Properties

Label 1960.2.q.i.361.1
Level $1960$
Weight $2$
Character 1960.361
Analytic conductor $15.651$
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] = [1960,2,Mod(361,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1960.q (of order \(3\), 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: \(15.6506787962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1960.361
Dual form 1960.2.q.i.961.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +(-2.00000 + 3.46410i) q^{11} +2.00000 q^{13} +(1.00000 - 1.73205i) q^{17} +(2.00000 + 3.46410i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-3.00000 - 5.19615i) q^{37} +6.00000 q^{41} -8.00000 q^{43} +(-1.50000 + 2.59808i) q^{45} +(2.00000 + 3.46410i) q^{47} +(-3.00000 + 5.19615i) q^{53} -4.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-4.00000 + 6.92820i) q^{67} +(-3.00000 + 5.19615i) q^{73} +(-4.50000 + 7.79423i) q^{81} +16.0000 q^{83} +2.00000 q^{85} +(-3.00000 - 5.19615i) q^{89} +(-2.00000 + 3.46410i) q^{95} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + 3 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{17} + 4 q^{19} - 4 q^{23} - q^{25} - 4 q^{29} - 8 q^{31} - 6 q^{37} + 12 q^{41} - 16 q^{43} - 3 q^{45} + 4 q^{47} - 6 q^{53} - 8 q^{55} - 4 q^{59} - 2 q^{61} + 2 q^{65} - 8 q^{67} - 6 q^{73} - 9 q^{81} + 32 q^{83} + 4 q^{85} - 6 q^{89} - 4 q^{95} + 28 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times\).

\(n\) \(981\) \(1081\) \(1177\) \(1471\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 1.73205i 0.242536 0.420084i −0.718900 0.695113i \(-0.755354\pi\)
0.961436 + 0.275029i \(0.0886875\pi\)
\(18\) 0 0
\(19\) 2.00000 + 3.46410i 0.458831 + 0.794719i 0.998899 0.0469020i \(-0.0149348\pi\)
−0.540068 + 0.841621i \(0.681602\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.00000 3.46410i −0.417029 0.722315i 0.578610 0.815604i \(-0.303595\pi\)
−0.995639 + 0.0932891i \(0.970262\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.00000 5.19615i −0.493197 0.854242i 0.506772 0.862080i \(-0.330838\pi\)
−0.999969 + 0.00783774i \(0.997505\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −1.50000 + 2.59808i −0.223607 + 0.387298i
\(46\) 0 0
\(47\) 2.00000 + 3.46410i 0.291730 + 0.505291i 0.974219 0.225605i \(-0.0724358\pi\)
−0.682489 + 0.730896i \(0.739102\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 + 5.19615i −0.412082 + 0.713746i −0.995117 0.0987002i \(-0.968532\pi\)
0.583036 + 0.812447i \(0.301865\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) −1.00000 1.73205i −0.128037 0.221766i 0.794879 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 + 1.73205i 0.124035 + 0.214834i
\(66\) 0 0
\(67\) −4.00000 + 6.92820i −0.488678 + 0.846415i −0.999915 0.0130248i \(-0.995854\pi\)
0.511237 + 0.859440i \(0.329187\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −3.00000 + 5.19615i −0.351123 + 0.608164i −0.986447 0.164083i \(-0.947534\pi\)
0.635323 + 0.772246i \(0.280867\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 5.19615i −0.317999 0.550791i 0.662071 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 + 3.46410i −0.205196 + 0.355409i
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −12.0000 −1.20605
\(100\) 0 0
\(101\) 3.00000 5.19615i 0.298511 0.517036i −0.677284 0.735721i \(-0.736843\pi\)
0.975796 + 0.218685i \(0.0701767\pi\)
\(102\) 0 0
\(103\) 2.00000 + 3.46410i 0.197066 + 0.341328i 0.947576 0.319531i \(-0.103525\pi\)
−0.750510 + 0.660859i \(0.770192\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(108\) 0 0
\(109\) −7.00000 + 12.1244i −0.670478 + 1.16130i 0.307290 + 0.951616i \(0.400578\pi\)
−0.977769 + 0.209687i \(0.932756\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 2.00000 3.46410i 0.186501 0.323029i
\(116\) 0 0
\(117\) 3.00000 + 5.19615i 0.277350 + 0.480384i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.50000 4.33013i −0.227273 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 + 10.3923i 0.524222 + 0.907980i 0.999602 + 0.0281993i \(0.00897729\pi\)
−0.475380 + 0.879781i \(0.657689\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.00000 + 8.66025i −0.427179 + 0.739895i −0.996621 0.0821359i \(-0.973826\pi\)
0.569442 + 0.822031i \(0.307159\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 + 6.92820i −0.334497 + 0.579365i
\(144\) 0 0
\(145\) −1.00000 1.73205i −0.0830455 0.143839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) 8.00000 13.8564i 0.651031 1.12762i −0.331842 0.943335i \(-0.607670\pi\)
0.982873 0.184284i \(-0.0589965\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −1.00000 + 1.73205i −0.0798087 + 0.138233i −0.903167 0.429289i \(-0.858764\pi\)
0.823359 + 0.567521i \(0.192098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.00000 13.8564i −0.626608 1.08532i −0.988227 0.152992i \(-0.951109\pi\)
0.361619 0.932326i \(-0.382224\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.00000 + 10.3923i −0.458831 + 0.794719i
\(172\) 0 0
\(173\) 7.00000 + 12.1244i 0.532200 + 0.921798i 0.999293 + 0.0375896i \(0.0119679\pi\)
−0.467093 + 0.884208i \(0.654699\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 + 17.3205i −0.747435 + 1.29460i 0.201613 + 0.979465i \(0.435382\pi\)
−0.949048 + 0.315130i \(0.897952\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) 4.00000 + 6.92820i 0.292509 + 0.506640i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.00000 6.92820i −0.289430 0.501307i 0.684244 0.729253i \(-0.260132\pi\)
−0.973674 + 0.227946i \(0.926799\pi\)
\(192\) 0 0
\(193\) 7.00000 12.1244i 0.503871 0.872730i −0.496119 0.868255i \(-0.665242\pi\)
0.999990 0.00447566i \(-0.00142465\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 4.00000 6.92820i 0.283552 0.491127i −0.688705 0.725042i \(-0.741820\pi\)
0.972257 + 0.233915i \(0.0751537\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 6.00000 10.3923i 0.417029 0.722315i
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.00000 6.92820i −0.272798 0.472500i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 3.46410i 0.134535 0.233021i
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −12.0000 + 20.7846i −0.796468 + 1.37952i 0.125435 + 0.992102i \(0.459967\pi\)
−0.921903 + 0.387421i \(0.873366\pi\)
\(228\) 0 0
\(229\) −13.0000 22.5167i −0.859064 1.48794i −0.872823 0.488037i \(-0.837713\pi\)
0.0137585 0.999905i \(-0.495620\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 + 5.19615i 0.196537 + 0.340411i 0.947403 0.320043i \(-0.103697\pi\)
−0.750867 + 0.660454i \(0.770364\pi\)
\(234\) 0 0
\(235\) −2.00000 + 3.46410i −0.130466 + 0.225973i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 1.00000 1.73205i 0.0644157 0.111571i −0.832019 0.554747i \(-0.812815\pi\)
0.896435 + 0.443176i \(0.146148\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000 + 6.92820i 0.254514 + 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.0000 25.9808i −0.935674 1.62064i −0.773427 0.633885i \(-0.781459\pi\)
−0.162247 0.986750i \(-0.551874\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.00000 5.19615i −0.185695 0.321634i
\(262\) 0 0
\(263\) 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i \(-0.712699\pi\)
0.989561 + 0.144112i \(0.0460326\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.00000 12.1244i 0.426798 0.739235i −0.569789 0.821791i \(-0.692975\pi\)
0.996586 + 0.0825561i \(0.0263084\pi\)
\(270\) 0 0
\(271\) 12.0000 + 20.7846i 0.728948 + 1.26258i 0.957328 + 0.289003i \(0.0933238\pi\)
−0.228380 + 0.973572i \(0.573343\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 3.46410i −0.120605 0.208893i
\(276\) 0 0
\(277\) 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i \(-0.736206\pi\)
0.976231 + 0.216731i \(0.0695395\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 4.00000 6.92820i 0.237775 0.411839i −0.722300 0.691580i \(-0.756915\pi\)
0.960076 + 0.279741i \(0.0902485\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26.0000 1.51894 0.759468 0.650545i \(-0.225459\pi\)
0.759468 + 0.650545i \(0.225459\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 6.92820i −0.231326 0.400668i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.00000 1.73205i 0.0572598 0.0991769i
\(306\) 0 0
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 27.7128i 0.907277 1.57145i 0.0894452 0.995992i \(-0.471491\pi\)
0.817832 0.575458i \(-0.195176\pi\)
\(312\) 0 0
\(313\) 13.0000 + 22.5167i 0.734803 + 1.27272i 0.954810 + 0.297218i \(0.0960589\pi\)
−0.220006 + 0.975499i \(0.570608\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 + 15.5885i 0.505490 + 0.875535i 0.999980 + 0.00635137i \(0.00202172\pi\)
−0.494489 + 0.869184i \(0.664645\pi\)
\(318\) 0 0
\(319\) 4.00000 6.92820i 0.223957 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) −1.00000 + 1.73205i −0.0554700 + 0.0960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.00000 + 10.3923i 0.329790 + 0.571213i 0.982470 0.186421i \(-0.0596888\pi\)
−0.652680 + 0.757634i \(0.726355\pi\)
\(332\) 0 0
\(333\) 9.00000 15.5885i 0.493197 0.854242i
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 27.7128i −0.866449 1.50073i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.00000 13.8564i 0.429463 0.743851i −0.567363 0.823468i \(-0.692036\pi\)
0.996826 + 0.0796169i \(0.0253697\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.00000 1.73205i 0.0532246 0.0921878i −0.838186 0.545385i \(-0.816383\pi\)
0.891410 + 0.453197i \(0.149717\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.0000 + 20.7846i 0.633336 + 1.09697i 0.986865 + 0.161546i \(0.0516481\pi\)
−0.353529 + 0.935423i \(0.615019\pi\)
\(360\) 0 0
\(361\) 1.50000 2.59808i 0.0789474 0.136741i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 10.0000 17.3205i 0.521996 0.904123i −0.477677 0.878536i \(-0.658521\pi\)
0.999673 0.0255875i \(-0.00814566\pi\)
\(368\) 0 0
\(369\) 9.00000 + 15.5885i 0.468521 + 0.811503i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.0000 19.0526i −0.569558 0.986504i −0.996610 0.0822766i \(-0.973781\pi\)
0.427051 0.904227i \(-0.359552\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −18.0000 31.1769i −0.919757 1.59307i −0.799783 0.600289i \(-0.795052\pi\)
−0.119974 0.992777i \(-0.538281\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.0000 20.7846i −0.609994 1.05654i
\(388\) 0 0
\(389\) −3.00000 + 5.19615i −0.152106 + 0.263455i −0.932002 0.362454i \(-0.881939\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.00000 1.73205i −0.0501886 0.0869291i 0.839840 0.542834i \(-0.182649\pi\)
−0.890028 + 0.455905i \(0.849316\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.00000 15.5885i −0.449439 0.778450i 0.548911 0.835881i \(-0.315043\pi\)
−0.998350 + 0.0574304i \(0.981709\pi\)
\(402\) 0 0
\(403\) −8.00000 + 13.8564i −0.398508 + 0.690237i
\(404\) 0 0
\(405\) −9.00000 −0.447214
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 5.00000 8.66025i 0.247234 0.428222i −0.715523 0.698589i \(-0.753812\pi\)
0.962757 + 0.270367i \(0.0871450\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 8.00000 + 13.8564i 0.392705 + 0.680184i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) −6.00000 + 10.3923i −0.291730 + 0.505291i
\(424\) 0 0
\(425\) 1.00000 + 1.73205i 0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000 34.6410i 0.963366 1.66860i 0.249424 0.968394i \(-0.419759\pi\)
0.713942 0.700205i \(-0.246908\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.00000 13.8564i 0.382692 0.662842i
\(438\) 0 0
\(439\) −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i \(-0.227810\pi\)
−0.945552 + 0.325471i \(0.894477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 20.7846i −0.570137 0.987507i −0.996551 0.0829786i \(-0.973557\pi\)
0.426414 0.904528i \(-0.359777\pi\)
\(444\) 0 0
\(445\) 3.00000 5.19615i 0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −12.0000 + 20.7846i −0.565058 + 0.978709i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.00000 8.66025i −0.233890 0.405110i 0.725059 0.688686i \(-0.241812\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.00000 + 6.92820i 0.185098 + 0.320599i 0.943610 0.331061i \(-0.107406\pi\)
−0.758512 + 0.651660i \(0.774073\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.0000 27.7128i 0.735681 1.27424i
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) 0 0
\(479\) 8.00000 13.8564i 0.365529 0.633115i −0.623332 0.781958i \(-0.714221\pi\)
0.988861 + 0.148842i \(0.0475547\pi\)
\(480\) 0 0
\(481\) −6.00000 10.3923i −0.273576 0.473848i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.00000 + 12.1244i 0.317854 + 0.550539i
\(486\) 0 0
\(487\) 10.0000 17.3205i 0.453143 0.784867i −0.545436 0.838152i \(-0.683636\pi\)
0.998579 + 0.0532853i \(0.0169693\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) −2.00000 + 3.46410i −0.0900755 + 0.156015i
\(494\) 0 0
\(495\) −6.00000 10.3923i −0.269680 0.467099i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 + 24.2487i 0.626726 + 1.08552i 0.988204 + 0.153141i \(0.0489388\pi\)
−0.361478 + 0.932381i \(0.617728\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.00000 1.73205i −0.0443242 0.0767718i 0.843012 0.537895i \(-0.180780\pi\)
−0.887336 + 0.461123i \(0.847447\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.00000 + 3.46410i −0.0881305 + 0.152647i
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.00000 8.66025i 0.219054 0.379413i −0.735465 0.677563i \(-0.763036\pi\)
0.954519 + 0.298150i \(0.0963696\pi\)
\(522\) 0 0
\(523\) −4.00000 6.92820i −0.174908 0.302949i 0.765222 0.643767i \(-0.222629\pi\)
−0.940129 + 0.340818i \(0.889296\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.00000 + 13.8564i 0.348485 + 0.603595i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.00000 + 1.73205i 0.0429934 + 0.0744667i 0.886721 0.462304i \(-0.152977\pi\)
−0.843728 + 0.536771i \(0.819644\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 3.00000 5.19615i 0.128037 0.221766i
\(550\) 0 0
\(551\) −4.00000 6.92820i −0.170406 0.295151i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.00000 + 12.1244i −0.296600 + 0.513725i −0.975356 0.220638i \(-0.929186\pi\)
0.678756 + 0.734364i \(0.262519\pi\)
\(558\) 0 0
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000 13.8564i 0.337160 0.583978i −0.646737 0.762713i \(-0.723867\pi\)
0.983897 + 0.178735i \(0.0572004\pi\)
\(564\) 0 0
\(565\) 9.00000 + 15.5885i 0.378633 + 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11.0000 + 19.0526i 0.461144 + 0.798725i 0.999018 0.0443003i \(-0.0141058\pi\)
−0.537874 + 0.843025i \(0.680772\pi\)
\(570\) 0 0
\(571\) −2.00000 + 3.46410i −0.0836974 + 0.144968i −0.904835 0.425762i \(-0.860006\pi\)
0.821138 + 0.570730i \(0.193340\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 1.00000 1.73205i 0.0416305 0.0721062i −0.844459 0.535620i \(-0.820078\pi\)
0.886090 + 0.463513i \(0.153411\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −12.0000 20.7846i −0.496989 0.860811i
\(584\) 0 0
\(585\) −3.00000 + 5.19615i −0.124035 + 0.214834i
\(586\) 0 0
\(587\) 48.0000 1.98117 0.990586 0.136892i \(-0.0437113\pi\)
0.990586 + 0.136892i \(0.0437113\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9.00000 + 15.5885i 0.369586 + 0.640141i 0.989501 0.144528i \(-0.0461663\pi\)
−0.619915 + 0.784669i \(0.712833\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 20.7846i 0.490307 0.849236i −0.509631 0.860393i \(-0.670218\pi\)
0.999938 + 0.0111569i \(0.00355143\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) −24.0000 −0.977356
\(604\) 0 0
\(605\) 2.50000 4.33013i 0.101639 0.176045i
\(606\) 0 0
\(607\) −6.00000 10.3923i −0.243532 0.421811i 0.718186 0.695852i \(-0.244973\pi\)
−0.961718 + 0.274041i \(0.911640\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 + 6.92820i 0.161823 + 0.280285i
\(612\) 0 0
\(613\) 21.0000 36.3731i 0.848182 1.46909i −0.0346469 0.999400i \(-0.511031\pi\)
0.882829 0.469695i \(-0.155636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) −2.00000 + 3.46410i −0.0803868 + 0.139234i −0.903416 0.428765i \(-0.858949\pi\)
0.823029 + 0.567999i \(0.192282\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.00000 10.3923i −0.238103 0.412406i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.00000 + 15.5885i −0.355479 + 0.615707i −0.987200 0.159489i \(-0.949015\pi\)
0.631721 + 0.775196i \(0.282349\pi\)
\(642\) 0 0
\(643\) 48.0000 1.89294 0.946468 0.322799i \(-0.104624\pi\)
0.946468 + 0.322799i \(0.104624\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000 10.3923i 0.235884 0.408564i −0.723645 0.690172i \(-0.757535\pi\)
0.959529 + 0.281609i \(0.0908680\pi\)
\(648\) 0 0
\(649\) −8.00000 13.8564i −0.314027 0.543912i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.0000 + 29.4449i 0.665261 + 1.15227i 0.979214 + 0.202828i \(0.0650132\pi\)
−0.313953 + 0.949439i \(0.601653\pi\)
\(654\) 0 0
\(655\) −6.00000 + 10.3923i −0.234439 + 0.406061i
\(656\) 0 0
\(657\) −18.0000 −0.702247
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −21.0000 + 36.3731i −0.816805 + 1.41475i 0.0912190 + 0.995831i \(0.470924\pi\)
−0.908024 + 0.418917i \(0.862410\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 + 6.92820i 0.154881 + 0.268261i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.0000 + 19.0526i 0.422764 + 0.732249i 0.996209 0.0869952i \(-0.0277265\pi\)
−0.573444 + 0.819244i \(0.694393\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 + 20.7846i −0.459167 + 0.795301i −0.998917 0.0465244i \(-0.985185\pi\)
0.539750 + 0.841825i \(0.318519\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.00000 + 10.3923i −0.228582 + 0.395915i
\(690\) 0 0
\(691\) 22.0000 + 38.1051i 0.836919 + 1.44959i 0.892458 + 0.451130i \(0.148979\pi\)
−0.0555386 + 0.998457i \(0.517688\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 10.3923i −0.227593 0.394203i
\(696\) 0 0
\(697\) 6.00000 10.3923i 0.227266 0.393637i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 0 0
\(703\) 12.0000 20.7846i 0.452589 0.783906i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19.0000 32.9090i −0.713560 1.23592i −0.963512 0.267664i \(-0.913748\pi\)
0.249952 0.968258i \(-0.419585\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −16.0000 27.7128i −0.596699 1.03351i −0.993305 0.115524i \(-0.963145\pi\)
0.396605 0.917989i \(-0.370188\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.00000 1.73205i 0.0371391 0.0643268i
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −8.00000 + 13.8564i −0.295891 + 0.512498i
\(732\) 0 0
\(733\) 15.0000 + 25.9808i 0.554038 + 0.959621i 0.997978 + 0.0635649i \(0.0202470\pi\)
−0.443940 + 0.896056i \(0.646420\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 27.7128i −0.589368 1.02081i
\(738\) 0 0
\(739\) −18.0000 + 31.1769i −0.662141 + 1.14686i 0.317911 + 0.948120i \(0.397019\pi\)
−0.980052 + 0.198741i \(0.936315\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −5.00000 + 8.66025i −0.183186 + 0.317287i
\(746\) 0 0
\(747\) 24.0000 + 41.5692i 0.878114 + 1.52094i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.0000 + 20.7846i 0.437886 + 0.758441i 0.997526 0.0702946i \(-0.0223939\pi\)
−0.559640 + 0.828736i \(0.689061\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.00000 5.19615i −0.108750 0.188360i 0.806514 0.591215i \(-0.201351\pi\)
−0.915264 + 0.402854i \(0.868018\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.00000 + 5.19615i 0.108465 + 0.187867i
\(766\) 0 0
\(767\) −4.00000 + 6.92820i −0.144432 + 0.250163i
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −21.0000 + 36.3731i −0.755318 + 1.30825i 0.189899 + 0.981804i \(0.439184\pi\)
−0.945216 + 0.326445i \(0.894149\pi\)
\(774\) 0 0
\(775\) −4.00000 6.92820i −0.143684 0.248868i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 + 20.7846i 0.429945 + 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 4.00000 6.92820i 0.142585 0.246964i −0.785885 0.618373i \(-0.787792\pi\)
0.928469 + 0.371409i \(0.121125\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 3.46410i −0.0710221 0.123014i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 9.00000 15.5885i 0.317999 0.550791i
\(802\) 0 0
\(803\) −12.0000 20.7846i −0.423471 0.733473i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −21.0000 + 36.3731i −0.738321 + 1.27881i 0.214930 + 0.976629i \(0.431048\pi\)
−0.953251 + 0.302180i \(0.902286\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 13.8564i 0.280228 0.485369i
\(816\) 0 0
\(817\) −16.0000 27.7128i −0.559769 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.0000 19.0526i −0.383903 0.664939i 0.607714 0.794156i \(-0.292087\pi\)
−0.991616 + 0.129217i \(0.958754\pi\)
\(822\) 0 0
\(823\) −2.00000 + 3.46410i −0.0697156 + 0.120751i −0.898776 0.438408i \(-0.855543\pi\)
0.829060 + 0.559159i \(0.188876\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) −1.00000 + 1.73205i −0.0347314 + 0.0601566i −0.882869 0.469620i \(-0.844391\pi\)
0.848137 + 0.529777i \(0.177724\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −6.00000 10.3923i −0.207639 0.359641i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.50000 7.79423i −0.154805 0.268130i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 + 20.7846i −0.411355 + 0.712487i
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) 0 0
\(857\) −3.00000 + 5.19615i −0.102478 + 0.177497i −0.912705 0.408619i \(-0.866010\pi\)
0.810227 + 0.586116i \(0.199344\pi\)
\(858\) 0 0
\(859\) −2.00000 3.46410i −0.0682391 0.118194i 0.829887 0.557931i \(-0.188405\pi\)
−0.898126 + 0.439738i \(0.855071\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 10.3923i −0.204242 0.353758i 0.745649 0.666339i \(-0.232140\pi\)
−0.949891 + 0.312581i \(0.898806\pi\)
\(864\) 0 0
\(865\) −7.00000 + 12.1244i −0.238007 + 0.412240i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 + 13.8564i −0.271070 + 0.469506i
\(872\) 0 0
\(873\) 21.0000 + 36.3731i 0.710742 + 1.23104i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −23.0000 39.8372i −0.776655 1.34521i −0.933860 0.357640i \(-0.883582\pi\)
0.157205 0.987566i \(-0.449752\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.0000 17.3205i −0.335767 0.581566i 0.647865 0.761755i \(-0.275662\pi\)
−0.983632 + 0.180190i \(0.942329\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −18.0000 31.1769i −0.603023 1.04447i
\(892\) 0 0
\(893\) −8.00000 + 13.8564i −0.267710 + 0.463687i
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 8.00000 13.8564i 0.266815 0.462137i
\(900\) 0 0
\(901\) 6.00000 + 10.3923i 0.199889 + 0.346218i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.00000 + 8.66025i 0.166206 + 0.287877i
\(906\) 0 0
\(907\) −24.0000 + 41.5692i −0.796907 + 1.38028i 0.124714 + 0.992193i \(0.460199\pi\)
−0.921621 + 0.388091i \(0.873135\pi\)
\(908\) 0 0
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) −32.0000 + 55.4256i −1.05905 + 1.83432i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.00000 6.92820i −0.131948 0.228540i 0.792480 0.609898i \(-0.208790\pi\)
−0.924427 + 0.381358i \(0.875456\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 0 0
\(927\) −6.00000 + 10.3923i −0.197066 + 0.341328i
\(928\) 0 0
\(929\) −7.00000 12.1244i −0.229663 0.397787i 0.728046 0.685529i \(-0.240429\pi\)
−0.957708 + 0.287742i \(0.907096\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.00000 + 6.92820i −0.130814 + 0.226576i
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23.0000 39.8372i 0.749779 1.29865i −0.198150 0.980172i \(-0.563493\pi\)
0.947929 0.318483i \(-0.103173\pi\)
\(942\) 0 0
\(943\) −12.0000 20.7846i −0.390774 0.676840i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.00000 + 6.92820i 0.129983 + 0.225136i 0.923670 0.383190i \(-0.125175\pi\)
−0.793687 + 0.608326i \(0.791841\pi\)
\(948\) 0 0
\(949\) −6.00000 + 10.3923i −0.194768 + 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 4.00000 6.92820i 0.129437 0.224191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −16.5000 28.5788i −0.532258 0.921898i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 14.0000 0.450676
\(966\) 0 0
\(967\) −36.0000 −1.15768 −0.578841 0.815440i \(-0.696495\pi\)
−0.578841 + 0.815440i \(0.696495\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.0000 + 17.3205i 0.320915 + 0.555842i 0.980677 0.195633i \(-0.0626762\pi\)
−0.659762 + 0.751475i \(0.729343\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00000 + 1.73205i −0.0319928 + 0.0554132i −0.881579 0.472037i \(-0.843519\pi\)
0.849586 + 0.527451i \(0.176852\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −42.0000 −1.34096
\(982\) 0 0
\(983\) 18.0000 31.1769i 0.574111 0.994389i −0.422027 0.906583i \(-0.638681\pi\)
0.996138 0.0878058i \(-0.0279855\pi\)
\(984\) 0 0
\(985\) 11.0000 + 19.0526i 0.350489 + 0.607065i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.0000 + 27.7128i 0.508770 + 0.881216i
\(990\) 0 0
\(991\) 20.0000 34.6410i 0.635321 1.10041i −0.351126 0.936328i \(-0.614201\pi\)
0.986447 0.164080i \(-0.0524655\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −21.0000 + 36.3731i −0.665077 + 1.15195i 0.314188 + 0.949361i \(0.398268\pi\)
−0.979265 + 0.202586i \(0.935066\pi\)
\(998\) 0 0
\(999\) 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 1960.2.q.i.361.1 2
7.2 even 3 inner 1960.2.q.i.961.1 2
7.3 odd 6 40.2.a.a.1.1 1
7.4 even 3 1960.2.a.g.1.1 1
7.5 odd 6 1960.2.q.h.961.1 2
7.6 odd 2 1960.2.q.h.361.1 2
21.17 even 6 360.2.a.a.1.1 1
28.3 even 6 80.2.a.a.1.1 1
28.11 odd 6 3920.2.a.s.1.1 1
35.3 even 12 200.2.c.b.49.2 2
35.4 even 6 9800.2.a.x.1.1 1
35.17 even 12 200.2.c.b.49.1 2
35.24 odd 6 200.2.a.c.1.1 1
56.3 even 6 320.2.a.d.1.1 1
56.45 odd 6 320.2.a.c.1.1 1
63.31 odd 6 3240.2.q.k.2161.1 2
63.38 even 6 3240.2.q.x.1081.1 2
63.52 odd 6 3240.2.q.k.1081.1 2
63.59 even 6 3240.2.q.x.2161.1 2
77.10 even 6 4840.2.a.f.1.1 1
84.59 odd 6 720.2.a.e.1.1 1
91.38 odd 6 6760.2.a.i.1.1 1
105.17 odd 12 1800.2.f.a.649.1 2
105.38 odd 12 1800.2.f.a.649.2 2
105.59 even 6 1800.2.a.v.1.1 1
112.3 even 12 1280.2.d.a.641.1 2
112.45 odd 12 1280.2.d.j.641.1 2
112.59 even 12 1280.2.d.a.641.2 2
112.101 odd 12 1280.2.d.j.641.2 2
140.3 odd 12 400.2.c.d.49.1 2
140.59 even 6 400.2.a.e.1.1 1
140.87 odd 12 400.2.c.d.49.2 2
168.59 odd 6 2880.2.a.bg.1.1 1
168.101 even 6 2880.2.a.t.1.1 1
280.3 odd 12 1600.2.c.m.449.1 2
280.59 even 6 1600.2.a.k.1.1 1
280.157 even 12 1600.2.c.k.449.1 2
280.213 even 12 1600.2.c.k.449.2 2
280.227 odd 12 1600.2.c.m.449.2 2
280.269 odd 6 1600.2.a.o.1.1 1
308.87 odd 6 9680.2.a.q.1.1 1
420.59 odd 6 3600.2.a.h.1.1 1
420.143 even 12 3600.2.f.t.2449.1 2
420.227 even 12 3600.2.f.t.2449.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.a.a.1.1 1 7.3 odd 6
80.2.a.a.1.1 1 28.3 even 6
200.2.a.c.1.1 1 35.24 odd 6
200.2.c.b.49.1 2 35.17 even 12
200.2.c.b.49.2 2 35.3 even 12
320.2.a.c.1.1 1 56.45 odd 6
320.2.a.d.1.1 1 56.3 even 6
360.2.a.a.1.1 1 21.17 even 6
400.2.a.e.1.1 1 140.59 even 6
400.2.c.d.49.1 2 140.3 odd 12
400.2.c.d.49.2 2 140.87 odd 12
720.2.a.e.1.1 1 84.59 odd 6
1280.2.d.a.641.1 2 112.3 even 12
1280.2.d.a.641.2 2 112.59 even 12
1280.2.d.j.641.1 2 112.45 odd 12
1280.2.d.j.641.2 2 112.101 odd 12
1600.2.a.k.1.1 1 280.59 even 6
1600.2.a.o.1.1 1 280.269 odd 6
1600.2.c.k.449.1 2 280.157 even 12
1600.2.c.k.449.2 2 280.213 even 12
1600.2.c.m.449.1 2 280.3 odd 12
1600.2.c.m.449.2 2 280.227 odd 12
1800.2.a.v.1.1 1 105.59 even 6
1800.2.f.a.649.1 2 105.17 odd 12
1800.2.f.a.649.2 2 105.38 odd 12
1960.2.a.g.1.1 1 7.4 even 3
1960.2.q.h.361.1 2 7.6 odd 2
1960.2.q.h.961.1 2 7.5 odd 6
1960.2.q.i.361.1 2 1.1 even 1 trivial
1960.2.q.i.961.1 2 7.2 even 3 inner
2880.2.a.t.1.1 1 168.101 even 6
2880.2.a.bg.1.1 1 168.59 odd 6
3240.2.q.k.1081.1 2 63.52 odd 6
3240.2.q.k.2161.1 2 63.31 odd 6
3240.2.q.x.1081.1 2 63.38 even 6
3240.2.q.x.2161.1 2 63.59 even 6
3600.2.a.h.1.1 1 420.59 odd 6
3600.2.f.t.2449.1 2 420.143 even 12
3600.2.f.t.2449.2 2 420.227 even 12
3920.2.a.s.1.1 1 28.11 odd 6
4840.2.a.f.1.1 1 77.10 even 6
6760.2.a.i.1.1 1 91.38 odd 6
9680.2.a.q.1.1 1 308.87 odd 6
9800.2.a.x.1.1 1 35.4 even 6