Properties

Label 2394.2.e.c.1063.7
Level $2394$
Weight $2$
Character 2394.1063
Analytic conductor $19.116$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2394,2,Mod(1063,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.1063");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.e (of order \(2\), degree \(1\), 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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 44x^{14} + 708x^{12} + 5378x^{10} + 20592x^{8} + 38856x^{6} + 33265x^{4} + 10216x^{2} + 900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 266)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1063.7
Root \(-0.584675i\) of defining polynomial
Character \(\chi\) \(=\) 2394.1063
Dual form 2394.2.e.c.1063.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.74394i q^{5} +(-2.24817 - 1.39489i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +2.74394i q^{5} +(-2.24817 - 1.39489i) q^{7} +1.00000i q^{8} +2.74394 q^{10} +1.52921 q^{11} -5.33820 q^{13} +(-1.39489 + 2.24817i) q^{14} +1.00000 q^{16} +2.78979i q^{17} +(4.01815 - 1.68953i) q^{19} -2.74394i q^{20} -1.52921i q^{22} -6.49634 q^{23} -2.52921 q^{25} +5.33820i q^{26} +(2.24817 + 1.39489i) q^{28} -6.88592i q^{29} +0.830650 q^{31} -1.00000i q^{32} +2.78979 q^{34} +(3.82751 - 6.16885i) q^{35} -10.3238i q^{37} +(-1.68953 - 4.01815i) q^{38} -2.74394 q^{40} +11.7026 q^{41} +0.668836 q^{43} -1.52921 q^{44} +6.49634i q^{46} +4.83118i q^{47} +(3.10854 + 6.27192i) q^{49} +2.52921i q^{50} +5.33820 q^{52} -2.02555i q^{53} +4.19606i q^{55} +(1.39489 - 2.24817i) q^{56} -6.88592 q^{58} +5.53373 q^{59} -7.03169i q^{61} -0.830650i q^{62} -1.00000 q^{64} -14.6477i q^{65} +3.43792i q^{67} -2.78979i q^{68} +(-6.16885 - 3.82751i) q^{70} +11.4635i q^{71} -12.5438i q^{73} -10.3238 q^{74} +(-4.01815 + 1.68953i) q^{76} +(-3.43792 - 2.13308i) q^{77} -4.32385i q^{79} +2.74394i q^{80} -11.7026i q^{82} -14.4465i q^{83} -7.65501 q^{85} -0.668836i q^{86} +1.52921i q^{88} +6.14458 q^{89} +(12.0012 + 7.44622i) q^{91} +6.49634 q^{92} +4.83118 q^{94} +(4.63597 + 11.0256i) q^{95} +5.31393 q^{97} +(6.27192 - 3.10854i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 6 q^{7} + 12 q^{11} + 16 q^{16} - 20 q^{23} - 28 q^{25} - 6 q^{28} - 8 q^{35} - 4 q^{43} - 12 q^{44} + 10 q^{49} - 16 q^{58} - 16 q^{64} - 12 q^{74} + 4 q^{77} + 16 q^{85} + 20 q^{92} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 2.74394i 1.22713i 0.789645 + 0.613564i \(0.210265\pi\)
−0.789645 + 0.613564i \(0.789735\pi\)
\(6\) 0 0
\(7\) −2.24817 1.39489i −0.849729 0.527220i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.74394 0.867710
\(11\) 1.52921 0.461074 0.230537 0.973064i \(-0.425952\pi\)
0.230537 + 0.973064i \(0.425952\pi\)
\(12\) 0 0
\(13\) −5.33820 −1.48055 −0.740275 0.672305i \(-0.765305\pi\)
−0.740275 + 0.672305i \(0.765305\pi\)
\(14\) −1.39489 + 2.24817i −0.372801 + 0.600849i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.78979i 0.676623i 0.941034 + 0.338311i \(0.109856\pi\)
−0.941034 + 0.338311i \(0.890144\pi\)
\(18\) 0 0
\(19\) 4.01815 1.68953i 0.921826 0.387605i
\(20\) 2.74394i 0.613564i
\(21\) 0 0
\(22\) 1.52921i 0.326028i
\(23\) −6.49634 −1.35458 −0.677290 0.735716i \(-0.736846\pi\)
−0.677290 + 0.735716i \(0.736846\pi\)
\(24\) 0 0
\(25\) −2.52921 −0.505842
\(26\) 5.33820i 1.04691i
\(27\) 0 0
\(28\) 2.24817 + 1.39489i 0.424864 + 0.263610i
\(29\) 6.88592i 1.27868i −0.768923 0.639342i \(-0.779207\pi\)
0.768923 0.639342i \(-0.220793\pi\)
\(30\) 0 0
\(31\) 0.830650 0.149189 0.0745946 0.997214i \(-0.476234\pi\)
0.0745946 + 0.997214i \(0.476234\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.78979 0.478445
\(35\) 3.82751 6.16885i 0.646966 1.04273i
\(36\) 0 0
\(37\) 10.3238i 1.69723i −0.529011 0.848615i \(-0.677437\pi\)
0.529011 0.848615i \(-0.322563\pi\)
\(38\) −1.68953 4.01815i −0.274078 0.651829i
\(39\) 0 0
\(40\) −2.74394 −0.433855
\(41\) 11.7026 1.82764 0.913818 0.406125i \(-0.133120\pi\)
0.913818 + 0.406125i \(0.133120\pi\)
\(42\) 0 0
\(43\) 0.668836 0.101997 0.0509983 0.998699i \(-0.483760\pi\)
0.0509983 + 0.998699i \(0.483760\pi\)
\(44\) −1.52921 −0.230537
\(45\) 0 0
\(46\) 6.49634i 0.957833i
\(47\) 4.83118i 0.704700i 0.935868 + 0.352350i \(0.114617\pi\)
−0.935868 + 0.352350i \(0.885383\pi\)
\(48\) 0 0
\(49\) 3.10854 + 6.27192i 0.444078 + 0.895988i
\(50\) 2.52921i 0.357684i
\(51\) 0 0
\(52\) 5.33820 0.740275
\(53\) 2.02555i 0.278231i −0.990276 0.139115i \(-0.955574\pi\)
0.990276 0.139115i \(-0.0444259\pi\)
\(54\) 0 0
\(55\) 4.19606i 0.565796i
\(56\) 1.39489 2.24817i 0.186401 0.300424i
\(57\) 0 0
\(58\) −6.88592 −0.904166
\(59\) 5.53373 0.720430 0.360215 0.932869i \(-0.382703\pi\)
0.360215 + 0.932869i \(0.382703\pi\)
\(60\) 0 0
\(61\) 7.03169i 0.900316i −0.892949 0.450158i \(-0.851368\pi\)
0.892949 0.450158i \(-0.148632\pi\)
\(62\) 0.830650i 0.105493i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 14.6477i 1.81682i
\(66\) 0 0
\(67\) 3.43792i 0.420009i 0.977700 + 0.210005i \(0.0673479\pi\)
−0.977700 + 0.210005i \(0.932652\pi\)
\(68\) 2.78979i 0.338311i
\(69\) 0 0
\(70\) −6.16885 3.82751i −0.737318 0.457474i
\(71\) 11.4635i 1.36046i 0.732997 + 0.680232i \(0.238121\pi\)
−0.732997 + 0.680232i \(0.761879\pi\)
\(72\) 0 0
\(73\) 12.5438i 1.46814i −0.679071 0.734072i \(-0.737617\pi\)
0.679071 0.734072i \(-0.262383\pi\)
\(74\) −10.3238 −1.20012
\(75\) 0 0
\(76\) −4.01815 + 1.68953i −0.460913 + 0.193802i
\(77\) −3.43792 2.13308i −0.391788 0.243087i
\(78\) 0 0
\(79\) 4.32385i 0.486471i −0.969967 0.243235i \(-0.921791\pi\)
0.969967 0.243235i \(-0.0782087\pi\)
\(80\) 2.74394i 0.306782i
\(81\) 0 0
\(82\) 11.7026i 1.29233i
\(83\) 14.4465i 1.58571i −0.609410 0.792856i \(-0.708594\pi\)
0.609410 0.792856i \(-0.291406\pi\)
\(84\) 0 0
\(85\) −7.65501 −0.830302
\(86\) 0.668836i 0.0721225i
\(87\) 0 0
\(88\) 1.52921i 0.163014i
\(89\) 6.14458 0.651325 0.325662 0.945486i \(-0.394413\pi\)
0.325662 + 0.945486i \(0.394413\pi\)
\(90\) 0 0
\(91\) 12.0012 + 7.44622i 1.25807 + 0.780576i
\(92\) 6.49634 0.677290
\(93\) 0 0
\(94\) 4.83118 0.498298
\(95\) 4.63597 + 11.0256i 0.475640 + 1.13120i
\(96\) 0 0
\(97\) 5.31393 0.539548 0.269774 0.962924i \(-0.413051\pi\)
0.269774 + 0.962924i \(0.413051\pi\)
\(98\) 6.27192 3.10854i 0.633559 0.314010i
\(99\) 0 0
\(100\) 2.52921 0.252921
\(101\) 8.95863i 0.891417i −0.895178 0.445709i \(-0.852952\pi\)
0.895178 0.445709i \(-0.147048\pi\)
\(102\) 0 0
\(103\) 12.7288 1.25420 0.627101 0.778938i \(-0.284241\pi\)
0.627101 + 0.778938i \(0.284241\pi\)
\(104\) 5.33820i 0.523453i
\(105\) 0 0
\(106\) −2.02555 −0.196739
\(107\) 1.15867i 0.112013i 0.998430 + 0.0560064i \(0.0178367\pi\)
−0.998430 + 0.0560064i \(0.982163\pi\)
\(108\) 0 0
\(109\) 2.02555i 0.194013i −0.995284 0.0970063i \(-0.969073\pi\)
0.995284 0.0970063i \(-0.0309267\pi\)
\(110\) 4.19606 0.400079
\(111\) 0 0
\(112\) −2.24817 1.39489i −0.212432 0.131805i
\(113\) 10.5966i 0.996844i −0.866935 0.498422i \(-0.833913\pi\)
0.866935 0.498422i \(-0.166087\pi\)
\(114\) 0 0
\(115\) 17.8256i 1.66224i
\(116\) 6.88592i 0.639342i
\(117\) 0 0
\(118\) 5.53373i 0.509421i
\(119\) 3.89146 6.27192i 0.356729 0.574946i
\(120\) 0 0
\(121\) −8.66152 −0.787411
\(122\) −7.03169 −0.636620
\(123\) 0 0
\(124\) −0.830650 −0.0745946
\(125\) 6.77970i 0.606395i
\(126\) 0 0
\(127\) 19.2353i 1.70686i −0.521209 0.853429i \(-0.674519\pi\)
0.521209 0.853429i \(-0.325481\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −14.6477 −1.28469
\(131\) 8.95863i 0.782719i 0.920238 + 0.391360i \(0.127995\pi\)
−0.920238 + 0.391360i \(0.872005\pi\)
\(132\) 0 0
\(133\) −11.3902 1.80653i −0.987655 0.156646i
\(134\) 3.43792 0.296991
\(135\) 0 0
\(136\) −2.78979 −0.239222
\(137\) 0.769087 0.0657075 0.0328538 0.999460i \(-0.489540\pi\)
0.0328538 + 0.999460i \(0.489540\pi\)
\(138\) 0 0
\(139\) 4.60077i 0.390232i −0.980780 0.195116i \(-0.937492\pi\)
0.980780 0.195116i \(-0.0625084\pi\)
\(140\) −3.82751 + 6.16885i −0.323483 + 0.521363i
\(141\) 0 0
\(142\) 11.4635 0.961993
\(143\) −8.16322 −0.682643
\(144\) 0 0
\(145\) 18.8946 1.56911
\(146\) −12.5438 −1.03814
\(147\) 0 0
\(148\) 10.3238i 0.848615i
\(149\) 1.65501 0.135584 0.0677919 0.997699i \(-0.478405\pi\)
0.0677919 + 0.997699i \(0.478405\pi\)
\(150\) 0 0
\(151\) 3.72075i 0.302790i 0.988473 + 0.151395i \(0.0483765\pi\)
−0.988473 + 0.151395i \(0.951623\pi\)
\(152\) 1.68953 + 4.01815i 0.137039 + 0.325915i
\(153\) 0 0
\(154\) −2.13308 + 3.43792i −0.171889 + 0.277036i
\(155\) 2.27925i 0.183074i
\(156\) 0 0
\(157\) 21.8613i 1.74472i −0.488860 0.872362i \(-0.662587\pi\)
0.488860 0.872362i \(-0.337413\pi\)
\(158\) −4.32385 −0.343987
\(159\) 0 0
\(160\) 2.74394 0.216928
\(161\) 14.6049 + 9.06171i 1.15103 + 0.714162i
\(162\) 0 0
\(163\) −9.52921 −0.746385 −0.373193 0.927754i \(-0.621737\pi\)
−0.373193 + 0.927754i \(0.621737\pi\)
\(164\) −11.7026 −0.913818
\(165\) 0 0
\(166\) −14.4465 −1.12127
\(167\) −16.4636 −1.27399 −0.636997 0.770866i \(-0.719824\pi\)
−0.636997 + 0.770866i \(0.719824\pi\)
\(168\) 0 0
\(169\) 15.4963 1.19203
\(170\) 7.65501i 0.587112i
\(171\) 0 0
\(172\) −0.668836 −0.0509983
\(173\) 7.59670 0.577567 0.288783 0.957394i \(-0.406749\pi\)
0.288783 + 0.957394i \(0.406749\pi\)
\(174\) 0 0
\(175\) 5.68609 + 3.52798i 0.429828 + 0.266690i
\(176\) 1.52921 0.115268
\(177\) 0 0
\(178\) 6.14458i 0.460556i
\(179\) 20.6477i 1.54328i −0.636059 0.771641i \(-0.719436\pi\)
0.636059 0.771641i \(-0.280564\pi\)
\(180\) 0 0
\(181\) −7.20564 −0.535591 −0.267795 0.963476i \(-0.586295\pi\)
−0.267795 + 0.963476i \(0.586295\pi\)
\(182\) 7.44622 12.0012i 0.551950 0.889586i
\(183\) 0 0
\(184\) 6.49634i 0.478917i
\(185\) 28.3280 2.08272
\(186\) 0 0
\(187\) 4.26617i 0.311973i
\(188\) 4.83118i 0.352350i
\(189\) 0 0
\(190\) 11.0256 4.63597i 0.799878 0.336329i
\(191\) −16.5474 −1.19733 −0.598665 0.800999i \(-0.704302\pi\)
−0.598665 + 0.800999i \(0.704302\pi\)
\(192\) 0 0
\(193\) 24.3684i 1.75408i −0.480419 0.877039i \(-0.659515\pi\)
0.480419 0.877039i \(-0.340485\pi\)
\(194\) 5.31393i 0.381518i
\(195\) 0 0
\(196\) −3.10854 6.27192i −0.222039 0.447994i
\(197\) 6.99268 0.498208 0.249104 0.968477i \(-0.419864\pi\)
0.249104 + 0.968477i \(0.419864\pi\)
\(198\) 0 0
\(199\) 20.0719i 1.42286i 0.702756 + 0.711431i \(0.251953\pi\)
−0.702756 + 0.711431i \(0.748047\pi\)
\(200\) 2.52921i 0.178842i
\(201\) 0 0
\(202\) −8.95863 −0.630327
\(203\) −9.60513 + 15.4807i −0.674148 + 1.08653i
\(204\) 0 0
\(205\) 32.1112i 2.24274i
\(206\) 12.7288i 0.886854i
\(207\) 0 0
\(208\) −5.33820 −0.370137
\(209\) 6.14458 2.58364i 0.425030 0.178714i
\(210\) 0 0
\(211\) 7.48902i 0.515566i 0.966203 + 0.257783i \(0.0829919\pi\)
−0.966203 + 0.257783i \(0.917008\pi\)
\(212\) 2.02555i 0.139115i
\(213\) 0 0
\(214\) 1.15867 0.0792050
\(215\) 1.83525i 0.125163i
\(216\) 0 0
\(217\) −1.86744 1.15867i −0.126770 0.0786556i
\(218\) −2.02555 −0.137188
\(219\) 0 0
\(220\) 4.19606i 0.282898i
\(221\) 14.8924i 1.00177i
\(222\) 0 0
\(223\) 13.5806 0.909426 0.454713 0.890638i \(-0.349742\pi\)
0.454713 + 0.890638i \(0.349742\pi\)
\(224\) −1.39489 + 2.24817i −0.0932003 + 0.150212i
\(225\) 0 0
\(226\) −10.5966 −0.704875
\(227\) 23.9027 1.58648 0.793240 0.608909i \(-0.208393\pi\)
0.793240 + 0.608909i \(0.208393\pi\)
\(228\) 0 0
\(229\) 15.0816i 0.996622i 0.866998 + 0.498311i \(0.166046\pi\)
−0.866998 + 0.498311i \(0.833954\pi\)
\(230\) −17.8256 −1.17538
\(231\) 0 0
\(232\) 6.88592 0.452083
\(233\) −12.1258 −0.794388 −0.397194 0.917735i \(-0.630016\pi\)
−0.397194 + 0.917735i \(0.630016\pi\)
\(234\) 0 0
\(235\) −13.2565 −0.864756
\(236\) −5.53373 −0.360215
\(237\) 0 0
\(238\) −6.27192 3.89146i −0.406548 0.252246i
\(239\) −9.55476 −0.618046 −0.309023 0.951055i \(-0.600002\pi\)
−0.309023 + 0.951055i \(0.600002\pi\)
\(240\) 0 0
\(241\) −8.77104 −0.564992 −0.282496 0.959268i \(-0.591162\pi\)
−0.282496 + 0.959268i \(0.591162\pi\)
\(242\) 8.66152i 0.556784i
\(243\) 0 0
\(244\) 7.03169i 0.450158i
\(245\) −17.2098 + 8.52966i −1.09949 + 0.544940i
\(246\) 0 0
\(247\) −21.4496 + 9.01904i −1.36481 + 0.573868i
\(248\) 0.830650i 0.0527463i
\(249\) 0 0
\(250\) 6.77970 0.428786
\(251\) 2.10882i 0.133108i 0.997783 + 0.0665538i \(0.0212004\pi\)
−0.997783 + 0.0665538i \(0.978800\pi\)
\(252\) 0 0
\(253\) −9.93426 −0.624562
\(254\) −19.2353 −1.20693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −22.7700 −1.42036 −0.710178 0.704022i \(-0.751385\pi\)
−0.710178 + 0.704022i \(0.751385\pi\)
\(258\) 0 0
\(259\) −14.4007 + 23.2098i −0.894814 + 1.44218i
\(260\) 14.6477i 0.908411i
\(261\) 0 0
\(262\) 8.95863 0.553466
\(263\) −22.3027 −1.37524 −0.687622 0.726069i \(-0.741345\pi\)
−0.687622 + 0.726069i \(0.741345\pi\)
\(264\) 0 0
\(265\) 5.55799 0.341425
\(266\) −1.80653 + 11.3902i −0.110766 + 0.698377i
\(267\) 0 0
\(268\) 3.43792i 0.210005i
\(269\) −3.88304 −0.236753 −0.118377 0.992969i \(-0.537769\pi\)
−0.118377 + 0.992969i \(0.537769\pi\)
\(270\) 0 0
\(271\) 19.7969i 1.20257i 0.799033 + 0.601287i \(0.205345\pi\)
−0.799033 + 0.601287i \(0.794655\pi\)
\(272\) 2.78979i 0.169156i
\(273\) 0 0
\(274\) 0.769087i 0.0464622i
\(275\) −3.86769 −0.233230
\(276\) 0 0
\(277\) 21.7061 1.30419 0.652097 0.758136i \(-0.273890\pi\)
0.652097 + 0.758136i \(0.273890\pi\)
\(278\) −4.60077 −0.275936
\(279\) 0 0
\(280\) 6.16885 + 3.82751i 0.368659 + 0.228737i
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0.703721i 0.0418319i −0.999781 0.0209159i \(-0.993342\pi\)
0.999781 0.0209159i \(-0.00665824\pi\)
\(284\) 11.4635i 0.680232i
\(285\) 0 0
\(286\) 8.16322i 0.482701i
\(287\) −26.3094 16.3238i −1.55299 0.963566i
\(288\) 0 0
\(289\) 9.21709 0.542182
\(290\) 18.8946i 1.10953i
\(291\) 0 0
\(292\) 12.5438i 0.734072i
\(293\) 11.5650 0.675636 0.337818 0.941211i \(-0.390311\pi\)
0.337818 + 0.941211i \(0.390311\pi\)
\(294\) 0 0
\(295\) 15.1842i 0.884059i
\(296\) 10.3238 0.600061
\(297\) 0 0
\(298\) 1.65501i 0.0958722i
\(299\) 34.6787 2.00552
\(300\) 0 0
\(301\) −1.50366 0.932956i −0.0866694 0.0537747i
\(302\) 3.72075 0.214105
\(303\) 0 0
\(304\) 4.01815 1.68953i 0.230456 0.0969012i
\(305\) 19.2945 1.10480
\(306\) 0 0
\(307\) −18.4686 −1.05406 −0.527030 0.849847i \(-0.676694\pi\)
−0.527030 + 0.849847i \(0.676694\pi\)
\(308\) 3.43792 + 2.13308i 0.195894 + 0.121544i
\(309\) 0 0
\(310\) 2.27925 0.129453
\(311\) 4.80842i 0.272661i 0.990663 + 0.136330i \(0.0435308\pi\)
−0.990663 + 0.136330i \(0.956469\pi\)
\(312\) 0 0
\(313\) 12.4521i 0.703837i 0.936031 + 0.351918i \(0.114471\pi\)
−0.936031 + 0.351918i \(0.885529\pi\)
\(314\) −21.8613 −1.23371
\(315\) 0 0
\(316\) 4.32385i 0.243235i
\(317\) 30.0767i 1.68927i 0.535340 + 0.844637i \(0.320183\pi\)
−0.535340 + 0.844637i \(0.679817\pi\)
\(318\) 0 0
\(319\) 10.5300i 0.589568i
\(320\) 2.74394i 0.153391i
\(321\) 0 0
\(322\) 9.06171 14.6049i 0.504989 0.813898i
\(323\) 4.71343 + 11.2098i 0.262262 + 0.623728i
\(324\) 0 0
\(325\) 13.5014 0.748924
\(326\) 9.52921i 0.527774i
\(327\) 0 0
\(328\) 11.7026i 0.646167i
\(329\) 6.73898 10.8613i 0.371532 0.598804i
\(330\) 0 0
\(331\) 10.8413i 0.595893i −0.954582 0.297947i \(-0.903698\pi\)
0.954582 0.297947i \(-0.0963018\pi\)
\(332\) 14.4465i 0.792856i
\(333\) 0 0
\(334\) 16.4636i 0.900850i
\(335\) −9.43346 −0.515405
\(336\) 0 0
\(337\) 8.27925i 0.451000i −0.974243 0.225500i \(-0.927598\pi\)
0.974243 0.225500i \(-0.0724015\pi\)
\(338\) 15.4963i 0.842890i
\(339\) 0 0
\(340\) 7.65501 0.415151
\(341\) 1.27024 0.0687872
\(342\) 0 0
\(343\) 1.76012 18.4364i 0.0950377 0.995474i
\(344\) 0.668836i 0.0360612i
\(345\) 0 0
\(346\) 7.59670i 0.408401i
\(347\) −1.32465 −0.0711112 −0.0355556 0.999368i \(-0.511320\pi\)
−0.0355556 + 0.999368i \(0.511320\pi\)
\(348\) 0 0
\(349\) 5.15328i 0.275849i −0.990443 0.137924i \(-0.955957\pi\)
0.990443 0.137924i \(-0.0440431\pi\)
\(350\) 3.52798 5.68609i 0.188578 0.303935i
\(351\) 0 0
\(352\) 1.52921i 0.0815071i
\(353\) 20.5237i 1.09236i −0.837666 0.546182i \(-0.816081\pi\)
0.837666 0.546182i \(-0.183919\pi\)
\(354\) 0 0
\(355\) −31.4551 −1.66946
\(356\) −6.14458 −0.325662
\(357\) 0 0
\(358\) −20.6477 −1.09126
\(359\) 16.2171 0.855905 0.427953 0.903801i \(-0.359235\pi\)
0.427953 + 0.903801i \(0.359235\pi\)
\(360\) 0 0
\(361\) 13.2910 13.5776i 0.699525 0.714608i
\(362\) 7.20564i 0.378720i
\(363\) 0 0
\(364\) −12.0012 7.44622i −0.629033 0.390288i
\(365\) 34.4195 1.80160
\(366\) 0 0
\(367\) 28.7403i 1.50023i −0.661306 0.750116i \(-0.729998\pi\)
0.661306 0.750116i \(-0.270002\pi\)
\(368\) −6.49634 −0.338645
\(369\) 0 0
\(370\) 28.3280i 1.47270i
\(371\) −2.82543 + 4.55378i −0.146689 + 0.236421i
\(372\) 0 0
\(373\) 6.88592i 0.356540i −0.983982 0.178270i \(-0.942950\pi\)
0.983982 0.178270i \(-0.0570500\pi\)
\(374\) 4.26617 0.220598
\(375\) 0 0
\(376\) −4.83118 −0.249149
\(377\) 36.7584i 1.89315i
\(378\) 0 0
\(379\) 8.56208i 0.439804i 0.975522 + 0.219902i \(0.0705738\pi\)
−0.975522 + 0.219902i \(0.929426\pi\)
\(380\) −4.63597 11.0256i −0.237820 0.565599i
\(381\) 0 0
\(382\) 16.5474i 0.846641i
\(383\) 0.0485251 0.00247952 0.00123976 0.999999i \(-0.499605\pi\)
0.00123976 + 0.999999i \(0.499605\pi\)
\(384\) 0 0
\(385\) 5.85306 9.43346i 0.298299 0.480773i
\(386\) −24.3684 −1.24032
\(387\) 0 0
\(388\) −5.31393 −0.269774
\(389\) −2.94158 −0.149144 −0.0745721 0.997216i \(-0.523759\pi\)
−0.0745721 + 0.997216i \(0.523759\pi\)
\(390\) 0 0
\(391\) 18.1234i 0.916540i
\(392\) −6.27192 + 3.10854i −0.316780 + 0.157005i
\(393\) 0 0
\(394\) 6.99268i 0.352286i
\(395\) 11.8644 0.596962
\(396\) 0 0
\(397\) 26.1275i 1.31130i 0.755064 + 0.655651i \(0.227606\pi\)
−0.755064 + 0.655651i \(0.772394\pi\)
\(398\) 20.0719 1.00612
\(399\) 0 0
\(400\) −2.52921 −0.126460
\(401\) 8.31734i 0.415348i −0.978198 0.207674i \(-0.933411\pi\)
0.978198 0.207674i \(-0.0665893\pi\)
\(402\) 0 0
\(403\) −4.43417 −0.220882
\(404\) 8.95863i 0.445709i
\(405\) 0 0
\(406\) 15.4807 + 9.60513i 0.768296 + 0.476695i
\(407\) 15.7873i 0.782548i
\(408\) 0 0
\(409\) 0.0822533 0.00406716 0.00203358 0.999998i \(-0.499353\pi\)
0.00203358 + 0.999998i \(0.499353\pi\)
\(410\) 32.1112 1.58586
\(411\) 0 0
\(412\) −12.7288 −0.627101
\(413\) −12.4408 7.71896i −0.612170 0.379825i
\(414\) 0 0
\(415\) 39.6404 1.94587
\(416\) 5.33820i 0.261727i
\(417\) 0 0
\(418\) −2.58364 6.14458i −0.126370 0.300541i
\(419\) 20.4384i 0.998480i 0.866464 + 0.499240i \(0.166387\pi\)
−0.866464 + 0.499240i \(0.833613\pi\)
\(420\) 0 0
\(421\) 11.2098i 0.546331i −0.961967 0.273165i \(-0.911929\pi\)
0.961967 0.273165i \(-0.0880706\pi\)
\(422\) 7.48902 0.364560
\(423\) 0 0
\(424\) 2.02555 0.0983694
\(425\) 7.05596i 0.342264i
\(426\) 0 0
\(427\) −9.80846 + 15.8084i −0.474665 + 0.765024i
\(428\) 1.15867i 0.0560064i
\(429\) 0 0
\(430\) 1.83525 0.0885035
\(431\) 15.6916i 0.755839i −0.925838 0.377920i \(-0.876640\pi\)
0.925838 0.377920i \(-0.123360\pi\)
\(432\) 0 0
\(433\) 30.1026 1.44664 0.723319 0.690514i \(-0.242615\pi\)
0.723319 + 0.690514i \(0.242615\pi\)
\(434\) −1.15867 + 1.86744i −0.0556179 + 0.0896402i
\(435\) 0 0
\(436\) 2.02555i 0.0970063i
\(437\) −26.1032 + 10.9758i −1.24869 + 0.525042i
\(438\) 0 0
\(439\) −32.1452 −1.53420 −0.767102 0.641525i \(-0.778302\pi\)
−0.767102 + 0.641525i \(0.778302\pi\)
\(440\) −4.19606 −0.200039
\(441\) 0 0
\(442\) −14.8924 −0.708361
\(443\) 29.3165 1.39287 0.696435 0.717620i \(-0.254768\pi\)
0.696435 + 0.717620i \(0.254768\pi\)
\(444\) 0 0
\(445\) 16.8604i 0.799258i
\(446\) 13.5806i 0.643061i
\(447\) 0 0
\(448\) 2.24817 + 1.39489i 0.106216 + 0.0659025i
\(449\) 14.3173i 0.675677i 0.941204 + 0.337838i \(0.109696\pi\)
−0.941204 + 0.337838i \(0.890304\pi\)
\(450\) 0 0
\(451\) 17.8957 0.842675
\(452\) 10.5966i 0.498422i
\(453\) 0 0
\(454\) 23.9027i 1.12181i
\(455\) −20.4320 + 32.9305i −0.957866 + 1.54381i
\(456\) 0 0
\(457\) −30.5564 −1.42937 −0.714684 0.699447i \(-0.753430\pi\)
−0.714684 + 0.699447i \(0.753430\pi\)
\(458\) 15.0816 0.704718
\(459\) 0 0
\(460\) 17.8256i 0.831122i
\(461\) 5.25748i 0.244865i −0.992477 0.122433i \(-0.960930\pi\)
0.992477 0.122433i \(-0.0390695\pi\)
\(462\) 0 0
\(463\) 13.4926 0.627054 0.313527 0.949579i \(-0.398489\pi\)
0.313527 + 0.949579i \(0.398489\pi\)
\(464\) 6.88592i 0.319671i
\(465\) 0 0
\(466\) 12.1258i 0.561717i
\(467\) 26.7842i 1.23943i 0.784829 + 0.619713i \(0.212751\pi\)
−0.784829 + 0.619713i \(0.787249\pi\)
\(468\) 0 0
\(469\) 4.79554 7.72904i 0.221437 0.356894i
\(470\) 13.2565i 0.611475i
\(471\) 0 0
\(472\) 5.53373i 0.254710i
\(473\) 1.02279 0.0470280
\(474\) 0 0
\(475\) −10.1627 + 4.27317i −0.466298 + 0.196067i
\(476\) −3.89146 + 6.27192i −0.178365 + 0.287473i
\(477\) 0 0
\(478\) 9.55476i 0.437025i
\(479\) 20.1864i 0.922340i −0.887312 0.461170i \(-0.847430\pi\)
0.887312 0.461170i \(-0.152570\pi\)
\(480\) 0 0
\(481\) 55.1107i 2.51283i
\(482\) 8.77104i 0.399510i
\(483\) 0 0
\(484\) 8.66152 0.393705
\(485\) 14.5811i 0.662094i
\(486\) 0 0
\(487\) 24.0576i 1.09015i −0.838386 0.545077i \(-0.816500\pi\)
0.838386 0.545077i \(-0.183500\pi\)
\(488\) 7.03169 0.318310
\(489\) 0 0
\(490\) 8.52966 + 17.2098i 0.385331 + 0.777458i
\(491\) 1.32465 0.0597808 0.0298904 0.999553i \(-0.490484\pi\)
0.0298904 + 0.999553i \(0.490484\pi\)
\(492\) 0 0
\(493\) 19.2103 0.865187
\(494\) 9.01904 + 21.4496i 0.405786 + 0.965065i
\(495\) 0 0
\(496\) 0.830650 0.0372973
\(497\) 15.9903 25.7718i 0.717264 1.15603i
\(498\) 0 0
\(499\) 25.8620 1.15774 0.578872 0.815419i \(-0.303493\pi\)
0.578872 + 0.815419i \(0.303493\pi\)
\(500\) 6.77970i 0.303198i
\(501\) 0 0
\(502\) 2.10882 0.0941213
\(503\) 19.9802i 0.890875i −0.895313 0.445437i \(-0.853048\pi\)
0.895313 0.445437i \(-0.146952\pi\)
\(504\) 0 0
\(505\) 24.5820 1.09388
\(506\) 9.93426i 0.441632i
\(507\) 0 0
\(508\) 19.2353i 0.853429i
\(509\) 25.3306 1.12276 0.561379 0.827559i \(-0.310271\pi\)
0.561379 + 0.827559i \(0.310271\pi\)
\(510\) 0 0
\(511\) −17.4973 + 28.2007i −0.774036 + 1.24752i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 22.7700i 1.00434i
\(515\) 34.9269i 1.53907i
\(516\) 0 0
\(517\) 7.38788i 0.324919i
\(518\) 23.2098 + 14.4007i 1.01978 + 0.632729i
\(519\) 0 0
\(520\) 14.6477 0.642344
\(521\) −15.1934 −0.665635 −0.332818 0.942991i \(-0.607999\pi\)
−0.332818 + 0.942991i \(0.607999\pi\)
\(522\) 0 0
\(523\) −15.1839 −0.663947 −0.331974 0.943289i \(-0.607715\pi\)
−0.331974 + 0.943289i \(0.607715\pi\)
\(524\) 8.95863i 0.391360i
\(525\) 0 0
\(526\) 22.3027i 0.972444i
\(527\) 2.31734i 0.100945i
\(528\) 0 0
\(529\) 19.2025 0.834889
\(530\) 5.55799i 0.241424i
\(531\) 0 0
\(532\) 11.3902 + 1.80653i 0.493827 + 0.0783232i
\(533\) −62.4706 −2.70590
\(534\) 0 0
\(535\) −3.17932 −0.137454
\(536\) −3.43792 −0.148496
\(537\) 0 0
\(538\) 3.88304i 0.167410i
\(539\) 4.75361 + 9.59107i 0.204753 + 0.413117i
\(540\) 0 0
\(541\) 32.3684 1.39163 0.695814 0.718222i \(-0.255044\pi\)
0.695814 + 0.718222i \(0.255044\pi\)
\(542\) 19.7969 0.850348
\(543\) 0 0
\(544\) 2.78979 0.119611
\(545\) 5.55799 0.238078
\(546\) 0 0
\(547\) 32.1022i 1.37259i 0.727323 + 0.686296i \(0.240764\pi\)
−0.727323 + 0.686296i \(0.759236\pi\)
\(548\) −0.769087 −0.0328538
\(549\) 0 0
\(550\) 3.86769i 0.164919i
\(551\) −11.6340 27.6686i −0.495624 1.17872i
\(552\) 0 0
\(553\) −6.03131 + 9.72075i −0.256477 + 0.413368i
\(554\) 21.7061i 0.922204i
\(555\) 0 0
\(556\) 4.60077i 0.195116i
\(557\) 29.4122 1.24624 0.623118 0.782128i \(-0.285866\pi\)
0.623118 + 0.782128i \(0.285866\pi\)
\(558\) 0 0
\(559\) −3.57038 −0.151011
\(560\) 3.82751 6.16885i 0.161742 0.260681i
\(561\) 0 0
\(562\) 0 0
\(563\) −18.2394 −0.768698 −0.384349 0.923188i \(-0.625574\pi\)
−0.384349 + 0.923188i \(0.625574\pi\)
\(564\) 0 0
\(565\) 29.0764 1.22325
\(566\) −0.703721 −0.0295796
\(567\) 0 0
\(568\) −11.4635 −0.480997
\(569\) 6.50741i 0.272805i 0.990654 + 0.136402i \(0.0435540\pi\)
−0.990654 + 0.136402i \(0.956446\pi\)
\(570\) 0 0
\(571\) 24.2557 1.01507 0.507534 0.861632i \(-0.330557\pi\)
0.507534 + 0.861632i \(0.330557\pi\)
\(572\) 8.16322 0.341321
\(573\) 0 0
\(574\) −16.3238 + 26.3094i −0.681344 + 1.09813i
\(575\) 16.4306 0.685204
\(576\) 0 0
\(577\) 0.341005i 0.0141962i 0.999975 + 0.00709812i \(0.00225942\pi\)
−0.999975 + 0.00709812i \(0.997741\pi\)
\(578\) 9.21709i 0.383380i
\(579\) 0 0
\(580\) −18.8946 −0.784554
\(581\) −20.1514 + 32.4782i −0.836019 + 1.34742i
\(582\) 0 0
\(583\) 3.09749i 0.128285i
\(584\) 12.5438 0.519068
\(585\) 0 0
\(586\) 11.5650i 0.477747i
\(587\) 16.9385i 0.699125i 0.936913 + 0.349563i \(0.113670\pi\)
−0.936913 + 0.349563i \(0.886330\pi\)
\(588\) 0 0
\(589\) 3.33767 1.40341i 0.137526 0.0578264i
\(590\) 15.1842 0.625124
\(591\) 0 0
\(592\) 10.3238i 0.424307i
\(593\) 0.183387i 0.00753082i −0.999993 0.00376541i \(-0.998801\pi\)
0.999993 0.00376541i \(-0.00119857\pi\)
\(594\) 0 0
\(595\) 17.2098 + 10.6779i 0.705532 + 0.437752i
\(596\) −1.65501 −0.0677919
\(597\) 0 0
\(598\) 34.6787i 1.41812i
\(599\) 12.5365i 0.512229i −0.966647 0.256114i \(-0.917558\pi\)
0.966647 0.256114i \(-0.0824423\pi\)
\(600\) 0 0
\(601\) −15.0774 −0.615021 −0.307510 0.951545i \(-0.599496\pi\)
−0.307510 + 0.951545i \(0.599496\pi\)
\(602\) −0.932956 + 1.50366i −0.0380244 + 0.0612845i
\(603\) 0 0
\(604\) 3.72075i 0.151395i
\(605\) 23.7667i 0.966253i
\(606\) 0 0
\(607\) −2.05236 −0.0833028 −0.0416514 0.999132i \(-0.513262\pi\)
−0.0416514 + 0.999132i \(0.513262\pi\)
\(608\) −1.68953 4.01815i −0.0685195 0.162957i
\(609\) 0 0
\(610\) 19.2945i 0.781213i
\(611\) 25.7898i 1.04334i
\(612\) 0 0
\(613\) −21.8685 −0.883262 −0.441631 0.897197i \(-0.645600\pi\)
−0.441631 + 0.897197i \(0.645600\pi\)
\(614\) 18.4686i 0.745333i
\(615\) 0 0
\(616\) 2.13308 3.43792i 0.0859444 0.138518i
\(617\) 1.59740 0.0643089 0.0321544 0.999483i \(-0.489763\pi\)
0.0321544 + 0.999483i \(0.489763\pi\)
\(618\) 0 0
\(619\) 17.3993i 0.699336i 0.936874 + 0.349668i \(0.113706\pi\)
−0.936874 + 0.349668i \(0.886294\pi\)
\(620\) 2.27925i 0.0915371i
\(621\) 0 0
\(622\) 4.80842 0.192800
\(623\) −13.8141 8.57104i −0.553449 0.343392i
\(624\) 0 0
\(625\) −31.2491 −1.24997
\(626\) 12.4521 0.497688
\(627\) 0 0
\(628\) 21.8613i 0.872362i
\(629\) 28.8013 1.14838
\(630\) 0 0
\(631\) −15.5382 −0.618565 −0.309282 0.950970i \(-0.600089\pi\)
−0.309282 + 0.950970i \(0.600089\pi\)
\(632\) 4.32385 0.171993
\(633\) 0 0
\(634\) 30.0767 1.19450
\(635\) 52.7806 2.09453
\(636\) 0 0
\(637\) −16.5940 33.4807i −0.657479 1.32655i
\(638\) −10.5300 −0.416887
\(639\) 0 0
\(640\) −2.74394 −0.108464
\(641\) 3.72075i 0.146961i −0.997297 0.0734803i \(-0.976589\pi\)
0.997297 0.0734803i \(-0.0234106\pi\)
\(642\) 0 0
\(643\) 14.9990i 0.591504i 0.955265 + 0.295752i \(0.0955701\pi\)
−0.955265 + 0.295752i \(0.904430\pi\)
\(644\) −14.6049 9.06171i −0.575513 0.357081i
\(645\) 0 0
\(646\) 11.2098 4.71343i 0.441042 0.185447i
\(647\) 10.5953i 0.416545i −0.978071 0.208272i \(-0.933216\pi\)
0.978071 0.208272i \(-0.0667840\pi\)
\(648\) 0 0
\(649\) 8.46223 0.332171
\(650\) 13.5014i 0.529569i
\(651\) 0 0
\(652\) 9.52921 0.373193
\(653\) 20.5129 0.802733 0.401366 0.915918i \(-0.368535\pi\)
0.401366 + 0.915918i \(0.368535\pi\)
\(654\) 0 0
\(655\) −24.5820 −0.960497
\(656\) 11.7026 0.456909
\(657\) 0 0
\(658\) −10.8613 6.73898i −0.423418 0.262713i
\(659\) 44.7154i 1.74186i −0.491404 0.870932i \(-0.663516\pi\)
0.491404 0.870932i \(-0.336484\pi\)
\(660\) 0 0
\(661\) 21.2948 0.828272 0.414136 0.910215i \(-0.364084\pi\)
0.414136 + 0.910215i \(0.364084\pi\)
\(662\) −10.8413 −0.421360
\(663\) 0 0
\(664\) 14.4465 0.560634
\(665\) 4.95702 31.2540i 0.192225 1.21198i
\(666\) 0 0
\(667\) 44.7333i 1.73208i
\(668\) 16.4636 0.636997
\(669\) 0 0
\(670\) 9.43346i 0.364446i
\(671\) 10.7529i 0.415112i
\(672\) 0 0
\(673\) 34.3815i 1.32531i −0.748926 0.662654i \(-0.769430\pi\)
0.748926 0.662654i \(-0.230570\pi\)
\(674\) −8.27925 −0.318905
\(675\) 0 0
\(676\) −15.4963 −0.596013
\(677\) −13.2476 −0.509145 −0.254573 0.967054i \(-0.581935\pi\)
−0.254573 + 0.967054i \(0.581935\pi\)
\(678\) 0 0
\(679\) −11.9466 7.41237i −0.458470 0.284461i
\(680\) 7.65501i 0.293556i
\(681\) 0 0
\(682\) 1.27024i 0.0486399i
\(683\) 40.0892i 1.53397i −0.641665 0.766985i \(-0.721756\pi\)
0.641665 0.766985i \(-0.278244\pi\)
\(684\) 0 0
\(685\) 2.11033i 0.0806315i
\(686\) −18.4364 1.76012i −0.703906 0.0672018i
\(687\) 0 0
\(688\) 0.668836 0.0254991
\(689\) 10.8128i 0.411934i
\(690\) 0 0
\(691\) 3.88304i 0.147718i 0.997269 + 0.0738589i \(0.0235314\pi\)
−0.997269 + 0.0738589i \(0.976469\pi\)
\(692\) −7.59670 −0.288783
\(693\) 0 0
\(694\) 1.32465i 0.0502832i
\(695\) 12.6242 0.478865
\(696\) 0 0
\(697\) 32.6477i 1.23662i
\(698\) −5.15328 −0.195054
\(699\) 0 0
\(700\) −5.68609 3.52798i −0.214914 0.133345i
\(701\) −42.4651 −1.60389 −0.801943 0.597401i \(-0.796200\pi\)
−0.801943 + 0.597401i \(0.796200\pi\)
\(702\) 0 0
\(703\) −17.4424 41.4827i −0.657854 1.56455i
\(704\) −1.52921 −0.0576342
\(705\) 0 0
\(706\) −20.5237 −0.772418
\(707\) −12.4963 + 20.1405i −0.469973 + 0.757463i
\(708\) 0 0
\(709\) 9.49259 0.356502 0.178251 0.983985i \(-0.442956\pi\)
0.178251 + 0.983985i \(0.442956\pi\)
\(710\) 31.4551i 1.18049i
\(711\) 0 0
\(712\) 6.14458i 0.230278i
\(713\) −5.39619 −0.202089
\(714\) 0 0
\(715\) 22.3994i 0.837690i
\(716\) 20.6477i 0.771641i
\(717\) 0 0
\(718\) 16.2171i 0.605216i
\(719\) 40.2152i 1.49977i 0.661567 + 0.749886i \(0.269892\pi\)
−0.661567 + 0.749886i \(0.730108\pi\)
\(720\) 0 0
\(721\) −28.6164 17.7553i −1.06573 0.661240i
\(722\) −13.5776 13.2910i −0.505304 0.494639i
\(723\) 0 0
\(724\) 7.20564 0.267795
\(725\) 17.4159i 0.646812i
\(726\) 0 0
\(727\) 15.8327i 0.587203i 0.955928 + 0.293601i \(0.0948538\pi\)
−0.955928 + 0.293601i \(0.905146\pi\)
\(728\) −7.44622 + 12.0012i −0.275975 + 0.444793i
\(729\) 0 0
\(730\) 34.4195i 1.27392i
\(731\) 1.86591i 0.0690132i
\(732\) 0 0
\(733\) 20.8446i 0.769913i −0.922935 0.384956i \(-0.874216\pi\)
0.922935 0.384956i \(-0.125784\pi\)
\(734\) −28.7403 −1.06082
\(735\) 0 0
\(736\) 6.49634i 0.239458i
\(737\) 5.25730i 0.193655i
\(738\) 0 0
\(739\) −34.9992 −1.28747 −0.643733 0.765250i \(-0.722615\pi\)
−0.643733 + 0.765250i \(0.722615\pi\)
\(740\) −28.3280 −1.04136
\(741\) 0 0
\(742\) 4.55378 + 2.82543i 0.167175 + 0.103725i
\(743\) 41.8985i 1.53711i 0.639786 + 0.768553i \(0.279023\pi\)
−0.639786 + 0.768553i \(0.720977\pi\)
\(744\) 0 0
\(745\) 4.54125i 0.166378i
\(746\) −6.88592 −0.252112
\(747\) 0 0
\(748\) 4.26617i 0.155987i
\(749\) 1.61622 2.60488i 0.0590554 0.0951804i
\(750\) 0 0
\(751\) 47.0226i 1.71588i −0.513749 0.857940i \(-0.671744\pi\)
0.513749 0.857940i \(-0.328256\pi\)
\(752\) 4.83118i 0.176175i
\(753\) 0 0
\(754\) 36.7584 1.33866
\(755\) −10.2095 −0.371562
\(756\) 0 0
\(757\) 12.1859 0.442903 0.221451 0.975171i \(-0.428921\pi\)
0.221451 + 0.975171i \(0.428921\pi\)
\(758\) 8.56208 0.310989
\(759\) 0 0
\(760\) −11.0256 + 4.63597i −0.399939 + 0.168164i
\(761\) 43.1626i 1.56464i 0.622876 + 0.782321i \(0.285964\pi\)
−0.622876 + 0.782321i \(0.714036\pi\)
\(762\) 0 0
\(763\) −2.82543 + 4.55378i −0.102287 + 0.164858i
\(764\) 16.5474 0.598665
\(765\) 0 0
\(766\) 0.0485251i 0.00175328i
\(767\) −29.5401 −1.06663
\(768\) 0 0
\(769\) 44.4328i 1.60229i −0.598472 0.801144i \(-0.704225\pi\)
0.598472 0.801144i \(-0.295775\pi\)
\(770\) −9.43346 5.85306i −0.339958 0.210929i
\(771\) 0 0
\(772\) 24.3684i 0.877039i
\(773\) −39.0961 −1.40619 −0.703095 0.711096i \(-0.748199\pi\)
−0.703095 + 0.711096i \(0.748199\pi\)
\(774\) 0 0
\(775\) −2.10089 −0.0754661
\(776\) 5.31393i 0.190759i
\(777\) 0 0
\(778\) 2.94158i 0.105461i
\(779\) 47.0226 19.7718i 1.68476 0.708400i
\(780\) 0 0
\(781\) 17.5300i 0.627274i
\(782\) −18.1234 −0.648092
\(783\) 0 0
\(784\) 3.10854 + 6.27192i 0.111019 + 0.223997i
\(785\) 59.9862 2.14100
\(786\) 0 0
\(787\) −4.65455 −0.165917 −0.0829584 0.996553i \(-0.526437\pi\)
−0.0829584 + 0.996553i \(0.526437\pi\)
\(788\) −6.99268 −0.249104
\(789\) 0 0
\(790\) 11.8644i 0.422116i
\(791\) −14.7811 + 23.8229i −0.525556 + 0.847047i
\(792\) 0 0
\(793\) 37.5366i 1.33296i
\(794\) 26.1275 0.927230
\(795\) 0 0
\(796\) 20.0719i 0.711431i
\(797\) 23.9027 0.846678 0.423339 0.905971i \(-0.360858\pi\)
0.423339 + 0.905971i \(0.360858\pi\)
\(798\) 0 0
\(799\) −13.4780 −0.476816
\(800\) 2.52921i 0.0894210i
\(801\) 0 0
\(802\) −8.31734 −0.293695
\(803\) 19.1821i 0.676923i
\(804\) 0 0
\(805\) −24.8648 + 40.0749i −0.876368 + 1.41246i
\(806\) 4.43417i 0.156187i
\(807\) 0 0
\(808\) 8.95863 0.315164
\(809\) −23.3657 −0.821494 −0.410747 0.911749i \(-0.634732\pi\)
−0.410747 + 0.911749i \(0.634732\pi\)
\(810\) 0 0
\(811\) 44.7337 1.57081 0.785406 0.618981i \(-0.212454\pi\)
0.785406 + 0.618981i \(0.212454\pi\)
\(812\) 9.60513 15.4807i 0.337074 0.543267i
\(813\) 0 0
\(814\) −15.7873 −0.553345
\(815\) 26.1476i 0.915910i
\(816\) 0 0
\(817\) 2.68748 1.13002i 0.0940231 0.0395344i
\(818\) 0.0822533i 0.00287592i
\(819\) 0 0
\(820\) 32.1112i 1.12137i
\(821\) 15.4269 0.538401 0.269201 0.963084i \(-0.413241\pi\)
0.269201 + 0.963084i \(0.413241\pi\)
\(822\) 0 0
\(823\) 44.0199 1.53444 0.767218 0.641386i \(-0.221640\pi\)
0.767218 + 0.641386i \(0.221640\pi\)
\(824\) 12.7288i 0.443427i
\(825\) 0 0
\(826\) −7.71896 + 12.4408i −0.268577 + 0.432870i
\(827\) 0.651261i 0.0226466i 0.999936 + 0.0113233i \(0.00360439\pi\)
−0.999936 + 0.0113233i \(0.996396\pi\)
\(828\) 0 0
\(829\) −9.76653 −0.339206 −0.169603 0.985513i \(-0.554248\pi\)
−0.169603 + 0.985513i \(0.554248\pi\)
\(830\) 39.6404i 1.37594i
\(831\) 0 0
\(832\) 5.33820 0.185069
\(833\) −17.4973 + 8.67217i −0.606246 + 0.300473i
\(834\) 0 0
\(835\) 45.1753i 1.56335i
\(836\) −6.14458 + 2.58364i −0.212515 + 0.0893572i
\(837\) 0 0
\(838\) 20.4384 0.706032
\(839\) −30.7589 −1.06192 −0.530958 0.847398i \(-0.678168\pi\)
−0.530958 + 0.847398i \(0.678168\pi\)
\(840\) 0 0
\(841\) −18.4159 −0.635032
\(842\) −11.2098 −0.386314
\(843\) 0 0
\(844\) 7.48902i 0.257783i
\(845\) 42.5210i 1.46277i
\(846\) 0 0
\(847\) 19.4726 + 12.0819i 0.669086 + 0.415139i
\(848\) 2.02555i 0.0695577i
\(849\) 0 0
\(850\) −7.05596 −0.242017
\(851\) 67.0672i 2.29904i
\(852\) 0 0
\(853\) 1.15311i 0.0394817i −0.999805 0.0197409i \(-0.993716\pi\)
0.999805 0.0197409i \(-0.00628412\pi\)
\(854\) 15.8084 + 9.80846i 0.540954 + 0.335639i
\(855\) 0 0
\(856\) −1.15867 −0.0396025
\(857\) 34.8849 1.19165 0.595823 0.803116i \(-0.296826\pi\)
0.595823 + 0.803116i \(0.296826\pi\)
\(858\) 0 0
\(859\) 39.4425i 1.34576i −0.739751 0.672880i \(-0.765057\pi\)
0.739751 0.672880i \(-0.234943\pi\)
\(860\) 1.83525i 0.0625814i
\(861\) 0 0
\(862\) −15.6916 −0.534459
\(863\) 45.6192i 1.55290i 0.630181 + 0.776448i \(0.282981\pi\)
−0.630181 + 0.776448i \(0.717019\pi\)
\(864\) 0 0
\(865\) 20.8449i 0.708748i
\(866\) 30.1026i 1.02293i
\(867\) 0 0
\(868\) 1.86744 + 1.15867i 0.0633852 + 0.0393278i
\(869\) 6.61207i 0.224299i
\(870\) 0 0
\(871\) 18.3523i 0.621844i
\(872\) 2.02555 0.0685938
\(873\) 0 0
\(874\) 10.9758 + 26.1032i 0.371261 + 0.882955i
\(875\) 9.45696 15.2419i 0.319704 0.515271i
\(876\) 0 0
\(877\) 21.4670i 0.724890i 0.932005 + 0.362445i \(0.118058\pi\)
−0.932005 + 0.362445i \(0.881942\pi\)
\(878\) 32.1452i 1.08485i
\(879\) 0 0
\(880\) 4.19606i 0.141449i
\(881\) 23.2218i 0.782361i −0.920314 0.391181i \(-0.872067\pi\)
0.920314 0.391181i \(-0.127933\pi\)
\(882\) 0 0
\(883\) −55.2588 −1.85961 −0.929803 0.368058i \(-0.880023\pi\)
−0.929803 + 0.368058i \(0.880023\pi\)
\(884\) 14.8924i 0.500887i
\(885\) 0 0
\(886\) 29.3165i 0.984908i
\(887\) −16.7872 −0.563661 −0.281830 0.959464i \(-0.590942\pi\)
−0.281830 + 0.959464i \(0.590942\pi\)
\(888\) 0 0
\(889\) −26.8312 + 43.2443i −0.899890 + 1.45037i
\(890\) 16.8604 0.565161
\(891\) 0 0
\(892\) −13.5806 −0.454713
\(893\) 8.16242 + 19.4124i 0.273145 + 0.649610i
\(894\) 0 0
\(895\) 56.6560 1.89380
\(896\) 1.39489 2.24817i 0.0466001 0.0751061i
\(897\) 0 0
\(898\) 14.3173 0.477776
\(899\) 5.71979i 0.190766i
\(900\) 0 0
\(901\) 5.65086 0.188257
\(902\) 17.8957i 0.595861i
\(903\) 0 0
\(904\) 10.5966 0.352437
\(905\) 19.7718i 0.657238i
\(906\) 0 0
\(907\) 39.5912i 1.31461i 0.753627 + 0.657303i \(0.228303\pi\)
−0.753627 + 0.657303i \(0.771697\pi\)
\(908\) −23.9027 −0.793240
\(909\) 0 0
\(910\) 32.9305 + 20.4320i 1.09164 + 0.677313i
\(911\) 24.6031i 0.815137i −0.913175 0.407569i \(-0.866377\pi\)
0.913175 0.407569i \(-0.133623\pi\)
\(912\) 0 0
\(913\) 22.0917i 0.731130i
\(914\) 30.5564i 1.01072i
\(915\) 0 0
\(916\) 15.0816i 0.498311i
\(917\) 12.4963 20.1405i 0.412666 0.665099i
\(918\) 0 0
\(919\) 20.7300 0.683820 0.341910 0.939733i \(-0.388926\pi\)
0.341910 + 0.939733i \(0.388926\pi\)
\(920\) 17.8256 0.587692
\(921\) 0 0
\(922\) −5.25748 −0.173146
\(923\) 61.1943i 2.01423i
\(924\) 0 0
\(925\) 26.1112i 0.858530i
\(926\) 13.4926i 0.443394i
\(927\) 0 0
\(928\) −6.88592 −0.226042
\(929\) 22.0690i 0.724060i 0.932166 + 0.362030i \(0.117916\pi\)
−0.932166 + 0.362030i \(0.882084\pi\)
\(930\) 0 0
\(931\) 23.0872 + 19.9495i 0.756651 + 0.653818i
\(932\) 12.1258 0.397194
\(933\) 0 0
\(934\) 26.7842 0.876406
\(935\) −11.7061 −0.382831
\(936\) 0 0
\(937\) 41.6688i 1.36126i 0.732627 + 0.680630i \(0.238294\pi\)
−0.732627 + 0.680630i \(0.761706\pi\)
\(938\) −7.72904 4.79554i −0.252362 0.156580i
\(939\) 0 0
\(940\) 13.2565 0.432378
\(941\) −20.6014 −0.671585 −0.335792 0.941936i \(-0.609004\pi\)
−0.335792 + 0.941936i \(0.609004\pi\)
\(942\) 0 0
\(943\) −76.0239 −2.47568
\(944\) 5.53373 0.180107
\(945\) 0 0
\(946\) 1.02279i 0.0332538i
\(947\) −35.2077 −1.14410 −0.572048 0.820220i \(-0.693851\pi\)
−0.572048 + 0.820220i \(0.693851\pi\)
\(948\) 0 0
\(949\) 66.9615i 2.17366i
\(950\) 4.27317 + 10.1627i 0.138640 + 0.329722i
\(951\) 0 0
\(952\) 6.27192 + 3.89146i 0.203274 + 0.126123i
\(953\) 8.06412i 0.261222i −0.991434 0.130611i \(-0.958306\pi\)
0.991434 0.130611i \(-0.0416940\pi\)
\(954\) 0 0
\(955\) 45.4052i 1.46928i
\(956\) 9.55476 0.309023
\(957\) 0 0
\(958\) −20.1864 −0.652193
\(959\) −1.72904 1.07279i −0.0558336 0.0346423i
\(960\) 0 0
\(961\) −30.3100 −0.977743
\(962\) 55.1107 1.77684
\(963\) 0 0
\(964\) 8.77104 0.282496
\(965\) 66.8655 2.15248
\(966\) 0 0
\(967\) 37.9723 1.22111 0.610554 0.791974i \(-0.290947\pi\)
0.610554 + 0.791974i \(0.290947\pi\)
\(968\) 8.66152i 0.278392i
\(969\) 0 0
\(970\) 14.5811 0.468171
\(971\) −46.3048 −1.48599 −0.742996 0.669296i \(-0.766596\pi\)
−0.742996 + 0.669296i \(0.766596\pi\)
\(972\) 0 0
\(973\) −6.41759 + 10.3433i −0.205738 + 0.331592i
\(974\) −24.0576 −0.770856
\(975\) 0 0
\(976\) 7.03169i 0.225079i
\(977\) 26.9780i 0.863104i 0.902088 + 0.431552i \(0.142034\pi\)
−0.902088 + 0.431552i \(0.857966\pi\)
\(978\) 0 0
\(979\) 9.39635 0.300309
\(980\) 17.2098 8.52966i 0.549746 0.272470i
\(981\) 0 0
\(982\) 1.32465i 0.0422714i
\(983\) −49.7145 −1.58565 −0.792824 0.609451i \(-0.791390\pi\)
−0.792824 + 0.609451i \(0.791390\pi\)
\(984\) 0 0
\(985\) 19.1875i 0.611365i
\(986\) 19.2103i 0.611779i
\(987\) 0 0
\(988\) 21.4496 9.01904i 0.682404 0.286934i
\(989\) −4.34499 −0.138163
\(990\) 0 0
\(991\) 42.3393i 1.34495i −0.740119 0.672476i \(-0.765231\pi\)
0.740119 0.672476i \(-0.234769\pi\)
\(992\) 0.830650i 0.0263732i
\(993\) 0 0
\(994\) −25.7718 15.9903i −0.817433 0.507182i
\(995\) −55.0762 −1.74603
\(996\) 0 0
\(997\) 15.3567i 0.486352i 0.969982 + 0.243176i \(0.0781893\pi\)
−0.969982 + 0.243176i \(0.921811\pi\)
\(998\) 25.8620i 0.818648i
\(999\) 0 0
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 2394.2.e.c.1063.7 16
3.2 odd 2 266.2.d.a.265.12 yes 16
7.6 odd 2 inner 2394.2.e.c.1063.2 16
12.11 even 2 2128.2.m.f.1329.9 16
19.18 odd 2 inner 2394.2.e.c.1063.15 16
21.20 even 2 266.2.d.a.265.13 yes 16
57.56 even 2 266.2.d.a.265.5 yes 16
84.83 odd 2 2128.2.m.f.1329.8 16
133.132 even 2 inner 2394.2.e.c.1063.10 16
228.227 odd 2 2128.2.m.f.1329.7 16
399.398 odd 2 266.2.d.a.265.4 16
1596.1595 even 2 2128.2.m.f.1329.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.d.a.265.4 16 399.398 odd 2
266.2.d.a.265.5 yes 16 57.56 even 2
266.2.d.a.265.12 yes 16 3.2 odd 2
266.2.d.a.265.13 yes 16 21.20 even 2
2128.2.m.f.1329.7 16 228.227 odd 2
2128.2.m.f.1329.8 16 84.83 odd 2
2128.2.m.f.1329.9 16 12.11 even 2
2128.2.m.f.1329.10 16 1596.1595 even 2
2394.2.e.c.1063.2 16 7.6 odd 2 inner
2394.2.e.c.1063.7 16 1.1 even 1 trivial
2394.2.e.c.1063.10 16 133.132 even 2 inner
2394.2.e.c.1063.15 16 19.18 odd 2 inner