Properties

Label 1728.2.i.n.1153.1
Level $1728$
Weight $2$
Character 1728.1153
Analytic conductor $13.798$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(577,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1728.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1153.1
Root \(-1.02187 + 0.977642i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1153
Dual form 1728.2.i.n.577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.18614 - 2.05446i) q^{5} +(-1.10489 + 1.91373i) q^{7} +O(q^{10})\) \(q+(-1.18614 - 2.05446i) q^{5} +(-1.10489 + 1.91373i) q^{7} +(2.96790 - 5.14055i) q^{11} +(2.18614 + 3.78651i) q^{13} -3.37228 q^{17} +3.72601 q^{19} +(1.10489 + 1.91373i) q^{23} +(-0.313859 + 0.543620i) q^{25} +(-0.186141 + 0.322405i) q^{29} +(-4.83090 - 8.36737i) q^{31} +5.24224 q^{35} -4.00000 q^{37} +(0.500000 + 0.866025i) q^{41} +(2.96790 - 5.14055i) q^{43} +(1.10489 - 1.91373i) q^{47} +(1.05842 + 1.83324i) q^{49} -4.00000 q^{53} -14.0814 q^{55} +(-5.17769 - 8.96801i) q^{59} +(7.55842 - 13.0916i) q^{61} +(5.18614 - 8.98266i) q^{65} +(-5.17769 - 8.96801i) q^{67} -4.41957 q^{71} +4.62772 q^{73} +(6.55842 + 11.3595i) q^{77} +(4.83090 - 8.36737i) q^{79} +(7.04069 - 12.1948i) q^{83} +(4.00000 + 6.92820i) q^{85} -1.25544 q^{89} -9.66181 q^{91} +(-4.41957 - 7.65492i) q^{95} +(-4.50000 + 7.79423i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} + 6 q^{13} - 4 q^{17} - 14 q^{25} + 10 q^{29} - 32 q^{37} + 4 q^{41} - 26 q^{49} - 32 q^{53} + 26 q^{61} + 30 q^{65} + 60 q^{73} + 18 q^{77} + 32 q^{85} - 56 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −1.18614 2.05446i −0.530458 0.918781i −0.999368 0.0355348i \(-0.988687\pi\)
0.468910 0.883246i \(-0.344647\pi\)
\(6\) 0 0
\(7\) −1.10489 + 1.91373i −0.417610 + 0.723322i −0.995699 0.0926519i \(-0.970466\pi\)
0.578088 + 0.815974i \(0.303799\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.96790 5.14055i 0.894855 1.54993i 0.0608712 0.998146i \(-0.480612\pi\)
0.833984 0.551789i \(-0.186055\pi\)
\(12\) 0 0
\(13\) 2.18614 + 3.78651i 0.606326 + 1.05019i 0.991840 + 0.127486i \(0.0406908\pi\)
−0.385514 + 0.922702i \(0.625976\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.37228 −0.817898 −0.408949 0.912557i \(-0.634105\pi\)
−0.408949 + 0.912557i \(0.634105\pi\)
\(18\) 0 0
\(19\) 3.72601 0.854805 0.427403 0.904061i \(-0.359429\pi\)
0.427403 + 0.904061i \(0.359429\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.10489 + 1.91373i 0.230386 + 0.399041i 0.957922 0.287029i \(-0.0926677\pi\)
−0.727536 + 0.686070i \(0.759334\pi\)
\(24\) 0 0
\(25\) −0.313859 + 0.543620i −0.0627719 + 0.108724i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.186141 + 0.322405i −0.0345655 + 0.0598691i −0.882791 0.469767i \(-0.844338\pi\)
0.848225 + 0.529636i \(0.177671\pi\)
\(30\) 0 0
\(31\) −4.83090 8.36737i −0.867656 1.50282i −0.864386 0.502830i \(-0.832292\pi\)
−0.00327038 0.999995i \(-0.501041\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.24224 0.886099
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.500000 + 0.866025i 0.0780869 + 0.135250i 0.902424 0.430848i \(-0.141786\pi\)
−0.824338 + 0.566099i \(0.808452\pi\)
\(42\) 0 0
\(43\) 2.96790 5.14055i 0.452600 0.783927i −0.545946 0.837820i \(-0.683830\pi\)
0.998547 + 0.0538934i \(0.0171631\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.10489 1.91373i 0.161165 0.279146i −0.774122 0.633037i \(-0.781808\pi\)
0.935287 + 0.353891i \(0.115141\pi\)
\(48\) 0 0
\(49\) 1.05842 + 1.83324i 0.151203 + 0.261892i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −14.0814 −1.89873
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.17769 8.96801i −0.674077 1.16754i −0.976738 0.214437i \(-0.931208\pi\)
0.302661 0.953098i \(-0.402125\pi\)
\(60\) 0 0
\(61\) 7.55842 13.0916i 0.967757 1.67620i 0.265738 0.964045i \(-0.414384\pi\)
0.702019 0.712159i \(-0.252282\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.18614 8.98266i 0.643262 1.11416i
\(66\) 0 0
\(67\) −5.17769 8.96801i −0.632555 1.09562i −0.987028 0.160551i \(-0.948673\pi\)
0.354473 0.935066i \(-0.384660\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.41957 −0.524507 −0.262253 0.964999i \(-0.584466\pi\)
−0.262253 + 0.964999i \(0.584466\pi\)
\(72\) 0 0
\(73\) 4.62772 0.541634 0.270817 0.962631i \(-0.412706\pi\)
0.270817 + 0.962631i \(0.412706\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.55842 + 11.3595i 0.747402 + 1.29454i
\(78\) 0 0
\(79\) 4.83090 8.36737i 0.543519 0.941403i −0.455179 0.890400i \(-0.650425\pi\)
0.998698 0.0510030i \(-0.0162418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.04069 12.1948i 0.772816 1.33856i −0.163198 0.986593i \(-0.552181\pi\)
0.936014 0.351963i \(-0.114486\pi\)
\(84\) 0 0
\(85\) 4.00000 + 6.92820i 0.433861 + 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.25544 −0.133076 −0.0665380 0.997784i \(-0.521195\pi\)
−0.0665380 + 0.997784i \(0.521195\pi\)
\(90\) 0 0
\(91\) −9.66181 −1.01283
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.41957 7.65492i −0.453439 0.785379i
\(96\) 0 0
\(97\) −4.50000 + 7.79423i −0.456906 + 0.791384i −0.998796 0.0490655i \(-0.984376\pi\)
0.541890 + 0.840450i \(0.317709\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.55842 13.0916i 0.752091 1.30266i −0.194716 0.980860i \(-0.562379\pi\)
0.946808 0.321800i \(-0.104288\pi\)
\(102\) 0 0
\(103\) 1.10489 + 1.91373i 0.108868 + 0.188566i 0.915312 0.402745i \(-0.131944\pi\)
−0.806444 + 0.591311i \(0.798611\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.72601 −0.360207 −0.180104 0.983648i \(-0.557643\pi\)
−0.180104 + 0.983648i \(0.557643\pi\)
\(108\) 0 0
\(109\) −17.4891 −1.67515 −0.837577 0.546319i \(-0.816029\pi\)
−0.837577 + 0.546319i \(0.816029\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.81386 + 10.0699i 0.546922 + 0.947296i 0.998483 + 0.0550564i \(0.0175339\pi\)
−0.451561 + 0.892240i \(0.649133\pi\)
\(114\) 0 0
\(115\) 2.62112 4.53991i 0.244420 0.423349i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.72601 6.45364i 0.341563 0.591604i
\(120\) 0 0
\(121\) −12.1168 20.9870i −1.10153 1.90791i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −10.3723 −0.927725
\(126\) 0 0
\(127\) 7.45202 0.661260 0.330630 0.943760i \(-0.392739\pi\)
0.330630 + 0.943760i \(0.392739\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.83090 + 8.36737i 0.422078 + 0.731061i 0.996143 0.0877494i \(-0.0279675\pi\)
−0.574065 + 0.818810i \(0.694634\pi\)
\(132\) 0 0
\(133\) −4.11684 + 7.13058i −0.356976 + 0.618300i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.24456 + 10.8159i −0.533509 + 0.924065i 0.465725 + 0.884930i \(0.345794\pi\)
−0.999234 + 0.0391351i \(0.987540\pi\)
\(138\) 0 0
\(139\) 1.45167 + 2.51437i 0.123129 + 0.213266i 0.921000 0.389562i \(-0.127374\pi\)
−0.797871 + 0.602829i \(0.794040\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 25.9530 2.17030
\(144\) 0 0
\(145\) 0.883156 0.0733421
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.81386 + 10.0699i 0.476290 + 0.824958i 0.999631 0.0271650i \(-0.00864795\pi\)
−0.523341 + 0.852123i \(0.675315\pi\)
\(150\) 0 0
\(151\) 1.10489 1.91373i 0.0899149 0.155737i −0.817560 0.575843i \(-0.804674\pi\)
0.907475 + 0.420106i \(0.138007\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.4603 + 19.8498i −0.920510 + 1.59437i
\(156\) 0 0
\(157\) −1.55842 2.69927i −0.124376 0.215425i 0.797113 0.603830i \(-0.206359\pi\)
−0.921489 + 0.388405i \(0.873026\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.88316 −0.384847
\(162\) 0 0
\(163\) −8.83915 −0.692335 −0.346168 0.938173i \(-0.612517\pi\)
−0.346168 + 0.938173i \(0.612517\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.83090 8.36737i −0.373827 0.647487i 0.616324 0.787493i \(-0.288621\pi\)
−0.990151 + 0.140006i \(0.955288\pi\)
\(168\) 0 0
\(169\) −3.05842 + 5.29734i −0.235263 + 0.407488i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.186141 0.322405i 0.0141520 0.0245120i −0.858863 0.512206i \(-0.828828\pi\)
0.873015 + 0.487694i \(0.162162\pi\)
\(174\) 0 0
\(175\) −0.693562 1.20128i −0.0524284 0.0908086i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.83915 0.660669 0.330334 0.943864i \(-0.392838\pi\)
0.330334 + 0.943864i \(0.392838\pi\)
\(180\) 0 0
\(181\) 8.23369 0.612005 0.306003 0.952031i \(-0.401008\pi\)
0.306003 + 0.952031i \(0.401008\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.74456 + 8.21782i 0.348827 + 0.604186i
\(186\) 0 0
\(187\) −10.0086 + 17.3354i −0.731901 + 1.26769i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.55691 14.8210i 0.619157 1.07241i −0.370483 0.928839i \(-0.620808\pi\)
0.989640 0.143572i \(-0.0458587\pi\)
\(192\) 0 0
\(193\) −0.500000 0.866025i −0.0359908 0.0623379i 0.847469 0.530845i \(-0.178125\pi\)
−0.883460 + 0.468507i \(0.844792\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0.744563 0.0530479 0.0265239 0.999648i \(-0.491556\pi\)
0.0265239 + 0.999648i \(0.491556\pi\)
\(198\) 0 0
\(199\) 20.7107 1.46815 0.734073 0.679071i \(-0.237617\pi\)
0.734073 + 0.679071i \(0.237617\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.411331 0.712446i −0.0288698 0.0500039i
\(204\) 0 0
\(205\) 1.18614 2.05446i 0.0828437 0.143489i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.0584 19.1537i 0.764927 1.32489i
\(210\) 0 0
\(211\) −2.62112 4.53991i −0.180445 0.312540i 0.761587 0.648063i \(-0.224421\pi\)
−0.942032 + 0.335522i \(0.891087\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.0814 −0.960342
\(216\) 0 0
\(217\) 21.3505 1.44937
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.37228 12.7692i −0.495913 0.858947i
\(222\) 0 0
\(223\) −1.10489 + 1.91373i −0.0739891 + 0.128153i −0.900646 0.434553i \(-0.856906\pi\)
0.826657 + 0.562706i \(0.190240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.48412 + 7.76673i −0.297622 + 0.515496i −0.975591 0.219594i \(-0.929527\pi\)
0.677970 + 0.735090i \(0.262860\pi\)
\(228\) 0 0
\(229\) −2.18614 3.78651i −0.144464 0.250219i 0.784709 0.619865i \(-0.212813\pi\)
−0.929173 + 0.369645i \(0.879479\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.86141 −0.187457 −0.0937285 0.995598i \(-0.529879\pi\)
−0.0937285 + 0.995598i \(0.529879\pi\)
\(234\) 0 0
\(235\) −5.24224 −0.341966
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.55691 + 14.8210i 0.553501 + 0.958692i 0.998018 + 0.0629214i \(0.0200418\pi\)
−0.444518 + 0.895770i \(0.646625\pi\)
\(240\) 0 0
\(241\) −0.127719 + 0.221215i −0.00822708 + 0.0142497i −0.870110 0.492858i \(-0.835952\pi\)
0.861883 + 0.507108i \(0.169285\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.51087 4.34896i 0.160414 0.277845i
\(246\) 0 0
\(247\) 8.14558 + 14.1086i 0.518291 + 0.897706i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −11.1780 −0.705551 −0.352776 0.935708i \(-0.614762\pi\)
−0.352776 + 0.935708i \(0.614762\pi\)
\(252\) 0 0
\(253\) 13.1168 0.824649
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.24456 5.61975i −0.202390 0.350550i 0.746908 0.664928i \(-0.231538\pi\)
−0.949298 + 0.314377i \(0.898204\pi\)
\(258\) 0 0
\(259\) 4.41957 7.65492i 0.274619 0.475654i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −8.55691 + 14.8210i −0.527642 + 0.913903i 0.471839 + 0.881685i \(0.343590\pi\)
−0.999481 + 0.0322179i \(0.989743\pi\)
\(264\) 0 0
\(265\) 4.74456 + 8.21782i 0.291456 + 0.504817i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.48913 0.578562 0.289281 0.957244i \(-0.406584\pi\)
0.289281 + 0.957244i \(0.406584\pi\)
\(270\) 0 0
\(271\) 14.9040 0.905356 0.452678 0.891674i \(-0.350469\pi\)
0.452678 + 0.891674i \(0.350469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.86301 + 3.22682i 0.112343 + 0.194585i
\(276\) 0 0
\(277\) 6.55842 11.3595i 0.394057 0.682527i −0.598923 0.800807i \(-0.704404\pi\)
0.992980 + 0.118279i \(0.0377378\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.9307 22.3966i 0.771381 1.33607i −0.165425 0.986222i \(-0.552900\pi\)
0.936806 0.349849i \(-0.113767\pi\)
\(282\) 0 0
\(283\) 2.62112 + 4.53991i 0.155809 + 0.269870i 0.933353 0.358959i \(-0.116868\pi\)
−0.777544 + 0.628828i \(0.783535\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.20979 −0.130440
\(288\) 0 0
\(289\) −5.62772 −0.331042
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 10.1861 + 17.6429i 0.595081 + 1.03071i 0.993535 + 0.113522i \(0.0362132\pi\)
−0.398455 + 0.917188i \(0.630453\pi\)
\(294\) 0 0
\(295\) −12.2829 + 21.2747i −0.715140 + 1.23866i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.83090 + 8.36737i −0.279378 + 0.483898i
\(300\) 0 0
\(301\) 6.55842 + 11.3595i 0.378021 + 0.654752i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −35.8614 −2.05342
\(306\) 0 0
\(307\) 28.8563 1.64692 0.823459 0.567376i \(-0.192041\pi\)
0.823459 + 0.567376i \(0.192041\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.34713 10.9935i −0.359913 0.623387i 0.628033 0.778186i \(-0.283860\pi\)
−0.987946 + 0.154800i \(0.950527\pi\)
\(312\) 0 0
\(313\) −9.61684 + 16.6569i −0.543576 + 0.941502i 0.455119 + 0.890431i \(0.349597\pi\)
−0.998695 + 0.0510708i \(0.983737\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.18614 + 7.25061i −0.235117 + 0.407235i −0.959307 0.282366i \(-0.908881\pi\)
0.724190 + 0.689601i \(0.242214\pi\)
\(318\) 0 0
\(319\) 1.10489 + 1.91373i 0.0618621 + 0.107148i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.5652 −0.699144
\(324\) 0 0
\(325\) −2.74456 −0.152241
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.44158 + 4.22894i 0.134609 + 0.233149i
\(330\) 0 0
\(331\) −16.7025 + 28.9296i −0.918052 + 1.59011i −0.115682 + 0.993286i \(0.536905\pi\)
−0.802370 + 0.596827i \(0.796428\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.2829 + 21.2747i −0.671088 + 1.16236i
\(336\) 0 0
\(337\) 4.50000 + 7.79423i 0.245131 + 0.424579i 0.962168 0.272456i \(-0.0878358\pi\)
−0.717038 + 0.697034i \(0.754502\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −57.3505 −3.10571
\(342\) 0 0
\(343\) −20.1463 −1.08780
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.17769 + 8.96801i 0.277953 + 0.481428i 0.970876 0.239583i \(-0.0770108\pi\)
−0.692923 + 0.721011i \(0.743677\pi\)
\(348\) 0 0
\(349\) −11.5584 + 20.0198i −0.618708 + 1.07163i 0.371014 + 0.928627i \(0.379010\pi\)
−0.989722 + 0.143007i \(0.954323\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.61684 + 16.6569i −0.511853 + 0.886555i 0.488053 + 0.872814i \(0.337707\pi\)
−0.999906 + 0.0137411i \(0.995626\pi\)
\(354\) 0 0
\(355\) 5.24224 + 9.07982i 0.278229 + 0.481907i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −22.3561 −1.17991 −0.589954 0.807437i \(-0.700854\pi\)
−0.589954 + 0.807437i \(0.700854\pi\)
\(360\) 0 0
\(361\) −5.11684 −0.269308
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.48913 9.50744i −0.287314 0.497642i
\(366\) 0 0
\(367\) 13.6700 23.6772i 0.713571 1.23594i −0.249937 0.968262i \(-0.580410\pi\)
0.963508 0.267679i \(-0.0862566\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.41957 7.65492i 0.229453 0.397424i
\(372\) 0 0
\(373\) −8.55842 14.8236i −0.443138 0.767538i 0.554782 0.831996i \(-0.312802\pi\)
−0.997920 + 0.0644576i \(0.979468\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.62772 −0.0838318
\(378\) 0 0
\(379\) 8.14558 0.418411 0.209205 0.977872i \(-0.432912\pi\)
0.209205 + 0.977872i \(0.432912\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.4603 + 19.8498i 0.585592 + 1.01428i 0.994801 + 0.101835i \(0.0324713\pi\)
−0.409209 + 0.912441i \(0.634195\pi\)
\(384\) 0 0
\(385\) 15.5584 26.9480i 0.792931 1.37340i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.67527 16.7581i 0.490555 0.849667i −0.509385 0.860538i \(-0.670127\pi\)
0.999941 + 0.0108715i \(0.00346057\pi\)
\(390\) 0 0
\(391\) −3.72601 6.45364i −0.188432 0.326375i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −22.9205 −1.15326
\(396\) 0 0
\(397\) 4.00000 0.200754 0.100377 0.994949i \(-0.467995\pi\)
0.100377 + 0.994949i \(0.467995\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.9891 + 24.2299i 0.698584 + 1.20998i 0.968958 + 0.247227i \(0.0795193\pi\)
−0.270374 + 0.962755i \(0.587147\pi\)
\(402\) 0 0
\(403\) 21.1221 36.5845i 1.05217 1.82240i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −11.8716 + 20.5622i −0.588453 + 1.01923i
\(408\) 0 0
\(409\) 9.12772 + 15.8097i 0.451337 + 0.781738i 0.998469 0.0553079i \(-0.0176140\pi\)
−0.547133 + 0.837046i \(0.684281\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 22.8832 1.12601
\(414\) 0 0
\(415\) −33.4050 −1.63979
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.62112 + 4.53991i 0.128050 + 0.221789i 0.922921 0.384989i \(-0.125795\pi\)
−0.794871 + 0.606778i \(0.792462\pi\)
\(420\) 0 0
\(421\) −11.1861 + 19.3750i −0.545179 + 0.944278i 0.453416 + 0.891299i \(0.350205\pi\)
−0.998596 + 0.0529792i \(0.983128\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.05842 1.83324i 0.0513410 0.0889252i
\(426\) 0 0
\(427\) 16.7025 + 28.9296i 0.808291 + 1.40000i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.45202 −0.358951 −0.179476 0.983762i \(-0.557440\pi\)
−0.179476 + 0.983762i \(0.557440\pi\)
\(432\) 0 0
\(433\) −18.1168 −0.870640 −0.435320 0.900276i \(-0.643365\pi\)
−0.435320 + 0.900276i \(0.643365\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.11684 + 7.13058i 0.196935 + 0.341102i
\(438\) 0 0
\(439\) −11.4603 + 19.8498i −0.546969 + 0.947377i 0.451512 + 0.892265i \(0.350885\pi\)
−0.998480 + 0.0551120i \(0.982448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.48412 7.76673i 0.213047 0.369008i −0.739620 0.673025i \(-0.764995\pi\)
0.952667 + 0.304017i \(0.0983279\pi\)
\(444\) 0 0
\(445\) 1.48913 + 2.57924i 0.0705913 + 0.122268i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.1168 0.854987 0.427493 0.904018i \(-0.359397\pi\)
0.427493 + 0.904018i \(0.359397\pi\)
\(450\) 0 0
\(451\) 5.93580 0.279506
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 11.4603 + 19.8498i 0.537265 + 0.930571i
\(456\) 0 0
\(457\) 12.9891 22.4978i 0.607606 1.05240i −0.384028 0.923321i \(-0.625463\pi\)
0.991634 0.129083i \(-0.0412032\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.5584 25.2159i 0.678053 1.17442i −0.297513 0.954718i \(-0.596157\pi\)
0.975566 0.219705i \(-0.0705095\pi\)
\(462\) 0 0
\(463\) 18.9123 + 32.7570i 0.878928 + 1.52235i 0.852518 + 0.522697i \(0.175074\pi\)
0.0264102 + 0.999651i \(0.491592\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.3083 1.68015 0.840075 0.542470i \(-0.182511\pi\)
0.840075 + 0.542470i \(0.182511\pi\)
\(468\) 0 0
\(469\) 22.8832 1.05665
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −17.6168 30.5133i −0.810023 1.40300i
\(474\) 0 0
\(475\) −1.16944 + 2.02554i −0.0536577 + 0.0929379i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.10489 + 1.91373i −0.0504839 + 0.0874406i −0.890163 0.455642i \(-0.849410\pi\)
0.839679 + 0.543083i \(0.182743\pi\)
\(480\) 0 0
\(481\) −8.74456 15.1460i −0.398718 0.690599i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.3505 0.969478
\(486\) 0 0
\(487\) 7.45202 0.337683 0.168842 0.985643i \(-0.445997\pi\)
0.168842 + 0.985643i \(0.445997\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.0645501 0.111804i −0.00291310 0.00504564i 0.864565 0.502521i \(-0.167594\pi\)
−0.867478 + 0.497475i \(0.834261\pi\)
\(492\) 0 0
\(493\) 0.627719 1.08724i 0.0282710 0.0489669i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.88316 8.45787i 0.219039 0.379388i
\(498\) 0 0
\(499\) 12.6297 + 21.8753i 0.565383 + 0.979273i 0.997014 + 0.0772222i \(0.0246051\pi\)
−0.431631 + 0.902050i \(0.642062\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.4845 −0.467479 −0.233740 0.972299i \(-0.575096\pi\)
−0.233740 + 0.972299i \(0.575096\pi\)
\(504\) 0 0
\(505\) −35.8614 −1.59581
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.67527 + 11.5619i 0.295876 + 0.512472i 0.975188 0.221377i \(-0.0710552\pi\)
−0.679312 + 0.733849i \(0.737722\pi\)
\(510\) 0 0
\(511\) −5.11313 + 8.85621i −0.226192 + 0.391776i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.62112 4.53991i 0.115500 0.200052i
\(516\) 0 0
\(517\) −6.55842 11.3595i −0.288439 0.499591i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 16.3505 0.716330 0.358165 0.933658i \(-0.383403\pi\)
0.358165 + 0.933658i \(0.383403\pi\)
\(522\) 0 0
\(523\) 19.3236 0.844963 0.422481 0.906372i \(-0.361159\pi\)
0.422481 + 0.906372i \(0.361159\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.2912 + 28.2171i 0.709654 + 1.22916i
\(528\) 0 0
\(529\) 9.05842 15.6896i 0.393844 0.682159i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.18614 + 3.78651i −0.0946923 + 0.164012i
\(534\) 0 0
\(535\) 4.41957 + 7.65492i 0.191075 + 0.330951i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.5652 0.541220
\(540\) 0 0
\(541\) −8.74456 −0.375958 −0.187979 0.982173i \(-0.560194\pi\)
−0.187979 + 0.982173i \(0.560194\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 20.7446 + 35.9306i 0.888599 + 1.53910i
\(546\) 0 0
\(547\) 0.758112 1.31309i 0.0324145 0.0561436i −0.849363 0.527809i \(-0.823014\pi\)
0.881778 + 0.471665i \(0.156347\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −0.693562 + 1.20128i −0.0295467 + 0.0511765i
\(552\) 0 0
\(553\) 10.6753 + 18.4901i 0.453958 + 0.786279i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.7446 −0.794233 −0.397116 0.917768i \(-0.629989\pi\)
−0.397116 + 0.917768i \(0.629989\pi\)
\(558\) 0 0
\(559\) 25.9530 1.09769
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.1135 19.2491i −0.468377 0.811254i 0.530969 0.847391i \(-0.321828\pi\)
−0.999347 + 0.0361375i \(0.988495\pi\)
\(564\) 0 0
\(565\) 13.7921 23.8886i 0.580238 1.00500i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −19.8723 + 34.4198i −0.833089 + 1.44295i 0.0624872 + 0.998046i \(0.480097\pi\)
−0.895577 + 0.444907i \(0.853237\pi\)
\(570\) 0 0
\(571\) −0.0645501 0.111804i −0.00270134 0.00467885i 0.864672 0.502338i \(-0.167527\pi\)
−0.867373 + 0.497659i \(0.834193\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.38712 −0.0578471
\(576\) 0 0
\(577\) 18.1168 0.754214 0.377107 0.926170i \(-0.376919\pi\)
0.377107 + 0.926170i \(0.376919\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.5584 + 26.9480i 0.645472 + 1.11799i
\(582\) 0 0
\(583\) −11.8716 + 20.5622i −0.491671 + 0.851599i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.87125 10.1693i 0.242332 0.419732i −0.719046 0.694963i \(-0.755421\pi\)
0.961378 + 0.275231i \(0.0887542\pi\)
\(588\) 0 0
\(589\) −18.0000 31.1769i −0.741677 1.28462i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 17.7228 0.727789 0.363894 0.931440i \(-0.381447\pi\)
0.363894 + 0.931440i \(0.381447\pi\)
\(594\) 0 0
\(595\) −17.6783 −0.724739
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13.6700 23.6772i −0.558543 0.967425i −0.997618 0.0689747i \(-0.978027\pi\)
0.439075 0.898450i \(-0.355306\pi\)
\(600\) 0 0
\(601\) 1.38316 2.39570i 0.0564201 0.0977225i −0.836436 0.548065i \(-0.815365\pi\)
0.892856 + 0.450342i \(0.148698\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −28.7446 + 49.7870i −1.16863 + 2.02413i
\(606\) 0 0
\(607\) −18.9123 32.7570i −0.767626 1.32957i −0.938847 0.344335i \(-0.888105\pi\)
0.171221 0.985233i \(-0.445229\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.66181 0.390875
\(612\) 0 0
\(613\) −36.4674 −1.47290 −0.736452 0.676490i \(-0.763500\pi\)
−0.736452 + 0.676490i \(0.763500\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.8723 32.6878i −0.759769 1.31596i −0.942968 0.332883i \(-0.891978\pi\)
0.183199 0.983076i \(-0.441355\pi\)
\(618\) 0 0
\(619\) 0.758112 1.31309i 0.0304711 0.0527775i −0.850388 0.526157i \(-0.823633\pi\)
0.880859 + 0.473379i \(0.156966\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.38712 2.40257i 0.0555740 0.0962569i
\(624\) 0 0
\(625\) 13.8723 + 24.0275i 0.554891 + 0.961100i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 13.4891 0.537847
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.83915 15.3098i −0.350771 0.607553i
\(636\) 0 0
\(637\) −4.62772 + 8.01544i −0.183357 + 0.317583i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.87228 11.9031i 0.271439 0.470146i −0.697792 0.716301i \(-0.745834\pi\)
0.969230 + 0.246155i \(0.0791672\pi\)
\(642\) 0 0
\(643\) −8.90370 15.4217i −0.351127 0.608171i 0.635320 0.772249i \(-0.280868\pi\)
−0.986447 + 0.164079i \(0.947535\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 40.0344 1.57391 0.786956 0.617009i \(-0.211656\pi\)
0.786956 + 0.617009i \(0.211656\pi\)
\(648\) 0 0
\(649\) −61.4674 −2.41281
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.5584 35.6082i −0.804513 1.39346i −0.916619 0.399762i \(-0.869093\pi\)
0.112106 0.993696i \(-0.464240\pi\)
\(654\) 0 0
\(655\) 11.4603 19.8498i 0.447790 0.775594i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −15.8798 + 27.5047i −0.618591 + 1.07143i 0.371153 + 0.928572i \(0.378963\pi\)
−0.989743 + 0.142858i \(0.954371\pi\)
\(660\) 0 0
\(661\) −1.55842 2.69927i −0.0606156 0.104989i 0.834125 0.551575i \(-0.185973\pi\)
−0.894741 + 0.446586i \(0.852640\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 19.5326 0.757443
\(666\) 0 0
\(667\) −0.822662 −0.0318536
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −44.8653 77.7089i −1.73200 2.99992i
\(672\) 0 0
\(673\) −11.6753 + 20.2222i −0.450048 + 0.779507i −0.998388 0.0567487i \(-0.981927\pi\)
0.548340 + 0.836255i \(0.315260\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.675266 + 1.16959i −0.0259526 + 0.0449512i −0.878710 0.477356i \(-0.841595\pi\)
0.852757 + 0.522307i \(0.174929\pi\)
\(678\) 0 0
\(679\) −9.94404 17.2236i −0.381617 0.660980i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.3083 −1.38930 −0.694650 0.719348i \(-0.744441\pi\)
−0.694650 + 0.719348i \(0.744441\pi\)
\(684\) 0 0
\(685\) 29.6277 1.13202
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.74456 15.1460i −0.333141 0.577018i
\(690\) 0 0
\(691\) −23.3319 + 40.4120i −0.887586 + 1.53734i −0.0448646 + 0.998993i \(0.514286\pi\)
−0.842721 + 0.538350i \(0.819048\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.44378 5.96480i 0.130630 0.226258i
\(696\) 0 0
\(697\) −1.68614 2.92048i −0.0638671 0.110621i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 34.7446 1.31228 0.656142 0.754637i \(-0.272187\pi\)
0.656142 + 0.754637i \(0.272187\pi\)
\(702\) 0 0
\(703\) −14.9040 −0.562117
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.7025 + 28.9296i 0.628162 + 1.08801i
\(708\) 0 0
\(709\) −6.30298 + 10.9171i −0.236714 + 0.410000i −0.959769 0.280790i \(-0.909404\pi\)
0.723056 + 0.690790i \(0.242737\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10.6753 18.4901i 0.399792 0.692460i
\(714\) 0 0
\(715\) −30.7839 53.3192i −1.15125 1.99403i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 32.5823 1.21512 0.607558 0.794275i \(-0.292149\pi\)
0.607558 + 0.794275i \(0.292149\pi\)
\(720\) 0 0
\(721\) −4.88316 −0.181858
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.116844 0.202380i −0.00433948 0.00751619i
\(726\) 0 0
\(727\) 17.3961 30.1309i 0.645184 1.11749i −0.339075 0.940759i \(-0.610114\pi\)
0.984259 0.176732i \(-0.0565527\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0086 + 17.3354i −0.370181 + 0.641172i
\(732\) 0 0
\(733\) 11.4416 + 19.8174i 0.422604 + 0.731972i 0.996193 0.0871711i \(-0.0277827\pi\)
−0.573589 + 0.819143i \(0.694449\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −61.4674 −2.26418
\(738\) 0 0
\(739\) −15.5976 −0.573767 −0.286884 0.957965i \(-0.592619\pi\)
−0.286884 + 0.957965i \(0.592619\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 13.7991 + 23.9008i 0.506242 + 0.876836i 0.999974 + 0.00722236i \(0.00229897\pi\)
−0.493732 + 0.869614i \(0.664368\pi\)
\(744\) 0 0
\(745\) 13.7921 23.8886i 0.505304 0.875212i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.11684 7.13058i 0.150426 0.260546i
\(750\) 0 0
\(751\) −11.4603 19.8498i −0.418191 0.724328i 0.577567 0.816344i \(-0.304002\pi\)
−0.995758 + 0.0920156i \(0.970669\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.24224 −0.190784
\(756\) 0 0
\(757\) 34.4674 1.25274 0.626369 0.779527i \(-0.284540\pi\)
0.626369 + 0.779527i \(0.284540\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.1861 + 17.6429i 0.369247 + 0.639555i 0.989448 0.144888i \(-0.0462822\pi\)
−0.620201 + 0.784443i \(0.712949\pi\)
\(762\) 0 0
\(763\) 19.3236 33.4695i 0.699562 1.21168i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 22.6383 39.2107i 0.817421 1.41582i
\(768\) 0 0
\(769\) 7.44158 + 12.8892i 0.268350 + 0.464796i 0.968436 0.249263i \(-0.0801882\pi\)
−0.700086 + 0.714059i \(0.746855\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.7446 −0.674195 −0.337098 0.941470i \(-0.609445\pi\)
−0.337098 + 0.941470i \(0.609445\pi\)
\(774\) 0 0
\(775\) 6.06490 0.217858
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.86301 + 3.22682i 0.0667491 + 0.115613i
\(780\) 0 0
\(781\) −13.1168 + 22.7190i −0.469358 + 0.812951i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.69702 + 6.40342i −0.131952 + 0.228548i
\(786\) 0 0
\(787\) −21.1221 36.5845i −0.752921 1.30410i −0.946402 0.322993i \(-0.895311\pi\)
0.193481 0.981104i \(-0.438022\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −25.6948 −0.913601
\(792\) 0 0
\(793\) 66.0951 2.34711
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −23.0475 39.9195i −0.816386 1.41402i −0.908328 0.418258i \(-0.862641\pi\)
0.0919424 0.995764i \(-0.470692\pi\)
\(798\) 0 0
\(799\) −3.72601 + 6.45364i −0.131817 + 0.228313i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.7346 23.7890i 0.484683 0.839496i
\(804\) 0 0
\(805\) 5.79211 + 10.0322i 0.204145 + 0.353590i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.6277 0.443967 0.221983 0.975050i \(-0.428747\pi\)
0.221983 + 0.975050i \(0.428747\pi\)
\(810\) 0 0
\(811\) −33.5341 −1.17754 −0.588771 0.808300i \(-0.700388\pi\)
−0.588771 + 0.808300i \(0.700388\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.4845 + 18.1596i 0.367255 + 0.636104i
\(816\) 0 0
\(817\) 11.0584 19.1537i 0.386885 0.670105i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.18614 + 7.25061i −0.146097 + 0.253048i −0.929782 0.368111i \(-0.880005\pi\)
0.783684 + 0.621159i \(0.213338\pi\)
\(822\) 0 0
\(823\) 11.4603 + 19.8498i 0.399480 + 0.691919i 0.993662 0.112411i \(-0.0358575\pi\)
−0.594182 + 0.804331i \(0.702524\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.83915 0.307367 0.153684 0.988120i \(-0.450886\pi\)
0.153684 + 0.988120i \(0.450886\pi\)
\(828\) 0 0
\(829\) 8.23369 0.285968 0.142984 0.989725i \(-0.454330\pi\)
0.142984 + 0.989725i \(0.454330\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.56930 6.18220i −0.123669 0.214201i
\(834\) 0 0
\(835\) −11.4603 + 19.8498i −0.396599 + 0.686929i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 4.83090 8.36737i 0.166781 0.288874i −0.770505 0.637434i \(-0.779996\pi\)
0.937286 + 0.348560i \(0.113329\pi\)
\(840\) 0 0
\(841\) 14.4307 + 24.9947i 0.497610 + 0.861887i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 14.5109 0.499189
\(846\) 0 0
\(847\) 53.5513 1.84004
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.41957 7.65492i −0.151501 0.262407i
\(852\) 0 0
\(853\) −10.5584 + 18.2877i −0.361513 + 0.626160i −0.988210 0.153104i \(-0.951073\pi\)
0.626697 + 0.779263i \(0.284407\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.0475 + 38.1875i −0.753130 + 1.30446i 0.193169 + 0.981166i \(0.438123\pi\)
−0.946299 + 0.323294i \(0.895210\pi\)
\(858\) 0 0
\(859\) −8.90370 15.4217i −0.303790 0.526180i 0.673201 0.739459i \(-0.264919\pi\)
−0.976991 + 0.213279i \(0.931586\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −17.6783 −0.601776 −0.300888 0.953659i \(-0.597283\pi\)
−0.300888 + 0.953659i \(0.597283\pi\)
\(864\) 0 0
\(865\) −0.883156 −0.0300282
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −28.6753 49.6670i −0.972742 1.68484i
\(870\) 0 0
\(871\) 22.6383 39.2107i 0.767069 1.32860i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.4603 19.8498i 0.387428 0.671044i
\(876\) 0 0
\(877\) −6.67527 11.5619i −0.225408 0.390418i 0.731034 0.682341i \(-0.239038\pi\)
−0.956442 + 0.291923i \(0.905705\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.23369 −0.142637 −0.0713183 0.997454i \(-0.522721\pi\)
−0.0713183 + 0.997454i \(0.522721\pi\)
\(882\) 0 0
\(883\) −21.4043 −0.720312 −0.360156 0.932892i \(-0.617277\pi\)
−0.360156 + 0.932892i \(0.617277\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −11.4603 19.8498i −0.384798 0.666490i 0.606943 0.794745i \(-0.292396\pi\)
−0.991741 + 0.128256i \(0.959062\pi\)
\(888\) 0 0
\(889\) −8.23369 + 14.2612i −0.276149 + 0.478304i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.11684 7.13058i 0.137765 0.238616i
\(894\) 0 0
\(895\) −10.4845 18.1596i −0.350457 0.607010i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.59691 0.119964
\(900\) 0 0
\(901\) 13.4891 0.449388
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.76631 16.9157i −0.324643 0.562299i
\(906\) 0 0
\(907\) 14.1459 24.5015i 0.469708 0.813558i −0.529692 0.848190i \(-0.677693\pi\)
0.999400 + 0.0346319i \(0.0110259\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.3643 45.6643i 0.873488 1.51293i 0.0151242 0.999886i \(-0.495186\pi\)
0.858364 0.513041i \(-0.171481\pi\)
\(912\) 0 0
\(913\) −41.7921 72.3861i −1.38312 2.39563i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −21.3505 −0.705057
\(918\) 0 0
\(919\) −25.1303 −0.828973 −0.414486 0.910056i \(-0.636039\pi\)
−0.414486 + 0.910056i \(0.636039\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −9.66181 16.7347i −0.318022 0.550831i
\(924\) 0 0
\(925\) 1.25544 2.17448i 0.0412785 0.0714965i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 18.1861 31.4993i 0.596668 1.03346i −0.396641 0.917974i \(-0.629824\pi\)
0.993309 0.115485i \(-0.0368424\pi\)
\(930\) 0 0
\(931\) 3.94369 + 6.83067i 0.129249 + 0.223866i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 47.4864 1.55297
\(936\) 0 0
\(937\) 12.2337 0.399657 0.199829 0.979831i \(-0.435961\pi\)
0.199829 + 0.979831i \(0.435961\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23.0475 + 39.9195i 0.751329 + 1.30134i 0.947179 + 0.320705i \(0.103920\pi\)
−0.195850 + 0.980634i \(0.562747\pi\)
\(942\) 0 0
\(943\) −1.10489 + 1.91373i −0.0359803 + 0.0623197i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.0493 + 29.5302i −0.554027 + 0.959603i 0.443951 + 0.896051i \(0.353576\pi\)
−0.997979 + 0.0635523i \(0.979757\pi\)
\(948\) 0 0
\(949\) 10.1168 + 17.5229i 0.328407 + 0.568817i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 31.6060 1.02382 0.511909 0.859040i \(-0.328939\pi\)
0.511909 + 0.859040i \(0.328939\pi\)
\(954\) 0 0
\(955\) −40.5988 −1.31375
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.7991 23.9008i −0.445598 0.771798i
\(960\) 0 0
\(961\) −31.1753 + 53.9971i −1.00565 + 1.74184i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.18614 + 2.05446i −0.0381832 + 0.0661353i
\(966\) 0 0
\(967\) 13.7991 + 23.9008i 0.443751 + 0.768599i 0.997964 0.0637757i \(-0.0203142\pi\)
−0.554214 + 0.832375i \(0.686981\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 49.1317 1.57671 0.788356 0.615220i \(-0.210933\pi\)
0.788356 + 0.615220i \(0.210933\pi\)
\(972\) 0 0
\(973\) −6.41578 −0.205680
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.38316 + 9.32390i 0.172222 + 0.298298i 0.939197 0.343380i \(-0.111572\pi\)
−0.766974 + 0.641678i \(0.778239\pi\)
\(978\) 0 0
\(979\) −3.72601 + 6.45364i −0.119084 + 0.206259i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.2994 35.1596i 0.647451 1.12142i −0.336279 0.941762i \(-0.609169\pi\)
0.983730 0.179655i \(-0.0574981\pi\)
\(984\) 0 0
\(985\) −0.883156 1.52967i −0.0281397 0.0487394i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.1168 0.417091
\(990\) 0 0
\(991\) −47.4864 −1.50845 −0.754227 0.656613i \(-0.771988\pi\)
−0.754227 + 0.656613i \(0.771988\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −24.5659 42.5493i −0.778790 1.34890i
\(996\) 0 0
\(997\) 20.6753 35.8106i 0.654792 1.13413i −0.327154 0.944971i \(-0.606089\pi\)
0.981946 0.189162i \(-0.0605772\pi\)
\(998\) 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 1728.2.i.n.1153.1 8
3.2 odd 2 576.2.i.n.385.4 8
4.3 odd 2 inner 1728.2.i.n.1153.2 8
8.3 odd 2 864.2.i.f.289.4 8
8.5 even 2 864.2.i.f.289.3 8
9.2 odd 6 5184.2.a.cf.1.2 4
9.4 even 3 inner 1728.2.i.n.577.1 8
9.5 odd 6 576.2.i.n.193.4 8
9.7 even 3 5184.2.a.cc.1.4 4
12.11 even 2 576.2.i.n.385.1 8
24.5 odd 2 288.2.i.f.97.1 8
24.11 even 2 288.2.i.f.97.4 yes 8
36.7 odd 6 5184.2.a.cc.1.3 4
36.11 even 6 5184.2.a.cf.1.1 4
36.23 even 6 576.2.i.n.193.1 8
36.31 odd 6 inner 1728.2.i.n.577.2 8
72.5 odd 6 288.2.i.f.193.1 yes 8
72.11 even 6 2592.2.a.u.1.3 4
72.13 even 6 864.2.i.f.577.3 8
72.29 odd 6 2592.2.a.u.1.4 4
72.43 odd 6 2592.2.a.x.1.1 4
72.59 even 6 288.2.i.f.193.4 yes 8
72.61 even 6 2592.2.a.x.1.2 4
72.67 odd 6 864.2.i.f.577.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.f.97.1 8 24.5 odd 2
288.2.i.f.97.4 yes 8 24.11 even 2
288.2.i.f.193.1 yes 8 72.5 odd 6
288.2.i.f.193.4 yes 8 72.59 even 6
576.2.i.n.193.1 8 36.23 even 6
576.2.i.n.193.4 8 9.5 odd 6
576.2.i.n.385.1 8 12.11 even 2
576.2.i.n.385.4 8 3.2 odd 2
864.2.i.f.289.3 8 8.5 even 2
864.2.i.f.289.4 8 8.3 odd 2
864.2.i.f.577.3 8 72.13 even 6
864.2.i.f.577.4 8 72.67 odd 6
1728.2.i.n.577.1 8 9.4 even 3 inner
1728.2.i.n.577.2 8 36.31 odd 6 inner
1728.2.i.n.1153.1 8 1.1 even 1 trivial
1728.2.i.n.1153.2 8 4.3 odd 2 inner
2592.2.a.u.1.3 4 72.11 even 6
2592.2.a.u.1.4 4 72.29 odd 6
2592.2.a.x.1.1 4 72.43 odd 6
2592.2.a.x.1.2 4 72.61 even 6
5184.2.a.cc.1.3 4 36.7 odd 6
5184.2.a.cc.1.4 4 9.7 even 3
5184.2.a.cf.1.1 4 36.11 even 6
5184.2.a.cf.1.2 4 9.2 odd 6