Properties

Label 2240.2.e.f.2239.1
Level $2240$
Weight $2$
Character 2240.2239
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(2239,2240)]
 
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, 0, 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("2240.2239");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6x^{14} + 28x^{12} + 16x^{10} - 40x^{8} + 610x^{6} + 1625x^{4} - 524x^{2} + 1444 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2239.1
Root \(0.328458 - 1.49331i\) of defining polynomial
Character \(\chi\) \(=\) 2240.2239
Dual form 2240.2.e.f.2239.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.13578i q^{3} +(-1.94442 - 1.10418i) q^{5} +(-2.35829 + 1.19935i) q^{7} -1.56155 q^{9} +O(q^{10})\) \(q-2.13578i q^{3} +(-1.94442 - 1.10418i) q^{5} +(-2.35829 + 1.19935i) q^{7} -1.56155 q^{9} +2.33205i q^{11} -1.09190 q^{13} +(-2.35829 + 4.15286i) q^{15} -4.98074 q^{17} -2.57501 q^{19} +(2.56155 + 5.03680i) q^{21} +6.04090 q^{23} +(2.56155 + 4.29400i) q^{25} -3.07221i q^{27} -0.561553 q^{29} +6.59603 q^{31} +4.98074 q^{33} +(5.90983 + 0.271945i) q^{35} +5.49966i q^{37} +2.33205i q^{39} +8.48528i q^{41} +1.32431 q^{43} +(3.03632 + 1.72424i) q^{45} -9.74247i q^{47} +(4.12311 - 5.65685i) q^{49} +10.6378i q^{51} +8.58800i q^{53} +(2.57501 - 4.53448i) q^{55} +5.49966i q^{57} +14.3211 q^{59} -0.620058i q^{61} +(3.68260 - 1.87285i) q^{63} +(2.12311 + 1.20565i) q^{65} -4.71659 q^{67} -12.9020i q^{69} -11.9473i q^{71} -9.96148 q^{73} +(9.17104 - 5.47091i) q^{75} +(-2.79695 - 5.49966i) q^{77} +10.6378i q^{79} -11.2462 q^{81} +3.86098i q^{83} +(9.68466 + 5.49966i) q^{85} +1.19935i q^{87} -2.82843i q^{89} +(2.57501 - 1.30957i) q^{91} -14.0877i q^{93} +(5.00691 + 2.84329i) q^{95} +14.9422 q^{97} -3.64162i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} + 8 q^{21} + 8 q^{25} + 24 q^{29} - 32 q^{65} - 48 q^{81} + 56 q^{85}+O(q^{100}) \) 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
\(3\) 2.13578i 1.23309i −0.787319 0.616546i \(-0.788531\pi\)
0.787319 0.616546i \(-0.211469\pi\)
\(4\) 0 0
\(5\) −1.94442 1.10418i −0.869572 0.493806i
\(6\) 0 0
\(7\) −2.35829 + 1.19935i −0.891352 + 0.453313i
\(8\) 0 0
\(9\) −1.56155 −0.520518
\(10\) 0 0
\(11\) 2.33205i 0.703139i 0.936162 + 0.351569i \(0.114352\pi\)
−0.936162 + 0.351569i \(0.885648\pi\)
\(12\) 0 0
\(13\) −1.09190 −0.302837 −0.151419 0.988470i \(-0.548384\pi\)
−0.151419 + 0.988470i \(0.548384\pi\)
\(14\) 0 0
\(15\) −2.35829 + 4.15286i −0.608909 + 1.07226i
\(16\) 0 0
\(17\) −4.98074 −1.20801 −0.604003 0.796982i \(-0.706429\pi\)
−0.604003 + 0.796982i \(0.706429\pi\)
\(18\) 0 0
\(19\) −2.57501 −0.590748 −0.295374 0.955382i \(-0.595444\pi\)
−0.295374 + 0.955382i \(0.595444\pi\)
\(20\) 0 0
\(21\) 2.56155 + 5.03680i 0.558977 + 1.09912i
\(22\) 0 0
\(23\) 6.04090 1.25961 0.629807 0.776752i \(-0.283134\pi\)
0.629807 + 0.776752i \(0.283134\pi\)
\(24\) 0 0
\(25\) 2.56155 + 4.29400i 0.512311 + 0.858800i
\(26\) 0 0
\(27\) 3.07221i 0.591246i
\(28\) 0 0
\(29\) −0.561553 −0.104278 −0.0521389 0.998640i \(-0.516604\pi\)
−0.0521389 + 0.998640i \(0.516604\pi\)
\(30\) 0 0
\(31\) 6.59603 1.18468 0.592341 0.805688i \(-0.298204\pi\)
0.592341 + 0.805688i \(0.298204\pi\)
\(32\) 0 0
\(33\) 4.98074 0.867035
\(34\) 0 0
\(35\) 5.90983 + 0.271945i 0.998943 + 0.0459670i
\(36\) 0 0
\(37\) 5.49966i 0.904138i 0.891983 + 0.452069i \(0.149314\pi\)
−0.891983 + 0.452069i \(0.850686\pi\)
\(38\) 0 0
\(39\) 2.33205i 0.373427i
\(40\) 0 0
\(41\) 8.48528i 1.32518i 0.748983 + 0.662589i \(0.230542\pi\)
−0.748983 + 0.662589i \(0.769458\pi\)
\(42\) 0 0
\(43\) 1.32431 0.201955 0.100977 0.994889i \(-0.467803\pi\)
0.100977 + 0.994889i \(0.467803\pi\)
\(44\) 0 0
\(45\) 3.03632 + 1.72424i 0.452627 + 0.257035i
\(46\) 0 0
\(47\) 9.74247i 1.42109i −0.703654 0.710543i \(-0.748450\pi\)
0.703654 0.710543i \(-0.251550\pi\)
\(48\) 0 0
\(49\) 4.12311 5.65685i 0.589015 0.808122i
\(50\) 0 0
\(51\) 10.6378i 1.48958i
\(52\) 0 0
\(53\) 8.58800i 1.17965i 0.807530 + 0.589826i \(0.200804\pi\)
−0.807530 + 0.589826i \(0.799196\pi\)
\(54\) 0 0
\(55\) 2.57501 4.53448i 0.347214 0.611430i
\(56\) 0 0
\(57\) 5.49966i 0.728447i
\(58\) 0 0
\(59\) 14.3211 1.86444 0.932222 0.361888i \(-0.117868\pi\)
0.932222 + 0.361888i \(0.117868\pi\)
\(60\) 0 0
\(61\) 0.620058i 0.0793903i −0.999212 0.0396951i \(-0.987361\pi\)
0.999212 0.0396951i \(-0.0126387\pi\)
\(62\) 0 0
\(63\) 3.68260 1.87285i 0.463964 0.235957i
\(64\) 0 0
\(65\) 2.12311 + 1.20565i 0.263339 + 0.149543i
\(66\) 0 0
\(67\) −4.71659 −0.576223 −0.288112 0.957597i \(-0.593027\pi\)
−0.288112 + 0.957597i \(0.593027\pi\)
\(68\) 0 0
\(69\) 12.9020i 1.55322i
\(70\) 0 0
\(71\) 11.9473i 1.41789i −0.705265 0.708943i \(-0.749172\pi\)
0.705265 0.708943i \(-0.250828\pi\)
\(72\) 0 0
\(73\) −9.96148 −1.16590 −0.582951 0.812507i \(-0.698102\pi\)
−0.582951 + 0.812507i \(0.698102\pi\)
\(74\) 0 0
\(75\) 9.17104 5.47091i 1.05898 0.631726i
\(76\) 0 0
\(77\) −2.79695 5.49966i −0.318742 0.626744i
\(78\) 0 0
\(79\) 10.6378i 1.19684i 0.801182 + 0.598421i \(0.204205\pi\)
−0.801182 + 0.598421i \(0.795795\pi\)
\(80\) 0 0
\(81\) −11.2462 −1.24958
\(82\) 0 0
\(83\) 3.86098i 0.423798i 0.977292 + 0.211899i \(0.0679648\pi\)
−0.977292 + 0.211899i \(0.932035\pi\)
\(84\) 0 0
\(85\) 9.68466 + 5.49966i 1.05045 + 0.596521i
\(86\) 0 0
\(87\) 1.19935i 0.128584i
\(88\) 0 0
\(89\) 2.82843i 0.299813i −0.988700 0.149906i \(-0.952103\pi\)
0.988700 0.149906i \(-0.0478972\pi\)
\(90\) 0 0
\(91\) 2.57501 1.30957i 0.269935 0.137280i
\(92\) 0 0
\(93\) 14.0877i 1.46082i
\(94\) 0 0
\(95\) 5.00691 + 2.84329i 0.513698 + 0.291715i
\(96\) 0 0
\(97\) 14.9422 1.51715 0.758576 0.651584i \(-0.225895\pi\)
0.758576 + 0.651584i \(0.225895\pi\)
\(98\) 0 0
\(99\) 3.64162i 0.365996i
\(100\) 0 0
\(101\) 15.1104i 1.50354i 0.659425 + 0.751770i \(0.270800\pi\)
−0.659425 + 0.751770i \(0.729200\pi\)
\(102\) 0 0
\(103\) 5.47091i 0.539065i 0.962991 + 0.269532i \(0.0868691\pi\)
−0.962991 + 0.269532i \(0.913131\pi\)
\(104\) 0 0
\(105\) 0.580814 12.6221i 0.0566816 1.23179i
\(106\) 0 0
\(107\) 10.0138 0.968072 0.484036 0.875048i \(-0.339170\pi\)
0.484036 + 0.875048i \(0.339170\pi\)
\(108\) 0 0
\(109\) −4.56155 −0.436918 −0.218459 0.975846i \(-0.570103\pi\)
−0.218459 + 0.975846i \(0.570103\pi\)
\(110\) 0 0
\(111\) 11.7460 1.11489
\(112\) 0 0
\(113\) 5.49966i 0.517364i −0.965963 0.258682i \(-0.916712\pi\)
0.965963 0.258682i \(-0.0832882\pi\)
\(114\) 0 0
\(115\) −11.7460 6.67026i −1.09532 0.622005i
\(116\) 0 0
\(117\) 1.70505 0.157632
\(118\) 0 0
\(119\) 11.7460 5.97366i 1.07676 0.547605i
\(120\) 0 0
\(121\) 5.56155 0.505596
\(122\) 0 0
\(123\) 18.1227 1.63407
\(124\) 0 0
\(125\) −0.239369 11.1778i −0.0214098 0.999771i
\(126\) 0 0
\(127\) 5.29723 0.470053 0.235026 0.971989i \(-0.424482\pi\)
0.235026 + 0.971989i \(0.424482\pi\)
\(128\) 0 0
\(129\) 2.82843i 0.249029i
\(130\) 0 0
\(131\) 2.57501 0.224980 0.112490 0.993653i \(-0.464117\pi\)
0.112490 + 0.993653i \(0.464117\pi\)
\(132\) 0 0
\(133\) 6.07263 3.08835i 0.526564 0.267794i
\(134\) 0 0
\(135\) −3.39228 + 5.97366i −0.291961 + 0.514131i
\(136\) 0 0
\(137\) 3.08835i 0.263855i 0.991259 + 0.131928i \(0.0421167\pi\)
−0.991259 + 0.131928i \(0.957883\pi\)
\(138\) 0 0
\(139\) 4.02102 0.341058 0.170529 0.985353i \(-0.445452\pi\)
0.170529 + 0.985353i \(0.445452\pi\)
\(140\) 0 0
\(141\) −20.8078 −1.75233
\(142\) 0 0
\(143\) 2.54635i 0.212937i
\(144\) 0 0
\(145\) 1.09190 + 0.620058i 0.0906770 + 0.0514930i
\(146\) 0 0
\(147\) −12.0818 8.80604i −0.996489 0.726310i
\(148\) 0 0
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 4.95118i 0.402922i 0.979497 + 0.201461i \(0.0645689\pi\)
−0.979497 + 0.201461i \(0.935431\pi\)
\(152\) 0 0
\(153\) 7.77769 0.628789
\(154\) 0 0
\(155\) −12.8255 7.28323i −1.03017 0.585003i
\(156\) 0 0
\(157\) −3.88884 −0.310364 −0.155182 0.987886i \(-0.549596\pi\)
−0.155182 + 0.987886i \(0.549596\pi\)
\(158\) 0 0
\(159\) 18.3421 1.45462
\(160\) 0 0
\(161\) −14.2462 + 7.24517i −1.12276 + 0.570999i
\(162\) 0 0
\(163\) 11.5012 0.900840 0.450420 0.892817i \(-0.351274\pi\)
0.450420 + 0.892817i \(0.351274\pi\)
\(164\) 0 0
\(165\) −9.68466 5.49966i −0.753950 0.428148i
\(166\) 0 0
\(167\) 0.673500i 0.0521170i 0.999660 + 0.0260585i \(0.00829562\pi\)
−0.999660 + 0.0260585i \(0.991704\pi\)
\(168\) 0 0
\(169\) −11.8078 −0.908290
\(170\) 0 0
\(171\) 4.02102 0.307495
\(172\) 0 0
\(173\) 11.0534 0.840372 0.420186 0.907438i \(-0.361965\pi\)
0.420186 + 0.907438i \(0.361965\pi\)
\(174\) 0 0
\(175\) −11.1909 7.05431i −0.845954 0.533256i
\(176\) 0 0
\(177\) 30.5866i 2.29903i
\(178\) 0 0
\(179\) 8.30571i 0.620798i 0.950606 + 0.310399i \(0.100463\pi\)
−0.950606 + 0.310399i \(0.899537\pi\)
\(180\) 0 0
\(181\) 3.44849i 0.256324i 0.991753 + 0.128162i \(0.0409077\pi\)
−0.991753 + 0.128162i \(0.959092\pi\)
\(182\) 0 0
\(183\) −1.32431 −0.0978956
\(184\) 0 0
\(185\) 6.07263 10.6937i 0.446469 0.786213i
\(186\) 0 0
\(187\) 11.6153i 0.849396i
\(188\) 0 0
\(189\) 3.68466 + 7.24517i 0.268019 + 0.527008i
\(190\) 0 0
\(191\) 19.9660i 1.44469i 0.691535 + 0.722343i \(0.256935\pi\)
−0.691535 + 0.722343i \(0.743065\pi\)
\(192\) 0 0
\(193\) 5.49966i 0.395874i 0.980215 + 0.197937i \(0.0634241\pi\)
−0.980215 + 0.197937i \(0.936576\pi\)
\(194\) 0 0
\(195\) 2.57501 4.53448i 0.184400 0.324721i
\(196\) 0 0
\(197\) 16.4990i 1.17550i −0.809042 0.587751i \(-0.800013\pi\)
0.809042 0.587751i \(-0.199987\pi\)
\(198\) 0 0
\(199\) −18.3421 −1.30024 −0.650118 0.759834i \(-0.725280\pi\)
−0.650118 + 0.759834i \(0.725280\pi\)
\(200\) 0 0
\(201\) 10.0736i 0.710536i
\(202\) 0 0
\(203\) 1.32431 0.673500i 0.0929481 0.0472704i
\(204\) 0 0
\(205\) 9.36932 16.4990i 0.654381 1.15234i
\(206\) 0 0
\(207\) −9.43318 −0.655651
\(208\) 0 0
\(209\) 6.00505i 0.415378i
\(210\) 0 0
\(211\) 0.287088i 0.0197640i 0.999951 + 0.00988198i \(0.00314558\pi\)
−0.999951 + 0.00988198i \(0.996854\pi\)
\(212\) 0 0
\(213\) −25.5169 −1.74839
\(214\) 0 0
\(215\) −2.57501 1.46228i −0.175614 0.0997266i
\(216\) 0 0
\(217\) −15.5554 + 7.91096i −1.05597 + 0.537031i
\(218\) 0 0
\(219\) 21.2755i 1.43767i
\(220\) 0 0
\(221\) 5.43845 0.365830
\(222\) 0 0
\(223\) 0.147647i 0.00988718i −0.999988 0.00494359i \(-0.998426\pi\)
0.999988 0.00494359i \(-0.00157360\pi\)
\(224\) 0 0
\(225\) −4.00000 6.70531i −0.266667 0.447021i
\(226\) 0 0
\(227\) 10.1530i 0.673881i 0.941526 + 0.336941i \(0.109392\pi\)
−0.941526 + 0.336941i \(0.890608\pi\)
\(228\) 0 0
\(229\) 9.45353i 0.624707i −0.949966 0.312354i \(-0.898883\pi\)
0.949966 0.312354i \(-0.101117\pi\)
\(230\) 0 0
\(231\) −11.7460 + 5.97366i −0.772833 + 0.393038i
\(232\) 0 0
\(233\) 10.9993i 0.720589i 0.932839 + 0.360294i \(0.117324\pi\)
−0.932839 + 0.360294i \(0.882676\pi\)
\(234\) 0 0
\(235\) −10.7575 + 18.9435i −0.701741 + 1.23574i
\(236\) 0 0
\(237\) 22.7199 1.47582
\(238\) 0 0
\(239\) 2.33205i 0.150848i −0.997152 0.0754238i \(-0.975969\pi\)
0.997152 0.0754238i \(-0.0240310\pi\)
\(240\) 0 0
\(241\) 16.0786i 1.03572i 0.855466 + 0.517858i \(0.173271\pi\)
−0.855466 + 0.517858i \(0.826729\pi\)
\(242\) 0 0
\(243\) 14.8028i 0.949601i
\(244\) 0 0
\(245\) −14.2633 + 6.44664i −0.911247 + 0.411861i
\(246\) 0 0
\(247\) 2.81164 0.178901
\(248\) 0 0
\(249\) 8.24621 0.522582
\(250\) 0 0
\(251\) 9.17104 0.578871 0.289435 0.957198i \(-0.406532\pi\)
0.289435 + 0.957198i \(0.406532\pi\)
\(252\) 0 0
\(253\) 14.0877i 0.885683i
\(254\) 0 0
\(255\) 11.7460 20.6843i 0.735566 1.29530i
\(256\) 0 0
\(257\) −6.55137 −0.408663 −0.204332 0.978902i \(-0.565502\pi\)
−0.204332 + 0.978902i \(0.565502\pi\)
\(258\) 0 0
\(259\) −6.59603 12.9698i −0.409857 0.805905i
\(260\) 0 0
\(261\) 0.876894 0.0542784
\(262\) 0 0
\(263\) 23.5829 1.45419 0.727093 0.686539i \(-0.240871\pi\)
0.727093 + 0.686539i \(0.240871\pi\)
\(264\) 0 0
\(265\) 9.48274 16.6987i 0.582520 1.02579i
\(266\) 0 0
\(267\) −6.04090 −0.369697
\(268\) 0 0
\(269\) 0.968253i 0.0590354i 0.999564 + 0.0295177i \(0.00939715\pi\)
−0.999564 + 0.0295177i \(0.990603\pi\)
\(270\) 0 0
\(271\) −6.59603 −0.400680 −0.200340 0.979726i \(-0.564205\pi\)
−0.200340 + 0.979726i \(0.564205\pi\)
\(272\) 0 0
\(273\) −2.79695 5.49966i −0.169279 0.332854i
\(274\) 0 0
\(275\) −10.0138 + 5.97366i −0.603856 + 0.360225i
\(276\) 0 0
\(277\) 19.5873i 1.17689i 0.808538 + 0.588444i \(0.200259\pi\)
−0.808538 + 0.588444i \(0.799741\pi\)
\(278\) 0 0
\(279\) −10.3000 −0.616648
\(280\) 0 0
\(281\) 17.0540 1.01735 0.508677 0.860957i \(-0.330135\pi\)
0.508677 + 0.860957i \(0.330135\pi\)
\(282\) 0 0
\(283\) 7.45904i 0.443394i 0.975116 + 0.221697i \(0.0711596\pi\)
−0.975116 + 0.221697i \(0.928840\pi\)
\(284\) 0 0
\(285\) 6.07263 10.6937i 0.359712 0.633437i
\(286\) 0 0
\(287\) −10.1768 20.0108i −0.600720 1.18120i
\(288\) 0 0
\(289\) 7.80776 0.459280
\(290\) 0 0
\(291\) 31.9133i 1.87079i
\(292\) 0 0
\(293\) 14.4635 0.844965 0.422483 0.906371i \(-0.361159\pi\)
0.422483 + 0.906371i \(0.361159\pi\)
\(294\) 0 0
\(295\) −27.8462 15.8131i −1.62127 0.920674i
\(296\) 0 0
\(297\) 7.16453 0.415728
\(298\) 0 0
\(299\) −6.59603 −0.381458
\(300\) 0 0
\(301\) −3.12311 + 1.58831i −0.180013 + 0.0915487i
\(302\) 0 0
\(303\) 32.2725 1.85400
\(304\) 0 0
\(305\) −0.684658 + 1.20565i −0.0392034 + 0.0690356i
\(306\) 0 0
\(307\) 13.8987i 0.793243i 0.917982 + 0.396622i \(0.129818\pi\)
−0.917982 + 0.396622i \(0.870182\pi\)
\(308\) 0 0
\(309\) 11.6847 0.664717
\(310\) 0 0
\(311\) −1.44600 −0.0819954 −0.0409977 0.999159i \(-0.513054\pi\)
−0.0409977 + 0.999159i \(0.513054\pi\)
\(312\) 0 0
\(313\) 7.16453 0.404963 0.202482 0.979286i \(-0.435099\pi\)
0.202482 + 0.979286i \(0.435099\pi\)
\(314\) 0 0
\(315\) −9.22851 0.424656i −0.519967 0.0239266i
\(316\) 0 0
\(317\) 24.4099i 1.37100i 0.728073 + 0.685499i \(0.240416\pi\)
−0.728073 + 0.685499i \(0.759584\pi\)
\(318\) 0 0
\(319\) 1.30957i 0.0733217i
\(320\) 0 0
\(321\) 21.3873i 1.19372i
\(322\) 0 0
\(323\) 12.8255 0.713628
\(324\) 0 0
\(325\) −2.79695 4.68860i −0.155147 0.260077i
\(326\) 0 0
\(327\) 9.74247i 0.538760i
\(328\) 0 0
\(329\) 11.6847 + 22.9756i 0.644196 + 1.26669i
\(330\) 0 0
\(331\) 8.30571i 0.456523i 0.973600 + 0.228262i \(0.0733041\pi\)
−0.973600 + 0.228262i \(0.926696\pi\)
\(332\) 0 0
\(333\) 8.58800i 0.470620i
\(334\) 0 0
\(335\) 9.17104 + 5.20798i 0.501067 + 0.284543i
\(336\) 0 0
\(337\) 30.5866i 1.66616i 0.553153 + 0.833080i \(0.313425\pi\)
−0.553153 + 0.833080i \(0.686575\pi\)
\(338\) 0 0
\(339\) −11.7460 −0.637958
\(340\) 0 0
\(341\) 15.3823i 0.832996i
\(342\) 0 0
\(343\) −2.93893 + 18.2856i −0.158687 + 0.987329i
\(344\) 0 0
\(345\) −14.2462 + 25.0870i −0.766990 + 1.35064i
\(346\) 0 0
\(347\) 1.32431 0.0710925 0.0355463 0.999368i \(-0.488683\pi\)
0.0355463 + 0.999368i \(0.488683\pi\)
\(348\) 0 0
\(349\) 18.8307i 1.00799i 0.863708 + 0.503993i \(0.168136\pi\)
−0.863708 + 0.503993i \(0.831864\pi\)
\(350\) 0 0
\(351\) 3.35453i 0.179051i
\(352\) 0 0
\(353\) −16.1685 −0.860564 −0.430282 0.902694i \(-0.641586\pi\)
−0.430282 + 0.902694i \(0.641586\pi\)
\(354\) 0 0
\(355\) −13.1921 + 23.2306i −0.700162 + 1.23295i
\(356\) 0 0
\(357\) −12.7584 25.0870i −0.675248 1.32774i
\(358\) 0 0
\(359\) 10.3507i 0.546288i −0.961973 0.273144i \(-0.911937\pi\)
0.961973 0.273144i \(-0.0880635\pi\)
\(360\) 0 0
\(361\) −12.3693 −0.651017
\(362\) 0 0
\(363\) 11.8782i 0.623446i
\(364\) 0 0
\(365\) 19.3693 + 10.9993i 1.01384 + 0.575730i
\(366\) 0 0
\(367\) 23.3783i 1.22034i 0.792272 + 0.610168i \(0.208898\pi\)
−0.792272 + 0.610168i \(0.791102\pi\)
\(368\) 0 0
\(369\) 13.2502i 0.689779i
\(370\) 0 0
\(371\) −10.3000 20.2530i −0.534752 1.05149i
\(372\) 0 0
\(373\) 11.6763i 0.604578i 0.953216 + 0.302289i \(0.0977508\pi\)
−0.953216 + 0.302289i \(0.902249\pi\)
\(374\) 0 0
\(375\) −23.8733 + 0.511240i −1.23281 + 0.0264003i
\(376\) 0 0
\(377\) 0.613157 0.0315792
\(378\) 0 0
\(379\) 24.9171i 1.27991i −0.768414 0.639954i \(-0.778954\pi\)
0.768414 0.639954i \(-0.221046\pi\)
\(380\) 0 0
\(381\) 11.3137i 0.579619i
\(382\) 0 0
\(383\) 18.6638i 0.953675i 0.878991 + 0.476838i \(0.158217\pi\)
−0.878991 + 0.476838i \(0.841783\pi\)
\(384\) 0 0
\(385\) −0.634188 + 13.7820i −0.0323212 + 0.702396i
\(386\) 0 0
\(387\) −2.06798 −0.105121
\(388\) 0 0
\(389\) 11.9309 0.604919 0.302460 0.953162i \(-0.402192\pi\)
0.302460 + 0.953162i \(0.402192\pi\)
\(390\) 0 0
\(391\) −30.0881 −1.52162
\(392\) 0 0
\(393\) 5.49966i 0.277421i
\(394\) 0 0
\(395\) 11.7460 20.6843i 0.591008 1.04074i
\(396\) 0 0
\(397\) −21.0149 −1.05471 −0.527353 0.849647i \(-0.676815\pi\)
−0.527353 + 0.849647i \(0.676815\pi\)
\(398\) 0 0
\(399\) −6.59603 12.9698i −0.330214 0.649303i
\(400\) 0 0
\(401\) 13.4384 0.671084 0.335542 0.942025i \(-0.391081\pi\)
0.335542 + 0.942025i \(0.391081\pi\)
\(402\) 0 0
\(403\) −7.20217 −0.358766
\(404\) 0 0
\(405\) 21.8674 + 12.4179i 1.08660 + 0.617050i
\(406\) 0 0
\(407\) −12.8255 −0.635734
\(408\) 0 0
\(409\) 30.2208i 1.49432i 0.664643 + 0.747161i \(0.268583\pi\)
−0.664643 + 0.747161i \(0.731417\pi\)
\(410\) 0 0
\(411\) 6.59603 0.325358
\(412\) 0 0
\(413\) −33.7733 + 17.1760i −1.66187 + 0.845176i
\(414\) 0 0
\(415\) 4.26324 7.50738i 0.209274 0.368523i
\(416\) 0 0
\(417\) 8.58800i 0.420556i
\(418\) 0 0
\(419\) −22.3631 −1.09251 −0.546254 0.837619i \(-0.683947\pi\)
−0.546254 + 0.837619i \(0.683947\pi\)
\(420\) 0 0
\(421\) −12.5616 −0.612213 −0.306106 0.951997i \(-0.599026\pi\)
−0.306106 + 0.951997i \(0.599026\pi\)
\(422\) 0 0
\(423\) 15.2134i 0.739700i
\(424\) 0 0
\(425\) −12.7584 21.3873i −0.618875 1.03744i
\(426\) 0 0
\(427\) 0.743668 + 1.46228i 0.0359886 + 0.0707647i
\(428\) 0 0
\(429\) −5.43845 −0.262571
\(430\) 0 0
\(431\) 11.6602i 0.561654i −0.959758 0.280827i \(-0.909391\pi\)
0.959758 0.280827i \(-0.0906087\pi\)
\(432\) 0 0
\(433\) 9.00400 0.432705 0.216352 0.976315i \(-0.430584\pi\)
0.216352 + 0.976315i \(0.430584\pi\)
\(434\) 0 0
\(435\) 1.32431 2.33205i 0.0634957 0.111813i
\(436\) 0 0
\(437\) −15.5554 −0.744114
\(438\) 0 0
\(439\) −31.5341 −1.50504 −0.752521 0.658568i \(-0.771162\pi\)
−0.752521 + 0.658568i \(0.771162\pi\)
\(440\) 0 0
\(441\) −6.43845 + 8.83348i −0.306593 + 0.420642i
\(442\) 0 0
\(443\) 17.5420 0.833448 0.416724 0.909033i \(-0.363178\pi\)
0.416724 + 0.909033i \(0.363178\pi\)
\(444\) 0 0
\(445\) −3.12311 + 5.49966i −0.148049 + 0.260709i
\(446\) 0 0
\(447\) 4.27156i 0.202038i
\(448\) 0 0
\(449\) 6.31534 0.298039 0.149020 0.988834i \(-0.452388\pi\)
0.149020 + 0.988834i \(0.452388\pi\)
\(450\) 0 0
\(451\) −19.7881 −0.931784
\(452\) 0 0
\(453\) 10.5746 0.496840
\(454\) 0 0
\(455\) −6.45291 0.296935i −0.302517 0.0139205i
\(456\) 0 0
\(457\) 10.3223i 0.482856i −0.970419 0.241428i \(-0.922384\pi\)
0.970419 0.241428i \(-0.0776157\pi\)
\(458\) 0 0
\(459\) 15.3019i 0.714229i
\(460\) 0 0
\(461\) 21.1154i 0.983444i −0.870752 0.491722i \(-0.836368\pi\)
0.870752 0.491722i \(-0.163632\pi\)
\(462\) 0 0
\(463\) 24.3266 1.13055 0.565277 0.824901i \(-0.308769\pi\)
0.565277 + 0.824901i \(0.308769\pi\)
\(464\) 0 0
\(465\) −15.5554 + 27.3924i −0.721363 + 1.27029i
\(466\) 0 0
\(467\) 23.1983i 1.07349i −0.843745 0.536744i \(-0.819654\pi\)
0.843745 0.536744i \(-0.180346\pi\)
\(468\) 0 0
\(469\) 11.1231 5.65685i 0.513617 0.261209i
\(470\) 0 0
\(471\) 8.30571i 0.382707i
\(472\) 0 0
\(473\) 3.08835i 0.142002i
\(474\) 0 0
\(475\) −6.59603 11.0571i −0.302646 0.507335i
\(476\) 0 0
\(477\) 13.4106i 0.614030i
\(478\) 0 0
\(479\) 30.0881 1.37476 0.687381 0.726297i \(-0.258760\pi\)
0.687381 + 0.726297i \(0.258760\pi\)
\(480\) 0 0
\(481\) 6.00505i 0.273807i
\(482\) 0 0
\(483\) 15.4741 + 30.4268i 0.704095 + 1.38447i
\(484\) 0 0
\(485\) −29.0540 16.4990i −1.31927 0.749179i
\(486\) 0 0
\(487\) −1.16128 −0.0526225 −0.0263112 0.999654i \(-0.508376\pi\)
−0.0263112 + 0.999654i \(0.508376\pi\)
\(488\) 0 0
\(489\) 24.5639i 1.11082i
\(490\) 0 0
\(491\) 14.2794i 0.644419i −0.946668 0.322210i \(-0.895574\pi\)
0.946668 0.322210i \(-0.104426\pi\)
\(492\) 0 0
\(493\) 2.79695 0.125968
\(494\) 0 0
\(495\) −4.02102 + 7.08084i −0.180731 + 0.318260i
\(496\) 0 0
\(497\) 14.3291 + 28.1753i 0.642746 + 1.26384i
\(498\) 0 0
\(499\) 25.6525i 1.14836i −0.818727 0.574182i \(-0.805320\pi\)
0.818727 0.574182i \(-0.194680\pi\)
\(500\) 0 0
\(501\) 1.43845 0.0642651
\(502\) 0 0
\(503\) 18.8114i 0.838761i 0.907811 + 0.419380i \(0.137753\pi\)
−0.907811 + 0.419380i \(0.862247\pi\)
\(504\) 0 0
\(505\) 16.6847 29.3810i 0.742458 1.30744i
\(506\) 0 0
\(507\) 25.2188i 1.12001i
\(508\) 0 0
\(509\) 28.0124i 1.24163i 0.783958 + 0.620814i \(0.213198\pi\)
−0.783958 + 0.620814i \(0.786802\pi\)
\(510\) 0 0
\(511\) 23.4921 11.9473i 1.03923 0.528519i
\(512\) 0 0
\(513\) 7.91096i 0.349278i
\(514\) 0 0
\(515\) 6.04090 10.6378i 0.266194 0.468756i
\(516\) 0 0
\(517\) 22.7199 0.999220
\(518\) 0 0
\(519\) 23.6076i 1.03626i
\(520\) 0 0
\(521\) 2.82843i 0.123916i 0.998079 + 0.0619578i \(0.0197344\pi\)
−0.998079 + 0.0619578i \(0.980266\pi\)
\(522\) 0 0
\(523\) 20.9472i 0.915958i 0.888963 + 0.457979i \(0.151426\pi\)
−0.888963 + 0.457979i \(0.848574\pi\)
\(524\) 0 0
\(525\) −15.0665 + 23.9013i −0.657554 + 1.04314i
\(526\) 0 0
\(527\) −32.8531 −1.43110
\(528\) 0 0
\(529\) 13.4924 0.586627
\(530\) 0 0
\(531\) −22.3631 −0.970476
\(532\) 0 0
\(533\) 9.26504i 0.401313i
\(534\) 0 0
\(535\) −19.4711 11.0571i −0.841808 0.478040i
\(536\) 0 0
\(537\) 17.7392 0.765502
\(538\) 0 0
\(539\) 13.1921 + 9.61528i 0.568222 + 0.414159i
\(540\) 0 0
\(541\) 19.4384 0.835724 0.417862 0.908510i \(-0.362780\pi\)
0.417862 + 0.908510i \(0.362780\pi\)
\(542\) 0 0
\(543\) 7.36520 0.316071
\(544\) 0 0
\(545\) 8.86958 + 5.03680i 0.379931 + 0.215753i
\(546\) 0 0
\(547\) −33.4337 −1.42952 −0.714762 0.699368i \(-0.753465\pi\)
−0.714762 + 0.699368i \(0.753465\pi\)
\(548\) 0 0
\(549\) 0.968253i 0.0413240i
\(550\) 0 0
\(551\) 1.44600 0.0616019
\(552\) 0 0
\(553\) −12.7584 25.0870i −0.542543 1.06681i
\(554\) 0 0
\(555\) −22.8393 12.9698i −0.969473 0.550538i
\(556\) 0 0
\(557\) 8.58800i 0.363885i −0.983309 0.181943i \(-0.941761\pi\)
0.983309 0.181943i \(-0.0582385\pi\)
\(558\) 0 0
\(559\) −1.44600 −0.0611595
\(560\) 0 0
\(561\) −24.8078 −1.04738
\(562\) 0 0
\(563\) 6.78554i 0.285977i 0.989724 + 0.142988i \(0.0456711\pi\)
−0.989724 + 0.142988i \(0.954329\pi\)
\(564\) 0 0
\(565\) −6.07263 + 10.6937i −0.255478 + 0.449885i
\(566\) 0 0
\(567\) 26.5219 13.4882i 1.11381 0.566450i
\(568\) 0 0
\(569\) −43.8617 −1.83878 −0.919390 0.393347i \(-0.871317\pi\)
−0.919390 + 0.393347i \(0.871317\pi\)
\(570\) 0 0
\(571\) 15.5889i 0.652377i 0.945305 + 0.326188i \(0.105764\pi\)
−0.945305 + 0.326188i \(0.894236\pi\)
\(572\) 0 0
\(573\) 42.6429 1.78143
\(574\) 0 0
\(575\) 15.4741 + 25.9396i 0.645313 + 1.08176i
\(576\) 0 0
\(577\) −36.0915 −1.50251 −0.751254 0.660013i \(-0.770551\pi\)
−0.751254 + 0.660013i \(0.770551\pi\)
\(578\) 0 0
\(579\) 11.7460 0.488149
\(580\) 0 0
\(581\) −4.63068 9.10534i −0.192113 0.377753i
\(582\) 0 0
\(583\) −20.0276 −0.829460
\(584\) 0 0
\(585\) −3.31534 1.88269i −0.137073 0.0778398i
\(586\) 0 0
\(587\) 2.80928i 0.115951i −0.998318 0.0579757i \(-0.981535\pi\)
0.998318 0.0579757i \(-0.0184646\pi\)
\(588\) 0 0
\(589\) −16.9848 −0.699848
\(590\) 0 0
\(591\) −35.2381 −1.44950
\(592\) 0 0
\(593\) 6.20705 0.254893 0.127447 0.991845i \(-0.459322\pi\)
0.127447 + 0.991845i \(0.459322\pi\)
\(594\) 0 0
\(595\) −29.4353 1.35448i −1.20673 0.0555285i
\(596\) 0 0
\(597\) 39.1746i 1.60331i
\(598\) 0 0
\(599\) 17.9210i 0.732232i 0.930569 + 0.366116i \(0.119313\pi\)
−0.930569 + 0.366116i \(0.880687\pi\)
\(600\) 0 0
\(601\) 42.2309i 1.72263i −0.508068 0.861317i \(-0.669640\pi\)
0.508068 0.861317i \(-0.330360\pi\)
\(602\) 0 0
\(603\) 7.36520 0.299934
\(604\) 0 0
\(605\) −10.8140 6.14098i −0.439652 0.249666i
\(606\) 0 0
\(607\) 44.9666i 1.82514i −0.408921 0.912570i \(-0.634095\pi\)
0.408921 0.912570i \(-0.365905\pi\)
\(608\) 0 0
\(609\) −1.43845 2.82843i −0.0582888 0.114614i
\(610\) 0 0
\(611\) 10.6378i 0.430358i
\(612\) 0 0
\(613\) 47.7626i 1.92911i −0.263874 0.964557i \(-0.585000\pi\)
0.263874 0.964557i \(-0.415000\pi\)
\(614\) 0 0
\(615\) −35.2381 20.0108i −1.42094 0.806913i
\(616\) 0 0
\(617\) 14.7647i 0.594404i −0.954815 0.297202i \(-0.903946\pi\)
0.954815 0.297202i \(-0.0960535\pi\)
\(618\) 0 0
\(619\) −26.0671 −1.04773 −0.523863 0.851803i \(-0.675510\pi\)
−0.523863 + 0.851803i \(0.675510\pi\)
\(620\) 0 0
\(621\) 18.5589i 0.744742i
\(622\) 0 0
\(623\) 3.39228 + 6.67026i 0.135909 + 0.267238i
\(624\) 0 0
\(625\) −11.8769 + 21.9986i −0.475076 + 0.879945i
\(626\) 0 0
\(627\) −12.8255 −0.512200
\(628\) 0 0
\(629\) 27.3924i 1.09220i
\(630\) 0 0
\(631\) 33.9582i 1.35186i −0.736968 0.675928i \(-0.763743\pi\)
0.736968 0.675928i \(-0.236257\pi\)
\(632\) 0 0
\(633\) 0.613157 0.0243708
\(634\) 0 0
\(635\) −10.3000 5.84912i −0.408745 0.232115i
\(636\) 0 0
\(637\) −4.50200 + 6.17669i −0.178376 + 0.244730i
\(638\) 0 0
\(639\) 18.6564i 0.738035i
\(640\) 0 0
\(641\) 39.8617 1.57444 0.787222 0.616670i \(-0.211519\pi\)
0.787222 + 0.616670i \(0.211519\pi\)
\(642\) 0 0
\(643\) 36.8341i 1.45260i 0.687380 + 0.726298i \(0.258761\pi\)
−0.687380 + 0.726298i \(0.741239\pi\)
\(644\) 0 0
\(645\) −3.12311 + 5.49966i −0.122972 + 0.216549i
\(646\) 0 0
\(647\) 36.5712i 1.43776i 0.695134 + 0.718881i \(0.255345\pi\)
−0.695134 + 0.718881i \(0.744655\pi\)
\(648\) 0 0
\(649\) 33.3974i 1.31096i
\(650\) 0 0
\(651\) 16.8961 + 33.2228i 0.662209 + 1.30211i
\(652\) 0 0
\(653\) 16.4990i 0.645654i −0.946458 0.322827i \(-0.895367\pi\)
0.946458 0.322827i \(-0.104633\pi\)
\(654\) 0 0
\(655\) −5.00691 2.84329i −0.195636 0.111096i
\(656\) 0 0
\(657\) 15.5554 0.606873
\(658\) 0 0
\(659\) 42.8381i 1.66874i 0.551207 + 0.834368i \(0.314167\pi\)
−0.551207 + 0.834368i \(0.685833\pi\)
\(660\) 0 0
\(661\) 1.51198i 0.0588092i −0.999568 0.0294046i \(-0.990639\pi\)
0.999568 0.0294046i \(-0.00936112\pi\)
\(662\) 0 0
\(663\) 11.6153i 0.451102i
\(664\) 0 0
\(665\) −15.2179 0.700260i −0.590124 0.0271549i
\(666\) 0 0
\(667\) −3.39228 −0.131350
\(668\) 0 0
\(669\) −0.315342 −0.0121918
\(670\) 0 0
\(671\) 1.44600 0.0558224
\(672\) 0 0
\(673\) 14.0877i 0.543039i 0.962433 + 0.271520i \(0.0875262\pi\)
−0.962433 + 0.271520i \(0.912474\pi\)
\(674\) 0 0
\(675\) 13.1921 7.86962i 0.507762 0.302902i
\(676\) 0 0
\(677\) 46.5317 1.78836 0.894179 0.447709i \(-0.147760\pi\)
0.894179 + 0.447709i \(0.147760\pi\)
\(678\) 0 0
\(679\) −35.2381 + 17.9210i −1.35232 + 0.687745i
\(680\) 0 0
\(681\) 21.6847 0.830958
\(682\) 0 0
\(683\) 20.1907 0.772574 0.386287 0.922379i \(-0.373757\pi\)
0.386287 + 0.922379i \(0.373757\pi\)
\(684\) 0 0
\(685\) 3.41011 6.00505i 0.130293 0.229441i
\(686\) 0 0
\(687\) −20.1907 −0.770322
\(688\) 0 0
\(689\) 9.37720i 0.357243i
\(690\) 0 0
\(691\) −37.8132 −1.43848 −0.719240 0.694761i \(-0.755510\pi\)
−0.719240 + 0.694761i \(0.755510\pi\)
\(692\) 0 0
\(693\) 4.36758 + 8.58800i 0.165911 + 0.326231i
\(694\) 0 0
\(695\) −7.81855 4.43994i −0.296575 0.168417i
\(696\) 0 0
\(697\) 42.2630i 1.60082i
\(698\) 0 0
\(699\) 23.4921 0.888553
\(700\) 0 0
\(701\) 30.6695 1.15837 0.579186 0.815196i \(-0.303371\pi\)
0.579186 + 0.815196i \(0.303371\pi\)
\(702\) 0 0
\(703\) 14.1617i 0.534118i
\(704\) 0 0
\(705\) 40.4591 + 22.9756i 1.52378 + 0.865312i
\(706\) 0 0
\(707\) −18.1227 35.6347i −0.681574 1.34018i
\(708\) 0 0
\(709\) −30.8078 −1.15701 −0.578505 0.815679i \(-0.696364\pi\)
−0.578505 + 0.815679i \(0.696364\pi\)
\(710\) 0 0
\(711\) 16.6114i 0.622977i
\(712\) 0 0
\(713\) 39.8459 1.49224
\(714\) 0 0
\(715\) −2.81164 + 4.95118i −0.105150 + 0.185164i
\(716\) 0 0
\(717\) −4.98074 −0.186009
\(718\) 0 0
\(719\) −27.1961 −1.01424 −0.507122 0.861874i \(-0.669291\pi\)
−0.507122 + 0.861874i \(0.669291\pi\)
\(720\) 0 0
\(721\) −6.56155 12.9020i −0.244365 0.480496i
\(722\) 0 0
\(723\) 34.3404 1.27713
\(724\) 0 0
\(725\) −1.43845 2.41131i −0.0534226 0.0895537i
\(726\) 0 0
\(727\) 32.8255i 1.21743i 0.793389 + 0.608715i \(0.208315\pi\)
−0.793389 + 0.608715i \(0.791685\pi\)
\(728\) 0 0
\(729\) −2.12311 −0.0786335
\(730\) 0 0
\(731\) −6.59603 −0.243963
\(732\) 0 0
\(733\) −24.4250 −0.902156 −0.451078 0.892484i \(-0.648960\pi\)
−0.451078 + 0.892484i \(0.648960\pi\)
\(734\) 0 0
\(735\) 13.7686 + 30.4632i 0.507863 + 1.12365i
\(736\) 0 0
\(737\) 10.9993i 0.405165i
\(738\) 0 0
\(739\) 49.5472i 1.82262i −0.411718 0.911311i \(-0.635071\pi\)
0.411718 0.911311i \(-0.364929\pi\)
\(740\) 0 0
\(741\) 6.00505i 0.220601i
\(742\) 0 0
\(743\) 9.43318 0.346070 0.173035 0.984916i \(-0.444643\pi\)
0.173035 + 0.984916i \(0.444643\pi\)
\(744\) 0 0
\(745\) −3.88884 2.20837i −0.142476 0.0809084i
\(746\) 0 0
\(747\) 6.02913i 0.220594i
\(748\) 0 0
\(749\) −23.6155 + 12.0101i −0.862893 + 0.438839i
\(750\) 0 0
\(751\) 37.5999i 1.37204i −0.727584 0.686019i \(-0.759357\pi\)
0.727584 0.686019i \(-0.240643\pi\)
\(752\) 0 0
\(753\) 19.5873i 0.713801i
\(754\) 0 0
\(755\) 5.46702 9.62719i 0.198965 0.350369i
\(756\) 0 0
\(757\) 25.7640i 0.936409i −0.883620 0.468204i \(-0.844901\pi\)
0.883620 0.468204i \(-0.155099\pi\)
\(758\) 0 0
\(759\) 30.0881 1.09213
\(760\) 0 0
\(761\) 43.1228i 1.56320i −0.623780 0.781600i \(-0.714404\pi\)
0.623780 0.781600i \(-0.285596\pi\)
\(762\) 0 0
\(763\) 10.7575 5.47091i 0.389447 0.198060i
\(764\) 0 0
\(765\) −15.1231 8.58800i −0.546777 0.310500i
\(766\) 0 0
\(767\) −15.6371 −0.564623
\(768\) 0 0
\(769\) 14.4903i 0.522535i −0.965266 0.261267i \(-0.915860\pi\)
0.965266 0.261267i \(-0.0841404\pi\)
\(770\) 0 0
\(771\) 13.9923i 0.503920i
\(772\) 0 0
\(773\) −15.6898 −0.564323 −0.282161 0.959367i \(-0.591051\pi\)
−0.282161 + 0.959367i \(0.591051\pi\)
\(774\) 0 0
\(775\) 16.8961 + 28.3234i 0.606925 + 1.01740i
\(776\) 0 0
\(777\) −27.7006 + 14.0877i −0.993755 + 0.505392i
\(778\) 0 0
\(779\) 21.8497i 0.782847i
\(780\) 0 0
\(781\) 27.8617 0.996971
\(782\) 0 0
\(783\) 1.72521i 0.0616538i
\(784\) 0 0
\(785\) 7.56155 + 4.29400i 0.269883 + 0.153259i
\(786\) 0 0
\(787\) 32.8578i 1.17126i 0.810580 + 0.585628i \(0.199152\pi\)
−0.810580 + 0.585628i \(0.800848\pi\)
\(788\) 0 0
\(789\) 50.3680i 1.79315i
\(790\) 0 0
\(791\) 6.59603 + 12.9698i 0.234528 + 0.461153i
\(792\) 0 0
\(793\) 0.677039i 0.0240423i
\(794\) 0 0
\(795\) −35.6647 20.2530i −1.26490 0.718301i
\(796\) 0 0
\(797\) −2.04937 −0.0725925 −0.0362963 0.999341i \(-0.511556\pi\)
−0.0362963 + 0.999341i \(0.511556\pi\)
\(798\) 0 0
\(799\) 48.5247i 1.71668i
\(800\) 0 0
\(801\) 4.41674i 0.156058i
\(802\) 0 0
\(803\) 23.2306i 0.819792i
\(804\) 0 0
\(805\) 35.7006 + 1.64279i 1.25828 + 0.0579007i
\(806\) 0 0
\(807\) 2.06798 0.0727962
\(808\) 0 0
\(809\) −27.0540 −0.951167 −0.475584 0.879671i \(-0.657763\pi\)
−0.475584 + 0.879671i \(0.657763\pi\)
\(810\) 0 0
\(811\) −9.17104 −0.322039 −0.161019 0.986951i \(-0.551478\pi\)
−0.161019 + 0.986951i \(0.551478\pi\)
\(812\) 0 0
\(813\) 14.0877i 0.494076i
\(814\) 0 0
\(815\) −22.3631 12.6994i −0.783345 0.444840i
\(816\) 0 0
\(817\) −3.41011 −0.119304
\(818\) 0 0
\(819\) −4.02102 + 2.04496i −0.140506 + 0.0714567i
\(820\) 0 0
\(821\) 35.9309 1.25400 0.626998 0.779021i \(-0.284283\pi\)
0.626998 + 0.779021i \(0.284283\pi\)
\(822\) 0 0
\(823\) 22.8393 0.796127 0.398064 0.917358i \(-0.369682\pi\)
0.398064 + 0.917358i \(0.369682\pi\)
\(824\) 0 0
\(825\) 12.7584 + 21.3873i 0.444191 + 0.744610i
\(826\) 0 0
\(827\) −4.71659 −0.164012 −0.0820059 0.996632i \(-0.526133\pi\)
−0.0820059 + 0.996632i \(0.526133\pi\)
\(828\) 0 0
\(829\) 43.3947i 1.50716i 0.657357 + 0.753579i \(0.271674\pi\)
−0.657357 + 0.753579i \(0.728326\pi\)
\(830\) 0 0
\(831\) 41.8342 1.45121
\(832\) 0 0
\(833\) −20.5361 + 28.1753i −0.711534 + 0.976217i
\(834\) 0 0
\(835\) 0.743668 1.30957i 0.0257357 0.0453195i
\(836\) 0 0
\(837\) 20.2644i 0.700438i
\(838\) 0 0
\(839\) 32.3461 1.11671 0.558356 0.829601i \(-0.311432\pi\)
0.558356 + 0.829601i \(0.311432\pi\)
\(840\) 0 0
\(841\) −28.6847 −0.989126
\(842\) 0 0
\(843\) 36.4235i 1.25449i
\(844\) 0 0
\(845\) 22.9593 + 13.0380i 0.789823 + 0.448519i
\(846\) 0 0
\(847\) −13.1158 + 6.67026i −0.450664 + 0.229193i
\(848\) 0 0
\(849\) 15.9309 0.546746
\(850\) 0 0
\(851\) 33.2228i 1.13886i
\(852\) 0 0
\(853\) 2.93137 0.100368 0.0501840 0.998740i \(-0.484019\pi\)
0.0501840 + 0.998740i \(0.484019\pi\)
\(854\) 0 0
\(855\) −7.81855 4.43994i −0.267389 0.151843i
\(856\) 0 0
\(857\) 5.59390 0.191084 0.0955419 0.995425i \(-0.469542\pi\)
0.0955419 + 0.995425i \(0.469542\pi\)
\(858\) 0 0
\(859\) 9.17104 0.312912 0.156456 0.987685i \(-0.449993\pi\)
0.156456 + 0.987685i \(0.449993\pi\)
\(860\) 0 0
\(861\) −42.7386 + 21.7355i −1.45653 + 0.740744i
\(862\) 0 0
\(863\) 30.7851 1.04794 0.523969 0.851737i \(-0.324451\pi\)
0.523969 + 0.851737i \(0.324451\pi\)
\(864\) 0 0
\(865\) −21.4924 12.2050i −0.730764 0.414981i
\(866\) 0 0
\(867\) 16.6757i 0.566335i
\(868\) 0 0
\(869\) −24.8078 −0.841546
\(870\) 0 0
\(871\) 5.15002 0.174502
\(872\) 0 0
\(873\) −23.3331 −0.789705
\(874\) 0 0
\(875\) 13.9706 + 26.0734i 0.472293 + 0.881442i
\(876\) 0 0
\(877\) 5.49966i 0.185710i −0.995680 0.0928551i \(-0.970401\pi\)
0.995680 0.0928551i \(-0.0295993\pi\)
\(878\) 0 0
\(879\) 30.8908i 1.04192i
\(880\) 0 0
\(881\) 30.7645i 1.03648i 0.855234 + 0.518241i \(0.173413\pi\)
−0.855234 + 0.518241i \(0.826587\pi\)
\(882\) 0 0
\(883\) 48.4902 1.63183 0.815913 0.578175i \(-0.196235\pi\)
0.815913 + 0.578175i \(0.196235\pi\)
\(884\) 0 0
\(885\) −33.7733 + 59.4733i −1.13528 + 1.99917i
\(886\) 0 0
\(887\) 31.7738i 1.06686i −0.845845 0.533429i \(-0.820903\pi\)
0.845845 0.533429i \(-0.179097\pi\)
\(888\) 0 0
\(889\) −12.4924 + 6.35324i −0.418982 + 0.213081i
\(890\) 0 0
\(891\) 26.2267i 0.878628i
\(892\) 0 0
\(893\) 25.0870i 0.839503i
\(894\) 0 0
\(895\) 9.17104 16.1498i 0.306554 0.539829i
\(896\) 0 0
\(897\) 14.0877i 0.470373i
\(898\) 0 0
\(899\) −3.70402 −0.123536
\(900\) 0 0
\(901\) 42.7746i 1.42503i
\(902\) 0 0
\(903\) 3.39228 + 6.67026i 0.112888 + 0.221972i
\(904\) 0 0
\(905\) 3.80776 6.70531i 0.126574 0.222892i
\(906\) 0 0
\(907\) 53.7874 1.78598 0.892991 0.450074i \(-0.148603\pi\)
0.892991 + 0.450074i \(0.148603\pi\)
\(908\) 0 0
\(909\) 23.5957i 0.782619i
\(910\) 0 0
\(911\) 56.0950i 1.85851i 0.369438 + 0.929255i \(0.379550\pi\)
−0.369438 + 0.929255i \(0.620450\pi\)
\(912\) 0 0
\(913\) −9.00400 −0.297989
\(914\) 0 0
\(915\) 2.57501 + 1.46228i 0.0851272 + 0.0483415i
\(916\) 0 0
\(917\) −6.07263 + 3.08835i −0.200536 + 0.101986i
\(918\) 0 0
\(919\) 39.1965i 1.29297i −0.762925 0.646487i \(-0.776238\pi\)
0.762925 0.646487i \(-0.223762\pi\)
\(920\) 0 0
\(921\) 29.6847 0.978143
\(922\) 0 0
\(923\) 13.0452i 0.429389i
\(924\) 0 0
\(925\) −23.6155 + 14.0877i −0.776474 + 0.463199i
\(926\) 0 0
\(927\) 8.54312i 0.280593i
\(928\) 0 0
\(929\) 28.9807i 0.950825i −0.879763 0.475412i \(-0.842299\pi\)
0.879763 0.475412i \(-0.157701\pi\)
\(930\) 0 0
\(931\) −10.6170 + 14.5665i −0.347960 + 0.477397i
\(932\) 0 0
\(933\) 3.08835i 0.101108i
\(934\) 0 0
\(935\) −12.8255 + 22.5851i −0.419437 + 0.738611i
\(936\) 0 0
\(937\) 49.4631 1.61589 0.807944 0.589259i \(-0.200580\pi\)
0.807944 + 0.589259i \(0.200580\pi\)
\(938\) 0 0
\(939\) 15.3019i 0.499357i
\(940\) 0 0
\(941\) 8.75714i 0.285475i 0.989761 + 0.142737i \(0.0455904\pi\)
−0.989761 + 0.142737i \(0.954410\pi\)
\(942\) 0 0
\(943\) 51.2587i 1.66921i
\(944\) 0 0
\(945\) 0.835470 18.1562i 0.0271778 0.590621i
\(946\) 0 0
\(947\) −52.6261 −1.71012 −0.855060 0.518529i \(-0.826480\pi\)
−0.855060 + 0.518529i \(0.826480\pi\)
\(948\) 0 0
\(949\) 10.8769 0.353079
\(950\) 0 0
\(951\) 52.1342 1.69057
\(952\) 0 0
\(953\) 31.2637i 1.01273i −0.862319 0.506365i \(-0.830989\pi\)
0.862319 0.506365i \(-0.169011\pi\)
\(954\) 0 0
\(955\) 22.0461 38.8222i 0.713395 1.25626i
\(956\) 0 0
\(957\) −2.79695 −0.0904125
\(958\) 0 0
\(959\) −3.70402 7.28323i −0.119609 0.235188i
\(960\) 0 0
\(961\) 12.5076 0.403470
\(962\) 0 0
\(963\) −15.6371 −0.503899
\(964\) 0 0
\(965\) 6.07263 10.6937i 0.195485 0.344241i
\(966\) 0 0
\(967\) 16.2177 0.521527 0.260764 0.965403i \(-0.416026\pi\)
0.260764 + 0.965403i \(0.416026\pi\)
\(968\) 0 0
\(969\) 27.3924i 0.879969i
\(970\) 0 0
\(971\) 36.3672 1.16708 0.583539 0.812085i \(-0.301668\pi\)
0.583539 + 0.812085i \(0.301668\pi\)
\(972\) 0 0
\(973\) −9.48274 + 4.82262i −0.304003 + 0.154606i
\(974\) 0 0
\(975\) −10.0138 + 5.97366i −0.320699 + 0.191310i
\(976\) 0 0
\(977\) 14.0877i 0.450704i 0.974277 + 0.225352i \(0.0723532\pi\)
−0.974277 + 0.225352i \(0.927647\pi\)
\(978\) 0 0
\(979\) 6.59603 0.210810
\(980\) 0 0
\(981\) 7.12311 0.227423
\(982\) 0 0
\(983\) 44.7361i 1.42686i −0.700727 0.713430i \(-0.747141\pi\)
0.700727 0.713430i \(-0.252859\pi\)
\(984\) 0 0
\(985\) −18.2179 + 32.0810i −0.580471 + 1.02218i
\(986\) 0 0
\(987\) 49.0708 24.9559i 1.56194 0.794353i
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 0.574176i 0.0182393i −0.999958 0.00911966i \(-0.997097\pi\)
0.999958 0.00911966i \(-0.00290292\pi\)
\(992\) 0 0
\(993\) 17.7392 0.562935
\(994\) 0 0
\(995\) 35.6647 + 20.2530i 1.13065 + 0.642065i
\(996\) 0 0
\(997\) −47.7580 −1.51251 −0.756256 0.654276i \(-0.772973\pi\)
−0.756256 + 0.654276i \(0.772973\pi\)
\(998\) 0 0
\(999\) 16.8961 0.534568
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.e.f.2239.1 16
4.3 odd 2 inner 2240.2.e.f.2239.13 16
5.4 even 2 inner 2240.2.e.f.2239.15 16
7.6 odd 2 inner 2240.2.e.f.2239.16 16
8.3 odd 2 140.2.c.b.139.11 yes 16
8.5 even 2 140.2.c.b.139.8 yes 16
20.19 odd 2 inner 2240.2.e.f.2239.3 16
28.27 even 2 inner 2240.2.e.f.2239.4 16
35.34 odd 2 inner 2240.2.e.f.2239.2 16
40.3 even 4 700.2.g.l.251.14 16
40.13 odd 4 700.2.g.l.251.15 16
40.19 odd 2 140.2.c.b.139.6 yes 16
40.27 even 4 700.2.g.l.251.3 16
40.29 even 2 140.2.c.b.139.9 yes 16
40.37 odd 4 700.2.g.l.251.2 16
56.3 even 6 980.2.s.f.19.2 32
56.5 odd 6 980.2.s.f.619.16 32
56.11 odd 6 980.2.s.f.19.1 32
56.13 odd 2 140.2.c.b.139.7 yes 16
56.19 even 6 980.2.s.f.619.11 32
56.27 even 2 140.2.c.b.139.12 yes 16
56.37 even 6 980.2.s.f.619.15 32
56.45 odd 6 980.2.s.f.19.5 32
56.51 odd 6 980.2.s.f.619.12 32
56.53 even 6 980.2.s.f.19.6 32
140.139 even 2 inner 2240.2.e.f.2239.14 16
280.13 even 4 700.2.g.l.251.16 16
280.19 even 6 980.2.s.f.619.6 32
280.27 odd 4 700.2.g.l.251.4 16
280.59 even 6 980.2.s.f.19.15 32
280.69 odd 2 140.2.c.b.139.10 yes 16
280.83 odd 4 700.2.g.l.251.13 16
280.109 even 6 980.2.s.f.19.11 32
280.139 even 2 140.2.c.b.139.5 16
280.149 even 6 980.2.s.f.619.2 32
280.179 odd 6 980.2.s.f.19.16 32
280.219 odd 6 980.2.s.f.619.5 32
280.229 odd 6 980.2.s.f.619.1 32
280.237 even 4 700.2.g.l.251.1 16
280.269 odd 6 980.2.s.f.19.12 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.c.b.139.5 16 280.139 even 2
140.2.c.b.139.6 yes 16 40.19 odd 2
140.2.c.b.139.7 yes 16 56.13 odd 2
140.2.c.b.139.8 yes 16 8.5 even 2
140.2.c.b.139.9 yes 16 40.29 even 2
140.2.c.b.139.10 yes 16 280.69 odd 2
140.2.c.b.139.11 yes 16 8.3 odd 2
140.2.c.b.139.12 yes 16 56.27 even 2
700.2.g.l.251.1 16 280.237 even 4
700.2.g.l.251.2 16 40.37 odd 4
700.2.g.l.251.3 16 40.27 even 4
700.2.g.l.251.4 16 280.27 odd 4
700.2.g.l.251.13 16 280.83 odd 4
700.2.g.l.251.14 16 40.3 even 4
700.2.g.l.251.15 16 40.13 odd 4
700.2.g.l.251.16 16 280.13 even 4
980.2.s.f.19.1 32 56.11 odd 6
980.2.s.f.19.2 32 56.3 even 6
980.2.s.f.19.5 32 56.45 odd 6
980.2.s.f.19.6 32 56.53 even 6
980.2.s.f.19.11 32 280.109 even 6
980.2.s.f.19.12 32 280.269 odd 6
980.2.s.f.19.15 32 280.59 even 6
980.2.s.f.19.16 32 280.179 odd 6
980.2.s.f.619.1 32 280.229 odd 6
980.2.s.f.619.2 32 280.149 even 6
980.2.s.f.619.5 32 280.219 odd 6
980.2.s.f.619.6 32 280.19 even 6
980.2.s.f.619.11 32 56.19 even 6
980.2.s.f.619.12 32 56.51 odd 6
980.2.s.f.619.15 32 56.37 even 6
980.2.s.f.619.16 32 56.5 odd 6
2240.2.e.f.2239.1 16 1.1 even 1 trivial
2240.2.e.f.2239.2 16 35.34 odd 2 inner
2240.2.e.f.2239.3 16 20.19 odd 2 inner
2240.2.e.f.2239.4 16 28.27 even 2 inner
2240.2.e.f.2239.13 16 4.3 odd 2 inner
2240.2.e.f.2239.14 16 140.139 even 2 inner
2240.2.e.f.2239.15 16 5.4 even 2 inner
2240.2.e.f.2239.16 16 7.6 odd 2 inner