) 0 0
\(929\) 11.0691i 0.363166i 0.983376 + 0.181583i \(0.0581222\pi\)
−0.983376 + 0.181583i \(0.941878\pi\)
\(930\) 0 0
\(931\) 1.56155i 0.0511778i
\(932\) 0 0
\(933\) −15.6155 −0.511229
\(934\) 0 0
\(935\) 36.4924 1.19343
\(936\) 0 0
\(937\) −31.8617 −1.04088 −0.520439 0.853899i \(-0.674232\pi\)
−0.520439 + 0.853899i \(0.674232\pi\)
\(938\) 0 0
\(939\) −25.1231 −0.819862
\(940\) 0 0
\(941\) 15.0691i 0.491240i −0.969366 0.245620i \(-0.921009\pi\)
0.969366 0.245620i \(-0.0789915\pi\)
\(942\) 0 0
\(943\) 86.8466i 2.82811i
\(944\) 0 0
\(945\) −3.56155 −0.115857
\(946\) 0 0
\(947\) 14.4924i 0.470940i −0.971882 0.235470i \(-0.924337\pi\)
0.971882 0.235470i \(-0.0756630\pi\)
\(948\) 0 0
\(949\) −22.9309 3.61553i −0.744368 0.117365i
\(950\) 0 0
\(951\) 6.00000i 0.194563i
\(952\) 0 0
\(953\) −46.6847 −1.51226 −0.756132 0.654419i \(-0.772913\pi\)
−0.756132 + 0.654419i \(0.772913\pi\)
\(954\) 0 0
\(955\) 56.9848i 1.84399i
\(956\) 0 0
\(957\) 38.7386i 1.25224i
\(958\) 0 0
\(959\) 19.3693 0.625468
\(960\) 0 0
\(961\) 28.5616 0.921340
\(962\) 0 0
\(963\) −16.4924 −0.531461
\(964\) 0 0
\(965\) −33.3693 −1.07420
\(966\) 0 0
\(967\) 32.4924i 1.04489i 0.852674 + 0.522443i \(0.174979\pi\)
−0.852674 + 0.522443i \(0.825021\pi\)
\(968\) 0 0
\(969\) 3.12311i 0.100329i
\(970\) 0 0
\(971\) 57.4773 1.84453 0.922267 0.386554i \(-0.126335\pi\)
0.922267 + 0.386554i \(0.126335\pi\)
\(972\) 0 0
\(973\) 22.2462i 0.713181i
\(974\) 0 0
\(975\) −4.31534 + 27.3693i −0.138202 + 0.876520i
\(976\) 0 0
\(977\) 36.6307i 1.17192i 0.810340 + 0.585960i \(0.199282\pi\)
−0.810340 + 0.585960i \(0.800718\pi\)
\(978\) 0 0
\(979\) 34.2462 1.09451
\(980\) 0 0
\(981\) 7.12311i 0.227423i
\(982\) 0 0
\(983\) 32.4384i 1.03463i 0.855796 + 0.517313i \(0.173068\pi\)
−0.855796 + 0.517313i \(0.826932\pi\)
\(984\) 0 0
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0.438447 0.0139559
\(988\) 0 0
\(989\) 48.3002 1.53586
\(990\) 0 0
\(991\) 18.2462 0.579610 0.289805 0.957086i \(-0.406410\pi\)
0.289805 + 0.957086i \(0.406410\pi\)
\(992\) 0 0
\(993\) 9.75379i 0.309527i
\(994\) 0 0
\(995\) 55.6155i 1.76313i
\(996\) 0 0
\(997\) −30.1080 −0.953528 −0.476764 0.879031i \(-0.658190\pi\)
−0.476764 + 0.879031i \(0.658190\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4368.2.h.l.337.4 4
4.3 odd 2 1092.2.e.e.337.4 yes 4
12.11 even 2 3276.2.e.e.2521.1 4
13.12 even 2 inner 4368.2.h.l.337.1 4
28.27 even 2 7644.2.e.i.4705.1 4
52.51 odd 2 1092.2.e.e.337.1 4
156.155 even 2 3276.2.e.e.2521.4 4
364.363 even 2 7644.2.e.i.4705.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1092.2.e.e.337.1 4 52.51 odd 2
1092.2.e.e.337.4 yes 4 4.3 odd 2
3276.2.e.e.2521.1 4 12.11 even 2
3276.2.e.e.2521.4 4 156.155 even 2
4368.2.h.l.337.1 4 13.12 even 2 inner
4368.2.h.l.337.4 4 1.1 even 1 trivial
7644.2.e.i.4705.1 4 28.27 even 2
7644.2.e.i.4705.4 4 364.363 even 2