Properties

Label 3240.2.q.d.2161.1
Level $3240$
Weight $2$
Character 3240.2161
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
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] = [3240,2,Mod(1081,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1081");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (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: \(25.8715302549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2161.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.2161
Dual form 3240.2.q.d.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(-1.00000 + 1.73205i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-2.00000 - 3.46410i) q^{13} +2.00000 q^{17} +4.00000 q^{19} +(4.00000 + 6.92820i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-5.00000 + 8.66025i) q^{29} +(-2.00000 - 3.46410i) q^{31} +2.00000 q^{35} +(4.00000 - 6.92820i) q^{43} +(4.00000 - 6.92820i) q^{47} +(1.50000 + 2.59808i) q^{49} -6.00000 q^{53} -2.00000 q^{55} +(-7.00000 - 12.1244i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-2.00000 + 3.46410i) q^{65} +(2.00000 + 3.46410i) q^{67} -12.0000 q^{71} +6.00000 q^{73} +(2.00000 + 3.46410i) q^{77} +(6.00000 - 10.3923i) q^{79} +(2.00000 - 3.46410i) q^{83} +(-1.00000 - 1.73205i) q^{85} +12.0000 q^{89} +8.00000 q^{91} +(-2.00000 - 3.46410i) q^{95} +(7.00000 - 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} - 2 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} + 8 q^{19} + 8 q^{23} - q^{25} - 10 q^{29} - 4 q^{31} + 4 q^{35} + 8 q^{43} + 8 q^{47} + 3 q^{49} - 12 q^{53} - 4 q^{55} - 14 q^{59} + 14 q^{61} - 4 q^{65} + 4 q^{67} - 24 q^{71} + 12 q^{73} + 4 q^{77} + 12 q^{79} + 4 q^{83} - 2 q^{85} + 24 q^{89} + 16 q^{91} - 4 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) −1.00000 + 1.73205i −0.377964 + 0.654654i −0.990766 0.135583i \(-0.956709\pi\)
0.612801 + 0.790237i \(0.290043\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 1.73205i 0.301511 0.522233i −0.674967 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) −2.00000 3.46410i −0.554700 0.960769i −0.997927 0.0643593i \(-0.979500\pi\)
0.443227 0.896410i \(-0.353834\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 + 6.92820i 0.834058 + 1.44463i 0.894795 + 0.446476i \(0.147321\pi\)
−0.0607377 + 0.998154i \(0.519345\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.00000 + 8.66025i −0.928477 + 1.60817i −0.142605 + 0.989780i \(0.545548\pi\)
−0.785872 + 0.618389i \(0.787786\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 4.00000 6.92820i 0.609994 1.05654i −0.381246 0.924473i \(-0.624505\pi\)
0.991241 0.132068i \(-0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 6.92820i 0.583460 1.01058i −0.411606 0.911362i \(-0.635032\pi\)
0.995066 0.0992202i \(-0.0316348\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.00000 12.1244i −0.911322 1.57846i −0.812198 0.583382i \(-0.801729\pi\)
−0.0991242 0.995075i \(-0.531604\pi\)
\(60\) 0 0
\(61\) 7.00000 12.1244i 0.896258 1.55236i 0.0640184 0.997949i \(-0.479608\pi\)
0.832240 0.554416i \(-0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 + 3.46410i −0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 + 3.46410i 0.227921 + 0.394771i
\(78\) 0 0
\(79\) 6.00000 10.3923i 0.675053 1.16923i −0.301401 0.953498i \(-0.597454\pi\)
0.976453 0.215728i \(-0.0692125\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.00000 3.46410i 0.219529 0.380235i −0.735135 0.677920i \(-0.762881\pi\)
0.954664 + 0.297686i \(0.0962148\pi\)
\(84\) 0 0
\(85\) −1.00000 1.73205i −0.108465 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 3.46410i −0.205196 0.355409i
\(96\) 0 0
\(97\) 7.00000 12.1244i 0.710742 1.23104i −0.253837 0.967247i \(-0.581693\pi\)
0.964579 0.263795i \(-0.0849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.00000 5.19615i 0.298511 0.517036i −0.677284 0.735721i \(-0.736843\pi\)
0.975796 + 0.218685i \(0.0701767\pi\)
\(102\) 0 0
\(103\) 7.00000 + 12.1244i 0.689730 + 1.19465i 0.971925 + 0.235291i \(0.0756043\pi\)
−0.282194 + 0.959357i \(0.591062\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.00000 + 15.5885i 0.846649 + 1.46644i 0.884182 + 0.467143i \(0.154717\pi\)
−0.0375328 + 0.999295i \(0.511950\pi\)
\(114\) 0 0
\(115\) 4.00000 6.92820i 0.373002 0.646058i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 + 3.46410i −0.183340 + 0.317554i
\(120\) 0 0
\(121\) 3.50000 + 6.06218i 0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.00000 15.5885i −0.786334 1.36197i −0.928199 0.372084i \(-0.878643\pi\)
0.141865 0.989886i \(-0.454690\pi\)
\(132\) 0 0
\(133\) −4.00000 + 6.92820i −0.346844 + 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00000 1.73205i 0.0854358 0.147979i −0.820141 0.572161i \(-0.806105\pi\)
0.905577 + 0.424182i \(0.139438\pi\)
\(138\) 0 0
\(139\) −6.00000 10.3923i −0.508913 0.881464i −0.999947 0.0103230i \(-0.996714\pi\)
0.491033 0.871141i \(-0.336619\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) 10.0000 0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 + 3.46410i −0.160644 + 0.278243i
\(156\) 0 0
\(157\) −2.00000 3.46410i −0.159617 0.276465i 0.775113 0.631822i \(-0.217693\pi\)
−0.934731 + 0.355357i \(0.884359\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.00000 + 1.73205i −0.0760286 + 0.131685i −0.901533 0.432710i \(-0.857557\pi\)
0.825505 + 0.564396i \(0.190891\pi\)
\(174\) 0 0
\(175\) −1.00000 1.73205i −0.0755929 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 3.46410i 0.146254 0.253320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.00000 3.46410i 0.144715 0.250654i −0.784552 0.620063i \(-0.787107\pi\)
0.929267 + 0.369410i \(0.120440\pi\)
\(192\) 0 0
\(193\) −11.0000 19.0526i −0.791797 1.37143i −0.924853 0.380325i \(-0.875812\pi\)
0.133056 0.991109i \(-0.457521\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.0000 17.3205i −0.701862 1.21566i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.00000 6.92820i 0.276686 0.479234i
\(210\) 0 0
\(211\) −10.0000 17.3205i −0.688428 1.19239i −0.972346 0.233544i \(-0.924968\pi\)
0.283918 0.958849i \(-0.408366\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 6.92820i −0.269069 0.466041i
\(222\) 0 0
\(223\) 11.0000 19.0526i 0.736614 1.27585i −0.217397 0.976083i \(-0.569757\pi\)
0.954011 0.299770i \(-0.0969101\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(228\) 0 0
\(229\) 1.00000 + 1.73205i 0.0660819 + 0.114457i 0.897173 0.441679i \(-0.145617\pi\)
−0.831092 + 0.556136i \(0.812283\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 + 10.3923i 0.388108 + 0.672222i 0.992195 0.124696i \(-0.0397955\pi\)
−0.604087 + 0.796918i \(0.706462\pi\)
\(240\) 0 0
\(241\) 5.00000 8.66025i 0.322078 0.557856i −0.658838 0.752285i \(-0.728952\pi\)
0.980917 + 0.194429i \(0.0622852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.50000 2.59808i 0.0958315 0.165985i
\(246\) 0 0
\(247\) −8.00000 13.8564i −0.509028 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0 0
\(265\) 3.00000 + 5.19615i 0.184289 + 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 + 1.73205i 0.0603023 + 0.104447i
\(276\) 0 0
\(277\) 8.00000 13.8564i 0.480673 0.832551i −0.519081 0.854725i \(-0.673726\pi\)
0.999754 + 0.0221745i \(0.00705893\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.0000 + 24.2487i −0.835170 + 1.44656i 0.0587220 + 0.998274i \(0.481297\pi\)
−0.893892 + 0.448282i \(0.852036\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.00000 + 1.73205i 0.0584206 + 0.101187i 0.893757 0.448552i \(-0.148060\pi\)
−0.835336 + 0.549740i \(0.814727\pi\)
\(294\) 0 0
\(295\) −7.00000 + 12.1244i −0.407556 + 0.705907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.0000 27.7128i 0.925304 1.60267i
\(300\) 0 0
\(301\) 8.00000 + 13.8564i 0.461112 + 0.798670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.0000 −0.801638
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.0000 17.3205i −0.567048 0.982156i −0.996856 0.0792356i \(-0.974752\pi\)
0.429808 0.902920i \(-0.358581\pi\)
\(312\) 0 0
\(313\) 5.00000 8.66025i 0.282617 0.489506i −0.689412 0.724370i \(-0.742131\pi\)
0.972028 + 0.234863i \(0.0754642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −15.0000 + 25.9808i −0.842484 + 1.45922i 0.0453045 + 0.998973i \(0.485574\pi\)
−0.887788 + 0.460252i \(0.847759\pi\)
\(318\) 0 0
\(319\) 10.0000 + 17.3205i 0.559893 + 0.969762i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.00000 + 13.8564i 0.441054 + 0.763928i
\(330\) 0 0
\(331\) 6.00000 10.3923i 0.329790 0.571213i −0.652680 0.757634i \(-0.726355\pi\)
0.982470 + 0.186421i \(0.0596888\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.00000 3.46410i 0.109272 0.189264i
\(336\) 0 0
\(337\) 1.00000 + 1.73205i 0.0544735 + 0.0943508i 0.891976 0.452082i \(-0.149319\pi\)
−0.837503 + 0.546433i \(0.815985\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.00000 + 13.8564i 0.429463 + 0.743851i 0.996826 0.0796169i \(-0.0253697\pi\)
−0.567363 + 0.823468i \(0.692036\pi\)
\(348\) 0 0
\(349\) 3.00000 5.19615i 0.160586 0.278144i −0.774493 0.632583i \(-0.781995\pi\)
0.935079 + 0.354439i \(0.115328\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.0000 + 22.5167i −0.691920 + 1.19844i 0.279288 + 0.960207i \(0.409902\pi\)
−0.971208 + 0.238233i \(0.923432\pi\)
\(354\) 0 0
\(355\) 6.00000 + 10.3923i 0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.00000 5.19615i −0.157027 0.271979i
\(366\) 0 0
\(367\) −13.0000 + 22.5167i −0.678594 + 1.17536i 0.296810 + 0.954937i \(0.404077\pi\)
−0.975404 + 0.220423i \(0.929256\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 10.3923i 0.311504 0.539542i
\(372\) 0 0
\(373\) 10.0000 + 17.3205i 0.517780 + 0.896822i 0.999787 + 0.0206542i \(0.00657489\pi\)
−0.482006 + 0.876168i \(0.660092\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 40.0000 2.06010
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 20.7846i −0.613171 1.06204i −0.990702 0.136047i \(-0.956560\pi\)
0.377531 0.925997i \(-0.376773\pi\)
\(384\) 0 0
\(385\) 2.00000 3.46410i 0.101929 0.176547i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 15.5885i 0.456318 0.790366i −0.542445 0.840091i \(-0.682501\pi\)
0.998763 + 0.0497253i \(0.0158346\pi\)
\(390\) 0 0
\(391\) 8.00000 + 13.8564i 0.404577 + 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 10.3923i −0.299626 0.518967i 0.676425 0.736512i \(-0.263528\pi\)
−0.976050 + 0.217545i \(0.930195\pi\)
\(402\) 0 0
\(403\) −8.00000 + 13.8564i −0.398508 + 0.690237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.00000 8.66025i −0.247234 0.428222i 0.715523 0.698589i \(-0.246188\pi\)
−0.962757 + 0.270367i \(0.912855\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 28.0000 1.37779
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.0000 + 25.9808i 0.732798 + 1.26924i 0.955683 + 0.294398i \(0.0951193\pi\)
−0.222885 + 0.974845i \(0.571547\pi\)
\(420\) 0 0
\(421\) −15.0000 + 25.9808i −0.731055 + 1.26622i 0.225377 + 0.974272i \(0.427639\pi\)
−0.956433 + 0.291953i \(0.905695\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 + 1.73205i −0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) 14.0000 + 24.2487i 0.677507 + 1.17348i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.0000 + 27.7128i 0.765384 + 1.32568i
\(438\) 0 0
\(439\) 16.0000 27.7128i 0.763638 1.32266i −0.177325 0.984152i \(-0.556744\pi\)
0.940963 0.338508i \(-0.109922\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000 20.7846i 0.570137 0.987507i −0.426414 0.904528i \(-0.640223\pi\)
0.996551 0.0829786i \(-0.0264433\pi\)
\(444\) 0 0
\(445\) −6.00000 10.3923i −0.284427 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.00000 6.92820i −0.187523 0.324799i
\(456\) 0 0
\(457\) −5.00000 + 8.66025i −0.233890 + 0.405110i −0.958950 0.283577i \(-0.908479\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −15.0000 + 25.9808i −0.698620 + 1.21004i 0.270326 + 0.962769i \(0.412869\pi\)
−0.968945 + 0.247276i \(0.920465\pi\)
\(462\) 0 0
\(463\) −15.0000 25.9808i −0.697109 1.20743i −0.969465 0.245232i \(-0.921136\pi\)
0.272355 0.962197i \(-0.412197\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 13.8564i −0.367840 0.637118i
\(474\) 0 0
\(475\) −2.00000 + 3.46410i −0.0917663 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.00000 + 3.46410i −0.0913823 + 0.158279i −0.908093 0.418769i \(-0.862462\pi\)
0.816711 + 0.577047i \(0.195795\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.00000 15.5885i −0.406164 0.703497i 0.588292 0.808649i \(-0.299801\pi\)
−0.994456 + 0.105151i \(0.966467\pi\)
\(492\) 0 0
\(493\) −10.0000 + 17.3205i −0.450377 + 0.780076i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 20.7846i 0.538274 0.932317i
\(498\) 0 0
\(499\) −10.0000 17.3205i −0.447661 0.775372i 0.550572 0.834788i \(-0.314410\pi\)
−0.998233 + 0.0594153i \(0.981076\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.00000 + 12.1244i 0.310270 + 0.537403i 0.978421 0.206623i \(-0.0662474\pi\)
−0.668151 + 0.744026i \(0.732914\pi\)
\(510\) 0 0
\(511\) −6.00000 + 10.3923i −0.265424 + 0.459728i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.00000 12.1244i 0.308457 0.534263i
\(516\) 0 0
\(517\) −8.00000 13.8564i −0.351840 0.609404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −20.0000 −0.876216 −0.438108 0.898922i \(-0.644351\pi\)
−0.438108 + 0.898922i \(0.644351\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 6.92820i −0.174243 0.301797i
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −6.00000 10.3923i −0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.00000 1.73205i −0.0428353 0.0741929i
\(546\) 0 0
\(547\) −4.00000 + 6.92820i −0.171028 + 0.296229i −0.938779 0.344519i \(-0.888042\pi\)
0.767752 + 0.640747i \(0.221375\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −20.0000 + 34.6410i −0.852029 + 1.47576i
\(552\) 0 0
\(553\) 12.0000 + 20.7846i 0.510292 + 0.883852i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 26.0000 1.10166 0.550828 0.834619i \(-0.314312\pi\)
0.550828 + 0.834619i \(0.314312\pi\)
\(558\) 0 0
\(559\) −32.0000 −1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.00000 + 6.92820i 0.168580 + 0.291989i 0.937921 0.346850i \(-0.112749\pi\)
−0.769341 + 0.638838i \(0.779415\pi\)
\(564\) 0 0
\(565\) 9.00000 15.5885i 0.378633 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.0000 + 17.3205i −0.419222 + 0.726113i −0.995861 0.0908852i \(-0.971030\pi\)
0.576640 + 0.816999i \(0.304364\pi\)
\(570\) 0 0
\(571\) −2.00000 3.46410i −0.0836974 0.144968i 0.821138 0.570730i \(-0.193340\pi\)
−0.904835 + 0.425762i \(0.860006\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) 26.0000 1.08239 0.541197 0.840896i \(-0.317971\pi\)
0.541197 + 0.840896i \(0.317971\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 + 6.92820i 0.165948 + 0.287430i
\(582\) 0 0
\(583\) −6.00000 + 10.3923i −0.248495 + 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 31.1769i 0.742940 1.28681i −0.208212 0.978084i \(-0.566764\pi\)
0.951151 0.308725i \(-0.0999023\pi\)
\(588\) 0 0
\(589\) −8.00000 13.8564i −0.329634 0.570943i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0000 20.7846i −0.490307 0.849236i 0.509631 0.860393i \(-0.329782\pi\)
−0.999938 + 0.0111569i \(0.996449\pi\)
\(600\) 0 0
\(601\) −11.0000 + 19.0526i −0.448699 + 0.777170i −0.998302 0.0582563i \(-0.981446\pi\)
0.549602 + 0.835426i \(0.314779\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.50000 6.06218i 0.142295 0.246463i
\(606\) 0 0
\(607\) −3.00000 5.19615i −0.121766 0.210905i 0.798698 0.601732i \(-0.205522\pi\)
−0.920464 + 0.390827i \(0.872189\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −48.0000 −1.93870 −0.969351 0.245680i \(-0.920989\pi\)
−0.969351 + 0.245680i \(0.920989\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 + 5.19615i 0.120775 + 0.209189i 0.920074 0.391745i \(-0.128129\pi\)
−0.799298 + 0.600935i \(0.794795\pi\)
\(618\) 0 0
\(619\) −22.0000 + 38.1051i −0.884255 + 1.53157i −0.0376891 + 0.999290i \(0.512000\pi\)
−0.846566 + 0.532284i \(0.821334\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 + 20.7846i −0.480770 + 0.832718i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.00000 15.5885i −0.357154 0.618609i
\(636\) 0 0
\(637\) 6.00000 10.3923i 0.237729 0.411758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 20.7846i 0.473972 0.820943i −0.525584 0.850741i \(-0.676153\pi\)
0.999556 + 0.0297987i \(0.00948663\pi\)
\(642\) 0 0
\(643\) −6.00000 10.3923i −0.236617 0.409832i 0.723124 0.690718i \(-0.242705\pi\)
−0.959741 + 0.280885i \(0.909372\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.00000 12.1244i −0.273931 0.474463i 0.695934 0.718106i \(-0.254991\pi\)
−0.969865 + 0.243643i \(0.921657\pi\)
\(654\) 0 0
\(655\) −9.00000 + 15.5885i −0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.00000 5.19615i 0.116863 0.202413i −0.801660 0.597781i \(-0.796049\pi\)
0.918523 + 0.395367i \(0.129383\pi\)
\(660\) 0 0
\(661\) −9.00000 15.5885i −0.350059 0.606321i 0.636200 0.771524i \(-0.280505\pi\)
−0.986260 + 0.165203i \(0.947172\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) −80.0000 −3.09761
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −14.0000 24.2487i −0.540464 0.936111i
\(672\) 0 0
\(673\) −15.0000 + 25.9808i −0.578208 + 1.00148i 0.417477 + 0.908687i \(0.362914\pi\)
−0.995685 + 0.0927975i \(0.970419\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.0000 22.5167i 0.499631 0.865386i −0.500369 0.865812i \(-0.666802\pi\)
1.00000 0.000426509i \(0.000135762\pi\)
\(678\) 0 0
\(679\) 14.0000 + 24.2487i 0.537271 + 0.930580i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 + 20.7846i 0.457164 + 0.791831i
\(690\) 0 0
\(691\) 14.0000 24.2487i 0.532585 0.922464i −0.466691 0.884420i \(-0.654554\pi\)
0.999276 0.0380440i \(-0.0121127\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 + 10.3923i −0.227593 + 0.394203i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 + 10.3923i 0.225653 + 0.390843i
\(708\) 0 0
\(709\) −7.00000 + 12.1244i −0.262891 + 0.455340i −0.967009 0.254743i \(-0.918009\pi\)
0.704118 + 0.710083i \(0.251342\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.0000 27.7128i 0.599205 1.03785i
\(714\) 0 0
\(715\) 4.00000 + 6.92820i 0.149592 + 0.259100i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.0000 1.49175 0.745874 0.666087i \(-0.232032\pi\)
0.745874 + 0.666087i \(0.232032\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.00000 8.66025i −0.185695 0.321634i
\(726\) 0 0
\(727\) −19.0000 + 32.9090i −0.704671 + 1.22053i 0.262139 + 0.965030i \(0.415572\pi\)
−0.966810 + 0.255496i \(0.917761\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.00000 13.8564i 0.295891 0.512498i
\(732\) 0 0
\(733\) 12.0000 + 20.7846i 0.443230 + 0.767697i 0.997927 0.0643554i \(-0.0204991\pi\)
−0.554697 + 0.832052i \(0.687166\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.0000 + 20.7846i 0.440237 + 0.762513i 0.997707 0.0676840i \(-0.0215610\pi\)
−0.557470 + 0.830197i \(0.688228\pi\)
\(744\) 0 0
\(745\) 5.00000 8.66025i 0.183186 0.317287i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 + 20.7846i −0.438470 + 0.759453i
\(750\) 0 0
\(751\) 18.0000 + 31.1769i 0.656829 + 1.13766i 0.981432 + 0.191811i \(0.0614361\pi\)
−0.324603 + 0.945851i \(0.605231\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 32.0000 1.16306 0.581530 0.813525i \(-0.302454\pi\)
0.581530 + 0.813525i \(0.302454\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 31.1769i −0.652499 1.13016i −0.982514 0.186187i \(-0.940387\pi\)
0.330015 0.943976i \(-0.392946\pi\)
\(762\) 0 0
\(763\) −2.00000 + 3.46410i −0.0724049 + 0.125409i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.0000 + 48.4974i −1.01102 + 1.75114i
\(768\) 0 0
\(769\) 25.0000 + 43.3013i 0.901523 + 1.56148i 0.825518 + 0.564376i \(0.190883\pi\)
0.0760054 + 0.997107i \(0.475783\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 4.00000 0.143684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 + 20.7846i −0.429394 + 0.743732i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.00000 + 3.46410i −0.0713831 + 0.123639i
\(786\) 0 0
\(787\) 14.0000 + 24.2487i 0.499046 + 0.864373i 0.999999 0.00110111i \(-0.000350496\pi\)
−0.500953 + 0.865474i \(0.667017\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) 0 0
\(793\) −56.0000 −1.98862
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.0000 + 25.9808i 0.531327 + 0.920286i 0.999331 + 0.0365596i \(0.0116399\pi\)
−0.468004 + 0.883726i \(0.655027\pi\)
\(798\) 0 0
\(799\) 8.00000 13.8564i 0.283020 0.490204i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00000 10.3923i 0.211735 0.366736i
\(804\) 0 0
\(805\) 8.00000 + 13.8564i 0.281963 + 0.488374i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 3.46410i −0.0700569 0.121342i
\(816\) 0 0
\(817\) 16.0000 27.7128i 0.559769 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.00000 12.1244i 0.244302 0.423143i −0.717633 0.696421i \(-0.754775\pi\)
0.961935 + 0.273278i \(0.0881079\pi\)
\(822\) 0 0
\(823\) 1.00000 + 1.73205i 0.0348578 + 0.0603755i 0.882928 0.469508i \(-0.155569\pi\)
−0.848070 + 0.529884i \(0.822235\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.00000 + 5.19615i 0.103944 + 0.180036i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −16.0000 + 27.7128i −0.552381 + 0.956753i 0.445721 + 0.895172i \(0.352947\pi\)
−0.998102 + 0.0615805i \(0.980386\pi\)
\(840\) 0 0
\(841\) −35.5000 61.4878i −1.22414 2.12027i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.00000 13.8564i 0.273915 0.474434i −0.695946 0.718094i \(-0.745015\pi\)
0.969861 + 0.243660i \(0.0783480\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.00000 + 15.5885i −0.307434 + 0.532492i −0.977800 0.209539i \(-0.932804\pi\)
0.670366 + 0.742030i \(0.266137\pi\)
\(858\) 0 0
\(859\) 2.00000 + 3.46410i 0.0682391 + 0.118194i 0.898126 0.439738i \(-0.144929\pi\)
−0.829887 + 0.557931i \(0.811595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.0000 20.7846i −0.407072 0.705070i
\(870\) 0 0
\(871\) 8.00000 13.8564i 0.271070 0.469506i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.00000 + 1.73205i −0.0338062 + 0.0585540i
\(876\) 0 0
\(877\) 10.0000 + 17.3205i 0.337676 + 0.584872i 0.983995 0.178195i \(-0.0570259\pi\)
−0.646319 + 0.763067i \(0.723693\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −36.0000 −1.21287 −0.606435 0.795133i \(-0.707401\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.0000 + 48.4974i 0.940148 + 1.62838i 0.765186 + 0.643809i \(0.222647\pi\)
0.174962 + 0.984575i \(0.444020\pi\)
\(888\) 0 0
\(889\) −18.0000 + 31.1769i −0.603701 + 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 16.0000 27.7128i 0.535420 0.927374i
\(894\) 0 0
\(895\) −7.00000 12.1244i −0.233984 0.405273i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 40.0000 1.33407
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.00000 1.73205i −0.0332411 0.0575753i
\(906\) 0 0
\(907\) −12.0000 + 20.7846i −0.398453 + 0.690142i −0.993535 0.113523i \(-0.963786\pi\)
0.595082 + 0.803665i \(0.297120\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 + 13.8564i −0.265052 + 0.459083i −0.967577 0.252575i \(-0.918722\pi\)
0.702525 + 0.711659i \(0.252056\pi\)
\(912\) 0 0
\(913\) −4.00000 6.92820i −0.132381 0.229290i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.0000 1.18882
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000 + 41.5692i 0.789970 + 1.36827i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.00000 + 3.46410i −0.0656179 + 0.113653i −0.896968 0.442096i \(-0.854235\pi\)
0.831350 + 0.555749i \(0.187569\pi\)
\(930\) 0 0
\(931\) 6.00000 + 10.3923i 0.196642 + 0.340594i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −11.0000 19.0526i −0.358590 0.621096i 0.629136 0.777295i \(-0.283409\pi\)
−0.987725 + 0.156200i \(0.950076\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000 48.4974i 0.909878 1.57595i 0.0956453 0.995415i \(-0.469509\pi\)
0.814232 0.580539i \(-0.197158\pi\)
\(948\) 0 0
\(949\) −12.0000 20.7846i −0.389536 0.674697i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) −4.00000 −0.129437
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.00000 + 3.46410i 0.0645834 + 0.111862i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −11.0000 + 19.0526i −0.354103 + 0.613324i
\(966\) 0 0
\(967\) −9.00000 15.5885i −0.289420 0.501291i 0.684251 0.729247i \(-0.260129\pi\)
−0.973672 + 0.227956i \(0.926796\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.00000 0.0641831 0.0320915 0.999485i \(-0.489783\pi\)
0.0320915 + 0.999485i \(0.489783\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.00000 1.73205i −0.0319928 0.0554132i 0.849586 0.527451i \(-0.176852\pi\)
−0.881579 + 0.472037i \(0.843519\pi\)
\(978\) 0 0
\(979\) 12.0000 20.7846i 0.383522 0.664279i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.0000 + 20.7846i −0.382741 + 0.662926i −0.991453 0.130465i \(-0.958353\pi\)
0.608712 + 0.793391i \(0.291686\pi\)
\(984\) 0 0
\(985\) 1.00000 + 1.73205i 0.0318626 + 0.0551877i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 6.92820i −0.126809 0.219639i
\(996\) 0 0
\(997\) −18.0000 + 31.1769i −0.570066 + 0.987383i 0.426493 + 0.904491i \(0.359749\pi\)
−0.996559 + 0.0828918i \(0.973584\pi\)
\(998\) 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 3240.2.q.d.2161.1 2
3.2 odd 2 3240.2.q.n.2161.1 2
9.2 odd 6 360.2.a.c.1.1 1
9.4 even 3 inner 3240.2.q.d.1081.1 2
9.5 odd 6 3240.2.q.n.1081.1 2
9.7 even 3 360.2.a.d.1.1 yes 1
36.7 odd 6 720.2.a.i.1.1 1
36.11 even 6 720.2.a.a.1.1 1
45.2 even 12 1800.2.f.h.649.2 2
45.7 odd 12 1800.2.f.d.649.2 2
45.29 odd 6 1800.2.a.i.1.1 1
45.34 even 6 1800.2.a.f.1.1 1
45.38 even 12 1800.2.f.h.649.1 2
45.43 odd 12 1800.2.f.d.649.1 2
72.11 even 6 2880.2.a.w.1.1 1
72.29 odd 6 2880.2.a.bd.1.1 1
72.43 odd 6 2880.2.a.e.1.1 1
72.61 even 6 2880.2.a.n.1.1 1
180.7 even 12 3600.2.f.q.2449.1 2
180.43 even 12 3600.2.f.q.2449.2 2
180.47 odd 12 3600.2.f.g.2449.1 2
180.79 odd 6 3600.2.a.bh.1.1 1
180.83 odd 12 3600.2.f.g.2449.2 2
180.119 even 6 3600.2.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.a.c.1.1 1 9.2 odd 6
360.2.a.d.1.1 yes 1 9.7 even 3
720.2.a.a.1.1 1 36.11 even 6
720.2.a.i.1.1 1 36.7 odd 6
1800.2.a.f.1.1 1 45.34 even 6
1800.2.a.i.1.1 1 45.29 odd 6
1800.2.f.d.649.1 2 45.43 odd 12
1800.2.f.d.649.2 2 45.7 odd 12
1800.2.f.h.649.1 2 45.38 even 12
1800.2.f.h.649.2 2 45.2 even 12
2880.2.a.e.1.1 1 72.43 odd 6
2880.2.a.n.1.1 1 72.61 even 6
2880.2.a.w.1.1 1 72.11 even 6
2880.2.a.bd.1.1 1 72.29 odd 6
3240.2.q.d.1081.1 2 9.4 even 3 inner
3240.2.q.d.2161.1 2 1.1 even 1 trivial
3240.2.q.n.1081.1 2 9.5 odd 6
3240.2.q.n.2161.1 2 3.2 odd 2
3600.2.a.bd.1.1 1 180.119 even 6
3600.2.a.bh.1.1 1 180.79 odd 6
3600.2.f.g.2449.1 2 180.47 odd 12
3600.2.f.g.2449.2 2 180.83 odd 12
3600.2.f.q.2449.1 2 180.7 even 12
3600.2.f.q.2449.2 2 180.43 even 12