Properties

Label 756.2.l.b.289.6
Level $756$
Weight $2$
Character 756.289
Analytic conductor $6.037$
Analytic rank $0$
Dimension $14$
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] = [756,2,Mod(289,756)]
 
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, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.l (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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.6
Root \(-0.473632 - 1.66604i\) of defining polynomial
Character \(\chi\) \(=\) 756.289
Dual form 756.2.l.b.361.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.90301 q^{5} +(-2.43415 - 1.03677i) q^{7} +O(q^{10})\) \(q+1.90301 q^{5} +(-2.43415 - 1.03677i) q^{7} +3.06586 q^{11} +(1.13161 + 1.96000i) q^{13} +(0.713726 + 1.23621i) q^{17} +(2.98444 - 5.16919i) q^{19} +7.15543 q^{23} -1.37856 q^{25} +(-0.468164 + 0.810884i) q^{29} +(4.11065 - 7.11985i) q^{31} +(-4.63221 - 1.97298i) q^{35} +(-1.41550 + 2.45171i) q^{37} +(5.31672 + 9.20883i) q^{41} +(2.98444 - 5.16919i) q^{43} +(-0.483340 - 0.837169i) q^{47} +(4.85021 + 5.04732i) q^{49} +(-5.45142 - 9.44213i) q^{53} +5.83436 q^{55} +(-5.68180 + 9.84117i) q^{59} +(-0.449718 - 0.778935i) q^{61} +(2.15346 + 3.72990i) q^{65} +(-0.813810 + 1.40956i) q^{67} +2.36378 q^{71} +(-0.996286 - 1.72562i) q^{73} +(-7.46279 - 3.17860i) q^{77} +(4.16945 + 7.22169i) q^{79} +(7.98203 - 13.8253i) q^{83} +(1.35822 + 2.35251i) q^{85} +(2.58992 - 4.48587i) q^{89} +(-0.722433 - 5.94416i) q^{91} +(5.67940 - 9.83701i) q^{95} +(0.922890 - 1.59849i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{5} - 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{5} - 3 q^{7} + 4 q^{11} + 2 q^{13} - 2 q^{17} + 7 q^{19} + 22 q^{23} + 18 q^{25} - q^{29} - q^{31} + 19 q^{35} + 10 q^{37} + 33 q^{41} + 7 q^{43} + 3 q^{47} - 13 q^{49} + 15 q^{53} - 28 q^{55} + 14 q^{59} - 10 q^{61} - 15 q^{65} + 6 q^{67} - 2 q^{71} + 21 q^{73} - 19 q^{77} - 10 q^{79} + 25 q^{83} + 8 q^{85} + 6 q^{89} + 2 q^{91} + 28 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 1.90301 0.851051 0.425525 0.904947i \(-0.360089\pi\)
0.425525 + 0.904947i \(0.360089\pi\)
\(6\) 0 0
\(7\) −2.43415 1.03677i −0.920024 0.391863i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.06586 0.924393 0.462196 0.886778i \(-0.347062\pi\)
0.462196 + 0.886778i \(0.347062\pi\)
\(12\) 0 0
\(13\) 1.13161 + 1.96000i 0.313851 + 0.543607i 0.979193 0.202933i \(-0.0650473\pi\)
−0.665341 + 0.746539i \(0.731714\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.713726 + 1.23621i 0.173104 + 0.299825i 0.939503 0.342539i \(-0.111287\pi\)
−0.766400 + 0.642364i \(0.777954\pi\)
\(18\) 0 0
\(19\) 2.98444 5.16919i 0.684677 1.18589i −0.288862 0.957371i \(-0.593277\pi\)
0.973538 0.228524i \(-0.0733898\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.15543 1.49201 0.746005 0.665940i \(-0.231969\pi\)
0.746005 + 0.665940i \(0.231969\pi\)
\(24\) 0 0
\(25\) −1.37856 −0.275713
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.468164 + 0.810884i −0.0869359 + 0.150577i −0.906214 0.422818i \(-0.861041\pi\)
0.819279 + 0.573396i \(0.194374\pi\)
\(30\) 0 0
\(31\) 4.11065 7.11985i 0.738294 1.27876i −0.214969 0.976621i \(-0.568965\pi\)
0.953263 0.302142i \(-0.0977016\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.63221 1.97298i −0.782987 0.333495i
\(36\) 0 0
\(37\) −1.41550 + 2.45171i −0.232706 + 0.403059i −0.958604 0.284744i \(-0.908091\pi\)
0.725897 + 0.687803i \(0.241425\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.31672 + 9.20883i 0.830332 + 1.43818i 0.897775 + 0.440454i \(0.145183\pi\)
−0.0674429 + 0.997723i \(0.521484\pi\)
\(42\) 0 0
\(43\) 2.98444 5.16919i 0.455122 0.788295i −0.543573 0.839362i \(-0.682929\pi\)
0.998695 + 0.0510671i \(0.0162622\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.483340 0.837169i −0.0705023 0.122114i 0.828619 0.559813i \(-0.189127\pi\)
−0.899122 + 0.437699i \(0.855794\pi\)
\(48\) 0 0
\(49\) 4.85021 + 5.04732i 0.692887 + 0.721046i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.45142 9.44213i −0.748810 1.29698i −0.948393 0.317096i \(-0.897292\pi\)
0.199583 0.979881i \(-0.436041\pi\)
\(54\) 0 0
\(55\) 5.83436 0.786705
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.68180 + 9.84117i −0.739708 + 1.28121i 0.212919 + 0.977070i \(0.431703\pi\)
−0.952627 + 0.304142i \(0.901630\pi\)
\(60\) 0 0
\(61\) −0.449718 0.778935i −0.0575805 0.0997324i 0.835798 0.549037i \(-0.185005\pi\)
−0.893379 + 0.449304i \(0.851672\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.15346 + 3.72990i 0.267104 + 0.462637i
\(66\) 0 0
\(67\) −0.813810 + 1.40956i −0.0994227 + 0.172205i −0.911446 0.411420i \(-0.865033\pi\)
0.812023 + 0.583625i \(0.198366\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.36378 0.280529 0.140264 0.990114i \(-0.455205\pi\)
0.140264 + 0.990114i \(0.455205\pi\)
\(72\) 0 0
\(73\) −0.996286 1.72562i −0.116606 0.201968i 0.801814 0.597573i \(-0.203868\pi\)
−0.918421 + 0.395605i \(0.870535\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.46279 3.17860i −0.850463 0.362235i
\(78\) 0 0
\(79\) 4.16945 + 7.22169i 0.469099 + 0.812504i 0.999376 0.0353209i \(-0.0112453\pi\)
−0.530277 + 0.847825i \(0.677912\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.98203 13.8253i 0.876141 1.51752i 0.0205995 0.999788i \(-0.493443\pi\)
0.855542 0.517734i \(-0.173224\pi\)
\(84\) 0 0
\(85\) 1.35822 + 2.35251i 0.147320 + 0.255166i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.58992 4.48587i 0.274531 0.475501i −0.695486 0.718540i \(-0.744811\pi\)
0.970017 + 0.243039i \(0.0781442\pi\)
\(90\) 0 0
\(91\) −0.722433 5.94416i −0.0757316 0.623118i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.67940 9.83701i 0.582694 1.00926i
\(96\) 0 0
\(97\) 0.922890 1.59849i 0.0937053 0.162302i −0.815362 0.578951i \(-0.803462\pi\)
0.909068 + 0.416649i \(0.136796\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.06350 0.802349 0.401174 0.916002i \(-0.368602\pi\)
0.401174 + 0.916002i \(0.368602\pi\)
\(102\) 0 0
\(103\) −17.7986 −1.75375 −0.876875 0.480718i \(-0.840376\pi\)
−0.876875 + 0.480718i \(0.840376\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.76005 + 15.1729i −0.846866 + 1.46682i 0.0371245 + 0.999311i \(0.488180\pi\)
−0.883991 + 0.467505i \(0.845153\pi\)
\(108\) 0 0
\(109\) 1.11441 + 1.93021i 0.106741 + 0.184881i 0.914448 0.404703i \(-0.132625\pi\)
−0.807707 + 0.589584i \(0.799292\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.59999 + 13.1636i 0.714947 + 1.23832i 0.962980 + 0.269573i \(0.0868824\pi\)
−0.248033 + 0.968751i \(0.579784\pi\)
\(114\) 0 0
\(115\) 13.6168 1.26978
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.455652 3.74909i −0.0417695 0.343679i
\(120\) 0 0
\(121\) −1.60048 −0.145498
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.1385 −1.08570
\(126\) 0 0
\(127\) −16.9303 −1.50232 −0.751161 0.660119i \(-0.770506\pi\)
−0.751161 + 0.660119i \(0.770506\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.38354 −0.208251 −0.104125 0.994564i \(-0.533204\pi\)
−0.104125 + 0.994564i \(0.533204\pi\)
\(132\) 0 0
\(133\) −12.6238 + 9.48844i −1.09463 + 0.822752i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.9820 −0.938254 −0.469127 0.883131i \(-0.655431\pi\)
−0.469127 + 0.883131i \(0.655431\pi\)
\(138\) 0 0
\(139\) −3.70422 6.41590i −0.314188 0.544190i 0.665076 0.746775i \(-0.268399\pi\)
−0.979265 + 0.202585i \(0.935066\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.46936 + 6.00910i 0.290122 + 0.502506i
\(144\) 0 0
\(145\) −0.890919 + 1.54312i −0.0739868 + 0.128149i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.6239 −1.44381 −0.721904 0.691993i \(-0.756733\pi\)
−0.721904 + 0.691993i \(0.756733\pi\)
\(150\) 0 0
\(151\) −20.3664 −1.65739 −0.828697 0.559697i \(-0.810917\pi\)
−0.828697 + 0.559697i \(0.810917\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 7.82259 13.5491i 0.628326 1.08829i
\(156\) 0 0
\(157\) −4.64118 + 8.03875i −0.370406 + 0.641562i −0.989628 0.143654i \(-0.954115\pi\)
0.619222 + 0.785216i \(0.287448\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −17.4174 7.41854i −1.37268 0.584663i
\(162\) 0 0
\(163\) −11.9069 + 20.6234i −0.932623 + 1.61535i −0.153803 + 0.988101i \(0.549152\pi\)
−0.778819 + 0.627248i \(0.784181\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.883505 + 1.53028i 0.0683676 + 0.118416i 0.898183 0.439622i \(-0.144888\pi\)
−0.829815 + 0.558038i \(0.811554\pi\)
\(168\) 0 0
\(169\) 3.93893 6.82242i 0.302994 0.524802i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0.180049 + 0.311855i 0.0136889 + 0.0237099i 0.872789 0.488098i \(-0.162309\pi\)
−0.859100 + 0.511808i \(0.828976\pi\)
\(174\) 0 0
\(175\) 3.35564 + 1.42926i 0.253662 + 0.108042i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.57701 + 6.19556i 0.267358 + 0.463078i 0.968179 0.250260i \(-0.0805160\pi\)
−0.700821 + 0.713337i \(0.747183\pi\)
\(180\) 0 0
\(181\) 11.0542 0.821650 0.410825 0.911714i \(-0.365241\pi\)
0.410825 + 0.911714i \(0.365241\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.69370 + 4.66563i −0.198045 + 0.343024i
\(186\) 0 0
\(187\) 2.18819 + 3.79005i 0.160016 + 0.277156i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.35083 7.53586i −0.314815 0.545276i 0.664583 0.747214i \(-0.268609\pi\)
−0.979398 + 0.201939i \(0.935276\pi\)
\(192\) 0 0
\(193\) 0.709644 1.22914i 0.0510813 0.0884754i −0.839354 0.543585i \(-0.817067\pi\)
0.890435 + 0.455110i \(0.150400\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.69424 −0.405698 −0.202849 0.979210i \(-0.565020\pi\)
−0.202849 + 0.979210i \(0.565020\pi\)
\(198\) 0 0
\(199\) 2.61327 + 4.52631i 0.185250 + 0.320862i 0.943661 0.330915i \(-0.107357\pi\)
−0.758411 + 0.651777i \(0.774024\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.98028 1.48844i 0.138989 0.104468i
\(204\) 0 0
\(205\) 10.1178 + 17.5245i 0.706655 + 1.22396i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.14987 15.8480i 0.632910 1.09623i
\(210\) 0 0
\(211\) −5.93079 10.2724i −0.408293 0.707183i 0.586406 0.810017i \(-0.300542\pi\)
−0.994699 + 0.102834i \(0.967209\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.67940 9.83701i 0.387332 0.670879i
\(216\) 0 0
\(217\) −17.3876 + 13.0690i −1.18035 + 0.887182i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.61531 + 2.79781i −0.108658 + 0.188201i
\(222\) 0 0
\(223\) 12.2950 21.2955i 0.823333 1.42605i −0.0798535 0.996807i \(-0.525445\pi\)
0.903187 0.429248i \(-0.141221\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.3180 −1.01669 −0.508344 0.861154i \(-0.669742\pi\)
−0.508344 + 0.861154i \(0.669742\pi\)
\(228\) 0 0
\(229\) 17.0459 1.12643 0.563214 0.826311i \(-0.309565\pi\)
0.563214 + 0.826311i \(0.309565\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.88255 + 17.1171i −0.647427 + 1.12138i 0.336308 + 0.941752i \(0.390822\pi\)
−0.983735 + 0.179625i \(0.942512\pi\)
\(234\) 0 0
\(235\) −0.919799 1.59314i −0.0600011 0.103925i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.35041 14.4633i −0.540143 0.935555i −0.998895 0.0469909i \(-0.985037\pi\)
0.458752 0.888564i \(-0.348297\pi\)
\(240\) 0 0
\(241\) 6.39995 0.412257 0.206129 0.978525i \(-0.433914\pi\)
0.206129 + 0.978525i \(0.433914\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 9.22999 + 9.60509i 0.589682 + 0.613647i
\(246\) 0 0
\(247\) 13.5088 0.859547
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 26.8346 1.69378 0.846891 0.531766i \(-0.178471\pi\)
0.846891 + 0.531766i \(0.178471\pi\)
\(252\) 0 0
\(253\) 21.9376 1.37920
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.17597 −0.447625 −0.223812 0.974632i \(-0.571850\pi\)
−0.223812 + 0.974632i \(0.571850\pi\)
\(258\) 0 0
\(259\) 5.98741 4.50030i 0.372039 0.279635i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −20.4137 −1.25876 −0.629382 0.777096i \(-0.716692\pi\)
−0.629382 + 0.777096i \(0.716692\pi\)
\(264\) 0 0
\(265\) −10.3741 17.9685i −0.637275 1.10379i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.37251 + 5.84136i 0.205626 + 0.356154i 0.950332 0.311238i \(-0.100744\pi\)
−0.744706 + 0.667392i \(0.767410\pi\)
\(270\) 0 0
\(271\) −1.04632 + 1.81228i −0.0635596 + 0.110088i −0.896054 0.443945i \(-0.853579\pi\)
0.832495 + 0.554033i \(0.186912\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.22649 −0.254867
\(276\) 0 0
\(277\) −23.5410 −1.41444 −0.707221 0.706992i \(-0.750052\pi\)
−0.707221 + 0.706992i \(0.750052\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.66048 + 16.7324i −0.576296 + 0.998173i 0.419604 + 0.907707i \(0.362169\pi\)
−0.995900 + 0.0904661i \(0.971164\pi\)
\(282\) 0 0
\(283\) 2.22658 3.85655i 0.132356 0.229248i −0.792228 0.610225i \(-0.791079\pi\)
0.924584 + 0.380977i \(0.124412\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.39426 27.9279i −0.200357 1.64853i
\(288\) 0 0
\(289\) 7.48119 12.9578i 0.440070 0.762224i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.7314 + 20.3193i 0.685354 + 1.18707i 0.973325 + 0.229429i \(0.0736858\pi\)
−0.287972 + 0.957639i \(0.592981\pi\)
\(294\) 0 0
\(295\) −10.8125 + 18.7278i −0.629529 + 1.09038i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.09714 + 14.0247i 0.468270 + 0.811067i
\(300\) 0 0
\(301\) −12.6238 + 9.48844i −0.727627 + 0.546904i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.855817 1.48232i −0.0490039 0.0848773i
\(306\) 0 0
\(307\) 7.79955 0.445144 0.222572 0.974916i \(-0.428555\pi\)
0.222572 + 0.974916i \(0.428555\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.49449 12.9808i 0.424974 0.736076i −0.571444 0.820641i \(-0.693617\pi\)
0.996418 + 0.0845650i \(0.0269501\pi\)
\(312\) 0 0
\(313\) −3.46332 5.99864i −0.195758 0.339063i 0.751391 0.659858i \(-0.229383\pi\)
−0.947149 + 0.320794i \(0.896050\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.42872 + 9.40282i 0.304907 + 0.528115i 0.977241 0.212133i \(-0.0680412\pi\)
−0.672333 + 0.740249i \(0.734708\pi\)
\(318\) 0 0
\(319\) −1.43533 + 2.48606i −0.0803629 + 0.139193i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.52027 0.474081
\(324\) 0 0
\(325\) −1.55999 2.70199i −0.0865328 0.149879i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.308570 + 2.53891i 0.0170120 + 0.139975i
\(330\) 0 0
\(331\) 4.02584 + 6.97297i 0.221280 + 0.383269i 0.955197 0.295971i \(-0.0956431\pi\)
−0.733917 + 0.679240i \(0.762310\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.54869 + 2.68240i −0.0846138 + 0.146555i
\(336\) 0 0
\(337\) 11.4293 + 19.7961i 0.622594 + 1.07836i 0.989001 + 0.147909i \(0.0472543\pi\)
−0.366407 + 0.930455i \(0.619412\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 12.6027 21.8285i 0.682474 1.18208i
\(342\) 0 0
\(343\) −6.57324 17.3145i −0.354922 0.934896i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.8892 22.3247i 0.691928 1.19845i −0.279278 0.960210i \(-0.590095\pi\)
0.971205 0.238244i \(-0.0765716\pi\)
\(348\) 0 0
\(349\) 6.90108 11.9530i 0.369406 0.639830i −0.620067 0.784549i \(-0.712894\pi\)
0.989473 + 0.144719i \(0.0462277\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.9028 1.32544 0.662721 0.748866i \(-0.269402\pi\)
0.662721 + 0.748866i \(0.269402\pi\)
\(354\) 0 0
\(355\) 4.49829 0.238744
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.6980 + 18.5295i −0.564620 + 0.977951i 0.432465 + 0.901651i \(0.357644\pi\)
−0.997085 + 0.0763002i \(0.975689\pi\)
\(360\) 0 0
\(361\) −8.31371 14.3998i −0.437564 0.757883i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.89594 3.28386i −0.0992380 0.171885i
\(366\) 0 0
\(367\) −11.5158 −0.601121 −0.300560 0.953763i \(-0.597174\pi\)
−0.300560 + 0.953763i \(0.597174\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.48026 + 28.6355i 0.180686 + 1.48668i
\(372\) 0 0
\(373\) −1.69398 −0.0877109 −0.0438555 0.999038i \(-0.513964\pi\)
−0.0438555 + 0.999038i \(0.513964\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.11911 −0.109140
\(378\) 0 0
\(379\) 8.50319 0.436780 0.218390 0.975862i \(-0.429920\pi\)
0.218390 + 0.975862i \(0.429920\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.7819 0.653124 0.326562 0.945176i \(-0.394110\pi\)
0.326562 + 0.945176i \(0.394110\pi\)
\(384\) 0 0
\(385\) −14.2017 6.04890i −0.723787 0.308280i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.55959 −0.180479 −0.0902393 0.995920i \(-0.528763\pi\)
−0.0902393 + 0.995920i \(0.528763\pi\)
\(390\) 0 0
\(391\) 5.10701 + 8.84560i 0.258273 + 0.447341i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.93448 + 13.7429i 0.399227 + 0.691482i
\(396\) 0 0
\(397\) 11.0411 19.1238i 0.554138 0.959795i −0.443832 0.896110i \(-0.646381\pi\)
0.997970 0.0636848i \(-0.0202852\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.59853 −0.129765 −0.0648823 0.997893i \(-0.520667\pi\)
−0.0648823 + 0.997893i \(0.520667\pi\)
\(402\) 0 0
\(403\) 18.6066 0.926859
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.33973 + 7.51662i −0.215112 + 0.372585i
\(408\) 0 0
\(409\) −1.51604 + 2.62585i −0.0749632 + 0.129840i −0.901070 0.433673i \(-0.857217\pi\)
0.826107 + 0.563513i \(0.190551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 24.0334 18.0642i 1.18261 0.888881i
\(414\) 0 0
\(415\) 15.1899 26.3096i 0.745641 1.29149i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 17.4979 + 30.3073i 0.854829 + 1.48061i 0.876804 + 0.480848i \(0.159671\pi\)
−0.0219749 + 0.999759i \(0.506995\pi\)
\(420\) 0 0
\(421\) 13.3264 23.0820i 0.649488 1.12495i −0.333757 0.942659i \(-0.608316\pi\)
0.983245 0.182288i \(-0.0583502\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.983916 1.70419i −0.0477269 0.0826655i
\(426\) 0 0
\(427\) 0.287106 + 2.36230i 0.0138940 + 0.114320i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 8.77241 + 15.1943i 0.422552 + 0.731882i 0.996188 0.0872286i \(-0.0278010\pi\)
−0.573636 + 0.819110i \(0.694468\pi\)
\(432\) 0 0
\(433\) −18.0202 −0.865997 −0.432998 0.901395i \(-0.642544\pi\)
−0.432998 + 0.901395i \(0.642544\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.3549 36.9878i 1.02154 1.76937i
\(438\) 0 0
\(439\) −18.7159 32.4169i −0.893263 1.54718i −0.835940 0.548820i \(-0.815077\pi\)
−0.0573222 0.998356i \(-0.518256\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.67162 + 6.35944i 0.174444 + 0.302146i 0.939969 0.341261i \(-0.110854\pi\)
−0.765525 + 0.643407i \(0.777521\pi\)
\(444\) 0 0
\(445\) 4.92863 8.53664i 0.233639 0.404675i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.3618 1.90479 0.952395 0.304866i \(-0.0986115\pi\)
0.952395 + 0.304866i \(0.0986115\pi\)
\(450\) 0 0
\(451\) 16.3003 + 28.2330i 0.767553 + 1.32944i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.37480 11.3118i −0.0644514 0.530305i
\(456\) 0 0
\(457\) 13.7360 + 23.7914i 0.642543 + 1.11292i 0.984863 + 0.173333i \(0.0554538\pi\)
−0.342321 + 0.939583i \(0.611213\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.36325 5.82532i 0.156642 0.271312i −0.777014 0.629484i \(-0.783266\pi\)
0.933656 + 0.358172i \(0.116600\pi\)
\(462\) 0 0
\(463\) 1.89569 + 3.28344i 0.0881004 + 0.152594i 0.906708 0.421759i \(-0.138587\pi\)
−0.818608 + 0.574353i \(0.805254\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.2166 + 17.6957i −0.472769 + 0.818860i −0.999514 0.0311635i \(-0.990079\pi\)
0.526746 + 0.850023i \(0.323412\pi\)
\(468\) 0 0
\(469\) 3.44233 2.58735i 0.158952 0.119473i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.14987 15.8480i 0.420712 0.728694i
\(474\) 0 0
\(475\) −4.11423 + 7.12606i −0.188774 + 0.326966i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.9079 −1.27514 −0.637572 0.770391i \(-0.720061\pi\)
−0.637572 + 0.770391i \(0.720061\pi\)
\(480\) 0 0
\(481\) −6.40715 −0.292141
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.75627 3.04194i 0.0797480 0.138128i
\(486\) 0 0
\(487\) −5.89480 10.2101i −0.267119 0.462663i 0.700998 0.713163i \(-0.252738\pi\)
−0.968117 + 0.250500i \(0.919405\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13.2596 22.9662i −0.598396 1.03645i −0.993058 0.117626i \(-0.962472\pi\)
0.394662 0.918826i \(-0.370862\pi\)
\(492\) 0 0
\(493\) −1.33656 −0.0601957
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −5.75380 2.45070i −0.258093 0.109929i
\(498\) 0 0
\(499\) 42.3329 1.89508 0.947540 0.319637i \(-0.103561\pi\)
0.947540 + 0.319637i \(0.103561\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −11.0768 −0.493890 −0.246945 0.969029i \(-0.579427\pi\)
−0.246945 + 0.969029i \(0.579427\pi\)
\(504\) 0 0
\(505\) 15.3449 0.682839
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.7262 −1.09597 −0.547984 0.836489i \(-0.684605\pi\)
−0.547984 + 0.836489i \(0.684605\pi\)
\(510\) 0 0
\(511\) 0.636042 + 5.23334i 0.0281368 + 0.231509i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −33.8709 −1.49253
\(516\) 0 0
\(517\) −1.48185 2.56665i −0.0651719 0.112881i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.42298 11.1249i −0.281396 0.487392i 0.690333 0.723492i \(-0.257464\pi\)
−0.971729 + 0.236100i \(0.924131\pi\)
\(522\) 0 0
\(523\) −1.70453 + 2.95234i −0.0745340 + 0.129097i −0.900884 0.434061i \(-0.857080\pi\)
0.826350 + 0.563158i \(0.190414\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.7355 0.511206
\(528\) 0 0
\(529\) 28.2002 1.22609
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0329 + 20.8416i −0.521202 + 0.902748i
\(534\) 0 0
\(535\) −16.6704 + 28.8741i −0.720726 + 1.24833i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.8701 + 15.4744i 0.640500 + 0.666530i
\(540\) 0 0
\(541\) −22.9553 + 39.7598i −0.986926 + 1.70941i −0.353884 + 0.935289i \(0.615139\pi\)
−0.633043 + 0.774117i \(0.718194\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.12073 + 3.67321i 0.0908421 + 0.157343i
\(546\) 0 0
\(547\) −12.5502 + 21.7376i −0.536608 + 0.929432i 0.462476 + 0.886632i \(0.346961\pi\)
−0.999084 + 0.0428004i \(0.986372\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.79441 + 4.84006i 0.119046 + 0.206194i
\(552\) 0 0
\(553\) −2.66183 21.9015i −0.113192 0.931345i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.836144 1.44824i −0.0354285 0.0613640i 0.847767 0.530368i \(-0.177946\pi\)
−0.883196 + 0.469004i \(0.844613\pi\)
\(558\) 0 0
\(559\) 13.5088 0.571363
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −11.3764 + 19.7046i −0.479460 + 0.830449i −0.999722 0.0235574i \(-0.992501\pi\)
0.520263 + 0.854006i \(0.325834\pi\)
\(564\) 0 0
\(565\) 14.4628 + 25.0504i 0.608456 + 1.05388i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.0292 22.5673i −0.546214 0.946071i −0.998529 0.0542125i \(-0.982735\pi\)
0.452315 0.891858i \(-0.350598\pi\)
\(570\) 0 0
\(571\) −6.24174 + 10.8110i −0.261209 + 0.452427i −0.966563 0.256428i \(-0.917454\pi\)
0.705355 + 0.708855i \(0.250788\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.86421 −0.411366
\(576\) 0 0
\(577\) −10.3756 17.9710i −0.431941 0.748143i 0.565100 0.825023i \(-0.308838\pi\)
−0.997040 + 0.0768793i \(0.975504\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −33.7631 + 25.3773i −1.40073 + 1.05283i
\(582\) 0 0
\(583\) −16.7133 28.9483i −0.692195 1.19892i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.67294 + 15.0220i −0.357971 + 0.620023i −0.987622 0.156855i \(-0.949865\pi\)
0.629651 + 0.776878i \(0.283198\pi\)
\(588\) 0 0
\(589\) −24.5359 42.4975i −1.01099 1.75108i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 14.0203 24.2839i 0.575745 0.997220i −0.420215 0.907425i \(-0.638045\pi\)
0.995960 0.0897956i \(-0.0286214\pi\)
\(594\) 0 0
\(595\) −0.867109 7.13455i −0.0355480 0.292488i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.6047 20.0999i 0.474155 0.821260i −0.525407 0.850851i \(-0.676087\pi\)
0.999562 + 0.0295906i \(0.00942035\pi\)
\(600\) 0 0
\(601\) −0.348014 + 0.602779i −0.0141958 + 0.0245878i −0.873036 0.487656i \(-0.837852\pi\)
0.858840 + 0.512244i \(0.171185\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.04572 −0.123826
\(606\) 0 0
\(607\) −1.71065 −0.0694333 −0.0347166 0.999397i \(-0.511053\pi\)
−0.0347166 + 0.999397i \(0.511053\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.09390 1.89469i 0.0442545 0.0766511i
\(612\) 0 0
\(613\) 1.77253 + 3.07010i 0.0715916 + 0.124000i 0.899599 0.436717i \(-0.143859\pi\)
−0.828007 + 0.560717i \(0.810525\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.58526 9.67395i −0.224854 0.389458i 0.731422 0.681925i \(-0.238857\pi\)
−0.956276 + 0.292467i \(0.905524\pi\)
\(618\) 0 0
\(619\) 19.2172 0.772403 0.386201 0.922415i \(-0.373787\pi\)
0.386201 + 0.922415i \(0.373787\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.9551 + 8.23414i −0.438906 + 0.329894i
\(624\) 0 0
\(625\) −16.2067 −0.648270
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.04111 −0.161130
\(630\) 0 0
\(631\) −23.1101 −0.920000 −0.460000 0.887919i \(-0.652151\pi\)
−0.460000 + 0.887919i \(0.652151\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −32.2185 −1.27855
\(636\) 0 0
\(637\) −4.40423 + 15.2180i −0.174502 + 0.602960i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 20.3938 0.805506 0.402753 0.915309i \(-0.368053\pi\)
0.402753 + 0.915309i \(0.368053\pi\)
\(642\) 0 0
\(643\) −1.31644 2.28015i −0.0519154 0.0899202i 0.838900 0.544286i \(-0.183199\pi\)
−0.890815 + 0.454366i \(0.849866\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.63856 6.30217i −0.143047 0.247764i 0.785596 0.618740i \(-0.212357\pi\)
−0.928642 + 0.370976i \(0.879023\pi\)
\(648\) 0 0
\(649\) −17.4196 + 30.1717i −0.683781 + 1.18434i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.8671 −0.894858 −0.447429 0.894319i \(-0.647660\pi\)
−0.447429 + 0.894319i \(0.647660\pi\)
\(654\) 0 0
\(655\) −4.53589 −0.177232
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.59545 6.22750i 0.140059 0.242589i −0.787460 0.616366i \(-0.788604\pi\)
0.927519 + 0.373777i \(0.121938\pi\)
\(660\) 0 0
\(661\) 17.2064 29.8023i 0.669250 1.15918i −0.308864 0.951106i \(-0.599949\pi\)
0.978114 0.208069i \(-0.0667179\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.0233 + 18.0566i −0.931583 + 0.700204i
\(666\) 0 0
\(667\) −3.34991 + 5.80222i −0.129709 + 0.224663i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.37877 2.38811i −0.0532270 0.0921919i
\(672\) 0 0
\(673\) 3.46705 6.00511i 0.133645 0.231480i −0.791434 0.611255i \(-0.790665\pi\)
0.925079 + 0.379775i \(0.123998\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.3014 + 29.9668i 0.664945 + 1.15172i 0.979300 + 0.202413i \(0.0648784\pi\)
−0.314355 + 0.949306i \(0.601788\pi\)
\(678\) 0 0
\(679\) −3.90373 + 2.93415i −0.149811 + 0.112602i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.1618 + 33.1892i 0.733206 + 1.26995i 0.955506 + 0.294971i \(0.0953101\pi\)
−0.222300 + 0.974978i \(0.571357\pi\)
\(684\) 0 0
\(685\) −20.8988 −0.798502
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.3377 21.3696i 0.470030 0.814116i
\(690\) 0 0
\(691\) 3.94953 + 6.84079i 0.150247 + 0.260236i 0.931318 0.364206i \(-0.118660\pi\)
−0.781071 + 0.624442i \(0.785326\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.04916 12.2095i −0.267390 0.463133i
\(696\) 0 0
\(697\) −7.58936 + 13.1452i −0.287467 + 0.497908i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.9291 0.979329 0.489664 0.871911i \(-0.337119\pi\)
0.489664 + 0.871911i \(0.337119\pi\)
\(702\) 0 0
\(703\) 8.44893 + 14.6340i 0.318657 + 0.551931i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.6278 8.36001i −0.738180 0.314410i
\(708\) 0 0
\(709\) −14.0523 24.3394i −0.527746 0.914084i −0.999477 0.0323408i \(-0.989704\pi\)
0.471731 0.881743i \(-0.343630\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 29.4134 50.9456i 1.10154 1.90793i
\(714\) 0 0
\(715\) 6.60221 + 11.4354i 0.246909 + 0.427658i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 5.89461 10.2098i 0.219832 0.380760i −0.734924 0.678149i \(-0.762782\pi\)
0.954756 + 0.297389i \(0.0961158\pi\)
\(720\) 0 0
\(721\) 43.3246 + 18.4531i 1.61349 + 0.687229i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.645394 1.11785i 0.0239693 0.0415161i
\(726\) 0 0
\(727\) −24.5207 + 42.4711i −0.909423 + 1.57517i −0.0945549 + 0.995520i \(0.530143\pi\)
−0.814868 + 0.579647i \(0.803191\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.52027 0.315134
\(732\) 0 0
\(733\) 15.0927 0.557461 0.278731 0.960369i \(-0.410086\pi\)
0.278731 + 0.960369i \(0.410086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.49503 + 4.32152i −0.0919056 + 0.159185i
\(738\) 0 0
\(739\) 15.9556 + 27.6359i 0.586937 + 1.01660i 0.994631 + 0.103486i \(0.0329996\pi\)
−0.407694 + 0.913119i \(0.633667\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.1582 21.0586i −0.446041 0.772565i 0.552083 0.833789i \(-0.313833\pi\)
−0.998124 + 0.0612238i \(0.980500\pi\)
\(744\) 0 0
\(745\) −33.5385 −1.22875
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 37.0541 27.8509i 1.35393 1.01765i
\(750\) 0 0
\(751\) −33.4463 −1.22047 −0.610237 0.792219i \(-0.708926\pi\)
−0.610237 + 0.792219i \(0.708926\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −38.7574 −1.41053
\(756\) 0 0
\(757\) −32.1248 −1.16759 −0.583797 0.811899i \(-0.698434\pi\)
−0.583797 + 0.811899i \(0.698434\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −33.7957 −1.22509 −0.612547 0.790435i \(-0.709855\pi\)
−0.612547 + 0.790435i \(0.709855\pi\)
\(762\) 0 0
\(763\) −0.711454 5.85382i −0.0257563 0.211923i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −25.7183 −0.928634
\(768\) 0 0
\(769\) 16.8957 + 29.2643i 0.609276 + 1.05530i 0.991360 + 0.131170i \(0.0418733\pi\)
−0.382084 + 0.924128i \(0.624793\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12.0231 + 20.8247i 0.432443 + 0.749012i 0.997083 0.0763245i \(-0.0243185\pi\)
−0.564640 + 0.825337i \(0.690985\pi\)
\(774\) 0 0
\(775\) −5.66679 + 9.81516i −0.203557 + 0.352571i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 63.4696 2.27404
\(780\) 0 0
\(781\) 7.24703 0.259319
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.83219 + 15.2978i −0.315234 + 0.546002i
\(786\) 0 0
\(787\) 3.22897 5.59274i 0.115100 0.199360i −0.802719 0.596357i \(-0.796614\pi\)
0.917820 + 0.396997i \(0.129948\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.85193 39.9216i −0.172515 1.41945i
\(792\) 0 0
\(793\) 1.01781 1.76290i 0.0361435 0.0626023i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.6987 32.3871i −0.662341 1.14721i −0.979999 0.199003i \(-0.936230\pi\)
0.317658 0.948205i \(-0.397104\pi\)
\(798\) 0 0
\(799\) 0.689944 1.19502i 0.0244085 0.0422767i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.05448 5.29051i −0.107790 0.186698i
\(804\) 0 0
\(805\) −33.1455 14.1175i −1.16822 0.497578i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.0048 + 34.6493i 0.703331 + 1.21821i 0.967290 + 0.253672i \(0.0816383\pi\)
−0.263959 + 0.964534i \(0.585028\pi\)
\(810\) 0 0
\(811\) −27.6946 −0.972489 −0.486245 0.873823i \(-0.661634\pi\)
−0.486245 + 0.873823i \(0.661634\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −22.6590 + 39.2465i −0.793709 + 1.37474i
\(816\) 0 0
\(817\) −17.8137 30.8543i −0.623223 1.07945i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −0.188732 0.326893i −0.00658679 0.0114087i 0.862713 0.505693i \(-0.168763\pi\)
−0.869300 + 0.494285i \(0.835430\pi\)
\(822\) 0 0
\(823\) −5.50313 + 9.53170i −0.191827 + 0.332254i −0.945856 0.324587i \(-0.894775\pi\)
0.754029 + 0.656841i \(0.228108\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.2473 −0.982254 −0.491127 0.871088i \(-0.663415\pi\)
−0.491127 + 0.871088i \(0.663415\pi\)
\(828\) 0 0
\(829\) −14.7833 25.6054i −0.513445 0.889313i −0.999878 0.0155953i \(-0.995036\pi\)
0.486433 0.873718i \(-0.338298\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.77783 + 9.59828i −0.0962460 + 0.332561i
\(834\) 0 0
\(835\) 1.68132 + 2.91212i 0.0581843 + 0.100778i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6.91508 + 11.9773i −0.238735 + 0.413501i −0.960352 0.278792i \(-0.910066\pi\)
0.721617 + 0.692293i \(0.243399\pi\)
\(840\) 0 0
\(841\) 14.0616 + 24.3555i 0.484884 + 0.839844i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.49581 12.9831i 0.257864 0.446633i
\(846\) 0 0
\(847\) 3.89581 + 1.65933i 0.133861 + 0.0570152i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.1285 + 17.5431i −0.347200 + 0.601369i
\(852\) 0 0
\(853\) 4.59367 7.95647i 0.157284 0.272424i −0.776604 0.629989i \(-0.783059\pi\)
0.933888 + 0.357565i \(0.116393\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.4460 −1.38161 −0.690805 0.723041i \(-0.742744\pi\)
−0.690805 + 0.723041i \(0.742744\pi\)
\(858\) 0 0
\(859\) −12.5128 −0.426933 −0.213466 0.976950i \(-0.568475\pi\)
−0.213466 + 0.976950i \(0.568475\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 11.8005 20.4391i 0.401694 0.695755i −0.592236 0.805764i \(-0.701755\pi\)
0.993931 + 0.110010i \(0.0350881\pi\)
\(864\) 0 0
\(865\) 0.342635 + 0.593462i 0.0116499 + 0.0201783i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 12.7830 + 22.1407i 0.433632 + 0.751073i
\(870\) 0 0
\(871\) −3.68365 −0.124816
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 29.5469 + 12.5848i 0.998866 + 0.425444i
\(876\) 0 0
\(877\) −27.5467 −0.930185 −0.465092 0.885262i \(-0.653979\pi\)
−0.465092 + 0.885262i \(0.653979\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −42.0894 −1.41803 −0.709014 0.705194i \(-0.750860\pi\)
−0.709014 + 0.705194i \(0.750860\pi\)
\(882\) 0 0
\(883\) 8.58158 0.288793 0.144397 0.989520i \(-0.453876\pi\)
0.144397 + 0.989520i \(0.453876\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.51907 −0.118159 −0.0590795 0.998253i \(-0.518817\pi\)
−0.0590795 + 0.998253i \(0.518817\pi\)
\(888\) 0 0
\(889\) 41.2110 + 17.5529i 1.38217 + 0.588704i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5.76998 −0.193085
\(894\) 0 0
\(895\) 6.80707 + 11.7902i 0.227535 + 0.394103i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.84891 + 6.66651i 0.128368 + 0.222341i
\(900\) 0 0
\(901\) 7.78163 13.4782i 0.259244 0.449023i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.0362 0.699266
\(906\) 0 0
\(907\) 37.2826 1.23795 0.618974 0.785412i \(-0.287549\pi\)
0.618974 + 0.785412i \(0.287549\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10.6458 18.4391i 0.352711 0.610914i −0.634012 0.773323i \(-0.718593\pi\)
0.986723 + 0.162409i \(0.0519265\pi\)
\(912\) 0 0
\(913\) 24.4718 42.3864i 0.809899 1.40279i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.80190 + 2.47118i 0.191595 + 0.0816056i
\(918\) 0 0
\(919\) 6.00453 10.4001i 0.198071 0.343069i −0.749832 0.661628i \(-0.769866\pi\)
0.947903 + 0.318559i \(0.103199\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.67487 + 4.63301i 0.0880444 + 0.152497i
\(924\) 0 0
\(925\) 1.95135 3.37984i 0.0641601 0.111129i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 10.2602 + 17.7712i 0.336626 + 0.583054i 0.983796 0.179292i \(-0.0573807\pi\)
−0.647170 + 0.762346i \(0.724047\pi\)
\(930\) 0 0
\(931\) 40.5657 10.0083i 1.32949 0.328008i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.16413 + 7.21249i 0.136182 + 0.235874i
\(936\) 0 0
\(937\) −10.9040 −0.356217 −0.178109 0.984011i \(-0.556998\pi\)
−0.178109 + 0.984011i \(0.556998\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −15.4807 + 26.8134i −0.504656 + 0.874090i 0.495329 + 0.868705i \(0.335047\pi\)
−0.999986 + 0.00538505i \(0.998286\pi\)
\(942\) 0 0
\(943\) 38.0434 + 65.8931i 1.23886 + 2.14578i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −16.0909 27.8702i −0.522884 0.905661i −0.999645 0.0266283i \(-0.991523\pi\)
0.476762 0.879032i \(-0.341810\pi\)
\(948\) 0 0
\(949\) 2.25481 3.90545i 0.0731942 0.126776i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −39.0934 −1.26636 −0.633179 0.774005i \(-0.718250\pi\)
−0.633179 + 0.774005i \(0.718250\pi\)
\(954\) 0 0
\(955\) −8.27967 14.3408i −0.267924 0.464057i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 26.7318 + 11.3858i 0.863216 + 0.367667i
\(960\) 0 0
\(961\) −18.2948 31.6876i −0.590156 1.02218i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.35046 2.33906i 0.0434728 0.0752971i
\(966\) 0 0
\(967\) −12.7235 22.0377i −0.409159 0.708684i 0.585637 0.810574i \(-0.300845\pi\)
−0.994796 + 0.101889i \(0.967511\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −8.81455 + 15.2673i −0.282872 + 0.489949i −0.972091 0.234604i \(-0.924621\pi\)
0.689219 + 0.724553i \(0.257954\pi\)
\(972\) 0 0
\(973\) 2.36482 + 19.4577i 0.0758128 + 0.623786i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.6269 23.6025i 0.435963 0.755109i −0.561411 0.827537i \(-0.689741\pi\)
0.997374 + 0.0724277i \(0.0230747\pi\)
\(978\) 0 0
\(979\) 7.94033 13.7531i 0.253774 0.439550i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.6741 0.659402 0.329701 0.944085i \(-0.393052\pi\)
0.329701 + 0.944085i \(0.393052\pi\)
\(984\) 0 0
\(985\) −10.8362 −0.345269
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.3549 36.9878i 0.679047 1.17614i
\(990\) 0 0
\(991\) −26.0081 45.0474i −0.826175 1.43098i −0.901018 0.433782i \(-0.857179\pi\)
0.0748425 0.997195i \(-0.476155\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.97307 + 8.61361i 0.157657 + 0.273070i
\(996\) 0 0
\(997\) −9.28538 −0.294071 −0.147035 0.989131i \(-0.546973\pi\)
−0.147035 + 0.989131i \(0.546973\pi\)
\(998\) 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.l.b.289.6 14
3.2 odd 2 252.2.l.b.205.4 yes 14
4.3 odd 2 3024.2.t.j.289.6 14
7.2 even 3 5292.2.j.h.3529.2 14
7.3 odd 6 5292.2.i.i.2125.6 14
7.4 even 3 756.2.i.b.613.2 14
7.5 odd 6 5292.2.j.g.3529.6 14
7.6 odd 2 5292.2.l.i.3313.2 14
9.2 odd 6 2268.2.k.e.1297.6 14
9.4 even 3 756.2.i.b.37.2 14
9.5 odd 6 252.2.i.b.121.2 yes 14
9.7 even 3 2268.2.k.f.1297.2 14
12.11 even 2 1008.2.t.j.961.4 14
21.2 odd 6 1764.2.j.g.1177.7 14
21.5 even 6 1764.2.j.h.1177.1 14
21.11 odd 6 252.2.i.b.25.2 14
21.17 even 6 1764.2.i.i.1537.6 14
21.20 even 2 1764.2.l.i.961.4 14
28.11 odd 6 3024.2.q.j.2881.2 14
36.23 even 6 1008.2.q.j.625.6 14
36.31 odd 6 3024.2.q.j.2305.2 14
63.4 even 3 inner 756.2.l.b.361.6 14
63.5 even 6 1764.2.j.h.589.1 14
63.11 odd 6 2268.2.k.e.1621.6 14
63.13 odd 6 5292.2.i.i.1549.6 14
63.23 odd 6 1764.2.j.g.589.7 14
63.25 even 3 2268.2.k.f.1621.2 14
63.31 odd 6 5292.2.l.i.361.2 14
63.32 odd 6 252.2.l.b.193.4 yes 14
63.40 odd 6 5292.2.j.g.1765.6 14
63.41 even 6 1764.2.i.i.373.6 14
63.58 even 3 5292.2.j.h.1765.2 14
63.59 even 6 1764.2.l.i.949.4 14
84.11 even 6 1008.2.q.j.529.6 14
252.67 odd 6 3024.2.t.j.1873.6 14
252.95 even 6 1008.2.t.j.193.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.i.b.25.2 14 21.11 odd 6
252.2.i.b.121.2 yes 14 9.5 odd 6
252.2.l.b.193.4 yes 14 63.32 odd 6
252.2.l.b.205.4 yes 14 3.2 odd 2
756.2.i.b.37.2 14 9.4 even 3
756.2.i.b.613.2 14 7.4 even 3
756.2.l.b.289.6 14 1.1 even 1 trivial
756.2.l.b.361.6 14 63.4 even 3 inner
1008.2.q.j.529.6 14 84.11 even 6
1008.2.q.j.625.6 14 36.23 even 6
1008.2.t.j.193.4 14 252.95 even 6
1008.2.t.j.961.4 14 12.11 even 2
1764.2.i.i.373.6 14 63.41 even 6
1764.2.i.i.1537.6 14 21.17 even 6
1764.2.j.g.589.7 14 63.23 odd 6
1764.2.j.g.1177.7 14 21.2 odd 6
1764.2.j.h.589.1 14 63.5 even 6
1764.2.j.h.1177.1 14 21.5 even 6
1764.2.l.i.949.4 14 63.59 even 6
1764.2.l.i.961.4 14 21.20 even 2
2268.2.k.e.1297.6 14 9.2 odd 6
2268.2.k.e.1621.6 14 63.11 odd 6
2268.2.k.f.1297.2 14 9.7 even 3
2268.2.k.f.1621.2 14 63.25 even 3
3024.2.q.j.2305.2 14 36.31 odd 6
3024.2.q.j.2881.2 14 28.11 odd 6
3024.2.t.j.289.6 14 4.3 odd 2
3024.2.t.j.1873.6 14 252.67 odd 6
5292.2.i.i.1549.6 14 63.13 odd 6
5292.2.i.i.2125.6 14 7.3 odd 6
5292.2.j.g.1765.6 14 63.40 odd 6
5292.2.j.g.3529.6 14 7.5 odd 6
5292.2.j.h.1765.2 14 63.58 even 3
5292.2.j.h.3529.2 14 7.2 even 3
5292.2.l.i.361.2 14 63.31 odd 6
5292.2.l.i.3313.2 14 7.6 odd 2