Properties

Label 6030.2.d.k.2411.3
Level $6030$
Weight $2$
Character 6030.2411
Analytic conductor $48.150$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6030,2,Mod(2411,6030)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6030, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("6030.2411");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6030 = 2 \cdot 3^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6030.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: \(48.1497924188\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2411.3
Character \(\chi\) \(=\) 6030.2411
Dual form 6030.2.d.k.2411.22

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.48375i q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -3.48375i q^{7} -1.00000 q^{8} +1.00000 q^{10} +3.99641 q^{11} +4.23077i q^{13} +3.48375i q^{14} +1.00000 q^{16} -2.08407i q^{17} +3.38304 q^{19} -1.00000 q^{20} -3.99641 q^{22} +3.40101i q^{23} +1.00000 q^{25} -4.23077i q^{26} -3.48375i q^{28} +7.87608i q^{29} -4.40560i q^{31} -1.00000 q^{32} +2.08407i q^{34} +3.48375i q^{35} +6.85166 q^{37} -3.38304 q^{38} +1.00000 q^{40} +3.37181 q^{41} +4.24695i q^{43} +3.99641 q^{44} -3.40101i q^{46} +0.604591i q^{47} -5.13654 q^{49} -1.00000 q^{50} +4.23077i q^{52} -12.4512 q^{53} -3.99641 q^{55} +3.48375i q^{56} -7.87608i q^{58} +6.23827i q^{59} -2.59685i q^{61} +4.40560i q^{62} +1.00000 q^{64} -4.23077i q^{65} +(-7.64991 - 2.91185i) q^{67} -2.08407i q^{68} -3.48375i q^{70} +8.66350i q^{71} +10.3997 q^{73} -6.85166 q^{74} +3.38304 q^{76} -13.9225i q^{77} +0.716620i q^{79} -1.00000 q^{80} -3.37181 q^{82} +14.3735i q^{83} +2.08407i q^{85} -4.24695i q^{86} -3.99641 q^{88} +18.2414i q^{89} +14.7389 q^{91} +3.40101i q^{92} -0.604591i q^{94} -3.38304 q^{95} -4.52609i q^{97} +5.13654 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{5} - 24 q^{8} + 24 q^{10} - 12 q^{11} + 24 q^{16} + 4 q^{19} - 24 q^{20} + 12 q^{22} + 24 q^{25} - 24 q^{32} - 16 q^{37} - 4 q^{38} + 24 q^{40} - 8 q^{41} - 12 q^{44} - 20 q^{49} - 24 q^{50} - 24 q^{53} + 12 q^{55} + 24 q^{64} - 32 q^{67} - 4 q^{73} + 16 q^{74} + 4 q^{76} - 24 q^{80} + 8 q^{82} + 12 q^{88} - 4 q^{95} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6030\mathbb{Z}\right)^\times\).

\(n\) \(1207\) \(3151\) \(4691\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.48375i 1.31674i −0.752697 0.658368i \(-0.771247\pi\)
0.752697 0.658368i \(-0.228753\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 3.99641 1.20496 0.602482 0.798133i \(-0.294179\pi\)
0.602482 + 0.798133i \(0.294179\pi\)
\(12\) 0 0
\(13\) 4.23077i 1.17340i 0.809803 + 0.586702i \(0.199574\pi\)
−0.809803 + 0.586702i \(0.800426\pi\)
\(14\) 3.48375i 0.931072i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.08407i 0.505460i −0.967537 0.252730i \(-0.918671\pi\)
0.967537 0.252730i \(-0.0813285\pi\)
\(18\) 0 0
\(19\) 3.38304 0.776122 0.388061 0.921634i \(-0.373145\pi\)
0.388061 + 0.921634i \(0.373145\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.99641 −0.852038
\(23\) 3.40101i 0.709161i 0.935026 + 0.354580i \(0.115376\pi\)
−0.935026 + 0.354580i \(0.884624\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.23077i 0.829722i
\(27\) 0 0
\(28\) 3.48375i 0.658368i
\(29\) 7.87608i 1.46255i 0.682082 + 0.731276i \(0.261075\pi\)
−0.682082 + 0.731276i \(0.738925\pi\)
\(30\) 0 0
\(31\) 4.40560i 0.791269i −0.918408 0.395634i \(-0.870525\pi\)
0.918408 0.395634i \(-0.129475\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.08407i 0.357415i
\(35\) 3.48375i 0.588862i
\(36\) 0 0
\(37\) 6.85166 1.12641 0.563203 0.826319i \(-0.309569\pi\)
0.563203 + 0.826319i \(0.309569\pi\)
\(38\) −3.38304 −0.548801
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.37181 0.526588 0.263294 0.964716i \(-0.415191\pi\)
0.263294 + 0.964716i \(0.415191\pi\)
\(42\) 0 0
\(43\) 4.24695i 0.647654i 0.946116 + 0.323827i \(0.104970\pi\)
−0.946116 + 0.323827i \(0.895030\pi\)
\(44\) 3.99641 0.602482
\(45\) 0 0
\(46\) 3.40101i 0.501452i
\(47\) 0.604591i 0.0881887i 0.999027 + 0.0440943i \(0.0140402\pi\)
−0.999027 + 0.0440943i \(0.985960\pi\)
\(48\) 0 0
\(49\) −5.13654 −0.733791
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 4.23077i 0.586702i
\(53\) −12.4512 −1.71031 −0.855153 0.518375i \(-0.826537\pi\)
−0.855153 + 0.518375i \(0.826537\pi\)
\(54\) 0 0
\(55\) −3.99641 −0.538876
\(56\) 3.48375i 0.465536i
\(57\) 0 0
\(58\) 7.87608i 1.03418i
\(59\) 6.23827i 0.812153i 0.913839 + 0.406076i \(0.133103\pi\)
−0.913839 + 0.406076i \(0.866897\pi\)
\(60\) 0 0
\(61\) 2.59685i 0.332492i −0.986084 0.166246i \(-0.946835\pi\)
0.986084 0.166246i \(-0.0531646\pi\)
\(62\) 4.40560i 0.559511i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.23077i 0.524762i
\(66\) 0 0
\(67\) −7.64991 2.91185i −0.934585 0.355739i
\(68\) 2.08407i 0.252730i
\(69\) 0 0
\(70\) 3.48375i 0.416388i
\(71\) 8.66350i 1.02817i 0.857740 + 0.514084i \(0.171868\pi\)
−0.857740 + 0.514084i \(0.828132\pi\)
\(72\) 0 0
\(73\) 10.3997 1.21719 0.608596 0.793480i \(-0.291733\pi\)
0.608596 + 0.793480i \(0.291733\pi\)
\(74\) −6.85166 −0.796489
\(75\) 0 0
\(76\) 3.38304 0.388061
\(77\) 13.9225i 1.58662i
\(78\) 0 0
\(79\) 0.716620i 0.0806260i 0.999187 + 0.0403130i \(0.0128355\pi\)
−0.999187 + 0.0403130i \(0.987164\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −3.37181 −0.372354
\(83\) 14.3735i 1.57769i 0.614590 + 0.788847i \(0.289322\pi\)
−0.614590 + 0.788847i \(0.710678\pi\)
\(84\) 0 0
\(85\) 2.08407i 0.226049i
\(86\) 4.24695i 0.457961i
\(87\) 0 0
\(88\) −3.99641 −0.426019
\(89\) 18.2414i 1.93358i 0.255568 + 0.966791i \(0.417738\pi\)
−0.255568 + 0.966791i \(0.582262\pi\)
\(90\) 0 0
\(91\) 14.7389 1.54506
\(92\) 3.40101i 0.354580i
\(93\) 0 0
\(94\) 0.604591i 0.0623588i
\(95\) −3.38304 −0.347092
\(96\) 0 0
\(97\) 4.52609i 0.459555i −0.973243 0.229777i \(-0.926200\pi\)
0.973243 0.229777i \(-0.0737998\pi\)
\(98\) 5.13654 0.518869
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 1.78667 0.177780 0.0888900 0.996041i \(-0.471668\pi\)
0.0888900 + 0.996041i \(0.471668\pi\)
\(102\) 0 0
\(103\) −4.68292 −0.461422 −0.230711 0.973022i \(-0.574105\pi\)
−0.230711 + 0.973022i \(0.574105\pi\)
\(104\) 4.23077i 0.414861i
\(105\) 0 0
\(106\) 12.4512 1.20937
\(107\) 9.29118i 0.898212i 0.893478 + 0.449106i \(0.148257\pi\)
−0.893478 + 0.449106i \(0.851743\pi\)
\(108\) 0 0
\(109\) 4.13442i 0.396005i −0.980201 0.198003i \(-0.936555\pi\)
0.980201 0.198003i \(-0.0634455\pi\)
\(110\) 3.99641 0.381043
\(111\) 0 0
\(112\) 3.48375i 0.329184i
\(113\) −11.0862 −1.04290 −0.521451 0.853281i \(-0.674609\pi\)
−0.521451 + 0.853281i \(0.674609\pi\)
\(114\) 0 0
\(115\) 3.40101i 0.317146i
\(116\) 7.87608i 0.731276i
\(117\) 0 0
\(118\) 6.23827i 0.574279i
\(119\) −7.26038 −0.665558
\(120\) 0 0
\(121\) 4.97130 0.451936
\(122\) 2.59685i 0.235107i
\(123\) 0 0
\(124\) 4.40560i 0.395634i
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −18.1578 −1.61125 −0.805623 0.592429i \(-0.798169\pi\)
−0.805623 + 0.592429i \(0.798169\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 4.23077i 0.371063i
\(131\) 17.7282i 1.54892i −0.632622 0.774461i \(-0.718021\pi\)
0.632622 0.774461i \(-0.281979\pi\)
\(132\) 0 0
\(133\) 11.7857i 1.02195i
\(134\) 7.64991 + 2.91185i 0.660852 + 0.251545i
\(135\) 0 0
\(136\) 2.08407i 0.178707i
\(137\) −3.48287 −0.297562 −0.148781 0.988870i \(-0.547535\pi\)
−0.148781 + 0.988870i \(0.547535\pi\)
\(138\) 0 0
\(139\) 12.1021i 1.02648i 0.858244 + 0.513242i \(0.171556\pi\)
−0.858244 + 0.513242i \(0.828444\pi\)
\(140\) 3.48375i 0.294431i
\(141\) 0 0
\(142\) 8.66350i 0.727025i
\(143\) 16.9079i 1.41391i
\(144\) 0 0
\(145\) 7.87608i 0.654073i
\(146\) −10.3997 −0.860685
\(147\) 0 0
\(148\) 6.85166 0.563203
\(149\) 3.95643i 0.324123i −0.986781 0.162062i \(-0.948186\pi\)
0.986781 0.162062i \(-0.0518144\pi\)
\(150\) 0 0
\(151\) −8.69934 −0.707942 −0.353971 0.935256i \(-0.615169\pi\)
−0.353971 + 0.935256i \(0.615169\pi\)
\(152\) −3.38304 −0.274400
\(153\) 0 0
\(154\) 13.9225i 1.12191i
\(155\) 4.40560i 0.353866i
\(156\) 0 0
\(157\) −12.4921 −0.996981 −0.498491 0.866895i \(-0.666112\pi\)
−0.498491 + 0.866895i \(0.666112\pi\)
\(158\) 0.716620i 0.0570112i
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 11.8483 0.933777
\(162\) 0 0
\(163\) 11.8292 0.926538 0.463269 0.886218i \(-0.346676\pi\)
0.463269 + 0.886218i \(0.346676\pi\)
\(164\) 3.37181 0.263294
\(165\) 0 0
\(166\) 14.3735i 1.11560i
\(167\) 16.5249i 1.27874i 0.768900 + 0.639369i \(0.220804\pi\)
−0.768900 + 0.639369i \(0.779196\pi\)
\(168\) 0 0
\(169\) −4.89938 −0.376876
\(170\) 2.08407i 0.159841i
\(171\) 0 0
\(172\) 4.24695i 0.323827i
\(173\) 19.1729i 1.45769i −0.684679 0.728844i \(-0.740058\pi\)
0.684679 0.728844i \(-0.259942\pi\)
\(174\) 0 0
\(175\) 3.48375i 0.263347i
\(176\) 3.99641 0.301241
\(177\) 0 0
\(178\) 18.2414i 1.36725i
\(179\) 15.8601 1.18544 0.592719 0.805409i \(-0.298054\pi\)
0.592719 + 0.805409i \(0.298054\pi\)
\(180\) 0 0
\(181\) 13.3117 0.989450 0.494725 0.869050i \(-0.335269\pi\)
0.494725 + 0.869050i \(0.335269\pi\)
\(182\) −14.7389 −1.09252
\(183\) 0 0
\(184\) 3.40101i 0.250726i
\(185\) −6.85166 −0.503744
\(186\) 0 0
\(187\) 8.32879i 0.609061i
\(188\) 0.604591i 0.0440943i
\(189\) 0 0
\(190\) 3.38304 0.245431
\(191\) 21.3256 1.54306 0.771532 0.636190i \(-0.219491\pi\)
0.771532 + 0.636190i \(0.219491\pi\)
\(192\) 0 0
\(193\) 8.68487 0.625151 0.312575 0.949893i \(-0.398808\pi\)
0.312575 + 0.949893i \(0.398808\pi\)
\(194\) 4.52609i 0.324954i
\(195\) 0 0
\(196\) −5.13654 −0.366896
\(197\) 21.1629 1.50779 0.753896 0.656994i \(-0.228172\pi\)
0.753896 + 0.656994i \(0.228172\pi\)
\(198\) 0 0
\(199\) 1.78351 0.126430 0.0632148 0.998000i \(-0.479865\pi\)
0.0632148 + 0.998000i \(0.479865\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −1.78667 −0.125709
\(203\) 27.4383 1.92579
\(204\) 0 0
\(205\) −3.37181 −0.235498
\(206\) 4.68292 0.326275
\(207\) 0 0
\(208\) 4.23077i 0.293351i
\(209\) 13.5200 0.935198
\(210\) 0 0
\(211\) 3.94725 0.271740 0.135870 0.990727i \(-0.456617\pi\)
0.135870 + 0.990727i \(0.456617\pi\)
\(212\) −12.4512 −0.855153
\(213\) 0 0
\(214\) 9.29118i 0.635132i
\(215\) 4.24695i 0.289640i
\(216\) 0 0
\(217\) −15.3480 −1.04189
\(218\) 4.13442i 0.280018i
\(219\) 0 0
\(220\) −3.99641 −0.269438
\(221\) 8.81720 0.593109
\(222\) 0 0
\(223\) 3.06080 0.204967 0.102483 0.994735i \(-0.467321\pi\)
0.102483 + 0.994735i \(0.467321\pi\)
\(224\) 3.48375i 0.232768i
\(225\) 0 0
\(226\) 11.0862 0.737443
\(227\) 0.309618i 0.0205501i −0.999947 0.0102750i \(-0.996729\pi\)
0.999947 0.0102750i \(-0.00327070\pi\)
\(228\) 0 0
\(229\) 3.45165i 0.228091i −0.993476 0.114046i \(-0.963619\pi\)
0.993476 0.114046i \(-0.0363810\pi\)
\(230\) 3.40101i 0.224256i
\(231\) 0 0
\(232\) 7.87608i 0.517090i
\(233\) 13.0421 0.854419 0.427210 0.904153i \(-0.359497\pi\)
0.427210 + 0.904153i \(0.359497\pi\)
\(234\) 0 0
\(235\) 0.604591i 0.0394392i
\(236\) 6.23827i 0.406076i
\(237\) 0 0
\(238\) 7.26038 0.470620
\(239\) 8.10257 0.524112 0.262056 0.965053i \(-0.415600\pi\)
0.262056 + 0.965053i \(0.415600\pi\)
\(240\) 0 0
\(241\) 19.6546 1.26606 0.633031 0.774127i \(-0.281811\pi\)
0.633031 + 0.774127i \(0.281811\pi\)
\(242\) −4.97130 −0.319567
\(243\) 0 0
\(244\) 2.59685i 0.166246i
\(245\) 5.13654 0.328161
\(246\) 0 0
\(247\) 14.3128i 0.910704i
\(248\) 4.40560i 0.279756i
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −8.55102 −0.539736 −0.269868 0.962897i \(-0.586980\pi\)
−0.269868 + 0.962897i \(0.586980\pi\)
\(252\) 0 0
\(253\) 13.5919i 0.854512i
\(254\) 18.1578 1.13932
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 3.41059i 0.212747i 0.994326 + 0.106373i \(0.0339239\pi\)
−0.994326 + 0.106373i \(0.966076\pi\)
\(258\) 0 0
\(259\) 23.8695i 1.48318i
\(260\) 4.23077i 0.262381i
\(261\) 0 0
\(262\) 17.7282i 1.09525i
\(263\) 25.1417i 1.55030i 0.631776 + 0.775151i \(0.282326\pi\)
−0.631776 + 0.775151i \(0.717674\pi\)
\(264\) 0 0
\(265\) 12.4512 0.764872
\(266\) 11.7857i 0.722625i
\(267\) 0 0
\(268\) −7.64991 2.91185i −0.467293 0.177869i
\(269\) 1.86659i 0.113808i 0.998380 + 0.0569041i \(0.0181229\pi\)
−0.998380 + 0.0569041i \(0.981877\pi\)
\(270\) 0 0
\(271\) 19.0863i 1.15941i 0.814825 + 0.579706i \(0.196833\pi\)
−0.814825 + 0.579706i \(0.803167\pi\)
\(272\) 2.08407i 0.126365i
\(273\) 0 0
\(274\) 3.48287 0.210408
\(275\) 3.99641 0.240993
\(276\) 0 0
\(277\) 12.0262 0.722585 0.361292 0.932453i \(-0.382336\pi\)
0.361292 + 0.932453i \(0.382336\pi\)
\(278\) 12.1021i 0.725834i
\(279\) 0 0
\(280\) 3.48375i 0.208194i
\(281\) −15.6985 −0.936495 −0.468247 0.883597i \(-0.655114\pi\)
−0.468247 + 0.883597i \(0.655114\pi\)
\(282\) 0 0
\(283\) −20.1358 −1.19695 −0.598475 0.801142i \(-0.704226\pi\)
−0.598475 + 0.801142i \(0.704226\pi\)
\(284\) 8.66350i 0.514084i
\(285\) 0 0
\(286\) 16.9079i 0.999784i
\(287\) 11.7466i 0.693378i
\(288\) 0 0
\(289\) 12.6567 0.744510
\(290\) 7.87608i 0.462500i
\(291\) 0 0
\(292\) 10.3997 0.608596
\(293\) 20.6521i 1.20651i −0.797550 0.603253i \(-0.793871\pi\)
0.797550 0.603253i \(-0.206129\pi\)
\(294\) 0 0
\(295\) 6.23827i 0.363206i
\(296\) −6.85166 −0.398245
\(297\) 0 0
\(298\) 3.95643i 0.229190i
\(299\) −14.3889 −0.832132
\(300\) 0 0
\(301\) 14.7953 0.852789
\(302\) 8.69934 0.500591
\(303\) 0 0
\(304\) 3.38304 0.194030
\(305\) 2.59685i 0.148695i
\(306\) 0 0
\(307\) 21.6799 1.23734 0.618669 0.785652i \(-0.287672\pi\)
0.618669 + 0.785652i \(0.287672\pi\)
\(308\) 13.9225i 0.793309i
\(309\) 0 0
\(310\) 4.40560i 0.250221i
\(311\) −13.1375 −0.744961 −0.372481 0.928040i \(-0.621493\pi\)
−0.372481 + 0.928040i \(0.621493\pi\)
\(312\) 0 0
\(313\) 16.0459i 0.906966i 0.891265 + 0.453483i \(0.149819\pi\)
−0.891265 + 0.453483i \(0.850181\pi\)
\(314\) 12.4921 0.704972
\(315\) 0 0
\(316\) 0.716620i 0.0403130i
\(317\) 6.77333i 0.380428i −0.981743 0.190214i \(-0.939082\pi\)
0.981743 0.190214i \(-0.0609182\pi\)
\(318\) 0 0
\(319\) 31.4761i 1.76232i
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −11.8483 −0.660280
\(323\) 7.05047i 0.392299i
\(324\) 0 0
\(325\) 4.23077i 0.234681i
\(326\) −11.8292 −0.655161
\(327\) 0 0
\(328\) −3.37181 −0.186177
\(329\) 2.10625 0.116121
\(330\) 0 0
\(331\) 7.91254i 0.434912i −0.976070 0.217456i \(-0.930224\pi\)
0.976070 0.217456i \(-0.0697759\pi\)
\(332\) 14.3735i 0.788847i
\(333\) 0 0
\(334\) 16.5249i 0.904204i
\(335\) 7.64991 + 2.91185i 0.417959 + 0.159091i
\(336\) 0 0
\(337\) 3.60081i 0.196148i 0.995179 + 0.0980742i \(0.0312683\pi\)
−0.995179 + 0.0980742i \(0.968732\pi\)
\(338\) 4.89938 0.266491
\(339\) 0 0
\(340\) 2.08407i 0.113024i
\(341\) 17.6066i 0.953449i
\(342\) 0 0
\(343\) 6.49184i 0.350526i
\(344\) 4.24695i 0.228980i
\(345\) 0 0
\(346\) 19.1729i 1.03074i
\(347\) 12.3427 0.662591 0.331295 0.943527i \(-0.392514\pi\)
0.331295 + 0.943527i \(0.392514\pi\)
\(348\) 0 0
\(349\) 0.685387 0.0366879 0.0183440 0.999832i \(-0.494161\pi\)
0.0183440 + 0.999832i \(0.494161\pi\)
\(350\) 3.48375i 0.186214i
\(351\) 0 0
\(352\) −3.99641 −0.213009
\(353\) −26.1784 −1.39334 −0.696669 0.717393i \(-0.745335\pi\)
−0.696669 + 0.717393i \(0.745335\pi\)
\(354\) 0 0
\(355\) 8.66350i 0.459811i
\(356\) 18.2414i 0.966791i
\(357\) 0 0
\(358\) −15.8601 −0.838231
\(359\) 13.3188i 0.702940i −0.936199 0.351470i \(-0.885682\pi\)
0.936199 0.351470i \(-0.114318\pi\)
\(360\) 0 0
\(361\) −7.55507 −0.397635
\(362\) −13.3117 −0.699647
\(363\) 0 0
\(364\) 14.7389 0.772531
\(365\) −10.3997 −0.544345
\(366\) 0 0
\(367\) 11.6296i 0.607060i −0.952822 0.303530i \(-0.901835\pi\)
0.952822 0.303530i \(-0.0981652\pi\)
\(368\) 3.40101i 0.177290i
\(369\) 0 0
\(370\) 6.85166 0.356201
\(371\) 43.3770i 2.25202i
\(372\) 0 0
\(373\) 4.54904i 0.235540i 0.993041 + 0.117770i \(0.0375746\pi\)
−0.993041 + 0.117770i \(0.962425\pi\)
\(374\) 8.32879i 0.430671i
\(375\) 0 0
\(376\) 0.604591i 0.0311794i
\(377\) −33.3219 −1.71616
\(378\) 0 0
\(379\) 22.5627i 1.15897i 0.814983 + 0.579484i \(0.196746\pi\)
−0.814983 + 0.579484i \(0.803254\pi\)
\(380\) −3.38304 −0.173546
\(381\) 0 0
\(382\) −21.3256 −1.09111
\(383\) 21.2816 1.08744 0.543719 0.839268i \(-0.317016\pi\)
0.543719 + 0.839268i \(0.317016\pi\)
\(384\) 0 0
\(385\) 13.9225i 0.709557i
\(386\) −8.68487 −0.442048
\(387\) 0 0
\(388\) 4.52609i 0.229777i
\(389\) 21.3579i 1.08289i −0.840737 0.541443i \(-0.817878\pi\)
0.840737 0.541443i \(-0.182122\pi\)
\(390\) 0 0
\(391\) 7.08794 0.358453
\(392\) 5.13654 0.259434
\(393\) 0 0
\(394\) −21.1629 −1.06617
\(395\) 0.716620i 0.0360571i
\(396\) 0 0
\(397\) −1.91628 −0.0961754 −0.0480877 0.998843i \(-0.515313\pi\)
−0.0480877 + 0.998843i \(0.515313\pi\)
\(398\) −1.78351 −0.0893993
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −6.89731 −0.344435 −0.172218 0.985059i \(-0.555093\pi\)
−0.172218 + 0.985059i \(0.555093\pi\)
\(402\) 0 0
\(403\) 18.6391 0.928477
\(404\) 1.78667 0.0888900
\(405\) 0 0
\(406\) −27.4383 −1.36174
\(407\) 27.3820 1.35728
\(408\) 0 0
\(409\) 32.9734i 1.63043i 0.579160 + 0.815214i \(0.303381\pi\)
−0.579160 + 0.815214i \(0.696619\pi\)
\(410\) 3.37181 0.166522
\(411\) 0 0
\(412\) −4.68292 −0.230711
\(413\) 21.7326 1.06939
\(414\) 0 0
\(415\) 14.3735i 0.705566i
\(416\) 4.23077i 0.207430i
\(417\) 0 0
\(418\) −13.5200 −0.661285
\(419\) 8.84340i 0.432028i 0.976390 + 0.216014i \(0.0693058\pi\)
−0.976390 + 0.216014i \(0.930694\pi\)
\(420\) 0 0
\(421\) 31.5517 1.53774 0.768868 0.639408i \(-0.220820\pi\)
0.768868 + 0.639408i \(0.220820\pi\)
\(422\) −3.94725 −0.192149
\(423\) 0 0
\(424\) 12.4512 0.604685
\(425\) 2.08407i 0.101092i
\(426\) 0 0
\(427\) −9.04677 −0.437804
\(428\) 9.29118i 0.449106i
\(429\) 0 0
\(430\) 4.24695i 0.204806i
\(431\) 26.2803i 1.26588i 0.774202 + 0.632939i \(0.218152\pi\)
−0.774202 + 0.632939i \(0.781848\pi\)
\(432\) 0 0
\(433\) 13.6961i 0.658193i 0.944296 + 0.329096i \(0.106744\pi\)
−0.944296 + 0.329096i \(0.893256\pi\)
\(434\) 15.3480 0.736728
\(435\) 0 0
\(436\) 4.13442i 0.198003i
\(437\) 11.5058i 0.550395i
\(438\) 0 0
\(439\) 15.2634 0.728481 0.364241 0.931305i \(-0.381329\pi\)
0.364241 + 0.931305i \(0.381329\pi\)
\(440\) 3.99641 0.190521
\(441\) 0 0
\(442\) −8.81720 −0.419391
\(443\) −22.5273 −1.07030 −0.535152 0.844756i \(-0.679746\pi\)
−0.535152 + 0.844756i \(0.679746\pi\)
\(444\) 0 0
\(445\) 18.2414i 0.864724i
\(446\) −3.06080 −0.144933
\(447\) 0 0
\(448\) 3.48375i 0.164592i
\(449\) 29.9612i 1.41396i 0.707235 + 0.706978i \(0.249942\pi\)
−0.707235 + 0.706978i \(0.750058\pi\)
\(450\) 0 0
\(451\) 13.4751 0.634520
\(452\) −11.0862 −0.521451
\(453\) 0 0
\(454\) 0.309618i 0.0145311i
\(455\) −14.7389 −0.690973
\(456\) 0 0
\(457\) 22.6809 1.06097 0.530485 0.847694i \(-0.322010\pi\)
0.530485 + 0.847694i \(0.322010\pi\)
\(458\) 3.45165i 0.161285i
\(459\) 0 0
\(460\) 3.40101i 0.158573i
\(461\) 10.5378i 0.490796i 0.969422 + 0.245398i \(0.0789186\pi\)
−0.969422 + 0.245398i \(0.921081\pi\)
\(462\) 0 0
\(463\) 20.6778i 0.960979i 0.877001 + 0.480489i \(0.159541\pi\)
−0.877001 + 0.480489i \(0.840459\pi\)
\(464\) 7.87608i 0.365638i
\(465\) 0 0
\(466\) −13.0421 −0.604166
\(467\) 5.30865i 0.245655i 0.992428 + 0.122827i \(0.0391962\pi\)
−0.992428 + 0.122827i \(0.960804\pi\)
\(468\) 0 0
\(469\) −10.1442 + 26.6504i −0.468414 + 1.23060i
\(470\) 0.604591i 0.0278877i
\(471\) 0 0
\(472\) 6.23827i 0.287139i
\(473\) 16.9726i 0.780399i
\(474\) 0 0
\(475\) 3.38304 0.155224
\(476\) −7.26038 −0.332779
\(477\) 0 0
\(478\) −8.10257 −0.370603
\(479\) 20.1076i 0.918737i −0.888246 0.459369i \(-0.848076\pi\)
0.888246 0.459369i \(-0.151924\pi\)
\(480\) 0 0
\(481\) 28.9878i 1.32173i
\(482\) −19.6546 −0.895241
\(483\) 0 0
\(484\) 4.97130 0.225968
\(485\) 4.52609i 0.205519i
\(486\) 0 0
\(487\) 22.4926i 1.01923i −0.860401 0.509617i \(-0.829787\pi\)
0.860401 0.509617i \(-0.170213\pi\)
\(488\) 2.59685i 0.117554i
\(489\) 0 0
\(490\) −5.13654 −0.232045
\(491\) 7.59955i 0.342963i 0.985187 + 0.171482i \(0.0548554\pi\)
−0.985187 + 0.171482i \(0.945145\pi\)
\(492\) 0 0
\(493\) 16.4143 0.739262
\(494\) 14.3128i 0.643965i
\(495\) 0 0
\(496\) 4.40560i 0.197817i
\(497\) 30.1815 1.35383
\(498\) 0 0
\(499\) 14.1638i 0.634058i 0.948416 + 0.317029i \(0.102685\pi\)
−0.948416 + 0.317029i \(0.897315\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 8.55102 0.381651
\(503\) −29.1111 −1.29800 −0.649000 0.760789i \(-0.724812\pi\)
−0.649000 + 0.760789i \(0.724812\pi\)
\(504\) 0 0
\(505\) −1.78667 −0.0795056
\(506\) 13.5919i 0.604231i
\(507\) 0 0
\(508\) −18.1578 −0.805623
\(509\) 32.9968i 1.46256i −0.682078 0.731280i \(-0.738923\pi\)
0.682078 0.731280i \(-0.261077\pi\)
\(510\) 0 0
\(511\) 36.2300i 1.60272i
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 3.41059i 0.150435i
\(515\) 4.68292 0.206354
\(516\) 0 0
\(517\) 2.41619i 0.106264i
\(518\) 23.8695i 1.04877i
\(519\) 0 0
\(520\) 4.23077i 0.185531i
\(521\) 1.17209 0.0513501 0.0256750 0.999670i \(-0.491826\pi\)
0.0256750 + 0.999670i \(0.491826\pi\)
\(522\) 0 0
\(523\) 33.2501 1.45393 0.726963 0.686676i \(-0.240931\pi\)
0.726963 + 0.686676i \(0.240931\pi\)
\(524\) 17.7282i 0.774461i
\(525\) 0 0
\(526\) 25.1417i 1.09623i
\(527\) −9.18156 −0.399955
\(528\) 0 0
\(529\) 11.4331 0.497091
\(530\) −12.4512 −0.540846
\(531\) 0 0
\(532\) 11.7857i 0.510973i
\(533\) 14.2653i 0.617901i
\(534\) 0 0
\(535\) 9.29118i 0.401693i
\(536\) 7.64991 + 2.91185i 0.330426 + 0.125773i
\(537\) 0 0
\(538\) 1.86659i 0.0804745i
\(539\) −20.5277 −0.884192
\(540\) 0 0
\(541\) 4.00668i 0.172261i −0.996284 0.0861304i \(-0.972550\pi\)
0.996284 0.0861304i \(-0.0274502\pi\)
\(542\) 19.0863i 0.819829i
\(543\) 0 0
\(544\) 2.08407i 0.0893536i
\(545\) 4.13442i 0.177099i
\(546\) 0 0
\(547\) 6.26255i 0.267767i −0.990997 0.133884i \(-0.957255\pi\)
0.990997 0.133884i \(-0.0427449\pi\)
\(548\) −3.48287 −0.148781
\(549\) 0 0
\(550\) −3.99641 −0.170408
\(551\) 26.6451i 1.13512i
\(552\) 0 0
\(553\) 2.49653 0.106163
\(554\) −12.0262 −0.510944
\(555\) 0 0
\(556\) 12.1021i 0.513242i
\(557\) 2.09611i 0.0888148i −0.999014 0.0444074i \(-0.985860\pi\)
0.999014 0.0444074i \(-0.0141400\pi\)
\(558\) 0 0
\(559\) −17.9679 −0.759960
\(560\) 3.48375i 0.147215i
\(561\) 0 0
\(562\) 15.6985 0.662202
\(563\) 9.67579 0.407786 0.203893 0.978993i \(-0.434641\pi\)
0.203893 + 0.978993i \(0.434641\pi\)
\(564\) 0 0
\(565\) 11.0862 0.466400
\(566\) 20.1358 0.846371
\(567\) 0 0
\(568\) 8.66350i 0.363512i
\(569\) 4.77375i 0.200126i −0.994981 0.100063i \(-0.968096\pi\)
0.994981 0.100063i \(-0.0319044\pi\)
\(570\) 0 0
\(571\) 35.3058 1.47750 0.738752 0.673978i \(-0.235416\pi\)
0.738752 + 0.673978i \(0.235416\pi\)
\(572\) 16.9079i 0.706954i
\(573\) 0 0
\(574\) 11.7466i 0.490292i
\(575\) 3.40101i 0.141832i
\(576\) 0 0
\(577\) 19.8356i 0.825769i −0.910783 0.412884i \(-0.864521\pi\)
0.910783 0.412884i \(-0.135479\pi\)
\(578\) −12.6567 −0.526448
\(579\) 0 0
\(580\) 7.87608i 0.327037i
\(581\) 50.0737 2.07741
\(582\) 0 0
\(583\) −49.7602 −2.06086
\(584\) −10.3997 −0.430342
\(585\) 0 0
\(586\) 20.6521i 0.853129i
\(587\) −41.6201 −1.71785 −0.858923 0.512104i \(-0.828866\pi\)
−0.858923 + 0.512104i \(0.828866\pi\)
\(588\) 0 0
\(589\) 14.9043i 0.614121i
\(590\) 6.23827i 0.256825i
\(591\) 0 0
\(592\) 6.85166 0.281601
\(593\) 31.4410 1.29113 0.645563 0.763707i \(-0.276623\pi\)
0.645563 + 0.763707i \(0.276623\pi\)
\(594\) 0 0
\(595\) 7.26038 0.297646
\(596\) 3.95643i 0.162062i
\(597\) 0 0
\(598\) 14.3889 0.588406
\(599\) −47.2767 −1.93167 −0.965837 0.259152i \(-0.916557\pi\)
−0.965837 + 0.259152i \(0.916557\pi\)
\(600\) 0 0
\(601\) −28.4147 −1.15906 −0.579529 0.814951i \(-0.696764\pi\)
−0.579529 + 0.814951i \(0.696764\pi\)
\(602\) −14.7953 −0.603013
\(603\) 0 0
\(604\) −8.69934 −0.353971
\(605\) −4.97130 −0.202112
\(606\) 0 0
\(607\) −22.3704 −0.907986 −0.453993 0.891005i \(-0.650001\pi\)
−0.453993 + 0.891005i \(0.650001\pi\)
\(608\) −3.38304 −0.137200
\(609\) 0 0
\(610\) 2.59685i 0.105143i
\(611\) −2.55788 −0.103481
\(612\) 0 0
\(613\) −28.3712 −1.14590 −0.572950 0.819590i \(-0.694201\pi\)
−0.572950 + 0.819590i \(0.694201\pi\)
\(614\) −21.6799 −0.874930
\(615\) 0 0
\(616\) 13.9225i 0.560954i
\(617\) 15.9077i 0.640418i 0.947347 + 0.320209i \(0.103753\pi\)
−0.947347 + 0.320209i \(0.896247\pi\)
\(618\) 0 0
\(619\) 28.2806 1.13669 0.568347 0.822789i \(-0.307583\pi\)
0.568347 + 0.822789i \(0.307583\pi\)
\(620\) 4.40560i 0.176933i
\(621\) 0 0
\(622\) 13.1375 0.526767
\(623\) 63.5485 2.54602
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 16.0459i 0.641322i
\(627\) 0 0
\(628\) −12.4921 −0.498491
\(629\) 14.2793i 0.569353i
\(630\) 0 0
\(631\) 17.4570i 0.694952i 0.937689 + 0.347476i \(0.112961\pi\)
−0.937689 + 0.347476i \(0.887039\pi\)
\(632\) 0.716620i 0.0285056i
\(633\) 0 0
\(634\) 6.77333i 0.269003i
\(635\) 18.1578 0.720571
\(636\) 0 0
\(637\) 21.7315i 0.861033i
\(638\) 31.4761i 1.24615i
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 39.8394 1.57356 0.786781 0.617232i \(-0.211746\pi\)
0.786781 + 0.617232i \(0.211746\pi\)
\(642\) 0 0
\(643\) 28.3510 1.11806 0.559028 0.829149i \(-0.311174\pi\)
0.559028 + 0.829149i \(0.311174\pi\)
\(644\) 11.8483 0.466888
\(645\) 0 0
\(646\) 7.05047i 0.277397i
\(647\) 28.3642 1.11511 0.557556 0.830140i \(-0.311739\pi\)
0.557556 + 0.830140i \(0.311739\pi\)
\(648\) 0 0
\(649\) 24.9307i 0.978614i
\(650\) 4.23077i 0.165944i
\(651\) 0 0
\(652\) 11.8292 0.463269
\(653\) −7.12955 −0.279001 −0.139500 0.990222i \(-0.544550\pi\)
−0.139500 + 0.990222i \(0.544550\pi\)
\(654\) 0 0
\(655\) 17.7282i 0.692699i
\(656\) 3.37181 0.131647
\(657\) 0 0
\(658\) −2.10625 −0.0821100
\(659\) 18.2907i 0.712503i −0.934390 0.356252i \(-0.884055\pi\)
0.934390 0.356252i \(-0.115945\pi\)
\(660\) 0 0
\(661\) 32.5538i 1.26620i 0.774072 + 0.633098i \(0.218217\pi\)
−0.774072 + 0.633098i \(0.781783\pi\)
\(662\) 7.91254i 0.307529i
\(663\) 0 0
\(664\) 14.3735i 0.557799i
\(665\) 11.7857i 0.457028i
\(666\) 0 0
\(667\) −26.7867 −1.03718
\(668\) 16.5249i 0.639369i
\(669\) 0 0
\(670\) −7.64991 2.91185i −0.295542 0.112495i
\(671\) 10.3781i 0.400641i
\(672\) 0 0
\(673\) 10.1435i 0.391003i 0.980703 + 0.195501i \(0.0626334\pi\)
−0.980703 + 0.195501i \(0.937367\pi\)
\(674\) 3.60081i 0.138698i
\(675\) 0 0
\(676\) −4.89938 −0.188438
\(677\) 23.0518 0.885952 0.442976 0.896533i \(-0.353923\pi\)
0.442976 + 0.896533i \(0.353923\pi\)
\(678\) 0 0
\(679\) −15.7678 −0.605112
\(680\) 2.08407i 0.0799203i
\(681\) 0 0
\(682\) 17.6066i 0.674191i
\(683\) −28.8993 −1.10580 −0.552901 0.833247i \(-0.686479\pi\)
−0.552901 + 0.833247i \(0.686479\pi\)
\(684\) 0 0
\(685\) 3.48287 0.133074
\(686\) 6.49184i 0.247859i
\(687\) 0 0
\(688\) 4.24695i 0.161914i
\(689\) 52.6782i 2.00688i
\(690\) 0 0
\(691\) 22.6111 0.860167 0.430083 0.902789i \(-0.358484\pi\)
0.430083 + 0.902789i \(0.358484\pi\)
\(692\) 19.1729i 0.728844i
\(693\) 0 0
\(694\) −12.3427 −0.468522
\(695\) 12.1021i 0.459057i
\(696\) 0 0
\(697\) 7.02708i 0.266170i
\(698\) −0.685387 −0.0259423
\(699\) 0 0
\(700\) 3.48375i 0.131674i
\(701\) 42.6689 1.61158 0.805791 0.592200i \(-0.201740\pi\)
0.805791 + 0.592200i \(0.201740\pi\)
\(702\) 0 0
\(703\) 23.1794 0.874228
\(704\) 3.99641 0.150620
\(705\) 0 0
\(706\) 26.1784 0.985238
\(707\) 6.22431i 0.234089i
\(708\) 0 0
\(709\) 25.2920 0.949859 0.474930 0.880024i \(-0.342473\pi\)
0.474930 + 0.880024i \(0.342473\pi\)
\(710\) 8.66350i 0.325135i
\(711\) 0 0
\(712\) 18.2414i 0.683625i
\(713\) 14.9835 0.561137
\(714\) 0 0
\(715\) 16.9079i 0.632319i
\(716\) 15.8601 0.592719
\(717\) 0 0
\(718\) 13.3188i 0.497053i
\(719\) 32.2768i 1.20372i −0.798601 0.601860i \(-0.794426\pi\)
0.798601 0.601860i \(-0.205574\pi\)
\(720\) 0 0
\(721\) 16.3141i 0.607570i
\(722\) 7.55507 0.281171
\(723\) 0 0
\(724\) 13.3117 0.494725
\(725\) 7.87608i 0.292510i
\(726\) 0 0
\(727\) 21.0542i 0.780856i 0.920634 + 0.390428i \(0.127673\pi\)
−0.920634 + 0.390428i \(0.872327\pi\)
\(728\) −14.7389 −0.546262
\(729\) 0 0
\(730\) 10.3997 0.384910
\(731\) 8.85093 0.327364
\(732\) 0 0
\(733\) 21.1508i 0.781221i −0.920556 0.390611i \(-0.872264\pi\)
0.920556 0.390611i \(-0.127736\pi\)
\(734\) 11.6296i 0.429256i
\(735\) 0 0
\(736\) 3.40101i 0.125363i
\(737\) −30.5722 11.6369i −1.12614 0.428652i
\(738\) 0 0
\(739\) 25.7851i 0.948519i 0.880385 + 0.474260i \(0.157284\pi\)
−0.880385 + 0.474260i \(0.842716\pi\)
\(740\) −6.85166 −0.251872
\(741\) 0 0
\(742\) 43.3770i 1.59242i
\(743\) 21.0941i 0.773869i 0.922107 + 0.386935i \(0.126466\pi\)
−0.922107 + 0.386935i \(0.873534\pi\)
\(744\) 0 0
\(745\) 3.95643i 0.144952i
\(746\) 4.54904i 0.166552i
\(747\) 0 0
\(748\) 8.32879i 0.304531i
\(749\) 32.3682 1.18271
\(750\) 0 0
\(751\) −37.2197 −1.35817 −0.679084 0.734061i \(-0.737623\pi\)
−0.679084 + 0.734061i \(0.737623\pi\)
\(752\) 0.604591i 0.0220472i
\(753\) 0 0
\(754\) 33.3219 1.21351
\(755\) 8.69934 0.316601
\(756\) 0 0
\(757\) 44.7365i 1.62597i −0.582281 0.812987i \(-0.697840\pi\)
0.582281 0.812987i \(-0.302160\pi\)
\(758\) 22.5627i 0.819514i
\(759\) 0 0
\(760\) 3.38304 0.122716
\(761\) 40.9089i 1.48295i −0.670982 0.741474i \(-0.734127\pi\)
0.670982 0.741474i \(-0.265873\pi\)
\(762\) 0 0
\(763\) −14.4033 −0.521434
\(764\) 21.3256 0.771532
\(765\) 0 0
\(766\) −21.2816 −0.768934
\(767\) −26.3926 −0.952983
\(768\) 0 0
\(769\) 24.5575i 0.885568i −0.896628 0.442784i \(-0.853991\pi\)
0.896628 0.442784i \(-0.146009\pi\)
\(770\) 13.9225i 0.501732i
\(771\) 0 0
\(772\) 8.68487 0.312575
\(773\) 5.23417i 0.188260i −0.995560 0.0941301i \(-0.969993\pi\)
0.995560 0.0941301i \(-0.0300070\pi\)
\(774\) 0 0
\(775\) 4.40560i 0.158254i
\(776\) 4.52609i 0.162477i
\(777\) 0 0
\(778\) 21.3579i 0.765717i
\(779\) 11.4070 0.408697
\(780\) 0 0
\(781\) 34.6229i 1.23890i
\(782\) −7.08794 −0.253464
\(783\) 0 0
\(784\) −5.13654 −0.183448
\(785\) 12.4921 0.445864
\(786\) 0 0
\(787\) 20.2364i 0.721348i 0.932692 + 0.360674i \(0.117453\pi\)
−0.932692 + 0.360674i \(0.882547\pi\)
\(788\) 21.1629 0.753896
\(789\) 0 0
\(790\) 0.716620i 0.0254962i
\(791\) 38.6216i 1.37323i
\(792\) 0 0
\(793\) 10.9866 0.390147
\(794\) 1.91628 0.0680063
\(795\) 0 0
\(796\) 1.78351 0.0632148
\(797\) 33.3752i 1.18221i 0.806595 + 0.591105i \(0.201308\pi\)
−0.806595 + 0.591105i \(0.798692\pi\)
\(798\) 0 0
\(799\) 1.26001 0.0445759
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 6.89731 0.243552
\(803\) 41.5614 1.46667
\(804\) 0 0
\(805\) −11.8483 −0.417598
\(806\) −18.6391 −0.656533
\(807\) 0 0
\(808\) −1.78667 −0.0628547
\(809\) 49.3188 1.73395 0.866977 0.498347i \(-0.166060\pi\)
0.866977 + 0.498347i \(0.166060\pi\)
\(810\) 0 0
\(811\) 30.3513i 1.06578i 0.846185 + 0.532889i \(0.178894\pi\)
−0.846185 + 0.532889i \(0.821106\pi\)
\(812\) 27.4383 0.962897
\(813\) 0 0
\(814\) −27.3820 −0.959740
\(815\) −11.8292 −0.414360
\(816\) 0 0
\(817\) 14.3676i 0.502658i
\(818\) 32.9734i 1.15289i
\(819\) 0 0
\(820\) −3.37181 −0.117749
\(821\) 38.7641i 1.35288i 0.736500 + 0.676438i \(0.236477\pi\)
−0.736500 + 0.676438i \(0.763523\pi\)
\(822\) 0 0
\(823\) −29.6749 −1.03440 −0.517201 0.855864i \(-0.673026\pi\)
−0.517201 + 0.855864i \(0.673026\pi\)
\(824\) 4.68292 0.163137
\(825\) 0 0
\(826\) −21.7326 −0.756173
\(827\) 45.0316i 1.56590i −0.622084 0.782951i \(-0.713714\pi\)
0.622084 0.782951i \(-0.286286\pi\)
\(828\) 0 0
\(829\) 10.3590 0.359783 0.179891 0.983686i \(-0.442425\pi\)
0.179891 + 0.983686i \(0.442425\pi\)
\(830\) 14.3735i 0.498911i
\(831\) 0 0
\(832\) 4.23077i 0.146675i
\(833\) 10.7049i 0.370903i
\(834\) 0 0
\(835\) 16.5249i 0.571869i
\(836\) 13.5200 0.467599
\(837\) 0 0
\(838\) 8.84340i 0.305490i
\(839\) 34.8547i 1.20332i −0.798753 0.601659i \(-0.794507\pi\)
0.798753 0.601659i \(-0.205493\pi\)
\(840\) 0 0
\(841\) −33.0327 −1.13906
\(842\) −31.5517 −1.08734
\(843\) 0 0
\(844\) 3.94725 0.135870
\(845\) 4.89938 0.168544
\(846\) 0 0
\(847\) 17.3188i 0.595080i
\(848\) −12.4512 −0.427577
\(849\) 0 0
\(850\) 2.08407i 0.0714829i
\(851\) 23.3026i 0.798802i
\(852\) 0 0
\(853\) 40.7371 1.39481 0.697405 0.716677i \(-0.254338\pi\)
0.697405 + 0.716677i \(0.254338\pi\)
\(854\) 9.04677 0.309574
\(855\) 0 0
\(856\) 9.29118i 0.317566i
\(857\) 24.7367 0.844990 0.422495 0.906365i \(-0.361154\pi\)
0.422495 + 0.906365i \(0.361154\pi\)
\(858\) 0 0
\(859\) 2.15510 0.0735310 0.0367655 0.999324i \(-0.488295\pi\)
0.0367655 + 0.999324i \(0.488295\pi\)
\(860\) 4.24695i 0.144820i
\(861\) 0 0
\(862\) 26.2803i 0.895111i
\(863\) 54.0311i 1.83924i −0.392811 0.919619i \(-0.628497\pi\)
0.392811 0.919619i \(-0.371503\pi\)
\(864\) 0 0
\(865\) 19.1729i 0.651898i
\(866\) 13.6961i 0.465413i
\(867\) 0 0
\(868\) −15.3480 −0.520946
\(869\) 2.86391i 0.0971514i
\(870\) 0 0
\(871\) 12.3193 32.3650i 0.417425 1.09665i
\(872\) 4.13442i 0.140009i
\(873\) 0 0
\(874\) 11.5058i 0.389188i
\(875\) 3.48375i 0.117772i
\(876\) 0 0
\(877\) −42.5051 −1.43530 −0.717648 0.696406i \(-0.754781\pi\)
−0.717648 + 0.696406i \(0.754781\pi\)
\(878\) −15.2634 −0.515114
\(879\) 0 0
\(880\) −3.99641 −0.134719
\(881\) 34.6532i 1.16750i 0.811935 + 0.583748i \(0.198414\pi\)
−0.811935 + 0.583748i \(0.801586\pi\)
\(882\) 0 0
\(883\) 30.7629i 1.03525i −0.855607 0.517627i \(-0.826816\pi\)
0.855607 0.517627i \(-0.173184\pi\)
\(884\) 8.81720 0.296555
\(885\) 0 0
\(886\) 22.5273 0.756819
\(887\) 4.64771i 0.156055i −0.996951 0.0780274i \(-0.975138\pi\)
0.996951 0.0780274i \(-0.0248622\pi\)
\(888\) 0 0
\(889\) 63.2573i 2.12158i
\(890\) 18.2414i 0.611452i
\(891\) 0 0
\(892\) 3.06080 0.102483
\(893\) 2.04535i 0.0684451i
\(894\) 0 0
\(895\) −15.8601 −0.530144
\(896\) 3.48375i 0.116384i
\(897\) 0 0
\(898\) 29.9612i 0.999818i
\(899\) 34.6989 1.15727
\(900\) 0 0
\(901\) 25.9492i 0.864492i
\(902\) −13.4751 −0.448673
\(903\) 0 0
\(904\) 11.0862 0.368722
\(905\) −13.3117 −0.442495
\(906\) 0 0
\(907\) −18.9881 −0.630488 −0.315244 0.949011i \(-0.602086\pi\)
−0.315244 + 0.949011i \(0.602086\pi\)
\(908\) 0.309618i 0.0102750i
\(909\) 0 0
\(910\) 14.7389 0.488591
\(911\) 19.4110i 0.643115i 0.946890 + 0.321558i \(0.104206\pi\)
−0.946890 + 0.321558i \(0.895794\pi\)
\(912\) 0 0
\(913\) 57.4423i 1.90106i
\(914\) −22.6809 −0.750219
\(915\) 0 0
\(916\) 3.45165i 0.114046i
\(917\) −61.7607 −2.03952
\(918\) 0 0
\(919\) 22.9752i 0.757882i −0.925421 0.378941i \(-0.876288\pi\)
0.925421 0.378941i \(-0.123712\pi\)
\(920\) 3.40101i 0.112128i
\(921\) 0 0
\(922\) 10.5378i 0.347045i
\(923\) −36.6532 −1.20646
\(924\) 0 0
\(925\) 6.85166 0.225281
\(926\) 20.6778i 0.679515i
\(927\) 0 0
\(928\) 7.87608i 0.258545i
\(929\) 19.4841 0.639253 0.319626 0.947544i \(-0.396443\pi\)
0.319626 + 0.947544i \(0.396443\pi\)
\(930\) 0 0
\(931\) −17.3771 −0.569511
\(932\) 13.0421 0.427210
\(933\) 0 0
\(934\) 5.30865i 0.173704i
\(935\) 8.32879i 0.272380i
\(936\) 0 0
\(937\) 41.1999i 1.34594i −0.739669 0.672971i \(-0.765018\pi\)
0.739669 0.672971i \(-0.234982\pi\)
\(938\) 10.1442 26.6504i 0.331219 0.870167i
\(939\) 0 0
\(940\) 0.604591i 0.0197196i
\(941\) −34.1662 −1.11379 −0.556893 0.830584i \(-0.688007\pi\)
−0.556893 + 0.830584i \(0.688007\pi\)
\(942\) 0 0
\(943\) 11.4676i 0.373436i
\(944\) 6.23827i 0.203038i
\(945\) 0 0
\(946\) 16.9726i 0.551826i
\(947\) 27.1731i 0.883007i −0.897259 0.441504i \(-0.854445\pi\)
0.897259 0.441504i \(-0.145555\pi\)
\(948\) 0 0
\(949\) 43.9987i 1.42826i
\(950\) −3.38304 −0.109760
\(951\) 0 0
\(952\) 7.26038 0.235310
\(953\) 19.9691i 0.646863i −0.946252 0.323432i \(-0.895163\pi\)
0.946252 0.323432i \(-0.104837\pi\)
\(954\) 0 0
\(955\) −21.3256 −0.690079
\(956\) 8.10257 0.262056
\(957\) 0 0
\(958\) 20.1076i 0.649645i
\(959\) 12.1335i 0.391810i
\(960\) 0 0
\(961\) 11.5907 0.373894
\(962\) 28.9878i 0.934603i
\(963\) 0 0
\(964\) 19.6546 0.633031
\(965\) −8.68487 −0.279576
\(966\) 0 0
\(967\) 5.96867 0.191939 0.0959697 0.995384i \(-0.469405\pi\)
0.0959697 + 0.995384i \(0.469405\pi\)
\(968\) −4.97130 −0.159784
\(969\) 0 0
\(970\) 4.52609i 0.145324i
\(971\) 11.5025i 0.369132i 0.982820 + 0.184566i \(0.0590880\pi\)
−0.982820 + 0.184566i \(0.940912\pi\)
\(972\) 0 0
\(973\) 42.1606 1.35161
\(974\) 22.4926i 0.720708i
\(975\) 0 0
\(976\) 2.59685i 0.0831230i
\(977\) 17.8647i 0.571544i −0.958298 0.285772i \(-0.907750\pi\)
0.958298 0.285772i \(-0.0922500\pi\)
\(978\) 0 0
\(979\) 72.9000i 2.32990i
\(980\) 5.13654 0.164081
\(981\) 0 0
\(982\) 7.59955i 0.242511i
\(983\) 42.4866 1.35511 0.677556 0.735472i \(-0.263039\pi\)
0.677556 + 0.735472i \(0.263039\pi\)
\(984\) 0 0
\(985\) −21.1629 −0.674305
\(986\) −16.4143 −0.522737
\(987\) 0 0
\(988\) 14.3128i 0.455352i
\(989\) −14.4439 −0.459291
\(990\) 0 0
\(991\) 31.3474i 0.995782i 0.867239 + 0.497891i \(0.165892\pi\)
−0.867239 + 0.497891i \(0.834108\pi\)
\(992\) 4.40560i 0.139878i
\(993\) 0 0
\(994\) −30.1815 −0.957299
\(995\) −1.78351 −0.0565411
\(996\) 0 0
\(997\) 50.4135 1.59661 0.798307 0.602251i \(-0.205729\pi\)
0.798307 + 0.602251i \(0.205729\pi\)
\(998\) 14.1638i 0.448347i
\(999\) 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 6030.2.d.k.2411.3 24
3.2 odd 2 6030.2.d.l.2411.3 yes 24
67.66 odd 2 6030.2.d.l.2411.22 yes 24
201.200 even 2 inner 6030.2.d.k.2411.22 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6030.2.d.k.2411.3 24 1.1 even 1 trivial
6030.2.d.k.2411.22 yes 24 201.200 even 2 inner
6030.2.d.l.2411.3 yes 24 3.2 odd 2
6030.2.d.l.2411.22 yes 24 67.66 odd 2