Properties

Label 538.2.a.d.1.4
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.49844\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.370477 q^{3} +1.00000 q^{4} +2.98038 q^{5} +0.370477 q^{6} -2.56529 q^{7} -1.00000 q^{8} -2.86275 q^{9} -2.98038 q^{10} -5.77573 q^{11} -0.370477 q^{12} -0.953732 q^{13} +2.56529 q^{14} -1.10416 q^{15} +1.00000 q^{16} +2.51668 q^{17} +2.86275 q^{18} -7.19922 q^{19} +2.98038 q^{20} +0.950379 q^{21} +5.77573 q^{22} -5.17399 q^{23} +0.370477 q^{24} +3.88265 q^{25} +0.953732 q^{26} +2.17201 q^{27} -2.56529 q^{28} +4.63236 q^{29} +1.10416 q^{30} +8.16921 q^{31} -1.00000 q^{32} +2.13977 q^{33} -2.51668 q^{34} -7.64552 q^{35} -2.86275 q^{36} -9.43775 q^{37} +7.19922 q^{38} +0.353336 q^{39} -2.98038 q^{40} +2.46810 q^{41} -0.950379 q^{42} +5.86496 q^{43} -5.77573 q^{44} -8.53207 q^{45} +5.17399 q^{46} -9.50969 q^{47} -0.370477 q^{48} -0.419304 q^{49} -3.88265 q^{50} -0.932373 q^{51} -0.953732 q^{52} -7.51226 q^{53} -2.17201 q^{54} -17.2139 q^{55} +2.56529 q^{56} +2.66714 q^{57} -4.63236 q^{58} -1.01485 q^{59} -1.10416 q^{60} -5.86716 q^{61} -8.16921 q^{62} +7.34377 q^{63} +1.00000 q^{64} -2.84248 q^{65} -2.13977 q^{66} +12.1288 q^{67} +2.51668 q^{68} +1.91685 q^{69} +7.64552 q^{70} -12.3537 q^{71} +2.86275 q^{72} +0.650878 q^{73} +9.43775 q^{74} -1.43843 q^{75} -7.19922 q^{76} +14.8164 q^{77} -0.353336 q^{78} +15.5158 q^{79} +2.98038 q^{80} +7.78356 q^{81} -2.46810 q^{82} -8.27171 q^{83} +0.950379 q^{84} +7.50067 q^{85} -5.86496 q^{86} -1.71618 q^{87} +5.77573 q^{88} +2.89852 q^{89} +8.53207 q^{90} +2.44660 q^{91} -5.17399 q^{92} -3.02650 q^{93} +9.50969 q^{94} -21.4564 q^{95} +0.370477 q^{96} +0.956247 q^{97} +0.419304 q^{98} +16.5345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} - 4 q^{3} + 7 q^{4} - 6 q^{5} + 4 q^{6} - 3 q^{7} - 7 q^{8} + 7 q^{9} + 6 q^{10} - 12 q^{11} - 4 q^{12} + 3 q^{13} + 3 q^{14} - 6 q^{15} + 7 q^{16} - 8 q^{17} - 7 q^{18} - 7 q^{19} - 6 q^{20}+ \cdots - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.370477 −0.213895 −0.106947 0.994265i \(-0.534108\pi\)
−0.106947 + 0.994265i \(0.534108\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.98038 1.33287 0.666433 0.745565i \(-0.267820\pi\)
0.666433 + 0.745565i \(0.267820\pi\)
\(6\) 0.370477 0.151247
\(7\) −2.56529 −0.969587 −0.484794 0.874629i \(-0.661105\pi\)
−0.484794 + 0.874629i \(0.661105\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.86275 −0.954249
\(10\) −2.98038 −0.942478
\(11\) −5.77573 −1.74145 −0.870724 0.491772i \(-0.836349\pi\)
−0.870724 + 0.491772i \(0.836349\pi\)
\(12\) −0.370477 −0.106947
\(13\) −0.953732 −0.264518 −0.132259 0.991215i \(-0.542223\pi\)
−0.132259 + 0.991215i \(0.542223\pi\)
\(14\) 2.56529 0.685602
\(15\) −1.10416 −0.285093
\(16\) 1.00000 0.250000
\(17\) 2.51668 0.610385 0.305193 0.952291i \(-0.401279\pi\)
0.305193 + 0.952291i \(0.401279\pi\)
\(18\) 2.86275 0.674756
\(19\) −7.19922 −1.65161 −0.825807 0.563953i \(-0.809280\pi\)
−0.825807 + 0.563953i \(0.809280\pi\)
\(20\) 2.98038 0.666433
\(21\) 0.950379 0.207390
\(22\) 5.77573 1.23139
\(23\) −5.17399 −1.07885 −0.539426 0.842033i \(-0.681359\pi\)
−0.539426 + 0.842033i \(0.681359\pi\)
\(24\) 0.370477 0.0756233
\(25\) 3.88265 0.776530
\(26\) 0.953732 0.187042
\(27\) 2.17201 0.418004
\(28\) −2.56529 −0.484794
\(29\) 4.63236 0.860208 0.430104 0.902779i \(-0.358477\pi\)
0.430104 + 0.902779i \(0.358477\pi\)
\(30\) 1.10416 0.201591
\(31\) 8.16921 1.46723 0.733617 0.679563i \(-0.237831\pi\)
0.733617 + 0.679563i \(0.237831\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.13977 0.372487
\(34\) −2.51668 −0.431608
\(35\) −7.64552 −1.29233
\(36\) −2.86275 −0.477124
\(37\) −9.43775 −1.55156 −0.775778 0.631006i \(-0.782642\pi\)
−0.775778 + 0.631006i \(0.782642\pi\)
\(38\) 7.19922 1.16787
\(39\) 0.353336 0.0565790
\(40\) −2.98038 −0.471239
\(41\) 2.46810 0.385453 0.192726 0.981253i \(-0.438267\pi\)
0.192726 + 0.981253i \(0.438267\pi\)
\(42\) −0.950379 −0.146647
\(43\) 5.86496 0.894398 0.447199 0.894435i \(-0.352422\pi\)
0.447199 + 0.894435i \(0.352422\pi\)
\(44\) −5.77573 −0.870724
\(45\) −8.53207 −1.27189
\(46\) 5.17399 0.762864
\(47\) −9.50969 −1.38713 −0.693565 0.720394i \(-0.743961\pi\)
−0.693565 + 0.720394i \(0.743961\pi\)
\(48\) −0.370477 −0.0534737
\(49\) −0.419304 −0.0599006
\(50\) −3.88265 −0.549089
\(51\) −0.932373 −0.130558
\(52\) −0.953732 −0.132259
\(53\) −7.51226 −1.03189 −0.515944 0.856622i \(-0.672559\pi\)
−0.515944 + 0.856622i \(0.672559\pi\)
\(54\) −2.17201 −0.295573
\(55\) −17.2139 −2.32112
\(56\) 2.56529 0.342801
\(57\) 2.66714 0.353272
\(58\) −4.63236 −0.608259
\(59\) −1.01485 −0.132122 −0.0660610 0.997816i \(-0.521043\pi\)
−0.0660610 + 0.997816i \(0.521043\pi\)
\(60\) −1.10416 −0.142547
\(61\) −5.86716 −0.751212 −0.375606 0.926779i \(-0.622565\pi\)
−0.375606 + 0.926779i \(0.622565\pi\)
\(62\) −8.16921 −1.03749
\(63\) 7.34377 0.925228
\(64\) 1.00000 0.125000
\(65\) −2.84248 −0.352566
\(66\) −2.13977 −0.263388
\(67\) 12.1288 1.48177 0.740883 0.671634i \(-0.234407\pi\)
0.740883 + 0.671634i \(0.234407\pi\)
\(68\) 2.51668 0.305193
\(69\) 1.91685 0.230761
\(70\) 7.64552 0.913815
\(71\) −12.3537 −1.46611 −0.733057 0.680167i \(-0.761907\pi\)
−0.733057 + 0.680167i \(0.761907\pi\)
\(72\) 2.86275 0.337378
\(73\) 0.650878 0.0761795 0.0380897 0.999274i \(-0.487873\pi\)
0.0380897 + 0.999274i \(0.487873\pi\)
\(74\) 9.43775 1.09712
\(75\) −1.43843 −0.166096
\(76\) −7.19922 −0.825807
\(77\) 14.8164 1.68849
\(78\) −0.353336 −0.0400074
\(79\) 15.5158 1.74567 0.872834 0.488018i \(-0.162280\pi\)
0.872834 + 0.488018i \(0.162280\pi\)
\(80\) 2.98038 0.333216
\(81\) 7.78356 0.864840
\(82\) −2.46810 −0.272556
\(83\) −8.27171 −0.907939 −0.453969 0.891017i \(-0.649992\pi\)
−0.453969 + 0.891017i \(0.649992\pi\)
\(84\) 0.950379 0.103695
\(85\) 7.50067 0.813562
\(86\) −5.86496 −0.632435
\(87\) −1.71618 −0.183994
\(88\) 5.77573 0.615695
\(89\) 2.89852 0.307242 0.153621 0.988130i \(-0.450907\pi\)
0.153621 + 0.988130i \(0.450907\pi\)
\(90\) 8.53207 0.899359
\(91\) 2.44660 0.256473
\(92\) −5.17399 −0.539426
\(93\) −3.02650 −0.313834
\(94\) 9.50969 0.980850
\(95\) −21.4564 −2.20138
\(96\) 0.370477 0.0378116
\(97\) 0.956247 0.0970922 0.0485461 0.998821i \(-0.484541\pi\)
0.0485461 + 0.998821i \(0.484541\pi\)
\(98\) 0.419304 0.0423561
\(99\) 16.5345 1.66178
\(100\) 3.88265 0.388265
\(101\) 5.92549 0.589608 0.294804 0.955558i \(-0.404746\pi\)
0.294804 + 0.955558i \(0.404746\pi\)
\(102\) 0.932373 0.0923187
\(103\) −7.57015 −0.745909 −0.372955 0.927850i \(-0.621655\pi\)
−0.372955 + 0.927850i \(0.621655\pi\)
\(104\) 0.953732 0.0935211
\(105\) 2.83249 0.276423
\(106\) 7.51226 0.729655
\(107\) 16.0455 1.55118 0.775589 0.631238i \(-0.217453\pi\)
0.775589 + 0.631238i \(0.217453\pi\)
\(108\) 2.17201 0.209002
\(109\) 2.50942 0.240359 0.120179 0.992752i \(-0.461653\pi\)
0.120179 + 0.992752i \(0.461653\pi\)
\(110\) 17.2139 1.64128
\(111\) 3.49647 0.331870
\(112\) −2.56529 −0.242397
\(113\) −13.4589 −1.26611 −0.633055 0.774107i \(-0.718199\pi\)
−0.633055 + 0.774107i \(0.718199\pi\)
\(114\) −2.66714 −0.249801
\(115\) −15.4205 −1.43796
\(116\) 4.63236 0.430104
\(117\) 2.73029 0.252416
\(118\) 1.01485 0.0934244
\(119\) −6.45602 −0.591822
\(120\) 1.10416 0.100796
\(121\) 22.3591 2.03264
\(122\) 5.86716 0.531187
\(123\) −0.914375 −0.0824464
\(124\) 8.16921 0.733617
\(125\) −3.33013 −0.297856
\(126\) −7.34377 −0.654235
\(127\) 2.35435 0.208915 0.104458 0.994529i \(-0.466689\pi\)
0.104458 + 0.994529i \(0.466689\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.17283 −0.191307
\(130\) 2.84248 0.249302
\(131\) 8.14252 0.711415 0.355708 0.934597i \(-0.384240\pi\)
0.355708 + 0.934597i \(0.384240\pi\)
\(132\) 2.13977 0.186243
\(133\) 18.4681 1.60138
\(134\) −12.1288 −1.04777
\(135\) 6.47342 0.557143
\(136\) −2.51668 −0.215804
\(137\) −13.5630 −1.15876 −0.579382 0.815056i \(-0.696706\pi\)
−0.579382 + 0.815056i \(0.696706\pi\)
\(138\) −1.91685 −0.163173
\(139\) −17.6145 −1.49405 −0.747023 0.664798i \(-0.768518\pi\)
−0.747023 + 0.664798i \(0.768518\pi\)
\(140\) −7.64552 −0.646165
\(141\) 3.52312 0.296700
\(142\) 12.3537 1.03670
\(143\) 5.50850 0.460644
\(144\) −2.86275 −0.238562
\(145\) 13.8062 1.14654
\(146\) −0.650878 −0.0538670
\(147\) 0.155342 0.0128124
\(148\) −9.43775 −0.775778
\(149\) 4.73354 0.387787 0.193893 0.981023i \(-0.437888\pi\)
0.193893 + 0.981023i \(0.437888\pi\)
\(150\) 1.43843 0.117447
\(151\) 18.5426 1.50897 0.754486 0.656316i \(-0.227886\pi\)
0.754486 + 0.656316i \(0.227886\pi\)
\(152\) 7.19922 0.583934
\(153\) −7.20463 −0.582460
\(154\) −14.8164 −1.19394
\(155\) 24.3473 1.95562
\(156\) 0.353336 0.0282895
\(157\) 6.62583 0.528799 0.264399 0.964413i \(-0.414826\pi\)
0.264399 + 0.964413i \(0.414826\pi\)
\(158\) −15.5158 −1.23437
\(159\) 2.78312 0.220716
\(160\) −2.98038 −0.235620
\(161\) 13.2728 1.04604
\(162\) −7.78356 −0.611534
\(163\) −7.35217 −0.575867 −0.287933 0.957650i \(-0.592968\pi\)
−0.287933 + 0.957650i \(0.592968\pi\)
\(164\) 2.46810 0.192726
\(165\) 6.37734 0.496475
\(166\) 8.27171 0.642010
\(167\) −18.4854 −1.43045 −0.715223 0.698896i \(-0.753675\pi\)
−0.715223 + 0.698896i \(0.753675\pi\)
\(168\) −0.950379 −0.0733234
\(169\) −12.0904 −0.930030
\(170\) −7.50067 −0.575275
\(171\) 20.6095 1.57605
\(172\) 5.86496 0.447199
\(173\) −15.2035 −1.15590 −0.577951 0.816071i \(-0.696148\pi\)
−0.577951 + 0.816071i \(0.696148\pi\)
\(174\) 1.71618 0.130103
\(175\) −9.96011 −0.752913
\(176\) −5.77573 −0.435362
\(177\) 0.375978 0.0282602
\(178\) −2.89852 −0.217253
\(179\) −8.92937 −0.667412 −0.333706 0.942677i \(-0.608299\pi\)
−0.333706 + 0.942677i \(0.608299\pi\)
\(180\) −8.53207 −0.635943
\(181\) 8.96265 0.666189 0.333094 0.942894i \(-0.391907\pi\)
0.333094 + 0.942894i \(0.391907\pi\)
\(182\) −2.44660 −0.181354
\(183\) 2.17365 0.160680
\(184\) 5.17399 0.381432
\(185\) −28.1280 −2.06802
\(186\) 3.02650 0.221914
\(187\) −14.5357 −1.06295
\(188\) −9.50969 −0.693565
\(189\) −5.57183 −0.405291
\(190\) 21.4564 1.55661
\(191\) −2.71179 −0.196218 −0.0981091 0.995176i \(-0.531279\pi\)
−0.0981091 + 0.995176i \(0.531279\pi\)
\(192\) −0.370477 −0.0267369
\(193\) −10.4616 −0.753042 −0.376521 0.926408i \(-0.622880\pi\)
−0.376521 + 0.926408i \(0.622880\pi\)
\(194\) −0.956247 −0.0686545
\(195\) 1.05307 0.0754122
\(196\) −0.419304 −0.0299503
\(197\) −8.17270 −0.582280 −0.291140 0.956680i \(-0.594035\pi\)
−0.291140 + 0.956680i \(0.594035\pi\)
\(198\) −16.5345 −1.17505
\(199\) 23.4576 1.66287 0.831433 0.555625i \(-0.187521\pi\)
0.831433 + 0.555625i \(0.187521\pi\)
\(200\) −3.88265 −0.274545
\(201\) −4.49343 −0.316942
\(202\) −5.92549 −0.416916
\(203\) −11.8833 −0.834047
\(204\) −0.932373 −0.0652792
\(205\) 7.35587 0.513757
\(206\) 7.57015 0.527437
\(207\) 14.8118 1.02949
\(208\) −0.953732 −0.0661294
\(209\) 41.5807 2.87620
\(210\) −2.83249 −0.195460
\(211\) 23.7236 1.63320 0.816600 0.577203i \(-0.195856\pi\)
0.816600 + 0.577203i \(0.195856\pi\)
\(212\) −7.51226 −0.515944
\(213\) 4.57676 0.313594
\(214\) −16.0455 −1.09685
\(215\) 17.4798 1.19211
\(216\) −2.17201 −0.147787
\(217\) −20.9564 −1.42261
\(218\) −2.50942 −0.169959
\(219\) −0.241135 −0.0162944
\(220\) −17.2139 −1.16056
\(221\) −2.40024 −0.161458
\(222\) −3.49647 −0.234667
\(223\) −4.63802 −0.310585 −0.155292 0.987869i \(-0.549632\pi\)
−0.155292 + 0.987869i \(0.549632\pi\)
\(224\) 2.56529 0.171400
\(225\) −11.1150 −0.741003
\(226\) 13.4589 0.895274
\(227\) 2.30295 0.152852 0.0764261 0.997075i \(-0.475649\pi\)
0.0764261 + 0.997075i \(0.475649\pi\)
\(228\) 2.66714 0.176636
\(229\) 1.98890 0.131430 0.0657150 0.997838i \(-0.479067\pi\)
0.0657150 + 0.997838i \(0.479067\pi\)
\(230\) 15.4205 1.01679
\(231\) −5.48913 −0.361159
\(232\) −4.63236 −0.304129
\(233\) 5.58928 0.366166 0.183083 0.983097i \(-0.441392\pi\)
0.183083 + 0.983097i \(0.441392\pi\)
\(234\) −2.73029 −0.178485
\(235\) −28.3425 −1.84886
\(236\) −1.01485 −0.0660610
\(237\) −5.74826 −0.373389
\(238\) 6.45602 0.418481
\(239\) −3.34553 −0.216404 −0.108202 0.994129i \(-0.534509\pi\)
−0.108202 + 0.994129i \(0.534509\pi\)
\(240\) −1.10416 −0.0712733
\(241\) 26.7139 1.72079 0.860395 0.509627i \(-0.170217\pi\)
0.860395 + 0.509627i \(0.170217\pi\)
\(242\) −22.3591 −1.43729
\(243\) −9.39967 −0.602989
\(244\) −5.86716 −0.375606
\(245\) −1.24968 −0.0798394
\(246\) 0.914375 0.0582984
\(247\) 6.86613 0.436881
\(248\) −8.16921 −0.518745
\(249\) 3.06448 0.194203
\(250\) 3.33013 0.210616
\(251\) −29.3076 −1.84988 −0.924940 0.380112i \(-0.875885\pi\)
−0.924940 + 0.380112i \(0.875885\pi\)
\(252\) 7.34377 0.462614
\(253\) 29.8836 1.87877
\(254\) −2.35435 −0.147725
\(255\) −2.77882 −0.174017
\(256\) 1.00000 0.0625000
\(257\) 10.3803 0.647504 0.323752 0.946142i \(-0.395056\pi\)
0.323752 + 0.946142i \(0.395056\pi\)
\(258\) 2.17283 0.135275
\(259\) 24.2105 1.50437
\(260\) −2.84248 −0.176283
\(261\) −13.2613 −0.820853
\(262\) −8.14252 −0.503046
\(263\) −8.83916 −0.545046 −0.272523 0.962149i \(-0.587858\pi\)
−0.272523 + 0.962149i \(0.587858\pi\)
\(264\) −2.13977 −0.131694
\(265\) −22.3894 −1.37537
\(266\) −18.4681 −1.13235
\(267\) −1.07383 −0.0657175
\(268\) 12.1288 0.740883
\(269\) −1.00000 −0.0609711
\(270\) −6.47342 −0.393960
\(271\) 7.39604 0.449277 0.224639 0.974442i \(-0.427880\pi\)
0.224639 + 0.974442i \(0.427880\pi\)
\(272\) 2.51668 0.152596
\(273\) −0.906407 −0.0548583
\(274\) 13.5630 0.819370
\(275\) −22.4251 −1.35229
\(276\) 1.91685 0.115381
\(277\) 13.5773 0.815779 0.407890 0.913031i \(-0.366265\pi\)
0.407890 + 0.913031i \(0.366265\pi\)
\(278\) 17.6145 1.05645
\(279\) −23.3864 −1.40011
\(280\) 7.64552 0.456907
\(281\) 13.5723 0.809653 0.404826 0.914394i \(-0.367332\pi\)
0.404826 + 0.914394i \(0.367332\pi\)
\(282\) −3.52312 −0.209799
\(283\) 20.5026 1.21875 0.609377 0.792880i \(-0.291419\pi\)
0.609377 + 0.792880i \(0.291419\pi\)
\(284\) −12.3537 −0.733057
\(285\) 7.94909 0.470864
\(286\) −5.50850 −0.325724
\(287\) −6.33139 −0.373730
\(288\) 2.86275 0.168689
\(289\) −10.6663 −0.627430
\(290\) −13.8062 −0.810727
\(291\) −0.354267 −0.0207675
\(292\) 0.650878 0.0380897
\(293\) −26.0133 −1.51971 −0.759857 0.650090i \(-0.774731\pi\)
−0.759857 + 0.650090i \(0.774731\pi\)
\(294\) −0.155342 −0.00905976
\(295\) −3.02463 −0.176101
\(296\) 9.43775 0.548558
\(297\) −12.5450 −0.727932
\(298\) −4.73354 −0.274207
\(299\) 4.93460 0.285376
\(300\) −1.43843 −0.0830479
\(301\) −15.0453 −0.867197
\(302\) −18.5426 −1.06700
\(303\) −2.19526 −0.126114
\(304\) −7.19922 −0.412903
\(305\) −17.4863 −1.00126
\(306\) 7.20463 0.411861
\(307\) −17.3993 −0.993031 −0.496516 0.868028i \(-0.665387\pi\)
−0.496516 + 0.868028i \(0.665387\pi\)
\(308\) 14.8164 0.844243
\(309\) 2.80457 0.159546
\(310\) −24.3473 −1.38284
\(311\) −13.4740 −0.764038 −0.382019 0.924155i \(-0.624771\pi\)
−0.382019 + 0.924155i \(0.624771\pi\)
\(312\) −0.353336 −0.0200037
\(313\) −9.89195 −0.559126 −0.279563 0.960127i \(-0.590190\pi\)
−0.279563 + 0.960127i \(0.590190\pi\)
\(314\) −6.62583 −0.373917
\(315\) 21.8872 1.23320
\(316\) 15.5158 0.872834
\(317\) 19.2200 1.07950 0.539751 0.841825i \(-0.318518\pi\)
0.539751 + 0.841825i \(0.318518\pi\)
\(318\) −2.78312 −0.156070
\(319\) −26.7553 −1.49801
\(320\) 2.98038 0.166608
\(321\) −5.94449 −0.331789
\(322\) −13.2728 −0.739663
\(323\) −18.1182 −1.00812
\(324\) 7.78356 0.432420
\(325\) −3.70301 −0.205406
\(326\) 7.35217 0.407199
\(327\) −0.929682 −0.0514115
\(328\) −2.46810 −0.136278
\(329\) 24.3951 1.34494
\(330\) −6.37734 −0.351061
\(331\) −10.6168 −0.583550 −0.291775 0.956487i \(-0.594246\pi\)
−0.291775 + 0.956487i \(0.594246\pi\)
\(332\) −8.27171 −0.453969
\(333\) 27.0179 1.48057
\(334\) 18.4854 1.01148
\(335\) 36.1483 1.97499
\(336\) 0.950379 0.0518474
\(337\) −4.73621 −0.257998 −0.128999 0.991645i \(-0.541176\pi\)
−0.128999 + 0.991645i \(0.541176\pi\)
\(338\) 12.0904 0.657631
\(339\) 4.98622 0.270814
\(340\) 7.50067 0.406781
\(341\) −47.1832 −2.55511
\(342\) −20.6095 −1.11444
\(343\) 19.0326 1.02767
\(344\) −5.86496 −0.316217
\(345\) 5.71292 0.307573
\(346\) 15.2035 0.817346
\(347\) −9.70333 −0.520902 −0.260451 0.965487i \(-0.583871\pi\)
−0.260451 + 0.965487i \(0.583871\pi\)
\(348\) −1.71618 −0.0919971
\(349\) −34.9874 −1.87283 −0.936416 0.350893i \(-0.885878\pi\)
−0.936416 + 0.350893i \(0.885878\pi\)
\(350\) 9.96011 0.532390
\(351\) −2.07152 −0.110569
\(352\) 5.77573 0.307847
\(353\) 17.9950 0.957777 0.478888 0.877876i \(-0.341040\pi\)
0.478888 + 0.877876i \(0.341040\pi\)
\(354\) −0.375978 −0.0199830
\(355\) −36.8187 −1.95413
\(356\) 2.89852 0.153621
\(357\) 2.39180 0.126588
\(358\) 8.92937 0.471932
\(359\) −8.64349 −0.456186 −0.228093 0.973639i \(-0.573249\pi\)
−0.228093 + 0.973639i \(0.573249\pi\)
\(360\) 8.53207 0.449679
\(361\) 32.8287 1.72783
\(362\) −8.96265 −0.471067
\(363\) −8.28351 −0.434772
\(364\) 2.44660 0.128236
\(365\) 1.93986 0.101537
\(366\) −2.17365 −0.113618
\(367\) −25.4599 −1.32899 −0.664497 0.747291i \(-0.731354\pi\)
−0.664497 + 0.747291i \(0.731354\pi\)
\(368\) −5.17399 −0.269713
\(369\) −7.06555 −0.367818
\(370\) 28.1280 1.46231
\(371\) 19.2711 1.00051
\(372\) −3.02650 −0.156917
\(373\) −5.63673 −0.291859 −0.145929 0.989295i \(-0.546617\pi\)
−0.145929 + 0.989295i \(0.546617\pi\)
\(374\) 14.5357 0.751622
\(375\) 1.23374 0.0637098
\(376\) 9.50969 0.490425
\(377\) −4.41803 −0.227540
\(378\) 5.57183 0.286584
\(379\) −16.2741 −0.835944 −0.417972 0.908460i \(-0.637259\pi\)
−0.417972 + 0.908460i \(0.637259\pi\)
\(380\) −21.4564 −1.10069
\(381\) −0.872233 −0.0446859
\(382\) 2.71179 0.138747
\(383\) 17.3292 0.885483 0.442742 0.896649i \(-0.354006\pi\)
0.442742 + 0.896649i \(0.354006\pi\)
\(384\) 0.370477 0.0189058
\(385\) 44.1585 2.25052
\(386\) 10.4616 0.532481
\(387\) −16.7899 −0.853478
\(388\) 0.956247 0.0485461
\(389\) 32.7406 1.66002 0.830008 0.557752i \(-0.188336\pi\)
0.830008 + 0.557752i \(0.188336\pi\)
\(390\) −1.05307 −0.0533245
\(391\) −13.0213 −0.658516
\(392\) 0.419304 0.0211781
\(393\) −3.01661 −0.152168
\(394\) 8.17270 0.411734
\(395\) 46.2430 2.32674
\(396\) 16.5345 0.830888
\(397\) −10.1894 −0.511392 −0.255696 0.966757i \(-0.582305\pi\)
−0.255696 + 0.966757i \(0.582305\pi\)
\(398\) −23.4576 −1.17582
\(399\) −6.84199 −0.342528
\(400\) 3.88265 0.194132
\(401\) 27.6214 1.37935 0.689673 0.724121i \(-0.257754\pi\)
0.689673 + 0.724121i \(0.257754\pi\)
\(402\) 4.49343 0.224112
\(403\) −7.79124 −0.388109
\(404\) 5.92549 0.294804
\(405\) 23.1979 1.15272
\(406\) 11.8833 0.589760
\(407\) 54.5099 2.70195
\(408\) 0.932373 0.0461593
\(409\) −32.3617 −1.60018 −0.800092 0.599877i \(-0.795216\pi\)
−0.800092 + 0.599877i \(0.795216\pi\)
\(410\) −7.35587 −0.363281
\(411\) 5.02477 0.247854
\(412\) −7.57015 −0.372955
\(413\) 2.60338 0.128104
\(414\) −14.8118 −0.727962
\(415\) −24.6528 −1.21016
\(416\) 0.953732 0.0467606
\(417\) 6.52578 0.319569
\(418\) −41.5807 −2.03378
\(419\) −20.7391 −1.01317 −0.506586 0.862189i \(-0.669093\pi\)
−0.506586 + 0.862189i \(0.669093\pi\)
\(420\) 2.83249 0.138211
\(421\) −2.74188 −0.133631 −0.0668155 0.997765i \(-0.521284\pi\)
−0.0668155 + 0.997765i \(0.521284\pi\)
\(422\) −23.7236 −1.15485
\(423\) 27.2238 1.32367
\(424\) 7.51226 0.364828
\(425\) 9.77140 0.473983
\(426\) −4.57676 −0.221745
\(427\) 15.0509 0.728366
\(428\) 16.0455 0.775589
\(429\) −2.04077 −0.0985294
\(430\) −17.4798 −0.842950
\(431\) −1.93294 −0.0931065 −0.0465533 0.998916i \(-0.514824\pi\)
−0.0465533 + 0.998916i \(0.514824\pi\)
\(432\) 2.17201 0.104501
\(433\) 4.35794 0.209429 0.104715 0.994502i \(-0.466607\pi\)
0.104715 + 0.994502i \(0.466607\pi\)
\(434\) 20.9564 1.00594
\(435\) −5.11487 −0.245239
\(436\) 2.50942 0.120179
\(437\) 37.2487 1.78185
\(438\) 0.241135 0.0115219
\(439\) 8.19465 0.391109 0.195555 0.980693i \(-0.437349\pi\)
0.195555 + 0.980693i \(0.437349\pi\)
\(440\) 17.2139 0.820638
\(441\) 1.20036 0.0571601
\(442\) 2.40024 0.114168
\(443\) −5.51425 −0.261990 −0.130995 0.991383i \(-0.541817\pi\)
−0.130995 + 0.991383i \(0.541817\pi\)
\(444\) 3.49647 0.165935
\(445\) 8.63867 0.409512
\(446\) 4.63802 0.219616
\(447\) −1.75367 −0.0829456
\(448\) −2.56529 −0.121198
\(449\) −19.1545 −0.903956 −0.451978 0.892029i \(-0.649281\pi\)
−0.451978 + 0.892029i \(0.649281\pi\)
\(450\) 11.1150 0.523968
\(451\) −14.2551 −0.671246
\(452\) −13.4589 −0.633055
\(453\) −6.86959 −0.322762
\(454\) −2.30295 −0.108083
\(455\) 7.29178 0.341844
\(456\) −2.66714 −0.124900
\(457\) −41.6610 −1.94882 −0.974409 0.224783i \(-0.927833\pi\)
−0.974409 + 0.224783i \(0.927833\pi\)
\(458\) −1.98890 −0.0929350
\(459\) 5.46627 0.255144
\(460\) −15.4205 −0.718982
\(461\) 28.7942 1.34108 0.670539 0.741874i \(-0.266063\pi\)
0.670539 + 0.741874i \(0.266063\pi\)
\(462\) 5.48913 0.255378
\(463\) 1.89638 0.0881321 0.0440660 0.999029i \(-0.485969\pi\)
0.0440660 + 0.999029i \(0.485969\pi\)
\(464\) 4.63236 0.215052
\(465\) −9.02012 −0.418298
\(466\) −5.58928 −0.258918
\(467\) −9.53499 −0.441227 −0.220613 0.975361i \(-0.570806\pi\)
−0.220613 + 0.975361i \(0.570806\pi\)
\(468\) 2.73029 0.126208
\(469\) −31.1138 −1.43670
\(470\) 28.3425 1.30734
\(471\) −2.45472 −0.113107
\(472\) 1.01485 0.0467122
\(473\) −33.8744 −1.55755
\(474\) 5.74826 0.264026
\(475\) −27.9520 −1.28253
\(476\) −6.45602 −0.295911
\(477\) 21.5057 0.984678
\(478\) 3.34553 0.153021
\(479\) 11.2845 0.515602 0.257801 0.966198i \(-0.417002\pi\)
0.257801 + 0.966198i \(0.417002\pi\)
\(480\) 1.10416 0.0503978
\(481\) 9.00108 0.410414
\(482\) −26.7139 −1.21678
\(483\) −4.91726 −0.223743
\(484\) 22.3591 1.01632
\(485\) 2.84998 0.129411
\(486\) 9.39967 0.426377
\(487\) 13.3515 0.605014 0.302507 0.953147i \(-0.402176\pi\)
0.302507 + 0.953147i \(0.402176\pi\)
\(488\) 5.86716 0.265594
\(489\) 2.72381 0.123175
\(490\) 1.24968 0.0564550
\(491\) 37.9516 1.71273 0.856365 0.516370i \(-0.172717\pi\)
0.856365 + 0.516370i \(0.172717\pi\)
\(492\) −0.914375 −0.0412232
\(493\) 11.6582 0.525058
\(494\) −6.86613 −0.308922
\(495\) 49.2789 2.21492
\(496\) 8.16921 0.366808
\(497\) 31.6908 1.42153
\(498\) −3.06448 −0.137323
\(499\) −27.6365 −1.23718 −0.618590 0.785714i \(-0.712296\pi\)
−0.618590 + 0.785714i \(0.712296\pi\)
\(500\) −3.33013 −0.148928
\(501\) 6.84842 0.305965
\(502\) 29.3076 1.30806
\(503\) 20.9442 0.933855 0.466927 0.884296i \(-0.345361\pi\)
0.466927 + 0.884296i \(0.345361\pi\)
\(504\) −7.34377 −0.327117
\(505\) 17.6602 0.785868
\(506\) −29.8836 −1.32849
\(507\) 4.47921 0.198929
\(508\) 2.35435 0.104458
\(509\) −25.5377 −1.13194 −0.565969 0.824426i \(-0.691498\pi\)
−0.565969 + 0.824426i \(0.691498\pi\)
\(510\) 2.77882 0.123048
\(511\) −1.66969 −0.0738627
\(512\) −1.00000 −0.0441942
\(513\) −15.6368 −0.690381
\(514\) −10.3803 −0.457854
\(515\) −22.5619 −0.994197
\(516\) −2.17283 −0.0956536
\(517\) 54.9254 2.41562
\(518\) −24.2105 −1.06375
\(519\) 5.63255 0.247242
\(520\) 2.84248 0.124651
\(521\) −38.8849 −1.70358 −0.851788 0.523886i \(-0.824482\pi\)
−0.851788 + 0.523886i \(0.824482\pi\)
\(522\) 13.2613 0.580430
\(523\) −25.0299 −1.09448 −0.547240 0.836976i \(-0.684321\pi\)
−0.547240 + 0.836976i \(0.684321\pi\)
\(524\) 8.14252 0.355708
\(525\) 3.68999 0.161044
\(526\) 8.83916 0.385406
\(527\) 20.5593 0.895578
\(528\) 2.13977 0.0931217
\(529\) 3.77022 0.163922
\(530\) 22.3894 0.972532
\(531\) 2.90525 0.126077
\(532\) 18.4681 0.800692
\(533\) −2.35391 −0.101959
\(534\) 1.07383 0.0464693
\(535\) 47.8217 2.06751
\(536\) −12.1288 −0.523883
\(537\) 3.30812 0.142756
\(538\) 1.00000 0.0431131
\(539\) 2.42179 0.104314
\(540\) 6.47342 0.278571
\(541\) 0.997803 0.0428989 0.0214494 0.999770i \(-0.493172\pi\)
0.0214494 + 0.999770i \(0.493172\pi\)
\(542\) −7.39604 −0.317687
\(543\) −3.32046 −0.142494
\(544\) −2.51668 −0.107902
\(545\) 7.47902 0.320366
\(546\) 0.906407 0.0387907
\(547\) −15.8011 −0.675608 −0.337804 0.941217i \(-0.609684\pi\)
−0.337804 + 0.941217i \(0.609684\pi\)
\(548\) −13.5630 −0.579382
\(549\) 16.7962 0.716843
\(550\) 22.4251 0.956211
\(551\) −33.3494 −1.42073
\(552\) −1.91685 −0.0815863
\(553\) −39.8026 −1.69258
\(554\) −13.5773 −0.576843
\(555\) 10.4208 0.442338
\(556\) −17.6145 −0.747023
\(557\) −36.3525 −1.54031 −0.770153 0.637860i \(-0.779820\pi\)
−0.770153 + 0.637860i \(0.779820\pi\)
\(558\) 23.3864 0.990025
\(559\) −5.59360 −0.236584
\(560\) −7.64552 −0.323082
\(561\) 5.38514 0.227361
\(562\) −13.5723 −0.572511
\(563\) −20.1622 −0.849736 −0.424868 0.905255i \(-0.639680\pi\)
−0.424868 + 0.905255i \(0.639680\pi\)
\(564\) 3.52312 0.148350
\(565\) −40.1127 −1.68755
\(566\) −20.5026 −0.861790
\(567\) −19.9671 −0.838538
\(568\) 12.3537 0.518349
\(569\) −16.0424 −0.672530 −0.336265 0.941767i \(-0.609164\pi\)
−0.336265 + 0.941767i \(0.609164\pi\)
\(570\) −7.94909 −0.332951
\(571\) −32.2187 −1.34831 −0.674156 0.738589i \(-0.735493\pi\)
−0.674156 + 0.738589i \(0.735493\pi\)
\(572\) 5.50850 0.230322
\(573\) 1.00466 0.0419701
\(574\) 6.33139 0.264267
\(575\) −20.0888 −0.837761
\(576\) −2.86275 −0.119281
\(577\) 42.6582 1.77589 0.887943 0.459954i \(-0.152134\pi\)
0.887943 + 0.459954i \(0.152134\pi\)
\(578\) 10.6663 0.443660
\(579\) 3.87578 0.161072
\(580\) 13.8062 0.573271
\(581\) 21.2193 0.880326
\(582\) 0.354267 0.0146849
\(583\) 43.3888 1.79698
\(584\) −0.650878 −0.0269335
\(585\) 8.13731 0.336436
\(586\) 26.0133 1.07460
\(587\) −13.8741 −0.572646 −0.286323 0.958133i \(-0.592433\pi\)
−0.286323 + 0.958133i \(0.592433\pi\)
\(588\) 0.155342 0.00640622
\(589\) −58.8119 −2.42330
\(590\) 3.02463 0.124522
\(591\) 3.02779 0.124547
\(592\) −9.43775 −0.387889
\(593\) 33.6435 1.38157 0.690786 0.723059i \(-0.257265\pi\)
0.690786 + 0.723059i \(0.257265\pi\)
\(594\) 12.5450 0.514726
\(595\) −19.2414 −0.788819
\(596\) 4.73354 0.193893
\(597\) −8.69051 −0.355679
\(598\) −4.93460 −0.201791
\(599\) 12.8677 0.525761 0.262881 0.964828i \(-0.415327\pi\)
0.262881 + 0.964828i \(0.415327\pi\)
\(600\) 1.43843 0.0587237
\(601\) 17.2785 0.704804 0.352402 0.935849i \(-0.385365\pi\)
0.352402 + 0.935849i \(0.385365\pi\)
\(602\) 15.0453 0.613201
\(603\) −34.7216 −1.41397
\(604\) 18.5426 0.754486
\(605\) 66.6384 2.70924
\(606\) 2.19526 0.0891762
\(607\) −36.7366 −1.49109 −0.745547 0.666453i \(-0.767812\pi\)
−0.745547 + 0.666453i \(0.767812\pi\)
\(608\) 7.19922 0.291967
\(609\) 4.40250 0.178398
\(610\) 17.4863 0.708001
\(611\) 9.06970 0.366921
\(612\) −7.20463 −0.291230
\(613\) −35.5029 −1.43395 −0.716975 0.697099i \(-0.754474\pi\)
−0.716975 + 0.697099i \(0.754474\pi\)
\(614\) 17.3993 0.702179
\(615\) −2.72518 −0.109890
\(616\) −14.8164 −0.596970
\(617\) 5.97403 0.240505 0.120253 0.992743i \(-0.461630\pi\)
0.120253 + 0.992743i \(0.461630\pi\)
\(618\) −2.80457 −0.112816
\(619\) 1.54631 0.0621513 0.0310757 0.999517i \(-0.490107\pi\)
0.0310757 + 0.999517i \(0.490107\pi\)
\(620\) 24.3473 0.977812
\(621\) −11.2380 −0.450965
\(622\) 13.4740 0.540256
\(623\) −7.43552 −0.297898
\(624\) 0.353336 0.0141447
\(625\) −29.3383 −1.17353
\(626\) 9.89195 0.395362
\(627\) −15.4047 −0.615205
\(628\) 6.62583 0.264399
\(629\) −23.7518 −0.947047
\(630\) −21.8872 −0.872007
\(631\) 33.5953 1.33741 0.668705 0.743528i \(-0.266849\pi\)
0.668705 + 0.743528i \(0.266849\pi\)
\(632\) −15.5158 −0.617187
\(633\) −8.78905 −0.349333
\(634\) −19.2200 −0.763324
\(635\) 7.01686 0.278456
\(636\) 2.78312 0.110358
\(637\) 0.399904 0.0158448
\(638\) 26.7553 1.05925
\(639\) 35.3655 1.39904
\(640\) −2.98038 −0.117810
\(641\) 22.6820 0.895884 0.447942 0.894063i \(-0.352157\pi\)
0.447942 + 0.894063i \(0.352157\pi\)
\(642\) 5.94449 0.234610
\(643\) 35.9742 1.41868 0.709341 0.704865i \(-0.248993\pi\)
0.709341 + 0.704865i \(0.248993\pi\)
\(644\) 13.2728 0.523021
\(645\) −6.47586 −0.254987
\(646\) 18.1182 0.712849
\(647\) −29.8868 −1.17497 −0.587485 0.809235i \(-0.699882\pi\)
−0.587485 + 0.809235i \(0.699882\pi\)
\(648\) −7.78356 −0.305767
\(649\) 5.86149 0.230084
\(650\) 3.70301 0.145244
\(651\) 7.76385 0.304289
\(652\) −7.35217 −0.287933
\(653\) −48.1187 −1.88303 −0.941516 0.336968i \(-0.890598\pi\)
−0.941516 + 0.336968i \(0.890598\pi\)
\(654\) 0.929682 0.0363534
\(655\) 24.2678 0.948220
\(656\) 2.46810 0.0963632
\(657\) −1.86330 −0.0726942
\(658\) −24.3951 −0.951019
\(659\) −27.3372 −1.06491 −0.532454 0.846459i \(-0.678730\pi\)
−0.532454 + 0.846459i \(0.678730\pi\)
\(660\) 6.37734 0.248237
\(661\) −20.6868 −0.804623 −0.402311 0.915503i \(-0.631793\pi\)
−0.402311 + 0.915503i \(0.631793\pi\)
\(662\) 10.6168 0.412632
\(663\) 0.889234 0.0345350
\(664\) 8.27171 0.321005
\(665\) 55.0418 2.13443
\(666\) −27.0179 −1.04692
\(667\) −23.9678 −0.928037
\(668\) −18.4854 −0.715223
\(669\) 1.71828 0.0664325
\(670\) −36.1483 −1.39653
\(671\) 33.8871 1.30820
\(672\) −0.950379 −0.0366617
\(673\) −3.68048 −0.141872 −0.0709360 0.997481i \(-0.522599\pi\)
−0.0709360 + 0.997481i \(0.522599\pi\)
\(674\) 4.73621 0.182432
\(675\) 8.43316 0.324593
\(676\) −12.0904 −0.465015
\(677\) −17.3358 −0.666268 −0.333134 0.942880i \(-0.608106\pi\)
−0.333134 + 0.942880i \(0.608106\pi\)
\(678\) −4.98622 −0.191495
\(679\) −2.45305 −0.0941393
\(680\) −7.50067 −0.287637
\(681\) −0.853190 −0.0326943
\(682\) 47.1832 1.80674
\(683\) 2.26489 0.0866636 0.0433318 0.999061i \(-0.486203\pi\)
0.0433318 + 0.999061i \(0.486203\pi\)
\(684\) 20.6095 0.788025
\(685\) −40.4228 −1.54448
\(686\) −19.0326 −0.726670
\(687\) −0.736840 −0.0281122
\(688\) 5.86496 0.223599
\(689\) 7.16468 0.272953
\(690\) −5.71292 −0.217487
\(691\) −34.9262 −1.32865 −0.664327 0.747442i \(-0.731282\pi\)
−0.664327 + 0.747442i \(0.731282\pi\)
\(692\) −15.2035 −0.577951
\(693\) −42.4156 −1.61124
\(694\) 9.70333 0.368333
\(695\) −52.4980 −1.99136
\(696\) 1.71618 0.0650517
\(697\) 6.21143 0.235275
\(698\) 34.9874 1.32429
\(699\) −2.07070 −0.0783210
\(700\) −9.96011 −0.376457
\(701\) −8.67601 −0.327688 −0.163844 0.986486i \(-0.552389\pi\)
−0.163844 + 0.986486i \(0.552389\pi\)
\(702\) 2.07152 0.0781844
\(703\) 67.9444 2.56257
\(704\) −5.77573 −0.217681
\(705\) 10.5002 0.395461
\(706\) −17.9950 −0.677251
\(707\) −15.2006 −0.571676
\(708\) 0.375978 0.0141301
\(709\) −52.4850 −1.97111 −0.985557 0.169342i \(-0.945836\pi\)
−0.985557 + 0.169342i \(0.945836\pi\)
\(710\) 36.8187 1.38178
\(711\) −44.4179 −1.66580
\(712\) −2.89852 −0.108626
\(713\) −42.2675 −1.58293
\(714\) −2.39180 −0.0895110
\(715\) 16.4174 0.613976
\(716\) −8.92937 −0.333706
\(717\) 1.23944 0.0462878
\(718\) 8.64349 0.322572
\(719\) 12.2225 0.455822 0.227911 0.973682i \(-0.426810\pi\)
0.227911 + 0.973682i \(0.426810\pi\)
\(720\) −8.53207 −0.317971
\(721\) 19.4196 0.723224
\(722\) −32.8287 −1.22176
\(723\) −9.89686 −0.368068
\(724\) 8.96265 0.333094
\(725\) 17.9858 0.667977
\(726\) 8.28351 0.307430
\(727\) 32.8117 1.21692 0.608460 0.793585i \(-0.291788\pi\)
0.608460 + 0.793585i \(0.291788\pi\)
\(728\) −2.44660 −0.0906769
\(729\) −19.8683 −0.735864
\(730\) −1.93986 −0.0717975
\(731\) 14.7602 0.545927
\(732\) 2.17365 0.0803402
\(733\) −11.8567 −0.437937 −0.218968 0.975732i \(-0.570269\pi\)
−0.218968 + 0.975732i \(0.570269\pi\)
\(734\) 25.4599 0.939741
\(735\) 0.462979 0.0170772
\(736\) 5.17399 0.190716
\(737\) −70.0525 −2.58042
\(738\) 7.06555 0.260087
\(739\) 23.7096 0.872173 0.436086 0.899905i \(-0.356364\pi\)
0.436086 + 0.899905i \(0.356364\pi\)
\(740\) −28.1280 −1.03401
\(741\) −2.54374 −0.0934466
\(742\) −19.2711 −0.707464
\(743\) 16.3166 0.598600 0.299300 0.954159i \(-0.403247\pi\)
0.299300 + 0.954159i \(0.403247\pi\)
\(744\) 3.02650 0.110957
\(745\) 14.1077 0.516867
\(746\) 5.63673 0.206375
\(747\) 23.6798 0.866399
\(748\) −14.5357 −0.531477
\(749\) −41.1613 −1.50400
\(750\) −1.23374 −0.0450496
\(751\) 30.5188 1.11365 0.556824 0.830630i \(-0.312020\pi\)
0.556824 + 0.830630i \(0.312020\pi\)
\(752\) −9.50969 −0.346783
\(753\) 10.8578 0.395680
\(754\) 4.41803 0.160895
\(755\) 55.2639 2.01126
\(756\) −5.57183 −0.202646
\(757\) 21.0756 0.766007 0.383003 0.923747i \(-0.374890\pi\)
0.383003 + 0.923747i \(0.374890\pi\)
\(758\) 16.2741 0.591101
\(759\) −11.0712 −0.401858
\(760\) 21.4564 0.778305
\(761\) −9.37156 −0.339719 −0.169859 0.985468i \(-0.554331\pi\)
−0.169859 + 0.985468i \(0.554331\pi\)
\(762\) 0.872233 0.0315977
\(763\) −6.43738 −0.233049
\(764\) −2.71179 −0.0981091
\(765\) −21.4725 −0.776340
\(766\) −17.3292 −0.626131
\(767\) 0.967894 0.0349486
\(768\) −0.370477 −0.0133684
\(769\) 26.9367 0.971363 0.485681 0.874136i \(-0.338572\pi\)
0.485681 + 0.874136i \(0.338572\pi\)
\(770\) −44.1585 −1.59136
\(771\) −3.84565 −0.138498
\(772\) −10.4616 −0.376521
\(773\) −14.7111 −0.529121 −0.264560 0.964369i \(-0.585227\pi\)
−0.264560 + 0.964369i \(0.585227\pi\)
\(774\) 16.7899 0.603500
\(775\) 31.7182 1.13935
\(776\) −0.956247 −0.0343273
\(777\) −8.96944 −0.321777
\(778\) −32.7406 −1.17381
\(779\) −17.7684 −0.636619
\(780\) 1.05307 0.0377061
\(781\) 71.3516 2.55316
\(782\) 13.0213 0.465641
\(783\) 10.0615 0.359570
\(784\) −0.419304 −0.0149751
\(785\) 19.7475 0.704818
\(786\) 3.01661 0.107599
\(787\) 3.73905 0.133283 0.0666414 0.997777i \(-0.478772\pi\)
0.0666414 + 0.997777i \(0.478772\pi\)
\(788\) −8.17270 −0.291140
\(789\) 3.27471 0.116583
\(790\) −46.2430 −1.64525
\(791\) 34.5260 1.22760
\(792\) −16.5345 −0.587526
\(793\) 5.59569 0.198709
\(794\) 10.1894 0.361609
\(795\) 8.29474 0.294184
\(796\) 23.4576 0.831433
\(797\) 7.11196 0.251919 0.125959 0.992035i \(-0.459799\pi\)
0.125959 + 0.992035i \(0.459799\pi\)
\(798\) 6.84199 0.242204
\(799\) −23.9329 −0.846685
\(800\) −3.88265 −0.137272
\(801\) −8.29772 −0.293185
\(802\) −27.6214 −0.975345
\(803\) −3.75929 −0.132663
\(804\) −4.49343 −0.158471
\(805\) 39.5579 1.39423
\(806\) 7.79124 0.274435
\(807\) 0.370477 0.0130414
\(808\) −5.92549 −0.208458
\(809\) −10.2508 −0.360398 −0.180199 0.983630i \(-0.557674\pi\)
−0.180199 + 0.983630i \(0.557674\pi\)
\(810\) −23.1979 −0.815093
\(811\) −17.6750 −0.620655 −0.310327 0.950630i \(-0.600439\pi\)
−0.310327 + 0.950630i \(0.600439\pi\)
\(812\) −11.8833 −0.417023
\(813\) −2.74006 −0.0960981
\(814\) −54.5099 −1.91057
\(815\) −21.9123 −0.767553
\(816\) −0.932373 −0.0326396
\(817\) −42.2231 −1.47720
\(818\) 32.3617 1.13150
\(819\) −7.00399 −0.244739
\(820\) 7.35587 0.256878
\(821\) 14.4456 0.504153 0.252077 0.967707i \(-0.418886\pi\)
0.252077 + 0.967707i \(0.418886\pi\)
\(822\) −5.02477 −0.175259
\(823\) −5.39437 −0.188036 −0.0940179 0.995571i \(-0.529971\pi\)
−0.0940179 + 0.995571i \(0.529971\pi\)
\(824\) 7.57015 0.263719
\(825\) 8.30799 0.289247
\(826\) −2.60338 −0.0905831
\(827\) −40.7336 −1.41645 −0.708223 0.705989i \(-0.750503\pi\)
−0.708223 + 0.705989i \(0.750503\pi\)
\(828\) 14.8118 0.514747
\(829\) −15.9379 −0.553547 −0.276773 0.960935i \(-0.589265\pi\)
−0.276773 + 0.960935i \(0.589265\pi\)
\(830\) 24.6528 0.855712
\(831\) −5.03007 −0.174491
\(832\) −0.953732 −0.0330647
\(833\) −1.05526 −0.0365625
\(834\) −6.52578 −0.225969
\(835\) −55.0936 −1.90659
\(836\) 41.5807 1.43810
\(837\) 17.7436 0.613309
\(838\) 20.7391 0.716421
\(839\) 47.0290 1.62362 0.811811 0.583921i \(-0.198482\pi\)
0.811811 + 0.583921i \(0.198482\pi\)
\(840\) −2.83249 −0.0977302
\(841\) −7.54122 −0.260042
\(842\) 2.74188 0.0944914
\(843\) −5.02821 −0.173181
\(844\) 23.7236 0.816600
\(845\) −36.0339 −1.23961
\(846\) −27.2238 −0.935975
\(847\) −57.3574 −1.97082
\(848\) −7.51226 −0.257972
\(849\) −7.59575 −0.260685
\(850\) −9.77140 −0.335156
\(851\) 48.8308 1.67390
\(852\) 4.57676 0.156797
\(853\) 5.50121 0.188358 0.0941790 0.995555i \(-0.469977\pi\)
0.0941790 + 0.995555i \(0.469977\pi\)
\(854\) −15.0509 −0.515032
\(855\) 61.4242 2.10066
\(856\) −16.0455 −0.548424
\(857\) −6.74343 −0.230351 −0.115176 0.993345i \(-0.536743\pi\)
−0.115176 + 0.993345i \(0.536743\pi\)
\(858\) 2.04077 0.0696708
\(859\) 38.3733 1.30928 0.654641 0.755940i \(-0.272820\pi\)
0.654641 + 0.755940i \(0.272820\pi\)
\(860\) 17.4798 0.596056
\(861\) 2.34563 0.0799390
\(862\) 1.93294 0.0658363
\(863\) 27.5099 0.936449 0.468224 0.883610i \(-0.344894\pi\)
0.468224 + 0.883610i \(0.344894\pi\)
\(864\) −2.17201 −0.0738934
\(865\) −45.3122 −1.54066
\(866\) −4.35794 −0.148089
\(867\) 3.95162 0.134204
\(868\) −20.9564 −0.711306
\(869\) −89.6153 −3.03999
\(870\) 5.11487 0.173410
\(871\) −11.5676 −0.391953
\(872\) −2.50942 −0.0849797
\(873\) −2.73749 −0.0926501
\(874\) −37.2487 −1.25996
\(875\) 8.54273 0.288797
\(876\) −0.241135 −0.00814720
\(877\) 40.4307 1.36525 0.682625 0.730769i \(-0.260838\pi\)
0.682625 + 0.730769i \(0.260838\pi\)
\(878\) −8.19465 −0.276556
\(879\) 9.63733 0.325059
\(880\) −17.2139 −0.580279
\(881\) 14.9852 0.504866 0.252433 0.967614i \(-0.418769\pi\)
0.252433 + 0.967614i \(0.418769\pi\)
\(882\) −1.20036 −0.0404183
\(883\) 45.8998 1.54465 0.772325 0.635227i \(-0.219094\pi\)
0.772325 + 0.635227i \(0.219094\pi\)
\(884\) −2.40024 −0.0807289
\(885\) 1.12056 0.0376671
\(886\) 5.51425 0.185255
\(887\) 38.4897 1.29236 0.646179 0.763186i \(-0.276366\pi\)
0.646179 + 0.763186i \(0.276366\pi\)
\(888\) −3.49647 −0.117334
\(889\) −6.03959 −0.202561
\(890\) −8.63867 −0.289569
\(891\) −44.9557 −1.50607
\(892\) −4.63802 −0.155292
\(893\) 68.4623 2.29100
\(894\) 1.75367 0.0586514
\(895\) −26.6129 −0.889570
\(896\) 2.56529 0.0857002
\(897\) −1.82816 −0.0610404
\(898\) 19.1545 0.639193
\(899\) 37.8427 1.26213
\(900\) −11.1150 −0.370501
\(901\) −18.9060 −0.629849
\(902\) 14.2551 0.474643
\(903\) 5.57393 0.185489
\(904\) 13.4589 0.447637
\(905\) 26.7121 0.887940
\(906\) 6.86959 0.228227
\(907\) −3.30242 −0.109655 −0.0548275 0.998496i \(-0.517461\pi\)
−0.0548275 + 0.998496i \(0.517461\pi\)
\(908\) 2.30295 0.0764261
\(909\) −16.9632 −0.562633
\(910\) −7.29178 −0.241720
\(911\) 0.593599 0.0196668 0.00983341 0.999952i \(-0.496870\pi\)
0.00983341 + 0.999952i \(0.496870\pi\)
\(912\) 2.66714 0.0883180
\(913\) 47.7752 1.58113
\(914\) 41.6610 1.37802
\(915\) 6.47828 0.214165
\(916\) 1.98890 0.0657150
\(917\) −20.8879 −0.689779
\(918\) −5.46627 −0.180414
\(919\) 7.37035 0.243125 0.121563 0.992584i \(-0.461209\pi\)
0.121563 + 0.992584i \(0.461209\pi\)
\(920\) 15.4205 0.508397
\(921\) 6.44604 0.212404
\(922\) −28.7942 −0.948285
\(923\) 11.7821 0.387813
\(924\) −5.48913 −0.180579
\(925\) −36.6435 −1.20483
\(926\) −1.89638 −0.0623188
\(927\) 21.6714 0.711783
\(928\) −4.63236 −0.152065
\(929\) 0.386422 0.0126781 0.00633905 0.999980i \(-0.497982\pi\)
0.00633905 + 0.999980i \(0.497982\pi\)
\(930\) 9.02012 0.295782
\(931\) 3.01866 0.0989327
\(932\) 5.58928 0.183083
\(933\) 4.99179 0.163424
\(934\) 9.53499 0.311994
\(935\) −43.3218 −1.41678
\(936\) −2.73029 −0.0892424
\(937\) −29.0617 −0.949405 −0.474702 0.880146i \(-0.657444\pi\)
−0.474702 + 0.880146i \(0.657444\pi\)
\(938\) 31.1138 1.01590
\(939\) 3.66474 0.119594
\(940\) −28.3425 −0.924429
\(941\) −22.0276 −0.718081 −0.359040 0.933322i \(-0.616896\pi\)
−0.359040 + 0.933322i \(0.616896\pi\)
\(942\) 2.45472 0.0799790
\(943\) −12.7699 −0.415847
\(944\) −1.01485 −0.0330305
\(945\) −16.6062 −0.540199
\(946\) 33.8744 1.10135
\(947\) −6.64851 −0.216047 −0.108024 0.994148i \(-0.534452\pi\)
−0.108024 + 0.994148i \(0.534452\pi\)
\(948\) −5.74826 −0.186695
\(949\) −0.620763 −0.0201508
\(950\) 27.9520 0.906884
\(951\) −7.12056 −0.230900
\(952\) 6.45602 0.209241
\(953\) −57.4933 −1.86239 −0.931195 0.364521i \(-0.881233\pi\)
−0.931195 + 0.364521i \(0.881233\pi\)
\(954\) −21.5057 −0.696273
\(955\) −8.08216 −0.261532
\(956\) −3.34553 −0.108202
\(957\) 9.91221 0.320416
\(958\) −11.2845 −0.364586
\(959\) 34.7929 1.12352
\(960\) −1.10416 −0.0356366
\(961\) 35.7360 1.15277
\(962\) −9.00108 −0.290207
\(963\) −45.9342 −1.48021
\(964\) 26.7139 0.860395
\(965\) −31.1795 −1.00370
\(966\) 4.91726 0.158210
\(967\) 44.0592 1.41685 0.708424 0.705788i \(-0.249407\pi\)
0.708424 + 0.705788i \(0.249407\pi\)
\(968\) −22.3591 −0.718647
\(969\) 6.71236 0.215632
\(970\) −2.84998 −0.0915072
\(971\) 25.8900 0.830850 0.415425 0.909627i \(-0.363633\pi\)
0.415425 + 0.909627i \(0.363633\pi\)
\(972\) −9.39967 −0.301494
\(973\) 45.1864 1.44861
\(974\) −13.3515 −0.427810
\(975\) 1.37188 0.0439353
\(976\) −5.86716 −0.187803
\(977\) 16.1799 0.517642 0.258821 0.965925i \(-0.416666\pi\)
0.258821 + 0.965925i \(0.416666\pi\)
\(978\) −2.72381 −0.0870979
\(979\) −16.7410 −0.535046
\(980\) −1.24968 −0.0399197
\(981\) −7.18383 −0.229362
\(982\) −37.9516 −1.21108
\(983\) −27.7004 −0.883504 −0.441752 0.897137i \(-0.645643\pi\)
−0.441752 + 0.897137i \(0.645643\pi\)
\(984\) 0.914375 0.0291492
\(985\) −24.3577 −0.776101
\(986\) −11.6582 −0.371272
\(987\) −9.03781 −0.287677
\(988\) 6.86613 0.218441
\(989\) −30.3453 −0.964923
\(990\) −49.2789 −1.56619
\(991\) −15.6306 −0.496523 −0.248262 0.968693i \(-0.579859\pi\)
−0.248262 + 0.968693i \(0.579859\pi\)
\(992\) −8.16921 −0.259373
\(993\) 3.93326 0.124818
\(994\) −31.6908 −1.00517
\(995\) 69.9126 2.21638
\(996\) 3.06448 0.0971017
\(997\) 59.0197 1.86917 0.934586 0.355737i \(-0.115770\pi\)
0.934586 + 0.355737i \(0.115770\pi\)
\(998\) 27.6365 0.874819
\(999\) −20.4989 −0.648557
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 538.2.a.d.1.4 7
3.2 odd 2 4842.2.a.o.1.1 7
4.3 odd 2 4304.2.a.i.1.4 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.d.1.4 7 1.1 even 1 trivial
4304.2.a.i.1.4 7 4.3 odd 2
4842.2.a.o.1.1 7 3.2 odd 2