Properties

Label 2736.2.s.y.577.3
Level $2736$
Weight $2$
Character 2736.577
Analytic conductor $21.847$
Analytic rank $0$
Dimension $6$
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] = [2736,2,Mod(577,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.s (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: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.2696112.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} + 18x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 577.3
Root \(1.17146 + 2.02903i\) of defining polynomial
Character \(\chi\) \(=\) 2736.577
Dual form 2736.2.s.y.1873.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.17146 + 2.02903i) q^{5} -3.83221 q^{7} +O(q^{10})\) \(q+(1.17146 + 2.02903i) q^{5} -3.83221 q^{7} +3.34292 q^{11} +(-3.08757 + 5.34782i) q^{13} +(2.59828 + 4.50035i) q^{17} +(3.01438 - 3.14857i) q^{19} +(-1.17146 + 2.02903i) q^{23} +(-0.244644 + 0.423736i) q^{25} +(0.0250961 - 0.0434676i) q^{29} +3.43910 q^{31} +(-4.48929 - 7.77568i) q^{35} -5.43910 q^{37} +(-3.64637 - 6.31569i) q^{41} +(-4.43049 - 7.67383i) q^{43} +(-5.36802 + 9.29768i) q^{47} +7.68585 q^{49} +(-1.59828 + 2.76830i) q^{53} +(3.91611 + 6.78289i) q^{55} +(-1.92682 - 3.33735i) q^{59} +(-1.46419 + 2.53606i) q^{61} -14.4679 q^{65} +(-5.75903 + 9.97493i) q^{67} +(-0.744644 - 1.28976i) q^{71} +(3.84292 + 6.65614i) q^{73} -12.8108 q^{77} +(0.0875674 + 0.151671i) q^{79} -7.00735 q^{83} +(-6.08757 + 10.5440i) q^{85} +(-6.77341 + 11.7319i) q^{89} +(11.8322 - 20.4940i) q^{91} +(9.91978 + 2.42785i) q^{95} +(1.84292 + 3.19204i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{5} + 4 q^{7} + 8 q^{11} + q^{13} + 11 q^{17} - q^{23} + 6 q^{25} - 3 q^{29} + 12 q^{31} - 12 q^{35} - 24 q^{37} - 19 q^{41} + 5 q^{43} - 17 q^{47} + 22 q^{49} - 5 q^{53} + 10 q^{55} - 13 q^{59} + 3 q^{61} - 42 q^{65} - 9 q^{67} + 3 q^{71} + 11 q^{73} - 20 q^{77} - 19 q^{79} + 24 q^{83} - 17 q^{85} + 3 q^{89} + 44 q^{91} + 13 q^{95} - q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.17146 + 2.02903i 0.523894 + 0.907410i 0.999613 + 0.0278132i \(0.00885437\pi\)
−0.475720 + 0.879597i \(0.657812\pi\)
\(6\) 0 0
\(7\) −3.83221 −1.44844 −0.724220 0.689569i \(-0.757800\pi\)
−0.724220 + 0.689569i \(0.757800\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.34292 1.00793 0.503965 0.863724i \(-0.331874\pi\)
0.503965 + 0.863724i \(0.331874\pi\)
\(12\) 0 0
\(13\) −3.08757 + 5.34782i −0.856337 + 1.48322i 0.0190619 + 0.999818i \(0.493932\pi\)
−0.875399 + 0.483401i \(0.839401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.59828 + 4.50035i 0.630175 + 1.09150i 0.987516 + 0.157521i \(0.0503503\pi\)
−0.357340 + 0.933974i \(0.616316\pi\)
\(18\) 0 0
\(19\) 3.01438 3.14857i 0.691547 0.722331i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.17146 + 2.02903i −0.244267 + 0.423082i −0.961925 0.273313i \(-0.911881\pi\)
0.717659 + 0.696395i \(0.245214\pi\)
\(24\) 0 0
\(25\) −0.244644 + 0.423736i −0.0489289 + 0.0847473i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.0250961 0.0434676i 0.00466022 0.00807174i −0.863686 0.504030i \(-0.831850\pi\)
0.868346 + 0.495959i \(0.165183\pi\)
\(30\) 0 0
\(31\) 3.43910 0.617680 0.308840 0.951114i \(-0.400059\pi\)
0.308840 + 0.951114i \(0.400059\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.48929 7.77568i −0.758828 1.31433i
\(36\) 0 0
\(37\) −5.43910 −0.894182 −0.447091 0.894488i \(-0.647540\pi\)
−0.447091 + 0.894488i \(0.647540\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.64637 6.31569i −0.569467 0.986345i −0.996619 0.0821653i \(-0.973816\pi\)
0.427152 0.904180i \(-0.359517\pi\)
\(42\) 0 0
\(43\) −4.43049 7.67383i −0.675643 1.17025i −0.976280 0.216510i \(-0.930533\pi\)
0.300637 0.953739i \(-0.402801\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.36802 + 9.29768i −0.783006 + 1.35621i 0.147177 + 0.989110i \(0.452981\pi\)
−0.930183 + 0.367096i \(0.880352\pi\)
\(48\) 0 0
\(49\) 7.68585 1.09798
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.59828 + 2.76830i −0.219540 + 0.380255i −0.954668 0.297674i \(-0.903789\pi\)
0.735127 + 0.677929i \(0.237122\pi\)
\(54\) 0 0
\(55\) 3.91611 + 6.78289i 0.528048 + 0.914605i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.92682 3.33735i −0.250850 0.434485i 0.712910 0.701256i \(-0.247377\pi\)
−0.963760 + 0.266770i \(0.914044\pi\)
\(60\) 0 0
\(61\) −1.46419 + 2.53606i −0.187471 + 0.324709i −0.944406 0.328781i \(-0.893362\pi\)
0.756936 + 0.653489i \(0.226696\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.4679 −1.79452
\(66\) 0 0
\(67\) −5.75903 + 9.97493i −0.703577 + 1.21863i 0.263625 + 0.964625i \(0.415082\pi\)
−0.967203 + 0.254007i \(0.918252\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.744644 1.28976i −0.0883730 0.153067i 0.818451 0.574577i \(-0.194833\pi\)
−0.906824 + 0.421510i \(0.861500\pi\)
\(72\) 0 0
\(73\) 3.84292 + 6.65614i 0.449780 + 0.779042i 0.998371 0.0570486i \(-0.0181690\pi\)
−0.548591 + 0.836091i \(0.684836\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.8108 −1.45992
\(78\) 0 0
\(79\) 0.0875674 + 0.151671i 0.00985210 + 0.0170643i 0.870909 0.491444i \(-0.163531\pi\)
−0.861057 + 0.508508i \(0.830197\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.00735 −0.769156 −0.384578 0.923092i \(-0.625653\pi\)
−0.384578 + 0.923092i \(0.625653\pi\)
\(84\) 0 0
\(85\) −6.08757 + 10.5440i −0.660289 + 1.14365i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.77341 + 11.7319i −0.717980 + 1.24358i 0.243818 + 0.969821i \(0.421600\pi\)
−0.961799 + 0.273758i \(0.911733\pi\)
\(90\) 0 0
\(91\) 11.8322 20.4940i 1.24035 2.14835i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 9.91978 + 2.42785i 1.01775 + 0.249092i
\(96\) 0 0
\(97\) 1.84292 + 3.19204i 0.187120 + 0.324102i 0.944289 0.329117i \(-0.106751\pi\)
−0.757169 + 0.653220i \(0.773418\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.80712 + 6.59412i −0.378822 + 0.656139i −0.990891 0.134665i \(-0.957004\pi\)
0.612069 + 0.790804i \(0.290337\pi\)
\(102\) 0 0
\(103\) −16.6430 −1.63988 −0.819942 0.572447i \(-0.805994\pi\)
−0.819942 + 0.572447i \(0.805994\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.29273 −0.608341 −0.304171 0.952618i \(-0.598379\pi\)
−0.304171 + 0.952618i \(0.598379\pi\)
\(108\) 0 0
\(109\) −5.57686 9.65940i −0.534166 0.925203i −0.999203 0.0399114i \(-0.987292\pi\)
0.465037 0.885291i \(-0.346041\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 17.7606 1.67078 0.835388 0.549660i \(-0.185243\pi\)
0.835388 + 0.549660i \(0.185243\pi\)
\(114\) 0 0
\(115\) −5.48929 −0.511879
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.95715 17.2463i −0.912771 1.58097i
\(120\) 0 0
\(121\) 0.175135 0.0159213
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.5682 0.945253
\(126\) 0 0
\(127\) 9.21251 15.9565i 0.817478 1.41591i −0.0900567 0.995937i \(-0.528705\pi\)
0.907535 0.419977i \(-0.137962\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.61266 + 2.79321i 0.140899 + 0.244044i 0.927835 0.372990i \(-0.121667\pi\)
−0.786936 + 0.617034i \(0.788334\pi\)
\(132\) 0 0
\(133\) −11.5518 + 12.0660i −1.00166 + 1.04625i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.98929 + 5.17760i −0.255392 + 0.442352i −0.965002 0.262243i \(-0.915538\pi\)
0.709610 + 0.704595i \(0.248871\pi\)
\(138\) 0 0
\(139\) −4.58389 + 7.93954i −0.388801 + 0.673423i −0.992289 0.123950i \(-0.960444\pi\)
0.603488 + 0.797372i \(0.293777\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.3215 + 17.8774i −0.863127 + 1.49498i
\(144\) 0 0
\(145\) 0.117596 0.00976584
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.07529 + 5.32656i 0.251937 + 0.436368i 0.964059 0.265688i \(-0.0855990\pi\)
−0.712122 + 0.702056i \(0.752266\pi\)
\(150\) 0 0
\(151\) −5.14637 −0.418805 −0.209403 0.977830i \(-0.567152\pi\)
−0.209403 + 0.977830i \(0.567152\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.02877 + 6.97803i 0.323599 + 0.560489i
\(156\) 0 0
\(157\) −2.20516 3.81946i −0.175991 0.304826i 0.764513 0.644609i \(-0.222980\pi\)
−0.940504 + 0.339783i \(0.889646\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.48929 7.77568i 0.353806 0.612809i
\(162\) 0 0
\(163\) −12.3215 −0.965094 −0.482547 0.875870i \(-0.660288\pi\)
−0.482547 + 0.875870i \(0.660288\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.89101 + 5.00738i −0.223713 + 0.387482i −0.955933 0.293586i \(-0.905151\pi\)
0.732220 + 0.681069i \(0.238485\pi\)
\(168\) 0 0
\(169\) −12.5661 21.7652i −0.966627 1.67425i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −0.912433 1.58038i −0.0693710 0.120154i 0.829254 0.558872i \(-0.188766\pi\)
−0.898625 + 0.438718i \(0.855433\pi\)
\(174\) 0 0
\(175\) 0.937529 1.62385i 0.0708705 0.122751i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.26396 0.617678 0.308839 0.951114i \(-0.400060\pi\)
0.308839 + 0.951114i \(0.400060\pi\)
\(180\) 0 0
\(181\) 10.8573 18.8054i 0.807017 1.39780i −0.107903 0.994161i \(-0.534414\pi\)
0.914921 0.403634i \(-0.132253\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.37169 11.0361i −0.468456 0.811390i
\(186\) 0 0
\(187\) 8.68585 + 15.0443i 0.635172 + 1.10015i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.58546 −0.331792 −0.165896 0.986143i \(-0.553052\pi\)
−0.165896 + 0.986143i \(0.553052\pi\)
\(192\) 0 0
\(193\) 6.08757 + 10.5440i 0.438193 + 0.758972i 0.997550 0.0699545i \(-0.0222854\pi\)
−0.559357 + 0.828927i \(0.688952\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.26817 0.304095 0.152047 0.988373i \(-0.451413\pi\)
0.152047 + 0.988373i \(0.451413\pi\)
\(198\) 0 0
\(199\) −1.43049 + 2.47768i −0.101405 + 0.175638i −0.912264 0.409604i \(-0.865667\pi\)
0.810859 + 0.585242i \(0.199000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.0961734 + 0.166577i −0.00675005 + 0.0116914i
\(204\) 0 0
\(205\) 8.54315 14.7972i 0.596680 1.03348i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0769 10.5254i 0.697031 0.728059i
\(210\) 0 0
\(211\) 9.13776 + 15.8271i 0.629069 + 1.08958i 0.987739 + 0.156115i \(0.0498972\pi\)
−0.358670 + 0.933465i \(0.616769\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.3803 17.9792i 0.707930 1.22617i
\(216\) 0 0
\(217\) −13.1793 −0.894672
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −32.0894 −2.15857
\(222\) 0 0
\(223\) −6.02510 10.4358i −0.403470 0.698831i 0.590672 0.806912i \(-0.298863\pi\)
−0.994142 + 0.108081i \(0.965529\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.71462 −0.578409 −0.289205 0.957267i \(-0.593391\pi\)
−0.289205 + 0.957267i \(0.593391\pi\)
\(228\) 0 0
\(229\) 15.5640 1.02850 0.514250 0.857640i \(-0.328070\pi\)
0.514250 + 0.857640i \(0.328070\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.2146 19.4243i −0.734694 1.27253i −0.954858 0.297064i \(-0.903992\pi\)
0.220164 0.975463i \(-0.429341\pi\)
\(234\) 0 0
\(235\) −25.1537 −1.64085
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.7434 1.98862 0.994312 0.106505i \(-0.0339662\pi\)
0.994312 + 0.106505i \(0.0339662\pi\)
\(240\) 0 0
\(241\) −2.47858 + 4.29302i −0.159659 + 0.276538i −0.934746 0.355317i \(-0.884373\pi\)
0.775087 + 0.631855i \(0.217706\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.00367 + 15.5948i 0.575224 + 0.996316i
\(246\) 0 0
\(247\) 7.53087 + 25.8418i 0.479178 + 1.64428i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −13.7664 + 23.8441i −0.868926 + 1.50502i −0.00583044 + 0.999983i \(0.501856\pi\)
−0.863095 + 0.505041i \(0.831477\pi\)
\(252\) 0 0
\(253\) −3.91611 + 6.78289i −0.246203 + 0.426437i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.94331 + 10.2941i −0.370733 + 0.642129i −0.989679 0.143305i \(-0.954227\pi\)
0.618945 + 0.785434i \(0.287560\pi\)
\(258\) 0 0
\(259\) 20.8438 1.29517
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −9.71094 16.8198i −0.598802 1.03716i −0.992998 0.118130i \(-0.962310\pi\)
0.394196 0.919026i \(-0.371023\pi\)
\(264\) 0 0
\(265\) −7.48929 −0.460063
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.24128 + 16.0064i 0.563451 + 0.975925i 0.997192 + 0.0748878i \(0.0238599\pi\)
−0.433741 + 0.901037i \(0.642807\pi\)
\(270\) 0 0
\(271\) 8.17146 + 14.1534i 0.496381 + 0.859757i 0.999991 0.00417390i \(-0.00132860\pi\)
−0.503610 + 0.863931i \(0.667995\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.817827 + 1.41652i −0.0493168 + 0.0854192i
\(276\) 0 0
\(277\) 15.7894 0.948691 0.474346 0.880339i \(-0.342685\pi\)
0.474346 + 0.880339i \(0.342685\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.21462 7.29993i 0.251423 0.435477i −0.712495 0.701677i \(-0.752435\pi\)
0.963918 + 0.266200i \(0.0857682\pi\)
\(282\) 0 0
\(283\) 0.905394 + 1.56819i 0.0538201 + 0.0932192i 0.891680 0.452666i \(-0.149527\pi\)
−0.837860 + 0.545885i \(0.816194\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 13.9736 + 24.2031i 0.824838 + 1.42866i
\(288\) 0 0
\(289\) −5.00211 + 8.66390i −0.294241 + 0.509641i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −24.2400 −1.41612 −0.708059 0.706154i \(-0.750429\pi\)
−0.708059 + 0.706154i \(0.750429\pi\)
\(294\) 0 0
\(295\) 4.51438 7.81914i 0.262838 0.455248i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.23393 12.5295i −0.418349 0.724602i
\(300\) 0 0
\(301\) 16.9786 + 29.4078i 0.978629 + 1.69504i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.86098 −0.392859
\(306\) 0 0
\(307\) 11.0016 + 19.0553i 0.627893 + 1.08754i 0.987974 + 0.154621i \(0.0494156\pi\)
−0.360081 + 0.932921i \(0.617251\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −32.4752 −1.84150 −0.920750 0.390153i \(-0.872422\pi\)
−0.920750 + 0.390153i \(0.872422\pi\)
\(312\) 0 0
\(313\) 4.62494 8.01064i 0.261417 0.452788i −0.705202 0.709007i \(-0.749143\pi\)
0.966619 + 0.256219i \(0.0824768\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.08757 5.34782i 0.173415 0.300364i −0.766197 0.642606i \(-0.777853\pi\)
0.939612 + 0.342243i \(0.111186\pi\)
\(318\) 0 0
\(319\) 0.0838942 0.145309i 0.00469717 0.00813574i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 22.0019 + 5.38493i 1.22422 + 0.299625i
\(324\) 0 0
\(325\) −1.51071 2.61663i −0.0837992 0.145144i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 20.5714 35.6307i 1.13414 1.96438i
\(330\) 0 0
\(331\) 10.3643 0.569676 0.284838 0.958576i \(-0.408060\pi\)
0.284838 + 0.958576i \(0.408060\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −26.9859 −1.47440
\(336\) 0 0
\(337\) −10.3108 17.8588i −0.561664 0.972831i −0.997351 0.0727331i \(-0.976828\pi\)
0.435687 0.900098i \(-0.356505\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 11.4966 0.622578
\(342\) 0 0
\(343\) −2.62831 −0.141915
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.9054 + 24.0848i 0.746481 + 1.29294i 0.949500 + 0.313768i \(0.101591\pi\)
−0.203019 + 0.979175i \(0.565075\pi\)
\(348\) 0 0
\(349\) 5.60688 0.300130 0.150065 0.988676i \(-0.452052\pi\)
0.150065 + 0.988676i \(0.452052\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 30.0105 1.59730 0.798648 0.601798i \(-0.205549\pi\)
0.798648 + 0.601798i \(0.205549\pi\)
\(354\) 0 0
\(355\) 1.74464 3.02181i 0.0925961 0.160381i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.48562 + 7.76931i 0.236742 + 0.410049i 0.959777 0.280762i \(-0.0905871\pi\)
−0.723036 + 0.690811i \(0.757254\pi\)
\(360\) 0 0
\(361\) −0.826971 18.9820i −0.0435248 0.999052i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.00367 + 15.5948i −0.471274 + 0.816270i
\(366\) 0 0
\(367\) −5.24621 + 9.08671i −0.273850 + 0.474322i −0.969844 0.243725i \(-0.921631\pi\)
0.695994 + 0.718047i \(0.254964\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.12494 10.6087i 0.317991 0.550777i
\(372\) 0 0
\(373\) −18.3074 −0.947922 −0.473961 0.880546i \(-0.657176\pi\)
−0.473961 + 0.880546i \(0.657176\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.154972 + 0.268419i 0.00798144 + 0.0138243i
\(378\) 0 0
\(379\) −24.8438 −1.27614 −0.638069 0.769979i \(-0.720267\pi\)
−0.638069 + 0.769979i \(0.720267\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.58600 2.74703i −0.0810408 0.140367i 0.822656 0.568539i \(-0.192491\pi\)
−0.903697 + 0.428172i \(0.859158\pi\)
\(384\) 0 0
\(385\) −15.0073 25.9935i −0.764845 1.32475i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5.69445 + 9.86308i −0.288720 + 0.500078i −0.973505 0.228668i \(-0.926563\pi\)
0.684784 + 0.728746i \(0.259896\pi\)
\(390\) 0 0
\(391\) −12.1751 −0.615723
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.205164 + 0.355354i −0.0103229 + 0.0178798i
\(396\) 0 0
\(397\) 9.44277 + 16.3554i 0.473919 + 0.820852i 0.999554 0.0298583i \(-0.00950560\pi\)
−0.525635 + 0.850710i \(0.676172\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0107 17.3391i −0.499911 0.865871i 0.500089 0.865974i \(-0.333301\pi\)
−1.00000 0.000102681i \(0.999967\pi\)
\(402\) 0 0
\(403\) −10.6184 + 18.3917i −0.528942 + 0.916155i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.1825 −0.901272
\(408\) 0 0
\(409\) −5.45715 + 9.45207i −0.269839 + 0.467375i −0.968820 0.247765i \(-0.920304\pi\)
0.698981 + 0.715140i \(0.253637\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.38397 + 12.7894i 0.363341 + 0.629326i
\(414\) 0 0
\(415\) −8.20884 14.2181i −0.402956 0.697940i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.4145 −0.753049 −0.376525 0.926407i \(-0.622881\pi\)
−0.376525 + 0.926407i \(0.622881\pi\)
\(420\) 0 0
\(421\) 2.98225 + 5.16541i 0.145346 + 0.251747i 0.929502 0.368817i \(-0.120237\pi\)
−0.784156 + 0.620564i \(0.786904\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.54262 −0.123335
\(426\) 0 0
\(427\) 5.61110 9.71870i 0.271540 0.470321i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.5095 25.1311i 0.698896 1.21052i −0.269954 0.962873i \(-0.587008\pi\)
0.968850 0.247650i \(-0.0796582\pi\)
\(432\) 0 0
\(433\) 9.19109 15.9194i 0.441695 0.765039i −0.556120 0.831102i \(-0.687711\pi\)
0.997815 + 0.0660630i \(0.0210438\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.85731 + 9.80471i 0.136684 + 0.469023i
\(438\) 0 0
\(439\) 3.80712 + 6.59412i 0.181704 + 0.314720i 0.942461 0.334317i \(-0.108505\pi\)
−0.760757 + 0.649037i \(0.775172\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.75168 + 6.49810i −0.178248 + 0.308734i −0.941280 0.337626i \(-0.890376\pi\)
0.763033 + 0.646360i \(0.223710\pi\)
\(444\) 0 0
\(445\) −31.7392 −1.50458
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.31836 −0.439761 −0.219880 0.975527i \(-0.570567\pi\)
−0.219880 + 0.975527i \(0.570567\pi\)
\(450\) 0 0
\(451\) −12.1895 21.1129i −0.573982 0.994166i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 55.4439 2.59925
\(456\) 0 0
\(457\) 22.4893 1.05200 0.526002 0.850483i \(-0.323690\pi\)
0.526002 + 0.850483i \(0.323690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.67303 + 2.89777i 0.0779207 + 0.134963i 0.902353 0.430998i \(-0.141839\pi\)
−0.824432 + 0.565961i \(0.808505\pi\)
\(462\) 0 0
\(463\) 26.0147 1.20901 0.604503 0.796603i \(-0.293372\pi\)
0.604503 + 0.796603i \(0.293372\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −11.3429 −0.524888 −0.262444 0.964947i \(-0.584528\pi\)
−0.262444 + 0.964947i \(0.584528\pi\)
\(468\) 0 0
\(469\) 22.0698 38.2260i 1.01909 1.76511i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14.8108 25.6530i −0.681001 1.17953i
\(474\) 0 0
\(475\) 0.596711 + 2.04758i 0.0273790 + 0.0939496i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.75199 9.96274i 0.262815 0.455209i −0.704174 0.710028i \(-0.748682\pi\)
0.966989 + 0.254818i \(0.0820157\pi\)
\(480\) 0 0
\(481\) 16.7936 29.0873i 0.765721 1.32627i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.31783 + 7.47870i −0.196062 + 0.339590i
\(486\) 0 0
\(487\) −22.3832 −1.01428 −0.507141 0.861863i \(-0.669298\pi\)
−0.507141 + 0.861863i \(0.669298\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.0160 + 19.0802i 0.497143 + 0.861077i 0.999995 0.00329588i \(-0.00104911\pi\)
−0.502852 + 0.864373i \(0.667716\pi\)
\(492\) 0 0
\(493\) 0.260826 0.0117470
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.85363 + 4.94264i 0.128003 + 0.221708i
\(498\) 0 0
\(499\) −0.605317 1.04844i −0.0270977 0.0469346i 0.852159 0.523284i \(-0.175293\pi\)
−0.879256 + 0.476349i \(0.841960\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 15.2462 26.4072i 0.679795 1.17744i −0.295247 0.955421i \(-0.595402\pi\)
0.975042 0.222019i \(-0.0712646\pi\)
\(504\) 0 0
\(505\) −17.8396 −0.793850
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.423144 + 0.732907i −0.0187555 + 0.0324855i −0.875251 0.483669i \(-0.839304\pi\)
0.856495 + 0.516155i \(0.172637\pi\)
\(510\) 0 0
\(511\) −14.7269 25.5077i −0.651479 1.12840i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −19.4966 33.7692i −0.859124 1.48805i
\(516\) 0 0
\(517\) −17.9449 + 31.0814i −0.789215 + 1.36696i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 42.6044 1.86653 0.933266 0.359187i \(-0.116946\pi\)
0.933266 + 0.359187i \(0.116946\pi\)
\(522\) 0 0
\(523\) −5.86224 + 10.1537i −0.256338 + 0.443990i −0.965258 0.261299i \(-0.915849\pi\)
0.708920 + 0.705289i \(0.249183\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.93573 + 15.4771i 0.389247 + 0.674195i
\(528\) 0 0
\(529\) 8.75536 + 15.1647i 0.380668 + 0.659336i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 45.0336 1.95062
\(534\) 0 0
\(535\) −7.37169 12.7681i −0.318706 0.552015i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 25.6932 1.10668
\(540\) 0 0
\(541\) 9.06614 15.7030i 0.389784 0.675126i −0.602636 0.798016i \(-0.705883\pi\)
0.992420 + 0.122890i \(0.0392163\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.0661 22.6312i 0.559692 0.969415i
\(546\) 0 0
\(547\) 21.7018 37.5886i 0.927902 1.60717i 0.141076 0.989999i \(-0.454944\pi\)
0.786826 0.617174i \(-0.211723\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.0612117 0.210045i −0.00260771 0.00894821i
\(552\) 0 0
\(553\) −0.335577 0.581236i −0.0142702 0.0247167i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.59828 7.96445i 0.194835 0.337465i −0.752011 0.659150i \(-0.770916\pi\)
0.946847 + 0.321686i \(0.104249\pi\)
\(558\) 0 0
\(559\) 54.7178 2.31431
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 41.3576 1.74302 0.871508 0.490382i \(-0.163143\pi\)
0.871508 + 0.490382i \(0.163143\pi\)
\(564\) 0 0
\(565\) 20.8059 + 36.0368i 0.875309 + 1.51608i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.22219 −0.0512369 −0.0256185 0.999672i \(-0.508156\pi\)
−0.0256185 + 0.999672i \(0.508156\pi\)
\(570\) 0 0
\(571\) 4.32150 0.180849 0.0904246 0.995903i \(-0.471178\pi\)
0.0904246 + 0.995903i \(0.471178\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.573183 0.992782i −0.0239034 0.0414019i
\(576\) 0 0
\(577\) 2.03863 0.0848695 0.0424347 0.999099i \(-0.486489\pi\)
0.0424347 + 0.999099i \(0.486489\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 26.8536 1.11408
\(582\) 0 0
\(583\) −5.34292 + 9.25421i −0.221281 + 0.383270i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.33011 + 7.49996i 0.178723 + 0.309557i 0.941443 0.337171i \(-0.109470\pi\)
−0.762721 + 0.646728i \(0.776137\pi\)
\(588\) 0 0
\(589\) 10.3668 10.8282i 0.427155 0.446170i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.48592 9.50190i 0.225280 0.390196i −0.731123 0.682245i \(-0.761004\pi\)
0.956403 + 0.292049i \(0.0943370\pi\)
\(594\) 0 0
\(595\) 23.3288 40.4067i 0.956389 1.65652i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −22.7109 + 39.3365i −0.927944 + 1.60725i −0.141187 + 0.989983i \(0.545092\pi\)
−0.786757 + 0.617263i \(0.788241\pi\)
\(600\) 0 0
\(601\) 29.8757 1.21865 0.609327 0.792919i \(-0.291440\pi\)
0.609327 + 0.792919i \(0.291440\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.205164 + 0.355354i 0.00834109 + 0.0144472i
\(606\) 0 0
\(607\) 34.7764 1.41153 0.705765 0.708446i \(-0.250604\pi\)
0.705765 + 0.708446i \(0.250604\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −33.1482 57.4144i −1.34103 2.32274i
\(612\) 0 0
\(613\) −7.38030 12.7831i −0.298087 0.516303i 0.677611 0.735421i \(-0.263015\pi\)
−0.975698 + 0.219118i \(0.929682\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.0181 + 22.5479i −0.524087 + 0.907746i 0.475520 + 0.879705i \(0.342260\pi\)
−0.999607 + 0.0280406i \(0.991073\pi\)
\(618\) 0 0
\(619\) 40.7497 1.63787 0.818933 0.573888i \(-0.194566\pi\)
0.818933 + 0.573888i \(0.194566\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 25.9572 44.9591i 1.03995 1.80125i
\(624\) 0 0
\(625\) 13.6035 + 23.5620i 0.544141 + 0.942479i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.1323 24.4778i −0.563491 0.975996i
\(630\) 0 0
\(631\) 15.9106 27.5580i 0.633392 1.09707i −0.353461 0.935449i \(-0.614995\pi\)
0.986853 0.161619i \(-0.0516714\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 43.1684 1.71309
\(636\) 0 0
\(637\) −23.7306 + 41.1025i −0.940239 + 1.62854i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.08233 7.07080i −0.161242 0.279280i 0.774072 0.633097i \(-0.218217\pi\)
−0.935314 + 0.353818i \(0.884883\pi\)
\(642\) 0 0
\(643\) 3.29851 + 5.71319i 0.130081 + 0.225306i 0.923707 0.383099i \(-0.125143\pi\)
−0.793627 + 0.608405i \(0.791810\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.60015 0.102223 0.0511113 0.998693i \(-0.483724\pi\)
0.0511113 + 0.998693i \(0.483724\pi\)
\(648\) 0 0
\(649\) −6.44120 11.1565i −0.252839 0.437930i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.39312 0.0936498 0.0468249 0.998903i \(-0.485090\pi\)
0.0468249 + 0.998903i \(0.485090\pi\)
\(654\) 0 0
\(655\) −3.77835 + 6.54429i −0.147632 + 0.255706i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −11.3417 + 19.6443i −0.441808 + 0.765235i −0.997824 0.0659373i \(-0.978996\pi\)
0.556015 + 0.831172i \(0.312330\pi\)
\(660\) 0 0
\(661\) −20.2750 + 35.1173i −0.788605 + 1.36590i 0.138216 + 0.990402i \(0.455863\pi\)
−0.926821 + 0.375502i \(0.877470\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −38.0147 9.30404i −1.47415 0.360795i
\(666\) 0 0
\(667\) 0.0587981 + 0.101841i 0.00227667 + 0.00394331i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.89468 + 8.47784i −0.188957 + 0.327283i
\(672\) 0 0
\(673\) −20.1004 −0.774813 −0.387406 0.921909i \(-0.626629\pi\)
−0.387406 + 0.921909i \(0.626629\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.3717 0.744515 0.372257 0.928130i \(-0.378584\pi\)
0.372257 + 0.928130i \(0.378584\pi\)
\(678\) 0 0
\(679\) −7.06247 12.2326i −0.271033 0.469443i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.1151 −0.616626 −0.308313 0.951285i \(-0.599764\pi\)
−0.308313 + 0.951285i \(0.599764\pi\)
\(684\) 0 0
\(685\) −14.0073 −0.535193
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.86959 17.0946i −0.376001 0.651253i
\(690\) 0 0
\(691\) −15.4145 −0.586397 −0.293198 0.956052i \(-0.594720\pi\)
−0.293198 + 0.956052i \(0.594720\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.4794 −0.814761
\(696\) 0 0
\(697\) 18.9485 32.8198i 0.717727 1.24314i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.9148 + 31.0294i 0.676634 + 1.17197i 0.975988 + 0.217823i \(0.0698956\pi\)
−0.299354 + 0.954142i \(0.596771\pi\)
\(702\) 0 0
\(703\) −16.3955 + 17.1254i −0.618369 + 0.645896i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.5897 25.2701i 0.548701 0.950378i
\(708\) 0 0
\(709\) −4.41400 + 7.64527i −0.165771 + 0.287124i −0.936929 0.349520i \(-0.886345\pi\)
0.771158 + 0.636644i \(0.219678\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.02877 + 6.97803i −0.150879 + 0.261329i
\(714\) 0 0
\(715\) −48.3650 −1.80875
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.13355 + 14.0877i 0.303330 + 0.525383i 0.976888 0.213751i \(-0.0685682\pi\)
−0.673558 + 0.739134i \(0.735235\pi\)
\(720\) 0 0
\(721\) 63.7795 2.37527
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.0122792 + 0.0212682i 0.000456038 + 0.000789882i
\(726\) 0 0
\(727\) 20.3803 + 35.2997i 0.755863 + 1.30919i 0.944944 + 0.327232i \(0.106116\pi\)
−0.189081 + 0.981962i \(0.560551\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 23.0233 39.8775i 0.851547 1.47492i
\(732\) 0 0
\(733\) 19.6827 0.726998 0.363499 0.931595i \(-0.381582\pi\)
0.363499 + 0.931595i \(0.381582\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.2520 + 33.3454i −0.709156 + 1.22829i
\(738\) 0 0
\(739\) −21.3775 37.0269i −0.786383 1.36206i −0.928169 0.372158i \(-0.878618\pi\)
0.141786 0.989897i \(-0.454715\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.34239 + 14.4494i 0.306052 + 0.530098i 0.977495 0.210958i \(-0.0676584\pi\)
−0.671443 + 0.741057i \(0.734325\pi\)
\(744\) 0 0
\(745\) −7.20516 + 12.4797i −0.263977 + 0.457221i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.1151 0.881146
\(750\) 0 0
\(751\) −15.4305 + 26.7264i −0.563067 + 0.975260i 0.434160 + 0.900836i \(0.357045\pi\)
−0.997227 + 0.0744242i \(0.976288\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.02877 10.4421i −0.219409 0.380028i
\(756\) 0 0
\(757\) 7.99139 + 13.8415i 0.290452 + 0.503078i 0.973917 0.226906i \(-0.0728610\pi\)
−0.683465 + 0.729984i \(0.739528\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.9975 −0.833658 −0.416829 0.908985i \(-0.636859\pi\)
−0.416829 + 0.908985i \(0.636859\pi\)
\(762\) 0 0
\(763\) 21.3717 + 37.0169i 0.773707 + 1.34010i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23.7967 0.859249
\(768\) 0 0
\(769\) −14.9731 + 25.9342i −0.539944 + 0.935211i 0.458962 + 0.888456i \(0.348221\pi\)
−0.998906 + 0.0467548i \(0.985112\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.75379 4.76970i 0.0990469 0.171554i −0.812244 0.583318i \(-0.801754\pi\)
0.911290 + 0.411764i \(0.135087\pi\)
\(774\) 0 0
\(775\) −0.841355 + 1.45727i −0.0302224 + 0.0523467i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −30.8769 7.55709i −1.10628 0.270761i
\(780\) 0 0
\(781\) −2.48929 4.31157i −0.0890737 0.154280i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.16653 8.94869i 0.184401 0.319392i
\(786\) 0 0
\(787\) 8.40719 0.299684 0.149842 0.988710i \(-0.452123\pi\)
0.149842 + 0.988710i \(0.452123\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −68.0624 −2.42002
\(792\) 0 0
\(793\) −9.04159 15.6605i −0.321076 0.556120i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.2186 −1.17666 −0.588332 0.808620i \(-0.700215\pi\)
−0.588332 + 0.808620i \(0.700215\pi\)
\(798\) 0 0
\(799\) −55.7904 −1.97372
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.8466 + 22.2510i 0.453347 + 0.785219i
\(804\) 0 0
\(805\) 21.0361 0.741426
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.8971 −0.453438 −0.226719 0.973960i \(-0.572800\pi\)
−0.226719 + 0.973960i \(0.572800\pi\)
\(810\) 0 0
\(811\) −1.33432 + 2.31110i −0.0468542 + 0.0811539i −0.888501 0.458874i \(-0.848253\pi\)
0.841647 + 0.540028i \(0.181586\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −14.4342 25.0007i −0.505607 0.875736i
\(816\) 0 0
\(817\) −37.5168 9.18219i −1.31255 0.321244i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16.8267 + 29.1448i −0.587257 + 1.01716i 0.407333 + 0.913280i \(0.366459\pi\)
−0.994590 + 0.103880i \(0.966874\pi\)
\(822\) 0 0
\(823\) 8.91978 15.4495i 0.310924 0.538536i −0.667639 0.744485i \(-0.732695\pi\)
0.978563 + 0.205949i \(0.0660282\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.87663 + 3.25041i −0.0652567 + 0.113028i −0.896808 0.442420i \(-0.854120\pi\)
0.831551 + 0.555448i \(0.187453\pi\)
\(828\) 0 0
\(829\) 2.77781 0.0964773 0.0482386 0.998836i \(-0.484639\pi\)
0.0482386 + 0.998836i \(0.484639\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 19.9700 + 34.5890i 0.691918 + 1.19844i
\(834\) 0 0
\(835\) −13.5468 −0.468807
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 28.1114 + 48.6904i 0.970513 + 1.68098i 0.694010 + 0.719966i \(0.255843\pi\)
0.276504 + 0.961013i \(0.410824\pi\)
\(840\) 0 0
\(841\) 14.4987 + 25.1126i 0.499957 + 0.865950i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 29.4415 50.9942i 1.01282 1.75425i
\(846\) 0 0
\(847\) −0.671153 −0.0230611
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.37169 11.0361i 0.218419 0.378312i
\(852\) 0 0
\(853\) 5.16232 + 8.94140i 0.176754 + 0.306148i 0.940767 0.339054i \(-0.110107\pi\)
−0.764013 + 0.645201i \(0.776774\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.5894 + 46.0543i 0.908278 + 1.57318i 0.816455 + 0.577408i \(0.195936\pi\)
0.0918226 + 0.995775i \(0.470731\pi\)
\(858\) 0 0
\(859\) −5.12651 + 8.87938i −0.174914 + 0.302960i −0.940132 0.340812i \(-0.889298\pi\)
0.765217 + 0.643772i \(0.222631\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.7679 −0.366545 −0.183273 0.983062i \(-0.558669\pi\)
−0.183273 + 0.983062i \(0.558669\pi\)
\(864\) 0 0
\(865\) 2.13776 3.70271i 0.0726860 0.125896i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0.292731 + 0.507025i 0.00993022 + 0.0171996i
\(870\) 0 0
\(871\) −35.5628 61.5965i −1.20500 2.08712i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −40.4998 −1.36914
\(876\) 0 0
\(877\) −11.3638 19.6827i −0.383729 0.664637i 0.607863 0.794042i \(-0.292027\pi\)
−0.991592 + 0.129404i \(0.958693\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.44644 0.284568 0.142284 0.989826i \(-0.454555\pi\)
0.142284 + 0.989826i \(0.454555\pi\)
\(882\) 0 0
\(883\) 22.5196 39.0051i 0.757846 1.31263i −0.186101 0.982531i \(-0.559585\pi\)
0.943947 0.330097i \(-0.107081\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 10.4231 18.0534i 0.349975 0.606174i −0.636270 0.771467i \(-0.719523\pi\)
0.986245 + 0.165292i \(0.0528568\pi\)
\(888\) 0 0
\(889\) −35.3043 + 61.1488i −1.18407 + 2.05087i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 13.0931 + 44.9284i 0.438144 + 1.50347i
\(894\) 0 0
\(895\) 9.68091 + 16.7678i 0.323597 + 0.560487i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0.0863077 0.149489i 0.00287852 0.00498575i
\(900\) 0 0
\(901\) −16.6111 −0.553396
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.8757 1.69116
\(906\) 0 0
\(907\) 12.2055 + 21.1405i 0.405276 + 0.701959i 0.994354 0.106118i \(-0.0338421\pi\)
−0.589078 + 0.808077i \(0.700509\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.2744 0.804248 0.402124 0.915585i \(-0.368272\pi\)
0.402124 + 0.915585i \(0.368272\pi\)
\(912\) 0 0
\(913\) −23.4250 −0.775255
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.18007 10.7042i −0.204084 0.353484i
\(918\) 0 0
\(919\) 31.8568 1.05086 0.525429 0.850837i \(-0.323905\pi\)
0.525429 + 0.850837i \(0.323905\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 9.19656 0.302708
\(924\) 0 0
\(925\) 1.33064 2.30474i 0.0437513 0.0757795i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25.1044 43.4820i −0.823648 1.42660i −0.902949 0.429749i \(-0.858602\pi\)
0.0793010 0.996851i \(-0.474731\pi\)
\(930\) 0 0
\(931\) 23.1681 24.1994i 0.759304 0.793104i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −20.3503 + 35.2477i −0.665525 + 1.15272i
\(936\) 0 0
\(937\) 2.32800 4.03222i 0.0760525 0.131727i −0.825491 0.564415i \(-0.809102\pi\)
0.901543 + 0.432689i \(0.142435\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.6987 + 20.2627i −0.381366 + 0.660544i −0.991258 0.131940i \(-0.957879\pi\)
0.609892 + 0.792484i \(0.291213\pi\)
\(942\) 0 0
\(943\) 17.0863 0.556407
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.7134 + 49.7330i 0.933059 + 1.61611i 0.778060 + 0.628190i \(0.216204\pi\)
0.154999 + 0.987915i \(0.450463\pi\)
\(948\) 0 0
\(949\) −47.4611 −1.54065
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.9966 34.6352i −0.647755 1.12194i −0.983658 0.180047i \(-0.942375\pi\)
0.335903 0.941896i \(-0.390958\pi\)
\(954\) 0 0
\(955\) −5.37169 9.30404i −0.173824 0.301072i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.4556 19.8417i 0.369920 0.640721i
\(960\) 0 0
\(961\) −19.1726 −0.618471
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.2627 + 24.7037i −0.459133 + 0.795241i
\(966\) 0 0
\(967\) −1.94120 3.36226i −0.0624248 0.108123i 0.833124 0.553086i \(-0.186550\pi\)
−0.895549 + 0.444963i \(0.853217\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.687414 + 1.19064i 0.0220602 + 0.0382093i 0.876845 0.480774i \(-0.159644\pi\)
−0.854785 + 0.518983i \(0.826311\pi\)
\(972\) 0 0
\(973\) 17.5665 30.4260i 0.563155 0.975412i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.2541 −0.647986 −0.323993 0.946059i \(-0.605025\pi\)
−0.323993 + 0.946059i \(0.605025\pi\)
\(978\) 0 0
\(979\) −22.6430 + 39.2188i −0.723673 + 1.25344i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −6.38030 11.0510i −0.203500 0.352472i 0.746154 0.665774i \(-0.231898\pi\)
−0.949654 + 0.313301i \(0.898565\pi\)
\(984\) 0 0
\(985\) 5.00000 + 8.66025i 0.159313 + 0.275939i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 20.7606 0.660149
\(990\) 0 0
\(991\) −9.47647 16.4137i −0.301030 0.521399i 0.675339 0.737507i \(-0.263997\pi\)
−0.976370 + 0.216108i \(0.930664\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.70306 −0.212501
\(996\) 0 0
\(997\) 6.03244 10.4485i 0.191049 0.330907i −0.754549 0.656244i \(-0.772144\pi\)
0.945598 + 0.325337i \(0.105478\pi\)
\(998\) 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 2736.2.s.y.577.3 6
3.2 odd 2 304.2.i.f.273.2 6
4.3 odd 2 1368.2.s.k.577.3 6
12.11 even 2 152.2.i.c.121.2 yes 6
19.11 even 3 inner 2736.2.s.y.1873.3 6
24.5 odd 2 1216.2.i.m.577.2 6
24.11 even 2 1216.2.i.n.577.2 6
57.11 odd 6 304.2.i.f.49.2 6
57.26 odd 6 5776.2.a.bk.1.2 3
57.50 even 6 5776.2.a.bq.1.2 3
76.11 odd 6 1368.2.s.k.505.3 6
228.11 even 6 152.2.i.c.49.2 6
228.83 even 6 2888.2.a.r.1.2 3
228.107 odd 6 2888.2.a.n.1.2 3
456.11 even 6 1216.2.i.n.961.2 6
456.125 odd 6 1216.2.i.m.961.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.i.c.49.2 6 228.11 even 6
152.2.i.c.121.2 yes 6 12.11 even 2
304.2.i.f.49.2 6 57.11 odd 6
304.2.i.f.273.2 6 3.2 odd 2
1216.2.i.m.577.2 6 24.5 odd 2
1216.2.i.m.961.2 6 456.125 odd 6
1216.2.i.n.577.2 6 24.11 even 2
1216.2.i.n.961.2 6 456.11 even 6
1368.2.s.k.505.3 6 76.11 odd 6
1368.2.s.k.577.3 6 4.3 odd 2
2736.2.s.y.577.3 6 1.1 even 1 trivial
2736.2.s.y.1873.3 6 19.11 even 3 inner
2888.2.a.n.1.2 3 228.107 odd 6
2888.2.a.r.1.2 3 228.83 even 6
5776.2.a.bk.1.2 3 57.26 odd 6
5776.2.a.bq.1.2 3 57.50 even 6