Properties

Label 740.2.a.f.1.3
Level $740$
Weight $2$
Character 740.1
Self dual yes
Analytic conductor $5.909$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(1,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.a (trivial)

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: yes
Analytic conductor: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.286164.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 11x^{2} - 3x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.51951\) of defining polynomial
Character \(\chi\) \(=\) 740.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.51951 q^{3} +1.00000 q^{5} +0.519509 q^{7} +3.34793 q^{9} +O(q^{10})\) \(q+2.51951 q^{3} +1.00000 q^{5} +0.519509 q^{7} +3.34793 q^{9} +6.16241 q^{11} -6.51034 q^{13} +2.51951 q^{15} -2.51034 q^{17} +6.68192 q^{19} +1.30891 q^{21} -3.03902 q^{23} +1.00000 q^{25} +0.876607 q^{27} -3.03902 q^{29} -4.33876 q^{31} +15.5263 q^{33} +0.519509 q^{35} +1.00000 q^{37} -16.4029 q^{39} +12.1624 q^{41} -2.34316 q^{43} +3.34793 q^{45} +9.54019 q^{47} -6.73011 q^{49} -6.32482 q^{51} -9.20143 q^{53} +6.16241 q^{55} +16.8352 q^{57} -9.37778 q^{59} +2.00000 q^{61} +1.73928 q^{63} -6.51034 q^{65} +0.357097 q^{67} -7.65684 q^{69} -2.87661 q^{71} -10.2404 q^{73} +2.51951 q^{75} +3.20143 q^{77} +0.681922 q^{79} -7.83516 q^{81} +6.17635 q^{83} -2.51034 q^{85} -7.65684 q^{87} -11.0207 q^{89} -3.38218 q^{91} -10.9315 q^{93} +6.68192 q^{95} +6.34316 q^{97} +20.6313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + 4 q^{5} - 5 q^{7} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + 4 q^{5} - 5 q^{7} + 13 q^{9} + 5 q^{11} - 6 q^{13} + 3 q^{15} + 10 q^{17} + 19 q^{21} + 2 q^{23} + 4 q^{25} + 9 q^{27} + 2 q^{29} - 4 q^{31} - 11 q^{33} - 5 q^{35} + 4 q^{37} + 2 q^{39} + 29 q^{41} + 4 q^{43} + 13 q^{45} - 9 q^{47} + q^{49} + 14 q^{51} - 3 q^{53} + 5 q^{55} + 8 q^{57} - 10 q^{59} + 8 q^{61} - 8 q^{63} - 6 q^{65} + 14 q^{67} - 44 q^{69} - 17 q^{71} + 7 q^{73} + 3 q^{75} - 21 q^{77} - 24 q^{79} + 28 q^{81} + 31 q^{83} + 10 q^{85} - 44 q^{87} - 4 q^{89} + 14 q^{91} + 18 q^{93} + 12 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.51951 1.45464 0.727320 0.686299i \(-0.240766\pi\)
0.727320 + 0.686299i \(0.240766\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0.519509 0.196356 0.0981780 0.995169i \(-0.468699\pi\)
0.0981780 + 0.995169i \(0.468699\pi\)
\(8\) 0 0
\(9\) 3.34793 1.11598
\(10\) 0 0
\(11\) 6.16241 1.85804 0.929019 0.370033i \(-0.120654\pi\)
0.929019 + 0.370033i \(0.120654\pi\)
\(12\) 0 0
\(13\) −6.51034 −1.80564 −0.902822 0.430015i \(-0.858508\pi\)
−0.902822 + 0.430015i \(0.858508\pi\)
\(14\) 0 0
\(15\) 2.51951 0.650535
\(16\) 0 0
\(17\) −2.51034 −0.608847 −0.304423 0.952537i \(-0.598464\pi\)
−0.304423 + 0.952537i \(0.598464\pi\)
\(18\) 0 0
\(19\) 6.68192 1.53294 0.766469 0.642281i \(-0.222012\pi\)
0.766469 + 0.642281i \(0.222012\pi\)
\(20\) 0 0
\(21\) 1.30891 0.285627
\(22\) 0 0
\(23\) −3.03902 −0.633679 −0.316840 0.948479i \(-0.602622\pi\)
−0.316840 + 0.948479i \(0.602622\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.876607 0.168703
\(28\) 0 0
\(29\) −3.03902 −0.564332 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(30\) 0 0
\(31\) −4.33876 −0.779264 −0.389632 0.920971i \(-0.627398\pi\)
−0.389632 + 0.920971i \(0.627398\pi\)
\(32\) 0 0
\(33\) 15.5263 2.70277
\(34\) 0 0
\(35\) 0.519509 0.0878131
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −16.4029 −2.62656
\(40\) 0 0
\(41\) 12.1624 1.89945 0.949725 0.313086i \(-0.101363\pi\)
0.949725 + 0.313086i \(0.101363\pi\)
\(42\) 0 0
\(43\) −2.34316 −0.357329 −0.178665 0.983910i \(-0.557178\pi\)
−0.178665 + 0.983910i \(0.557178\pi\)
\(44\) 0 0
\(45\) 3.34793 0.499080
\(46\) 0 0
\(47\) 9.54019 1.39158 0.695790 0.718245i \(-0.255055\pi\)
0.695790 + 0.718245i \(0.255055\pi\)
\(48\) 0 0
\(49\) −6.73011 −0.961444
\(50\) 0 0
\(51\) −6.32482 −0.885653
\(52\) 0 0
\(53\) −9.20143 −1.26391 −0.631957 0.775004i \(-0.717748\pi\)
−0.631957 + 0.775004i \(0.717748\pi\)
\(54\) 0 0
\(55\) 6.16241 0.830939
\(56\) 0 0
\(57\) 16.8352 2.22987
\(58\) 0 0
\(59\) −9.37778 −1.22088 −0.610441 0.792062i \(-0.709008\pi\)
−0.610441 + 0.792062i \(0.709008\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 1.73928 0.219129
\(64\) 0 0
\(65\) −6.51034 −0.807508
\(66\) 0 0
\(67\) 0.357097 0.0436264 0.0218132 0.999762i \(-0.493056\pi\)
0.0218132 + 0.999762i \(0.493056\pi\)
\(68\) 0 0
\(69\) −7.65684 −0.921775
\(70\) 0 0
\(71\) −2.87661 −0.341390 −0.170695 0.985324i \(-0.554601\pi\)
−0.170695 + 0.985324i \(0.554601\pi\)
\(72\) 0 0
\(73\) −10.2404 −1.19855 −0.599277 0.800542i \(-0.704545\pi\)
−0.599277 + 0.800542i \(0.704545\pi\)
\(74\) 0 0
\(75\) 2.51951 0.290928
\(76\) 0 0
\(77\) 3.20143 0.364837
\(78\) 0 0
\(79\) 0.681922 0.0767222 0.0383611 0.999264i \(-0.487786\pi\)
0.0383611 + 0.999264i \(0.487786\pi\)
\(80\) 0 0
\(81\) −7.83516 −0.870574
\(82\) 0 0
\(83\) 6.17635 0.677942 0.338971 0.940797i \(-0.389921\pi\)
0.338971 + 0.940797i \(0.389921\pi\)
\(84\) 0 0
\(85\) −2.51034 −0.272285
\(86\) 0 0
\(87\) −7.65684 −0.820899
\(88\) 0 0
\(89\) −11.0207 −1.16819 −0.584095 0.811685i \(-0.698550\pi\)
−0.584095 + 0.811685i \(0.698550\pi\)
\(90\) 0 0
\(91\) −3.38218 −0.354549
\(92\) 0 0
\(93\) −10.9315 −1.13355
\(94\) 0 0
\(95\) 6.68192 0.685551
\(96\) 0 0
\(97\) 6.34316 0.644051 0.322025 0.946731i \(-0.395636\pi\)
0.322025 + 0.946731i \(0.395636\pi\)
\(98\) 0 0
\(99\) 20.6313 2.07352
\(100\) 0 0
\(101\) 19.4466 1.93501 0.967507 0.252845i \(-0.0813664\pi\)
0.967507 + 0.252845i \(0.0813664\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 1.30891 0.127736
\(106\) 0 0
\(107\) −15.0251 −1.45253 −0.726265 0.687415i \(-0.758745\pi\)
−0.726265 + 0.687415i \(0.758745\pi\)
\(108\) 0 0
\(109\) 15.7070 1.50446 0.752229 0.658902i \(-0.228979\pi\)
0.752229 + 0.658902i \(0.228979\pi\)
\(110\) 0 0
\(111\) 2.51951 0.239141
\(112\) 0 0
\(113\) −2.18552 −0.205596 −0.102798 0.994702i \(-0.532780\pi\)
−0.102798 + 0.994702i \(0.532780\pi\)
\(114\) 0 0
\(115\) −3.03902 −0.283390
\(116\) 0 0
\(117\) −21.7961 −2.01505
\(118\) 0 0
\(119\) −1.30414 −0.119551
\(120\) 0 0
\(121\) 26.9753 2.45230
\(122\) 0 0
\(123\) 30.6433 2.76301
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.17635 0.193119 0.0965597 0.995327i \(-0.469216\pi\)
0.0965597 + 0.995327i \(0.469216\pi\)
\(128\) 0 0
\(129\) −5.90362 −0.519785
\(130\) 0 0
\(131\) −12.7416 −1.11324 −0.556620 0.830767i \(-0.687902\pi\)
−0.556620 + 0.830767i \(0.687902\pi\)
\(132\) 0 0
\(133\) 3.47132 0.301002
\(134\) 0 0
\(135\) 0.876607 0.0754463
\(136\) 0 0
\(137\) −4.26513 −0.364394 −0.182197 0.983262i \(-0.558321\pi\)
−0.182197 + 0.983262i \(0.558321\pi\)
\(138\) 0 0
\(139\) −11.6290 −0.986356 −0.493178 0.869928i \(-0.664165\pi\)
−0.493178 + 0.869928i \(0.664165\pi\)
\(140\) 0 0
\(141\) 24.0366 2.02425
\(142\) 0 0
\(143\) −40.1194 −3.35495
\(144\) 0 0
\(145\) −3.03902 −0.252377
\(146\) 0 0
\(147\) −16.9566 −1.39855
\(148\) 0 0
\(149\) −17.3591 −1.42211 −0.711056 0.703136i \(-0.751783\pi\)
−0.711056 + 0.703136i \(0.751783\pi\)
\(150\) 0 0
\(151\) 0.343164 0.0279263 0.0139631 0.999903i \(-0.495555\pi\)
0.0139631 + 0.999903i \(0.495555\pi\)
\(152\) 0 0
\(153\) −8.40444 −0.679458
\(154\) 0 0
\(155\) −4.33876 −0.348497
\(156\) 0 0
\(157\) −4.16241 −0.332197 −0.166098 0.986109i \(-0.553117\pi\)
−0.166098 + 0.986109i \(0.553117\pi\)
\(158\) 0 0
\(159\) −23.1831 −1.83854
\(160\) 0 0
\(161\) −1.57880 −0.124427
\(162\) 0 0
\(163\) 19.3455 1.51526 0.757628 0.652686i \(-0.226358\pi\)
0.757628 + 0.652686i \(0.226358\pi\)
\(164\) 0 0
\(165\) 15.5263 1.20872
\(166\) 0 0
\(167\) −0.979321 −0.0757821 −0.0378911 0.999282i \(-0.512064\pi\)
−0.0378911 + 0.999282i \(0.512064\pi\)
\(168\) 0 0
\(169\) 29.3845 2.26035
\(170\) 0 0
\(171\) 22.3706 1.71072
\(172\) 0 0
\(173\) −22.9363 −1.74381 −0.871907 0.489671i \(-0.837117\pi\)
−0.871907 + 0.489671i \(0.837117\pi\)
\(174\) 0 0
\(175\) 0.519509 0.0392712
\(176\) 0 0
\(177\) −23.6274 −1.77594
\(178\) 0 0
\(179\) −16.7600 −1.25270 −0.626349 0.779543i \(-0.715452\pi\)
−0.626349 + 0.779543i \(0.715452\pi\)
\(180\) 0 0
\(181\) −11.4466 −0.850822 −0.425411 0.905000i \(-0.639871\pi\)
−0.425411 + 0.905000i \(0.639871\pi\)
\(182\) 0 0
\(183\) 5.03902 0.372495
\(184\) 0 0
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) −15.4697 −1.13126
\(188\) 0 0
\(189\) 0.455405 0.0331259
\(190\) 0 0
\(191\) −10.4351 −0.755060 −0.377530 0.925997i \(-0.623226\pi\)
−0.377530 + 0.925997i \(0.623226\pi\)
\(192\) 0 0
\(193\) 20.4029 1.46863 0.734315 0.678809i \(-0.237503\pi\)
0.734315 + 0.678809i \(0.237503\pi\)
\(194\) 0 0
\(195\) −16.4029 −1.17463
\(196\) 0 0
\(197\) −11.1051 −0.791202 −0.395601 0.918422i \(-0.629464\pi\)
−0.395601 + 0.918422i \(0.629464\pi\)
\(198\) 0 0
\(199\) −0.622223 −0.0441083 −0.0220541 0.999757i \(-0.507021\pi\)
−0.0220541 + 0.999757i \(0.507021\pi\)
\(200\) 0 0
\(201\) 0.899710 0.0634606
\(202\) 0 0
\(203\) −1.57880 −0.110810
\(204\) 0 0
\(205\) 12.1624 0.849460
\(206\) 0 0
\(207\) −10.1744 −0.707171
\(208\) 0 0
\(209\) 41.1768 2.84826
\(210\) 0 0
\(211\) −18.3001 −1.25983 −0.629917 0.776662i \(-0.716911\pi\)
−0.629917 + 0.776662i \(0.716911\pi\)
\(212\) 0 0
\(213\) −7.24764 −0.496600
\(214\) 0 0
\(215\) −2.34316 −0.159802
\(216\) 0 0
\(217\) −2.25403 −0.153013
\(218\) 0 0
\(219\) −25.8009 −1.74346
\(220\) 0 0
\(221\) 16.3432 1.09936
\(222\) 0 0
\(223\) −0.784636 −0.0525431 −0.0262715 0.999655i \(-0.508363\pi\)
−0.0262715 + 0.999655i \(0.508363\pi\)
\(224\) 0 0
\(225\) 3.34793 0.223195
\(226\) 0 0
\(227\) 6.32482 0.419793 0.209897 0.977724i \(-0.432687\pi\)
0.209897 + 0.977724i \(0.432687\pi\)
\(228\) 0 0
\(229\) 2.08437 0.137739 0.0688697 0.997626i \(-0.478061\pi\)
0.0688697 + 0.997626i \(0.478061\pi\)
\(230\) 0 0
\(231\) 8.06604 0.530706
\(232\) 0 0
\(233\) 26.4626 1.73362 0.866810 0.498639i \(-0.166167\pi\)
0.866810 + 0.498639i \(0.166167\pi\)
\(234\) 0 0
\(235\) 9.54019 0.622333
\(236\) 0 0
\(237\) 1.71811 0.111603
\(238\) 0 0
\(239\) 18.6636 1.20725 0.603623 0.797270i \(-0.293723\pi\)
0.603623 + 0.797270i \(0.293723\pi\)
\(240\) 0 0
\(241\) −0.0596981 −0.00384549 −0.00192274 0.999998i \(-0.500612\pi\)
−0.00192274 + 0.999998i \(0.500612\pi\)
\(242\) 0 0
\(243\) −22.3706 −1.43507
\(244\) 0 0
\(245\) −6.73011 −0.429971
\(246\) 0 0
\(247\) −43.5016 −2.76794
\(248\) 0 0
\(249\) 15.5614 0.986161
\(250\) 0 0
\(251\) 3.05295 0.192701 0.0963503 0.995347i \(-0.469283\pi\)
0.0963503 + 0.995347i \(0.469283\pi\)
\(252\) 0 0
\(253\) −18.7277 −1.17740
\(254\) 0 0
\(255\) −6.32482 −0.396076
\(256\) 0 0
\(257\) −20.9132 −1.30453 −0.652265 0.757991i \(-0.726181\pi\)
−0.652265 + 0.757991i \(0.726181\pi\)
\(258\) 0 0
\(259\) 0.519509 0.0322807
\(260\) 0 0
\(261\) −10.1744 −0.629780
\(262\) 0 0
\(263\) −7.91122 −0.487827 −0.243913 0.969797i \(-0.578431\pi\)
−0.243913 + 0.969797i \(0.578431\pi\)
\(264\) 0 0
\(265\) −9.20143 −0.565239
\(266\) 0 0
\(267\) −27.7667 −1.69929
\(268\) 0 0
\(269\) −0.606717 −0.0369922 −0.0184961 0.999829i \(-0.505888\pi\)
−0.0184961 + 0.999829i \(0.505888\pi\)
\(270\) 0 0
\(271\) 20.5931 1.25094 0.625472 0.780247i \(-0.284906\pi\)
0.625472 + 0.780247i \(0.284906\pi\)
\(272\) 0 0
\(273\) −8.52144 −0.515741
\(274\) 0 0
\(275\) 6.16241 0.371607
\(276\) 0 0
\(277\) 21.2842 1.27885 0.639423 0.768855i \(-0.279173\pi\)
0.639423 + 0.768855i \(0.279173\pi\)
\(278\) 0 0
\(279\) −14.5258 −0.869640
\(280\) 0 0
\(281\) −10.3432 −0.617021 −0.308511 0.951221i \(-0.599831\pi\)
−0.308511 + 0.951221i \(0.599831\pi\)
\(282\) 0 0
\(283\) 1.67518 0.0995789 0.0497894 0.998760i \(-0.484145\pi\)
0.0497894 + 0.998760i \(0.484145\pi\)
\(284\) 0 0
\(285\) 16.8352 0.997229
\(286\) 0 0
\(287\) 6.31849 0.372969
\(288\) 0 0
\(289\) −10.6982 −0.629306
\(290\) 0 0
\(291\) 15.9817 0.936862
\(292\) 0 0
\(293\) 9.96332 0.582063 0.291032 0.956713i \(-0.406001\pi\)
0.291032 + 0.956713i \(0.406001\pi\)
\(294\) 0 0
\(295\) −9.37778 −0.545995
\(296\) 0 0
\(297\) 5.40201 0.313456
\(298\) 0 0
\(299\) 19.7850 1.14420
\(300\) 0 0
\(301\) −1.21730 −0.0701637
\(302\) 0 0
\(303\) 48.9960 2.81475
\(304\) 0 0
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 26.2265 1.49683 0.748413 0.663233i \(-0.230816\pi\)
0.748413 + 0.663233i \(0.230816\pi\)
\(308\) 0 0
\(309\) 5.03902 0.286660
\(310\) 0 0
\(311\) 1.74881 0.0991658 0.0495829 0.998770i \(-0.484211\pi\)
0.0495829 + 0.998770i \(0.484211\pi\)
\(312\) 0 0
\(313\) −21.7762 −1.23087 −0.615433 0.788189i \(-0.711019\pi\)
−0.615433 + 0.788189i \(0.711019\pi\)
\(314\) 0 0
\(315\) 1.73928 0.0979973
\(316\) 0 0
\(317\) 4.42120 0.248319 0.124160 0.992262i \(-0.460376\pi\)
0.124160 + 0.992262i \(0.460376\pi\)
\(318\) 0 0
\(319\) −18.7277 −1.04855
\(320\) 0 0
\(321\) −37.8558 −2.11291
\(322\) 0 0
\(323\) −16.7739 −0.933324
\(324\) 0 0
\(325\) −6.51034 −0.361129
\(326\) 0 0
\(327\) 39.5740 2.18844
\(328\) 0 0
\(329\) 4.95622 0.273245
\(330\) 0 0
\(331\) 32.4168 1.78179 0.890894 0.454211i \(-0.150079\pi\)
0.890894 + 0.454211i \(0.150079\pi\)
\(332\) 0 0
\(333\) 3.34793 0.183465
\(334\) 0 0
\(335\) 0.357097 0.0195103
\(336\) 0 0
\(337\) 29.8789 1.62761 0.813805 0.581138i \(-0.197392\pi\)
0.813805 + 0.581138i \(0.197392\pi\)
\(338\) 0 0
\(339\) −5.50643 −0.299068
\(340\) 0 0
\(341\) −26.7372 −1.44790
\(342\) 0 0
\(343\) −7.13292 −0.385142
\(344\) 0 0
\(345\) −7.65684 −0.412230
\(346\) 0 0
\(347\) 2.88294 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(348\) 0 0
\(349\) 31.5494 1.68880 0.844399 0.535714i \(-0.179958\pi\)
0.844399 + 0.535714i \(0.179958\pi\)
\(350\) 0 0
\(351\) −5.70701 −0.304617
\(352\) 0 0
\(353\) −2.68633 −0.142979 −0.0714894 0.997441i \(-0.522775\pi\)
−0.0714894 + 0.997441i \(0.522775\pi\)
\(354\) 0 0
\(355\) −2.87661 −0.152674
\(356\) 0 0
\(357\) −3.28581 −0.173903
\(358\) 0 0
\(359\) 21.9570 1.15885 0.579423 0.815027i \(-0.303278\pi\)
0.579423 + 0.815027i \(0.303278\pi\)
\(360\) 0 0
\(361\) 25.6481 1.34990
\(362\) 0 0
\(363\) 67.9646 3.56722
\(364\) 0 0
\(365\) −10.2404 −0.536010
\(366\) 0 0
\(367\) −25.3499 −1.32325 −0.661627 0.749833i \(-0.730134\pi\)
−0.661627 + 0.749833i \(0.730134\pi\)
\(368\) 0 0
\(369\) 40.7189 2.11974
\(370\) 0 0
\(371\) −4.78023 −0.248177
\(372\) 0 0
\(373\) 23.6322 1.22363 0.611813 0.791002i \(-0.290440\pi\)
0.611813 + 0.791002i \(0.290440\pi\)
\(374\) 0 0
\(375\) 2.51951 0.130107
\(376\) 0 0
\(377\) 19.7850 1.01898
\(378\) 0 0
\(379\) −18.8216 −0.966800 −0.483400 0.875400i \(-0.660598\pi\)
−0.483400 + 0.875400i \(0.660598\pi\)
\(380\) 0 0
\(381\) 5.48332 0.280919
\(382\) 0 0
\(383\) 11.0485 0.564554 0.282277 0.959333i \(-0.408910\pi\)
0.282277 + 0.959333i \(0.408910\pi\)
\(384\) 0 0
\(385\) 3.20143 0.163160
\(386\) 0 0
\(387\) −7.84474 −0.398771
\(388\) 0 0
\(389\) 29.4235 1.49183 0.745916 0.666040i \(-0.232012\pi\)
0.745916 + 0.666040i \(0.232012\pi\)
\(390\) 0 0
\(391\) 7.62897 0.385814
\(392\) 0 0
\(393\) −32.1026 −1.61936
\(394\) 0 0
\(395\) 0.681922 0.0343112
\(396\) 0 0
\(397\) 17.2014 0.863315 0.431658 0.902038i \(-0.357929\pi\)
0.431658 + 0.902038i \(0.357929\pi\)
\(398\) 0 0
\(399\) 8.74603 0.437849
\(400\) 0 0
\(401\) 18.6496 0.931319 0.465660 0.884964i \(-0.345817\pi\)
0.465660 + 0.884964i \(0.345817\pi\)
\(402\) 0 0
\(403\) 28.2468 1.40707
\(404\) 0 0
\(405\) −7.83516 −0.389332
\(406\) 0 0
\(407\) 6.16241 0.305459
\(408\) 0 0
\(409\) 11.0390 0.545844 0.272922 0.962036i \(-0.412010\pi\)
0.272922 + 0.962036i \(0.412010\pi\)
\(410\) 0 0
\(411\) −10.7460 −0.530062
\(412\) 0 0
\(413\) −4.87184 −0.239728
\(414\) 0 0
\(415\) 6.17635 0.303185
\(416\) 0 0
\(417\) −29.2993 −1.43479
\(418\) 0 0
\(419\) −9.95698 −0.486430 −0.243215 0.969972i \(-0.578202\pi\)
−0.243215 + 0.969972i \(0.578202\pi\)
\(420\) 0 0
\(421\) 22.8153 1.11195 0.555974 0.831200i \(-0.312346\pi\)
0.555974 + 0.831200i \(0.312346\pi\)
\(422\) 0 0
\(423\) 31.9399 1.55297
\(424\) 0 0
\(425\) −2.51034 −0.121769
\(426\) 0 0
\(427\) 1.03902 0.0502816
\(428\) 0 0
\(429\) −101.081 −4.88025
\(430\) 0 0
\(431\) 8.67239 0.417735 0.208867 0.977944i \(-0.433022\pi\)
0.208867 + 0.977944i \(0.433022\pi\)
\(432\) 0 0
\(433\) −5.38852 −0.258956 −0.129478 0.991582i \(-0.541330\pi\)
−0.129478 + 0.991582i \(0.541330\pi\)
\(434\) 0 0
\(435\) −7.65684 −0.367117
\(436\) 0 0
\(437\) −20.3065 −0.971391
\(438\) 0 0
\(439\) 0.603884 0.0288218 0.0144109 0.999896i \(-0.495413\pi\)
0.0144109 + 0.999896i \(0.495413\pi\)
\(440\) 0 0
\(441\) −22.5319 −1.07295
\(442\) 0 0
\(443\) 27.2655 1.29542 0.647712 0.761885i \(-0.275726\pi\)
0.647712 + 0.761885i \(0.275726\pi\)
\(444\) 0 0
\(445\) −11.0207 −0.522430
\(446\) 0 0
\(447\) −43.7364 −2.06866
\(448\) 0 0
\(449\) 9.28581 0.438224 0.219112 0.975700i \(-0.429684\pi\)
0.219112 + 0.975700i \(0.429684\pi\)
\(450\) 0 0
\(451\) 74.9498 3.52925
\(452\) 0 0
\(453\) 0.864604 0.0406227
\(454\) 0 0
\(455\) −3.38218 −0.158559
\(456\) 0 0
\(457\) 12.2651 0.573738 0.286869 0.957970i \(-0.407385\pi\)
0.286869 + 0.957970i \(0.407385\pi\)
\(458\) 0 0
\(459\) −2.20058 −0.102714
\(460\) 0 0
\(461\) 1.81291 0.0844357 0.0422178 0.999108i \(-0.486558\pi\)
0.0422178 + 0.999108i \(0.486558\pi\)
\(462\) 0 0
\(463\) −23.7349 −1.10305 −0.551527 0.834157i \(-0.685954\pi\)
−0.551527 + 0.834157i \(0.685954\pi\)
\(464\) 0 0
\(465\) −10.9315 −0.506938
\(466\) 0 0
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) 0.185515 0.00856630
\(470\) 0 0
\(471\) −10.4872 −0.483226
\(472\) 0 0
\(473\) −14.4395 −0.663931
\(474\) 0 0
\(475\) 6.68192 0.306588
\(476\) 0 0
\(477\) −30.8057 −1.41050
\(478\) 0 0
\(479\) −9.62456 −0.439758 −0.219879 0.975527i \(-0.570566\pi\)
−0.219879 + 0.975527i \(0.570566\pi\)
\(480\) 0 0
\(481\) −6.51034 −0.296846
\(482\) 0 0
\(483\) −3.97780 −0.180996
\(484\) 0 0
\(485\) 6.34316 0.288028
\(486\) 0 0
\(487\) −31.7165 −1.43721 −0.718607 0.695417i \(-0.755220\pi\)
−0.718607 + 0.695417i \(0.755220\pi\)
\(488\) 0 0
\(489\) 48.7412 2.20415
\(490\) 0 0
\(491\) −5.02068 −0.226580 −0.113290 0.993562i \(-0.536139\pi\)
−0.113290 + 0.993562i \(0.536139\pi\)
\(492\) 0 0
\(493\) 7.62897 0.343591
\(494\) 0 0
\(495\) 20.6313 0.927308
\(496\) 0 0
\(497\) −1.49442 −0.0670341
\(498\) 0 0
\(499\) 34.8794 1.56141 0.780707 0.624897i \(-0.214859\pi\)
0.780707 + 0.624897i \(0.214859\pi\)
\(500\) 0 0
\(501\) −2.46741 −0.110236
\(502\) 0 0
\(503\) 20.2158 0.901377 0.450688 0.892681i \(-0.351179\pi\)
0.450688 + 0.892681i \(0.351179\pi\)
\(504\) 0 0
\(505\) 19.4466 0.865364
\(506\) 0 0
\(507\) 74.0346 3.28799
\(508\) 0 0
\(509\) 4.18075 0.185309 0.0926543 0.995698i \(-0.470465\pi\)
0.0926543 + 0.995698i \(0.470465\pi\)
\(510\) 0 0
\(511\) −5.32001 −0.235343
\(512\) 0 0
\(513\) 5.85742 0.258611
\(514\) 0 0
\(515\) 2.00000 0.0881305
\(516\) 0 0
\(517\) 58.7906 2.58561
\(518\) 0 0
\(519\) −57.7882 −2.53662
\(520\) 0 0
\(521\) −6.26036 −0.274271 −0.137136 0.990552i \(-0.543790\pi\)
−0.137136 + 0.990552i \(0.543790\pi\)
\(522\) 0 0
\(523\) 30.3432 1.32681 0.663407 0.748259i \(-0.269110\pi\)
0.663407 + 0.748259i \(0.269110\pi\)
\(524\) 0 0
\(525\) 1.30891 0.0571255
\(526\) 0 0
\(527\) 10.8918 0.474452
\(528\) 0 0
\(529\) −13.7644 −0.598451
\(530\) 0 0
\(531\) −31.3961 −1.36248
\(532\) 0 0
\(533\) −79.1814 −3.42973
\(534\) 0 0
\(535\) −15.0251 −0.649591
\(536\) 0 0
\(537\) −42.2269 −1.82222
\(538\) 0 0
\(539\) −41.4737 −1.78640
\(540\) 0 0
\(541\) −16.0502 −0.690051 −0.345025 0.938593i \(-0.612130\pi\)
−0.345025 + 0.938593i \(0.612130\pi\)
\(542\) 0 0
\(543\) −28.8399 −1.23764
\(544\) 0 0
\(545\) 15.7070 0.672814
\(546\) 0 0
\(547\) 21.0804 0.901332 0.450666 0.892693i \(-0.351187\pi\)
0.450666 + 0.892693i \(0.351187\pi\)
\(548\) 0 0
\(549\) 6.69586 0.285772
\(550\) 0 0
\(551\) −20.3065 −0.865085
\(552\) 0 0
\(553\) 0.354265 0.0150649
\(554\) 0 0
\(555\) 2.51951 0.106947
\(556\) 0 0
\(557\) 0.959408 0.0406514 0.0203257 0.999793i \(-0.493530\pi\)
0.0203257 + 0.999793i \(0.493530\pi\)
\(558\) 0 0
\(559\) 15.2548 0.645209
\(560\) 0 0
\(561\) −38.9762 −1.64558
\(562\) 0 0
\(563\) −41.3957 −1.74462 −0.872310 0.488954i \(-0.837379\pi\)
−0.872310 + 0.488954i \(0.837379\pi\)
\(564\) 0 0
\(565\) −2.18552 −0.0919453
\(566\) 0 0
\(567\) −4.07044 −0.170942
\(568\) 0 0
\(569\) −26.4626 −1.10937 −0.554684 0.832061i \(-0.687161\pi\)
−0.554684 + 0.832061i \(0.687161\pi\)
\(570\) 0 0
\(571\) 17.4482 0.730185 0.365093 0.930971i \(-0.381037\pi\)
0.365093 + 0.930971i \(0.381037\pi\)
\(572\) 0 0
\(573\) −26.2914 −1.09834
\(574\) 0 0
\(575\) −3.03902 −0.126736
\(576\) 0 0
\(577\) 30.2954 1.26121 0.630607 0.776103i \(-0.282806\pi\)
0.630607 + 0.776103i \(0.282806\pi\)
\(578\) 0 0
\(579\) 51.4052 2.13633
\(580\) 0 0
\(581\) 3.20867 0.133118
\(582\) 0 0
\(583\) −56.7030 −2.34840
\(584\) 0 0
\(585\) −21.7961 −0.901160
\(586\) 0 0
\(587\) 21.2858 0.878559 0.439280 0.898350i \(-0.355234\pi\)
0.439280 + 0.898350i \(0.355234\pi\)
\(588\) 0 0
\(589\) −28.9912 −1.19456
\(590\) 0 0
\(591\) −27.9793 −1.15091
\(592\) 0 0
\(593\) −17.5629 −0.721223 −0.360612 0.932716i \(-0.617432\pi\)
−0.360612 + 0.932716i \(0.617432\pi\)
\(594\) 0 0
\(595\) −1.30414 −0.0534647
\(596\) 0 0
\(597\) −1.56770 −0.0641616
\(598\) 0 0
\(599\) −12.8901 −0.526675 −0.263338 0.964704i \(-0.584823\pi\)
−0.263338 + 0.964704i \(0.584823\pi\)
\(600\) 0 0
\(601\) −13.8424 −0.564641 −0.282321 0.959320i \(-0.591104\pi\)
−0.282321 + 0.959320i \(0.591104\pi\)
\(602\) 0 0
\(603\) 1.19554 0.0486860
\(604\) 0 0
\(605\) 26.9753 1.09670
\(606\) 0 0
\(607\) 24.7460 1.00441 0.502205 0.864749i \(-0.332522\pi\)
0.502205 + 0.864749i \(0.332522\pi\)
\(608\) 0 0
\(609\) −3.97780 −0.161189
\(610\) 0 0
\(611\) −62.1099 −2.51270
\(612\) 0 0
\(613\) −11.9753 −0.483679 −0.241839 0.970316i \(-0.577751\pi\)
−0.241839 + 0.970316i \(0.577751\pi\)
\(614\) 0 0
\(615\) 30.6433 1.23566
\(616\) 0 0
\(617\) −25.5167 −1.02726 −0.513632 0.858010i \(-0.671700\pi\)
−0.513632 + 0.858010i \(0.671700\pi\)
\(618\) 0 0
\(619\) −24.0660 −0.967296 −0.483648 0.875263i \(-0.660688\pi\)
−0.483648 + 0.875263i \(0.660688\pi\)
\(620\) 0 0
\(621\) −2.66402 −0.106904
\(622\) 0 0
\(623\) −5.72535 −0.229381
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 103.745 4.14318
\(628\) 0 0
\(629\) −2.51034 −0.100094
\(630\) 0 0
\(631\) −24.5545 −0.977500 −0.488750 0.872424i \(-0.662547\pi\)
−0.488750 + 0.872424i \(0.662547\pi\)
\(632\) 0 0
\(633\) −46.1074 −1.83260
\(634\) 0 0
\(635\) 2.17635 0.0863656
\(636\) 0 0
\(637\) 43.8153 1.73603
\(638\) 0 0
\(639\) −9.63067 −0.380983
\(640\) 0 0
\(641\) −5.20300 −0.205506 −0.102753 0.994707i \(-0.532765\pi\)
−0.102753 + 0.994707i \(0.532765\pi\)
\(642\) 0 0
\(643\) −0.0875650 −0.00345323 −0.00172661 0.999999i \(-0.500550\pi\)
−0.00172661 + 0.999999i \(0.500550\pi\)
\(644\) 0 0
\(645\) −5.90362 −0.232455
\(646\) 0 0
\(647\) −5.81606 −0.228653 −0.114326 0.993443i \(-0.536471\pi\)
−0.114326 + 0.993443i \(0.536471\pi\)
\(648\) 0 0
\(649\) −57.7897 −2.26845
\(650\) 0 0
\(651\) −5.67904 −0.222579
\(652\) 0 0
\(653\) 12.5716 0.491965 0.245983 0.969274i \(-0.420889\pi\)
0.245983 + 0.969274i \(0.420889\pi\)
\(654\) 0 0
\(655\) −12.7416 −0.497856
\(656\) 0 0
\(657\) −34.2843 −1.33756
\(658\) 0 0
\(659\) 21.7730 0.848157 0.424079 0.905625i \(-0.360598\pi\)
0.424079 + 0.905625i \(0.360598\pi\)
\(660\) 0 0
\(661\) −9.29462 −0.361519 −0.180759 0.983527i \(-0.557856\pi\)
−0.180759 + 0.983527i \(0.557856\pi\)
\(662\) 0 0
\(663\) 41.1768 1.59917
\(664\) 0 0
\(665\) 3.47132 0.134612
\(666\) 0 0
\(667\) 9.23563 0.357605
\(668\) 0 0
\(669\) −1.97690 −0.0764312
\(670\) 0 0
\(671\) 12.3248 0.475795
\(672\) 0 0
\(673\) 34.2221 1.31917 0.659583 0.751632i \(-0.270733\pi\)
0.659583 + 0.751632i \(0.270733\pi\)
\(674\) 0 0
\(675\) 0.876607 0.0337406
\(676\) 0 0
\(677\) 15.4761 0.594794 0.297397 0.954754i \(-0.403881\pi\)
0.297397 + 0.954754i \(0.403881\pi\)
\(678\) 0 0
\(679\) 3.29533 0.126463
\(680\) 0 0
\(681\) 15.9355 0.610648
\(682\) 0 0
\(683\) 12.3750 0.473516 0.236758 0.971569i \(-0.423915\pi\)
0.236758 + 0.971569i \(0.423915\pi\)
\(684\) 0 0
\(685\) −4.26513 −0.162962
\(686\) 0 0
\(687\) 5.25160 0.200361
\(688\) 0 0
\(689\) 59.9044 2.28218
\(690\) 0 0
\(691\) 1.69819 0.0646024 0.0323012 0.999478i \(-0.489716\pi\)
0.0323012 + 0.999478i \(0.489716\pi\)
\(692\) 0 0
\(693\) 10.7182 0.407149
\(694\) 0 0
\(695\) −11.6290 −0.441112
\(696\) 0 0
\(697\) −30.5318 −1.15647
\(698\) 0 0
\(699\) 66.6727 2.52179
\(700\) 0 0
\(701\) 33.3869 1.26100 0.630502 0.776187i \(-0.282849\pi\)
0.630502 + 0.776187i \(0.282849\pi\)
\(702\) 0 0
\(703\) 6.68192 0.252013
\(704\) 0 0
\(705\) 24.0366 0.905271
\(706\) 0 0
\(707\) 10.1027 0.379952
\(708\) 0 0
\(709\) −19.9403 −0.748874 −0.374437 0.927252i \(-0.622164\pi\)
−0.374437 + 0.927252i \(0.622164\pi\)
\(710\) 0 0
\(711\) 2.28302 0.0856201
\(712\) 0 0
\(713\) 13.1856 0.493803
\(714\) 0 0
\(715\) −40.1194 −1.50038
\(716\) 0 0
\(717\) 47.0231 1.75611
\(718\) 0 0
\(719\) 43.8694 1.63605 0.818027 0.575180i \(-0.195068\pi\)
0.818027 + 0.575180i \(0.195068\pi\)
\(720\) 0 0
\(721\) 1.03902 0.0386951
\(722\) 0 0
\(723\) −0.150410 −0.00559380
\(724\) 0 0
\(725\) −3.03902 −0.112866
\(726\) 0 0
\(727\) −41.4052 −1.53563 −0.767817 0.640669i \(-0.778657\pi\)
−0.767817 + 0.640669i \(0.778657\pi\)
\(728\) 0 0
\(729\) −32.8574 −1.21694
\(730\) 0 0
\(731\) 5.88214 0.217559
\(732\) 0 0
\(733\) 25.6919 0.948950 0.474475 0.880269i \(-0.342638\pi\)
0.474475 + 0.880269i \(0.342638\pi\)
\(734\) 0 0
\(735\) −16.9566 −0.625453
\(736\) 0 0
\(737\) 2.20058 0.0810594
\(738\) 0 0
\(739\) −46.4195 −1.70757 −0.853785 0.520625i \(-0.825699\pi\)
−0.853785 + 0.520625i \(0.825699\pi\)
\(740\) 0 0
\(741\) −109.603 −4.02635
\(742\) 0 0
\(743\) 9.86501 0.361912 0.180956 0.983491i \(-0.442081\pi\)
0.180956 + 0.983491i \(0.442081\pi\)
\(744\) 0 0
\(745\) −17.3591 −0.635987
\(746\) 0 0
\(747\) 20.6780 0.756567
\(748\) 0 0
\(749\) −7.80567 −0.285213
\(750\) 0 0
\(751\) 37.5860 1.37153 0.685765 0.727823i \(-0.259468\pi\)
0.685765 + 0.727823i \(0.259468\pi\)
\(752\) 0 0
\(753\) 7.69194 0.280310
\(754\) 0 0
\(755\) 0.343164 0.0124890
\(756\) 0 0
\(757\) 47.1672 1.71432 0.857161 0.515049i \(-0.172226\pi\)
0.857161 + 0.515049i \(0.172226\pi\)
\(758\) 0 0
\(759\) −47.1846 −1.71269
\(760\) 0 0
\(761\) −30.0167 −1.08810 −0.544052 0.839052i \(-0.683111\pi\)
−0.544052 + 0.839052i \(0.683111\pi\)
\(762\) 0 0
\(763\) 8.15994 0.295410
\(764\) 0 0
\(765\) −8.40444 −0.303863
\(766\) 0 0
\(767\) 61.0525 2.20448
\(768\) 0 0
\(769\) −32.3662 −1.16715 −0.583577 0.812058i \(-0.698347\pi\)
−0.583577 + 0.812058i \(0.698347\pi\)
\(770\) 0 0
\(771\) −52.6910 −1.89762
\(772\) 0 0
\(773\) −40.6338 −1.46150 −0.730748 0.682648i \(-0.760828\pi\)
−0.730748 + 0.682648i \(0.760828\pi\)
\(774\) 0 0
\(775\) −4.33876 −0.155853
\(776\) 0 0
\(777\) 1.30891 0.0469568
\(778\) 0 0
\(779\) 81.2683 2.91174
\(780\) 0 0
\(781\) −17.7268 −0.634316
\(782\) 0 0
\(783\) −2.66402 −0.0952044
\(784\) 0 0
\(785\) −4.16241 −0.148563
\(786\) 0 0
\(787\) −13.7089 −0.488671 −0.244335 0.969691i \(-0.578570\pi\)
−0.244335 + 0.969691i \(0.578570\pi\)
\(788\) 0 0
\(789\) −19.9324 −0.709612
\(790\) 0 0
\(791\) −1.13540 −0.0403700
\(792\) 0 0
\(793\) −13.0207 −0.462378
\(794\) 0 0
\(795\) −23.1831 −0.822220
\(796\) 0 0
\(797\) 18.3321 0.649355 0.324677 0.945825i \(-0.394744\pi\)
0.324677 + 0.945825i \(0.394744\pi\)
\(798\) 0 0
\(799\) −23.9491 −0.847259
\(800\) 0 0
\(801\) −36.8964 −1.30367
\(802\) 0 0
\(803\) −63.1059 −2.22696
\(804\) 0 0
\(805\) −1.57880 −0.0556453
\(806\) 0 0
\(807\) −1.52863 −0.0538103
\(808\) 0 0
\(809\) 18.3248 0.644267 0.322133 0.946694i \(-0.395600\pi\)
0.322133 + 0.946694i \(0.395600\pi\)
\(810\) 0 0
\(811\) −9.45775 −0.332106 −0.166053 0.986117i \(-0.553102\pi\)
−0.166053 + 0.986117i \(0.553102\pi\)
\(812\) 0 0
\(813\) 51.8846 1.81967
\(814\) 0 0
\(815\) 19.3455 0.677643
\(816\) 0 0
\(817\) −15.6568 −0.547763
\(818\) 0 0
\(819\) −11.3233 −0.395668
\(820\) 0 0
\(821\) −32.4689 −1.13317 −0.566586 0.824003i \(-0.691736\pi\)
−0.566586 + 0.824003i \(0.691736\pi\)
\(822\) 0 0
\(823\) 32.0370 1.11674 0.558369 0.829593i \(-0.311427\pi\)
0.558369 + 0.829593i \(0.311427\pi\)
\(824\) 0 0
\(825\) 15.5263 0.540555
\(826\) 0 0
\(827\) 2.20543 0.0766903 0.0383451 0.999265i \(-0.487791\pi\)
0.0383451 + 0.999265i \(0.487791\pi\)
\(828\) 0 0
\(829\) 26.9745 0.936862 0.468431 0.883500i \(-0.344819\pi\)
0.468431 + 0.883500i \(0.344819\pi\)
\(830\) 0 0
\(831\) 53.6258 1.86026
\(832\) 0 0
\(833\) 16.8949 0.585372
\(834\) 0 0
\(835\) −0.979321 −0.0338908
\(836\) 0 0
\(837\) −3.80338 −0.131464
\(838\) 0 0
\(839\) −20.8424 −0.719560 −0.359780 0.933037i \(-0.617148\pi\)
−0.359780 + 0.933037i \(0.617148\pi\)
\(840\) 0 0
\(841\) −19.7644 −0.681530
\(842\) 0 0
\(843\) −26.0597 −0.897544
\(844\) 0 0
\(845\) 29.3845 1.01086
\(846\) 0 0
\(847\) 14.0139 0.481524
\(848\) 0 0
\(849\) 4.22062 0.144851
\(850\) 0 0
\(851\) −3.03902 −0.104176
\(852\) 0 0
\(853\) −48.7093 −1.66778 −0.833888 0.551933i \(-0.813890\pi\)
−0.833888 + 0.551933i \(0.813890\pi\)
\(854\) 0 0
\(855\) 22.3706 0.765058
\(856\) 0 0
\(857\) −28.4212 −0.970850 −0.485425 0.874278i \(-0.661335\pi\)
−0.485425 + 0.874278i \(0.661335\pi\)
\(858\) 0 0
\(859\) 16.2926 0.555895 0.277947 0.960596i \(-0.410346\pi\)
0.277947 + 0.960596i \(0.410346\pi\)
\(860\) 0 0
\(861\) 15.9195 0.542535
\(862\) 0 0
\(863\) 28.1333 0.957670 0.478835 0.877905i \(-0.341059\pi\)
0.478835 + 0.877905i \(0.341059\pi\)
\(864\) 0 0
\(865\) −22.9363 −0.779858
\(866\) 0 0
\(867\) −26.9542 −0.915413
\(868\) 0 0
\(869\) 4.20228 0.142553
\(870\) 0 0
\(871\) −2.32482 −0.0787737
\(872\) 0 0
\(873\) 21.2365 0.718745
\(874\) 0 0
\(875\) 0.519509 0.0175626
\(876\) 0 0
\(877\) −25.0899 −0.847226 −0.423613 0.905843i \(-0.639238\pi\)
−0.423613 + 0.905843i \(0.639238\pi\)
\(878\) 0 0
\(879\) 25.1027 0.846692
\(880\) 0 0
\(881\) 29.2197 0.984436 0.492218 0.870472i \(-0.336186\pi\)
0.492218 + 0.870472i \(0.336186\pi\)
\(882\) 0 0
\(883\) 26.9745 0.907763 0.453882 0.891062i \(-0.350039\pi\)
0.453882 + 0.891062i \(0.350039\pi\)
\(884\) 0 0
\(885\) −23.6274 −0.794226
\(886\) 0 0
\(887\) −31.2839 −1.05041 −0.525205 0.850976i \(-0.676011\pi\)
−0.525205 + 0.850976i \(0.676011\pi\)
\(888\) 0 0
\(889\) 1.13063 0.0379202
\(890\) 0 0
\(891\) −48.2835 −1.61756
\(892\) 0 0
\(893\) 63.7468 2.13321
\(894\) 0 0
\(895\) −16.7600 −0.560224
\(896\) 0 0
\(897\) 49.8486 1.66440
\(898\) 0 0
\(899\) 13.1856 0.439763
\(900\) 0 0
\(901\) 23.0987 0.769530
\(902\) 0 0
\(903\) −3.06699 −0.102063
\(904\) 0 0
\(905\) −11.4466 −0.380499
\(906\) 0 0
\(907\) 17.4697 0.580073 0.290037 0.957016i \(-0.406332\pi\)
0.290037 + 0.957016i \(0.406332\pi\)
\(908\) 0 0
\(909\) 65.1060 2.15943
\(910\) 0 0
\(911\) −37.6659 −1.24793 −0.623964 0.781453i \(-0.714479\pi\)
−0.623964 + 0.781453i \(0.714479\pi\)
\(912\) 0 0
\(913\) 38.0612 1.25964
\(914\) 0 0
\(915\) 5.03902 0.166585
\(916\) 0 0
\(917\) −6.61939 −0.218592
\(918\) 0 0
\(919\) −43.6119 −1.43862 −0.719312 0.694687i \(-0.755543\pi\)
−0.719312 + 0.694687i \(0.755543\pi\)
\(920\) 0 0
\(921\) 66.0780 2.17734
\(922\) 0 0
\(923\) 18.7277 0.616429
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 6.69586 0.219921
\(928\) 0 0
\(929\) 22.8367 0.749249 0.374625 0.927177i \(-0.377772\pi\)
0.374625 + 0.927177i \(0.377772\pi\)
\(930\) 0 0
\(931\) −44.9701 −1.47383
\(932\) 0 0
\(933\) 4.40614 0.144250
\(934\) 0 0
\(935\) −15.4697 −0.505915
\(936\) 0 0
\(937\) −14.2309 −0.464904 −0.232452 0.972608i \(-0.574675\pi\)
−0.232452 + 0.972608i \(0.574675\pi\)
\(938\) 0 0
\(939\) −54.8654 −1.79047
\(940\) 0 0
\(941\) 16.4140 0.535080 0.267540 0.963547i \(-0.413789\pi\)
0.267540 + 0.963547i \(0.413789\pi\)
\(942\) 0 0
\(943\) −36.9618 −1.20364
\(944\) 0 0
\(945\) 0.455405 0.0148143
\(946\) 0 0
\(947\) −33.9506 −1.10325 −0.551624 0.834093i \(-0.685992\pi\)
−0.551624 + 0.834093i \(0.685992\pi\)
\(948\) 0 0
\(949\) 66.6688 2.16416
\(950\) 0 0
\(951\) 11.1393 0.361215
\(952\) 0 0
\(953\) −41.5358 −1.34548 −0.672738 0.739881i \(-0.734882\pi\)
−0.672738 + 0.739881i \(0.734882\pi\)
\(954\) 0 0
\(955\) −10.4351 −0.337673
\(956\) 0 0
\(957\) −47.1846 −1.52526
\(958\) 0 0
\(959\) −2.21577 −0.0715510
\(960\) 0 0
\(961\) −12.1752 −0.392748
\(962\) 0 0
\(963\) −50.3029 −1.62099
\(964\) 0 0
\(965\) 20.4029 0.656791
\(966\) 0 0
\(967\) −16.2341 −0.522054 −0.261027 0.965332i \(-0.584061\pi\)
−0.261027 + 0.965332i \(0.584061\pi\)
\(968\) 0 0
\(969\) −42.2620 −1.35765
\(970\) 0 0
\(971\) 30.7277 0.986098 0.493049 0.870001i \(-0.335882\pi\)
0.493049 + 0.870001i \(0.335882\pi\)
\(972\) 0 0
\(973\) −6.04136 −0.193677
\(974\) 0 0
\(975\) −16.4029 −0.525312
\(976\) 0 0
\(977\) −20.8654 −0.667544 −0.333772 0.942654i \(-0.608322\pi\)
−0.333772 + 0.942654i \(0.608322\pi\)
\(978\) 0 0
\(979\) −67.9140 −2.17054
\(980\) 0 0
\(981\) 52.5859 1.67894
\(982\) 0 0
\(983\) −20.2360 −0.645430 −0.322715 0.946496i \(-0.604595\pi\)
−0.322715 + 0.946496i \(0.604595\pi\)
\(984\) 0 0
\(985\) −11.1051 −0.353836
\(986\) 0 0
\(987\) 12.4872 0.397473
\(988\) 0 0
\(989\) 7.12092 0.226432
\(990\) 0 0
\(991\) −9.33642 −0.296581 −0.148291 0.988944i \(-0.547377\pi\)
−0.148291 + 0.988944i \(0.547377\pi\)
\(992\) 0 0
\(993\) 81.6744 2.59186
\(994\) 0 0
\(995\) −0.622223 −0.0197258
\(996\) 0 0
\(997\) −13.9506 −0.441821 −0.220911 0.975294i \(-0.570903\pi\)
−0.220911 + 0.975294i \(0.570903\pi\)
\(998\) 0 0
\(999\) 0.876607 0.0277346
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 740.2.a.f.1.3 4
3.2 odd 2 6660.2.a.r.1.3 4
4.3 odd 2 2960.2.a.v.1.2 4
5.2 odd 4 3700.2.d.i.149.3 8
5.3 odd 4 3700.2.d.i.149.6 8
5.4 even 2 3700.2.a.j.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
740.2.a.f.1.3 4 1.1 even 1 trivial
2960.2.a.v.1.2 4 4.3 odd 2
3700.2.a.j.1.2 4 5.4 even 2
3700.2.d.i.149.3 8 5.2 odd 4
3700.2.d.i.149.6 8 5.3 odd 4
6660.2.a.r.1.3 4 3.2 odd 2