Properties

Label 1680.2.f.g.881.2
Level $1680$
Weight $2$
Character 1680.881
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
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] = [1680,2,Mod(881,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1680.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.f (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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.2
Root \(1.68614 - 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 1680.881
Dual form 1680.2.f.g.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.68614 + 0.396143i) q^{3} +1.00000 q^{5} +(2.00000 - 1.73205i) q^{7} +(2.68614 - 1.33591i) q^{9} +O(q^{10})\) \(q+(-1.68614 + 0.396143i) q^{3} +1.00000 q^{5} +(2.00000 - 1.73205i) q^{7} +(2.68614 - 1.33591i) q^{9} +0.792287i q^{11} +5.84096i q^{13} +(-1.68614 + 0.396143i) q^{15} -1.37228 q^{17} +3.46410i q^{19} +(-2.68614 + 3.71277i) q^{21} +1.87953i q^{23} +1.00000 q^{25} +(-4.00000 + 3.31662i) q^{27} -4.25639i q^{29} -3.46410i q^{31} +(-0.313859 - 1.33591i) q^{33} +(2.00000 - 1.73205i) q^{35} +4.74456 q^{37} +(-2.31386 - 9.84868i) q^{39} -6.00000 q^{41} +6.74456 q^{43} +(2.68614 - 1.33591i) q^{45} +7.37228 q^{47} +(1.00000 - 6.92820i) q^{49} +(2.31386 - 0.543620i) q^{51} +8.51278i q^{53} +0.792287i q^{55} +(-1.37228 - 5.84096i) q^{57} -2.74456 q^{59} +6.92820i q^{61} +(3.05842 - 7.32435i) q^{63} +5.84096i q^{65} +6.74456 q^{67} +(-0.744563 - 3.16915i) q^{69} +13.5615i q^{71} +6.92820i q^{73} +(-1.68614 + 0.396143i) q^{75} +(1.37228 + 1.58457i) q^{77} -3.37228 q^{79} +(5.43070 - 7.17687i) q^{81} -5.48913 q^{83} -1.37228 q^{85} +(1.68614 + 7.17687i) q^{87} +3.25544 q^{89} +(10.1168 + 11.6819i) q^{91} +(1.37228 + 5.84096i) q^{93} +3.46410i q^{95} -1.08724i q^{97} +(1.05842 + 2.12819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 4 q^{5} + 8 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 4 q^{5} + 8 q^{7} + 5 q^{9} - q^{15} + 6 q^{17} - 5 q^{21} + 4 q^{25} - 16 q^{27} - 7 q^{33} + 8 q^{35} - 4 q^{37} - 15 q^{39} - 24 q^{41} + 4 q^{43} + 5 q^{45} + 18 q^{47} + 4 q^{49} + 15 q^{51} + 6 q^{57} + 12 q^{59} - 5 q^{63} + 4 q^{67} + 20 q^{69} - q^{75} - 6 q^{77} - 2 q^{79} - 7 q^{81} + 24 q^{83} + 6 q^{85} + q^{87} + 36 q^{89} + 6 q^{91} - 6 q^{93} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) −1.68614 + 0.396143i −0.973494 + 0.228714i
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 0 0
\(9\) 2.68614 1.33591i 0.895380 0.445302i
\(10\) 0 0
\(11\) 0.792287i 0.238884i 0.992841 + 0.119442i \(0.0381105\pi\)
−0.992841 + 0.119442i \(0.961890\pi\)
\(12\) 0 0
\(13\) 5.84096i 1.61999i 0.586436 + 0.809996i \(0.300531\pi\)
−0.586436 + 0.809996i \(0.699469\pi\)
\(14\) 0 0
\(15\) −1.68614 + 0.396143i −0.435360 + 0.102284i
\(16\) 0 0
\(17\) −1.37228 −0.332827 −0.166414 0.986056i \(-0.553219\pi\)
−0.166414 + 0.986056i \(0.553219\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −2.68614 + 3.71277i −0.586164 + 0.810192i
\(22\) 0 0
\(23\) 1.87953i 0.391909i 0.980613 + 0.195954i \(0.0627804\pi\)
−0.980613 + 0.195954i \(0.937220\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 + 3.31662i −0.769800 + 0.638285i
\(28\) 0 0
\(29\) 4.25639i 0.790392i −0.918597 0.395196i \(-0.870677\pi\)
0.918597 0.395196i \(-0.129323\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) −0.313859 1.33591i −0.0546359 0.232552i
\(34\) 0 0
\(35\) 2.00000 1.73205i 0.338062 0.292770i
\(36\) 0 0
\(37\) 4.74456 0.780001 0.390001 0.920815i \(-0.372475\pi\)
0.390001 + 0.920815i \(0.372475\pi\)
\(38\) 0 0
\(39\) −2.31386 9.84868i −0.370514 1.57705i
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.74456 1.02854 0.514268 0.857629i \(-0.328064\pi\)
0.514268 + 0.857629i \(0.328064\pi\)
\(44\) 0 0
\(45\) 2.68614 1.33591i 0.400426 0.199145i
\(46\) 0 0
\(47\) 7.37228 1.07536 0.537679 0.843150i \(-0.319301\pi\)
0.537679 + 0.843150i \(0.319301\pi\)
\(48\) 0 0
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 0 0
\(51\) 2.31386 0.543620i 0.324005 0.0761221i
\(52\) 0 0
\(53\) 8.51278i 1.16932i 0.811278 + 0.584660i \(0.198772\pi\)
−0.811278 + 0.584660i \(0.801228\pi\)
\(54\) 0 0
\(55\) 0.792287i 0.106832i
\(56\) 0 0
\(57\) −1.37228 5.84096i −0.181763 0.773654i
\(58\) 0 0
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 3.05842 7.32435i 0.385325 0.922781i
\(64\) 0 0
\(65\) 5.84096i 0.724482i
\(66\) 0 0
\(67\) 6.74456 0.823979 0.411990 0.911188i \(-0.364834\pi\)
0.411990 + 0.911188i \(0.364834\pi\)
\(68\) 0 0
\(69\) −0.744563 3.16915i −0.0896348 0.381521i
\(70\) 0 0
\(71\) 13.5615i 1.60945i 0.593649 + 0.804724i \(0.297687\pi\)
−0.593649 + 0.804724i \(0.702313\pi\)
\(72\) 0 0
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 0 0
\(75\) −1.68614 + 0.396143i −0.194699 + 0.0457427i
\(76\) 0 0
\(77\) 1.37228 + 1.58457i 0.156386 + 0.180579i
\(78\) 0 0
\(79\) −3.37228 −0.379411 −0.189706 0.981841i \(-0.560753\pi\)
−0.189706 + 0.981841i \(0.560753\pi\)
\(80\) 0 0
\(81\) 5.43070 7.17687i 0.603411 0.797430i
\(82\) 0 0
\(83\) −5.48913 −0.602510 −0.301255 0.953544i \(-0.597406\pi\)
−0.301255 + 0.953544i \(0.597406\pi\)
\(84\) 0 0
\(85\) −1.37228 −0.148845
\(86\) 0 0
\(87\) 1.68614 + 7.17687i 0.180773 + 0.769441i
\(88\) 0 0
\(89\) 3.25544 0.345076 0.172538 0.985003i \(-0.444803\pi\)
0.172538 + 0.985003i \(0.444803\pi\)
\(90\) 0 0
\(91\) 10.1168 + 11.6819i 1.06053 + 1.22460i
\(92\) 0 0
\(93\) 1.37228 + 5.84096i 0.142299 + 0.605680i
\(94\) 0 0
\(95\) 3.46410i 0.355409i
\(96\) 0 0
\(97\) 1.08724i 0.110393i −0.998476 0.0551963i \(-0.982422\pi\)
0.998476 0.0551963i \(-0.0175785\pi\)
\(98\) 0 0
\(99\) 1.05842 + 2.12819i 0.106375 + 0.213892i
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 16.2333i 1.59951i 0.600326 + 0.799756i \(0.295038\pi\)
−0.600326 + 0.799756i \(0.704962\pi\)
\(104\) 0 0
\(105\) −2.68614 + 3.71277i −0.262140 + 0.362329i
\(106\) 0 0
\(107\) 6.63325i 0.641260i 0.947204 + 0.320630i \(0.103895\pi\)
−0.947204 + 0.320630i \(0.896105\pi\)
\(108\) 0 0
\(109\) 0.116844 0.0111916 0.00559581 0.999984i \(-0.498219\pi\)
0.00559581 + 0.999984i \(0.498219\pi\)
\(110\) 0 0
\(111\) −8.00000 + 1.87953i −0.759326 + 0.178397i
\(112\) 0 0
\(113\) 10.0974i 0.949879i −0.880018 0.474939i \(-0.842470\pi\)
0.880018 0.474939i \(-0.157530\pi\)
\(114\) 0 0
\(115\) 1.87953i 0.175267i
\(116\) 0 0
\(117\) 7.80298 + 15.6896i 0.721386 + 1.45051i
\(118\) 0 0
\(119\) −2.74456 + 2.37686i −0.251594 + 0.217886i
\(120\) 0 0
\(121\) 10.3723 0.942935
\(122\) 0 0
\(123\) 10.1168 2.37686i 0.912205 0.214314i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.7446 −0.953426 −0.476713 0.879059i \(-0.658172\pi\)
−0.476713 + 0.879059i \(0.658172\pi\)
\(128\) 0 0
\(129\) −11.3723 + 2.67181i −1.00127 + 0.235240i
\(130\) 0 0
\(131\) 17.4891 1.52803 0.764016 0.645197i \(-0.223225\pi\)
0.764016 + 0.645197i \(0.223225\pi\)
\(132\) 0 0
\(133\) 6.00000 + 6.92820i 0.520266 + 0.600751i
\(134\) 0 0
\(135\) −4.00000 + 3.31662i −0.344265 + 0.285450i
\(136\) 0 0
\(137\) 13.2665i 1.13343i 0.823913 + 0.566717i \(0.191787\pi\)
−0.823913 + 0.566717i \(0.808213\pi\)
\(138\) 0 0
\(139\) 1.28962i 0.109384i 0.998503 + 0.0546921i \(0.0174177\pi\)
−0.998503 + 0.0546921i \(0.982582\pi\)
\(140\) 0 0
\(141\) −12.4307 + 2.92048i −1.04685 + 0.245949i
\(142\) 0 0
\(143\) −4.62772 −0.386989
\(144\) 0 0
\(145\) 4.25639i 0.353474i
\(146\) 0 0
\(147\) 1.05842 + 12.0781i 0.0872972 + 0.996182i
\(148\) 0 0
\(149\) 10.0974i 0.827207i −0.910457 0.413604i \(-0.864270\pi\)
0.910457 0.413604i \(-0.135730\pi\)
\(150\) 0 0
\(151\) −3.37228 −0.274432 −0.137216 0.990541i \(-0.543816\pi\)
−0.137216 + 0.990541i \(0.543816\pi\)
\(152\) 0 0
\(153\) −3.68614 + 1.83324i −0.298007 + 0.148209i
\(154\) 0 0
\(155\) 3.46410i 0.278243i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −3.37228 14.3537i −0.267439 1.13833i
\(160\) 0 0
\(161\) 3.25544 + 3.75906i 0.256564 + 0.296255i
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 0 0
\(165\) −0.313859 1.33591i −0.0244339 0.104000i
\(166\) 0 0
\(167\) 22.1168 1.71145 0.855726 0.517429i \(-0.173111\pi\)
0.855726 + 0.517429i \(0.173111\pi\)
\(168\) 0 0
\(169\) −21.1168 −1.62437
\(170\) 0 0
\(171\) 4.62772 + 9.30506i 0.353890 + 0.711576i
\(172\) 0 0
\(173\) 16.1168 1.22534 0.612670 0.790338i \(-0.290095\pi\)
0.612670 + 0.790338i \(0.290095\pi\)
\(174\) 0 0
\(175\) 2.00000 1.73205i 0.151186 0.130931i
\(176\) 0 0
\(177\) 4.62772 1.08724i 0.347841 0.0817220i
\(178\) 0 0
\(179\) 6.63325i 0.495792i 0.968787 + 0.247896i \(0.0797392\pi\)
−0.968787 + 0.247896i \(0.920261\pi\)
\(180\) 0 0
\(181\) 18.6101i 1.38328i −0.722242 0.691640i \(-0.756889\pi\)
0.722242 0.691640i \(-0.243111\pi\)
\(182\) 0 0
\(183\) −2.74456 11.6819i −0.202884 0.863553i
\(184\) 0 0
\(185\) 4.74456 0.348827
\(186\) 0 0
\(187\) 1.08724i 0.0795069i
\(188\) 0 0
\(189\) −2.25544 + 13.5615i −0.164059 + 0.986451i
\(190\) 0 0
\(191\) 15.6434i 1.13191i −0.824435 0.565957i \(-0.808507\pi\)
0.824435 0.565957i \(-0.191493\pi\)
\(192\) 0 0
\(193\) 22.2337 1.60042 0.800208 0.599723i \(-0.204722\pi\)
0.800208 + 0.599723i \(0.204722\pi\)
\(194\) 0 0
\(195\) −2.31386 9.84868i −0.165699 0.705279i
\(196\) 0 0
\(197\) 22.3692i 1.59374i 0.604152 + 0.796869i \(0.293512\pi\)
−0.604152 + 0.796869i \(0.706488\pi\)
\(198\) 0 0
\(199\) 12.9715i 0.919528i −0.888041 0.459764i \(-0.847934\pi\)
0.888041 0.459764i \(-0.152066\pi\)
\(200\) 0 0
\(201\) −11.3723 + 2.67181i −0.802139 + 0.188455i
\(202\) 0 0
\(203\) −7.37228 8.51278i −0.517433 0.597480i
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 2.51087 + 5.04868i 0.174518 + 0.350907i
\(208\) 0 0
\(209\) −2.74456 −0.189845
\(210\) 0 0
\(211\) −6.11684 −0.421101 −0.210550 0.977583i \(-0.567526\pi\)
−0.210550 + 0.977583i \(0.567526\pi\)
\(212\) 0 0
\(213\) −5.37228 22.8665i −0.368103 1.56679i
\(214\) 0 0
\(215\) 6.74456 0.459975
\(216\) 0 0
\(217\) −6.00000 6.92820i −0.407307 0.470317i
\(218\) 0 0
\(219\) −2.74456 11.6819i −0.185460 0.789391i
\(220\) 0 0
\(221\) 8.01544i 0.539177i
\(222\) 0 0
\(223\) 20.9870i 1.40539i −0.711490 0.702696i \(-0.751979\pi\)
0.711490 0.702696i \(-0.248021\pi\)
\(224\) 0 0
\(225\) 2.68614 1.33591i 0.179076 0.0890605i
\(226\) 0 0
\(227\) 15.6060 1.03580 0.517902 0.855440i \(-0.326713\pi\)
0.517902 + 0.855440i \(0.326713\pi\)
\(228\) 0 0
\(229\) 4.75372i 0.314135i −0.987588 0.157067i \(-0.949796\pi\)
0.987588 0.157067i \(-0.0502040\pi\)
\(230\) 0 0
\(231\) −2.94158 2.12819i −0.193542 0.140025i
\(232\) 0 0
\(233\) 3.75906i 0.246264i 0.992390 + 0.123132i \(0.0392938\pi\)
−0.992390 + 0.123132i \(0.960706\pi\)
\(234\) 0 0
\(235\) 7.37228 0.480915
\(236\) 0 0
\(237\) 5.68614 1.33591i 0.369355 0.0867765i
\(238\) 0 0
\(239\) 15.6434i 1.01188i −0.862567 0.505942i \(-0.831145\pi\)
0.862567 0.505942i \(-0.168855\pi\)
\(240\) 0 0
\(241\) 23.3639i 1.50500i 0.658593 + 0.752499i \(0.271152\pi\)
−0.658593 + 0.752499i \(0.728848\pi\)
\(242\) 0 0
\(243\) −6.31386 + 14.2525i −0.405034 + 0.914302i
\(244\) 0 0
\(245\) 1.00000 6.92820i 0.0638877 0.442627i
\(246\) 0 0
\(247\) −20.2337 −1.28744
\(248\) 0 0
\(249\) 9.25544 2.17448i 0.586540 0.137802i
\(250\) 0 0
\(251\) −17.4891 −1.10390 −0.551952 0.833876i \(-0.686117\pi\)
−0.551952 + 0.833876i \(0.686117\pi\)
\(252\) 0 0
\(253\) −1.48913 −0.0936205
\(254\) 0 0
\(255\) 2.31386 0.543620i 0.144899 0.0340428i
\(256\) 0 0
\(257\) 23.4891 1.46521 0.732606 0.680653i \(-0.238304\pi\)
0.732606 + 0.680653i \(0.238304\pi\)
\(258\) 0 0
\(259\) 9.48913 8.21782i 0.589626 0.510631i
\(260\) 0 0
\(261\) −5.68614 11.4333i −0.351963 0.707701i
\(262\) 0 0
\(263\) 13.5615i 0.836235i 0.908393 + 0.418118i \(0.137310\pi\)
−0.908393 + 0.418118i \(0.862690\pi\)
\(264\) 0 0
\(265\) 8.51278i 0.522936i
\(266\) 0 0
\(267\) −5.48913 + 1.28962i −0.335929 + 0.0789235i
\(268\) 0 0
\(269\) −8.74456 −0.533165 −0.266583 0.963812i \(-0.585895\pi\)
−0.266583 + 0.963812i \(0.585895\pi\)
\(270\) 0 0
\(271\) 15.1460i 0.920056i −0.887905 0.460028i \(-0.847839\pi\)
0.887905 0.460028i \(-0.152161\pi\)
\(272\) 0 0
\(273\) −21.6861 15.6896i −1.31250 0.949581i
\(274\) 0 0
\(275\) 0.792287i 0.0477767i
\(276\) 0 0
\(277\) −6.23369 −0.374546 −0.187273 0.982308i \(-0.559965\pi\)
−0.187273 + 0.982308i \(0.559965\pi\)
\(278\) 0 0
\(279\) −4.62772 9.30506i −0.277054 0.557080i
\(280\) 0 0
\(281\) 4.84630i 0.289106i 0.989497 + 0.144553i \(0.0461744\pi\)
−0.989497 + 0.144553i \(0.953826\pi\)
\(282\) 0 0
\(283\) 9.30506i 0.553129i 0.960995 + 0.276564i \(0.0891959\pi\)
−0.960995 + 0.276564i \(0.910804\pi\)
\(284\) 0 0
\(285\) −1.37228 5.84096i −0.0812869 0.345989i
\(286\) 0 0
\(287\) −12.0000 + 10.3923i −0.708338 + 0.613438i
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) 0.430703 + 1.83324i 0.0252483 + 0.107466i
\(292\) 0 0
\(293\) −28.1168 −1.64260 −0.821302 0.570494i \(-0.806752\pi\)
−0.821302 + 0.570494i \(0.806752\pi\)
\(294\) 0 0
\(295\) −2.74456 −0.159795
\(296\) 0 0
\(297\) −2.62772 3.16915i −0.152476 0.183893i
\(298\) 0 0
\(299\) −10.9783 −0.634889
\(300\) 0 0
\(301\) 13.4891 11.6819i 0.777500 0.673335i
\(302\) 0 0
\(303\) −10.1168 + 2.37686i −0.581198 + 0.136547i
\(304\) 0 0
\(305\) 6.92820i 0.396708i
\(306\) 0 0
\(307\) 7.13058i 0.406964i 0.979079 + 0.203482i \(0.0652258\pi\)
−0.979079 + 0.203482i \(0.934774\pi\)
\(308\) 0 0
\(309\) −6.43070 27.3716i −0.365830 1.55711i
\(310\) 0 0
\(311\) −20.2337 −1.14735 −0.573674 0.819084i \(-0.694482\pi\)
−0.573674 + 0.819084i \(0.694482\pi\)
\(312\) 0 0
\(313\) 24.4511i 1.38206i −0.722827 0.691029i \(-0.757158\pi\)
0.722827 0.691029i \(-0.242842\pi\)
\(314\) 0 0
\(315\) 3.05842 7.32435i 0.172323 0.412680i
\(316\) 0 0
\(317\) 8.51278i 0.478125i 0.971004 + 0.239063i \(0.0768401\pi\)
−0.971004 + 0.239063i \(0.923160\pi\)
\(318\) 0 0
\(319\) 3.37228 0.188812
\(320\) 0 0
\(321\) −2.62772 11.1846i −0.146665 0.624263i
\(322\) 0 0
\(323\) 4.75372i 0.264504i
\(324\) 0 0
\(325\) 5.84096i 0.323998i
\(326\) 0 0
\(327\) −0.197015 + 0.0462870i −0.0108950 + 0.00255968i
\(328\) 0 0
\(329\) 14.7446 12.7692i 0.812894 0.703987i
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 12.7446 6.33830i 0.698398 0.347336i
\(334\) 0 0
\(335\) 6.74456 0.368495
\(336\) 0 0
\(337\) −18.2337 −0.993252 −0.496626 0.867965i \(-0.665428\pi\)
−0.496626 + 0.867965i \(0.665428\pi\)
\(338\) 0 0
\(339\) 4.00000 + 17.0256i 0.217250 + 0.924701i
\(340\) 0 0
\(341\) 2.74456 0.148626
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) −0.744563 3.16915i −0.0400859 0.170621i
\(346\) 0 0
\(347\) 18.3152i 0.983210i 0.870818 + 0.491605i \(0.163590\pi\)
−0.870818 + 0.491605i \(0.836410\pi\)
\(348\) 0 0
\(349\) 4.75372i 0.254461i 0.991873 + 0.127230i \(0.0406088\pi\)
−0.991873 + 0.127230i \(0.959391\pi\)
\(350\) 0 0
\(351\) −19.3723 23.3639i −1.03402 1.24707i
\(352\) 0 0
\(353\) 7.88316 0.419578 0.209789 0.977747i \(-0.432722\pi\)
0.209789 + 0.977747i \(0.432722\pi\)
\(354\) 0 0
\(355\) 13.5615i 0.719767i
\(356\) 0 0
\(357\) 3.68614 5.09496i 0.195091 0.269654i
\(358\) 0 0
\(359\) 9.80240i 0.517351i −0.965964 0.258675i \(-0.916714\pi\)
0.965964 0.258675i \(-0.0832860\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −17.4891 + 4.10891i −0.917941 + 0.215662i
\(364\) 0 0
\(365\) 6.92820i 0.362639i
\(366\) 0 0
\(367\) 25.3360i 1.32253i −0.750154 0.661263i \(-0.770021\pi\)
0.750154 0.661263i \(-0.229979\pi\)
\(368\) 0 0
\(369\) −16.1168 + 8.01544i −0.839009 + 0.417267i
\(370\) 0 0
\(371\) 14.7446 + 17.0256i 0.765500 + 0.883923i
\(372\) 0 0
\(373\) −24.7446 −1.28122 −0.640612 0.767864i \(-0.721319\pi\)
−0.640612 + 0.767864i \(0.721319\pi\)
\(374\) 0 0
\(375\) −1.68614 + 0.396143i −0.0870719 + 0.0204568i
\(376\) 0 0
\(377\) 24.8614 1.28043
\(378\) 0 0
\(379\) −1.48913 −0.0764912 −0.0382456 0.999268i \(-0.512177\pi\)
−0.0382456 + 0.999268i \(0.512177\pi\)
\(380\) 0 0
\(381\) 18.1168 4.25639i 0.928154 0.218061i
\(382\) 0 0
\(383\) 17.4891 0.893653 0.446826 0.894621i \(-0.352554\pi\)
0.446826 + 0.894621i \(0.352554\pi\)
\(384\) 0 0
\(385\) 1.37228 + 1.58457i 0.0699379 + 0.0807574i
\(386\) 0 0
\(387\) 18.1168 9.01011i 0.920931 0.458010i
\(388\) 0 0
\(389\) 13.7638i 0.697854i −0.937150 0.348927i \(-0.886546\pi\)
0.937150 0.348927i \(-0.113454\pi\)
\(390\) 0 0
\(391\) 2.57924i 0.130438i
\(392\) 0 0
\(393\) −29.4891 + 6.92820i −1.48753 + 0.349482i
\(394\) 0 0
\(395\) −3.37228 −0.169678
\(396\) 0 0
\(397\) 3.66648i 0.184015i 0.995758 + 0.0920077i \(0.0293284\pi\)
−0.995758 + 0.0920077i \(0.970672\pi\)
\(398\) 0 0
\(399\) −12.8614 9.30506i −0.643876 0.465836i
\(400\) 0 0
\(401\) 2.67181i 0.133424i 0.997772 + 0.0667120i \(0.0212509\pi\)
−0.997772 + 0.0667120i \(0.978749\pi\)
\(402\) 0 0
\(403\) 20.2337 1.00791
\(404\) 0 0
\(405\) 5.43070 7.17687i 0.269854 0.356622i
\(406\) 0 0
\(407\) 3.75906i 0.186329i
\(408\) 0 0
\(409\) 35.0458i 1.73290i −0.499262 0.866451i \(-0.666396\pi\)
0.499262 0.866451i \(-0.333604\pi\)
\(410\) 0 0
\(411\) −5.25544 22.3692i −0.259232 1.10339i
\(412\) 0 0
\(413\) −5.48913 + 4.75372i −0.270102 + 0.233915i
\(414\) 0 0
\(415\) −5.48913 −0.269451
\(416\) 0 0
\(417\) −0.510875 2.17448i −0.0250176 0.106485i
\(418\) 0 0
\(419\) 2.74456 0.134081 0.0670403 0.997750i \(-0.478644\pi\)
0.0670403 + 0.997750i \(0.478644\pi\)
\(420\) 0 0
\(421\) −25.6060 −1.24796 −0.623979 0.781441i \(-0.714485\pi\)
−0.623979 + 0.781441i \(0.714485\pi\)
\(422\) 0 0
\(423\) 19.8030 9.84868i 0.962854 0.478859i
\(424\) 0 0
\(425\) −1.37228 −0.0665654
\(426\) 0 0
\(427\) 12.0000 + 13.8564i 0.580721 + 0.670559i
\(428\) 0 0
\(429\) 7.80298 1.83324i 0.376732 0.0885097i
\(430\) 0 0
\(431\) 31.6742i 1.52569i −0.646579 0.762847i \(-0.723801\pi\)
0.646579 0.762847i \(-0.276199\pi\)
\(432\) 0 0
\(433\) 2.57924i 0.123950i −0.998078 0.0619752i \(-0.980260\pi\)
0.998078 0.0619752i \(-0.0197400\pi\)
\(434\) 0 0
\(435\) 1.68614 + 7.17687i 0.0808443 + 0.344105i
\(436\) 0 0
\(437\) −6.51087 −0.311457
\(438\) 0 0
\(439\) 1.28962i 0.0615502i −0.999526 0.0307751i \(-0.990202\pi\)
0.999526 0.0307751i \(-0.00979757\pi\)
\(440\) 0 0
\(441\) −6.56930 19.9460i −0.312824 0.949811i
\(442\) 0 0
\(443\) 6.63325i 0.315155i 0.987507 + 0.157578i \(0.0503684\pi\)
−0.987507 + 0.157578i \(0.949632\pi\)
\(444\) 0 0
\(445\) 3.25544 0.154323
\(446\) 0 0
\(447\) 4.00000 + 17.0256i 0.189194 + 0.805281i
\(448\) 0 0
\(449\) 28.2101i 1.33132i 0.746256 + 0.665660i \(0.231850\pi\)
−0.746256 + 0.665660i \(0.768150\pi\)
\(450\) 0 0
\(451\) 4.75372i 0.223844i
\(452\) 0 0
\(453\) 5.68614 1.33591i 0.267158 0.0627664i
\(454\) 0 0
\(455\) 10.1168 + 11.6819i 0.474285 + 0.547657i
\(456\) 0 0
\(457\) −32.9783 −1.54266 −0.771329 0.636437i \(-0.780408\pi\)
−0.771329 + 0.636437i \(0.780408\pi\)
\(458\) 0 0
\(459\) 5.48913 4.55134i 0.256210 0.212438i
\(460\) 0 0
\(461\) −8.74456 −0.407275 −0.203637 0.979046i \(-0.565276\pi\)
−0.203637 + 0.979046i \(0.565276\pi\)
\(462\) 0 0
\(463\) −36.4674 −1.69478 −0.847391 0.530969i \(-0.821828\pi\)
−0.847391 + 0.530969i \(0.821828\pi\)
\(464\) 0 0
\(465\) 1.37228 + 5.84096i 0.0636380 + 0.270868i
\(466\) 0 0
\(467\) 13.8832 0.642436 0.321218 0.947005i \(-0.395908\pi\)
0.321218 + 0.947005i \(0.395908\pi\)
\(468\) 0 0
\(469\) 13.4891 11.6819i 0.622870 0.539421i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.34363i 0.245700i
\(474\) 0 0
\(475\) 3.46410i 0.158944i
\(476\) 0 0
\(477\) 11.3723 + 22.8665i 0.520701 + 1.04699i
\(478\) 0 0
\(479\) −18.5109 −0.845783 −0.422892 0.906180i \(-0.638985\pi\)
−0.422892 + 0.906180i \(0.638985\pi\)
\(480\) 0 0
\(481\) 27.7128i 1.26360i
\(482\) 0 0
\(483\) −6.97825 5.04868i −0.317521 0.229723i
\(484\) 0 0
\(485\) 1.08724i 0.0493691i
\(486\) 0 0
\(487\) −14.5109 −0.657550 −0.328775 0.944408i \(-0.606636\pi\)
−0.328775 + 0.944408i \(0.606636\pi\)
\(488\) 0 0
\(489\) 13.4891 3.16915i 0.609999 0.143314i
\(490\) 0 0
\(491\) 10.8896i 0.491442i −0.969341 0.245721i \(-0.920975\pi\)
0.969341 0.245721i \(-0.0790248\pi\)
\(492\) 0 0
\(493\) 5.84096i 0.263064i
\(494\) 0 0
\(495\) 1.05842 + 2.12819i 0.0475725 + 0.0956552i
\(496\) 0 0
\(497\) 23.4891 + 27.1229i 1.05363 + 1.21663i
\(498\) 0 0
\(499\) 10.3505 0.463353 0.231677 0.972793i \(-0.425579\pi\)
0.231677 + 0.972793i \(0.425579\pi\)
\(500\) 0 0
\(501\) −37.2921 + 8.76144i −1.66609 + 0.391432i
\(502\) 0 0
\(503\) 27.6060 1.23089 0.615445 0.788180i \(-0.288976\pi\)
0.615445 + 0.788180i \(0.288976\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) 35.6060 8.36530i 1.58132 0.371516i
\(508\) 0 0
\(509\) −38.2337 −1.69468 −0.847339 0.531052i \(-0.821797\pi\)
−0.847339 + 0.531052i \(0.821797\pi\)
\(510\) 0 0
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 0 0
\(513\) −11.4891 13.8564i −0.507257 0.611775i
\(514\) 0 0
\(515\) 16.2333i 0.715323i
\(516\) 0 0
\(517\) 5.84096i 0.256885i
\(518\) 0 0
\(519\) −27.1753 + 6.38458i −1.19286 + 0.280252i
\(520\) 0 0
\(521\) 34.4674 1.51004 0.755022 0.655700i \(-0.227626\pi\)
0.755022 + 0.655700i \(0.227626\pi\)
\(522\) 0 0
\(523\) 10.3923i 0.454424i −0.973845 0.227212i \(-0.927039\pi\)
0.973845 0.227212i \(-0.0729610\pi\)
\(524\) 0 0
\(525\) −2.68614 + 3.71277i −0.117233 + 0.162038i
\(526\) 0 0
\(527\) 4.75372i 0.207075i
\(528\) 0 0
\(529\) 19.4674 0.846408
\(530\) 0 0
\(531\) −7.37228 + 3.66648i −0.319930 + 0.159112i
\(532\) 0 0
\(533\) 35.0458i 1.51800i
\(534\) 0 0
\(535\) 6.63325i 0.286780i
\(536\) 0 0
\(537\) −2.62772 11.1846i −0.113394 0.482651i
\(538\) 0 0
\(539\) 5.48913 + 0.792287i 0.236433 + 0.0341262i
\(540\) 0 0
\(541\) 18.6277 0.800868 0.400434 0.916326i \(-0.368859\pi\)
0.400434 + 0.916326i \(0.368859\pi\)
\(542\) 0 0
\(543\) 7.37228 + 31.3793i 0.316375 + 1.34661i
\(544\) 0 0
\(545\) 0.116844 0.00500505
\(546\) 0 0
\(547\) −42.9783 −1.83762 −0.918809 0.394703i \(-0.870847\pi\)
−0.918809 + 0.394703i \(0.870847\pi\)
\(548\) 0 0
\(549\) 9.25544 + 18.6101i 0.395012 + 0.794261i
\(550\) 0 0
\(551\) 14.7446 0.628139
\(552\) 0 0
\(553\) −6.74456 + 5.84096i −0.286808 + 0.248383i
\(554\) 0 0
\(555\) −8.00000 + 1.87953i −0.339581 + 0.0797815i
\(556\) 0 0
\(557\) 30.8820i 1.30851i −0.756274 0.654255i \(-0.772982\pi\)
0.756274 0.654255i \(-0.227018\pi\)
\(558\) 0 0
\(559\) 39.3947i 1.66622i
\(560\) 0 0
\(561\) 0.430703 + 1.83324i 0.0181843 + 0.0773995i
\(562\) 0 0
\(563\) −5.48913 −0.231339 −0.115670 0.993288i \(-0.536901\pi\)
−0.115670 + 0.993288i \(0.536901\pi\)
\(564\) 0 0
\(565\) 10.0974i 0.424799i
\(566\) 0 0
\(567\) −1.56930 23.7600i −0.0659043 0.997826i
\(568\) 0 0
\(569\) 10.6873i 0.448033i 0.974585 + 0.224017i \(0.0719170\pi\)
−0.974585 + 0.224017i \(0.928083\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 6.19702 + 26.3769i 0.258884 + 1.10191i
\(574\) 0 0
\(575\) 1.87953i 0.0783817i
\(576\) 0 0
\(577\) 12.7692i 0.531587i −0.964030 0.265794i \(-0.914366\pi\)
0.964030 0.265794i \(-0.0856340\pi\)
\(578\) 0 0
\(579\) −37.4891 + 8.80773i −1.55799 + 0.366037i
\(580\) 0 0
\(581\) −10.9783 + 9.50744i −0.455455 + 0.394435i
\(582\) 0 0
\(583\) −6.74456 −0.279331
\(584\) 0 0
\(585\) 7.80298 + 15.6896i 0.322614 + 0.648687i
\(586\) 0 0
\(587\) 5.48913 0.226560 0.113280 0.993563i \(-0.463864\pi\)
0.113280 + 0.993563i \(0.463864\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) −8.86141 37.7176i −0.364510 1.55149i
\(592\) 0 0
\(593\) −1.37228 −0.0563528 −0.0281764 0.999603i \(-0.508970\pi\)
−0.0281764 + 0.999603i \(0.508970\pi\)
\(594\) 0 0
\(595\) −2.74456 + 2.37686i −0.112516 + 0.0974418i
\(596\) 0 0
\(597\) 5.13859 + 21.8719i 0.210309 + 0.895155i
\(598\) 0 0
\(599\) 1.78695i 0.0730129i −0.999333 0.0365065i \(-0.988377\pi\)
0.999333 0.0365065i \(-0.0116230\pi\)
\(600\) 0 0
\(601\) 2.17448i 0.0886989i 0.999016 + 0.0443495i \(0.0141215\pi\)
−0.999016 + 0.0443495i \(0.985878\pi\)
\(602\) 0 0
\(603\) 18.1168 9.01011i 0.737775 0.366920i
\(604\) 0 0
\(605\) 10.3723 0.421693
\(606\) 0 0
\(607\) 20.9870i 0.851836i −0.904762 0.425918i \(-0.859951\pi\)
0.904762 0.425918i \(-0.140049\pi\)
\(608\) 0 0
\(609\) 15.8030 + 11.4333i 0.640369 + 0.463299i
\(610\) 0 0
\(611\) 43.0612i 1.74207i
\(612\) 0 0
\(613\) −4.51087 −0.182193 −0.0910963 0.995842i \(-0.529037\pi\)
−0.0910963 + 0.995842i \(0.529037\pi\)
\(614\) 0 0
\(615\) 10.1168 2.37686i 0.407951 0.0958443i
\(616\) 0 0
\(617\) 37.8102i 1.52218i −0.648646 0.761090i \(-0.724665\pi\)
0.648646 0.761090i \(-0.275335\pi\)
\(618\) 0 0
\(619\) 1.28962i 0.0518342i 0.999664 + 0.0259171i \(0.00825060\pi\)
−0.999664 + 0.0259171i \(0.991749\pi\)
\(620\) 0 0
\(621\) −6.23369 7.51811i −0.250149 0.301691i
\(622\) 0 0
\(623\) 6.51087 5.63858i 0.260853 0.225905i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.62772 1.08724i 0.184813 0.0434202i
\(628\) 0 0
\(629\) −6.51087 −0.259606
\(630\) 0 0
\(631\) 5.88316 0.234205 0.117102 0.993120i \(-0.462639\pi\)
0.117102 + 0.993120i \(0.462639\pi\)
\(632\) 0 0
\(633\) 10.3139 2.42315i 0.409939 0.0963115i
\(634\) 0 0
\(635\) −10.7446 −0.426385
\(636\) 0 0
\(637\) 40.4674 + 5.84096i 1.60338 + 0.231427i
\(638\) 0 0
\(639\) 18.1168 + 36.4280i 0.716691 + 1.44107i
\(640\) 0 0
\(641\) 29.7021i 1.17316i 0.809890 + 0.586582i \(0.199527\pi\)
−0.809890 + 0.586582i \(0.800473\pi\)
\(642\) 0 0
\(643\) 39.5971i 1.56156i −0.624807 0.780779i \(-0.714822\pi\)
0.624807 0.780779i \(-0.285178\pi\)
\(644\) 0 0
\(645\) −11.3723 + 2.67181i −0.447783 + 0.105203i
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 2.17448i 0.0853559i
\(650\) 0 0
\(651\) 12.8614 + 9.30506i 0.504078 + 0.364694i
\(652\) 0 0
\(653\) 33.0564i 1.29360i −0.762660 0.646799i \(-0.776107\pi\)
0.762660 0.646799i \(-0.223893\pi\)
\(654\) 0 0
\(655\) 17.4891 0.683357
\(656\) 0 0
\(657\) 9.25544 + 18.6101i 0.361089 + 0.726050i
\(658\) 0 0
\(659\) 20.3971i 0.794558i −0.917698 0.397279i \(-0.869955\pi\)
0.917698 0.397279i \(-0.130045\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i −0.990881 0.134738i \(-0.956981\pi\)
0.990881 0.134738i \(-0.0430193\pi\)
\(662\) 0 0
\(663\) 3.17527 + 13.5152i 0.123317 + 0.524886i
\(664\) 0 0
\(665\) 6.00000 + 6.92820i 0.232670 + 0.268664i
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) 8.31386 + 35.3870i 0.321432 + 1.36814i
\(670\) 0 0
\(671\) −5.48913 −0.211905
\(672\) 0 0
\(673\) 22.2337 0.857046 0.428523 0.903531i \(-0.359034\pi\)
0.428523 + 0.903531i \(0.359034\pi\)
\(674\) 0 0
\(675\) −4.00000 + 3.31662i −0.153960 + 0.127657i
\(676\) 0 0
\(677\) 21.6060 0.830385 0.415192 0.909734i \(-0.363714\pi\)
0.415192 + 0.909734i \(0.363714\pi\)
\(678\) 0 0
\(679\) −1.88316 2.17448i −0.0722689 0.0834489i
\(680\) 0 0
\(681\) −26.3139 + 6.18220i −1.00835 + 0.236903i
\(682\) 0 0
\(683\) 51.7764i 1.98117i −0.136906 0.990584i \(-0.543716\pi\)
0.136906 0.990584i \(-0.456284\pi\)
\(684\) 0 0
\(685\) 13.2665i 0.506887i
\(686\) 0 0
\(687\) 1.88316 + 8.01544i 0.0718469 + 0.305808i
\(688\) 0 0
\(689\) −49.7228 −1.89429
\(690\) 0 0
\(691\) 38.5099i 1.46498i 0.680775 + 0.732492i \(0.261643\pi\)
−0.680775 + 0.732492i \(0.738357\pi\)
\(692\) 0 0
\(693\) 5.80298 + 2.42315i 0.220437 + 0.0920478i
\(694\) 0 0
\(695\) 1.28962i 0.0489181i
\(696\) 0 0
\(697\) 8.23369 0.311873
\(698\) 0 0
\(699\) −1.48913 6.33830i −0.0563239 0.239736i
\(700\) 0 0
\(701\) 45.8256i 1.73081i −0.501074 0.865405i \(-0.667061\pi\)
0.501074 0.865405i \(-0.332939\pi\)
\(702\) 0 0
\(703\) 16.4356i 0.619882i
\(704\) 0 0
\(705\) −12.4307 + 2.92048i −0.468167 + 0.109992i
\(706\) 0 0
\(707\) 12.0000 10.3923i 0.451306 0.390843i
\(708\) 0 0
\(709\) 24.1168 0.905727 0.452864 0.891580i \(-0.350402\pi\)
0.452864 + 0.891580i \(0.350402\pi\)
\(710\) 0 0
\(711\) −9.05842 + 4.50506i −0.339717 + 0.168953i
\(712\) 0 0
\(713\) 6.51087 0.243834
\(714\) 0 0
\(715\) −4.62772 −0.173067
\(716\) 0 0
\(717\) 6.19702 + 26.3769i 0.231432 + 0.985064i
\(718\) 0 0
\(719\) −40.4674 −1.50918 −0.754589 0.656197i \(-0.772164\pi\)
−0.754589 + 0.656197i \(0.772164\pi\)
\(720\) 0 0
\(721\) 28.1168 + 32.4665i 1.04713 + 1.20912i
\(722\) 0 0
\(723\) −9.25544 39.3947i −0.344213 1.46511i
\(724\) 0 0
\(725\) 4.25639i 0.158078i
\(726\) 0 0
\(727\) 3.46410i 0.128476i −0.997935 0.0642382i \(-0.979538\pi\)
0.997935 0.0642382i \(-0.0204617\pi\)
\(728\) 0 0
\(729\) 5.00000 26.5330i 0.185185 0.982704i
\(730\) 0 0
\(731\) −9.25544 −0.342325
\(732\) 0 0
\(733\) 10.1899i 0.376373i −0.982133 0.188187i \(-0.939739\pi\)
0.982133 0.188187i \(-0.0602610\pi\)
\(734\) 0 0
\(735\) 1.05842 + 12.0781i 0.0390405 + 0.445506i
\(736\) 0 0
\(737\) 5.34363i 0.196835i
\(738\) 0 0
\(739\) 8.62772 0.317376 0.158688 0.987329i \(-0.449274\pi\)
0.158688 + 0.987329i \(0.449274\pi\)
\(740\) 0 0
\(741\) 34.1168 8.01544i 1.25331 0.294455i
\(742\) 0 0
\(743\) 36.9253i 1.35466i 0.735680 + 0.677329i \(0.236863\pi\)
−0.735680 + 0.677329i \(0.763137\pi\)
\(744\) 0 0
\(745\) 10.0974i 0.369938i
\(746\) 0 0
\(747\) −14.7446 + 7.33296i −0.539475 + 0.268299i
\(748\) 0 0
\(749\) 11.4891 + 13.2665i 0.419804 + 0.484747i
\(750\) 0 0
\(751\) 22.3505 0.815582 0.407791 0.913075i \(-0.366299\pi\)
0.407791 + 0.913075i \(0.366299\pi\)
\(752\) 0 0
\(753\) 29.4891 6.92820i 1.07464 0.252478i
\(754\) 0 0
\(755\) −3.37228 −0.122730
\(756\) 0 0
\(757\) −54.2337 −1.97116 −0.985578 0.169219i \(-0.945875\pi\)
−0.985578 + 0.169219i \(0.945875\pi\)
\(758\) 0 0
\(759\) 2.51087 0.589907i 0.0911390 0.0214123i
\(760\) 0 0
\(761\) −26.2337 −0.950970 −0.475485 0.879724i \(-0.657727\pi\)
−0.475485 + 0.879724i \(0.657727\pi\)
\(762\) 0 0
\(763\) 0.233688 0.202380i 0.00846007 0.00732664i
\(764\) 0 0
\(765\) −3.68614 + 1.83324i −0.133273 + 0.0662810i
\(766\) 0 0
\(767\) 16.0309i 0.578842i
\(768\) 0 0
\(769\) 21.1894i 0.764108i 0.924140 + 0.382054i \(0.124783\pi\)
−0.924140 + 0.382054i \(0.875217\pi\)
\(770\) 0 0
\(771\) −39.6060 + 9.30506i −1.42637 + 0.335114i
\(772\) 0 0
\(773\) −46.6277 −1.67708 −0.838541 0.544838i \(-0.816591\pi\)
−0.838541 + 0.544838i \(0.816591\pi\)
\(774\) 0 0
\(775\) 3.46410i 0.124434i
\(776\) 0 0
\(777\) −12.7446 + 17.6155i −0.457209 + 0.631951i
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) −10.7446 −0.384471
\(782\) 0 0
\(783\) 14.1168 + 17.0256i 0.504495 + 0.608444i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.5253i 1.65845i 0.558916 + 0.829224i \(0.311217\pi\)
−0.558916 + 0.829224i \(0.688783\pi\)
\(788\) 0 0
\(789\) −5.37228 22.8665i −0.191258 0.814070i
\(790\) 0 0
\(791\) −17.4891 20.1947i −0.621842 0.718041i
\(792\) 0 0
\(793\) −40.4674 −1.43704
\(794\) 0 0
\(795\) −3.37228 14.3537i −0.119602 0.509075i
\(796\) 0 0
\(797\) −42.8614 −1.51823 −0.759114 0.650957i \(-0.774368\pi\)
−0.759114 + 0.650957i \(0.774368\pi\)
\(798\) 0 0
\(799\) −10.1168 −0.357908
\(800\) 0 0
\(801\) 8.74456 4.34896i 0.308974 0.153663i
\(802\) 0 0
\(803\) −5.48913 −0.193707
\(804\) 0 0
\(805\) 3.25544 + 3.75906i 0.114739 + 0.132489i
\(806\) 0 0
\(807\) 14.7446 3.46410i 0.519033 0.121942i
\(808\) 0 0
\(809\) 36.7229i 1.29111i −0.763714 0.645555i \(-0.776626\pi\)
0.763714 0.645555i \(-0.223374\pi\)
\(810\) 0 0
\(811\) 1.28962i 0.0452847i 0.999744 + 0.0226423i \(0.00720790\pi\)
−0.999744 + 0.0226423i \(0.992792\pi\)
\(812\) 0 0
\(813\) 6.00000 + 25.5383i 0.210429 + 0.895668i
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 23.3639i 0.817398i
\(818\) 0 0
\(819\) 42.7812 + 17.8641i 1.49490 + 0.624223i
\(820\) 0 0
\(821\) 11.7745i 0.410933i 0.978664 + 0.205466i \(0.0658711\pi\)
−0.978664 + 0.205466i \(0.934129\pi\)
\(822\) 0 0
\(823\) 48.2337 1.68132 0.840660 0.541563i \(-0.182167\pi\)
0.840660 + 0.541563i \(0.182167\pi\)
\(824\) 0 0
\(825\) −0.313859 1.33591i −0.0109272 0.0465103i
\(826\) 0 0
\(827\) 18.3152i 0.636881i 0.947943 + 0.318441i \(0.103159\pi\)
−0.947943 + 0.318441i \(0.896841\pi\)
\(828\) 0 0
\(829\) 32.4665i 1.12761i 0.825908 + 0.563805i \(0.190663\pi\)
−0.825908 + 0.563805i \(0.809337\pi\)
\(830\) 0 0
\(831\) 10.5109 2.46943i 0.364618 0.0856637i
\(832\) 0 0
\(833\) −1.37228 + 9.50744i −0.0475467 + 0.329413i
\(834\) 0 0
\(835\) 22.1168 0.765385
\(836\) 0 0
\(837\) 11.4891 + 13.8564i 0.397122 + 0.478947i
\(838\) 0 0
\(839\) 53.4891 1.84665 0.923325 0.384020i \(-0.125461\pi\)
0.923325 + 0.384020i \(0.125461\pi\)
\(840\) 0 0
\(841\) 10.8832 0.375281
\(842\) 0 0
\(843\) −1.91983 8.17154i −0.0661224 0.281443i
\(844\) 0 0
\(845\) −21.1168 −0.726442
\(846\) 0 0
\(847\) 20.7446 17.9653i 0.712792 0.617296i
\(848\) 0 0
\(849\) −3.68614 15.6896i −0.126508 0.538467i
\(850\) 0 0
\(851\) 8.91754i 0.305689i
\(852\) 0 0
\(853\) 46.7277i 1.59993i −0.600049 0.799963i \(-0.704852\pi\)
0.600049 0.799963i \(-0.295148\pi\)
\(854\) 0 0
\(855\) 4.62772 + 9.30506i 0.158265 + 0.318226i
\(856\) 0 0
\(857\) −22.4674 −0.767471 −0.383735 0.923443i \(-0.625363\pi\)
−0.383735 + 0.923443i \(0.625363\pi\)
\(858\) 0 0
\(859\) 45.4381i 1.55033i −0.631760 0.775164i \(-0.717667\pi\)
0.631760 0.775164i \(-0.282333\pi\)
\(860\) 0 0
\(861\) 16.1168 22.2766i 0.549261 0.759185i
\(862\) 0 0
\(863\) 28.0078i 0.953395i −0.879067 0.476698i \(-0.841834\pi\)
0.879067 0.476698i \(-0.158166\pi\)
\(864\) 0 0
\(865\) 16.1168 0.547989
\(866\) 0 0
\(867\) 25.4891 5.98844i 0.865656 0.203378i
\(868\) 0 0
\(869\) 2.67181i 0.0906351i
\(870\) 0 0
\(871\) 39.3947i 1.33484i
\(872\) 0 0
\(873\) −1.45245 2.92048i −0.0491581 0.0988433i
\(874\) 0 0
\(875\) 2.00000 1.73205i 0.0676123 0.0585540i
\(876\) 0 0
\(877\) 30.4674 1.02881 0.514405 0.857547i \(-0.328013\pi\)
0.514405 + 0.857547i \(0.328013\pi\)
\(878\) 0 0
\(879\) 47.4090 11.1383i 1.59906 0.375686i
\(880\) 0 0
\(881\) −2.23369 −0.0752549 −0.0376274 0.999292i \(-0.511980\pi\)
−0.0376274 + 0.999292i \(0.511980\pi\)
\(882\) 0 0
\(883\) −26.5109 −0.892162 −0.446081 0.894993i \(-0.647181\pi\)
−0.446081 + 0.894993i \(0.647181\pi\)
\(884\) 0 0
\(885\) 4.62772 1.08724i 0.155559 0.0365472i
\(886\) 0 0
\(887\) −41.4891 −1.39307 −0.696534 0.717524i \(-0.745276\pi\)
−0.696534 + 0.717524i \(0.745276\pi\)
\(888\) 0 0
\(889\) −21.4891 + 18.6101i −0.720722 + 0.624164i
\(890\) 0 0
\(891\) 5.68614 + 4.30268i 0.190493 + 0.144145i
\(892\) 0 0
\(893\) 25.5383i 0.854608i
\(894\) 0 0
\(895\) 6.63325i 0.221725i
\(896\) 0 0
\(897\) 18.5109 4.34896i 0.618060 0.145208i
\(898\) 0 0
\(899\) −14.7446 −0.491759
\(900\) 0 0
\(901\) 11.6819i 0.389181i
\(902\) 0 0
\(903\) −18.1168 + 25.0410i −0.602891 + 0.833312i
\(904\) 0 0
\(905\) 18.6101i 0.618622i
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 0 0
\(909\) 16.1168 8.01544i 0.534562 0.265855i
\(910\) 0 0
\(911\) 23.6588i 0.783851i −0.919997 0.391926i \(-0.871809\pi\)
0.919997 0.391926i \(-0.128191\pi\)
\(912\) 0 0
\(913\) 4.34896i 0.143930i
\(914\) 0 0
\(915\) −2.74456 11.6819i −0.0907324 0.386193i
\(916\) 0 0
\(917\) 34.9783 30.2921i 1.15508 1.00033i
\(918\) 0 0
\(919\) 11.3723 0.375137 0.187568 0.982252i \(-0.439939\pi\)
0.187568 + 0.982252i \(0.439939\pi\)
\(920\) 0 0
\(921\) −2.82473 12.0232i −0.0930782 0.396177i
\(922\) 0 0
\(923\) −79.2119 −2.60729
\(924\) 0 0
\(925\) 4.74456 0.156000
\(926\) 0 0
\(927\) 21.6861 + 43.6048i 0.712266 + 1.43217i
\(928\) 0 0
\(929\) 7.02175 0.230376 0.115188 0.993344i \(-0.463253\pi\)
0.115188 + 0.993344i \(0.463253\pi\)
\(930\) 0 0
\(931\) 24.0000 + 3.46410i 0.786568 + 0.113531i
\(932\) 0 0
\(933\) 34.1168 8.01544i 1.11694 0.262414i
\(934\) 0 0
\(935\) 1.08724i 0.0355566i
\(936\) 0 0
\(937\) 49.9894i 1.63308i −0.577287 0.816542i \(-0.695888\pi\)
0.577287 0.816542i \(-0.304112\pi\)
\(938\) 0 0
\(939\) 9.68614 + 41.2280i 0.316095 + 1.34542i
\(940\) 0 0
\(941\) −27.2554 −0.888502 −0.444251 0.895902i \(-0.646530\pi\)
−0.444251 + 0.895902i \(0.646530\pi\)
\(942\) 0 0
\(943\) 11.2772i 0.367235i
\(944\) 0 0
\(945\) −2.25544 + 13.5615i −0.0733694 + 0.441154i
\(946\) 0 0
\(947\) 48.2025i 1.56637i 0.621789 + 0.783185i \(0.286406\pi\)
−0.621789 + 0.783185i \(0.713594\pi\)
\(948\) 0 0
\(949\) −40.4674 −1.31363
\(950\) 0 0
\(951\) −3.37228 14.3537i −0.109354 0.465452i
\(952\) 0 0
\(953\) 38.8048i 1.25701i 0.777805 + 0.628506i \(0.216333\pi\)
−0.777805 + 0.628506i \(0.783667\pi\)
\(954\) 0 0
\(955\) 15.6434i 0.506207i
\(956\) 0 0
\(957\) −5.68614 + 1.33591i −0.183807 + 0.0431838i
\(958\) 0 0
\(959\) 22.9783 + 26.5330i 0.742006 + 0.856795i
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 8.86141 + 17.8178i 0.285555 + 0.574172i
\(964\) 0 0
\(965\) 22.2337 0.715728
\(966\) 0 0
\(967\) 24.2337 0.779303 0.389651 0.920962i \(-0.372595\pi\)
0.389651 + 0.920962i \(0.372595\pi\)
\(968\) 0 0
\(969\) 1.88316 + 8.01544i 0.0604957 + 0.257493i
\(970\) 0 0
\(971\) 43.2119 1.38674 0.693369 0.720583i \(-0.256126\pi\)
0.693369 + 0.720583i \(0.256126\pi\)
\(972\) 0 0
\(973\) 2.23369 + 2.57924i 0.0716087 + 0.0826867i
\(974\) 0 0
\(975\) −2.31386 9.84868i −0.0741028 0.315410i
\(976\) 0 0
\(977\) 38.8048i 1.24148i 0.784018 + 0.620738i \(0.213167\pi\)
−0.784018 + 0.620738i \(0.786833\pi\)
\(978\) 0 0
\(979\) 2.57924i 0.0824329i
\(980\) 0 0
\(981\) 0.313859 0.156093i 0.0100208 0.00498366i
\(982\) 0 0
\(983\) −11.1386 −0.355266 −0.177633 0.984097i \(-0.556844\pi\)
−0.177633 + 0.984097i \(0.556844\pi\)
\(984\) 0 0
\(985\) 22.3692i 0.712741i
\(986\) 0 0
\(987\) −19.8030 + 27.3716i −0.630336 + 0.871247i
\(988\) 0 0
\(989\) 12.6766i 0.403092i
\(990\) 0 0
\(991\) −2.51087 −0.0797606 −0.0398803 0.999204i \(-0.512698\pi\)
−0.0398803 + 0.999204i \(0.512698\pi\)
\(992\) 0 0
\(993\) −6.74456 + 1.58457i −0.214032 + 0.0502849i
\(994\) 0 0
\(995\) 12.9715i 0.411226i
\(996\) 0 0
\(997\) 21.8719i 0.692688i −0.938107 0.346344i \(-0.887423\pi\)
0.938107 0.346344i \(-0.112577\pi\)
\(998\) 0 0
\(999\) −18.9783 + 15.7359i −0.600445 + 0.497863i
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 1680.2.f.g.881.2 4
3.2 odd 2 1680.2.f.h.881.4 4
4.3 odd 2 105.2.b.d.41.1 yes 4
7.6 odd 2 1680.2.f.h.881.3 4
12.11 even 2 105.2.b.c.41.4 yes 4
20.3 even 4 525.2.g.d.524.2 8
20.7 even 4 525.2.g.d.524.7 8
20.19 odd 2 525.2.b.e.251.4 4
21.20 even 2 inner 1680.2.f.g.881.1 4
28.3 even 6 735.2.s.i.656.2 4
28.11 odd 6 735.2.s.j.656.2 4
28.19 even 6 735.2.s.h.521.1 4
28.23 odd 6 735.2.s.g.521.1 4
28.27 even 2 105.2.b.c.41.1 4
60.23 odd 4 525.2.g.e.524.7 8
60.47 odd 4 525.2.g.e.524.2 8
60.59 even 2 525.2.b.g.251.1 4
84.11 even 6 735.2.s.h.656.1 4
84.23 even 6 735.2.s.i.521.2 4
84.47 odd 6 735.2.s.j.521.2 4
84.59 odd 6 735.2.s.g.656.1 4
84.83 odd 2 105.2.b.d.41.4 yes 4
140.27 odd 4 525.2.g.e.524.8 8
140.83 odd 4 525.2.g.e.524.1 8
140.139 even 2 525.2.b.g.251.4 4
420.83 even 4 525.2.g.d.524.8 8
420.167 even 4 525.2.g.d.524.1 8
420.419 odd 2 525.2.b.e.251.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.b.c.41.1 4 28.27 even 2
105.2.b.c.41.4 yes 4 12.11 even 2
105.2.b.d.41.1 yes 4 4.3 odd 2
105.2.b.d.41.4 yes 4 84.83 odd 2
525.2.b.e.251.1 4 420.419 odd 2
525.2.b.e.251.4 4 20.19 odd 2
525.2.b.g.251.1 4 60.59 even 2
525.2.b.g.251.4 4 140.139 even 2
525.2.g.d.524.1 8 420.167 even 4
525.2.g.d.524.2 8 20.3 even 4
525.2.g.d.524.7 8 20.7 even 4
525.2.g.d.524.8 8 420.83 even 4
525.2.g.e.524.1 8 140.83 odd 4
525.2.g.e.524.2 8 60.47 odd 4
525.2.g.e.524.7 8 60.23 odd 4
525.2.g.e.524.8 8 140.27 odd 4
735.2.s.g.521.1 4 28.23 odd 6
735.2.s.g.656.1 4 84.59 odd 6
735.2.s.h.521.1 4 28.19 even 6
735.2.s.h.656.1 4 84.11 even 6
735.2.s.i.521.2 4 84.23 even 6
735.2.s.i.656.2 4 28.3 even 6
735.2.s.j.521.2 4 84.47 odd 6
735.2.s.j.656.2 4 28.11 odd 6
1680.2.f.g.881.1 4 21.20 even 2 inner
1680.2.f.g.881.2 4 1.1 even 1 trivial
1680.2.f.h.881.3 4 7.6 odd 2
1680.2.f.h.881.4 4 3.2 odd 2