Properties

Label 920.2.q.a.873.34
Level $920$
Weight $2$
Character 920.873
Analytic conductor $7.346$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [920,2,Mod(137,920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("920.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.q (of order \(4\), degree \(2\), 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: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 873.34
Character \(\chi\) \(=\) 920.873
Dual form 920.2.q.a.137.34

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.79740 + 1.79740i) q^{3} +(0.878232 + 2.05638i) q^{5} +(-1.78887 - 1.78887i) q^{7} +3.46127i q^{9} +O(q^{10})\) \(q+(1.79740 + 1.79740i) q^{3} +(0.878232 + 2.05638i) q^{5} +(-1.78887 - 1.78887i) q^{7} +3.46127i q^{9} +0.939750i q^{11} +(2.75254 + 2.75254i) q^{13} +(-2.11760 + 5.27467i) q^{15} +(-0.360642 - 0.360642i) q^{17} +0.0967405 q^{19} -6.43060i q^{21} +(-1.60123 + 4.52063i) q^{23} +(-3.45742 + 3.61196i) q^{25} +(-0.829091 + 0.829091i) q^{27} +7.97211i q^{29} +1.56913 q^{31} +(-1.68910 + 1.68910i) q^{33} +(2.10755 - 5.24963i) q^{35} +(-0.364907 - 0.364907i) q^{37} +9.89483i q^{39} -0.742731 q^{41} +(4.73328 - 4.73328i) q^{43} +(-7.11770 + 3.03980i) q^{45} +(2.67229 - 2.67229i) q^{47} -0.599920i q^{49} -1.29644i q^{51} +(-6.26494 + 6.26494i) q^{53} +(-1.93248 + 0.825319i) q^{55} +(0.173881 + 0.173881i) q^{57} -14.3575i q^{59} -11.0607i q^{61} +(6.19175 - 6.19175i) q^{63} +(-3.24291 + 8.07766i) q^{65} +(-2.54924 - 2.54924i) q^{67} +(-11.0034 + 5.24732i) q^{69} -8.30693 q^{71} +(-4.23791 - 4.23791i) q^{73} +(-12.7065 + 0.277784i) q^{75} +(1.68109 - 1.68109i) q^{77} +3.19554 q^{79} +7.40341 q^{81} +(6.25214 - 6.25214i) q^{83} +(0.424891 - 1.05835i) q^{85} +(-14.3290 + 14.3290i) q^{87} +16.8361 q^{89} -9.84786i q^{91} +(2.82034 + 2.82034i) q^{93} +(0.0849607 + 0.198935i) q^{95} +(6.66643 + 6.66643i) q^{97} -3.25273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q+O(q^{10}) \) Copy content Toggle raw display \( 72 q + 20 q^{23} - 16 q^{25} + 24 q^{27} + 16 q^{31} + 32 q^{41} + 32 q^{47} + 40 q^{55} - 16 q^{73} + 8 q^{75} + 48 q^{77} - 40 q^{81} - 40 q^{85} + 88 q^{87} - 72 q^{93} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/920\mathbb{Z}\right)^\times\).

\(n\) \(231\) \(281\) \(461\) \(737\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.79740 + 1.79740i 1.03773 + 1.03773i 0.999260 + 0.0384679i \(0.0122477\pi\)
0.0384679 + 0.999260i \(0.487752\pi\)
\(4\) 0 0
\(5\) 0.878232 + 2.05638i 0.392758 + 0.919642i
\(6\) 0 0
\(7\) −1.78887 1.78887i −0.676128 0.676128i 0.282994 0.959122i \(-0.408672\pi\)
−0.959122 + 0.282994i \(0.908672\pi\)
\(8\) 0 0
\(9\) 3.46127i 1.15376i
\(10\) 0 0
\(11\) 0.939750i 0.283345i 0.989914 + 0.141673i \(0.0452480\pi\)
−0.989914 + 0.141673i \(0.954752\pi\)
\(12\) 0 0
\(13\) 2.75254 + 2.75254i 0.763418 + 0.763418i 0.976939 0.213520i \(-0.0684929\pi\)
−0.213520 + 0.976939i \(0.568493\pi\)
\(14\) 0 0
\(15\) −2.11760 + 5.27467i −0.546763 + 1.36191i
\(16\) 0 0
\(17\) −0.360642 0.360642i −0.0874686 0.0874686i 0.662019 0.749487i \(-0.269700\pi\)
−0.749487 + 0.662019i \(0.769700\pi\)
\(18\) 0 0
\(19\) 0.0967405 0.0221938 0.0110969 0.999938i \(-0.496468\pi\)
0.0110969 + 0.999938i \(0.496468\pi\)
\(20\) 0 0
\(21\) 6.43060i 1.40327i
\(22\) 0 0
\(23\) −1.60123 + 4.52063i −0.333879 + 0.942616i
\(24\) 0 0
\(25\) −3.45742 + 3.61196i −0.691483 + 0.722393i
\(26\) 0 0
\(27\) −0.829091 + 0.829091i −0.159559 + 0.159559i
\(28\) 0 0
\(29\) 7.97211i 1.48038i 0.672396 + 0.740191i \(0.265265\pi\)
−0.672396 + 0.740191i \(0.734735\pi\)
\(30\) 0 0
\(31\) 1.56913 0.281823 0.140912 0.990022i \(-0.454997\pi\)
0.140912 + 0.990022i \(0.454997\pi\)
\(32\) 0 0
\(33\) −1.68910 + 1.68910i −0.294035 + 0.294035i
\(34\) 0 0
\(35\) 2.10755 5.24963i 0.356241 0.887350i
\(36\) 0 0
\(37\) −0.364907 0.364907i −0.0599903 0.0599903i 0.676475 0.736465i \(-0.263507\pi\)
−0.736465 + 0.676475i \(0.763507\pi\)
\(38\) 0 0
\(39\) 9.89483i 1.58444i
\(40\) 0 0
\(41\) −0.742731 −0.115995 −0.0579975 0.998317i \(-0.518472\pi\)
−0.0579975 + 0.998317i \(0.518472\pi\)
\(42\) 0 0
\(43\) 4.73328 4.73328i 0.721818 0.721818i −0.247157 0.968975i \(-0.579496\pi\)
0.968975 + 0.247157i \(0.0794964\pi\)
\(44\) 0 0
\(45\) −7.11770 + 3.03980i −1.06104 + 0.453147i
\(46\) 0 0
\(47\) 2.67229 2.67229i 0.389793 0.389793i −0.484821 0.874614i \(-0.661115\pi\)
0.874614 + 0.484821i \(0.161115\pi\)
\(48\) 0 0
\(49\) 0.599920i 0.0857029i
\(50\) 0 0
\(51\) 1.29644i 0.181537i
\(52\) 0 0
\(53\) −6.26494 + 6.26494i −0.860556 + 0.860556i −0.991403 0.130846i \(-0.958231\pi\)
0.130846 + 0.991403i \(0.458231\pi\)
\(54\) 0 0
\(55\) −1.93248 + 0.825319i −0.260576 + 0.111286i
\(56\) 0 0
\(57\) 0.173881 + 0.173881i 0.0230311 + 0.0230311i
\(58\) 0 0
\(59\) 14.3575i 1.86919i −0.355712 0.934596i \(-0.615762\pi\)
0.355712 0.934596i \(-0.384238\pi\)
\(60\) 0 0
\(61\) 11.0607i 1.41618i −0.706124 0.708089i \(-0.749558\pi\)
0.706124 0.708089i \(-0.250442\pi\)
\(62\) 0 0
\(63\) 6.19175 6.19175i 0.780087 0.780087i
\(64\) 0 0
\(65\) −3.24291 + 8.07766i −0.402233 + 1.00191i
\(66\) 0 0
\(67\) −2.54924 2.54924i −0.311439 0.311439i 0.534028 0.845467i \(-0.320678\pi\)
−0.845467 + 0.534028i \(0.820678\pi\)
\(68\) 0 0
\(69\) −11.0034 + 5.24732i −1.32465 + 0.631703i
\(70\) 0 0
\(71\) −8.30693 −0.985851 −0.492926 0.870071i \(-0.664073\pi\)
−0.492926 + 0.870071i \(0.664073\pi\)
\(72\) 0 0
\(73\) −4.23791 4.23791i −0.496010 0.496010i 0.414183 0.910193i \(-0.364067\pi\)
−0.910193 + 0.414183i \(0.864067\pi\)
\(74\) 0 0
\(75\) −12.7065 + 0.277784i −1.46722 + 0.0320757i
\(76\) 0 0
\(77\) 1.68109 1.68109i 0.191577 0.191577i
\(78\) 0 0
\(79\) 3.19554 0.359526 0.179763 0.983710i \(-0.442467\pi\)
0.179763 + 0.983710i \(0.442467\pi\)
\(80\) 0 0
\(81\) 7.40341 0.822601
\(82\) 0 0
\(83\) 6.25214 6.25214i 0.686262 0.686262i −0.275142 0.961404i \(-0.588725\pi\)
0.961404 + 0.275142i \(0.0887248\pi\)
\(84\) 0 0
\(85\) 0.424891 1.05835i 0.0460859 0.114794i
\(86\) 0 0
\(87\) −14.3290 + 14.3290i −1.53623 + 1.53623i
\(88\) 0 0
\(89\) 16.8361 1.78462 0.892309 0.451425i \(-0.149084\pi\)
0.892309 + 0.451425i \(0.149084\pi\)
\(90\) 0 0
\(91\) 9.84786i 1.03234i
\(92\) 0 0
\(93\) 2.82034 + 2.82034i 0.292456 + 0.292456i
\(94\) 0 0
\(95\) 0.0849607 + 0.198935i 0.00871678 + 0.0204103i
\(96\) 0 0
\(97\) 6.66643 + 6.66643i 0.676873 + 0.676873i 0.959291 0.282418i \(-0.0911366\pi\)
−0.282418 + 0.959291i \(0.591137\pi\)
\(98\) 0 0
\(99\) −3.25273 −0.326912
\(100\) 0 0
\(101\) 6.37689 0.634524 0.317262 0.948338i \(-0.397237\pi\)
0.317262 + 0.948338i \(0.397237\pi\)
\(102\) 0 0
\(103\) 0.118775 0.118775i 0.0117032 0.0117032i −0.701231 0.712934i \(-0.747366\pi\)
0.712934 + 0.701231i \(0.247366\pi\)
\(104\) 0 0
\(105\) 13.2238 5.64757i 1.29051 0.551146i
\(106\) 0 0
\(107\) 7.23266 + 7.23266i 0.699208 + 0.699208i 0.964240 0.265032i \(-0.0853825\pi\)
−0.265032 + 0.964240i \(0.585382\pi\)
\(108\) 0 0
\(109\) 10.9817 1.05185 0.525927 0.850529i \(-0.323718\pi\)
0.525927 + 0.850529i \(0.323718\pi\)
\(110\) 0 0
\(111\) 1.31176i 0.124507i
\(112\) 0 0
\(113\) −11.3798 + 11.3798i −1.07052 + 1.07052i −0.0732051 + 0.997317i \(0.523323\pi\)
−0.997317 + 0.0732051i \(0.976677\pi\)
\(114\) 0 0
\(115\) −10.7024 + 0.677422i −0.998003 + 0.0631700i
\(116\) 0 0
\(117\) −9.52731 + 9.52731i −0.880800 + 0.880800i
\(118\) 0 0
\(119\) 1.29028i 0.118280i
\(120\) 0 0
\(121\) 10.1169 0.919716
\(122\) 0 0
\(123\) −1.33498 1.33498i −0.120371 0.120371i
\(124\) 0 0
\(125\) −10.4640 3.93762i −0.935928 0.352192i
\(126\) 0 0
\(127\) −0.0958360 + 0.0958360i −0.00850407 + 0.00850407i −0.711346 0.702842i \(-0.751914\pi\)
0.702842 + 0.711346i \(0.251914\pi\)
\(128\) 0 0
\(129\) 17.0152 1.49810
\(130\) 0 0
\(131\) 4.25882 0.372094 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(132\) 0 0
\(133\) −0.173056 0.173056i −0.0150058 0.0150058i
\(134\) 0 0
\(135\) −2.43306 0.976793i −0.209405 0.0840690i
\(136\) 0 0
\(137\) −1.57409 1.57409i −0.134483 0.134483i 0.636661 0.771144i \(-0.280315\pi\)
−0.771144 + 0.636661i \(0.780315\pi\)
\(138\) 0 0
\(139\) 13.6617i 1.15877i −0.815053 0.579387i \(-0.803292\pi\)
0.815053 0.579387i \(-0.196708\pi\)
\(140\) 0 0
\(141\) 9.60632 0.808998
\(142\) 0 0
\(143\) −2.58670 + 2.58670i −0.216311 + 0.216311i
\(144\) 0 0
\(145\) −16.3937 + 7.00136i −1.36142 + 0.581431i
\(146\) 0 0
\(147\) 1.07829 1.07829i 0.0889363 0.0889363i
\(148\) 0 0
\(149\) 0.460684 0.0377407 0.0188704 0.999822i \(-0.493993\pi\)
0.0188704 + 0.999822i \(0.493993\pi\)
\(150\) 0 0
\(151\) 2.79803 0.227700 0.113850 0.993498i \(-0.463682\pi\)
0.113850 + 0.993498i \(0.463682\pi\)
\(152\) 0 0
\(153\) 1.24828 1.24828i 0.100918 0.100918i
\(154\) 0 0
\(155\) 1.37806 + 3.22672i 0.110688 + 0.259177i
\(156\) 0 0
\(157\) 16.4704 + 16.4704i 1.31448 + 1.31448i 0.918075 + 0.396406i \(0.129743\pi\)
0.396406 + 0.918075i \(0.370257\pi\)
\(158\) 0 0
\(159\) −22.5212 −1.78605
\(160\) 0 0
\(161\) 10.9512 5.22241i 0.863074 0.411584i
\(162\) 0 0
\(163\) −7.58729 7.58729i −0.594282 0.594282i 0.344503 0.938785i \(-0.388047\pi\)
−0.938785 + 0.344503i \(0.888047\pi\)
\(164\) 0 0
\(165\) −4.95687 1.99002i −0.385892 0.154923i
\(166\) 0 0
\(167\) 10.4325 10.4325i 0.807290 0.807290i −0.176933 0.984223i \(-0.556618\pi\)
0.984223 + 0.176933i \(0.0566176\pi\)
\(168\) 0 0
\(169\) 2.15300i 0.165615i
\(170\) 0 0
\(171\) 0.334845i 0.0256063i
\(172\) 0 0
\(173\) 7.02168 + 7.02168i 0.533849 + 0.533849i 0.921715 0.387867i \(-0.126788\pi\)
−0.387867 + 0.921715i \(0.626788\pi\)
\(174\) 0 0
\(175\) 12.6462 0.276465i 0.955960 0.0208988i
\(176\) 0 0
\(177\) 25.8062 25.8062i 1.93971 1.93971i
\(178\) 0 0
\(179\) 4.58904i 0.343001i −0.985184 0.171500i \(-0.945139\pi\)
0.985184 0.171500i \(-0.0548615\pi\)
\(180\) 0 0
\(181\) 14.4761i 1.07600i −0.842944 0.538001i \(-0.819180\pi\)
0.842944 0.538001i \(-0.180820\pi\)
\(182\) 0 0
\(183\) 19.8805 19.8805i 1.46961 1.46961i
\(184\) 0 0
\(185\) 0.429915 1.07086i 0.0316079 0.0787312i
\(186\) 0 0
\(187\) 0.338914 0.338914i 0.0247838 0.0247838i
\(188\) 0 0
\(189\) 2.96626 0.215764
\(190\) 0 0
\(191\) 17.2791i 1.25027i 0.780516 + 0.625136i \(0.214956\pi\)
−0.780516 + 0.625136i \(0.785044\pi\)
\(192\) 0 0
\(193\) 2.77664 + 2.77664i 0.199867 + 0.199867i 0.799943 0.600076i \(-0.204863\pi\)
−0.600076 + 0.799943i \(0.704863\pi\)
\(194\) 0 0
\(195\) −20.3476 + 8.68996i −1.45712 + 0.622301i
\(196\) 0 0
\(197\) −13.8758 + 13.8758i −0.988612 + 0.988612i −0.999936 0.0113241i \(-0.996395\pi\)
0.0113241 + 0.999936i \(0.496395\pi\)
\(198\) 0 0
\(199\) −25.4300 −1.80269 −0.901344 0.433104i \(-0.857418\pi\)
−0.901344 + 0.433104i \(0.857418\pi\)
\(200\) 0 0
\(201\) 9.16399i 0.646378i
\(202\) 0 0
\(203\) 14.2610 14.2610i 1.00093 1.00093i
\(204\) 0 0
\(205\) −0.652290 1.52734i −0.0455579 0.106674i
\(206\) 0 0
\(207\) −15.6471 5.54229i −1.08755 0.385216i
\(208\) 0 0
\(209\) 0.0909119i 0.00628850i
\(210\) 0 0
\(211\) −14.8824 −1.02455 −0.512273 0.858822i \(-0.671197\pi\)
−0.512273 + 0.858822i \(0.671197\pi\)
\(212\) 0 0
\(213\) −14.9309 14.9309i −1.02305 1.02305i
\(214\) 0 0
\(215\) 13.8903 + 5.57651i 0.947314 + 0.380315i
\(216\) 0 0
\(217\) −2.80696 2.80696i −0.190549 0.190549i
\(218\) 0 0
\(219\) 15.2344i 1.02945i
\(220\) 0 0
\(221\) 1.98537i 0.133550i
\(222\) 0 0
\(223\) −6.07979 6.07979i −0.407133 0.407133i 0.473604 0.880738i \(-0.342953\pi\)
−0.880738 + 0.473604i \(0.842953\pi\)
\(224\) 0 0
\(225\) −12.5020 11.9671i −0.833466 0.797804i
\(226\) 0 0
\(227\) 21.0463 + 21.0463i 1.39689 + 1.39689i 0.808783 + 0.588107i \(0.200127\pi\)
0.588107 + 0.808783i \(0.299873\pi\)
\(228\) 0 0
\(229\) −19.7303 −1.30382 −0.651908 0.758298i \(-0.726031\pi\)
−0.651908 + 0.758298i \(0.726031\pi\)
\(230\) 0 0
\(231\) 6.04316 0.397611
\(232\) 0 0
\(233\) −4.55466 4.55466i −0.298386 0.298386i 0.541995 0.840381i \(-0.317669\pi\)
−0.840381 + 0.541995i \(0.817669\pi\)
\(234\) 0 0
\(235\) 7.84213 + 3.14835i 0.511564 + 0.205376i
\(236\) 0 0
\(237\) 5.74365 + 5.74365i 0.373090 + 0.373090i
\(238\) 0 0
\(239\) 24.7095i 1.59832i −0.601117 0.799161i \(-0.705277\pi\)
0.601117 0.799161i \(-0.294723\pi\)
\(240\) 0 0
\(241\) 10.4670i 0.674239i 0.941462 + 0.337119i \(0.109453\pi\)
−0.941462 + 0.337119i \(0.890547\pi\)
\(242\) 0 0
\(243\) 15.7941 + 15.7941i 1.01319 + 1.01319i
\(244\) 0 0
\(245\) 1.23367 0.526869i 0.0788160 0.0336604i
\(246\) 0 0
\(247\) 0.266283 + 0.266283i 0.0169432 + 0.0169432i
\(248\) 0 0
\(249\) 22.4752 1.42431
\(250\) 0 0
\(251\) 23.2669i 1.46859i −0.678829 0.734297i \(-0.737512\pi\)
0.678829 0.734297i \(-0.262488\pi\)
\(252\) 0 0
\(253\) −4.24826 1.50475i −0.267086 0.0946031i
\(254\) 0 0
\(255\) 2.66597 1.13857i 0.166949 0.0713001i
\(256\) 0 0
\(257\) −3.11820 + 3.11820i −0.194508 + 0.194508i −0.797641 0.603133i \(-0.793919\pi\)
0.603133 + 0.797641i \(0.293919\pi\)
\(258\) 0 0
\(259\) 1.30554i 0.0811222i
\(260\) 0 0
\(261\) −27.5936 −1.70800
\(262\) 0 0
\(263\) 2.94773 2.94773i 0.181765 0.181765i −0.610360 0.792124i \(-0.708975\pi\)
0.792124 + 0.610360i \(0.208975\pi\)
\(264\) 0 0
\(265\) −18.3852 7.38104i −1.12939 0.453414i
\(266\) 0 0
\(267\) 30.2611 + 30.2611i 1.85195 + 1.85195i
\(268\) 0 0
\(269\) 22.4439i 1.36843i −0.729280 0.684215i \(-0.760145\pi\)
0.729280 0.684215i \(-0.239855\pi\)
\(270\) 0 0
\(271\) −19.7051 −1.19700 −0.598500 0.801123i \(-0.704236\pi\)
−0.598500 + 0.801123i \(0.704236\pi\)
\(272\) 0 0
\(273\) 17.7005 17.7005i 1.07128 1.07128i
\(274\) 0 0
\(275\) −3.39434 3.24910i −0.204686 0.195928i
\(276\) 0 0
\(277\) −11.9674 + 11.9674i −0.719051 + 0.719051i −0.968411 0.249360i \(-0.919780\pi\)
0.249360 + 0.968411i \(0.419780\pi\)
\(278\) 0 0
\(279\) 5.43117i 0.325156i
\(280\) 0 0
\(281\) 13.1566i 0.784859i −0.919782 0.392429i \(-0.871635\pi\)
0.919782 0.392429i \(-0.128365\pi\)
\(282\) 0 0
\(283\) 8.88725 8.88725i 0.528292 0.528292i −0.391771 0.920063i \(-0.628137\pi\)
0.920063 + 0.391771i \(0.128137\pi\)
\(284\) 0 0
\(285\) −0.204858 + 0.510274i −0.0121347 + 0.0302260i
\(286\) 0 0
\(287\) 1.32865 + 1.32865i 0.0784275 + 0.0784275i
\(288\) 0 0
\(289\) 16.7399i 0.984698i
\(290\) 0 0
\(291\) 23.9644i 1.40482i
\(292\) 0 0
\(293\) 21.5379 21.5379i 1.25826 1.25826i 0.306336 0.951923i \(-0.400897\pi\)
0.951923 0.306336i \(-0.0991032\pi\)
\(294\) 0 0
\(295\) 29.5246 12.6092i 1.71899 0.734139i
\(296\) 0 0
\(297\) −0.779138 0.779138i −0.0452102 0.0452102i
\(298\) 0 0
\(299\) −16.8507 + 8.03577i −0.974500 + 0.464721i
\(300\) 0 0
\(301\) −16.9344 −0.976083
\(302\) 0 0
\(303\) 11.4618 + 11.4618i 0.658463 + 0.658463i
\(304\) 0 0
\(305\) 22.7450 9.71386i 1.30238 0.556214i
\(306\) 0 0
\(307\) −4.95084 + 4.95084i −0.282560 + 0.282560i −0.834129 0.551569i \(-0.814029\pi\)
0.551569 + 0.834129i \(0.314029\pi\)
\(308\) 0 0
\(309\) 0.426971 0.0242896
\(310\) 0 0
\(311\) 22.9472 1.30121 0.650607 0.759414i \(-0.274514\pi\)
0.650607 + 0.759414i \(0.274514\pi\)
\(312\) 0 0
\(313\) −0.758211 + 0.758211i −0.0428566 + 0.0428566i −0.728210 0.685354i \(-0.759648\pi\)
0.685354 + 0.728210i \(0.259648\pi\)
\(314\) 0 0
\(315\) 18.1704 + 7.29481i 1.02379 + 0.411016i
\(316\) 0 0
\(317\) −1.33215 + 1.33215i −0.0748213 + 0.0748213i −0.743527 0.668706i \(-0.766849\pi\)
0.668706 + 0.743527i \(0.266849\pi\)
\(318\) 0 0
\(319\) −7.49178 −0.419459
\(320\) 0 0
\(321\) 25.9999i 1.45117i
\(322\) 0 0
\(323\) −0.0348887 0.0348887i −0.00194126 0.00194126i
\(324\) 0 0
\(325\) −19.4588 + 0.425400i −1.07938 + 0.0235969i
\(326\) 0 0
\(327\) 19.7385 + 19.7385i 1.09154 + 1.09154i
\(328\) 0 0
\(329\) −9.56072 −0.527100
\(330\) 0 0
\(331\) 0.465793 0.0256023 0.0128011 0.999918i \(-0.495925\pi\)
0.0128011 + 0.999918i \(0.495925\pi\)
\(332\) 0 0
\(333\) 1.26304 1.26304i 0.0692142 0.0692142i
\(334\) 0 0
\(335\) 3.00339 7.48103i 0.164092 0.408733i
\(336\) 0 0
\(337\) −15.4997 15.4997i −0.844323 0.844323i 0.145094 0.989418i \(-0.453651\pi\)
−0.989418 + 0.145094i \(0.953651\pi\)
\(338\) 0 0
\(339\) −40.9081 −2.22182
\(340\) 0 0
\(341\) 1.47459i 0.0798533i
\(342\) 0 0
\(343\) −13.5952 + 13.5952i −0.734074 + 0.734074i
\(344\) 0 0
\(345\) −20.4540 18.0188i −1.10121 0.970102i
\(346\) 0 0
\(347\) 4.11658 4.11658i 0.220990 0.220990i −0.587925 0.808915i \(-0.700055\pi\)
0.808915 + 0.587925i \(0.200055\pi\)
\(348\) 0 0
\(349\) 15.3428i 0.821280i 0.911798 + 0.410640i \(0.134695\pi\)
−0.911798 + 0.410640i \(0.865305\pi\)
\(350\) 0 0
\(351\) −4.56422 −0.243620
\(352\) 0 0
\(353\) 0.359155 + 0.359155i 0.0191159 + 0.0191159i 0.716600 0.697484i \(-0.245697\pi\)
−0.697484 + 0.716600i \(0.745697\pi\)
\(354\) 0 0
\(355\) −7.29542 17.0822i −0.387200 0.906630i
\(356\) 0 0
\(357\) −2.31915 + 2.31915i −0.122742 + 0.122742i
\(358\) 0 0
\(359\) 17.7510 0.936862 0.468431 0.883500i \(-0.344819\pi\)
0.468431 + 0.883500i \(0.344819\pi\)
\(360\) 0 0
\(361\) −18.9906 −0.999507
\(362\) 0 0
\(363\) 18.1840 + 18.1840i 0.954414 + 0.954414i
\(364\) 0 0
\(365\) 4.99290 12.4366i 0.261340 0.650963i
\(366\) 0 0
\(367\) −5.58372 5.58372i −0.291468 0.291468i 0.546192 0.837660i \(-0.316077\pi\)
−0.837660 + 0.546192i \(0.816077\pi\)
\(368\) 0 0
\(369\) 2.57079i 0.133830i
\(370\) 0 0
\(371\) 22.4143 1.16369
\(372\) 0 0
\(373\) −1.41486 + 1.41486i −0.0732588 + 0.0732588i −0.742787 0.669528i \(-0.766496\pi\)
0.669528 + 0.742787i \(0.266496\pi\)
\(374\) 0 0
\(375\) −11.7305 25.8854i −0.605759 1.33672i
\(376\) 0 0
\(377\) −21.9436 + 21.9436i −1.13015 + 1.13015i
\(378\) 0 0
\(379\) −31.7521 −1.63099 −0.815497 0.578761i \(-0.803537\pi\)
−0.815497 + 0.578761i \(0.803537\pi\)
\(380\) 0 0
\(381\) −0.344511 −0.0176498
\(382\) 0 0
\(383\) −12.5180 + 12.5180i −0.639638 + 0.639638i −0.950466 0.310828i \(-0.899394\pi\)
0.310828 + 0.950466i \(0.399394\pi\)
\(384\) 0 0
\(385\) 4.93334 + 1.98057i 0.251426 + 0.100939i
\(386\) 0 0
\(387\) 16.3832 + 16.3832i 0.832803 + 0.832803i
\(388\) 0 0
\(389\) −26.9742 −1.36765 −0.683823 0.729647i \(-0.739684\pi\)
−0.683823 + 0.729647i \(0.739684\pi\)
\(390\) 0 0
\(391\) 2.20780 1.05286i 0.111653 0.0532453i
\(392\) 0 0
\(393\) 7.65478 + 7.65478i 0.386133 + 0.386133i
\(394\) 0 0
\(395\) 2.80642 + 6.57124i 0.141207 + 0.330635i
\(396\) 0 0
\(397\) −12.5053 + 12.5053i −0.627621 + 0.627621i −0.947469 0.319848i \(-0.896368\pi\)
0.319848 + 0.947469i \(0.396368\pi\)
\(398\) 0 0
\(399\) 0.622100i 0.0311439i
\(400\) 0 0
\(401\) 28.3962i 1.41804i −0.705189 0.709020i \(-0.749138\pi\)
0.705189 0.709020i \(-0.250862\pi\)
\(402\) 0 0
\(403\) 4.31909 + 4.31909i 0.215149 + 0.215149i
\(404\) 0 0
\(405\) 6.50191 + 15.2242i 0.323083 + 0.756498i
\(406\) 0 0
\(407\) 0.342921 0.342921i 0.0169980 0.0169980i
\(408\) 0 0
\(409\) 10.4636i 0.517392i −0.965959 0.258696i \(-0.916707\pi\)
0.965959 0.258696i \(-0.0832928\pi\)
\(410\) 0 0
\(411\) 5.65852i 0.279114i
\(412\) 0 0
\(413\) −25.6837 + 25.6837i −1.26381 + 1.26381i
\(414\) 0 0
\(415\) 18.3476 + 7.36596i 0.900650 + 0.361581i
\(416\) 0 0
\(417\) 24.5556 24.5556i 1.20249 1.20249i
\(418\) 0 0
\(419\) 4.01808 0.196296 0.0981480 0.995172i \(-0.468708\pi\)
0.0981480 + 0.995172i \(0.468708\pi\)
\(420\) 0 0
\(421\) 8.50020i 0.414274i −0.978312 0.207137i \(-0.933585\pi\)
0.978312 0.207137i \(-0.0664146\pi\)
\(422\) 0 0
\(423\) 9.24951 + 9.24951i 0.449727 + 0.449727i
\(424\) 0 0
\(425\) 2.54952 0.0557365i 0.123670 0.00270362i
\(426\) 0 0
\(427\) −19.7861 + 19.7861i −0.957516 + 0.957516i
\(428\) 0 0
\(429\) −9.29866 −0.448944
\(430\) 0 0
\(431\) 2.40012i 0.115610i 0.998328 + 0.0578048i \(0.0184101\pi\)
−0.998328 + 0.0578048i \(0.981590\pi\)
\(432\) 0 0
\(433\) 13.4482 13.4482i 0.646281 0.646281i −0.305811 0.952092i \(-0.598928\pi\)
0.952092 + 0.305811i \(0.0989277\pi\)
\(434\) 0 0
\(435\) −42.0502 16.8818i −2.01615 0.809418i
\(436\) 0 0
\(437\) −0.154904 + 0.437328i −0.00741005 + 0.0209202i
\(438\) 0 0
\(439\) 25.8506i 1.23378i 0.787049 + 0.616891i \(0.211608\pi\)
−0.787049 + 0.616891i \(0.788392\pi\)
\(440\) 0 0
\(441\) 2.07649 0.0988804
\(442\) 0 0
\(443\) 16.8114 + 16.8114i 0.798735 + 0.798735i 0.982896 0.184161i \(-0.0589568\pi\)
−0.184161 + 0.982896i \(0.558957\pi\)
\(444\) 0 0
\(445\) 14.7860 + 34.6214i 0.700922 + 1.64121i
\(446\) 0 0
\(447\) 0.828032 + 0.828032i 0.0391646 + 0.0391646i
\(448\) 0 0
\(449\) 33.1684i 1.56532i 0.622452 + 0.782658i \(0.286136\pi\)
−0.622452 + 0.782658i \(0.713864\pi\)
\(450\) 0 0
\(451\) 0.697981i 0.0328666i
\(452\) 0 0
\(453\) 5.02916 + 5.02916i 0.236291 + 0.236291i
\(454\) 0 0
\(455\) 20.2510 8.64871i 0.949380 0.405458i
\(456\) 0 0
\(457\) 8.09465 + 8.09465i 0.378652 + 0.378652i 0.870616 0.491964i \(-0.163721\pi\)
−0.491964 + 0.870616i \(0.663721\pi\)
\(458\) 0 0
\(459\) 0.598011 0.0279127
\(460\) 0 0
\(461\) 7.29745 0.339876 0.169938 0.985455i \(-0.445643\pi\)
0.169938 + 0.985455i \(0.445643\pi\)
\(462\) 0 0
\(463\) −24.9813 24.9813i −1.16098 1.16098i −0.984261 0.176720i \(-0.943451\pi\)
−0.176720 0.984261i \(-0.556549\pi\)
\(464\) 0 0
\(465\) −3.32279 + 8.27662i −0.154090 + 0.383819i
\(466\) 0 0
\(467\) 0.603738 + 0.603738i 0.0279377 + 0.0279377i 0.720938 0.693000i \(-0.243711\pi\)
−0.693000 + 0.720938i \(0.743711\pi\)
\(468\) 0 0
\(469\) 9.12049i 0.421145i
\(470\) 0 0
\(471\) 59.2077i 2.72815i
\(472\) 0 0
\(473\) 4.44810 + 4.44810i 0.204524 + 0.204524i
\(474\) 0 0
\(475\) −0.334472 + 0.349423i −0.0153466 + 0.0160326i
\(476\) 0 0
\(477\) −21.6847 21.6847i −0.992873 0.992873i
\(478\) 0 0
\(479\) 17.4344 0.796598 0.398299 0.917256i \(-0.369601\pi\)
0.398299 + 0.917256i \(0.369601\pi\)
\(480\) 0 0
\(481\) 2.00884i 0.0915954i
\(482\) 0 0
\(483\) 29.0704 + 10.2969i 1.32275 + 0.468524i
\(484\) 0 0
\(485\) −7.85405 + 19.5634i −0.356634 + 0.888328i
\(486\) 0 0
\(487\) 16.3954 16.3954i 0.742947 0.742947i −0.230197 0.973144i \(-0.573937\pi\)
0.973144 + 0.230197i \(0.0739370\pi\)
\(488\) 0 0
\(489\) 27.2747i 1.23341i
\(490\) 0 0
\(491\) −32.2651 −1.45610 −0.728051 0.685522i \(-0.759574\pi\)
−0.728051 + 0.685522i \(0.759574\pi\)
\(492\) 0 0
\(493\) 2.87508 2.87508i 0.129487 0.129487i
\(494\) 0 0
\(495\) −2.85665 6.68886i −0.128397 0.300642i
\(496\) 0 0
\(497\) 14.8600 + 14.8600i 0.666561 + 0.666561i
\(498\) 0 0
\(499\) 38.4458i 1.72107i 0.509391 + 0.860535i \(0.329871\pi\)
−0.509391 + 0.860535i \(0.670129\pi\)
\(500\) 0 0
\(501\) 37.5026 1.67549
\(502\) 0 0
\(503\) −25.5636 + 25.5636i −1.13982 + 1.13982i −0.151344 + 0.988481i \(0.548360\pi\)
−0.988481 + 0.151344i \(0.951640\pi\)
\(504\) 0 0
\(505\) 5.60039 + 13.1133i 0.249214 + 0.583535i
\(506\) 0 0
\(507\) −3.86980 + 3.86980i −0.171864 + 0.171864i
\(508\) 0 0
\(509\) 41.7856i 1.85211i 0.377384 + 0.926057i \(0.376824\pi\)
−0.377384 + 0.926057i \(0.623176\pi\)
\(510\) 0 0
\(511\) 15.1621i 0.670732i
\(512\) 0 0
\(513\) −0.0802067 + 0.0802067i −0.00354121 + 0.00354121i
\(514\) 0 0
\(515\) 0.348559 + 0.139935i 0.0153593 + 0.00616626i
\(516\) 0 0
\(517\) 2.51128 + 2.51128i 0.110446 + 0.110446i
\(518\) 0 0
\(519\) 25.2415i 1.10798i
\(520\) 0 0
\(521\) 18.1834i 0.796630i −0.917249 0.398315i \(-0.869595\pi\)
0.917249 0.398315i \(-0.130405\pi\)
\(522\) 0 0
\(523\) −16.4301 + 16.4301i −0.718439 + 0.718439i −0.968286 0.249846i \(-0.919620\pi\)
0.249846 + 0.968286i \(0.419620\pi\)
\(524\) 0 0
\(525\) 23.2271 + 22.2333i 1.01371 + 0.970339i
\(526\) 0 0
\(527\) −0.565893 0.565893i −0.0246507 0.0246507i
\(528\) 0 0
\(529\) −17.8721 14.4771i −0.777049 0.629440i
\(530\) 0 0
\(531\) 49.6953 2.15659
\(532\) 0 0
\(533\) −2.04440 2.04440i −0.0885528 0.0885528i
\(534\) 0 0
\(535\) −8.52116 + 21.2251i −0.368402 + 0.917640i
\(536\) 0 0
\(537\) 8.24832 8.24832i 0.355941 0.355941i
\(538\) 0 0
\(539\) 0.563775 0.0242835
\(540\) 0 0
\(541\) 34.2384 1.47202 0.736012 0.676968i \(-0.236707\pi\)
0.736012 + 0.676968i \(0.236707\pi\)
\(542\) 0 0
\(543\) 26.0193 26.0193i 1.11660 1.11660i
\(544\) 0 0
\(545\) 9.64447 + 22.5825i 0.413124 + 0.967330i
\(546\) 0 0
\(547\) −14.3746 + 14.3746i −0.614616 + 0.614616i −0.944145 0.329530i \(-0.893110\pi\)
0.329530 + 0.944145i \(0.393110\pi\)
\(548\) 0 0
\(549\) 38.2841 1.63393
\(550\) 0 0
\(551\) 0.771226i 0.0328553i
\(552\) 0 0
\(553\) −5.71639 5.71639i −0.243085 0.243085i
\(554\) 0 0
\(555\) 2.69749 1.15203i 0.114502 0.0489011i
\(556\) 0 0
\(557\) 14.8272 + 14.8272i 0.628248 + 0.628248i 0.947627 0.319379i \(-0.103474\pi\)
−0.319379 + 0.947627i \(0.603474\pi\)
\(558\) 0 0
\(559\) 26.0571 1.10210
\(560\) 0 0
\(561\) 1.21832 0.0514377
\(562\) 0 0
\(563\) −26.4159 + 26.4159i −1.11330 + 1.11330i −0.120596 + 0.992702i \(0.538481\pi\)
−0.992702 + 0.120596i \(0.961519\pi\)
\(564\) 0 0
\(565\) −33.3953 13.4071i −1.40495 0.564042i
\(566\) 0 0
\(567\) −13.2437 13.2437i −0.556183 0.556183i
\(568\) 0 0
\(569\) 7.60059 0.318633 0.159317 0.987228i \(-0.449071\pi\)
0.159317 + 0.987228i \(0.449071\pi\)
\(570\) 0 0
\(571\) 20.8517i 0.872616i −0.899797 0.436308i \(-0.856286\pi\)
0.899797 0.436308i \(-0.143714\pi\)
\(572\) 0 0
\(573\) −31.0574 + 31.0574i −1.29744 + 1.29744i
\(574\) 0 0
\(575\) −10.7922 21.4133i −0.450067 0.892995i
\(576\) 0 0
\(577\) 18.6561 18.6561i 0.776663 0.776663i −0.202598 0.979262i \(-0.564939\pi\)
0.979262 + 0.202598i \(0.0649386\pi\)
\(578\) 0 0
\(579\) 9.98143i 0.414814i
\(580\) 0 0
\(581\) −22.3685 −0.928001
\(582\) 0 0
\(583\) −5.88748 5.88748i −0.243834 0.243834i
\(584\) 0 0
\(585\) −27.9590 11.2246i −1.15596 0.464080i
\(586\) 0 0
\(587\) −17.2734 + 17.2734i −0.712950 + 0.712950i −0.967151 0.254201i \(-0.918187\pi\)
0.254201 + 0.967151i \(0.418187\pi\)
\(588\) 0 0
\(589\) 0.151798 0.00625473
\(590\) 0 0
\(591\) −49.8807 −2.05182
\(592\) 0 0
\(593\) 7.17002 + 7.17002i 0.294438 + 0.294438i 0.838830 0.544393i \(-0.183240\pi\)
−0.544393 + 0.838830i \(0.683240\pi\)
\(594\) 0 0
\(595\) −2.65331 + 1.13317i −0.108775 + 0.0464553i
\(596\) 0 0
\(597\) −45.7079 45.7079i −1.87070 1.87070i
\(598\) 0 0
\(599\) 36.9740i 1.51072i −0.655311 0.755359i \(-0.727462\pi\)
0.655311 0.755359i \(-0.272538\pi\)
\(600\) 0 0
\(601\) −17.7821 −0.725348 −0.362674 0.931916i \(-0.618136\pi\)
−0.362674 + 0.931916i \(0.618136\pi\)
\(602\) 0 0
\(603\) 8.82361 8.82361i 0.359325 0.359325i
\(604\) 0 0
\(605\) 8.88496 + 20.8042i 0.361225 + 0.845809i
\(606\) 0 0
\(607\) −25.3666 + 25.3666i −1.02960 + 1.02960i −0.0300509 + 0.999548i \(0.509567\pi\)
−0.999548 + 0.0300509i \(0.990433\pi\)
\(608\) 0 0
\(609\) 51.2655 2.07738
\(610\) 0 0
\(611\) 14.7112 0.595150
\(612\) 0 0
\(613\) 13.0720 13.0720i 0.527975 0.527975i −0.391993 0.919968i \(-0.628214\pi\)
0.919968 + 0.391993i \(0.128214\pi\)
\(614\) 0 0
\(615\) 1.57281 3.91766i 0.0634218 0.157975i
\(616\) 0 0
\(617\) −8.31615 8.31615i −0.334795 0.334795i 0.519609 0.854404i \(-0.326078\pi\)
−0.854404 + 0.519609i \(0.826078\pi\)
\(618\) 0 0
\(619\) −10.2423 −0.411673 −0.205836 0.978586i \(-0.565991\pi\)
−0.205836 + 0.978586i \(0.565991\pi\)
\(620\) 0 0
\(621\) −2.42045 5.07557i −0.0971291 0.203676i
\(622\) 0 0
\(623\) −30.1174 30.1174i −1.20663 1.20663i
\(624\) 0 0
\(625\) −1.09256 24.9761i −0.0437023 0.999045i
\(626\) 0 0
\(627\) −0.163405 + 0.163405i −0.00652576 + 0.00652576i
\(628\) 0 0
\(629\) 0.263202i 0.0104945i
\(630\) 0 0
\(631\) 32.8600i 1.30814i −0.756435 0.654068i \(-0.773061\pi\)
0.756435 0.654068i \(-0.226939\pi\)
\(632\) 0 0
\(633\) −26.7496 26.7496i −1.06320 1.06320i
\(634\) 0 0
\(635\) −0.281242 0.112909i −0.0111607 0.00448066i
\(636\) 0 0
\(637\) 1.65131 1.65131i 0.0654272 0.0654272i
\(638\) 0 0
\(639\) 28.7526i 1.13743i
\(640\) 0 0
\(641\) 25.9918i 1.02662i 0.858204 + 0.513308i \(0.171580\pi\)
−0.858204 + 0.513308i \(0.828420\pi\)
\(642\) 0 0
\(643\) 20.7155 20.7155i 0.816940 0.816940i −0.168723 0.985663i \(-0.553964\pi\)
0.985663 + 0.168723i \(0.0539644\pi\)
\(644\) 0 0
\(645\) 14.9433 + 34.9897i 0.588391 + 1.37772i
\(646\) 0 0
\(647\) −1.48231 + 1.48231i −0.0582757 + 0.0582757i −0.735644 0.677368i \(-0.763120\pi\)
0.677368 + 0.735644i \(0.263120\pi\)
\(648\) 0 0
\(649\) 13.4925 0.529626
\(650\) 0 0
\(651\) 10.0904i 0.395475i
\(652\) 0 0
\(653\) −25.7750 25.7750i −1.00865 1.00865i −0.999962 0.00869217i \(-0.997233\pi\)
−0.00869217 0.999962i \(-0.502767\pi\)
\(654\) 0 0
\(655\) 3.74023 + 8.75775i 0.146143 + 0.342194i
\(656\) 0 0
\(657\) 14.6686 14.6686i 0.572275 0.572275i
\(658\) 0 0
\(659\) 34.6005 1.34785 0.673923 0.738802i \(-0.264608\pi\)
0.673923 + 0.738802i \(0.264608\pi\)
\(660\) 0 0
\(661\) 27.6795i 1.07661i −0.842750 0.538305i \(-0.819065\pi\)
0.842750 0.538305i \(-0.180935\pi\)
\(662\) 0 0
\(663\) 3.56850 3.56850i 0.138589 0.138589i
\(664\) 0 0
\(665\) 0.203886 0.507852i 0.00790634 0.0196937i
\(666\) 0 0
\(667\) −36.0389 12.7652i −1.39543 0.494269i
\(668\) 0 0
\(669\) 21.8556i 0.844987i
\(670\) 0 0
\(671\) 10.3943 0.401267
\(672\) 0 0
\(673\) 28.8217 + 28.8217i 1.11099 + 1.11099i 0.993016 + 0.117977i \(0.0376410\pi\)
0.117977 + 0.993016i \(0.462359\pi\)
\(674\) 0 0
\(675\) −0.128134 5.86116i −0.00493189 0.225596i
\(676\) 0 0
\(677\) −15.0623 15.0623i −0.578890 0.578890i 0.355707 0.934597i \(-0.384240\pi\)
−0.934597 + 0.355707i \(0.884240\pi\)
\(678\) 0 0
\(679\) 23.8507i 0.915305i
\(680\) 0 0
\(681\) 75.6570i 2.89918i
\(682\) 0 0
\(683\) −21.2791 21.2791i −0.814220 0.814220i 0.171043 0.985264i \(-0.445286\pi\)
−0.985264 + 0.171043i \(0.945286\pi\)
\(684\) 0 0
\(685\) 1.85451 4.61934i 0.0708572 0.176496i
\(686\) 0 0
\(687\) −35.4632 35.4632i −1.35301 1.35301i
\(688\) 0 0
\(689\) −34.4891 −1.31393
\(690\) 0 0
\(691\) −30.3837 −1.15585 −0.577925 0.816090i \(-0.696137\pi\)
−0.577925 + 0.816090i \(0.696137\pi\)
\(692\) 0 0
\(693\) 5.81870 + 5.81870i 0.221034 + 0.221034i
\(694\) 0 0
\(695\) 28.0938 11.9982i 1.06566 0.455117i
\(696\) 0 0
\(697\) 0.267860 + 0.267860i 0.0101459 + 0.0101459i
\(698\) 0 0
\(699\) 16.3731i 0.619287i
\(700\) 0 0
\(701\) 49.3304i 1.86318i −0.363506 0.931592i \(-0.618420\pi\)
0.363506 0.931592i \(-0.381580\pi\)
\(702\) 0 0
\(703\) −0.0353013 0.0353013i −0.00133141 0.00133141i
\(704\) 0 0
\(705\) 8.43658 + 19.7543i 0.317740 + 0.743989i
\(706\) 0 0
\(707\) −11.4074 11.4074i −0.429019 0.429019i
\(708\) 0 0
\(709\) 18.7528 0.704274 0.352137 0.935948i \(-0.385455\pi\)
0.352137 + 0.935948i \(0.385455\pi\)
\(710\) 0 0
\(711\) 11.0606i 0.414806i
\(712\) 0 0
\(713\) −2.51253 + 7.09343i −0.0940950 + 0.265651i
\(714\) 0 0
\(715\) −7.59097 3.04752i −0.283886 0.113971i
\(716\) 0 0
\(717\) 44.4127 44.4127i 1.65862 1.65862i
\(718\) 0 0
\(719\) 18.2558i 0.680828i 0.940276 + 0.340414i \(0.110567\pi\)
−0.940276 + 0.340414i \(0.889433\pi\)
\(720\) 0 0
\(721\) −0.424945 −0.0158258
\(722\) 0 0
\(723\) −18.8134 + 18.8134i −0.699676 + 0.699676i
\(724\) 0 0
\(725\) −28.7950 27.5629i −1.06942 1.02366i
\(726\) 0 0
\(727\) 16.5725 + 16.5725i 0.614638 + 0.614638i 0.944151 0.329513i \(-0.106884\pi\)
−0.329513 + 0.944151i \(0.606884\pi\)
\(728\) 0 0
\(729\) 34.5664i 1.28024i
\(730\) 0 0
\(731\) −3.41404 −0.126273
\(732\) 0 0
\(733\) −2.29943 + 2.29943i −0.0849315 + 0.0849315i −0.748296 0.663365i \(-0.769128\pi\)
0.663365 + 0.748296i \(0.269128\pi\)
\(734\) 0 0
\(735\) 3.16438 + 1.27039i 0.116720 + 0.0468591i
\(736\) 0 0
\(737\) 2.39565 2.39565i 0.0882448 0.0882448i
\(738\) 0 0
\(739\) 2.44271i 0.0898567i −0.998990 0.0449284i \(-0.985694\pi\)
0.998990 0.0449284i \(-0.0143060\pi\)
\(740\) 0 0
\(741\) 0.957231i 0.0351648i
\(742\) 0 0
\(743\) −35.8272 + 35.8272i −1.31437 + 1.31437i −0.396216 + 0.918157i \(0.629677\pi\)
−0.918157 + 0.396216i \(0.870323\pi\)
\(744\) 0 0
\(745\) 0.404588 + 0.947343i 0.0148229 + 0.0347079i
\(746\) 0 0
\(747\) 21.6404 + 21.6404i 0.791780 + 0.791780i
\(748\) 0 0
\(749\) 25.8765i 0.945507i
\(750\) 0 0
\(751\) 24.1199i 0.880147i 0.897962 + 0.440073i \(0.145048\pi\)
−0.897962 + 0.440073i \(0.854952\pi\)
\(752\) 0 0
\(753\) 41.8198 41.8198i 1.52400 1.52400i
\(754\) 0 0
\(755\) 2.45732 + 5.75381i 0.0894309 + 0.209403i
\(756\) 0 0
\(757\) −31.9249 31.9249i −1.16033 1.16033i −0.984403 0.175926i \(-0.943708\pi\)
−0.175926 0.984403i \(-0.556292\pi\)
\(758\) 0 0
\(759\) −4.93116 10.3404i −0.178990 0.375334i
\(760\) 0 0
\(761\) 41.7758 1.51437 0.757186 0.653199i \(-0.226574\pi\)
0.757186 + 0.653199i \(0.226574\pi\)
\(762\) 0 0
\(763\) −19.6448 19.6448i −0.711188 0.711188i
\(764\) 0 0
\(765\) 3.66323 + 1.47066i 0.132444 + 0.0531719i
\(766\) 0 0
\(767\) 39.5197 39.5197i 1.42697 1.42697i
\(768\) 0 0
\(769\) −31.5466 −1.13760 −0.568800 0.822476i \(-0.692592\pi\)
−0.568800 + 0.822476i \(0.692592\pi\)
\(770\) 0 0
\(771\) −11.2093 −0.403693
\(772\) 0 0
\(773\) −25.3692 + 25.3692i −0.912465 + 0.912465i −0.996466 0.0840007i \(-0.973230\pi\)
0.0840007 + 0.996466i \(0.473230\pi\)
\(774\) 0 0
\(775\) −5.42512 + 5.66762i −0.194876 + 0.203587i
\(776\) 0 0
\(777\) −2.34657 + 2.34657i −0.0841827 + 0.0841827i
\(778\) 0 0
\(779\) −0.0718522 −0.00257437
\(780\) 0 0
\(781\) 7.80643i 0.279336i
\(782\) 0 0
\(783\) −6.60960 6.60960i −0.236208 0.236208i
\(784\) 0 0
\(785\) −19.4046 + 48.3343i −0.692580 + 1.72513i
\(786\) 0 0
\(787\) 4.98641 + 4.98641i 0.177746 + 0.177746i 0.790373 0.612626i \(-0.209887\pi\)
−0.612626 + 0.790373i \(0.709887\pi\)
\(788\) 0 0
\(789\) 10.5965 0.377245
\(790\) 0 0
\(791\) 40.7139 1.44762
\(792\) 0 0
\(793\) 30.4451 30.4451i 1.08114 1.08114i
\(794\) 0 0
\(795\) −19.7788 46.3122i −0.701483 1.64252i
\(796\) 0 0
\(797\) 0.771307 + 0.771307i 0.0273211 + 0.0273211i 0.720635 0.693314i \(-0.243850\pi\)
−0.693314 + 0.720635i \(0.743850\pi\)
\(798\) 0 0
\(799\) −1.92748 −0.0681893
\(800\) 0 0
\(801\) 58.2742i 2.05902i
\(802\) 0 0
\(803\) 3.98258 3.98258i 0.140542 0.140542i
\(804\) 0 0
\(805\) 20.3569 + 17.9333i 0.717488 + 0.632066i
\(806\) 0 0
\(807\) 40.3407 40.3407i 1.42006 1.42006i
\(808\) 0 0
\(809\) 31.1320i 1.09454i 0.836955 + 0.547272i \(0.184334\pi\)
−0.836955 + 0.547272i \(0.815666\pi\)
\(810\) 0 0
\(811\) 32.0746 1.12629 0.563146 0.826357i \(-0.309591\pi\)
0.563146 + 0.826357i \(0.309591\pi\)
\(812\) 0 0
\(813\) −35.4179 35.4179i −1.24216 1.24216i
\(814\) 0 0
\(815\) 8.93896 22.2658i 0.313118 0.779936i
\(816\) 0 0
\(817\) 0.457900 0.457900i 0.0160199 0.0160199i
\(818\) 0 0
\(819\) 34.0861 1.19107
\(820\) 0 0
\(821\) −41.3373 −1.44268 −0.721340 0.692581i \(-0.756474\pi\)
−0.721340 + 0.692581i \(0.756474\pi\)
\(822\) 0 0
\(823\) 11.7246 + 11.7246i 0.408695 + 0.408695i 0.881283 0.472589i \(-0.156680\pi\)
−0.472589 + 0.881283i \(0.656680\pi\)
\(824\) 0 0
\(825\) −0.261047 11.9409i −0.00908850 0.415729i
\(826\) 0 0
\(827\) 14.7516 + 14.7516i 0.512963 + 0.512963i 0.915433 0.402470i \(-0.131848\pi\)
−0.402470 + 0.915433i \(0.631848\pi\)
\(828\) 0 0
\(829\) 10.9157i 0.379117i 0.981869 + 0.189558i \(0.0607056\pi\)
−0.981869 + 0.189558i \(0.939294\pi\)
\(830\) 0 0
\(831\) −43.0203 −1.49236
\(832\) 0 0
\(833\) −0.216357 + 0.216357i −0.00749631 + 0.00749631i
\(834\) 0 0
\(835\) 30.6153 + 12.2910i 1.05949 + 0.425349i
\(836\) 0 0
\(837\) −1.30095 + 1.30095i −0.0449673 + 0.0449673i
\(838\) 0 0
\(839\) 0.812978 0.0280671 0.0140336 0.999902i \(-0.495533\pi\)
0.0140336 + 0.999902i \(0.495533\pi\)
\(840\) 0 0
\(841\) −34.5545 −1.19153
\(842\) 0 0
\(843\) 23.6477 23.6477i 0.814470 0.814470i
\(844\) 0 0
\(845\) −4.42739 + 1.89084i −0.152307 + 0.0650467i
\(846\) 0 0
\(847\) −18.0977 18.0977i −0.621845 0.621845i
\(848\) 0 0
\(849\) 31.9478 1.09645
\(850\) 0 0
\(851\) 2.23391 1.06531i 0.0765773 0.0365183i
\(852\) 0 0
\(853\) 11.1821 + 11.1821i 0.382866 + 0.382866i 0.872134 0.489267i \(-0.162736\pi\)
−0.489267 + 0.872134i \(0.662736\pi\)
\(854\) 0 0
\(855\) −0.688570 + 0.294072i −0.0235486 + 0.0100571i
\(856\) 0 0
\(857\) −18.0181 + 18.0181i −0.615487 + 0.615487i −0.944370 0.328884i \(-0.893328\pi\)
0.328884 + 0.944370i \(0.393328\pi\)
\(858\) 0 0
\(859\) 0.153928i 0.00525197i −0.999997 0.00262598i \(-0.999164\pi\)
0.999997 0.00262598i \(-0.000835878\pi\)
\(860\) 0 0
\(861\) 4.77621i 0.162773i
\(862\) 0 0
\(863\) 25.8948 + 25.8948i 0.881469 + 0.881469i 0.993684 0.112215i \(-0.0357945\pi\)
−0.112215 + 0.993684i \(0.535795\pi\)
\(864\) 0 0
\(865\) −8.27260 + 20.6059i −0.281277 + 0.700623i
\(866\) 0 0
\(867\) 30.0882 30.0882i 1.02185 1.02185i
\(868\) 0 0
\(869\) 3.00300i 0.101870i
\(870\) 0 0
\(871\) 14.0338i 0.475517i
\(872\) 0 0
\(873\) −23.0743 + 23.0743i −0.780948 + 0.780948i
\(874\) 0 0
\(875\) 11.6748 + 25.7626i 0.394680 + 0.870933i
\(876\) 0 0
\(877\) 21.4985 21.4985i 0.725954 0.725954i −0.243857 0.969811i \(-0.578413\pi\)
0.969811 + 0.243857i \(0.0784128\pi\)
\(878\) 0 0
\(879\) 77.4245 2.61146
\(880\) 0 0
\(881\) 0.629018i 0.0211922i −0.999944 0.0105961i \(-0.996627\pi\)
0.999944 0.0105961i \(-0.00337290\pi\)
\(882\) 0 0
\(883\) 5.22919 + 5.22919i 0.175976 + 0.175976i 0.789599 0.613623i \(-0.210288\pi\)
−0.613623 + 0.789599i \(0.710288\pi\)
\(884\) 0 0
\(885\) 75.7312 + 30.4035i 2.54568 + 1.02200i
\(886\) 0 0
\(887\) −11.8883 + 11.8883i −0.399171 + 0.399171i −0.877941 0.478770i \(-0.841083\pi\)
0.478770 + 0.877941i \(0.341083\pi\)
\(888\) 0 0
\(889\) 0.342875 0.0114997
\(890\) 0 0
\(891\) 6.95735i 0.233080i
\(892\) 0 0
\(893\) 0.258518 0.258518i 0.00865099 0.00865099i
\(894\) 0 0
\(895\) 9.43681 4.03024i 0.315438 0.134716i
\(896\) 0 0
\(897\) −44.7308 15.8439i −1.49352 0.529012i
\(898\) 0 0
\(899\) 12.5092i 0.417206i
\(900\) 0 0
\(901\) 4.51881 0.150543
\(902\) 0 0
\(903\) −30.4378 30.4378i −1.01291 1.01291i
\(904\) 0 0
\(905\) 29.7684 12.7134i 0.989536 0.422608i
\(906\) 0 0
\(907\) −5.72698 5.72698i −0.190161 0.190161i 0.605605 0.795766i \(-0.292931\pi\)
−0.795766 + 0.605605i \(0.792931\pi\)
\(908\) 0 0
\(909\) 22.0722i 0.732087i
\(910\) 0 0
\(911\) 31.0071i 1.02731i −0.857997 0.513655i \(-0.828291\pi\)
0.857997 0.513655i \(-0.171709\pi\)
\(912\) 0 0
\(913\) 5.87545 + 5.87545i 0.194449 + 0.194449i
\(914\) 0 0
\(915\) 58.3415 + 23.4222i 1.92871 + 0.774313i
\(916\) 0 0
\(917\) −7.61845 7.61845i −0.251583 0.251583i
\(918\) 0 0
\(919\) −4.62320 −0.152505 −0.0762527 0.997089i \(-0.524296\pi\)
−0.0762527 + 0.997089i \(0.524296\pi\)
\(920\) 0 0
\(921\) −17.7973 −0.586440
\(922\) 0 0
\(923\) −22.8652 22.8652i −0.752617 0.752617i
\(924\) 0 0
\(925\) 2.57966 0.0563955i 0.0848188 0.00185427i
\(926\) 0 0
\(927\) 0.411112 + 0.411112i 0.0135027 + 0.0135027i
\(928\) 0 0
\(929\) 39.1876i 1.28570i −0.765990 0.642852i \(-0.777751\pi\)
0.765990 0.642852i \(-0.222249\pi\)
\(930\) 0 0
\(931\) 0.0580366i 0.00190207i
\(932\) 0 0
\(933\) 41.2452 + 41.2452i 1.35031 + 1.35031i
\(934\) 0 0
\(935\) 0.994581 + 0.399291i 0.0325263 + 0.0130582i
\(936\) 0 0
\(937\) −23.3368 23.3368i −0.762378 0.762378i 0.214373 0.976752i \(-0.431229\pi\)
−0.976752 + 0.214373i \(0.931229\pi\)
\(938\) 0 0
\(939\) −2.72561 −0.0889471
\(940\) 0 0
\(941\) 43.2393i 1.40956i −0.709425 0.704781i \(-0.751045\pi\)
0.709425 0.704781i \(-0.248955\pi\)
\(942\) 0 0
\(943\) 1.18928 3.35761i 0.0387284 0.109339i
\(944\) 0 0
\(945\) 2.60507 + 6.09977i 0.0847429 + 0.198426i
\(946\) 0 0
\(947\) 23.7242 23.7242i 0.770932 0.770932i −0.207338 0.978269i \(-0.566480\pi\)
0.978269 + 0.207338i \(0.0664799\pi\)
\(948\) 0 0
\(949\) 23.3301i 0.757327i
\(950\) 0 0
\(951\) −4.78882 −0.155288
\(952\) 0 0
\(953\) −34.9642 + 34.9642i −1.13260 + 1.13260i −0.142859 + 0.989743i \(0.545629\pi\)
−0.989743 + 0.142859i \(0.954371\pi\)
\(954\) 0 0
\(955\) −35.5324 + 15.1751i −1.14980 + 0.491054i
\(956\) 0 0
\(957\) −13.4657 13.4657i −0.435285 0.435285i
\(958\) 0 0
\(959\) 5.63166i 0.181856i
\(960\) 0 0
\(961\) −28.5378 −0.920576
\(962\) 0 0
\(963\) −25.0342 + 25.0342i −0.806716 + 0.806716i
\(964\) 0 0
\(965\) −3.27129 + 8.14836i −0.105307 + 0.262305i
\(966\) 0 0
\(967\) −3.13933 + 3.13933i −0.100954 + 0.100954i −0.755780 0.654826i \(-0.772742\pi\)
0.654826 + 0.755780i \(0.272742\pi\)
\(968\) 0 0
\(969\) 0.125418i 0.00402900i
\(970\) 0 0
\(971\) 55.3061i 1.77486i 0.460945 + 0.887429i \(0.347511\pi\)
−0.460945 + 0.887429i \(0.652489\pi\)
\(972\) 0 0
\(973\) −24.4390 + 24.4390i −0.783479 + 0.783479i
\(974\) 0 0
\(975\) −35.7398 34.2105i −1.14459 1.09561i
\(976\) 0 0
\(977\) 36.9868 + 36.9868i 1.18331 + 1.18331i 0.978881 + 0.204433i \(0.0655351\pi\)
0.204433 + 0.978881i \(0.434465\pi\)
\(978\) 0 0
\(979\) 15.8217i 0.505663i
\(980\) 0 0
\(981\) 38.0106i 1.21359i
\(982\) 0 0
\(983\) −38.1670 + 38.1670i −1.21734 + 1.21734i −0.248777 + 0.968561i \(0.580029\pi\)
−0.968561 + 0.248777i \(0.919971\pi\)
\(984\) 0 0
\(985\) −40.7202 16.3478i −1.29745 0.520884i
\(986\) 0 0
\(987\) −17.1844 17.1844i −0.546986 0.546986i
\(988\) 0 0
\(989\) 13.8183 + 28.9764i 0.439397 + 0.921397i
\(990\) 0 0
\(991\) 22.3418 0.709711 0.354856 0.934921i \(-0.384530\pi\)
0.354856 + 0.934921i \(0.384530\pi\)
\(992\) 0 0
\(993\) 0.837215 + 0.837215i 0.0265682 + 0.0265682i
\(994\) 0 0
\(995\) −22.3335 52.2939i −0.708019 1.65783i
\(996\) 0 0
\(997\) 43.1162 43.1162i 1.36550 1.36550i 0.498766 0.866737i \(-0.333787\pi\)
0.866737 0.498766i \(-0.166213\pi\)
\(998\) 0 0
\(999\) 0.605081 0.0191439
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 920.2.q.a.873.34 yes 72
5.2 odd 4 inner 920.2.q.a.137.33 72
23.22 odd 2 inner 920.2.q.a.873.33 yes 72
115.22 even 4 inner 920.2.q.a.137.34 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.q.a.137.33 72 5.2 odd 4 inner
920.2.q.a.137.34 yes 72 115.22 even 4 inner
920.2.q.a.873.33 yes 72 23.22 odd 2 inner
920.2.q.a.873.34 yes 72 1.1 even 1 trivial