Properties

Label 2268.2.w.i.269.4
Level $2268$
Weight $2$
Character 2268.269
Analytic conductor $18.110$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(269,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.w (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{9} \)
Twist minimal: no (minimal twist has level 756)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.4
Root \(-0.385418 + 0.222521i\) of defining polynomial
Character \(\chi\) \(=\) 2268.269
Dual form 2268.2.w.i.1349.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.385418 - 0.667563i) q^{5} +(2.37047 - 1.17511i) q^{7} +(4.68157 - 2.70291i) q^{11} +(4.57338 - 2.64044i) q^{13} +(-2.85433 + 4.94385i) q^{17} +(0.535344 - 0.309081i) q^{19} +(5.83782 + 3.37047i) q^{23} +(2.20291 + 3.81555i) q^{25} +(-6.99408 - 4.03803i) q^{29} -0.580455i q^{31} +(0.129163 - 2.03534i) q^{35} +(-4.53803 - 7.86010i) q^{37} +(4.29615 + 7.44116i) q^{41} +(-1.70291 + 2.94952i) q^{43} -0.770835 q^{47} +(4.23825 - 5.57111i) q^{49} +(-6.63798 - 3.83244i) q^{53} -4.16699i q^{55} -13.7885 q^{59} -4.12928i q^{61} -4.07069i q^{65} +13.5526 q^{67} +2.25906i q^{71} +(1.96197 + 1.13274i) q^{73} +(7.92132 - 11.9085i) q^{77} -8.07069 q^{79} +(1.92709 - 3.33781i) q^{83} +(2.20022 + 3.81089i) q^{85} +(2.59808 + 4.50000i) q^{89} +(7.73825 - 11.6333i) q^{91} -0.476501i q^{95} +(12.1468 + 7.01293i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{19} - 24 q^{37} + 6 q^{43} + 54 q^{73} - 48 q^{79} + 6 q^{85} + 42 q^{91} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 0.385418 0.667563i 0.172364 0.298543i −0.766882 0.641788i \(-0.778193\pi\)
0.939246 + 0.343245i \(0.111526\pi\)
\(6\) 0 0
\(7\) 2.37047 1.17511i 0.895953 0.444148i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.68157 2.70291i 1.41155 0.814957i 0.416013 0.909359i \(-0.363427\pi\)
0.995534 + 0.0944018i \(0.0300938\pi\)
\(12\) 0 0
\(13\) 4.57338 2.64044i 1.26843 0.732326i 0.293736 0.955887i \(-0.405101\pi\)
0.974690 + 0.223560i \(0.0717680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.85433 + 4.94385i −0.692277 + 1.19906i 0.278813 + 0.960345i \(0.410059\pi\)
−0.971090 + 0.238713i \(0.923274\pi\)
\(18\) 0 0
\(19\) 0.535344 0.309081i 0.122816 0.0709080i −0.437333 0.899300i \(-0.644077\pi\)
0.560150 + 0.828391i \(0.310744\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.83782 + 3.37047i 1.21727 + 0.702791i 0.964333 0.264691i \(-0.0852700\pi\)
0.252937 + 0.967483i \(0.418603\pi\)
\(24\) 0 0
\(25\) 2.20291 + 3.81555i 0.440581 + 0.763109i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.99408 4.03803i −1.29877 0.749844i −0.318576 0.947897i \(-0.603205\pi\)
−0.980191 + 0.198053i \(0.936538\pi\)
\(30\) 0 0
\(31\) 0.580455i 0.104253i −0.998640 0.0521264i \(-0.983400\pi\)
0.998640 0.0521264i \(-0.0165999\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.129163 2.03534i 0.0218326 0.344036i
\(36\) 0 0
\(37\) −4.53803 7.86010i −0.746048 1.29219i −0.949704 0.313150i \(-0.898616\pi\)
0.203656 0.979043i \(-0.434718\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.29615 + 7.44116i 0.670947 + 1.16211i 0.977636 + 0.210304i \(0.0674453\pi\)
−0.306690 + 0.951810i \(0.599221\pi\)
\(42\) 0 0
\(43\) −1.70291 + 2.94952i −0.259691 + 0.449798i −0.966159 0.257947i \(-0.916954\pi\)
0.706468 + 0.707745i \(0.250287\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.770835 −0.112438 −0.0562189 0.998418i \(-0.517904\pi\)
−0.0562189 + 0.998418i \(0.517904\pi\)
\(48\) 0 0
\(49\) 4.23825 5.57111i 0.605464 0.795872i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.63798 3.83244i −0.911796 0.526426i −0.0307875 0.999526i \(-0.509802\pi\)
−0.881009 + 0.473100i \(0.843135\pi\)
\(54\) 0 0
\(55\) 4.16699i 0.561877i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −13.7885 −1.79510 −0.897552 0.440908i \(-0.854657\pi\)
−0.897552 + 0.440908i \(0.854657\pi\)
\(60\) 0 0
\(61\) 4.12928i 0.528701i −0.964427 0.264350i \(-0.914842\pi\)
0.964427 0.264350i \(-0.0851575\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.07069i 0.504907i
\(66\) 0 0
\(67\) 13.5526 1.65571 0.827855 0.560943i \(-0.189561\pi\)
0.827855 + 0.560943i \(0.189561\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.25906i 0.268101i 0.990974 + 0.134051i \(0.0427985\pi\)
−0.990974 + 0.134051i \(0.957202\pi\)
\(72\) 0 0
\(73\) 1.96197 + 1.13274i 0.229631 + 0.132577i 0.610402 0.792092i \(-0.291008\pi\)
−0.380771 + 0.924669i \(0.624341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.92132 11.9085i 0.902718 1.35710i
\(78\) 0 0
\(79\) −8.07069 −0.908023 −0.454012 0.890996i \(-0.650008\pi\)
−0.454012 + 0.890996i \(0.650008\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.92709 3.33781i 0.211525 0.366373i −0.740667 0.671873i \(-0.765490\pi\)
0.952192 + 0.305500i \(0.0988236\pi\)
\(84\) 0 0
\(85\) 2.20022 + 3.81089i 0.238647 + 0.413349i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.59808 + 4.50000i 0.275396 + 0.476999i 0.970235 0.242166i \(-0.0778579\pi\)
−0.694839 + 0.719165i \(0.744525\pi\)
\(90\) 0 0
\(91\) 7.73825 11.6333i 0.811189 1.21950i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.476501i 0.0488880i
\(96\) 0 0
\(97\) 12.1468 + 7.01293i 1.23332 + 0.712055i 0.967720 0.252029i \(-0.0810979\pi\)
0.265596 + 0.964084i \(0.414431\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.35241 + 11.0027i 0.632088 + 1.09481i 0.987124 + 0.159956i \(0.0511352\pi\)
−0.355036 + 0.934853i \(0.615531\pi\)
\(102\) 0 0
\(103\) 2.57069 + 1.48419i 0.253297 + 0.146241i 0.621273 0.783594i \(-0.286616\pi\)
−0.367976 + 0.929835i \(0.619949\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.26891 + 2.46466i −0.412691 + 0.238267i −0.691945 0.721950i \(-0.743246\pi\)
0.279254 + 0.960217i \(0.409913\pi\)
\(108\) 0 0
\(109\) 3.40581 5.89904i 0.326218 0.565026i −0.655540 0.755160i \(-0.727559\pi\)
0.981758 + 0.190134i \(0.0608924\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.48172 3.16487i 0.515677 0.297726i −0.219487 0.975615i \(-0.570438\pi\)
0.735164 + 0.677889i \(0.237105\pi\)
\(114\) 0 0
\(115\) 4.50000 2.59808i 0.419627 0.242272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.956559 + 15.0734i −0.0876876 + 1.38177i
\(120\) 0 0
\(121\) 9.11141 15.7814i 0.828310 1.43467i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 7.25033 0.648489
\(126\) 0 0
\(127\) −4.41119 −0.391430 −0.195715 0.980661i \(-0.562703\pi\)
−0.195715 + 0.980661i \(0.562703\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.39600 + 7.61410i −0.384080 + 0.665247i −0.991641 0.129026i \(-0.958815\pi\)
0.607561 + 0.794273i \(0.292148\pi\)
\(132\) 0 0
\(133\) 0.905813 1.36175i 0.0785440 0.118079i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.91281 2.25906i 0.334294 0.193005i −0.323452 0.946245i \(-0.604843\pi\)
0.657746 + 0.753240i \(0.271510\pi\)
\(138\) 0 0
\(139\) 6.64944 3.83906i 0.563998 0.325624i −0.190750 0.981639i \(-0.561092\pi\)
0.754749 + 0.656014i \(0.227759\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.2737 24.7228i 1.19363 2.06743i
\(144\) 0 0
\(145\) −5.39128 + 3.11266i −0.447721 + 0.258492i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.40674 4.27628i −0.606784 0.350327i 0.164922 0.986307i \(-0.447263\pi\)
−0.771706 + 0.635980i \(0.780596\pi\)
\(150\) 0 0
\(151\) 3.53803 + 6.12805i 0.287921 + 0.498694i 0.973313 0.229480i \(-0.0737026\pi\)
−0.685392 + 0.728174i \(0.740369\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.387490 0.223717i −0.0311239 0.0179694i
\(156\) 0 0
\(157\) 10.6465i 0.849682i −0.905268 0.424841i \(-0.860330\pi\)
0.905268 0.424841i \(-0.139670\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.7990 + 1.12953i 1.40276 + 0.0890195i
\(162\) 0 0
\(163\) −7.53534 13.0516i −0.590214 1.02228i −0.994203 0.107517i \(-0.965710\pi\)
0.403990 0.914764i \(-0.367623\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.69215 15.0553i −0.672619 1.16501i −0.977159 0.212511i \(-0.931836\pi\)
0.304540 0.952500i \(-0.401497\pi\)
\(168\) 0 0
\(169\) 7.44385 12.8931i 0.572603 0.991778i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −19.7554 −1.50198 −0.750989 0.660314i \(-0.770423\pi\)
−0.750989 + 0.660314i \(0.770423\pi\)
\(174\) 0 0
\(175\) 9.70560 + 6.45599i 0.733674 + 0.488027i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.1444 8.74363i −1.13195 0.653529i −0.187522 0.982260i \(-0.560046\pi\)
−0.944424 + 0.328731i \(0.893379\pi\)
\(180\) 0 0
\(181\) 4.11997i 0.306235i 0.988208 + 0.153118i \(0.0489313\pi\)
−0.988208 + 0.153118i \(0.951069\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.99615 −0.514367
\(186\) 0 0
\(187\) 30.8600i 2.25670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.3696i 1.03975i 0.854244 + 0.519873i \(0.174021\pi\)
−0.854244 + 0.519873i \(0.825979\pi\)
\(192\) 0 0
\(193\) 2.07069 0.149051 0.0745257 0.997219i \(-0.476256\pi\)
0.0745257 + 0.997219i \(0.476256\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.2881i 1.44547i 0.691126 + 0.722735i \(0.257115\pi\)
−0.691126 + 0.722735i \(0.742885\pi\)
\(198\) 0 0
\(199\) 5.42394 + 3.13151i 0.384493 + 0.221987i 0.679771 0.733424i \(-0.262079\pi\)
−0.295279 + 0.955411i \(0.595412\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −21.3244 1.35325i −1.49668 0.0949794i
\(204\) 0 0
\(205\) 6.62325 0.462588
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.67083 2.89397i 0.115574 0.200180i
\(210\) 0 0
\(211\) −1.86778 3.23509i −0.128583 0.222713i 0.794545 0.607206i \(-0.207710\pi\)
−0.923128 + 0.384493i \(0.874376\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.31266 + 2.27359i 0.0895227 + 0.155058i
\(216\) 0 0
\(217\) −0.682096 1.37595i −0.0463037 0.0934056i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 30.1468i 2.02789i
\(222\) 0 0
\(223\) −13.7201 7.92132i −0.918768 0.530451i −0.0355260 0.999369i \(-0.511311\pi\)
−0.883242 + 0.468918i \(0.844644\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.95640 + 3.38859i 0.129851 + 0.224909i 0.923619 0.383312i \(-0.125217\pi\)
−0.793768 + 0.608221i \(0.791883\pi\)
\(228\) 0 0
\(229\) −4.39397 2.53686i −0.290362 0.167640i 0.347743 0.937590i \(-0.386948\pi\)
−0.638105 + 0.769949i \(0.720281\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.56891 + 0.905813i −0.102783 + 0.0593418i −0.550510 0.834828i \(-0.685567\pi\)
0.447727 + 0.894170i \(0.352234\pi\)
\(234\) 0 0
\(235\) −0.297093 + 0.514581i −0.0193802 + 0.0335676i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.43813 4.29440i 0.481132 0.277782i −0.239756 0.970833i \(-0.577067\pi\)
0.720888 + 0.693051i \(0.243734\pi\)
\(240\) 0 0
\(241\) 20.6521 11.9235i 1.33032 0.768061i 0.344972 0.938613i \(-0.387888\pi\)
0.985349 + 0.170552i \(0.0545551\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.08557 4.97650i −0.133242 0.317937i
\(246\) 0 0
\(247\) 1.63222 2.82709i 0.103856 0.179883i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.25033 −0.457637 −0.228818 0.973469i \(-0.573486\pi\)
−0.228818 + 0.973469i \(0.573486\pi\)
\(252\) 0 0
\(253\) 36.4403 2.29098
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.2737 24.7228i 0.890370 1.54217i 0.0509387 0.998702i \(-0.483779\pi\)
0.839432 0.543465i \(-0.182888\pi\)
\(258\) 0 0
\(259\) −19.9937 13.2995i −1.24235 0.826388i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.95640 1.12953i 0.120637 0.0696498i −0.438467 0.898747i \(-0.644478\pi\)
0.559104 + 0.829097i \(0.311145\pi\)
\(264\) 0 0
\(265\) −5.11679 + 2.95418i −0.314322 + 0.181474i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.33605 14.4385i 0.508258 0.880329i −0.491696 0.870767i \(-0.663623\pi\)
0.999954 0.00956210i \(-0.00304376\pi\)
\(270\) 0 0
\(271\) −27.2528 + 15.7344i −1.65549 + 0.955797i −0.680730 + 0.732534i \(0.738337\pi\)
−0.974758 + 0.223263i \(0.928329\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 20.6261 + 11.9085i 1.24380 + 0.718110i
\(276\) 0 0
\(277\) −9.51991 16.4890i −0.571996 0.990726i −0.996361 0.0852348i \(-0.972836\pi\)
0.424365 0.905491i \(-0.360497\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.58142 3.79978i −0.392614 0.226676i 0.290678 0.956821i \(-0.406119\pi\)
−0.683292 + 0.730145i \(0.739453\pi\)
\(282\) 0 0
\(283\) 30.9824i 1.84171i 0.389903 + 0.920856i \(0.372509\pi\)
−0.389903 + 0.920856i \(0.627491\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.9281 + 12.5906i 1.11729 + 0.743199i
\(288\) 0 0
\(289\) −7.79440 13.5003i −0.458494 0.794136i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.46891 4.27628i −0.144235 0.249823i 0.784852 0.619683i \(-0.212739\pi\)
−0.929087 + 0.369860i \(0.879406\pi\)
\(294\) 0 0
\(295\) −5.31431 + 9.20466i −0.309411 + 0.535916i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 35.5981 2.05869
\(300\) 0 0
\(301\) −0.570688 + 8.99284i −0.0328939 + 0.518339i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.75656 1.59150i −0.157840 0.0911289i
\(306\) 0 0
\(307\) 7.95736i 0.454151i 0.973877 + 0.227075i \(0.0729164\pi\)
−0.973877 + 0.227075i \(0.927084\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.87565 0.559997 0.279998 0.960000i \(-0.409666\pi\)
0.279998 + 0.960000i \(0.409666\pi\)
\(312\) 0 0
\(313\) 16.8248i 0.950993i 0.879718 + 0.475496i \(0.157732\pi\)
−0.879718 + 0.475496i \(0.842268\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.7700i 1.39122i −0.718419 0.695611i \(-0.755134\pi\)
0.718419 0.695611i \(-0.244866\pi\)
\(318\) 0 0
\(319\) −43.6577 −2.44436
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.52888i 0.196352i
\(324\) 0 0
\(325\) 20.1494 + 11.6333i 1.11769 + 0.645299i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.82724 + 0.905813i −0.100739 + 0.0499391i
\(330\) 0 0
\(331\) −18.0054 −0.989665 −0.494833 0.868988i \(-0.664771\pi\)
−0.494833 + 0.868988i \(0.664771\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.22340 9.04719i 0.285385 0.494301i
\(336\) 0 0
\(337\) −2.60872 4.51844i −0.142106 0.246135i 0.786183 0.617993i \(-0.212054\pi\)
−0.928290 + 0.371858i \(0.878721\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.56891 2.71744i −0.0849615 0.147158i
\(342\) 0 0
\(343\) 3.50000 18.1865i 0.188982 0.981981i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.2828i 0.874104i 0.899436 + 0.437052i \(0.143977\pi\)
−0.899436 + 0.437052i \(0.856023\pi\)
\(348\) 0 0
\(349\) 15.0734 + 8.70262i 0.806859 + 0.465840i 0.845864 0.533399i \(-0.179085\pi\)
−0.0390047 + 0.999239i \(0.512419\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.9447 + 25.8850i 0.795427 + 1.37772i 0.922568 + 0.385835i \(0.126087\pi\)
−0.127141 + 0.991885i \(0.540580\pi\)
\(354\) 0 0
\(355\) 1.50807 + 0.870682i 0.0800398 + 0.0462110i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −20.2700 + 11.7029i −1.06981 + 0.617656i −0.928130 0.372257i \(-0.878584\pi\)
−0.141681 + 0.989912i \(0.545251\pi\)
\(360\) 0 0
\(361\) −9.30894 + 16.1236i −0.489944 + 0.848608i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.51235 0.873158i 0.0791602 0.0457032i
\(366\) 0 0
\(367\) −17.3696 + 10.0283i −0.906684 + 0.523474i −0.879363 0.476152i \(-0.842031\pi\)
−0.0273213 + 0.999627i \(0.508698\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −20.2386 1.28435i −1.05074 0.0666800i
\(372\) 0 0
\(373\) 3.99462 6.91889i 0.206834 0.358247i −0.743882 0.668311i \(-0.767017\pi\)
0.950715 + 0.310065i \(0.100351\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −42.6487 −2.19652
\(378\) 0 0
\(379\) −34.5816 −1.77634 −0.888170 0.459516i \(-0.848023\pi\)
−0.888170 + 0.459516i \(0.848023\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.01058 6.94653i 0.204931 0.354951i −0.745180 0.666864i \(-0.767636\pi\)
0.950111 + 0.311913i \(0.100970\pi\)
\(384\) 0 0
\(385\) −4.89666 9.87772i −0.249557 0.503415i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.9446 9.20560i 0.808421 0.466742i −0.0379861 0.999278i \(-0.512094\pi\)
0.846407 + 0.532536i \(0.178761\pi\)
\(390\) 0 0
\(391\) −33.3262 + 19.2409i −1.68538 + 0.973052i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.11058 + 5.38769i −0.156511 + 0.271084i
\(396\) 0 0
\(397\) −21.9973 + 12.7002i −1.10401 + 0.637402i −0.937272 0.348598i \(-0.886658\pi\)
−0.166741 + 0.986001i \(0.553324\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.43813 4.29440i −0.371442 0.214452i 0.302646 0.953103i \(-0.402130\pi\)
−0.674088 + 0.738651i \(0.735463\pi\)
\(402\) 0 0
\(403\) −1.53266 2.65464i −0.0763470 0.132237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −42.4902 24.5318i −2.10616 1.21599i
\(408\) 0 0
\(409\) 3.51112i 0.173614i 0.996225 + 0.0868069i \(0.0276663\pi\)
−0.996225 + 0.0868069i \(0.972334\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −32.6851 + 16.2029i −1.60833 + 0.797293i
\(414\) 0 0
\(415\) −1.48547 2.57290i −0.0729187 0.126299i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.7862 + 25.6105i 0.722355 + 1.25116i 0.960054 + 0.279816i \(0.0902735\pi\)
−0.237699 + 0.971339i \(0.576393\pi\)
\(420\) 0 0
\(421\) 11.5172 19.9484i 0.561315 0.972226i −0.436067 0.899914i \(-0.643629\pi\)
0.997382 0.0723120i \(-0.0230377\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −25.1513 −1.22002
\(426\) 0 0
\(427\) −4.85235 9.78834i −0.234822 0.473691i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.2386 + 11.6848i 0.974861 + 0.562836i 0.900715 0.434411i \(-0.143043\pi\)
0.0741463 + 0.997247i \(0.476377\pi\)
\(432\) 0 0
\(433\) 37.9199i 1.82231i 0.412059 + 0.911157i \(0.364810\pi\)
−0.412059 + 0.911157i \(0.635190\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.16699 0.199334
\(438\) 0 0
\(439\) 3.13783i 0.149760i −0.997193 0.0748802i \(-0.976143\pi\)
0.997193 0.0748802i \(-0.0238574\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.2282i 1.15112i −0.817761 0.575558i \(-0.804785\pi\)
0.817761 0.575558i \(-0.195215\pi\)
\(444\) 0 0
\(445\) 4.00538 0.189873
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 25.1812i 1.18837i −0.804327 0.594187i \(-0.797474\pi\)
0.804327 0.594187i \(-0.202526\pi\)
\(450\) 0 0
\(451\) 40.2255 + 23.2242i 1.89415 + 1.09359i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.78349 9.64944i −0.224253 0.452373i
\(456\) 0 0
\(457\) −26.2121 −1.22615 −0.613074 0.790025i \(-0.710067\pi\)
−0.613074 + 0.790025i \(0.710067\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.899998 + 1.55884i −0.0419171 + 0.0726026i −0.886223 0.463259i \(-0.846680\pi\)
0.844306 + 0.535862i \(0.180013\pi\)
\(462\) 0 0
\(463\) 10.2228 + 17.7064i 0.475095 + 0.822888i 0.999593 0.0285234i \(-0.00908051\pi\)
−0.524499 + 0.851411i \(0.675747\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.25271 9.09797i −0.243067 0.421004i 0.718520 0.695507i \(-0.244820\pi\)
−0.961586 + 0.274503i \(0.911487\pi\)
\(468\) 0 0
\(469\) 32.1259 15.9257i 1.48344 0.735381i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18.4112i 0.846547i
\(474\) 0 0
\(475\) 2.35862 + 1.36175i 0.108221 + 0.0624815i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.98349 5.16756i −0.136319 0.236112i 0.789781 0.613388i \(-0.210194\pi\)
−0.926101 + 0.377276i \(0.876861\pi\)
\(480\) 0 0
\(481\) −41.5083 23.9648i −1.89261 1.09270i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.36314 5.40581i 0.425158 0.245465i
\(486\) 0 0
\(487\) −12.0199 + 20.8191i −0.544674 + 0.943403i 0.453953 + 0.891026i \(0.350013\pi\)
−0.998627 + 0.0523777i \(0.983320\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 18.6697 10.7790i 0.842553 0.486448i −0.0155783 0.999879i \(-0.504959\pi\)
0.858131 + 0.513430i \(0.171626\pi\)
\(492\) 0 0
\(493\) 39.9268 23.0518i 1.79821 1.03820i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.65464 + 5.35504i 0.119077 + 0.240206i
\(498\) 0 0
\(499\) −14.4257 + 24.9861i −0.645784 + 1.11853i 0.338336 + 0.941025i \(0.390136\pi\)
−0.984120 + 0.177505i \(0.943197\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.7805 0.926555 0.463278 0.886213i \(-0.346673\pi\)
0.463278 + 0.886213i \(0.346673\pi\)
\(504\) 0 0
\(505\) 9.79331 0.435797
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.7027 + 22.0018i −0.563039 + 0.975212i 0.434190 + 0.900821i \(0.357035\pi\)
−0.997229 + 0.0743909i \(0.976299\pi\)
\(510\) 0 0
\(511\) 5.98188 + 0.379611i 0.264623 + 0.0167930i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.98158 1.14406i 0.0873187 0.0504135i
\(516\) 0 0
\(517\) −3.60872 + 2.08350i −0.158711 + 0.0916320i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.02709 + 1.77897i −0.0449976 + 0.0779381i −0.887647 0.460524i \(-0.847661\pi\)
0.842649 + 0.538463i \(0.180995\pi\)
\(522\) 0 0
\(523\) −9.81790 + 5.66837i −0.429307 + 0.247860i −0.699051 0.715071i \(-0.746394\pi\)
0.269744 + 0.962932i \(0.413061\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.86968 + 1.65681i 0.125005 + 0.0721717i
\(528\) 0 0
\(529\) 11.2201 + 19.4338i 0.487832 + 0.844949i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 39.2959 + 22.6875i 1.70209 + 0.982704i
\(534\) 0 0
\(535\) 3.79969i 0.164275i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.78349 37.5371i 0.206040 1.61684i
\(540\) 0 0
\(541\) −9.39128 16.2662i −0.403763 0.699337i 0.590414 0.807101i \(-0.298965\pi\)
−0.994177 + 0.107763i \(0.965631\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.62532 4.54719i −0.112456 0.194780i
\(546\) 0 0
\(547\) 3.45928 5.99165i 0.147908 0.256184i −0.782546 0.622593i \(-0.786079\pi\)
0.930454 + 0.366408i \(0.119413\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.99231 −0.212680
\(552\) 0 0
\(553\) −19.1313 + 9.48392i −0.813546 + 0.403297i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00704 5.20022i −0.381641 0.220340i 0.296891 0.954911i \(-0.404050\pi\)
−0.678532 + 0.734571i \(0.737383\pi\)
\(558\) 0 0
\(559\) 17.9857i 0.760714i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.02148 −0.253775 −0.126887 0.991917i \(-0.540499\pi\)
−0.126887 + 0.991917i \(0.540499\pi\)
\(564\) 0 0
\(565\) 4.87919i 0.205269i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 29.3534i 1.23056i 0.788309 + 0.615280i \(0.210957\pi\)
−0.788309 + 0.615280i \(0.789043\pi\)
\(570\) 0 0
\(571\) 17.2282 0.720977 0.360489 0.932764i \(-0.382610\pi\)
0.360489 + 0.932764i \(0.382610\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 29.6993i 1.23855i
\(576\) 0 0
\(577\) 1.85594 + 1.07153i 0.0772636 + 0.0446082i 0.538134 0.842859i \(-0.319129\pi\)
−0.460870 + 0.887467i \(0.652463\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0.645816 10.1767i 0.0267930 0.422201i
\(582\) 0 0
\(583\) −41.4349 −1.71606
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.79200 + 15.2282i −0.362885 + 0.628535i −0.988434 0.151650i \(-0.951541\pi\)
0.625550 + 0.780184i \(0.284875\pi\)
\(588\) 0 0
\(589\) −0.179407 0.310743i −0.00739235 0.0128039i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.1469 + 21.0391i 0.498815 + 0.863973i 0.999999 0.00136757i \(-0.000435311\pi\)
−0.501184 + 0.865341i \(0.667102\pi\)
\(594\) 0 0
\(595\) 9.69375 + 6.44811i 0.397405 + 0.264347i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 37.8116i 1.54494i −0.635051 0.772471i \(-0.719021\pi\)
0.635051 0.772471i \(-0.280979\pi\)
\(600\) 0 0
\(601\) −2.51344 1.45114i −0.102525 0.0591931i 0.447861 0.894103i \(-0.352186\pi\)
−0.550386 + 0.834910i \(0.685519\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.02339 12.1649i −0.285542 0.494572i
\(606\) 0 0
\(607\) −9.67403 5.58530i −0.392657 0.226701i 0.290654 0.956828i \(-0.406127\pi\)
−0.683311 + 0.730128i \(0.739461\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.52532 + 2.03534i −0.142619 + 0.0823412i
\(612\) 0 0
\(613\) −8.12953 + 14.0808i −0.328349 + 0.568717i −0.982184 0.187920i \(-0.939825\pi\)
0.653836 + 0.756637i \(0.273159\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.62501 2.67025i 0.186196 0.107500i −0.404005 0.914757i \(-0.632382\pi\)
0.590200 + 0.807257i \(0.299049\pi\)
\(618\) 0 0
\(619\) −18.3669 + 10.6041i −0.738227 + 0.426216i −0.821424 0.570317i \(-0.806820\pi\)
0.0831972 + 0.996533i \(0.473487\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.4466 + 7.61410i 0.458600 + 0.305052i
\(624\) 0 0
\(625\) −8.22013 + 14.2377i −0.328805 + 0.569507i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 51.8122 2.06589
\(630\) 0 0
\(631\) 44.7338 1.78082 0.890411 0.455157i \(-0.150417\pi\)
0.890411 + 0.455157i \(0.150417\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.70015 + 2.94475i −0.0674684 + 0.116859i
\(636\) 0 0
\(637\) 4.67294 36.6696i 0.185149 1.45290i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.6508 13.6548i 0.934152 0.539333i 0.0460296 0.998940i \(-0.485343\pi\)
0.888122 + 0.459607i \(0.152010\pi\)
\(642\) 0 0
\(643\) −30.9429 + 17.8649i −1.22027 + 0.704524i −0.964976 0.262340i \(-0.915506\pi\)
−0.255295 + 0.966863i \(0.582173\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.4537 + 35.4268i −0.804117 + 1.39277i 0.112768 + 0.993621i \(0.464028\pi\)
−0.916885 + 0.399150i \(0.869305\pi\)
\(648\) 0 0
\(649\) −64.5517 + 37.2689i −2.53387 + 1.46293i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −22.5825 13.0380i −0.883723 0.510218i −0.0118388 0.999930i \(-0.503768\pi\)
−0.871884 + 0.489712i \(0.837102\pi\)
\(654\) 0 0
\(655\) 3.38859 + 5.86921i 0.132403 + 0.229329i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 35.3830 + 20.4284i 1.37833 + 0.795778i 0.991958 0.126566i \(-0.0403956\pi\)
0.386370 + 0.922344i \(0.373729\pi\)
\(660\) 0 0
\(661\) 7.25780i 0.282296i 0.989989 + 0.141148i \(0.0450793\pi\)
−0.989989 + 0.141148i \(0.954921\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.559939 1.12953i −0.0217135 0.0438013i
\(666\) 0 0
\(667\) −27.2201 47.1466i −1.05397 1.82553i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.1611 19.3315i −0.430868 0.746286i
\(672\) 0 0
\(673\) −17.6233 + 30.5244i −0.679326 + 1.17663i 0.295858 + 0.955232i \(0.404395\pi\)
−0.975184 + 0.221396i \(0.928939\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.9634 −0.421359 −0.210680 0.977555i \(-0.567568\pi\)
−0.210680 + 0.977555i \(0.567568\pi\)
\(678\) 0 0
\(679\) 37.0344 + 2.35021i 1.42125 + 0.0901929i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.70628 4.44922i −0.294873 0.170245i 0.345265 0.938505i \(-0.387789\pi\)
−0.640137 + 0.768261i \(0.721123\pi\)
\(684\) 0 0
\(685\) 3.48273i 0.133068i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −40.4773 −1.54206
\(690\) 0 0
\(691\) 8.40607i 0.319782i 0.987135 + 0.159891i \(0.0511143\pi\)
−0.987135 + 0.159891i \(0.948886\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.91856i 0.224504i
\(696\) 0 0
\(697\) −49.0506 −1.85792
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.3351i 1.07020i −0.844788 0.535101i \(-0.820273\pi\)
0.844788 0.535101i \(-0.179727\pi\)
\(702\) 0 0
\(703\) −4.85881 2.80524i −0.183254 0.105802i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.9875 + 18.6168i 1.05258 + 0.700156i
\(708\) 0 0
\(709\) −24.0868 −0.904599 −0.452300 0.891866i \(-0.649396\pi\)
−0.452300 + 0.891866i \(0.649396\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.95640 3.38859i 0.0732679 0.126904i
\(714\) 0 0
\(715\) −11.0027 19.0572i −0.411477 0.712699i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.57083 4.45281i −0.0958759 0.166062i 0.814098 0.580728i \(-0.197232\pi\)
−0.909974 + 0.414666i \(0.863899\pi\)
\(720\) 0 0
\(721\) 7.83781 + 0.497389i 0.291895 + 0.0185237i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 35.5816i 1.32147i
\(726\) 0 0
\(727\) −5.99193 3.45945i −0.222229 0.128304i 0.384753 0.923019i \(-0.374287\pi\)
−0.606982 + 0.794716i \(0.707620\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9.72132 16.8378i −0.359556 0.622769i
\(732\) 0 0
\(733\) −14.8940 8.59904i −0.550121 0.317613i 0.199050 0.979989i \(-0.436215\pi\)
−0.749171 + 0.662377i \(0.769548\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 63.4473 36.6313i 2.33711 1.34933i
\(738\) 0 0
\(739\) 10.6114 18.3795i 0.390347 0.676101i −0.602148 0.798384i \(-0.705688\pi\)
0.992495 + 0.122284i \(0.0390217\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.1891 + 16.8523i −1.07085 + 0.618253i −0.928412 0.371552i \(-0.878826\pi\)
−0.142433 + 0.989804i \(0.545492\pi\)
\(744\) 0 0
\(745\) −5.70937 + 3.29631i −0.209175 + 0.120767i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −7.22309 + 10.8588i −0.263926 + 0.396772i
\(750\) 0 0
\(751\) −2.41119 + 4.17630i −0.0879856 + 0.152395i −0.906660 0.421863i \(-0.861376\pi\)
0.818674 + 0.574259i \(0.194710\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.45448 0.198509
\(756\) 0 0
\(757\) −7.30426 −0.265478 −0.132739 0.991151i \(-0.542377\pi\)
−0.132739 + 0.991151i \(0.542377\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.91925 + 13.7165i −0.287072 + 0.497224i −0.973110 0.230342i \(-0.926015\pi\)
0.686037 + 0.727566i \(0.259349\pi\)
\(762\) 0 0
\(763\) 1.14138 17.9857i 0.0413206 0.651126i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −63.0598 + 36.4076i −2.27696 + 1.31460i
\(768\) 0 0
\(769\) 40.9810 23.6604i 1.47781 0.853215i 0.478126 0.878291i \(-0.341316\pi\)
0.999686 + 0.0250761i \(0.00798279\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 10.5173 18.2165i 0.378282 0.655203i −0.612531 0.790447i \(-0.709848\pi\)
0.990812 + 0.135244i \(0.0431817\pi\)
\(774\) 0 0
\(775\) 2.21475 1.27869i 0.0795562 0.0459318i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 4.59984 + 2.65572i 0.164806 + 0.0951510i
\(780\) 0 0
\(781\) 6.10603 + 10.5760i 0.218491 + 0.378437i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.10720 4.10334i −0.253667 0.146455i
\(786\) 0 0
\(787\) 39.0152i 1.39074i 0.718652 + 0.695370i \(0.244760\pi\)
−0.718652 + 0.695370i \(0.755240\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.27519 13.9438i 0.329788 0.495786i
\(792\) 0 0
\(793\) −10.9031 18.8848i −0.387181 0.670618i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.8426 + 27.4403i 0.561175 + 0.971984i 0.997394 + 0.0721433i \(0.0229839\pi\)
−0.436219 + 0.899840i \(0.643683\pi\)
\(798\) 0 0
\(799\) 2.20022 3.81089i 0.0778381 0.134820i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.2468 0.432180
\(804\) 0 0
\(805\) 7.61410 11.4466i 0.268362 0.403441i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.93123 + 1.11500i 0.0678985 + 0.0392012i 0.533565 0.845759i \(-0.320852\pi\)
−0.465666 + 0.884960i \(0.654185\pi\)
\(810\) 0 0
\(811\) 55.3535i 1.94373i 0.235548 + 0.971863i \(0.424312\pi\)
−0.235548 + 0.971863i \(0.575688\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.6170 −0.406926
\(816\) 0 0
\(817\) 2.10534i 0.0736566i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 45.9821i 1.60479i 0.596797 + 0.802393i \(0.296440\pi\)
−0.596797 + 0.802393i \(0.703560\pi\)
\(822\) 0 0
\(823\) 42.2989 1.47445 0.737223 0.675649i \(-0.236137\pi\)
0.737223 + 0.675649i \(0.236137\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.1105i 0.734084i 0.930204 + 0.367042i \(0.119630\pi\)
−0.930204 + 0.367042i \(0.880370\pi\)
\(828\) 0 0
\(829\) 47.3615 + 27.3442i 1.64493 + 0.949703i 0.979043 + 0.203652i \(0.0652812\pi\)
0.665890 + 0.746050i \(0.268052\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 15.4453 + 36.8550i 0.535149 + 1.27695i
\(834\) 0 0
\(835\) −13.4004 −0.463741
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.6825 + 37.5553i −0.748564 + 1.29655i 0.199947 + 0.979807i \(0.435923\pi\)
−0.948511 + 0.316745i \(0.897410\pi\)
\(840\) 0 0
\(841\) 18.1114 + 31.3699i 0.624531 + 1.08172i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.73798 9.93847i −0.197392 0.341894i
\(846\) 0 0
\(847\) 3.05347 48.1163i 0.104918 1.65329i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 61.1812i 2.09726i
\(852\) 0 0
\(853\) 23.0353 + 13.2995i 0.788715 + 0.455365i 0.839510 0.543344i \(-0.182842\pi\)
−0.0507949 + 0.998709i \(0.516175\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.8155 25.6613i −0.506090 0.876573i −0.999975 0.00704593i \(-0.997757\pi\)
0.493886 0.869527i \(-0.335576\pi\)
\(858\) 0 0
\(859\) −4.19285 2.42074i −0.143058 0.0825947i 0.426763 0.904364i \(-0.359654\pi\)
−0.569821 + 0.821769i \(0.692987\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −19.0258 + 10.9846i −0.647647 + 0.373919i −0.787554 0.616246i \(-0.788653\pi\)
0.139907 + 0.990165i \(0.455320\pi\)
\(864\) 0 0
\(865\) −7.61410 + 13.1880i −0.258887 + 0.448406i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −37.7835 + 21.8143i −1.28172 + 0.740000i
\(870\) 0 0
\(871\) 61.9810 35.7847i 2.10015 1.21252i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 17.1867 8.51991i 0.581016 0.288026i
\(876\) 0 0
\(877\) 21.2989 36.8907i 0.719212 1.24571i −0.242100 0.970251i \(-0.577836\pi\)
0.961312 0.275461i \(-0.0888304\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −41.2523 −1.38982 −0.694912 0.719095i \(-0.744557\pi\)
−0.694912 + 0.719095i \(0.744557\pi\)
\(882\) 0 0
\(883\) 4.21206 0.141747 0.0708736 0.997485i \(-0.477421\pi\)
0.0708736 + 0.997485i \(0.477421\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.79585 3.11051i 0.0602988 0.104441i −0.834300 0.551310i \(-0.814128\pi\)
0.894599 + 0.446870i \(0.147461\pi\)
\(888\) 0 0
\(889\) −10.4566 + 5.18362i −0.350703 + 0.173853i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −0.412662 + 0.238250i −0.0138092 + 0.00797275i
\(894\) 0 0
\(895\) −11.6738 + 6.73989i −0.390213 + 0.225290i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.34389 + 4.05974i −0.0781733 + 0.135400i
\(900\) 0 0
\(901\) 37.8940 21.8781i 1.26243 0.728865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.75034 + 1.58791i 0.0914244 + 0.0527839i
\(906\) 0 0
\(907\) −2.92394 5.06440i −0.0970877 0.168161i 0.813390 0.581718i \(-0.197619\pi\)
−0.910478 + 0.413558i \(0.864286\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.58142 + 3.79978i 0.218052 + 0.125892i 0.605048 0.796189i \(-0.293154\pi\)
−0.386996 + 0.922081i \(0.626487\pi\)
\(912\) 0 0
\(913\) 20.8350i 0.689536i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.47321 + 23.2148i −0.0486498 + 0.766619i
\(918\) 0 0
\(919\) −13.3062 23.0471i −0.438933 0.760254i 0.558675 0.829387i \(-0.311310\pi\)
−0.997607 + 0.0691331i \(0.977977\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.96492 + 10.3315i 0.196338 + 0.340067i
\(924\) 0 0
\(925\) 19.9937 34.6301i 0.657390 1.13863i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 52.4113 1.71956 0.859779 0.510667i \(-0.170601\pi\)
0.859779 + 0.510667i \(0.170601\pi\)
\(930\) 0 0
\(931\) 0.546998 4.29242i 0.0179271 0.140678i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 20.6010 + 11.8940i 0.673723 + 0.388974i
\(936\) 0 0
\(937\) 49.0719i 1.60311i −0.597922 0.801554i \(-0.704007\pi\)
0.597922 0.801554i \(-0.295993\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.4638 0.895294 0.447647 0.894210i \(-0.352262\pi\)
0.447647 + 0.894210i \(0.352262\pi\)
\(942\) 0 0
\(943\) 57.9202i 1.88614i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 58.9512i 1.91566i −0.287341 0.957828i \(-0.592771\pi\)
0.287341 0.957828i \(-0.407229\pi\)
\(948\) 0 0
\(949\) 11.9638 0.388360
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 16.6649i 0.539828i −0.962884 0.269914i \(-0.913005\pi\)
0.962884 0.269914i \(-0.0869953\pi\)
\(954\) 0 0
\(955\) 9.59259 + 5.53828i 0.310409 + 0.179215i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.62056 9.95300i 0.213789 0.321399i
\(960\) 0 0
\(961\) 30.6631 0.989131
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0.798079 1.38231i 0.0256911 0.0444983i
\(966\) 0 0
\(967\) −11.6729 20.2181i −0.375376 0.650171i 0.615007 0.788522i \(-0.289153\pi\)
−0.990383 + 0.138351i \(0.955820\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −7.66507 13.2763i −0.245984 0.426056i 0.716424 0.697665i \(-0.245778\pi\)
−0.962408 + 0.271609i \(0.912444\pi\)
\(972\) 0 0
\(973\) 11.2510 16.9142i 0.360690 0.542243i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 19.7463i 0.631741i −0.948802 0.315870i \(-0.897704\pi\)
0.948802 0.315870i \(-0.102296\pi\)
\(978\) 0 0
\(979\) 24.3262 + 14.0447i 0.777467 + 0.448871i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −11.4194 19.7790i −0.364222 0.630851i 0.624429 0.781082i \(-0.285332\pi\)
−0.988651 + 0.150230i \(0.951998\pi\)
\(984\) 0 0
\(985\) 13.5436 + 7.81940i 0.431535 + 0.249147i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −19.8825 + 11.4792i −0.632228 + 0.365017i
\(990\) 0 0
\(991\) −3.92125 + 6.79180i −0.124563 + 0.215749i −0.921562 0.388232i \(-0.873086\pi\)
0.796999 + 0.603980i \(0.206419\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.18096 2.41388i 0.132545 0.0765251i
\(996\) 0 0
\(997\) −0.608720 + 0.351445i −0.0192783 + 0.0111304i −0.509608 0.860407i \(-0.670210\pi\)
0.490330 + 0.871537i \(0.336876\pi\)
\(998\) 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 2268.2.w.i.269.4 12
3.2 odd 2 inner 2268.2.w.i.269.3 12
7.5 odd 6 2268.2.bm.i.593.4 12
9.2 odd 6 756.2.t.e.269.3 12
9.4 even 3 2268.2.bm.i.1025.3 12
9.5 odd 6 2268.2.bm.i.1025.4 12
9.7 even 3 756.2.t.e.269.4 yes 12
21.5 even 6 2268.2.bm.i.593.3 12
63.5 even 6 inner 2268.2.w.i.1349.4 12
63.11 odd 6 5292.2.f.e.2645.7 12
63.25 even 3 5292.2.f.e.2645.6 12
63.38 even 6 5292.2.f.e.2645.5 12
63.40 odd 6 inner 2268.2.w.i.1349.3 12
63.47 even 6 756.2.t.e.593.4 yes 12
63.52 odd 6 5292.2.f.e.2645.8 12
63.61 odd 6 756.2.t.e.593.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.t.e.269.3 12 9.2 odd 6
756.2.t.e.269.4 yes 12 9.7 even 3
756.2.t.e.593.3 yes 12 63.61 odd 6
756.2.t.e.593.4 yes 12 63.47 even 6
2268.2.w.i.269.3 12 3.2 odd 2 inner
2268.2.w.i.269.4 12 1.1 even 1 trivial
2268.2.w.i.1349.3 12 63.40 odd 6 inner
2268.2.w.i.1349.4 12 63.5 even 6 inner
2268.2.bm.i.593.3 12 21.5 even 6
2268.2.bm.i.593.4 12 7.5 odd 6
2268.2.bm.i.1025.3 12 9.4 even 3
2268.2.bm.i.1025.4 12 9.5 odd 6
5292.2.f.e.2645.5 12 63.38 even 6
5292.2.f.e.2645.6 12 63.25 even 3
5292.2.f.e.2645.7 12 63.11 odd 6
5292.2.f.e.2645.8 12 63.52 odd 6