Properties

Label 1152.2.p.f.191.5
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 191.5
Character \(\chi\) \(=\) 1152.191
Dual form 1152.2.p.f.959.5

$q$-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 0.669418 0.742886i \(-0.266544\pi\)
−0.308649 + 0.951176i \(0.599877\pi\)
\(8\) 0 0
\(9\) −1.82953 2.37757i −0.609843 0.792522i
\(10\) 0 0
\(11\) 2.15418 + 1.24372i 0.649509 + 0.374994i 0.788268 0.615332i \(-0.210978\pi\)
−0.138759 + 0.990326i \(0.544311\pi\)
\(12\) 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 \(-0.234666\pi\)
0.740337 + 0.672236i \(0.234666\pi\)
\(20\) 0 0
\(21\) −1.58656 + 1.06167i −0.346215 + 0.231675i
\(22\) 0 0
\(23\) 1.52981 + 2.64970i 0.318987 + 0.552501i 0.980277 0.197629i \(-0.0633241\pi\)
−0.661290 + 0.750130i \(0.729991\pi\)
\(24\) 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 −0.944169 0.329460i \(-0.893133\pi\)
−0.186764 + 0.982405i \(0.559800\pi\)
\(32\) 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 −0.994436 0.105345i \(-0.966405\pi\)
0.588449 + 0.808534i \(0.299739\pi\)
\(44\) 0 0
\(45\) 1.78859 4.33477i 0.266628 0.646189i
\(46\) 0 0
\(47\) −1.73458 + 3.00438i −0.253015 + 0.438234i −0.964354 0.264614i \(-0.914755\pi\)
0.711340 + 0.702848i \(0.248089\pi\)
\(48\) 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 −0.380488 0.924786i \(-0.624244\pi\)
0.610644 + 0.791905i \(0.290911\pi\)
\(60\) 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 0.787875 0.615835i \(-0.211181\pi\)
−0.927266 + 0.374403i \(0.877848\pi\)
\(68\) 0 0
\(69\) −5.28782 + 0.350205i −0.636579 + 0.0421597i
\(70\) 0 0
\(71\) −13.1244 −1.55758 −0.778788 0.627287i \(-0.784165\pi\)
−0.778788 + 0.627287i \(0.784165\pi\)
\(72\) 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 0.301496 0.953467i \(-0.402514\pi\)
−0.674979 + 0.737837i \(0.735847\pi\)
\(80\) 0 0
\(81\) −2.30564 + 8.69966i −0.256182 + 0.966628i
\(82\) 0 0
\(83\) 11.2198 + 6.47775i 1.23153 + 0.711026i 0.967349 0.253447i \(-0.0815643\pi\)
0.264183 + 0.964472i \(0.414898\pi\)
\(84\) 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 0.394028 0.919098i \(-0.371081\pi\)
0.992977 + 0.118311i \(0.0377480\pi\)
\(104\) 0 0
\(105\) −2.67712 1.31794i −0.261261 0.128618i
\(106\) 0 0
\(107\) 5.80067i 0.560772i −0.959887 0.280386i \(-0.909538\pi\)
0.959887 0.280386i \(-0.0904624\pi\)
\(108\) 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 \(0.310228\pi\)
−0.561490 + 0.827484i \(0.689772\pi\)
\(128\) 0 0
\(129\) −5.12883 7.66451i −0.451568 0.674823i
\(130\) 0 0
\(131\) −7.75570 + 4.47775i −0.677618 + 0.391223i −0.798957 0.601388i \(-0.794615\pi\)
0.121339 + 0.992611i \(0.461281\pi\)
\(132\) 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 0.821912 0.569615i \(-0.192908\pi\)
−0.904257 + 0.426989i \(0.859574\pi\)
\(140\) 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 0.0396516 0.999214i \(-0.487375\pi\)
−0.845519 + 0.533946i \(0.820709\pi\)
\(152\) 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 \(-0.570919\pi\)
−0.220961 + 0.975283i \(0.570919\pi\)
\(164\) 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 \(-0.810238\pi\)
−0.0724922 0.997369i \(-0.523095\pi\)
\(168\) 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 \(-0.869334\pi\)
0.916922 0.399066i \(-0.130666\pi\)
\(180\) 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 \(-0.611385\pi\)
0.984957 0.172801i \(-0.0552818\pi\)
\(192\) 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 \(-0.921306\pi\)
0.969596 0.244713i \(-0.0786938\pi\)
\(200\) 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 0.707973 0.706240i \(-0.249610\pi\)
−0.965608 + 0.260003i \(0.916277\pi\)
\(212\) 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 −0.207816 0.978168i \(-0.566636\pi\)
−0.951026 + 0.309110i \(0.899969\pi\)
\(224\) 0 0
\(225\) −7.60324 + 1.01154i −0.506883 + 0.0674361i
\(226\) 0 0
\(227\) −18.3301 10.5829i −1.21661 0.702409i −0.252417 0.967618i \(-0.581226\pi\)
−0.964191 + 0.265209i \(0.914559\pi\)
\(228\) 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 \(0.119115\pi\)
−0.148832 + 0.988862i \(0.547551\pi\)
\(240\) 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 \(0.187736\pi\)
−0.831057 + 0.556187i \(0.812264\pi\)
\(252\) 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 −0.892783 0.450487i \(-0.851250\pi\)
0.836525 + 0.547929i \(0.184584\pi\)
\(264\) 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 \(0.0785356\pi\)
−0.969717 + 0.244231i \(0.921464\pi\)
\(272\) 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 \(-0.741167\pi\)
−0.285522 0.958372i \(-0.592167\pi\)
\(284\) 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 \(-0.397288\pi\)
0.317110 + 0.948389i \(0.397288\pi\)
\(308\) 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 \(-0.963902\pi\)
0.398787 0.917044i \(-0.369431\pi\)
\(312\) 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 \(0.426956\pi\)
−0.957056 + 0.289903i \(0.906377\pi\)
\(332\) 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 −0.783872 0.620922i \(-0.786758\pi\)
0.145798 + 0.989314i \(0.453425\pi\)
\(348\) 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 \(-0.452686\pi\)
0.148095 + 0.988973i \(0.452686\pi\)
\(360\) 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 −0.154677 0.987965i \(-0.549434\pi\)
−0.932941 + 0.360029i \(0.882767\pi\)
\(368\) 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 \(-0.213170\pi\)
0.784013 + 0.620745i \(0.213170\pi\)
\(380\) 0 0
\(381\) −28.9820 14.2678i −1.48479 0.730961i
\(382\) 0 0
\(383\) −8.10743 14.0425i −0.414270 0.717537i 0.581081 0.813846i \(-0.302630\pi\)
−0.995352 + 0.0963082i \(0.969297\pi\)
\(384\) 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 −0.299168 0.954200i \(-0.596709\pi\)
0.676778 + 0.736188i \(0.263376\pi\)
\(420\) 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 \(-0.332510\pi\)
0.502238 + 0.864730i \(0.332510\pi\)
\(432\) 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 0.932530 0.361092i \(-0.117596\pi\)
0.153550 + 0.988141i \(0.450929\pi\)
\(440\) 0 0
\(441\) −6.61985 + 16.0436i −0.315231 + 0.763982i
\(442\) 0 0
\(443\) −25.6178 14.7905i −1.21714 0.702716i −0.252835 0.967509i \(-0.581363\pi\)
−0.964305 + 0.264793i \(0.914696\pi\)
\(444\) 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 −0.771486 0.636246i \(-0.780486\pi\)
0.165262 + 0.986250i \(0.447153\pi\)
\(464\) 0 0
\(465\) 16.3599 10.9475i 0.758672 0.507677i
\(466\) 0 0
\(467\) 22.9660i 1.06274i 0.847140 + 0.531369i \(0.178322\pi\)
−0.847140 + 0.531369i \(0.821678\pi\)
\(468\) 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 \(0.412495\pi\)
−0.969235 + 0.246137i \(0.920839\pi\)
\(480\) 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 \(0.193853\pi\)
−0.820217 + 0.572053i \(0.806147\pi\)
\(488\) 0 0
\(489\) 4.31623 8.76753i 0.195187 0.396481i
\(490\) 0 0
\(491\) 28.5831 16.5025i 1.28994 0.744746i 0.311295 0.950313i \(-0.399237\pi\)
0.978643 + 0.205568i \(0.0659041\pi\)
\(492\) 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 0.606364 0.795187i \(-0.292627\pi\)
−0.991834 + 0.127533i \(0.959294\pi\)
\(500\) 0 0
\(501\) 40.2011 2.66246i 1.79605 0.118950i
\(502\) 0 0
\(503\) −39.7159 −1.77084 −0.885421 0.464789i \(-0.846130\pi\)
−0.885421 + 0.464789i \(0.846130\pi\)
\(504\) 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 \(-0.340947\pi\)
0.479145 + 0.877736i \(0.340947\pi\)
\(524\) 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 −0.740316 0.672259i \(-0.765324\pi\)
0.952352 + 0.305003i \(0.0986573\pi\)
\(548\) 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 −0.175310 0.984513i \(-0.556093\pi\)
0.764958 + 0.644080i \(0.222760\pi\)
\(564\) 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 0.782748 0.622338i \(-0.213817\pi\)
−0.930335 + 0.366711i \(0.880484\pi\)
\(572\) 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 0.865470 0.500960i \(-0.167020\pi\)
−0.00110929 + 0.999999i \(0.500353\pi\)
\(588\) 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 0.876858 0.480750i \(-0.159636\pi\)
−0.854771 + 0.519006i \(0.826302\pi\)
\(600\) 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 −0.430527 0.902578i \(-0.641672\pi\)
0.566391 + 0.824136i \(0.308339\pi\)
\(608\) 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 \(0.439956\pi\)
−0.944422 + 0.328735i \(0.893378\pi\)
\(620\) 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 \(-0.991472\pi\)
0.999641 0.0267870i \(-0.00852759\pi\)
\(632\) 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 0.999532 0.0305983i \(-0.00974125\pi\)
−0.526265 + 0.850321i \(0.676408\pi\)
\(644\) 0 0
\(645\) 6.36686 12.9329i 0.250695 0.509234i
\(646\) 0 0
\(647\) 8.55443 0.336309 0.168155 0.985761i \(-0.446219\pi\)
0.168155 + 0.985761i \(0.446219\pi\)
\(648\) 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 0.692361 0.721551i \(-0.256570\pi\)
−0.278701 + 0.960378i \(0.589904\pi\)
\(660\) 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 \(0.141635\pi\)
−0.902629 + 0.430420i \(0.858365\pi\)
\(684\) 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 −0.748462 0.663177i \(-0.769207\pi\)
0.948560 + 0.316599i \(0.102541\pi\)
\(692\) 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 \(-0.515002\pi\)
−0.0471140 + 0.998890i \(0.515002\pi\)
\(720\) 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 −0.199423 0.979914i \(-0.563907\pi\)
−0.948341 + 0.317252i \(0.897240\pi\)
\(728\) 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 \(-0.233138\pi\)
0.743556 + 0.668674i \(0.233138\pi\)
\(740\) 0 0
\(741\) 4.68298 + 70.7094i 0.172034 + 2.59757i
\(742\) 0 0
\(743\) −9.83837 17.0406i −0.360935 0.625158i 0.627180 0.778874i \(-0.284209\pi\)
−0.988115 + 0.153717i \(0.950876\pi\)
\(744\) 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 0.0646065 0.997911i \(-0.479421\pi\)
0.896519 + 0.443005i \(0.146087\pi\)
\(752\) 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 0.936571 0.350478i \(-0.113981\pi\)
−0.771808 + 0.635855i \(0.780647\pi\)
\(788\) 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 \(-0.778845\pi\)
−0.768196 + 0.640215i \(0.778845\pi\)
\(812\) 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 −0.684265 0.729233i \(-0.739877\pi\)
0.289402 + 0.957208i \(0.406544\pi\)
\(824\) 0 0
\(825\) 0.727938 + 10.9913i 0.0253436 + 0.382668i
\(826\) 0 0
\(827\) 5.42534i 0.188658i −0.995541 0.0943289i \(-0.969929\pi\)
0.995541 0.0943289i \(-0.0300705\pi\)
\(828\) 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 −0.973415 0.229050i \(-0.926438\pi\)
0.685071 + 0.728477i \(0.259771\pi\)
\(840\) 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 \(-0.916450\pi\)
0.258162 0.966102i \(-0.416883\pi\)
\(860\) 0 0
\(861\) −7.25663 + 14.7403i −0.247305 + 0.502349i
\(862\) 0 0
\(863\) 1.67319 0.0569560 0.0284780 0.999594i \(-0.490934\pi\)
0.0284780 + 0.999594i \(0.490934\pi\)
\(864\) 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 \(-0.571620\pi\)
−0.223107 + 0.974794i \(0.571620\pi\)
\(884\) 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 \(0.0242633\pi\)
−0.432599 + 0.901587i \(0.642403\pi\)
\(888\) 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 −0.987251 0.159173i \(-0.949117\pi\)
0.631473 + 0.775398i \(0.282451\pi\)
\(908\) 0 0
\(909\) 54.4478 7.24378i 1.80592 0.240261i
\(910\) 0 0
\(911\) 14.6331 25.3452i 0.484816 0.839725i −0.515032 0.857171i \(-0.672220\pi\)
0.999848 + 0.0174455i \(0.00555335\pi\)
\(912\) 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 \(0.238056\pi\)
−0.733136 + 0.680082i \(0.761944\pi\)
\(920\) 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 0.234622 0.972087i \(-0.424615\pi\)
−0.724541 + 0.689232i \(0.757948\pi\)
\(948\) 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 0.0737845 0.997274i \(-0.476492\pi\)
0.900557 + 0.434738i \(0.143159\pi\)
\(968\) 0 0
\(969\) 8.77190 + 4.31839i 0.281794 + 0.138727i
\(970\) 0 0
\(971\) 40.1769i 1.28934i −0.764461 0.644670i \(-0.776995\pi\)
0.764461 0.644670i \(-0.223005\pi\)
\(972\) 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 \(-0.539714\pi\)
0.921514 0.388344i \(-0.126953\pi\)
\(984\) 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 \(-0.725200\pi\)
0.649925 0.759999i \(-0.274800\pi\)
\(992\) 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.f.191.5 24
3.2 odd 2 3456.2.p.f.575.4 24
4.3 odd 2 1152.2.p.g.191.8 yes 24
8.3 odd 2 1152.2.p.g.191.5 yes 24
8.5 even 2 inner 1152.2.p.f.191.8 yes 24
9.4 even 3 3456.2.p.g.2879.9 24
9.5 odd 6 1152.2.p.g.959.5 yes 24
12.11 even 2 3456.2.p.g.575.4 24
24.5 odd 2 3456.2.p.f.575.9 24
24.11 even 2 3456.2.p.g.575.9 24
36.23 even 6 inner 1152.2.p.f.959.8 yes 24
36.31 odd 6 3456.2.p.f.2879.9 24
72.5 odd 6 1152.2.p.g.959.8 yes 24
72.13 even 6 3456.2.p.g.2879.4 24
72.59 even 6 inner 1152.2.p.f.959.5 yes 24
72.67 odd 6 3456.2.p.f.2879.4 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.p.f.191.5 24 1.1 even 1 trivial
1152.2.p.f.191.8 yes 24 8.5 even 2 inner
1152.2.p.f.959.5 yes 24 72.59 even 6 inner
1152.2.p.f.959.8 yes 24 36.23 even 6 inner
1152.2.p.g.191.5 yes 24 8.3 odd 2
1152.2.p.g.191.8 yes 24 4.3 odd 2
1152.2.p.g.959.5 yes 24 9.5 odd 6
1152.2.p.g.959.8 yes 24 72.5 odd 6
3456.2.p.f.575.4 24 3.2 odd 2
3456.2.p.f.575.9 24 24.5 odd 2
3456.2.p.f.2879.4 24 72.67 odd 6
3456.2.p.f.2879.9 24 36.31 odd 6
3456.2.p.g.575.4 24 12.11 even 2
3456.2.p.g.575.9 24 24.11 even 2
3456.2.p.g.2879.4 24 72.13 even 6
3456.2.p.g.2879.9 24 9.4 even 3