Properties

Label 1728.2.r.e.1441.3
Level $1728$
Weight $2$
Character 1728.1441
Analytic conductor $13.798$
Analytic rank $0$
Dimension $12$
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(289,1728)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.r (of order \(6\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1441.3
Root \(-1.50511 - 0.403293i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1441
Dual form 1728.2.r.e.289.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{5} +(-0.495361 - 0.857990i) q^{7} +O(q^{10})\) \(q+(-1.50000 - 0.866025i) q^{5} +(-0.495361 - 0.857990i) q^{7} +(-1.81937 + 1.05042i) q^{11} +(5.50924 + 3.18076i) q^{13} -3.81681 q^{17} +2.00000i q^{19} +(3.55142 - 6.15125i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(7.22522 - 4.17148i) q^{29} +(1.07804 - 1.86723i) q^{31} +1.71598i q^{35} +4.62947i q^{37} +(0.408405 - 0.707378i) q^{41} +(1.97802 - 1.14201i) q^{43} +(-3.39278 - 5.87646i) q^{47} +(3.00924 - 5.21215i) q^{49} -3.14681i q^{53} +3.63875 q^{55} +(-10.3210 - 5.95882i) q^{59} +(4.22522 - 2.43943i) q^{61} +(-5.50924 - 9.54228i) q^{65} +(-11.8944 - 6.86723i) q^{67} -13.5391 q^{71} +10.0185 q^{73} +(1.80249 + 1.04067i) q^{77} +(-4.54214 - 7.86723i) q^{79} +(3.71007 - 2.14201i) q^{83} +(5.72522 + 3.30545i) q^{85} +14.0185 q^{89} -6.30249i q^{91} +(1.73205 - 3.00000i) q^{95} +(-6.22522 - 10.7824i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{5} + 6 q^{13} - 12 q^{25} + 18 q^{29} - 18 q^{41} - 24 q^{49} - 18 q^{61} - 6 q^{65} - 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) −1.50000 0.866025i −0.670820 0.387298i 0.125567 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) −0.495361 0.857990i −0.187229 0.324290i 0.757097 0.653303i \(-0.226617\pi\)
−0.944325 + 0.329013i \(0.893284\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.81937 + 1.05042i −0.548561 + 0.316712i −0.748542 0.663088i \(-0.769246\pi\)
0.199980 + 0.979800i \(0.435912\pi\)
\(12\) 0 0
\(13\) 5.50924 + 3.18076i 1.52799 + 0.882184i 0.999446 + 0.0332758i \(0.0105940\pi\)
0.528541 + 0.848908i \(0.322739\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.81681 −0.925712 −0.462856 0.886433i \(-0.653175\pi\)
−0.462856 + 0.886433i \(0.653175\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.55142 6.15125i 0.740523 1.28262i −0.211734 0.977327i \(-0.567911\pi\)
0.952257 0.305296i \(-0.0987555\pi\)
\(24\) 0 0
\(25\) −1.00000 1.73205i −0.200000 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.22522 4.17148i 1.34169 0.774624i 0.354634 0.935005i \(-0.384606\pi\)
0.987055 + 0.160381i \(0.0512723\pi\)
\(30\) 0 0
\(31\) 1.07804 1.86723i 0.193622 0.335364i −0.752826 0.658220i \(-0.771310\pi\)
0.946448 + 0.322856i \(0.104643\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.71598i 0.290054i
\(36\) 0 0
\(37\) 4.62947i 0.761080i 0.924765 + 0.380540i \(0.124262\pi\)
−0.924765 + 0.380540i \(0.875738\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.408405 0.707378i 0.0637822 0.110474i −0.832371 0.554219i \(-0.813017\pi\)
0.896153 + 0.443745i \(0.146350\pi\)
\(42\) 0 0
\(43\) 1.97802 1.14201i 0.301645 0.174155i −0.341537 0.939868i \(-0.610947\pi\)
0.643182 + 0.765714i \(0.277614\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.39278 5.87646i −0.494887 0.857170i 0.505095 0.863064i \(-0.331457\pi\)
−0.999983 + 0.00589362i \(0.998124\pi\)
\(48\) 0 0
\(49\) 3.00924 5.21215i 0.429891 0.744593i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.14681i 0.432247i −0.976366 0.216124i \(-0.930659\pi\)
0.976366 0.216124i \(-0.0693414\pi\)
\(54\) 0 0
\(55\) 3.63875 0.490648
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.3210 5.95882i −1.34368 0.775772i −0.356332 0.934359i \(-0.615973\pi\)
−0.987345 + 0.158587i \(0.949306\pi\)
\(60\) 0 0
\(61\) 4.22522 2.43943i 0.540983 0.312337i −0.204494 0.978868i \(-0.565555\pi\)
0.745477 + 0.666531i \(0.232222\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.50924 9.54228i −0.683337 1.18357i
\(66\) 0 0
\(67\) −11.8944 6.86723i −1.45313 0.838965i −0.454473 0.890761i \(-0.650172\pi\)
−0.998658 + 0.0517956i \(0.983506\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.5391 −1.60680 −0.803399 0.595442i \(-0.796977\pi\)
−0.803399 + 0.595442i \(0.796977\pi\)
\(72\) 0 0
\(73\) 10.0185 1.17257 0.586287 0.810104i \(-0.300589\pi\)
0.586287 + 0.810104i \(0.300589\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.80249 + 1.04067i 0.205413 + 0.118595i
\(78\) 0 0
\(79\) −4.54214 7.86723i −0.511031 0.885132i −0.999918 0.0127849i \(-0.995930\pi\)
0.488887 0.872347i \(-0.337403\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.71007 2.14201i 0.407233 0.235116i −0.282367 0.959306i \(-0.591120\pi\)
0.689600 + 0.724190i \(0.257786\pi\)
\(84\) 0 0
\(85\) 5.72522 + 3.30545i 0.620987 + 0.358527i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.0185 1.48595 0.742977 0.669316i \(-0.233413\pi\)
0.742977 + 0.669316i \(0.233413\pi\)
\(90\) 0 0
\(91\) 6.30249i 0.660681i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.73205 3.00000i 0.177705 0.307794i
\(96\) 0 0
\(97\) −6.22522 10.7824i −0.632075 1.09479i −0.987127 0.159939i \(-0.948870\pi\)
0.355052 0.934847i \(-0.384463\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8025 7.96887i 1.37340 0.792932i 0.382045 0.924144i \(-0.375220\pi\)
0.991354 + 0.131211i \(0.0418867\pi\)
\(102\) 0 0
\(103\) 8.58893 14.8765i 0.846292 1.46582i −0.0382019 0.999270i \(-0.512163\pi\)
0.884494 0.466551i \(-0.154504\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 16.2017i 1.56627i 0.621849 + 0.783137i \(0.286382\pi\)
−0.621849 + 0.783137i \(0.713618\pi\)
\(108\) 0 0
\(109\) 0.816078i 0.0781661i −0.999236 0.0390830i \(-0.987556\pi\)
0.999236 0.0390830i \(-0.0124437\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.21598 + 5.57024i −0.302534 + 0.524004i −0.976709 0.214567i \(-0.931166\pi\)
0.674175 + 0.738571i \(0.264499\pi\)
\(114\) 0 0
\(115\) −10.6543 + 6.15125i −0.993516 + 0.573607i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.89070 + 3.27478i 0.173320 + 0.300199i
\(120\) 0 0
\(121\) −3.29326 + 5.70409i −0.299387 + 0.518553i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) 16.1871 1.43638 0.718188 0.695849i \(-0.244972\pi\)
0.718188 + 0.695849i \(0.244972\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.68223 + 3.85799i 0.583830 + 0.337074i 0.762654 0.646807i \(-0.223896\pi\)
−0.178824 + 0.983881i \(0.557229\pi\)
\(132\) 0 0
\(133\) 1.71598 0.990721i 0.148794 0.0859064i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.30757 7.46094i −0.368021 0.637431i 0.621235 0.783624i \(-0.286631\pi\)
−0.989256 + 0.146193i \(0.953298\pi\)
\(138\) 0 0
\(139\) 4.95018 + 2.85799i 0.419869 + 0.242412i 0.695021 0.718989i \(-0.255395\pi\)
−0.275152 + 0.961401i \(0.588728\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −13.3645 −1.11759
\(144\) 0 0
\(145\) −14.4504 −1.20004
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.22522 + 0.707378i 0.100374 + 0.0579507i 0.549346 0.835595i \(-0.314877\pi\)
−0.448973 + 0.893545i \(0.648210\pi\)
\(150\) 0 0
\(151\) 1.64473 + 2.84875i 0.133846 + 0.231828i 0.925156 0.379587i \(-0.123934\pi\)
−0.791310 + 0.611415i \(0.790601\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.23413 + 1.86723i −0.259772 + 0.149979i
\(156\) 0 0
\(157\) −5.78402 3.33941i −0.461615 0.266514i 0.251108 0.967959i \(-0.419205\pi\)
−0.712723 + 0.701445i \(0.752538\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.03694 −0.554589
\(162\) 0 0
\(163\) 7.43196i 0.582116i −0.956705 0.291058i \(-0.905993\pi\)
0.956705 0.291058i \(-0.0940073\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.95018 8.57397i 0.383057 0.663474i −0.608441 0.793599i \(-0.708205\pi\)
0.991498 + 0.130126i \(0.0415380\pi\)
\(168\) 0 0
\(169\) 13.7345 + 23.7888i 1.05650 + 1.82991i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.5000 6.06218i 0.798300 0.460899i −0.0445762 0.999006i \(-0.514194\pi\)
0.842876 + 0.538107i \(0.180860\pi\)
\(174\) 0 0
\(175\) −0.990721 + 1.71598i −0.0748915 + 0.129716i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000i 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 4.62947i 0.344106i −0.985088 0.172053i \(-0.944960\pi\)
0.985088 0.172053i \(-0.0550400\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00924 6.94420i 0.294765 0.510548i
\(186\) 0 0
\(187\) 6.94420 4.00924i 0.507810 0.293184i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.58893 + 14.8765i 0.621473 + 1.07642i 0.989212 + 0.146494i \(0.0467988\pi\)
−0.367739 + 0.929929i \(0.619868\pi\)
\(192\) 0 0
\(193\) 5.79326 10.0342i 0.417008 0.722278i −0.578629 0.815591i \(-0.696412\pi\)
0.995637 + 0.0933122i \(0.0297455\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.6698i 1.25892i 0.777033 + 0.629460i \(0.216724\pi\)
−0.777033 + 0.629460i \(0.783276\pi\)
\(198\) 0 0
\(199\) −16.5044 −1.16997 −0.584984 0.811045i \(-0.698899\pi\)
−0.584984 + 0.811045i \(0.698899\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.15818 4.13277i −0.502405 0.290064i
\(204\) 0 0
\(205\) −1.22522 + 0.707378i −0.0855728 + 0.0494055i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.10083 3.63875i −0.145317 0.251697i
\(210\) 0 0
\(211\) 10.1623 + 5.86723i 0.699604 + 0.403916i 0.807200 0.590278i \(-0.200982\pi\)
−0.107596 + 0.994195i \(0.534315\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.95604 −0.269800
\(216\) 0 0
\(217\) −2.13608 −0.145007
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −21.0277 12.1404i −1.41448 0.816648i
\(222\) 0 0
\(223\) 0.670004 + 1.16048i 0.0448668 + 0.0777116i 0.887587 0.460641i \(-0.152380\pi\)
−0.842720 + 0.538352i \(0.819047\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.37678 + 1.94958i −0.224125 + 0.129398i −0.607859 0.794045i \(-0.707971\pi\)
0.383734 + 0.923444i \(0.374638\pi\)
\(228\) 0 0
\(229\) −1.93196 1.11542i −0.127667 0.0737089i 0.434806 0.900524i \(-0.356817\pi\)
−0.562474 + 0.826815i \(0.690150\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.26724 0.607117 0.303559 0.952813i \(-0.401825\pi\)
0.303559 + 0.952813i \(0.401825\pi\)
\(234\) 0 0
\(235\) 11.7529i 0.766676i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.4796 18.1512i 0.677871 1.17411i −0.297750 0.954644i \(-0.596236\pi\)
0.975621 0.219463i \(-0.0704304\pi\)
\(240\) 0 0
\(241\) −7.51847 13.0224i −0.484307 0.838845i 0.515530 0.856871i \(-0.327595\pi\)
−0.999838 + 0.0180266i \(0.994262\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.02771 + 5.21215i −0.576759 + 0.332992i
\(246\) 0 0
\(247\) −6.36152 + 11.0185i −0.404774 + 0.701089i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.164719i 0.0103969i 0.999986 + 0.00519847i \(0.00165473\pi\)
−0.999986 + 0.00519847i \(0.998345\pi\)
\(252\) 0 0
\(253\) 14.9219i 0.938130i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.4176 + 23.2400i −0.836969 + 1.44967i 0.0554474 + 0.998462i \(0.482341\pi\)
−0.892417 + 0.451212i \(0.850992\pi\)
\(258\) 0 0
\(259\) 3.97204 2.29326i 0.246810 0.142496i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.98769 + 17.2992i 0.615867 + 1.06671i 0.990232 + 0.139432i \(0.0445276\pi\)
−0.374364 + 0.927282i \(0.622139\pi\)
\(264\) 0 0
\(265\) −2.72522 + 4.72021i −0.167409 + 0.289960i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.666581i 0.0406422i −0.999794 0.0203211i \(-0.993531\pi\)
0.999794 0.0203211i \(-0.00646884\pi\)
\(270\) 0 0
\(271\) −17.0352 −1.03482 −0.517408 0.855739i \(-0.673103\pi\)
−0.517408 + 0.855739i \(0.673103\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.63875 + 2.10083i 0.219425 + 0.126685i
\(276\) 0 0
\(277\) −3.21598 + 1.85675i −0.193229 + 0.111561i −0.593494 0.804839i \(-0.702252\pi\)
0.400264 + 0.916400i \(0.368918\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 13.5277 + 23.4307i 0.806995 + 1.39776i 0.914936 + 0.403600i \(0.132241\pi\)
−0.107940 + 0.994157i \(0.534426\pi\)
\(282\) 0 0
\(283\) −0.994145 0.573970i −0.0590958 0.0341190i 0.470161 0.882581i \(-0.344196\pi\)
−0.529257 + 0.848462i \(0.677529\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.809231 −0.0477674
\(288\) 0 0
\(289\) −2.43196 −0.143056
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −22.8025 13.1650i −1.33214 0.769109i −0.346509 0.938047i \(-0.612633\pi\)
−0.985627 + 0.168938i \(0.945966\pi\)
\(294\) 0 0
\(295\) 10.3210 + 17.8765i 0.600911 + 1.04081i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 39.1313 22.5924i 2.26302 1.30655i
\(300\) 0 0
\(301\) −1.95967 1.13141i −0.112953 0.0652136i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.45043 −0.483870
\(306\) 0 0
\(307\) 7.43196i 0.424164i 0.977252 + 0.212082i \(0.0680244\pi\)
−0.977252 + 0.212082i \(0.931976\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.96618 + 8.60168i −0.281606 + 0.487756i −0.971781 0.235887i \(-0.924201\pi\)
0.690174 + 0.723643i \(0.257534\pi\)
\(312\) 0 0
\(313\) 4.51847 + 7.82622i 0.255399 + 0.442364i 0.965004 0.262236i \(-0.0844598\pi\)
−0.709605 + 0.704600i \(0.751126\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.52771 + 4.34612i −0.422798 + 0.244103i −0.696274 0.717776i \(-0.745160\pi\)
0.273476 + 0.961879i \(0.411827\pi\)
\(318\) 0 0
\(319\) −8.76357 + 15.1790i −0.490666 + 0.849858i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.63362i 0.424746i
\(324\) 0 0
\(325\) 12.7230i 0.705747i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.36130 + 5.82194i −0.185314 + 0.320974i
\(330\) 0 0
\(331\) −11.1462 + 6.43527i −0.612651 + 0.353714i −0.774002 0.633183i \(-0.781748\pi\)
0.161351 + 0.986897i \(0.448415\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.8944 + 20.6017i 0.649860 + 1.12559i
\(336\) 0 0
\(337\) 1.21598 2.10614i 0.0662386 0.114729i −0.831004 0.556266i \(-0.812233\pi\)
0.897243 + 0.441538i \(0.145567\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.52957i 0.245290i
\(342\) 0 0
\(343\) −12.8977 −0.696409
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.2373 11.6840i −1.08640 0.627232i −0.153782 0.988105i \(-0.549146\pi\)
−0.932615 + 0.360873i \(0.882479\pi\)
\(348\) 0 0
\(349\) −8.66641 + 5.00355i −0.463902 + 0.267834i −0.713684 0.700468i \(-0.752975\pi\)
0.249781 + 0.968302i \(0.419641\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.6924 18.5198i −0.569100 0.985711i −0.996655 0.0817214i \(-0.973958\pi\)
0.427555 0.903989i \(-0.359375\pi\)
\(354\) 0 0
\(355\) 20.3087 + 11.7252i 1.07787 + 0.622310i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 23.9634 1.26474 0.632370 0.774666i \(-0.282082\pi\)
0.632370 + 0.774666i \(0.282082\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.0277 8.67625i −0.786586 0.454136i
\(366\) 0 0
\(367\) 8.25564 + 14.2992i 0.430941 + 0.746411i 0.996955 0.0779844i \(-0.0248484\pi\)
−0.566014 + 0.824396i \(0.691515\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.69993 + 1.55880i −0.140173 + 0.0809291i
\(372\) 0 0
\(373\) −21.2437 12.2650i −1.09996 0.635060i −0.163747 0.986502i \(-0.552358\pi\)
−0.936210 + 0.351442i \(0.885691\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 53.0739 2.73344
\(378\) 0 0
\(379\) 10.8824i 0.558991i −0.960147 0.279495i \(-0.909833\pi\)
0.960147 0.279495i \(-0.0901672\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.66073 + 2.87646i −0.0848591 + 0.146980i −0.905331 0.424706i \(-0.860377\pi\)
0.820472 + 0.571687i \(0.193711\pi\)
\(384\) 0 0
\(385\) −1.80249 3.12201i −0.0918635 0.159112i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.19751 0.691382i 0.0607161 0.0350545i −0.469335 0.883020i \(-0.655506\pi\)
0.530051 + 0.847966i \(0.322173\pi\)
\(390\) 0 0
\(391\) −13.5551 + 23.4781i −0.685511 + 1.18734i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.7345i 0.791686i
\(396\) 0 0
\(397\) 19.9685i 1.00219i −0.865392 0.501096i \(-0.832930\pi\)
0.865392 0.501096i \(-0.167070\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.326046 + 0.564728i −0.0162819 + 0.0282012i −0.874052 0.485833i \(-0.838516\pi\)
0.857770 + 0.514034i \(0.171850\pi\)
\(402\) 0 0
\(403\) 11.8784 6.85799i 0.591705 0.341621i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.86286 8.42272i −0.241043 0.417499i
\(408\) 0 0
\(409\) −7.51847 + 13.0224i −0.371764 + 0.643915i −0.989837 0.142206i \(-0.954580\pi\)
0.618073 + 0.786121i \(0.287914\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.8071i 0.580988i
\(414\) 0 0
\(415\) −7.42014 −0.364240
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −11.4024 6.58321i −0.557046 0.321611i 0.194913 0.980821i \(-0.437558\pi\)
−0.751959 + 0.659210i \(0.770891\pi\)
\(420\) 0 0
\(421\) 3.36130 1.94065i 0.163820 0.0945813i −0.415849 0.909434i \(-0.636515\pi\)
0.579668 + 0.814853i \(0.303182\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.81681 + 6.61091i 0.185142 + 0.320676i
\(426\) 0 0
\(427\) −4.18601 2.41679i −0.202575 0.116957i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.81339 −0.183684 −0.0918422 0.995774i \(-0.529276\pi\)
−0.0918422 + 0.995774i \(0.529276\pi\)
\(432\) 0 0
\(433\) −10.0185 −0.481457 −0.240728 0.970592i \(-0.577386\pi\)
−0.240728 + 0.970592i \(0.577386\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.3025 + 7.10285i 0.588508 + 0.339775i
\(438\) 0 0
\(439\) −12.2117 21.1512i −0.582832 1.00949i −0.995142 0.0984504i \(-0.968611\pi\)
0.412310 0.911043i \(-0.364722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.3799 12.3437i 1.01579 0.586466i 0.102907 0.994691i \(-0.467186\pi\)
0.912881 + 0.408225i \(0.133852\pi\)
\(444\) 0 0
\(445\) −21.0277 12.1404i −0.996809 0.575508i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −21.6521 −1.02183 −0.510913 0.859633i \(-0.670692\pi\)
−0.510913 + 0.859633i \(0.670692\pi\)
\(450\) 0 0
\(451\) 1.71598i 0.0808023i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.45812 + 9.45374i −0.255880 + 0.443198i
\(456\) 0 0
\(457\) 11.2437 + 19.4746i 0.525957 + 0.910985i 0.999543 + 0.0302371i \(0.00962623\pi\)
−0.473585 + 0.880748i \(0.657040\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.6756 + 12.5144i −1.00954 + 0.582856i −0.911056 0.412283i \(-0.864731\pi\)
−0.0984799 + 0.995139i \(0.531398\pi\)
\(462\) 0 0
\(463\) −9.73830 + 16.8672i −0.452577 + 0.783886i −0.998545 0.0539194i \(-0.982829\pi\)
0.545968 + 0.837806i \(0.316162\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.40332i 0.111212i 0.998453 + 0.0556062i \(0.0177091\pi\)
−0.998453 + 0.0556062i \(0.982291\pi\)
\(468\) 0 0
\(469\) 13.6070i 0.628314i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.39917 + 4.15548i −0.110314 + 0.191069i
\(474\) 0 0
\(475\) 3.46410 2.00000i 0.158944 0.0917663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.6543 + 18.4537i 0.486806 + 0.843173i 0.999885 0.0151687i \(-0.00482852\pi\)
−0.513079 + 0.858341i \(0.671495\pi\)
\(480\) 0 0
\(481\) −14.7252 + 25.5048i −0.671412 + 1.16292i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.5648i 0.979206i
\(486\) 0 0
\(487\) 40.7851 1.84815 0.924075 0.382210i \(-0.124837\pi\)
0.924075 + 0.382210i \(0.124837\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.6776 + 17.7117i 1.38446 + 0.799320i 0.992684 0.120740i \(-0.0385267\pi\)
0.391778 + 0.920060i \(0.371860\pi\)
\(492\) 0 0
\(493\) −27.5773 + 15.9217i −1.24202 + 0.717079i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.70674 + 11.6164i 0.300839 + 0.521068i
\(498\) 0 0
\(499\) 3.74206 + 2.16048i 0.167518 + 0.0967164i 0.581415 0.813607i \(-0.302499\pi\)
−0.413897 + 0.910324i \(0.635833\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.24550 −0.323061 −0.161530 0.986868i \(-0.551643\pi\)
−0.161530 + 0.986868i \(0.551643\pi\)
\(504\) 0 0
\(505\) −27.6050 −1.22841
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.22522 + 4.17148i 0.320252 + 0.184898i 0.651505 0.758644i \(-0.274138\pi\)
−0.331253 + 0.943542i \(0.607471\pi\)
\(510\) 0 0
\(511\) −4.96276 8.59575i −0.219539 0.380253i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.7668 + 14.8765i −1.13542 + 0.655535i
\(516\) 0 0
\(517\) 12.3454 + 7.12765i 0.542952 + 0.313474i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −16.9714 −0.743529 −0.371764 0.928327i \(-0.621247\pi\)
−0.371764 + 0.928327i \(0.621247\pi\)
\(522\) 0 0
\(523\) 40.0554i 1.75150i 0.482764 + 0.875750i \(0.339633\pi\)
−0.482764 + 0.875750i \(0.660367\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.11469 + 7.12685i −0.179239 + 0.310450i
\(528\) 0 0
\(529\) −13.7252 23.7728i −0.596748 1.03360i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.50000 2.59808i 0.194917 0.112535i
\(534\) 0 0
\(535\) 14.0310 24.3025i 0.606615 1.05069i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.6438i 0.544606i
\(540\) 0 0
\(541\) 25.5956i 1.10044i −0.835020 0.550220i \(-0.814544\pi\)
0.835020 0.550220i \(-0.185456\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.706744 + 1.22412i −0.0302736 + 0.0524354i
\(546\) 0 0
\(547\) 29.2469 16.8857i 1.25051 0.721980i 0.279296 0.960205i \(-0.409899\pi\)
0.971210 + 0.238225i \(0.0765655\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.34296 + 14.4504i 0.355422 + 0.615609i
\(552\) 0 0
\(553\) −4.50000 + 7.79423i −0.191359 + 0.331444i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.3364i 0.776937i 0.921462 + 0.388469i \(0.126996\pi\)
−0.921462 + 0.388469i \(0.873004\pi\)
\(558\) 0 0
\(559\) 14.5298 0.614546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 35.8258 + 20.6840i 1.50988 + 0.871728i 0.999934 + 0.0115205i \(0.00366716\pi\)
0.509944 + 0.860208i \(0.329666\pi\)
\(564\) 0 0
\(565\) 9.64794 5.57024i 0.405892 0.234342i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.683190 1.18332i −0.0286408 0.0496073i 0.851350 0.524599i \(-0.175785\pi\)
−0.879991 + 0.474991i \(0.842451\pi\)
\(570\) 0 0
\(571\) −15.3585 8.86723i −0.642733 0.371082i 0.142934 0.989732i \(-0.454346\pi\)
−0.785666 + 0.618650i \(0.787680\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.2057 −0.592418
\(576\) 0 0
\(577\) 20.5865 0.857028 0.428514 0.903535i \(-0.359037\pi\)
0.428514 + 0.903535i \(0.359037\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.67565 2.12214i −0.152491 0.0880410i
\(582\) 0 0
\(583\) 3.30545 + 5.72522i 0.136898 + 0.237114i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.5450 7.24284i 0.517786 0.298944i −0.218242 0.975895i \(-0.570032\pi\)
0.736028 + 0.676951i \(0.236699\pi\)
\(588\) 0 0
\(589\) 3.73445 + 2.15609i 0.153875 + 0.0888400i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −4.58651 −0.188345 −0.0941727 0.995556i \(-0.530021\pi\)
−0.0941727 + 0.995556i \(0.530021\pi\)
\(594\) 0 0
\(595\) 6.54957i 0.268506i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.31144 + 2.27148i −0.0535839 + 0.0928101i −0.891573 0.452877i \(-0.850398\pi\)
0.837989 + 0.545687i \(0.183731\pi\)
\(600\) 0 0
\(601\) −3.50924 6.07817i −0.143145 0.247934i 0.785535 0.618818i \(-0.212388\pi\)
−0.928679 + 0.370884i \(0.879055\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.87977 5.70409i 0.401670 0.231904i
\(606\) 0 0
\(607\) 18.5892 32.1974i 0.754512 1.30685i −0.191105 0.981570i \(-0.561207\pi\)
0.945617 0.325283i \(-0.105460\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 43.1664i 1.74633i
\(612\) 0 0
\(613\) 10.7096i 0.432557i −0.976332 0.216278i \(-0.930608\pi\)
0.976332 0.216278i \(-0.0693919\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 16.6101 28.7695i 0.668696 1.15822i −0.309573 0.950876i \(-0.600186\pi\)
0.978269 0.207340i \(-0.0664806\pi\)
\(618\) 0 0
\(619\) −22.8106 + 13.1697i −0.916836 + 0.529336i −0.882624 0.470079i \(-0.844225\pi\)
−0.0342118 + 0.999415i \(0.510892\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.94420 12.0277i −0.278213 0.481880i
\(624\) 0 0
\(625\) 5.50000 9.52628i 0.220000 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.6698i 0.704541i
\(630\) 0 0
\(631\) −21.6327 −0.861183 −0.430592 0.902547i \(-0.641695\pi\)
−0.430592 + 0.902547i \(0.641695\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −24.2807 14.0185i −0.963550 0.556306i
\(636\) 0 0
\(637\) 33.1572 19.1433i 1.31374 0.758485i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.88485 + 3.26466i 0.0744471 + 0.128946i 0.900846 0.434139i \(-0.142947\pi\)
−0.826399 + 0.563086i \(0.809614\pi\)
\(642\) 0 0
\(643\) −29.2469 16.8857i −1.15338 0.665907i −0.203675 0.979039i \(-0.565289\pi\)
−0.949710 + 0.313132i \(0.898622\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15.7020 −0.617311 −0.308656 0.951174i \(-0.599879\pi\)
−0.308656 + 0.951174i \(0.599879\pi\)
\(648\) 0 0
\(649\) 25.0369 0.982786
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.4009 + 15.8199i 1.07228 + 0.619080i 0.928803 0.370574i \(-0.120839\pi\)
0.143475 + 0.989654i \(0.454172\pi\)
\(654\) 0 0
\(655\) −6.68223 11.5740i −0.261097 0.452232i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.4389 + 18.1512i −1.22468 + 0.707072i −0.965913 0.258866i \(-0.916651\pi\)
−0.258772 + 0.965939i \(0.583318\pi\)
\(660\) 0 0
\(661\) −12.6479 7.30229i −0.491948 0.284026i 0.233434 0.972373i \(-0.425004\pi\)
−0.725382 + 0.688346i \(0.758337\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.43196 −0.133086
\(666\) 0 0
\(667\) 59.2588i 2.29451i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.12483 + 8.87646i −0.197842 + 0.342672i
\(672\) 0 0
\(673\) 1.79326 + 3.10601i 0.0691249 + 0.119728i 0.898516 0.438940i \(-0.144646\pi\)
−0.829391 + 0.558668i \(0.811313\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.25292 0.723375i 0.0481537 0.0278016i −0.475730 0.879591i \(-0.657816\pi\)
0.523884 + 0.851790i \(0.324483\pi\)
\(678\) 0 0
\(679\) −6.16745 + 10.6823i −0.236685 + 0.409951i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 16.9714i 0.649391i 0.945819 + 0.324696i \(0.105262\pi\)
−0.945819 + 0.324696i \(0.894738\pi\)
\(684\) 0 0
\(685\) 14.9219i 0.570136i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0092 17.3365i 0.381322 0.660468i
\(690\) 0 0
\(691\) −8.44628 + 4.87646i −0.321312 + 0.185509i −0.651977 0.758239i \(-0.726060\pi\)
0.330665 + 0.943748i \(0.392727\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.95018 8.57397i −0.187771 0.325229i
\(696\) 0 0
\(697\) −1.55880 + 2.69993i −0.0590439 + 0.102267i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.3398i 1.25923i −0.776908 0.629614i \(-0.783213\pi\)
0.776908 0.629614i \(-0.216787\pi\)
\(702\) 0 0
\(703\) −9.25893 −0.349207
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.6744 7.89493i −0.514280 0.296919i
\(708\) 0 0
\(709\) −44.5369 + 25.7134i −1.67262 + 0.965688i −0.706457 + 0.707756i \(0.749708\pi\)
−0.966163 + 0.257931i \(0.916959\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.65717 13.2626i −0.286763 0.496689i
\(714\) 0 0
\(715\) 20.0467 + 11.5740i 0.749704 + 0.432842i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.1178 0.824854 0.412427 0.910991i \(-0.364681\pi\)
0.412427 + 0.910991i \(0.364681\pi\)
\(720\) 0 0
\(721\) −17.0185 −0.633801
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −14.4504 8.34296i −0.536676 0.309850i
\(726\) 0 0
\(727\) 14.2930 + 24.7562i 0.530099 + 0.918158i 0.999383 + 0.0351108i \(0.0111784\pi\)
−0.469285 + 0.883047i \(0.655488\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.54973 + 4.35884i −0.279237 + 0.161217i
\(732\) 0 0
\(733\) 39.4167 + 22.7572i 1.45589 + 0.840558i 0.998805 0.0488654i \(-0.0155605\pi\)
0.457084 + 0.889424i \(0.348894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 28.8538 1.06284
\(738\) 0 0
\(739\) 19.9815i 0.735032i 0.930017 + 0.367516i \(0.119792\pi\)
−0.930017 + 0.367516i \(0.880208\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.8386 32.6294i 0.691121 1.19706i −0.280350 0.959898i \(-0.590451\pi\)
0.971471 0.237158i \(-0.0762160\pi\)
\(744\) 0 0
\(745\) −1.22522 2.12214i −0.0448884 0.0777490i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.9009 8.02567i 0.507926 0.293251i
\(750\) 0 0
\(751\) 6.25820 10.8395i 0.228365 0.395540i −0.728959 0.684558i \(-0.759995\pi\)
0.957324 + 0.289018i \(0.0933287\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.69751i 0.207354i
\(756\) 0 0
\(757\) 12.5872i 0.457491i 0.973486 + 0.228745i \(0.0734623\pi\)
−0.973486 + 0.228745i \(0.926538\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5.94120 + 10.2905i −0.215368 + 0.373029i −0.953386 0.301752i \(-0.902428\pi\)
0.738018 + 0.674781i \(0.235762\pi\)
\(762\) 0 0
\(763\) −0.700187 + 0.404253i −0.0253485 + 0.0146349i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −37.9071 65.6571i −1.36875 2.37074i
\(768\) 0 0
\(769\) 22.5369 39.0351i 0.812703 1.40764i −0.0982627 0.995161i \(-0.531329\pi\)
0.910966 0.412482i \(-0.135338\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 34.0384i 1.22428i −0.790751 0.612138i \(-0.790310\pi\)
0.790751 0.612138i \(-0.209690\pi\)
\(774\) 0 0
\(775\) −4.31217 −0.154898
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.41476 + 0.816810i 0.0506889 + 0.0292653i
\(780\) 0 0
\(781\) 24.6327 14.2217i 0.881427 0.508892i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.78402 + 10.0182i 0.206441 + 0.357565i
\(786\) 0 0
\(787\) 3.67808 + 2.12354i 0.131109 + 0.0756960i 0.564120 0.825693i \(-0.309215\pi\)
−0.433011 + 0.901389i \(0.642549\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.37228 0.226572
\(792\) 0 0
\(793\) 31.0369 1.10215
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37.2806 + 21.5240i 1.32055 + 0.762419i 0.983816 0.179182i \(-0.0573451\pi\)
0.336732 + 0.941601i \(0.390678\pi\)
\(798\) 0 0
\(799\) 12.9496 + 22.4293i 0.458123 + 0.793493i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −18.2273 + 10.5236i −0.643229 + 0.371368i
\(804\) 0 0
\(805\) 10.5554 + 6.09417i 0.372029 + 0.214791i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −25.2488 −0.887699 −0.443850 0.896101i \(-0.646388\pi\)
−0.443850 + 0.896101i \(0.646388\pi\)
\(810\) 0 0
\(811\) 36.9378i 1.29706i 0.761188 + 0.648531i \(0.224616\pi\)
−0.761188 + 0.648531i \(0.775384\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.43627 + 11.1479i −0.225453 + 0.390495i
\(816\) 0 0
\(817\) 2.28402 + 3.95604i 0.0799078 + 0.138404i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −10.7748 + 6.22082i −0.376043 + 0.217108i −0.676095 0.736814i \(-0.736329\pi\)
0.300052 + 0.953923i \(0.402996\pi\)
\(822\) 0 0
\(823\) 0.760749 1.31766i 0.0265180 0.0459306i −0.852462 0.522789i \(-0.824891\pi\)
0.878980 + 0.476859i \(0.158225\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.0369i 0.349019i −0.984656 0.174509i \(-0.944166\pi\)
0.984656 0.174509i \(-0.0558339\pi\)
\(828\) 0 0
\(829\) 43.7824i 1.52063i 0.649556 + 0.760314i \(0.274955\pi\)
−0.649556 + 0.760314i \(0.725045\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −11.4857 + 19.8938i −0.397955 + 0.689279i
\(834\) 0 0
\(835\) −14.8506 + 8.57397i −0.513925 + 0.296714i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.0626 36.4814i −0.727161 1.25948i −0.958078 0.286506i \(-0.907506\pi\)
0.230918 0.972973i \(-0.425827\pi\)
\(840\) 0 0
\(841\) 20.3025 35.1649i 0.700086 1.21258i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 47.5775i 1.63672i
\(846\) 0 0
\(847\) 6.52540 0.224215
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 28.4770 + 16.4412i 0.976178 + 0.563597i
\(852\) 0 0
\(853\) 22.7748 13.1490i 0.779794 0.450214i −0.0565634 0.998399i \(-0.518014\pi\)
0.836357 + 0.548185i \(0.184681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.24369 16.0105i −0.315758 0.546909i 0.663840 0.747875i \(-0.268926\pi\)
−0.979598 + 0.200965i \(0.935592\pi\)
\(858\) 0 0
\(859\) −39.8954 23.0336i −1.36121 0.785898i −0.371429 0.928461i \(-0.621132\pi\)
−0.989786 + 0.142564i \(0.954465\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 46.0812 1.56862 0.784311 0.620368i \(-0.213017\pi\)
0.784311 + 0.620368i \(0.213017\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 16.5277 + 9.54228i 0.560664 + 0.323700i
\(870\) 0 0
\(871\) −43.6860 75.6663i −1.48024 2.56386i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.4026 6.00593i 0.351671 0.203037i
\(876\) 0 0
\(877\) 28.5396 + 16.4773i 0.963713 + 0.556400i 0.897314 0.441393i \(-0.145516\pi\)
0.0663989 + 0.997793i \(0.478849\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −21.9815 −0.740577 −0.370288 0.928917i \(-0.620741\pi\)
−0.370288 + 0.928917i \(0.620741\pi\)
\(882\) 0 0
\(883\) 10.0000i 0.336527i −0.985742 0.168263i \(-0.946184\pi\)
0.985742 0.168263i \(-0.0538159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.67672 2.90417i 0.0562988 0.0975124i −0.836502 0.547963i \(-0.815403\pi\)
0.892801 + 0.450451i \(0.148737\pi\)
\(888\) 0 0
\(889\) −8.01847 13.8884i −0.268931 0.465802i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 11.7529 6.78555i 0.393297 0.227070i
\(894\) 0 0
\(895\) −5.19615 + 9.00000i −0.173688 + 0.300837i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.9881i 0.599938i
\(900\) 0 0
\(901\) 12.0108i 0.400137i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00924 + 6.94420i −0.133271 + 0.230833i
\(906\) 0 0
\(907\) 8.87423 5.12354i 0.294664 0.170124i −0.345379 0.938463i \(-0.612250\pi\)
0.640043 + 0.768339i \(0.278916\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −12.7036 22.0033i −0.420890 0.729002i 0.575137 0.818057i \(-0.304949\pi\)
−0.996027 + 0.0890549i \(0.971615\pi\)
\(912\) 0 0
\(913\) −4.50000 + 7.79423i −0.148928 + 0.257951i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.64439i 0.252440i
\(918\) 0 0
\(919\) 44.1134 1.45517 0.727584 0.686019i \(-0.240643\pi\)
0.727584 + 0.686019i \(0.240643\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −74.5902 43.0646i −2.45517 1.41749i
\(924\) 0 0
\(925\) 8.01847 4.62947i 0.263646 0.152216i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −17.8117 30.8508i −0.584384 1.01218i −0.994952 0.100353i \(-0.968003\pi\)
0.410568 0.911830i \(-0.365330\pi\)
\(930\) 0 0
\(931\) 10.4243 + 6.01847i 0.341643 + 0.197247i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.8884 −0.454199
\(936\) 0 0
\(937\) −15.4135 −0.503537 −0.251768 0.967788i \(-0.581012\pi\)
−0.251768 + 0.967788i \(0.581012\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.55541 4.36212i −0.246299 0.142201i 0.371769 0.928325i \(-0.378751\pi\)
−0.618069 + 0.786124i \(0.712085\pi\)
\(942\) 0 0
\(943\) −2.90084 5.02440i −0.0944643 0.163617i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −13.7691 + 7.94958i −0.447435 + 0.258327i −0.706746 0.707467i \(-0.749838\pi\)
0.259311 + 0.965794i \(0.416504\pi\)
\(948\) 0 0
\(949\) 55.1941 + 31.8663i 1.79168 + 1.03443i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −25.6890 −0.832149 −0.416075 0.909330i \(-0.636595\pi\)
−0.416075 + 0.909330i \(0.636595\pi\)
\(954\) 0 0
\(955\) 29.7529i 0.962782i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.26761 + 7.39171i −0.137808 + 0.238691i
\(960\) 0 0
\(961\) 13.1756 + 22.8209i 0.425021 + 0.736158i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −17.3798 + 10.0342i −0.559475 + 0.323013i
\(966\) 0 0
\(967\) 15.3585 26.6017i 0.493896 0.855452i −0.506080 0.862487i \(-0.668906\pi\)
0.999975 + 0.00703449i \(0.00223917\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.0924i 1.28662i 0.765604 + 0.643312i \(0.222440\pi\)
−0.765604 + 0.643312i \(0.777560\pi\)
\(972\) 0 0
\(973\) 5.66294i 0.181546i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.4361 42.3246i 0.781780 1.35408i −0.149123 0.988819i \(-0.547645\pi\)
0.930904 0.365265i \(-0.119022\pi\)
\(978\) 0 0
\(979\) −25.5048 + 14.7252i −0.815138 + 0.470620i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.60077 + 9.70081i 0.178637 + 0.309408i 0.941414 0.337254i \(-0.109498\pi\)
−0.762777 + 0.646662i \(0.776165\pi\)
\(984\) 0 0
\(985\) 15.3025 26.5047i 0.487578 0.844510i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.2230i 0.515863i
\(990\) 0 0
\(991\) −15.7020 −0.498792 −0.249396 0.968402i \(-0.580232\pi\)
−0.249396 + 0.968402i \(0.580232\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 24.7566 + 14.2933i 0.784838 + 0.453127i
\(996\) 0 0
\(997\) 21.0984 12.1811i 0.668192 0.385781i −0.127199 0.991877i \(-0.540599\pi\)
0.795391 + 0.606096i \(0.207265\pi\)
\(998\) 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.r.e.1441.3 12
3.2 odd 2 576.2.r.f.481.2 yes 12
4.3 odd 2 inner 1728.2.r.e.1441.4 12
8.3 odd 2 1728.2.r.f.1441.4 12
8.5 even 2 1728.2.r.f.1441.3 12
9.2 odd 6 576.2.r.e.97.5 yes 12
9.4 even 3 5184.2.d.r.2593.10 12
9.5 odd 6 5184.2.d.q.2593.4 12
9.7 even 3 1728.2.r.f.289.3 12
12.11 even 2 576.2.r.f.481.5 yes 12
24.5 odd 2 576.2.r.e.481.5 yes 12
24.11 even 2 576.2.r.e.481.2 yes 12
36.7 odd 6 1728.2.r.f.289.4 12
36.11 even 6 576.2.r.e.97.2 12
36.23 even 6 5184.2.d.q.2593.3 12
36.31 odd 6 5184.2.d.r.2593.9 12
72.5 odd 6 5184.2.d.q.2593.10 12
72.11 even 6 576.2.r.f.97.5 yes 12
72.13 even 6 5184.2.d.r.2593.4 12
72.29 odd 6 576.2.r.f.97.2 yes 12
72.43 odd 6 inner 1728.2.r.e.289.4 12
72.59 even 6 5184.2.d.q.2593.9 12
72.61 even 6 inner 1728.2.r.e.289.3 12
72.67 odd 6 5184.2.d.r.2593.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.e.97.2 12 36.11 even 6
576.2.r.e.97.5 yes 12 9.2 odd 6
576.2.r.e.481.2 yes 12 24.11 even 2
576.2.r.e.481.5 yes 12 24.5 odd 2
576.2.r.f.97.2 yes 12 72.29 odd 6
576.2.r.f.97.5 yes 12 72.11 even 6
576.2.r.f.481.2 yes 12 3.2 odd 2
576.2.r.f.481.5 yes 12 12.11 even 2
1728.2.r.e.289.3 12 72.61 even 6 inner
1728.2.r.e.289.4 12 72.43 odd 6 inner
1728.2.r.e.1441.3 12 1.1 even 1 trivial
1728.2.r.e.1441.4 12 4.3 odd 2 inner
1728.2.r.f.289.3 12 9.7 even 3
1728.2.r.f.289.4 12 36.7 odd 6
1728.2.r.f.1441.3 12 8.5 even 2
1728.2.r.f.1441.4 12 8.3 odd 2
5184.2.d.q.2593.3 12 36.23 even 6
5184.2.d.q.2593.4 12 9.5 odd 6
5184.2.d.q.2593.9 12 72.59 even 6
5184.2.d.q.2593.10 12 72.5 odd 6
5184.2.d.r.2593.3 12 72.67 odd 6
5184.2.d.r.2593.4 12 72.13 even 6
5184.2.d.r.2593.9 12 36.31 odd 6
5184.2.d.r.2593.10 12 9.4 even 3