Properties

Label 7616.2.a.bp.1.3
Level $7616$
Weight $2$
Character 7616.1
Self dual yes
Analytic conductor $60.814$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7616,2,Mod(1,7616)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7616, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7616.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7616.a (trivial)

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: yes
Analytic conductor: \(60.8140661794\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 952)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.48718\) of defining polynomial
Character \(\chi\) \(=\) 7616.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.48718 q^{3} -4.02435 q^{5} +1.00000 q^{7} +3.18609 q^{9} +O(q^{10})\) \(q+2.48718 q^{3} -4.02435 q^{5} +1.00000 q^{7} +3.18609 q^{9} -1.23607 q^{11} +6.47214 q^{13} -10.0093 q^{15} +1.00000 q^{17} +6.97437 q^{19} +2.48718 q^{21} -7.07433 q^{23} +11.1954 q^{25} +0.462835 q^{27} +6.31040 q^{29} -4.88824 q^{31} -3.07433 q^{33} -4.02435 q^{35} +2.63387 q^{37} +16.0974 q^{39} -10.1954 q^{41} -2.69762 q^{43} -12.8219 q^{45} +3.67652 q^{47} +1.00000 q^{49} +2.48718 q^{51} +1.39047 q^{53} +4.97437 q^{55} +17.3465 q^{57} +5.07433 q^{59} -3.93369 q^{61} +3.18609 q^{63} -26.0461 q^{65} -5.63259 q^{67} -17.5952 q^{69} +6.84431 q^{71} +0.246650 q^{73} +27.8450 q^{75} -1.23607 q^{77} +11.6765 q^{79} -8.40711 q^{81} -6.65089 q^{83} -4.02435 q^{85} +15.6951 q^{87} +2.27872 q^{89} +6.47214 q^{91} -12.1580 q^{93} -28.0673 q^{95} +10.7069 q^{97} -3.93822 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{3} + q^{5} + 4 q^{7} + 7 q^{9} + 4 q^{11} + 8 q^{13} - 12 q^{15} + 4 q^{17} + 14 q^{19} + 3 q^{21} - 8 q^{23} + 11 q^{25} + 12 q^{27} - 4 q^{29} - 5 q^{31} + 8 q^{33} + q^{35} + 4 q^{37} - 4 q^{39} - 7 q^{41} + 19 q^{43} + 2 q^{45} - 8 q^{47} + 4 q^{49} + 3 q^{51} - 5 q^{53} + 6 q^{55} + 44 q^{57} + 23 q^{61} + 7 q^{63} - 8 q^{65} + 15 q^{67} + 2 q^{69} - 2 q^{71} - 5 q^{73} + 10 q^{75} + 4 q^{77} + 24 q^{79} - 8 q^{81} + 10 q^{83} + q^{85} - 16 q^{87} - 16 q^{89} + 8 q^{91} + 20 q^{93} - 22 q^{95} - 15 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.48718 1.43598 0.717988 0.696055i \(-0.245063\pi\)
0.717988 + 0.696055i \(0.245063\pi\)
\(4\) 0 0
\(5\) −4.02435 −1.79974 −0.899872 0.436154i \(-0.856340\pi\)
−0.899872 + 0.436154i \(0.856340\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 3.18609 1.06203
\(10\) 0 0
\(11\) −1.23607 −0.372689 −0.186344 0.982485i \(-0.559664\pi\)
−0.186344 + 0.982485i \(0.559664\pi\)
\(12\) 0 0
\(13\) 6.47214 1.79505 0.897524 0.440966i \(-0.145364\pi\)
0.897524 + 0.440966i \(0.145364\pi\)
\(14\) 0 0
\(15\) −10.0093 −2.58439
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 6.97437 1.60003 0.800015 0.599980i \(-0.204825\pi\)
0.800015 + 0.599980i \(0.204825\pi\)
\(20\) 0 0
\(21\) 2.48718 0.542748
\(22\) 0 0
\(23\) −7.07433 −1.47510 −0.737550 0.675293i \(-0.764017\pi\)
−0.737550 + 0.675293i \(0.764017\pi\)
\(24\) 0 0
\(25\) 11.1954 2.23908
\(26\) 0 0
\(27\) 0.462835 0.0890727
\(28\) 0 0
\(29\) 6.31040 1.17181 0.585906 0.810379i \(-0.300739\pi\)
0.585906 + 0.810379i \(0.300739\pi\)
\(30\) 0 0
\(31\) −4.88824 −0.877954 −0.438977 0.898498i \(-0.644659\pi\)
−0.438977 + 0.898498i \(0.644659\pi\)
\(32\) 0 0
\(33\) −3.07433 −0.535172
\(34\) 0 0
\(35\) −4.02435 −0.680239
\(36\) 0 0
\(37\) 2.63387 0.433006 0.216503 0.976282i \(-0.430535\pi\)
0.216503 + 0.976282i \(0.430535\pi\)
\(38\) 0 0
\(39\) 16.0974 2.57765
\(40\) 0 0
\(41\) −10.1954 −1.59225 −0.796126 0.605131i \(-0.793121\pi\)
−0.796126 + 0.605131i \(0.793121\pi\)
\(42\) 0 0
\(43\) −2.69762 −0.411384 −0.205692 0.978617i \(-0.565944\pi\)
−0.205692 + 0.978617i \(0.565944\pi\)
\(44\) 0 0
\(45\) −12.8219 −1.91138
\(46\) 0 0
\(47\) 3.67652 0.536276 0.268138 0.963381i \(-0.413592\pi\)
0.268138 + 0.963381i \(0.413592\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.48718 0.348276
\(52\) 0 0
\(53\) 1.39047 0.190996 0.0954982 0.995430i \(-0.469556\pi\)
0.0954982 + 0.995430i \(0.469556\pi\)
\(54\) 0 0
\(55\) 4.97437 0.670744
\(56\) 0 0
\(57\) 17.3465 2.29761
\(58\) 0 0
\(59\) 5.07433 0.660621 0.330311 0.943872i \(-0.392846\pi\)
0.330311 + 0.943872i \(0.392846\pi\)
\(60\) 0 0
\(61\) −3.93369 −0.503657 −0.251829 0.967772i \(-0.581032\pi\)
−0.251829 + 0.967772i \(0.581032\pi\)
\(62\) 0 0
\(63\) 3.18609 0.401409
\(64\) 0 0
\(65\) −26.0461 −3.23063
\(66\) 0 0
\(67\) −5.63259 −0.688131 −0.344065 0.938946i \(-0.611804\pi\)
−0.344065 + 0.938946i \(0.611804\pi\)
\(68\) 0 0
\(69\) −17.5952 −2.11821
\(70\) 0 0
\(71\) 6.84431 0.812270 0.406135 0.913813i \(-0.366876\pi\)
0.406135 + 0.913813i \(0.366876\pi\)
\(72\) 0 0
\(73\) 0.246650 0.0288682 0.0144341 0.999896i \(-0.495405\pi\)
0.0144341 + 0.999896i \(0.495405\pi\)
\(74\) 0 0
\(75\) 27.8450 3.21526
\(76\) 0 0
\(77\) −1.23607 −0.140863
\(78\) 0 0
\(79\) 11.6765 1.31371 0.656856 0.754016i \(-0.271886\pi\)
0.656856 + 0.754016i \(0.271886\pi\)
\(80\) 0 0
\(81\) −8.40711 −0.934123
\(82\) 0 0
\(83\) −6.65089 −0.730030 −0.365015 0.931002i \(-0.618936\pi\)
−0.365015 + 0.931002i \(0.618936\pi\)
\(84\) 0 0
\(85\) −4.02435 −0.436502
\(86\) 0 0
\(87\) 15.6951 1.68269
\(88\) 0 0
\(89\) 2.27872 0.241543 0.120772 0.992680i \(-0.461463\pi\)
0.120772 + 0.992680i \(0.461463\pi\)
\(90\) 0 0
\(91\) 6.47214 0.678464
\(92\) 0 0
\(93\) −12.1580 −1.26072
\(94\) 0 0
\(95\) −28.0673 −2.87964
\(96\) 0 0
\(97\) 10.7069 1.08712 0.543562 0.839369i \(-0.317075\pi\)
0.543562 + 0.839369i \(0.317075\pi\)
\(98\) 0 0
\(99\) −3.93822 −0.395806
\(100\) 0 0
\(101\) 16.7931 1.67097 0.835486 0.549512i \(-0.185187\pi\)
0.835486 + 0.549512i \(0.185187\pi\)
\(102\) 0 0
\(103\) −9.57656 −0.943607 −0.471803 0.881704i \(-0.656397\pi\)
−0.471803 + 0.881704i \(0.656397\pi\)
\(104\) 0 0
\(105\) −10.0093 −0.976808
\(106\) 0 0
\(107\) 16.9613 1.63971 0.819855 0.572571i \(-0.194054\pi\)
0.819855 + 0.572571i \(0.194054\pi\)
\(108\) 0 0
\(109\) 19.6082 1.87813 0.939065 0.343741i \(-0.111694\pi\)
0.939065 + 0.343741i \(0.111694\pi\)
\(110\) 0 0
\(111\) 6.55093 0.621787
\(112\) 0 0
\(113\) −13.4952 −1.26952 −0.634761 0.772709i \(-0.718901\pi\)
−0.634761 + 0.772709i \(0.718901\pi\)
\(114\) 0 0
\(115\) 28.4696 2.65480
\(116\) 0 0
\(117\) 20.6208 1.90639
\(118\) 0 0
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −9.47214 −0.861103
\(122\) 0 0
\(123\) −25.3578 −2.28644
\(124\) 0 0
\(125\) −24.9324 −2.23002
\(126\) 0 0
\(127\) −4.36484 −0.387317 −0.193659 0.981069i \(-0.562035\pi\)
−0.193659 + 0.981069i \(0.562035\pi\)
\(128\) 0 0
\(129\) −6.70948 −0.590737
\(130\) 0 0
\(131\) 10.6826 0.933341 0.466670 0.884431i \(-0.345453\pi\)
0.466670 + 0.884431i \(0.345453\pi\)
\(132\) 0 0
\(133\) 6.97437 0.604755
\(134\) 0 0
\(135\) −1.86261 −0.160308
\(136\) 0 0
\(137\) 18.1954 1.55454 0.777268 0.629169i \(-0.216605\pi\)
0.777268 + 0.629169i \(0.216605\pi\)
\(138\) 0 0
\(139\) 13.8430 1.17415 0.587075 0.809532i \(-0.300279\pi\)
0.587075 + 0.809532i \(0.300279\pi\)
\(140\) 0 0
\(141\) 9.14419 0.770080
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −25.3952 −2.10896
\(146\) 0 0
\(147\) 2.48718 0.205140
\(148\) 0 0
\(149\) 3.42987 0.280986 0.140493 0.990082i \(-0.455131\pi\)
0.140493 + 0.990082i \(0.455131\pi\)
\(150\) 0 0
\(151\) 9.16976 0.746224 0.373112 0.927786i \(-0.378291\pi\)
0.373112 + 0.927786i \(0.378291\pi\)
\(152\) 0 0
\(153\) 3.18609 0.257580
\(154\) 0 0
\(155\) 19.6720 1.58009
\(156\) 0 0
\(157\) 6.70215 0.534890 0.267445 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(158\) 0 0
\(159\) 3.45837 0.274266
\(160\) 0 0
\(161\) −7.07433 −0.557535
\(162\) 0 0
\(163\) −14.4104 −1.12871 −0.564353 0.825533i \(-0.690874\pi\)
−0.564353 + 0.825533i \(0.690874\pi\)
\(164\) 0 0
\(165\) 12.3722 0.963173
\(166\) 0 0
\(167\) 3.25112 0.251579 0.125789 0.992057i \(-0.459854\pi\)
0.125789 + 0.992057i \(0.459854\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 0 0
\(171\) 22.2210 1.69928
\(172\) 0 0
\(173\) −4.25717 −0.323666 −0.161833 0.986818i \(-0.551741\pi\)
−0.161833 + 0.986818i \(0.551741\pi\)
\(174\) 0 0
\(175\) 11.1954 0.846292
\(176\) 0 0
\(177\) 12.6208 0.948637
\(178\) 0 0
\(179\) −2.72772 −0.203879 −0.101940 0.994791i \(-0.532505\pi\)
−0.101940 + 0.994791i \(0.532505\pi\)
\(180\) 0 0
\(181\) −3.15471 −0.234488 −0.117244 0.993103i \(-0.537406\pi\)
−0.117244 + 0.993103i \(0.537406\pi\)
\(182\) 0 0
\(183\) −9.78381 −0.723240
\(184\) 0 0
\(185\) −10.5996 −0.779300
\(186\) 0 0
\(187\) −1.23607 −0.0903902
\(188\) 0 0
\(189\) 0.462835 0.0336663
\(190\) 0 0
\(191\) −3.53263 −0.255612 −0.127806 0.991799i \(-0.540794\pi\)
−0.127806 + 0.991799i \(0.540794\pi\)
\(192\) 0 0
\(193\) 16.2275 1.16808 0.584039 0.811726i \(-0.301472\pi\)
0.584039 + 0.811726i \(0.301472\pi\)
\(194\) 0 0
\(195\) −64.7816 −4.63910
\(196\) 0 0
\(197\) −17.7569 −1.26513 −0.632563 0.774509i \(-0.717997\pi\)
−0.632563 + 0.774509i \(0.717997\pi\)
\(198\) 0 0
\(199\) 15.1486 1.07386 0.536928 0.843628i \(-0.319585\pi\)
0.536928 + 0.843628i \(0.319585\pi\)
\(200\) 0 0
\(201\) −14.0093 −0.988140
\(202\) 0 0
\(203\) 6.31040 0.442903
\(204\) 0 0
\(205\) 41.0298 2.86565
\(206\) 0 0
\(207\) −22.5394 −1.56660
\(208\) 0 0
\(209\) −8.62079 −0.596313
\(210\) 0 0
\(211\) 10.5826 0.728537 0.364269 0.931294i \(-0.381319\pi\)
0.364269 + 0.931294i \(0.381319\pi\)
\(212\) 0 0
\(213\) 17.0231 1.16640
\(214\) 0 0
\(215\) 10.8562 0.740385
\(216\) 0 0
\(217\) −4.88824 −0.331835
\(218\) 0 0
\(219\) 0.613463 0.0414540
\(220\) 0 0
\(221\) 6.47214 0.435363
\(222\) 0 0
\(223\) 3.95130 0.264599 0.132299 0.991210i \(-0.457764\pi\)
0.132299 + 0.991210i \(0.457764\pi\)
\(224\) 0 0
\(225\) 35.6695 2.37797
\(226\) 0 0
\(227\) 14.1031 0.936059 0.468029 0.883713i \(-0.344964\pi\)
0.468029 + 0.883713i \(0.344964\pi\)
\(228\) 0 0
\(229\) 17.5766 1.16149 0.580746 0.814085i \(-0.302761\pi\)
0.580746 + 0.814085i \(0.302761\pi\)
\(230\) 0 0
\(231\) −3.07433 −0.202276
\(232\) 0 0
\(233\) 15.5766 1.02045 0.510227 0.860040i \(-0.329561\pi\)
0.510227 + 0.860040i \(0.329561\pi\)
\(234\) 0 0
\(235\) −14.7956 −0.965159
\(236\) 0 0
\(237\) 29.0417 1.88646
\(238\) 0 0
\(239\) 5.16976 0.334404 0.167202 0.985923i \(-0.446527\pi\)
0.167202 + 0.985923i \(0.446527\pi\)
\(240\) 0 0
\(241\) −2.45830 −0.158353 −0.0791766 0.996861i \(-0.525229\pi\)
−0.0791766 + 0.996861i \(0.525229\pi\)
\(242\) 0 0
\(243\) −22.2985 −1.43045
\(244\) 0 0
\(245\) −4.02435 −0.257106
\(246\) 0 0
\(247\) 45.1391 2.87213
\(248\) 0 0
\(249\) −16.5420 −1.04831
\(250\) 0 0
\(251\) −17.3863 −1.09741 −0.548707 0.836015i \(-0.684880\pi\)
−0.548707 + 0.836015i \(0.684880\pi\)
\(252\) 0 0
\(253\) 8.74435 0.549753
\(254\) 0 0
\(255\) −10.0093 −0.626807
\(256\) 0 0
\(257\) −10.2787 −0.641169 −0.320584 0.947220i \(-0.603879\pi\)
−0.320584 + 0.947220i \(0.603879\pi\)
\(258\) 0 0
\(259\) 2.63387 0.163661
\(260\) 0 0
\(261\) 20.1055 1.24450
\(262\) 0 0
\(263\) 10.1487 0.625793 0.312897 0.949787i \(-0.398701\pi\)
0.312897 + 0.949787i \(0.398701\pi\)
\(264\) 0 0
\(265\) −5.59576 −0.343745
\(266\) 0 0
\(267\) 5.66759 0.346851
\(268\) 0 0
\(269\) 2.47055 0.150632 0.0753161 0.997160i \(-0.476003\pi\)
0.0753161 + 0.997160i \(0.476003\pi\)
\(270\) 0 0
\(271\) −21.0864 −1.28091 −0.640455 0.767996i \(-0.721254\pi\)
−0.640455 + 0.767996i \(0.721254\pi\)
\(272\) 0 0
\(273\) 16.0974 0.974259
\(274\) 0 0
\(275\) −13.8383 −0.834479
\(276\) 0 0
\(277\) −25.2335 −1.51613 −0.758067 0.652176i \(-0.773856\pi\)
−0.758067 + 0.652176i \(0.773856\pi\)
\(278\) 0 0
\(279\) −15.5744 −0.932413
\(280\) 0 0
\(281\) 27.5391 1.64285 0.821423 0.570319i \(-0.193180\pi\)
0.821423 + 0.570319i \(0.193180\pi\)
\(282\) 0 0
\(283\) 26.9385 1.60133 0.800665 0.599113i \(-0.204480\pi\)
0.800665 + 0.599113i \(0.204480\pi\)
\(284\) 0 0
\(285\) −69.8086 −4.13510
\(286\) 0 0
\(287\) −10.1954 −0.601815
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 26.6301 1.56108
\(292\) 0 0
\(293\) −9.20439 −0.537726 −0.268863 0.963178i \(-0.586648\pi\)
−0.268863 + 0.963178i \(0.586648\pi\)
\(294\) 0 0
\(295\) −20.4209 −1.18895
\(296\) 0 0
\(297\) −0.572096 −0.0331964
\(298\) 0 0
\(299\) −45.7860 −2.64787
\(300\) 0 0
\(301\) −2.69762 −0.155488
\(302\) 0 0
\(303\) 41.7674 2.39948
\(304\) 0 0
\(305\) 15.8305 0.906454
\(306\) 0 0
\(307\) −5.84878 −0.333807 −0.166904 0.985973i \(-0.553377\pi\)
−0.166904 + 0.985973i \(0.553377\pi\)
\(308\) 0 0
\(309\) −23.8187 −1.35500
\(310\) 0 0
\(311\) −30.5282 −1.73109 −0.865547 0.500828i \(-0.833029\pi\)
−0.865547 + 0.500828i \(0.833029\pi\)
\(312\) 0 0
\(313\) −17.2880 −0.977173 −0.488586 0.872515i \(-0.662487\pi\)
−0.488586 + 0.872515i \(0.662487\pi\)
\(314\) 0 0
\(315\) −12.8219 −0.722434
\(316\) 0 0
\(317\) −4.99395 −0.280488 −0.140244 0.990117i \(-0.544789\pi\)
−0.140244 + 0.990117i \(0.544789\pi\)
\(318\) 0 0
\(319\) −7.80008 −0.436721
\(320\) 0 0
\(321\) 42.1859 2.35459
\(322\) 0 0
\(323\) 6.97437 0.388064
\(324\) 0 0
\(325\) 72.4581 4.01925
\(326\) 0 0
\(327\) 48.7693 2.69695
\(328\) 0 0
\(329\) 3.67652 0.202693
\(330\) 0 0
\(331\) 27.5211 1.51270 0.756349 0.654168i \(-0.226981\pi\)
0.756349 + 0.654168i \(0.226981\pi\)
\(332\) 0 0
\(333\) 8.39176 0.459865
\(334\) 0 0
\(335\) 22.6675 1.23846
\(336\) 0 0
\(337\) −19.6439 −1.07007 −0.535035 0.844830i \(-0.679701\pi\)
−0.535035 + 0.844830i \(0.679701\pi\)
\(338\) 0 0
\(339\) −33.5651 −1.82300
\(340\) 0 0
\(341\) 6.04220 0.327203
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 70.8091 3.81223
\(346\) 0 0
\(347\) 3.53838 0.189950 0.0949751 0.995480i \(-0.469723\pi\)
0.0949751 + 0.995480i \(0.469723\pi\)
\(348\) 0 0
\(349\) 15.9699 0.854849 0.427425 0.904051i \(-0.359421\pi\)
0.427425 + 0.904051i \(0.359421\pi\)
\(350\) 0 0
\(351\) 2.99553 0.159890
\(352\) 0 0
\(353\) −17.3440 −0.923127 −0.461564 0.887107i \(-0.652711\pi\)
−0.461564 + 0.887107i \(0.652711\pi\)
\(354\) 0 0
\(355\) −27.5439 −1.46188
\(356\) 0 0
\(357\) 2.48718 0.131636
\(358\) 0 0
\(359\) −15.2579 −0.805279 −0.402639 0.915359i \(-0.631907\pi\)
−0.402639 + 0.915359i \(0.631907\pi\)
\(360\) 0 0
\(361\) 29.6418 1.56010
\(362\) 0 0
\(363\) −23.5590 −1.23652
\(364\) 0 0
\(365\) −0.992604 −0.0519553
\(366\) 0 0
\(367\) 20.8162 1.08660 0.543298 0.839540i \(-0.317175\pi\)
0.543298 + 0.839540i \(0.317175\pi\)
\(368\) 0 0
\(369\) −32.4834 −1.69102
\(370\) 0 0
\(371\) 1.39047 0.0721899
\(372\) 0 0
\(373\) 18.0929 0.936813 0.468407 0.883513i \(-0.344828\pi\)
0.468407 + 0.883513i \(0.344828\pi\)
\(374\) 0 0
\(375\) −62.0115 −3.20226
\(376\) 0 0
\(377\) 40.8417 2.10346
\(378\) 0 0
\(379\) −32.2265 −1.65536 −0.827681 0.561198i \(-0.810341\pi\)
−0.827681 + 0.561198i \(0.810341\pi\)
\(380\) 0 0
\(381\) −10.8562 −0.556179
\(382\) 0 0
\(383\) −5.77648 −0.295164 −0.147582 0.989050i \(-0.547149\pi\)
−0.147582 + 0.989050i \(0.547149\pi\)
\(384\) 0 0
\(385\) 4.97437 0.253517
\(386\) 0 0
\(387\) −8.59486 −0.436901
\(388\) 0 0
\(389\) 16.6070 0.842006 0.421003 0.907059i \(-0.361678\pi\)
0.421003 + 0.907059i \(0.361678\pi\)
\(390\) 0 0
\(391\) −7.07433 −0.357764
\(392\) 0 0
\(393\) 26.5695 1.34026
\(394\) 0 0
\(395\) −46.9904 −2.36434
\(396\) 0 0
\(397\) 9.80083 0.491890 0.245945 0.969284i \(-0.420902\pi\)
0.245945 + 0.969284i \(0.420902\pi\)
\(398\) 0 0
\(399\) 17.3465 0.868413
\(400\) 0 0
\(401\) 26.4696 1.32183 0.660914 0.750462i \(-0.270169\pi\)
0.660914 + 0.750462i \(0.270169\pi\)
\(402\) 0 0
\(403\) −31.6374 −1.57597
\(404\) 0 0
\(405\) 33.8331 1.68118
\(406\) 0 0
\(407\) −3.25565 −0.161376
\(408\) 0 0
\(409\) 22.9296 1.13380 0.566898 0.823788i \(-0.308143\pi\)
0.566898 + 0.823788i \(0.308143\pi\)
\(410\) 0 0
\(411\) 45.2553 2.23228
\(412\) 0 0
\(413\) 5.07433 0.249691
\(414\) 0 0
\(415\) 26.7655 1.31387
\(416\) 0 0
\(417\) 34.4302 1.68605
\(418\) 0 0
\(419\) 5.11334 0.249803 0.124902 0.992169i \(-0.460138\pi\)
0.124902 + 0.992169i \(0.460138\pi\)
\(420\) 0 0
\(421\) 22.2928 1.08648 0.543242 0.839576i \(-0.317197\pi\)
0.543242 + 0.839576i \(0.317197\pi\)
\(422\) 0 0
\(423\) 11.7137 0.569541
\(424\) 0 0
\(425\) 11.1954 0.543056
\(426\) 0 0
\(427\) −3.93369 −0.190365
\(428\) 0 0
\(429\) −19.8975 −0.960659
\(430\) 0 0
\(431\) −16.4510 −0.792415 −0.396208 0.918161i \(-0.629674\pi\)
−0.396208 + 0.918161i \(0.629674\pi\)
\(432\) 0 0
\(433\) 10.1999 0.490177 0.245088 0.969501i \(-0.421183\pi\)
0.245088 + 0.969501i \(0.421183\pi\)
\(434\) 0 0
\(435\) −63.1627 −3.02842
\(436\) 0 0
\(437\) −49.3390 −2.36020
\(438\) 0 0
\(439\) −18.4074 −0.878538 −0.439269 0.898356i \(-0.644762\pi\)
−0.439269 + 0.898356i \(0.644762\pi\)
\(440\) 0 0
\(441\) 3.18609 0.151718
\(442\) 0 0
\(443\) 1.43047 0.0679635 0.0339817 0.999422i \(-0.489181\pi\)
0.0339817 + 0.999422i \(0.489181\pi\)
\(444\) 0 0
\(445\) −9.17035 −0.434716
\(446\) 0 0
\(447\) 8.53073 0.403490
\(448\) 0 0
\(449\) 3.72778 0.175925 0.0879625 0.996124i \(-0.471964\pi\)
0.0879625 + 0.996124i \(0.471964\pi\)
\(450\) 0 0
\(451\) 12.6022 0.593414
\(452\) 0 0
\(453\) 22.8069 1.07156
\(454\) 0 0
\(455\) −26.0461 −1.22106
\(456\) 0 0
\(457\) 19.7601 0.924338 0.462169 0.886792i \(-0.347071\pi\)
0.462169 + 0.886792i \(0.347071\pi\)
\(458\) 0 0
\(459\) 0.462835 0.0216033
\(460\) 0 0
\(461\) 4.40227 0.205034 0.102517 0.994731i \(-0.467310\pi\)
0.102517 + 0.994731i \(0.467310\pi\)
\(462\) 0 0
\(463\) −4.51887 −0.210010 −0.105005 0.994472i \(-0.533486\pi\)
−0.105005 + 0.994472i \(0.533486\pi\)
\(464\) 0 0
\(465\) 48.9279 2.26898
\(466\) 0 0
\(467\) 1.51893 0.0702877 0.0351438 0.999382i \(-0.488811\pi\)
0.0351438 + 0.999382i \(0.488811\pi\)
\(468\) 0 0
\(469\) −5.63259 −0.260089
\(470\) 0 0
\(471\) 16.6695 0.768090
\(472\) 0 0
\(473\) 3.33444 0.153318
\(474\) 0 0
\(475\) 78.0808 3.58259
\(476\) 0 0
\(477\) 4.43018 0.202844
\(478\) 0 0
\(479\) 39.8418 1.82042 0.910209 0.414148i \(-0.135920\pi\)
0.910209 + 0.414148i \(0.135920\pi\)
\(480\) 0 0
\(481\) 17.0468 0.777267
\(482\) 0 0
\(483\) −17.5952 −0.800608
\(484\) 0 0
\(485\) −43.0884 −1.95654
\(486\) 0 0
\(487\) −9.19532 −0.416680 −0.208340 0.978056i \(-0.566806\pi\)
−0.208340 + 0.978056i \(0.566806\pi\)
\(488\) 0 0
\(489\) −35.8412 −1.62080
\(490\) 0 0
\(491\) 26.5282 1.19720 0.598600 0.801048i \(-0.295724\pi\)
0.598600 + 0.801048i \(0.295724\pi\)
\(492\) 0 0
\(493\) 6.31040 0.284206
\(494\) 0 0
\(495\) 15.8488 0.712350
\(496\) 0 0
\(497\) 6.84431 0.307009
\(498\) 0 0
\(499\) 28.9312 1.29514 0.647569 0.762007i \(-0.275786\pi\)
0.647569 + 0.762007i \(0.275786\pi\)
\(500\) 0 0
\(501\) 8.08613 0.361262
\(502\) 0 0
\(503\) −31.1977 −1.39103 −0.695517 0.718509i \(-0.744825\pi\)
−0.695517 + 0.718509i \(0.744825\pi\)
\(504\) 0 0
\(505\) −67.5811 −3.00732
\(506\) 0 0
\(507\) 71.8511 3.19102
\(508\) 0 0
\(509\) −3.87888 −0.171928 −0.0859641 0.996298i \(-0.527397\pi\)
−0.0859641 + 0.996298i \(0.527397\pi\)
\(510\) 0 0
\(511\) 0.246650 0.0109111
\(512\) 0 0
\(513\) 3.22798 0.142519
\(514\) 0 0
\(515\) 38.5394 1.69825
\(516\) 0 0
\(517\) −4.54443 −0.199864
\(518\) 0 0
\(519\) −10.5884 −0.464777
\(520\) 0 0
\(521\) 44.1050 1.93227 0.966137 0.258030i \(-0.0830734\pi\)
0.966137 + 0.258030i \(0.0830734\pi\)
\(522\) 0 0
\(523\) −0.735418 −0.0321576 −0.0160788 0.999871i \(-0.505118\pi\)
−0.0160788 + 0.999871i \(0.505118\pi\)
\(524\) 0 0
\(525\) 27.8450 1.21526
\(526\) 0 0
\(527\) −4.88824 −0.212935
\(528\) 0 0
\(529\) 27.0461 1.17592
\(530\) 0 0
\(531\) 16.1673 0.701599
\(532\) 0 0
\(533\) −65.9859 −2.85817
\(534\) 0 0
\(535\) −68.2582 −2.95106
\(536\) 0 0
\(537\) −6.78434 −0.292766
\(538\) 0 0
\(539\) −1.23607 −0.0532412
\(540\) 0 0
\(541\) −5.95682 −0.256104 −0.128052 0.991767i \(-0.540872\pi\)
−0.128052 + 0.991767i \(0.540872\pi\)
\(542\) 0 0
\(543\) −7.84635 −0.336719
\(544\) 0 0
\(545\) −78.9104 −3.38015
\(546\) 0 0
\(547\) −39.8844 −1.70533 −0.852667 0.522455i \(-0.825016\pi\)
−0.852667 + 0.522455i \(0.825016\pi\)
\(548\) 0 0
\(549\) −12.5331 −0.534899
\(550\) 0 0
\(551\) 44.0110 1.87493
\(552\) 0 0
\(553\) 11.6765 0.496536
\(554\) 0 0
\(555\) −26.3632 −1.11906
\(556\) 0 0
\(557\) −29.5740 −1.25309 −0.626545 0.779385i \(-0.715532\pi\)
−0.626545 + 0.779385i \(0.715532\pi\)
\(558\) 0 0
\(559\) −17.4594 −0.738453
\(560\) 0 0
\(561\) −3.07433 −0.129798
\(562\) 0 0
\(563\) 43.7900 1.84553 0.922763 0.385367i \(-0.125925\pi\)
0.922763 + 0.385367i \(0.125925\pi\)
\(564\) 0 0
\(565\) 54.3094 2.28481
\(566\) 0 0
\(567\) −8.40711 −0.353065
\(568\) 0 0
\(569\) −44.3088 −1.85752 −0.928761 0.370678i \(-0.879125\pi\)
−0.928761 + 0.370678i \(0.879125\pi\)
\(570\) 0 0
\(571\) −25.9543 −1.08615 −0.543076 0.839684i \(-0.682740\pi\)
−0.543076 + 0.839684i \(0.682740\pi\)
\(572\) 0 0
\(573\) −8.78631 −0.367053
\(574\) 0 0
\(575\) −79.1999 −3.30286
\(576\) 0 0
\(577\) −11.1442 −0.463939 −0.231969 0.972723i \(-0.574517\pi\)
−0.231969 + 0.972723i \(0.574517\pi\)
\(578\) 0 0
\(579\) 40.3607 1.67733
\(580\) 0 0
\(581\) −6.65089 −0.275926
\(582\) 0 0
\(583\) −1.71872 −0.0711822
\(584\) 0 0
\(585\) −82.9853 −3.43102
\(586\) 0 0
\(587\) 19.6676 0.811768 0.405884 0.913925i \(-0.366964\pi\)
0.405884 + 0.913925i \(0.366964\pi\)
\(588\) 0 0
\(589\) −34.0924 −1.40475
\(590\) 0 0
\(591\) −44.1647 −1.81669
\(592\) 0 0
\(593\) −28.1396 −1.15555 −0.577777 0.816194i \(-0.696080\pi\)
−0.577777 + 0.816194i \(0.696080\pi\)
\(594\) 0 0
\(595\) −4.02435 −0.164982
\(596\) 0 0
\(597\) 37.6774 1.54203
\(598\) 0 0
\(599\) −2.43434 −0.0994645 −0.0497322 0.998763i \(-0.515837\pi\)
−0.0497322 + 0.998763i \(0.515837\pi\)
\(600\) 0 0
\(601\) −15.8462 −0.646381 −0.323190 0.946334i \(-0.604755\pi\)
−0.323190 + 0.946334i \(0.604755\pi\)
\(602\) 0 0
\(603\) −17.9459 −0.730815
\(604\) 0 0
\(605\) 38.1192 1.54977
\(606\) 0 0
\(607\) −18.1413 −0.736334 −0.368167 0.929760i \(-0.620014\pi\)
−0.368167 + 0.929760i \(0.620014\pi\)
\(608\) 0 0
\(609\) 15.6951 0.635999
\(610\) 0 0
\(611\) 23.7950 0.962641
\(612\) 0 0
\(613\) −14.9702 −0.604641 −0.302320 0.953206i \(-0.597761\pi\)
−0.302320 + 0.953206i \(0.597761\pi\)
\(614\) 0 0
\(615\) 102.049 4.11500
\(616\) 0 0
\(617\) 7.43440 0.299298 0.149649 0.988739i \(-0.452186\pi\)
0.149649 + 0.988739i \(0.452186\pi\)
\(618\) 0 0
\(619\) −12.9548 −0.520697 −0.260348 0.965515i \(-0.583837\pi\)
−0.260348 + 0.965515i \(0.583837\pi\)
\(620\) 0 0
\(621\) −3.27425 −0.131391
\(622\) 0 0
\(623\) 2.27872 0.0912948
\(624\) 0 0
\(625\) 44.3598 1.77439
\(626\) 0 0
\(627\) −21.4415 −0.856291
\(628\) 0 0
\(629\) 2.63387 0.105019
\(630\) 0 0
\(631\) −0.899738 −0.0358180 −0.0179090 0.999840i \(-0.505701\pi\)
−0.0179090 + 0.999840i \(0.505701\pi\)
\(632\) 0 0
\(633\) 26.3209 1.04616
\(634\) 0 0
\(635\) 17.5657 0.697072
\(636\) 0 0
\(637\) 6.47214 0.256435
\(638\) 0 0
\(639\) 21.8066 0.862655
\(640\) 0 0
\(641\) −14.8322 −0.585837 −0.292919 0.956137i \(-0.594626\pi\)
−0.292919 + 0.956137i \(0.594626\pi\)
\(642\) 0 0
\(643\) −17.5168 −0.690793 −0.345397 0.938457i \(-0.612256\pi\)
−0.345397 + 0.938457i \(0.612256\pi\)
\(644\) 0 0
\(645\) 27.0013 1.06318
\(646\) 0 0
\(647\) −1.05829 −0.0416057 −0.0208028 0.999784i \(-0.506622\pi\)
−0.0208028 + 0.999784i \(0.506622\pi\)
\(648\) 0 0
\(649\) −6.27222 −0.246206
\(650\) 0 0
\(651\) −12.1580 −0.476508
\(652\) 0 0
\(653\) −35.2848 −1.38080 −0.690400 0.723428i \(-0.742565\pi\)
−0.690400 + 0.723428i \(0.742565\pi\)
\(654\) 0 0
\(655\) −42.9904 −1.67977
\(656\) 0 0
\(657\) 0.785848 0.0306588
\(658\) 0 0
\(659\) −20.1047 −0.783169 −0.391585 0.920142i \(-0.628073\pi\)
−0.391585 + 0.920142i \(0.628073\pi\)
\(660\) 0 0
\(661\) −12.6541 −0.492186 −0.246093 0.969246i \(-0.579147\pi\)
−0.246093 + 0.969246i \(0.579147\pi\)
\(662\) 0 0
\(663\) 16.0974 0.625171
\(664\) 0 0
\(665\) −28.0673 −1.08840
\(666\) 0 0
\(667\) −44.6418 −1.72854
\(668\) 0 0
\(669\) 9.82762 0.379958
\(670\) 0 0
\(671\) 4.86231 0.187707
\(672\) 0 0
\(673\) −21.6342 −0.833937 −0.416968 0.908921i \(-0.636907\pi\)
−0.416968 + 0.908921i \(0.636907\pi\)
\(674\) 0 0
\(675\) 5.18162 0.199441
\(676\) 0 0
\(677\) −27.9965 −1.07599 −0.537996 0.842948i \(-0.680818\pi\)
−0.537996 + 0.842948i \(0.680818\pi\)
\(678\) 0 0
\(679\) 10.7069 0.410894
\(680\) 0 0
\(681\) 35.0771 1.34416
\(682\) 0 0
\(683\) −40.2225 −1.53907 −0.769536 0.638603i \(-0.779513\pi\)
−0.769536 + 0.638603i \(0.779513\pi\)
\(684\) 0 0
\(685\) −73.2246 −2.79777
\(686\) 0 0
\(687\) 43.7162 1.66788
\(688\) 0 0
\(689\) 8.99934 0.342848
\(690\) 0 0
\(691\) −30.7414 −1.16946 −0.584729 0.811229i \(-0.698799\pi\)
−0.584729 + 0.811229i \(0.698799\pi\)
\(692\) 0 0
\(693\) −3.93822 −0.149601
\(694\) 0 0
\(695\) −55.7092 −2.11317
\(696\) 0 0
\(697\) −10.1954 −0.386178
\(698\) 0 0
\(699\) 38.7418 1.46535
\(700\) 0 0
\(701\) −24.3696 −0.920428 −0.460214 0.887808i \(-0.652227\pi\)
−0.460214 + 0.887808i \(0.652227\pi\)
\(702\) 0 0
\(703\) 18.3696 0.692823
\(704\) 0 0
\(705\) −36.7994 −1.38595
\(706\) 0 0
\(707\) 16.7931 0.631568
\(708\) 0 0
\(709\) 22.2957 0.837334 0.418667 0.908140i \(-0.362497\pi\)
0.418667 + 0.908140i \(0.362497\pi\)
\(710\) 0 0
\(711\) 37.2024 1.39520
\(712\) 0 0
\(713\) 34.5810 1.29507
\(714\) 0 0
\(715\) 32.1948 1.20402
\(716\) 0 0
\(717\) 12.8581 0.480196
\(718\) 0 0
\(719\) 14.2466 0.531310 0.265655 0.964068i \(-0.414412\pi\)
0.265655 + 0.964068i \(0.414412\pi\)
\(720\) 0 0
\(721\) −9.57656 −0.356650
\(722\) 0 0
\(723\) −6.11426 −0.227392
\(724\) 0 0
\(725\) 70.6474 2.62378
\(726\) 0 0
\(727\) −26.2762 −0.974529 −0.487264 0.873255i \(-0.662005\pi\)
−0.487264 + 0.873255i \(0.662005\pi\)
\(728\) 0 0
\(729\) −30.2393 −1.11997
\(730\) 0 0
\(731\) −2.69762 −0.0997752
\(732\) 0 0
\(733\) 29.7464 1.09871 0.549354 0.835590i \(-0.314874\pi\)
0.549354 + 0.835590i \(0.314874\pi\)
\(734\) 0 0
\(735\) −10.0093 −0.369199
\(736\) 0 0
\(737\) 6.96227 0.256458
\(738\) 0 0
\(739\) 20.0628 0.738021 0.369010 0.929425i \(-0.379697\pi\)
0.369010 + 0.929425i \(0.379697\pi\)
\(740\) 0 0
\(741\) 112.269 4.12431
\(742\) 0 0
\(743\) 10.0948 0.370344 0.185172 0.982706i \(-0.440716\pi\)
0.185172 + 0.982706i \(0.440716\pi\)
\(744\) 0 0
\(745\) −13.8030 −0.505703
\(746\) 0 0
\(747\) −21.1903 −0.775314
\(748\) 0 0
\(749\) 16.9613 0.619752
\(750\) 0 0
\(751\) 2.58103 0.0941831 0.0470916 0.998891i \(-0.485005\pi\)
0.0470916 + 0.998891i \(0.485005\pi\)
\(752\) 0 0
\(753\) −43.2430 −1.57586
\(754\) 0 0
\(755\) −36.9023 −1.34301
\(756\) 0 0
\(757\) −42.6072 −1.54859 −0.774293 0.632828i \(-0.781894\pi\)
−0.774293 + 0.632828i \(0.781894\pi\)
\(758\) 0 0
\(759\) 21.7488 0.789432
\(760\) 0 0
\(761\) −6.81361 −0.246993 −0.123497 0.992345i \(-0.539411\pi\)
−0.123497 + 0.992345i \(0.539411\pi\)
\(762\) 0 0
\(763\) 19.6082 0.709866
\(764\) 0 0
\(765\) −12.8219 −0.463578
\(766\) 0 0
\(767\) 32.8417 1.18585
\(768\) 0 0
\(769\) −9.62782 −0.347188 −0.173594 0.984817i \(-0.555538\pi\)
−0.173594 + 0.984817i \(0.555538\pi\)
\(770\) 0 0
\(771\) −25.5651 −0.920703
\(772\) 0 0
\(773\) 32.6842 1.17557 0.587784 0.809018i \(-0.300001\pi\)
0.587784 + 0.809018i \(0.300001\pi\)
\(774\) 0 0
\(775\) −54.7258 −1.96581
\(776\) 0 0
\(777\) 6.55093 0.235013
\(778\) 0 0
\(779\) −71.1064 −2.54765
\(780\) 0 0
\(781\) −8.46003 −0.302724
\(782\) 0 0
\(783\) 2.92067 0.104376
\(784\) 0 0
\(785\) −26.9718 −0.962665
\(786\) 0 0
\(787\) −40.6653 −1.44956 −0.724782 0.688979i \(-0.758059\pi\)
−0.724782 + 0.688979i \(0.758059\pi\)
\(788\) 0 0
\(789\) 25.2416 0.898624
\(790\) 0 0
\(791\) −13.4952 −0.479834
\(792\) 0 0
\(793\) −25.4594 −0.904089
\(794\) 0 0
\(795\) −13.9177 −0.493609
\(796\) 0 0
\(797\) −19.0019 −0.673082 −0.336541 0.941669i \(-0.609257\pi\)
−0.336541 + 0.941669i \(0.609257\pi\)
\(798\) 0 0
\(799\) 3.67652 0.130066
\(800\) 0 0
\(801\) 7.26019 0.256526
\(802\) 0 0
\(803\) −0.304876 −0.0107588
\(804\) 0 0
\(805\) 28.4696 1.00342
\(806\) 0 0
\(807\) 6.14472 0.216304
\(808\) 0 0
\(809\) −7.95386 −0.279643 −0.139821 0.990177i \(-0.544653\pi\)
−0.139821 + 0.990177i \(0.544653\pi\)
\(810\) 0 0
\(811\) −40.6097 −1.42600 −0.712999 0.701165i \(-0.752664\pi\)
−0.712999 + 0.701165i \(0.752664\pi\)
\(812\) 0 0
\(813\) −52.4458 −1.83936
\(814\) 0 0
\(815\) 57.9923 2.03138
\(816\) 0 0
\(817\) −18.8142 −0.658226
\(818\) 0 0
\(819\) 20.6208 0.720549
\(820\) 0 0
\(821\) −50.7051 −1.76962 −0.884810 0.465951i \(-0.845712\pi\)
−0.884810 + 0.465951i \(0.845712\pi\)
\(822\) 0 0
\(823\) −46.3922 −1.61713 −0.808564 0.588408i \(-0.799755\pi\)
−0.808564 + 0.588408i \(0.799755\pi\)
\(824\) 0 0
\(825\) −34.4183 −1.19829
\(826\) 0 0
\(827\) −34.6166 −1.20374 −0.601869 0.798595i \(-0.705577\pi\)
−0.601869 + 0.798595i \(0.705577\pi\)
\(828\) 0 0
\(829\) −2.04614 −0.0710653 −0.0355326 0.999369i \(-0.511313\pi\)
−0.0355326 + 0.999369i \(0.511313\pi\)
\(830\) 0 0
\(831\) −62.7604 −2.17713
\(832\) 0 0
\(833\) 1.00000 0.0346479
\(834\) 0 0
\(835\) −13.0836 −0.452778
\(836\) 0 0
\(837\) −2.26245 −0.0782017
\(838\) 0 0
\(839\) −2.11462 −0.0730049 −0.0365024 0.999334i \(-0.511622\pi\)
−0.0365024 + 0.999334i \(0.511622\pi\)
\(840\) 0 0
\(841\) 10.8211 0.373142
\(842\) 0 0
\(843\) 68.4949 2.35909
\(844\) 0 0
\(845\) −116.258 −3.99938
\(846\) 0 0
\(847\) −9.47214 −0.325466
\(848\) 0 0
\(849\) 67.0011 2.29947
\(850\) 0 0
\(851\) −18.6329 −0.638727
\(852\) 0 0
\(853\) −22.7338 −0.778392 −0.389196 0.921155i \(-0.627247\pi\)
−0.389196 + 0.921155i \(0.627247\pi\)
\(854\) 0 0
\(855\) −89.4249 −3.05827
\(856\) 0 0
\(857\) 2.03743 0.0695973 0.0347986 0.999394i \(-0.488921\pi\)
0.0347986 + 0.999394i \(0.488921\pi\)
\(858\) 0 0
\(859\) −15.8648 −0.541301 −0.270650 0.962678i \(-0.587239\pi\)
−0.270650 + 0.962678i \(0.587239\pi\)
\(860\) 0 0
\(861\) −25.3578 −0.864192
\(862\) 0 0
\(863\) 14.8857 0.506714 0.253357 0.967373i \(-0.418465\pi\)
0.253357 + 0.967373i \(0.418465\pi\)
\(864\) 0 0
\(865\) 17.1323 0.582517
\(866\) 0 0
\(867\) 2.48718 0.0844692
\(868\) 0 0
\(869\) −14.4330 −0.489605
\(870\) 0 0
\(871\) −36.4549 −1.23523
\(872\) 0 0
\(873\) 34.1132 1.15456
\(874\) 0 0
\(875\) −24.9324 −0.842869
\(876\) 0 0
\(877\) 3.66844 0.123874 0.0619372 0.998080i \(-0.480272\pi\)
0.0619372 + 0.998080i \(0.480272\pi\)
\(878\) 0 0
\(879\) −22.8930 −0.772162
\(880\) 0 0
\(881\) −39.9695 −1.34661 −0.673304 0.739366i \(-0.735126\pi\)
−0.673304 + 0.739366i \(0.735126\pi\)
\(882\) 0 0
\(883\) −17.1028 −0.575556 −0.287778 0.957697i \(-0.592917\pi\)
−0.287778 + 0.957697i \(0.592917\pi\)
\(884\) 0 0
\(885\) −50.7905 −1.70730
\(886\) 0 0
\(887\) 26.5622 0.891871 0.445936 0.895065i \(-0.352871\pi\)
0.445936 + 0.895065i \(0.352871\pi\)
\(888\) 0 0
\(889\) −4.36484 −0.146392
\(890\) 0 0
\(891\) 10.3918 0.348137
\(892\) 0 0
\(893\) 25.6414 0.858058
\(894\) 0 0
\(895\) 10.9773 0.366931
\(896\) 0 0
\(897\) −113.878 −3.80229
\(898\) 0 0
\(899\) −30.8467 −1.02880
\(900\) 0 0
\(901\) 1.39047 0.0463234
\(902\) 0 0
\(903\) −6.70948 −0.223278
\(904\) 0 0
\(905\) 12.6957 0.422018
\(906\) 0 0
\(907\) 42.8523 1.42289 0.711443 0.702744i \(-0.248042\pi\)
0.711443 + 0.702744i \(0.248042\pi\)
\(908\) 0 0
\(909\) 53.5041 1.77462
\(910\) 0 0
\(911\) 37.1141 1.22964 0.614822 0.788666i \(-0.289228\pi\)
0.614822 + 0.788666i \(0.289228\pi\)
\(912\) 0 0
\(913\) 8.22096 0.272074
\(914\) 0 0
\(915\) 39.3735 1.30165
\(916\) 0 0
\(917\) 10.6826 0.352770
\(918\) 0 0
\(919\) −23.7655 −0.783950 −0.391975 0.919976i \(-0.628208\pi\)
−0.391975 + 0.919976i \(0.628208\pi\)
\(920\) 0 0
\(921\) −14.5470 −0.479340
\(922\) 0 0
\(923\) 44.2973 1.45806
\(924\) 0 0
\(925\) 29.4872 0.969535
\(926\) 0 0
\(927\) −30.5118 −1.00214
\(928\) 0 0
\(929\) 24.7467 0.811912 0.405956 0.913893i \(-0.366939\pi\)
0.405956 + 0.913893i \(0.366939\pi\)
\(930\) 0 0
\(931\) 6.97437 0.228576
\(932\) 0 0
\(933\) −75.9292 −2.48581
\(934\) 0 0
\(935\) 4.97437 0.162679
\(936\) 0 0
\(937\) 26.9577 0.880671 0.440336 0.897833i \(-0.354859\pi\)
0.440336 + 0.897833i \(0.354859\pi\)
\(938\) 0 0
\(939\) −42.9983 −1.40320
\(940\) 0 0
\(941\) 30.6992 1.00077 0.500383 0.865804i \(-0.333193\pi\)
0.500383 + 0.865804i \(0.333193\pi\)
\(942\) 0 0
\(943\) 72.1255 2.34873
\(944\) 0 0
\(945\) −1.86261 −0.0605907
\(946\) 0 0
\(947\) 28.3892 0.922525 0.461262 0.887264i \(-0.347397\pi\)
0.461262 + 0.887264i \(0.347397\pi\)
\(948\) 0 0
\(949\) 1.59635 0.0518197
\(950\) 0 0
\(951\) −12.4209 −0.402774
\(952\) 0 0
\(953\) 39.1278 1.26747 0.633737 0.773549i \(-0.281520\pi\)
0.633737 + 0.773549i \(0.281520\pi\)
\(954\) 0 0
\(955\) 14.2166 0.460037
\(956\) 0 0
\(957\) −19.4002 −0.627121
\(958\) 0 0
\(959\) 18.1954 0.587560
\(960\) 0 0
\(961\) −7.10510 −0.229197
\(962\) 0 0
\(963\) 54.0402 1.74142
\(964\) 0 0
\(965\) −65.3050 −2.10224
\(966\) 0 0
\(967\) −39.8719 −1.28219 −0.641097 0.767460i \(-0.721520\pi\)
−0.641097 + 0.767460i \(0.721520\pi\)
\(968\) 0 0
\(969\) 17.3465 0.557251
\(970\) 0 0
\(971\) 16.8891 0.541996 0.270998 0.962580i \(-0.412646\pi\)
0.270998 + 0.962580i \(0.412646\pi\)
\(972\) 0 0
\(973\) 13.8430 0.443787
\(974\) 0 0
\(975\) 180.217 5.77155
\(976\) 0 0
\(977\) 21.4487 0.686205 0.343103 0.939298i \(-0.388522\pi\)
0.343103 + 0.939298i \(0.388522\pi\)
\(978\) 0 0
\(979\) −2.81665 −0.0900205
\(980\) 0 0
\(981\) 62.4736 1.99463
\(982\) 0 0
\(983\) 48.8004 1.55649 0.778245 0.627960i \(-0.216110\pi\)
0.778245 + 0.627960i \(0.216110\pi\)
\(984\) 0 0
\(985\) 71.4600 2.27690
\(986\) 0 0
\(987\) 9.14419 0.291063
\(988\) 0 0
\(989\) 19.0839 0.606832
\(990\) 0 0
\(991\) −5.00203 −0.158895 −0.0794474 0.996839i \(-0.525316\pi\)
−0.0794474 + 0.996839i \(0.525316\pi\)
\(992\) 0 0
\(993\) 68.4502 2.17220
\(994\) 0 0
\(995\) −60.9632 −1.93266
\(996\) 0 0
\(997\) 54.7966 1.73543 0.867713 0.497066i \(-0.165589\pi\)
0.867713 + 0.497066i \(0.165589\pi\)
\(998\) 0 0
\(999\) 1.21905 0.0385690
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 7616.2.a.bp.1.3 4
4.3 odd 2 7616.2.a.bj.1.2 4
8.3 odd 2 952.2.a.g.1.3 4
8.5 even 2 1904.2.a.q.1.2 4
24.11 even 2 8568.2.a.bj.1.1 4
56.27 even 2 6664.2.a.o.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
952.2.a.g.1.3 4 8.3 odd 2
1904.2.a.q.1.2 4 8.5 even 2
6664.2.a.o.1.2 4 56.27 even 2
7616.2.a.bj.1.2 4 4.3 odd 2
7616.2.a.bp.1.3 4 1.1 even 1 trivial
8568.2.a.bj.1.1 4 24.11 even 2