Properties

Label 1521.2.b.l.1351.1
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,2,Mod(1351,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), 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: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.1
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.l.1351.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.24698i q^{2} -3.04892 q^{4} -1.44504i q^{5} +2.04892i q^{7} +2.35690i q^{8} +O(q^{10})\) \(q-2.24698i q^{2} -3.04892 q^{4} -1.44504i q^{5} +2.04892i q^{7} +2.35690i q^{8} -3.24698 q^{10} +2.55496i q^{11} +4.60388 q^{14} -0.801938 q^{16} -5.29590 q^{17} +5.85086i q^{19} +4.40581i q^{20} +5.74094 q^{22} -1.89008 q^{23} +2.91185 q^{25} -6.24698i q^{28} -2.26875 q^{29} +4.26875i q^{31} +6.51573i q^{32} +11.8998i q^{34} +2.96077 q^{35} +5.35690i q^{37} +13.1468 q^{38} +3.40581 q^{40} +1.27413i q^{41} -6.13706 q^{43} -7.78986i q^{44} +4.24698i q^{46} +2.95108i q^{47} +2.80194 q^{49} -6.54288i q^{50} -5.52111 q^{53} +3.69202 q^{55} -4.82908 q^{56} +5.09783i q^{58} +12.2078i q^{59} +8.56465 q^{61} +9.59179 q^{62} +13.0368 q^{64} -0.576728i q^{67} +16.1468 q^{68} -6.65279i q^{70} -4.59419i q^{71} -10.5526i q^{73} +12.0368 q^{74} -17.8388i q^{76} -5.23490 q^{77} -15.7778 q^{79} +1.15883i q^{80} +2.86294 q^{82} +7.72348i q^{83} +7.65279i q^{85} +13.7899i q^{86} -6.02177 q^{88} -6.61356i q^{89} +5.76271 q^{92} +6.63102 q^{94} +8.45473 q^{95} -11.9269i q^{97} -6.29590i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62} + 22 q^{64} + 42 q^{68} + 16 q^{74} + 16 q^{77} - 10 q^{79} + 28 q^{82} - 30 q^{88} + 10 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.24698i − 1.58885i −0.607359 0.794427i \(-0.707771\pi\)
0.607359 0.794427i \(-0.292229\pi\)
\(3\) 0 0
\(4\) −3.04892 −1.52446
\(5\) − 1.44504i − 0.646242i −0.946358 0.323121i \(-0.895268\pi\)
0.946358 0.323121i \(-0.104732\pi\)
\(6\) 0 0
\(7\) 2.04892i 0.774418i 0.921992 + 0.387209i \(0.126561\pi\)
−0.921992 + 0.387209i \(0.873439\pi\)
\(8\) 2.35690i 0.833289i
\(9\) 0 0
\(10\) −3.24698 −1.02679
\(11\) 2.55496i 0.770349i 0.922844 + 0.385174i \(0.125859\pi\)
−0.922844 + 0.385174i \(0.874141\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.60388 1.23044
\(15\) 0 0
\(16\) −0.801938 −0.200484
\(17\) −5.29590 −1.28444 −0.642222 0.766519i \(-0.721987\pi\)
−0.642222 + 0.766519i \(0.721987\pi\)
\(18\) 0 0
\(19\) 5.85086i 1.34228i 0.741331 + 0.671139i \(0.234195\pi\)
−0.741331 + 0.671139i \(0.765805\pi\)
\(20\) 4.40581i 0.985170i
\(21\) 0 0
\(22\) 5.74094 1.22397
\(23\) −1.89008 −0.394110 −0.197055 0.980392i \(-0.563138\pi\)
−0.197055 + 0.980392i \(0.563138\pi\)
\(24\) 0 0
\(25\) 2.91185 0.582371
\(26\) 0 0
\(27\) 0 0
\(28\) − 6.24698i − 1.18057i
\(29\) −2.26875 −0.421296 −0.210648 0.977562i \(-0.567557\pi\)
−0.210648 + 0.977562i \(0.567557\pi\)
\(30\) 0 0
\(31\) 4.26875i 0.766690i 0.923605 + 0.383345i \(0.125228\pi\)
−0.923605 + 0.383345i \(0.874772\pi\)
\(32\) 6.51573i 1.15183i
\(33\) 0 0
\(34\) 11.8998i 2.04079i
\(35\) 2.96077 0.500462
\(36\) 0 0
\(37\) 5.35690i 0.880668i 0.897834 + 0.440334i \(0.145140\pi\)
−0.897834 + 0.440334i \(0.854860\pi\)
\(38\) 13.1468 2.13268
\(39\) 0 0
\(40\) 3.40581 0.538506
\(41\) 1.27413i 0.198985i 0.995038 + 0.0994926i \(0.0317220\pi\)
−0.995038 + 0.0994926i \(0.968278\pi\)
\(42\) 0 0
\(43\) −6.13706 −0.935893 −0.467947 0.883757i \(-0.655006\pi\)
−0.467947 + 0.883757i \(0.655006\pi\)
\(44\) − 7.78986i − 1.17437i
\(45\) 0 0
\(46\) 4.24698i 0.626183i
\(47\) 2.95108i 0.430460i 0.976563 + 0.215230i \(0.0690501\pi\)
−0.976563 + 0.215230i \(0.930950\pi\)
\(48\) 0 0
\(49\) 2.80194 0.400277
\(50\) − 6.54288i − 0.925302i
\(51\) 0 0
\(52\) 0 0
\(53\) −5.52111 −0.758382 −0.379191 0.925318i \(-0.623798\pi\)
−0.379191 + 0.925318i \(0.623798\pi\)
\(54\) 0 0
\(55\) 3.69202 0.497832
\(56\) −4.82908 −0.645314
\(57\) 0 0
\(58\) 5.09783i 0.669378i
\(59\) 12.2078i 1.58931i 0.607059 + 0.794657i \(0.292349\pi\)
−0.607059 + 0.794657i \(0.707651\pi\)
\(60\) 0 0
\(61\) 8.56465 1.09659 0.548295 0.836285i \(-0.315277\pi\)
0.548295 + 0.836285i \(0.315277\pi\)
\(62\) 9.59179 1.21816
\(63\) 0 0
\(64\) 13.0368 1.62960
\(65\) 0 0
\(66\) 0 0
\(67\) − 0.576728i − 0.0704586i −0.999379 0.0352293i \(-0.988784\pi\)
0.999379 0.0352293i \(-0.0112162\pi\)
\(68\) 16.1468 1.95808
\(69\) 0 0
\(70\) − 6.65279i − 0.795161i
\(71\) − 4.59419i − 0.545230i −0.962123 0.272615i \(-0.912112\pi\)
0.962123 0.272615i \(-0.0878885\pi\)
\(72\) 0 0
\(73\) − 10.5526i − 1.23508i −0.786538 0.617542i \(-0.788128\pi\)
0.786538 0.617542i \(-0.211872\pi\)
\(74\) 12.0368 1.39925
\(75\) 0 0
\(76\) − 17.8388i − 2.04625i
\(77\) −5.23490 −0.596572
\(78\) 0 0
\(79\) −15.7778 −1.77514 −0.887569 0.460674i \(-0.847608\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(80\) 1.15883i 0.129562i
\(81\) 0 0
\(82\) 2.86294 0.316158
\(83\) 7.72348i 0.847762i 0.905718 + 0.423881i \(0.139333\pi\)
−0.905718 + 0.423881i \(0.860667\pi\)
\(84\) 0 0
\(85\) 7.65279i 0.830062i
\(86\) 13.7899i 1.48700i
\(87\) 0 0
\(88\) −6.02177 −0.641923
\(89\) − 6.61356i − 0.701036i −0.936556 0.350518i \(-0.886005\pi\)
0.936556 0.350518i \(-0.113995\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.76271 0.600804
\(93\) 0 0
\(94\) 6.63102 0.683938
\(95\) 8.45473 0.867437
\(96\) 0 0
\(97\) − 11.9269i − 1.21100i −0.795847 0.605498i \(-0.792974\pi\)
0.795847 0.605498i \(-0.207026\pi\)
\(98\) − 6.29590i − 0.635982i
\(99\) 0 0
\(100\) −8.87800 −0.887800
\(101\) 13.0640 1.29991 0.649957 0.759971i \(-0.274787\pi\)
0.649957 + 0.759971i \(0.274787\pi\)
\(102\) 0 0
\(103\) −9.16852 −0.903401 −0.451701 0.892170i \(-0.649182\pi\)
−0.451701 + 0.892170i \(0.649182\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.4058i 1.20496i
\(107\) 6.89977 0.667026 0.333513 0.942745i \(-0.391766\pi\)
0.333513 + 0.942745i \(0.391766\pi\)
\(108\) 0 0
\(109\) − 0.121998i − 0.0116853i −0.999983 0.00584264i \(-0.998140\pi\)
0.999983 0.00584264i \(-0.00185978\pi\)
\(110\) − 8.29590i − 0.790983i
\(111\) 0 0
\(112\) − 1.64310i − 0.155259i
\(113\) −7.30798 −0.687477 −0.343738 0.939065i \(-0.611693\pi\)
−0.343738 + 0.939065i \(0.611693\pi\)
\(114\) 0 0
\(115\) 2.73125i 0.254690i
\(116\) 6.91723 0.642249
\(117\) 0 0
\(118\) 27.4306 2.52519
\(119\) − 10.8509i − 0.994696i
\(120\) 0 0
\(121\) 4.47219 0.406563
\(122\) − 19.2446i − 1.74232i
\(123\) 0 0
\(124\) − 13.0151i − 1.16879i
\(125\) − 11.4330i − 1.02260i
\(126\) 0 0
\(127\) 18.9705 1.68336 0.841678 0.539980i \(-0.181568\pi\)
0.841678 + 0.539980i \(0.181568\pi\)
\(128\) − 16.2620i − 1.43738i
\(129\) 0 0
\(130\) 0 0
\(131\) −3.25667 −0.284536 −0.142268 0.989828i \(-0.545440\pi\)
−0.142268 + 0.989828i \(0.545440\pi\)
\(132\) 0 0
\(133\) −11.9879 −1.03948
\(134\) −1.29590 −0.111948
\(135\) 0 0
\(136\) − 12.4819i − 1.07031i
\(137\) − 0.792249i − 0.0676864i −0.999427 0.0338432i \(-0.989225\pi\)
0.999427 0.0338432i \(-0.0107747\pi\)
\(138\) 0 0
\(139\) −11.3394 −0.961799 −0.480899 0.876776i \(-0.659690\pi\)
−0.480899 + 0.876776i \(0.659690\pi\)
\(140\) −9.02715 −0.762933
\(141\) 0 0
\(142\) −10.3230 −0.866291
\(143\) 0 0
\(144\) 0 0
\(145\) 3.27844i 0.272260i
\(146\) −23.7114 −1.96237
\(147\) 0 0
\(148\) − 16.3327i − 1.34254i
\(149\) 8.40581i 0.688631i 0.938854 + 0.344316i \(0.111889\pi\)
−0.938854 + 0.344316i \(0.888111\pi\)
\(150\) 0 0
\(151\) 14.1293i 1.14983i 0.818215 + 0.574913i \(0.194964\pi\)
−0.818215 + 0.574913i \(0.805036\pi\)
\(152\) −13.7899 −1.11851
\(153\) 0 0
\(154\) 11.7627i 0.947866i
\(155\) 6.16852 0.495468
\(156\) 0 0
\(157\) −9.43296 −0.752832 −0.376416 0.926451i \(-0.622844\pi\)
−0.376416 + 0.926451i \(0.622844\pi\)
\(158\) 35.4523i 2.82044i
\(159\) 0 0
\(160\) 9.41550 0.744361
\(161\) − 3.87263i − 0.305206i
\(162\) 0 0
\(163\) 8.70410i 0.681758i 0.940107 + 0.340879i \(0.110725\pi\)
−0.940107 + 0.340879i \(0.889275\pi\)
\(164\) − 3.88471i − 0.303345i
\(165\) 0 0
\(166\) 17.3545 1.34697
\(167\) 23.8538i 1.84587i 0.384961 + 0.922933i \(0.374215\pi\)
−0.384961 + 0.922933i \(0.625785\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 17.1957 1.31885
\(171\) 0 0
\(172\) 18.7114 1.42673
\(173\) −18.8552 −1.43353 −0.716766 0.697314i \(-0.754378\pi\)
−0.716766 + 0.697314i \(0.754378\pi\)
\(174\) 0 0
\(175\) 5.96615i 0.450998i
\(176\) − 2.04892i − 0.154443i
\(177\) 0 0
\(178\) −14.8605 −1.11384
\(179\) 6.02177 0.450088 0.225044 0.974349i \(-0.427747\pi\)
0.225044 + 0.974349i \(0.427747\pi\)
\(180\) 0 0
\(181\) 4.77777 0.355129 0.177565 0.984109i \(-0.443178\pi\)
0.177565 + 0.984109i \(0.443178\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) − 4.45473i − 0.328407i
\(185\) 7.74094 0.569125
\(186\) 0 0
\(187\) − 13.5308i − 0.989470i
\(188\) − 8.99761i − 0.656218i
\(189\) 0 0
\(190\) − 18.9976i − 1.37823i
\(191\) −18.4306 −1.33359 −0.666795 0.745242i \(-0.732334\pi\)
−0.666795 + 0.745242i \(0.732334\pi\)
\(192\) 0 0
\(193\) 6.05429i 0.435798i 0.975971 + 0.217899i \(0.0699203\pi\)
−0.975971 + 0.217899i \(0.930080\pi\)
\(194\) −26.7995 −1.92410
\(195\) 0 0
\(196\) −8.54288 −0.610205
\(197\) − 11.4155i − 0.813321i −0.913579 0.406660i \(-0.866693\pi\)
0.913579 0.406660i \(-0.133307\pi\)
\(198\) 0 0
\(199\) 13.9051 0.985710 0.492855 0.870111i \(-0.335953\pi\)
0.492855 + 0.870111i \(0.335953\pi\)
\(200\) 6.86294i 0.485283i
\(201\) 0 0
\(202\) − 29.3545i − 2.06538i
\(203\) − 4.64848i − 0.326259i
\(204\) 0 0
\(205\) 1.84117 0.128593
\(206\) 20.6015i 1.43537i
\(207\) 0 0
\(208\) 0 0
\(209\) −14.9487 −1.03402
\(210\) 0 0
\(211\) −13.2446 −0.911795 −0.455897 0.890032i \(-0.650682\pi\)
−0.455897 + 0.890032i \(0.650682\pi\)
\(212\) 16.8334 1.15612
\(213\) 0 0
\(214\) − 15.5036i − 1.05981i
\(215\) 8.86831i 0.604814i
\(216\) 0 0
\(217\) −8.74632 −0.593739
\(218\) −0.274127 −0.0185662
\(219\) 0 0
\(220\) −11.2567 −0.758924
\(221\) 0 0
\(222\) 0 0
\(223\) 7.33513i 0.491196i 0.969372 + 0.245598i \(0.0789844\pi\)
−0.969372 + 0.245598i \(0.921016\pi\)
\(224\) −13.3502 −0.891997
\(225\) 0 0
\(226\) 16.4209i 1.09230i
\(227\) − 8.67456i − 0.575751i −0.957668 0.287875i \(-0.907051\pi\)
0.957668 0.287875i \(-0.0929489\pi\)
\(228\) 0 0
\(229\) − 13.6866i − 0.904439i −0.891907 0.452219i \(-0.850632\pi\)
0.891907 0.452219i \(-0.149368\pi\)
\(230\) 6.13706 0.404666
\(231\) 0 0
\(232\) − 5.34721i − 0.351061i
\(233\) −5.08815 −0.333336 −0.166668 0.986013i \(-0.553301\pi\)
−0.166668 + 0.986013i \(0.553301\pi\)
\(234\) 0 0
\(235\) 4.26444 0.278181
\(236\) − 37.2204i − 2.42284i
\(237\) 0 0
\(238\) −24.3817 −1.58043
\(239\) − 10.9239i − 0.706611i −0.935508 0.353305i \(-0.885058\pi\)
0.935508 0.353305i \(-0.114942\pi\)
\(240\) 0 0
\(241\) 11.9148i 0.767502i 0.923437 + 0.383751i \(0.125368\pi\)
−0.923437 + 0.383751i \(0.874632\pi\)
\(242\) − 10.0489i − 0.645969i
\(243\) 0 0
\(244\) −26.1129 −1.67171
\(245\) − 4.04892i − 0.258676i
\(246\) 0 0
\(247\) 0 0
\(248\) −10.0610 −0.638874
\(249\) 0 0
\(250\) −25.6896 −1.62475
\(251\) 22.3478 1.41058 0.705290 0.708919i \(-0.250817\pi\)
0.705290 + 0.708919i \(0.250817\pi\)
\(252\) 0 0
\(253\) − 4.82908i − 0.303602i
\(254\) − 42.6262i − 2.67461i
\(255\) 0 0
\(256\) −10.4668 −0.654176
\(257\) −18.6601 −1.16398 −0.581992 0.813194i \(-0.697727\pi\)
−0.581992 + 0.813194i \(0.697727\pi\)
\(258\) 0 0
\(259\) −10.9758 −0.682005
\(260\) 0 0
\(261\) 0 0
\(262\) 7.31767i 0.452087i
\(263\) −14.3991 −0.887887 −0.443944 0.896055i \(-0.646421\pi\)
−0.443944 + 0.896055i \(0.646421\pi\)
\(264\) 0 0
\(265\) 7.97823i 0.490099i
\(266\) 26.9366i 1.65159i
\(267\) 0 0
\(268\) 1.75840i 0.107411i
\(269\) −0.652793 −0.0398015 −0.0199007 0.999802i \(-0.506335\pi\)
−0.0199007 + 0.999802i \(0.506335\pi\)
\(270\) 0 0
\(271\) − 1.99569i − 0.121229i −0.998161 0.0606147i \(-0.980694\pi\)
0.998161 0.0606147i \(-0.0193061\pi\)
\(272\) 4.24698 0.257511
\(273\) 0 0
\(274\) −1.78017 −0.107544
\(275\) 7.43967i 0.448629i
\(276\) 0 0
\(277\) −11.7845 −0.708061 −0.354030 0.935234i \(-0.615189\pi\)
−0.354030 + 0.935234i \(0.615189\pi\)
\(278\) 25.4795i 1.52816i
\(279\) 0 0
\(280\) 6.97823i 0.417029i
\(281\) 6.47219i 0.386098i 0.981189 + 0.193049i \(0.0618377\pi\)
−0.981189 + 0.193049i \(0.938162\pi\)
\(282\) 0 0
\(283\) −6.58104 −0.391202 −0.195601 0.980684i \(-0.562666\pi\)
−0.195601 + 0.980684i \(0.562666\pi\)
\(284\) 14.0073i 0.831180i
\(285\) 0 0
\(286\) 0 0
\(287\) −2.61058 −0.154098
\(288\) 0 0
\(289\) 11.0465 0.649796
\(290\) 7.36658 0.432581
\(291\) 0 0
\(292\) 32.1739i 1.88284i
\(293\) 24.3381i 1.42185i 0.703269 + 0.710924i \(0.251723\pi\)
−0.703269 + 0.710924i \(0.748277\pi\)
\(294\) 0 0
\(295\) 17.6407 1.02708
\(296\) −12.6256 −0.733851
\(297\) 0 0
\(298\) 18.8877 1.09413
\(299\) 0 0
\(300\) 0 0
\(301\) − 12.5743i − 0.724773i
\(302\) 31.7482 1.82691
\(303\) 0 0
\(304\) − 4.69202i − 0.269106i
\(305\) − 12.3763i − 0.708663i
\(306\) 0 0
\(307\) − 14.0737i − 0.803227i −0.915809 0.401613i \(-0.868450\pi\)
0.915809 0.401613i \(-0.131550\pi\)
\(308\) 15.9608 0.909449
\(309\) 0 0
\(310\) − 13.8605i − 0.787226i
\(311\) −29.7700 −1.68810 −0.844051 0.536263i \(-0.819836\pi\)
−0.844051 + 0.536263i \(0.819836\pi\)
\(312\) 0 0
\(313\) −7.47889 −0.422732 −0.211366 0.977407i \(-0.567791\pi\)
−0.211366 + 0.977407i \(0.567791\pi\)
\(314\) 21.1957i 1.19614i
\(315\) 0 0
\(316\) 48.1051 2.70613
\(317\) 30.0301i 1.68666i 0.537396 + 0.843330i \(0.319408\pi\)
−0.537396 + 0.843330i \(0.680592\pi\)
\(318\) 0 0
\(319\) − 5.79656i − 0.324545i
\(320\) − 18.8388i − 1.05312i
\(321\) 0 0
\(322\) −8.70171 −0.484927
\(323\) − 30.9855i − 1.72408i
\(324\) 0 0
\(325\) 0 0
\(326\) 19.5579 1.08321
\(327\) 0 0
\(328\) −3.00298 −0.165812
\(329\) −6.04652 −0.333356
\(330\) 0 0
\(331\) 15.7168i 0.863872i 0.901904 + 0.431936i \(0.142169\pi\)
−0.901904 + 0.431936i \(0.857831\pi\)
\(332\) − 23.5483i − 1.29238i
\(333\) 0 0
\(334\) 53.5991 2.93281
\(335\) −0.833397 −0.0455333
\(336\) 0 0
\(337\) −1.95407 −0.106445 −0.0532224 0.998583i \(-0.516949\pi\)
−0.0532224 + 0.998583i \(0.516949\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 23.3327i − 1.26540i
\(341\) −10.9065 −0.590619
\(342\) 0 0
\(343\) 20.0834i 1.08440i
\(344\) − 14.4644i − 0.779869i
\(345\) 0 0
\(346\) 42.3672i 2.27767i
\(347\) 17.1250 0.919317 0.459659 0.888096i \(-0.347972\pi\)
0.459659 + 0.888096i \(0.347972\pi\)
\(348\) 0 0
\(349\) − 10.4668i − 0.560276i −0.959960 0.280138i \(-0.909620\pi\)
0.959960 0.280138i \(-0.0903802\pi\)
\(350\) 13.4058 0.716571
\(351\) 0 0
\(352\) −16.6474 −0.887310
\(353\) 15.5308i 0.826621i 0.910590 + 0.413310i \(0.135628\pi\)
−0.910590 + 0.413310i \(0.864372\pi\)
\(354\) 0 0
\(355\) −6.63879 −0.352351
\(356\) 20.1642i 1.06870i
\(357\) 0 0
\(358\) − 13.5308i − 0.715125i
\(359\) 21.4263i 1.13083i 0.824805 + 0.565417i \(0.191285\pi\)
−0.824805 + 0.565417i \(0.808715\pi\)
\(360\) 0 0
\(361\) −15.2325 −0.801711
\(362\) − 10.7356i − 0.564249i
\(363\) 0 0
\(364\) 0 0
\(365\) −15.2489 −0.798164
\(366\) 0 0
\(367\) 34.3032 1.79061 0.895306 0.445452i \(-0.146957\pi\)
0.895306 + 0.445452i \(0.146957\pi\)
\(368\) 1.51573 0.0790129
\(369\) 0 0
\(370\) − 17.3937i − 0.904257i
\(371\) − 11.3123i − 0.587305i
\(372\) 0 0
\(373\) −12.5961 −0.652202 −0.326101 0.945335i \(-0.605735\pi\)
−0.326101 + 0.945335i \(0.605735\pi\)
\(374\) −30.4034 −1.57212
\(375\) 0 0
\(376\) −6.95539 −0.358697
\(377\) 0 0
\(378\) 0 0
\(379\) − 16.5386i − 0.849529i −0.905304 0.424765i \(-0.860357\pi\)
0.905304 0.424765i \(-0.139643\pi\)
\(380\) −25.7778 −1.32237
\(381\) 0 0
\(382\) 41.4131i 2.11888i
\(383\) 7.53617i 0.385080i 0.981289 + 0.192540i \(0.0616726\pi\)
−0.981289 + 0.192540i \(0.938327\pi\)
\(384\) 0 0
\(385\) 7.56465i 0.385530i
\(386\) 13.6039 0.692419
\(387\) 0 0
\(388\) 36.3642i 1.84611i
\(389\) 35.5555 1.80274 0.901369 0.433052i \(-0.142563\pi\)
0.901369 + 0.433052i \(0.142563\pi\)
\(390\) 0 0
\(391\) 10.0097 0.506212
\(392\) 6.60388i 0.333546i
\(393\) 0 0
\(394\) −25.6504 −1.29225
\(395\) 22.7995i 1.14717i
\(396\) 0 0
\(397\) 1.35152i 0.0678308i 0.999425 + 0.0339154i \(0.0107977\pi\)
−0.999425 + 0.0339154i \(0.989202\pi\)
\(398\) − 31.2446i − 1.56615i
\(399\) 0 0
\(400\) −2.33513 −0.116756
\(401\) − 0.579121i − 0.0289199i −0.999895 0.0144600i \(-0.995397\pi\)
0.999895 0.0144600i \(-0.00460291\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −39.8310 −1.98167
\(405\) 0 0
\(406\) −10.4450 −0.518379
\(407\) −13.6866 −0.678422
\(408\) 0 0
\(409\) 15.1575i 0.749490i 0.927128 + 0.374745i \(0.122270\pi\)
−0.927128 + 0.374745i \(0.877730\pi\)
\(410\) − 4.13706i − 0.204315i
\(411\) 0 0
\(412\) 27.9541 1.37720
\(413\) −25.0127 −1.23079
\(414\) 0 0
\(415\) 11.1608 0.547860
\(416\) 0 0
\(417\) 0 0
\(418\) 33.5894i 1.64291i
\(419\) 35.7235 1.74521 0.872603 0.488430i \(-0.162430\pi\)
0.872603 + 0.488430i \(0.162430\pi\)
\(420\) 0 0
\(421\) 35.0465i 1.70806i 0.520221 + 0.854032i \(0.325849\pi\)
−0.520221 + 0.854032i \(0.674151\pi\)
\(422\) 29.7603i 1.44871i
\(423\) 0 0
\(424\) − 13.0127i − 0.631951i
\(425\) −15.4209 −0.748022
\(426\) 0 0
\(427\) 17.5483i 0.849220i
\(428\) −21.0368 −1.01685
\(429\) 0 0
\(430\) 19.9269 0.960961
\(431\) − 34.2814i − 1.65128i −0.564199 0.825639i \(-0.690815\pi\)
0.564199 0.825639i \(-0.309185\pi\)
\(432\) 0 0
\(433\) −13.7385 −0.660232 −0.330116 0.943940i \(-0.607088\pi\)
−0.330116 + 0.943940i \(0.607088\pi\)
\(434\) 19.6528i 0.943364i
\(435\) 0 0
\(436\) 0.371961i 0.0178137i
\(437\) − 11.0586i − 0.529005i
\(438\) 0 0
\(439\) −10.2403 −0.488742 −0.244371 0.969682i \(-0.578581\pi\)
−0.244371 + 0.969682i \(0.578581\pi\)
\(440\) 8.70171i 0.414838i
\(441\) 0 0
\(442\) 0 0
\(443\) −12.1763 −0.578513 −0.289257 0.957252i \(-0.593408\pi\)
−0.289257 + 0.957252i \(0.593408\pi\)
\(444\) 0 0
\(445\) −9.55688 −0.453039
\(446\) 16.4819 0.780440
\(447\) 0 0
\(448\) 26.7114i 1.26199i
\(449\) 12.9051i 0.609032i 0.952507 + 0.304516i \(0.0984947\pi\)
−0.952507 + 0.304516i \(0.901505\pi\)
\(450\) 0 0
\(451\) −3.25534 −0.153288
\(452\) 22.2814 1.04803
\(453\) 0 0
\(454\) −19.4916 −0.914785
\(455\) 0 0
\(456\) 0 0
\(457\) − 4.65710i − 0.217850i −0.994050 0.108925i \(-0.965259\pi\)
0.994050 0.108925i \(-0.0347409\pi\)
\(458\) −30.7536 −1.43702
\(459\) 0 0
\(460\) − 8.32736i − 0.388265i
\(461\) − 31.5405i − 1.46899i −0.678616 0.734493i \(-0.737420\pi\)
0.678616 0.734493i \(-0.262580\pi\)
\(462\) 0 0
\(463\) 17.6504i 0.820284i 0.912022 + 0.410142i \(0.134521\pi\)
−0.912022 + 0.410142i \(0.865479\pi\)
\(464\) 1.81940 0.0844633
\(465\) 0 0
\(466\) 11.4330i 0.529622i
\(467\) −32.1726 −1.48877 −0.744385 0.667751i \(-0.767257\pi\)
−0.744385 + 0.667751i \(0.767257\pi\)
\(468\) 0 0
\(469\) 1.18167 0.0545644
\(470\) − 9.58211i − 0.441990i
\(471\) 0 0
\(472\) −28.7724 −1.32436
\(473\) − 15.6799i − 0.720964i
\(474\) 0 0
\(475\) 17.0368i 0.781704i
\(476\) 33.0834i 1.51637i
\(477\) 0 0
\(478\) −24.5459 −1.12270
\(479\) 34.8998i 1.59461i 0.603576 + 0.797306i \(0.293742\pi\)
−0.603576 + 0.797306i \(0.706258\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 26.7724 1.21945
\(483\) 0 0
\(484\) −13.6353 −0.619788
\(485\) −17.2349 −0.782596
\(486\) 0 0
\(487\) − 41.8351i − 1.89573i −0.318676 0.947864i \(-0.603238\pi\)
0.318676 0.947864i \(-0.396762\pi\)
\(488\) 20.1860i 0.913776i
\(489\) 0 0
\(490\) −9.09783 −0.410998
\(491\) 21.8455 0.985873 0.492936 0.870065i \(-0.335924\pi\)
0.492936 + 0.870065i \(0.335924\pi\)
\(492\) 0 0
\(493\) 12.0151 0.541131
\(494\) 0 0
\(495\) 0 0
\(496\) − 3.42327i − 0.153709i
\(497\) 9.41311 0.422236
\(498\) 0 0
\(499\) − 23.5472i − 1.05412i −0.849829 0.527058i \(-0.823295\pi\)
0.849829 0.527058i \(-0.176705\pi\)
\(500\) 34.8582i 1.55890i
\(501\) 0 0
\(502\) − 50.2150i − 2.24121i
\(503\) 7.08682 0.315986 0.157993 0.987440i \(-0.449498\pi\)
0.157993 + 0.987440i \(0.449498\pi\)
\(504\) 0 0
\(505\) − 18.8780i − 0.840060i
\(506\) −10.8509 −0.482379
\(507\) 0 0
\(508\) −57.8394 −2.56621
\(509\) 7.61894i 0.337704i 0.985641 + 0.168852i \(0.0540059\pi\)
−0.985641 + 0.168852i \(0.945994\pi\)
\(510\) 0 0
\(511\) 21.6213 0.956471
\(512\) − 9.00538i − 0.397985i
\(513\) 0 0
\(514\) 41.9288i 1.84940i
\(515\) 13.2489i 0.583816i
\(516\) 0 0
\(517\) −7.53989 −0.331604
\(518\) 24.6625i 1.08361i
\(519\) 0 0
\(520\) 0 0
\(521\) 39.5133 1.73111 0.865555 0.500813i \(-0.166966\pi\)
0.865555 + 0.500813i \(0.166966\pi\)
\(522\) 0 0
\(523\) −15.8194 −0.691734 −0.345867 0.938284i \(-0.612415\pi\)
−0.345867 + 0.938284i \(0.612415\pi\)
\(524\) 9.92931 0.433764
\(525\) 0 0
\(526\) 32.3545i 1.41072i
\(527\) − 22.6069i − 0.984770i
\(528\) 0 0
\(529\) −19.4276 −0.844678
\(530\) 17.9269 0.778696
\(531\) 0 0
\(532\) 36.5502 1.58465
\(533\) 0 0
\(534\) 0 0
\(535\) − 9.97046i − 0.431061i
\(536\) 1.35929 0.0587123
\(537\) 0 0
\(538\) 1.46681i 0.0632388i
\(539\) 7.15883i 0.308353i
\(540\) 0 0
\(541\) 34.4819i 1.48249i 0.671234 + 0.741246i \(0.265765\pi\)
−0.671234 + 0.741246i \(0.734235\pi\)
\(542\) −4.48427 −0.192616
\(543\) 0 0
\(544\) − 34.5066i − 1.47946i
\(545\) −0.176292 −0.00755152
\(546\) 0 0
\(547\) 36.8582 1.57594 0.787970 0.615713i \(-0.211132\pi\)
0.787970 + 0.615713i \(0.211132\pi\)
\(548\) 2.41550i 0.103185i
\(549\) 0 0
\(550\) 16.7168 0.712806
\(551\) − 13.2741i − 0.565497i
\(552\) 0 0
\(553\) − 32.3274i − 1.37470i
\(554\) 26.4795i 1.12501i
\(555\) 0 0
\(556\) 34.5730 1.46622
\(557\) − 1.27652i − 0.0540879i −0.999634 0.0270439i \(-0.991391\pi\)
0.999634 0.0270439i \(-0.00860940\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.37435 −0.100335
\(561\) 0 0
\(562\) 14.5429 0.613454
\(563\) −9.12737 −0.384673 −0.192336 0.981329i \(-0.561607\pi\)
−0.192336 + 0.981329i \(0.561607\pi\)
\(564\) 0 0
\(565\) 10.5603i 0.444277i
\(566\) 14.7875i 0.621563i
\(567\) 0 0
\(568\) 10.8280 0.454334
\(569\) −5.72156 −0.239860 −0.119930 0.992782i \(-0.538267\pi\)
−0.119930 + 0.992782i \(0.538267\pi\)
\(570\) 0 0
\(571\) −7.60148 −0.318112 −0.159056 0.987270i \(-0.550845\pi\)
−0.159056 + 0.987270i \(0.550845\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 5.86592i 0.244839i
\(575\) −5.50365 −0.229518
\(576\) 0 0
\(577\) − 45.1564i − 1.87989i −0.341330 0.939944i \(-0.610877\pi\)
0.341330 0.939944i \(-0.389123\pi\)
\(578\) − 24.8213i − 1.03243i
\(579\) 0 0
\(580\) − 9.99569i − 0.415048i
\(581\) −15.8248 −0.656522
\(582\) 0 0
\(583\) − 14.1062i − 0.584219i
\(584\) 24.8713 1.02918
\(585\) 0 0
\(586\) 54.6872 2.25911
\(587\) − 32.4040i − 1.33746i −0.743507 0.668728i \(-0.766839\pi\)
0.743507 0.668728i \(-0.233161\pi\)
\(588\) 0 0
\(589\) −24.9758 −1.02911
\(590\) − 39.6383i − 1.63188i
\(591\) 0 0
\(592\) − 4.29590i − 0.176560i
\(593\) − 36.6848i − 1.50647i −0.657754 0.753233i \(-0.728493\pi\)
0.657754 0.753233i \(-0.271507\pi\)
\(594\) 0 0
\(595\) −15.6799 −0.642815
\(596\) − 25.6286i − 1.04979i
\(597\) 0 0
\(598\) 0 0
\(599\) 9.99223 0.408271 0.204136 0.978943i \(-0.434562\pi\)
0.204136 + 0.978943i \(0.434562\pi\)
\(600\) 0 0
\(601\) −1.81163 −0.0738978 −0.0369489 0.999317i \(-0.511764\pi\)
−0.0369489 + 0.999317i \(0.511764\pi\)
\(602\) −28.2543 −1.15156
\(603\) 0 0
\(604\) − 43.0790i − 1.75286i
\(605\) − 6.46250i − 0.262738i
\(606\) 0 0
\(607\) 11.2161 0.455248 0.227624 0.973749i \(-0.426904\pi\)
0.227624 + 0.973749i \(0.426904\pi\)
\(608\) −38.1226 −1.54608
\(609\) 0 0
\(610\) −27.8092 −1.12596
\(611\) 0 0
\(612\) 0 0
\(613\) 20.8944i 0.843917i 0.906615 + 0.421958i \(0.138657\pi\)
−0.906615 + 0.421958i \(0.861343\pi\)
\(614\) −31.6233 −1.27621
\(615\) 0 0
\(616\) − 12.3381i − 0.497117i
\(617\) 12.0992i 0.487094i 0.969889 + 0.243547i \(0.0783110\pi\)
−0.969889 + 0.243547i \(0.921689\pi\)
\(618\) 0 0
\(619\) − 10.5526i − 0.424143i −0.977254 0.212072i \(-0.931979\pi\)
0.977254 0.212072i \(-0.0680210\pi\)
\(620\) −18.8073 −0.755320
\(621\) 0 0
\(622\) 66.8926i 2.68215i
\(623\) 13.5506 0.542895
\(624\) 0 0
\(625\) −1.96184 −0.0784735
\(626\) 16.8049i 0.671660i
\(627\) 0 0
\(628\) 28.7603 1.14766
\(629\) − 28.3696i − 1.13117i
\(630\) 0 0
\(631\) − 13.8514i − 0.551417i −0.961241 0.275709i \(-0.911087\pi\)
0.961241 0.275709i \(-0.0889125\pi\)
\(632\) − 37.1866i − 1.47920i
\(633\) 0 0
\(634\) 67.4771 2.67986
\(635\) − 27.4131i − 1.08786i
\(636\) 0 0
\(637\) 0 0
\(638\) −13.0248 −0.515655
\(639\) 0 0
\(640\) −23.4993 −0.928893
\(641\) 34.9608 1.38087 0.690434 0.723396i \(-0.257420\pi\)
0.690434 + 0.723396i \(0.257420\pi\)
\(642\) 0 0
\(643\) − 33.3980i − 1.31709i −0.752541 0.658545i \(-0.771172\pi\)
0.752541 0.658545i \(-0.228828\pi\)
\(644\) 11.8073i 0.465273i
\(645\) 0 0
\(646\) −69.6238 −2.73931
\(647\) 2.32842 0.0915397 0.0457698 0.998952i \(-0.485426\pi\)
0.0457698 + 0.998952i \(0.485426\pi\)
\(648\) 0 0
\(649\) −31.1903 −1.22433
\(650\) 0 0
\(651\) 0 0
\(652\) − 26.5381i − 1.03931i
\(653\) −14.5714 −0.570221 −0.285111 0.958495i \(-0.592030\pi\)
−0.285111 + 0.958495i \(0.592030\pi\)
\(654\) 0 0
\(655\) 4.70602i 0.183879i
\(656\) − 1.02177i − 0.0398934i
\(657\) 0 0
\(658\) 13.5864i 0.529654i
\(659\) −11.1395 −0.433932 −0.216966 0.976179i \(-0.569616\pi\)
−0.216966 + 0.976179i \(0.569616\pi\)
\(660\) 0 0
\(661\) − 13.8498i − 0.538694i −0.963043 0.269347i \(-0.913192\pi\)
0.963043 0.269347i \(-0.0868079\pi\)
\(662\) 35.3153 1.37257
\(663\) 0 0
\(664\) −18.2034 −0.706430
\(665\) 17.3230i 0.671759i
\(666\) 0 0
\(667\) 4.28813 0.166037
\(668\) − 72.7284i − 2.81395i
\(669\) 0 0
\(670\) 1.87263i 0.0723458i
\(671\) 21.8823i 0.844757i
\(672\) 0 0
\(673\) 6.52973 0.251703 0.125851 0.992049i \(-0.459834\pi\)
0.125851 + 0.992049i \(0.459834\pi\)
\(674\) 4.39075i 0.169125i
\(675\) 0 0
\(676\) 0 0
\(677\) 11.3104 0.434693 0.217346 0.976095i \(-0.430260\pi\)
0.217346 + 0.976095i \(0.430260\pi\)
\(678\) 0 0
\(679\) 24.4373 0.937816
\(680\) −18.0368 −0.691681
\(681\) 0 0
\(682\) 24.5066i 0.938407i
\(683\) − 14.1793i − 0.542555i −0.962501 0.271277i \(-0.912554\pi\)
0.962501 0.271277i \(-0.0874461\pi\)
\(684\) 0 0
\(685\) −1.14483 −0.0437418
\(686\) 45.1269 1.72295
\(687\) 0 0
\(688\) 4.92154 0.187632
\(689\) 0 0
\(690\) 0 0
\(691\) − 30.7952i − 1.17151i −0.810490 0.585753i \(-0.800799\pi\)
0.810490 0.585753i \(-0.199201\pi\)
\(692\) 57.4878 2.18536
\(693\) 0 0
\(694\) − 38.4795i − 1.46066i
\(695\) 16.3860i 0.621555i
\(696\) 0 0
\(697\) − 6.74764i − 0.255585i
\(698\) −23.5187 −0.890196
\(699\) 0 0
\(700\) − 18.1903i − 0.687528i
\(701\) 6.73184 0.254258 0.127129 0.991886i \(-0.459424\pi\)
0.127129 + 0.991886i \(0.459424\pi\)
\(702\) 0 0
\(703\) −31.3424 −1.18210
\(704\) 33.3086i 1.25536i
\(705\) 0 0
\(706\) 34.8974 1.31338
\(707\) 26.7670i 1.00668i
\(708\) 0 0
\(709\) − 47.6252i − 1.78860i −0.447467 0.894300i \(-0.647674\pi\)
0.447467 0.894300i \(-0.352326\pi\)
\(710\) 14.9172i 0.559834i
\(711\) 0 0
\(712\) 15.5875 0.584166
\(713\) − 8.06829i − 0.302160i
\(714\) 0 0
\(715\) 0 0
\(716\) −18.3599 −0.686141
\(717\) 0 0
\(718\) 48.1444 1.79673
\(719\) −5.99330 −0.223512 −0.111756 0.993736i \(-0.535648\pi\)
−0.111756 + 0.993736i \(0.535648\pi\)
\(720\) 0 0
\(721\) − 18.7855i − 0.699610i
\(722\) 34.2271i 1.27380i
\(723\) 0 0
\(724\) −14.5670 −0.541380
\(725\) −6.60627 −0.245351
\(726\) 0 0
\(727\) 24.1226 0.894657 0.447329 0.894370i \(-0.352375\pi\)
0.447329 + 0.894370i \(0.352375\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 34.2640i 1.26817i
\(731\) 32.5013 1.20210
\(732\) 0 0
\(733\) − 36.0646i − 1.33208i −0.745918 0.666038i \(-0.767989\pi\)
0.745918 0.666038i \(-0.232011\pi\)
\(734\) − 77.0786i − 2.84502i
\(735\) 0 0
\(736\) − 12.3153i − 0.453947i
\(737\) 1.47352 0.0542777
\(738\) 0 0
\(739\) 27.5254i 1.01254i 0.862375 + 0.506269i \(0.168976\pi\)
−0.862375 + 0.506269i \(0.831024\pi\)
\(740\) −23.6015 −0.867608
\(741\) 0 0
\(742\) −25.4185 −0.933142
\(743\) − 10.4692i − 0.384078i −0.981387 0.192039i \(-0.938490\pi\)
0.981387 0.192039i \(-0.0615100\pi\)
\(744\) 0 0
\(745\) 12.1468 0.445023
\(746\) 28.3032i 1.03625i
\(747\) 0 0
\(748\) 41.2543i 1.50841i
\(749\) 14.1371i 0.516557i
\(750\) 0 0
\(751\) −4.06770 −0.148433 −0.0742163 0.997242i \(-0.523646\pi\)
−0.0742163 + 0.997242i \(0.523646\pi\)
\(752\) − 2.36658i − 0.0863005i
\(753\) 0 0
\(754\) 0 0
\(755\) 20.4174 0.743066
\(756\) 0 0
\(757\) 20.4336 0.742670 0.371335 0.928499i \(-0.378900\pi\)
0.371335 + 0.928499i \(0.378900\pi\)
\(758\) −37.1618 −1.34978
\(759\) 0 0
\(760\) 19.9269i 0.722825i
\(761\) 27.0237i 0.979608i 0.871833 + 0.489804i \(0.162932\pi\)
−0.871833 + 0.489804i \(0.837068\pi\)
\(762\) 0 0
\(763\) 0.249964 0.00904929
\(764\) 56.1933 2.03300
\(765\) 0 0
\(766\) 16.9336 0.611837
\(767\) 0 0
\(768\) 0 0
\(769\) 37.9407i 1.36818i 0.729400 + 0.684088i \(0.239799\pi\)
−0.729400 + 0.684088i \(0.760201\pi\)
\(770\) 16.9976 0.612551
\(771\) 0 0
\(772\) − 18.4590i − 0.664355i
\(773\) − 16.3375i − 0.587620i −0.955864 0.293810i \(-0.905077\pi\)
0.955864 0.293810i \(-0.0949232\pi\)
\(774\) 0 0
\(775\) 12.4300i 0.446498i
\(776\) 28.1105 1.00911
\(777\) 0 0
\(778\) − 79.8926i − 2.86429i
\(779\) −7.45473 −0.267093
\(780\) 0 0
\(781\) 11.7380 0.420017
\(782\) − 22.4916i − 0.804297i
\(783\) 0 0
\(784\) −2.24698 −0.0802493
\(785\) 13.6310i 0.486512i
\(786\) 0 0
\(787\) − 18.6907i − 0.666251i −0.942882 0.333126i \(-0.891897\pi\)
0.942882 0.333126i \(-0.108103\pi\)
\(788\) 34.8049i 1.23987i
\(789\) 0 0
\(790\) 51.2301 1.82269
\(791\) − 14.9734i − 0.532394i
\(792\) 0 0
\(793\) 0 0
\(794\) 3.03684 0.107773
\(795\) 0 0
\(796\) −42.3957 −1.50267
\(797\) 29.2519 1.03615 0.518077 0.855334i \(-0.326648\pi\)
0.518077 + 0.855334i \(0.326648\pi\)
\(798\) 0 0
\(799\) − 15.6286i − 0.552901i
\(800\) 18.9729i 0.670792i
\(801\) 0 0
\(802\) −1.30127 −0.0459496
\(803\) 26.9614 0.951446
\(804\) 0 0
\(805\) −5.59611 −0.197237
\(806\) 0 0
\(807\) 0 0
\(808\) 30.7904i 1.08320i
\(809\) 6.65087 0.233832 0.116916 0.993142i \(-0.462699\pi\)
0.116916 + 0.993142i \(0.462699\pi\)
\(810\) 0 0
\(811\) − 3.89200i − 0.136667i −0.997663 0.0683333i \(-0.978232\pi\)
0.997663 0.0683333i \(-0.0217681\pi\)
\(812\) 14.1728i 0.497369i
\(813\) 0 0
\(814\) 30.7536i 1.07791i
\(815\) 12.5778 0.440581
\(816\) 0 0
\(817\) − 35.9071i − 1.25623i
\(818\) 34.0586 1.19083
\(819\) 0 0
\(820\) −5.61356 −0.196034
\(821\) 45.9982i 1.60535i 0.596418 + 0.802674i \(0.296590\pi\)
−0.596418 + 0.802674i \(0.703410\pi\)
\(822\) 0 0
\(823\) −7.95300 −0.277224 −0.138612 0.990347i \(-0.544264\pi\)
−0.138612 + 0.990347i \(0.544264\pi\)
\(824\) − 21.6093i − 0.752794i
\(825\) 0 0
\(826\) 56.2030i 1.95555i
\(827\) − 27.9648i − 0.972432i −0.873839 0.486216i \(-0.838377\pi\)
0.873839 0.486216i \(-0.161623\pi\)
\(828\) 0 0
\(829\) −27.6310 −0.959665 −0.479833 0.877360i \(-0.659303\pi\)
−0.479833 + 0.877360i \(0.659303\pi\)
\(830\) − 25.0780i − 0.870470i
\(831\) 0 0
\(832\) 0 0
\(833\) −14.8388 −0.514133
\(834\) 0 0
\(835\) 34.4698 1.19288
\(836\) 45.5773 1.57632
\(837\) 0 0
\(838\) − 80.2699i − 2.77288i
\(839\) − 28.6848i − 0.990311i −0.868804 0.495155i \(-0.835111\pi\)
0.868804 0.495155i \(-0.164889\pi\)
\(840\) 0 0
\(841\) −23.8528 −0.822509
\(842\) 78.7488 2.71386
\(843\) 0 0
\(844\) 40.3817 1.38999
\(845\) 0 0
\(846\) 0 0
\(847\) 9.16315i 0.314849i
\(848\) 4.42758 0.152044
\(849\) 0 0
\(850\) 34.6504i 1.18850i
\(851\) − 10.1250i − 0.347080i
\(852\) 0 0
\(853\) − 43.2078i − 1.47941i −0.672934 0.739703i \(-0.734966\pi\)
0.672934 0.739703i \(-0.265034\pi\)
\(854\) 39.4306 1.34929
\(855\) 0 0
\(856\) 16.2620i 0.555825i
\(857\) 35.1685 1.20133 0.600667 0.799499i \(-0.294902\pi\)
0.600667 + 0.799499i \(0.294902\pi\)
\(858\) 0 0
\(859\) 27.3793 0.934168 0.467084 0.884213i \(-0.345305\pi\)
0.467084 + 0.884213i \(0.345305\pi\)
\(860\) − 27.0388i − 0.922014i
\(861\) 0 0
\(862\) −77.0297 −2.62364
\(863\) − 41.3913i − 1.40898i −0.709715 0.704489i \(-0.751176\pi\)
0.709715 0.704489i \(-0.248824\pi\)
\(864\) 0 0
\(865\) 27.2465i 0.926409i
\(866\) 30.8702i 1.04901i
\(867\) 0 0
\(868\) 26.6668 0.905130
\(869\) − 40.3116i − 1.36748i
\(870\) 0 0
\(871\) 0 0
\(872\) 0.287536 0.00973721
\(873\) 0 0
\(874\) −24.8485 −0.840512
\(875\) 23.4252 0.791916
\(876\) 0 0
\(877\) 24.7472i 0.835653i 0.908527 + 0.417826i \(0.137208\pi\)
−0.908527 + 0.417826i \(0.862792\pi\)
\(878\) 23.0097i 0.776539i
\(879\) 0 0
\(880\) −2.96077 −0.0998076
\(881\) −28.5875 −0.963137 −0.481568 0.876409i \(-0.659933\pi\)
−0.481568 + 0.876409i \(0.659933\pi\)
\(882\) 0 0
\(883\) −9.61702 −0.323639 −0.161819 0.986820i \(-0.551736\pi\)
−0.161819 + 0.986820i \(0.551736\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 27.3599i 0.919173i
\(887\) −15.9661 −0.536091 −0.268045 0.963406i \(-0.586378\pi\)
−0.268045 + 0.963406i \(0.586378\pi\)
\(888\) 0 0
\(889\) 38.8689i 1.30362i
\(890\) 21.4741i 0.719814i
\(891\) 0 0
\(892\) − 22.3642i − 0.748809i
\(893\) −17.2664 −0.577797
\(894\) 0 0
\(895\) − 8.70171i − 0.290866i
\(896\) 33.3196 1.11313
\(897\) 0 0
\(898\) 28.9976 0.967663
\(899\) − 9.68473i − 0.323004i
\(900\) 0 0
\(901\) 29.2392 0.974099
\(902\) 7.31468i 0.243552i
\(903\) 0 0
\(904\) − 17.2241i − 0.572867i
\(905\) − 6.90408i − 0.229500i
\(906\) 0 0
\(907\) 28.8364 0.957496 0.478748 0.877952i \(-0.341091\pi\)
0.478748 + 0.877952i \(0.341091\pi\)
\(908\) 26.4480i 0.877709i
\(909\) 0 0
\(910\) 0 0
\(911\) −38.5633 −1.27766 −0.638830 0.769348i \(-0.720581\pi\)
−0.638830 + 0.769348i \(0.720581\pi\)
\(912\) 0 0
\(913\) −19.7332 −0.653073
\(914\) −10.4644 −0.346132
\(915\) 0 0
\(916\) 41.7294i 1.37878i
\(917\) − 6.67264i − 0.220350i
\(918\) 0 0
\(919\) 8.87502 0.292760 0.146380 0.989228i \(-0.453238\pi\)
0.146380 + 0.989228i \(0.453238\pi\)
\(920\) −6.43727 −0.212231
\(921\) 0 0
\(922\) −70.8708 −2.33401
\(923\) 0 0
\(924\) 0 0
\(925\) 15.5985i 0.512875i
\(926\) 39.6601 1.30331
\(927\) 0 0
\(928\) − 14.7826i − 0.485261i
\(929\) 24.2295i 0.794945i 0.917614 + 0.397472i \(0.130113\pi\)
−0.917614 + 0.397472i \(0.869887\pi\)
\(930\) 0 0
\(931\) 16.3937i 0.537283i
\(932\) 15.5133 0.508156
\(933\) 0 0
\(934\) 72.2911i 2.36544i
\(935\) −19.5526 −0.639437
\(936\) 0 0
\(937\) 17.2644 0.564005 0.282002 0.959414i \(-0.409001\pi\)
0.282002 + 0.959414i \(0.409001\pi\)
\(938\) − 2.65519i − 0.0866949i
\(939\) 0 0
\(940\) −13.0019 −0.424076
\(941\) 4.34050i 0.141496i 0.997494 + 0.0707482i \(0.0225387\pi\)
−0.997494 + 0.0707482i \(0.977461\pi\)
\(942\) 0 0
\(943\) − 2.40821i − 0.0784220i
\(944\) − 9.78986i − 0.318633i
\(945\) 0 0
\(946\) −35.2325 −1.14551
\(947\) 45.0146i 1.46278i 0.681961 + 0.731389i \(0.261128\pi\)
−0.681961 + 0.731389i \(0.738872\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 38.2814 1.24201
\(951\) 0 0
\(952\) 25.5743 0.828869
\(953\) −46.8859 −1.51878 −0.759391 0.650634i \(-0.774503\pi\)
−0.759391 + 0.650634i \(0.774503\pi\)
\(954\) 0 0
\(955\) 26.6329i 0.861822i
\(956\) 33.3062i 1.07720i
\(957\) 0 0
\(958\) 78.4191 2.53361
\(959\) 1.62325 0.0524176
\(960\) 0 0
\(961\) 12.7778 0.412186
\(962\) 0 0
\(963\) 0 0
\(964\) − 36.3274i − 1.17003i
\(965\) 8.74871 0.281631
\(966\) 0 0
\(967\) − 6.29457i − 0.202420i −0.994865 0.101210i \(-0.967729\pi\)
0.994865 0.101210i \(-0.0322714\pi\)
\(968\) 10.5405i 0.338784i
\(969\) 0 0
\(970\) 38.7265i 1.24343i
\(971\) 41.8068 1.34165 0.670823 0.741618i \(-0.265941\pi\)
0.670823 + 0.741618i \(0.265941\pi\)
\(972\) 0 0
\(973\) − 23.2336i − 0.744834i
\(974\) −94.0025 −3.01203
\(975\) 0 0
\(976\) −6.86831 −0.219849
\(977\) 23.7530i 0.759926i 0.925002 + 0.379963i \(0.124063\pi\)
−0.925002 + 0.379963i \(0.875937\pi\)
\(978\) 0 0
\(979\) 16.8974 0.540043
\(980\) 12.3448i 0.394341i
\(981\) 0 0
\(982\) − 49.0863i − 1.56641i
\(983\) 55.7251i 1.77736i 0.458532 + 0.888678i \(0.348375\pi\)
−0.458532 + 0.888678i \(0.651625\pi\)
\(984\) 0 0
\(985\) −16.4959 −0.525602
\(986\) − 26.9976i − 0.859779i
\(987\) 0 0
\(988\) 0 0
\(989\) 11.5996 0.368845
\(990\) 0 0
\(991\) −35.5512 −1.12932 −0.564661 0.825323i \(-0.690993\pi\)
−0.564661 + 0.825323i \(0.690993\pi\)
\(992\) −27.8140 −0.883096
\(993\) 0 0
\(994\) − 21.1511i − 0.670871i
\(995\) − 20.0935i − 0.637007i
\(996\) 0 0
\(997\) 6.61058 0.209359 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(998\) −52.9101 −1.67484
\(999\) 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 1521.2.b.l.1351.1 6
3.2 odd 2 169.2.b.b.168.6 6
12.11 even 2 2704.2.f.o.337.4 6
13.5 odd 4 1521.2.a.o.1.1 3
13.8 odd 4 1521.2.a.r.1.3 3
13.12 even 2 inner 1521.2.b.l.1351.6 6
39.2 even 12 169.2.c.b.22.1 6
39.5 even 4 169.2.a.c.1.3 yes 3
39.8 even 4 169.2.a.b.1.1 3
39.11 even 12 169.2.c.c.22.3 6
39.17 odd 6 169.2.e.b.23.1 12
39.20 even 12 169.2.c.c.146.3 6
39.23 odd 6 169.2.e.b.147.6 12
39.29 odd 6 169.2.e.b.147.1 12
39.32 even 12 169.2.c.b.146.1 6
39.35 odd 6 169.2.e.b.23.6 12
39.38 odd 2 169.2.b.b.168.1 6
156.47 odd 4 2704.2.a.z.1.2 3
156.83 odd 4 2704.2.a.ba.1.2 3
156.155 even 2 2704.2.f.o.337.3 6
195.44 even 4 4225.2.a.bb.1.1 3
195.164 even 4 4225.2.a.bg.1.3 3
273.83 odd 4 8281.2.a.bj.1.3 3
273.125 odd 4 8281.2.a.bf.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.1 3 39.8 even 4
169.2.a.c.1.3 yes 3 39.5 even 4
169.2.b.b.168.1 6 39.38 odd 2
169.2.b.b.168.6 6 3.2 odd 2
169.2.c.b.22.1 6 39.2 even 12
169.2.c.b.146.1 6 39.32 even 12
169.2.c.c.22.3 6 39.11 even 12
169.2.c.c.146.3 6 39.20 even 12
169.2.e.b.23.1 12 39.17 odd 6
169.2.e.b.23.6 12 39.35 odd 6
169.2.e.b.147.1 12 39.29 odd 6
169.2.e.b.147.6 12 39.23 odd 6
1521.2.a.o.1.1 3 13.5 odd 4
1521.2.a.r.1.3 3 13.8 odd 4
1521.2.b.l.1351.1 6 1.1 even 1 trivial
1521.2.b.l.1351.6 6 13.12 even 2 inner
2704.2.a.z.1.2 3 156.47 odd 4
2704.2.a.ba.1.2 3 156.83 odd 4
2704.2.f.o.337.3 6 156.155 even 2
2704.2.f.o.337.4 6 12.11 even 2
4225.2.a.bb.1.1 3 195.44 even 4
4225.2.a.bg.1.3 3 195.164 even 4
8281.2.a.bf.1.1 3 273.125 odd 4
8281.2.a.bj.1.3 3 273.83 odd 4