Properties

Label 1152.2.p.g.191.8
Level $1152$
Weight $2$
Character 1152.191
Analytic conductor $9.199$
Analytic rank $0$
Dimension $24$
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] = [1152,2,Mod(191,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.p (of order \(6\), 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: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 191.8
Character \(\chi\) \(=\) 1152.191
Dual form 1152.2.p.g.959.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.765007 - 1.55395i) q^{3} +(0.781546 + 1.35368i) q^{5} +(-0.954503 - 0.551083i) q^{7} +(-1.82953 - 2.37757i) q^{9} +O(q^{10})\) \(q+(0.765007 - 1.55395i) q^{3} +(0.781546 + 1.35368i) q^{5} +(-0.954503 - 0.551083i) q^{7} +(-1.82953 - 2.37757i) q^{9} +(-2.15418 - 1.24372i) q^{11} +(5.48986 - 3.16957i) q^{13} +(2.70144 - 0.178912i) q^{15} -0.874622i q^{17} -6.45410 q^{19} +(-1.58656 + 1.06167i) q^{21} +(-1.52981 - 2.64970i) q^{23} +(1.27837 - 2.21421i) q^{25} +(-5.09422 + 1.02415i) q^{27} +(-0.767807 + 1.32988i) q^{29} +(6.29677 - 3.63544i) q^{31} +(-3.58063 + 2.39604i) q^{33} -1.72279i q^{35} +1.74638i q^{37} +(-0.725582 - 10.9557i) q^{39} +(7.45339 - 4.30322i) q^{41} +(2.66223 - 4.61112i) q^{43} +(1.78859 - 4.33477i) q^{45} +(1.73458 - 3.00438i) q^{47} +(-2.89262 - 5.01016i) q^{49} +(-1.35912 - 0.669092i) q^{51} -11.1315 q^{53} -3.88808i q^{55} +(-4.93743 + 10.0294i) q^{57} +(-1.76786 + 1.02067i) q^{59} +(6.23086 + 3.59739i) q^{61} +(0.436057 + 3.27762i) q^{63} +(8.58115 + 4.95433i) q^{65} +(1.14096 + 1.97621i) q^{67} +(-5.28782 + 0.350205i) q^{69} +13.1244 q^{71} -4.92960 q^{73} +(-2.46281 - 3.68041i) q^{75} +(1.37078 + 2.37426i) q^{77} +(3.31959 + 1.91656i) q^{79} +(-2.30564 + 8.69966i) q^{81} +(-11.2198 - 6.47775i) q^{83} +(1.18396 - 0.683557i) q^{85} +(1.47919 + 2.21050i) q^{87} +13.3358i q^{89} -6.98679 q^{91} +(-0.832229 - 12.5660i) q^{93} +(-5.04418 - 8.73677i) q^{95} +(4.65272 - 8.05875i) q^{97} +(0.984118 + 7.39712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 12 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 12 q^{7} - 4 q^{9} + 20 q^{15} - 12 q^{23} - 12 q^{25} - 36 q^{31} + 4 q^{33} + 20 q^{39} - 12 q^{41} + 12 q^{47} + 12 q^{49} + 4 q^{57} - 92 q^{63} - 48 q^{65} + 24 q^{73} + 84 q^{79} - 20 q^{81} + 68 q^{87} + 24 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.765007 1.55395i 0.441677 0.897174i
\(4\) 0 0
\(5\) 0.781546 + 1.35368i 0.349518 + 0.605383i 0.986164 0.165774i \(-0.0530121\pi\)
−0.636646 + 0.771156i \(0.719679\pi\)
\(6\) 0 0
\(7\) −0.954503 0.551083i −0.360768 0.208290i 0.308649 0.951176i \(-0.400123\pi\)
−0.669418 + 0.742886i \(0.733456\pi\)
\(8\) 0 0
\(9\) −1.82953 2.37757i −0.609843 0.792522i
\(10\) 0 0
\(11\) −2.15418 1.24372i −0.649509 0.374994i 0.138759 0.990326i \(-0.455689\pi\)
−0.788268 + 0.615332i \(0.789022\pi\)
\(12\) 0 0
\(13\) 5.48986 3.16957i 1.52261 0.879081i 0.522971 0.852351i \(-0.324824\pi\)
0.999643 0.0267307i \(-0.00850965\pi\)
\(14\) 0 0
\(15\) 2.70144 0.178912i 0.697508 0.0461950i
\(16\) 0 0
\(17\) 0.874622i 0.212127i −0.994359 0.106064i \(-0.966175\pi\)
0.994359 0.106064i \(-0.0338247\pi\)
\(18\) 0 0
\(19\) −6.45410 −1.48067 −0.740337 0.672236i \(-0.765334\pi\)
−0.740337 + 0.672236i \(0.765334\pi\)
\(20\) 0 0
\(21\) −1.58656 + 1.06167i −0.346215 + 0.231675i
\(22\) 0 0
\(23\) −1.52981 2.64970i −0.318987 0.552501i 0.661290 0.750130i \(-0.270009\pi\)
−0.980277 + 0.197629i \(0.936676\pi\)
\(24\) 0 0
\(25\) 1.27837 2.21421i 0.255675 0.442841i
\(26\) 0 0
\(27\) −5.09422 + 1.02415i −0.980384 + 0.197097i
\(28\) 0 0
\(29\) −0.767807 + 1.32988i −0.142578 + 0.246953i −0.928467 0.371415i \(-0.878873\pi\)
0.785889 + 0.618368i \(0.212206\pi\)
\(30\) 0 0
\(31\) 6.29677 3.63544i 1.13093 0.652945i 0.186764 0.982405i \(-0.440200\pi\)
0.944169 + 0.329460i \(0.106867\pi\)
\(32\) 0 0
\(33\) −3.58063 + 2.39604i −0.623309 + 0.417097i
\(34\) 0 0
\(35\) 1.72279i 0.291204i
\(36\) 0 0
\(37\) 1.74638i 0.287103i 0.989643 + 0.143551i \(0.0458522\pi\)
−0.989643 + 0.143551i \(0.954148\pi\)
\(38\) 0 0
\(39\) −0.725582 10.9557i −0.116186 1.75432i
\(40\) 0 0
\(41\) 7.45339 4.30322i 1.16402 0.672049i 0.211759 0.977322i \(-0.432081\pi\)
0.952265 + 0.305273i \(0.0987476\pi\)
\(42\) 0 0
\(43\) 2.66223 4.61112i 0.405986 0.703189i −0.588449 0.808534i \(-0.700261\pi\)
0.994436 + 0.105345i \(0.0335947\pi\)
\(44\) 0 0
\(45\) 1.78859 4.33477i 0.266628 0.646189i
\(46\) 0 0
\(47\) 1.73458 3.00438i 0.253015 0.438234i −0.711340 0.702848i \(-0.751911\pi\)
0.964354 + 0.264614i \(0.0852446\pi\)
\(48\) 0 0
\(49\) −2.89262 5.01016i −0.413231 0.715737i
\(50\) 0 0
\(51\) −1.35912 0.669092i −0.190315 0.0936916i
\(52\) 0 0
\(53\) −11.1315 −1.52903 −0.764514 0.644607i \(-0.777021\pi\)
−0.764514 + 0.644607i \(0.777021\pi\)
\(54\) 0 0
\(55\) 3.88808i 0.524269i
\(56\) 0 0
\(57\) −4.93743 + 10.0294i −0.653979 + 1.32842i
\(58\) 0 0
\(59\) −1.76786 + 1.02067i −0.230155 + 0.132880i −0.610644 0.791905i \(-0.709089\pi\)
0.380488 + 0.924786i \(0.375756\pi\)
\(60\) 0 0
\(61\) 6.23086 + 3.59739i 0.797780 + 0.460599i 0.842694 0.538392i \(-0.180968\pi\)
−0.0449142 + 0.998991i \(0.514301\pi\)
\(62\) 0 0
\(63\) 0.436057 + 3.27762i 0.0549380 + 0.412941i
\(64\) 0 0
\(65\) 8.58115 + 4.95433i 1.06436 + 0.614509i
\(66\) 0 0
\(67\) 1.14096 + 1.97621i 0.139391 + 0.241432i 0.927266 0.374403i \(-0.122152\pi\)
−0.787875 + 0.615835i \(0.788819\pi\)
\(68\) 0 0
\(69\) −5.28782 + 0.350205i −0.636579 + 0.0421597i
\(70\) 0 0
\(71\) 13.1244 1.55758 0.778788 0.627287i \(-0.215835\pi\)
0.778788 + 0.627287i \(0.215835\pi\)
\(72\) 0 0
\(73\) −4.92960 −0.576966 −0.288483 0.957485i \(-0.593151\pi\)
−0.288483 + 0.957485i \(0.593151\pi\)
\(74\) 0 0
\(75\) −2.46281 3.68041i −0.284380 0.424977i
\(76\) 0 0
\(77\) 1.37078 + 2.37426i 0.156215 + 0.270572i
\(78\) 0 0
\(79\) 3.31959 + 1.91656i 0.373483 + 0.215630i 0.674979 0.737837i \(-0.264153\pi\)
−0.301496 + 0.953467i \(0.597486\pi\)
\(80\) 0 0
\(81\) −2.30564 + 8.69966i −0.256182 + 0.966628i
\(82\) 0 0
\(83\) −11.2198 6.47775i −1.23153 0.711026i −0.264183 0.964472i \(-0.585102\pi\)
−0.967349 + 0.253447i \(0.918436\pi\)
\(84\) 0 0
\(85\) 1.18396 0.683557i 0.128418 0.0741422i
\(86\) 0 0
\(87\) 1.47919 + 2.21050i 0.158586 + 0.236991i
\(88\) 0 0
\(89\) 13.3358i 1.41359i 0.707416 + 0.706797i \(0.249861\pi\)
−0.707416 + 0.706797i \(0.750139\pi\)
\(90\) 0 0
\(91\) −6.98679 −0.732414
\(92\) 0 0
\(93\) −0.832229 12.5660i −0.0862982 1.30303i
\(94\) 0 0
\(95\) −5.04418 8.73677i −0.517522 0.896374i
\(96\) 0 0
\(97\) 4.65272 8.05875i 0.472412 0.818242i −0.527089 0.849810i \(-0.676717\pi\)
0.999502 + 0.0315680i \(0.0100501\pi\)
\(98\) 0 0
\(99\) 0.984118 + 7.39712i 0.0989076 + 0.743438i
\(100\) 0 0
\(101\) −9.15459 + 15.8562i −0.910916 + 1.57775i −0.0981434 + 0.995172i \(0.531290\pi\)
−0.812773 + 0.582581i \(0.802043\pi\)
\(102\) 0 0
\(103\) −14.0766 + 8.12710i −1.38700 + 0.800787i −0.992977 0.118311i \(-0.962252\pi\)
−0.394028 + 0.919098i \(0.628919\pi\)
\(104\) 0 0
\(105\) −2.67712 1.31794i −0.261261 0.128618i
\(106\) 0 0
\(107\) 5.80067i 0.560772i 0.959887 + 0.280386i \(0.0904624\pi\)
−0.959887 + 0.280386i \(0.909538\pi\)
\(108\) 0 0
\(109\) 7.34771i 0.703783i −0.936041 0.351892i \(-0.885539\pi\)
0.936041 0.351892i \(-0.114461\pi\)
\(110\) 0 0
\(111\) 2.71379 + 1.33599i 0.257581 + 0.126807i
\(112\) 0 0
\(113\) −0.260651 + 0.150487i −0.0245200 + 0.0141566i −0.512210 0.858860i \(-0.671173\pi\)
0.487690 + 0.873017i \(0.337840\pi\)
\(114\) 0 0
\(115\) 2.39123 4.14173i 0.222983 0.386218i
\(116\) 0 0
\(117\) −17.5797 7.25368i −1.62525 0.670603i
\(118\) 0 0
\(119\) −0.481989 + 0.834830i −0.0441839 + 0.0765287i
\(120\) 0 0
\(121\) −2.40634 4.16791i −0.218758 0.378901i
\(122\) 0 0
\(123\) −0.985097 14.8742i −0.0888232 1.34116i
\(124\) 0 0
\(125\) 11.8119 1.05649
\(126\) 0 0
\(127\) 18.6505i 1.65497i −0.561490 0.827484i \(-0.689772\pi\)
0.561490 0.827484i \(-0.310228\pi\)
\(128\) 0 0
\(129\) −5.12883 7.66451i −0.451568 0.674823i
\(130\) 0 0
\(131\) 7.75570 4.47775i 0.677618 0.391223i −0.121339 0.992611i \(-0.538719\pi\)
0.798957 + 0.601388i \(0.205385\pi\)
\(132\) 0 0
\(133\) 6.16046 + 3.55675i 0.534180 + 0.308409i
\(134\) 0 0
\(135\) −5.36773 6.09552i −0.461981 0.524618i
\(136\) 0 0
\(137\) 8.59826 + 4.96421i 0.734599 + 0.424121i 0.820102 0.572217i \(-0.193917\pi\)
−0.0855034 + 0.996338i \(0.527250\pi\)
\(138\) 0 0
\(139\) 0.970830 + 1.68153i 0.0823448 + 0.142625i 0.904257 0.426989i \(-0.140426\pi\)
−0.821912 + 0.569615i \(0.807092\pi\)
\(140\) 0 0
\(141\) −3.34170 4.99383i −0.281422 0.420556i
\(142\) 0 0
\(143\) −15.7682 −1.31860
\(144\) 0 0
\(145\) −2.40030 −0.199334
\(146\) 0 0
\(147\) −9.99841 + 0.662181i −0.824655 + 0.0546158i
\(148\) 0 0
\(149\) 9.65962 + 16.7310i 0.791347 + 1.37065i 0.925133 + 0.379643i \(0.123953\pi\)
−0.133786 + 0.991010i \(0.542713\pi\)
\(150\) 0 0
\(151\) 9.90266 + 5.71730i 0.805867 + 0.465267i 0.845519 0.533946i \(-0.179291\pi\)
−0.0396516 + 0.999214i \(0.512625\pi\)
\(152\) 0 0
\(153\) −2.07947 + 1.60015i −0.168115 + 0.129364i
\(154\) 0 0
\(155\) 9.84243 + 5.68253i 0.790563 + 0.456432i
\(156\) 0 0
\(157\) 15.2905 8.82799i 1.22032 0.704551i 0.255332 0.966854i \(-0.417815\pi\)
0.964986 + 0.262303i \(0.0844820\pi\)
\(158\) 0 0
\(159\) −8.51566 + 17.2978i −0.675336 + 1.37180i
\(160\) 0 0
\(161\) 3.37220i 0.265767i
\(162\) 0 0
\(163\) 5.64209 0.441922 0.220961 0.975283i \(-0.429081\pi\)
0.220961 + 0.975283i \(0.429081\pi\)
\(164\) 0 0
\(165\) −6.04189 2.97441i −0.470361 0.231557i
\(166\) 0 0
\(167\) 11.6305 + 20.1446i 0.899993 + 1.55883i 0.827501 + 0.561465i \(0.189762\pi\)
0.0724922 + 0.997369i \(0.476905\pi\)
\(168\) 0 0
\(169\) 13.5924 23.5427i 1.04557 1.81098i
\(170\) 0 0
\(171\) 11.8080 + 15.3451i 0.902979 + 1.17347i
\(172\) 0 0
\(173\) −5.69231 + 9.85937i −0.432778 + 0.749594i −0.997111 0.0759531i \(-0.975800\pi\)
0.564333 + 0.825547i \(0.309133\pi\)
\(174\) 0 0
\(175\) −2.44042 + 1.40898i −0.184479 + 0.106509i
\(176\) 0 0
\(177\) 0.233654 + 3.52799i 0.0175625 + 0.265180i
\(178\) 0 0
\(179\) 10.6783i 0.798133i 0.916922 + 0.399066i \(0.130666\pi\)
−0.916922 + 0.399066i \(0.869334\pi\)
\(180\) 0 0
\(181\) 5.74638i 0.427125i 0.976929 + 0.213562i \(0.0685067\pi\)
−0.976929 + 0.213562i \(0.931493\pi\)
\(182\) 0 0
\(183\) 10.3568 6.93043i 0.765598 0.512312i
\(184\) 0 0
\(185\) −2.36403 + 1.36487i −0.173807 + 0.100348i
\(186\) 0 0
\(187\) −1.08778 + 1.88409i −0.0795464 + 0.137778i
\(188\) 0 0
\(189\) 5.42684 + 1.82979i 0.394745 + 0.133097i
\(190\) 0 0
\(191\) −8.87439 + 15.3709i −0.642129 + 1.11220i 0.342828 + 0.939398i \(0.388615\pi\)
−0.984957 + 0.172801i \(0.944718\pi\)
\(192\) 0 0
\(193\) −10.5823 18.3291i −0.761732 1.31936i −0.941957 0.335733i \(-0.891016\pi\)
0.180225 0.983625i \(-0.442317\pi\)
\(194\) 0 0
\(195\) 14.2634 9.54460i 1.02143 0.683503i
\(196\) 0 0
\(197\) −5.51688 −0.393061 −0.196531 0.980498i \(-0.562968\pi\)
−0.196531 + 0.980498i \(0.562968\pi\)
\(198\) 0 0
\(199\) 6.90420i 0.489426i 0.969596 + 0.244713i \(0.0786938\pi\)
−0.969596 + 0.244713i \(0.921306\pi\)
\(200\) 0 0
\(201\) 3.94377 0.261191i 0.278172 0.0184230i
\(202\) 0 0
\(203\) 1.46575 0.846250i 0.102875 0.0593951i
\(204\) 0 0
\(205\) 11.6503 + 6.72632i 0.813694 + 0.469786i
\(206\) 0 0
\(207\) −3.50102 + 8.48493i −0.243338 + 0.589743i
\(208\) 0 0
\(209\) 13.9033 + 8.02707i 0.961711 + 0.555244i
\(210\) 0 0
\(211\) 3.74236 + 6.48196i 0.257635 + 0.446237i 0.965608 0.260003i \(-0.0837235\pi\)
−0.707973 + 0.706240i \(0.750390\pi\)
\(212\) 0 0
\(213\) 10.0402 20.3946i 0.687946 1.39742i
\(214\) 0 0
\(215\) 8.32262 0.567598
\(216\) 0 0
\(217\) −8.01372 −0.544007
\(218\) 0 0
\(219\) −3.77117 + 7.66035i −0.254832 + 0.517639i
\(220\) 0 0
\(221\) −2.77218 4.80155i −0.186477 0.322987i
\(222\) 0 0
\(223\) 17.3052 + 9.99117i 1.15884 + 0.669058i 0.951026 0.309110i \(-0.100031\pi\)
0.207816 + 0.978168i \(0.433364\pi\)
\(224\) 0 0
\(225\) −7.60324 + 1.01154i −0.506883 + 0.0674361i
\(226\) 0 0
\(227\) 18.3301 + 10.5829i 1.21661 + 0.702409i 0.964191 0.265209i \(-0.0854410\pi\)
0.252417 + 0.967618i \(0.418774\pi\)
\(228\) 0 0
\(229\) 3.97745 2.29638i 0.262838 0.151749i −0.362791 0.931871i \(-0.618176\pi\)
0.625628 + 0.780121i \(0.284843\pi\)
\(230\) 0 0
\(231\) 4.73814 0.313801i 0.311747 0.0206466i
\(232\) 0 0
\(233\) 19.1408i 1.25396i −0.779037 0.626979i \(-0.784291\pi\)
0.779037 0.626979i \(-0.215709\pi\)
\(234\) 0 0
\(235\) 5.42262 0.353733
\(236\) 0 0
\(237\) 5.51775 3.69229i 0.358416 0.239840i
\(238\) 0 0
\(239\) −12.0889 20.9385i −0.781964 1.35440i −0.930796 0.365539i \(-0.880885\pi\)
0.148832 0.988862i \(-0.452449\pi\)
\(240\) 0 0
\(241\) −5.04705 + 8.74174i −0.325109 + 0.563105i −0.981534 0.191286i \(-0.938734\pi\)
0.656426 + 0.754391i \(0.272068\pi\)
\(242\) 0 0
\(243\) 11.7550 + 10.2381i 0.754084 + 0.656778i
\(244\) 0 0
\(245\) 4.52142 7.83133i 0.288863 0.500325i
\(246\) 0 0
\(247\) −35.4321 + 20.4568i −2.25449 + 1.30163i
\(248\) 0 0
\(249\) −18.6493 + 12.4795i −1.18185 + 0.790856i
\(250\) 0 0
\(251\) 17.6233i 1.11237i −0.831057 0.556187i \(-0.812264\pi\)
0.831057 0.556187i \(-0.187736\pi\)
\(252\) 0 0
\(253\) 7.61058i 0.478473i
\(254\) 0 0
\(255\) −0.156481 2.36273i −0.00979920 0.147960i
\(256\) 0 0
\(257\) 16.6994 9.64142i 1.04168 0.601415i 0.121372 0.992607i \(-0.461271\pi\)
0.920309 + 0.391192i \(0.127937\pi\)
\(258\) 0 0
\(259\) 0.962399 1.66692i 0.0598005 0.103578i
\(260\) 0 0
\(261\) 4.56660 0.607544i 0.282666 0.0376061i
\(262\) 0 0
\(263\) 0.912358 1.58025i 0.0562584 0.0974424i −0.836525 0.547929i \(-0.815416\pi\)
0.892783 + 0.450487i \(0.148750\pi\)
\(264\) 0 0
\(265\) −8.69977 15.0684i −0.534422 0.925647i
\(266\) 0 0
\(267\) 20.7232 + 10.2020i 1.26824 + 0.624352i
\(268\) 0 0
\(269\) −2.48843 −0.151722 −0.0758610 0.997118i \(-0.524171\pi\)
−0.0758610 + 0.997118i \(0.524171\pi\)
\(270\) 0 0
\(271\) 8.04111i 0.488463i −0.969717 0.244231i \(-0.921464\pi\)
0.969717 0.244231i \(-0.0785356\pi\)
\(272\) 0 0
\(273\) −5.34494 + 10.8571i −0.323490 + 0.657103i
\(274\) 0 0
\(275\) −5.50769 + 3.17987i −0.332126 + 0.191753i
\(276\) 0 0
\(277\) 4.27917 + 2.47058i 0.257110 + 0.148443i 0.623016 0.782209i \(-0.285907\pi\)
−0.365905 + 0.930652i \(0.619241\pi\)
\(278\) 0 0
\(279\) −20.1636 8.31984i −1.20717 0.498096i
\(280\) 0 0
\(281\) 12.4468 + 7.18614i 0.742511 + 0.428689i 0.822982 0.568068i \(-0.192309\pi\)
−0.0804703 + 0.996757i \(0.525642\pi\)
\(282\) 0 0
\(283\) 16.3639 + 28.3432i 0.972736 + 1.68483i 0.687214 + 0.726455i \(0.258833\pi\)
0.285522 + 0.958372i \(0.407833\pi\)
\(284\) 0 0
\(285\) −17.4353 + 1.15472i −1.03278 + 0.0683996i
\(286\) 0 0
\(287\) −9.48571 −0.559924
\(288\) 0 0
\(289\) 16.2350 0.955002
\(290\) 0 0
\(291\) −8.96354 13.3951i −0.525452 0.785234i
\(292\) 0 0
\(293\) −2.68074 4.64317i −0.156610 0.271257i 0.777034 0.629459i \(-0.216723\pi\)
−0.933644 + 0.358202i \(0.883390\pi\)
\(294\) 0 0
\(295\) −2.76332 1.59540i −0.160887 0.0928880i
\(296\) 0 0
\(297\) 12.2476 + 4.12957i 0.710679 + 0.239622i
\(298\) 0 0
\(299\) −16.7969 9.69767i −0.971387 0.560831i
\(300\) 0 0
\(301\) −5.08222 + 2.93422i −0.292934 + 0.169126i
\(302\) 0 0
\(303\) 17.6365 + 26.3559i 1.01319 + 1.51411i
\(304\) 0 0
\(305\) 11.2461i 0.643950i
\(306\) 0 0
\(307\) −11.1124 −0.634219 −0.317110 0.948389i \(-0.602712\pi\)
−0.317110 + 0.948389i \(0.602712\pi\)
\(308\) 0 0
\(309\) 1.86047 + 28.0916i 0.105838 + 1.59807i
\(310\) 0 0
\(311\) 10.4892 + 18.1679i 0.594790 + 1.03021i 0.993576 + 0.113163i \(0.0360981\pi\)
−0.398787 + 0.917044i \(0.630569\pi\)
\(312\) 0 0
\(313\) −11.3679 + 19.6898i −0.642553 + 1.11293i 0.342308 + 0.939588i \(0.388791\pi\)
−0.984861 + 0.173347i \(0.944542\pi\)
\(314\) 0 0
\(315\) −4.09604 + 3.15189i −0.230785 + 0.177589i
\(316\) 0 0
\(317\) −14.6224 + 25.3267i −0.821276 + 1.42249i 0.0834573 + 0.996511i \(0.473404\pi\)
−0.904733 + 0.425980i \(0.859930\pi\)
\(318\) 0 0
\(319\) 3.30799 1.90987i 0.185212 0.106932i
\(320\) 0 0
\(321\) 9.01396 + 4.43755i 0.503110 + 0.247680i
\(322\) 0 0
\(323\) 5.64490i 0.314091i
\(324\) 0 0
\(325\) 16.2076i 0.899035i
\(326\) 0 0
\(327\) −11.4180 5.62105i −0.631416 0.310845i
\(328\) 0 0
\(329\) −3.31133 + 1.91180i −0.182559 + 0.105401i
\(330\) 0 0
\(331\) 13.2737 22.9908i 0.729591 1.26369i −0.227465 0.973786i \(-0.573044\pi\)
0.957056 0.289903i \(-0.0936229\pi\)
\(332\) 0 0
\(333\) 4.15213 3.19505i 0.227535 0.175088i
\(334\) 0 0
\(335\) −1.78343 + 3.08899i −0.0974392 + 0.168770i
\(336\) 0 0
\(337\) 4.28240 + 7.41733i 0.233277 + 0.404048i 0.958771 0.284181i \(-0.0917217\pi\)
−0.725493 + 0.688229i \(0.758388\pi\)
\(338\) 0 0
\(339\) 0.0344497 + 0.520163i 0.00187105 + 0.0282514i
\(340\) 0 0
\(341\) −18.0858 −0.979402
\(342\) 0 0
\(343\) 14.0914i 0.760866i
\(344\) 0 0
\(345\) −4.60674 6.88430i −0.248019 0.370638i
\(346\) 0 0
\(347\) 11.8860 6.86239i 0.638075 0.368393i −0.145798 0.989314i \(-0.546575\pi\)
0.783872 + 0.620922i \(0.213242\pi\)
\(348\) 0 0
\(349\) −11.2390 6.48885i −0.601611 0.347340i 0.168064 0.985776i \(-0.446248\pi\)
−0.769675 + 0.638436i \(0.779582\pi\)
\(350\) 0 0
\(351\) −24.7205 + 21.7689i −1.31948 + 1.16194i
\(352\) 0 0
\(353\) −22.9788 13.2668i −1.22304 0.706120i −0.257472 0.966286i \(-0.582889\pi\)
−0.965564 + 0.260166i \(0.916223\pi\)
\(354\) 0 0
\(355\) 10.2573 + 17.7662i 0.544401 + 0.942930i
\(356\) 0 0
\(357\) 0.928560 + 1.38764i 0.0491446 + 0.0734416i
\(358\) 0 0
\(359\) −5.61198 −0.296189 −0.148095 0.988973i \(-0.547314\pi\)
−0.148095 + 0.988973i \(0.547314\pi\)
\(360\) 0 0
\(361\) 22.6555 1.19239
\(362\) 0 0
\(363\) −8.31759 + 0.550863i −0.436560 + 0.0289128i
\(364\) 0 0
\(365\) −3.85270 6.67308i −0.201660 0.349285i
\(366\) 0 0
\(367\) 20.8358 + 12.0295i 1.08762 + 0.627937i 0.932941 0.360029i \(-0.117233\pi\)
0.154677 + 0.987965i \(0.450566\pi\)
\(368\) 0 0
\(369\) −23.8674 9.84806i −1.24249 0.512670i
\(370\) 0 0
\(371\) 10.6250 + 6.13437i 0.551625 + 0.318481i
\(372\) 0 0
\(373\) 14.8743 8.58769i 0.770163 0.444654i −0.0627699 0.998028i \(-0.519993\pi\)
0.832933 + 0.553374i \(0.186660\pi\)
\(374\) 0 0
\(375\) 9.03617 18.3551i 0.466626 0.947853i
\(376\) 0 0
\(377\) 9.73448i 0.501351i
\(378\) 0 0
\(379\) −30.5262 −1.56803 −0.784013 0.620745i \(-0.786830\pi\)
−0.784013 + 0.620745i \(0.786830\pi\)
\(380\) 0 0
\(381\) −28.9820 14.2678i −1.48479 0.730961i
\(382\) 0 0
\(383\) 8.10743 + 14.0425i 0.414270 + 0.717537i 0.995352 0.0963082i \(-0.0307034\pi\)
−0.581081 + 0.813846i \(0.697370\pi\)
\(384\) 0 0
\(385\) −2.14266 + 3.71119i −0.109200 + 0.189140i
\(386\) 0 0
\(387\) −15.8339 + 2.10655i −0.804881 + 0.107082i
\(388\) 0 0
\(389\) 12.8869 22.3207i 0.653391 1.13171i −0.328903 0.944364i \(-0.606679\pi\)
0.982295 0.187343i \(-0.0599877\pi\)
\(390\) 0 0
\(391\) −2.31749 + 1.33800i −0.117200 + 0.0676657i
\(392\) 0 0
\(393\) −1.02505 15.4775i −0.0517070 0.780736i
\(394\) 0 0
\(395\) 5.99153i 0.301466i
\(396\) 0 0
\(397\) 9.36711i 0.470122i −0.971981 0.235061i \(-0.924471\pi\)
0.971981 0.235061i \(-0.0755290\pi\)
\(398\) 0 0
\(399\) 10.2398 6.85213i 0.512631 0.343035i
\(400\) 0 0
\(401\) −4.16389 + 2.40402i −0.207935 + 0.120051i −0.600351 0.799736i \(-0.704973\pi\)
0.392416 + 0.919788i \(0.371639\pi\)
\(402\) 0 0
\(403\) 23.0456 39.9161i 1.14798 1.98836i
\(404\) 0 0
\(405\) −13.5785 + 3.67809i −0.674720 + 0.182766i
\(406\) 0 0
\(407\) 2.17200 3.76201i 0.107662 0.186476i
\(408\) 0 0
\(409\) 18.1532 + 31.4422i 0.897616 + 1.55472i 0.830534 + 0.556968i \(0.188036\pi\)
0.0670820 + 0.997747i \(0.478631\pi\)
\(410\) 0 0
\(411\) 14.2919 9.56363i 0.704966 0.471739i
\(412\) 0 0
\(413\) 2.24990 0.110710
\(414\) 0 0
\(415\) 20.2506i 0.994065i
\(416\) 0 0
\(417\) 3.35570 0.222244i 0.164330 0.0108833i
\(418\) 0 0
\(419\) −7.72947 + 4.46261i −0.377609 + 0.218013i −0.676778 0.736188i \(-0.736624\pi\)
0.299168 + 0.954200i \(0.403291\pi\)
\(420\) 0 0
\(421\) −0.670604 0.387173i −0.0326832 0.0188697i 0.483569 0.875306i \(-0.339340\pi\)
−0.516253 + 0.856436i \(0.672673\pi\)
\(422\) 0 0
\(423\) −10.3166 + 1.37253i −0.501610 + 0.0667345i
\(424\) 0 0
\(425\) −1.93659 1.11809i −0.0939386 0.0542355i
\(426\) 0 0
\(427\) −3.96492 6.86744i −0.191876 0.332339i
\(428\) 0 0
\(429\) −12.0628 + 24.5030i −0.582396 + 1.18302i
\(430\) 0 0
\(431\) −20.8535 −1.00448 −0.502238 0.864730i \(-0.667490\pi\)
−0.502238 + 0.864730i \(0.667490\pi\)
\(432\) 0 0
\(433\) −5.02013 −0.241252 −0.120626 0.992698i \(-0.538490\pi\)
−0.120626 + 0.992698i \(0.538490\pi\)
\(434\) 0 0
\(435\) −1.83625 + 3.72996i −0.0880414 + 0.178838i
\(436\) 0 0
\(437\) 9.87354 + 17.1015i 0.472315 + 0.818074i
\(438\) 0 0
\(439\) −22.7559 13.1381i −1.08608 0.627049i −0.153550 0.988141i \(-0.549071\pi\)
−0.932530 + 0.361092i \(0.882404\pi\)
\(440\) 0 0
\(441\) −6.61985 + 16.0436i −0.315231 + 0.763982i
\(442\) 0 0
\(443\) 25.6178 + 14.7905i 1.21714 + 0.702716i 0.964305 0.264793i \(-0.0853037\pi\)
0.252835 + 0.967509i \(0.418637\pi\)
\(444\) 0 0
\(445\) −18.0524 + 10.4226i −0.855765 + 0.494076i
\(446\) 0 0
\(447\) 33.3888 2.21129i 1.57923 0.104591i
\(448\) 0 0
\(449\) 1.43585i 0.0677617i 0.999426 + 0.0338809i \(0.0107867\pi\)
−0.999426 + 0.0338809i \(0.989213\pi\)
\(450\) 0 0
\(451\) −21.4079 −1.00806
\(452\) 0 0
\(453\) 16.4600 11.0145i 0.773359 0.517505i
\(454\) 0 0
\(455\) −5.46049 9.45785i −0.255992 0.443391i
\(456\) 0 0
\(457\) 2.66816 4.62138i 0.124811 0.216179i −0.796848 0.604180i \(-0.793501\pi\)
0.921659 + 0.388001i \(0.126834\pi\)
\(458\) 0 0
\(459\) 0.895741 + 4.45552i 0.0418096 + 0.207966i
\(460\) 0 0
\(461\) −4.71373 + 8.16442i −0.219540 + 0.380255i −0.954668 0.297674i \(-0.903789\pi\)
0.735127 + 0.677929i \(0.237122\pi\)
\(462\) 0 0
\(463\) 13.0444 7.53117i 0.606223 0.350003i −0.165262 0.986250i \(-0.552847\pi\)
0.771486 + 0.636246i \(0.219514\pi\)
\(464\) 0 0
\(465\) 16.3599 10.9475i 0.758672 0.507677i
\(466\) 0 0
\(467\) 22.9660i 1.06274i −0.847140 0.531369i \(-0.821678\pi\)
0.847140 0.531369i \(-0.178322\pi\)
\(468\) 0 0
\(469\) 2.51506i 0.116135i
\(470\) 0 0
\(471\) −2.02091 30.5142i −0.0931188 1.40602i
\(472\) 0 0
\(473\) −11.4698 + 6.62212i −0.527384 + 0.304485i
\(474\) 0 0
\(475\) −8.25075 + 14.2907i −0.378571 + 0.655703i
\(476\) 0 0
\(477\) 20.3654 + 26.4658i 0.932467 + 1.21179i
\(478\) 0 0
\(479\) 15.2716 26.4512i 0.697779 1.20859i −0.271456 0.962451i \(-0.587505\pi\)
0.969235 0.246137i \(-0.0791614\pi\)
\(480\) 0 0
\(481\) 5.53527 + 9.58737i 0.252387 + 0.437146i
\(482\) 0 0
\(483\) 5.24024 + 2.57976i 0.238439 + 0.117383i
\(484\) 0 0
\(485\) 14.5453 0.660466
\(486\) 0 0
\(487\) 25.2482i 1.14411i −0.820217 0.572053i \(-0.806147\pi\)
0.820217 0.572053i \(-0.193853\pi\)
\(488\) 0 0
\(489\) 4.31623 8.76753i 0.195187 0.396481i
\(490\) 0 0
\(491\) −28.5831 + 16.5025i −1.28994 + 0.744746i −0.978643 0.205568i \(-0.934096\pi\)
−0.311295 + 0.950313i \(0.600763\pi\)
\(492\) 0 0
\(493\) 1.16314 + 0.671541i 0.0523853 + 0.0302447i
\(494\) 0 0
\(495\) −9.24417 + 7.11336i −0.415495 + 0.319722i
\(496\) 0 0
\(497\) −12.5273 7.23262i −0.561924 0.324427i
\(498\) 0 0
\(499\) 8.61075 + 14.9143i 0.385470 + 0.667654i 0.991834 0.127533i \(-0.0407060\pi\)
−0.606364 + 0.795187i \(0.707373\pi\)
\(500\) 0 0
\(501\) 40.2011 2.66246i 1.79605 0.118950i
\(502\) 0 0
\(503\) 39.7159 1.77084 0.885421 0.464789i \(-0.153870\pi\)
0.885421 + 0.464789i \(0.153870\pi\)
\(504\) 0 0
\(505\) −28.6189 −1.27353
\(506\) 0 0
\(507\) −26.1859 39.1322i −1.16296 1.73792i
\(508\) 0 0
\(509\) −16.7472 29.0071i −0.742309 1.28572i −0.951442 0.307829i \(-0.900398\pi\)
0.209133 0.977887i \(-0.432936\pi\)
\(510\) 0 0
\(511\) 4.70532 + 2.71662i 0.208151 + 0.120176i
\(512\) 0 0
\(513\) 32.8787 6.60995i 1.45163 0.291836i
\(514\) 0 0
\(515\) −22.0029 12.7034i −0.969566 0.559779i
\(516\) 0 0
\(517\) −7.47320 + 4.31465i −0.328671 + 0.189758i
\(518\) 0 0
\(519\) 10.9663 + 16.3881i 0.481368 + 0.719356i
\(520\) 0 0
\(521\) 19.2115i 0.841671i −0.907137 0.420836i \(-0.861737\pi\)
0.907137 0.420836i \(-0.138263\pi\)
\(522\) 0 0
\(523\) −21.9153 −0.958290 −0.479145 0.877736i \(-0.659053\pi\)
−0.479145 + 0.877736i \(0.659053\pi\)
\(524\) 0 0
\(525\) 0.322545 + 4.87018i 0.0140770 + 0.212552i
\(526\) 0 0
\(527\) −3.17964 5.50729i −0.138507 0.239901i
\(528\) 0 0
\(529\) 6.81938 11.8115i 0.296495 0.513544i
\(530\) 0 0
\(531\) 5.66106 + 2.33585i 0.245669 + 0.101367i
\(532\) 0 0
\(533\) 27.2787 47.2481i 1.18157 2.04654i
\(534\) 0 0
\(535\) −7.85223 + 4.53349i −0.339481 + 0.196000i
\(536\) 0 0
\(537\) 16.5935 + 8.16896i 0.716064 + 0.352517i
\(538\) 0 0
\(539\) 14.3904i 0.619837i
\(540\) 0 0
\(541\) 18.1171i 0.778913i 0.921045 + 0.389456i \(0.127337\pi\)
−0.921045 + 0.389456i \(0.872663\pi\)
\(542\) 0 0
\(543\) 8.92959 + 4.39602i 0.383205 + 0.188651i
\(544\) 0 0
\(545\) 9.94643 5.74257i 0.426058 0.245985i
\(546\) 0 0
\(547\) −4.95910 + 8.58941i −0.212036 + 0.367257i −0.952352 0.305003i \(-0.901343\pi\)
0.740316 + 0.672259i \(0.234676\pi\)
\(548\) 0 0
\(549\) −2.84652 21.3958i −0.121486 0.913151i
\(550\) 0 0
\(551\) 4.95551 8.58319i 0.211112 0.365656i
\(552\) 0 0
\(553\) −2.11237 3.65873i −0.0898271 0.155585i
\(554\) 0 0
\(555\) 0.312448 + 4.71773i 0.0132627 + 0.200256i
\(556\) 0 0
\(557\) 9.59817 0.406687 0.203344 0.979107i \(-0.434819\pi\)
0.203344 + 0.979107i \(0.434819\pi\)
\(558\) 0 0
\(559\) 33.7525i 1.42758i
\(560\) 0 0
\(561\) 2.09563 + 3.13170i 0.0884775 + 0.132221i
\(562\) 0 0
\(563\) −13.9909 + 8.07768i −0.589648 + 0.340433i −0.764958 0.644080i \(-0.777240\pi\)
0.175310 + 0.984513i \(0.443907\pi\)
\(564\) 0 0
\(565\) −0.407422 0.235225i −0.0171404 0.00989599i
\(566\) 0 0
\(567\) 6.99497 7.03325i 0.293761 0.295369i
\(568\) 0 0
\(569\) −22.7099 13.1116i −0.952048 0.549665i −0.0583315 0.998297i \(-0.518578\pi\)
−0.893717 + 0.448632i \(0.851911\pi\)
\(570\) 0 0
\(571\) 3.52667 + 6.10837i 0.147587 + 0.255628i 0.930335 0.366711i \(-0.119516\pi\)
−0.782748 + 0.622338i \(0.786183\pi\)
\(572\) 0 0
\(573\) 17.0967 + 25.5492i 0.714223 + 1.06733i
\(574\) 0 0
\(575\) −7.82266 −0.326227
\(576\) 0 0
\(577\) −19.1200 −0.795978 −0.397989 0.917390i \(-0.630292\pi\)
−0.397989 + 0.917390i \(0.630292\pi\)
\(578\) 0 0
\(579\) −36.5781 + 2.42252i −1.52013 + 0.100676i
\(580\) 0 0
\(581\) 7.13956 + 12.3661i 0.296199 + 0.513031i
\(582\) 0 0
\(583\) 23.9792 + 13.8444i 0.993118 + 0.573377i
\(584\) 0 0
\(585\) −3.92023 29.4664i −0.162081 1.21828i
\(586\) 0 0
\(587\) −20.9418 12.0908i −0.864361 0.499039i 0.00110929 0.999999i \(-0.499647\pi\)
−0.865470 + 0.500960i \(0.832980\pi\)
\(588\) 0 0
\(589\) −40.6400 + 23.4635i −1.67454 + 0.966798i
\(590\) 0 0
\(591\) −4.22045 + 8.57296i −0.173606 + 0.352644i
\(592\) 0 0
\(593\) 15.8938i 0.652681i −0.945252 0.326340i \(-0.894184\pi\)
0.945252 0.326340i \(-0.105816\pi\)
\(594\) 0 0
\(595\) −1.50679 −0.0617722
\(596\) 0 0
\(597\) 10.7288 + 5.28176i 0.439100 + 0.216168i
\(598\) 0 0
\(599\) −0.540573 0.936300i −0.0220872 0.0382562i 0.854771 0.519006i \(-0.173698\pi\)
−0.876858 + 0.480750i \(0.840364\pi\)
\(600\) 0 0
\(601\) −7.09881 + 12.2955i −0.289566 + 0.501544i −0.973706 0.227807i \(-0.926844\pi\)
0.684140 + 0.729351i \(0.260178\pi\)
\(602\) 0 0
\(603\) 2.61113 6.32824i 0.106334 0.257706i
\(604\) 0 0
\(605\) 3.76133 6.51482i 0.152920 0.264865i
\(606\) 0 0
\(607\) −3.34733 + 1.93258i −0.135864 + 0.0784411i −0.566391 0.824136i \(-0.691661\pi\)
0.430527 + 0.902578i \(0.358328\pi\)
\(608\) 0 0
\(609\) −0.193725 2.92509i −0.00785011 0.118531i
\(610\) 0 0
\(611\) 21.9915i 0.889682i
\(612\) 0 0
\(613\) 5.87436i 0.237263i 0.992938 + 0.118632i \(0.0378507\pi\)
−0.992938 + 0.118632i \(0.962149\pi\)
\(614\) 0 0
\(615\) 19.3649 12.9584i 0.780870 0.522532i
\(616\) 0 0
\(617\) 21.7236 12.5421i 0.874559 0.504927i 0.00569811 0.999984i \(-0.498186\pi\)
0.868860 + 0.495057i \(0.164853\pi\)
\(618\) 0 0
\(619\) 18.8316 32.6172i 0.756904 1.31100i −0.187518 0.982261i \(-0.560044\pi\)
0.944422 0.328735i \(-0.106622\pi\)
\(620\) 0 0
\(621\) 10.5069 + 11.9314i 0.421626 + 0.478792i
\(622\) 0 0
\(623\) 7.34914 12.7291i 0.294437 0.509980i
\(624\) 0 0
\(625\) 2.83966 + 4.91844i 0.113586 + 0.196737i
\(626\) 0 0
\(627\) 23.1098 15.4643i 0.922916 0.617584i
\(628\) 0 0
\(629\) 1.52742 0.0609022
\(630\) 0 0
\(631\) 1.34576i 0.0535740i 0.999641 + 0.0267870i \(0.00852759\pi\)
−0.999641 + 0.0267870i \(0.991472\pi\)
\(632\) 0 0
\(633\) 12.9356 0.856706i 0.514143 0.0340510i
\(634\) 0 0
\(635\) 25.2468 14.5762i 1.00189 0.578441i
\(636\) 0 0
\(637\) −31.7601 18.3367i −1.25838 0.726527i
\(638\) 0 0
\(639\) −24.0114 31.2041i −0.949878 1.23441i
\(640\) 0 0
\(641\) −12.5197 7.22823i −0.494497 0.285498i 0.231941 0.972730i \(-0.425492\pi\)
−0.726438 + 0.687232i \(0.758826\pi\)
\(642\) 0 0
\(643\) −12.0008 20.7861i −0.473267 0.819722i 0.526265 0.850321i \(-0.323592\pi\)
−0.999532 + 0.0305983i \(0.990259\pi\)
\(644\) 0 0
\(645\) 6.36686 12.9329i 0.250695 0.509234i
\(646\) 0 0
\(647\) −8.55443 −0.336309 −0.168155 0.985761i \(-0.553781\pi\)
−0.168155 + 0.985761i \(0.553781\pi\)
\(648\) 0 0
\(649\) 5.07771 0.199317
\(650\) 0 0
\(651\) −6.13055 + 12.4529i −0.240275 + 0.488069i
\(652\) 0 0
\(653\) −6.58668 11.4085i −0.257757 0.446448i 0.707884 0.706329i \(-0.249650\pi\)
−0.965641 + 0.259881i \(0.916317\pi\)
\(654\) 0 0
\(655\) 12.1229 + 6.99914i 0.473679 + 0.273479i
\(656\) 0 0
\(657\) 9.01884 + 11.7204i 0.351859 + 0.457258i
\(658\) 0 0
\(659\) −10.6191 6.13093i −0.413661 0.238827i 0.278701 0.960378i \(-0.410096\pi\)
−0.692361 + 0.721551i \(0.743430\pi\)
\(660\) 0 0
\(661\) −12.6734 + 7.31697i −0.492937 + 0.284597i −0.725792 0.687914i \(-0.758526\pi\)
0.232855 + 0.972511i \(0.425193\pi\)
\(662\) 0 0
\(663\) −9.58211 + 0.634610i −0.372138 + 0.0246462i
\(664\) 0 0
\(665\) 11.1190i 0.431178i
\(666\) 0 0
\(667\) 4.69838 0.181922
\(668\) 0 0
\(669\) 28.7644 19.2482i 1.11210 0.744177i
\(670\) 0 0
\(671\) −8.94826 15.4988i −0.345444 0.598326i
\(672\) 0 0
\(673\) −6.35891 + 11.0140i −0.245118 + 0.424557i −0.962165 0.272468i \(-0.912160\pi\)
0.717047 + 0.697025i \(0.245493\pi\)
\(674\) 0 0
\(675\) −4.24465 + 12.5889i −0.163376 + 0.484547i
\(676\) 0 0
\(677\) 13.3262 23.0816i 0.512167 0.887099i −0.487734 0.872992i \(-0.662176\pi\)
0.999901 0.0141062i \(-0.00449029\pi\)
\(678\) 0 0
\(679\) −8.88207 + 5.12807i −0.340863 + 0.196797i
\(680\) 0 0
\(681\) 30.4679 20.3881i 1.16753 0.781272i
\(682\) 0 0
\(683\) 22.4974i 0.860841i −0.902629 0.430420i \(-0.858365\pi\)
0.902629 0.430420i \(-0.141635\pi\)
\(684\) 0 0
\(685\) 15.5190i 0.592951i
\(686\) 0 0
\(687\) −0.525691 7.93752i −0.0200564 0.302835i
\(688\) 0 0
\(689\) −61.1103 + 35.2821i −2.32812 + 1.34414i
\(690\) 0 0
\(691\) −5.25993 + 9.11047i −0.200097 + 0.346579i −0.948560 0.316599i \(-0.897459\pi\)
0.748462 + 0.663177i \(0.230793\pi\)
\(692\) 0 0
\(693\) 3.13708 7.60290i 0.119168 0.288810i
\(694\) 0 0
\(695\) −1.51750 + 2.62838i −0.0575619 + 0.0997002i
\(696\) 0 0
\(697\) −3.76369 6.51890i −0.142560 0.246921i
\(698\) 0 0
\(699\) −29.7439 14.6429i −1.12502 0.553844i
\(700\) 0 0
\(701\) 30.8786 1.16627 0.583134 0.812376i \(-0.301826\pi\)
0.583134 + 0.812376i \(0.301826\pi\)
\(702\) 0 0
\(703\) 11.2713i 0.425105i
\(704\) 0 0
\(705\) 4.14834 8.42648i 0.156235 0.317360i
\(706\) 0 0
\(707\) 17.4762 10.0899i 0.657259 0.379469i
\(708\) 0 0
\(709\) −8.71948 5.03419i −0.327467 0.189063i 0.327249 0.944938i \(-0.393878\pi\)
−0.654716 + 0.755875i \(0.727212\pi\)
\(710\) 0 0
\(711\) −1.51652 11.3989i −0.0568741 0.427494i
\(712\) 0 0
\(713\) −19.2657 11.1230i −0.721506 0.416561i
\(714\) 0 0
\(715\) −12.3236 21.3450i −0.460875 0.798259i
\(716\) 0 0
\(717\) −41.7855 + 2.76740i −1.56051 + 0.103350i
\(718\) 0 0
\(719\) 2.52664 0.0942280 0.0471140 0.998890i \(-0.484998\pi\)
0.0471140 + 0.998890i \(0.484998\pi\)
\(720\) 0 0
\(721\) 17.9148 0.667183
\(722\) 0 0
\(723\) 9.72322 + 14.5304i 0.361610 + 0.540390i
\(724\) 0 0
\(725\) 1.96309 + 3.40017i 0.0729072 + 0.126279i
\(726\) 0 0
\(727\) 30.9471 + 17.8673i 1.14776 + 0.662662i 0.948341 0.317252i \(-0.102760\pi\)
0.199423 + 0.979914i \(0.436093\pi\)
\(728\) 0 0
\(729\) 24.9022 10.4345i 0.922305 0.386462i
\(730\) 0 0
\(731\) −4.03299 2.32845i −0.149165 0.0861207i
\(732\) 0 0
\(733\) 3.78054 2.18269i 0.139637 0.0806196i −0.428554 0.903516i \(-0.640977\pi\)
0.568191 + 0.822897i \(0.307643\pi\)
\(734\) 0 0
\(735\) −8.71059 13.0171i −0.321295 0.480143i
\(736\) 0 0
\(737\) 5.67614i 0.209083i
\(738\) 0 0
\(739\) −40.4265 −1.48711 −0.743556 0.668674i \(-0.766862\pi\)
−0.743556 + 0.668674i \(0.766862\pi\)
\(740\) 0 0
\(741\) 4.68298 + 70.7094i 0.172034 + 2.59757i
\(742\) 0 0
\(743\) 9.83837 + 17.0406i 0.360935 + 0.625158i 0.988115 0.153717i \(-0.0491243\pi\)
−0.627180 + 0.778874i \(0.715791\pi\)
\(744\) 0 0
\(745\) −15.0989 + 26.1520i −0.553180 + 0.958136i
\(746\) 0 0
\(747\) 5.12567 + 38.5270i 0.187538 + 1.40963i
\(748\) 0 0
\(749\) 3.19665 5.53676i 0.116803 0.202309i
\(750\) 0 0
\(751\) −26.3391 + 15.2069i −0.961126 + 0.554906i −0.896519 0.443005i \(-0.853913\pi\)
−0.0646065 + 0.997911i \(0.520579\pi\)
\(752\) 0 0
\(753\) −27.3858 13.4820i −0.997994 0.491310i
\(754\) 0 0
\(755\) 17.8733i 0.650477i
\(756\) 0 0
\(757\) 5.71127i 0.207580i −0.994599 0.103790i \(-0.966903\pi\)
0.994599 0.103790i \(-0.0330969\pi\)
\(758\) 0 0
\(759\) 11.8265 + 5.82214i 0.429274 + 0.211330i
\(760\) 0 0
\(761\) −32.8500 + 18.9660i −1.19081 + 0.687516i −0.958491 0.285123i \(-0.907966\pi\)
−0.232322 + 0.972639i \(0.574632\pi\)
\(762\) 0 0
\(763\) −4.04920 + 7.01342i −0.146591 + 0.253903i
\(764\) 0 0
\(765\) −3.79128 1.56434i −0.137074 0.0565590i
\(766\) 0 0
\(767\) −6.47019 + 11.2067i −0.233625 + 0.404651i
\(768\) 0 0
\(769\) 20.7986 + 36.0243i 0.750018 + 1.29907i 0.947813 + 0.318826i \(0.103289\pi\)
−0.197795 + 0.980243i \(0.563378\pi\)
\(770\) 0 0
\(771\) −2.20712 33.3258i −0.0794876 1.20020i
\(772\) 0 0
\(773\) −20.0762 −0.722090 −0.361045 0.932548i \(-0.617580\pi\)
−0.361045 + 0.932548i \(0.617580\pi\)
\(774\) 0 0
\(775\) 18.5898i 0.667765i
\(776\) 0 0
\(777\) −1.85408 2.77073i −0.0665146 0.0993993i
\(778\) 0 0
\(779\) −48.1049 + 27.7734i −1.72354 + 0.995085i
\(780\) 0 0
\(781\) −28.2723 16.3230i −1.01166 0.584083i
\(782\) 0 0
\(783\) 2.54939 7.56106i 0.0911077 0.270210i
\(784\) 0 0
\(785\) 23.9005 + 13.7990i 0.853045 + 0.492506i
\(786\) 0 0
\(787\) −4.62218 8.00585i −0.164763 0.285378i 0.771808 0.635855i \(-0.219353\pi\)
−0.936571 + 0.350478i \(0.886019\pi\)
\(788\) 0 0
\(789\) −1.75767 2.62666i −0.0625748 0.0935116i
\(790\) 0 0
\(791\) 0.331723 0.0117947
\(792\) 0 0
\(793\) 45.6088 1.61961
\(794\) 0 0
\(795\) −30.0710 + 1.99156i −1.06651 + 0.0706334i
\(796\) 0 0
\(797\) −5.18050 8.97288i −0.183503 0.317836i 0.759568 0.650428i \(-0.225410\pi\)
−0.943071 + 0.332592i \(0.892077\pi\)
\(798\) 0 0
\(799\) −2.62770 1.51710i −0.0929613 0.0536713i
\(800\) 0 0
\(801\) 31.7068 24.3983i 1.12030 0.862071i
\(802\) 0 0
\(803\) 10.6192 + 6.13102i 0.374745 + 0.216359i
\(804\) 0 0
\(805\) −4.56487 + 2.63553i −0.160891 + 0.0928902i
\(806\) 0 0
\(807\) −1.90366 + 3.86689i −0.0670121 + 0.136121i
\(808\) 0 0
\(809\) 12.1122i 0.425841i 0.977070 + 0.212921i \(0.0682976\pi\)
−0.977070 + 0.212921i \(0.931702\pi\)
\(810\) 0 0
\(811\) 43.7534 1.53639 0.768196 0.640215i \(-0.221155\pi\)
0.768196 + 0.640215i \(0.221155\pi\)
\(812\) 0 0
\(813\) −12.4955 6.15150i −0.438236 0.215743i
\(814\) 0 0
\(815\) 4.40955 + 7.63756i 0.154460 + 0.267532i
\(816\) 0 0
\(817\) −17.1823 + 29.7606i −0.601133 + 1.04119i
\(818\) 0 0
\(819\) 12.7825 + 16.6115i 0.446658 + 0.580454i
\(820\) 0 0
\(821\) 1.09747 1.90088i 0.0383021 0.0663411i −0.846239 0.532804i \(-0.821138\pi\)
0.884541 + 0.466463i \(0.154472\pi\)
\(822\) 0 0
\(823\) 11.3278 6.54013i 0.394864 0.227975i −0.289402 0.957208i \(-0.593456\pi\)
0.684265 + 0.729233i \(0.260123\pi\)
\(824\) 0 0
\(825\) 0.727938 + 10.9913i 0.0253436 + 0.382668i
\(826\) 0 0
\(827\) 5.42534i 0.188658i 0.995541 + 0.0943289i \(0.0300705\pi\)
−0.995541 + 0.0943289i \(0.969929\pi\)
\(828\) 0 0
\(829\) 1.91806i 0.0666169i −0.999445 0.0333085i \(-0.989396\pi\)
0.999445 0.0333085i \(-0.0106044\pi\)
\(830\) 0 0
\(831\) 7.11275 4.75961i 0.246739 0.165109i
\(832\) 0 0
\(833\) −4.38199 + 2.52995i −0.151827 + 0.0876574i
\(834\) 0 0
\(835\) −18.1795 + 31.4878i −0.629127 + 1.08968i
\(836\) 0 0
\(837\) −28.3539 + 24.9686i −0.980055 + 0.863040i
\(838\) 0 0
\(839\) 8.35203 14.4661i 0.288344 0.499427i −0.685071 0.728477i \(-0.740229\pi\)
0.973415 + 0.229050i \(0.0735620\pi\)
\(840\) 0 0
\(841\) 13.3209 + 23.0726i 0.459343 + 0.795605i
\(842\) 0 0
\(843\) 20.6888 13.8442i 0.712559 0.476820i
\(844\) 0 0
\(845\) 42.4923 1.46178
\(846\) 0 0
\(847\) 5.30437i 0.182260i
\(848\) 0 0
\(849\) 56.5625 3.74605i 1.94122 0.128564i
\(850\) 0 0
\(851\) 4.62738 2.67162i 0.158625 0.0915820i
\(852\) 0 0
\(853\) −38.1750 22.0403i −1.30709 0.754646i −0.325477 0.945550i \(-0.605525\pi\)
−0.981609 + 0.190904i \(0.938858\pi\)
\(854\) 0 0
\(855\) −11.5438 + 27.9770i −0.394789 + 0.956795i
\(856\) 0 0
\(857\) −35.6626 20.5898i −1.21821 0.703334i −0.253676 0.967289i \(-0.581640\pi\)
−0.964535 + 0.263955i \(0.914973\pi\)
\(858\) 0 0
\(859\) 20.7385 + 35.9201i 0.707587 + 1.22558i 0.965750 + 0.259476i \(0.0835498\pi\)
−0.258162 + 0.966102i \(0.583117\pi\)
\(860\) 0 0
\(861\) −7.25663 + 14.7403i −0.247305 + 0.502349i
\(862\) 0 0
\(863\) −1.67319 −0.0569560 −0.0284780 0.999594i \(-0.509066\pi\)
−0.0284780 + 0.999594i \(0.509066\pi\)
\(864\) 0 0
\(865\) −17.7952 −0.605055
\(866\) 0 0
\(867\) 12.4199 25.2285i 0.421802 0.856803i
\(868\) 0 0
\(869\) −4.76732 8.25724i −0.161720 0.280108i
\(870\) 0 0
\(871\) 12.5275 + 7.23273i 0.424477 + 0.245072i
\(872\) 0 0
\(873\) −27.6725 + 3.68157i −0.936572 + 0.124602i
\(874\) 0 0
\(875\) −11.2745 6.50932i −0.381147 0.220055i
\(876\) 0 0
\(877\) 6.16758 3.56085i 0.208264 0.120241i −0.392240 0.919863i \(-0.628300\pi\)
0.600504 + 0.799621i \(0.294966\pi\)
\(878\) 0 0
\(879\) −9.26605 + 0.613678i −0.312536 + 0.0206988i
\(880\) 0 0
\(881\) 24.8828i 0.838323i −0.907912 0.419162i \(-0.862324\pi\)
0.907912 0.419162i \(-0.137676\pi\)
\(882\) 0 0
\(883\) 13.2594 0.446213 0.223107 0.974794i \(-0.428380\pi\)
0.223107 + 0.974794i \(0.428380\pi\)
\(884\) 0 0
\(885\) −4.59314 + 3.07357i −0.154397 + 0.103317i
\(886\) 0 0
\(887\) −16.8122 29.1195i −0.564497 0.977738i −0.997096 0.0761517i \(-0.975737\pi\)
0.432599 0.901587i \(-0.357597\pi\)
\(888\) 0 0
\(889\) −10.2780 + 17.8020i −0.344713 + 0.597060i
\(890\) 0 0
\(891\) 15.7867 15.8731i 0.528873 0.531767i
\(892\) 0 0
\(893\) −11.1952 + 19.3906i −0.374632 + 0.648882i
\(894\) 0 0
\(895\) −14.4550 + 8.34557i −0.483176 + 0.278962i
\(896\) 0 0
\(897\) −27.9194 + 18.6827i −0.932202 + 0.623798i
\(898\) 0 0
\(899\) 11.1653i 0.372382i
\(900\) 0 0
\(901\) 9.73584i 0.324348i
\(902\) 0 0
\(903\) 0.671705 + 10.1422i 0.0223529 + 0.337512i
\(904\) 0 0
\(905\) −7.77874 + 4.49106i −0.258574 + 0.149288i
\(906\) 0 0
\(907\) 10.7148 18.5585i 0.355778 0.616226i −0.631473 0.775398i \(-0.717549\pi\)
0.987251 + 0.159173i \(0.0508826\pi\)
\(908\) 0 0
\(909\) 54.4478 7.24378i 1.80592 0.240261i
\(910\) 0 0
\(911\) −14.6331 + 25.3452i −0.484816 + 0.839725i −0.999848 0.0174455i \(-0.994447\pi\)
0.515032 + 0.857171i \(0.327780\pi\)
\(912\) 0 0
\(913\) 16.1130 + 27.9085i 0.533261 + 0.923636i
\(914\) 0 0
\(915\) 17.4759 + 8.60334i 0.577735 + 0.284418i
\(916\) 0 0
\(917\) −9.87045 −0.325951
\(918\) 0 0
\(919\) 41.2334i 1.36016i −0.733136 0.680082i \(-0.761944\pi\)
0.733136 0.680082i \(-0.238056\pi\)
\(920\) 0 0
\(921\) −8.50107 + 17.2682i −0.280120 + 0.569005i
\(922\) 0 0
\(923\) 72.0510 41.5987i 2.37159 1.36924i
\(924\) 0 0
\(925\) 3.86684 + 2.23252i 0.127141 + 0.0734049i
\(926\) 0 0
\(927\) 45.0762 + 18.5992i 1.48050 + 0.610877i
\(928\) 0 0
\(929\) −14.4159 8.32305i −0.472972 0.273070i 0.244511 0.969646i \(-0.421373\pi\)
−0.717483 + 0.696576i \(0.754706\pi\)
\(930\) 0 0
\(931\) 18.6692 + 32.3361i 0.611860 + 1.05977i
\(932\) 0 0
\(933\) 36.2563 2.40121i 1.18698 0.0786120i
\(934\) 0 0
\(935\) −3.40060 −0.111212
\(936\) 0 0
\(937\) −6.27457 −0.204981 −0.102491 0.994734i \(-0.532681\pi\)
−0.102491 + 0.994734i \(0.532681\pi\)
\(938\) 0 0
\(939\) 21.9005 + 32.7280i 0.714695 + 1.06804i
\(940\) 0 0
\(941\) −23.9585 41.4974i −0.781025 1.35278i −0.931345 0.364138i \(-0.881364\pi\)
0.150320 0.988637i \(-0.451970\pi\)
\(942\) 0 0
\(943\) −22.8045 13.1662i −0.742616 0.428750i
\(944\) 0 0
\(945\) 1.76439 + 8.77625i 0.0573954 + 0.285492i
\(946\) 0 0
\(947\) 15.0765 + 8.70441i 0.489920 + 0.282855i 0.724541 0.689232i \(-0.242052\pi\)
−0.234622 + 0.972087i \(0.575385\pi\)
\(948\) 0 0
\(949\) −27.0628 + 15.6247i −0.878496 + 0.507200i
\(950\) 0 0
\(951\) 28.1703 + 42.0976i 0.913484 + 1.36511i
\(952\) 0 0
\(953\) 4.44379i 0.143948i −0.997406 0.0719742i \(-0.977070\pi\)
0.997406 0.0719742i \(-0.0229299\pi\)
\(954\) 0 0
\(955\) −27.7430 −0.897742
\(956\) 0 0
\(957\) −0.437209 6.60151i −0.0141330 0.213397i
\(958\) 0 0
\(959\) −5.47138 9.47670i −0.176680 0.306019i
\(960\) 0 0
\(961\) 10.9329 18.9363i 0.352673 0.610848i
\(962\) 0 0
\(963\) 13.7915 10.6125i 0.444424 0.341983i
\(964\) 0 0
\(965\) 16.5411 28.6501i 0.532478 0.922278i
\(966\) 0 0
\(967\) −30.2987 + 17.4930i −0.974342 + 0.562536i −0.900557 0.434738i \(-0.856841\pi\)
−0.0737845 + 0.997274i \(0.523508\pi\)
\(968\) 0 0
\(969\) 8.77190 + 4.31839i 0.281794 + 0.138727i
\(970\) 0 0
\(971\) 40.1769i 1.28934i 0.764461 + 0.644670i \(0.223005\pi\)
−0.764461 + 0.644670i \(0.776995\pi\)
\(972\) 0 0
\(973\) 2.14003i 0.0686063i
\(974\) 0 0
\(975\) −25.1858 12.3989i −0.806591 0.397083i
\(976\) 0 0
\(977\) 19.2404 11.1084i 0.615554 0.355390i −0.159582 0.987185i \(-0.551015\pi\)
0.775136 + 0.631794i \(0.217681\pi\)
\(978\) 0 0
\(979\) 16.5860 28.7277i 0.530090 0.918143i
\(980\) 0 0
\(981\) −17.4697 + 13.4429i −0.557764 + 0.429198i
\(982\) 0 0
\(983\) −24.9905 + 43.2848i −0.797073 + 1.38057i 0.124441 + 0.992227i \(0.460286\pi\)
−0.921514 + 0.388344i \(0.873047\pi\)
\(984\) 0 0
\(985\) −4.31169 7.46807i −0.137382 0.237952i
\(986\) 0 0
\(987\) 0.437650 + 6.60818i 0.0139306 + 0.210341i
\(988\) 0 0
\(989\) −16.2908 −0.518017
\(990\) 0 0
\(991\) 47.8498i 1.52000i 0.649925 + 0.759999i \(0.274800\pi\)
−0.649925 + 0.759999i \(0.725200\pi\)
\(992\) 0 0
\(993\) −25.5721 38.2149i −0.811506 1.21271i
\(994\) 0 0
\(995\) −9.34606 + 5.39595i −0.296290 + 0.171063i
\(996\) 0 0
\(997\) −4.23642 2.44590i −0.134169 0.0774623i 0.431413 0.902154i \(-0.358015\pi\)
−0.565582 + 0.824692i \(0.691348\pi\)
\(998\) 0 0
\(999\) −1.78855 8.89644i −0.0565871 0.281471i
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 1152.2.p.g.191.8 yes 24
3.2 odd 2 3456.2.p.g.575.4 24
4.3 odd 2 1152.2.p.f.191.5 24
8.3 odd 2 1152.2.p.f.191.8 yes 24
8.5 even 2 inner 1152.2.p.g.191.5 yes 24
9.4 even 3 3456.2.p.f.2879.9 24
9.5 odd 6 1152.2.p.f.959.8 yes 24
12.11 even 2 3456.2.p.f.575.4 24
24.5 odd 2 3456.2.p.g.575.9 24
24.11 even 2 3456.2.p.f.575.9 24
36.23 even 6 inner 1152.2.p.g.959.5 yes 24
36.31 odd 6 3456.2.p.g.2879.9 24
72.5 odd 6 1152.2.p.f.959.5 yes 24
72.13 even 6 3456.2.p.f.2879.4 24
72.59 even 6 inner 1152.2.p.g.959.8 yes 24
72.67 odd 6 3456.2.p.g.2879.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.p.f.191.5 24 4.3 odd 2
1152.2.p.f.191.8 yes 24 8.3 odd 2
1152.2.p.f.959.5 yes 24 72.5 odd 6
1152.2.p.f.959.8 yes 24 9.5 odd 6
1152.2.p.g.191.5 yes 24 8.5 even 2 inner
1152.2.p.g.191.8 yes 24 1.1 even 1 trivial
1152.2.p.g.959.5 yes 24 36.23 even 6 inner
1152.2.p.g.959.8 yes 24 72.59 even 6 inner
3456.2.p.f.575.4 24 12.11 even 2
3456.2.p.f.575.9 24 24.11 even 2
3456.2.p.f.2879.4 24 72.13 even 6
3456.2.p.f.2879.9 24 9.4 even 3
3456.2.p.g.575.4 24 3.2 odd 2
3456.2.p.g.575.9 24 24.5 odd 2
3456.2.p.g.2879.4 24 72.67 odd 6
3456.2.p.g.2879.9 24 36.31 odd 6