Properties

Label 1024.2.e.n.769.2
Level $1024$
Weight $2$
Character 1024.769
Analytic conductor $8.177$
Analytic rank $0$
Dimension $4$
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] = [1024,2,Mod(257,1024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1024, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1024.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.e (of order \(4\), degree \(2\), not 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: \(8.17668116698\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 769.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1024.769
Dual form 1024.2.e.n.257.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.00000i) q^{3} +(1.41421 + 1.41421i) q^{5} -2.82843i q^{7} +1.00000i q^{9} +O(q^{10})\) \(q+(1.00000 - 1.00000i) q^{3} +(1.41421 + 1.41421i) q^{5} -2.82843i q^{7} +1.00000i q^{9} +(-3.00000 - 3.00000i) q^{11} +(4.24264 - 4.24264i) q^{13} +2.82843 q^{15} +(-3.00000 + 3.00000i) q^{19} +(-2.82843 - 2.82843i) q^{21} -8.48528i q^{23} -1.00000i q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.41421 + 1.41421i) q^{29} +5.65685 q^{31} -6.00000 q^{33} +(4.00000 - 4.00000i) q^{35} +(4.24264 + 4.24264i) q^{37} -8.48528i q^{39} -6.00000i q^{41} +(3.00000 + 3.00000i) q^{43} +(-1.41421 + 1.41421i) q^{45} -1.00000 q^{49} +(1.41421 + 1.41421i) q^{53} -8.48528i q^{55} +6.00000i q^{57} +(1.00000 + 1.00000i) q^{59} +(-4.24264 + 4.24264i) q^{61} +2.82843 q^{63} +12.0000 q^{65} +(-9.00000 + 9.00000i) q^{67} +(-8.48528 - 8.48528i) q^{69} -8.48528i q^{71} +12.0000i q^{73} +(-1.00000 - 1.00000i) q^{75} +(-8.48528 + 8.48528i) q^{77} +5.65685 q^{79} +5.00000 q^{81} +(3.00000 - 3.00000i) q^{83} +2.82843i q^{87} -12.0000i q^{89} +(-12.0000 - 12.0000i) q^{91} +(5.65685 - 5.65685i) q^{93} -8.48528 q^{95} -8.00000 q^{97} +(3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 12 q^{11} - 12 q^{19} + 16 q^{27} - 24 q^{33} + 16 q^{35} + 12 q^{43} - 4 q^{49} + 4 q^{59} + 48 q^{65} - 36 q^{67} - 4 q^{75} + 20 q^{81} + 12 q^{83} - 48 q^{91} - 32 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(1\)

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
\(3\) 1.00000 1.00000i 0.577350 0.577350i −0.356822 0.934172i \(-0.616140\pi\)
0.934172 + 0.356822i \(0.116140\pi\)
\(4\) 0 0
\(5\) 1.41421 + 1.41421i 0.632456 + 0.632456i 0.948683 0.316228i \(-0.102416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i −0.845154 0.534522i \(-0.820491\pi\)
0.845154 0.534522i \(-0.179509\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −3.00000 3.00000i −0.904534 0.904534i 0.0912903 0.995824i \(-0.470901\pi\)
−0.995824 + 0.0912903i \(0.970901\pi\)
\(12\) 0 0
\(13\) 4.24264 4.24264i 1.17670 1.17670i 0.196116 0.980581i \(-0.437167\pi\)
0.980581 0.196116i \(-0.0628330\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.00000 + 3.00000i −0.688247 + 0.688247i −0.961844 0.273597i \(-0.911786\pi\)
0.273597 + 0.961844i \(0.411786\pi\)
\(20\) 0 0
\(21\) −2.82843 2.82843i −0.617213 0.617213i
\(22\) 0 0
\(23\) 8.48528i 1.76930i −0.466252 0.884652i \(-0.654396\pi\)
0.466252 0.884652i \(-0.345604\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 4.00000 + 4.00000i 0.769800 + 0.769800i
\(28\) 0 0
\(29\) −1.41421 + 1.41421i −0.262613 + 0.262613i −0.826115 0.563502i \(-0.809454\pi\)
0.563502 + 0.826115i \(0.309454\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 4.00000 4.00000i 0.676123 0.676123i
\(36\) 0 0
\(37\) 4.24264 + 4.24264i 0.697486 + 0.697486i 0.963868 0.266382i \(-0.0858282\pi\)
−0.266382 + 0.963868i \(0.585828\pi\)
\(38\) 0 0
\(39\) 8.48528i 1.35873i
\(40\) 0 0
\(41\) 6.00000i 0.937043i −0.883452 0.468521i \(-0.844787\pi\)
0.883452 0.468521i \(-0.155213\pi\)
\(42\) 0 0
\(43\) 3.00000 + 3.00000i 0.457496 + 0.457496i 0.897833 0.440337i \(-0.145141\pi\)
−0.440337 + 0.897833i \(0.645141\pi\)
\(44\) 0 0
\(45\) −1.41421 + 1.41421i −0.210819 + 0.210819i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.41421 + 1.41421i 0.194257 + 0.194257i 0.797533 0.603276i \(-0.206138\pi\)
−0.603276 + 0.797533i \(0.706138\pi\)
\(54\) 0 0
\(55\) 8.48528i 1.14416i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 1.00000 + 1.00000i 0.130189 + 0.130189i 0.769199 0.639010i \(-0.220656\pi\)
−0.639010 + 0.769199i \(0.720656\pi\)
\(60\) 0 0
\(61\) −4.24264 + 4.24264i −0.543214 + 0.543214i −0.924470 0.381255i \(-0.875492\pi\)
0.381255 + 0.924470i \(0.375492\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −9.00000 + 9.00000i −1.09952 + 1.09952i −0.105059 + 0.994466i \(0.533503\pi\)
−0.994466 + 0.105059i \(0.966497\pi\)
\(68\) 0 0
\(69\) −8.48528 8.48528i −1.02151 1.02151i
\(70\) 0 0
\(71\) 8.48528i 1.00702i −0.863990 0.503509i \(-0.832042\pi\)
0.863990 0.503509i \(-0.167958\pi\)
\(72\) 0 0
\(73\) 12.0000i 1.40449i 0.711934 + 0.702247i \(0.247820\pi\)
−0.711934 + 0.702247i \(0.752180\pi\)
\(74\) 0 0
\(75\) −1.00000 1.00000i −0.115470 0.115470i
\(76\) 0 0
\(77\) −8.48528 + 8.48528i −0.966988 + 0.966988i
\(78\) 0 0
\(79\) 5.65685 0.636446 0.318223 0.948016i \(-0.396914\pi\)
0.318223 + 0.948016i \(0.396914\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) 0 0
\(83\) 3.00000 3.00000i 0.329293 0.329293i −0.523025 0.852318i \(-0.675196\pi\)
0.852318 + 0.523025i \(0.175196\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.82843i 0.303239i
\(88\) 0 0
\(89\) 12.0000i 1.27200i −0.771690 0.635999i \(-0.780588\pi\)
0.771690 0.635999i \(-0.219412\pi\)
\(90\) 0 0
\(91\) −12.0000 12.0000i −1.25794 1.25794i
\(92\) 0 0
\(93\) 5.65685 5.65685i 0.586588 0.586588i
\(94\) 0 0
\(95\) −8.48528 −0.870572
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) 3.00000 3.00000i 0.301511 0.301511i
\(100\) 0 0
\(101\) −7.07107 7.07107i −0.703598 0.703598i 0.261583 0.965181i \(-0.415755\pi\)
−0.965181 + 0.261583i \(0.915755\pi\)
\(102\) 0 0
\(103\) 2.82843i 0.278693i 0.990244 + 0.139347i \(0.0445002\pi\)
−0.990244 + 0.139347i \(0.955500\pi\)
\(104\) 0 0
\(105\) 8.00000i 0.780720i
\(106\) 0 0
\(107\) 5.00000 + 5.00000i 0.483368 + 0.483368i 0.906206 0.422837i \(-0.138966\pi\)
−0.422837 + 0.906206i \(0.638966\pi\)
\(108\) 0 0
\(109\) −4.24264 + 4.24264i −0.406371 + 0.406371i −0.880471 0.474100i \(-0.842774\pi\)
0.474100 + 0.880471i \(0.342774\pi\)
\(110\) 0 0
\(111\) 8.48528 0.805387
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 12.0000 12.0000i 1.11901 1.11901i
\(116\) 0 0
\(117\) 4.24264 + 4.24264i 0.392232 + 0.392232i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.00000i 0.636364i
\(122\) 0 0
\(123\) −6.00000 6.00000i −0.541002 0.541002i
\(124\) 0 0
\(125\) 8.48528 8.48528i 0.758947 0.758947i
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) −1.00000 + 1.00000i −0.0873704 + 0.0873704i −0.749441 0.662071i \(-0.769678\pi\)
0.662071 + 0.749441i \(0.269678\pi\)
\(132\) 0 0
\(133\) 8.48528 + 8.48528i 0.735767 + 0.735767i
\(134\) 0 0
\(135\) 11.3137i 0.973729i
\(136\) 0 0
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 3.00000 + 3.00000i 0.254457 + 0.254457i 0.822795 0.568338i \(-0.192414\pi\)
−0.568338 + 0.822795i \(0.692414\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −25.4558 −2.12872
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) −1.00000 + 1.00000i −0.0824786 + 0.0824786i
\(148\) 0 0
\(149\) 15.5563 + 15.5563i 1.27443 + 1.27443i 0.943741 + 0.330684i \(0.107280\pi\)
0.330684 + 0.943741i \(0.392720\pi\)
\(150\) 0 0
\(151\) 14.1421i 1.15087i 0.817847 + 0.575435i \(0.195167\pi\)
−0.817847 + 0.575435i \(0.804833\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 + 8.00000i 0.642575 + 0.642575i
\(156\) 0 0
\(157\) −4.24264 + 4.24264i −0.338600 + 0.338600i −0.855840 0.517241i \(-0.826959\pi\)
0.517241 + 0.855840i \(0.326959\pi\)
\(158\) 0 0
\(159\) 2.82843 0.224309
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) −15.0000 + 15.0000i −1.17489 + 1.17489i −0.193862 + 0.981029i \(0.562101\pi\)
−0.981029 + 0.193862i \(0.937899\pi\)
\(164\) 0 0
\(165\) −8.48528 8.48528i −0.660578 0.660578i
\(166\) 0 0
\(167\) 8.48528i 0.656611i 0.944572 + 0.328305i \(0.106478\pi\)
−0.944572 + 0.328305i \(0.893522\pi\)
\(168\) 0 0
\(169\) 23.0000i 1.76923i
\(170\) 0 0
\(171\) −3.00000 3.00000i −0.229416 0.229416i
\(172\) 0 0
\(173\) 9.89949 9.89949i 0.752645 0.752645i −0.222327 0.974972i \(-0.571365\pi\)
0.974972 + 0.222327i \(0.0713654\pi\)
\(174\) 0 0
\(175\) −2.82843 −0.213809
\(176\) 0 0
\(177\) 2.00000 0.150329
\(178\) 0 0
\(179\) −13.0000 + 13.0000i −0.971666 + 0.971666i −0.999609 0.0279439i \(-0.991104\pi\)
0.0279439 + 0.999609i \(0.491104\pi\)
\(180\) 0 0
\(181\) −12.7279 12.7279i −0.946059 0.946059i 0.0525588 0.998618i \(-0.483262\pi\)
−0.998618 + 0.0525588i \(0.983262\pi\)
\(182\) 0 0
\(183\) 8.48528i 0.627250i
\(184\) 0 0
\(185\) 12.0000i 0.882258i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 11.3137 11.3137i 0.822951 0.822951i
\(190\) 0 0
\(191\) −16.9706 −1.22795 −0.613973 0.789327i \(-0.710430\pi\)
−0.613973 + 0.789327i \(0.710430\pi\)
\(192\) 0 0
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) 0 0
\(195\) 12.0000 12.0000i 0.859338 0.859338i
\(196\) 0 0
\(197\) 7.07107 + 7.07107i 0.503793 + 0.503793i 0.912614 0.408822i \(-0.134060\pi\)
−0.408822 + 0.912614i \(0.634060\pi\)
\(198\) 0 0
\(199\) 19.7990i 1.40351i 0.712417 + 0.701757i \(0.247601\pi\)
−0.712417 + 0.701757i \(0.752399\pi\)
\(200\) 0 0
\(201\) 18.0000i 1.26962i
\(202\) 0 0
\(203\) 4.00000 + 4.00000i 0.280745 + 0.280745i
\(204\) 0 0
\(205\) 8.48528 8.48528i 0.592638 0.592638i
\(206\) 0 0
\(207\) 8.48528 0.589768
\(208\) 0 0
\(209\) 18.0000 1.24509
\(210\) 0 0
\(211\) 3.00000 3.00000i 0.206529 0.206529i −0.596262 0.802790i \(-0.703348\pi\)
0.802790 + 0.596262i \(0.203348\pi\)
\(212\) 0 0
\(213\) −8.48528 8.48528i −0.581402 0.581402i
\(214\) 0 0
\(215\) 8.48528i 0.578691i
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) 0 0
\(219\) 12.0000 + 12.0000i 0.810885 + 0.810885i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.65685 −0.378811 −0.189405 0.981899i \(-0.560656\pi\)
−0.189405 + 0.981899i \(0.560656\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 9.00000 9.00000i 0.597351 0.597351i −0.342256 0.939607i \(-0.611191\pi\)
0.939607 + 0.342256i \(0.111191\pi\)
\(228\) 0 0
\(229\) 12.7279 + 12.7279i 0.841085 + 0.841085i 0.989000 0.147915i \(-0.0472563\pi\)
−0.147915 + 0.989000i \(0.547256\pi\)
\(230\) 0 0
\(231\) 16.9706i 1.11658i
\(232\) 0 0
\(233\) 12.0000i 0.786146i 0.919507 + 0.393073i \(0.128588\pi\)
−0.919507 + 0.393073i \(0.871412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.65685 5.65685i 0.367452 0.367452i
\(238\) 0 0
\(239\) −16.9706 −1.09773 −0.548867 0.835910i \(-0.684941\pi\)
−0.548867 + 0.835910i \(0.684941\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) −7.00000 + 7.00000i −0.449050 + 0.449050i
\(244\) 0 0
\(245\) −1.41421 1.41421i −0.0903508 0.0903508i
\(246\) 0 0
\(247\) 25.4558i 1.61972i
\(248\) 0 0
\(249\) 6.00000i 0.380235i
\(250\) 0 0
\(251\) 9.00000 + 9.00000i 0.568075 + 0.568075i 0.931589 0.363514i \(-0.118423\pi\)
−0.363514 + 0.931589i \(0.618423\pi\)
\(252\) 0 0
\(253\) −25.4558 + 25.4558i −1.60040 + 1.60040i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 12.0000 12.0000i 0.745644 0.745644i
\(260\) 0 0
\(261\) −1.41421 1.41421i −0.0875376 0.0875376i
\(262\) 0 0
\(263\) 8.48528i 0.523225i −0.965173 0.261612i \(-0.915746\pi\)
0.965173 0.261612i \(-0.0842542\pi\)
\(264\) 0 0
\(265\) 4.00000i 0.245718i
\(266\) 0 0
\(267\) −12.0000 12.0000i −0.734388 0.734388i
\(268\) 0 0
\(269\) −9.89949 + 9.89949i −0.603583 + 0.603583i −0.941261 0.337679i \(-0.890358\pi\)
0.337679 + 0.941261i \(0.390358\pi\)
\(270\) 0 0
\(271\) 22.6274 1.37452 0.687259 0.726413i \(-0.258814\pi\)
0.687259 + 0.726413i \(0.258814\pi\)
\(272\) 0 0
\(273\) −24.0000 −1.45255
\(274\) 0 0
\(275\) −3.00000 + 3.00000i −0.180907 + 0.180907i
\(276\) 0 0
\(277\) −4.24264 4.24264i −0.254916 0.254916i 0.568067 0.822982i \(-0.307691\pi\)
−0.822982 + 0.568067i \(0.807691\pi\)
\(278\) 0 0
\(279\) 5.65685i 0.338667i
\(280\) 0 0
\(281\) 12.0000i 0.715860i 0.933748 + 0.357930i \(0.116517\pi\)
−0.933748 + 0.357930i \(0.883483\pi\)
\(282\) 0 0
\(283\) −9.00000 9.00000i −0.534994 0.534994i 0.387060 0.922055i \(-0.373491\pi\)
−0.922055 + 0.387060i \(0.873491\pi\)
\(284\) 0 0
\(285\) −8.48528 + 8.48528i −0.502625 + 0.502625i
\(286\) 0 0
\(287\) −16.9706 −1.00174
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −8.00000 + 8.00000i −0.468968 + 0.468968i
\(292\) 0 0
\(293\) 7.07107 + 7.07107i 0.413096 + 0.413096i 0.882816 0.469720i \(-0.155645\pi\)
−0.469720 + 0.882816i \(0.655645\pi\)
\(294\) 0 0
\(295\) 2.82843i 0.164677i
\(296\) 0 0
\(297\) 24.0000i 1.39262i
\(298\) 0 0
\(299\) −36.0000 36.0000i −2.08193 2.08193i
\(300\) 0 0
\(301\) 8.48528 8.48528i 0.489083 0.489083i
\(302\) 0 0
\(303\) −14.1421 −0.812444
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −3.00000 + 3.00000i −0.171219 + 0.171219i −0.787515 0.616296i \(-0.788633\pi\)
0.616296 + 0.787515i \(0.288633\pi\)
\(308\) 0 0
\(309\) 2.82843 + 2.82843i 0.160904 + 0.160904i
\(310\) 0 0
\(311\) 8.48528i 0.481156i 0.970630 + 0.240578i \(0.0773370\pi\)
−0.970630 + 0.240578i \(0.922663\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 0 0
\(315\) 4.00000 + 4.00000i 0.225374 + 0.225374i
\(316\) 0 0
\(317\) −7.07107 + 7.07107i −0.397151 + 0.397151i −0.877227 0.480076i \(-0.840609\pi\)
0.480076 + 0.877227i \(0.340609\pi\)
\(318\) 0 0
\(319\) 8.48528 0.475085
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.24264 4.24264i −0.235339 0.235339i
\(326\) 0 0
\(327\) 8.48528i 0.469237i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 21.0000 + 21.0000i 1.15426 + 1.15426i 0.985689 + 0.168576i \(0.0539168\pi\)
0.168576 + 0.985689i \(0.446083\pi\)
\(332\) 0 0
\(333\) −4.24264 + 4.24264i −0.232495 + 0.232495i
\(334\) 0 0
\(335\) −25.4558 −1.39080
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) −6.00000 + 6.00000i −0.325875 + 0.325875i
\(340\) 0 0
\(341\) −16.9706 16.9706i −0.919007 0.919007i
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 24.0000i 1.29212i
\(346\) 0 0
\(347\) −15.0000 15.0000i −0.805242 0.805242i 0.178667 0.983910i \(-0.442821\pi\)
−0.983910 + 0.178667i \(0.942821\pi\)
\(348\) 0 0
\(349\) 4.24264 4.24264i 0.227103 0.227103i −0.584378 0.811481i \(-0.698662\pi\)
0.811481 + 0.584378i \(0.198662\pi\)
\(350\) 0 0
\(351\) 33.9411 1.81164
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 12.0000 12.0000i 0.636894 0.636894i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.48528i 0.447836i −0.974608 0.223918i \(-0.928115\pi\)
0.974608 0.223918i \(-0.0718848\pi\)
\(360\) 0 0
\(361\) 1.00000i 0.0526316i
\(362\) 0 0
\(363\) 7.00000 + 7.00000i 0.367405 + 0.367405i
\(364\) 0 0
\(365\) −16.9706 + 16.9706i −0.888280 + 0.888280i
\(366\) 0 0
\(367\) 11.3137 0.590571 0.295285 0.955409i \(-0.404585\pi\)
0.295285 + 0.955409i \(0.404585\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 4.00000 4.00000i 0.207670 0.207670i
\(372\) 0 0
\(373\) −4.24264 4.24264i −0.219676 0.219676i 0.588686 0.808362i \(-0.299645\pi\)
−0.808362 + 0.588686i \(0.799645\pi\)
\(374\) 0 0
\(375\) 16.9706i 0.876356i
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 9.00000 + 9.00000i 0.462299 + 0.462299i 0.899408 0.437109i \(-0.143998\pi\)
−0.437109 + 0.899408i \(0.643998\pi\)
\(380\) 0 0
\(381\) 5.65685 5.65685i 0.289809 0.289809i
\(382\) 0 0
\(383\) −33.9411 −1.73431 −0.867155 0.498038i \(-0.834054\pi\)
−0.867155 + 0.498038i \(0.834054\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) −3.00000 + 3.00000i −0.152499 + 0.152499i
\(388\) 0 0
\(389\) −18.3848 18.3848i −0.932145 0.932145i 0.0656946 0.997840i \(-0.479074\pi\)
−0.997840 + 0.0656946i \(0.979074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 2.00000i 0.100887i
\(394\) 0 0
\(395\) 8.00000 + 8.00000i 0.402524 + 0.402524i
\(396\) 0 0
\(397\) 12.7279 12.7279i 0.638796 0.638796i −0.311462 0.950258i \(-0.600819\pi\)
0.950258 + 0.311462i \(0.100819\pi\)
\(398\) 0 0
\(399\) 16.9706 0.849591
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 24.0000 24.0000i 1.19553 1.19553i
\(404\) 0 0
\(405\) 7.07107 + 7.07107i 0.351364 + 0.351364i
\(406\) 0 0
\(407\) 25.4558i 1.26180i
\(408\) 0 0
\(409\) 22.0000i 1.08783i 0.839140 + 0.543915i \(0.183059\pi\)
−0.839140 + 0.543915i \(0.816941\pi\)
\(410\) 0 0
\(411\) 6.00000 + 6.00000i 0.295958 + 0.295958i
\(412\) 0 0
\(413\) 2.82843 2.82843i 0.139178 0.139178i
\(414\) 0 0
\(415\) 8.48528 0.416526
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) −9.00000 + 9.00000i −0.439679 + 0.439679i −0.891904 0.452225i \(-0.850630\pi\)
0.452225 + 0.891904i \(0.350630\pi\)
\(420\) 0 0
\(421\) −4.24264 4.24264i −0.206774 0.206774i 0.596121 0.802895i \(-0.296708\pi\)
−0.802895 + 0.596121i \(0.796708\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0000 + 12.0000i 0.580721 + 0.580721i
\(428\) 0 0
\(429\) −25.4558 + 25.4558i −1.22902 + 1.22902i
\(430\) 0 0
\(431\) 16.9706 0.817443 0.408722 0.912659i \(-0.365975\pi\)
0.408722 + 0.912659i \(0.365975\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) −4.00000 + 4.00000i −0.191785 + 0.191785i
\(436\) 0 0
\(437\) 25.4558 + 25.4558i 1.21772 + 1.21772i
\(438\) 0 0
\(439\) 2.82843i 0.134993i −0.997719 0.0674967i \(-0.978499\pi\)
0.997719 0.0674967i \(-0.0215012\pi\)
\(440\) 0 0
\(441\) 1.00000i 0.0476190i
\(442\) 0 0
\(443\) −9.00000 9.00000i −0.427603 0.427603i 0.460208 0.887811i \(-0.347775\pi\)
−0.887811 + 0.460208i \(0.847775\pi\)
\(444\) 0 0
\(445\) 16.9706 16.9706i 0.804482 0.804482i
\(446\) 0 0
\(447\) 31.1127 1.47158
\(448\) 0 0
\(449\) 24.0000 1.13263 0.566315 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) −18.0000 + 18.0000i −0.847587 + 0.847587i
\(452\) 0 0
\(453\) 14.1421 + 14.1421i 0.664455 + 0.664455i
\(454\) 0 0
\(455\) 33.9411i 1.59118i
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.3848 18.3848i 0.856264 0.856264i −0.134631 0.990896i \(-0.542985\pi\)
0.990896 + 0.134631i \(0.0429850\pi\)
\(462\) 0 0
\(463\) −5.65685 −0.262896 −0.131448 0.991323i \(-0.541963\pi\)
−0.131448 + 0.991323i \(0.541963\pi\)
\(464\) 0 0
\(465\) 16.0000 0.741982
\(466\) 0 0
\(467\) −3.00000 + 3.00000i −0.138823 + 0.138823i −0.773103 0.634280i \(-0.781297\pi\)
0.634280 + 0.773103i \(0.281297\pi\)
\(468\) 0 0
\(469\) 25.4558 + 25.4558i 1.17544 + 1.17544i
\(470\) 0 0
\(471\) 8.48528i 0.390981i
\(472\) 0 0
\(473\) 18.0000i 0.827641i
\(474\) 0 0
\(475\) 3.00000 + 3.00000i 0.137649 + 0.137649i
\(476\) 0 0
\(477\) −1.41421 + 1.41421i −0.0647524 + 0.0647524i
\(478\) 0 0
\(479\) −16.9706 −0.775405 −0.387702 0.921785i \(-0.626731\pi\)
−0.387702 + 0.921785i \(0.626731\pi\)
\(480\) 0 0
\(481\) 36.0000 1.64146
\(482\) 0 0
\(483\) −24.0000 + 24.0000i −1.09204 + 1.09204i
\(484\) 0 0
\(485\) −11.3137 11.3137i −0.513729 0.513729i
\(486\) 0 0
\(487\) 36.7696i 1.66619i −0.553132 0.833094i \(-0.686567\pi\)
0.553132 0.833094i \(-0.313433\pi\)
\(488\) 0 0
\(489\) 30.0000i 1.35665i
\(490\) 0 0
\(491\) 19.0000 + 19.0000i 0.857458 + 0.857458i 0.991038 0.133580i \(-0.0426473\pi\)
−0.133580 + 0.991038i \(0.542647\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.48528 0.381385
\(496\) 0 0
\(497\) −24.0000 −1.07655
\(498\) 0 0
\(499\) 3.00000 3.00000i 0.134298 0.134298i −0.636762 0.771060i \(-0.719727\pi\)
0.771060 + 0.636762i \(0.219727\pi\)
\(500\) 0 0
\(501\) 8.48528 + 8.48528i 0.379094 + 0.379094i
\(502\) 0 0
\(503\) 42.4264i 1.89170i −0.324604 0.945850i \(-0.605231\pi\)
0.324604 0.945850i \(-0.394769\pi\)
\(504\) 0 0
\(505\) 20.0000i 0.889988i
\(506\) 0 0
\(507\) −23.0000 23.0000i −1.02147 1.02147i
\(508\) 0 0
\(509\) 24.0416 24.0416i 1.06563 1.06563i 0.0679369 0.997690i \(-0.478358\pi\)
0.997690 0.0679369i \(-0.0216417\pi\)
\(510\) 0 0
\(511\) 33.9411 1.50147
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) −4.00000 + 4.00000i −0.176261 + 0.176261i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 19.7990i 0.869079i
\(520\) 0 0
\(521\) 42.0000i 1.84005i −0.391856 0.920027i \(-0.628167\pi\)
0.391856 0.920027i \(-0.371833\pi\)
\(522\) 0 0
\(523\) −3.00000 3.00000i −0.131181 0.131181i 0.638468 0.769649i \(-0.279569\pi\)
−0.769649 + 0.638468i \(0.779569\pi\)
\(524\) 0 0
\(525\) −2.82843 + 2.82843i −0.123443 + 0.123443i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −49.0000 −2.13043
\(530\) 0 0
\(531\) −1.00000 + 1.00000i −0.0433963 + 0.0433963i
\(532\) 0 0
\(533\) −25.4558 25.4558i −1.10262 1.10262i
\(534\) 0 0
\(535\) 14.1421i 0.611418i
\(536\) 0 0
\(537\) 26.0000i 1.12198i
\(538\) 0 0
\(539\) 3.00000 + 3.00000i 0.129219 + 0.129219i
\(540\) 0 0
\(541\) 29.6985 29.6985i 1.27684 1.27684i 0.334410 0.942428i \(-0.391463\pi\)
0.942428 0.334410i \(-0.108537\pi\)
\(542\) 0 0
\(543\) −25.4558 −1.09241
\(544\) 0 0
\(545\) −12.0000 −0.514024
\(546\) 0 0
\(547\) 15.0000 15.0000i 0.641354 0.641354i −0.309535 0.950888i \(-0.600173\pi\)
0.950888 + 0.309535i \(0.100173\pi\)
\(548\) 0 0
\(549\) −4.24264 4.24264i −0.181071 0.181071i
\(550\) 0 0
\(551\) 8.48528i 0.361485i
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) 12.0000 + 12.0000i 0.509372 + 0.509372i
\(556\) 0 0
\(557\) −1.41421 + 1.41421i −0.0599222 + 0.0599222i −0.736433 0.676511i \(-0.763491\pi\)
0.676511 + 0.736433i \(0.263491\pi\)
\(558\) 0 0
\(559\) 25.4558 1.07667
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 21.0000 21.0000i 0.885044 0.885044i −0.108998 0.994042i \(-0.534764\pi\)
0.994042 + 0.108998i \(0.0347641\pi\)
\(564\) 0 0
\(565\) −8.48528 8.48528i −0.356978 0.356978i
\(566\) 0 0
\(567\) 14.1421i 0.593914i
\(568\) 0 0
\(569\) 6.00000i 0.251533i −0.992060 0.125767i \(-0.959861\pi\)
0.992060 0.125767i \(-0.0401390\pi\)
\(570\) 0 0
\(571\) −9.00000 9.00000i −0.376638 0.376638i 0.493250 0.869888i \(-0.335809\pi\)
−0.869888 + 0.493250i \(0.835809\pi\)
\(572\) 0 0
\(573\) −16.9706 + 16.9706i −0.708955 + 0.708955i
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 24.0000 24.0000i 0.997406 0.997406i
\(580\) 0 0
\(581\) −8.48528 8.48528i −0.352029 0.352029i
\(582\) 0 0
\(583\) 8.48528i 0.351424i
\(584\) 0 0
\(585\) 12.0000i 0.496139i
\(586\) 0 0
\(587\) −11.0000 11.0000i −0.454019 0.454019i 0.442667 0.896686i \(-0.354032\pi\)
−0.896686 + 0.442667i \(0.854032\pi\)
\(588\) 0 0
\(589\) −16.9706 + 16.9706i −0.699260 + 0.699260i
\(590\) 0 0
\(591\) 14.1421 0.581730
\(592\) 0 0
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 19.7990 + 19.7990i 0.810319 + 0.810319i
\(598\) 0 0
\(599\) 25.4558i 1.04010i −0.854137 0.520049i \(-0.825914\pi\)
0.854137 0.520049i \(-0.174086\pi\)
\(600\) 0 0
\(601\) 12.0000i 0.489490i −0.969587 0.244745i \(-0.921296\pi\)
0.969587 0.244745i \(-0.0787043\pi\)
\(602\) 0 0
\(603\) −9.00000 9.00000i −0.366508 0.366508i
\(604\) 0 0
\(605\) −9.89949 + 9.89949i −0.402472 + 0.402472i
\(606\) 0 0
\(607\) 22.6274 0.918419 0.459209 0.888328i \(-0.348133\pi\)
0.459209 + 0.888328i \(0.348133\pi\)
\(608\) 0 0
\(609\) 8.00000 0.324176
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −4.24264 4.24264i −0.171359 0.171359i 0.616217 0.787576i \(-0.288664\pi\)
−0.787576 + 0.616217i \(0.788664\pi\)
\(614\) 0 0
\(615\) 16.9706i 0.684319i
\(616\) 0 0
\(617\) 36.0000i 1.44931i 0.689114 + 0.724653i \(0.258000\pi\)
−0.689114 + 0.724653i \(0.742000\pi\)
\(618\) 0 0
\(619\) −27.0000 27.0000i −1.08522 1.08522i −0.996013 0.0892087i \(-0.971566\pi\)
−0.0892087 0.996013i \(-0.528434\pi\)
\(620\) 0 0
\(621\) 33.9411 33.9411i 1.36201 1.36201i
\(622\) 0 0
\(623\) −33.9411 −1.35982
\(624\) 0 0
\(625\) 19.0000 0.760000
\(626\) 0 0
\(627\) 18.0000 18.0000i 0.718851 0.718851i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 31.1127i 1.23858i −0.785164 0.619288i \(-0.787421\pi\)
0.785164 0.619288i \(-0.212579\pi\)
\(632\) 0 0
\(633\) 6.00000i 0.238479i
\(634\) 0 0
\(635\) 8.00000 + 8.00000i 0.317470 + 0.317470i
\(636\) 0 0
\(637\) −4.24264 + 4.24264i −0.168100 + 0.168100i
\(638\) 0 0
\(639\) 8.48528 0.335673
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −9.00000 + 9.00000i −0.354925 + 0.354925i −0.861938 0.507013i \(-0.830750\pi\)
0.507013 + 0.861938i \(0.330750\pi\)
\(644\) 0 0
\(645\) 8.48528 + 8.48528i 0.334108 + 0.334108i
\(646\) 0 0
\(647\) 25.4558i 1.00077i 0.865802 + 0.500386i \(0.166809\pi\)
−0.865802 + 0.500386i \(0.833191\pi\)
\(648\) 0 0
\(649\) 6.00000i 0.235521i
\(650\) 0 0
\(651\) −16.0000 16.0000i −0.627089 0.627089i
\(652\) 0 0
\(653\) 1.41421 1.41421i 0.0553425 0.0553425i −0.678894 0.734236i \(-0.737540\pi\)
0.734236 + 0.678894i \(0.237540\pi\)
\(654\) 0 0
\(655\) −2.82843 −0.110516
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −29.0000 + 29.0000i −1.12968 + 1.12968i −0.139451 + 0.990229i \(0.544534\pi\)
−0.990229 + 0.139451i \(0.955466\pi\)
\(660\) 0 0
\(661\) −12.7279 12.7279i −0.495059 0.495059i 0.414837 0.909896i \(-0.363839\pi\)
−0.909896 + 0.414837i \(0.863839\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 24.0000i 0.930680i
\(666\) 0 0
\(667\) 12.0000 + 12.0000i 0.464642 + 0.464642i
\(668\) 0 0
\(669\) −5.65685 + 5.65685i −0.218707 + 0.218707i
\(670\) 0 0
\(671\) 25.4558 0.982712
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) 4.00000 4.00000i 0.153960 0.153960i
\(676\) 0 0
\(677\) 26.8701 + 26.8701i 1.03270 + 1.03270i 0.999447 + 0.0332533i \(0.0105868\pi\)
0.0332533 + 0.999447i \(0.489413\pi\)
\(678\) 0 0
\(679\) 22.6274i 0.868361i
\(680\) 0 0
\(681\) 18.0000i 0.689761i
\(682\) 0 0
\(683\) 27.0000 + 27.0000i 1.03313 + 1.03313i 0.999432 + 0.0336941i \(0.0107272\pi\)
0.0336941 + 0.999432i \(0.489273\pi\)
\(684\) 0 0
\(685\) −8.48528 + 8.48528i −0.324206 + 0.324206i
\(686\) 0 0
\(687\) 25.4558 0.971201
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −21.0000 + 21.0000i −0.798878 + 0.798878i −0.982919 0.184041i \(-0.941082\pi\)
0.184041 + 0.982919i \(0.441082\pi\)
\(692\) 0 0
\(693\) −8.48528 8.48528i −0.322329 0.322329i
\(694\) 0 0
\(695\) 8.48528i 0.321865i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 12.0000 + 12.0000i 0.453882 + 0.453882i
\(700\) 0 0
\(701\) −32.5269 + 32.5269i −1.22852 + 1.22852i −0.264003 + 0.964522i \(0.585043\pi\)
−0.964522 + 0.264003i \(0.914957\pi\)
\(702\) 0 0
\(703\) −25.4558 −0.960085
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.0000 + 20.0000i −0.752177 + 0.752177i
\(708\) 0 0
\(709\) −12.7279 12.7279i −0.478007 0.478007i 0.426487 0.904494i \(-0.359751\pi\)
−0.904494 + 0.426487i \(0.859751\pi\)
\(710\) 0 0
\(711\) 5.65685i 0.212149i
\(712\) 0 0
\(713\) 48.0000i 1.79761i
\(714\) 0 0
\(715\) −36.0000 36.0000i −1.34632 1.34632i
\(716\) 0 0
\(717\) −16.9706 + 16.9706i −0.633777 + 0.633777i
\(718\) 0 0
\(719\) −16.9706 −0.632895 −0.316448 0.948610i \(-0.602490\pi\)
−0.316448 + 0.948610i \(0.602490\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.41421 + 1.41421i 0.0525226 + 0.0525226i
\(726\) 0 0
\(727\) 2.82843i 0.104901i −0.998624 0.0524503i \(-0.983297\pi\)
0.998624 0.0524503i \(-0.0167031\pi\)
\(728\) 0 0
\(729\) 29.0000i 1.07407i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −29.6985 + 29.6985i −1.09694 + 1.09694i −0.102172 + 0.994767i \(0.532579\pi\)
−0.994767 + 0.102172i \(0.967421\pi\)
\(734\) 0 0
\(735\) −2.82843 −0.104328
\(736\) 0 0
\(737\) 54.0000 1.98912
\(738\) 0 0
\(739\) 33.0000 33.0000i 1.21392 1.21392i 0.244200 0.969725i \(-0.421475\pi\)
0.969725 0.244200i \(-0.0785252\pi\)
\(740\) 0 0
\(741\) 25.4558 + 25.4558i 0.935144 + 0.935144i
\(742\) 0 0
\(743\) 25.4558i 0.933884i −0.884288 0.466942i \(-0.845356\pi\)
0.884288 0.466942i \(-0.154644\pi\)
\(744\) 0 0
\(745\) 44.0000i 1.61204i
\(746\) 0 0
\(747\) 3.00000 + 3.00000i 0.109764 + 0.109764i
\(748\) 0 0
\(749\) 14.1421 14.1421i 0.516742 0.516742i
\(750\) 0 0
\(751\) 5.65685 0.206422 0.103211 0.994660i \(-0.467088\pi\)
0.103211 + 0.994660i \(0.467088\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) −20.0000 + 20.0000i −0.727875 + 0.727875i
\(756\) 0 0
\(757\) −29.6985 29.6985i −1.07941 1.07941i −0.996562 0.0828476i \(-0.973599\pi\)
−0.0828476 0.996562i \(-0.526401\pi\)
\(758\) 0 0
\(759\) 50.9117i 1.84798i
\(760\) 0 0
\(761\) 42.0000i 1.52250i −0.648459 0.761249i \(-0.724586\pi\)
0.648459 0.761249i \(-0.275414\pi\)
\(762\) 0 0
\(763\) 12.0000 + 12.0000i 0.434429 + 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.48528 0.306386
\(768\) 0 0
\(769\) −24.0000 −0.865462 −0.432731 0.901523i \(-0.642450\pi\)
−0.432731 + 0.901523i \(0.642450\pi\)
\(770\) 0 0
\(771\) 6.00000 6.00000i 0.216085 0.216085i
\(772\) 0 0
\(773\) 9.89949 + 9.89949i 0.356060 + 0.356060i 0.862358 0.506298i \(-0.168987\pi\)
−0.506298 + 0.862358i \(0.668987\pi\)
\(774\) 0 0
\(775\) 5.65685i 0.203200i
\(776\) 0 0
\(777\) 24.0000i 0.860995i
\(778\) 0 0
\(779\) 18.0000 + 18.0000i 0.644917 + 0.644917i
\(780\) 0 0
\(781\) −25.4558 + 25.4558i −0.910882 + 0.910882i
\(782\) 0 0
\(783\) −11.3137 −0.404319
\(784\) 0 0
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) −21.0000 + 21.0000i −0.748569 + 0.748569i −0.974210 0.225641i \(-0.927552\pi\)
0.225641 + 0.974210i \(0.427552\pi\)
\(788\) 0 0
\(789\) −8.48528 8.48528i −0.302084 0.302084i
\(790\) 0 0
\(791\) 16.9706i 0.603404i
\(792\) 0 0
\(793\) 36.0000i 1.27840i
\(794\) 0 0
\(795\) 4.00000 + 4.00000i 0.141865 + 0.141865i
\(796\) 0 0
\(797\) 1.41421 1.41421i 0.0500940 0.0500940i −0.681616 0.731710i \(-0.738723\pi\)
0.731710 + 0.681616i \(0.238723\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 12.0000 0.423999
\(802\) 0 0
\(803\) 36.0000 36.0000i 1.27041 1.27041i
\(804\) 0 0
\(805\) −33.9411 33.9411i −1.19627 1.19627i
\(806\) 0 0
\(807\) 19.7990i 0.696957i
\(808\) 0 0
\(809\) 42.0000i 1.47664i 0.674450 + 0.738321i \(0.264381\pi\)
−0.674450 + 0.738321i \(0.735619\pi\)
\(810\) 0 0
\(811\) −3.00000 3.00000i −0.105344 0.105344i 0.652470 0.757814i \(-0.273733\pi\)
−0.757814 + 0.652470i \(0.773733\pi\)
\(812\) 0 0
\(813\) 22.6274 22.6274i 0.793578 0.793578i
\(814\) 0 0
\(815\) −42.4264 −1.48613
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) 0 0
\(819\) 12.0000 12.0000i 0.419314 0.419314i
\(820\) 0 0
\(821\) −26.8701 26.8701i −0.937771 0.937771i 0.0604026 0.998174i \(-0.480762\pi\)
−0.998174 + 0.0604026i \(0.980762\pi\)
\(822\) 0 0
\(823\) 19.7990i 0.690149i 0.938575 + 0.345075i \(0.112146\pi\)
−0.938575 + 0.345075i \(0.887854\pi\)
\(824\) 0 0
\(825\) 6.00000i 0.208893i
\(826\) 0 0
\(827\) 31.0000 + 31.0000i 1.07798 + 1.07798i 0.996691 + 0.0812847i \(0.0259023\pi\)
0.0812847 + 0.996691i \(0.474098\pi\)
\(828\) 0 0
\(829\) 12.7279 12.7279i 0.442059 0.442059i −0.450644 0.892704i \(-0.648806\pi\)
0.892704 + 0.450644i \(0.148806\pi\)
\(830\) 0 0
\(831\) −8.48528 −0.294351
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −12.0000 + 12.0000i −0.415277 + 0.415277i
\(836\) 0 0
\(837\) 22.6274 + 22.6274i 0.782118 + 0.782118i
\(838\) 0 0
\(839\) 8.48528i 0.292944i 0.989215 + 0.146472i \(0.0467919\pi\)
−0.989215 + 0.146472i \(0.953208\pi\)
\(840\) 0 0
\(841\) 25.0000i 0.862069i
\(842\) 0 0
\(843\) 12.0000 + 12.0000i 0.413302 + 0.413302i
\(844\) 0 0
\(845\) 32.5269 32.5269i 1.11896 1.11896i
\(846\) 0 0
\(847\) 19.7990 0.680301
\(848\) 0 0
\(849\) −18.0000 −0.617758
\(850\) 0 0
\(851\) 36.0000 36.0000i 1.23406 1.23406i
\(852\) 0 0
\(853\) 12.7279 + 12.7279i 0.435796 + 0.435796i 0.890594 0.454799i \(-0.150289\pi\)
−0.454799 + 0.890594i \(0.650289\pi\)
\(854\) 0 0
\(855\) 8.48528i 0.290191i
\(856\) 0 0
\(857\) 54.0000i 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) 0 0
\(859\) 33.0000 + 33.0000i 1.12595 + 1.12595i 0.990830 + 0.135116i \(0.0431406\pi\)
0.135116 + 0.990830i \(0.456859\pi\)
\(860\) 0 0
\(861\) −16.9706 + 16.9706i −0.578355 + 0.578355i
\(862\) 0 0
\(863\) −50.9117 −1.73305 −0.866527 0.499130i \(-0.833653\pi\)
−0.866527 + 0.499130i \(0.833653\pi\)
\(864\) 0 0
\(865\) 28.0000 0.952029
\(866\) 0 0
\(867\) −17.0000 + 17.0000i −0.577350 + 0.577350i
\(868\) 0 0
\(869\) −16.9706 16.9706i −0.575687 0.575687i
\(870\) 0 0
\(871\) 76.3675i 2.58762i
\(872\) 0 0
\(873\) 8.00000i 0.270759i
\(874\) 0 0
\(875\) −24.0000 24.0000i −0.811348 0.811348i
\(876\) 0 0
\(877\) −4.24264 + 4.24264i −0.143264 + 0.143264i −0.775101 0.631837i \(-0.782301\pi\)
0.631837 + 0.775101i \(0.282301\pi\)
\(878\) 0 0
\(879\) 14.1421 0.477002
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) −27.0000 + 27.0000i −0.908622 + 0.908622i −0.996161 0.0875388i \(-0.972100\pi\)
0.0875388 + 0.996161i \(0.472100\pi\)
\(884\) 0 0
\(885\) 2.82843 + 2.82843i 0.0950765 + 0.0950765i
\(886\) 0 0
\(887\) 42.4264i 1.42454i 0.701906 + 0.712270i \(0.252333\pi\)
−0.701906 + 0.712270i \(0.747667\pi\)
\(888\) 0 0
\(889\) 16.0000i 0.536623i
\(890\) 0 0
\(891\) −15.0000 15.0000i −0.502519 0.502519i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −36.7696 −1.22907
\(896\) 0 0
\(897\) −72.0000 −2.40401
\(898\) 0 0
\(899\) −8.00000 + 8.00000i −0.266815 + 0.266815i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 16.9706i 0.564745i
\(904\) 0 0
\(905\) 36.0000i 1.19668i
\(906\) 0 0
\(907\) 3.00000 + 3.00000i 0.0996134 + 0.0996134i 0.755157 0.655544i \(-0.227561\pi\)
−0.655544 + 0.755157i \(0.727561\pi\)
\(908\) 0 0
\(909\) 7.07107 7.07107i 0.234533 0.234533i
\(910\) 0 0
\(911\) 33.9411 1.12452 0.562260 0.826961i \(-0.309932\pi\)
0.562260 + 0.826961i \(0.309932\pi\)
\(912\) 0 0
\(913\) −18.0000 −0.595713
\(914\) 0 0
\(915\) −12.0000 + 12.0000i −0.396708 + 0.396708i
\(916\) 0 0
\(917\) 2.82843 + 2.82843i 0.0934029 + 0.0934029i
\(918\) 0 0
\(919\) 2.82843i 0.0933012i 0.998911 + 0.0466506i \(0.0148547\pi\)
−0.998911 + 0.0466506i \(0.985145\pi\)
\(920\) 0 0
\(921\) 6.00000i 0.197707i
\(922\) 0 0
\(923\) −36.0000 36.0000i −1.18495 1.18495i
\(924\) 0 0
\(925\) 4.24264 4.24264i 0.139497 0.139497i
\(926\) 0 0
\(927\) −2.82843 −0.0928977
\(928\) 0 0
\(929\) −24.0000 −0.787414 −0.393707 0.919236i \(-0.628808\pi\)
−0.393707 + 0.919236i \(0.628808\pi\)
\(930\) 0 0
\(931\) 3.00000 3.00000i 0.0983210 0.0983210i
\(932\) 0 0
\(933\) 8.48528 + 8.48528i 0.277796 + 0.277796i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20.0000i 0.653372i −0.945133 0.326686i \(-0.894068\pi\)
0.945133 0.326686i \(-0.105932\pi\)
\(938\) 0 0
\(939\) 6.00000 + 6.00000i 0.195803 + 0.195803i
\(940\) 0 0
\(941\) −15.5563 + 15.5563i −0.507122 + 0.507122i −0.913642 0.406520i \(-0.866742\pi\)
0.406520 + 0.913642i \(0.366742\pi\)
\(942\) 0 0
\(943\) −50.9117 −1.65791
\(944\) 0 0
\(945\) 32.0000 1.04096
\(946\) 0 0
\(947\) 29.0000 29.0000i 0.942373 0.942373i −0.0560543 0.998428i \(-0.517852\pi\)
0.998428 + 0.0560543i \(0.0178520\pi\)
\(948\) 0 0
\(949\) 50.9117 + 50.9117i 1.65266 + 1.65266i
\(950\) 0 0
\(951\) 14.1421i 0.458590i
\(952\) 0 0
\(953\) 6.00000i 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 0 0
\(955\) −24.0000 24.0000i −0.776622 0.776622i
\(956\) 0 0
\(957\) 8.48528 8.48528i 0.274290 0.274290i
\(958\) 0 0
\(959\) 16.9706 0.548008
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −5.00000 + 5.00000i −0.161123 + 0.161123i
\(964\) 0 0
\(965\) 33.9411 + 33.9411i 1.09260 + 1.09260i
\(966\) 0 0
\(967\) 31.1127i 1.00052i 0.865876 + 0.500258i \(0.166762\pi\)
−0.865876 + 0.500258i \(0.833238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.0000 21.0000i −0.673922 0.673922i 0.284696 0.958618i \(-0.408107\pi\)
−0.958618 + 0.284696i \(0.908107\pi\)
\(972\) 0 0
\(973\) 8.48528 8.48528i 0.272026 0.272026i
\(974\) 0 0
\(975\) −8.48528 −0.271746
\(976\) 0 0
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) −36.0000 + 36.0000i −1.15056 + 1.15056i
\(980\) 0 0
\(981\) −4.24264 4.24264i −0.135457 0.135457i
\(982\) 0 0
\(983\) 42.4264i 1.35319i 0.736354 + 0.676596i \(0.236546\pi\)
−0.736354 + 0.676596i \(0.763454\pi\)
\(984\) 0 0
\(985\) 20.0000i 0.637253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 25.4558 25.4558i 0.809449 0.809449i
\(990\) 0 0
\(991\) 45.2548 1.43757 0.718784 0.695234i \(-0.244699\pi\)
0.718784 + 0.695234i \(0.244699\pi\)
\(992\) 0 0
\(993\) 42.0000 1.33283
\(994\) 0 0
\(995\) −28.0000 + 28.0000i −0.887660 + 0.887660i
\(996\) 0 0
\(997\) −12.7279 12.7279i −0.403097 0.403097i 0.476226 0.879323i \(-0.342005\pi\)
−0.879323 + 0.476226i \(0.842005\pi\)
\(998\) 0 0
\(999\) 33.9411i 1.07385i
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 1024.2.e.n.769.2 4
4.3 odd 2 1024.2.e.h.769.2 4
8.3 odd 2 inner 1024.2.e.n.769.1 4
8.5 even 2 1024.2.e.h.769.1 4
16.3 odd 4 1024.2.e.h.257.2 4
16.5 even 4 1024.2.e.h.257.1 4
16.11 odd 4 inner 1024.2.e.n.257.1 4
16.13 even 4 inner 1024.2.e.n.257.2 4
32.3 odd 8 512.2.a.b.1.1 2
32.5 even 8 512.2.b.e.257.4 4
32.11 odd 8 512.2.b.e.257.3 4
32.13 even 8 512.2.a.e.1.1 yes 2
32.19 odd 8 512.2.a.e.1.2 yes 2
32.21 even 8 512.2.b.e.257.1 4
32.27 odd 8 512.2.b.e.257.2 4
32.29 even 8 512.2.a.b.1.2 yes 2
96.5 odd 8 4608.2.d.j.2305.2 4
96.11 even 8 4608.2.d.j.2305.3 4
96.29 odd 8 4608.2.a.p.1.1 2
96.35 even 8 4608.2.a.p.1.2 2
96.53 odd 8 4608.2.d.j.2305.4 4
96.59 even 8 4608.2.d.j.2305.1 4
96.77 odd 8 4608.2.a.c.1.1 2
96.83 even 8 4608.2.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.b.1.1 2 32.3 odd 8
512.2.a.b.1.2 yes 2 32.29 even 8
512.2.a.e.1.1 yes 2 32.13 even 8
512.2.a.e.1.2 yes 2 32.19 odd 8
512.2.b.e.257.1 4 32.21 even 8
512.2.b.e.257.2 4 32.27 odd 8
512.2.b.e.257.3 4 32.11 odd 8
512.2.b.e.257.4 4 32.5 even 8
1024.2.e.h.257.1 4 16.5 even 4
1024.2.e.h.257.2 4 16.3 odd 4
1024.2.e.h.769.1 4 8.5 even 2
1024.2.e.h.769.2 4 4.3 odd 2
1024.2.e.n.257.1 4 16.11 odd 4 inner
1024.2.e.n.257.2 4 16.13 even 4 inner
1024.2.e.n.769.1 4 8.3 odd 2 inner
1024.2.e.n.769.2 4 1.1 even 1 trivial
4608.2.a.c.1.1 2 96.77 odd 8
4608.2.a.c.1.2 2 96.83 even 8
4608.2.a.p.1.1 2 96.29 odd 8
4608.2.a.p.1.2 2 96.35 even 8
4608.2.d.j.2305.1 4 96.59 even 8
4608.2.d.j.2305.2 4 96.5 odd 8
4608.2.d.j.2305.3 4 96.11 even 8
4608.2.d.j.2305.4 4 96.53 odd 8