Properties

Label 1920.2.bl.a.289.2
Level $1920$
Weight $2$
Character 1920.289
Analytic conductor $15.331$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,2,Mod(289,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1920.bl (of order \(4\), degree \(2\), 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: \(15.3312771881\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 289.2
Character \(\chi\) \(=\) 1920.289
Dual form 1920.2.bl.a.1249.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.98421 - 1.03097i) q^{5} -3.91927 q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.98421 - 1.03097i) q^{5} -3.91927 q^{7} -1.00000i q^{9} +(2.93423 - 2.93423i) q^{11} +(0.732828 - 0.732828i) q^{13} +(-0.674045 + 2.13206i) q^{15} +2.89302i q^{17} +(-1.67534 - 1.67534i) q^{19} +(2.77134 - 2.77134i) q^{21} -1.73133 q^{23} +(2.87420 - 4.09133i) q^{25} +(0.707107 + 0.707107i) q^{27} +(4.99003 + 4.99003i) q^{29} -10.8149 q^{31} +4.14963i q^{33} +(-7.77666 + 4.04065i) q^{35} +(-6.41524 - 6.41524i) q^{37} +1.03637i q^{39} +0.00577512i q^{41} +(2.23930 + 2.23930i) q^{43} +(-1.03097 - 1.98421i) q^{45} -11.6451i q^{47} +8.36066 q^{49} +(-2.04567 - 2.04567i) q^{51} +(-5.55765 - 5.55765i) q^{53} +(2.79703 - 8.84724i) q^{55} +2.36929 q^{57} +(-3.83290 + 3.83290i) q^{59} +(-9.30595 - 9.30595i) q^{61} +3.91927i q^{63} +(0.698563 - 2.20961i) q^{65} +(3.85243 - 3.85243i) q^{67} +(1.22423 - 1.22423i) q^{69} +1.15355i q^{71} -7.98713 q^{73} +(0.860637 + 4.92537i) q^{75} +(-11.5000 + 11.5000i) q^{77} +0.843960 q^{79} -1.00000 q^{81} +(5.20069 - 5.20069i) q^{83} +(2.98261 + 5.74036i) q^{85} -7.05696 q^{87} -5.40896i q^{89} +(-2.87215 + 2.87215i) q^{91} +(7.64726 - 7.64726i) q^{93} +(-5.05147 - 1.59701i) q^{95} -2.24154i q^{97} +(-2.93423 - 2.93423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 8 q^{19} - 48 q^{31} - 24 q^{35} + 48 q^{49} - 8 q^{51} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{69} + 16 q^{75} - 96 q^{79} - 48 q^{81} + 32 q^{91} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 1.98421 1.03097i 0.887367 0.461064i
\(6\) 0 0
\(7\) −3.91927 −1.48134 −0.740672 0.671867i \(-0.765493\pi\)
−0.740672 + 0.671867i \(0.765493\pi\)
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 2.93423 2.93423i 0.884703 0.884703i −0.109305 0.994008i \(-0.534862\pi\)
0.994008 + 0.109305i \(0.0348625\pi\)
\(12\) 0 0
\(13\) 0.732828 0.732828i 0.203250 0.203250i −0.598141 0.801391i \(-0.704094\pi\)
0.801391 + 0.598141i \(0.204094\pi\)
\(14\) 0 0
\(15\) −0.674045 + 2.13206i −0.174038 + 0.550495i
\(16\) 0 0
\(17\) 2.89302i 0.701659i 0.936439 + 0.350830i \(0.114100\pi\)
−0.936439 + 0.350830i \(0.885900\pi\)
\(18\) 0 0
\(19\) −1.67534 1.67534i −0.384350 0.384350i 0.488316 0.872667i \(-0.337611\pi\)
−0.872667 + 0.488316i \(0.837611\pi\)
\(20\) 0 0
\(21\) 2.77134 2.77134i 0.604756 0.604756i
\(22\) 0 0
\(23\) −1.73133 −0.361006 −0.180503 0.983574i \(-0.557773\pi\)
−0.180503 + 0.983574i \(0.557773\pi\)
\(24\) 0 0
\(25\) 2.87420 4.09133i 0.574841 0.818265i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 4.99003 + 4.99003i 0.926625 + 0.926625i 0.997486 0.0708617i \(-0.0225749\pi\)
−0.0708617 + 0.997486i \(0.522575\pi\)
\(30\) 0 0
\(31\) −10.8149 −1.94241 −0.971203 0.238254i \(-0.923425\pi\)
−0.971203 + 0.238254i \(0.923425\pi\)
\(32\) 0 0
\(33\) 4.14963i 0.722357i
\(34\) 0 0
\(35\) −7.77666 + 4.04065i −1.31450 + 0.682994i
\(36\) 0 0
\(37\) −6.41524 6.41524i −1.05466 1.05466i −0.998417 0.0562413i \(-0.982088\pi\)
−0.0562413 0.998417i \(-0.517912\pi\)
\(38\) 0 0
\(39\) 1.03637i 0.165953i
\(40\) 0 0
\(41\) 0.00577512i 0.000901923i 1.00000 0.000450961i \(0.000143545\pi\)
−1.00000 0.000450961i \(0.999856\pi\)
\(42\) 0 0
\(43\) 2.23930 + 2.23930i 0.341490 + 0.341490i 0.856927 0.515437i \(-0.172370\pi\)
−0.515437 + 0.856927i \(0.672370\pi\)
\(44\) 0 0
\(45\) −1.03097 1.98421i −0.153688 0.295789i
\(46\) 0 0
\(47\) 11.6451i 1.69861i −0.527900 0.849307i \(-0.677020\pi\)
0.527900 0.849307i \(-0.322980\pi\)
\(48\) 0 0
\(49\) 8.36066 1.19438
\(50\) 0 0
\(51\) −2.04567 2.04567i −0.286451 0.286451i
\(52\) 0 0
\(53\) −5.55765 5.55765i −0.763402 0.763402i 0.213533 0.976936i \(-0.431503\pi\)
−0.976936 + 0.213533i \(0.931503\pi\)
\(54\) 0 0
\(55\) 2.79703 8.84724i 0.377152 1.19296i
\(56\) 0 0
\(57\) 2.36929 0.313821
\(58\) 0 0
\(59\) −3.83290 + 3.83290i −0.499001 + 0.499001i −0.911127 0.412126i \(-0.864786\pi\)
0.412126 + 0.911127i \(0.364786\pi\)
\(60\) 0 0
\(61\) −9.30595 9.30595i −1.19150 1.19150i −0.976645 0.214859i \(-0.931071\pi\)
−0.214859 0.976645i \(-0.568929\pi\)
\(62\) 0 0
\(63\) 3.91927i 0.493781i
\(64\) 0 0
\(65\) 0.698563 2.20961i 0.0866461 0.274068i
\(66\) 0 0
\(67\) 3.85243 3.85243i 0.470649 0.470649i −0.431476 0.902124i \(-0.642007\pi\)
0.902124 + 0.431476i \(0.142007\pi\)
\(68\) 0 0
\(69\) 1.22423 1.22423i 0.147380 0.147380i
\(70\) 0 0
\(71\) 1.15355i 0.136901i 0.997655 + 0.0684504i \(0.0218055\pi\)
−0.997655 + 0.0684504i \(0.978195\pi\)
\(72\) 0 0
\(73\) −7.98713 −0.934823 −0.467411 0.884040i \(-0.654813\pi\)
−0.467411 + 0.884040i \(0.654813\pi\)
\(74\) 0 0
\(75\) 0.860637 + 4.92537i 0.0993778 + 0.568733i
\(76\) 0 0
\(77\) −11.5000 + 11.5000i −1.31055 + 1.31055i
\(78\) 0 0
\(79\) 0.843960 0.0949529 0.0474765 0.998872i \(-0.484882\pi\)
0.0474765 + 0.998872i \(0.484882\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 5.20069 5.20069i 0.570850 0.570850i −0.361516 0.932366i \(-0.617741\pi\)
0.932366 + 0.361516i \(0.117741\pi\)
\(84\) 0 0
\(85\) 2.98261 + 5.74036i 0.323510 + 0.622629i
\(86\) 0 0
\(87\) −7.05696 −0.756586
\(88\) 0 0
\(89\) 5.40896i 0.573349i −0.958028 0.286674i \(-0.907450\pi\)
0.958028 0.286674i \(-0.0925498\pi\)
\(90\) 0 0
\(91\) −2.87215 + 2.87215i −0.301083 + 0.301083i
\(92\) 0 0
\(93\) 7.64726 7.64726i 0.792984 0.792984i
\(94\) 0 0
\(95\) −5.05147 1.59701i −0.518270 0.163850i
\(96\) 0 0
\(97\) 2.24154i 0.227594i −0.993504 0.113797i \(-0.963699\pi\)
0.993504 0.113797i \(-0.0363013\pi\)
\(98\) 0 0
\(99\) −2.93423 2.93423i −0.294901 0.294901i
\(100\) 0 0
\(101\) 7.50655 7.50655i 0.746930 0.746930i −0.226972 0.973901i \(-0.572882\pi\)
0.973901 + 0.226972i \(0.0728824\pi\)
\(102\) 0 0
\(103\) 3.91110 0.385372 0.192686 0.981260i \(-0.438280\pi\)
0.192686 + 0.981260i \(0.438280\pi\)
\(104\) 0 0
\(105\) 2.64176 8.35610i 0.257810 0.815472i
\(106\) 0 0
\(107\) 2.60233 + 2.60233i 0.251577 + 0.251577i 0.821617 0.570040i \(-0.193072\pi\)
−0.570040 + 0.821617i \(0.693072\pi\)
\(108\) 0 0
\(109\) 2.06635 + 2.06635i 0.197920 + 0.197920i 0.799108 0.601188i \(-0.205306\pi\)
−0.601188 + 0.799108i \(0.705306\pi\)
\(110\) 0 0
\(111\) 9.07252 0.861125
\(112\) 0 0
\(113\) 8.13610i 0.765380i −0.923877 0.382690i \(-0.874998\pi\)
0.923877 0.382690i \(-0.125002\pi\)
\(114\) 0 0
\(115\) −3.43532 + 1.78495i −0.320345 + 0.166447i
\(116\) 0 0
\(117\) −0.732828 0.732828i −0.0677500 0.0677500i
\(118\) 0 0
\(119\) 11.3385i 1.03940i
\(120\) 0 0
\(121\) 6.21941i 0.565401i
\(122\) 0 0
\(123\) −0.00408363 0.00408363i −0.000368208 0.000368208i
\(124\) 0 0
\(125\) 1.48500 11.0813i 0.132822 0.991140i
\(126\) 0 0
\(127\) 16.0862i 1.42742i 0.700442 + 0.713709i \(0.252986\pi\)
−0.700442 + 0.713709i \(0.747014\pi\)
\(128\) 0 0
\(129\) −3.16685 −0.278826
\(130\) 0 0
\(131\) −13.0407 13.0407i −1.13937 1.13937i −0.988563 0.150808i \(-0.951813\pi\)
−0.150808 0.988563i \(-0.548187\pi\)
\(132\) 0 0
\(133\) 6.56612 + 6.56612i 0.569355 + 0.569355i
\(134\) 0 0
\(135\) 2.13206 + 0.674045i 0.183498 + 0.0580125i
\(136\) 0 0
\(137\) −6.51614 −0.556711 −0.278356 0.960478i \(-0.589789\pi\)
−0.278356 + 0.960478i \(0.589789\pi\)
\(138\) 0 0
\(139\) 15.0953 15.0953i 1.28036 1.28036i 0.339902 0.940461i \(-0.389606\pi\)
0.940461 0.339902i \(-0.110394\pi\)
\(140\) 0 0
\(141\) 8.23434 + 8.23434i 0.693456 + 0.693456i
\(142\) 0 0
\(143\) 4.30057i 0.359632i
\(144\) 0 0
\(145\) 15.0458 + 4.75671i 1.24949 + 0.395023i
\(146\) 0 0
\(147\) −5.91188 + 5.91188i −0.487604 + 0.487604i
\(148\) 0 0
\(149\) −4.27402 + 4.27402i −0.350141 + 0.350141i −0.860162 0.510021i \(-0.829638\pi\)
0.510021 + 0.860162i \(0.329638\pi\)
\(150\) 0 0
\(151\) 8.39998i 0.683581i 0.939776 + 0.341790i \(0.111033\pi\)
−0.939776 + 0.341790i \(0.888967\pi\)
\(152\) 0 0
\(153\) 2.89302 0.233886
\(154\) 0 0
\(155\) −21.4590 + 11.1498i −1.72363 + 0.895573i
\(156\) 0 0
\(157\) 5.68893 5.68893i 0.454026 0.454026i −0.442662 0.896689i \(-0.645966\pi\)
0.896689 + 0.442662i \(0.145966\pi\)
\(158\) 0 0
\(159\) 7.85971 0.623315
\(160\) 0 0
\(161\) 6.78553 0.534775
\(162\) 0 0
\(163\) −5.63869 + 5.63869i −0.441656 + 0.441656i −0.892568 0.450912i \(-0.851099\pi\)
0.450912 + 0.892568i \(0.351099\pi\)
\(164\) 0 0
\(165\) 4.27814 + 8.23374i 0.333053 + 0.640996i
\(166\) 0 0
\(167\) −6.45583 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(168\) 0 0
\(169\) 11.9259i 0.917379i
\(170\) 0 0
\(171\) −1.67534 + 1.67534i −0.128117 + 0.128117i
\(172\) 0 0
\(173\) −7.33664 + 7.33664i −0.557795 + 0.557795i −0.928679 0.370884i \(-0.879055\pi\)
0.370884 + 0.928679i \(0.379055\pi\)
\(174\) 0 0
\(175\) −11.2648 + 16.0350i −0.851537 + 1.21213i
\(176\) 0 0
\(177\) 5.42054i 0.407432i
\(178\) 0 0
\(179\) 1.04377 + 1.04377i 0.0780153 + 0.0780153i 0.745038 0.667022i \(-0.232431\pi\)
−0.667022 + 0.745038i \(0.732431\pi\)
\(180\) 0 0
\(181\) 12.1024 12.1024i 0.899567 0.899567i −0.0958310 0.995398i \(-0.530551\pi\)
0.995398 + 0.0958310i \(0.0305508\pi\)
\(182\) 0 0
\(183\) 13.1606 0.972859
\(184\) 0 0
\(185\) −19.3431 6.11528i −1.42213 0.449604i
\(186\) 0 0
\(187\) 8.48877 + 8.48877i 0.620761 + 0.620761i
\(188\) 0 0
\(189\) −2.77134 2.77134i −0.201585 0.201585i
\(190\) 0 0
\(191\) −21.2273 −1.53596 −0.767978 0.640477i \(-0.778737\pi\)
−0.767978 + 0.640477i \(0.778737\pi\)
\(192\) 0 0
\(193\) 19.2782i 1.38767i −0.720132 0.693837i \(-0.755919\pi\)
0.720132 0.693837i \(-0.244081\pi\)
\(194\) 0 0
\(195\) 1.06847 + 2.05639i 0.0765148 + 0.147261i
\(196\) 0 0
\(197\) −9.48350 9.48350i −0.675672 0.675672i 0.283346 0.959018i \(-0.408556\pi\)
−0.959018 + 0.283346i \(0.908556\pi\)
\(198\) 0 0
\(199\) 8.65421i 0.613481i −0.951793 0.306740i \(-0.900762\pi\)
0.951793 0.306740i \(-0.0992383\pi\)
\(200\) 0 0
\(201\) 5.44815i 0.384283i
\(202\) 0 0
\(203\) −19.5572 19.5572i −1.37265 1.37265i
\(204\) 0 0
\(205\) 0.00595398 + 0.0114591i 0.000415844 + 0.000800336i
\(206\) 0 0
\(207\) 1.73133i 0.120335i
\(208\) 0 0
\(209\) −9.83169 −0.680072
\(210\) 0 0
\(211\) 0.875946 + 0.875946i 0.0603026 + 0.0603026i 0.736615 0.676312i \(-0.236423\pi\)
−0.676312 + 0.736615i \(0.736423\pi\)
\(212\) 0 0
\(213\) −0.815681 0.815681i −0.0558895 0.0558895i
\(214\) 0 0
\(215\) 6.75190 + 2.13460i 0.460476 + 0.145578i
\(216\) 0 0
\(217\) 42.3863 2.87737
\(218\) 0 0
\(219\) 5.64775 5.64775i 0.381640 0.381640i
\(220\) 0 0
\(221\) 2.12008 + 2.12008i 0.142612 + 0.142612i
\(222\) 0 0
\(223\) 10.4375i 0.698946i −0.936946 0.349473i \(-0.886361\pi\)
0.936946 0.349473i \(-0.113639\pi\)
\(224\) 0 0
\(225\) −4.09133 2.87420i −0.272755 0.191614i
\(226\) 0 0
\(227\) 13.7218 13.7218i 0.910747 0.910747i −0.0855842 0.996331i \(-0.527276\pi\)
0.996331 + 0.0855842i \(0.0272757\pi\)
\(228\) 0 0
\(229\) −8.43878 + 8.43878i −0.557651 + 0.557651i −0.928638 0.370987i \(-0.879019\pi\)
0.370987 + 0.928638i \(0.379019\pi\)
\(230\) 0 0
\(231\) 16.2635i 1.07006i
\(232\) 0 0
\(233\) −28.6100 −1.87431 −0.937153 0.348919i \(-0.886549\pi\)
−0.937153 + 0.348919i \(0.886549\pi\)
\(234\) 0 0
\(235\) −12.0058 23.1064i −0.783169 1.50729i
\(236\) 0 0
\(237\) −0.596770 + 0.596770i −0.0387644 + 0.0387644i
\(238\) 0 0
\(239\) 21.4924 1.39023 0.695115 0.718899i \(-0.255354\pi\)
0.695115 + 0.718899i \(0.255354\pi\)
\(240\) 0 0
\(241\) −19.5943 −1.26218 −0.631091 0.775709i \(-0.717393\pi\)
−0.631091 + 0.775709i \(0.717393\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 16.5893 8.61959i 1.05985 0.550685i
\(246\) 0 0
\(247\) −2.45548 −0.156238
\(248\) 0 0
\(249\) 7.35489i 0.466097i
\(250\) 0 0
\(251\) 9.58072 9.58072i 0.604730 0.604730i −0.336834 0.941564i \(-0.609356\pi\)
0.941564 + 0.336834i \(0.109356\pi\)
\(252\) 0 0
\(253\) −5.08011 + 5.08011i −0.319384 + 0.319384i
\(254\) 0 0
\(255\) −6.16807 1.95002i −0.386260 0.122115i
\(256\) 0 0
\(257\) 13.2730i 0.827946i 0.910289 + 0.413973i \(0.135859\pi\)
−0.910289 + 0.413973i \(0.864141\pi\)
\(258\) 0 0
\(259\) 25.1430 + 25.1430i 1.56231 + 1.56231i
\(260\) 0 0
\(261\) 4.99003 4.99003i 0.308875 0.308875i
\(262\) 0 0
\(263\) 13.0606 0.805349 0.402675 0.915343i \(-0.368081\pi\)
0.402675 + 0.915343i \(0.368081\pi\)
\(264\) 0 0
\(265\) −16.7573 5.29780i −1.02940 0.325441i
\(266\) 0 0
\(267\) 3.82471 + 3.82471i 0.234069 + 0.234069i
\(268\) 0 0
\(269\) 12.4067 + 12.4067i 0.756451 + 0.756451i 0.975675 0.219224i \(-0.0703525\pi\)
−0.219224 + 0.975675i \(0.570353\pi\)
\(270\) 0 0
\(271\) 2.11193 0.128291 0.0641453 0.997941i \(-0.479568\pi\)
0.0641453 + 0.997941i \(0.479568\pi\)
\(272\) 0 0
\(273\) 4.06183i 0.245833i
\(274\) 0 0
\(275\) −3.57132 20.4385i −0.215359 1.23249i
\(276\) 0 0
\(277\) 13.2116 + 13.2116i 0.793808 + 0.793808i 0.982111 0.188303i \(-0.0602988\pi\)
−0.188303 + 0.982111i \(0.560299\pi\)
\(278\) 0 0
\(279\) 10.8149i 0.647469i
\(280\) 0 0
\(281\) 3.78786i 0.225965i 0.993597 + 0.112982i \(0.0360404\pi\)
−0.993597 + 0.112982i \(0.963960\pi\)
\(282\) 0 0
\(283\) 3.16274 + 3.16274i 0.188005 + 0.188005i 0.794833 0.606828i \(-0.207558\pi\)
−0.606828 + 0.794833i \(0.707558\pi\)
\(284\) 0 0
\(285\) 4.70118 2.44267i 0.278474 0.144691i
\(286\) 0 0
\(287\) 0.0226343i 0.00133606i
\(288\) 0 0
\(289\) 8.63046 0.507674
\(290\) 0 0
\(291\) 1.58501 + 1.58501i 0.0929147 + 0.0929147i
\(292\) 0 0
\(293\) −8.51339 8.51339i −0.497357 0.497357i 0.413257 0.910614i \(-0.364391\pi\)
−0.910614 + 0.413257i \(0.864391\pi\)
\(294\) 0 0
\(295\) −3.65368 + 11.5569i −0.212726 + 0.672868i
\(296\) 0 0
\(297\) 4.14963 0.240786
\(298\) 0 0
\(299\) −1.26876 + 1.26876i −0.0733745 + 0.0733745i
\(300\) 0 0
\(301\) −8.77642 8.77642i −0.505864 0.505864i
\(302\) 0 0
\(303\) 10.6159i 0.609866i
\(304\) 0 0
\(305\) −28.0591 8.87083i −1.60666 0.507942i
\(306\) 0 0
\(307\) −7.12093 + 7.12093i −0.406413 + 0.406413i −0.880486 0.474073i \(-0.842783\pi\)
0.474073 + 0.880486i \(0.342783\pi\)
\(308\) 0 0
\(309\) −2.76557 + 2.76557i −0.157328 + 0.157328i
\(310\) 0 0
\(311\) 21.2980i 1.20770i 0.797099 + 0.603849i \(0.206367\pi\)
−0.797099 + 0.603849i \(0.793633\pi\)
\(312\) 0 0
\(313\) 7.49789 0.423806 0.211903 0.977291i \(-0.432034\pi\)
0.211903 + 0.977291i \(0.432034\pi\)
\(314\) 0 0
\(315\) 4.04065 + 7.77666i 0.227665 + 0.438165i
\(316\) 0 0
\(317\) −2.35354 + 2.35354i −0.132188 + 0.132188i −0.770105 0.637917i \(-0.779796\pi\)
0.637917 + 0.770105i \(0.279796\pi\)
\(318\) 0 0
\(319\) 29.2838 1.63958
\(320\) 0 0
\(321\) −3.68025 −0.205412
\(322\) 0 0
\(323\) 4.84680 4.84680i 0.269683 0.269683i
\(324\) 0 0
\(325\) −0.891943 5.10453i −0.0494761 0.283149i
\(326\) 0 0
\(327\) −2.92226 −0.161601
\(328\) 0 0
\(329\) 45.6403i 2.51623i
\(330\) 0 0
\(331\) −22.6845 + 22.6845i −1.24685 + 1.24685i −0.289753 + 0.957102i \(0.593573\pi\)
−0.957102 + 0.289753i \(0.906427\pi\)
\(332\) 0 0
\(333\) −6.41524 + 6.41524i −0.351553 + 0.351553i
\(334\) 0 0
\(335\) 3.67230 11.6158i 0.200639 0.634637i
\(336\) 0 0
\(337\) 8.90357i 0.485008i −0.970150 0.242504i \(-0.922031\pi\)
0.970150 0.242504i \(-0.0779688\pi\)
\(338\) 0 0
\(339\) 5.75309 + 5.75309i 0.312465 + 0.312465i
\(340\) 0 0
\(341\) −31.7333 + 31.7333i −1.71845 + 1.71845i
\(342\) 0 0
\(343\) −5.33279 −0.287944
\(344\) 0 0
\(345\) 1.16699 3.69129i 0.0628287 0.198732i
\(346\) 0 0
\(347\) −8.87783 8.87783i −0.476587 0.476587i 0.427451 0.904038i \(-0.359412\pi\)
−0.904038 + 0.427451i \(0.859412\pi\)
\(348\) 0 0
\(349\) 8.23040 + 8.23040i 0.440563 + 0.440563i 0.892201 0.451638i \(-0.149160\pi\)
−0.451638 + 0.892201i \(0.649160\pi\)
\(350\) 0 0
\(351\) 1.03637 0.0553176
\(352\) 0 0
\(353\) 34.7125i 1.84756i 0.382922 + 0.923781i \(0.374918\pi\)
−0.382922 + 0.923781i \(0.625082\pi\)
\(354\) 0 0
\(355\) 1.18927 + 2.28888i 0.0631200 + 0.121481i
\(356\) 0 0
\(357\) 8.01753 + 8.01753i 0.424333 + 0.424333i
\(358\) 0 0
\(359\) 2.81082i 0.148350i 0.997245 + 0.0741748i \(0.0236323\pi\)
−0.997245 + 0.0741748i \(0.976368\pi\)
\(360\) 0 0
\(361\) 13.3864i 0.704550i
\(362\) 0 0
\(363\) 4.39778 + 4.39778i 0.230824 + 0.230824i
\(364\) 0 0
\(365\) −15.8482 + 8.23449i −0.829531 + 0.431013i
\(366\) 0 0
\(367\) 18.2835i 0.954392i 0.878797 + 0.477196i \(0.158347\pi\)
−0.878797 + 0.477196i \(0.841653\pi\)
\(368\) 0 0
\(369\) 0.00577512 0.000300641
\(370\) 0 0
\(371\) 21.7819 + 21.7819i 1.13086 + 1.13086i
\(372\) 0 0
\(373\) 23.4176 + 23.4176i 1.21252 + 1.21252i 0.970195 + 0.242324i \(0.0779098\pi\)
0.242324 + 0.970195i \(0.422090\pi\)
\(374\) 0 0
\(375\) 6.78560 + 8.88570i 0.350407 + 0.458856i
\(376\) 0 0
\(377\) 7.31366 0.376673
\(378\) 0 0
\(379\) 6.20944 6.20944i 0.318957 0.318957i −0.529409 0.848367i \(-0.677586\pi\)
0.848367 + 0.529409i \(0.177586\pi\)
\(380\) 0 0
\(381\) −11.3746 11.3746i −0.582741 0.582741i
\(382\) 0 0
\(383\) 14.4005i 0.735832i −0.929859 0.367916i \(-0.880071\pi\)
0.929859 0.367916i \(-0.119929\pi\)
\(384\) 0 0
\(385\) −10.9623 + 34.6747i −0.558692 + 1.76719i
\(386\) 0 0
\(387\) 2.23930 2.23930i 0.113830 0.113830i
\(388\) 0 0
\(389\) −15.4753 + 15.4753i −0.784630 + 0.784630i −0.980608 0.195978i \(-0.937212\pi\)
0.195978 + 0.980608i \(0.437212\pi\)
\(390\) 0 0
\(391\) 5.00875i 0.253304i
\(392\) 0 0
\(393\) 18.4423 0.930292
\(394\) 0 0
\(395\) 1.67460 0.870098i 0.0842581 0.0437794i
\(396\) 0 0
\(397\) 15.0440 15.0440i 0.755036 0.755036i −0.220378 0.975414i \(-0.570729\pi\)
0.975414 + 0.220378i \(0.0707292\pi\)
\(398\) 0 0
\(399\) −9.28590 −0.464876
\(400\) 0 0
\(401\) 28.7953 1.43797 0.718985 0.695026i \(-0.244607\pi\)
0.718985 + 0.695026i \(0.244607\pi\)
\(402\) 0 0
\(403\) −7.92543 + 7.92543i −0.394794 + 0.394794i
\(404\) 0 0
\(405\) −1.98421 + 1.03097i −0.0985963 + 0.0512293i
\(406\) 0 0
\(407\) −37.6476 −1.86612
\(408\) 0 0
\(409\) 23.3562i 1.15489i 0.816430 + 0.577444i \(0.195950\pi\)
−0.816430 + 0.577444i \(0.804050\pi\)
\(410\) 0 0
\(411\) 4.60760 4.60760i 0.227276 0.227276i
\(412\) 0 0
\(413\) 15.0222 15.0222i 0.739192 0.739192i
\(414\) 0 0
\(415\) 4.95752 15.6810i 0.243355 0.769751i
\(416\) 0 0
\(417\) 21.3479i 1.04541i
\(418\) 0 0
\(419\) 22.4332 + 22.4332i 1.09594 + 1.09594i 0.994881 + 0.101055i \(0.0322217\pi\)
0.101055 + 0.994881i \(0.467778\pi\)
\(420\) 0 0
\(421\) −14.7632 + 14.7632i −0.719513 + 0.719513i −0.968505 0.248993i \(-0.919900\pi\)
0.248993 + 0.968505i \(0.419900\pi\)
\(422\) 0 0
\(423\) −11.6451 −0.566205
\(424\) 0 0
\(425\) 11.8363 + 8.31511i 0.574144 + 0.403342i
\(426\) 0 0
\(427\) 36.4725 + 36.4725i 1.76503 + 1.76503i
\(428\) 0 0
\(429\) 3.04096 + 3.04096i 0.146819 + 0.146819i
\(430\) 0 0
\(431\) 5.92220 0.285262 0.142631 0.989776i \(-0.454444\pi\)
0.142631 + 0.989776i \(0.454444\pi\)
\(432\) 0 0
\(433\) 5.27026i 0.253272i 0.991949 + 0.126636i \(0.0404181\pi\)
−0.991949 + 0.126636i \(0.959582\pi\)
\(434\) 0 0
\(435\) −14.0025 + 7.27551i −0.671369 + 0.348834i
\(436\) 0 0
\(437\) 2.90057 + 2.90057i 0.138753 + 0.138753i
\(438\) 0 0
\(439\) 17.8427i 0.851587i 0.904820 + 0.425793i \(0.140005\pi\)
−0.904820 + 0.425793i \(0.859995\pi\)
\(440\) 0 0
\(441\) 8.36066i 0.398127i
\(442\) 0 0
\(443\) 5.89855 + 5.89855i 0.280248 + 0.280248i 0.833208 0.552960i \(-0.186502\pi\)
−0.552960 + 0.833208i \(0.686502\pi\)
\(444\) 0 0
\(445\) −5.57648 10.7325i −0.264350 0.508771i
\(446\) 0 0
\(447\) 6.04438i 0.285889i
\(448\) 0 0
\(449\) 8.12803 0.383586 0.191793 0.981435i \(-0.438570\pi\)
0.191793 + 0.981435i \(0.438570\pi\)
\(450\) 0 0
\(451\) 0.0169455 + 0.0169455i 0.000797934 + 0.000797934i
\(452\) 0 0
\(453\) −5.93968 5.93968i −0.279071 0.279071i
\(454\) 0 0
\(455\) −2.73786 + 8.66005i −0.128353 + 0.405990i
\(456\) 0 0
\(457\) −6.35488 −0.297269 −0.148634 0.988892i \(-0.547488\pi\)
−0.148634 + 0.988892i \(0.547488\pi\)
\(458\) 0 0
\(459\) −2.04567 + 2.04567i −0.0954838 + 0.0954838i
\(460\) 0 0
\(461\) −2.13085 2.13085i −0.0992438 0.0992438i 0.655742 0.754985i \(-0.272356\pi\)
−0.754985 + 0.655742i \(0.772356\pi\)
\(462\) 0 0
\(463\) 18.4540i 0.857630i −0.903392 0.428815i \(-0.858931\pi\)
0.903392 0.428815i \(-0.141069\pi\)
\(464\) 0 0
\(465\) 7.28970 23.0579i 0.338052 1.06928i
\(466\) 0 0
\(467\) 5.63243 5.63243i 0.260638 0.260638i −0.564675 0.825313i \(-0.690999\pi\)
0.825313 + 0.564675i \(0.190999\pi\)
\(468\) 0 0
\(469\) −15.0987 + 15.0987i −0.697193 + 0.697193i
\(470\) 0 0
\(471\) 8.04537i 0.370711i
\(472\) 0 0
\(473\) 13.1412 0.604235
\(474\) 0 0
\(475\) −11.6697 + 2.03910i −0.535441 + 0.0935604i
\(476\) 0 0
\(477\) −5.55765 + 5.55765i −0.254467 + 0.254467i
\(478\) 0 0
\(479\) 21.4860 0.981720 0.490860 0.871239i \(-0.336683\pi\)
0.490860 + 0.871239i \(0.336683\pi\)
\(480\) 0 0
\(481\) −9.40253 −0.428718
\(482\) 0 0
\(483\) −4.79810 + 4.79810i −0.218321 + 0.218321i
\(484\) 0 0
\(485\) −2.31096 4.44769i −0.104935 0.201959i
\(486\) 0 0
\(487\) 36.3297 1.64626 0.823128 0.567856i \(-0.192227\pi\)
0.823128 + 0.567856i \(0.192227\pi\)
\(488\) 0 0
\(489\) 7.97431i 0.360611i
\(490\) 0 0
\(491\) −1.63599 + 1.63599i −0.0738313 + 0.0738313i −0.743058 0.669227i \(-0.766625\pi\)
0.669227 + 0.743058i \(0.266625\pi\)
\(492\) 0 0
\(493\) −14.4362 + 14.4362i −0.650175 + 0.650175i
\(494\) 0 0
\(495\) −8.84724 2.79703i −0.397654 0.125717i
\(496\) 0 0
\(497\) 4.52106i 0.202797i
\(498\) 0 0
\(499\) −18.7955 18.7955i −0.841402 0.841402i 0.147639 0.989041i \(-0.452833\pi\)
−0.989041 + 0.147639i \(0.952833\pi\)
\(500\) 0 0
\(501\) 4.56496 4.56496i 0.203947 0.203947i
\(502\) 0 0
\(503\) 13.8118 0.615838 0.307919 0.951413i \(-0.400367\pi\)
0.307919 + 0.951413i \(0.400367\pi\)
\(504\) 0 0
\(505\) 7.15557 22.6336i 0.318419 1.00718i
\(506\) 0 0
\(507\) −8.43290 8.43290i −0.374518 0.374518i
\(508\) 0 0
\(509\) 5.40184 + 5.40184i 0.239432 + 0.239432i 0.816615 0.577183i \(-0.195848\pi\)
−0.577183 + 0.816615i \(0.695848\pi\)
\(510\) 0 0
\(511\) 31.3037 1.38479
\(512\) 0 0
\(513\) 2.36929i 0.104607i
\(514\) 0 0
\(515\) 7.76046 4.03223i 0.341967 0.177681i
\(516\) 0 0
\(517\) −34.1694 34.1694i −1.50277 1.50277i
\(518\) 0 0
\(519\) 10.3756i 0.455437i
\(520\) 0 0
\(521\) 33.6969i 1.47629i −0.674642 0.738145i \(-0.735702\pi\)
0.674642 0.738145i \(-0.264298\pi\)
\(522\) 0 0
\(523\) 4.31481 + 4.31481i 0.188673 + 0.188673i 0.795122 0.606449i \(-0.207407\pi\)
−0.606449 + 0.795122i \(0.707407\pi\)
\(524\) 0 0
\(525\) −3.37307 19.3039i −0.147213 0.842489i
\(526\) 0 0
\(527\) 31.2876i 1.36291i
\(528\) 0 0
\(529\) −20.0025 −0.869674
\(530\) 0 0
\(531\) 3.83290 + 3.83290i 0.166334 + 0.166334i
\(532\) 0 0
\(533\) 0.00423217 + 0.00423217i 0.000183316 + 0.000183316i
\(534\) 0 0
\(535\) 7.84650 + 2.48065i 0.339234 + 0.107248i
\(536\) 0 0
\(537\) −1.47612 −0.0636993
\(538\) 0 0
\(539\) 24.5321 24.5321i 1.05667 1.05667i
\(540\) 0 0
\(541\) 22.8010 + 22.8010i 0.980290 + 0.980290i 0.999809 0.0195198i \(-0.00621375\pi\)
−0.0195198 + 0.999809i \(0.506214\pi\)
\(542\) 0 0
\(543\) 17.1154i 0.734493i
\(544\) 0 0
\(545\) 6.23041 + 1.96973i 0.266882 + 0.0843740i
\(546\) 0 0
\(547\) 5.56014 5.56014i 0.237735 0.237735i −0.578177 0.815911i \(-0.696236\pi\)
0.815911 + 0.578177i \(0.196236\pi\)
\(548\) 0 0
\(549\) −9.30595 + 9.30595i −0.397168 + 0.397168i
\(550\) 0 0
\(551\) 16.7200i 0.712297i
\(552\) 0 0
\(553\) −3.30771 −0.140658
\(554\) 0 0
\(555\) 18.0018 9.35349i 0.764134 0.397034i
\(556\) 0 0
\(557\) −12.2628 + 12.2628i −0.519591 + 0.519591i −0.917447 0.397857i \(-0.869754\pi\)
0.397857 + 0.917447i \(0.369754\pi\)
\(558\) 0 0
\(559\) 3.28204 0.138816
\(560\) 0 0
\(561\) −12.0049 −0.506849
\(562\) 0 0
\(563\) −0.327630 + 0.327630i −0.0138079 + 0.0138079i −0.713977 0.700169i \(-0.753108\pi\)
0.700169 + 0.713977i \(0.253108\pi\)
\(564\) 0 0
\(565\) −8.38807 16.1438i −0.352889 0.679173i
\(566\) 0 0
\(567\) 3.91927 0.164594
\(568\) 0 0
\(569\) 35.6010i 1.49247i 0.665681 + 0.746237i \(0.268141\pi\)
−0.665681 + 0.746237i \(0.731859\pi\)
\(570\) 0 0
\(571\) −3.57427 + 3.57427i −0.149579 + 0.149579i −0.777930 0.628351i \(-0.783730\pi\)
0.628351 + 0.777930i \(0.283730\pi\)
\(572\) 0 0
\(573\) 15.0100 15.0100i 0.627051 0.627051i
\(574\) 0 0
\(575\) −4.97618 + 7.08342i −0.207521 + 0.295399i
\(576\) 0 0
\(577\) 8.80260i 0.366457i −0.983070 0.183229i \(-0.941345\pi\)
0.983070 0.183229i \(-0.0586548\pi\)
\(578\) 0 0
\(579\) 13.6317 + 13.6317i 0.566515 + 0.566515i
\(580\) 0 0
\(581\) −20.3829 + 20.3829i −0.845625 + 0.845625i
\(582\) 0 0
\(583\) −32.6149 −1.35077
\(584\) 0 0
\(585\) −2.20961 0.698563i −0.0913561 0.0288820i
\(586\) 0 0
\(587\) −17.9887 17.9887i −0.742475 0.742475i 0.230579 0.973054i \(-0.425938\pi\)
−0.973054 + 0.230579i \(0.925938\pi\)
\(588\) 0 0
\(589\) 18.1186 + 18.1186i 0.746564 + 0.746564i
\(590\) 0 0
\(591\) 13.4117 0.551684
\(592\) 0 0
\(593\) 25.4791i 1.04630i −0.852240 0.523150i \(-0.824757\pi\)
0.852240 0.523150i \(-0.175243\pi\)
\(594\) 0 0
\(595\) −11.6897 22.4980i −0.479229 0.922328i
\(596\) 0 0
\(597\) 6.11945 + 6.11945i 0.250452 + 0.250452i
\(598\) 0 0
\(599\) 14.6977i 0.600532i 0.953856 + 0.300266i \(0.0970754\pi\)
−0.953856 + 0.300266i \(0.902925\pi\)
\(600\) 0 0
\(601\) 12.2790i 0.500869i 0.968134 + 0.250434i \(0.0805735\pi\)
−0.968134 + 0.250434i \(0.919427\pi\)
\(602\) 0 0
\(603\) −3.85243 3.85243i −0.156883 0.156883i
\(604\) 0 0
\(605\) −6.41202 12.3406i −0.260686 0.501718i
\(606\) 0 0
\(607\) 26.2195i 1.06421i −0.846677 0.532107i \(-0.821400\pi\)
0.846677 0.532107i \(-0.178600\pi\)
\(608\) 0 0
\(609\) 27.6581 1.12076
\(610\) 0 0
\(611\) −8.53386 8.53386i −0.345243 0.345243i
\(612\) 0 0
\(613\) −23.4055 23.4055i −0.945340 0.945340i 0.0532416 0.998582i \(-0.483045\pi\)
−0.998582 + 0.0532416i \(0.983045\pi\)
\(614\) 0 0
\(615\) −0.0123129 0.00389269i −0.000496503 0.000156968i
\(616\) 0 0
\(617\) −27.5735 −1.11007 −0.555033 0.831828i \(-0.687294\pi\)
−0.555033 + 0.831828i \(0.687294\pi\)
\(618\) 0 0
\(619\) 12.2452 12.2452i 0.492176 0.492176i −0.416815 0.908991i \(-0.636854\pi\)
0.908991 + 0.416815i \(0.136854\pi\)
\(620\) 0 0
\(621\) −1.22423 1.22423i −0.0491268 0.0491268i
\(622\) 0 0
\(623\) 21.1992i 0.849327i
\(624\) 0 0
\(625\) −8.47792 23.5186i −0.339117 0.940744i
\(626\) 0 0
\(627\) 6.95205 6.95205i 0.277638 0.277638i
\(628\) 0 0
\(629\) 18.5594 18.5594i 0.740011 0.740011i
\(630\) 0 0
\(631\) 25.5846i 1.01851i −0.860617 0.509253i \(-0.829922\pi\)
0.860617 0.509253i \(-0.170078\pi\)
\(632\) 0 0
\(633\) −1.23877 −0.0492369
\(634\) 0 0
\(635\) 16.5844 + 31.9184i 0.658131 + 1.26664i
\(636\) 0 0
\(637\) 6.12692 6.12692i 0.242758 0.242758i
\(638\) 0 0
\(639\) 1.15355 0.0456336
\(640\) 0 0
\(641\) −3.52020 −0.139040 −0.0695198 0.997581i \(-0.522147\pi\)
−0.0695198 + 0.997581i \(0.522147\pi\)
\(642\) 0 0
\(643\) 19.0869 19.0869i 0.752715 0.752715i −0.222270 0.974985i \(-0.571347\pi\)
0.974985 + 0.222270i \(0.0713466\pi\)
\(644\) 0 0
\(645\) −6.28370 + 3.26493i −0.247421 + 0.128556i
\(646\) 0 0
\(647\) 5.02456 0.197536 0.0987678 0.995111i \(-0.468510\pi\)
0.0987678 + 0.995111i \(0.468510\pi\)
\(648\) 0 0
\(649\) 22.4932i 0.882935i
\(650\) 0 0
\(651\) −29.9717 + 29.9717i −1.17468 + 1.17468i
\(652\) 0 0
\(653\) 6.07220 6.07220i 0.237623 0.237623i −0.578242 0.815865i \(-0.696261\pi\)
0.815865 + 0.578242i \(0.196261\pi\)
\(654\) 0 0
\(655\) −39.3201 12.4310i −1.53636 0.485718i
\(656\) 0 0
\(657\) 7.98713i 0.311608i
\(658\) 0 0
\(659\) 5.88182 + 5.88182i 0.229123 + 0.229123i 0.812326 0.583203i \(-0.198201\pi\)
−0.583203 + 0.812326i \(0.698201\pi\)
\(660\) 0 0
\(661\) 24.0275 24.0275i 0.934562 0.934562i −0.0634242 0.997987i \(-0.520202\pi\)
0.997987 + 0.0634242i \(0.0202021\pi\)
\(662\) 0 0
\(663\) −2.99825 −0.116442
\(664\) 0 0
\(665\) 19.7981 + 6.25911i 0.767736 + 0.242718i
\(666\) 0 0
\(667\) −8.63936 8.63936i −0.334517 0.334517i
\(668\) 0 0
\(669\) 7.38042 + 7.38042i 0.285344 + 0.285344i
\(670\) 0 0
\(671\) −54.6116 −2.10826
\(672\) 0 0
\(673\) 36.5327i 1.40823i 0.710084 + 0.704117i \(0.248657\pi\)
−0.710084 + 0.704117i \(0.751343\pi\)
\(674\) 0 0
\(675\) 4.92537 0.860637i 0.189578 0.0331259i
\(676\) 0 0
\(677\) −20.5460 20.5460i −0.789645 0.789645i 0.191791 0.981436i \(-0.438571\pi\)
−0.981436 + 0.191791i \(0.938571\pi\)
\(678\) 0 0
\(679\) 8.78518i 0.337144i
\(680\) 0 0
\(681\) 19.4055i 0.743622i
\(682\) 0 0
\(683\) 18.2074 + 18.2074i 0.696688 + 0.696688i 0.963695 0.267007i \(-0.0860346\pi\)
−0.267007 + 0.963695i \(0.586035\pi\)
\(684\) 0 0
\(685\) −12.9294 + 6.71794i −0.494007 + 0.256679i
\(686\) 0 0
\(687\) 11.9342i 0.455320i
\(688\) 0 0
\(689\) −8.14561 −0.310323
\(690\) 0 0
\(691\) −11.5350 11.5350i −0.438811 0.438811i 0.452801 0.891612i \(-0.350425\pi\)
−0.891612 + 0.452801i \(0.850425\pi\)
\(692\) 0 0
\(693\) 11.5000 + 11.5000i 0.436850 + 0.436850i
\(694\) 0 0
\(695\) 14.3894 45.5149i 0.545823 1.72648i
\(696\) 0 0
\(697\) −0.0167075 −0.000632842
\(698\) 0 0
\(699\) 20.2304 20.2304i 0.765182 0.765182i
\(700\) 0 0
\(701\) −16.5440 16.5440i −0.624859 0.624859i 0.321911 0.946770i \(-0.395675\pi\)
−0.946770 + 0.321911i \(0.895675\pi\)
\(702\) 0 0
\(703\) 21.4955i 0.810717i
\(704\) 0 0
\(705\) 24.8280 + 7.84932i 0.935078 + 0.295623i
\(706\) 0 0
\(707\) −29.4202 + 29.4202i −1.10646 + 1.10646i
\(708\) 0 0
\(709\) 8.47518 8.47518i 0.318292 0.318292i −0.529819 0.848111i \(-0.677740\pi\)
0.848111 + 0.529819i \(0.177740\pi\)
\(710\) 0 0
\(711\) 0.843960i 0.0316510i
\(712\) 0 0
\(713\) 18.7240 0.701221
\(714\) 0 0
\(715\) −4.43376 8.53325i −0.165813 0.319125i
\(716\) 0 0
\(717\) −15.1974 + 15.1974i −0.567559 + 0.567559i
\(718\) 0 0
\(719\) −11.5303 −0.430008 −0.215004 0.976613i \(-0.568976\pi\)
−0.215004 + 0.976613i \(0.568976\pi\)
\(720\) 0 0
\(721\) −15.3287 −0.570869
\(722\) 0 0
\(723\) 13.8553 13.8553i 0.515284 0.515284i
\(724\) 0 0
\(725\) 34.7582 6.07348i 1.29089 0.225564i
\(726\) 0 0
\(727\) 32.2855 1.19740 0.598702 0.800972i \(-0.295684\pi\)
0.598702 + 0.800972i \(0.295684\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −6.47833 + 6.47833i −0.239610 + 0.239610i
\(732\) 0 0
\(733\) 33.6309 33.6309i 1.24219 1.24219i 0.283094 0.959092i \(-0.408639\pi\)
0.959092 0.283094i \(-0.0913609\pi\)
\(734\) 0 0
\(735\) −5.63546 + 17.8254i −0.207867 + 0.657500i
\(736\) 0 0
\(737\) 22.6078i 0.832769i
\(738\) 0 0
\(739\) 11.6806 + 11.6806i 0.429676 + 0.429676i 0.888518 0.458842i \(-0.151736\pi\)
−0.458842 + 0.888518i \(0.651736\pi\)
\(740\) 0 0
\(741\) 1.73628 1.73628i 0.0637840 0.0637840i
\(742\) 0 0
\(743\) −11.7641 −0.431582 −0.215791 0.976440i \(-0.569233\pi\)
−0.215791 + 0.976440i \(0.569233\pi\)
\(744\) 0 0
\(745\) −4.07418 + 12.8870i −0.149266 + 0.472142i
\(746\) 0 0
\(747\) −5.20069 5.20069i −0.190283 0.190283i
\(748\) 0 0
\(749\) −10.1992 10.1992i −0.372672 0.372672i
\(750\) 0 0
\(751\) 9.89062 0.360914 0.180457 0.983583i \(-0.442242\pi\)
0.180457 + 0.983583i \(0.442242\pi\)
\(752\) 0 0
\(753\) 13.5492i 0.493760i
\(754\) 0 0
\(755\) 8.66012 + 16.6673i 0.315174 + 0.606587i
\(756\) 0 0
\(757\) 19.7078 + 19.7078i 0.716291 + 0.716291i 0.967844 0.251553i \(-0.0809413\pi\)
−0.251553 + 0.967844i \(0.580941\pi\)
\(758\) 0 0
\(759\) 7.18436i 0.260776i
\(760\) 0 0
\(761\) 48.2102i 1.74762i −0.486270 0.873808i \(-0.661643\pi\)
0.486270 0.873808i \(-0.338357\pi\)
\(762\) 0 0
\(763\) −8.09857 8.09857i −0.293188 0.293188i
\(764\) 0 0
\(765\) 5.74036 2.98261i 0.207543 0.107837i
\(766\) 0 0
\(767\) 5.61771i 0.202844i
\(768\) 0 0
\(769\) −30.7814 −1.11000 −0.555002 0.831849i \(-0.687283\pi\)
−0.555002 + 0.831849i \(0.687283\pi\)
\(770\) 0 0
\(771\) −9.38542 9.38542i −0.338008 0.338008i
\(772\) 0 0
\(773\) 31.7802 + 31.7802i 1.14306 + 1.14306i 0.987888 + 0.155167i \(0.0495916\pi\)
0.155167 + 0.987888i \(0.450408\pi\)
\(774\) 0 0
\(775\) −31.0841 + 44.2471i −1.11657 + 1.58940i
\(776\) 0 0
\(777\) −35.5576 −1.27562
\(778\) 0 0
\(779\) 0.00967532 0.00967532i 0.000346654 0.000346654i
\(780\) 0 0
\(781\) 3.38477 + 3.38477i 0.121117 + 0.121117i
\(782\) 0 0
\(783\) 7.05696i 0.252195i
\(784\) 0 0
\(785\) 5.42294 17.1532i 0.193553 0.612223i
\(786\) 0 0
\(787\) 21.8749 21.8749i 0.779754 0.779754i −0.200035 0.979789i \(-0.564105\pi\)
0.979789 + 0.200035i \(0.0641054\pi\)
\(788\) 0 0
\(789\) −9.23522 + 9.23522i −0.328782 + 0.328782i
\(790\) 0 0
\(791\) 31.8876i 1.13379i
\(792\) 0 0
\(793\) −13.6393 −0.484346
\(794\) 0 0
\(795\) 15.5953 8.10312i 0.553110 0.287388i
\(796\) 0 0
\(797\) −16.0370 + 16.0370i −0.568058 + 0.568058i −0.931584 0.363526i \(-0.881573\pi\)
0.363526 + 0.931584i \(0.381573\pi\)
\(798\) 0 0
\(799\) 33.6895 1.19185
\(800\) 0 0
\(801\) −5.40896 −0.191116
\(802\) 0 0
\(803\) −23.4361 + 23.4361i −0.827041 + 0.827041i
\(804\) 0 0
\(805\) 13.4639 6.99568i 0.474542 0.246565i
\(806\) 0 0
\(807\) −17.5457 −0.617639
\(808\) 0 0
\(809\) 37.0836i 1.30379i −0.758309 0.651895i \(-0.773974\pi\)
0.758309 0.651895i \(-0.226026\pi\)
\(810\) 0 0
\(811\) 11.8858 11.8858i 0.417369 0.417369i −0.466927 0.884296i \(-0.654639\pi\)
0.884296 + 0.466927i \(0.154639\pi\)
\(812\) 0 0
\(813\) −1.49336 + 1.49336i −0.0523744 + 0.0523744i
\(814\) 0 0
\(815\) −5.37504 + 17.0017i −0.188279 + 0.595543i
\(816\) 0 0
\(817\) 7.50320i 0.262504i
\(818\) 0 0
\(819\) 2.87215 + 2.87215i 0.100361 + 0.100361i
\(820\) 0 0
\(821\) 18.0568 18.0568i 0.630186 0.630186i −0.317928 0.948115i \(-0.602987\pi\)
0.948115 + 0.317928i \(0.102987\pi\)
\(822\) 0 0
\(823\) −9.66750 −0.336988 −0.168494 0.985703i \(-0.553890\pi\)
−0.168494 + 0.985703i \(0.553890\pi\)
\(824\) 0 0
\(825\) 16.9775 + 11.9269i 0.591080 + 0.415240i
\(826\) 0 0
\(827\) −15.9204 15.9204i −0.553607 0.553607i 0.373873 0.927480i \(-0.378030\pi\)
−0.927480 + 0.373873i \(0.878030\pi\)
\(828\) 0 0
\(829\) −6.21199 6.21199i −0.215751 0.215751i 0.590954 0.806705i \(-0.298751\pi\)
−0.806705 + 0.590954i \(0.798751\pi\)
\(830\) 0 0
\(831\) −18.6840 −0.648141
\(832\) 0 0
\(833\) 24.1875i 0.838048i
\(834\) 0 0
\(835\) −12.8097 + 6.65576i −0.443299 + 0.230332i
\(836\) 0 0
\(837\) −7.64726 7.64726i −0.264328 0.264328i
\(838\) 0 0
\(839\) 11.4138i 0.394047i −0.980399 0.197023i \(-0.936873\pi\)
0.980399 0.197023i \(-0.0631275\pi\)
\(840\) 0 0
\(841\) 20.8007i 0.717266i
\(842\) 0 0
\(843\) −2.67842 2.67842i −0.0922497 0.0922497i
\(844\) 0 0
\(845\) 12.2953 + 23.6636i 0.422970 + 0.814052i
\(846\) 0 0
\(847\) 24.3755i 0.837553i
\(848\) 0 0
\(849\) −4.47278 −0.153506
\(850\) 0 0
\(851\) 11.1069 + 11.1069i 0.380739 + 0.380739i
\(852\) 0 0
\(853\) 12.1530 + 12.1530i 0.416110 + 0.416110i 0.883861 0.467751i \(-0.154935\pi\)
−0.467751 + 0.883861i \(0.654935\pi\)
\(854\) 0 0
\(855\) −1.59701 + 5.05147i −0.0546166 + 0.172757i
\(856\) 0 0
\(857\) −23.6445 −0.807681 −0.403840 0.914829i \(-0.632325\pi\)
−0.403840 + 0.914829i \(0.632325\pi\)
\(858\) 0 0
\(859\) −18.9186 + 18.9186i −0.645494 + 0.645494i −0.951901 0.306407i \(-0.900873\pi\)
0.306407 + 0.951901i \(0.400873\pi\)
\(860\) 0 0
\(861\) 0.0160048 + 0.0160048i 0.000545443 + 0.000545443i
\(862\) 0 0
\(863\) 32.1371i 1.09396i 0.837146 + 0.546980i \(0.184223\pi\)
−0.837146 + 0.546980i \(0.815777\pi\)
\(864\) 0 0
\(865\) −6.99361 + 22.1213i −0.237790 + 0.752148i
\(866\) 0 0
\(867\) −6.10266 + 6.10266i −0.207257 + 0.207257i
\(868\) 0 0
\(869\) 2.47637 2.47637i 0.0840052 0.0840052i
\(870\) 0 0
\(871\) 5.64633i 0.191319i
\(872\) 0 0
\(873\) −2.24154 −0.0758645
\(874\) 0 0
\(875\) −5.82010 + 43.4305i −0.196755 + 1.46822i
\(876\) 0 0
\(877\) 33.5598 33.5598i 1.13323 1.13323i 0.143597 0.989636i \(-0.454133\pi\)
0.989636 0.143597i \(-0.0458670\pi\)
\(878\) 0 0
\(879\) 12.0397 0.406091
\(880\) 0 0
\(881\) 3.11390 0.104910 0.0524550 0.998623i \(-0.483295\pi\)
0.0524550 + 0.998623i \(0.483295\pi\)
\(882\) 0 0
\(883\) −36.1043 + 36.1043i −1.21501 + 1.21501i −0.245647 + 0.969359i \(0.579000\pi\)
−0.969359 + 0.245647i \(0.921000\pi\)
\(884\) 0 0
\(885\) −5.58841 10.7555i −0.187852 0.361542i
\(886\) 0 0
\(887\) 38.4317 1.29041 0.645206 0.764009i \(-0.276772\pi\)
0.645206 + 0.764009i \(0.276772\pi\)
\(888\) 0 0
\(889\) 63.0461i 2.11450i
\(890\) 0 0
\(891\) −2.93423 + 2.93423i −0.0983004 + 0.0983004i
\(892\) 0 0
\(893\) −19.5096 + 19.5096i −0.652863 + 0.652863i
\(894\) 0 0
\(895\) 3.14717 + 0.994971i 0.105198 + 0.0332582i
\(896\) 0 0
\(897\) 1.79430i 0.0599100i
\(898\) 0 0
\(899\) −53.9664 53.9664i −1.79988 1.79988i
\(900\) 0 0
\(901\) 16.0784 16.0784i 0.535649 0.535649i
\(902\) 0 0
\(903\) 12.4117 0.413037
\(904\) 0 0
\(905\) 11.5366 36.4910i 0.383488 1.21300i
\(906\) 0 0
\(907\) −37.6279 37.6279i −1.24941 1.24941i −0.955979 0.293436i \(-0.905201\pi\)
−0.293436 0.955979i \(-0.594799\pi\)
\(908\) 0 0
\(909\) −7.50655 7.50655i −0.248977 0.248977i
\(910\) 0 0
\(911\) −11.8631 −0.393042 −0.196521 0.980500i \(-0.562964\pi\)
−0.196521 + 0.980500i \(0.562964\pi\)
\(912\) 0 0
\(913\) 30.5200i 1.01007i
\(914\) 0 0
\(915\) 26.1134 13.5682i 0.863283 0.448550i
\(916\) 0 0
\(917\) 51.1100 + 51.1100i 1.68780 + 1.68780i
\(918\) 0 0
\(919\) 46.4906i 1.53358i −0.641895 0.766792i \(-0.721852\pi\)
0.641895 0.766792i \(-0.278148\pi\)
\(920\) 0 0
\(921\) 10.0705i 0.331835i
\(922\) 0 0
\(923\) 0.845351 + 0.845351i 0.0278251 + 0.0278251i
\(924\) 0 0
\(925\) −44.6855 + 7.80814i −1.46925 + 0.256730i
\(926\) 0 0
\(927\) 3.91110i 0.128457i
\(928\) 0 0
\(929\) 3.18351 0.104448 0.0522238 0.998635i \(-0.483369\pi\)
0.0522238 + 0.998635i \(0.483369\pi\)
\(930\) 0 0
\(931\) −14.0070 14.0070i −0.459060 0.459060i
\(932\) 0 0
\(933\) −15.0599 15.0599i −0.493040 0.493040i
\(934\) 0 0
\(935\) 25.5952 + 8.09187i 0.837053 + 0.264632i
\(936\) 0 0
\(937\) −27.2455 −0.890072 −0.445036 0.895513i \(-0.646809\pi\)
−0.445036 + 0.895513i \(0.646809\pi\)
\(938\) 0 0
\(939\) −5.30181 + 5.30181i −0.173018 + 0.173018i
\(940\) 0 0
\(941\) −4.15297 4.15297i −0.135383 0.135383i 0.636168 0.771551i \(-0.280519\pi\)
−0.771551 + 0.636168i \(0.780519\pi\)
\(942\) 0 0
\(943\) 0.00999862i 0.000325600i
\(944\) 0 0
\(945\) −8.35610 2.64176i −0.271824 0.0859365i
\(946\) 0 0
\(947\) −18.3519 + 18.3519i −0.596358 + 0.596358i −0.939341 0.342984i \(-0.888562\pi\)
0.342984 + 0.939341i \(0.388562\pi\)
\(948\) 0 0
\(949\) −5.85319 + 5.85319i −0.190003 + 0.190003i
\(950\) 0 0
\(951\) 3.32841i 0.107931i
\(952\) 0 0
\(953\) 52.0669 1.68661 0.843307 0.537433i \(-0.180606\pi\)
0.843307 + 0.537433i \(0.180606\pi\)
\(954\) 0 0
\(955\) −42.1195 + 21.8847i −1.36296 + 0.708173i
\(956\) 0 0
\(957\) −20.7067 + 20.7067i −0.669354 + 0.669354i
\(958\) 0 0
\(959\) 25.5385 0.824681
\(960\) 0 0
\(961\) 85.9611 2.77294
\(962\) 0 0
\(963\) 2.60233 2.60233i 0.0838589 0.0838589i
\(964\) 0 0
\(965\) −19.8752 38.2520i −0.639806 1.23138i
\(966\) 0 0
\(967\) 32.8074 1.05502 0.527508 0.849550i \(-0.323127\pi\)
0.527508 + 0.849550i \(0.323127\pi\)
\(968\) 0 0
\(969\) 6.85441i 0.220195i
\(970\) 0 0
\(971\) 25.4682 25.4682i 0.817314 0.817314i −0.168404 0.985718i \(-0.553861\pi\)
0.985718 + 0.168404i \(0.0538613\pi\)
\(972\) 0 0
\(973\) −59.1623 + 59.1623i −1.89666 + 1.89666i
\(974\) 0 0
\(975\) 4.24015 + 2.97875i 0.135793 + 0.0953964i
\(976\) 0 0
\(977\) 2.69863i 0.0863367i 0.999068 + 0.0431684i \(0.0137452\pi\)
−0.999068 + 0.0431684i \(0.986255\pi\)
\(978\) 0 0
\(979\) −15.8711 15.8711i −0.507244 0.507244i
\(980\) 0 0
\(981\) 2.06635 2.06635i 0.0659734 0.0659734i
\(982\) 0 0
\(983\) −25.2037 −0.803875 −0.401937 0.915667i \(-0.631663\pi\)
−0.401937 + 0.915667i \(0.631663\pi\)
\(984\) 0 0
\(985\) −28.5945 9.04009i −0.911097 0.288041i
\(986\) 0 0
\(987\) −32.2726 32.2726i −1.02725 1.02725i
\(988\) 0 0
\(989\) −3.87696 3.87696i −0.123280 0.123280i
\(990\) 0 0
\(991\) −9.06410 −0.287931 −0.143965 0.989583i \(-0.545985\pi\)
−0.143965 + 0.989583i \(0.545985\pi\)
\(992\) 0 0
\(993\) 32.0808i 1.01805i
\(994\) 0 0
\(995\) −8.92223 17.1718i −0.282854 0.544382i
\(996\) 0 0
\(997\) 25.5035 + 25.5035i 0.807703 + 0.807703i 0.984286 0.176583i \(-0.0565044\pi\)
−0.176583 + 0.984286i \(0.556504\pi\)
\(998\) 0 0
\(999\) 9.07252i 0.287042i
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 1920.2.bl.a.289.2 48
4.3 odd 2 1920.2.bl.b.289.23 48
5.4 even 2 inner 1920.2.bl.a.289.23 48
8.3 odd 2 960.2.bl.a.529.2 48
8.5 even 2 240.2.bl.a.109.3 48
16.3 odd 4 960.2.bl.a.49.17 48
16.5 even 4 inner 1920.2.bl.a.1249.23 48
16.11 odd 4 1920.2.bl.b.1249.2 48
16.13 even 4 240.2.bl.a.229.22 yes 48
20.19 odd 2 1920.2.bl.b.289.2 48
24.5 odd 2 720.2.bm.h.109.22 48
40.19 odd 2 960.2.bl.a.529.17 48
40.29 even 2 240.2.bl.a.109.22 yes 48
48.29 odd 4 720.2.bm.h.469.3 48
80.19 odd 4 960.2.bl.a.49.2 48
80.29 even 4 240.2.bl.a.229.3 yes 48
80.59 odd 4 1920.2.bl.b.1249.23 48
80.69 even 4 inner 1920.2.bl.a.1249.2 48
120.29 odd 2 720.2.bm.h.109.3 48
240.29 odd 4 720.2.bm.h.469.22 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.bl.a.109.3 48 8.5 even 2
240.2.bl.a.109.22 yes 48 40.29 even 2
240.2.bl.a.229.3 yes 48 80.29 even 4
240.2.bl.a.229.22 yes 48 16.13 even 4
720.2.bm.h.109.3 48 120.29 odd 2
720.2.bm.h.109.22 48 24.5 odd 2
720.2.bm.h.469.3 48 48.29 odd 4
720.2.bm.h.469.22 48 240.29 odd 4
960.2.bl.a.49.2 48 80.19 odd 4
960.2.bl.a.49.17 48 16.3 odd 4
960.2.bl.a.529.2 48 8.3 odd 2
960.2.bl.a.529.17 48 40.19 odd 2
1920.2.bl.a.289.2 48 1.1 even 1 trivial
1920.2.bl.a.289.23 48 5.4 even 2 inner
1920.2.bl.a.1249.2 48 80.69 even 4 inner
1920.2.bl.a.1249.23 48 16.5 even 4 inner
1920.2.bl.b.289.2 48 20.19 odd 2
1920.2.bl.b.289.23 48 4.3 odd 2
1920.2.bl.b.1249.2 48 16.11 odd 4
1920.2.bl.b.1249.23 48 80.59 odd 4