Properties

Label 2240.2.n.b.1119.3
Level $2240$
Weight $2$
Character 2240.1119
Analytic conductor $17.886$
Analytic rank $0$
Dimension $8$
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(1119,2240)]
 
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([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1119");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (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: \(8\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1119.3
Root \(-2.21837 - 1.28078i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1119
Dual form 2240.2.n.b.1119.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +(2.00000 + 1.00000i) q^{5} +(-2.00000 - 1.73205i) q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} +(2.00000 + 1.00000i) q^{5} +(-2.00000 - 1.73205i) q^{7} +3.56155 q^{9} +0.972638 q^{11} -0.561553i q^{13} +(-5.12311 - 2.56155i) q^{15} -4.43674 q^{17} -1.12311i q^{19} +(5.12311 + 4.43674i) q^{21} +1.12311 q^{23} +(3.00000 + 4.00000i) q^{25} -1.43845 q^{27} +4.43674i q^{29} -8.87348 q^{31} -2.49146 q^{33} +(-2.26795 - 5.46410i) q^{35} +8.87348 q^{37} +1.43845i q^{39} -8.87348i q^{41} -1.94528i q^{43} +(7.12311 + 3.56155i) q^{45} +0.972638i q^{47} +(1.00000 + 6.92820i) q^{49} +11.3649 q^{51} +(1.94528 + 0.972638i) q^{55} +2.87689i q^{57} +4.00000i q^{59} -1.12311 q^{61} +(-7.12311 - 6.16879i) q^{63} +(0.561553 - 1.12311i) q^{65} +6.92820i q^{67} -2.87689 q^{69} +14.2462i q^{71} -6.92820 q^{73} +(-7.68466 - 10.2462i) q^{75} +(-1.94528 - 1.68466i) q^{77} -7.68466i q^{79} -7.00000 q^{81} +4.00000 q^{83} +(-8.87348 - 4.43674i) q^{85} -11.3649i q^{87} +8.87348i q^{89} +(-0.972638 + 1.12311i) q^{91} +22.7299 q^{93} +(1.12311 - 2.24621i) q^{95} -9.41967 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{5} - 16 q^{7} + 12 q^{9} - 8 q^{15} + 8 q^{21} - 24 q^{23} + 24 q^{25} - 28 q^{27} - 32 q^{35} + 24 q^{45} + 8 q^{49} + 24 q^{61} - 24 q^{63} - 12 q^{65} - 56 q^{69} - 12 q^{75} - 56 q^{81} + 32 q^{83} - 24 q^{95}+O(q^{100}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 2.00000 + 1.00000i 0.894427 + 0.447214i
\(6\) 0 0
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) 0.972638 0.293261 0.146631 0.989191i \(-0.453157\pi\)
0.146631 + 0.989191i \(0.453157\pi\)
\(12\) 0 0
\(13\) 0.561553i 0.155747i −0.996963 0.0778734i \(-0.975187\pi\)
0.996963 0.0778734i \(-0.0248130\pi\)
\(14\) 0 0
\(15\) −5.12311 2.56155i −1.32278 0.661390i
\(16\) 0 0
\(17\) −4.43674 −1.07607 −0.538034 0.842923i \(-0.680833\pi\)
−0.538034 + 0.842923i \(0.680833\pi\)
\(18\) 0 0
\(19\) 1.12311i 0.257658i −0.991667 0.128829i \(-0.958878\pi\)
0.991667 0.128829i \(-0.0411218\pi\)
\(20\) 0 0
\(21\) 5.12311 + 4.43674i 1.11795 + 0.968176i
\(22\) 0 0
\(23\) 1.12311 0.234184 0.117092 0.993121i \(-0.462643\pi\)
0.117092 + 0.993121i \(0.462643\pi\)
\(24\) 0 0
\(25\) 3.00000 + 4.00000i 0.600000 + 0.800000i
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) 4.43674i 0.823882i 0.911211 + 0.411941i \(0.135149\pi\)
−0.911211 + 0.411941i \(0.864851\pi\)
\(30\) 0 0
\(31\) −8.87348 −1.59372 −0.796862 0.604161i \(-0.793508\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(32\) 0 0
\(33\) −2.49146 −0.433708
\(34\) 0 0
\(35\) −2.26795 5.46410i −0.383353 0.923602i
\(36\) 0 0
\(37\) 8.87348 1.45879 0.729396 0.684092i \(-0.239801\pi\)
0.729396 + 0.684092i \(0.239801\pi\)
\(38\) 0 0
\(39\) 1.43845i 0.230336i
\(40\) 0 0
\(41\) 8.87348i 1.38580i −0.721031 0.692902i \(-0.756332\pi\)
0.721031 0.692902i \(-0.243668\pi\)
\(42\) 0 0
\(43\) 1.94528i 0.296652i −0.988939 0.148326i \(-0.952612\pi\)
0.988939 0.148326i \(-0.0473885\pi\)
\(44\) 0 0
\(45\) 7.12311 + 3.56155i 1.06185 + 0.530925i
\(46\) 0 0
\(47\) 0.972638i 0.141874i 0.997481 + 0.0709369i \(0.0225989\pi\)
−0.997481 + 0.0709369i \(0.977401\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 11.3649 1.59141
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 1.94528 + 0.972638i 0.262301 + 0.131150i
\(56\) 0 0
\(57\) 2.87689i 0.381054i
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) −1.12311 −0.143799 −0.0718995 0.997412i \(-0.522906\pi\)
−0.0718995 + 0.997412i \(0.522906\pi\)
\(62\) 0 0
\(63\) −7.12311 6.16879i −0.897427 0.777195i
\(64\) 0 0
\(65\) 0.561553 1.12311i 0.0696521 0.139304i
\(66\) 0 0
\(67\) 6.92820i 0.846415i 0.906033 + 0.423207i \(0.139096\pi\)
−0.906033 + 0.423207i \(0.860904\pi\)
\(68\) 0 0
\(69\) −2.87689 −0.346337
\(70\) 0 0
\(71\) 14.2462i 1.69071i 0.534202 + 0.845357i \(0.320612\pi\)
−0.534202 + 0.845357i \(0.679388\pi\)
\(72\) 0 0
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 0 0
\(75\) −7.68466 10.2462i −0.887348 1.18313i
\(76\) 0 0
\(77\) −1.94528 1.68466i −0.221685 0.191985i
\(78\) 0 0
\(79\) 7.68466i 0.864592i −0.901732 0.432296i \(-0.857704\pi\)
0.901732 0.432296i \(-0.142296\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −8.87348 4.43674i −0.962464 0.481232i
\(86\) 0 0
\(87\) 11.3649i 1.21845i
\(88\) 0 0
\(89\) 8.87348i 0.940587i 0.882510 + 0.470293i \(0.155852\pi\)
−0.882510 + 0.470293i \(0.844148\pi\)
\(90\) 0 0
\(91\) −0.972638 + 1.12311i −0.101960 + 0.117733i
\(92\) 0 0
\(93\) 22.7299 2.35698
\(94\) 0 0
\(95\) 1.12311 2.24621i 0.115228 0.230456i
\(96\) 0 0
\(97\) −9.41967 −0.956422 −0.478211 0.878245i \(-0.658715\pi\)
−0.478211 + 0.878245i \(0.658715\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
\(100\) 0 0
\(101\) −11.3693 −1.13129 −0.565645 0.824649i \(-0.691373\pi\)
−0.565645 + 0.824649i \(0.691373\pi\)
\(102\) 0 0
\(103\) 14.8290i 1.46115i −0.682833 0.730575i \(-0.739252\pi\)
0.682833 0.730575i \(-0.260748\pi\)
\(104\) 0 0
\(105\) 5.80947 + 13.9966i 0.566946 + 1.36593i
\(106\) 0 0
\(107\) 10.8188i 1.04589i 0.852367 + 0.522944i \(0.175166\pi\)
−0.852367 + 0.522944i \(0.824834\pi\)
\(108\) 0 0
\(109\) 18.2931i 1.75217i −0.482161 0.876083i \(-0.660148\pi\)
0.482161 0.876083i \(-0.339852\pi\)
\(110\) 0 0
\(111\) −22.7299 −2.15743
\(112\) 0 0
\(113\) 15.3693i 1.44582i 0.690940 + 0.722912i \(0.257197\pi\)
−0.690940 + 0.722912i \(0.742803\pi\)
\(114\) 0 0
\(115\) 2.24621 + 1.12311i 0.209460 + 0.104730i
\(116\) 0 0
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 8.87348 + 7.68466i 0.813431 + 0.704451i
\(120\) 0 0
\(121\) −10.0540 −0.913998
\(122\) 0 0
\(123\) 22.7299i 2.04948i
\(124\) 0 0
\(125\) 2.00000 + 11.0000i 0.178885 + 0.983870i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 4.98293i 0.438722i
\(130\) 0 0
\(131\) 19.3693i 1.69231i 0.532941 + 0.846153i \(0.321087\pi\)
−0.532941 + 0.846153i \(0.678913\pi\)
\(132\) 0 0
\(133\) −1.94528 + 2.24621i −0.168677 + 0.194771i
\(134\) 0 0
\(135\) −2.87689 1.43845i −0.247604 0.123802i
\(136\) 0 0
\(137\) 13.1231i 1.12118i 0.828093 + 0.560591i \(0.189426\pi\)
−0.828093 + 0.560591i \(0.810574\pi\)
\(138\) 0 0
\(139\) 14.2462i 1.20835i 0.796852 + 0.604174i \(0.206497\pi\)
−0.796852 + 0.604174i \(0.793503\pi\)
\(140\) 0 0
\(141\) 2.49146i 0.209819i
\(142\) 0 0
\(143\) 0.546188i 0.0456745i
\(144\) 0 0
\(145\) −4.43674 + 8.87348i −0.368451 + 0.736902i
\(146\) 0 0
\(147\) −2.56155 17.7470i −0.211273 1.46374i
\(148\) 0 0
\(149\) 10.8188i 0.886307i −0.896446 0.443153i \(-0.853860\pi\)
0.896446 0.443153i \(-0.146140\pi\)
\(150\) 0 0
\(151\) 15.0540i 1.22508i 0.790441 + 0.612538i \(0.209851\pi\)
−0.790441 + 0.612538i \(0.790149\pi\)
\(152\) 0 0
\(153\) −15.8017 −1.27749
\(154\) 0 0
\(155\) −17.7470 8.87348i −1.42547 0.712735i
\(156\) 0 0
\(157\) 20.2462i 1.61582i 0.589303 + 0.807912i \(0.299402\pi\)
−0.589303 + 0.807912i \(0.700598\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.24621 1.94528i −0.177026 0.153309i
\(162\) 0 0
\(163\) 24.6752i 1.93271i 0.257216 + 0.966354i \(0.417195\pi\)
−0.257216 + 0.966354i \(0.582805\pi\)
\(164\) 0 0
\(165\) −4.98293 2.49146i −0.387920 0.193960i
\(166\) 0 0
\(167\) 2.91791i 0.225795i −0.993607 0.112897i \(-0.963987\pi\)
0.993607 0.112897i \(-0.0360132\pi\)
\(168\) 0 0
\(169\) 12.6847 0.975743
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) 9.68466i 0.736311i 0.929764 + 0.368155i \(0.120011\pi\)
−0.929764 + 0.368155i \(0.879989\pi\)
\(174\) 0 0
\(175\) 0.928203 13.1962i 0.0701656 0.997535i
\(176\) 0 0
\(177\) 10.2462i 0.770152i
\(178\) 0 0
\(179\) −21.2111 −1.58539 −0.792695 0.609619i \(-0.791322\pi\)
−0.792695 + 0.609619i \(0.791322\pi\)
\(180\) 0 0
\(181\) −14.2462 −1.05891 −0.529456 0.848337i \(-0.677604\pi\)
−0.529456 + 0.848337i \(0.677604\pi\)
\(182\) 0 0
\(183\) 2.87689 0.212666
\(184\) 0 0
\(185\) 17.7470 + 8.87348i 1.30478 + 0.652391i
\(186\) 0 0
\(187\) −4.31534 −0.315569
\(188\) 0 0
\(189\) 2.87689 + 2.49146i 0.209263 + 0.181227i
\(190\) 0 0
\(191\) 5.43845i 0.393512i 0.980452 + 0.196756i \(0.0630407\pi\)
−0.980452 + 0.196756i \(0.936959\pi\)
\(192\) 0 0
\(193\) 7.36932i 0.530455i −0.964186 0.265228i \(-0.914553\pi\)
0.964186 0.265228i \(-0.0854471\pi\)
\(194\) 0 0
\(195\) −1.43845 + 2.87689i −0.103009 + 0.206019i
\(196\) 0 0
\(197\) −22.7299 −1.61944 −0.809719 0.586818i \(-0.800380\pi\)
−0.809719 + 0.586818i \(0.800380\pi\)
\(198\) 0 0
\(199\) 17.7470 1.25805 0.629024 0.777386i \(-0.283455\pi\)
0.629024 + 0.777386i \(0.283455\pi\)
\(200\) 0 0
\(201\) 17.7470i 1.25177i
\(202\) 0 0
\(203\) 7.68466 8.87348i 0.539357 0.622796i
\(204\) 0 0
\(205\) 8.87348 17.7470i 0.619751 1.23950i
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 1.09238i 0.0755612i
\(210\) 0 0
\(211\) −14.8290 −1.02087 −0.510437 0.859915i \(-0.670516\pi\)
−0.510437 + 0.859915i \(0.670516\pi\)
\(212\) 0 0
\(213\) 36.4924i 2.50042i
\(214\) 0 0
\(215\) 1.94528 3.89055i 0.132667 0.265333i
\(216\) 0 0
\(217\) 17.7470 + 15.3693i 1.20474 + 1.04334i
\(218\) 0 0
\(219\) 17.7470 1.19923
\(220\) 0 0
\(221\) 2.49146i 0.167594i
\(222\) 0 0
\(223\) 0.972638i 0.0651327i 0.999470 + 0.0325663i \(0.0103680\pi\)
−0.999470 + 0.0325663i \(0.989632\pi\)
\(224\) 0 0
\(225\) 10.6847 + 14.2462i 0.712311 + 0.949747i
\(226\) 0 0
\(227\) −15.6847 −1.04103 −0.520514 0.853853i \(-0.674260\pi\)
−0.520514 + 0.853853i \(0.674260\pi\)
\(228\) 0 0
\(229\) 1.12311 0.0742169 0.0371085 0.999311i \(-0.488185\pi\)
0.0371085 + 0.999311i \(0.488185\pi\)
\(230\) 0 0
\(231\) 4.98293 + 4.31534i 0.327853 + 0.283929i
\(232\) 0 0
\(233\) 28.4924i 1.86660i −0.359097 0.933300i \(-0.616915\pi\)
0.359097 0.933300i \(-0.383085\pi\)
\(234\) 0 0
\(235\) −0.972638 + 1.94528i −0.0634479 + 0.126896i
\(236\) 0 0
\(237\) 19.6847i 1.27866i
\(238\) 0 0
\(239\) 7.68466i 0.497079i −0.968622 0.248540i \(-0.920049\pi\)
0.968622 0.248540i \(-0.0799506\pi\)
\(240\) 0 0
\(241\) 4.98293i 0.320979i 0.987038 + 0.160489i \(0.0513072\pi\)
−0.987038 + 0.160489i \(0.948693\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) −4.92820 + 14.8564i −0.314851 + 0.949141i
\(246\) 0 0
\(247\) −0.630683 −0.0401294
\(248\) 0 0
\(249\) −10.2462 −0.649327
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 1.09238 0.0686770
\(254\) 0 0
\(255\) 22.7299 + 11.3649i 1.42340 + 0.711700i
\(256\) 0 0
\(257\) 6.92820 0.432169 0.216085 0.976375i \(-0.430671\pi\)
0.216085 + 0.976375i \(0.430671\pi\)
\(258\) 0 0
\(259\) −17.7470 15.3693i −1.10274 0.955003i
\(260\) 0 0
\(261\) 15.8017i 0.978100i
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 22.7299i 1.39105i
\(268\) 0 0
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) −22.7299 −1.38074 −0.690371 0.723455i \(-0.742553\pi\)
−0.690371 + 0.723455i \(0.742553\pi\)
\(272\) 0 0
\(273\) 2.49146 2.87689i 0.150790 0.174118i
\(274\) 0 0
\(275\) 2.91791 + 3.89055i 0.175957 + 0.234609i
\(276\) 0 0
\(277\) −13.8564 −0.832551 −0.416275 0.909239i \(-0.636665\pi\)
−0.416275 + 0.909239i \(0.636665\pi\)
\(278\) 0 0
\(279\) −31.6034 −1.89204
\(280\) 0 0
\(281\) −25.6847 −1.53222 −0.766109 0.642711i \(-0.777810\pi\)
−0.766109 + 0.642711i \(0.777810\pi\)
\(282\) 0 0
\(283\) 5.43845 0.323282 0.161641 0.986850i \(-0.448321\pi\)
0.161641 + 0.986850i \(0.448321\pi\)
\(284\) 0 0
\(285\) −2.87689 + 5.75379i −0.170413 + 0.340825i
\(286\) 0 0
\(287\) −15.3693 + 17.7470i −0.907222 + 1.04757i
\(288\) 0 0
\(289\) 2.68466 0.157921
\(290\) 0 0
\(291\) 24.1290 1.41447
\(292\) 0 0
\(293\) 14.3153i 0.836311i −0.908375 0.418156i \(-0.862677\pi\)
0.908375 0.418156i \(-0.137323\pi\)
\(294\) 0 0
\(295\) −4.00000 + 8.00000i −0.232889 + 0.465778i
\(296\) 0 0
\(297\) −1.39909 −0.0811833
\(298\) 0 0
\(299\) 0.630683i 0.0364733i
\(300\) 0 0
\(301\) −3.36932 + 3.89055i −0.194204 + 0.224248i
\(302\) 0 0
\(303\) 29.1231 1.67308
\(304\) 0 0
\(305\) −2.24621 1.12311i −0.128618 0.0643088i
\(306\) 0 0
\(307\) −9.93087 −0.566785 −0.283392 0.959004i \(-0.591460\pi\)
−0.283392 + 0.959004i \(0.591460\pi\)
\(308\) 0 0
\(309\) 37.9854i 2.16091i
\(310\) 0 0
\(311\) 18.8393 1.06828 0.534140 0.845396i \(-0.320635\pi\)
0.534140 + 0.845396i \(0.320635\pi\)
\(312\) 0 0
\(313\) −18.2931 −1.03399 −0.516995 0.855988i \(-0.672949\pi\)
−0.516995 + 0.855988i \(0.672949\pi\)
\(314\) 0 0
\(315\) −8.07742 19.4607i −0.455111 1.09649i
\(316\) 0 0
\(317\) 17.7470 0.996768 0.498384 0.866956i \(-0.333927\pi\)
0.498384 + 0.866956i \(0.333927\pi\)
\(318\) 0 0
\(319\) 4.31534i 0.241613i
\(320\) 0 0
\(321\) 27.7128i 1.54678i
\(322\) 0 0
\(323\) 4.98293i 0.277257i
\(324\) 0 0
\(325\) 2.24621 1.68466i 0.124597 0.0934480i
\(326\) 0 0
\(327\) 46.8589i 2.59130i
\(328\) 0 0
\(329\) 1.68466 1.94528i 0.0928782 0.107247i
\(330\) 0 0
\(331\) 17.3205 0.952021 0.476011 0.879440i \(-0.342082\pi\)
0.476011 + 0.879440i \(0.342082\pi\)
\(332\) 0 0
\(333\) 31.6034 1.73185
\(334\) 0 0
\(335\) −6.92820 + 13.8564i −0.378528 + 0.757056i
\(336\) 0 0
\(337\) 22.7386i 1.23865i 0.785134 + 0.619326i \(0.212594\pi\)
−0.785134 + 0.619326i \(0.787406\pi\)
\(338\) 0 0
\(339\) 39.3693i 2.13825i
\(340\) 0 0
\(341\) −8.63068 −0.467378
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) −5.75379 2.87689i −0.309774 0.154887i
\(346\) 0 0
\(347\) 15.8017i 0.848279i −0.905597 0.424139i \(-0.860577\pi\)
0.905597 0.424139i \(-0.139423\pi\)
\(348\) 0 0
\(349\) 1.12311 0.0601185 0.0300592 0.999548i \(-0.490430\pi\)
0.0300592 + 0.999548i \(0.490430\pi\)
\(350\) 0 0
\(351\) 0.807764i 0.0431153i
\(352\) 0 0
\(353\) 18.2931 0.973646 0.486823 0.873501i \(-0.338156\pi\)
0.486823 + 0.873501i \(0.338156\pi\)
\(354\) 0 0
\(355\) −14.2462 + 28.4924i −0.756110 + 1.51222i
\(356\) 0 0
\(357\) −22.7299 19.6847i −1.20299 1.04182i
\(358\) 0 0
\(359\) 12.0000i 0.633336i −0.948536 0.316668i \(-0.897436\pi\)
0.948536 0.316668i \(-0.102564\pi\)
\(360\) 0 0
\(361\) 17.7386 0.933612
\(362\) 0 0
\(363\) 25.7538 1.35172
\(364\) 0 0
\(365\) −13.8564 6.92820i −0.725277 0.362639i
\(366\) 0 0
\(367\) 36.4666i 1.90354i −0.306814 0.951769i \(-0.599263\pi\)
0.306814 0.951769i \(-0.400737\pi\)
\(368\) 0 0
\(369\) 31.6034i 1.64521i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.6204 1.37835 0.689177 0.724593i \(-0.257972\pi\)
0.689177 + 0.724593i \(0.257972\pi\)
\(374\) 0 0
\(375\) −5.12311 28.1771i −0.264556 1.45506i
\(376\) 0 0
\(377\) 2.49146 0.128317
\(378\) 0 0
\(379\) 28.1393 1.44542 0.722708 0.691153i \(-0.242897\pi\)
0.722708 + 0.691153i \(0.242897\pi\)
\(380\) 0 0
\(381\) 30.7386 1.57479
\(382\) 0 0
\(383\) 0.426450i 0.0217906i −0.999941 0.0108953i \(-0.996532\pi\)
0.999941 0.0108953i \(-0.00346815\pi\)
\(384\) 0 0
\(385\) −2.20589 5.31459i −0.112423 0.270857i
\(386\) 0 0
\(387\) 6.92820i 0.352180i
\(388\) 0 0
\(389\) 0.546188i 0.0276928i 0.999904 + 0.0138464i \(0.00440759\pi\)
−0.999904 + 0.0138464i \(0.995592\pi\)
\(390\) 0 0
\(391\) −4.98293 −0.251997
\(392\) 0 0
\(393\) 49.6155i 2.50277i
\(394\) 0 0
\(395\) 7.68466 15.3693i 0.386657 0.773314i
\(396\) 0 0
\(397\) 3.93087i 0.197285i −0.995123 0.0986423i \(-0.968550\pi\)
0.995123 0.0986423i \(-0.0314500\pi\)
\(398\) 0 0
\(399\) 4.98293 5.75379i 0.249458 0.288050i
\(400\) 0 0
\(401\) −2.80776 −0.140213 −0.0701065 0.997540i \(-0.522334\pi\)
−0.0701065 + 0.997540i \(0.522334\pi\)
\(402\) 0 0
\(403\) 4.98293i 0.248217i
\(404\) 0 0
\(405\) −14.0000 7.00000i −0.695666 0.347833i
\(406\) 0 0
\(407\) 8.63068 0.427807
\(408\) 0 0
\(409\) 31.6034i 1.56269i 0.624102 + 0.781343i \(0.285465\pi\)
−0.624102 + 0.781343i \(0.714535\pi\)
\(410\) 0 0
\(411\) 33.6155i 1.65813i
\(412\) 0 0
\(413\) 6.92820 8.00000i 0.340915 0.393654i
\(414\) 0 0
\(415\) 8.00000 + 4.00000i 0.392705 + 0.196352i
\(416\) 0 0
\(417\) 36.4924i 1.78704i
\(418\) 0 0
\(419\) 26.7386i 1.30627i −0.757242 0.653134i \(-0.773454\pi\)
0.757242 0.653134i \(-0.226546\pi\)
\(420\) 0 0
\(421\) 4.43674i 0.216233i −0.994138 0.108117i \(-0.965518\pi\)
0.994138 0.108117i \(-0.0344820\pi\)
\(422\) 0 0
\(423\) 3.46410i 0.168430i
\(424\) 0 0
\(425\) −13.3102 17.7470i −0.645640 0.860854i
\(426\) 0 0
\(427\) 2.24621 + 1.94528i 0.108702 + 0.0941385i
\(428\) 0 0
\(429\) 1.39909i 0.0675486i
\(430\) 0 0
\(431\) 16.3153i 0.785882i −0.919564 0.392941i \(-0.871458\pi\)
0.919564 0.392941i \(-0.128542\pi\)
\(432\) 0 0
\(433\) −3.03765 −0.145980 −0.0729901 0.997333i \(-0.523254\pi\)
−0.0729901 + 0.997333i \(0.523254\pi\)
\(434\) 0 0
\(435\) 11.3649 22.7299i 0.544907 1.08981i
\(436\) 0 0
\(437\) 1.26137i 0.0603393i
\(438\) 0 0
\(439\) −13.8564 −0.661330 −0.330665 0.943748i \(-0.607273\pi\)
−0.330665 + 0.943748i \(0.607273\pi\)
\(440\) 0 0
\(441\) 3.56155 + 24.6752i 0.169598 + 1.17501i
\(442\) 0 0
\(443\) 29.6581i 1.40910i 0.709655 + 0.704549i \(0.248851\pi\)
−0.709655 + 0.704549i \(0.751149\pi\)
\(444\) 0 0
\(445\) −8.87348 + 17.7470i −0.420643 + 0.841287i
\(446\) 0 0
\(447\) 27.7128i 1.31077i
\(448\) 0 0
\(449\) 29.0540 1.37114 0.685571 0.728006i \(-0.259553\pi\)
0.685571 + 0.728006i \(0.259553\pi\)
\(450\) 0 0
\(451\) 8.63068i 0.406403i
\(452\) 0 0
\(453\) 38.5616i 1.81178i
\(454\) 0 0
\(455\) −3.06838 + 1.27357i −0.143848 + 0.0597060i
\(456\) 0 0
\(457\) 0.630683i 0.0295021i 0.999891 + 0.0147511i \(0.00469558\pi\)
−0.999891 + 0.0147511i \(0.995304\pi\)
\(458\) 0 0
\(459\) 6.38202 0.297887
\(460\) 0 0
\(461\) −20.0000 −0.931493 −0.465746 0.884918i \(-0.654214\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) −11.3693 −0.528377 −0.264188 0.964471i \(-0.585104\pi\)
−0.264188 + 0.964471i \(0.585104\pi\)
\(464\) 0 0
\(465\) 45.4598 + 22.7299i 2.10815 + 1.05407i
\(466\) 0 0
\(467\) 31.0540 1.43701 0.718503 0.695524i \(-0.244828\pi\)
0.718503 + 0.695524i \(0.244828\pi\)
\(468\) 0 0
\(469\) 12.0000 13.8564i 0.554109 0.639829i
\(470\) 0 0
\(471\) 51.8617i 2.38966i
\(472\) 0 0
\(473\) 1.89205i 0.0869965i
\(474\) 0 0
\(475\) 4.49242 3.36932i 0.206126 0.154595i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.8393 0.860791 0.430396 0.902640i \(-0.358374\pi\)
0.430396 + 0.902640i \(0.358374\pi\)
\(480\) 0 0
\(481\) 4.98293i 0.227202i
\(482\) 0 0
\(483\) 5.75379 + 4.98293i 0.261806 + 0.226731i
\(484\) 0 0
\(485\) −18.8393 9.41967i −0.855450 0.427725i
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) 63.2067i 2.85831i
\(490\) 0 0
\(491\) −28.6855 −1.29456 −0.647278 0.762254i \(-0.724093\pi\)
−0.647278 + 0.762254i \(0.724093\pi\)
\(492\) 0 0
\(493\) 19.6847i 0.886552i
\(494\) 0 0
\(495\) 6.92820 + 3.46410i 0.311400 + 0.155700i
\(496\) 0 0
\(497\) 24.6752 28.4924i 1.10683 1.27806i
\(498\) 0 0
\(499\) −14.8290 −0.663839 −0.331920 0.943308i \(-0.607696\pi\)
−0.331920 + 0.943308i \(0.607696\pi\)
\(500\) 0 0
\(501\) 7.47439i 0.333931i
\(502\) 0 0
\(503\) 30.6307i 1.36576i 0.730532 + 0.682878i \(0.239272\pi\)
−0.730532 + 0.682878i \(0.760728\pi\)
\(504\) 0 0
\(505\) −22.7386 11.3693i −1.01186 0.505928i
\(506\) 0 0
\(507\) −32.4924 −1.44304
\(508\) 0 0
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 0 0
\(511\) 13.8564 + 12.0000i 0.612971 + 0.530849i
\(512\) 0 0
\(513\) 1.61553i 0.0713273i
\(514\) 0 0
\(515\) 14.8290 29.6581i 0.653446 1.30689i
\(516\) 0 0
\(517\) 0.946025i 0.0416061i
\(518\) 0 0
\(519\) 24.8078i 1.08894i
\(520\) 0 0
\(521\) 22.7299i 0.995814i 0.867230 + 0.497907i \(0.165898\pi\)
−0.867230 + 0.497907i \(0.834102\pi\)
\(522\) 0 0
\(523\) 7.50758 0.328283 0.164142 0.986437i \(-0.447515\pi\)
0.164142 + 0.986437i \(0.447515\pi\)
\(524\) 0 0
\(525\) −2.37764 + 33.8026i −0.103769 + 1.47527i
\(526\) 0 0
\(527\) 39.3693 1.71495
\(528\) 0 0
\(529\) −21.7386 −0.945158
\(530\) 0 0
\(531\) 14.2462i 0.618233i
\(532\) 0 0
\(533\) −4.98293 −0.215835
\(534\) 0 0
\(535\) −10.8188 + 21.6375i −0.467736 + 0.935471i
\(536\) 0 0
\(537\) 54.3333 2.34465
\(538\) 0 0
\(539\) 0.972638 + 6.73863i 0.0418945 + 0.290254i
\(540\) 0 0
\(541\) 8.32729i 0.358018i −0.983847 0.179009i \(-0.942711\pi\)
0.983847 0.179009i \(-0.0572892\pi\)
\(542\) 0 0
\(543\) 36.4924 1.56604
\(544\) 0 0
\(545\) 18.2931 36.5863i 0.783592 1.56718i
\(546\) 0 0
\(547\) 15.8017i 0.675631i −0.941212 0.337816i \(-0.890312\pi\)
0.941212 0.337816i \(-0.109688\pi\)
\(548\) 0 0
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) 4.98293 0.212280
\(552\) 0 0
\(553\) −13.3102 + 15.3693i −0.566008 + 0.653570i
\(554\) 0 0
\(555\) −45.4598 22.7299i −1.92966 0.964830i
\(556\) 0 0
\(557\) 13.8564 0.587115 0.293557 0.955941i \(-0.405161\pi\)
0.293557 + 0.955941i \(0.405161\pi\)
\(558\) 0 0
\(559\) −1.09238 −0.0462025
\(560\) 0 0
\(561\) 11.0540 0.466699
\(562\) 0 0
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 0 0
\(565\) −15.3693 + 30.7386i −0.646592 + 1.29318i
\(566\) 0 0
\(567\) 14.0000 + 12.1244i 0.587945 + 0.509175i
\(568\) 0 0
\(569\) −32.2462 −1.35183 −0.675916 0.736979i \(-0.736252\pi\)
−0.675916 + 0.736979i \(0.736252\pi\)
\(570\) 0 0
\(571\) −7.35465 −0.307783 −0.153891 0.988088i \(-0.549181\pi\)
−0.153891 + 0.988088i \(0.549181\pi\)
\(572\) 0 0
\(573\) 13.9309i 0.581970i
\(574\) 0 0
\(575\) 3.36932 + 4.49242i 0.140510 + 0.187347i
\(576\) 0 0
\(577\) −39.9307 −1.66233 −0.831167 0.556022i \(-0.812327\pi\)
−0.831167 + 0.556022i \(0.812327\pi\)
\(578\) 0 0
\(579\) 18.8769i 0.784497i
\(580\) 0 0
\(581\) −8.00000 6.92820i −0.331896 0.287430i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 2.00000 4.00000i 0.0826898 0.165380i
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 9.96585i 0.410636i
\(590\) 0 0
\(591\) 58.2238 2.39501
\(592\) 0 0
\(593\) 0.546188 0.0224292 0.0112146 0.999937i \(-0.496430\pi\)
0.0112146 + 0.999937i \(0.496430\pi\)
\(594\) 0 0
\(595\) 10.0623 + 24.2428i 0.412514 + 0.993858i
\(596\) 0 0
\(597\) −45.4598 −1.86054
\(598\) 0 0
\(599\) 5.43845i 0.222209i −0.993809 0.111104i \(-0.964561\pi\)
0.993809 0.111104i \(-0.0354388\pi\)
\(600\) 0 0
\(601\) 9.96585i 0.406516i −0.979125 0.203258i \(-0.934847\pi\)
0.979125 0.203258i \(-0.0651530\pi\)
\(602\) 0 0
\(603\) 24.6752i 1.00485i
\(604\) 0 0
\(605\) −20.1080 10.0540i −0.817504 0.408752i
\(606\) 0 0
\(607\) 16.7743i 0.680849i 0.940272 + 0.340424i \(0.110571\pi\)
−0.940272 + 0.340424i \(0.889429\pi\)
\(608\) 0 0
\(609\) −19.6847 + 22.7299i −0.797663 + 0.921061i
\(610\) 0 0
\(611\) 0.546188 0.0220964
\(612\) 0 0
\(613\) 9.96585 0.402517 0.201259 0.979538i \(-0.435497\pi\)
0.201259 + 0.979538i \(0.435497\pi\)
\(614\) 0 0
\(615\) −22.7299 + 45.4598i −0.916557 + 1.83311i
\(616\) 0 0
\(617\) 30.7386i 1.23749i −0.785592 0.618745i \(-0.787641\pi\)
0.785592 0.618745i \(-0.212359\pi\)
\(618\) 0 0
\(619\) 29.6155i 1.19035i 0.803597 + 0.595174i \(0.202917\pi\)
−0.803597 + 0.595174i \(0.797083\pi\)
\(620\) 0 0
\(621\) −1.61553 −0.0648289
\(622\) 0 0
\(623\) 15.3693 17.7470i 0.615759 0.711017i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) 0 0
\(627\) 2.79818i 0.111748i
\(628\) 0 0
\(629\) −39.3693 −1.56976
\(630\) 0 0
\(631\) 16.3153i 0.649503i 0.945799 + 0.324752i \(0.105281\pi\)
−0.945799 + 0.324752i \(0.894719\pi\)
\(632\) 0 0
\(633\) 37.9854 1.50978
\(634\) 0 0
\(635\) −24.0000 12.0000i −0.952411 0.476205i
\(636\) 0 0
\(637\) 3.89055 0.561553i 0.154149 0.0222495i
\(638\) 0 0
\(639\) 50.7386i 2.00719i
\(640\) 0 0
\(641\) 8.24621 0.325706 0.162853 0.986650i \(-0.447930\pi\)
0.162853 + 0.986650i \(0.447930\pi\)
\(642\) 0 0
\(643\) 25.3002 0.997742 0.498871 0.866676i \(-0.333748\pi\)
0.498871 + 0.866676i \(0.333748\pi\)
\(644\) 0 0
\(645\) −4.98293 + 9.96585i −0.196203 + 0.392405i
\(646\) 0 0
\(647\) 32.0298i 1.25922i −0.776911 0.629611i \(-0.783214\pi\)
0.776911 0.629611i \(-0.216786\pi\)
\(648\) 0 0
\(649\) 3.89055i 0.152718i
\(650\) 0 0
\(651\) −45.4598 39.3693i −1.78171 1.54301i
\(652\) 0 0
\(653\) −12.7640 −0.499495 −0.249748 0.968311i \(-0.580348\pi\)
−0.249748 + 0.968311i \(0.580348\pi\)
\(654\) 0 0
\(655\) −19.3693 + 38.7386i −0.756822 + 1.51364i
\(656\) 0 0
\(657\) −24.6752 −0.962670
\(658\) 0 0
\(659\) −2.91791 −0.113666 −0.0568329 0.998384i \(-0.518100\pi\)
−0.0568329 + 0.998384i \(0.518100\pi\)
\(660\) 0 0
\(661\) 38.2462 1.48761 0.743803 0.668399i \(-0.233020\pi\)
0.743803 + 0.668399i \(0.233020\pi\)
\(662\) 0 0
\(663\) 6.38202i 0.247857i
\(664\) 0 0
\(665\) −6.13676 + 2.54715i −0.237973 + 0.0987741i
\(666\) 0 0
\(667\) 4.98293i 0.192940i
\(668\) 0 0
\(669\) 2.49146i 0.0963255i
\(670\) 0 0
\(671\) −1.09238 −0.0421707
\(672\) 0 0
\(673\) 15.3693i 0.592444i 0.955119 + 0.296222i \(0.0957268\pi\)
−0.955119 + 0.296222i \(0.904273\pi\)
\(674\) 0 0
\(675\) −4.31534 5.75379i −0.166098 0.221463i
\(676\) 0 0
\(677\) 17.6847i 0.679677i −0.940484 0.339838i \(-0.889628\pi\)
0.940484 0.339838i \(-0.110372\pi\)
\(678\) 0 0
\(679\) 18.8393 + 16.3153i 0.722987 + 0.626125i
\(680\) 0 0
\(681\) 40.1771 1.53959
\(682\) 0 0
\(683\) 29.6581i 1.13484i −0.823430 0.567418i \(-0.807943\pi\)
0.823430 0.567418i \(-0.192057\pi\)
\(684\) 0 0
\(685\) −13.1231 + 26.2462i −0.501408 + 1.00282i
\(686\) 0 0
\(687\) −2.87689 −0.109760
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 7.50758i 0.285602i 0.989751 + 0.142801i \(0.0456108\pi\)
−0.989751 + 0.142801i \(0.954389\pi\)
\(692\) 0 0
\(693\) −6.92820 6.00000i −0.263181 0.227921i
\(694\) 0 0
\(695\) −14.2462 + 28.4924i −0.540390 + 1.08078i
\(696\) 0 0
\(697\) 39.3693i 1.49122i
\(698\) 0 0
\(699\) 72.9848i 2.76054i
\(700\) 0 0
\(701\) 41.0230i 1.54942i −0.632318 0.774709i \(-0.717896\pi\)
0.632318 0.774709i \(-0.282104\pi\)
\(702\) 0 0
\(703\) 9.96585i 0.375869i
\(704\) 0 0
\(705\) 2.49146 4.98293i 0.0938339 0.187668i
\(706\) 0 0
\(707\) 22.7386 + 19.6922i 0.855174 + 0.740603i
\(708\) 0 0
\(709\) 0.546188i 0.0205125i 0.999947 + 0.0102563i \(0.00326472\pi\)
−0.999947 + 0.0102563i \(0.996735\pi\)
\(710\) 0 0
\(711\) 27.3693i 1.02643i
\(712\) 0 0
\(713\) −9.96585 −0.373224
\(714\) 0 0
\(715\) 0.546188 1.09238i 0.0204263 0.0408525i
\(716\) 0 0
\(717\) 19.6847i 0.735137i
\(718\) 0 0
\(719\) 8.87348 0.330925 0.165462 0.986216i \(-0.447088\pi\)
0.165462 + 0.986216i \(0.447088\pi\)
\(720\) 0 0
\(721\) −25.6847 + 29.6581i −0.956547 + 1.10452i
\(722\) 0 0
\(723\) 12.7640i 0.474699i
\(724\) 0 0
\(725\) −17.7470 + 13.3102i −0.659105 + 0.494329i
\(726\) 0 0
\(727\) 14.2829i 0.529722i −0.964287 0.264861i \(-0.914674\pi\)
0.964287 0.264861i \(-0.0853260\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 8.63068i 0.319217i
\(732\) 0 0
\(733\) 31.3002i 1.15610i 0.816002 + 0.578049i \(0.196186\pi\)
−0.816002 + 0.578049i \(0.803814\pi\)
\(734\) 0 0
\(735\) 12.6239 38.0555i 0.465638 1.40370i
\(736\) 0 0
\(737\) 6.73863i 0.248221i
\(738\) 0 0
\(739\) −20.6649 −0.760170 −0.380085 0.924952i \(-0.624105\pi\)
−0.380085 + 0.924952i \(0.624105\pi\)
\(740\) 0 0
\(741\) 1.61553 0.0593479
\(742\) 0 0
\(743\) 38.2462 1.40312 0.701559 0.712612i \(-0.252488\pi\)
0.701559 + 0.712612i \(0.252488\pi\)
\(744\) 0 0
\(745\) 10.8188 21.6375i 0.396369 0.792737i
\(746\) 0 0
\(747\) 14.2462 0.521242
\(748\) 0 0
\(749\) 18.7386 21.6375i 0.684695 0.790617i
\(750\) 0 0
\(751\) 15.6847i 0.572341i 0.958179 + 0.286171i \(0.0923824\pi\)
−0.958179 + 0.286171i \(0.907618\pi\)
\(752\) 0 0
\(753\) 51.2311i 1.86696i
\(754\) 0 0
\(755\) −15.0540 + 30.1080i −0.547870 + 1.09574i
\(756\) 0 0
\(757\) 13.8564 0.503620 0.251810 0.967777i \(-0.418974\pi\)
0.251810 + 0.967777i \(0.418974\pi\)
\(758\) 0 0
\(759\) −2.79818 −0.101567
\(760\) 0 0
\(761\) 30.5110i 1.10602i 0.833174 + 0.553011i \(0.186521\pi\)
−0.833174 + 0.553011i \(0.813479\pi\)
\(762\) 0 0
\(763\) −31.6847 + 36.5863i −1.14706 + 1.32451i
\(764\) 0 0
\(765\) −31.6034 15.8017i −1.14262 0.571311i
\(766\) 0 0
\(767\) 2.24621 0.0811060
\(768\) 0 0
\(769\) 3.89055i 0.140297i −0.997537 0.0701484i \(-0.977653\pi\)
0.997537 0.0701484i \(-0.0223473\pi\)
\(770\) 0 0
\(771\) −17.7470 −0.639141
\(772\) 0 0
\(773\) 2.94602i 0.105961i −0.998596 0.0529806i \(-0.983128\pi\)
0.998596 0.0529806i \(-0.0168721\pi\)
\(774\) 0 0
\(775\) −26.6204 35.4939i −0.956234 1.27498i
\(776\) 0 0
\(777\) 45.4598 + 39.3693i 1.63086 + 1.41237i
\(778\) 0 0
\(779\) −9.96585 −0.357064
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) 6.38202i 0.228075i
\(784\) 0 0
\(785\) −20.2462 + 40.4924i −0.722618 + 1.44524i
\(786\) 0 0
\(787\) −44.8078 −1.59722 −0.798612 0.601846i \(-0.794432\pi\)
−0.798612 + 0.601846i \(0.794432\pi\)
\(788\) 0 0
\(789\) −30.7386 −1.09432
\(790\) 0 0
\(791\) 26.6204 30.7386i 0.946514 1.09294i
\(792\) 0 0
\(793\) 0.630683i 0.0223962i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0540i 0.745770i 0.927878 + 0.372885i \(0.121631\pi\)
−0.927878 + 0.372885i \(0.878369\pi\)
\(798\) 0 0
\(799\) 4.31534i 0.152666i
\(800\) 0 0
\(801\) 31.6034i 1.11665i
\(802\) 0 0
\(803\) −6.73863 −0.237801
\(804\) 0 0
\(805\) −2.54715 6.13676i −0.0897751 0.216292i
\(806\) 0 0
\(807\) −51.2311 −1.80342
\(808\) 0 0
\(809\) 31.3002 1.10046 0.550228 0.835014i \(-0.314541\pi\)
0.550228 + 0.835014i \(0.314541\pi\)
\(810\) 0 0
\(811\) 14.2462i 0.500252i −0.968213 0.250126i \(-0.919528\pi\)
0.968213 0.250126i \(-0.0804721\pi\)
\(812\) 0 0
\(813\) 58.2238 2.04200
\(814\) 0 0
\(815\) −24.6752 + 49.3503i −0.864333 + 1.72867i
\(816\) 0 0
\(817\) −2.18475 −0.0764347
\(818\) 0 0
\(819\) −3.46410 + 4.00000i −0.121046 + 0.139771i
\(820\) 0 0
\(821\) 18.2931i 0.638435i 0.947681 + 0.319218i \(0.103420\pi\)
−0.947681 + 0.319218i \(0.896580\pi\)
\(822\) 0 0
\(823\) −35.3693 −1.23290 −0.616448 0.787395i \(-0.711429\pi\)
−0.616448 + 0.787395i \(0.711429\pi\)
\(824\) 0 0
\(825\) −7.47439 9.96585i −0.260225 0.346967i
\(826\) 0 0
\(827\) 34.6410i 1.20459i 0.798275 + 0.602293i \(0.205746\pi\)
−0.798275 + 0.602293i \(0.794254\pi\)
\(828\) 0 0
\(829\) −29.6155 −1.02859 −0.514295 0.857613i \(-0.671946\pi\)
−0.514295 + 0.857613i \(0.671946\pi\)
\(830\) 0 0
\(831\) 35.4939 1.23127
\(832\) 0 0
\(833\) −4.43674 30.7386i −0.153724 1.06503i
\(834\) 0 0
\(835\) 2.91791 5.83583i 0.100979 0.201957i
\(836\) 0 0
\(837\) 12.7640 0.441189
\(838\) 0 0
\(839\) −32.6957 −1.12878 −0.564391 0.825507i \(-0.690889\pi\)
−0.564391 + 0.825507i \(0.690889\pi\)
\(840\) 0 0
\(841\) 9.31534 0.321219
\(842\) 0 0
\(843\) 65.7926 2.26602
\(844\) 0 0
\(845\) 25.3693 + 12.6847i 0.872731 + 0.436366i
\(846\) 0 0
\(847\) 20.1080 + 17.4140i 0.690917 + 0.598352i
\(848\) 0 0
\(849\) −13.9309 −0.478106
\(850\) 0 0
\(851\) 9.96585 0.341625
\(852\) 0 0
\(853\) 22.4924i 0.770126i 0.922890 + 0.385063i \(0.125820\pi\)
−0.922890 + 0.385063i \(0.874180\pi\)
\(854\) 0 0
\(855\) 4.00000 8.00000i 0.136797 0.273594i
\(856\) 0 0
\(857\) −32.4563 −1.10868 −0.554342 0.832289i \(-0.687030\pi\)
−0.554342 + 0.832289i \(0.687030\pi\)
\(858\) 0 0
\(859\) 9.75379i 0.332795i 0.986059 + 0.166397i \(0.0532135\pi\)
−0.986059 + 0.166397i \(0.946787\pi\)
\(860\) 0 0
\(861\) 39.3693 45.4598i 1.34170 1.54926i
\(862\) 0 0
\(863\) −3.36932 −0.114693 −0.0573464 0.998354i \(-0.518264\pi\)
−0.0573464 + 0.998354i \(0.518264\pi\)
\(864\) 0 0
\(865\) −9.68466 + 19.3693i −0.329288 + 0.658577i
\(866\) 0 0
\(867\) −6.87689 −0.233552
\(868\) 0 0
\(869\) 7.47439i 0.253551i
\(870\) 0 0
\(871\) 3.89055 0.131826
\(872\) 0 0
\(873\) −33.5486 −1.13545
\(874\) 0 0
\(875\) 15.0526 25.4641i 0.508869 0.860844i
\(876\) 0 0
\(877\) −26.6204 −0.898908 −0.449454 0.893303i \(-0.648381\pi\)
−0.449454 + 0.893303i \(0.648381\pi\)
\(878\) 0 0
\(879\) 36.6695i 1.23683i
\(880\) 0 0
\(881\) 13.8564i 0.466834i −0.972377 0.233417i \(-0.925009\pi\)
0.972377 0.233417i \(-0.0749907\pi\)
\(882\) 0 0
\(883\) 15.8017i 0.531769i 0.964005 + 0.265884i \(0.0856640\pi\)
−0.964005 + 0.265884i \(0.914336\pi\)
\(884\) 0 0
\(885\) 10.2462 20.4924i 0.344423 0.688845i
\(886\) 0 0
\(887\) 32.0298i 1.07546i 0.843118 + 0.537728i \(0.180717\pi\)
−0.843118 + 0.537728i \(0.819283\pi\)
\(888\) 0 0
\(889\) 24.0000 + 20.7846i 0.804934 + 0.697093i
\(890\) 0 0
\(891\) −6.80847 −0.228092
\(892\) 0 0
\(893\) 1.09238 0.0365549
\(894\) 0 0
\(895\) −42.4221 21.2111i −1.41802 0.709008i
\(896\) 0 0
\(897\) 1.61553i 0.0539409i
\(898\) 0 0
\(899\) 39.3693i 1.31304i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 8.63068 9.96585i 0.287211 0.331643i
\(904\) 0 0
\(905\) −28.4924 14.2462i −0.947120 0.473560i
\(906\) 0 0
\(907\) 33.5486i 1.11396i −0.830524 0.556982i \(-0.811959\pi\)
0.830524 0.556982i \(-0.188041\pi\)
\(908\) 0 0
\(909\) −40.4924 −1.34305
\(910\) 0 0
\(911\) 18.7386i 0.620839i −0.950600 0.310419i \(-0.899531\pi\)
0.950600 0.310419i \(-0.100469\pi\)
\(912\) 0 0
\(913\) 3.89055 0.128758
\(914\) 0 0
\(915\) 5.75379 + 2.87689i 0.190214 + 0.0951072i
\(916\) 0 0
\(917\) 33.5486 38.7386i 1.10787 1.27926i
\(918\) 0 0
\(919\) 7.68466i 0.253493i 0.991935 + 0.126747i \(0.0404536\pi\)
−0.991935 + 0.126747i \(0.959546\pi\)
\(920\) 0 0
\(921\) 25.4384 0.838225
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 26.6204 + 35.4939i 0.875275 + 1.16703i
\(926\) 0 0
\(927\) 52.8144i 1.73465i
\(928\) 0 0
\(929\) 9.96585i 0.326969i 0.986546 + 0.163485i \(0.0522734\pi\)
−0.986546 + 0.163485i \(0.947727\pi\)
\(930\) 0 0
\(931\) 7.78110 1.12311i 0.255015 0.0368083i
\(932\) 0 0
\(933\) −48.2579 −1.57989
\(934\) 0 0
\(935\) −8.63068 4.31534i −0.282254 0.141127i
\(936\) 0 0
\(937\) −31.0572 −1.01459 −0.507297 0.861771i \(-0.669355\pi\)
−0.507297 + 0.861771i \(0.669355\pi\)
\(938\) 0 0
\(939\) 46.8589 1.52918
\(940\) 0 0
\(941\) −19.3693 −0.631422 −0.315711 0.948855i \(-0.602243\pi\)
−0.315711 + 0.948855i \(0.602243\pi\)
\(942\) 0 0
\(943\) 9.96585i 0.324533i
\(944\) 0 0
\(945\) 3.26233 + 7.85982i 0.106123 + 0.255680i
\(946\) 0 0
\(947\) 19.6922i 0.639912i −0.947432 0.319956i \(-0.896332\pi\)
0.947432 0.319956i \(-0.103668\pi\)
\(948\) 0 0
\(949\) 3.89055i 0.126293i
\(950\) 0 0
\(951\) −45.4598 −1.47413
\(952\) 0 0
\(953\) 17.6155i 0.570623i −0.958435 0.285311i \(-0.907903\pi\)
0.958435 0.285311i \(-0.0920970\pi\)
\(954\) 0 0
\(955\) −5.43845 + 10.8769i −0.175984 + 0.351968i
\(956\) 0 0
\(957\) 11.0540i 0.357324i
\(958\) 0 0
\(959\) 22.7299 26.2462i 0.733986 0.847534i
\(960\) 0 0
\(961\) 47.7386 1.53996
\(962\) 0 0
\(963\) 38.5316i 1.24166i
\(964\) 0 0
\(965\) 7.36932 14.7386i 0.237227 0.474453i
\(966\) 0 0
\(967\) −5.26137 −0.169194 −0.0845971 0.996415i \(-0.526960\pi\)
−0.0845971 + 0.996415i \(0.526960\pi\)
\(968\) 0 0
\(969\) 12.7640i 0.410040i
\(970\) 0 0
\(971\) 12.6307i 0.405338i −0.979247 0.202669i \(-0.935038\pi\)
0.979247 0.202669i \(-0.0649615\pi\)
\(972\) 0 0
\(973\) 24.6752 28.4924i 0.791049 0.913425i
\(974\) 0 0
\(975\) −5.75379 + 4.31534i −0.184269 + 0.138202i
\(976\) 0 0
\(977\) 37.1231i 1.18767i −0.804586 0.593837i \(-0.797613\pi\)
0.804586 0.593837i \(-0.202387\pi\)
\(978\) 0 0
\(979\) 8.63068i 0.275838i
\(980\) 0 0
\(981\) 65.1520i 2.08014i
\(982\) 0 0
\(983\) 4.86319i 0.155112i 0.996988 + 0.0775558i \(0.0247116\pi\)
−0.996988 + 0.0775558i \(0.975288\pi\)
\(984\) 0 0
\(985\) −45.4598 22.7299i −1.44847 0.724234i
\(986\) 0 0
\(987\) −4.31534 + 4.98293i −0.137359 + 0.158608i
\(988\) 0 0
\(989\) 2.18475i 0.0694710i
\(990\) 0 0
\(991\) 49.4773i 1.57170i −0.618419 0.785849i \(-0.712226\pi\)
0.618419 0.785849i \(-0.287774\pi\)
\(992\) 0 0
\(993\) −44.3674 −1.40796
\(994\) 0 0
\(995\) 35.4939 + 17.7470i 1.12523 + 0.562616i
\(996\) 0 0
\(997\) 54.1771i 1.71581i 0.513812 + 0.857903i \(0.328233\pi\)
−0.513812 + 0.857903i \(0.671767\pi\)
\(998\) 0 0
\(999\) −12.7640 −0.403836
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.n.b.1119.3 yes 8
4.3 odd 2 2240.2.n.h.1119.8 yes 8
5.4 even 2 2240.2.n.h.1119.6 yes 8
7.6 odd 2 2240.2.n.g.1119.6 yes 8
8.3 odd 2 2240.2.n.a.1119.2 yes 8
8.5 even 2 2240.2.n.g.1119.5 yes 8
20.19 odd 2 inner 2240.2.n.b.1119.1 yes 8
28.27 even 2 2240.2.n.a.1119.1 8
35.34 odd 2 2240.2.n.a.1119.3 yes 8
40.19 odd 2 2240.2.n.g.1119.7 yes 8
40.29 even 2 2240.2.n.a.1119.4 yes 8
56.13 odd 2 inner 2240.2.n.b.1119.4 yes 8
56.27 even 2 2240.2.n.h.1119.7 yes 8
140.139 even 2 2240.2.n.g.1119.8 yes 8
280.69 odd 2 2240.2.n.h.1119.5 yes 8
280.139 even 2 inner 2240.2.n.b.1119.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.n.a.1119.1 8 28.27 even 2
2240.2.n.a.1119.2 yes 8 8.3 odd 2
2240.2.n.a.1119.3 yes 8 35.34 odd 2
2240.2.n.a.1119.4 yes 8 40.29 even 2
2240.2.n.b.1119.1 yes 8 20.19 odd 2 inner
2240.2.n.b.1119.2 yes 8 280.139 even 2 inner
2240.2.n.b.1119.3 yes 8 1.1 even 1 trivial
2240.2.n.b.1119.4 yes 8 56.13 odd 2 inner
2240.2.n.g.1119.5 yes 8 8.5 even 2
2240.2.n.g.1119.6 yes 8 7.6 odd 2
2240.2.n.g.1119.7 yes 8 40.19 odd 2
2240.2.n.g.1119.8 yes 8 140.139 even 2
2240.2.n.h.1119.5 yes 8 280.69 odd 2
2240.2.n.h.1119.6 yes 8 5.4 even 2
2240.2.n.h.1119.7 yes 8 56.27 even 2
2240.2.n.h.1119.8 yes 8 4.3 odd 2