Properties

Label 60.6.d.a.49.5
Level $60$
Weight $6$
Character 60.49
Analytic conductor $9.623$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,6,Mod(49,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.d (of order \(2\), degree \(1\), 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: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 373x^{4} + 33732x^{2} + 186624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4}\cdot 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.5
Root \(15.1546i\) of defining polynomial
Character \(\chi\) \(=\) 60.49
Dual form 60.6.d.a.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.00000i q^{3} +(-20.3651 - 52.0602i) q^{5} +121.510i q^{7} -81.0000 q^{9} +O(q^{10})\) \(q+9.00000i q^{3} +(-20.3651 - 52.0602i) q^{5} +121.510i q^{7} -81.0000 q^{9} -699.623 q^{11} -436.743i q^{13} +(468.542 - 183.286i) q^{15} -1610.09i q^{17} -2778.55 q^{19} -1093.59 q^{21} +1469.98i q^{23} +(-2295.52 + 2120.42i) q^{25} -729.000i q^{27} -2233.05 q^{29} +4632.47 q^{31} -6296.60i q^{33} +(6325.85 - 2474.58i) q^{35} +5109.62i q^{37} +3930.68 q^{39} +7683.61 q^{41} +20417.9i q^{43} +(1649.58 + 4216.87i) q^{45} -21018.3i q^{47} +2042.21 q^{49} +14490.8 q^{51} +6033.60i q^{53} +(14247.9 + 36422.5i) q^{55} -25006.9i q^{57} +27762.7 q^{59} -48757.3 q^{61} -9842.35i q^{63} +(-22736.9 + 8894.32i) q^{65} -11289.5i q^{67} -13229.9 q^{69} -15381.5 q^{71} +24443.1i q^{73} +(-19083.8 - 20659.7i) q^{75} -85011.4i q^{77} -23718.9 q^{79} +6561.00 q^{81} -18970.7i q^{83} +(-83821.4 + 32789.6i) q^{85} -20097.4i q^{87} -57532.2 q^{89} +53068.8 q^{91} +41692.2i q^{93} +(56585.5 + 144652. i) q^{95} -173854. i q^{97} +56669.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 38 q^{5} - 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 38 q^{5} - 486 q^{9} + 296 q^{11} + 396 q^{15} - 6000 q^{19} + 1584 q^{21} + 6054 q^{25} - 15924 q^{29} + 264 q^{31} + 20096 q^{35} + 16340 q^{41} + 3078 q^{45} - 27654 q^{49} + 15624 q^{51} - 26088 q^{55} + 92456 q^{59} + 6252 q^{61} - 52440 q^{65} + 18936 q^{69} - 160800 q^{71} - 29448 q^{75} + 128952 q^{79} + 39366 q^{81} - 177864 q^{85} + 76060 q^{89} - 98400 q^{91} + 232800 q^{95} - 23976 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 9.00000i 0.577350i
\(4\) 0 0
\(5\) −20.3651 52.0602i −0.364302 0.931281i
\(6\) 0 0
\(7\) 121.510i 0.937278i 0.883390 + 0.468639i \(0.155256\pi\)
−0.883390 + 0.468639i \(0.844744\pi\)
\(8\) 0 0
\(9\) −81.0000 −0.333333
\(10\) 0 0
\(11\) −699.623 −1.74334 −0.871670 0.490093i \(-0.836963\pi\)
−0.871670 + 0.490093i \(0.836963\pi\)
\(12\) 0 0
\(13\) 436.743i 0.716749i −0.933578 0.358375i \(-0.883331\pi\)
0.933578 0.358375i \(-0.116669\pi\)
\(14\) 0 0
\(15\) 468.542 183.286i 0.537675 0.210330i
\(16\) 0 0
\(17\) 1610.09i 1.35122i −0.737258 0.675611i \(-0.763880\pi\)
0.737258 0.675611i \(-0.236120\pi\)
\(18\) 0 0
\(19\) −2778.55 −1.76577 −0.882884 0.469590i \(-0.844402\pi\)
−0.882884 + 0.469590i \(0.844402\pi\)
\(20\) 0 0
\(21\) −1093.59 −0.541138
\(22\) 0 0
\(23\) 1469.98i 0.579420i 0.957115 + 0.289710i \(0.0935588\pi\)
−0.957115 + 0.289710i \(0.906441\pi\)
\(24\) 0 0
\(25\) −2295.52 + 2120.42i −0.734567 + 0.678536i
\(26\) 0 0
\(27\) 729.000i 0.192450i
\(28\) 0 0
\(29\) −2233.05 −0.493063 −0.246532 0.969135i \(-0.579291\pi\)
−0.246532 + 0.969135i \(0.579291\pi\)
\(30\) 0 0
\(31\) 4632.47 0.865781 0.432891 0.901446i \(-0.357494\pi\)
0.432891 + 0.901446i \(0.357494\pi\)
\(32\) 0 0
\(33\) 6296.60i 1.00652i
\(34\) 0 0
\(35\) 6325.85 2474.58i 0.872869 0.341453i
\(36\) 0 0
\(37\) 5109.62i 0.613598i 0.951774 + 0.306799i \(0.0992579\pi\)
−0.951774 + 0.306799i \(0.900742\pi\)
\(38\) 0 0
\(39\) 3930.68 0.413815
\(40\) 0 0
\(41\) 7683.61 0.713848 0.356924 0.934133i \(-0.383826\pi\)
0.356924 + 0.934133i \(0.383826\pi\)
\(42\) 0 0
\(43\) 20417.9i 1.68399i 0.539484 + 0.841996i \(0.318619\pi\)
−0.539484 + 0.841996i \(0.681381\pi\)
\(44\) 0 0
\(45\) 1649.58 + 4216.87i 0.121434 + 0.310427i
\(46\) 0 0
\(47\) 21018.3i 1.38788i −0.720030 0.693942i \(-0.755872\pi\)
0.720030 0.693942i \(-0.244128\pi\)
\(48\) 0 0
\(49\) 2042.21 0.121510
\(50\) 0 0
\(51\) 14490.8 0.780129
\(52\) 0 0
\(53\) 6033.60i 0.295044i 0.989059 + 0.147522i \(0.0471297\pi\)
−0.989059 + 0.147522i \(0.952870\pi\)
\(54\) 0 0
\(55\) 14247.9 + 36422.5i 0.635103 + 1.62354i
\(56\) 0 0
\(57\) 25006.9i 1.01947i
\(58\) 0 0
\(59\) 27762.7 1.03832 0.519160 0.854677i \(-0.326245\pi\)
0.519160 + 0.854677i \(0.326245\pi\)
\(60\) 0 0
\(61\) −48757.3 −1.67770 −0.838852 0.544359i \(-0.816773\pi\)
−0.838852 + 0.544359i \(0.816773\pi\)
\(62\) 0 0
\(63\) 9842.35i 0.312426i
\(64\) 0 0
\(65\) −22736.9 + 8894.32i −0.667495 + 0.261113i
\(66\) 0 0
\(67\) 11289.5i 0.307246i −0.988130 0.153623i \(-0.950906\pi\)
0.988130 0.153623i \(-0.0490942\pi\)
\(68\) 0 0
\(69\) −13229.9 −0.334528
\(70\) 0 0
\(71\) −15381.5 −0.362121 −0.181061 0.983472i \(-0.557953\pi\)
−0.181061 + 0.983472i \(0.557953\pi\)
\(72\) 0 0
\(73\) 24443.1i 0.536845i 0.963301 + 0.268422i \(0.0865023\pi\)
−0.963301 + 0.268422i \(0.913498\pi\)
\(74\) 0 0
\(75\) −19083.8 20659.7i −0.391753 0.424103i
\(76\) 0 0
\(77\) 85011.4i 1.63399i
\(78\) 0 0
\(79\) −23718.9 −0.427589 −0.213794 0.976879i \(-0.568582\pi\)
−0.213794 + 0.976879i \(0.568582\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) 18970.7i 0.302266i −0.988513 0.151133i \(-0.951708\pi\)
0.988513 0.151133i \(-0.0482921\pi\)
\(84\) 0 0
\(85\) −83821.4 + 32789.6i −1.25837 + 0.492254i
\(86\) 0 0
\(87\) 20097.4i 0.284670i
\(88\) 0 0
\(89\) −57532.2 −0.769902 −0.384951 0.922937i \(-0.625782\pi\)
−0.384951 + 0.922937i \(0.625782\pi\)
\(90\) 0 0
\(91\) 53068.8 0.671793
\(92\) 0 0
\(93\) 41692.2i 0.499859i
\(94\) 0 0
\(95\) 56585.5 + 144652.i 0.643274 + 1.64443i
\(96\) 0 0
\(97\) 173854.i 1.87610i −0.346499 0.938050i \(-0.612629\pi\)
0.346499 0.938050i \(-0.387371\pi\)
\(98\) 0 0
\(99\) 56669.4 0.581113
\(100\) 0 0
\(101\) −77773.0 −0.758622 −0.379311 0.925269i \(-0.623839\pi\)
−0.379311 + 0.925269i \(0.623839\pi\)
\(102\) 0 0
\(103\) 123836.i 1.15015i 0.818102 + 0.575073i \(0.195026\pi\)
−0.818102 + 0.575073i \(0.804974\pi\)
\(104\) 0 0
\(105\) 22271.2 + 56932.7i 0.197138 + 0.503951i
\(106\) 0 0
\(107\) 92949.4i 0.784851i −0.919784 0.392426i \(-0.871636\pi\)
0.919784 0.392426i \(-0.128364\pi\)
\(108\) 0 0
\(109\) 53772.7 0.433506 0.216753 0.976226i \(-0.430453\pi\)
0.216753 + 0.976226i \(0.430453\pi\)
\(110\) 0 0
\(111\) −45986.5 −0.354261
\(112\) 0 0
\(113\) 37540.9i 0.276572i 0.990392 + 0.138286i \(0.0441594\pi\)
−0.990392 + 0.138286i \(0.955841\pi\)
\(114\) 0 0
\(115\) 76527.6 29936.4i 0.539602 0.211084i
\(116\) 0 0
\(117\) 35376.1i 0.238916i
\(118\) 0 0
\(119\) 195642. 1.26647
\(120\) 0 0
\(121\) 328421. 2.03923
\(122\) 0 0
\(123\) 69152.5i 0.412140i
\(124\) 0 0
\(125\) 157138. + 76322.6i 0.899512 + 0.436896i
\(126\) 0 0
\(127\) 154211.i 0.848409i −0.905566 0.424204i \(-0.860554\pi\)
0.905566 0.424204i \(-0.139446\pi\)
\(128\) 0 0
\(129\) −183761. −0.972253
\(130\) 0 0
\(131\) −136783. −0.696391 −0.348196 0.937422i \(-0.613206\pi\)
−0.348196 + 0.937422i \(0.613206\pi\)
\(132\) 0 0
\(133\) 337623.i 1.65502i
\(134\) 0 0
\(135\) −37951.9 + 14846.2i −0.179225 + 0.0701100i
\(136\) 0 0
\(137\) 303547.i 1.38173i 0.722982 + 0.690867i \(0.242771\pi\)
−0.722982 + 0.690867i \(0.757229\pi\)
\(138\) 0 0
\(139\) −139766. −0.613572 −0.306786 0.951779i \(-0.599254\pi\)
−0.306786 + 0.951779i \(0.599254\pi\)
\(140\) 0 0
\(141\) 189165. 0.801296
\(142\) 0 0
\(143\) 305555.i 1.24954i
\(144\) 0 0
\(145\) 45476.3 + 116253.i 0.179624 + 0.459180i
\(146\) 0 0
\(147\) 18379.9i 0.0701537i
\(148\) 0 0
\(149\) −160751. −0.593184 −0.296592 0.955004i \(-0.595850\pi\)
−0.296592 + 0.955004i \(0.595850\pi\)
\(150\) 0 0
\(151\) −274592. −0.980045 −0.490022 0.871710i \(-0.663011\pi\)
−0.490022 + 0.871710i \(0.663011\pi\)
\(152\) 0 0
\(153\) 130417.i 0.450408i
\(154\) 0 0
\(155\) −94340.8 241167.i −0.315406 0.806285i
\(156\) 0 0
\(157\) 333596.i 1.08012i −0.841627 0.540059i \(-0.818402\pi\)
0.841627 0.540059i \(-0.181598\pi\)
\(158\) 0 0
\(159\) −54302.4 −0.170344
\(160\) 0 0
\(161\) −178618. −0.543077
\(162\) 0 0
\(163\) 159791.i 0.471067i −0.971866 0.235533i \(-0.924316\pi\)
0.971866 0.235533i \(-0.0756837\pi\)
\(164\) 0 0
\(165\) −327802. + 128231.i −0.937351 + 0.366677i
\(166\) 0 0
\(167\) 549868.i 1.52569i −0.646579 0.762847i \(-0.723801\pi\)
0.646579 0.762847i \(-0.276199\pi\)
\(168\) 0 0
\(169\) 180549. 0.486271
\(170\) 0 0
\(171\) 225062. 0.588590
\(172\) 0 0
\(173\) 85278.5i 0.216633i −0.994116 0.108316i \(-0.965454\pi\)
0.994116 0.108316i \(-0.0345459\pi\)
\(174\) 0 0
\(175\) −257654. 278930.i −0.635977 0.688494i
\(176\) 0 0
\(177\) 249864.i 0.599474i
\(178\) 0 0
\(179\) −378124. −0.882066 −0.441033 0.897491i \(-0.645388\pi\)
−0.441033 + 0.897491i \(0.645388\pi\)
\(180\) 0 0
\(181\) 74902.5 0.169942 0.0849708 0.996383i \(-0.472920\pi\)
0.0849708 + 0.996383i \(0.472920\pi\)
\(182\) 0 0
\(183\) 438816.i 0.968623i
\(184\) 0 0
\(185\) 266007. 104058.i 0.571432 0.223535i
\(186\) 0 0
\(187\) 1.12645e6i 2.35564i
\(188\) 0 0
\(189\) 88581.1 0.180379
\(190\) 0 0
\(191\) −810880. −1.60832 −0.804161 0.594412i \(-0.797385\pi\)
−0.804161 + 0.594412i \(0.797385\pi\)
\(192\) 0 0
\(193\) 120513.i 0.232884i −0.993197 0.116442i \(-0.962851\pi\)
0.993197 0.116442i \(-0.0371489\pi\)
\(194\) 0 0
\(195\) −80048.9 204632.i −0.150754 0.385378i
\(196\) 0 0
\(197\) 1.04311e6i 1.91499i 0.288456 + 0.957493i \(0.406858\pi\)
−0.288456 + 0.957493i \(0.593142\pi\)
\(198\) 0 0
\(199\) −87715.4 −0.157016 −0.0785079 0.996913i \(-0.525016\pi\)
−0.0785079 + 0.996913i \(0.525016\pi\)
\(200\) 0 0
\(201\) 101605. 0.177389
\(202\) 0 0
\(203\) 271338.i 0.462137i
\(204\) 0 0
\(205\) −156478. 400010.i −0.260056 0.664793i
\(206\) 0 0
\(207\) 119069.i 0.193140i
\(208\) 0 0
\(209\) 1.94393e6 3.07834
\(210\) 0 0
\(211\) 530246. 0.819919 0.409959 0.912104i \(-0.365543\pi\)
0.409959 + 0.912104i \(0.365543\pi\)
\(212\) 0 0
\(213\) 138434.i 0.209071i
\(214\) 0 0
\(215\) 1.06296e6 415813.i 1.56827 0.613482i
\(216\) 0 0
\(217\) 562893.i 0.811478i
\(218\) 0 0
\(219\) −219988. −0.309948
\(220\) 0 0
\(221\) −703193. −0.968488
\(222\) 0 0
\(223\) 1.27008e6i 1.71028i 0.518397 + 0.855140i \(0.326529\pi\)
−0.518397 + 0.855140i \(0.673471\pi\)
\(224\) 0 0
\(225\) 185937. 171754.i 0.244856 0.226179i
\(226\) 0 0
\(227\) 127005.i 0.163590i 0.996649 + 0.0817948i \(0.0260652\pi\)
−0.996649 + 0.0817948i \(0.973935\pi\)
\(228\) 0 0
\(229\) −548288. −0.690908 −0.345454 0.938436i \(-0.612275\pi\)
−0.345454 + 0.938436i \(0.612275\pi\)
\(230\) 0 0
\(231\) 765103. 0.943387
\(232\) 0 0
\(233\) 794504.i 0.958752i −0.877610 0.479376i \(-0.840863\pi\)
0.877610 0.479376i \(-0.159137\pi\)
\(234\) 0 0
\(235\) −1.09422e6 + 428041.i −1.29251 + 0.505610i
\(236\) 0 0
\(237\) 213470.i 0.246868i
\(238\) 0 0
\(239\) 584107. 0.661451 0.330725 0.943727i \(-0.392707\pi\)
0.330725 + 0.943727i \(0.392707\pi\)
\(240\) 0 0
\(241\) 1.64168e6 1.82073 0.910365 0.413805i \(-0.135801\pi\)
0.910365 + 0.413805i \(0.135801\pi\)
\(242\) 0 0
\(243\) 59049.0i 0.0641500i
\(244\) 0 0
\(245\) −41589.9 106318.i −0.0442663 0.113160i
\(246\) 0 0
\(247\) 1.21351e6i 1.26561i
\(248\) 0 0
\(249\) 170737. 0.174513
\(250\) 0 0
\(251\) −569849. −0.570920 −0.285460 0.958391i \(-0.592146\pi\)
−0.285460 + 0.958391i \(0.592146\pi\)
\(252\) 0 0
\(253\) 1.02843e6i 1.01013i
\(254\) 0 0
\(255\) −295107. 754393.i −0.284203 0.726519i
\(256\) 0 0
\(257\) 499302.i 0.471553i 0.971807 + 0.235776i \(0.0757633\pi\)
−0.971807 + 0.235776i \(0.924237\pi\)
\(258\) 0 0
\(259\) −620872. −0.575112
\(260\) 0 0
\(261\) 180877. 0.164354
\(262\) 0 0
\(263\) 716870.i 0.639074i −0.947574 0.319537i \(-0.896473\pi\)
0.947574 0.319537i \(-0.103527\pi\)
\(264\) 0 0
\(265\) 314110. 122875.i 0.274769 0.107485i
\(266\) 0 0
\(267\) 517789.i 0.444503i
\(268\) 0 0
\(269\) −2.19836e6 −1.85233 −0.926163 0.377124i \(-0.876913\pi\)
−0.926163 + 0.377124i \(0.876913\pi\)
\(270\) 0 0
\(271\) −253384. −0.209583 −0.104792 0.994494i \(-0.533418\pi\)
−0.104792 + 0.994494i \(0.533418\pi\)
\(272\) 0 0
\(273\) 477619.i 0.387860i
\(274\) 0 0
\(275\) 1.60600e6 1.48350e6i 1.28060 1.18292i
\(276\) 0 0
\(277\) 1.15596e6i 0.905199i 0.891714 + 0.452600i \(0.149503\pi\)
−0.891714 + 0.452600i \(0.850497\pi\)
\(278\) 0 0
\(279\) −375230. −0.288594
\(280\) 0 0
\(281\) −2.37453e6 −1.79396 −0.896979 0.442074i \(-0.854243\pi\)
−0.896979 + 0.442074i \(0.854243\pi\)
\(282\) 0 0
\(283\) 1.56745e6i 1.16340i 0.813405 + 0.581698i \(0.197611\pi\)
−0.813405 + 0.581698i \(0.802389\pi\)
\(284\) 0 0
\(285\) −1.30187e6 + 509269.i −0.949410 + 0.371394i
\(286\) 0 0
\(287\) 933639.i 0.669074i
\(288\) 0 0
\(289\) −1.17252e6 −0.825803
\(290\) 0 0
\(291\) 1.56469e6 1.08317
\(292\) 0 0
\(293\) 1.74262e6i 1.18586i −0.805254 0.592930i \(-0.797971\pi\)
0.805254 0.592930i \(-0.202029\pi\)
\(294\) 0 0
\(295\) −565390. 1.44533e6i −0.378262 0.966967i
\(296\) 0 0
\(297\) 510025.i 0.335506i
\(298\) 0 0
\(299\) 642005. 0.415298
\(300\) 0 0
\(301\) −2.48099e6 −1.57837
\(302\) 0 0
\(303\) 699957.i 0.437991i
\(304\) 0 0
\(305\) 992949. + 2.53832e6i 0.611192 + 1.56241i
\(306\) 0 0
\(307\) 3.02826e6i 1.83378i −0.399137 0.916891i \(-0.630690\pi\)
0.399137 0.916891i \(-0.369310\pi\)
\(308\) 0 0
\(309\) −1.11452e6 −0.664037
\(310\) 0 0
\(311\) 139928. 0.0820356 0.0410178 0.999158i \(-0.486940\pi\)
0.0410178 + 0.999158i \(0.486940\pi\)
\(312\) 0 0
\(313\) 2.76332e6i 1.59430i −0.603782 0.797149i \(-0.706340\pi\)
0.603782 0.797149i \(-0.293660\pi\)
\(314\) 0 0
\(315\) −512394. + 200441.i −0.290956 + 0.113818i
\(316\) 0 0
\(317\) 1.81991e6i 1.01719i −0.861006 0.508595i \(-0.830165\pi\)
0.861006 0.508595i \(-0.169835\pi\)
\(318\) 0 0
\(319\) 1.56229e6 0.859577
\(320\) 0 0
\(321\) 836545. 0.453134
\(322\) 0 0
\(323\) 4.47370e6i 2.38595i
\(324\) 0 0
\(325\) 926079. + 1.00255e6i 0.486340 + 0.526501i
\(326\) 0 0
\(327\) 483954.i 0.250285i
\(328\) 0 0
\(329\) 2.55395e6 1.30083
\(330\) 0 0
\(331\) 2.72290e6 1.36603 0.683017 0.730403i \(-0.260668\pi\)
0.683017 + 0.730403i \(0.260668\pi\)
\(332\) 0 0
\(333\) 413879.i 0.204533i
\(334\) 0 0
\(335\) −587732. + 229911.i −0.286132 + 0.111931i
\(336\) 0 0
\(337\) 1.95014e6i 0.935385i −0.883891 0.467693i \(-0.845085\pi\)
0.883891 0.467693i \(-0.154915\pi\)
\(338\) 0 0
\(339\) −337868. −0.159679
\(340\) 0 0
\(341\) −3.24098e6 −1.50935
\(342\) 0 0
\(343\) 2.29038e6i 1.05117i
\(344\) 0 0
\(345\) 269428. + 688749.i 0.121869 + 0.311539i
\(346\) 0 0
\(347\) 1.57423e6i 0.701852i −0.936403 0.350926i \(-0.885867\pi\)
0.936403 0.350926i \(-0.114133\pi\)
\(348\) 0 0
\(349\) 723241. 0.317848 0.158924 0.987291i \(-0.449197\pi\)
0.158924 + 0.987291i \(0.449197\pi\)
\(350\) 0 0
\(351\) −318385. −0.137938
\(352\) 0 0
\(353\) 1.94949e6i 0.832694i 0.909206 + 0.416347i \(0.136690\pi\)
−0.909206 + 0.416347i \(0.863310\pi\)
\(354\) 0 0
\(355\) 313247. + 800766.i 0.131922 + 0.337236i
\(356\) 0 0
\(357\) 1.76078e6i 0.731198i
\(358\) 0 0
\(359\) 1.57967e6 0.646891 0.323445 0.946247i \(-0.395159\pi\)
0.323445 + 0.946247i \(0.395159\pi\)
\(360\) 0 0
\(361\) 5.24423e6 2.11794
\(362\) 0 0
\(363\) 2.95579e6i 1.17735i
\(364\) 0 0
\(365\) 1.27251e6 497787.i 0.499953 0.195574i
\(366\) 0 0
\(367\) 549351.i 0.212904i −0.994318 0.106452i \(-0.966051\pi\)
0.994318 0.106452i \(-0.0339491\pi\)
\(368\) 0 0
\(369\) −622372. −0.237949
\(370\) 0 0
\(371\) −733145. −0.276538
\(372\) 0 0
\(373\) 4.11946e6i 1.53309i −0.642189 0.766546i \(-0.721974\pi\)
0.642189 0.766546i \(-0.278026\pi\)
\(374\) 0 0
\(375\) −686904. + 1.41424e6i −0.252242 + 0.519333i
\(376\) 0 0
\(377\) 975266.i 0.353403i
\(378\) 0 0
\(379\) 481024. 0.172016 0.0860079 0.996294i \(-0.472589\pi\)
0.0860079 + 0.996294i \(0.472589\pi\)
\(380\) 0 0
\(381\) 1.38790e6 0.489829
\(382\) 0 0
\(383\) 74246.2i 0.0258629i −0.999916 0.0129314i \(-0.995884\pi\)
0.999916 0.0129314i \(-0.00411632\pi\)
\(384\) 0 0
\(385\) −4.42571e6 + 1.73127e6i −1.52171 + 0.595268i
\(386\) 0 0
\(387\) 1.65385e6i 0.561331i
\(388\) 0 0
\(389\) 373796. 0.125245 0.0626225 0.998037i \(-0.480054\pi\)
0.0626225 + 0.998037i \(0.480054\pi\)
\(390\) 0 0
\(391\) 2.36680e6 0.782925
\(392\) 0 0
\(393\) 1.23105e6i 0.402062i
\(394\) 0 0
\(395\) 483038. + 1.23481e6i 0.155772 + 0.398205i
\(396\) 0 0
\(397\) 3.50136e6i 1.11496i −0.830189 0.557482i \(-0.811767\pi\)
0.830189 0.557482i \(-0.188233\pi\)
\(398\) 0 0
\(399\) 3.03860e6 0.955524
\(400\) 0 0
\(401\) −1.45598e6 −0.452162 −0.226081 0.974109i \(-0.572591\pi\)
−0.226081 + 0.974109i \(0.572591\pi\)
\(402\) 0 0
\(403\) 2.02320e6i 0.620548i
\(404\) 0 0
\(405\) −133616. 341567.i −0.0404781 0.103476i
\(406\) 0 0
\(407\) 3.57480e6i 1.06971i
\(408\) 0 0
\(409\) −55982.6 −0.0165480 −0.00827398 0.999966i \(-0.502634\pi\)
−0.00827398 + 0.999966i \(0.502634\pi\)
\(410\) 0 0
\(411\) −2.73192e6 −0.797745
\(412\) 0 0
\(413\) 3.37345e6i 0.973194i
\(414\) 0 0
\(415\) −987620. + 386341.i −0.281494 + 0.110116i
\(416\) 0 0
\(417\) 1.25790e6i 0.354246i
\(418\) 0 0
\(419\) −3.15797e6 −0.878765 −0.439382 0.898300i \(-0.644803\pi\)
−0.439382 + 0.898300i \(0.644803\pi\)
\(420\) 0 0
\(421\) −4.18745e6 −1.15145 −0.575725 0.817644i \(-0.695280\pi\)
−0.575725 + 0.817644i \(0.695280\pi\)
\(422\) 0 0
\(423\) 1.70248e6i 0.462628i
\(424\) 0 0
\(425\) 3.41407e6 + 3.69599e6i 0.916853 + 0.992564i
\(426\) 0 0
\(427\) 5.92453e6i 1.57248i
\(428\) 0 0
\(429\) −2.74999e6 −0.721421
\(430\) 0 0
\(431\) −5.00857e6 −1.29873 −0.649367 0.760475i \(-0.724966\pi\)
−0.649367 + 0.760475i \(0.724966\pi\)
\(432\) 0 0
\(433\) 6.43666e6i 1.64984i 0.565253 + 0.824918i \(0.308779\pi\)
−0.565253 + 0.824918i \(0.691221\pi\)
\(434\) 0 0
\(435\) −1.04627e6 + 409286.i −0.265108 + 0.103706i
\(436\) 0 0
\(437\) 4.08442e6i 1.02312i
\(438\) 0 0
\(439\) −4.41638e6 −1.09372 −0.546859 0.837225i \(-0.684177\pi\)
−0.546859 + 0.837225i \(0.684177\pi\)
\(440\) 0 0
\(441\) −165419. −0.0405032
\(442\) 0 0
\(443\) 1.58430e6i 0.383555i −0.981438 0.191778i \(-0.938575\pi\)
0.981438 0.191778i \(-0.0614252\pi\)
\(444\) 0 0
\(445\) 1.17165e6 + 2.99513e6i 0.280477 + 0.716995i
\(446\) 0 0
\(447\) 1.44676e6i 0.342475i
\(448\) 0 0
\(449\) 654015. 0.153099 0.0765494 0.997066i \(-0.475610\pi\)
0.0765494 + 0.997066i \(0.475610\pi\)
\(450\) 0 0
\(451\) −5.37563e6 −1.24448
\(452\) 0 0
\(453\) 2.47133e6i 0.565829i
\(454\) 0 0
\(455\) −1.08075e6 2.76277e6i −0.244736 0.625628i
\(456\) 0 0
\(457\) 4.54972e6i 1.01905i 0.860457 + 0.509523i \(0.170178\pi\)
−0.860457 + 0.509523i \(0.829822\pi\)
\(458\) 0 0
\(459\) −1.17375e6 −0.260043
\(460\) 0 0
\(461\) 6.13272e6 1.34400 0.672002 0.740549i \(-0.265434\pi\)
0.672002 + 0.740549i \(0.265434\pi\)
\(462\) 0 0
\(463\) 4.80795e6i 1.04233i 0.853455 + 0.521167i \(0.174503\pi\)
−0.853455 + 0.521167i \(0.825497\pi\)
\(464\) 0 0
\(465\) 2.17050e6 849067.i 0.465509 0.182100i
\(466\) 0 0
\(467\) 5.52182e6i 1.17163i 0.810445 + 0.585815i \(0.199225\pi\)
−0.810445 + 0.585815i \(0.800775\pi\)
\(468\) 0 0
\(469\) 1.37179e6 0.287975
\(470\) 0 0
\(471\) 3.00236e6 0.623607
\(472\) 0 0
\(473\) 1.42848e7i 2.93577i
\(474\) 0 0
\(475\) 6.37822e6 5.89170e6i 1.29708 1.19814i
\(476\) 0 0
\(477\) 488722.i 0.0983480i
\(478\) 0 0
\(479\) −4.10086e6 −0.816650 −0.408325 0.912837i \(-0.633887\pi\)
−0.408325 + 0.912837i \(0.633887\pi\)
\(480\) 0 0
\(481\) 2.23159e6 0.439796
\(482\) 0 0
\(483\) 1.60757e6i 0.313546i
\(484\) 0 0
\(485\) −9.05089e6 + 3.54057e6i −1.74718 + 0.683468i
\(486\) 0 0
\(487\) 1.67945e6i 0.320882i 0.987045 + 0.160441i \(0.0512917\pi\)
−0.987045 + 0.160441i \(0.948708\pi\)
\(488\) 0 0
\(489\) 1.43812e6 0.271971
\(490\) 0 0
\(491\) 3.85817e6 0.722234 0.361117 0.932521i \(-0.382396\pi\)
0.361117 + 0.932521i \(0.382396\pi\)
\(492\) 0 0
\(493\) 3.59540e6i 0.666238i
\(494\) 0 0
\(495\) −1.15408e6 2.95022e6i −0.211701 0.541180i
\(496\) 0 0
\(497\) 1.86902e6i 0.339408i
\(498\) 0 0
\(499\) −3.53067e6 −0.634755 −0.317378 0.948299i \(-0.602802\pi\)
−0.317378 + 0.948299i \(0.602802\pi\)
\(500\) 0 0
\(501\) 4.94881e6 0.880860
\(502\) 0 0
\(503\) 8.46451e6i 1.49170i 0.666113 + 0.745851i \(0.267957\pi\)
−0.666113 + 0.745851i \(0.732043\pi\)
\(504\) 0 0
\(505\) 1.58386e6 + 4.04888e6i 0.276368 + 0.706490i
\(506\) 0 0
\(507\) 1.62494e6i 0.280749i
\(508\) 0 0
\(509\) 8.12223e6 1.38957 0.694785 0.719217i \(-0.255499\pi\)
0.694785 + 0.719217i \(0.255499\pi\)
\(510\) 0 0
\(511\) −2.97009e6 −0.503173
\(512\) 0 0
\(513\) 2.02556e6i 0.339822i
\(514\) 0 0
\(515\) 6.44691e6 2.52193e6i 1.07111 0.419001i
\(516\) 0 0
\(517\) 1.47049e7i 2.41956i
\(518\) 0 0
\(519\) 767506. 0.125073
\(520\) 0 0
\(521\) 7.20656e6 1.16314 0.581572 0.813495i \(-0.302438\pi\)
0.581572 + 0.813495i \(0.302438\pi\)
\(522\) 0 0
\(523\) 2.08544e6i 0.333384i −0.986009 0.166692i \(-0.946691\pi\)
0.986009 0.166692i \(-0.0533085\pi\)
\(524\) 0 0
\(525\) 2.51037e6 2.31888e6i 0.397502 0.367181i
\(526\) 0 0
\(527\) 7.45868e6i 1.16986i
\(528\) 0 0
\(529\) 4.27549e6 0.664273
\(530\) 0 0
\(531\) −2.24877e6 −0.346106
\(532\) 0 0
\(533\) 3.35576e6i 0.511650i
\(534\) 0 0
\(535\) −4.83896e6 + 1.89293e6i −0.730917 + 0.285923i
\(536\) 0 0
\(537\) 3.40311e6i 0.509261i
\(538\) 0 0
\(539\) −1.42878e6 −0.211833
\(540\) 0 0
\(541\) −1.54682e6 −0.227220 −0.113610 0.993525i \(-0.536241\pi\)
−0.113610 + 0.993525i \(0.536241\pi\)
\(542\) 0 0
\(543\) 674122.i 0.0981159i
\(544\) 0 0
\(545\) −1.09509e6 2.79941e6i −0.157927 0.403716i
\(546\) 0 0
\(547\) 882763.i 0.126147i 0.998009 + 0.0630734i \(0.0200902\pi\)
−0.998009 + 0.0630734i \(0.979910\pi\)
\(548\) 0 0
\(549\) 3.94934e6 0.559235
\(550\) 0 0
\(551\) 6.20463e6 0.870636
\(552\) 0 0
\(553\) 2.88209e6i 0.400770i
\(554\) 0 0
\(555\) 936522. + 2.39407e6i 0.129058 + 0.329916i
\(556\) 0 0
\(557\) 1.32281e7i 1.80659i 0.429022 + 0.903294i \(0.358858\pi\)
−0.429022 + 0.903294i \(0.641142\pi\)
\(558\) 0 0
\(559\) 8.91737e6 1.20700
\(560\) 0 0
\(561\) −1.01381e7 −1.36003
\(562\) 0 0
\(563\) 1.71554e6i 0.228102i 0.993475 + 0.114051i \(0.0363828\pi\)
−0.993475 + 0.114051i \(0.963617\pi\)
\(564\) 0 0
\(565\) 1.95439e6 764526.i 0.257567 0.100756i
\(566\) 0 0
\(567\) 797230.i 0.104142i
\(568\) 0 0
\(569\) −1.06624e7 −1.38062 −0.690312 0.723512i \(-0.742527\pi\)
−0.690312 + 0.723512i \(0.742527\pi\)
\(570\) 0 0
\(571\) 2.13322e6 0.273807 0.136904 0.990584i \(-0.456285\pi\)
0.136904 + 0.990584i \(0.456285\pi\)
\(572\) 0 0
\(573\) 7.29792e6i 0.928565i
\(574\) 0 0
\(575\) −3.11699e6 3.37438e6i −0.393157 0.425623i
\(576\) 0 0
\(577\) 2.90217e6i 0.362897i 0.983400 + 0.181448i \(0.0580785\pi\)
−0.983400 + 0.181448i \(0.941921\pi\)
\(578\) 0 0
\(579\) 1.08461e6 0.134456
\(580\) 0 0
\(581\) 2.30514e6 0.283307
\(582\) 0 0
\(583\) 4.22124e6i 0.514362i
\(584\) 0 0
\(585\) 1.84169e6 720440.i 0.222498 0.0870378i
\(586\) 0 0
\(587\) 1.38744e7i 1.66195i 0.556310 + 0.830975i \(0.312217\pi\)
−0.556310 + 0.830975i \(0.687783\pi\)
\(588\) 0 0
\(589\) −1.28715e7 −1.52877
\(590\) 0 0
\(591\) −9.38801e6 −1.10562
\(592\) 0 0
\(593\) 1.05925e6i 0.123697i 0.998086 + 0.0618487i \(0.0196996\pi\)
−0.998086 + 0.0618487i \(0.980300\pi\)
\(594\) 0 0
\(595\) −3.98428e6 1.01852e7i −0.461379 1.17944i
\(596\) 0 0
\(597\) 789439.i 0.0906531i
\(598\) 0 0
\(599\) 821292. 0.0935256 0.0467628 0.998906i \(-0.485110\pi\)
0.0467628 + 0.998906i \(0.485110\pi\)
\(600\) 0 0
\(601\) −8.26399e6 −0.933262 −0.466631 0.884452i \(-0.654532\pi\)
−0.466631 + 0.884452i \(0.654532\pi\)
\(602\) 0 0
\(603\) 914447.i 0.102415i
\(604\) 0 0
\(605\) −6.68833e6 1.70976e7i −0.742898 1.89910i
\(606\) 0 0
\(607\) 252816.i 0.0278505i −0.999903 0.0139252i \(-0.995567\pi\)
0.999903 0.0139252i \(-0.00443269\pi\)
\(608\) 0 0
\(609\) 2.44205e6 0.266815
\(610\) 0 0
\(611\) −9.17960e6 −0.994765
\(612\) 0 0
\(613\) 5.25712e6i 0.565062i −0.959258 0.282531i \(-0.908826\pi\)
0.959258 0.282531i \(-0.0911741\pi\)
\(614\) 0 0
\(615\) 3.60009e6 1.40830e6i 0.383818 0.150144i
\(616\) 0 0
\(617\) 6.60527e6i 0.698518i −0.937026 0.349259i \(-0.886433\pi\)
0.937026 0.349259i \(-0.113567\pi\)
\(618\) 0 0
\(619\) 1.53364e7 1.60878 0.804392 0.594100i \(-0.202491\pi\)
0.804392 + 0.594100i \(0.202491\pi\)
\(620\) 0 0
\(621\) 1.07162e6 0.111509
\(622\) 0 0
\(623\) 6.99076e6i 0.721613i
\(624\) 0 0
\(625\) 773229. 9.73497e6i 0.0791786 0.996860i
\(626\) 0 0
\(627\) 1.74954e7i 1.77728i
\(628\) 0 0
\(629\) 8.22692e6 0.829108
\(630\) 0 0
\(631\) −4.10137e6 −0.410068 −0.205034 0.978755i \(-0.565730\pi\)
−0.205034 + 0.978755i \(0.565730\pi\)
\(632\) 0 0
\(633\) 4.77221e6i 0.473380i
\(634\) 0 0
\(635\) −8.02824e6 + 3.14052e6i −0.790107 + 0.309077i
\(636\) 0 0
\(637\) 891922.i 0.0870920i
\(638\) 0 0
\(639\) 1.24590e6 0.120707
\(640\) 0 0
\(641\) −2.80263e6 −0.269414 −0.134707 0.990885i \(-0.543009\pi\)
−0.134707 + 0.990885i \(0.543009\pi\)
\(642\) 0 0
\(643\) 1.05243e7i 1.00384i −0.864914 0.501920i \(-0.832627\pi\)
0.864914 0.501920i \(-0.167373\pi\)
\(644\) 0 0
\(645\) 3.74232e6 + 9.56664e6i 0.354194 + 0.905441i
\(646\) 0 0
\(647\) 8.43484e6i 0.792166i 0.918215 + 0.396083i \(0.129631\pi\)
−0.918215 + 0.396083i \(0.870369\pi\)
\(648\) 0 0
\(649\) −1.94234e7 −1.81014
\(650\) 0 0
\(651\) −5.06604e6 −0.468507
\(652\) 0 0
\(653\) 5.31348e6i 0.487637i 0.969821 + 0.243818i \(0.0784001\pi\)
−0.969821 + 0.243818i \(0.921600\pi\)
\(654\) 0 0
\(655\) 2.78560e6 + 7.12094e6i 0.253697 + 0.648536i
\(656\) 0 0
\(657\) 1.97989e6i 0.178948i
\(658\) 0 0
\(659\) −1.80659e7 −1.62049 −0.810246 0.586091i \(-0.800666\pi\)
−0.810246 + 0.586091i \(0.800666\pi\)
\(660\) 0 0
\(661\) 7.01234e6 0.624251 0.312125 0.950041i \(-0.398959\pi\)
0.312125 + 0.950041i \(0.398959\pi\)
\(662\) 0 0
\(663\) 6.32874e6i 0.559157i
\(664\) 0 0
\(665\) −1.75767e7 + 6.87573e6i −1.54128 + 0.602927i
\(666\) 0 0
\(667\) 3.28254e6i 0.285691i
\(668\) 0 0
\(669\) −1.14307e7 −0.987431
\(670\) 0 0
\(671\) 3.41117e7 2.92481
\(672\) 0 0
\(673\) 3.39367e6i 0.288823i 0.989518 + 0.144412i \(0.0461289\pi\)
−0.989518 + 0.144412i \(0.953871\pi\)
\(674\) 0 0
\(675\) 1.54579e6 + 1.67344e6i 0.130584 + 0.141368i
\(676\) 0 0
\(677\) 6.77774e6i 0.568347i 0.958773 + 0.284173i \(0.0917191\pi\)
−0.958773 + 0.284173i \(0.908281\pi\)
\(678\) 0 0
\(679\) 2.11251e7 1.75843
\(680\) 0 0
\(681\) −1.14304e6 −0.0944484
\(682\) 0 0
\(683\) 9.91859e6i 0.813576i 0.913523 + 0.406788i \(0.133351\pi\)
−0.913523 + 0.406788i \(0.866649\pi\)
\(684\) 0 0
\(685\) 1.58027e7 6.18178e6i 1.28678 0.503369i
\(686\) 0 0
\(687\) 4.93460e6i 0.398896i
\(688\) 0 0
\(689\) 2.63513e6 0.211473
\(690\) 0 0
\(691\) 46476.2 0.00370285 0.00185142 0.999998i \(-0.499411\pi\)
0.00185142 + 0.999998i \(0.499411\pi\)
\(692\) 0 0
\(693\) 6.88593e6i 0.544665i
\(694\) 0 0
\(695\) 2.84636e6 + 7.27625e6i 0.223526 + 0.571407i
\(696\) 0 0
\(697\) 1.23713e7i 0.964567i
\(698\) 0 0
\(699\) 7.15054e6 0.553536
\(700\) 0 0
\(701\) 1.23937e7 0.952589 0.476294 0.879286i \(-0.341980\pi\)
0.476294 + 0.879286i \(0.341980\pi\)
\(702\) 0 0
\(703\) 1.41973e7i 1.08347i
\(704\) 0 0
\(705\) −3.85237e6 9.84796e6i −0.291914 0.746231i
\(706\) 0 0
\(707\) 9.45023e6i 0.711040i
\(708\) 0 0
\(709\) 9.73930e6 0.727632 0.363816 0.931471i \(-0.381474\pi\)
0.363816 + 0.931471i \(0.381474\pi\)
\(710\) 0 0
\(711\) 1.92123e6 0.142530
\(712\) 0 0
\(713\) 6.80966e6i 0.501651i
\(714\) 0 0
\(715\) 1.59072e7 6.22267e6i 1.16367 0.455210i
\(716\) 0 0
\(717\) 5.25696e6i 0.381889i
\(718\) 0 0
\(719\) −1.67144e7 −1.20578 −0.602891 0.797823i \(-0.705985\pi\)
−0.602891 + 0.797823i \(0.705985\pi\)
\(720\) 0 0
\(721\) −1.50473e7 −1.07801
\(722\) 0 0
\(723\) 1.47751e7i 1.05120i
\(724\) 0 0
\(725\) 5.12601e6 4.73500e6i 0.362188 0.334561i
\(726\) 0 0
\(727\) 1.21859e7i 0.855109i 0.903990 + 0.427555i \(0.140625\pi\)
−0.903990 + 0.427555i \(0.859375\pi\)
\(728\) 0 0
\(729\) −531441. −0.0370370
\(730\) 0 0
\(731\) 3.28746e7 2.27545
\(732\) 0 0
\(733\) 1.90166e6i 0.130729i 0.997861 + 0.0653647i \(0.0208211\pi\)
−0.997861 + 0.0653647i \(0.979179\pi\)
\(734\) 0 0
\(735\) 956862. 374309.i 0.0653327 0.0255571i
\(736\) 0 0
\(737\) 7.89837e6i 0.535635i
\(738\) 0 0
\(739\) 9.24528e6 0.622743 0.311372 0.950288i \(-0.399212\pi\)
0.311372 + 0.950288i \(0.399212\pi\)
\(740\) 0 0
\(741\) −1.09216e7 −0.730702
\(742\) 0 0
\(743\) 8.62223e6i 0.572991i −0.958082 0.286495i \(-0.907510\pi\)
0.958082 0.286495i \(-0.0924903\pi\)
\(744\) 0 0
\(745\) 3.27372e6 + 8.36875e6i 0.216098 + 0.552420i
\(746\) 0 0
\(747\) 1.53663e6i 0.100755i
\(748\) 0 0
\(749\) 1.12943e7 0.735624
\(750\) 0 0
\(751\) −2.53525e7 −1.64029 −0.820146 0.572154i \(-0.806108\pi\)
−0.820146 + 0.572154i \(0.806108\pi\)
\(752\) 0 0
\(753\) 5.12864e6i 0.329621i
\(754\) 0 0
\(755\) 5.59211e6 + 1.42953e7i 0.357033 + 0.912697i
\(756\) 0 0
\(757\) 156645.i 0.00993522i −0.999988 0.00496761i \(-0.998419\pi\)
0.999988 0.00496761i \(-0.00158125\pi\)
\(758\) 0 0
\(759\) 9.25591e6 0.583196
\(760\) 0 0
\(761\) −1.56741e7 −0.981118 −0.490559 0.871408i \(-0.663207\pi\)
−0.490559 + 0.871408i \(0.663207\pi\)
\(762\) 0 0
\(763\) 6.53394e6i 0.406316i
\(764\) 0 0
\(765\) 6.78953e6 2.65596e6i 0.419456 0.164085i
\(766\) 0 0
\(767\) 1.21251e7i 0.744214i
\(768\) 0 0
\(769\) −4.84095e6 −0.295199 −0.147600 0.989047i \(-0.547155\pi\)
−0.147600 + 0.989047i \(0.547155\pi\)
\(770\) 0 0
\(771\) −4.49372e6 −0.272251
\(772\) 0 0
\(773\) 1.85997e6i 0.111958i −0.998432 0.0559792i \(-0.982172\pi\)
0.998432 0.0559792i \(-0.0178281\pi\)
\(774\) 0 0
\(775\) −1.06339e7 + 9.82280e6i −0.635975 + 0.587464i
\(776\) 0 0
\(777\) 5.58784e6i 0.332041i
\(778\) 0 0
\(779\) −2.13493e7 −1.26049
\(780\) 0 0
\(781\) 1.07613e7 0.631300
\(782\) 0 0
\(783\) 1.62789e6i 0.0948901i
\(784\) 0 0
\(785\) −1.73670e7 + 6.79372e6i −1.00589 + 0.393490i
\(786\) 0 0
\(787\) 1.38288e7i 0.795881i −0.917411 0.397940i \(-0.869725\pi\)
0.917411 0.397940i \(-0.130275\pi\)
\(788\) 0 0
\(789\) 6.45183e6 0.368970
\(790\) 0 0
\(791\) −4.56161e6 −0.259225
\(792\) 0 0
\(793\) 2.12944e7i 1.20249i
\(794\) 0 0
\(795\) 1.10588e6 + 2.82699e6i 0.0620567 + 0.158638i
\(796\) 0 0
\(797\) 9.71374e6i 0.541677i −0.962625 0.270839i \(-0.912699\pi\)
0.962625 0.270839i \(-0.0873010\pi\)
\(798\) 0 0
\(799\) −3.38413e7 −1.87534
\(800\) 0 0
\(801\) 4.66011e6 0.256634
\(802\) 0 0
\(803\) 1.71009e7i 0.935903i
\(804\) 0 0
\(805\) 3.63759e6 + 9.29891e6i 0.197844 + 0.505757i
\(806\) 0 0
\(807\) 1.97852e7i 1.06944i
\(808\) 0 0
\(809\) −1.71666e6 −0.0922175 −0.0461088 0.998936i \(-0.514682\pi\)
−0.0461088 + 0.998936i \(0.514682\pi\)
\(810\) 0 0
\(811\) 1.83595e7 0.980188 0.490094 0.871670i \(-0.336962\pi\)
0.490094 + 0.871670i \(0.336962\pi\)
\(812\) 0 0
\(813\) 2.28046e6i 0.121003i
\(814\) 0 0
\(815\) −8.31873e6 + 3.25416e6i −0.438695 + 0.171611i
\(816\) 0 0
\(817\) 5.67321e7i 2.97354i
\(818\) 0 0
\(819\) −4.29857e6 −0.223931
\(820\) 0 0
\(821\) −1.04696e7 −0.542092 −0.271046 0.962566i \(-0.587370\pi\)
−0.271046 + 0.962566i \(0.587370\pi\)
\(822\) 0 0
\(823\) 3.62514e7i 1.86563i 0.360355 + 0.932815i \(0.382656\pi\)
−0.360355 + 0.932815i \(0.617344\pi\)
\(824\) 0 0
\(825\) 1.33515e7 + 1.44540e7i 0.682958 + 0.739355i
\(826\) 0 0
\(827\) 2.44656e7i 1.24392i −0.783050 0.621959i \(-0.786337\pi\)
0.783050 0.621959i \(-0.213663\pi\)
\(828\) 0 0
\(829\) 1.38073e7 0.697789 0.348894 0.937162i \(-0.386557\pi\)
0.348894 + 0.937162i \(0.386557\pi\)
\(830\) 0 0
\(831\) −1.04037e7 −0.522617
\(832\) 0 0
\(833\) 3.28814e6i 0.164187i
\(834\) 0 0
\(835\) −2.86262e7 + 1.11981e7i −1.42085 + 0.555814i
\(836\) 0 0
\(837\) 3.37707e6i 0.166620i
\(838\) 0 0
\(839\) 1.90471e7 0.934164 0.467082 0.884214i \(-0.345305\pi\)
0.467082 + 0.884214i \(0.345305\pi\)
\(840\) 0 0
\(841\) −1.55247e7 −0.756889
\(842\) 0 0
\(843\) 2.13708e7i 1.03574i
\(844\) 0 0
\(845\) −3.67690e6 9.39941e6i −0.177150 0.452855i
\(846\) 0 0
\(847\) 3.99066e7i 1.91133i
\(848\) 0 0
\(849\) −1.41070e7 −0.671687
\(850\) 0 0
\(851\) −7.51106e6 −0.355531
\(852\) 0 0
\(853\) 9.14111e6i 0.430156i 0.976597 + 0.215078i \(0.0690006\pi\)
−0.976597 + 0.215078i \(0.930999\pi\)
\(854\) 0 0
\(855\) −4.58342e6 1.17168e7i −0.214425 0.548142i
\(856\) 0 0
\(857\) 3.39276e7i 1.57798i −0.614407 0.788989i \(-0.710605\pi\)
0.614407 0.788989i \(-0.289395\pi\)
\(858\) 0 0
\(859\) −2.26635e7 −1.04796 −0.523979 0.851731i \(-0.675553\pi\)
−0.523979 + 0.851731i \(0.675553\pi\)
\(860\) 0 0
\(861\) −8.40275e6 −0.386290
\(862\) 0 0
\(863\) 2.01189e7i 0.919555i −0.888034 0.459777i \(-0.847929\pi\)
0.888034 0.459777i \(-0.152071\pi\)
\(864\) 0 0
\(865\) −4.43961e6 + 1.73671e6i −0.201746 + 0.0789198i
\(866\) 0 0
\(867\) 1.05527e7i 0.476778i
\(868\) 0 0
\(869\) 1.65943e7 0.745433
\(870\) 0 0
\(871\) −4.93059e6 −0.220218
\(872\) 0 0
\(873\) 1.40822e7i 0.625367i
\(874\) 0 0
\(875\) −9.27400e6 + 1.90939e7i −0.409493 + 0.843093i
\(876\) 0 0
\(877\) 2.99763e7i 1.31607i 0.752988 + 0.658035i \(0.228612\pi\)
−0.752988 + 0.658035i \(0.771388\pi\)
\(878\) 0 0
\(879\) 1.56836e7 0.684657
\(880\) 0 0
\(881\) 3.74388e7 1.62511 0.812554 0.582886i \(-0.198077\pi\)
0.812554 + 0.582886i \(0.198077\pi\)
\(882\) 0 0
\(883\) 2.86458e7i 1.23640i −0.786021 0.618200i \(-0.787862\pi\)
0.786021 0.618200i \(-0.212138\pi\)
\(884\) 0 0
\(885\) 1.30080e7 5.08851e6i 0.558279 0.218390i
\(886\) 0 0
\(887\) 2.81325e7i 1.20060i 0.799775 + 0.600300i \(0.204952\pi\)
−0.799775 + 0.600300i \(0.795048\pi\)
\(888\) 0 0
\(889\) 1.87382e7 0.795195
\(890\) 0 0
\(891\) −4.59022e6 −0.193704
\(892\) 0 0
\(893\) 5.84004e7i 2.45068i
\(894\) 0 0
\(895\) 7.70053e6 + 1.96852e7i 0.321339 + 0.821451i
\(896\) 0 0
\(897\) 5.77804e6i 0.239773i
\(898\) 0 0
\(899\) −1.03445e7 −0.426885
\(900\) 0 0
\(901\) 9.71462e6 0.398670
\(902\) 0 0
\(903\) 2.23289e7i 0.911272i
\(904\) 0 0
\(905\) −1.52540e6 3.89944e6i −0.0619102 0.158263i
\(906\) 0 0
\(907\) 2.20623e7i 0.890498i −0.895407 0.445249i \(-0.853115\pi\)
0.895407 0.445249i \(-0.146885\pi\)
\(908\) 0 0
\(909\) 6.29961e6 0.252874
\(910\) 0 0
\(911\) −1.25081e7 −0.499340 −0.249670 0.968331i \(-0.580322\pi\)
−0.249670 + 0.968331i \(0.580322\pi\)
\(912\) 0 0
\(913\) 1.32724e7i 0.526952i
\(914\) 0 0
\(915\) −2.28448e7 + 8.93654e6i −0.902060 + 0.352872i
\(916\) 0 0
\(917\) 1.66205e7i 0.652712i
\(918\) 0 0
\(919\) 3.06659e6 0.119775 0.0598877 0.998205i \(-0.480926\pi\)
0.0598877 + 0.998205i \(0.480926\pi\)
\(920\) 0 0
\(921\) 2.72544e7 1.05873
\(922\) 0 0
\(923\) 6.71777e6i 0.259550i
\(924\) 0 0
\(925\) −1.08346e7 1.17292e7i −0.416348 0.450729i
\(926\) 0 0
\(927\) 1.00307e7i 0.383382i
\(928\) 0 0
\(929\) 4.08272e7 1.55206 0.776032 0.630693i \(-0.217229\pi\)
0.776032 + 0.630693i \(0.217229\pi\)
\(930\) 0 0
\(931\) −5.67439e6 −0.214558
\(932\) 0 0
\(933\) 1.25935e6i 0.0473633i
\(934\) 0 0
\(935\) 5.86433e7 2.29404e7i 2.19376 0.858166i
\(936\) 0 0
\(937\) 3.07349e6i 0.114362i 0.998364 + 0.0571812i \(0.0182113\pi\)
−0.998364 + 0.0571812i \(0.981789\pi\)
\(938\) 0 0
\(939\) 2.48698e7 0.920469
\(940\) 0 0
\(941\) 2.32976e6 0.0857702 0.0428851 0.999080i \(-0.486345\pi\)
0.0428851 + 0.999080i \(0.486345\pi\)
\(942\) 0 0
\(943\) 1.12948e7i 0.413617i
\(944\) 0 0
\(945\) −1.80397e6 4.61155e6i −0.0657126 0.167984i
\(946\) 0 0
\(947\) 2.11096e7i 0.764902i −0.923976 0.382451i \(-0.875080\pi\)
0.923976 0.382451i \(-0.124920\pi\)
\(948\) 0 0
\(949\) 1.06753e7 0.384783
\(950\) 0 0
\(951\) 1.63792e7 0.587275
\(952\) 0 0
\(953\) 2.66559e7i 0.950738i 0.879787 + 0.475369i \(0.157685\pi\)
−0.879787 + 0.475369i \(0.842315\pi\)
\(954\) 0 0
\(955\) 1.65137e7 + 4.22145e7i 0.585916 + 1.49780i
\(956\) 0 0
\(957\) 1.40606e7i 0.496277i
\(958\) 0 0
\(959\) −3.68841e7 −1.29507
\(960\) 0 0
\(961\) −7.16939e6 −0.250423
\(962\) 0 0
\(963\) 7.52890e6i 0.261617i
\(964\) 0 0
\(965\) −6.27391e6 + 2.45426e6i −0.216880 + 0.0848402i
\(966\) 0 0
\(967\) 4.22706e6i 0.145369i 0.997355 + 0.0726846i \(0.0231566\pi\)
−0.997355 + 0.0726846i \(0.976843\pi\)
\(968\) 0 0
\(969\) −4.02633e7 −1.37753
\(970\) 0 0
\(971\) 755839. 0.0257265 0.0128633 0.999917i \(-0.495905\pi\)
0.0128633 + 0.999917i \(0.495905\pi\)
\(972\) 0 0
\(973\) 1.69831e7i 0.575087i
\(974\) 0 0
\(975\) −9.02297e6 + 8.33471e6i −0.303975 + 0.280788i
\(976\) 0 0
\(977\) 239167.i 0.00801613i 0.999992 + 0.00400806i \(0.00127581\pi\)
−0.999992 + 0.00400806i \(0.998724\pi\)
\(978\) 0 0
\(979\) 4.02508e7 1.34220
\(980\) 0 0
\(981\) −4.35559e6 −0.144502
\(982\) 0 0
\(983\) 3.78483e7i 1.24929i −0.780910 0.624644i \(-0.785244\pi\)
0.780910 0.624644i \(-0.214756\pi\)
\(984\) 0 0
\(985\) 5.43046e7 2.12431e7i 1.78339 0.697634i
\(986\) 0 0
\(987\) 2.29855e7i 0.751037i
\(988\) 0 0
\(989\) −3.00140e7 −0.975738
\(990\) 0 0
\(991\) −2.04454e7 −0.661321 −0.330661 0.943750i \(-0.607272\pi\)
−0.330661 + 0.943750i \(0.607272\pi\)
\(992\) 0 0
\(993\) 2.45061e7i 0.788680i
\(994\) 0 0
\(995\) 1.78634e6 + 4.56648e6i 0.0572012 + 0.146226i
\(996\) 0 0
\(997\) 1.20062e6i 0.0382532i 0.999817 + 0.0191266i \(0.00608855\pi\)
−0.999817 + 0.0191266i \(0.993911\pi\)
\(998\) 0 0
\(999\) 3.72491e6 0.118087
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 60.6.d.a.49.5 yes 6
3.2 odd 2 180.6.d.d.109.4 6
4.3 odd 2 240.6.f.d.49.2 6
5.2 odd 4 300.6.a.j.1.1 3
5.3 odd 4 300.6.a.i.1.3 3
5.4 even 2 inner 60.6.d.a.49.2 6
12.11 even 2 720.6.f.m.289.4 6
15.2 even 4 900.6.a.x.1.1 3
15.8 even 4 900.6.a.w.1.3 3
15.14 odd 2 180.6.d.d.109.3 6
20.19 odd 2 240.6.f.d.49.5 6
60.59 even 2 720.6.f.m.289.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.6.d.a.49.2 6 5.4 even 2 inner
60.6.d.a.49.5 yes 6 1.1 even 1 trivial
180.6.d.d.109.3 6 15.14 odd 2
180.6.d.d.109.4 6 3.2 odd 2
240.6.f.d.49.2 6 4.3 odd 2
240.6.f.d.49.5 6 20.19 odd 2
300.6.a.i.1.3 3 5.3 odd 4
300.6.a.j.1.1 3 5.2 odd 4
720.6.f.m.289.3 6 60.59 even 2
720.6.f.m.289.4 6 12.11 even 2
900.6.a.w.1.3 3 15.8 even 4
900.6.a.x.1.1 3 15.2 even 4