Properties

Label 756.2.j.a.505.3
Level $756$
Weight $2$
Character 756.505
Analytic conductor $6.037$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(253,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.253");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.j (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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 505.3
Root \(0.500000 - 2.05195i\) of defining polynomial
Character \(\chi\) \(=\) 756.505
Dual form 756.2.j.a.253.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.02704 + 1.77889i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-2.52704 + 4.37697i) q^{11} +(0.500000 + 0.866025i) q^{13} -0.273346 q^{17} -5.38151 q^{19} +(-2.66372 - 4.61369i) q^{23} +(0.390369 - 0.676139i) q^{25} +(-4.16372 + 7.21177i) q^{29} +(5.08113 + 8.80077i) q^{31} -2.05408 q^{35} +8.16225 q^{37} +(-2.52704 - 4.37697i) q^{41} +(-2.30039 + 3.98439i) q^{43} +(0.690757 - 1.19643i) q^{47} +(-0.500000 - 0.866025i) q^{49} -3.43560 q^{53} -10.3815 q^{55} +(0.890369 + 1.54216i) q^{59} +(-0.390369 + 0.676139i) q^{61} +(-1.02704 + 1.77889i) q^{65} +(4.19076 + 7.25860i) q^{67} +7.78074 q^{71} +9.38151 q^{73} +(-2.52704 - 4.37697i) q^{77} +(-6.47150 + 11.2090i) q^{79} +(-2.86333 + 4.95943i) q^{83} +(-0.280738 - 0.486253i) q^{85} +13.8171 q^{89} -1.00000 q^{91} +(-5.52704 - 9.57312i) q^{95} +(1.10963 - 1.92194i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59}+ \cdots + 15 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(4\) 0 0
\(5\) 1.02704 + 1.77889i 0.459307 + 0.795543i 0.998924 0.0463670i \(-0.0147644\pi\)
−0.539617 + 0.841910i \(0.681431\pi\)
\(6\) 0 0
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.52704 + 4.37697i −0.761932 + 1.31970i 0.179922 + 0.983681i \(0.442416\pi\)
−0.941854 + 0.336024i \(0.890918\pi\)
\(12\) 0 0
\(13\) 0.500000 + 0.866025i 0.138675 + 0.240192i 0.926995 0.375073i \(-0.122382\pi\)
−0.788320 + 0.615265i \(0.789049\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.273346 −0.0662962 −0.0331481 0.999450i \(-0.510553\pi\)
−0.0331481 + 0.999450i \(0.510553\pi\)
\(18\) 0 0
\(19\) −5.38151 −1.23460 −0.617302 0.786726i \(-0.711774\pi\)
−0.617302 + 0.786726i \(0.711774\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.66372 4.61369i −0.555423 0.962021i −0.997870 0.0652265i \(-0.979223\pi\)
0.442447 0.896794i \(-0.354110\pi\)
\(24\) 0 0
\(25\) 0.390369 0.676139i 0.0780738 0.135228i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.16372 + 7.21177i −0.773183 + 1.33919i 0.162628 + 0.986687i \(0.448003\pi\)
−0.935810 + 0.352504i \(0.885330\pi\)
\(30\) 0 0
\(31\) 5.08113 + 8.80077i 0.912597 + 1.58066i 0.810382 + 0.585903i \(0.199260\pi\)
0.102216 + 0.994762i \(0.467407\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.05408 −0.347204
\(36\) 0 0
\(37\) 8.16225 1.34187 0.670933 0.741518i \(-0.265894\pi\)
0.670933 + 0.741518i \(0.265894\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.52704 4.37697i −0.394658 0.683567i 0.598400 0.801198i \(-0.295803\pi\)
−0.993057 + 0.117631i \(0.962470\pi\)
\(42\) 0 0
\(43\) −2.30039 + 3.98439i −0.350806 + 0.607614i −0.986391 0.164417i \(-0.947426\pi\)
0.635585 + 0.772031i \(0.280759\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.690757 1.19643i 0.100757 0.174517i −0.811240 0.584714i \(-0.801207\pi\)
0.911997 + 0.410197i \(0.134540\pi\)
\(48\) 0 0
\(49\) −0.500000 0.866025i −0.0714286 0.123718i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.43560 −0.471916 −0.235958 0.971763i \(-0.575823\pi\)
−0.235958 + 0.971763i \(0.575823\pi\)
\(54\) 0 0
\(55\) −10.3815 −1.39984
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.890369 + 1.54216i 0.115916 + 0.200773i 0.918146 0.396243i \(-0.129686\pi\)
−0.802229 + 0.597016i \(0.796353\pi\)
\(60\) 0 0
\(61\) −0.390369 + 0.676139i −0.0499816 + 0.0865707i −0.889934 0.456090i \(-0.849250\pi\)
0.839952 + 0.542660i \(0.182583\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.02704 + 1.77889i −0.127389 + 0.220644i
\(66\) 0 0
\(67\) 4.19076 + 7.25860i 0.511982 + 0.886780i 0.999904 + 0.0138919i \(0.00442207\pi\)
−0.487921 + 0.872888i \(0.662245\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.78074 0.923404 0.461702 0.887035i \(-0.347239\pi\)
0.461702 + 0.887035i \(0.347239\pi\)
\(72\) 0 0
\(73\) 9.38151 1.09802 0.549012 0.835815i \(-0.315004\pi\)
0.549012 + 0.835815i \(0.315004\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.52704 4.37697i −0.287983 0.498801i
\(78\) 0 0
\(79\) −6.47150 + 11.2090i −0.728100 + 1.26111i 0.229585 + 0.973289i \(0.426263\pi\)
−0.957685 + 0.287818i \(0.907070\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.86333 + 4.95943i −0.314291 + 0.544368i −0.979287 0.202479i \(-0.935100\pi\)
0.664996 + 0.746847i \(0.268433\pi\)
\(84\) 0 0
\(85\) −0.280738 0.486253i −0.0304503 0.0527415i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.8171 1.46461 0.732306 0.680976i \(-0.238444\pi\)
0.732306 + 0.680976i \(0.238444\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.52704 9.57312i −0.567063 0.982181i
\(96\) 0 0
\(97\) 1.10963 1.92194i 0.112666 0.195143i −0.804178 0.594388i \(-0.797394\pi\)
0.916844 + 0.399245i \(0.130728\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.36333 2.36135i 0.135656 0.234963i −0.790192 0.612860i \(-0.790019\pi\)
0.925848 + 0.377896i \(0.123352\pi\)
\(102\) 0 0
\(103\) −8.99115 15.5731i −0.885924 1.53447i −0.844651 0.535317i \(-0.820192\pi\)
−0.0412728 0.999148i \(-0.513141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.10817 −0.107131 −0.0535653 0.998564i \(-0.517059\pi\)
−0.0535653 + 0.998564i \(0.517059\pi\)
\(108\) 0 0
\(109\) 3.38151 0.323890 0.161945 0.986800i \(-0.448223\pi\)
0.161945 + 0.986800i \(0.448223\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.43560 + 16.3429i 0.887626 + 1.53741i 0.842673 + 0.538425i \(0.180980\pi\)
0.0449531 + 0.998989i \(0.485686\pi\)
\(114\) 0 0
\(115\) 5.47150 9.47691i 0.510220 0.883726i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.136673 0.236725i 0.0125288 0.0217005i
\(120\) 0 0
\(121\) −7.27188 12.5953i −0.661080 1.14502i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.8741 1.06205
\(126\) 0 0
\(127\) 17.1623 1.52290 0.761452 0.648221i \(-0.224487\pi\)
0.761452 + 0.648221i \(0.224487\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.94445 15.4922i −0.781481 1.35356i −0.931079 0.364817i \(-0.881132\pi\)
0.149599 0.988747i \(-0.452202\pi\)
\(132\) 0 0
\(133\) 2.69076 4.66053i 0.233318 0.404119i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.24630 + 3.89071i −0.191915 + 0.332406i −0.945885 0.324503i \(-0.894803\pi\)
0.753970 + 0.656909i \(0.228136\pi\)
\(138\) 0 0
\(139\) −9.07227 15.7136i −0.769500 1.33281i −0.937834 0.347083i \(-0.887172\pi\)
0.168334 0.985730i \(-0.446161\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.05408 −0.422644
\(144\) 0 0
\(145\) −17.1052 −1.42051
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.25370 3.90352i −0.184630 0.319788i 0.758822 0.651298i \(-0.225775\pi\)
−0.943452 + 0.331510i \(0.892442\pi\)
\(150\) 0 0
\(151\) 5.49115 9.51094i 0.446863 0.773990i −0.551317 0.834296i \(-0.685874\pi\)
0.998180 + 0.0603064i \(0.0192078\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −10.4371 + 18.0775i −0.838325 + 1.45202i
\(156\) 0 0
\(157\) −2.08998 3.61995i −0.166799 0.288904i 0.770494 0.637447i \(-0.220010\pi\)
−0.937293 + 0.348544i \(0.886676\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 5.32743 0.419860
\(162\) 0 0
\(163\) −5.61849 −0.440074 −0.220037 0.975492i \(-0.570618\pi\)
−0.220037 + 0.975492i \(0.570618\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.44592 9.43260i −0.421418 0.729917i 0.574661 0.818392i \(-0.305134\pi\)
−0.996078 + 0.0884750i \(0.971801\pi\)
\(168\) 0 0
\(169\) 6.00000 10.3923i 0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.30039 12.6446i 0.555038 0.961354i −0.442862 0.896590i \(-0.646037\pi\)
0.997901 0.0647648i \(-0.0206297\pi\)
\(174\) 0 0
\(175\) 0.390369 + 0.676139i 0.0295091 + 0.0511113i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 23.4897 1.75570 0.877851 0.478934i \(-0.158977\pi\)
0.877851 + 0.478934i \(0.158977\pi\)
\(180\) 0 0
\(181\) −1.39922 −0.104003 −0.0520017 0.998647i \(-0.516560\pi\)
−0.0520017 + 0.998647i \(0.516560\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.38298 + 14.5197i 0.616329 + 1.06751i
\(186\) 0 0
\(187\) 0.690757 1.19643i 0.0505132 0.0874914i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.6819 20.2336i 0.845273 1.46406i −0.0401112 0.999195i \(-0.512771\pi\)
0.885384 0.464860i \(-0.153895\pi\)
\(192\) 0 0
\(193\) 7.27188 + 12.5953i 0.523442 + 0.906628i 0.999628 + 0.0272830i \(0.00868552\pi\)
−0.476186 + 0.879345i \(0.657981\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.3422 1.23558 0.617791 0.786343i \(-0.288028\pi\)
0.617791 + 0.786343i \(0.288028\pi\)
\(198\) 0 0
\(199\) −11.5438 −0.818316 −0.409158 0.912464i \(-0.634178\pi\)
−0.409158 + 0.912464i \(0.634178\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.16372 7.21177i −0.292236 0.506167i
\(204\) 0 0
\(205\) 5.19076 8.99066i 0.362538 0.627935i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 13.5993 23.5547i 0.940684 1.62931i
\(210\) 0 0
\(211\) 12.2630 + 21.2402i 0.844222 + 1.46223i 0.886295 + 0.463120i \(0.153270\pi\)
−0.0420736 + 0.999115i \(0.513396\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.45038 −0.644511
\(216\) 0 0
\(217\) −10.1623 −0.689859
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −0.136673 0.236725i −0.00919363 0.0159238i
\(222\) 0 0
\(223\) −4.28074 + 7.41446i −0.286659 + 0.496509i −0.973010 0.230762i \(-0.925878\pi\)
0.686351 + 0.727271i \(0.259211\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.5993 + 18.3586i −0.703501 + 1.21850i 0.263729 + 0.964597i \(0.415048\pi\)
−0.967230 + 0.253903i \(0.918286\pi\)
\(228\) 0 0
\(229\) 2.28074 + 3.95035i 0.150715 + 0.261047i 0.931491 0.363765i \(-0.118509\pi\)
−0.780775 + 0.624812i \(0.785176\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −13.5074 −0.884899 −0.442449 0.896794i \(-0.645890\pi\)
−0.442449 + 0.896794i \(0.645890\pi\)
\(234\) 0 0
\(235\) 2.83775 0.185114
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.82743 11.8255i −0.441630 0.764925i 0.556181 0.831061i \(-0.312266\pi\)
−0.997811 + 0.0661361i \(0.978933\pi\)
\(240\) 0 0
\(241\) −1.60963 + 2.78796i −0.103685 + 0.179588i −0.913200 0.407511i \(-0.866397\pi\)
0.809515 + 0.587099i \(0.199730\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.02704 1.77889i 0.0656153 0.113649i
\(246\) 0 0
\(247\) −2.69076 4.66053i −0.171209 0.296542i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.38151 −0.276559 −0.138279 0.990393i \(-0.544157\pi\)
−0.138279 + 0.990393i \(0.544157\pi\)
\(252\) 0 0
\(253\) 26.9253 1.69278
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.72665 + 9.91886i 0.357219 + 0.618721i 0.987495 0.157650i \(-0.0503917\pi\)
−0.630276 + 0.776371i \(0.717058\pi\)
\(258\) 0 0
\(259\) −4.08113 + 7.06872i −0.253589 + 0.439229i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.41741 + 5.91913i −0.210727 + 0.364989i −0.951942 0.306278i \(-0.900916\pi\)
0.741216 + 0.671267i \(0.234250\pi\)
\(264\) 0 0
\(265\) −3.52850 6.11155i −0.216754 0.375429i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.67257 −0.589747 −0.294873 0.955536i \(-0.595278\pi\)
−0.294873 + 0.955536i \(0.595278\pi\)
\(270\) 0 0
\(271\) −12.8377 −0.779838 −0.389919 0.920849i \(-0.627497\pi\)
−0.389919 + 0.920849i \(0.627497\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.97296 + 3.41726i 0.118974 + 0.206069i
\(276\) 0 0
\(277\) −5.79153 + 10.0312i −0.347980 + 0.602718i −0.985890 0.167392i \(-0.946465\pi\)
0.637911 + 0.770110i \(0.279799\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.46410 + 4.26795i −0.146996 + 0.254605i −0.930116 0.367266i \(-0.880294\pi\)
0.783120 + 0.621871i \(0.213627\pi\)
\(282\) 0 0
\(283\) 9.30039 + 16.1087i 0.552851 + 0.957565i 0.998067 + 0.0621426i \(0.0197934\pi\)
−0.445217 + 0.895423i \(0.646873\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.05408 0.298333
\(288\) 0 0
\(289\) −16.9253 −0.995605
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.3801 21.4429i −0.723250 1.25271i −0.959690 0.281060i \(-0.909314\pi\)
0.236440 0.971646i \(-0.424019\pi\)
\(294\) 0 0
\(295\) −1.82889 + 3.16774i −0.106482 + 0.184433i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.66372 4.61369i 0.154047 0.266817i
\(300\) 0 0
\(301\) −2.30039 3.98439i −0.132592 0.229656i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.60370 −0.0918277
\(306\) 0 0
\(307\) 21.9430 1.25235 0.626176 0.779681i \(-0.284619\pi\)
0.626176 + 0.779681i \(0.284619\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 13.5811 + 23.5232i 0.770115 + 1.33388i 0.937499 + 0.347987i \(0.113135\pi\)
−0.167384 + 0.985892i \(0.553532\pi\)
\(312\) 0 0
\(313\) 4.27188 7.39912i 0.241461 0.418223i −0.719670 0.694317i \(-0.755707\pi\)
0.961131 + 0.276094i \(0.0890400\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.199612 0.345738i 0.0112113 0.0194186i −0.860365 0.509678i \(-0.829765\pi\)
0.871577 + 0.490259i \(0.163098\pi\)
\(318\) 0 0
\(319\) −21.0438 36.4489i −1.17822 2.04075i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.47102 0.0818495
\(324\) 0 0
\(325\) 0.780738 0.0433076
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.690757 + 1.19643i 0.0380827 + 0.0659611i
\(330\) 0 0
\(331\) 2.80924 4.86575i 0.154410 0.267446i −0.778434 0.627726i \(-0.783986\pi\)
0.932844 + 0.360281i \(0.117319\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.60817 + 14.9098i −0.470314 + 0.814609i
\(336\) 0 0
\(337\) 14.4911 + 25.0994i 0.789383 + 1.36725i 0.926345 + 0.376675i \(0.122933\pi\)
−0.136962 + 0.990576i \(0.543734\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −51.3609 −2.78135
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.2345 + 29.8511i 0.925198 + 1.60249i 0.791243 + 0.611502i \(0.209434\pi\)
0.133955 + 0.990987i \(0.457232\pi\)
\(348\) 0 0
\(349\) −8.78074 + 15.2087i −0.470022 + 0.814102i −0.999412 0.0342762i \(-0.989087\pi\)
0.529390 + 0.848378i \(0.322421\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.4445 28.4826i 0.875250 1.51598i 0.0187537 0.999824i \(-0.494030\pi\)
0.856496 0.516153i \(-0.172636\pi\)
\(354\) 0 0
\(355\) 7.99115 + 13.8411i 0.424126 + 0.734608i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.96362 −0.156414 −0.0782071 0.996937i \(-0.524920\pi\)
−0.0782071 + 0.996937i \(0.524920\pi\)
\(360\) 0 0
\(361\) 9.96070 0.524247
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.63521 + 16.6887i 0.504330 + 0.873525i
\(366\) 0 0
\(367\) −6.68190 + 11.5734i −0.348792 + 0.604126i −0.986035 0.166537i \(-0.946742\pi\)
0.637243 + 0.770663i \(0.280075\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.71780 2.97532i 0.0891837 0.154471i
\(372\) 0 0
\(373\) −2.30039 3.98439i −0.119110 0.206304i 0.800305 0.599592i \(-0.204671\pi\)
−0.919415 + 0.393289i \(0.871337\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.32743 −0.428884
\(378\) 0 0
\(379\) −2.21926 −0.113996 −0.0569979 0.998374i \(-0.518153\pi\)
−0.0569979 + 0.998374i \(0.518153\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.5167 26.8758i −0.792868 1.37329i −0.924184 0.381947i \(-0.875254\pi\)
0.131317 0.991340i \(-0.458079\pi\)
\(384\) 0 0
\(385\) 5.19076 8.99066i 0.264545 0.458206i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.4174 + 21.5076i −0.629588 + 1.09048i 0.358047 + 0.933704i \(0.383443\pi\)
−0.987634 + 0.156774i \(0.949890\pi\)
\(390\) 0 0
\(391\) 0.728116 + 1.26113i 0.0368224 + 0.0637783i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −26.5860 −1.33769
\(396\) 0 0
\(397\) 17.7237 0.889528 0.444764 0.895648i \(-0.353287\pi\)
0.444764 + 0.895648i \(0.353287\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.0885 + 26.1341i 0.753485 + 1.30507i 0.946124 + 0.323804i \(0.104962\pi\)
−0.192640 + 0.981270i \(0.561705\pi\)
\(402\) 0 0
\(403\) −5.08113 + 8.80077i −0.253109 + 0.438398i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −20.6264 + 35.7259i −1.02241 + 1.77087i
\(408\) 0 0
\(409\) −8.38151 14.5172i −0.414439 0.717830i 0.580930 0.813953i \(-0.302689\pi\)
−0.995369 + 0.0961236i \(0.969356\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.78074 −0.0876244
\(414\) 0 0
\(415\) −11.7630 −0.577424
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.44445 + 2.50187i 0.0705662 + 0.122224i 0.899150 0.437641i \(-0.144186\pi\)
−0.828583 + 0.559866i \(0.810853\pi\)
\(420\) 0 0
\(421\) 0.0899807 0.155851i 0.00438539 0.00759572i −0.863824 0.503793i \(-0.831937\pi\)
0.868210 + 0.496197i \(0.165271\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.106706 + 0.184820i −0.00517600 + 0.00896509i
\(426\) 0 0
\(427\) −0.390369 0.676139i −0.0188913 0.0327207i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.76595 0.229568 0.114784 0.993390i \(-0.463382\pi\)
0.114784 + 0.993390i \(0.463382\pi\)
\(432\) 0 0
\(433\) −27.7630 −1.33421 −0.667103 0.744965i \(-0.732466\pi\)
−0.667103 + 0.744965i \(0.732466\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.3348 + 24.8286i 0.685728 + 1.18771i
\(438\) 0 0
\(439\) 2.32889 4.03376i 0.111152 0.192521i −0.805083 0.593162i \(-0.797879\pi\)
0.916235 + 0.400641i \(0.131213\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.38151 + 2.39285i −0.0656377 + 0.113688i −0.896977 0.442078i \(-0.854242\pi\)
0.831339 + 0.555766i \(0.187575\pi\)
\(444\) 0 0
\(445\) 14.1908 + 24.5791i 0.672706 + 1.16516i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.9430 −0.941168 −0.470584 0.882355i \(-0.655957\pi\)
−0.470584 + 0.882355i \(0.655957\pi\)
\(450\) 0 0
\(451\) 25.5438 1.20281
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.02704 1.77889i −0.0481485 0.0833956i
\(456\) 0 0
\(457\) −13.6908 + 23.7131i −0.640427 + 1.10925i 0.344911 + 0.938635i \(0.387909\pi\)
−0.985338 + 0.170616i \(0.945424\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.02558 + 5.24046i −0.140915 + 0.244072i −0.927842 0.372975i \(-0.878338\pi\)
0.786926 + 0.617047i \(0.211671\pi\)
\(462\) 0 0
\(463\) 8.77188 + 15.1933i 0.407664 + 0.706095i 0.994628 0.103519i \(-0.0330101\pi\)
−0.586964 + 0.809613i \(0.699677\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 23.6156 1.09280 0.546399 0.837525i \(-0.315998\pi\)
0.546399 + 0.837525i \(0.315998\pi\)
\(468\) 0 0
\(469\) −8.38151 −0.387022
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −11.6264 20.1374i −0.534580 0.925920i
\(474\) 0 0
\(475\) −2.10078 + 3.63865i −0.0963902 + 0.166953i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.1264 33.1278i 0.873906 1.51365i 0.0159814 0.999872i \(-0.494913\pi\)
0.857924 0.513776i \(-0.171754\pi\)
\(480\) 0 0
\(481\) 4.08113 + 7.06872i 0.186083 + 0.322306i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.55855 0.206993
\(486\) 0 0
\(487\) −6.57918 −0.298131 −0.149066 0.988827i \(-0.547627\pi\)
−0.149066 + 0.988827i \(0.547627\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.02704 1.77889i −0.0463498 0.0802801i 0.841920 0.539603i \(-0.181426\pi\)
−0.888270 + 0.459323i \(0.848092\pi\)
\(492\) 0 0
\(493\) 1.13814 1.97131i 0.0512591 0.0887833i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.89037 + 6.73832i −0.174507 + 0.302255i
\(498\) 0 0
\(499\) 8.16225 + 14.1374i 0.365393 + 0.632879i 0.988839 0.148987i \(-0.0476014\pi\)
−0.623446 + 0.781866i \(0.714268\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.60078 0.249726 0.124863 0.992174i \(-0.460151\pi\)
0.124863 + 0.992174i \(0.460151\pi\)
\(504\) 0 0
\(505\) 5.60078 0.249231
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.336285 + 0.582462i 0.0149056 + 0.0258172i 0.873382 0.487036i \(-0.161922\pi\)
−0.858476 + 0.512853i \(0.828589\pi\)
\(510\) 0 0
\(511\) −4.69076 + 8.12463i −0.207507 + 0.359412i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 18.4686 31.9885i 0.813822 1.40958i
\(516\) 0 0
\(517\) 3.49115 + 6.04684i 0.153540 + 0.265940i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −26.4533 −1.15894 −0.579470 0.814993i \(-0.696740\pi\)
−0.579470 + 0.814993i \(0.696740\pi\)
\(522\) 0 0
\(523\) 27.3068 1.19404 0.597021 0.802225i \(-0.296351\pi\)
0.597021 + 0.802225i \(0.296351\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.38891 2.40566i −0.0605017 0.104792i
\(528\) 0 0
\(529\) −2.69076 + 4.66053i −0.116989 + 0.202632i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.52704 4.37697i 0.109458 0.189587i
\(534\) 0 0
\(535\) −1.13814 1.97131i −0.0492059 0.0852271i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.05408 0.217695
\(540\) 0 0
\(541\) −19.3245 −0.830825 −0.415413 0.909633i \(-0.636363\pi\)
−0.415413 + 0.909633i \(0.636363\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.47296 + 6.01534i 0.148765 + 0.257669i
\(546\) 0 0
\(547\) 9.17111 15.8848i 0.392128 0.679186i −0.600602 0.799548i \(-0.705072\pi\)
0.992730 + 0.120362i \(0.0384056\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22.4071 38.8102i 0.954574 1.65337i
\(552\) 0 0
\(553\) −6.47150 11.2090i −0.275196 0.476653i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.19863 −0.389758 −0.194879 0.980827i \(-0.562431\pi\)
−0.194879 + 0.980827i \(0.562431\pi\)
\(558\) 0 0
\(559\) −4.60078 −0.194592
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 16.5811 + 28.7194i 0.698811 + 1.21038i 0.968879 + 0.247535i \(0.0796206\pi\)
−0.270068 + 0.962841i \(0.587046\pi\)
\(564\) 0 0
\(565\) −19.3815 + 33.5698i −0.815386 + 1.41229i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.1008 22.6912i 0.549213 0.951265i −0.449116 0.893474i \(-0.648261\pi\)
0.998329 0.0577914i \(-0.0184058\pi\)
\(570\) 0 0
\(571\) −4.89037 8.47037i −0.204656 0.354474i 0.745367 0.666654i \(-0.232274\pi\)
−0.950023 + 0.312180i \(0.898941\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.15933 −0.173456
\(576\) 0 0
\(577\) 36.3068 1.51147 0.755736 0.654877i \(-0.227279\pi\)
0.755736 + 0.654877i \(0.227279\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.86333 4.95943i −0.118791 0.205752i
\(582\) 0 0
\(583\) 8.68190 15.0375i 0.359568 0.622789i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0737 + 20.9123i −0.498336 + 0.863144i −0.999998 0.00191995i \(-0.999389\pi\)
0.501662 + 0.865064i \(0.332722\pi\)
\(588\) 0 0
\(589\) −27.3442 47.3615i −1.12670 1.95150i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −41.4897 −1.70378 −0.851889 0.523723i \(-0.824543\pi\)
−0.851889 + 0.523723i \(0.824543\pi\)
\(594\) 0 0
\(595\) 0.561476 0.0230183
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.3422 19.6453i −0.463430 0.802685i 0.535699 0.844409i \(-0.320048\pi\)
−0.999129 + 0.0417243i \(0.986715\pi\)
\(600\) 0 0
\(601\) 20.1249 34.8573i 0.820912 1.42186i −0.0840927 0.996458i \(-0.526799\pi\)
0.905004 0.425403i \(-0.139867\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 14.9371 25.8717i 0.607278 1.05184i
\(606\) 0 0
\(607\) −8.66225 15.0035i −0.351590 0.608972i 0.634938 0.772563i \(-0.281026\pi\)
−0.986528 + 0.163591i \(0.947692\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.38151 0.0558901
\(612\) 0 0
\(613\) −32.9646 −1.33143 −0.665713 0.746207i \(-0.731873\pi\)
−0.665713 + 0.746207i \(0.731873\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.4700 23.3308i −0.542283 0.939262i −0.998772 0.0495330i \(-0.984227\pi\)
0.456489 0.889729i \(-0.349107\pi\)
\(618\) 0 0
\(619\) 0.991146 1.71671i 0.0398375 0.0690006i −0.845419 0.534103i \(-0.820649\pi\)
0.885257 + 0.465103i \(0.153983\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.90856 + 11.9660i −0.276785 + 0.479407i
\(624\) 0 0
\(625\) 10.2434 + 17.7421i 0.409735 + 0.709682i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.23112 −0.0889606
\(630\) 0 0
\(631\) −25.4868 −1.01461 −0.507306 0.861766i \(-0.669359\pi\)
−0.507306 + 0.861766i \(0.669359\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.6264 + 30.5297i 0.699481 + 1.21154i
\(636\) 0 0
\(637\) 0.500000 0.866025i 0.0198107 0.0343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.0423 + 34.7143i −0.791623 + 1.37113i 0.133338 + 0.991071i \(0.457430\pi\)
−0.924961 + 0.380061i \(0.875903\pi\)
\(642\) 0 0
\(643\) −3.50885 6.07751i −0.138376 0.239674i 0.788506 0.615027i \(-0.210855\pi\)
−0.926882 + 0.375353i \(0.877521\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.8142 0.464464 0.232232 0.972660i \(-0.425397\pi\)
0.232232 + 0.972660i \(0.425397\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.136673 + 0.236725i 0.00534843 + 0.00926375i 0.868687 0.495361i \(-0.164964\pi\)
−0.863339 + 0.504625i \(0.831631\pi\)
\(654\) 0 0
\(655\) 18.3727 31.8224i 0.717879 1.24340i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.39970 12.8167i 0.288251 0.499266i −0.685141 0.728410i \(-0.740259\pi\)
0.973392 + 0.229144i \(0.0735928\pi\)
\(660\) 0 0
\(661\) 4.50885 + 7.80956i 0.175374 + 0.303757i 0.940291 0.340372i \(-0.110553\pi\)
−0.764917 + 0.644129i \(0.777220\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.0541 0.428659
\(666\) 0 0
\(667\) 44.3638 1.71777
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.97296 3.41726i −0.0761652 0.131922i
\(672\) 0 0
\(673\) −11.9803 + 20.7506i −0.461809 + 0.799876i −0.999051 0.0435519i \(-0.986133\pi\)
0.537243 + 0.843428i \(0.319466\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.32889 + 5.76581i −0.127940 + 0.221598i −0.922878 0.385092i \(-0.874170\pi\)
0.794938 + 0.606690i \(0.207503\pi\)
\(678\) 0 0
\(679\) 1.10963 + 1.92194i 0.0425837 + 0.0737572i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 30.0728 1.15070 0.575351 0.817907i \(-0.304865\pi\)
0.575351 + 0.817907i \(0.304865\pi\)
\(684\) 0 0
\(685\) −9.22820 −0.352591
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.71780 2.97532i −0.0654429 0.113351i
\(690\) 0 0
\(691\) 1.63814 2.83733i 0.0623176 0.107937i −0.833183 0.552997i \(-0.813484\pi\)
0.895501 + 0.445060i \(0.146817\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.6352 32.2771i 0.706874 1.22434i
\(696\) 0 0
\(697\) 0.690757 + 1.19643i 0.0261643 + 0.0453179i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.2891 1.21954 0.609771 0.792578i \(-0.291261\pi\)
0.609771 + 0.792578i \(0.291261\pi\)
\(702\) 0 0
\(703\) −43.9253 −1.65667
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.36333 + 2.36135i 0.0512732 + 0.0888078i
\(708\) 0 0
\(709\) 24.5438 42.5111i 0.921761 1.59654i 0.125071 0.992148i \(-0.460084\pi\)
0.796690 0.604388i \(-0.206582\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 27.0693 46.8855i 1.01376 1.75588i
\(714\) 0 0
\(715\) −5.19076 8.99066i −0.194123 0.336231i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 41.9617 1.56491 0.782453 0.622710i \(-0.213968\pi\)
0.782453 + 0.622710i \(0.213968\pi\)
\(720\) 0 0
\(721\) 17.9823 0.669696
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.25077 + 5.63050i 0.120731 + 0.209112i
\(726\) 0 0
\(727\) 14.2434 24.6703i 0.528258 0.914969i −0.471200 0.882027i \(-0.656179\pi\)
0.999457 0.0329425i \(-0.0104878\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.628802 1.08912i 0.0232571 0.0402825i
\(732\) 0 0
\(733\) −14.2630 24.7043i −0.526817 0.912474i −0.999512 0.0312475i \(-0.990052\pi\)
0.472695 0.881226i \(-0.343281\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −42.3609 −1.56038
\(738\) 0 0
\(739\) 7.85934 0.289110 0.144555 0.989497i \(-0.453825\pi\)
0.144555 + 0.989497i \(0.453825\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −3.37364 5.84332i −0.123767 0.214371i 0.797483 0.603341i \(-0.206164\pi\)
−0.921250 + 0.388970i \(0.872831\pi\)
\(744\) 0 0
\(745\) 4.62928 8.01815i 0.169604 0.293762i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.554084 0.959702i 0.0202458 0.0350667i
\(750\) 0 0
\(751\) −11.0900 19.2084i −0.404679 0.700925i 0.589605 0.807692i \(-0.299283\pi\)
−0.994284 + 0.106767i \(0.965950\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 22.5586 0.820990
\(756\) 0 0
\(757\) −20.3815 −0.740779 −0.370389 0.928877i \(-0.620776\pi\)
−0.370389 + 0.928877i \(0.620776\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.3274 + 35.2081i 0.736869 + 1.27629i 0.953899 + 0.300129i \(0.0970298\pi\)
−0.217030 + 0.976165i \(0.569637\pi\)
\(762\) 0 0
\(763\) −1.69076 + 2.92848i −0.0612095 + 0.106018i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −0.890369 + 1.54216i −0.0321494 + 0.0556843i
\(768\) 0 0
\(769\) 16.9518 + 29.3615i 0.611299 + 1.05880i 0.991022 + 0.133701i \(0.0426861\pi\)
−0.379723 + 0.925100i \(0.623981\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.74825 0.314653 0.157326 0.987547i \(-0.449713\pi\)
0.157326 + 0.987547i \(0.449713\pi\)
\(774\) 0 0
\(775\) 7.93406 0.285000
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.5993 + 23.5547i 0.487246 + 0.843935i
\(780\) 0 0
\(781\) −19.6623 + 34.0560i −0.703571 + 1.21862i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.29300 7.43569i 0.153224 0.265391i
\(786\) 0 0
\(787\) −4.64260 8.04122i −0.165491 0.286639i 0.771339 0.636425i \(-0.219588\pi\)
−0.936830 + 0.349786i \(0.886254\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.8712 −0.670983
\(792\) 0 0
\(793\) −0.780738 −0.0277248
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.2271 + 19.4460i 0.397685 + 0.688811i 0.993440 0.114355i \(-0.0364802\pi\)
−0.595754 + 0.803167i \(0.703147\pi\)
\(798\) 0 0
\(799\) −0.188816 + 0.327039i −0.00667983 + 0.0115698i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −23.7075 + 41.0626i −0.836619 + 1.44907i
\(804\) 0 0
\(805\) 5.47150 + 9.47691i 0.192845 + 0.334017i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.40215 0.189929 0.0949647 0.995481i \(-0.469726\pi\)
0.0949647 + 0.995481i \(0.469726\pi\)
\(810\) 0 0
\(811\) −0.0177088 −0.000621841 −0.000310920 1.00000i \(-0.500099\pi\)
−0.000310920 1.00000i \(0.500099\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.77042 9.99466i −0.202129 0.350098i
\(816\) 0 0
\(817\) 12.3796 21.4420i 0.433106 0.750162i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.528505 + 0.915397i −0.0184449 + 0.0319476i −0.875101 0.483941i \(-0.839205\pi\)
0.856656 + 0.515889i \(0.172538\pi\)
\(822\) 0 0
\(823\) −6.76303 11.7139i −0.235744 0.408321i 0.723744 0.690068i \(-0.242419\pi\)
−0.959489 + 0.281747i \(0.909086\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 49.3068 1.71457 0.857283 0.514846i \(-0.172151\pi\)
0.857283 + 0.514846i \(0.172151\pi\)
\(828\) 0 0
\(829\) −53.7060 −1.86529 −0.932644 0.360799i \(-0.882504\pi\)
−0.932644 + 0.360799i \(0.882504\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.136673 + 0.236725i 0.00473544 + 0.00820203i
\(834\) 0 0
\(835\) 11.1864 19.3754i 0.387120 0.670512i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18.7163 + 32.4176i −0.646160 + 1.11918i 0.337873 + 0.941192i \(0.390293\pi\)
−0.984032 + 0.177990i \(0.943041\pi\)
\(840\) 0 0
\(841\) −20.1730 34.9407i −0.695622 1.20485i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 24.6490 0.847952
\(846\) 0 0
\(847\) 14.5438 0.499730
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −21.7419 37.6581i −0.745303 1.29090i
\(852\) 0 0
\(853\) 10.3092 17.8561i 0.352982 0.611382i −0.633789 0.773506i \(-0.718501\pi\)
0.986770 + 0.162124i \(0.0518344\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.4445 18.0903i 0.356776 0.617954i −0.630644 0.776072i \(-0.717209\pi\)
0.987420 + 0.158118i \(0.0505427\pi\)
\(858\) 0 0
\(859\) −13.4430 23.2839i −0.458669 0.794438i 0.540222 0.841523i \(-0.318340\pi\)
−0.998891 + 0.0470847i \(0.985007\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.8142 −0.912766 −0.456383 0.889784i \(-0.650855\pi\)
−0.456383 + 0.889784i \(0.650855\pi\)
\(864\) 0 0
\(865\) 29.9912 1.01973
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −32.7075 56.6510i −1.10953 1.92175i
\(870\) 0 0
\(871\) −4.19076 + 7.25860i −0.141998 + 0.245948i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.93706 + 10.2833i −0.200709 + 0.347639i
\(876\) 0 0
\(877\) 18.9538 + 32.8289i 0.640024 + 1.10855i 0.985427 + 0.170099i \(0.0544089\pi\)
−0.345403 + 0.938454i \(0.612258\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.47782 −0.251934 −0.125967 0.992034i \(-0.540203\pi\)
−0.125967 + 0.992034i \(0.540203\pi\)
\(882\) 0 0
\(883\) 5.07472 0.170778 0.0853889 0.996348i \(-0.472787\pi\)
0.0853889 + 0.996348i \(0.472787\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.26157 + 16.0415i 0.310973 + 0.538621i 0.978573 0.205899i \(-0.0660118\pi\)
−0.667600 + 0.744520i \(0.732678\pi\)
\(888\) 0 0
\(889\) −8.58113 + 14.8629i −0.287802 + 0.498487i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.71732 + 6.43859i −0.124395 + 0.215459i
\(894\) 0 0
\(895\) 24.1249 + 41.7855i 0.806406 + 1.39674i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −84.6255 −2.82242
\(900\) 0 0
\(901\) 0.939108 0.0312862
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.43706 2.48906i −0.0477695 0.0827393i
\(906\) 0 0
\(907\) 24.8245 42.9973i 0.824284 1.42770i −0.0781810 0.996939i \(-0.524911\pi\)
0.902465 0.430763i \(-0.141755\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 11.9808 20.7514i 0.396943 0.687525i −0.596404 0.802684i \(-0.703405\pi\)
0.993347 + 0.115159i \(0.0367379\pi\)
\(912\) 0 0
\(913\) −14.4715 25.0654i −0.478937 0.829543i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.8889 0.590744
\(918\) 0 0
\(919\) −18.8377 −0.621400 −0.310700 0.950508i \(-0.600563\pi\)
−0.310700 + 0.950508i \(0.600563\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.89037 + 6.73832i 0.128053 + 0.221794i
\(924\) 0 0
\(925\) 3.18629 5.51882i 0.104765 0.181458i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.45185 12.9070i 0.244487 0.423464i −0.717500 0.696558i \(-0.754714\pi\)
0.961987 + 0.273094i \(0.0880471\pi\)
\(930\) 0 0
\(931\) 2.69076 + 4.66053i 0.0881860 + 0.152743i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.83775 0.0928043
\(936\) 0 0
\(937\) 3.94299 0.128812 0.0644059 0.997924i \(-0.479485\pi\)
0.0644059 + 0.997924i \(0.479485\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −21.4056 37.0756i −0.697804 1.20863i −0.969226 0.246171i \(-0.920827\pi\)
0.271423 0.962460i \(-0.412506\pi\)
\(942\) 0 0
\(943\) −13.4626 + 23.3180i −0.438404 + 0.759338i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.5919 + 23.5419i −0.441678 + 0.765009i −0.997814 0.0660823i \(-0.978950\pi\)
0.556136 + 0.831091i \(0.312283\pi\)
\(948\) 0 0
\(949\) 4.69076 + 8.12463i 0.152268 + 0.263737i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.12295 −0.0363760 −0.0181880 0.999835i \(-0.505790\pi\)
−0.0181880 + 0.999835i \(0.505790\pi\)
\(954\) 0 0
\(955\) 47.9912 1.55296
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.24630 3.89071i −0.0725369 0.125638i
\(960\) 0 0
\(961\) −36.1357 + 62.5889i −1.16567 + 2.01900i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.9371 + 25.8717i −0.480841 + 0.832841i
\(966\) 0 0
\(967\) 4.75223 + 8.23111i 0.152822 + 0.264695i 0.932264 0.361780i \(-0.117831\pi\)
−0.779442 + 0.626474i \(0.784497\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.5979 0.789383 0.394691 0.918814i \(-0.370852\pi\)
0.394691 + 0.918814i \(0.370852\pi\)
\(972\) 0 0
\(973\) 18.1445 0.581687
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.7016 21.9997i −0.406359 0.703834i 0.588120 0.808774i \(-0.299868\pi\)
−0.994479 + 0.104940i \(0.966535\pi\)
\(978\) 0 0
\(979\) −34.9164 + 60.4770i −1.11593 + 1.93285i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11.4267 19.7917i 0.364457 0.631257i −0.624232 0.781239i \(-0.714588\pi\)
0.988689 + 0.149982i \(0.0479214\pi\)
\(984\) 0 0
\(985\) 17.8112 + 30.8499i 0.567512 + 0.982959i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.5103 0.779383
\(990\) 0 0
\(991\) 24.4690 0.777285 0.388642 0.921389i \(-0.372944\pi\)
0.388642 + 0.921389i \(0.372944\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −11.8559 20.5351i −0.375858 0.651006i
\(996\) 0 0
\(997\) −13.0000 + 22.5167i −0.411714 + 0.713110i −0.995077 0.0991016i \(-0.968403\pi\)
0.583363 + 0.812211i \(0.301736\pi\)
\(998\) 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 756.2.j.a.505.3 6
3.2 odd 2 252.2.j.b.169.2 yes 6
4.3 odd 2 3024.2.r.i.2017.3 6
7.2 even 3 5292.2.i.d.2125.3 6
7.3 odd 6 5292.2.l.d.3313.3 6
7.4 even 3 5292.2.l.g.3313.1 6
7.5 odd 6 5292.2.i.g.2125.1 6
7.6 odd 2 5292.2.j.e.3529.1 6
9.2 odd 6 2268.2.a.g.1.3 3
9.4 even 3 inner 756.2.j.a.253.3 6
9.5 odd 6 252.2.j.b.85.2 6
9.7 even 3 2268.2.a.j.1.1 3
12.11 even 2 1008.2.r.g.673.2 6
21.2 odd 6 1764.2.i.f.1537.2 6
21.5 even 6 1764.2.i.e.1537.2 6
21.11 odd 6 1764.2.l.d.961.1 6
21.17 even 6 1764.2.l.g.961.3 6
21.20 even 2 1764.2.j.d.1177.2 6
36.7 odd 6 9072.2.a.bz.1.1 3
36.11 even 6 9072.2.a.bt.1.3 3
36.23 even 6 1008.2.r.g.337.2 6
36.31 odd 6 3024.2.r.i.1009.3 6
63.4 even 3 5292.2.i.d.1549.3 6
63.5 even 6 1764.2.l.g.949.3 6
63.13 odd 6 5292.2.j.e.1765.1 6
63.23 odd 6 1764.2.l.d.949.1 6
63.31 odd 6 5292.2.i.g.1549.1 6
63.32 odd 6 1764.2.i.f.373.2 6
63.40 odd 6 5292.2.l.d.361.3 6
63.41 even 6 1764.2.j.d.589.2 6
63.58 even 3 5292.2.l.g.361.1 6
63.59 even 6 1764.2.i.e.373.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.b.85.2 6 9.5 odd 6
252.2.j.b.169.2 yes 6 3.2 odd 2
756.2.j.a.253.3 6 9.4 even 3 inner
756.2.j.a.505.3 6 1.1 even 1 trivial
1008.2.r.g.337.2 6 36.23 even 6
1008.2.r.g.673.2 6 12.11 even 2
1764.2.i.e.373.2 6 63.59 even 6
1764.2.i.e.1537.2 6 21.5 even 6
1764.2.i.f.373.2 6 63.32 odd 6
1764.2.i.f.1537.2 6 21.2 odd 6
1764.2.j.d.589.2 6 63.41 even 6
1764.2.j.d.1177.2 6 21.20 even 2
1764.2.l.d.949.1 6 63.23 odd 6
1764.2.l.d.961.1 6 21.11 odd 6
1764.2.l.g.949.3 6 63.5 even 6
1764.2.l.g.961.3 6 21.17 even 6
2268.2.a.g.1.3 3 9.2 odd 6
2268.2.a.j.1.1 3 9.7 even 3
3024.2.r.i.1009.3 6 36.31 odd 6
3024.2.r.i.2017.3 6 4.3 odd 2
5292.2.i.d.1549.3 6 63.4 even 3
5292.2.i.d.2125.3 6 7.2 even 3
5292.2.i.g.1549.1 6 63.31 odd 6
5292.2.i.g.2125.1 6 7.5 odd 6
5292.2.j.e.1765.1 6 63.13 odd 6
5292.2.j.e.3529.1 6 7.6 odd 2
5292.2.l.d.361.3 6 63.40 odd 6
5292.2.l.d.3313.3 6 7.3 odd 6
5292.2.l.g.361.1 6 63.58 even 3
5292.2.l.g.3313.1 6 7.4 even 3
9072.2.a.bt.1.3 3 36.11 even 6
9072.2.a.bz.1.1 3 36.7 odd 6