Properties

Label 72.2.l.a.11.1
Level $72$
Weight $2$
Character 72.11
Analytic conductor $0.575$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [72,2,Mod(11,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 72.l (of order \(6\), degree \(2\), 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: \(0.574922894553\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 11.1
Root \(-1.22474 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 72.11
Dual form 72.2.l.a.59.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 - 0.707107i) q^{2} +(1.72474 - 0.158919i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 - 1.02494i) q^{6} -2.82843i q^{8} +(2.94949 - 0.548188i) q^{9} +O(q^{10})\) \(q+(-1.22474 - 0.707107i) q^{2} +(1.72474 - 0.158919i) q^{3} +(1.00000 + 1.73205i) q^{4} +(-2.22474 - 1.02494i) q^{6} -2.82843i q^{8} +(2.94949 - 0.548188i) q^{9} +(-3.27526 - 1.89097i) q^{11} +(2.00000 + 2.82843i) q^{12} +(-2.00000 + 3.46410i) q^{16} +8.02458i q^{17} +(-4.00000 - 1.41421i) q^{18} -8.34847 q^{19} +(2.67423 + 4.63191i) q^{22} +(-0.449490 - 4.87832i) q^{24} +(2.50000 - 4.33013i) q^{25} +(5.00000 - 1.41421i) q^{27} +(4.89898 - 2.82843i) q^{32} +(-5.94949 - 2.74094i) q^{33} +(5.67423 - 9.82806i) q^{34} +(3.89898 + 4.56048i) q^{36} +(10.2247 + 5.90326i) q^{38} +(-0.398979 + 0.230351i) q^{41} +(1.17423 - 2.03383i) q^{43} -7.56388i q^{44} +(-2.89898 + 6.29253i) q^{48} +(-3.50000 - 6.06218i) q^{49} +(-6.12372 + 3.53553i) q^{50} +(1.27526 + 13.8404i) q^{51} +(-7.12372 - 1.80348i) q^{54} +(-14.3990 + 1.32673i) q^{57} +(10.6237 - 6.13361i) q^{59} -8.00000 q^{64} +(5.34847 + 7.56388i) q^{66} +(7.17423 + 12.4261i) q^{67} +(-13.8990 + 8.02458i) q^{68} +(-1.55051 - 8.34242i) q^{72} +13.6969 q^{73} +(3.62372 - 7.86566i) q^{75} +(-8.34847 - 14.4600i) q^{76} +(8.39898 - 3.23375i) q^{81} +0.651531 q^{82} +(-2.44949 - 1.41421i) q^{83} +(-2.87628 + 1.66062i) q^{86} +(-5.34847 + 9.26382i) q^{88} -5.65685i q^{89} +(8.00000 - 5.65685i) q^{96} +(-9.84847 + 17.0580i) q^{97} +9.89949i q^{98} +(-10.6969 - 3.78194i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 4 q^{4} - 4 q^{6} + 2 q^{9} - 18 q^{11} + 8 q^{12} - 8 q^{16} - 16 q^{18} - 4 q^{19} - 4 q^{22} + 8 q^{24} + 10 q^{25} + 20 q^{27} - 14 q^{33} + 8 q^{34} - 4 q^{36} + 36 q^{38} + 18 q^{41} - 10 q^{43} + 8 q^{48} - 14 q^{49} + 10 q^{51} - 4 q^{54} - 38 q^{57} + 18 q^{59} - 32 q^{64} - 8 q^{66} + 14 q^{67} - 36 q^{68} - 16 q^{72} - 4 q^{73} - 10 q^{75} - 4 q^{76} + 14 q^{81} + 32 q^{82} - 36 q^{86} + 8 q^{88} + 32 q^{96} - 10 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22474 0.707107i −0.866025 0.500000i
\(3\) 1.72474 0.158919i 0.995782 0.0917517i
\(4\) 1.00000 + 1.73205i 0.500000 + 0.866025i
\(5\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) −2.22474 1.02494i −0.908248 0.418432i
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 2.82843i 1.00000i
\(9\) 2.94949 0.548188i 0.983163 0.182729i
\(10\) 0 0
\(11\) −3.27526 1.89097i −0.987527 0.570149i −0.0829925 0.996550i \(-0.526448\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 2.00000 + 2.82843i 0.577350 + 0.816497i
\(13\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 8.02458i 1.94625i 0.230285 + 0.973123i \(0.426034\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) −4.00000 1.41421i −0.942809 0.333333i
\(19\) −8.34847 −1.91527 −0.957635 0.287984i \(-0.907015\pi\)
−0.957635 + 0.287984i \(0.907015\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.67423 + 4.63191i 0.570149 + 0.987527i
\(23\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) −0.449490 4.87832i −0.0917517 0.995782i
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 5.00000 1.41421i 0.962250 0.272166i
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 4.89898 2.82843i 0.866025 0.500000i
\(33\) −5.94949 2.74094i −1.03567 0.477137i
\(34\) 5.67423 9.82806i 0.973123 1.68550i
\(35\) 0 0
\(36\) 3.89898 + 4.56048i 0.649830 + 0.760080i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 10.2247 + 5.90326i 1.65867 + 0.957635i
\(39\) 0 0
\(40\) 0 0
\(41\) −0.398979 + 0.230351i −0.0623101 + 0.0359748i −0.530831 0.847477i \(-0.678120\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 1.17423 2.03383i 0.179069 0.310157i −0.762493 0.646997i \(-0.776025\pi\)
0.941562 + 0.336840i \(0.109358\pi\)
\(44\) 7.56388i 1.14030i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) −2.89898 + 6.29253i −0.418432 + 0.908248i
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) −6.12372 + 3.53553i −0.866025 + 0.500000i
\(51\) 1.27526 + 13.8404i 0.178571 + 1.93804i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −7.12372 1.80348i −0.969416 0.245423i
\(55\) 0 0
\(56\) 0 0
\(57\) −14.3990 + 1.32673i −1.90719 + 0.175729i
\(58\) 0 0
\(59\) 10.6237 6.13361i 1.38309 0.798528i 0.390567 0.920575i \(-0.372279\pi\)
0.992524 + 0.122047i \(0.0389457\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 5.34847 + 7.56388i 0.658351 + 0.931049i
\(67\) 7.17423 + 12.4261i 0.876472 + 1.51809i 0.855186 + 0.518321i \(0.173443\pi\)
0.0212861 + 0.999773i \(0.493224\pi\)
\(68\) −13.8990 + 8.02458i −1.68550 + 0.973123i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.55051 8.34242i −0.182729 0.983163i
\(73\) 13.6969 1.60311 0.801553 0.597924i \(-0.204008\pi\)
0.801553 + 0.597924i \(0.204008\pi\)
\(74\) 0 0
\(75\) 3.62372 7.86566i 0.418432 0.908248i
\(76\) −8.34847 14.4600i −0.957635 1.65867i
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(80\) 0 0
\(81\) 8.39898 3.23375i 0.933220 0.359306i
\(82\) 0.651531 0.0719495
\(83\) −2.44949 1.41421i −0.268866 0.155230i 0.359506 0.933143i \(-0.382945\pi\)
−0.628372 + 0.777913i \(0.716279\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.87628 + 1.66062i −0.310157 + 0.179069i
\(87\) 0 0
\(88\) −5.34847 + 9.26382i −0.570149 + 0.987527i
\(89\) 5.65685i 0.599625i −0.953998 0.299813i \(-0.903076\pi\)
0.953998 0.299813i \(-0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 8.00000 5.65685i 0.816497 0.577350i
\(97\) −9.84847 + 17.0580i −0.999961 + 1.73198i −0.492287 + 0.870433i \(0.663839\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.89949i 1.00000i
\(99\) −10.6969 3.78194i −1.07508 0.380099i
\(100\) 10.0000 1.00000
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 8.22474 17.8526i 0.814371 1.76767i
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.70334i 0.454689i −0.973814 0.227345i \(-0.926996\pi\)
0.973814 0.227345i \(-0.0730044\pi\)
\(108\) 7.44949 + 7.24604i 0.716827 + 0.697251i
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.79796 5.65685i 0.921714 0.532152i 0.0375328 0.999295i \(-0.488050\pi\)
0.884182 + 0.467143i \(0.154717\pi\)
\(114\) 18.5732 + 8.55671i 1.73954 + 0.801410i
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −17.3485 −1.59706
\(119\) 0 0
\(120\) 0 0
\(121\) 1.65153 + 2.86054i 0.150139 + 0.260049i
\(122\) 0 0
\(123\) −0.651531 + 0.460702i −0.0587466 + 0.0415401i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 9.79796 + 5.65685i 0.866025 + 0.500000i
\(129\) 1.70204 3.69445i 0.149856 0.325278i
\(130\) 0 0
\(131\) −12.2474 + 7.07107i −1.07006 + 0.617802i −0.928199 0.372084i \(-0.878643\pi\)
−0.141865 + 0.989886i \(0.545310\pi\)
\(132\) −1.20204 13.0458i −0.104624 1.13549i
\(133\) 0 0
\(134\) 20.2918i 1.75294i
\(135\) 0 0
\(136\) 22.6969 1.94625
\(137\) −14.2980 8.25493i −1.22156 0.705266i −0.256307 0.966595i \(-0.582506\pi\)
−0.965250 + 0.261329i \(0.915839\pi\)
\(138\) 0 0
\(139\) −1.82577 3.16232i −0.154859 0.268224i 0.778148 0.628080i \(-0.216159\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −4.00000 + 11.3137i −0.333333 + 0.942809i
\(145\) 0 0
\(146\) −16.7753 9.68520i −1.38833 0.801553i
\(147\) −7.00000 9.89949i −0.577350 0.816497i
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) −10.0000 + 7.07107i −0.816497 + 0.577350i
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 23.6130i 1.91527i
\(153\) 4.39898 + 23.6684i 0.355636 + 1.91348i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −12.5732 1.97846i −0.987845 0.155442i
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −0.797959 0.460702i −0.0623101 0.0359748i
\(165\) 0 0
\(166\) 2.00000 + 3.46410i 0.155230 + 0.268866i
\(167\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(168\) 0 0
\(169\) −6.50000 + 11.2583i −0.500000 + 0.866025i
\(170\) 0 0
\(171\) −24.6237 + 4.57653i −1.88302 + 0.349976i
\(172\) 4.69694 0.358138
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 13.1010 7.56388i 0.987527 0.570149i
\(177\) 17.3485 12.2672i 1.30399 0.922061i
\(178\) −4.00000 + 6.92820i −0.299813 + 0.519291i
\(179\) 19.7990i 1.47985i 0.672692 + 0.739923i \(0.265138\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.1742 26.2825i 1.10965 1.92197i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) −13.7980 + 1.27135i −0.995782 + 0.0917517i
\(193\) −12.8485 22.2542i −0.924853 1.60189i −0.791797 0.610784i \(-0.790854\pi\)
−0.133056 0.991109i \(-0.542479\pi\)
\(194\) 24.1237 13.9278i 1.73198 0.999961i
\(195\) 0 0
\(196\) 7.00000 12.1244i 0.500000 0.866025i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 10.4268 + 12.1958i 0.740999 + 0.866717i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −12.2474 7.07107i −0.866025 0.500000i
\(201\) 14.3485 + 20.2918i 1.01206 + 1.43127i
\(202\) 0 0
\(203\) 0 0
\(204\) −22.6969 + 16.0492i −1.58910 + 1.12367i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 27.3434 + 15.7867i 1.89138 + 1.09199i
\(210\) 0 0
\(211\) −7.00000 12.1244i −0.481900 0.834675i 0.517884 0.855451i \(-0.326720\pi\)
−0.999784 + 0.0207756i \(0.993386\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −3.32577 + 5.76039i −0.227345 + 0.393772i
\(215\) 0 0
\(216\) −4.00000 14.1421i −0.272166 0.962250i
\(217\) 0 0
\(218\) 0 0
\(219\) 23.6237 2.17670i 1.59634 0.147088i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) 5.00000 14.1421i 0.333333 0.942809i
\(226\) −16.0000 −1.06430
\(227\) 23.7247 + 13.6975i 1.57467 + 0.909134i 0.995585 + 0.0938647i \(0.0299221\pi\)
0.579082 + 0.815270i \(0.303411\pi\)
\(228\) −16.6969 23.6130i −1.10578 1.56381i
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.1523i 1.51676i −0.651813 0.758380i \(-0.725991\pi\)
0.651813 0.758380i \(-0.274009\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 21.2474 + 12.2672i 1.38309 + 0.798528i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −0.848469 + 1.46959i −0.0546547 + 0.0946647i −0.892058 0.451920i \(-0.850739\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 4.67123i 0.300278i
\(243\) 13.9722 6.91215i 0.896317 0.443415i
\(244\) 0 0
\(245\) 0 0
\(246\) 1.12372 0.103540i 0.0716460 0.00660149i
\(247\) 0 0
\(248\) 0 0
\(249\) −4.44949 2.04989i −0.281975 0.129906i
\(250\) 0 0
\(251\) 20.7525i 1.30989i 0.755678 + 0.654943i \(0.227307\pi\)
−0.755678 + 0.654943i \(0.772693\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) −27.3990 + 15.8188i −1.70910 + 0.986750i −0.773427 + 0.633885i \(0.781459\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) −4.69694 + 3.32124i −0.292419 + 0.206771i
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 20.0000 1.23560
\(263\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) −7.75255 + 16.8277i −0.477137 + 1.03567i
\(265\) 0 0
\(266\) 0 0
\(267\) −0.898979 9.75663i −0.0550167 0.597096i
\(268\) −14.3485 + 24.8523i −0.876472 + 1.51809i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −27.7980 16.0492i −1.68550 0.973123i
\(273\) 0 0
\(274\) 11.6742 + 20.2204i 0.705266 + 1.22156i
\(275\) −16.3763 + 9.45485i −0.987527 + 0.570149i
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 5.16404i 0.309719i
\(279\) 0 0
\(280\) 0 0
\(281\) −24.4949 14.1421i −1.46124 0.843649i −0.462174 0.886789i \(-0.652930\pi\)
−0.999069 + 0.0431402i \(0.986264\pi\)
\(282\) 0 0
\(283\) 11.0000 + 19.0526i 0.653882 + 1.13256i 0.982173 + 0.187980i \(0.0601941\pi\)
−0.328291 + 0.944577i \(0.606473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 12.8990 11.0280i 0.760080 0.649830i
\(289\) −47.3939 −2.78788
\(290\) 0 0
\(291\) −14.2753 + 30.9859i −0.836830 + 1.81642i
\(292\) 13.6969 + 23.7238i 0.801553 + 1.38833i
\(293\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(294\) 1.57321 + 17.0741i 0.0917517 + 0.995782i
\(295\) 0 0
\(296\) 0 0
\(297\) −19.0505 4.82294i −1.10542 0.279855i
\(298\) 0 0
\(299\) 0 0
\(300\) 17.2474 1.58919i 0.995782 0.0917517i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 16.6969 28.9199i 0.957635 1.65867i
\(305\) 0 0
\(306\) 11.3485 32.0983i 0.648749 1.83494i
\(307\) 9.65153 0.550842 0.275421 0.961324i \(-0.411183\pi\)
0.275421 + 0.961324i \(0.411183\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 12.1969 21.1257i 0.689412 1.19410i −0.282617 0.959233i \(-0.591202\pi\)
0.972028 0.234863i \(-0.0754642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −0.747449 8.11207i −0.0417185 0.452771i
\(322\) 0 0
\(323\) 66.9930i 3.72759i
\(324\) 14.0000 + 11.3137i 0.777778 + 0.628539i
\(325\) 0 0
\(326\) −2.44949 1.41421i −0.135665 0.0783260i
\(327\) 0 0
\(328\) 0.651531 + 1.12848i 0.0359748 + 0.0623101i
\(329\) 0 0
\(330\) 0 0
\(331\) −13.0000 + 22.5167i −0.714545 + 1.23763i 0.248590 + 0.968609i \(0.420033\pi\)
−0.963135 + 0.269019i \(0.913301\pi\)
\(332\) 5.65685i 0.310460i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.1969 + 31.5180i 0.991250 + 1.71690i 0.609936 + 0.792451i \(0.291195\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 15.9217 9.19239i 0.866025 0.500000i
\(339\) 16.0000 11.3137i 0.869001 0.614476i
\(340\) 0 0
\(341\) 0 0
\(342\) 33.3939 + 11.8065i 1.80573 + 0.638423i
\(343\) 0 0
\(344\) −5.75255 3.32124i −0.310157 0.179069i
\(345\) 0 0
\(346\) 0 0
\(347\) −11.4217 + 6.59431i −0.613148 + 0.354001i −0.774197 0.632945i \(-0.781846\pi\)
0.161048 + 0.986947i \(0.448512\pi\)
\(348\) 0 0
\(349\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −21.3939 −1.14030
\(353\) 12.7020 + 7.33353i 0.676061 + 0.390324i 0.798369 0.602168i \(-0.205696\pi\)
−0.122308 + 0.992492i \(0.539030\pi\)
\(354\) −29.9217 + 2.75699i −1.59032 + 0.146533i
\(355\) 0 0
\(356\) 9.79796 5.65685i 0.519291 0.299813i
\(357\) 0 0
\(358\) 14.0000 24.2487i 0.739923 1.28158i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 50.6969 2.66826
\(362\) 0 0
\(363\) 3.30306 + 4.67123i 0.173366 + 0.245176i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 0 0
\(369\) −1.05051 + 0.898133i −0.0546874 + 0.0467550i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) −37.1691 + 21.4596i −1.92197 + 1.10965i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −26.3485 −1.35343 −0.676715 0.736245i \(-0.736597\pi\)
−0.676715 + 0.736245i \(0.736597\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 17.7980 + 8.19955i 0.908248 + 0.418432i
\(385\) 0 0
\(386\) 36.3410i 1.84971i
\(387\) 2.34847 6.64247i 0.119379 0.337656i
\(388\) −39.3939 −1.99992
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −17.1464 + 9.89949i −0.866025 + 0.500000i
\(393\) −20.0000 + 14.1421i −1.00887 + 0.713376i
\(394\) 0 0
\(395\) 0 0
\(396\) −4.14643 22.3096i −0.208366 1.12110i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 10.0000 + 17.3205i 0.500000 + 0.866025i
\(401\) 21.6464 12.4976i 1.08097 0.624099i 0.149813 0.988714i \(-0.452133\pi\)
0.931158 + 0.364615i \(0.118800\pi\)
\(402\) −3.22474 34.9982i −0.160836 1.74555i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 39.1464 3.60697i 1.93804 0.178571i
\(409\) 9.19694 + 15.9296i 0.454759 + 0.787666i 0.998674 0.0514740i \(-0.0163919\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −25.9722 11.9654i −1.28111 0.590211i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.65153 5.16404i −0.178816 0.252884i
\(418\) −22.3258 38.6694i −1.09199 1.89138i
\(419\) 31.8434 18.3848i 1.55565 0.898155i 0.557986 0.829851i \(-0.311574\pi\)
0.997665 0.0683046i \(-0.0217590\pi\)
\(420\) 0 0
\(421\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 19.7990i 0.963800i
\(423\) 0 0
\(424\) 0 0
\(425\) 34.7474 + 20.0614i 1.68550 + 0.973123i
\(426\) 0 0
\(427\) 0 0
\(428\) 8.14643 4.70334i 0.393772 0.227345i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) −5.10102 + 20.1489i −0.245423 + 0.969416i
\(433\) −4.30306 −0.206792 −0.103396 0.994640i \(-0.532971\pi\)
−0.103396 + 0.994640i \(0.532971\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −30.4722 14.0386i −1.45602 0.670790i
\(439\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) −13.6464 15.9617i −0.649830 0.760080i
\(442\) 0 0
\(443\) −30.2753 17.4794i −1.43842 0.830473i −0.440681 0.897664i \(-0.645263\pi\)
−0.997740 + 0.0671913i \(0.978596\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 39.2015i 1.85003i 0.379927 + 0.925016i \(0.375949\pi\)
−0.379927 + 0.925016i \(0.624051\pi\)
\(450\) −16.1237 + 13.7850i −0.760080 + 0.649830i
\(451\) 1.74235 0.0820439
\(452\) 19.5959 + 11.3137i 0.921714 + 0.532152i
\(453\) 0 0
\(454\) −19.3712 33.5519i −0.909134 1.57467i
\(455\) 0 0
\(456\) 3.75255 + 40.7265i 0.175729 + 1.90719i
\(457\) 21.1969 36.7142i 0.991551 1.71742i 0.383437 0.923567i \(-0.374740\pi\)
0.608114 0.793849i \(-0.291926\pi\)
\(458\) 0 0
\(459\) 11.3485 + 40.1229i 0.529701 + 1.87278i
\(460\) 0 0
\(461\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −16.3712 + 28.3557i −0.758380 + 1.31355i
\(467\) 10.4244i 0.482384i −0.970477 0.241192i \(-0.922462\pi\)
0.970477 0.241192i \(-0.0775384\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −17.3485 30.0484i −0.798528 1.38309i
\(473\) −7.69184 + 4.44088i −0.353671 + 0.204192i
\(474\) 0 0
\(475\) −20.8712 + 36.1499i −0.957635 + 1.65867i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.07832 1.19992i 0.0946647 0.0546547i
\(483\) 0 0
\(484\) −3.30306 + 5.72107i −0.150139 + 0.260049i
\(485\) 0 0
\(486\) −22.0000 1.41421i −0.997940 0.0641500i
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 3.44949 0.317837i 0.155991 0.0143731i
\(490\) 0 0
\(491\) 37.6237 21.7221i 1.69793 0.980303i 0.750218 0.661190i \(-0.229948\pi\)
0.947717 0.319113i \(-0.103385\pi\)
\(492\) −1.44949 0.667783i −0.0653480 0.0301060i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 4.00000 + 5.65685i 0.179244 + 0.253490i
\(499\) −14.8712 25.7576i −0.665725 1.15307i −0.979088 0.203436i \(-0.934789\pi\)
0.313363 0.949633i \(-0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 14.6742 25.4165i 0.654943 1.13439i
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.42168 + 20.4507i −0.418432 + 0.908248i
\(508\) 0 0
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) −41.7423 + 11.8065i −1.84297 + 0.521271i
\(514\) 44.7423 1.97350
\(515\) 0 0
\(516\) 8.10102 0.746431i 0.356628 0.0328598i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.4313i 0.763678i −0.924229 0.381839i \(-0.875291\pi\)
0.924229 0.381839i \(-0.124709\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) −24.4949 14.1421i −1.07006 0.617802i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 21.3939 15.1278i 0.931049 0.658351i
\(529\) 11.5000 19.9186i 0.500000 0.866025i
\(530\) 0 0
\(531\) 27.9722 23.9148i 1.21389 1.03781i
\(532\) 0 0
\(533\) 0 0
\(534\) −5.79796 + 12.5851i −0.250902 + 0.544609i
\(535\) 0 0
\(536\) 35.1464 20.2918i 1.51809 0.876472i
\(537\) 3.14643 + 34.1482i 0.135778 + 1.47360i
\(538\) 0 0
\(539\) 26.4736i 1.14030i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 22.6969 + 39.3123i 0.973123 + 1.68550i
\(545\) 0 0
\(546\) 0 0
\(547\) −7.82577 + 13.5546i −0.334606 + 0.579554i −0.983409 0.181402i \(-0.941936\pi\)
0.648803 + 0.760956i \(0.275270\pi\)
\(548\) 33.0197i 1.41053i
\(549\) 0 0
\(550\) 26.7423 1.14030
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 3.65153 6.32464i 0.154859 0.268224i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 21.9949 47.7422i 0.928625 2.01568i
\(562\) 20.0000 + 34.6410i 0.843649 + 1.46124i
\(563\) −38.4217 + 22.1828i −1.61928 + 0.934892i −0.632175 + 0.774826i \(0.717837\pi\)
−0.987106 + 0.160066i \(0.948829\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 31.1127i 1.30776i
\(567\) 0 0
\(568\) 0 0
\(569\) −41.2980 23.8434i −1.73130 0.999567i −0.880366 0.474295i \(-0.842703\pi\)
−0.850935 0.525271i \(-0.823964\pi\)
\(570\) 0 0
\(571\) −23.8712 41.3461i −0.998978 1.73028i −0.538642 0.842535i \(-0.681062\pi\)
−0.460336 0.887745i \(-0.652271\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −23.5959 + 4.38551i −0.983163 + 0.182729i
\(577\) −12.3939 −0.515964 −0.257982 0.966150i \(-0.583058\pi\)
−0.257982 + 0.966150i \(0.583058\pi\)
\(578\) 58.0454 + 33.5125i 2.41437 + 1.39394i
\(579\) −25.6969 36.3410i −1.06793 1.51028i
\(580\) 0 0
\(581\) 0 0
\(582\) 39.3939 27.8557i 1.63293 1.15465i
\(583\) 0 0
\(584\) 38.7408i 1.60311i
\(585\) 0 0
\(586\) 0 0
\(587\) −25.3207 14.6189i −1.04510 0.603386i −0.123823 0.992304i \(-0.539516\pi\)
−0.921272 + 0.388918i \(0.872849\pi\)
\(588\) 10.1464 22.0239i 0.418432 0.908248i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 45.2548i 1.85839i 0.369586 + 0.929197i \(0.379500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 19.9217 + 19.3776i 0.817397 + 0.795073i
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) −22.2474 10.2494i −0.908248 0.418432i
\(601\) −18.8485 + 32.6465i −0.768845 + 1.33168i 0.169344 + 0.985557i \(0.445835\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) 27.9722 + 32.7179i 1.13912 + 1.33238i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) −40.8990 + 23.6130i −1.65867 + 0.957635i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −36.5959 + 31.2877i −1.47930 + 1.26473i
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −11.8207 6.82466i −0.477043 0.275421i
\(615\) 0 0
\(616\) 0 0
\(617\) −5.35357 + 3.09089i −0.215527 + 0.124434i −0.603877 0.797077i \(-0.706378\pi\)
0.388351 + 0.921512i \(0.373045\pi\)
\(618\) 0 0
\(619\) −11.8712 + 20.5615i −0.477143 + 0.826435i −0.999657 0.0261952i \(-0.991661\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) −29.8763 + 17.2491i −1.19410 + 0.689412i
\(627\) 49.6691 + 22.8827i 1.98359 + 0.913845i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −14.0000 19.7990i −0.556450 0.786939i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −19.2526 11.1155i −0.760430 0.439034i 0.0690201 0.997615i \(-0.478013\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) −4.82066 + 10.4637i −0.190256 + 0.412971i
\(643\) 16.1742 + 28.0146i 0.637850 + 1.10479i 0.985904 + 0.167313i \(0.0535092\pi\)
−0.348054 + 0.937474i \(0.613157\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −47.3712 + 82.0493i −1.86379 + 3.22819i
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −9.14643 23.7559i −0.359306 0.933220i
\(649\) −46.3939 −1.82112
\(650\) 0 0
\(651\) 0 0
\(652\) 2.00000 + 3.46410i 0.0783260 + 0.135665i
\(653\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.84281i 0.0719495i
\(657\) 40.3990 7.50850i 1.57611 0.292934i
\(658\) 0 0
\(659\) 41.6413 + 24.0416i 1.62212 + 0.936529i 0.986353 + 0.164644i \(0.0526477\pi\)
0.635763 + 0.771885i \(0.280686\pi\)
\(660\) 0 0
\(661\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 31.8434 18.3848i 1.23763 0.714545i
\(663\) 0 0
\(664\) −4.00000 + 6.92820i −0.155230 + 0.268866i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.00000 8.66025i 0.192736 0.333828i −0.753420 0.657539i \(-0.771597\pi\)
0.946156 + 0.323711i \(0.104931\pi\)
\(674\) 51.4687i 1.98250i
\(675\) 6.37628 25.1862i 0.245423 0.969416i
\(676\) −26.0000 −1.00000
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) −27.5959 + 2.54270i −1.05981 + 0.0976517i
\(679\) 0 0
\(680\) 0 0
\(681\) 43.0959 + 19.8544i 1.65144 + 0.760821i
\(682\) 0 0
\(683\) 51.9294i 1.98702i 0.113728 + 0.993512i \(0.463721\pi\)
−0.113728 + 0.993512i \(0.536279\pi\)
\(684\) −32.5505 38.0730i −1.24460 1.45576i
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.69694 + 8.13534i 0.179069 + 0.310157i
\(689\) 0 0
\(690\) 0 0
\(691\) 23.0000 39.8372i 0.874961 1.51548i 0.0181572 0.999835i \(-0.494220\pi\)
0.856804 0.515642i \(-0.172447\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 18.6515 0.708002
\(695\) 0 0
\(696\) 0 0
\(697\) −1.84847 3.20164i −0.0700158 0.121271i
\(698\) 0 0
\(699\) −3.67934 39.9319i −0.139165 1.51036i
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 26.2020 + 15.1278i 0.987527 + 0.570149i
\(705\) 0 0
\(706\) −10.3712 17.9634i −0.390324 0.676061i
\(707\) 0 0
\(708\) 38.5959 + 17.7812i 1.45052 + 0.668259i
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −16.0000 −0.599625
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −34.2929 + 19.7990i −1.28158 + 0.739923i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −62.0908 35.8481i −2.31078 1.33413i
\(723\) −1.22985 + 2.66951i −0.0457385 + 0.0992801i
\(724\) 0 0
\(725\) 0 0
\(726\) −0.742346 8.05669i −0.0275510 0.299012i
\(727\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 23.0000 14.1421i 0.851852 0.523783i
\(730\) 0 0
\(731\) 16.3207 + 9.42274i 0.603642 + 0.348513i
\(732\) 0 0
\(733\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 54.2650i 1.99888i
\(738\) 1.92168 0.357161i 0.0707381 0.0131473i
\(739\) 53.7423 1.97694 0.988472 0.151403i \(-0.0483792\pi\)
0.988472 + 0.151403i \(0.0483792\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.00000 2.82843i −0.292705 0.103487i
\(748\) 60.6969 2.21930
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) 0 0
\(753\) 3.29796 + 35.7928i 0.120184 + 1.30436i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 32.2702 + 18.6312i 1.17210 + 0.676715i
\(759\) 0 0
\(760\) 0 0
\(761\) 9.79796 5.65685i 0.355176 0.205061i −0.311787 0.950152i \(-0.600927\pi\)
0.666962 + 0.745091i \(0.267594\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 22.6274i −0.577350 0.816497i
\(769\) 11.0000 + 19.0526i 0.396670 + 0.687053i 0.993313 0.115454i \(-0.0368323\pi\)
−0.596643 + 0.802507i \(0.703499\pi\)
\(770\) 0 0
\(771\) −44.7423 + 31.6376i −1.61136 + 1.13940i
\(772\) 25.6969 44.5084i 0.924853 1.60189i
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) −7.57321 + 6.47472i −0.272214 + 0.232729i
\(775\) 0 0
\(776\) 48.2474 + 27.8557i 1.73198 + 0.999961i
\(777\) 0 0
\(778\) 0 0
\(779\) 3.33087 1.92308i 0.119341 0.0689014i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 0 0
\(786\) 34.4949 3.17837i 1.23039 0.113369i
\(787\) −25.0000 43.3013i −0.891154 1.54352i −0.838494 0.544911i \(-0.816563\pi\)
−0.0526599 0.998613i \(-0.516770\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −10.6969 + 30.2555i −0.380099 + 1.07508i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 28.2843i 1.00000i
\(801\) −3.10102 16.6848i −0.109569 0.589530i
\(802\) −35.3485 −1.24820
\(803\) −44.8610 25.9005i −1.58311 0.914009i
\(804\) −20.7980 + 45.1441i −0.733487 + 1.59211i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 33.4804i 1.17711i 0.808458 + 0.588555i \(0.200303\pi\)
−0.808458 + 0.588555i \(0.799697\pi\)
\(810\) 0 0
\(811\) 17.7423 0.623018 0.311509 0.950243i \(-0.399166\pi\)
0.311509 + 0.950243i \(0.399166\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −50.4949 23.2631i −1.76767 0.814371i
\(817\) −9.80306 + 16.9794i −0.342966 + 0.594034i
\(818\) 26.0129i 0.909519i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 23.3485 + 33.0197i 0.814371 + 1.15170i
\(823\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) 0 0
\(825\) −26.7423 + 18.9097i −0.931049 + 0.658351i
\(826\) 0 0
\(827\) 19.7990i 0.688478i 0.938882 + 0.344239i \(0.111863\pi\)
−0.938882 + 0.344239i \(0.888137\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 48.6464 28.0860i 1.68550 0.973123i
\(834\) 0.820663 + 8.90666i 0.0284172 + 0.308412i
\(835\) 0 0
\(836\) 63.1468i 2.18398i
\(837\) 0 0
\(838\) −52.0000 −1.79631
\(839\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(840\) 0 0
\(841\) 14.5000 + 25.1147i 0.500000 + 0.866025i
\(842\) 0 0
\(843\) −44.4949 20.4989i −1.53249 0.706019i
\(844\) 14.0000 24.2487i 0.481900 0.834675i
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 22.0000 + 31.1127i 0.755038 + 1.06779i
\(850\) −28.3712 49.1403i −0.973123 1.68550i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −13.3031 −0.454689
\(857\) 19.5959 + 11.3137i 0.669384 + 0.386469i 0.795843 0.605503i \(-0.207028\pi\)
−0.126459 + 0.991972i \(0.540361\pi\)
\(858\) 0 0
\(859\) −10.8258 18.7508i −0.369370 0.639768i 0.620097 0.784525i \(-0.287093\pi\)
−0.989467 + 0.144757i \(0.953760\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 20.4949 21.0703i 0.697251 0.716827i
\(865\) 0 0
\(866\) 5.27015 + 3.04272i 0.179087 + 0.103396i
\(867\) −81.7423 + 7.53177i −2.77612 + 0.255792i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −19.6969 + 55.7114i −0.666640 + 1.88554i
\(874\) 0 0
\(875\) 0 0
\(876\) 27.3939 + 38.7408i 0.925553 + 1.30893i
\(877\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 56.5685i 1.90584i −0.303218 0.952921i \(-0.598061\pi\)
0.303218 0.952921i \(-0.401939\pi\)
\(882\) 5.42679 + 29.1985i 0.182729 + 0.983163i
\(883\) −52.4393 −1.76472 −0.882361 0.470573i \(-0.844047\pi\)
−0.882361 + 0.470573i \(0.844047\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.7196 + 42.8157i 0.830473 + 1.43842i
\(887\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −33.6237 5.29085i −1.12644 0.177250i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 27.7196 48.0118i 0.925016 1.60218i
\(899\) 0 0
\(900\) 29.4949 5.48188i 0.983163 0.182729i
\(901\) 0 0
\(902\) −2.13393 1.23202i −0.0710521 0.0410219i
\(903\) 0 0
\(904\) −16.0000 27.7128i −0.532152 0.921714i
\(905\) 0 0
\(906\) 0 0
\(907\) 23.2196 40.2176i 0.770996 1.33540i −0.166022 0.986122i \(-0.553092\pi\)
0.937018 0.349281i \(-0.113574\pi\)
\(908\) 54.7900i 1.81827i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(912\) 24.2020 52.5330i 0.801410 1.73954i
\(913\) 5.34847 + 9.26382i 0.177008 + 0.306588i
\(914\) −51.9217 + 29.9770i −1.71742 + 0.991551i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 14.4722 57.1649i 0.477654 1.88672i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 16.6464 1.53381i 0.548518 0.0505407i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24.4949 14.1421i −0.803652 0.463988i 0.0410949 0.999155i \(-0.486915\pi\)
−0.844746 + 0.535167i \(0.820249\pi\)
\(930\) 0 0
\(931\) 29.2196 + 50.6099i 0.957635 + 1.65867i
\(932\) 40.1010 23.1523i 1.31355 0.758380i
\(933\) 0 0
\(934\) −7.37117 + 12.7672i −0.241192 + 0.417757i
\(935\) 0 0
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) 17.6793 38.3748i 0.576943 1.25231i
\(940\) 0 0
\(941\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 49.0689i 1.59706i
\(945\) 0 0
\(946\) 12.5607 0.408384
\(947\) 45.7702 + 26.4254i 1.48733 + 0.858710i 0.999896 0.0144491i \(-0.00459946\pi\)
0.487435 + 0.873160i \(0.337933\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 51.1237 29.5163i 1.65867 0.957635i
\(951\) 0 0
\(952\) 0 0
\(953\) 13.7456i 0.445265i 0.974902 + 0.222633i \(0.0714650\pi\)
−0.974902 + 0.222633i \(0.928535\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −15.5000 + 26.8468i −0.500000 + 0.866025i
\(962\) 0 0
\(963\) −2.57832 13.8725i −0.0830851 0.447034i
\(964\) −3.39388 −0.109309
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 8.09082 4.67123i 0.260049 0.150139i
\(969\) −10.6464 115.546i −0.342013 3.71186i
\(970\) 0 0
\(971\) 31.1127i 0.998454i −0.866471 0.499227i \(-0.833617\pi\)
0.866471 0.499227i \(-0.166383\pi\)
\(972\) 25.9444 + 17.2884i 0.832167 + 0.554526i
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.4444 + 12.9583i −0.718060 + 0.414572i −0.814038 0.580812i \(-0.802735\pi\)
0.0959785 + 0.995383i \(0.469402\pi\)
\(978\) −4.44949 2.04989i −0.142279 0.0655482i
\(979\) −10.6969 + 18.5276i −0.341876 + 0.592146i
\(980\) 0 0
\(981\) 0 0
\(982\) −61.4393 −1.96061
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 1.30306 + 1.84281i 0.0415401 + 0.0587466i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −18.8434 + 40.9014i −0.597976 + 1.29797i
\(994\) 0 0
\(995\) 0 0
\(996\) −0.898979 9.75663i −0.0284853 0.309151i
\(997\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 42.0620i 1.33145i
\(999\) 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 72.2.l.a.11.1 4
3.2 odd 2 216.2.l.a.35.2 4
4.3 odd 2 288.2.p.a.47.1 4
8.3 odd 2 CM 72.2.l.a.11.1 4
8.5 even 2 288.2.p.a.47.1 4
9.2 odd 6 648.2.f.a.323.1 4
9.4 even 3 216.2.l.a.179.2 4
9.5 odd 6 inner 72.2.l.a.59.1 yes 4
9.7 even 3 648.2.f.a.323.4 4
12.11 even 2 864.2.p.a.143.2 4
24.5 odd 2 864.2.p.a.143.2 4
24.11 even 2 216.2.l.a.35.2 4
36.7 odd 6 2592.2.f.a.1295.2 4
36.11 even 6 2592.2.f.a.1295.3 4
36.23 even 6 288.2.p.a.239.1 4
36.31 odd 6 864.2.p.a.719.2 4
72.5 odd 6 288.2.p.a.239.1 4
72.11 even 6 648.2.f.a.323.1 4
72.13 even 6 864.2.p.a.719.2 4
72.29 odd 6 2592.2.f.a.1295.3 4
72.43 odd 6 648.2.f.a.323.4 4
72.59 even 6 inner 72.2.l.a.59.1 yes 4
72.61 even 6 2592.2.f.a.1295.2 4
72.67 odd 6 216.2.l.a.179.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.2.l.a.11.1 4 1.1 even 1 trivial
72.2.l.a.11.1 4 8.3 odd 2 CM
72.2.l.a.59.1 yes 4 9.5 odd 6 inner
72.2.l.a.59.1 yes 4 72.59 even 6 inner
216.2.l.a.35.2 4 3.2 odd 2
216.2.l.a.35.2 4 24.11 even 2
216.2.l.a.179.2 4 9.4 even 3
216.2.l.a.179.2 4 72.67 odd 6
288.2.p.a.47.1 4 4.3 odd 2
288.2.p.a.47.1 4 8.5 even 2
288.2.p.a.239.1 4 36.23 even 6
288.2.p.a.239.1 4 72.5 odd 6
648.2.f.a.323.1 4 9.2 odd 6
648.2.f.a.323.1 4 72.11 even 6
648.2.f.a.323.4 4 9.7 even 3
648.2.f.a.323.4 4 72.43 odd 6
864.2.p.a.143.2 4 12.11 even 2
864.2.p.a.143.2 4 24.5 odd 2
864.2.p.a.719.2 4 36.31 odd 6
864.2.p.a.719.2 4 72.13 even 6
2592.2.f.a.1295.2 4 36.7 odd 6
2592.2.f.a.1295.2 4 72.61 even 6
2592.2.f.a.1295.3 4 36.11 even 6
2592.2.f.a.1295.3 4 72.29 odd 6