Properties

Label 5082.2.a.bu.1.1
Level $5082$
Weight $2$
Character 5082.1
Self dual yes
Analytic conductor $40.580$
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] = [5082,2,Mod(1,5082)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5082, 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("5082.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5082.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: \(40.5799743072\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 462)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 5082.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.46410 q^{10} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -3.46410 q^{15} +1.00000 q^{16} -3.46410 q^{17} +1.00000 q^{18} -5.46410 q^{19} -3.46410 q^{20} -1.00000 q^{21} +6.92820 q^{23} +1.00000 q^{24} +7.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +6.00000 q^{29} -3.46410 q^{30} +5.46410 q^{31} +1.00000 q^{32} -3.46410 q^{34} +3.46410 q^{35} +1.00000 q^{36} -4.92820 q^{37} -5.46410 q^{38} -2.00000 q^{39} -3.46410 q^{40} -3.46410 q^{41} -1.00000 q^{42} +10.9282 q^{43} -3.46410 q^{45} +6.92820 q^{46} +9.46410 q^{47} +1.00000 q^{48} +1.00000 q^{49} +7.00000 q^{50} -3.46410 q^{51} -2.00000 q^{52} -0.928203 q^{53} +1.00000 q^{54} -1.00000 q^{56} -5.46410 q^{57} +6.00000 q^{58} +6.92820 q^{59} -3.46410 q^{60} -2.00000 q^{61} +5.46410 q^{62} -1.00000 q^{63} +1.00000 q^{64} +6.92820 q^{65} +14.9282 q^{67} -3.46410 q^{68} +6.92820 q^{69} +3.46410 q^{70} -12.0000 q^{71} +1.00000 q^{72} +0.535898 q^{73} -4.92820 q^{74} +7.00000 q^{75} -5.46410 q^{76} -2.00000 q^{78} +10.9282 q^{79} -3.46410 q^{80} +1.00000 q^{81} -3.46410 q^{82} -4.39230 q^{83} -1.00000 q^{84} +12.0000 q^{85} +10.9282 q^{86} +6.00000 q^{87} -0.928203 q^{89} -3.46410 q^{90} +2.00000 q^{91} +6.92820 q^{92} +5.46410 q^{93} +9.46410 q^{94} +18.9282 q^{95} +1.00000 q^{96} +2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{18} - 4 q^{19} - 2 q^{21} + 2 q^{24} + 14 q^{25} - 4 q^{26} + 2 q^{27} - 2 q^{28} + 12 q^{29} + 4 q^{31} + 2 q^{32} + 2 q^{36} + 4 q^{37} - 4 q^{38} - 4 q^{39} - 2 q^{42} + 8 q^{43} + 12 q^{47} + 2 q^{48} + 2 q^{49} + 14 q^{50} - 4 q^{52} + 12 q^{53} + 2 q^{54} - 2 q^{56} - 4 q^{57} + 12 q^{58} - 4 q^{61} + 4 q^{62} - 2 q^{63} + 2 q^{64} + 16 q^{67} - 24 q^{71} + 2 q^{72} + 8 q^{73} + 4 q^{74} + 14 q^{75} - 4 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 2 q^{84} + 24 q^{85} + 8 q^{86} + 12 q^{87} + 12 q^{89} + 4 q^{91} + 4 q^{93} + 12 q^{94} + 24 q^{95} + 2 q^{96} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.46410 −1.09545
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.46410 −0.894427
\(16\) 1.00000 0.250000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 1.00000 0.235702
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) −3.46410 −0.774597
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 1.00000 0.204124
\(25\) 7.00000 1.40000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −3.46410 −0.632456
\(31\) 5.46410 0.981382 0.490691 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.46410 −0.594089
\(35\) 3.46410 0.585540
\(36\) 1.00000 0.166667
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) −5.46410 −0.886394
\(39\) −2.00000 −0.320256
\(40\) −3.46410 −0.547723
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) −1.00000 −0.154303
\(43\) 10.9282 1.66654 0.833268 0.552870i \(-0.186467\pi\)
0.833268 + 0.552870i \(0.186467\pi\)
\(44\) 0 0
\(45\) −3.46410 −0.516398
\(46\) 6.92820 1.02151
\(47\) 9.46410 1.38048 0.690241 0.723580i \(-0.257505\pi\)
0.690241 + 0.723580i \(0.257505\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 7.00000 0.989949
\(51\) −3.46410 −0.485071
\(52\) −2.00000 −0.277350
\(53\) −0.928203 −0.127499 −0.0637493 0.997966i \(-0.520306\pi\)
−0.0637493 + 0.997966i \(0.520306\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −5.46410 −0.723738
\(58\) 6.00000 0.787839
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) −3.46410 −0.447214
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 5.46410 0.693942
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 6.92820 0.859338
\(66\) 0 0
\(67\) 14.9282 1.82377 0.911885 0.410445i \(-0.134627\pi\)
0.911885 + 0.410445i \(0.134627\pi\)
\(68\) −3.46410 −0.420084
\(69\) 6.92820 0.834058
\(70\) 3.46410 0.414039
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 0.535898 0.0627222 0.0313611 0.999508i \(-0.490016\pi\)
0.0313611 + 0.999508i \(0.490016\pi\)
\(74\) −4.92820 −0.572892
\(75\) 7.00000 0.808290
\(76\) −5.46410 −0.626775
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 10.9282 1.22952 0.614759 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) −3.46410 −0.387298
\(81\) 1.00000 0.111111
\(82\) −3.46410 −0.382546
\(83\) −4.39230 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(84\) −1.00000 −0.109109
\(85\) 12.0000 1.30158
\(86\) 10.9282 1.17842
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) −3.46410 −0.365148
\(91\) 2.00000 0.209657
\(92\) 6.92820 0.722315
\(93\) 5.46410 0.566601
\(94\) 9.46410 0.976148
\(95\) 18.9282 1.94199
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 7.00000 0.700000
\(101\) 19.8564 1.97579 0.987893 0.155136i \(-0.0495815\pi\)
0.987893 + 0.155136i \(0.0495815\pi\)
\(102\) −3.46410 −0.342997
\(103\) −13.4641 −1.32666 −0.663329 0.748328i \(-0.730857\pi\)
−0.663329 + 0.748328i \(0.730857\pi\)
\(104\) −2.00000 −0.196116
\(105\) 3.46410 0.338062
\(106\) −0.928203 −0.0901551
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 1.00000 0.0962250
\(109\) −15.8564 −1.51877 −0.759384 0.650643i \(-0.774500\pi\)
−0.759384 + 0.650643i \(0.774500\pi\)
\(110\) 0 0
\(111\) −4.92820 −0.467764
\(112\) −1.00000 −0.0944911
\(113\) 7.85641 0.739069 0.369534 0.929217i \(-0.379517\pi\)
0.369534 + 0.929217i \(0.379517\pi\)
\(114\) −5.46410 −0.511760
\(115\) −24.0000 −2.23801
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 6.92820 0.637793
\(119\) 3.46410 0.317554
\(120\) −3.46410 −0.316228
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) −3.46410 −0.312348
\(124\) 5.46410 0.490691
\(125\) −6.92820 −0.619677
\(126\) −1.00000 −0.0890871
\(127\) 10.9282 0.969721 0.484861 0.874591i \(-0.338870\pi\)
0.484861 + 0.874591i \(0.338870\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.9282 0.962175
\(130\) 6.92820 0.607644
\(131\) 4.39230 0.383757 0.191879 0.981419i \(-0.438542\pi\)
0.191879 + 0.981419i \(0.438542\pi\)
\(132\) 0 0
\(133\) 5.46410 0.473798
\(134\) 14.9282 1.28960
\(135\) −3.46410 −0.298142
\(136\) −3.46410 −0.297044
\(137\) 7.85641 0.671218 0.335609 0.942001i \(-0.391058\pi\)
0.335609 + 0.942001i \(0.391058\pi\)
\(138\) 6.92820 0.589768
\(139\) 13.4641 1.14201 0.571005 0.820947i \(-0.306554\pi\)
0.571005 + 0.820947i \(0.306554\pi\)
\(140\) 3.46410 0.292770
\(141\) 9.46410 0.797021
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −20.7846 −1.72607
\(146\) 0.535898 0.0443513
\(147\) 1.00000 0.0824786
\(148\) −4.92820 −0.405096
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 7.00000 0.571548
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −5.46410 −0.443197
\(153\) −3.46410 −0.280056
\(154\) 0 0
\(155\) −18.9282 −1.52035
\(156\) −2.00000 −0.160128
\(157\) −2.39230 −0.190927 −0.0954634 0.995433i \(-0.530433\pi\)
−0.0954634 + 0.995433i \(0.530433\pi\)
\(158\) 10.9282 0.869401
\(159\) −0.928203 −0.0736113
\(160\) −3.46410 −0.273861
\(161\) −6.92820 −0.546019
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −3.46410 −0.270501
\(165\) 0 0
\(166\) −4.39230 −0.340909
\(167\) −18.9282 −1.46471 −0.732354 0.680924i \(-0.761578\pi\)
−0.732354 + 0.680924i \(0.761578\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 12.0000 0.920358
\(171\) −5.46410 −0.417850
\(172\) 10.9282 0.833268
\(173\) 0.928203 0.0705700 0.0352850 0.999377i \(-0.488766\pi\)
0.0352850 + 0.999377i \(0.488766\pi\)
\(174\) 6.00000 0.454859
\(175\) −7.00000 −0.529150
\(176\) 0 0
\(177\) 6.92820 0.520756
\(178\) −0.928203 −0.0695718
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) −3.46410 −0.258199
\(181\) 6.39230 0.475136 0.237568 0.971371i \(-0.423650\pi\)
0.237568 + 0.971371i \(0.423650\pi\)
\(182\) 2.00000 0.148250
\(183\) −2.00000 −0.147844
\(184\) 6.92820 0.510754
\(185\) 17.0718 1.25514
\(186\) 5.46410 0.400647
\(187\) 0 0
\(188\) 9.46410 0.690241
\(189\) −1.00000 −0.0727393
\(190\) 18.9282 1.37320
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 1.00000 0.0721688
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 2.00000 0.143592
\(195\) 6.92820 0.496139
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 0.392305 0.0278098 0.0139049 0.999903i \(-0.495574\pi\)
0.0139049 + 0.999903i \(0.495574\pi\)
\(200\) 7.00000 0.494975
\(201\) 14.9282 1.05295
\(202\) 19.8564 1.39709
\(203\) −6.00000 −0.421117
\(204\) −3.46410 −0.242536
\(205\) 12.0000 0.838116
\(206\) −13.4641 −0.938088
\(207\) 6.92820 0.481543
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 3.46410 0.239046
\(211\) −16.7846 −1.15550 −0.577750 0.816214i \(-0.696069\pi\)
−0.577750 + 0.816214i \(0.696069\pi\)
\(212\) −0.928203 −0.0637493
\(213\) −12.0000 −0.822226
\(214\) 6.92820 0.473602
\(215\) −37.8564 −2.58179
\(216\) 1.00000 0.0680414
\(217\) −5.46410 −0.370927
\(218\) −15.8564 −1.07393
\(219\) 0.535898 0.0362127
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) −4.92820 −0.330759
\(223\) −8.39230 −0.561990 −0.280995 0.959709i \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.00000 0.466667
\(226\) 7.85641 0.522600
\(227\) 4.39230 0.291528 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(228\) −5.46410 −0.361869
\(229\) −21.3205 −1.40890 −0.704449 0.709754i \(-0.748806\pi\)
−0.704449 + 0.709754i \(0.748806\pi\)
\(230\) −24.0000 −1.58251
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 7.85641 0.514690 0.257345 0.966320i \(-0.417152\pi\)
0.257345 + 0.966320i \(0.417152\pi\)
\(234\) −2.00000 −0.130744
\(235\) −32.7846 −2.13863
\(236\) 6.92820 0.450988
\(237\) 10.9282 0.709863
\(238\) 3.46410 0.224544
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −3.46410 −0.223607
\(241\) −9.60770 −0.618886 −0.309443 0.950918i \(-0.600143\pi\)
−0.309443 + 0.950918i \(0.600143\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) −3.46410 −0.221313
\(246\) −3.46410 −0.220863
\(247\) 10.9282 0.695345
\(248\) 5.46410 0.346971
\(249\) −4.39230 −0.278351
\(250\) −6.92820 −0.438178
\(251\) 30.9282 1.95217 0.976085 0.217387i \(-0.0697534\pi\)
0.976085 + 0.217387i \(0.0697534\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 10.9282 0.685696
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 12.9282 0.806439 0.403220 0.915103i \(-0.367891\pi\)
0.403220 + 0.915103i \(0.367891\pi\)
\(258\) 10.9282 0.680360
\(259\) 4.92820 0.306224
\(260\) 6.92820 0.429669
\(261\) 6.00000 0.371391
\(262\) 4.39230 0.271357
\(263\) 5.07180 0.312740 0.156370 0.987699i \(-0.450021\pi\)
0.156370 + 0.987699i \(0.450021\pi\)
\(264\) 0 0
\(265\) 3.21539 0.197520
\(266\) 5.46410 0.335026
\(267\) −0.928203 −0.0568051
\(268\) 14.9282 0.911885
\(269\) 24.2487 1.47847 0.739235 0.673448i \(-0.235187\pi\)
0.739235 + 0.673448i \(0.235187\pi\)
\(270\) −3.46410 −0.210819
\(271\) 24.7846 1.50556 0.752779 0.658273i \(-0.228713\pi\)
0.752779 + 0.658273i \(0.228713\pi\)
\(272\) −3.46410 −0.210042
\(273\) 2.00000 0.121046
\(274\) 7.85641 0.474623
\(275\) 0 0
\(276\) 6.92820 0.417029
\(277\) 11.8564 0.712382 0.356191 0.934413i \(-0.384075\pi\)
0.356191 + 0.934413i \(0.384075\pi\)
\(278\) 13.4641 0.807523
\(279\) 5.46410 0.327127
\(280\) 3.46410 0.207020
\(281\) 7.85641 0.468674 0.234337 0.972155i \(-0.424708\pi\)
0.234337 + 0.972155i \(0.424708\pi\)
\(282\) 9.46410 0.563579
\(283\) 13.4641 0.800358 0.400179 0.916437i \(-0.368948\pi\)
0.400179 + 0.916437i \(0.368948\pi\)
\(284\) −12.0000 −0.712069
\(285\) 18.9282 1.12121
\(286\) 0 0
\(287\) 3.46410 0.204479
\(288\) 1.00000 0.0589256
\(289\) −5.00000 −0.294118
\(290\) −20.7846 −1.22051
\(291\) 2.00000 0.117242
\(292\) 0.535898 0.0313611
\(293\) 24.9282 1.45632 0.728161 0.685407i \(-0.240375\pi\)
0.728161 + 0.685407i \(0.240375\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) −4.92820 −0.286446
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −13.8564 −0.801337
\(300\) 7.00000 0.404145
\(301\) −10.9282 −0.629891
\(302\) 16.0000 0.920697
\(303\) 19.8564 1.14072
\(304\) −5.46410 −0.313388
\(305\) 6.92820 0.396708
\(306\) −3.46410 −0.198030
\(307\) −10.5359 −0.601315 −0.300658 0.953732i \(-0.597206\pi\)
−0.300658 + 0.953732i \(0.597206\pi\)
\(308\) 0 0
\(309\) −13.4641 −0.765946
\(310\) −18.9282 −1.07505
\(311\) 28.3923 1.60998 0.804990 0.593288i \(-0.202171\pi\)
0.804990 + 0.593288i \(0.202171\pi\)
\(312\) −2.00000 −0.113228
\(313\) −16.9282 −0.956839 −0.478419 0.878132i \(-0.658790\pi\)
−0.478419 + 0.878132i \(0.658790\pi\)
\(314\) −2.39230 −0.135006
\(315\) 3.46410 0.195180
\(316\) 10.9282 0.614759
\(317\) 12.9282 0.726120 0.363060 0.931766i \(-0.381732\pi\)
0.363060 + 0.931766i \(0.381732\pi\)
\(318\) −0.928203 −0.0520511
\(319\) 0 0
\(320\) −3.46410 −0.193649
\(321\) 6.92820 0.386695
\(322\) −6.92820 −0.386094
\(323\) 18.9282 1.05319
\(324\) 1.00000 0.0555556
\(325\) −14.0000 −0.776580
\(326\) −4.00000 −0.221540
\(327\) −15.8564 −0.876861
\(328\) −3.46410 −0.191273
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −9.07180 −0.498631 −0.249316 0.968422i \(-0.580206\pi\)
−0.249316 + 0.968422i \(0.580206\pi\)
\(332\) −4.39230 −0.241059
\(333\) −4.92820 −0.270064
\(334\) −18.9282 −1.03571
\(335\) −51.7128 −2.82537
\(336\) −1.00000 −0.0545545
\(337\) −20.9282 −1.14003 −0.570016 0.821634i \(-0.693063\pi\)
−0.570016 + 0.821634i \(0.693063\pi\)
\(338\) −9.00000 −0.489535
\(339\) 7.85641 0.426701
\(340\) 12.0000 0.650791
\(341\) 0 0
\(342\) −5.46410 −0.295465
\(343\) −1.00000 −0.0539949
\(344\) 10.9282 0.589209
\(345\) −24.0000 −1.29212
\(346\) 0.928203 0.0499005
\(347\) 20.7846 1.11578 0.557888 0.829916i \(-0.311612\pi\)
0.557888 + 0.829916i \(0.311612\pi\)
\(348\) 6.00000 0.321634
\(349\) −10.7846 −0.577287 −0.288643 0.957437i \(-0.593204\pi\)
−0.288643 + 0.957437i \(0.593204\pi\)
\(350\) −7.00000 −0.374166
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) 12.9282 0.688099 0.344049 0.938952i \(-0.388201\pi\)
0.344049 + 0.938952i \(0.388201\pi\)
\(354\) 6.92820 0.368230
\(355\) 41.5692 2.20627
\(356\) −0.928203 −0.0491947
\(357\) 3.46410 0.183340
\(358\) −20.7846 −1.09850
\(359\) 5.07180 0.267679 0.133840 0.991003i \(-0.457269\pi\)
0.133840 + 0.991003i \(0.457269\pi\)
\(360\) −3.46410 −0.182574
\(361\) 10.8564 0.571390
\(362\) 6.39230 0.335972
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −1.85641 −0.0971688
\(366\) −2.00000 −0.104542
\(367\) 19.3205 1.00852 0.504261 0.863551i \(-0.331765\pi\)
0.504261 + 0.863551i \(0.331765\pi\)
\(368\) 6.92820 0.361158
\(369\) −3.46410 −0.180334
\(370\) 17.0718 0.887520
\(371\) 0.928203 0.0481899
\(372\) 5.46410 0.283300
\(373\) −29.7128 −1.53847 −0.769236 0.638965i \(-0.779363\pi\)
−0.769236 + 0.638965i \(0.779363\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 9.46410 0.488074
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) −17.8564 −0.917222 −0.458611 0.888637i \(-0.651653\pi\)
−0.458611 + 0.888637i \(0.651653\pi\)
\(380\) 18.9282 0.970996
\(381\) 10.9282 0.559869
\(382\) 20.7846 1.06343
\(383\) 14.5359 0.742750 0.371375 0.928483i \(-0.378886\pi\)
0.371375 + 0.928483i \(0.378886\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) 10.9282 0.555512
\(388\) 2.00000 0.101535
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 6.92820 0.350823
\(391\) −24.0000 −1.21373
\(392\) 1.00000 0.0505076
\(393\) 4.39230 0.221562
\(394\) −18.0000 −0.906827
\(395\) −37.8564 −1.90476
\(396\) 0 0
\(397\) −2.39230 −0.120066 −0.0600332 0.998196i \(-0.519121\pi\)
−0.0600332 + 0.998196i \(0.519121\pi\)
\(398\) 0.392305 0.0196645
\(399\) 5.46410 0.273547
\(400\) 7.00000 0.350000
\(401\) 4.14359 0.206921 0.103461 0.994634i \(-0.467008\pi\)
0.103461 + 0.994634i \(0.467008\pi\)
\(402\) 14.9282 0.744551
\(403\) −10.9282 −0.544373
\(404\) 19.8564 0.987893
\(405\) −3.46410 −0.172133
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) −3.46410 −0.171499
\(409\) 9.32051 0.460869 0.230435 0.973088i \(-0.425985\pi\)
0.230435 + 0.973088i \(0.425985\pi\)
\(410\) 12.0000 0.592638
\(411\) 7.85641 0.387528
\(412\) −13.4641 −0.663329
\(413\) −6.92820 −0.340915
\(414\) 6.92820 0.340503
\(415\) 15.2154 0.746894
\(416\) −2.00000 −0.0980581
\(417\) 13.4641 0.659340
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 3.46410 0.169031
\(421\) −37.7128 −1.83801 −0.919005 0.394246i \(-0.871006\pi\)
−0.919005 + 0.394246i \(0.871006\pi\)
\(422\) −16.7846 −0.817062
\(423\) 9.46410 0.460160
\(424\) −0.928203 −0.0450775
\(425\) −24.2487 −1.17624
\(426\) −12.0000 −0.581402
\(427\) 2.00000 0.0967868
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) −37.8564 −1.82560
\(431\) 18.9282 0.911739 0.455870 0.890047i \(-0.349328\pi\)
0.455870 + 0.890047i \(0.349328\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −5.46410 −0.262285
\(435\) −20.7846 −0.996546
\(436\) −15.8564 −0.759384
\(437\) −37.8564 −1.81092
\(438\) 0.535898 0.0256062
\(439\) 24.7846 1.18290 0.591452 0.806340i \(-0.298555\pi\)
0.591452 + 0.806340i \(0.298555\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 6.92820 0.329541
\(443\) −20.7846 −0.987507 −0.493753 0.869602i \(-0.664375\pi\)
−0.493753 + 0.869602i \(0.664375\pi\)
\(444\) −4.92820 −0.233882
\(445\) 3.21539 0.152424
\(446\) −8.39230 −0.397387
\(447\) 6.00000 0.283790
\(448\) −1.00000 −0.0472456
\(449\) −19.8564 −0.937082 −0.468541 0.883442i \(-0.655220\pi\)
−0.468541 + 0.883442i \(0.655220\pi\)
\(450\) 7.00000 0.329983
\(451\) 0 0
\(452\) 7.85641 0.369534
\(453\) 16.0000 0.751746
\(454\) 4.39230 0.206141
\(455\) −6.92820 −0.324799
\(456\) −5.46410 −0.255880
\(457\) −15.8564 −0.741731 −0.370866 0.928687i \(-0.620939\pi\)
−0.370866 + 0.928687i \(0.620939\pi\)
\(458\) −21.3205 −0.996242
\(459\) −3.46410 −0.161690
\(460\) −24.0000 −1.11901
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) −18.9282 −0.877774
\(466\) 7.85641 0.363941
\(467\) −15.7128 −0.727102 −0.363551 0.931574i \(-0.618436\pi\)
−0.363551 + 0.931574i \(0.618436\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −14.9282 −0.689320
\(470\) −32.7846 −1.51224
\(471\) −2.39230 −0.110232
\(472\) 6.92820 0.318896
\(473\) 0 0
\(474\) 10.9282 0.501949
\(475\) −38.2487 −1.75497
\(476\) 3.46410 0.158777
\(477\) −0.928203 −0.0424995
\(478\) 24.0000 1.09773
\(479\) 13.8564 0.633115 0.316558 0.948573i \(-0.397473\pi\)
0.316558 + 0.948573i \(0.397473\pi\)
\(480\) −3.46410 −0.158114
\(481\) 9.85641 0.449413
\(482\) −9.60770 −0.437619
\(483\) −6.92820 −0.315244
\(484\) 0 0
\(485\) −6.92820 −0.314594
\(486\) 1.00000 0.0453609
\(487\) 40.7846 1.84813 0.924064 0.382239i \(-0.124847\pi\)
0.924064 + 0.382239i \(0.124847\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −4.00000 −0.180886
\(490\) −3.46410 −0.156492
\(491\) 1.85641 0.0837785 0.0418892 0.999122i \(-0.486662\pi\)
0.0418892 + 0.999122i \(0.486662\pi\)
\(492\) −3.46410 −0.156174
\(493\) −20.7846 −0.936092
\(494\) 10.9282 0.491683
\(495\) 0 0
\(496\) 5.46410 0.245345
\(497\) 12.0000 0.538274
\(498\) −4.39230 −0.196824
\(499\) −9.07180 −0.406109 −0.203055 0.979167i \(-0.565087\pi\)
−0.203055 + 0.979167i \(0.565087\pi\)
\(500\) −6.92820 −0.309839
\(501\) −18.9282 −0.845650
\(502\) 30.9282 1.38039
\(503\) −18.9282 −0.843967 −0.421983 0.906604i \(-0.638666\pi\)
−0.421983 + 0.906604i \(0.638666\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −68.7846 −3.06087
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 10.9282 0.484861
\(509\) −41.3205 −1.83150 −0.915750 0.401749i \(-0.868402\pi\)
−0.915750 + 0.401749i \(0.868402\pi\)
\(510\) 12.0000 0.531369
\(511\) −0.535898 −0.0237067
\(512\) 1.00000 0.0441942
\(513\) −5.46410 −0.241246
\(514\) 12.9282 0.570239
\(515\) 46.6410 2.05525
\(516\) 10.9282 0.481087
\(517\) 0 0
\(518\) 4.92820 0.216533
\(519\) 0.928203 0.0407436
\(520\) 6.92820 0.303822
\(521\) 36.9282 1.61785 0.808927 0.587909i \(-0.200049\pi\)
0.808927 + 0.587909i \(0.200049\pi\)
\(522\) 6.00000 0.262613
\(523\) −24.3923 −1.06660 −0.533301 0.845926i \(-0.679048\pi\)
−0.533301 + 0.845926i \(0.679048\pi\)
\(524\) 4.39230 0.191879
\(525\) −7.00000 −0.305505
\(526\) 5.07180 0.221141
\(527\) −18.9282 −0.824525
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 3.21539 0.139668
\(531\) 6.92820 0.300658
\(532\) 5.46410 0.236899
\(533\) 6.92820 0.300094
\(534\) −0.928203 −0.0401673
\(535\) −24.0000 −1.03761
\(536\) 14.9282 0.644800
\(537\) −20.7846 −0.896922
\(538\) 24.2487 1.04544
\(539\) 0 0
\(540\) −3.46410 −0.149071
\(541\) 25.7128 1.10548 0.552740 0.833354i \(-0.313582\pi\)
0.552740 + 0.833354i \(0.313582\pi\)
\(542\) 24.7846 1.06459
\(543\) 6.39230 0.274320
\(544\) −3.46410 −0.148522
\(545\) 54.9282 2.35287
\(546\) 2.00000 0.0855921
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 7.85641 0.335609
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −32.7846 −1.39667
\(552\) 6.92820 0.294884
\(553\) −10.9282 −0.464714
\(554\) 11.8564 0.503730
\(555\) 17.0718 0.724657
\(556\) 13.4641 0.571005
\(557\) −21.7128 −0.920001 −0.460001 0.887919i \(-0.652151\pi\)
−0.460001 + 0.887919i \(0.652151\pi\)
\(558\) 5.46410 0.231314
\(559\) −21.8564 −0.924427
\(560\) 3.46410 0.146385
\(561\) 0 0
\(562\) 7.85641 0.331403
\(563\) 18.2487 0.769091 0.384546 0.923106i \(-0.374358\pi\)
0.384546 + 0.923106i \(0.374358\pi\)
\(564\) 9.46410 0.398511
\(565\) −27.2154 −1.14496
\(566\) 13.4641 0.565938
\(567\) −1.00000 −0.0419961
\(568\) −12.0000 −0.503509
\(569\) 31.8564 1.33549 0.667745 0.744390i \(-0.267260\pi\)
0.667745 + 0.744390i \(0.267260\pi\)
\(570\) 18.9282 0.792815
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 0 0
\(573\) 20.7846 0.868290
\(574\) 3.46410 0.144589
\(575\) 48.4974 2.02248
\(576\) 1.00000 0.0416667
\(577\) −30.7846 −1.28158 −0.640790 0.767716i \(-0.721393\pi\)
−0.640790 + 0.767716i \(0.721393\pi\)
\(578\) −5.00000 −0.207973
\(579\) −26.0000 −1.08052
\(580\) −20.7846 −0.863034
\(581\) 4.39230 0.182224
\(582\) 2.00000 0.0829027
\(583\) 0 0
\(584\) 0.535898 0.0221756
\(585\) 6.92820 0.286446
\(586\) 24.9282 1.02977
\(587\) −20.7846 −0.857873 −0.428936 0.903335i \(-0.641112\pi\)
−0.428936 + 0.903335i \(0.641112\pi\)
\(588\) 1.00000 0.0412393
\(589\) −29.8564 −1.23021
\(590\) −24.0000 −0.988064
\(591\) −18.0000 −0.740421
\(592\) −4.92820 −0.202548
\(593\) 20.5359 0.843308 0.421654 0.906757i \(-0.361450\pi\)
0.421654 + 0.906757i \(0.361450\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 6.00000 0.245770
\(597\) 0.392305 0.0160560
\(598\) −13.8564 −0.566631
\(599\) 1.85641 0.0758507 0.0379254 0.999281i \(-0.487925\pi\)
0.0379254 + 0.999281i \(0.487925\pi\)
\(600\) 7.00000 0.285774
\(601\) −18.3923 −0.750238 −0.375119 0.926977i \(-0.622398\pi\)
−0.375119 + 0.926977i \(0.622398\pi\)
\(602\) −10.9282 −0.445400
\(603\) 14.9282 0.607923
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 19.8564 0.806611
\(607\) 19.7128 0.800118 0.400059 0.916489i \(-0.368990\pi\)
0.400059 + 0.916489i \(0.368990\pi\)
\(608\) −5.46410 −0.221599
\(609\) −6.00000 −0.243132
\(610\) 6.92820 0.280515
\(611\) −18.9282 −0.765753
\(612\) −3.46410 −0.140028
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) −10.5359 −0.425194
\(615\) 12.0000 0.483887
\(616\) 0 0
\(617\) −19.8564 −0.799389 −0.399694 0.916648i \(-0.630884\pi\)
−0.399694 + 0.916648i \(0.630884\pi\)
\(618\) −13.4641 −0.541606
\(619\) −31.7128 −1.27465 −0.637323 0.770597i \(-0.719958\pi\)
−0.637323 + 0.770597i \(0.719958\pi\)
\(620\) −18.9282 −0.760175
\(621\) 6.92820 0.278019
\(622\) 28.3923 1.13843
\(623\) 0.928203 0.0371877
\(624\) −2.00000 −0.0800641
\(625\) −11.0000 −0.440000
\(626\) −16.9282 −0.676587
\(627\) 0 0
\(628\) −2.39230 −0.0954634
\(629\) 17.0718 0.680697
\(630\) 3.46410 0.138013
\(631\) −0.784610 −0.0312348 −0.0156174 0.999878i \(-0.504971\pi\)
−0.0156174 + 0.999878i \(0.504971\pi\)
\(632\) 10.9282 0.434701
\(633\) −16.7846 −0.667128
\(634\) 12.9282 0.513445
\(635\) −37.8564 −1.50229
\(636\) −0.928203 −0.0368057
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −3.46410 −0.136931
\(641\) 31.8564 1.25825 0.629126 0.777303i \(-0.283413\pi\)
0.629126 + 0.777303i \(0.283413\pi\)
\(642\) 6.92820 0.273434
\(643\) 14.9282 0.588711 0.294355 0.955696i \(-0.404895\pi\)
0.294355 + 0.955696i \(0.404895\pi\)
\(644\) −6.92820 −0.273009
\(645\) −37.8564 −1.49059
\(646\) 18.9282 0.744720
\(647\) 0.679492 0.0267136 0.0133568 0.999911i \(-0.495748\pi\)
0.0133568 + 0.999911i \(0.495748\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −14.0000 −0.549125
\(651\) −5.46410 −0.214155
\(652\) −4.00000 −0.156652
\(653\) −33.7128 −1.31928 −0.659642 0.751580i \(-0.729292\pi\)
−0.659642 + 0.751580i \(0.729292\pi\)
\(654\) −15.8564 −0.620035
\(655\) −15.2154 −0.594514
\(656\) −3.46410 −0.135250
\(657\) 0.535898 0.0209074
\(658\) −9.46410 −0.368949
\(659\) 17.0718 0.665023 0.332511 0.943099i \(-0.392104\pi\)
0.332511 + 0.943099i \(0.392104\pi\)
\(660\) 0 0
\(661\) 44.2487 1.72108 0.860538 0.509387i \(-0.170128\pi\)
0.860538 + 0.509387i \(0.170128\pi\)
\(662\) −9.07180 −0.352585
\(663\) 6.92820 0.269069
\(664\) −4.39230 −0.170454
\(665\) −18.9282 −0.734004
\(666\) −4.92820 −0.190964
\(667\) 41.5692 1.60957
\(668\) −18.9282 −0.732354
\(669\) −8.39230 −0.324465
\(670\) −51.7128 −1.99784
\(671\) 0 0
\(672\) −1.00000 −0.0385758
\(673\) −20.9282 −0.806723 −0.403361 0.915041i \(-0.632158\pi\)
−0.403361 + 0.915041i \(0.632158\pi\)
\(674\) −20.9282 −0.806124
\(675\) 7.00000 0.269430
\(676\) −9.00000 −0.346154
\(677\) −2.78461 −0.107021 −0.0535106 0.998567i \(-0.517041\pi\)
−0.0535106 + 0.998567i \(0.517041\pi\)
\(678\) 7.85641 0.301723
\(679\) −2.00000 −0.0767530
\(680\) 12.0000 0.460179
\(681\) 4.39230 0.168313
\(682\) 0 0
\(683\) 3.21539 0.123033 0.0615167 0.998106i \(-0.480406\pi\)
0.0615167 + 0.998106i \(0.480406\pi\)
\(684\) −5.46410 −0.208925
\(685\) −27.2154 −1.03985
\(686\) −1.00000 −0.0381802
\(687\) −21.3205 −0.813428
\(688\) 10.9282 0.416634
\(689\) 1.85641 0.0707235
\(690\) −24.0000 −0.913664
\(691\) −33.0718 −1.25811 −0.629055 0.777361i \(-0.716558\pi\)
−0.629055 + 0.777361i \(0.716558\pi\)
\(692\) 0.928203 0.0352850
\(693\) 0 0
\(694\) 20.7846 0.788973
\(695\) −46.6410 −1.76919
\(696\) 6.00000 0.227429
\(697\) 12.0000 0.454532
\(698\) −10.7846 −0.408203
\(699\) 7.85641 0.297157
\(700\) −7.00000 −0.264575
\(701\) −45.7128 −1.72655 −0.863275 0.504735i \(-0.831590\pi\)
−0.863275 + 0.504735i \(0.831590\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 26.9282 1.01562
\(704\) 0 0
\(705\) −32.7846 −1.23474
\(706\) 12.9282 0.486559
\(707\) −19.8564 −0.746777
\(708\) 6.92820 0.260378
\(709\) 46.7846 1.75703 0.878516 0.477712i \(-0.158534\pi\)
0.878516 + 0.477712i \(0.158534\pi\)
\(710\) 41.5692 1.56007
\(711\) 10.9282 0.409840
\(712\) −0.928203 −0.0347859
\(713\) 37.8564 1.41773
\(714\) 3.46410 0.129641
\(715\) 0 0
\(716\) −20.7846 −0.776757
\(717\) 24.0000 0.896296
\(718\) 5.07180 0.189278
\(719\) −37.1769 −1.38646 −0.693232 0.720714i \(-0.743814\pi\)
−0.693232 + 0.720714i \(0.743814\pi\)
\(720\) −3.46410 −0.129099
\(721\) 13.4641 0.501429
\(722\) 10.8564 0.404034
\(723\) −9.60770 −0.357314
\(724\) 6.39230 0.237568
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) 33.1769 1.23046 0.615232 0.788346i \(-0.289062\pi\)
0.615232 + 0.788346i \(0.289062\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) −1.85641 −0.0687087
\(731\) −37.8564 −1.40017
\(732\) −2.00000 −0.0739221
\(733\) −12.1436 −0.448534 −0.224267 0.974528i \(-0.571999\pi\)
−0.224267 + 0.974528i \(0.571999\pi\)
\(734\) 19.3205 0.713133
\(735\) −3.46410 −0.127775
\(736\) 6.92820 0.255377
\(737\) 0 0
\(738\) −3.46410 −0.127515
\(739\) −8.00000 −0.294285 −0.147142 0.989115i \(-0.547008\pi\)
−0.147142 + 0.989115i \(0.547008\pi\)
\(740\) 17.0718 0.627572
\(741\) 10.9282 0.401458
\(742\) 0.928203 0.0340754
\(743\) 41.5692 1.52503 0.762513 0.646972i \(-0.223965\pi\)
0.762513 + 0.646972i \(0.223965\pi\)
\(744\) 5.46410 0.200324
\(745\) −20.7846 −0.761489
\(746\) −29.7128 −1.08786
\(747\) −4.39230 −0.160706
\(748\) 0 0
\(749\) −6.92820 −0.253151
\(750\) −6.92820 −0.252982
\(751\) −24.7846 −0.904403 −0.452202 0.891916i \(-0.649361\pi\)
−0.452202 + 0.891916i \(0.649361\pi\)
\(752\) 9.46410 0.345120
\(753\) 30.9282 1.12709
\(754\) −12.0000 −0.437014
\(755\) −55.4256 −2.01715
\(756\) −1.00000 −0.0363696
\(757\) 22.7846 0.828121 0.414060 0.910249i \(-0.364110\pi\)
0.414060 + 0.910249i \(0.364110\pi\)
\(758\) −17.8564 −0.648574
\(759\) 0 0
\(760\) 18.9282 0.686598
\(761\) 25.6077 0.928278 0.464139 0.885762i \(-0.346364\pi\)
0.464139 + 0.885762i \(0.346364\pi\)
\(762\) 10.9282 0.395887
\(763\) 15.8564 0.574040
\(764\) 20.7846 0.751961
\(765\) 12.0000 0.433861
\(766\) 14.5359 0.525203
\(767\) −13.8564 −0.500326
\(768\) 1.00000 0.0360844
\(769\) −28.5359 −1.02903 −0.514515 0.857481i \(-0.672028\pi\)
−0.514515 + 0.857481i \(0.672028\pi\)
\(770\) 0 0
\(771\) 12.9282 0.465598
\(772\) −26.0000 −0.935760
\(773\) −32.5359 −1.17023 −0.585117 0.810949i \(-0.698952\pi\)
−0.585117 + 0.810949i \(0.698952\pi\)
\(774\) 10.9282 0.392806
\(775\) 38.2487 1.37393
\(776\) 2.00000 0.0717958
\(777\) 4.92820 0.176798
\(778\) −30.0000 −1.07555
\(779\) 18.9282 0.678173
\(780\) 6.92820 0.248069
\(781\) 0 0
\(782\) −24.0000 −0.858238
\(783\) 6.00000 0.214423
\(784\) 1.00000 0.0357143
\(785\) 8.28719 0.295782
\(786\) 4.39230 0.156668
\(787\) −33.1769 −1.18263 −0.591315 0.806441i \(-0.701391\pi\)
−0.591315 + 0.806441i \(0.701391\pi\)
\(788\) −18.0000 −0.641223
\(789\) 5.07180 0.180561
\(790\) −37.8564 −1.34687
\(791\) −7.85641 −0.279342
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) −2.39230 −0.0848997
\(795\) 3.21539 0.114038
\(796\) 0.392305 0.0139049
\(797\) −32.5359 −1.15248 −0.576240 0.817280i \(-0.695481\pi\)
−0.576240 + 0.817280i \(0.695481\pi\)
\(798\) 5.46410 0.193427
\(799\) −32.7846 −1.15984
\(800\) 7.00000 0.247487
\(801\) −0.928203 −0.0327964
\(802\) 4.14359 0.146315
\(803\) 0 0
\(804\) 14.9282 0.526477
\(805\) 24.0000 0.845889
\(806\) −10.9282 −0.384930
\(807\) 24.2487 0.853595
\(808\) 19.8564 0.698546
\(809\) −11.0718 −0.389264 −0.194632 0.980876i \(-0.562351\pi\)
−0.194632 + 0.980876i \(0.562351\pi\)
\(810\) −3.46410 −0.121716
\(811\) 17.1769 0.603163 0.301582 0.953440i \(-0.402485\pi\)
0.301582 + 0.953440i \(0.402485\pi\)
\(812\) −6.00000 −0.210559
\(813\) 24.7846 0.869234
\(814\) 0 0
\(815\) 13.8564 0.485369
\(816\) −3.46410 −0.121268
\(817\) −59.7128 −2.08909
\(818\) 9.32051 0.325884
\(819\) 2.00000 0.0698857
\(820\) 12.0000 0.419058
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 7.85641 0.274024
\(823\) 40.7846 1.42166 0.710831 0.703363i \(-0.248319\pi\)
0.710831 + 0.703363i \(0.248319\pi\)
\(824\) −13.4641 −0.469044
\(825\) 0 0
\(826\) −6.92820 −0.241063
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 6.92820 0.240772
\(829\) 44.2487 1.53682 0.768411 0.639957i \(-0.221048\pi\)
0.768411 + 0.639957i \(0.221048\pi\)
\(830\) 15.2154 0.528134
\(831\) 11.8564 0.411294
\(832\) −2.00000 −0.0693375
\(833\) −3.46410 −0.120024
\(834\) 13.4641 0.466224
\(835\) 65.5692 2.26912
\(836\) 0 0
\(837\) 5.46410 0.188867
\(838\) −36.0000 −1.24360
\(839\) −33.4641 −1.15531 −0.577655 0.816281i \(-0.696032\pi\)
−0.577655 + 0.816281i \(0.696032\pi\)
\(840\) 3.46410 0.119523
\(841\) 7.00000 0.241379
\(842\) −37.7128 −1.29967
\(843\) 7.85641 0.270589
\(844\) −16.7846 −0.577750
\(845\) 31.1769 1.07252
\(846\) 9.46410 0.325383
\(847\) 0 0
\(848\) −0.928203 −0.0318746
\(849\) 13.4641 0.462087
\(850\) −24.2487 −0.831724
\(851\) −34.1436 −1.17043
\(852\) −12.0000 −0.411113
\(853\) 49.7128 1.70213 0.851067 0.525057i \(-0.175956\pi\)
0.851067 + 0.525057i \(0.175956\pi\)
\(854\) 2.00000 0.0684386
\(855\) 18.9282 0.647331
\(856\) 6.92820 0.236801
\(857\) 5.32051 0.181745 0.0908725 0.995863i \(-0.471034\pi\)
0.0908725 + 0.995863i \(0.471034\pi\)
\(858\) 0 0
\(859\) 52.7846 1.80099 0.900494 0.434869i \(-0.143205\pi\)
0.900494 + 0.434869i \(0.143205\pi\)
\(860\) −37.8564 −1.29089
\(861\) 3.46410 0.118056
\(862\) 18.9282 0.644697
\(863\) 44.7846 1.52449 0.762243 0.647291i \(-0.224098\pi\)
0.762243 + 0.647291i \(0.224098\pi\)
\(864\) 1.00000 0.0340207
\(865\) −3.21539 −0.109327
\(866\) 2.00000 0.0679628
\(867\) −5.00000 −0.169809
\(868\) −5.46410 −0.185464
\(869\) 0 0
\(870\) −20.7846 −0.704664
\(871\) −29.8564 −1.01165
\(872\) −15.8564 −0.536966
\(873\) 2.00000 0.0676897
\(874\) −37.8564 −1.28051
\(875\) 6.92820 0.234216
\(876\) 0.535898 0.0181063
\(877\) 46.0000 1.55331 0.776655 0.629926i \(-0.216915\pi\)
0.776655 + 0.629926i \(0.216915\pi\)
\(878\) 24.7846 0.836440
\(879\) 24.9282 0.840807
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 1.00000 0.0336718
\(883\) −41.8564 −1.40858 −0.704290 0.709912i \(-0.748735\pi\)
−0.704290 + 0.709912i \(0.748735\pi\)
\(884\) 6.92820 0.233021
\(885\) −24.0000 −0.806751
\(886\) −20.7846 −0.698273
\(887\) −10.1436 −0.340589 −0.170294 0.985393i \(-0.554472\pi\)
−0.170294 + 0.985393i \(0.554472\pi\)
\(888\) −4.92820 −0.165380
\(889\) −10.9282 −0.366520
\(890\) 3.21539 0.107780
\(891\) 0 0
\(892\) −8.39230 −0.280995
\(893\) −51.7128 −1.73050
\(894\) 6.00000 0.200670
\(895\) 72.0000 2.40669
\(896\) −1.00000 −0.0334077
\(897\) −13.8564 −0.462652
\(898\) −19.8564 −0.662617
\(899\) 32.7846 1.09343
\(900\) 7.00000 0.233333
\(901\) 3.21539 0.107120
\(902\) 0 0
\(903\) −10.9282 −0.363668
\(904\) 7.85641 0.261300
\(905\) −22.1436 −0.736078
\(906\) 16.0000 0.531564
\(907\) 14.9282 0.495683 0.247841 0.968801i \(-0.420279\pi\)
0.247841 + 0.968801i \(0.420279\pi\)
\(908\) 4.39230 0.145764
\(909\) 19.8564 0.658595
\(910\) −6.92820 −0.229668
\(911\) −48.4974 −1.60679 −0.803396 0.595446i \(-0.796976\pi\)
−0.803396 + 0.595446i \(0.796976\pi\)
\(912\) −5.46410 −0.180934
\(913\) 0 0
\(914\) −15.8564 −0.524483
\(915\) 6.92820 0.229039
\(916\) −21.3205 −0.704449
\(917\) −4.39230 −0.145047
\(918\) −3.46410 −0.114332
\(919\) −21.8564 −0.720976 −0.360488 0.932764i \(-0.617390\pi\)
−0.360488 + 0.932764i \(0.617390\pi\)
\(920\) −24.0000 −0.791257
\(921\) −10.5359 −0.347170
\(922\) 6.00000 0.197599
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) −34.4974 −1.13427
\(926\) 8.00000 0.262896
\(927\) −13.4641 −0.442219
\(928\) 6.00000 0.196960
\(929\) 40.6410 1.33339 0.666694 0.745331i \(-0.267709\pi\)
0.666694 + 0.745331i \(0.267709\pi\)
\(930\) −18.9282 −0.620680
\(931\) −5.46410 −0.179079
\(932\) 7.85641 0.257345
\(933\) 28.3923 0.929522
\(934\) −15.7128 −0.514139
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 14.3923 0.470176 0.235088 0.971974i \(-0.424462\pi\)
0.235088 + 0.971974i \(0.424462\pi\)
\(938\) −14.9282 −0.487423
\(939\) −16.9282 −0.552431
\(940\) −32.7846 −1.06932
\(941\) 11.0718 0.360930 0.180465 0.983581i \(-0.442240\pi\)
0.180465 + 0.983581i \(0.442240\pi\)
\(942\) −2.39230 −0.0779455
\(943\) −24.0000 −0.781548
\(944\) 6.92820 0.225494
\(945\) 3.46410 0.112687
\(946\) 0 0
\(947\) 49.8564 1.62012 0.810058 0.586350i \(-0.199436\pi\)
0.810058 + 0.586350i \(0.199436\pi\)
\(948\) 10.9282 0.354932
\(949\) −1.07180 −0.0347920
\(950\) −38.2487 −1.24095
\(951\) 12.9282 0.419226
\(952\) 3.46410 0.112272
\(953\) 26.7846 0.867639 0.433819 0.901000i \(-0.357166\pi\)
0.433819 + 0.901000i \(0.357166\pi\)
\(954\) −0.928203 −0.0300517
\(955\) −72.0000 −2.32987
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 13.8564 0.447680
\(959\) −7.85641 −0.253697
\(960\) −3.46410 −0.111803
\(961\) −1.14359 −0.0368901
\(962\) 9.85641 0.317783
\(963\) 6.92820 0.223258
\(964\) −9.60770 −0.309443
\(965\) 90.0666 2.89935
\(966\) −6.92820 −0.222911
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 18.9282 0.608061
\(970\) −6.92820 −0.222451
\(971\) −15.7128 −0.504248 −0.252124 0.967695i \(-0.581129\pi\)
−0.252124 + 0.967695i \(0.581129\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.4641 −0.431639
\(974\) 40.7846 1.30682
\(975\) −14.0000 −0.448359
\(976\) −2.00000 −0.0640184
\(977\) −47.5692 −1.52187 −0.760937 0.648826i \(-0.775260\pi\)
−0.760937 + 0.648826i \(0.775260\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) −3.46410 −0.110657
\(981\) −15.8564 −0.506256
\(982\) 1.85641 0.0592403
\(983\) 23.3205 0.743809 0.371904 0.928271i \(-0.378705\pi\)
0.371904 + 0.928271i \(0.378705\pi\)
\(984\) −3.46410 −0.110432
\(985\) 62.3538 1.98676
\(986\) −20.7846 −0.661917
\(987\) −9.46410 −0.301246
\(988\) 10.9282 0.347672
\(989\) 75.7128 2.40753
\(990\) 0 0
\(991\) −2.14359 −0.0680935 −0.0340467 0.999420i \(-0.510840\pi\)
−0.0340467 + 0.999420i \(0.510840\pi\)
\(992\) 5.46410 0.173485
\(993\) −9.07180 −0.287885
\(994\) 12.0000 0.380617
\(995\) −1.35898 −0.0430827
\(996\) −4.39230 −0.139176
\(997\) −24.6410 −0.780389 −0.390194 0.920732i \(-0.627592\pi\)
−0.390194 + 0.920732i \(0.627592\pi\)
\(998\) −9.07180 −0.287163
\(999\) −4.92820 −0.155921
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 5082.2.a.bu.1.1 2
11.10 odd 2 462.2.a.h.1.1 2
33.32 even 2 1386.2.a.p.1.2 2
44.43 even 2 3696.2.a.bc.1.1 2
77.76 even 2 3234.2.a.x.1.2 2
231.230 odd 2 9702.2.a.dd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.h.1.1 2 11.10 odd 2
1386.2.a.p.1.2 2 33.32 even 2
3234.2.a.x.1.2 2 77.76 even 2
3696.2.a.bc.1.1 2 44.43 even 2
5082.2.a.bu.1.1 2 1.1 even 1 trivial
9702.2.a.dd.1.1 2 231.230 odd 2