Properties

Label 2175.2.a.r.1.2
Level $2175$
Weight $2$
Character 2175.1
Self dual yes
Analytic conductor $17.367$
Analytic rank $0$
Dimension $2$
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] = [2175,2,Mod(1,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 2175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +5.00000 q^{11} -5.79129 q^{12} -4.58258 q^{13} -2.79129 q^{14} +17.9564 q^{16} +3.00000 q^{17} +2.79129 q^{18} -5.58258 q^{19} +1.00000 q^{21} +13.9564 q^{22} +4.00000 q^{23} -10.5826 q^{24} -12.7913 q^{26} -1.00000 q^{27} -5.79129 q^{28} +1.00000 q^{29} +4.00000 q^{31} +28.9564 q^{32} -5.00000 q^{33} +8.37386 q^{34} +5.79129 q^{36} +4.00000 q^{37} -15.5826 q^{38} +4.58258 q^{39} +9.16515 q^{41} +2.79129 q^{42} +0.417424 q^{43} +28.9564 q^{44} +11.1652 q^{46} -1.41742 q^{47} -17.9564 q^{48} -6.00000 q^{49} -3.00000 q^{51} -26.5390 q^{52} -9.58258 q^{53} -2.79129 q^{54} -10.5826 q^{56} +5.58258 q^{57} +2.79129 q^{58} +1.58258 q^{59} -14.7477 q^{61} +11.1652 q^{62} -1.00000 q^{63} +44.9129 q^{64} -13.9564 q^{66} -14.1652 q^{67} +17.3739 q^{68} -4.00000 q^{69} -0.417424 q^{71} +10.5826 q^{72} -4.00000 q^{73} +11.1652 q^{74} -32.3303 q^{76} -5.00000 q^{77} +12.7913 q^{78} -1.58258 q^{79} +1.00000 q^{81} +25.5826 q^{82} +2.41742 q^{83} +5.79129 q^{84} +1.16515 q^{86} -1.00000 q^{87} +52.9129 q^{88} +10.5826 q^{89} +4.58258 q^{91} +23.1652 q^{92} -4.00000 q^{93} -3.95644 q^{94} -28.9564 q^{96} -2.41742 q^{97} -16.7477 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26} - 2 q^{27} - 7 q^{28} + 2 q^{29} + 8 q^{31} + 35 q^{32} - 10 q^{33} + 3 q^{34} + 7 q^{36} + 8 q^{37} - 22 q^{38} + q^{42} + 10 q^{43} + 35 q^{44} + 4 q^{46} - 12 q^{47} - 13 q^{48} - 12 q^{49} - 6 q^{51} - 21 q^{52} - 10 q^{53} - q^{54} - 12 q^{56} + 2 q^{57} + q^{58} - 6 q^{59} - 2 q^{61} + 4 q^{62} - 2 q^{63} + 44 q^{64} - 5 q^{66} - 10 q^{67} + 21 q^{68} - 8 q^{69} - 10 q^{71} + 12 q^{72} - 8 q^{73} + 4 q^{74} - 28 q^{76} - 10 q^{77} + 21 q^{78} + 6 q^{79} + 2 q^{81} + 42 q^{82} + 14 q^{83} + 7 q^{84} - 16 q^{86} - 2 q^{87} + 60 q^{88} + 12 q^{89} + 28 q^{92} - 8 q^{93} + 15 q^{94} - 35 q^{96} - 14 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.79129 1.97374 0.986869 0.161521i \(-0.0516399\pi\)
0.986869 + 0.161521i \(0.0516399\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.79129 2.89564
\(5\) 0 0
\(6\) −2.79129 −1.13954
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 10.5826 3.74151
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −5.79129 −1.67180
\(13\) −4.58258 −1.27098 −0.635489 0.772110i \(-0.719201\pi\)
−0.635489 + 0.772110i \(0.719201\pi\)
\(14\) −2.79129 −0.746003
\(15\) 0 0
\(16\) 17.9564 4.48911
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.79129 0.657913
\(19\) −5.58258 −1.28073 −0.640365 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 13.9564 2.97552
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −10.5826 −2.16016
\(25\) 0 0
\(26\) −12.7913 −2.50858
\(27\) −1.00000 −0.192450
\(28\) −5.79129 −1.09445
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 28.9564 5.11882
\(33\) −5.00000 −0.870388
\(34\) 8.37386 1.43611
\(35\) 0 0
\(36\) 5.79129 0.965215
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −15.5826 −2.52783
\(39\) 4.58258 0.733799
\(40\) 0 0
\(41\) 9.16515 1.43136 0.715678 0.698430i \(-0.246118\pi\)
0.715678 + 0.698430i \(0.246118\pi\)
\(42\) 2.79129 0.430705
\(43\) 0.417424 0.0636566 0.0318283 0.999493i \(-0.489867\pi\)
0.0318283 + 0.999493i \(0.489867\pi\)
\(44\) 28.9564 4.36535
\(45\) 0 0
\(46\) 11.1652 1.64621
\(47\) −1.41742 −0.206753 −0.103376 0.994642i \(-0.532965\pi\)
−0.103376 + 0.994642i \(0.532965\pi\)
\(48\) −17.9564 −2.59179
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) −26.5390 −3.68030
\(53\) −9.58258 −1.31627 −0.658134 0.752901i \(-0.728654\pi\)
−0.658134 + 0.752901i \(0.728654\pi\)
\(54\) −2.79129 −0.379846
\(55\) 0 0
\(56\) −10.5826 −1.41416
\(57\) 5.58258 0.739430
\(58\) 2.79129 0.366514
\(59\) 1.58258 0.206034 0.103017 0.994680i \(-0.467150\pi\)
0.103017 + 0.994680i \(0.467150\pi\)
\(60\) 0 0
\(61\) −14.7477 −1.88825 −0.944126 0.329583i \(-0.893092\pi\)
−0.944126 + 0.329583i \(0.893092\pi\)
\(62\) 11.1652 1.41798
\(63\) −1.00000 −0.125988
\(64\) 44.9129 5.61411
\(65\) 0 0
\(66\) −13.9564 −1.71792
\(67\) −14.1652 −1.73055 −0.865274 0.501299i \(-0.832856\pi\)
−0.865274 + 0.501299i \(0.832856\pi\)
\(68\) 17.3739 2.10689
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −0.417424 −0.0495392 −0.0247696 0.999693i \(-0.507885\pi\)
−0.0247696 + 0.999693i \(0.507885\pi\)
\(72\) 10.5826 1.24717
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 11.1652 1.29792
\(75\) 0 0
\(76\) −32.3303 −3.70854
\(77\) −5.00000 −0.569803
\(78\) 12.7913 1.44833
\(79\) −1.58258 −0.178054 −0.0890268 0.996029i \(-0.528376\pi\)
−0.0890268 + 0.996029i \(0.528376\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 25.5826 2.82512
\(83\) 2.41742 0.265347 0.132673 0.991160i \(-0.457644\pi\)
0.132673 + 0.991160i \(0.457644\pi\)
\(84\) 5.79129 0.631881
\(85\) 0 0
\(86\) 1.16515 0.125642
\(87\) −1.00000 −0.107211
\(88\) 52.9129 5.64053
\(89\) 10.5826 1.12175 0.560875 0.827900i \(-0.310465\pi\)
0.560875 + 0.827900i \(0.310465\pi\)
\(90\) 0 0
\(91\) 4.58258 0.480384
\(92\) 23.1652 2.41513
\(93\) −4.00000 −0.414781
\(94\) −3.95644 −0.408076
\(95\) 0 0
\(96\) −28.9564 −2.95535
\(97\) −2.41742 −0.245452 −0.122726 0.992441i \(-0.539164\pi\)
−0.122726 + 0.992441i \(0.539164\pi\)
\(98\) −16.7477 −1.69178
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 8.58258 0.853998 0.426999 0.904252i \(-0.359571\pi\)
0.426999 + 0.904252i \(0.359571\pi\)
\(102\) −8.37386 −0.829136
\(103\) 3.16515 0.311872 0.155936 0.987767i \(-0.450161\pi\)
0.155936 + 0.987767i \(0.450161\pi\)
\(104\) −48.4955 −4.75537
\(105\) 0 0
\(106\) −26.7477 −2.59797
\(107\) 13.1652 1.27272 0.636362 0.771391i \(-0.280439\pi\)
0.636362 + 0.771391i \(0.280439\pi\)
\(108\) −5.79129 −0.557267
\(109\) −4.16515 −0.398949 −0.199475 0.979903i \(-0.563924\pi\)
−0.199475 + 0.979903i \(0.563924\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −17.9564 −1.69672
\(113\) −4.16515 −0.391824 −0.195912 0.980621i \(-0.562767\pi\)
−0.195912 + 0.980621i \(0.562767\pi\)
\(114\) 15.5826 1.45944
\(115\) 0 0
\(116\) 5.79129 0.537708
\(117\) −4.58258 −0.423659
\(118\) 4.41742 0.406657
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −41.1652 −3.72692
\(123\) −9.16515 −0.826394
\(124\) 23.1652 2.08029
\(125\) 0 0
\(126\) −2.79129 −0.248668
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 67.4519 5.96196
\(129\) −0.417424 −0.0367522
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) −28.9564 −2.52033
\(133\) 5.58258 0.484071
\(134\) −39.5390 −3.41565
\(135\) 0 0
\(136\) 31.7477 2.72235
\(137\) 20.3303 1.73693 0.868467 0.495746i \(-0.165105\pi\)
0.868467 + 0.495746i \(0.165105\pi\)
\(138\) −11.1652 −0.950441
\(139\) −18.5826 −1.57615 −0.788077 0.615577i \(-0.788923\pi\)
−0.788077 + 0.615577i \(0.788923\pi\)
\(140\) 0 0
\(141\) 1.41742 0.119369
\(142\) −1.16515 −0.0977773
\(143\) −22.9129 −1.91607
\(144\) 17.9564 1.49637
\(145\) 0 0
\(146\) −11.1652 −0.924035
\(147\) 6.00000 0.494872
\(148\) 23.1652 1.90416
\(149\) −10.7477 −0.880488 −0.440244 0.897878i \(-0.645108\pi\)
−0.440244 + 0.897878i \(0.645108\pi\)
\(150\) 0 0
\(151\) −11.1652 −0.908607 −0.454304 0.890847i \(-0.650112\pi\)
−0.454304 + 0.890847i \(0.650112\pi\)
\(152\) −59.0780 −4.79186
\(153\) 3.00000 0.242536
\(154\) −13.9564 −1.12464
\(155\) 0 0
\(156\) 26.5390 2.12482
\(157\) 16.7477 1.33661 0.668307 0.743886i \(-0.267019\pi\)
0.668307 + 0.743886i \(0.267019\pi\)
\(158\) −4.41742 −0.351431
\(159\) 9.58258 0.759948
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 2.79129 0.219304
\(163\) 1.58258 0.123957 0.0619784 0.998077i \(-0.480259\pi\)
0.0619784 + 0.998077i \(0.480259\pi\)
\(164\) 53.0780 4.14470
\(165\) 0 0
\(166\) 6.74773 0.523725
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 10.5826 0.816463
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) −5.58258 −0.426910
\(172\) 2.41742 0.184327
\(173\) −15.1652 −1.15299 −0.576493 0.817102i \(-0.695579\pi\)
−0.576493 + 0.817102i \(0.695579\pi\)
\(174\) −2.79129 −0.211607
\(175\) 0 0
\(176\) 89.7822 6.76759
\(177\) −1.58258 −0.118954
\(178\) 29.5390 2.21404
\(179\) −22.7477 −1.70024 −0.850122 0.526585i \(-0.823472\pi\)
−0.850122 + 0.526585i \(0.823472\pi\)
\(180\) 0 0
\(181\) −2.16515 −0.160934 −0.0804672 0.996757i \(-0.525641\pi\)
−0.0804672 + 0.996757i \(0.525641\pi\)
\(182\) 12.7913 0.948153
\(183\) 14.7477 1.09018
\(184\) 42.3303 3.12063
\(185\) 0 0
\(186\) −11.1652 −0.818669
\(187\) 15.0000 1.09691
\(188\) −8.20871 −0.598682
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) −44.9129 −3.24131
\(193\) −16.3303 −1.17548 −0.587740 0.809050i \(-0.699982\pi\)
−0.587740 + 0.809050i \(0.699982\pi\)
\(194\) −6.74773 −0.484459
\(195\) 0 0
\(196\) −34.7477 −2.48198
\(197\) −20.3303 −1.44847 −0.724237 0.689551i \(-0.757808\pi\)
−0.724237 + 0.689551i \(0.757808\pi\)
\(198\) 13.9564 0.991841
\(199\) 22.5826 1.60084 0.800418 0.599442i \(-0.204611\pi\)
0.800418 + 0.599442i \(0.204611\pi\)
\(200\) 0 0
\(201\) 14.1652 0.999133
\(202\) 23.9564 1.68557
\(203\) −1.00000 −0.0701862
\(204\) −17.3739 −1.21641
\(205\) 0 0
\(206\) 8.83485 0.615553
\(207\) 4.00000 0.278019
\(208\) −82.2867 −5.70556
\(209\) −27.9129 −1.93077
\(210\) 0 0
\(211\) −16.3303 −1.12422 −0.562112 0.827061i \(-0.690011\pi\)
−0.562112 + 0.827061i \(0.690011\pi\)
\(212\) −55.4955 −3.81144
\(213\) 0.417424 0.0286014
\(214\) 36.7477 2.51202
\(215\) 0 0
\(216\) −10.5826 −0.720053
\(217\) −4.00000 −0.271538
\(218\) −11.6261 −0.787421
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −13.7477 −0.924772
\(222\) −11.1652 −0.749356
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) −28.9564 −1.93473
\(225\) 0 0
\(226\) −11.6261 −0.773359
\(227\) −1.58258 −0.105039 −0.0525196 0.998620i \(-0.516725\pi\)
−0.0525196 + 0.998620i \(0.516725\pi\)
\(228\) 32.3303 2.14113
\(229\) 1.16515 0.0769954 0.0384977 0.999259i \(-0.487743\pi\)
0.0384977 + 0.999259i \(0.487743\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) 10.5826 0.694780
\(233\) −5.16515 −0.338380 −0.169190 0.985583i \(-0.554115\pi\)
−0.169190 + 0.985583i \(0.554115\pi\)
\(234\) −12.7913 −0.836193
\(235\) 0 0
\(236\) 9.16515 0.596601
\(237\) 1.58258 0.102799
\(238\) −8.37386 −0.542797
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) −7.33030 −0.472186 −0.236093 0.971730i \(-0.575867\pi\)
−0.236093 + 0.971730i \(0.575867\pi\)
\(242\) 39.0780 2.51203
\(243\) −1.00000 −0.0641500
\(244\) −85.4083 −5.46771
\(245\) 0 0
\(246\) −25.5826 −1.63109
\(247\) 25.5826 1.62778
\(248\) 42.3303 2.68798
\(249\) −2.41742 −0.153198
\(250\) 0 0
\(251\) −2.16515 −0.136663 −0.0683316 0.997663i \(-0.521768\pi\)
−0.0683316 + 0.997663i \(0.521768\pi\)
\(252\) −5.79129 −0.364817
\(253\) 20.0000 1.25739
\(254\) −5.58258 −0.350282
\(255\) 0 0
\(256\) 98.4519 6.15324
\(257\) 14.7477 0.919938 0.459969 0.887935i \(-0.347861\pi\)
0.459969 + 0.887935i \(0.347861\pi\)
\(258\) −1.16515 −0.0725392
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −41.8693 −2.58670
\(263\) 6.33030 0.390343 0.195172 0.980769i \(-0.437474\pi\)
0.195172 + 0.980769i \(0.437474\pi\)
\(264\) −52.9129 −3.25656
\(265\) 0 0
\(266\) 15.5826 0.955429
\(267\) −10.5826 −0.647643
\(268\) −82.0345 −5.01105
\(269\) 13.4174 0.818075 0.409037 0.912518i \(-0.365865\pi\)
0.409037 + 0.912518i \(0.365865\pi\)
\(270\) 0 0
\(271\) −17.1652 −1.04271 −0.521354 0.853340i \(-0.674573\pi\)
−0.521354 + 0.853340i \(0.674573\pi\)
\(272\) 53.8693 3.26631
\(273\) −4.58258 −0.277350
\(274\) 56.7477 3.42826
\(275\) 0 0
\(276\) −23.1652 −1.39438
\(277\) −20.9129 −1.25653 −0.628267 0.777998i \(-0.716235\pi\)
−0.628267 + 0.777998i \(0.716235\pi\)
\(278\) −51.8693 −3.11091
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −9.58258 −0.571649 −0.285824 0.958282i \(-0.592267\pi\)
−0.285824 + 0.958282i \(0.592267\pi\)
\(282\) 3.95644 0.235603
\(283\) 19.1652 1.13925 0.569625 0.821905i \(-0.307088\pi\)
0.569625 + 0.821905i \(0.307088\pi\)
\(284\) −2.41742 −0.143448
\(285\) 0 0
\(286\) −63.9564 −3.78182
\(287\) −9.16515 −0.541002
\(288\) 28.9564 1.70627
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 2.41742 0.141712
\(292\) −23.1652 −1.35564
\(293\) −11.8348 −0.691399 −0.345700 0.938345i \(-0.612358\pi\)
−0.345700 + 0.938345i \(0.612358\pi\)
\(294\) 16.7477 0.976747
\(295\) 0 0
\(296\) 42.3303 2.46040
\(297\) −5.00000 −0.290129
\(298\) −30.0000 −1.73785
\(299\) −18.3303 −1.06007
\(300\) 0 0
\(301\) −0.417424 −0.0240599
\(302\) −31.1652 −1.79335
\(303\) −8.58258 −0.493056
\(304\) −100.243 −5.74934
\(305\) 0 0
\(306\) 8.37386 0.478702
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −28.9564 −1.64995
\(309\) −3.16515 −0.180059
\(310\) 0 0
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) 48.4955 2.74551
\(313\) −1.41742 −0.0801176 −0.0400588 0.999197i \(-0.512755\pi\)
−0.0400588 + 0.999197i \(0.512755\pi\)
\(314\) 46.7477 2.63813
\(315\) 0 0
\(316\) −9.16515 −0.515580
\(317\) 25.0000 1.40414 0.702070 0.712108i \(-0.252259\pi\)
0.702070 + 0.712108i \(0.252259\pi\)
\(318\) 26.7477 1.49994
\(319\) 5.00000 0.279946
\(320\) 0 0
\(321\) −13.1652 −0.734807
\(322\) −11.1652 −0.622210
\(323\) −16.7477 −0.931868
\(324\) 5.79129 0.321738
\(325\) 0 0
\(326\) 4.41742 0.244659
\(327\) 4.16515 0.230333
\(328\) 96.9909 5.35543
\(329\) 1.41742 0.0781451
\(330\) 0 0
\(331\) 28.3303 1.55717 0.778587 0.627537i \(-0.215937\pi\)
0.778587 + 0.627537i \(0.215937\pi\)
\(332\) 14.0000 0.768350
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 0 0
\(336\) 17.9564 0.979604
\(337\) 2.83485 0.154424 0.0772120 0.997015i \(-0.475398\pi\)
0.0772120 + 0.997015i \(0.475398\pi\)
\(338\) 22.3303 1.21461
\(339\) 4.16515 0.226220
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) −15.5826 −0.842609
\(343\) 13.0000 0.701934
\(344\) 4.41742 0.238172
\(345\) 0 0
\(346\) −42.3303 −2.27569
\(347\) 11.1652 0.599377 0.299688 0.954037i \(-0.403117\pi\)
0.299688 + 0.954037i \(0.403117\pi\)
\(348\) −5.79129 −0.310446
\(349\) 32.3303 1.73060 0.865301 0.501253i \(-0.167127\pi\)
0.865301 + 0.501253i \(0.167127\pi\)
\(350\) 0 0
\(351\) 4.58258 0.244600
\(352\) 144.782 7.71692
\(353\) 33.1652 1.76520 0.882601 0.470122i \(-0.155790\pi\)
0.882601 + 0.470122i \(0.155790\pi\)
\(354\) −4.41742 −0.234783
\(355\) 0 0
\(356\) 61.2867 3.24819
\(357\) 3.00000 0.158777
\(358\) −63.4955 −3.35584
\(359\) −8.83485 −0.466285 −0.233143 0.972443i \(-0.574901\pi\)
−0.233143 + 0.972443i \(0.574901\pi\)
\(360\) 0 0
\(361\) 12.1652 0.640271
\(362\) −6.04356 −0.317643
\(363\) −14.0000 −0.734809
\(364\) 26.5390 1.39102
\(365\) 0 0
\(366\) 41.1652 2.15174
\(367\) −17.4955 −0.913255 −0.456628 0.889658i \(-0.650943\pi\)
−0.456628 + 0.889658i \(0.650943\pi\)
\(368\) 71.8258 3.74418
\(369\) 9.16515 0.477119
\(370\) 0 0
\(371\) 9.58258 0.497503
\(372\) −23.1652 −1.20106
\(373\) 8.33030 0.431327 0.215663 0.976468i \(-0.430809\pi\)
0.215663 + 0.976468i \(0.430809\pi\)
\(374\) 41.8693 2.16501
\(375\) 0 0
\(376\) −15.0000 −0.773566
\(377\) −4.58258 −0.236015
\(378\) 2.79129 0.143568
\(379\) −26.0000 −1.33553 −0.667765 0.744372i \(-0.732749\pi\)
−0.667765 + 0.744372i \(0.732749\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −11.1652 −0.571259
\(383\) 5.58258 0.285256 0.142628 0.989776i \(-0.454445\pi\)
0.142628 + 0.989776i \(0.454445\pi\)
\(384\) −67.4519 −3.44214
\(385\) 0 0
\(386\) −45.5826 −2.32009
\(387\) 0.417424 0.0212189
\(388\) −14.0000 −0.710742
\(389\) 2.58258 0.130942 0.0654709 0.997854i \(-0.479145\pi\)
0.0654709 + 0.997854i \(0.479145\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) −63.4955 −3.20700
\(393\) 15.0000 0.756650
\(394\) −56.7477 −2.85891
\(395\) 0 0
\(396\) 28.9564 1.45512
\(397\) −29.1652 −1.46376 −0.731878 0.681435i \(-0.761356\pi\)
−0.731878 + 0.681435i \(0.761356\pi\)
\(398\) 63.0345 3.15963
\(399\) −5.58258 −0.279478
\(400\) 0 0
\(401\) 21.5826 1.07778 0.538891 0.842375i \(-0.318843\pi\)
0.538891 + 0.842375i \(0.318843\pi\)
\(402\) 39.5390 1.97203
\(403\) −18.3303 −0.913097
\(404\) 49.7042 2.47287
\(405\) 0 0
\(406\) −2.79129 −0.138529
\(407\) 20.0000 0.991363
\(408\) −31.7477 −1.57175
\(409\) 24.7477 1.22370 0.611848 0.790975i \(-0.290426\pi\)
0.611848 + 0.790975i \(0.290426\pi\)
\(410\) 0 0
\(411\) −20.3303 −1.00282
\(412\) 18.3303 0.903069
\(413\) −1.58258 −0.0778735
\(414\) 11.1652 0.548737
\(415\) 0 0
\(416\) −132.695 −6.50591
\(417\) 18.5826 0.909993
\(418\) −77.9129 −3.81084
\(419\) −18.8348 −0.920143 −0.460071 0.887882i \(-0.652176\pi\)
−0.460071 + 0.887882i \(0.652176\pi\)
\(420\) 0 0
\(421\) 13.5826 0.661974 0.330987 0.943635i \(-0.392618\pi\)
0.330987 + 0.943635i \(0.392618\pi\)
\(422\) −45.5826 −2.21893
\(423\) −1.41742 −0.0689175
\(424\) −101.408 −4.92482
\(425\) 0 0
\(426\) 1.16515 0.0564518
\(427\) 14.7477 0.713693
\(428\) 76.2432 3.68535
\(429\) 22.9129 1.10624
\(430\) 0 0
\(431\) −20.8348 −1.00358 −0.501790 0.864990i \(-0.667325\pi\)
−0.501790 + 0.864990i \(0.667325\pi\)
\(432\) −17.9564 −0.863930
\(433\) 39.0780 1.87797 0.938985 0.343958i \(-0.111768\pi\)
0.938985 + 0.343958i \(0.111768\pi\)
\(434\) −11.1652 −0.535944
\(435\) 0 0
\(436\) −24.1216 −1.15521
\(437\) −22.3303 −1.06820
\(438\) 11.1652 0.533492
\(439\) 28.9129 1.37994 0.689968 0.723840i \(-0.257624\pi\)
0.689968 + 0.723840i \(0.257624\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −38.3739 −1.82526
\(443\) 8.58258 0.407770 0.203885 0.978995i \(-0.434643\pi\)
0.203885 + 0.978995i \(0.434643\pi\)
\(444\) −23.1652 −1.09937
\(445\) 0 0
\(446\) −19.5390 −0.925199
\(447\) 10.7477 0.508350
\(448\) −44.9129 −2.12193
\(449\) 34.0780 1.60824 0.804121 0.594466i \(-0.202636\pi\)
0.804121 + 0.594466i \(0.202636\pi\)
\(450\) 0 0
\(451\) 45.8258 2.15785
\(452\) −24.1216 −1.13458
\(453\) 11.1652 0.524585
\(454\) −4.41742 −0.207320
\(455\) 0 0
\(456\) 59.0780 2.76658
\(457\) −23.7477 −1.11087 −0.555436 0.831559i \(-0.687449\pi\)
−0.555436 + 0.831559i \(0.687449\pi\)
\(458\) 3.25227 0.151969
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 9.16515 0.426864 0.213432 0.976958i \(-0.431536\pi\)
0.213432 + 0.976958i \(0.431536\pi\)
\(462\) 13.9564 0.649312
\(463\) −18.1652 −0.844206 −0.422103 0.906548i \(-0.638708\pi\)
−0.422103 + 0.906548i \(0.638708\pi\)
\(464\) 17.9564 0.833607
\(465\) 0 0
\(466\) −14.4174 −0.667874
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) −26.5390 −1.22677
\(469\) 14.1652 0.654086
\(470\) 0 0
\(471\) −16.7477 −0.771695
\(472\) 16.7477 0.770877
\(473\) 2.08712 0.0959659
\(474\) 4.41742 0.202899
\(475\) 0 0
\(476\) −17.3739 −0.796330
\(477\) −9.58258 −0.438756
\(478\) −72.5735 −3.31943
\(479\) −0.834849 −0.0381452 −0.0190726 0.999818i \(-0.506071\pi\)
−0.0190726 + 0.999818i \(0.506071\pi\)
\(480\) 0 0
\(481\) −18.3303 −0.835790
\(482\) −20.4610 −0.931972
\(483\) 4.00000 0.182006
\(484\) 81.0780 3.68536
\(485\) 0 0
\(486\) −2.79129 −0.126615
\(487\) 34.3303 1.55565 0.777827 0.628478i \(-0.216322\pi\)
0.777827 + 0.628478i \(0.216322\pi\)
\(488\) −156.069 −7.06491
\(489\) −1.58258 −0.0715665
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) −53.0780 −2.39294
\(493\) 3.00000 0.135113
\(494\) 71.4083 3.21281
\(495\) 0 0
\(496\) 71.8258 3.22507
\(497\) 0.417424 0.0187240
\(498\) −6.74773 −0.302373
\(499\) −22.5826 −1.01093 −0.505467 0.862846i \(-0.668680\pi\)
−0.505467 + 0.862846i \(0.668680\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.04356 −0.269737
\(503\) −22.9129 −1.02163 −0.510817 0.859689i \(-0.670657\pi\)
−0.510817 + 0.859689i \(0.670657\pi\)
\(504\) −10.5826 −0.471385
\(505\) 0 0
\(506\) 55.8258 2.48176
\(507\) −8.00000 −0.355292
\(508\) −11.5826 −0.513894
\(509\) −0.747727 −0.0331424 −0.0165712 0.999863i \(-0.505275\pi\)
−0.0165712 + 0.999863i \(0.505275\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 139.904 6.18293
\(513\) 5.58258 0.246477
\(514\) 41.1652 1.81572
\(515\) 0 0
\(516\) −2.41742 −0.106421
\(517\) −7.08712 −0.311691
\(518\) −11.1652 −0.490569
\(519\) 15.1652 0.665676
\(520\) 0 0
\(521\) −21.0780 −0.923445 −0.461723 0.887024i \(-0.652768\pi\)
−0.461723 + 0.887024i \(0.652768\pi\)
\(522\) 2.79129 0.122171
\(523\) −3.33030 −0.145624 −0.0728120 0.997346i \(-0.523197\pi\)
−0.0728120 + 0.997346i \(0.523197\pi\)
\(524\) −86.8693 −3.79490
\(525\) 0 0
\(526\) 17.6697 0.770435
\(527\) 12.0000 0.522728
\(528\) −89.7822 −3.90727
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 1.58258 0.0686779
\(532\) 32.3303 1.40170
\(533\) −42.0000 −1.81922
\(534\) −29.5390 −1.27828
\(535\) 0 0
\(536\) −149.904 −6.47486
\(537\) 22.7477 0.981637
\(538\) 37.4519 1.61467
\(539\) −30.0000 −1.29219
\(540\) 0 0
\(541\) 23.4955 1.01015 0.505074 0.863076i \(-0.331465\pi\)
0.505074 + 0.863076i \(0.331465\pi\)
\(542\) −47.9129 −2.05803
\(543\) 2.16515 0.0929155
\(544\) 86.8693 3.72449
\(545\) 0 0
\(546\) −12.7913 −0.547417
\(547\) 14.1652 0.605658 0.302829 0.953045i \(-0.402069\pi\)
0.302829 + 0.953045i \(0.402069\pi\)
\(548\) 117.739 5.02955
\(549\) −14.7477 −0.629418
\(550\) 0 0
\(551\) −5.58258 −0.237826
\(552\) −42.3303 −1.80170
\(553\) 1.58258 0.0672980
\(554\) −58.3739 −2.48007
\(555\) 0 0
\(556\) −107.617 −4.56398
\(557\) −4.74773 −0.201168 −0.100584 0.994929i \(-0.532071\pi\)
−0.100584 + 0.994929i \(0.532071\pi\)
\(558\) 11.1652 0.472659
\(559\) −1.91288 −0.0809061
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) −26.7477 −1.12828
\(563\) 5.41742 0.228317 0.114159 0.993463i \(-0.463583\pi\)
0.114159 + 0.993463i \(0.463583\pi\)
\(564\) 8.20871 0.345649
\(565\) 0 0
\(566\) 53.4955 2.24858
\(567\) −1.00000 −0.0419961
\(568\) −4.41742 −0.185351
\(569\) 19.4174 0.814021 0.407010 0.913424i \(-0.366571\pi\)
0.407010 + 0.913424i \(0.366571\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −132.695 −5.54826
\(573\) 4.00000 0.167102
\(574\) −25.5826 −1.06780
\(575\) 0 0
\(576\) 44.9129 1.87137
\(577\) 0.834849 0.0347552 0.0173776 0.999849i \(-0.494468\pi\)
0.0173776 + 0.999849i \(0.494468\pi\)
\(578\) −22.3303 −0.928818
\(579\) 16.3303 0.678664
\(580\) 0 0
\(581\) −2.41742 −0.100292
\(582\) 6.74773 0.279702
\(583\) −47.9129 −1.98435
\(584\) −42.3303 −1.75164
\(585\) 0 0
\(586\) −33.0345 −1.36464
\(587\) 2.41742 0.0997778 0.0498889 0.998755i \(-0.484113\pi\)
0.0498889 + 0.998755i \(0.484113\pi\)
\(588\) 34.7477 1.43297
\(589\) −22.3303 −0.920104
\(590\) 0 0
\(591\) 20.3303 0.836277
\(592\) 71.8258 2.95202
\(593\) −17.5826 −0.722030 −0.361015 0.932560i \(-0.617570\pi\)
−0.361015 + 0.932560i \(0.617570\pi\)
\(594\) −13.9564 −0.572640
\(595\) 0 0
\(596\) −62.2432 −2.54958
\(597\) −22.5826 −0.924243
\(598\) −51.1652 −2.09230
\(599\) −18.1652 −0.742208 −0.371104 0.928591i \(-0.621021\pi\)
−0.371104 + 0.928591i \(0.621021\pi\)
\(600\) 0 0
\(601\) 4.33030 0.176637 0.0883184 0.996092i \(-0.471851\pi\)
0.0883184 + 0.996092i \(0.471851\pi\)
\(602\) −1.16515 −0.0474880
\(603\) −14.1652 −0.576850
\(604\) −64.6606 −2.63100
\(605\) 0 0
\(606\) −23.9564 −0.973164
\(607\) 5.58258 0.226590 0.113295 0.993561i \(-0.463860\pi\)
0.113295 + 0.993561i \(0.463860\pi\)
\(608\) −161.652 −6.55583
\(609\) 1.00000 0.0405220
\(610\) 0 0
\(611\) 6.49545 0.262778
\(612\) 17.3739 0.702297
\(613\) 16.2523 0.656423 0.328212 0.944604i \(-0.393554\pi\)
0.328212 + 0.944604i \(0.393554\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −52.9129 −2.13192
\(617\) 39.4955 1.59003 0.795014 0.606592i \(-0.207464\pi\)
0.795014 + 0.606592i \(0.207464\pi\)
\(618\) −8.83485 −0.355390
\(619\) −5.16515 −0.207605 −0.103802 0.994598i \(-0.533101\pi\)
−0.103802 + 0.994598i \(0.533101\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −8.37386 −0.335761
\(623\) −10.5826 −0.423982
\(624\) 82.2867 3.29411
\(625\) 0 0
\(626\) −3.95644 −0.158131
\(627\) 27.9129 1.11473
\(628\) 96.9909 3.87036
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −34.9129 −1.38986 −0.694930 0.719078i \(-0.744565\pi\)
−0.694930 + 0.719078i \(0.744565\pi\)
\(632\) −16.7477 −0.666189
\(633\) 16.3303 0.649071
\(634\) 69.7822 2.77141
\(635\) 0 0
\(636\) 55.4955 2.20054
\(637\) 27.4955 1.08941
\(638\) 13.9564 0.552541
\(639\) −0.417424 −0.0165131
\(640\) 0 0
\(641\) 2.58258 0.102006 0.0510028 0.998699i \(-0.483758\pi\)
0.0510028 + 0.998699i \(0.483758\pi\)
\(642\) −36.7477 −1.45032
\(643\) 29.6606 1.16970 0.584850 0.811141i \(-0.301153\pi\)
0.584850 + 0.811141i \(0.301153\pi\)
\(644\) −23.1652 −0.912835
\(645\) 0 0
\(646\) −46.7477 −1.83926
\(647\) 2.74773 0.108024 0.0540121 0.998540i \(-0.482799\pi\)
0.0540121 + 0.998540i \(0.482799\pi\)
\(648\) 10.5826 0.415723
\(649\) 7.91288 0.310608
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 9.16515 0.358935
\(653\) −7.83485 −0.306601 −0.153301 0.988180i \(-0.548990\pi\)
−0.153301 + 0.988180i \(0.548990\pi\)
\(654\) 11.6261 0.454618
\(655\) 0 0
\(656\) 164.573 6.42552
\(657\) −4.00000 −0.156055
\(658\) 3.95644 0.154238
\(659\) −0.165151 −0.00643338 −0.00321669 0.999995i \(-0.501024\pi\)
−0.00321669 + 0.999995i \(0.501024\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 79.0780 3.07345
\(663\) 13.7477 0.533917
\(664\) 25.5826 0.992796
\(665\) 0 0
\(666\) 11.1652 0.432641
\(667\) 4.00000 0.154881
\(668\) 0 0
\(669\) 7.00000 0.270636
\(670\) 0 0
\(671\) −73.7386 −2.84665
\(672\) 28.9564 1.11702
\(673\) 25.7477 0.992502 0.496251 0.868179i \(-0.334710\pi\)
0.496251 + 0.868179i \(0.334710\pi\)
\(674\) 7.91288 0.304793
\(675\) 0 0
\(676\) 46.3303 1.78193
\(677\) 46.8258 1.79966 0.899830 0.436241i \(-0.143690\pi\)
0.899830 + 0.436241i \(0.143690\pi\)
\(678\) 11.6261 0.446499
\(679\) 2.41742 0.0927722
\(680\) 0 0
\(681\) 1.58258 0.0606444
\(682\) 55.8258 2.13768
\(683\) 11.1652 0.427223 0.213611 0.976919i \(-0.431477\pi\)
0.213611 + 0.976919i \(0.431477\pi\)
\(684\) −32.3303 −1.23618
\(685\) 0 0
\(686\) 36.2867 1.38543
\(687\) −1.16515 −0.0444533
\(688\) 7.49545 0.285762
\(689\) 43.9129 1.67295
\(690\) 0 0
\(691\) −2.91288 −0.110811 −0.0554056 0.998464i \(-0.517645\pi\)
−0.0554056 + 0.998464i \(0.517645\pi\)
\(692\) −87.8258 −3.33863
\(693\) −5.00000 −0.189934
\(694\) 31.1652 1.18301
\(695\) 0 0
\(696\) −10.5826 −0.401131
\(697\) 27.4955 1.04146
\(698\) 90.2432 3.41575
\(699\) 5.16515 0.195364
\(700\) 0 0
\(701\) 41.0780 1.55150 0.775748 0.631043i \(-0.217373\pi\)
0.775748 + 0.631043i \(0.217373\pi\)
\(702\) 12.7913 0.482776
\(703\) −22.3303 −0.842203
\(704\) 224.564 8.46359
\(705\) 0 0
\(706\) 92.5735 3.48405
\(707\) −8.58258 −0.322781
\(708\) −9.16515 −0.344447
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −1.58258 −0.0593512
\(712\) 111.991 4.19704
\(713\) 16.0000 0.599205
\(714\) 8.37386 0.313384
\(715\) 0 0
\(716\) −131.739 −4.92330
\(717\) 26.0000 0.970988
\(718\) −24.6606 −0.920326
\(719\) −37.9129 −1.41391 −0.706956 0.707258i \(-0.749932\pi\)
−0.706956 + 0.707258i \(0.749932\pi\)
\(720\) 0 0
\(721\) −3.16515 −0.117876
\(722\) 33.9564 1.26373
\(723\) 7.33030 0.272617
\(724\) −12.5390 −0.466009
\(725\) 0 0
\(726\) −39.0780 −1.45032
\(727\) −46.3303 −1.71830 −0.859148 0.511727i \(-0.829006\pi\)
−0.859148 + 0.511727i \(0.829006\pi\)
\(728\) 48.4955 1.79736
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.25227 0.0463170
\(732\) 85.4083 3.15678
\(733\) 47.5826 1.75750 0.878751 0.477280i \(-0.158377\pi\)
0.878751 + 0.477280i \(0.158377\pi\)
\(734\) −48.8348 −1.80253
\(735\) 0 0
\(736\) 115.826 4.26939
\(737\) −70.8258 −2.60890
\(738\) 25.5826 0.941708
\(739\) 46.7477 1.71964 0.859821 0.510595i \(-0.170575\pi\)
0.859821 + 0.510595i \(0.170575\pi\)
\(740\) 0 0
\(741\) −25.5826 −0.939799
\(742\) 26.7477 0.981940
\(743\) −2.25227 −0.0826279 −0.0413139 0.999146i \(-0.513154\pi\)
−0.0413139 + 0.999146i \(0.513154\pi\)
\(744\) −42.3303 −1.55190
\(745\) 0 0
\(746\) 23.2523 0.851326
\(747\) 2.41742 0.0884489
\(748\) 86.8693 3.17626
\(749\) −13.1652 −0.481044
\(750\) 0 0
\(751\) 37.4955 1.36823 0.684114 0.729375i \(-0.260189\pi\)
0.684114 + 0.729375i \(0.260189\pi\)
\(752\) −25.4519 −0.928135
\(753\) 2.16515 0.0789025
\(754\) −12.7913 −0.465831
\(755\) 0 0
\(756\) 5.79129 0.210627
\(757\) −34.3303 −1.24776 −0.623878 0.781522i \(-0.714444\pi\)
−0.623878 + 0.781522i \(0.714444\pi\)
\(758\) −72.5735 −2.63599
\(759\) −20.0000 −0.725954
\(760\) 0 0
\(761\) 45.5826 1.65237 0.826184 0.563401i \(-0.190507\pi\)
0.826184 + 0.563401i \(0.190507\pi\)
\(762\) 5.58258 0.202235
\(763\) 4.16515 0.150789
\(764\) −23.1652 −0.838086
\(765\) 0 0
\(766\) 15.5826 0.563021
\(767\) −7.25227 −0.261864
\(768\) −98.4519 −3.55258
\(769\) 16.4174 0.592027 0.296014 0.955184i \(-0.404343\pi\)
0.296014 + 0.955184i \(0.404343\pi\)
\(770\) 0 0
\(771\) −14.7477 −0.531126
\(772\) −94.5735 −3.40377
\(773\) −13.1652 −0.473518 −0.236759 0.971568i \(-0.576085\pi\)
−0.236759 + 0.971568i \(0.576085\pi\)
\(774\) 1.16515 0.0418805
\(775\) 0 0
\(776\) −25.5826 −0.918361
\(777\) 4.00000 0.143499
\(778\) 7.20871 0.258445
\(779\) −51.1652 −1.83318
\(780\) 0 0
\(781\) −2.08712 −0.0746831
\(782\) 33.4955 1.19779
\(783\) −1.00000 −0.0357371
\(784\) −107.739 −3.84781
\(785\) 0 0
\(786\) 41.8693 1.49343
\(787\) 24.0000 0.855508 0.427754 0.903895i \(-0.359305\pi\)
0.427754 + 0.903895i \(0.359305\pi\)
\(788\) −117.739 −4.19427
\(789\) −6.33030 −0.225365
\(790\) 0 0
\(791\) 4.16515 0.148096
\(792\) 52.9129 1.88018
\(793\) 67.5826 2.39993
\(794\) −81.4083 −2.88907
\(795\) 0 0
\(796\) 130.782 4.63545
\(797\) 4.33030 0.153387 0.0766936 0.997055i \(-0.475564\pi\)
0.0766936 + 0.997055i \(0.475564\pi\)
\(798\) −15.5826 −0.551617
\(799\) −4.25227 −0.150435
\(800\) 0 0
\(801\) 10.5826 0.373917
\(802\) 60.2432 2.12726
\(803\) −20.0000 −0.705785
\(804\) 82.0345 2.89313
\(805\) 0 0
\(806\) −51.1652 −1.80222
\(807\) −13.4174 −0.472316
\(808\) 90.8258 3.19524
\(809\) −20.0780 −0.705906 −0.352953 0.935641i \(-0.614822\pi\)
−0.352953 + 0.935641i \(0.614822\pi\)
\(810\) 0 0
\(811\) 23.4174 0.822297 0.411148 0.911568i \(-0.365128\pi\)
0.411148 + 0.911568i \(0.365128\pi\)
\(812\) −5.79129 −0.203234
\(813\) 17.1652 0.602008
\(814\) 55.8258 1.95669
\(815\) 0 0
\(816\) −53.8693 −1.88580
\(817\) −2.33030 −0.0815270
\(818\) 69.0780 2.41526
\(819\) 4.58258 0.160128
\(820\) 0 0
\(821\) 23.4955 0.819997 0.409999 0.912086i \(-0.365529\pi\)
0.409999 + 0.912086i \(0.365529\pi\)
\(822\) −56.7477 −1.97930
\(823\) 13.0780 0.455871 0.227936 0.973676i \(-0.426802\pi\)
0.227936 + 0.973676i \(0.426802\pi\)
\(824\) 33.4955 1.16687
\(825\) 0 0
\(826\) −4.41742 −0.153702
\(827\) −47.8258 −1.66306 −0.831532 0.555476i \(-0.812536\pi\)
−0.831532 + 0.555476i \(0.812536\pi\)
\(828\) 23.1652 0.805045
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 20.9129 0.725460
\(832\) −205.817 −7.13541
\(833\) −18.0000 −0.623663
\(834\) 51.8693 1.79609
\(835\) 0 0
\(836\) −161.652 −5.59083
\(837\) −4.00000 −0.138260
\(838\) −52.5735 −1.81612
\(839\) 6.49545 0.224248 0.112124 0.993694i \(-0.464235\pi\)
0.112124 + 0.993694i \(0.464235\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 37.9129 1.30656
\(843\) 9.58258 0.330041
\(844\) −94.5735 −3.25535
\(845\) 0 0
\(846\) −3.95644 −0.136025
\(847\) −14.0000 −0.481046
\(848\) −172.069 −5.90887
\(849\) −19.1652 −0.657746
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 2.41742 0.0828196
\(853\) 6.74773 0.231038 0.115519 0.993305i \(-0.463147\pi\)
0.115519 + 0.993305i \(0.463147\pi\)
\(854\) 41.1652 1.40864
\(855\) 0 0
\(856\) 139.321 4.76190
\(857\) −26.6606 −0.910709 −0.455354 0.890310i \(-0.650487\pi\)
−0.455354 + 0.890310i \(0.650487\pi\)
\(858\) 63.9564 2.18344
\(859\) −38.4174 −1.31079 −0.655393 0.755288i \(-0.727497\pi\)
−0.655393 + 0.755288i \(0.727497\pi\)
\(860\) 0 0
\(861\) 9.16515 0.312348
\(862\) −58.1561 −1.98080
\(863\) 15.5826 0.530437 0.265219 0.964188i \(-0.414556\pi\)
0.265219 + 0.964188i \(0.414556\pi\)
\(864\) −28.9564 −0.985118
\(865\) 0 0
\(866\) 109.078 3.70662
\(867\) 8.00000 0.271694
\(868\) −23.1652 −0.786276
\(869\) −7.91288 −0.268426
\(870\) 0 0
\(871\) 64.9129 2.19949
\(872\) −44.0780 −1.49267
\(873\) −2.41742 −0.0818174
\(874\) −62.3303 −2.10835
\(875\) 0 0
\(876\) 23.1652 0.782678
\(877\) −56.3303 −1.90214 −0.951070 0.308977i \(-0.900013\pi\)
−0.951070 + 0.308977i \(0.900013\pi\)
\(878\) 80.7042 2.72363
\(879\) 11.8348 0.399180
\(880\) 0 0
\(881\) 32.0780 1.08074 0.540368 0.841429i \(-0.318285\pi\)
0.540368 + 0.841429i \(0.318285\pi\)
\(882\) −16.7477 −0.563925
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) −79.6170 −2.67781
\(885\) 0 0
\(886\) 23.9564 0.804832
\(887\) −0.912878 −0.0306515 −0.0153257 0.999883i \(-0.504879\pi\)
−0.0153257 + 0.999883i \(0.504879\pi\)
\(888\) −42.3303 −1.42051
\(889\) 2.00000 0.0670778
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) −40.5390 −1.35735
\(893\) 7.91288 0.264794
\(894\) 30.0000 1.00335
\(895\) 0 0
\(896\) −67.4519 −2.25341
\(897\) 18.3303 0.612031
\(898\) 95.1216 3.17425
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −28.7477 −0.957726
\(902\) 127.913 4.25903
\(903\) 0.417424 0.0138910
\(904\) −44.0780 −1.46601
\(905\) 0 0
\(906\) 31.1652 1.03539
\(907\) −14.4174 −0.478723 −0.239361 0.970931i \(-0.576938\pi\)
−0.239361 + 0.970931i \(0.576938\pi\)
\(908\) −9.16515 −0.304156
\(909\) 8.58258 0.284666
\(910\) 0 0
\(911\) 40.8258 1.35262 0.676309 0.736618i \(-0.263578\pi\)
0.676309 + 0.736618i \(0.263578\pi\)
\(912\) 100.243 3.31938
\(913\) 12.0871 0.400025
\(914\) −66.2867 −2.19257
\(915\) 0 0
\(916\) 6.74773 0.222951
\(917\) 15.0000 0.495344
\(918\) −8.37386 −0.276379
\(919\) −26.9129 −0.887774 −0.443887 0.896083i \(-0.646401\pi\)
−0.443887 + 0.896083i \(0.646401\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 25.5826 0.842517
\(923\) 1.91288 0.0629632
\(924\) 28.9564 0.952597
\(925\) 0 0
\(926\) −50.7042 −1.66624
\(927\) 3.16515 0.103957
\(928\) 28.9564 0.950542
\(929\) 52.3303 1.71690 0.858451 0.512896i \(-0.171427\pi\)
0.858451 + 0.512896i \(0.171427\pi\)
\(930\) 0 0
\(931\) 33.4955 1.09777
\(932\) −29.9129 −0.979829
\(933\) 3.00000 0.0982156
\(934\) 22.3303 0.730670
\(935\) 0 0
\(936\) −48.4955 −1.58512
\(937\) 9.41742 0.307654 0.153827 0.988098i \(-0.450840\pi\)
0.153827 + 0.988098i \(0.450840\pi\)
\(938\) 39.5390 1.29099
\(939\) 1.41742 0.0462559
\(940\) 0 0
\(941\) −25.9129 −0.844736 −0.422368 0.906425i \(-0.638801\pi\)
−0.422368 + 0.906425i \(0.638801\pi\)
\(942\) −46.7477 −1.52312
\(943\) 36.6606 1.19383
\(944\) 28.4174 0.924908
\(945\) 0 0
\(946\) 5.82576 0.189412
\(947\) 9.08712 0.295292 0.147646 0.989040i \(-0.452830\pi\)
0.147646 + 0.989040i \(0.452830\pi\)
\(948\) 9.16515 0.297670
\(949\) 18.3303 0.595027
\(950\) 0 0
\(951\) −25.0000 −0.810681
\(952\) −31.7477 −1.02895
\(953\) 28.4174 0.920531 0.460265 0.887781i \(-0.347754\pi\)
0.460265 + 0.887781i \(0.347754\pi\)
\(954\) −26.7477 −0.865990
\(955\) 0 0
\(956\) −150.573 −4.86989
\(957\) −5.00000 −0.161627
\(958\) −2.33030 −0.0752887
\(959\) −20.3303 −0.656500
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −51.1652 −1.64963
\(963\) 13.1652 0.424241
\(964\) −42.4519 −1.36728
\(965\) 0 0
\(966\) 11.1652 0.359233
\(967\) −36.7477 −1.18173 −0.590864 0.806771i \(-0.701213\pi\)
−0.590864 + 0.806771i \(0.701213\pi\)
\(968\) 148.156 4.76192
\(969\) 16.7477 0.538015
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) −5.79129 −0.185756
\(973\) 18.5826 0.595730
\(974\) 95.8258 3.07046
\(975\) 0 0
\(976\) −264.817 −8.47657
\(977\) 57.4955 1.83944 0.919721 0.392572i \(-0.128415\pi\)
0.919721 + 0.392572i \(0.128415\pi\)
\(978\) −4.41742 −0.141254
\(979\) 52.9129 1.69110
\(980\) 0 0
\(981\) −4.16515 −0.132983
\(982\) 44.6606 1.42518
\(983\) −36.8348 −1.17485 −0.587425 0.809279i \(-0.699858\pi\)
−0.587425 + 0.809279i \(0.699858\pi\)
\(984\) −96.9909 −3.09196
\(985\) 0 0
\(986\) 8.37386 0.266678
\(987\) −1.41742 −0.0451171
\(988\) 148.156 4.71347
\(989\) 1.66970 0.0530933
\(990\) 0 0
\(991\) −48.0780 −1.52725 −0.763624 0.645661i \(-0.776582\pi\)
−0.763624 + 0.645661i \(0.776582\pi\)
\(992\) 115.826 3.67747
\(993\) −28.3303 −0.899035
\(994\) 1.16515 0.0369564
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) 19.1652 0.606966 0.303483 0.952837i \(-0.401850\pi\)
0.303483 + 0.952837i \(0.401850\pi\)
\(998\) −63.0345 −1.99532
\(999\) −4.00000 −0.126554
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 2175.2.a.r.1.2 2
3.2 odd 2 6525.2.a.t.1.1 2
5.2 odd 4 2175.2.c.f.349.4 4
5.3 odd 4 2175.2.c.f.349.1 4
5.4 even 2 435.2.a.f.1.1 2
15.14 odd 2 1305.2.a.m.1.2 2
20.19 odd 2 6960.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.1 2 5.4 even 2
1305.2.a.m.1.2 2 15.14 odd 2
2175.2.a.r.1.2 2 1.1 even 1 trivial
2175.2.c.f.349.1 4 5.3 odd 4
2175.2.c.f.349.4 4 5.2 odd 4
6525.2.a.t.1.1 2 3.2 odd 2
6960.2.a.bw.1.2 2 20.19 odd 2