Properties

Label 507.2.x.a.98.1
Level $507$
Weight $2$
Character 507.98
Analytic conductor $4.048$
Analytic rank $0$
Dimension $48$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(2,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(156))
 
chi = DirichletCharacter(H, H._module([78, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.x (of order \(156\), degree \(48\), 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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{156}]$

Embedding invariants

Embedding label 98.1
Character \(\chi\) \(=\) 507.98
Dual form 507.2.x.a.119.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.64291 - 0.548485i) q^{3} +(-1.44240 + 1.38545i) q^{4} +(-0.527444 - 5.22047i) q^{7} +(2.39833 - 1.80223i) q^{9} +O(q^{10})\) \(q+(1.64291 - 0.548485i) q^{3} +(-1.44240 + 1.38545i) q^{4} +(-0.527444 - 5.22047i) q^{7} +(2.39833 - 1.80223i) q^{9} +(-1.60985 + 3.06731i) q^{12} +(2.59808 + 2.50000i) q^{13} +(0.161064 - 3.99676i) q^{16} +(1.98272 - 7.39960i) q^{19} +(-3.72990 - 8.28749i) q^{21} +(-4.11492 + 2.84032i) q^{25} +(2.95175 - 4.27635i) q^{27} +(7.99349 + 6.79929i) q^{28} +(1.48545 - 0.272220i) q^{31} +(-0.962468 + 5.92230i) q^{36} +(5.91139 + 2.10674i) q^{37} +(5.63963 + 2.68228i) q^{39} +(11.5365 + 5.47415i) q^{43} +(-1.92755 - 6.65466i) q^{48} +(-20.1166 + 4.10684i) q^{49} +(-7.21110 - 0.00651091i) q^{52} +(-0.801139 - 13.2444i) q^{57} +(-11.3705 + 1.84789i) q^{61} +(-10.6735 - 11.5698i) q^{63} +(5.30498 + 5.98809i) q^{64} +(-9.94094 + 0.200222i) q^{67} +(2.09927 + 2.67951i) q^{73} +(-5.20258 + 6.92338i) q^{75} +(7.39189 + 13.4202i) q^{76} +(-0.441406 - 0.108797i) q^{79} +(2.50396 - 8.64466i) q^{81} +(16.8619 + 6.78634i) q^{84} +(11.6808 - 14.8818i) q^{91} +(2.29117 - 1.26198i) q^{93} +(2.18324 + 9.69385i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 10 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 10 q^{7} + 6 q^{9} - 8 q^{16} - 14 q^{19} - 18 q^{21} + 20 q^{28} + 14 q^{31} + 2 q^{37} + 24 q^{39} + 6 q^{43} - 18 q^{49} - 28 q^{52} - 12 q^{57} - 24 q^{63} - 32 q^{67} + 34 q^{73} + 30 q^{75} + 28 q^{76} + 18 q^{81} + 12 q^{84} - 2 q^{91} - 6 q^{93} + 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(-1\) \(e\left(\frac{59}{156}\right)\)

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 −0.373361 0.927686i \(-0.621795\pi\)
0.373361 + 0.927686i \(0.378205\pi\)
\(3\) 1.64291 0.548485i 0.948536 0.316668i
\(4\) −1.44240 + 1.38545i −0.721202 + 0.692724i
\(5\) 0 0 −0.297503 0.954721i \(-0.596154\pi\)
0.297503 + 0.954721i \(0.403846\pi\)
\(6\) 0 0
\(7\) −0.527444 5.22047i −0.199355 1.97315i −0.228369 0.973575i \(-0.573339\pi\)
0.0290142 0.999579i \(-0.490763\pi\)
\(8\) 0 0
\(9\) 2.39833 1.80223i 0.799443 0.600742i
\(10\) 0 0
\(11\) 0 0 0.990080 0.140502i \(-0.0448718\pi\)
−0.990080 + 0.140502i \(0.955128\pi\)
\(12\) −1.60985 + 3.06731i −0.464723 + 0.885456i
\(13\) 2.59808 + 2.50000i 0.720577 + 0.693375i
\(14\) 0 0
\(15\) 0 0
\(16\) 0.161064 3.99676i 0.0402659 0.999189i
\(17\) 0 0 −0.774605 0.632445i \(-0.782051\pi\)
0.774605 + 0.632445i \(0.217949\pi\)
\(18\) 0 0
\(19\) 1.98272 7.39960i 0.454867 1.69759i −0.233613 0.972330i \(-0.575055\pi\)
0.688479 0.725256i \(-0.258279\pi\)
\(20\) 0 0
\(21\) −3.72990 8.28749i −0.813930 1.80848i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −4.11492 + 2.84032i −0.822984 + 0.568065i
\(26\) 0 0
\(27\) 2.95175 4.27635i 0.568065 0.822984i
\(28\) 7.99349 + 6.79929i 1.51063 + 1.28495i
\(29\) 0 0 −0.919979 0.391967i \(-0.871795\pi\)
0.919979 + 0.391967i \(0.128205\pi\)
\(30\) 0 0
\(31\) 1.48545 0.272220i 0.266796 0.0488921i −0.0451919 0.998978i \(-0.514390\pi\)
0.311987 + 0.950086i \(0.399005\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.962468 + 5.92230i −0.160411 + 0.987050i
\(37\) 5.91139 + 2.10674i 0.971826 + 0.346346i 0.773504 0.633791i \(-0.218502\pi\)
0.198322 + 0.980137i \(0.436451\pi\)
\(38\) 0 0
\(39\) 5.63963 + 2.68228i 0.903063 + 0.429508i
\(40\) 0 0
\(41\) 0 0 −0.894635 0.446798i \(-0.852564\pi\)
0.894635 + 0.446798i \(0.147436\pi\)
\(42\) 0 0
\(43\) 11.5365 + 5.47415i 1.75930 + 0.834800i 0.977399 + 0.211401i \(0.0678026\pi\)
0.781904 + 0.623399i \(0.214249\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.855781 0.517338i \(-0.173077\pi\)
−0.855781 + 0.517338i \(0.826923\pi\)
\(48\) −1.92755 6.65466i −0.278217 0.960518i
\(49\) −20.1166 + 4.10684i −2.87380 + 0.586691i
\(50\) 0 0
\(51\) 0 0
\(52\) −7.21110 0.00651091i −1.00000 0.000902901i
\(53\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.801139 13.2444i −0.106113 1.75426i
\(58\) 0 0
\(59\) 0 0 −0.735006 0.678061i \(-0.762821\pi\)
0.735006 + 0.678061i \(0.237179\pi\)
\(60\) 0 0
\(61\) −11.3705 + 1.84789i −1.45585 + 0.236598i −0.836261 0.548331i \(-0.815263\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(62\) 0 0
\(63\) −10.6735 11.5698i −1.34473 1.45766i
\(64\) 5.30498 + 5.98809i 0.663123 + 0.748511i
\(65\) 0 0
\(66\) 0 0
\(67\) −9.94094 + 0.200222i −1.21448 + 0.0244610i −0.623237 0.782033i \(-0.714183\pi\)
−0.591242 + 0.806494i \(0.701362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.834256 0.551377i \(-0.185897\pi\)
−0.834256 + 0.551377i \(0.814103\pi\)
\(72\) 0 0
\(73\) 2.09927 + 2.67951i 0.245700 + 0.313614i 0.894258 0.447552i \(-0.147704\pi\)
−0.648557 + 0.761166i \(0.724627\pi\)
\(74\) 0 0
\(75\) −5.20258 + 6.92338i −0.600742 + 0.799443i
\(76\) 7.39189 + 13.4202i 0.847908 + 1.53940i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.441406 0.108797i −0.0496620 0.0122406i 0.214406 0.976745i \(-0.431218\pi\)
−0.264068 + 0.964504i \(0.585064\pi\)
\(80\) 0 0
\(81\) 2.50396 8.64466i 0.278217 0.960518i
\(82\) 0 0
\(83\) 0 0 −0.998176 0.0603785i \(-0.980769\pi\)
0.998176 + 0.0603785i \(0.0192308\pi\)
\(84\) 16.8619 + 6.78634i 1.83979 + 0.740450i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(90\) 0 0
\(91\) 11.6808 14.8818i 1.22449 1.56004i
\(92\) 0 0
\(93\) 2.29117 1.26198i 0.237583 0.130862i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.18324 + 9.69385i 0.221674 + 0.984261i 0.953175 + 0.302420i \(0.0977946\pi\)
−0.731501 + 0.681841i \(0.761180\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.00026 9.79791i 0.200026 0.979791i
\(101\) 0 0 −0.0804666 0.996757i \(-0.525641\pi\)
0.0804666 + 0.996757i \(0.474359\pi\)
\(102\) 0 0
\(103\) −13.1592 + 14.8537i −1.29661 + 1.46358i −0.490970 + 0.871177i \(0.663357\pi\)
−0.805645 + 0.592399i \(0.798181\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(108\) 1.66704 + 10.2577i 0.160411 + 0.987050i
\(109\) −3.73419 + 20.3768i −0.357671 + 1.95175i −0.0556241 + 0.998452i \(0.517715\pi\)
−0.302046 + 0.953293i \(0.597670\pi\)
\(110\) 0 0
\(111\) 10.8674 + 0.218882i 1.03149 + 0.0207754i
\(112\) −20.9499 + 1.26724i −1.97958 + 0.119743i
\(113\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.7366 + 1.31350i 0.992600 + 0.121433i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 10.5657 3.06039i 0.960518 0.278217i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.76548 + 2.45067i −0.158545 + 0.220077i
\(125\) 0 0
\(126\) 0 0
\(127\) −8.18527 7.86205i −0.726325 0.697644i 0.235327 0.971916i \(-0.424384\pi\)
−0.961652 + 0.274272i \(0.911563\pi\)
\(128\) 0 0
\(129\) 21.9560 + 2.66594i 1.93312 + 0.234723i
\(130\) 0 0
\(131\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(132\) 0 0
\(133\) −39.6752 6.44785i −3.44028 0.559099i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.335705 0.941967i \(-0.608974\pi\)
0.335705 + 0.941967i \(0.391026\pi\)
\(138\) 0 0
\(139\) 19.0229 + 12.0294i 1.61351 + 1.02032i 0.966098 + 0.258175i \(0.0831210\pi\)
0.647407 + 0.762144i \(0.275853\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −6.81678 9.87581i −0.568065 0.822984i
\(145\) 0 0
\(146\) 0 0
\(147\) −30.7973 + 17.7808i −2.54012 + 1.46654i
\(148\) −11.4454 + 5.15115i −0.940805 + 0.423422i
\(149\) 0 0 0.482459 0.875918i \(-0.339744\pi\)
−0.482459 + 0.875918i \(0.660256\pi\)
\(150\) 0 0
\(151\) 4.29671 7.10763i 0.349662 0.578411i −0.630602 0.776107i \(-0.717192\pi\)
0.980263 + 0.197696i \(0.0633457\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −11.8508 + 3.94448i −0.948822 + 0.315811i
\(157\) 4.68563 + 2.45921i 0.373954 + 0.196266i 0.641221 0.767356i \(-0.278428\pi\)
−0.267267 + 0.963623i \(0.586121\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 19.4443 16.5394i 1.52300 1.29547i 0.746156 0.665771i \(-0.231897\pi\)
0.776842 0.629696i \(-0.216821\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.927686 0.373361i \(-0.121795\pi\)
−0.927686 + 0.373361i \(0.878205\pi\)
\(168\) 0 0
\(169\) 0.500000 + 12.9904i 0.0384615 + 0.999260i
\(170\) 0 0
\(171\) −8.58056 21.3200i −0.656172 1.63038i
\(172\) −24.2245 + 8.08732i −1.84710 + 0.616653i
\(173\) 0 0 0.721202 0.692724i \(-0.243590\pi\)
−0.721202 + 0.692724i \(0.756410\pi\)
\(174\) 0 0
\(175\) 16.9982 + 19.9837i 1.28495 + 1.51063i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.774605 0.632445i \(-0.217949\pi\)
−0.774605 + 0.632445i \(0.782051\pi\)
\(180\) 0 0
\(181\) 8.11496 15.4618i 0.603180 1.14926i −0.372080 0.928201i \(-0.621355\pi\)
0.975260 0.221062i \(-0.0709525\pi\)
\(182\) 0 0
\(183\) −17.6673 + 9.27249i −1.30600 + 0.685442i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −23.8815 13.1540i −1.73712 0.956813i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 12.0000 + 6.92820i 0.866025 + 0.500000i
\(193\) 23.5348 + 2.37781i 1.69407 + 0.171158i 0.899770 0.436365i \(-0.143734\pi\)
0.794302 + 0.607524i \(0.207837\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 23.3265 33.7943i 1.66618 2.41388i
\(197\) 0 0 −0.761712 0.647915i \(-0.775641\pi\)
0.761712 + 0.647915i \(0.224359\pi\)
\(198\) 0 0
\(199\) −10.8515 + 17.1602i −0.769239 + 1.21645i 0.202241 + 0.979336i \(0.435177\pi\)
−0.971481 + 0.237119i \(0.923797\pi\)
\(200\) 0 0
\(201\) −16.2223 + 5.78140i −1.14423 + 0.407789i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 10.4103 9.98122i 0.721828 0.692073i
\(209\) 0 0
\(210\) 0 0
\(211\) 2.81244 2.92806i 0.193616 0.201576i −0.617317 0.786714i \(-0.711780\pi\)
0.810933 + 0.585139i \(0.198960\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.20461 7.61120i −0.149659 0.516682i
\(218\) 0 0
\(219\) 4.91859 + 3.25079i 0.332367 + 0.219668i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −26.4841 5.96471i −1.77351 0.399427i −0.795159 0.606401i \(-0.792613\pi\)
−0.978346 + 0.206974i \(0.933638\pi\)
\(224\) 0 0
\(225\) −4.75002 + 14.2280i −0.316668 + 0.948536i
\(226\) 0 0
\(227\) 0 0 0.0201371 0.999797i \(-0.493590\pi\)
−0.0201371 + 0.999797i \(0.506410\pi\)
\(228\) 19.5050 + 17.9939i 1.29175 + 1.19167i
\(229\) 14.5776 + 2.67145i 0.963316 + 0.176534i 0.638823 0.769354i \(-0.279422\pi\)
0.324493 + 0.945888i \(0.394806\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.784865 + 0.0633608i −0.0509824 + 0.00411573i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 30.2023 6.80213i 1.94550 0.438164i 0.956079 0.293108i \(-0.0946894\pi\)
0.989423 0.145056i \(-0.0463362\pi\)
\(242\) 0 0
\(243\) −0.627684 15.5758i −0.0402659 0.999189i
\(244\) 13.8407 18.4187i 0.886063 1.17914i
\(245\) 0 0
\(246\) 0 0
\(247\) 23.6503 14.2679i 1.50483 0.907847i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.0804666 0.996757i \(-0.474359\pi\)
−0.0804666 + 0.996757i \(0.525641\pi\)
\(252\) 31.4249 + 1.90085i 1.97958 + 0.119743i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −15.9481 1.28747i −0.996757 0.0804666i
\(257\) 0 0 −0.960518 0.278217i \(-0.910256\pi\)
0.960518 + 0.278217i \(0.0897436\pi\)
\(258\) 0 0
\(259\) 7.88025 31.9714i 0.489655 1.98661i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.799443 0.600742i \(-0.794872\pi\)
0.799443 + 0.600742i \(0.205128\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 14.0615 14.0615i 0.858940 0.858940i
\(269\) 0 0 0.200026 0.979791i \(-0.435897\pi\)
−0.200026 + 0.979791i \(0.564103\pi\)
\(270\) 0 0
\(271\) 0.570747 + 28.3374i 0.0346704 + 1.72137i 0.526073 + 0.850439i \(0.323664\pi\)
−0.491403 + 0.870933i \(0.663516\pi\)
\(272\) 0 0
\(273\) 11.0282 30.8563i 0.667455 1.86751i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.26128 32.3740i −0.316120 1.94516i −0.326446 0.945216i \(-0.605851\pi\)
0.0103260 0.999947i \(-0.496713\pi\)
\(278\) 0 0
\(279\) 3.07201 3.33000i 0.183916 0.199362i
\(280\) 0 0
\(281\) 0 0 0.998176 0.0603785i \(-0.0192308\pi\)
−0.998176 + 0.0603785i \(0.980769\pi\)
\(282\) 0 0
\(283\) −10.5454 + 5.00387i −0.626860 + 0.297449i −0.715498 0.698615i \(-0.753800\pi\)
0.0886374 + 0.996064i \(0.471749\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.40044 + 16.6564i 0.200026 + 0.979791i
\(290\) 0 0
\(291\) 8.90380 + 14.7287i 0.521950 + 0.863411i
\(292\) −6.74032 0.956520i −0.394448 0.0559761i
\(293\) 0 0 0.584522 0.811378i \(-0.301282\pi\)
−0.584522 + 0.811378i \(0.698718\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −2.08776 17.1942i −0.120537 0.992709i
\(301\) 22.4928 63.1134i 1.29646 3.63780i
\(302\) 0 0
\(303\) 0 0
\(304\) −29.2551 9.11625i −1.67789 0.522853i
\(305\) 0 0
\(306\) 0 0
\(307\) −2.30456 12.5756i −0.131528 0.717726i −0.981584 0.191033i \(-0.938816\pi\)
0.850056 0.526693i \(-0.176568\pi\)
\(308\) 0 0
\(309\) −13.4724 + 31.6209i −0.766419 + 1.79885i
\(310\) 0 0
\(311\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(312\) 0 0
\(313\) −18.8900 27.3669i −1.06773 1.54687i −0.815111 0.579304i \(-0.803324\pi\)
−0.252616 0.967567i \(-0.581291\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.787418 0.454616i 0.0442957 0.0255741i
\(317\) 0 0 0.911900 0.410413i \(-0.134615\pi\)
−0.911900 + 0.410413i \(0.865385\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 8.36502 + 15.9382i 0.464723 + 0.885456i
\(325\) −17.7917 2.90792i −0.986905 0.161302i
\(326\) 0 0
\(327\) 5.04143 + 35.5255i 0.278792 + 1.96456i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 24.2052 2.44554i 1.33044 0.134419i 0.590468 0.807061i \(-0.298943\pi\)
0.739968 + 0.672642i \(0.234841\pi\)
\(332\) 0 0
\(333\) 17.9743 5.60101i 0.984984 0.306933i
\(334\) 0 0
\(335\) 0 0
\(336\) −33.7238 + 13.5727i −1.83979 + 0.740450i
\(337\) 31.3370i 1.70703i 0.521065 + 0.853517i \(0.325535\pi\)
−0.521065 + 0.853517i \(0.674465\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 21.1229 + 67.7858i 1.14053 + 3.66009i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(348\) 0 0
\(349\) −36.7163 + 5.21041i −1.96538 + 0.278907i −0.965551 + 0.260214i \(0.916207\pi\)
−0.999825 + 0.0186925i \(0.994050\pi\)
\(350\) 0 0
\(351\) 18.3597 3.73090i 0.979971 0.199141i
\(352\) 0 0
\(353\) 0 0 −0.584522 0.811378i \(-0.698718\pi\)
0.584522 + 0.811378i \(0.301282\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.410413 0.911900i \(-0.634615\pi\)
0.410413 + 0.911900i \(0.365385\pi\)
\(360\) 0 0
\(361\) −34.3685 19.8427i −1.80887 1.04435i
\(362\) 0 0
\(363\) 15.6799 10.8231i 0.822984 0.568065i
\(364\) 3.76946 + 37.6488i 0.197573 + 1.97333i
\(365\) 0 0
\(366\) 0 0
\(367\) 35.0835 + 14.9477i 1.83135 + 0.780264i 0.953554 + 0.301222i \(0.0973945\pi\)
0.877792 + 0.479042i \(0.159016\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.55637 + 4.99458i −0.0806943 + 0.258957i
\(373\) −29.6280 + 12.6233i −1.53408 + 0.653609i −0.983456 0.181146i \(-0.942019\pi\)
−0.550622 + 0.834755i \(0.685609\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −19.3675 9.67250i −0.994840 0.496843i −0.126050 0.992024i \(-0.540230\pi\)
−0.868790 + 0.495181i \(0.835102\pi\)
\(380\) 0 0
\(381\) −17.7599 8.42718i −0.909867 0.431737i
\(382\) 0 0
\(383\) 0 0 −0.811378 0.584522i \(-0.801282\pi\)
0.811378 + 0.584522i \(0.198718\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 37.5340 7.66263i 1.90796 0.389513i
\(388\) −16.5794 10.9577i −0.841693 0.556293i
\(389\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −24.6775 22.7656i −1.23853 1.14257i −0.984502 0.175376i \(-0.943886\pi\)
−0.254026 0.967197i \(-0.581755\pi\)
\(398\) 0 0
\(399\) −68.7195 + 11.1680i −3.44028 + 0.559099i
\(400\) 10.6893 + 16.9038i 0.534466 + 0.845190i
\(401\) 0 0 −0.678061 0.735006i \(-0.737179\pi\)
0.678061 + 0.735006i \(0.262821\pi\)
\(402\) 0 0
\(403\) 4.53987 + 3.00639i 0.226147 + 0.149759i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.99095 + 4.62046i −0.345680 + 0.228467i −0.712254 0.701922i \(-0.752326\pi\)
0.366574 + 0.930389i \(0.380531\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.59810 39.6564i −0.0787327 1.95373i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 37.8510 + 9.32943i 1.85357 + 0.456864i
\(418\) 0 0
\(419\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(420\) 0 0
\(421\) 29.0205 + 1.75542i 1.41437 + 0.0855539i 0.749798 0.661667i \(-0.230151\pi\)
0.664576 + 0.747221i \(0.268612\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 15.6442 + 58.3849i 0.757075 + 2.82544i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.875918 0.482459i \(-0.160256\pi\)
−0.875918 + 0.482459i \(0.839744\pi\)
\(432\) −16.6161 12.4862i −0.799443 0.600742i
\(433\) −26.5497 + 1.06992i −1.27590 + 0.0514169i −0.668940 0.743317i \(-0.733252\pi\)
−0.606958 + 0.794734i \(0.707611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −22.8448 34.5651i −1.09407 1.65537i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.652051 + 8.07710i 0.0311207 + 0.385499i 0.993503 + 0.113802i \(0.0363029\pi\)
−0.962383 + 0.271697i \(0.912415\pi\)
\(440\) 0 0
\(441\) −40.8448 + 46.1042i −1.94499 + 2.19544i
\(442\) 0 0
\(443\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(444\) −15.9785 + 14.7405i −0.758304 + 0.699554i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 28.4626 30.8529i 1.34473 1.45766i
\(449\) 0 0 −0.999797 0.0201371i \(-0.993590\pi\)
0.999797 + 0.0201371i \(0.00641026\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 3.16070 14.0339i 0.148503 0.659371i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.5940 29.6466i 0.916570 1.38681i −0.00558406 0.999984i \(-0.501777\pi\)
0.922154 0.386823i \(-0.126428\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.990080 0.140502i \(-0.955128\pi\)
0.990080 + 0.140502i \(0.0448718\pi\)
\(462\) 0 0
\(463\) −27.4861 21.5340i −1.27739 1.00077i −0.999190 0.0402476i \(-0.987185\pi\)
−0.278200 0.960523i \(-0.589738\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(468\) −17.3063 + 12.9804i −0.799985 + 0.600020i
\(469\) 6.28854 + 51.7908i 0.290378 + 2.39148i
\(470\) 0 0
\(471\) 9.04692 + 1.47027i 0.416860 + 0.0677464i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 12.8585 + 36.0803i 0.589991 + 1.65548i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.647915 0.761712i \(-0.275641\pi\)
−0.647915 + 0.761712i \(0.724359\pi\)
\(480\) 0 0
\(481\) 10.0914 + 20.2519i 0.460128 + 0.923409i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 12.4124 22.5351i 0.562460 1.02116i −0.430486 0.902597i \(-0.641658\pi\)
0.992946 0.118565i \(-0.0378293\pi\)
\(488\) 0 0
\(489\) 22.8737 37.8378i 1.03439 1.71108i
\(490\) 0 0
\(491\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.848743 5.98085i −0.0381097 0.268548i
\(497\) 0 0
\(498\) 0 0
\(499\) 16.1232 35.8243i 0.721774 1.60372i −0.0733768 0.997304i \(-0.523378\pi\)
0.795151 0.606412i \(-0.207392\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.692724 0.721202i \(-0.743590\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.94649 + 21.0678i 0.352916 + 0.935655i
\(508\) 22.6989 1.00710
\(509\) 0 0 −0.373361 0.927686i \(-0.621795\pi\)
0.373361 + 0.927686i \(0.378205\pi\)
\(510\) 0 0
\(511\) 12.8811 12.3725i 0.569826 0.547325i
\(512\) 0 0
\(513\) −25.7908 30.3206i −1.13869 1.33869i
\(514\) 0 0
\(515\) 0 0
\(516\) −35.3630 + 26.5735i −1.55677 + 1.16984i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(522\) 0 0
\(523\) −1.76126 + 43.7052i −0.0770145 + 1.91110i 0.253567 + 0.967318i \(0.418396\pi\)
−0.330581 + 0.943778i \(0.607245\pi\)
\(524\) 0 0
\(525\) 38.8874 + 23.5082i 1.69718 + 1.02598i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 66.1609 45.6676i 2.86844 1.97994i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.65262 + 8.51255i −0.114045 + 0.365983i −0.994042 0.108996i \(-0.965237\pi\)
0.879997 + 0.474979i \(0.157544\pi\)
\(542\) 0 0
\(543\) 4.85163 29.8533i 0.208203 1.28113i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.44792 + 44.8677i −0.232936 + 1.91840i 0.139273 + 0.990254i \(0.455523\pi\)
−0.372209 + 0.928149i \(0.621400\pi\)
\(548\) 0 0
\(549\) −23.9399 + 24.9241i −1.02173 + 1.06374i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.335154 + 2.36173i −0.0142522 + 0.100431i
\(554\) 0 0
\(555\) 0 0
\(556\) −44.1049 + 9.00408i −1.87046 + 0.381858i
\(557\) 0 0 −0.834256 0.551377i \(-0.814103\pi\)
0.834256 + 0.551377i \(0.185897\pi\)
\(558\) 0 0
\(559\) 16.2874 + 43.0636i 0.688884 + 1.82139i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.316668 0.948536i \(-0.397436\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −46.4499 8.51227i −1.95071 0.357482i
\(568\) 0 0
\(569\) 0 0 −0.534466 0.845190i \(-0.679487\pi\)
0.534466 + 0.845190i \(0.320513\pi\)
\(570\) 0 0
\(571\) 11.8609 + 13.3882i 0.496363 + 0.560279i 0.942293 0.334790i \(-0.108665\pi\)
−0.445929 + 0.895068i \(0.647127\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 23.5150 + 4.80062i 0.979791 + 0.200026i
\(577\) −30.2280 30.2280i −1.25841 1.25841i −0.951853 0.306554i \(-0.900824\pi\)
−0.306554 0.951853i \(-0.599176\pi\)
\(578\) 0 0
\(579\) 39.9698 9.00195i 1.66109 0.374108i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(588\) 19.7878 68.3153i 0.816033 2.81727i
\(589\) 0.930920 11.5315i 0.0383579 0.475148i
\(590\) 0 0
\(591\) 0 0
\(592\) 9.37223 23.2871i 0.385196 0.957092i
\(593\) 0 0 0.0603785 0.998176i \(-0.480769\pi\)
−0.0603785 + 0.998176i \(0.519231\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.41588 + 34.1446i −0.344439 + 1.39744i
\(598\) 0 0
\(599\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(600\) 0 0
\(601\) −7.65049 5.74897i −0.312070 0.234505i 0.433154 0.901320i \(-0.357401\pi\)
−0.745223 + 0.666815i \(0.767657\pi\)
\(602\) 0 0
\(603\) −23.4808 + 18.3960i −0.956211 + 0.749144i
\(604\) 3.64966 + 16.2050i 0.148503 + 0.659371i
\(605\) 0 0
\(606\) 0 0
\(607\) 7.00386 34.3072i 0.284278 1.39248i −0.547235 0.836979i \(-0.684320\pi\)
0.831513 0.555506i \(-0.187475\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −34.7265 + 32.0360i −1.40259 + 1.29392i −0.498300 + 0.867005i \(0.666042\pi\)
−0.904290 + 0.426919i \(0.859599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.678061 0.735006i \(-0.262821\pi\)
−0.678061 + 0.735006i \(0.737179\pi\)
\(618\) 0 0
\(619\) −34.6321 + 2.09486i −1.39198 + 0.0841994i −0.739271 0.673408i \(-0.764830\pi\)
−0.652711 + 0.757607i \(0.726368\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 11.6287 22.1082i 0.465522 0.885036i
\(625\) 8.86512 23.3754i 0.354605 0.935016i
\(626\) 0 0
\(627\) 0 0
\(628\) −10.1657 + 2.94453i −0.405655 + 0.117499i
\(629\) 0 0
\(630\) 0 0
\(631\) −26.5337 + 36.8316i −1.05629 + 1.46624i −0.180484 + 0.983578i \(0.557766\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 3.01459 6.35312i 0.119819 0.252514i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −62.5316 39.6217i −2.47759 1.56987i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 −0.391967 0.919979i \(-0.628205\pi\)
0.391967 + 0.919979i \(0.371795\pi\)
\(642\) 0 0
\(643\) −0.855787 1.71356i −0.0337490 0.0675763i 0.877122 0.480268i \(-0.159461\pi\)
−0.910871 + 0.412692i \(0.864589\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −7.79661 11.2953i −0.305573 0.442699i
\(652\) −5.13208 + 50.7957i −0.200988 + 1.98931i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.86382 + 2.64300i 0.384824 + 0.103113i
\(658\) 0 0
\(659\) 0 0 0.632445 0.774605i \(-0.282051\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(660\) 0 0
\(661\) −39.0968 + 28.1656i −1.52069 + 1.09551i −0.556315 + 0.830971i \(0.687785\pi\)
−0.964374 + 0.264543i \(0.914779\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −46.7826 + 4.72662i −1.80872 + 0.182742i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −6.00393 17.9840i −0.231435 0.693231i −0.998655 0.0518477i \(-0.983489\pi\)
0.767220 0.641384i \(-0.221639\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) −18.7187 18.0447i −0.719950 0.694026i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 49.4549 16.5105i 1.89791 0.633614i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.100522 0.994935i \(-0.532051\pi\)
0.100522 + 0.994935i \(0.467949\pi\)
\(684\) 41.9144 + 18.8641i 1.60264 + 0.721288i
\(685\) 0 0
\(686\) 0 0
\(687\) 25.4150 3.60665i 0.969643 0.137602i
\(688\) 23.7370 45.2270i 0.904963 1.72426i
\(689\) 0 0
\(690\) 0 0
\(691\) −30.4607 42.2826i −1.15878 1.60851i −0.686681 0.726959i \(-0.740933\pi\)
−0.472098 0.881546i \(-0.656503\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −52.2047 5.27444i −1.97315 0.199355i
\(701\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(702\) 0 0
\(703\) 27.3096 39.5649i 1.03000 1.49222i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.97439 + 1.98489i −0.149262 + 0.0745442i −0.519872 0.854244i \(-0.674020\pi\)
0.370610 + 0.928788i \(0.379149\pi\)
\(710\) 0 0
\(711\) −1.25471 + 0.534583i −0.0470554 + 0.0200484i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.903450 0.428693i \(-0.858974\pi\)
0.903450 + 0.428693i \(0.141026\pi\)
\(720\) 0 0
\(721\) 84.4839 + 60.8628i 3.14635 + 2.26665i
\(722\) 0 0
\(723\) 45.8889 27.7408i 1.70663 1.03169i
\(724\) 9.71642 + 33.5450i 0.361108 + 1.24669i
\(725\) 0 0
\(726\) 0 0
\(727\) −3.43881 1.30417i −0.127538 0.0483689i 0.290010 0.957024i \(-0.406341\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) −9.57433 25.2454i −0.354605 0.935016i
\(730\) 0 0
\(731\) 0 0
\(732\) 12.6368 37.8518i 0.467069 1.39904i
\(733\) −1.71595 28.3680i −0.0633799 1.04780i −0.881329 0.472504i \(-0.843350\pi\)
0.817949 0.575291i \(-0.195111\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 24.5392 + 26.6001i 0.902690 + 0.978499i 0.999859 0.0167685i \(-0.00533782\pi\)
−0.0971693 + 0.995268i \(0.530979\pi\)
\(740\) 0 0
\(741\) 31.0296 36.4128i 1.13990 1.33766i
\(742\) 0 0
\(743\) 0 0 0.999797 0.0201371i \(-0.00641026\pi\)
−0.999797 + 0.0201371i \(0.993590\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23.7306 + 31.5797i −0.865942 + 1.15236i 0.121218 + 0.992626i \(0.461320\pi\)
−0.987160 + 0.159734i \(0.948936\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 52.6709 14.1131i 1.91562 0.513290i
\(757\) −11.5482 + 39.8690i −0.419727 + 1.44906i 0.420730 + 0.907186i \(0.361774\pi\)
−0.840457 + 0.541879i \(0.817713\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.373361 0.927686i \(-0.378205\pi\)
−0.373361 + 0.927686i \(0.621795\pi\)
\(762\) 0 0
\(763\) 108.346 + 8.74661i 3.92240 + 0.316649i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −26.9075 + 6.63211i −0.970942 + 0.239316i
\(769\) 47.4373 26.1287i 1.71063 0.942224i 0.751295 0.659967i \(-0.229430\pi\)
0.959339 0.282257i \(-0.0910831\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −37.2410 + 29.1765i −1.34033 + 1.05008i
\(773\) 0 0 −0.219715 0.975564i \(-0.570513\pi\)
0.219715 + 0.975564i \(0.429487\pi\)
\(774\) 0 0
\(775\) −5.33934 + 5.33934i −0.191795 + 0.191795i
\(776\) 0 0
\(777\) −4.58928 56.8485i −0.164640 2.03943i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 13.1740 + 81.0627i 0.470499 + 2.89509i
\(785\) 0 0
\(786\) 0 0
\(787\) −8.92279 0.179715i −0.318063 0.00640615i −0.139151 0.990271i \(-0.544437\pi\)
−0.178912 + 0.983865i \(0.557258\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −34.1612 23.6254i −1.21310 0.838961i
\(794\) 0 0
\(795\) 0 0
\(796\) −8.12239 39.7861i −0.287890 1.41018i
\(797\) 0 0 0.960518 0.278217i \(-0.0897436\pi\)
−0.960518 + 0.278217i \(0.910256\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 15.3893 30.8143i 0.542737 1.08674i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.987050 0.160411i \(-0.948718\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(810\) 0 0
\(811\) −17.3352 5.40187i −0.608721 0.189685i −0.0223803 0.999750i \(-0.507124\pi\)
−0.586341 + 0.810064i \(0.699432\pi\)
\(812\) 0 0
\(813\) 16.4803 + 46.2428i 0.577990 + 1.62181i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 63.3802 74.5120i 2.21739 2.60685i
\(818\) 0 0
\(819\) 1.19412 56.7430i 0.0417260 1.98276i
\(820\) 0 0
\(821\) 0 0 0.100522 0.994935i \(-0.467949\pi\)
−0.100522 + 0.994935i \(0.532051\pi\)
\(822\) 0 0
\(823\) 42.1737 24.3490i 1.47008 0.848753i 0.470647 0.882322i \(-0.344021\pi\)
0.999436 + 0.0335690i \(0.0106873\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.517338 0.855781i \(-0.326923\pi\)
−0.517338 + 0.855781i \(0.673077\pi\)
\(828\) 0 0
\(829\) 57.3466 + 2.31099i 1.99173 + 0.0802640i 0.999147 0.0412960i \(-0.0131487\pi\)
0.992584 + 0.121560i \(0.0387897\pi\)
\(830\) 0 0
\(831\) −26.4005 50.3019i −0.915822 1.74495i
\(832\) −1.18747 + 28.8200i −0.0411681 + 0.999152i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.22059 7.15585i 0.111320 0.247342i
\(838\) 0 0
\(839\) 0 0 0.761712 0.647915i \(-0.224359\pi\)
−0.761712 + 0.647915i \(0.775641\pi\)
\(840\) 0 0
\(841\) 20.0890 + 20.9149i 0.692724 + 0.721202i
\(842\) 0 0
\(843\) 0 0
\(844\) 8.11993i 0.279500i
\(845\) 0 0
\(846\) 0 0
\(847\) −21.5495 53.5438i −0.740450 1.83979i
\(848\) 0 0
\(849\) −14.5807 + 14.0049i −0.500407 + 0.480648i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −43.1585 19.4240i −1.47772 0.665067i −0.498317 0.866995i \(-0.666048\pi\)
−0.979400 + 0.201928i \(0.935279\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(858\) 0 0
\(859\) 30.2494 15.8761i 1.03210 0.541686i 0.138409 0.990375i \(-0.455801\pi\)
0.893688 + 0.448689i \(0.148109\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.855781 0.517338i \(-0.826923\pi\)
0.855781 + 0.517338i \(0.173077\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.7224 + 25.5000i 0.500000 + 0.866025i
\(868\) 13.7249 + 7.92405i 0.465852 + 0.268960i
\(869\) 0 0
\(870\) 0 0
\(871\) −26.3279 24.3321i −0.892086 0.824463i
\(872\) 0 0
\(873\) 22.7066 + 19.3143i 0.768503 + 0.653692i
\(874\) 0 0
\(875\) 0 0
\(876\) −11.5984 + 2.12549i −0.391874 + 0.0718135i
\(877\) −54.8026 + 19.5309i −1.85055 + 0.659511i −0.862357 + 0.506300i \(0.831013\pi\)
−0.988193 + 0.153211i \(0.951038\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.160411 0.987050i \(-0.448718\pi\)
−0.160411 + 0.987050i \(0.551282\pi\)
\(882\) 0 0
\(883\) −40.2610 + 4.88857i −1.35489 + 0.164514i −0.765553 0.643373i \(-0.777534\pi\)
−0.589338 + 0.807887i \(0.700611\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.692724 0.721202i \(-0.256410\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(888\) 0 0
\(889\) −36.7264 + 46.8778i −1.23176 + 1.57223i
\(890\) 0 0
\(891\) 0 0
\(892\) 46.4646 28.0888i 1.55575 0.940483i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −12.8608 27.1035i −0.428693 0.903450i
\(901\) 0 0
\(902\) 0 0
\(903\) 2.33691 116.027i 0.0777676 3.86113i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 31.9527 + 50.5291i 1.06097 + 1.67779i 0.633446 + 0.773787i \(0.281640\pi\)
0.427525 + 0.904004i \(0.359386\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(912\) −53.0637 + 1.06876i −1.75711 + 0.0353903i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −24.7280 + 16.3432i −0.817035 + 0.539996i
\(917\) 0 0
\(918\) 0 0
\(919\) −2.43632 60.4566i −0.0803667 1.99428i −0.104147 0.994562i \(-0.533211\pi\)
0.0237804 0.999717i \(-0.492430\pi\)
\(920\) 0 0
\(921\) −10.6837 19.3966i −0.352040 0.639138i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −30.3087 + 8.12119i −0.996544 + 0.267023i
\(926\) 0 0
\(927\) −4.79041 + 59.3398i −0.157338 + 1.94898i
\(928\) 0 0
\(929\) 0 0 −0.927686 0.373361i \(-0.878205\pi\)
0.927686 + 0.373361i \(0.121795\pi\)
\(930\) 0 0
\(931\) −9.49661 + 156.998i −0.311239 + 5.14539i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −31.5471 + 7.77567i −1.03060 + 0.254020i −0.718131 0.695908i \(-0.755002\pi\)
−0.312469 + 0.949928i \(0.601156\pi\)
\(938\) 0 0
\(939\) −46.0450 34.6006i −1.50262 1.12915i
\(940\) 0 0
\(941\) 0 0 0.787183 0.616719i \(-0.211538\pi\)
−0.787183 + 0.616719i \(0.788462\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.0201371 0.999797i \(-0.506410\pi\)
0.0201371 + 0.999797i \(0.493590\pi\)
\(948\) 1.04431 1.17878i 0.0339176 0.0382850i
\(949\) −1.24473 + 12.2097i −0.0404058 + 0.396345i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −26.8530 + 10.1840i −0.866227 + 0.328516i
\(962\) 0 0
\(963\) 0 0
\(964\) −34.1400 + 51.6552i −1.09957 + 1.66370i
\(965\) 0 0
\(966\) 0 0
\(967\) −24.6157 40.7194i −0.791588 1.30945i −0.946883 0.321578i \(-0.895787\pi\)
0.155295 0.987868i \(-0.450367\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.428693 0.903450i \(-0.358974\pi\)
−0.428693 + 0.903450i \(0.641026\pi\)
\(972\) 22.4849 + 21.5970i 0.721202 + 0.692724i
\(973\) 52.7655 105.654i 1.69159 3.38710i
\(974\) 0 0
\(975\) −30.8251 + 4.98101i −0.987195 + 0.159520i
\(976\) 5.55419 + 45.7429i 0.177785 + 1.46419i
\(977\) 0 0 0.335705 0.941967i \(-0.391026\pi\)
−0.335705 + 0.941967i \(0.608974\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 27.7678 + 55.6002i 0.886559 + 1.77518i
\(982\) 0 0
\(983\) 0 0 −0.180255 0.983620i \(-0.557692\pi\)
0.180255 + 0.983620i \(0.442308\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −14.3458 + 53.3464i −0.456399 + 1.69717i
\(989\) 0 0
\(990\) 0 0
\(991\) −25.0043 + 43.3087i −0.794288 + 1.37575i 0.129003 + 0.991644i \(0.458822\pi\)
−0.923291 + 0.384102i \(0.874511\pi\)
\(992\) 0 0
\(993\) 38.4256 17.2940i 1.21940 0.548808i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 32.9167 40.3156i 1.04248 1.27681i 0.0825460 0.996587i \(-0.473695\pi\)
0.959936 0.280221i \(-0.0904077\pi\)
\(998\) 0 0
\(999\) 26.4581 19.0606i 0.837097 0.603050i
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 507.2.x.a.98.1 48
3.2 odd 2 CM 507.2.x.a.98.1 48
169.119 odd 156 inner 507.2.x.a.119.1 yes 48
507.119 even 156 inner 507.2.x.a.119.1 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.x.a.98.1 48 1.1 even 1 trivial
507.2.x.a.98.1 48 3.2 odd 2 CM
507.2.x.a.119.1 yes 48 169.119 odd 156 inner
507.2.x.a.119.1 yes 48 507.119 even 156 inner