Properties

Label 2240.2.l.d.1569.9
Level $2240$
Weight $2$
Character 2240.1569
Analytic conductor $17.886$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1569,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.l (of order \(2\), degree \(1\), 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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1569.9
Character \(\chi\) \(=\) 2240.1569
Dual form 2240.2.l.d.1569.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.09381 q^{3} +(2.20510 - 0.370875i) q^{5} +1.00000i q^{7} -1.80358 q^{9} +O(q^{10})\) \(q-1.09381 q^{3} +(2.20510 - 0.370875i) q^{5} +1.00000i q^{7} -1.80358 q^{9} +1.07385i q^{11} -3.77197 q^{13} +(-2.41196 + 0.405667i) q^{15} -5.08737i q^{17} +0.279623i q^{19} -1.09381i q^{21} +4.13057i q^{23} +(4.72490 - 1.63563i) q^{25} +5.25421 q^{27} -4.87012i q^{29} +10.1243 q^{31} -1.17459i q^{33} +(0.370875 + 2.20510i) q^{35} +2.19379 q^{37} +4.12582 q^{39} +11.0451 q^{41} -1.86758 q^{43} +(-3.97706 + 0.668903i) q^{45} +4.78644i q^{47} -1.00000 q^{49} +5.56462i q^{51} +5.30210 q^{53} +(0.398266 + 2.36795i) q^{55} -0.305855i q^{57} +5.22153i q^{59} -14.5860i q^{61} -1.80358i q^{63} +(-8.31755 + 1.39893i) q^{65} -9.74025 q^{67} -4.51806i q^{69} +4.18240 q^{71} +12.4860i q^{73} +(-5.16815 + 1.78907i) q^{75} -1.07385 q^{77} +12.4280 q^{79} -0.336372 q^{81} -2.13284 q^{83} +(-1.88678 - 11.2181i) q^{85} +5.32699i q^{87} +8.20284 q^{89} -3.77197i q^{91} -11.0741 q^{93} +(0.103705 + 0.616596i) q^{95} +9.92547i q^{97} -1.93678i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 4 q^{5} + 12 q^{9} + 4 q^{13} + 24 q^{37} + 48 q^{45} - 24 q^{49} - 88 q^{53} + 36 q^{65} - 20 q^{77} + 16 q^{81} + 56 q^{85} - 40 q^{89} - 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.09381 −0.631512 −0.315756 0.948840i \(-0.602258\pi\)
−0.315756 + 0.948840i \(0.602258\pi\)
\(4\) 0 0
\(5\) 2.20510 0.370875i 0.986149 0.165860i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.80358 −0.601193
\(10\) 0 0
\(11\) 1.07385i 0.323779i 0.986809 + 0.161889i \(0.0517588\pi\)
−0.986809 + 0.161889i \(0.948241\pi\)
\(12\) 0 0
\(13\) −3.77197 −1.04616 −0.523078 0.852285i \(-0.675216\pi\)
−0.523078 + 0.852285i \(0.675216\pi\)
\(14\) 0 0
\(15\) −2.41196 + 0.405667i −0.622765 + 0.104743i
\(16\) 0 0
\(17\) 5.08737i 1.23387i −0.787015 0.616934i \(-0.788375\pi\)
0.787015 0.616934i \(-0.211625\pi\)
\(18\) 0 0
\(19\) 0.279623i 0.0641499i 0.999485 + 0.0320750i \(0.0102115\pi\)
−0.999485 + 0.0320750i \(0.989788\pi\)
\(20\) 0 0
\(21\) 1.09381i 0.238689i
\(22\) 0 0
\(23\) 4.13057i 0.861283i 0.902523 + 0.430642i \(0.141713\pi\)
−0.902523 + 0.430642i \(0.858287\pi\)
\(24\) 0 0
\(25\) 4.72490 1.63563i 0.944981 0.327126i
\(26\) 0 0
\(27\) 5.25421 1.01117
\(28\) 0 0
\(29\) 4.87012i 0.904359i −0.891927 0.452180i \(-0.850647\pi\)
0.891927 0.452180i \(-0.149353\pi\)
\(30\) 0 0
\(31\) 10.1243 1.81838 0.909191 0.416378i \(-0.136701\pi\)
0.909191 + 0.416378i \(0.136701\pi\)
\(32\) 0 0
\(33\) 1.17459i 0.204470i
\(34\) 0 0
\(35\) 0.370875 + 2.20510i 0.0626894 + 0.372729i
\(36\) 0 0
\(37\) 2.19379 0.360657 0.180329 0.983606i \(-0.442284\pi\)
0.180329 + 0.983606i \(0.442284\pi\)
\(38\) 0 0
\(39\) 4.12582 0.660660
\(40\) 0 0
\(41\) 11.0451 1.72495 0.862476 0.506097i \(-0.168912\pi\)
0.862476 + 0.506097i \(0.168912\pi\)
\(42\) 0 0
\(43\) −1.86758 −0.284804 −0.142402 0.989809i \(-0.545483\pi\)
−0.142402 + 0.989809i \(0.545483\pi\)
\(44\) 0 0
\(45\) −3.97706 + 0.668903i −0.592866 + 0.0997141i
\(46\) 0 0
\(47\) 4.78644i 0.698174i 0.937090 + 0.349087i \(0.113508\pi\)
−0.937090 + 0.349087i \(0.886492\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 5.56462i 0.779202i
\(52\) 0 0
\(53\) 5.30210 0.728299 0.364150 0.931340i \(-0.381360\pi\)
0.364150 + 0.931340i \(0.381360\pi\)
\(54\) 0 0
\(55\) 0.398266 + 2.36795i 0.0537021 + 0.319294i
\(56\) 0 0
\(57\) 0.305855i 0.0405114i
\(58\) 0 0
\(59\) 5.22153i 0.679785i 0.940464 + 0.339893i \(0.110391\pi\)
−0.940464 + 0.339893i \(0.889609\pi\)
\(60\) 0 0
\(61\) 14.5860i 1.86755i −0.357863 0.933774i \(-0.616495\pi\)
0.357863 0.933774i \(-0.383505\pi\)
\(62\) 0 0
\(63\) 1.80358i 0.227229i
\(64\) 0 0
\(65\) −8.31755 + 1.39893i −1.03167 + 0.173516i
\(66\) 0 0
\(67\) −9.74025 −1.18996 −0.594980 0.803740i \(-0.702840\pi\)
−0.594980 + 0.803740i \(0.702840\pi\)
\(68\) 0 0
\(69\) 4.51806i 0.543911i
\(70\) 0 0
\(71\) 4.18240 0.496360 0.248180 0.968714i \(-0.420168\pi\)
0.248180 + 0.968714i \(0.420168\pi\)
\(72\) 0 0
\(73\) 12.4860i 1.46138i 0.682709 + 0.730690i \(0.260802\pi\)
−0.682709 + 0.730690i \(0.739198\pi\)
\(74\) 0 0
\(75\) −5.16815 + 1.78907i −0.596767 + 0.206584i
\(76\) 0 0
\(77\) −1.07385 −0.122377
\(78\) 0 0
\(79\) 12.4280 1.39826 0.699131 0.714994i \(-0.253571\pi\)
0.699131 + 0.714994i \(0.253571\pi\)
\(80\) 0 0
\(81\) −0.336372 −0.0373747
\(82\) 0 0
\(83\) −2.13284 −0.234110 −0.117055 0.993125i \(-0.537345\pi\)
−0.117055 + 0.993125i \(0.537345\pi\)
\(84\) 0 0
\(85\) −1.88678 11.2181i −0.204650 1.21678i
\(86\) 0 0
\(87\) 5.32699i 0.571114i
\(88\) 0 0
\(89\) 8.20284 0.869500 0.434750 0.900551i \(-0.356837\pi\)
0.434750 + 0.900551i \(0.356837\pi\)
\(90\) 0 0
\(91\) 3.77197i 0.395410i
\(92\) 0 0
\(93\) −11.0741 −1.14833
\(94\) 0 0
\(95\) 0.103705 + 0.616596i 0.0106399 + 0.0632614i
\(96\) 0 0
\(97\) 9.92547i 1.00778i 0.863768 + 0.503889i \(0.168098\pi\)
−0.863768 + 0.503889i \(0.831902\pi\)
\(98\) 0 0
\(99\) 1.93678i 0.194654i
\(100\) 0 0
\(101\) 3.68390i 0.366562i 0.983061 + 0.183281i \(0.0586718\pi\)
−0.983061 + 0.183281i \(0.941328\pi\)
\(102\) 0 0
\(103\) 17.1850i 1.69329i −0.532157 0.846646i \(-0.678618\pi\)
0.532157 0.846646i \(-0.321382\pi\)
\(104\) 0 0
\(105\) −0.405667 2.41196i −0.0395891 0.235383i
\(106\) 0 0
\(107\) 9.64783 0.932691 0.466346 0.884603i \(-0.345570\pi\)
0.466346 + 0.884603i \(0.345570\pi\)
\(108\) 0 0
\(109\) 1.25100i 0.119824i 0.998204 + 0.0599121i \(0.0190820\pi\)
−0.998204 + 0.0599121i \(0.980918\pi\)
\(110\) 0 0
\(111\) −2.39959 −0.227759
\(112\) 0 0
\(113\) 12.9327i 1.21661i 0.793704 + 0.608305i \(0.208150\pi\)
−0.793704 + 0.608305i \(0.791850\pi\)
\(114\) 0 0
\(115\) 1.53193 + 9.10831i 0.142853 + 0.849354i
\(116\) 0 0
\(117\) 6.80304 0.628941
\(118\) 0 0
\(119\) 5.08737 0.466358
\(120\) 0 0
\(121\) 9.84684 0.895167
\(122\) 0 0
\(123\) −12.0812 −1.08933
\(124\) 0 0
\(125\) 9.81225 5.35908i 0.877635 0.479330i
\(126\) 0 0
\(127\) 8.66938i 0.769283i −0.923066 0.384641i \(-0.874325\pi\)
0.923066 0.384641i \(-0.125675\pi\)
\(128\) 0 0
\(129\) 2.04278 0.179857
\(130\) 0 0
\(131\) 15.7893i 1.37952i −0.724040 0.689758i \(-0.757717\pi\)
0.724040 0.689758i \(-0.242283\pi\)
\(132\) 0 0
\(133\) −0.279623 −0.0242464
\(134\) 0 0
\(135\) 11.5860 1.94865i 0.997167 0.167714i
\(136\) 0 0
\(137\) 14.1779i 1.21130i −0.795730 0.605652i \(-0.792913\pi\)
0.795730 0.605652i \(-0.207087\pi\)
\(138\) 0 0
\(139\) 18.8789i 1.60129i 0.599141 + 0.800644i \(0.295509\pi\)
−0.599141 + 0.800644i \(0.704491\pi\)
\(140\) 0 0
\(141\) 5.23546i 0.440905i
\(142\) 0 0
\(143\) 4.05054i 0.338723i
\(144\) 0 0
\(145\) −1.80621 10.7391i −0.149997 0.891833i
\(146\) 0 0
\(147\) 1.09381 0.0902160
\(148\) 0 0
\(149\) 5.60627i 0.459283i −0.973275 0.229642i \(-0.926245\pi\)
0.973275 0.229642i \(-0.0737554\pi\)
\(150\) 0 0
\(151\) 12.8279 1.04392 0.521959 0.852971i \(-0.325201\pi\)
0.521959 + 0.852971i \(0.325201\pi\)
\(152\) 0 0
\(153\) 9.17547i 0.741792i
\(154\) 0 0
\(155\) 22.3251 3.75486i 1.79320 0.301598i
\(156\) 0 0
\(157\) 5.49231 0.438334 0.219167 0.975687i \(-0.429666\pi\)
0.219167 + 0.975687i \(0.429666\pi\)
\(158\) 0 0
\(159\) −5.79949 −0.459930
\(160\) 0 0
\(161\) −4.13057 −0.325535
\(162\) 0 0
\(163\) −17.7379 −1.38934 −0.694670 0.719329i \(-0.744450\pi\)
−0.694670 + 0.719329i \(0.744450\pi\)
\(164\) 0 0
\(165\) −0.435627 2.59009i −0.0339135 0.201638i
\(166\) 0 0
\(167\) 3.70432i 0.286649i 0.989676 + 0.143325i \(0.0457793\pi\)
−0.989676 + 0.143325i \(0.954221\pi\)
\(168\) 0 0
\(169\) 1.22774 0.0944413
\(170\) 0 0
\(171\) 0.504322i 0.0385665i
\(172\) 0 0
\(173\) 8.57366 0.651843 0.325922 0.945397i \(-0.394325\pi\)
0.325922 + 0.945397i \(0.394325\pi\)
\(174\) 0 0
\(175\) 1.63563 + 4.72490i 0.123642 + 0.357169i
\(176\) 0 0
\(177\) 5.71136i 0.429292i
\(178\) 0 0
\(179\) 17.5839i 1.31428i 0.753768 + 0.657141i \(0.228234\pi\)
−0.753768 + 0.657141i \(0.771766\pi\)
\(180\) 0 0
\(181\) 5.02458i 0.373474i 0.982410 + 0.186737i \(0.0597913\pi\)
−0.982410 + 0.186737i \(0.940209\pi\)
\(182\) 0 0
\(183\) 15.9543i 1.17938i
\(184\) 0 0
\(185\) 4.83752 0.813623i 0.355662 0.0598188i
\(186\) 0 0
\(187\) 5.46309 0.399501
\(188\) 0 0
\(189\) 5.25421i 0.382187i
\(190\) 0 0
\(191\) 0.983348 0.0711525 0.0355763 0.999367i \(-0.488673\pi\)
0.0355763 + 0.999367i \(0.488673\pi\)
\(192\) 0 0
\(193\) 6.55980i 0.472185i −0.971731 0.236092i \(-0.924133\pi\)
0.971731 0.236092i \(-0.0758668\pi\)
\(194\) 0 0
\(195\) 9.09783 1.53016i 0.651509 0.109577i
\(196\) 0 0
\(197\) 5.24696 0.373831 0.186915 0.982376i \(-0.440151\pi\)
0.186915 + 0.982376i \(0.440151\pi\)
\(198\) 0 0
\(199\) −6.69606 −0.474671 −0.237336 0.971428i \(-0.576274\pi\)
−0.237336 + 0.971428i \(0.576274\pi\)
\(200\) 0 0
\(201\) 10.6540 0.751474
\(202\) 0 0
\(203\) 4.87012 0.341816
\(204\) 0 0
\(205\) 24.3555 4.09635i 1.70106 0.286101i
\(206\) 0 0
\(207\) 7.44981i 0.517797i
\(208\) 0 0
\(209\) −0.300274 −0.0207704
\(210\) 0 0
\(211\) 5.43877i 0.374420i −0.982320 0.187210i \(-0.940056\pi\)
0.982320 0.187210i \(-0.0599445\pi\)
\(212\) 0 0
\(213\) −4.57476 −0.313457
\(214\) 0 0
\(215\) −4.11820 + 0.692641i −0.280859 + 0.0472377i
\(216\) 0 0
\(217\) 10.1243i 0.687284i
\(218\) 0 0
\(219\) 13.6574i 0.922879i
\(220\) 0 0
\(221\) 19.1894i 1.29082i
\(222\) 0 0
\(223\) 1.81776i 0.121726i 0.998146 + 0.0608630i \(0.0193853\pi\)
−0.998146 + 0.0608630i \(0.980615\pi\)
\(224\) 0 0
\(225\) −8.52173 + 2.94999i −0.568115 + 0.196666i
\(226\) 0 0
\(227\) 11.2918 0.749463 0.374732 0.927133i \(-0.377735\pi\)
0.374732 + 0.927133i \(0.377735\pi\)
\(228\) 0 0
\(229\) 16.4723i 1.08852i −0.838916 0.544262i \(-0.816810\pi\)
0.838916 0.544262i \(-0.183190\pi\)
\(230\) 0 0
\(231\) 1.17459 0.0772825
\(232\) 0 0
\(233\) 4.41351i 0.289139i −0.989495 0.144569i \(-0.953820\pi\)
0.989495 0.144569i \(-0.0461797\pi\)
\(234\) 0 0
\(235\) 1.77517 + 10.5546i 0.115800 + 0.688504i
\(236\) 0 0
\(237\) −13.5939 −0.883019
\(238\) 0 0
\(239\) 11.8480 0.766385 0.383192 0.923669i \(-0.374825\pi\)
0.383192 + 0.923669i \(0.374825\pi\)
\(240\) 0 0
\(241\) −7.85453 −0.505955 −0.252977 0.967472i \(-0.581410\pi\)
−0.252977 + 0.967472i \(0.581410\pi\)
\(242\) 0 0
\(243\) −15.3947 −0.987570
\(244\) 0 0
\(245\) −2.20510 + 0.370875i −0.140878 + 0.0236944i
\(246\) 0 0
\(247\) 1.05473i 0.0671108i
\(248\) 0 0
\(249\) 2.33292 0.147843
\(250\) 0 0
\(251\) 23.3008i 1.47073i 0.677671 + 0.735365i \(0.262989\pi\)
−0.677671 + 0.735365i \(0.737011\pi\)
\(252\) 0 0
\(253\) −4.43563 −0.278865
\(254\) 0 0
\(255\) 2.06378 + 12.2705i 0.129239 + 0.768410i
\(256\) 0 0
\(257\) 2.07004i 0.129126i −0.997914 0.0645629i \(-0.979435\pi\)
0.997914 0.0645629i \(-0.0205653\pi\)
\(258\) 0 0
\(259\) 2.19379i 0.136316i
\(260\) 0 0
\(261\) 8.78365i 0.543694i
\(262\) 0 0
\(263\) 30.4778i 1.87934i −0.342084 0.939669i \(-0.611133\pi\)
0.342084 0.939669i \(-0.388867\pi\)
\(264\) 0 0
\(265\) 11.6916 1.96642i 0.718212 0.120796i
\(266\) 0 0
\(267\) −8.97236 −0.549099
\(268\) 0 0
\(269\) 22.1810i 1.35240i −0.736718 0.676200i \(-0.763625\pi\)
0.736718 0.676200i \(-0.236375\pi\)
\(270\) 0 0
\(271\) 4.03998 0.245411 0.122706 0.992443i \(-0.460843\pi\)
0.122706 + 0.992443i \(0.460843\pi\)
\(272\) 0 0
\(273\) 4.12582i 0.249706i
\(274\) 0 0
\(275\) 1.75643 + 5.07385i 0.105917 + 0.305965i
\(276\) 0 0
\(277\) 16.0433 0.963946 0.481973 0.876186i \(-0.339920\pi\)
0.481973 + 0.876186i \(0.339920\pi\)
\(278\) 0 0
\(279\) −18.2600 −1.09320
\(280\) 0 0
\(281\) 6.48827 0.387058 0.193529 0.981095i \(-0.438007\pi\)
0.193529 + 0.981095i \(0.438007\pi\)
\(282\) 0 0
\(283\) 7.48606 0.445000 0.222500 0.974933i \(-0.428578\pi\)
0.222500 + 0.974933i \(0.428578\pi\)
\(284\) 0 0
\(285\) −0.113434 0.674439i −0.00671925 0.0399503i
\(286\) 0 0
\(287\) 11.0451i 0.651971i
\(288\) 0 0
\(289\) −8.88131 −0.522430
\(290\) 0 0
\(291\) 10.8566i 0.636424i
\(292\) 0 0
\(293\) −19.3728 −1.13177 −0.565885 0.824484i \(-0.691465\pi\)
−0.565885 + 0.824484i \(0.691465\pi\)
\(294\) 0 0
\(295\) 1.93654 + 11.5140i 0.112749 + 0.670370i
\(296\) 0 0
\(297\) 5.64225i 0.327396i
\(298\) 0 0
\(299\) 15.5804i 0.901036i
\(300\) 0 0
\(301\) 1.86758i 0.107646i
\(302\) 0 0
\(303\) 4.02949i 0.231488i
\(304\) 0 0
\(305\) −5.40959 32.1636i −0.309752 1.84168i
\(306\) 0 0
\(307\) 20.9673 1.19667 0.598335 0.801246i \(-0.295829\pi\)
0.598335 + 0.801246i \(0.295829\pi\)
\(308\) 0 0
\(309\) 18.7972i 1.06933i
\(310\) 0 0
\(311\) −15.9522 −0.904566 −0.452283 0.891875i \(-0.649390\pi\)
−0.452283 + 0.891875i \(0.649390\pi\)
\(312\) 0 0
\(313\) 10.8486i 0.613199i 0.951839 + 0.306599i \(0.0991912\pi\)
−0.951839 + 0.306599i \(0.900809\pi\)
\(314\) 0 0
\(315\) −0.668903 3.97706i −0.0376884 0.224082i
\(316\) 0 0
\(317\) −10.4714 −0.588132 −0.294066 0.955785i \(-0.595009\pi\)
−0.294066 + 0.955785i \(0.595009\pi\)
\(318\) 0 0
\(319\) 5.22980 0.292812
\(320\) 0 0
\(321\) −10.5529 −0.589006
\(322\) 0 0
\(323\) 1.42255 0.0791525
\(324\) 0 0
\(325\) −17.8222 + 6.16955i −0.988597 + 0.342225i
\(326\) 0 0
\(327\) 1.36836i 0.0756704i
\(328\) 0 0
\(329\) −4.78644 −0.263885
\(330\) 0 0
\(331\) 28.2958i 1.55528i −0.628710 0.777640i \(-0.716417\pi\)
0.628710 0.777640i \(-0.283583\pi\)
\(332\) 0 0
\(333\) −3.95667 −0.216824
\(334\) 0 0
\(335\) −21.4782 + 3.61242i −1.17348 + 0.197367i
\(336\) 0 0
\(337\) 22.5822i 1.23013i −0.788477 0.615064i \(-0.789130\pi\)
0.788477 0.615064i \(-0.210870\pi\)
\(338\) 0 0
\(339\) 14.1460i 0.768303i
\(340\) 0 0
\(341\) 10.8720i 0.588754i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −1.67564 9.96276i −0.0902133 0.536377i
\(346\) 0 0
\(347\) −28.5947 −1.53505 −0.767523 0.641022i \(-0.778511\pi\)
−0.767523 + 0.641022i \(0.778511\pi\)
\(348\) 0 0
\(349\) 15.5209i 0.830812i 0.909636 + 0.415406i \(0.136361\pi\)
−0.909636 + 0.415406i \(0.863639\pi\)
\(350\) 0 0
\(351\) −19.8187 −1.05784
\(352\) 0 0
\(353\) 29.7066i 1.58112i 0.612383 + 0.790561i \(0.290211\pi\)
−0.612383 + 0.790561i \(0.709789\pi\)
\(354\) 0 0
\(355\) 9.22261 1.55115i 0.489485 0.0823265i
\(356\) 0 0
\(357\) −5.56462 −0.294511
\(358\) 0 0
\(359\) −22.4817 −1.18654 −0.593270 0.805003i \(-0.702163\pi\)
−0.593270 + 0.805003i \(0.702163\pi\)
\(360\) 0 0
\(361\) 18.9218 0.995885
\(362\) 0 0
\(363\) −10.7706 −0.565309
\(364\) 0 0
\(365\) 4.63076 + 27.5329i 0.242385 + 1.44114i
\(366\) 0 0
\(367\) 23.8919i 1.24715i 0.781764 + 0.623574i \(0.214320\pi\)
−0.781764 + 0.623574i \(0.785680\pi\)
\(368\) 0 0
\(369\) −19.9207 −1.03703
\(370\) 0 0
\(371\) 5.30210i 0.275271i
\(372\) 0 0
\(373\) −3.52111 −0.182316 −0.0911581 0.995836i \(-0.529057\pi\)
−0.0911581 + 0.995836i \(0.529057\pi\)
\(374\) 0 0
\(375\) −10.7327 + 5.86181i −0.554237 + 0.302703i
\(376\) 0 0
\(377\) 18.3699i 0.946100i
\(378\) 0 0
\(379\) 2.35864i 0.121155i −0.998163 0.0605775i \(-0.980706\pi\)
0.998163 0.0605775i \(-0.0192942\pi\)
\(380\) 0 0
\(381\) 9.48266i 0.485811i
\(382\) 0 0
\(383\) 8.63272i 0.441111i −0.975374 0.220556i \(-0.929213\pi\)
0.975374 0.220556i \(-0.0707871\pi\)
\(384\) 0 0
\(385\) −2.36795 + 0.398266i −0.120682 + 0.0202975i
\(386\) 0 0
\(387\) 3.36833 0.171222
\(388\) 0 0
\(389\) 16.3195i 0.827430i 0.910406 + 0.413715i \(0.135769\pi\)
−0.910406 + 0.413715i \(0.864231\pi\)
\(390\) 0 0
\(391\) 21.0137 1.06271
\(392\) 0 0
\(393\) 17.2705i 0.871181i
\(394\) 0 0
\(395\) 27.4050 4.60925i 1.37889 0.231916i
\(396\) 0 0
\(397\) −0.404753 −0.0203139 −0.0101570 0.999948i \(-0.503233\pi\)
−0.0101570 + 0.999948i \(0.503233\pi\)
\(398\) 0 0
\(399\) 0.305855 0.0153119
\(400\) 0 0
\(401\) −25.0436 −1.25062 −0.625308 0.780378i \(-0.715027\pi\)
−0.625308 + 0.780378i \(0.715027\pi\)
\(402\) 0 0
\(403\) −38.1886 −1.90231
\(404\) 0 0
\(405\) −0.741733 + 0.124752i −0.0368570 + 0.00619898i
\(406\) 0 0
\(407\) 2.35581i 0.116773i
\(408\) 0 0
\(409\) 13.8580 0.685236 0.342618 0.939475i \(-0.388686\pi\)
0.342618 + 0.939475i \(0.388686\pi\)
\(410\) 0 0
\(411\) 15.5080i 0.764952i
\(412\) 0 0
\(413\) −5.22153 −0.256935
\(414\) 0 0
\(415\) −4.70312 + 0.791018i −0.230867 + 0.0388295i
\(416\) 0 0
\(417\) 20.6499i 1.01123i
\(418\) 0 0
\(419\) 26.9313i 1.31568i −0.753158 0.657839i \(-0.771471\pi\)
0.753158 0.657839i \(-0.228529\pi\)
\(420\) 0 0
\(421\) 34.5126i 1.68204i −0.541002 0.841022i \(-0.681955\pi\)
0.541002 0.841022i \(-0.318045\pi\)
\(422\) 0 0
\(423\) 8.63272i 0.419737i
\(424\) 0 0
\(425\) −8.32106 24.0373i −0.403631 1.16598i
\(426\) 0 0
\(427\) 14.5860 0.705867
\(428\) 0 0
\(429\) 4.43052i 0.213908i
\(430\) 0 0
\(431\) −17.0970 −0.823532 −0.411766 0.911290i \(-0.635088\pi\)
−0.411766 + 0.911290i \(0.635088\pi\)
\(432\) 0 0
\(433\) 26.2904i 1.26344i 0.775197 + 0.631719i \(0.217650\pi\)
−0.775197 + 0.631719i \(0.782350\pi\)
\(434\) 0 0
\(435\) 1.97565 + 11.7465i 0.0947252 + 0.563203i
\(436\) 0 0
\(437\) −1.15500 −0.0552513
\(438\) 0 0
\(439\) −0.881339 −0.0420640 −0.0210320 0.999779i \(-0.506695\pi\)
−0.0210320 + 0.999779i \(0.506695\pi\)
\(440\) 0 0
\(441\) 1.80358 0.0858847
\(442\) 0 0
\(443\) −38.3910 −1.82401 −0.912005 0.410178i \(-0.865466\pi\)
−0.912005 + 0.410178i \(0.865466\pi\)
\(444\) 0 0
\(445\) 18.0881 3.04223i 0.857456 0.144216i
\(446\) 0 0
\(447\) 6.13219i 0.290043i
\(448\) 0 0
\(449\) −42.1553 −1.98943 −0.994716 0.102667i \(-0.967262\pi\)
−0.994716 + 0.102667i \(0.967262\pi\)
\(450\) 0 0
\(451\) 11.8608i 0.558503i
\(452\) 0 0
\(453\) −14.0313 −0.659247
\(454\) 0 0
\(455\) −1.39893 8.31755i −0.0655828 0.389933i
\(456\) 0 0
\(457\) 21.6832i 1.01430i −0.861858 0.507149i \(-0.830699\pi\)
0.861858 0.507149i \(-0.169301\pi\)
\(458\) 0 0
\(459\) 26.7301i 1.24765i
\(460\) 0 0
\(461\) 33.8367i 1.57593i 0.615718 + 0.787966i \(0.288866\pi\)
−0.615718 + 0.787966i \(0.711134\pi\)
\(462\) 0 0
\(463\) 23.1430i 1.07555i −0.843089 0.537773i \(-0.819266\pi\)
0.843089 0.537773i \(-0.180734\pi\)
\(464\) 0 0
\(465\) −24.4195 + 4.10711i −1.13243 + 0.190463i
\(466\) 0 0
\(467\) −20.0911 −0.929706 −0.464853 0.885388i \(-0.653893\pi\)
−0.464853 + 0.885388i \(0.653893\pi\)
\(468\) 0 0
\(469\) 9.74025i 0.449763i
\(470\) 0 0
\(471\) −6.00755 −0.276813
\(472\) 0 0
\(473\) 2.00551i 0.0922135i
\(474\) 0 0
\(475\) 0.457360 + 1.32119i 0.0209851 + 0.0606204i
\(476\) 0 0
\(477\) −9.56275 −0.437848
\(478\) 0 0
\(479\) −26.3459 −1.20378 −0.601888 0.798580i \(-0.705585\pi\)
−0.601888 + 0.798580i \(0.705585\pi\)
\(480\) 0 0
\(481\) −8.27491 −0.377303
\(482\) 0 0
\(483\) 4.51806 0.205579
\(484\) 0 0
\(485\) 3.68111 + 21.8866i 0.167151 + 0.993820i
\(486\) 0 0
\(487\) 0.00797075i 0.000361189i −1.00000 0.000180595i \(-0.999943\pi\)
1.00000 0.000180595i \(-5.74850e-5\pi\)
\(488\) 0 0
\(489\) 19.4019 0.877385
\(490\) 0 0
\(491\) 19.3750i 0.874379i −0.899369 0.437190i \(-0.855974\pi\)
0.899369 0.437190i \(-0.144026\pi\)
\(492\) 0 0
\(493\) −24.7761 −1.11586
\(494\) 0 0
\(495\) −0.718303 4.27078i −0.0322853 0.191957i
\(496\) 0 0
\(497\) 4.18240i 0.187606i
\(498\) 0 0
\(499\) 20.6514i 0.924484i 0.886754 + 0.462242i \(0.152955\pi\)
−0.886754 + 0.462242i \(0.847045\pi\)
\(500\) 0 0
\(501\) 4.05183i 0.181022i
\(502\) 0 0
\(503\) 15.6811i 0.699184i 0.936902 + 0.349592i \(0.113680\pi\)
−0.936902 + 0.349592i \(0.886320\pi\)
\(504\) 0 0
\(505\) 1.36627 + 8.12336i 0.0607981 + 0.361485i
\(506\) 0 0
\(507\) −1.34291 −0.0596408
\(508\) 0 0
\(509\) 28.3123i 1.25492i 0.778648 + 0.627461i \(0.215906\pi\)
−0.778648 + 0.627461i \(0.784094\pi\)
\(510\) 0 0
\(511\) −12.4860 −0.552350
\(512\) 0 0
\(513\) 1.46920i 0.0648666i
\(514\) 0 0
\(515\) −6.37351 37.8947i −0.280850 1.66984i
\(516\) 0 0
\(517\) −5.13994 −0.226054
\(518\) 0 0
\(519\) −9.37796 −0.411647
\(520\) 0 0
\(521\) 8.94766 0.392004 0.196002 0.980604i \(-0.437204\pi\)
0.196002 + 0.980604i \(0.437204\pi\)
\(522\) 0 0
\(523\) −33.7287 −1.47485 −0.737425 0.675429i \(-0.763959\pi\)
−0.737425 + 0.675429i \(0.763959\pi\)
\(524\) 0 0
\(525\) −1.78907 5.16815i −0.0780815 0.225557i
\(526\) 0 0
\(527\) 51.5062i 2.24364i
\(528\) 0 0
\(529\) 5.93839 0.258191
\(530\) 0 0
\(531\) 9.41743i 0.408682i
\(532\) 0 0
\(533\) −41.6617 −1.80457
\(534\) 0 0
\(535\) 21.2744 3.57814i 0.919773 0.154697i
\(536\) 0 0
\(537\) 19.2335i 0.829985i
\(538\) 0 0
\(539\) 1.07385i 0.0462541i
\(540\) 0 0
\(541\) 31.4226i 1.35096i −0.737378 0.675481i \(-0.763936\pi\)
0.737378 0.675481i \(-0.236064\pi\)
\(542\) 0 0
\(543\) 5.49594i 0.235853i
\(544\) 0 0
\(545\) 0.463966 + 2.75858i 0.0198741 + 0.118165i
\(546\) 0 0
\(547\) 34.8319 1.48930 0.744652 0.667453i \(-0.232615\pi\)
0.744652 + 0.667453i \(0.232615\pi\)
\(548\) 0 0
\(549\) 26.3070i 1.12276i
\(550\) 0 0
\(551\) 1.36180 0.0580146
\(552\) 0 0
\(553\) 12.4280i 0.528493i
\(554\) 0 0
\(555\) −5.29134 + 0.889950i −0.224605 + 0.0377763i
\(556\) 0 0
\(557\) 30.1339 1.27682 0.638408 0.769698i \(-0.279593\pi\)
0.638408 + 0.769698i \(0.279593\pi\)
\(558\) 0 0
\(559\) 7.04447 0.297949
\(560\) 0 0
\(561\) −5.97558 −0.252289
\(562\) 0 0
\(563\) 13.7846 0.580951 0.290476 0.956882i \(-0.406186\pi\)
0.290476 + 0.956882i \(0.406186\pi\)
\(564\) 0 0
\(565\) 4.79643 + 28.5179i 0.201787 + 1.19976i
\(566\) 0 0
\(567\) 0.336372i 0.0141263i
\(568\) 0 0
\(569\) 25.4658 1.06758 0.533791 0.845616i \(-0.320767\pi\)
0.533791 + 0.845616i \(0.320767\pi\)
\(570\) 0 0
\(571\) 15.8001i 0.661213i 0.943769 + 0.330606i \(0.107253\pi\)
−0.943769 + 0.330606i \(0.892747\pi\)
\(572\) 0 0
\(573\) −1.07560 −0.0449337
\(574\) 0 0
\(575\) 6.75609 + 19.5165i 0.281749 + 0.813896i
\(576\) 0 0
\(577\) 18.1131i 0.754057i 0.926202 + 0.377028i \(0.123054\pi\)
−0.926202 + 0.377028i \(0.876946\pi\)
\(578\) 0 0
\(579\) 7.17518i 0.298190i
\(580\) 0 0
\(581\) 2.13284i 0.0884851i
\(582\) 0 0
\(583\) 5.69368i 0.235808i
\(584\) 0 0
\(585\) 15.0014 2.52308i 0.620230 0.104316i
\(586\) 0 0
\(587\) 37.8724 1.56316 0.781581 0.623804i \(-0.214414\pi\)
0.781581 + 0.623804i \(0.214414\pi\)
\(588\) 0 0
\(589\) 2.83100i 0.116649i
\(590\) 0 0
\(591\) −5.73918 −0.236078
\(592\) 0 0
\(593\) 18.1567i 0.745604i −0.927911 0.372802i \(-0.878397\pi\)
0.927911 0.372802i \(-0.121603\pi\)
\(594\) 0 0
\(595\) 11.2181 1.88678i 0.459899 0.0773504i
\(596\) 0 0
\(597\) 7.32422 0.299761
\(598\) 0 0
\(599\) −21.8947 −0.894592 −0.447296 0.894386i \(-0.647613\pi\)
−0.447296 + 0.894386i \(0.647613\pi\)
\(600\) 0 0
\(601\) −29.1613 −1.18951 −0.594757 0.803906i \(-0.702752\pi\)
−0.594757 + 0.803906i \(0.702752\pi\)
\(602\) 0 0
\(603\) 17.5673 0.715395
\(604\) 0 0
\(605\) 21.7132 3.65195i 0.882768 0.148473i
\(606\) 0 0
\(607\) 28.0842i 1.13990i 0.821679 + 0.569950i \(0.193038\pi\)
−0.821679 + 0.569950i \(0.806962\pi\)
\(608\) 0 0
\(609\) −5.32699 −0.215861
\(610\) 0 0
\(611\) 18.0543i 0.730399i
\(612\) 0 0
\(613\) 17.1906 0.694321 0.347160 0.937806i \(-0.387146\pi\)
0.347160 + 0.937806i \(0.387146\pi\)
\(614\) 0 0
\(615\) −26.6403 + 4.48063i −1.07424 + 0.180676i
\(616\) 0 0
\(617\) 14.5274i 0.584851i 0.956288 + 0.292425i \(0.0944623\pi\)
−0.956288 + 0.292425i \(0.905538\pi\)
\(618\) 0 0
\(619\) 12.9968i 0.522387i 0.965287 + 0.261193i \(0.0841160\pi\)
−0.965287 + 0.261193i \(0.915884\pi\)
\(620\) 0 0
\(621\) 21.7029i 0.870906i
\(622\) 0 0
\(623\) 8.20284i 0.328640i
\(624\) 0 0
\(625\) 19.6494 15.4564i 0.785977 0.618256i
\(626\) 0 0
\(627\) 0.328443 0.0131168
\(628\) 0 0
\(629\) 11.1606i 0.445003i
\(630\) 0 0
\(631\) −15.6079 −0.621341 −0.310671 0.950518i \(-0.600554\pi\)
−0.310671 + 0.950518i \(0.600554\pi\)
\(632\) 0 0
\(633\) 5.94898i 0.236451i
\(634\) 0 0
\(635\) −3.21526 19.1168i −0.127594 0.758628i
\(636\) 0 0
\(637\) 3.77197 0.149451
\(638\) 0 0
\(639\) −7.54329 −0.298408
\(640\) 0 0
\(641\) −24.6906 −0.975222 −0.487611 0.873061i \(-0.662132\pi\)
−0.487611 + 0.873061i \(0.662132\pi\)
\(642\) 0 0
\(643\) 24.8495 0.979969 0.489984 0.871731i \(-0.337002\pi\)
0.489984 + 0.871731i \(0.337002\pi\)
\(644\) 0 0
\(645\) 4.50453 0.757618i 0.177366 0.0298312i
\(646\) 0 0
\(647\) 1.76410i 0.0693539i −0.999399 0.0346770i \(-0.988960\pi\)
0.999399 0.0346770i \(-0.0110402\pi\)
\(648\) 0 0
\(649\) −5.60716 −0.220100
\(650\) 0 0
\(651\) 11.0741i 0.434028i
\(652\) 0 0
\(653\) 37.2584 1.45803 0.729017 0.684496i \(-0.239978\pi\)
0.729017 + 0.684496i \(0.239978\pi\)
\(654\) 0 0
\(655\) −5.85586 34.8169i −0.228807 1.36041i
\(656\) 0 0
\(657\) 22.5196i 0.878571i
\(658\) 0 0
\(659\) 35.5909i 1.38642i 0.720734 + 0.693212i \(0.243805\pi\)
−0.720734 + 0.693212i \(0.756195\pi\)
\(660\) 0 0
\(661\) 23.9986i 0.933438i 0.884406 + 0.466719i \(0.154564\pi\)
−0.884406 + 0.466719i \(0.845436\pi\)
\(662\) 0 0
\(663\) 20.9896i 0.815167i
\(664\) 0 0
\(665\) −0.616596 + 0.103705i −0.0239106 + 0.00402152i
\(666\) 0 0
\(667\) 20.1164 0.778910
\(668\) 0 0
\(669\) 1.98828i 0.0768714i
\(670\) 0 0
\(671\) 15.6632 0.604673
\(672\) 0 0
\(673\) 28.6776i 1.10544i 0.833367 + 0.552720i \(0.186410\pi\)
−0.833367 + 0.552720i \(0.813590\pi\)
\(674\) 0 0
\(675\) 24.8256 8.59395i 0.955538 0.330781i
\(676\) 0 0
\(677\) 33.2371 1.27741 0.638703 0.769454i \(-0.279471\pi\)
0.638703 + 0.769454i \(0.279471\pi\)
\(678\) 0 0
\(679\) −9.92547 −0.380905
\(680\) 0 0
\(681\) −12.3511 −0.473295
\(682\) 0 0
\(683\) −36.6848 −1.40370 −0.701852 0.712322i \(-0.747643\pi\)
−0.701852 + 0.712322i \(0.747643\pi\)
\(684\) 0 0
\(685\) −5.25825 31.2637i −0.200907 1.19453i
\(686\) 0 0
\(687\) 18.0176i 0.687415i
\(688\) 0 0
\(689\) −19.9993 −0.761914
\(690\) 0 0
\(691\) 24.0123i 0.913471i 0.889602 + 0.456736i \(0.150981\pi\)
−0.889602 + 0.456736i \(0.849019\pi\)
\(692\) 0 0
\(693\) 1.93678 0.0735721
\(694\) 0 0
\(695\) 7.00172 + 41.6298i 0.265590 + 1.57911i
\(696\) 0 0
\(697\) 56.1904i 2.12836i
\(698\) 0 0
\(699\) 4.82754i 0.182595i
\(700\) 0 0
\(701\) 24.8234i 0.937566i 0.883313 + 0.468783i \(0.155307\pi\)
−0.883313 + 0.468783i \(0.844693\pi\)
\(702\) 0 0
\(703\) 0.613435i 0.0231361i
\(704\) 0 0
\(705\) −1.94170 11.5447i −0.0731288 0.434798i
\(706\) 0 0
\(707\) −3.68390 −0.138547
\(708\) 0 0
\(709\) 6.61054i 0.248264i 0.992266 + 0.124132i \(0.0396146\pi\)
−0.992266 + 0.124132i \(0.960385\pi\)
\(710\) 0 0
\(711\) −22.4149 −0.840625
\(712\) 0 0
\(713\) 41.8192i 1.56614i
\(714\) 0 0
\(715\) −1.50225 8.93183i −0.0561808 0.334032i
\(716\) 0 0
\(717\) −12.9595 −0.483981
\(718\) 0 0
\(719\) 25.9038 0.966050 0.483025 0.875607i \(-0.339538\pi\)
0.483025 + 0.875607i \(0.339538\pi\)
\(720\) 0 0
\(721\) 17.1850 0.640004
\(722\) 0 0
\(723\) 8.59137 0.319516
\(724\) 0 0
\(725\) −7.96573 23.0109i −0.295840 0.854602i
\(726\) 0 0
\(727\) 7.35460i 0.272767i 0.990656 + 0.136383i \(0.0435479\pi\)
−0.990656 + 0.136383i \(0.956452\pi\)
\(728\) 0 0
\(729\) 17.8480 0.661037
\(730\) 0 0
\(731\) 9.50109i 0.351410i
\(732\) 0 0
\(733\) 8.12528 0.300114 0.150057 0.988677i \(-0.452054\pi\)
0.150057 + 0.988677i \(0.452054\pi\)
\(734\) 0 0
\(735\) 2.41196 0.405667i 0.0889664 0.0149633i
\(736\) 0 0
\(737\) 10.4596i 0.385284i
\(738\) 0 0
\(739\) 33.7064i 1.23991i −0.784637 0.619955i \(-0.787151\pi\)
0.784637 0.619955i \(-0.212849\pi\)
\(740\) 0 0
\(741\) 1.15367i 0.0423813i
\(742\) 0 0
\(743\) 1.46383i 0.0537028i −0.999639 0.0268514i \(-0.991452\pi\)
0.999639 0.0268514i \(-0.00854809\pi\)
\(744\) 0 0
\(745\) −2.07923 12.3624i −0.0761769 0.452922i
\(746\) 0 0
\(747\) 3.84674 0.140745
\(748\) 0 0
\(749\) 9.64783i 0.352524i
\(750\) 0 0
\(751\) −20.5655 −0.750444 −0.375222 0.926935i \(-0.622434\pi\)
−0.375222 + 0.926935i \(0.622434\pi\)
\(752\) 0 0
\(753\) 25.4866i 0.928784i
\(754\) 0 0
\(755\) 28.2867 4.75754i 1.02946 0.173145i
\(756\) 0 0
\(757\) 20.6397 0.750162 0.375081 0.926992i \(-0.377615\pi\)
0.375081 + 0.926992i \(0.377615\pi\)
\(758\) 0 0
\(759\) 4.85174 0.176107
\(760\) 0 0
\(761\) 1.18178 0.0428396 0.0214198 0.999771i \(-0.493181\pi\)
0.0214198 + 0.999771i \(0.493181\pi\)
\(762\) 0 0
\(763\) −1.25100 −0.0452893
\(764\) 0 0
\(765\) 3.40295 + 20.2328i 0.123034 + 0.731518i
\(766\) 0 0
\(767\) 19.6954i 0.711161i
\(768\) 0 0
\(769\) −6.15898 −0.222098 −0.111049 0.993815i \(-0.535421\pi\)
−0.111049 + 0.993815i \(0.535421\pi\)
\(770\) 0 0
\(771\) 2.26424i 0.0815445i
\(772\) 0 0
\(773\) −32.3637 −1.16404 −0.582022 0.813173i \(-0.697738\pi\)
−0.582022 + 0.813173i \(0.697738\pi\)
\(774\) 0 0
\(775\) 47.8365 16.5597i 1.71834 0.594841i
\(776\) 0 0
\(777\) 2.39959i 0.0860849i
\(778\) 0 0
\(779\) 3.08846i 0.110656i
\(780\) 0 0
\(781\) 4.49129i 0.160711i
\(782\) 0 0
\(783\) 25.5886i 0.914463i
\(784\) 0 0
\(785\) 12.1111 2.03696i 0.432263 0.0727023i
\(786\) 0 0
\(787\) 14.1116 0.503024 0.251512 0.967854i \(-0.419072\pi\)
0.251512 + 0.967854i \(0.419072\pi\)
\(788\) 0 0
\(789\) 33.3369i 1.18682i
\(790\) 0 0
\(791\) −12.9327 −0.459835
\(792\) 0 0
\(793\) 55.0180i 1.95375i
\(794\) 0 0
\(795\) −12.7884 + 2.15089i −0.453559 + 0.0762842i
\(796\) 0 0
\(797\) −41.3283 −1.46392 −0.731961 0.681347i \(-0.761395\pi\)
−0.731961 + 0.681347i \(0.761395\pi\)
\(798\) 0 0
\(799\) 24.3504 0.861455
\(800\) 0 0
\(801\) −14.7945 −0.522737
\(802\) 0 0
\(803\) −13.4082 −0.473164
\(804\) 0 0
\(805\) −9.10831 + 1.53193i −0.321026 + 0.0539933i
\(806\) 0 0
\(807\) 24.2618i 0.854057i
\(808\) 0 0
\(809\) 27.8798 0.980201 0.490100 0.871666i \(-0.336960\pi\)
0.490100 + 0.871666i \(0.336960\pi\)
\(810\) 0 0
\(811\) 28.8990i 1.01478i 0.861716 + 0.507391i \(0.169390\pi\)
−0.861716 + 0.507391i \(0.830610\pi\)
\(812\) 0 0
\(813\) −4.41897 −0.154980
\(814\) 0 0
\(815\) −39.1138 + 6.57855i −1.37010 + 0.230437i
\(816\) 0 0
\(817\) 0.522219i 0.0182701i
\(818\) 0 0
\(819\) 6.80304i 0.237717i
\(820\) 0 0
\(821\) 18.2968i 0.638564i −0.947660 0.319282i \(-0.896558\pi\)
0.947660 0.319282i \(-0.103442\pi\)
\(822\) 0 0
\(823\) 56.5497i 1.97120i −0.169102 0.985599i \(-0.554087\pi\)
0.169102 0.985599i \(-0.445913\pi\)
\(824\) 0 0
\(825\) −1.92120 5.54984i −0.0668876 0.193220i
\(826\) 0 0
\(827\) 2.93726 0.102138 0.0510692 0.998695i \(-0.483737\pi\)
0.0510692 + 0.998695i \(0.483737\pi\)
\(828\) 0 0
\(829\) 42.5110i 1.47647i −0.674544 0.738234i \(-0.735660\pi\)
0.674544 0.738234i \(-0.264340\pi\)
\(830\) 0 0
\(831\) −17.5483 −0.608743
\(832\) 0 0
\(833\) 5.08737i 0.176267i
\(834\) 0 0
\(835\) 1.37384 + 8.16839i 0.0475438 + 0.282679i
\(836\) 0 0
\(837\) 53.1953 1.83870
\(838\) 0 0
\(839\) 2.39641 0.0827331 0.0413666 0.999144i \(-0.486829\pi\)
0.0413666 + 0.999144i \(0.486829\pi\)
\(840\) 0 0
\(841\) 5.28191 0.182135
\(842\) 0 0
\(843\) −7.09694 −0.244431
\(844\) 0 0
\(845\) 2.70728 0.455337i 0.0931332 0.0156641i
\(846\) 0 0
\(847\) 9.84684i 0.338341i
\(848\) 0 0
\(849\) −8.18833 −0.281023
\(850\) 0 0
\(851\) 9.06161i 0.310628i
\(852\) 0 0
\(853\) −3.83044 −0.131152 −0.0655759 0.997848i \(-0.520888\pi\)
−0.0655759 + 0.997848i \(0.520888\pi\)
\(854\) 0 0
\(855\) −0.187041 1.11208i −0.00639665 0.0380323i
\(856\) 0 0
\(857\) 55.4188i 1.89307i 0.322602 + 0.946535i \(0.395442\pi\)
−0.322602 + 0.946535i \(0.604558\pi\)
\(858\) 0 0
\(859\) 1.13603i 0.0387607i 0.999812 + 0.0193803i \(0.00616934\pi\)
−0.999812 + 0.0193803i \(0.993831\pi\)
\(860\) 0 0
\(861\) 12.0812i 0.411727i
\(862\) 0 0
\(863\) 45.8120i 1.55946i −0.626117 0.779729i \(-0.715357\pi\)
0.626117 0.779729i \(-0.284643\pi\)
\(864\) 0 0
\(865\) 18.9057 3.17976i 0.642815 0.108115i
\(866\) 0 0
\(867\) 9.71448 0.329921
\(868\) 0 0
\(869\) 13.3459i 0.452728i
\(870\) 0 0
\(871\) 36.7399 1.24488
\(872\) 0 0
\(873\) 17.9014i 0.605869i
\(874\) 0 0
\(875\) 5.35908 + 9.81225i 0.181170 + 0.331715i
\(876\) 0 0
\(877\) −49.4124 −1.66854 −0.834269 0.551358i \(-0.814110\pi\)
−0.834269 + 0.551358i \(0.814110\pi\)
\(878\) 0 0
\(879\) 21.1902 0.714726
\(880\) 0 0
\(881\) −49.6973 −1.67435 −0.837173 0.546939i \(-0.815793\pi\)
−0.837173 + 0.546939i \(0.815793\pi\)
\(882\) 0 0
\(883\) −45.8249 −1.54213 −0.771066 0.636756i \(-0.780276\pi\)
−0.771066 + 0.636756i \(0.780276\pi\)
\(884\) 0 0
\(885\) −2.11820 12.5941i −0.0712026 0.423346i
\(886\) 0 0
\(887\) 31.4562i 1.05619i −0.849184 0.528097i \(-0.822906\pi\)
0.849184 0.528097i \(-0.177094\pi\)
\(888\) 0 0
\(889\) 8.66938 0.290762
\(890\) 0 0
\(891\) 0.361214i 0.0121011i
\(892\) 0 0
\(893\) −1.33840 −0.0447878
\(894\) 0 0
\(895\) 6.52143 + 38.7742i 0.217987 + 1.29608i
\(896\) 0 0
\(897\) 17.0420i 0.569015i
\(898\) 0 0
\(899\) 49.3067i 1.64447i
\(900\) 0 0
\(901\) 26.9737i 0.898625i
\(902\) 0 0
\(903\) 2.04278i 0.0679796i
\(904\) 0 0
\(905\) 1.86349 + 11.0797i 0.0619446 + 0.368301i
\(906\) 0 0
\(907\) −40.2056 −1.33501 −0.667503 0.744607i \(-0.732637\pi\)
−0.667503 + 0.744607i \(0.732637\pi\)
\(908\) 0 0
\(909\) 6.64420i 0.220374i
\(910\) 0 0
\(911\) 4.19808 0.139089 0.0695444 0.997579i \(-0.477845\pi\)
0.0695444 + 0.997579i \(0.477845\pi\)
\(912\) 0 0
\(913\) 2.29036i 0.0757998i
\(914\) 0 0
\(915\) 5.91707 + 35.1809i 0.195612 + 1.16304i
\(916\) 0 0
\(917\) 15.7893 0.521408
\(918\) 0 0
\(919\) 35.1340 1.15896 0.579482 0.814985i \(-0.303255\pi\)
0.579482 + 0.814985i \(0.303255\pi\)
\(920\) 0 0
\(921\) −22.9343 −0.755711
\(922\) 0 0
\(923\) −15.7759 −0.519270
\(924\) 0 0
\(925\) 10.3655 3.58824i 0.340814 0.117980i
\(926\) 0 0
\(927\) 30.9946i 1.01799i
\(928\) 0 0
\(929\) 7.36911 0.241773 0.120886 0.992666i \(-0.461426\pi\)
0.120886 + 0.992666i \(0.461426\pi\)
\(930\) 0 0
\(931\) 0.279623i 0.00916427i
\(932\) 0 0
\(933\) 17.4487 0.571244
\(934\) 0 0
\(935\) 12.0466 2.02612i 0.393967 0.0662613i
\(936\) 0 0
\(937\) 8.99362i 0.293809i 0.989151 + 0.146904i \(0.0469310\pi\)
−0.989151 + 0.146904i \(0.953069\pi\)
\(938\) 0 0
\(939\) 11.8663i 0.387242i
\(940\) 0 0
\(941\) 41.4187i 1.35021i −0.737722 0.675105i \(-0.764098\pi\)
0.737722 0.675105i \(-0.235902\pi\)
\(942\) 0 0
\(943\) 45.6225i 1.48567i
\(944\) 0 0
\(945\) 1.94865 + 11.5860i 0.0633897 + 0.376894i
\(946\) 0 0
\(947\) −28.8177 −0.936449 −0.468225 0.883609i \(-0.655106\pi\)
−0.468225 + 0.883609i \(0.655106\pi\)
\(948\) 0 0
\(949\) 47.0969i 1.52883i
\(950\) 0 0
\(951\) 11.4537 0.371412
\(952\) 0 0
\(953\) 7.39709i 0.239615i −0.992797 0.119808i \(-0.961772\pi\)
0.992797 0.119808i \(-0.0382278\pi\)
\(954\) 0 0
\(955\) 2.16838 0.364699i 0.0701670 0.0118014i
\(956\) 0 0
\(957\) −5.72041 −0.184915
\(958\) 0 0
\(959\) 14.1779 0.457830
\(960\) 0 0
\(961\) 71.5020 2.30652
\(962\) 0 0
\(963\) −17.4006 −0.560727
\(964\) 0 0
\(965\) −2.43287 14.4650i −0.0783168 0.465645i
\(966\) 0 0
\(967\) 23.8031i 0.765457i 0.923861 + 0.382728i \(0.125015\pi\)
−0.923861 + 0.382728i \(0.874985\pi\)
\(968\) 0 0
\(969\) −1.55600 −0.0499858
\(970\) 0 0
\(971\) 29.5025i 0.946781i −0.880853 0.473390i \(-0.843030\pi\)
0.880853 0.473390i \(-0.156970\pi\)
\(972\) 0 0
\(973\) −18.8789 −0.605230
\(974\) 0 0
\(975\) 19.4941 6.74832i 0.624311 0.216119i
\(976\) 0 0
\(977\) 6.48739i 0.207550i 0.994601 + 0.103775i \(0.0330922\pi\)
−0.994601 + 0.103775i \(0.966908\pi\)
\(978\) 0 0
\(979\) 8.80865i 0.281526i
\(980\) 0 0
\(981\) 2.25628i 0.0720374i
\(982\) 0 0
\(983\) 14.7636i 0.470885i 0.971888 + 0.235442i \(0.0756539\pi\)
−0.971888 + 0.235442i \(0.924346\pi\)
\(984\) 0 0
\(985\) 11.5701 1.94597i 0.368653 0.0620037i
\(986\) 0 0
\(987\) 5.23546 0.166647
\(988\) 0 0
\(989\) 7.71419i 0.245297i
\(990\) 0 0
\(991\) 51.7476 1.64382 0.821909 0.569619i \(-0.192909\pi\)
0.821909 + 0.569619i \(0.192909\pi\)
\(992\) 0 0
\(993\) 30.9503i 0.982178i
\(994\) 0 0
\(995\) −14.7655 + 2.48340i −0.468097 + 0.0787292i
\(996\) 0 0
\(997\) 54.8514 1.73716 0.868580 0.495549i \(-0.165033\pi\)
0.868580 + 0.495549i \(0.165033\pi\)
\(998\) 0 0
\(999\) 11.5266 0.364687
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 2240.2.l.d.1569.9 yes 24
4.3 odd 2 inner 2240.2.l.d.1569.16 yes 24
5.4 even 2 2240.2.l.c.1569.16 yes 24
8.3 odd 2 2240.2.l.c.1569.10 yes 24
8.5 even 2 2240.2.l.c.1569.15 yes 24
20.19 odd 2 2240.2.l.c.1569.9 24
40.19 odd 2 inner 2240.2.l.d.1569.15 yes 24
40.29 even 2 inner 2240.2.l.d.1569.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.l.c.1569.9 24 20.19 odd 2
2240.2.l.c.1569.10 yes 24 8.3 odd 2
2240.2.l.c.1569.15 yes 24 8.5 even 2
2240.2.l.c.1569.16 yes 24 5.4 even 2
2240.2.l.d.1569.9 yes 24 1.1 even 1 trivial
2240.2.l.d.1569.10 yes 24 40.29 even 2 inner
2240.2.l.d.1569.15 yes 24 40.19 odd 2 inner
2240.2.l.d.1569.16 yes 24 4.3 odd 2 inner