Properties

Label 2790.2.d.g.559.2
Level $2790$
Weight $2$
Character 2790.559
Analytic conductor $22.278$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2790,2,Mod(559,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2790.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.d (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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 559.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2790.559
Dual form 2790.2.d.g.559.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +5.00000 q^{11} -4.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +5.00000 q^{19} +(-1.00000 + 2.00000i) q^{20} +5.00000i q^{22} +9.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +4.00000 q^{26} +1.00000i q^{28} -2.00000 q^{29} +1.00000 q^{31} +1.00000i q^{32} +(-2.00000 - 1.00000i) q^{35} -8.00000i q^{37} +5.00000i q^{38} +(-2.00000 - 1.00000i) q^{40} -6.00000 q^{41} +1.00000i q^{43} -5.00000 q^{44} -9.00000 q^{46} -12.0000i q^{47} +6.00000 q^{49} +(4.00000 - 3.00000i) q^{50} +4.00000i q^{52} +13.0000i q^{53} +(5.00000 - 10.0000i) q^{55} -1.00000 q^{56} -2.00000i q^{58} +10.0000 q^{59} -14.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} +(-8.00000 - 4.00000i) q^{65} -14.0000i q^{67} +(1.00000 - 2.00000i) q^{70} +9.00000 q^{71} -9.00000i q^{73} +8.00000 q^{74} -5.00000 q^{76} -5.00000i q^{77} -5.00000 q^{79} +(1.00000 - 2.00000i) q^{80} -6.00000i q^{82} -6.00000i q^{83} -1.00000 q^{86} -5.00000i q^{88} +3.00000 q^{89} -4.00000 q^{91} -9.00000i q^{92} +12.0000 q^{94} +(5.00000 - 10.0000i) q^{95} +18.0000i q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 2 q^{5} + 4 q^{10} + 10 q^{11} + 2 q^{14} + 2 q^{16} + 10 q^{19} - 2 q^{20} - 6 q^{25} + 8 q^{26} - 4 q^{29} + 2 q^{31} - 4 q^{35} - 4 q^{40} - 12 q^{41} - 10 q^{44} - 18 q^{46} + 12 q^{49} + 8 q^{50} + 10 q^{55} - 2 q^{56} + 20 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{65} + 2 q^{70} + 18 q^{71} + 16 q^{74} - 10 q^{76} - 10 q^{79} + 2 q^{80} - 2 q^{86} + 6 q^{89} - 8 q^{91} + 24 q^{94} + 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times\).

\(n\) \(1117\) \(1801\) \(2171\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) 1.00000 2.00000i 0.447214 0.894427i
\(6\) 0 0
\(7\) 1.00000i 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.00000 + 1.00000i 0.632456 + 0.316228i
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 4.00000i 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −1.00000 + 2.00000i −0.223607 + 0.447214i
\(21\) 0 0
\(22\) 5.00000i 1.06600i
\(23\) 9.00000i 1.87663i 0.345782 + 0.938315i \(0.387614\pi\)
−0.345782 + 0.938315i \(0.612386\pi\)
\(24\) 0 0
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000i 0.188982i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 1.00000i −0.338062 0.169031i
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 5.00000i 0.811107i
\(39\) 0 0
\(40\) −2.00000 1.00000i −0.316228 0.158114i
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) 12.0000i 1.75038i −0.483779 0.875190i \(-0.660736\pi\)
0.483779 0.875190i \(-0.339264\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 4.00000 3.00000i 0.565685 0.424264i
\(51\) 0 0
\(52\) 4.00000i 0.554700i
\(53\) 13.0000i 1.78569i 0.450367 + 0.892844i \(0.351293\pi\)
−0.450367 + 0.892844i \(0.648707\pi\)
\(54\) 0 0
\(55\) 5.00000 10.0000i 0.674200 1.34840i
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000i 0.262613i
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) −8.00000 4.00000i −0.992278 0.496139i
\(66\) 0 0
\(67\) 14.0000i 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 1.00000 2.00000i 0.119523 0.239046i
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 9.00000i 1.05337i −0.850060 0.526685i \(-0.823435\pi\)
0.850060 0.526685i \(-0.176565\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −5.00000 −0.573539
\(77\) 5.00000i 0.569803i
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 1.00000 2.00000i 0.111803 0.223607i
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) 6.00000i 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) 5.00000i 0.533002i
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 9.00000i 0.938315i
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 5.00000 10.0000i 0.512989 1.02598i
\(96\) 0 0
\(97\) 18.0000i 1.82762i 0.406138 + 0.913812i \(0.366875\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 6.00000i 0.606092i
\(99\) 0 0
\(100\) 3.00000 + 4.00000i 0.300000 + 0.400000i
\(101\) 7.00000 0.696526 0.348263 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) 11.0000i 1.06341i 0.846930 + 0.531705i \(0.178449\pi\)
−0.846930 + 0.531705i \(0.821551\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 10.0000 + 5.00000i 0.953463 + 0.476731i
\(111\) 0 0
\(112\) 1.00000i 0.0944911i
\(113\) 13.0000i 1.22294i −0.791269 0.611469i \(-0.790579\pi\)
0.791269 0.611469i \(-0.209421\pi\)
\(114\) 0 0
\(115\) 18.0000 + 9.00000i 1.67851 + 0.839254i
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 10.0000i 0.920575i
\(119\) 0 0
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 14.0000i 1.26750i
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) −11.0000 + 2.00000i −0.983870 + 0.178885i
\(126\) 0 0
\(127\) 20.0000i 1.77471i −0.461084 0.887357i \(-0.652539\pi\)
0.461084 0.887357i \(-0.347461\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 4.00000 8.00000i 0.350823 0.701646i
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 5.00000i 0.433555i
\(134\) 14.0000 1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 2.00000 + 1.00000i 0.169031 + 0.0845154i
\(141\) 0 0
\(142\) 9.00000i 0.755263i
\(143\) 20.0000i 1.67248i
\(144\) 0 0
\(145\) −2.00000 + 4.00000i −0.166091 + 0.332182i
\(146\) 9.00000 0.744845
\(147\) 0 0
\(148\) 8.00000i 0.657596i
\(149\) 7.00000 0.573462 0.286731 0.958011i \(-0.407431\pi\)
0.286731 + 0.958011i \(0.407431\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 5.00000i 0.405554i
\(153\) 0 0
\(154\) 5.00000 0.402911
\(155\) 1.00000 2.00000i 0.0803219 0.160644i
\(156\) 0 0
\(157\) 13.0000i 1.03751i −0.854922 0.518756i \(-0.826395\pi\)
0.854922 0.518756i \(-0.173605\pi\)
\(158\) 5.00000i 0.397779i
\(159\) 0 0
\(160\) 2.00000 + 1.00000i 0.158114 + 0.0790569i
\(161\) 9.00000 0.709299
\(162\) 0 0
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 9.00000i 0.696441i −0.937413 0.348220i \(-0.886786\pi\)
0.937413 0.348220i \(-0.113214\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 1.00000i 0.0762493i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) −4.00000 + 3.00000i −0.302372 + 0.226779i
\(176\) 5.00000 0.376889
\(177\) 0 0
\(178\) 3.00000i 0.224860i
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 4.00000i 0.296500i
\(183\) 0 0
\(184\) 9.00000 0.663489
\(185\) −16.0000 8.00000i −1.17634 0.588172i
\(186\) 0 0
\(187\) 0 0
\(188\) 12.0000i 0.875190i
\(189\) 0 0
\(190\) 10.0000 + 5.00000i 0.725476 + 0.362738i
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 22.0000i 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) −18.0000 −1.29232
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 10.0000i 0.712470i −0.934396 0.356235i \(-0.884060\pi\)
0.934396 0.356235i \(-0.115940\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) −4.00000 + 3.00000i −0.282843 + 0.212132i
\(201\) 0 0
\(202\) 7.00000i 0.492518i
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) −6.00000 + 12.0000i −0.419058 + 0.838116i
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 4.00000i 0.277350i
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 13.0000i 0.892844i
\(213\) 0 0
\(214\) −11.0000 −0.751945
\(215\) 2.00000 + 1.00000i 0.136399 + 0.0681994i
\(216\) 0 0
\(217\) 1.00000i 0.0678844i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) −5.00000 + 10.0000i −0.337100 + 0.674200i
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 13.0000 0.864747
\(227\) 15.0000i 0.995585i 0.867296 + 0.497792i \(0.165856\pi\)
−0.867296 + 0.497792i \(0.834144\pi\)
\(228\) 0 0
\(229\) 19.0000 1.25556 0.627778 0.778393i \(-0.283965\pi\)
0.627778 + 0.778393i \(0.283965\pi\)
\(230\) −9.00000 + 18.0000i −0.593442 + 1.18688i
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 1.00000i 0.0655122i 0.999463 + 0.0327561i \(0.0104285\pi\)
−0.999463 + 0.0327561i \(0.989572\pi\)
\(234\) 0 0
\(235\) −24.0000 12.0000i −1.56559 0.782794i
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 14.0000i 0.899954i
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 6.00000 12.0000i 0.383326 0.766652i
\(246\) 0 0
\(247\) 20.0000i 1.27257i
\(248\) 1.00000i 0.0635001i
\(249\) 0 0
\(250\) −2.00000 11.0000i −0.126491 0.695701i
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 45.0000i 2.82913i
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.00000i 0.436648i −0.975876 0.218324i \(-0.929941\pi\)
0.975876 0.218324i \(-0.0700590\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 8.00000 + 4.00000i 0.496139 + 0.248069i
\(261\) 0 0
\(262\) 6.00000i 0.370681i
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 26.0000 + 13.0000i 1.59717 + 0.798584i
\(266\) 5.00000 0.306570
\(267\) 0 0
\(268\) 14.0000i 0.855186i
\(269\) −22.0000 −1.34136 −0.670682 0.741745i \(-0.733998\pi\)
−0.670682 + 0.741745i \(0.733998\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −15.0000 20.0000i −0.904534 1.20605i
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) 16.0000i 0.959616i
\(279\) 0 0
\(280\) −1.00000 + 2.00000i −0.0597614 + 0.119523i
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 26.0000i 1.54554i 0.634686 + 0.772770i \(0.281129\pi\)
−0.634686 + 0.772770i \(0.718871\pi\)
\(284\) −9.00000 −0.534052
\(285\) 0 0
\(286\) 20.0000 1.18262
\(287\) 6.00000i 0.354169i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) −4.00000 2.00000i −0.234888 0.117444i
\(291\) 0 0
\(292\) 9.00000i 0.526685i
\(293\) 14.0000i 0.817889i 0.912559 + 0.408944i \(0.134103\pi\)
−0.912559 + 0.408944i \(0.865897\pi\)
\(294\) 0 0
\(295\) 10.0000 20.0000i 0.582223 1.16445i
\(296\) −8.00000 −0.464991
\(297\) 0 0
\(298\) 7.00000i 0.405499i
\(299\) 36.0000 2.08193
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) −14.0000 + 28.0000i −0.801638 + 1.60328i
\(306\) 0 0
\(307\) 8.00000i 0.456584i 0.973593 + 0.228292i \(0.0733141\pi\)
−0.973593 + 0.228292i \(0.926686\pi\)
\(308\) 5.00000i 0.284901i
\(309\) 0 0
\(310\) 2.00000 + 1.00000i 0.113592 + 0.0567962i
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 34.0000i 1.90963i 0.297200 + 0.954815i \(0.403947\pi\)
−0.297200 + 0.954815i \(0.596053\pi\)
\(318\) 0 0
\(319\) −10.0000 −0.559893
\(320\) −1.00000 + 2.00000i −0.0559017 + 0.111803i
\(321\) 0 0
\(322\) 9.00000i 0.501550i
\(323\) 0 0
\(324\) 0 0
\(325\) −16.0000 + 12.0000i −0.887520 + 0.665640i
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −24.0000 −1.31916 −0.659580 0.751635i \(-0.729266\pi\)
−0.659580 + 0.751635i \(0.729266\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 0 0
\(334\) 9.00000 0.492458
\(335\) −28.0000 14.0000i −1.52980 0.764902i
\(336\) 0 0
\(337\) 2.00000i 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) 3.00000i 0.163178i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.00000 0.270765
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 1.00000 0.0539164
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 18.0000i 0.966291i −0.875540 0.483145i \(-0.839494\pi\)
0.875540 0.483145i \(-0.160506\pi\)
\(348\) 0 0
\(349\) −4.00000 −0.214115 −0.107058 0.994253i \(-0.534143\pi\)
−0.107058 + 0.994253i \(0.534143\pi\)
\(350\) −3.00000 4.00000i −0.160357 0.213809i
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) 4.00000i 0.212899i 0.994318 + 0.106449i \(0.0339482\pi\)
−0.994318 + 0.106449i \(0.966052\pi\)
\(354\) 0 0
\(355\) 9.00000 18.0000i 0.477670 0.955341i
\(356\) −3.00000 −0.159000
\(357\) 0 0
\(358\) 16.0000i 0.845626i
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 17.0000i 0.893500i
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −18.0000 9.00000i −0.942163 0.471082i
\(366\) 0 0
\(367\) 18.0000i 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) 9.00000i 0.469157i
\(369\) 0 0
\(370\) 8.00000 16.0000i 0.415900 0.831800i
\(371\) 13.0000 0.674926
\(372\) 0 0
\(373\) 1.00000i 0.0517780i 0.999665 + 0.0258890i \(0.00824165\pi\)
−0.999665 + 0.0258890i \(0.991758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 8.00000i 0.412021i
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) −5.00000 + 10.0000i −0.256495 + 0.512989i
\(381\) 0 0
\(382\) 8.00000i 0.409316i
\(383\) 12.0000i 0.613171i −0.951843 0.306586i \(-0.900813\pi\)
0.951843 0.306586i \(-0.0991866\pi\)
\(384\) 0 0
\(385\) −10.0000 5.00000i −0.509647 0.254824i
\(386\) 22.0000 1.11977
\(387\) 0 0
\(388\) 18.0000i 0.913812i
\(389\) −16.0000 −0.811232 −0.405616 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000i 0.303046i
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) −5.00000 + 10.0000i −0.251577 + 0.503155i
\(396\) 0 0
\(397\) 3.00000i 0.150566i 0.997162 + 0.0752828i \(0.0239860\pi\)
−0.997162 + 0.0752828i \(0.976014\pi\)
\(398\) 11.0000i 0.551380i
\(399\) 0 0
\(400\) −3.00000 4.00000i −0.150000 0.200000i
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) −7.00000 −0.348263
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 40.0000i 1.98273i
\(408\) 0 0
\(409\) 28.0000 1.38451 0.692255 0.721653i \(-0.256617\pi\)
0.692255 + 0.721653i \(0.256617\pi\)
\(410\) −12.0000 6.00000i −0.592638 0.296319i
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) 10.0000i 0.492068i
\(414\) 0 0
\(415\) −12.0000 6.00000i −0.589057 0.294528i
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 25.0000i 1.22279i
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 13.0000i 0.632830i
\(423\) 0 0
\(424\) 13.0000 0.631336
\(425\) 0 0
\(426\) 0 0
\(427\) 14.0000i 0.677507i
\(428\) 11.0000i 0.531705i
\(429\) 0 0
\(430\) −1.00000 + 2.00000i −0.0482243 + 0.0964486i
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 0 0
\(433\) 11.0000i 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) 1.00000 0.0480015
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 45.0000i 2.15264i
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) −10.0000 5.00000i −0.476731 0.238366i
\(441\) 0 0
\(442\) 0 0
\(443\) 13.0000i 0.617649i −0.951119 0.308824i \(-0.900064\pi\)
0.951119 0.308824i \(-0.0999355\pi\)
\(444\) 0 0
\(445\) 3.00000 6.00000i 0.142214 0.284427i
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) 1.00000i 0.0472456i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −30.0000 −1.41264
\(452\) 13.0000i 0.611469i
\(453\) 0 0
\(454\) −15.0000 −0.703985
\(455\) −4.00000 + 8.00000i −0.187523 + 0.375046i
\(456\) 0 0
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) 19.0000i 0.887812i
\(459\) 0 0
\(460\) −18.0000 9.00000i −0.839254 0.419627i
\(461\) −36.0000 −1.67669 −0.838344 0.545142i \(-0.816476\pi\)
−0.838344 + 0.545142i \(0.816476\pi\)
\(462\) 0 0
\(463\) 30.0000i 1.39422i 0.716965 + 0.697109i \(0.245531\pi\)
−0.716965 + 0.697109i \(0.754469\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −1.00000 −0.0463241
\(467\) 20.0000i 0.925490i 0.886492 + 0.462745i \(0.153135\pi\)
−0.886492 + 0.462745i \(0.846865\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 12.0000 24.0000i 0.553519 1.10704i
\(471\) 0 0
\(472\) 10.0000i 0.460287i
\(473\) 5.00000i 0.229900i
\(474\) 0 0
\(475\) −15.0000 20.0000i −0.688247 0.917663i
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000i 0.274434i
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) −32.0000 −1.45907
\(482\) 24.0000i 1.09317i
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 36.0000 + 18.0000i 1.63468 + 0.817338i
\(486\) 0 0
\(487\) 24.0000i 1.08754i −0.839233 0.543772i \(-0.816996\pi\)
0.839233 0.543772i \(-0.183004\pi\)
\(488\) 14.0000i 0.633750i
\(489\) 0 0
\(490\) 12.0000 + 6.00000i 0.542105 + 0.271052i
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 20.0000 0.899843
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 9.00000i 0.403705i
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 11.0000 2.00000i 0.491935 0.0894427i
\(501\) 0 0
\(502\) 28.0000i 1.24970i
\(503\) 22.0000i 0.980932i −0.871460 0.490466i \(-0.836827\pi\)
0.871460 0.490466i \(-0.163173\pi\)
\(504\) 0 0
\(505\) 7.00000 14.0000i 0.311496 0.622992i
\(506\) −45.0000 −2.00049
\(507\) 0 0
\(508\) 20.0000i 0.887357i
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −9.00000 −0.398137
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 7.00000 0.308757
\(515\) 16.0000 + 8.00000i 0.705044 + 0.352522i
\(516\) 0 0
\(517\) 60.0000i 2.63880i
\(518\) 8.00000i 0.351500i
\(519\) 0 0
\(520\) −4.00000 + 8.00000i −0.175412 + 0.350823i
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 17.0000i 0.743358i 0.928361 + 0.371679i \(0.121218\pi\)
−0.928361 + 0.371679i \(0.878782\pi\)
\(524\) −6.00000 −0.262111
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) −58.0000 −2.52174
\(530\) −13.0000 + 26.0000i −0.564684 + 1.12937i
\(531\) 0 0
\(532\) 5.00000i 0.216777i
\(533\) 24.0000i 1.03956i
\(534\) 0 0
\(535\) 22.0000 + 11.0000i 0.951143 + 0.475571i
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 22.0000i 0.948487i
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 25.0000i 1.07384i
\(543\) 0 0
\(544\) 0 0
\(545\) 2.00000 4.00000i 0.0856706 0.171341i
\(546\) 0 0
\(547\) 16.0000i 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 0 0
\(550\) 20.0000 15.0000i 0.852803 0.639602i
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 5.00000i 0.212622i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) 7.00000i 0.296600i 0.988942 + 0.148300i \(0.0473800\pi\)
−0.988942 + 0.148300i \(0.952620\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −2.00000 1.00000i −0.0845154 0.0422577i
\(561\) 0 0
\(562\) 12.0000i 0.506189i
\(563\) 44.0000i 1.85438i 0.374593 + 0.927189i \(0.377783\pi\)
−0.374593 + 0.927189i \(0.622217\pi\)
\(564\) 0 0
\(565\) −26.0000 13.0000i −1.09383 0.546914i
\(566\) −26.0000 −1.09286
\(567\) 0 0
\(568\) 9.00000i 0.377632i
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 20.0000i 0.836242i
\(573\) 0 0
\(574\) −6.00000 −0.250435
\(575\) 36.0000 27.0000i 1.50130 1.12598i
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 17.0000i 0.707107i
\(579\) 0 0
\(580\) 2.00000 4.00000i 0.0830455 0.166091i
\(581\) −6.00000 −0.248922
\(582\) 0 0
\(583\) 65.0000i 2.69202i
\(584\) −9.00000 −0.372423
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 6.00000i 0.247647i −0.992304 0.123823i \(-0.960484\pi\)
0.992304 0.123823i \(-0.0395156\pi\)
\(588\) 0 0
\(589\) 5.00000 0.206021
\(590\) 20.0000 + 10.0000i 0.823387 + 0.411693i
\(591\) 0 0
\(592\) 8.00000i 0.328798i
\(593\) 10.0000i 0.410651i 0.978694 + 0.205325i \(0.0658253\pi\)
−0.978694 + 0.205325i \(0.934175\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.00000 −0.286731
\(597\) 0 0
\(598\) 36.0000i 1.47215i
\(599\) −39.0000 −1.59350 −0.796748 0.604311i \(-0.793448\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) −12.0000 −0.489490 −0.244745 0.969587i \(-0.578704\pi\)
−0.244745 + 0.969587i \(0.578704\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) 0 0
\(604\) 0 0
\(605\) 14.0000 28.0000i 0.569181 1.13836i
\(606\) 0 0
\(607\) 13.0000i 0.527654i −0.964570 0.263827i \(-0.915015\pi\)
0.964570 0.263827i \(-0.0849848\pi\)
\(608\) 5.00000i 0.202777i
\(609\) 0 0
\(610\) −28.0000 14.0000i −1.13369 0.566843i
\(611\) −48.0000 −1.94187
\(612\) 0 0
\(613\) 4.00000i 0.161558i 0.996732 + 0.0807792i \(0.0257409\pi\)
−0.996732 + 0.0807792i \(0.974259\pi\)
\(614\) −8.00000 −0.322854
\(615\) 0 0
\(616\) −5.00000 −0.201456
\(617\) 9.00000i 0.362326i 0.983453 + 0.181163i \(0.0579862\pi\)
−0.983453 + 0.181163i \(0.942014\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) −1.00000 + 2.00000i −0.0401610 + 0.0803219i
\(621\) 0 0
\(622\) 8.00000i 0.320771i
\(623\) 3.00000i 0.120192i
\(624\) 0 0
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 13.0000i 0.518756i
\(629\) 0 0
\(630\) 0 0
\(631\) 31.0000 1.23409 0.617045 0.786928i \(-0.288330\pi\)
0.617045 + 0.786928i \(0.288330\pi\)
\(632\) 5.00000i 0.198889i
\(633\) 0 0
\(634\) −34.0000 −1.35031
\(635\) −40.0000 20.0000i −1.58735 0.793676i
\(636\) 0 0
\(637\) 24.0000i 0.950915i
\(638\) 10.0000i 0.395904i
\(639\) 0 0
\(640\) −2.00000 1.00000i −0.0790569 0.0395285i
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 5.00000i 0.197181i −0.995128 0.0985904i \(-0.968567\pi\)
0.995128 0.0985904i \(-0.0314334\pi\)
\(644\) −9.00000 −0.354650
\(645\) 0 0
\(646\) 0 0
\(647\) 5.00000i 0.196570i 0.995158 + 0.0982851i \(0.0313357\pi\)
−0.995158 + 0.0982851i \(0.968664\pi\)
\(648\) 0 0
\(649\) 50.0000 1.96267
\(650\) −12.0000 16.0000i −0.470679 0.627572i
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 18.0000i 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 0 0
\(655\) 6.00000 12.0000i 0.234439 0.468879i
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) 12.0000i 0.467809i
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 24.0000i 0.932786i
\(663\) 0 0
\(664\) −6.00000 −0.232845
\(665\) −10.0000 5.00000i −0.387783 0.193892i
\(666\) 0 0
\(667\) 18.0000i 0.696963i
\(668\) 9.00000i 0.348220i
\(669\) 0 0
\(670\) 14.0000 28.0000i 0.540867 1.08173i
\(671\) −70.0000 −2.70232
\(672\) 0 0
\(673\) 38.0000i 1.46479i 0.680879 + 0.732396i \(0.261598\pi\)
−0.680879 + 0.732396i \(0.738402\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 51.0000i 1.96009i 0.198778 + 0.980045i \(0.436303\pi\)
−0.198778 + 0.980045i \(0.563697\pi\)
\(678\) 0 0
\(679\) 18.0000 0.690777
\(680\) 0 0
\(681\) 0 0
\(682\) 5.00000i 0.191460i
\(683\) 3.00000i 0.114792i −0.998351 0.0573959i \(-0.981720\pi\)
0.998351 0.0573959i \(-0.0182797\pi\)
\(684\) 0 0
\(685\) 4.00000 + 2.00000i 0.152832 + 0.0764161i
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) 1.00000i 0.0381246i
\(689\) 52.0000 1.98104
\(690\) 0 0
\(691\) 27.0000 1.02713 0.513564 0.858051i \(-0.328325\pi\)
0.513564 + 0.858051i \(0.328325\pi\)
\(692\) 6.00000i 0.228086i
\(693\) 0 0
\(694\) 18.0000 0.683271
\(695\) −16.0000 + 32.0000i −0.606915 + 1.21383i
\(696\) 0 0
\(697\) 0 0
\(698\) 4.00000i 0.151402i
\(699\) 0 0
\(700\) 4.00000 3.00000i 0.151186 0.113389i
\(701\) −29.0000 −1.09531 −0.547657 0.836703i \(-0.684480\pi\)
−0.547657 + 0.836703i \(0.684480\pi\)
\(702\) 0 0
\(703\) 40.0000i 1.50863i
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −4.00000 −0.150542
\(707\) 7.00000i 0.263262i
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 18.0000 + 9.00000i 0.675528 + 0.337764i
\(711\) 0 0
\(712\) 3.00000i 0.112430i
\(713\) 9.00000i 0.337053i
\(714\) 0 0
\(715\) −40.0000 20.0000i −1.49592 0.747958i
\(716\) 16.0000 0.597948
\(717\) 0 0
\(718\) 9.00000i 0.335877i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 6.00000i 0.223297i
\(723\) 0 0
\(724\) −17.0000 −0.631800
\(725\) 6.00000 + 8.00000i 0.222834 + 0.297113i
\(726\) 0 0
\(727\) 1.00000i 0.0370879i −0.999828 0.0185440i \(-0.994097\pi\)
0.999828 0.0185440i \(-0.00590307\pi\)
\(728\) 4.00000i 0.148250i
\(729\) 0 0
\(730\) 9.00000 18.0000i 0.333105 0.666210i
\(731\) 0 0
\(732\) 0 0
\(733\) 14.0000i 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 70.0000i 2.57848i
\(738\) 0 0
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 16.0000 + 8.00000i 0.588172 + 0.294086i
\(741\) 0 0
\(742\) 13.0000i 0.477245i
\(743\) 43.0000i 1.57752i 0.614703 + 0.788759i \(0.289276\pi\)
−0.614703 + 0.788759i \(0.710724\pi\)
\(744\) 0 0
\(745\) 7.00000 14.0000i 0.256460 0.512920i
\(746\) −1.00000 −0.0366126
\(747\) 0 0
\(748\) 0 0
\(749\) 11.0000 0.401931
\(750\) 0 0
\(751\) 18.0000 0.656829 0.328415 0.944534i \(-0.393486\pi\)
0.328415 + 0.944534i \(0.393486\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) 11.0000i 0.399538i
\(759\) 0 0
\(760\) −10.0000 5.00000i −0.362738 0.181369i
\(761\) −15.0000 −0.543750 −0.271875 0.962333i \(-0.587644\pi\)
−0.271875 + 0.962333i \(0.587644\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 40.0000i 1.44432i
\(768\) 0 0
\(769\) 35.0000 1.26213 0.631066 0.775729i \(-0.282618\pi\)
0.631066 + 0.775729i \(0.282618\pi\)
\(770\) 5.00000 10.0000i 0.180187 0.360375i
\(771\) 0 0
\(772\) 22.0000i 0.791797i
\(773\) 27.0000i 0.971123i 0.874203 + 0.485561i \(0.161385\pi\)
−0.874203 + 0.485561i \(0.838615\pi\)
\(774\) 0 0
\(775\) −3.00000 4.00000i −0.107763 0.143684i
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 16.0000i 0.573628i
\(779\) −30.0000 −1.07486
\(780\) 0 0
\(781\) 45.0000 1.61023
\(782\) 0 0
\(783\) 0 0
\(784\) 6.00000 0.214286
\(785\) −26.0000 13.0000i −0.927980 0.463990i
\(786\) 0 0
\(787\) 23.0000i 0.819861i 0.912117 + 0.409931i \(0.134447\pi\)
−0.912117 + 0.409931i \(0.865553\pi\)
\(788\) 10.0000i 0.356235i
\(789\) 0 0
\(790\) −10.0000 5.00000i −0.355784 0.177892i
\(791\) −13.0000 −0.462227
\(792\) 0 0
\(793\) 56.0000i 1.98862i
\(794\) −3.00000 −0.106466
\(795\) 0 0
\(796\) −11.0000 −0.389885
\(797\) 2.00000i 0.0708436i −0.999372 0.0354218i \(-0.988723\pi\)
0.999372 0.0354218i \(-0.0112775\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 4.00000 3.00000i 0.141421 0.106066i
\(801\) 0 0
\(802\) 25.0000i 0.882781i
\(803\) 45.0000i 1.58802i
\(804\) 0 0
\(805\) 9.00000 18.0000i 0.317208 0.634417i
\(806\) 4.00000 0.140894
\(807\) 0 0
\(808\) 7.00000i 0.246259i
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) 55.0000 1.93131 0.965656 0.259825i \(-0.0836650\pi\)
0.965656 + 0.259825i \(0.0836650\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) 40.0000 1.40200
\(815\) 8.00000 + 4.00000i 0.280228 + 0.140114i
\(816\) 0 0
\(817\) 5.00000i 0.174928i
\(818\) 28.0000i 0.978997i
\(819\) 0 0
\(820\) 6.00000 12.0000i 0.209529 0.419058i
\(821\) 2.00000 0.0698005 0.0349002 0.999391i \(-0.488889\pi\)
0.0349002 + 0.999391i \(0.488889\pi\)
\(822\) 0 0
\(823\) 2.00000i 0.0697156i −0.999392 0.0348578i \(-0.988902\pi\)
0.999392 0.0348578i \(-0.0110978\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) 46.0000i 1.59958i −0.600282 0.799788i \(-0.704945\pi\)
0.600282 0.799788i \(-0.295055\pi\)
\(828\) 0 0
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 6.00000 12.0000i 0.208263 0.416526i
\(831\) 0 0
\(832\) 4.00000i 0.138675i
\(833\) 0 0
\(834\) 0 0
\(835\) −18.0000 9.00000i −0.622916 0.311458i
\(836\) −25.0000 −0.864643
\(837\) 0 0
\(838\) 30.0000i 1.03633i
\(839\) −21.0000 −0.725001 −0.362500 0.931984i \(-0.618077\pi\)
−0.362500 + 0.931984i \(0.618077\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000i 0.896019i
\(843\) 0 0
\(844\) 13.0000 0.447478
\(845\) −3.00000 + 6.00000i −0.103203 + 0.206406i
\(846\) 0 0
\(847\) 14.0000i 0.481046i
\(848\) 13.0000i 0.446422i
\(849\) 0 0
\(850\) 0 0
\(851\) 72.0000 2.46813
\(852\) 0 0
\(853\) 21.0000i 0.719026i 0.933140 + 0.359513i \(0.117057\pi\)
−0.933140 + 0.359513i \(0.882943\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) 11.0000 0.375972
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −2.00000 1.00000i −0.0681994 0.0340997i
\(861\) 0 0
\(862\) 32.0000i 1.08992i
\(863\) 31.0000i 1.05525i 0.849477 + 0.527626i \(0.176918\pi\)
−0.849477 + 0.527626i \(0.823082\pi\)
\(864\) 0 0
\(865\) 12.0000 + 6.00000i 0.408012 + 0.204006i
\(866\) 11.0000 0.373795
\(867\) 0 0
\(868\) 1.00000i 0.0339422i
\(869\) −25.0000 −0.848067
\(870\) 0 0
\(871\) −56.0000 −1.89749
\(872\) 2.00000i 0.0677285i
\(873\) 0 0
\(874\) −45.0000 −1.52215
\(875\) 2.00000 + 11.0000i 0.0676123 + 0.371868i
\(876\) 0 0
\(877\) 6.00000i 0.202606i 0.994856 + 0.101303i \(0.0323011\pi\)
−0.994856 + 0.101303i \(0.967699\pi\)
\(878\) 20.0000i 0.674967i
\(879\) 0 0
\(880\) 5.00000 10.0000i 0.168550 0.337100i
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 23.0000i 0.774012i 0.922077 + 0.387006i \(0.126491\pi\)
−0.922077 + 0.387006i \(0.873509\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 13.0000 0.436744
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 6.00000 + 3.00000i 0.201120 + 0.100560i
\(891\) 0 0
\(892\) 16.0000i 0.535720i
\(893\) 60.0000i 2.00782i
\(894\) 0 0
\(895\) −16.0000 + 32.0000i −0.534821 + 1.06964i
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 6.00000i 0.200223i
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) 30.0000i 0.998891i
\(903\) 0 0
\(904\) −13.0000 −0.432374
\(905\) 17.0000 34.0000i 0.565099 1.13020i
\(906\) 0 0
\(907\) 4.00000i 0.132818i −0.997792 0.0664089i \(-0.978846\pi\)
0.997792 0.0664089i \(-0.0211542\pi\)
\(908\) 15.0000i 0.497792i
\(909\) 0 0
\(910\) −8.00000 4.00000i −0.265197 0.132599i
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) 0 0
\(913\) 30.0000i 0.992855i
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −19.0000 −0.627778
\(917\) 6.00000i 0.198137i
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 9.00000 18.0000i 0.296721 0.593442i
\(921\) 0 0
\(922\) 36.0000i 1.18560i
\(923\) 36.0000i 1.18495i
\(924\) 0 0
\(925\) −32.0000 + 24.0000i −1.05215 + 0.789115i
\(926\) −30.0000 −0.985861
\(927\) 0 0
\(928\) 2.00000i 0.0656532i
\(929\) −11.0000 −0.360898 −0.180449 0.983584i \(-0.557755\pi\)
−0.180449 + 0.983584i \(0.557755\pi\)
\(930\) 0 0
\(931\) 30.0000 0.983210
\(932\) 1.00000i 0.0327561i
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 54.0000i 1.76410i 0.471153 + 0.882052i \(0.343838\pi\)
−0.471153 + 0.882052i \(0.656162\pi\)
\(938\) 14.0000i 0.457116i
\(939\) 0 0
\(940\) 24.0000 + 12.0000i 0.782794 + 0.391397i
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) 54.0000i 1.75848i
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) −5.00000 −0.162564
\(947\) 10.0000i 0.324956i −0.986712 0.162478i \(-0.948051\pi\)
0.986712 0.162478i \(-0.0519487\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 20.0000 15.0000i 0.648886 0.486664i
\(951\) 0 0
\(952\) 0 0
\(953\) 52.0000i 1.68445i 0.539130 + 0.842223i \(0.318753\pi\)
−0.539130 + 0.842223i \(0.681247\pi\)
\(954\) 0 0
\(955\) −8.00000 + 16.0000i −0.258874 + 0.517748i
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 21.0000i 0.678479i
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 32.0000i 1.03172i
\(963\) 0 0
\(964\) −24.0000 −0.772988
\(965\) −44.0000 22.0000i −1.41641 0.708205i
\(966\) 0 0
\(967\) 20.0000i 0.643157i 0.946883 + 0.321578i \(0.104213\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 14.0000i 0.449977i
\(969\) 0 0
\(970\) −18.0000 + 36.0000i −0.577945 + 1.15589i
\(971\) −2.00000 −0.0641831 −0.0320915 0.999485i \(-0.510217\pi\)
−0.0320915 + 0.999485i \(0.510217\pi\)
\(972\) 0 0
\(973\) 16.0000i 0.512936i
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 22.0000i 0.703842i 0.936030 + 0.351921i \(0.114471\pi\)
−0.936030 + 0.351921i \(0.885529\pi\)
\(978\) 0 0
\(979\) 15.0000 0.479402
\(980\) −6.00000 + 12.0000i −0.191663 + 0.383326i
\(981\) 0 0
\(982\) 15.0000i 0.478669i
\(983\) 52.0000i 1.65854i 0.558846 + 0.829271i \(0.311244\pi\)
−0.558846 + 0.829271i \(0.688756\pi\)
\(984\) 0 0
\(985\) −20.0000 10.0000i −0.637253 0.318626i
\(986\) 0 0
\(987\) 0 0
\(988\) 20.0000i 0.636285i
\(989\) −9.00000 −0.286183
\(990\) 0 0
\(991\) −3.00000 −0.0952981 −0.0476491 0.998864i \(-0.515173\pi\)
−0.0476491 + 0.998864i \(0.515173\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 0 0
\(994\) 9.00000 0.285463
\(995\) 11.0000 22.0000i 0.348723 0.697447i
\(996\) 0 0
\(997\) 18.0000i 0.570066i 0.958518 + 0.285033i \(0.0920045\pi\)
−0.958518 + 0.285033i \(0.907995\pi\)
\(998\) 6.00000i 0.189927i
\(999\) 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 2790.2.d.g.559.2 2
3.2 odd 2 930.2.d.b.559.1 2
5.4 even 2 inner 2790.2.d.g.559.1 2
15.2 even 4 4650.2.a.bq.1.1 1
15.8 even 4 4650.2.a.f.1.1 1
15.14 odd 2 930.2.d.b.559.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.b.559.1 2 3.2 odd 2
930.2.d.b.559.2 yes 2 15.14 odd 2
2790.2.d.g.559.1 2 5.4 even 2 inner
2790.2.d.g.559.2 2 1.1 even 1 trivial
4650.2.a.f.1.1 1 15.8 even 4
4650.2.a.bq.1.1 1 15.2 even 4