Properties

Label 975.2.a.l.1.2
Level $975$
Weight $2$
Character 975.1
Self dual yes
Analytic conductor $7.785$
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] = [975,2,Mod(1,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, 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("975.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 975.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -2.41421 q^{6} +2.82843 q^{7} +4.41421 q^{8} +1.00000 q^{9} -2.00000 q^{11} -3.82843 q^{12} +1.00000 q^{13} +6.82843 q^{14} +3.00000 q^{16} +3.65685 q^{17} +2.41421 q^{18} +2.82843 q^{19} -2.82843 q^{21} -4.82843 q^{22} +4.00000 q^{23} -4.41421 q^{24} +2.41421 q^{26} -1.00000 q^{27} +10.8284 q^{28} +2.00000 q^{29} -6.82843 q^{31} -1.58579 q^{32} +2.00000 q^{33} +8.82843 q^{34} +3.82843 q^{36} -3.65685 q^{37} +6.82843 q^{38} -1.00000 q^{39} +10.8284 q^{41} -6.82843 q^{42} -9.65685 q^{43} -7.65685 q^{44} +9.65685 q^{46} +0.343146 q^{47} -3.00000 q^{48} +1.00000 q^{49} -3.65685 q^{51} +3.82843 q^{52} +2.00000 q^{53} -2.41421 q^{54} +12.4853 q^{56} -2.82843 q^{57} +4.82843 q^{58} -3.65685 q^{59} -9.31371 q^{61} -16.4853 q^{62} +2.82843 q^{63} -9.82843 q^{64} +4.82843 q^{66} -1.17157 q^{67} +14.0000 q^{68} -4.00000 q^{69} +2.00000 q^{71} +4.41421 q^{72} -11.6569 q^{73} -8.82843 q^{74} +10.8284 q^{76} -5.65685 q^{77} -2.41421 q^{78} +11.3137 q^{79} +1.00000 q^{81} +26.1421 q^{82} +7.65685 q^{83} -10.8284 q^{84} -23.3137 q^{86} -2.00000 q^{87} -8.82843 q^{88} +9.17157 q^{89} +2.82843 q^{91} +15.3137 q^{92} +6.82843 q^{93} +0.828427 q^{94} +1.58579 q^{96} +7.65685 q^{97} +2.41421 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 6 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} + 2 q^{13} + 8 q^{14} + 6 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{22} + 8 q^{23} - 6 q^{24} + 2 q^{26} - 2 q^{27} + 16 q^{28} + 4 q^{29} - 8 q^{31} - 6 q^{32} + 4 q^{33} + 12 q^{34} + 2 q^{36} + 4 q^{37} + 8 q^{38} - 2 q^{39} + 16 q^{41} - 8 q^{42} - 8 q^{43} - 4 q^{44} + 8 q^{46} + 12 q^{47} - 6 q^{48} + 2 q^{49} + 4 q^{51} + 2 q^{52} + 4 q^{53} - 2 q^{54} + 8 q^{56} + 4 q^{58} + 4 q^{59} + 4 q^{61} - 16 q^{62} - 14 q^{64} + 4 q^{66} - 8 q^{67} + 28 q^{68} - 8 q^{69} + 4 q^{71} + 6 q^{72} - 12 q^{73} - 12 q^{74} + 16 q^{76} - 2 q^{78} + 2 q^{81} + 24 q^{82} + 4 q^{83} - 16 q^{84} - 24 q^{86} - 4 q^{87} - 12 q^{88} + 24 q^{89} + 8 q^{92} + 8 q^{93} - 4 q^{94} + 6 q^{96} + 4 q^{97} + 2 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 1.70711 0.853553 0.521005i \(-0.174443\pi\)
0.853553 + 0.521005i \(0.174443\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) 0 0
\(6\) −2.41421 −0.985599
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) 4.41421 1.56066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −3.82843 −1.10517
\(13\) 1.00000 0.277350
\(14\) 6.82843 1.82497
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 3.65685 0.886917 0.443459 0.896295i \(-0.353751\pi\)
0.443459 + 0.896295i \(0.353751\pi\)
\(18\) 2.41421 0.569036
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) −2.82843 −0.617213
\(22\) −4.82843 −1.02942
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −4.41421 −0.901048
\(25\) 0 0
\(26\) 2.41421 0.473466
\(27\) −1.00000 −0.192450
\(28\) 10.8284 2.04638
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −6.82843 −1.22642 −0.613211 0.789919i \(-0.710122\pi\)
−0.613211 + 0.789919i \(0.710122\pi\)
\(32\) −1.58579 −0.280330
\(33\) 2.00000 0.348155
\(34\) 8.82843 1.51406
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) −3.65685 −0.601183 −0.300592 0.953753i \(-0.597184\pi\)
−0.300592 + 0.953753i \(0.597184\pi\)
\(38\) 6.82843 1.10772
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 10.8284 1.69112 0.845558 0.533883i \(-0.179268\pi\)
0.845558 + 0.533883i \(0.179268\pi\)
\(42\) −6.82843 −1.05365
\(43\) −9.65685 −1.47266 −0.736328 0.676625i \(-0.763442\pi\)
−0.736328 + 0.676625i \(0.763442\pi\)
\(44\) −7.65685 −1.15431
\(45\) 0 0
\(46\) 9.65685 1.42383
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) −3.00000 −0.433013
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −3.65685 −0.512062
\(52\) 3.82843 0.530907
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) 12.4853 1.66842
\(57\) −2.82843 −0.374634
\(58\) 4.82843 0.634004
\(59\) −3.65685 −0.476082 −0.238041 0.971255i \(-0.576505\pi\)
−0.238041 + 0.971255i \(0.576505\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) −16.4853 −2.09363
\(63\) 2.82843 0.356348
\(64\) −9.82843 −1.22855
\(65\) 0 0
\(66\) 4.82843 0.594338
\(67\) −1.17157 −0.143130 −0.0715652 0.997436i \(-0.522799\pi\)
−0.0715652 + 0.997436i \(0.522799\pi\)
\(68\) 14.0000 1.69775
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 4.41421 0.520220
\(73\) −11.6569 −1.36433 −0.682166 0.731198i \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) −8.82843 −1.02628
\(75\) 0 0
\(76\) 10.8284 1.24211
\(77\) −5.65685 −0.644658
\(78\) −2.41421 −0.273356
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 26.1421 2.88692
\(83\) 7.65685 0.840449 0.420224 0.907420i \(-0.361951\pi\)
0.420224 + 0.907420i \(0.361951\pi\)
\(84\) −10.8284 −1.18148
\(85\) 0 0
\(86\) −23.3137 −2.51398
\(87\) −2.00000 −0.214423
\(88\) −8.82843 −0.941113
\(89\) 9.17157 0.972185 0.486092 0.873907i \(-0.338422\pi\)
0.486092 + 0.873907i \(0.338422\pi\)
\(90\) 0 0
\(91\) 2.82843 0.296500
\(92\) 15.3137 1.59656
\(93\) 6.82843 0.708075
\(94\) 0.828427 0.0854457
\(95\) 0 0
\(96\) 1.58579 0.161849
\(97\) 7.65685 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(98\) 2.41421 0.243872
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) −8.82843 −0.874145
\(103\) −13.6569 −1.34565 −0.672825 0.739802i \(-0.734919\pi\)
−0.672825 + 0.739802i \(0.734919\pi\)
\(104\) 4.41421 0.432849
\(105\) 0 0
\(106\) 4.82843 0.468978
\(107\) −11.3137 −1.09374 −0.546869 0.837218i \(-0.684180\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) −3.82843 −0.368391
\(109\) −17.3137 −1.65835 −0.829176 0.558987i \(-0.811190\pi\)
−0.829176 + 0.558987i \(0.811190\pi\)
\(110\) 0 0
\(111\) 3.65685 0.347093
\(112\) 8.48528 0.801784
\(113\) −17.3137 −1.62874 −0.814368 0.580348i \(-0.802916\pi\)
−0.814368 + 0.580348i \(0.802916\pi\)
\(114\) −6.82843 −0.639541
\(115\) 0 0
\(116\) 7.65685 0.710921
\(117\) 1.00000 0.0924500
\(118\) −8.82843 −0.812723
\(119\) 10.3431 0.948155
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −22.4853 −2.03572
\(123\) −10.8284 −0.976366
\(124\) −26.1421 −2.34763
\(125\) 0 0
\(126\) 6.82843 0.608325
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) −20.5563 −1.81694
\(129\) 9.65685 0.850239
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 7.65685 0.666444
\(133\) 8.00000 0.693688
\(134\) −2.82843 −0.244339
\(135\) 0 0
\(136\) 16.1421 1.38418
\(137\) 5.17157 0.441837 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(138\) −9.65685 −0.822046
\(139\) 15.3137 1.29889 0.649446 0.760408i \(-0.275001\pi\)
0.649446 + 0.760408i \(0.275001\pi\)
\(140\) 0 0
\(141\) −0.343146 −0.0288981
\(142\) 4.82843 0.405193
\(143\) −2.00000 −0.167248
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −28.1421 −2.32906
\(147\) −1.00000 −0.0824786
\(148\) −14.0000 −1.15079
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 0 0
\(151\) −20.4853 −1.66707 −0.833534 0.552468i \(-0.813686\pi\)
−0.833534 + 0.552468i \(0.813686\pi\)
\(152\) 12.4853 1.01269
\(153\) 3.65685 0.295639
\(154\) −13.6569 −1.10050
\(155\) 0 0
\(156\) −3.82843 −0.306519
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 27.3137 2.17296
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) 2.41421 0.189679
\(163\) −13.1716 −1.03168 −0.515839 0.856686i \(-0.672520\pi\)
−0.515839 + 0.856686i \(0.672520\pi\)
\(164\) 41.4558 3.23716
\(165\) 0 0
\(166\) 18.4853 1.43474
\(167\) −7.65685 −0.592505 −0.296253 0.955110i \(-0.595737\pi\)
−0.296253 + 0.955110i \(0.595737\pi\)
\(168\) −12.4853 −0.963260
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) −36.9706 −2.81898
\(173\) 0.343146 0.0260889 0.0130444 0.999915i \(-0.495848\pi\)
0.0130444 + 0.999915i \(0.495848\pi\)
\(174\) −4.82843 −0.366042
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 3.65685 0.274866
\(178\) 22.1421 1.65962
\(179\) −0.686292 −0.0512958 −0.0256479 0.999671i \(-0.508165\pi\)
−0.0256479 + 0.999671i \(0.508165\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 6.82843 0.506157
\(183\) 9.31371 0.688489
\(184\) 17.6569 1.30168
\(185\) 0 0
\(186\) 16.4853 1.20876
\(187\) −7.31371 −0.534831
\(188\) 1.31371 0.0958120
\(189\) −2.82843 −0.205738
\(190\) 0 0
\(191\) −19.3137 −1.39749 −0.698745 0.715370i \(-0.746258\pi\)
−0.698745 + 0.715370i \(0.746258\pi\)
\(192\) 9.82843 0.709306
\(193\) 17.3137 1.24627 0.623134 0.782115i \(-0.285859\pi\)
0.623134 + 0.782115i \(0.285859\pi\)
\(194\) 18.4853 1.32717
\(195\) 0 0
\(196\) 3.82843 0.273459
\(197\) 16.4853 1.17453 0.587264 0.809396i \(-0.300205\pi\)
0.587264 + 0.809396i \(0.300205\pi\)
\(198\) −4.82843 −0.343141
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) 1.17157 0.0826364
\(202\) −8.82843 −0.621166
\(203\) 5.65685 0.397033
\(204\) −14.0000 −0.980196
\(205\) 0 0
\(206\) −32.9706 −2.29717
\(207\) 4.00000 0.278019
\(208\) 3.00000 0.208013
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 7.65685 0.525875
\(213\) −2.00000 −0.137038
\(214\) −27.3137 −1.86713
\(215\) 0 0
\(216\) −4.41421 −0.300349
\(217\) −19.3137 −1.31110
\(218\) −41.7990 −2.83098
\(219\) 11.6569 0.787697
\(220\) 0 0
\(221\) 3.65685 0.245987
\(222\) 8.82843 0.592525
\(223\) −4.48528 −0.300357 −0.150178 0.988659i \(-0.547985\pi\)
−0.150178 + 0.988659i \(0.547985\pi\)
\(224\) −4.48528 −0.299685
\(225\) 0 0
\(226\) −41.7990 −2.78043
\(227\) −5.31371 −0.352683 −0.176342 0.984329i \(-0.556426\pi\)
−0.176342 + 0.984329i \(0.556426\pi\)
\(228\) −10.8284 −0.717130
\(229\) 21.3137 1.40845 0.704225 0.709977i \(-0.251295\pi\)
0.704225 + 0.709977i \(0.251295\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 8.82843 0.579615
\(233\) 26.9706 1.76690 0.883450 0.468525i \(-0.155214\pi\)
0.883450 + 0.468525i \(0.155214\pi\)
\(234\) 2.41421 0.157822
\(235\) 0 0
\(236\) −14.0000 −0.911322
\(237\) −11.3137 −0.734904
\(238\) 24.9706 1.61860
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 11.6569 0.750884 0.375442 0.926846i \(-0.377491\pi\)
0.375442 + 0.926846i \(0.377491\pi\)
\(242\) −16.8995 −1.08634
\(243\) −1.00000 −0.0641500
\(244\) −35.6569 −2.28270
\(245\) 0 0
\(246\) −26.1421 −1.66676
\(247\) 2.82843 0.179969
\(248\) −30.1421 −1.91403
\(249\) −7.65685 −0.485233
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 10.8284 0.682127
\(253\) −8.00000 −0.502956
\(254\) 13.6569 0.856907
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) 15.6569 0.976648 0.488324 0.872662i \(-0.337608\pi\)
0.488324 + 0.872662i \(0.337608\pi\)
\(258\) 23.3137 1.45145
\(259\) −10.3431 −0.642692
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −19.3137 −1.19320
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 8.82843 0.543352
\(265\) 0 0
\(266\) 19.3137 1.18420
\(267\) −9.17157 −0.561291
\(268\) −4.48528 −0.273982
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 11.7990 0.716738 0.358369 0.933580i \(-0.383333\pi\)
0.358369 + 0.933580i \(0.383333\pi\)
\(272\) 10.9706 0.665188
\(273\) −2.82843 −0.171184
\(274\) 12.4853 0.754263
\(275\) 0 0
\(276\) −15.3137 −0.921777
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 36.9706 2.21735
\(279\) −6.82843 −0.408807
\(280\) 0 0
\(281\) 26.8284 1.60045 0.800225 0.599700i \(-0.204713\pi\)
0.800225 + 0.599700i \(0.204713\pi\)
\(282\) −0.828427 −0.0493321
\(283\) 4.97056 0.295469 0.147735 0.989027i \(-0.452802\pi\)
0.147735 + 0.989027i \(0.452802\pi\)
\(284\) 7.65685 0.454351
\(285\) 0 0
\(286\) −4.82843 −0.285511
\(287\) 30.6274 1.80788
\(288\) −1.58579 −0.0934434
\(289\) −3.62742 −0.213377
\(290\) 0 0
\(291\) −7.65685 −0.448853
\(292\) −44.6274 −2.61162
\(293\) 26.1421 1.52724 0.763620 0.645666i \(-0.223420\pi\)
0.763620 + 0.645666i \(0.223420\pi\)
\(294\) −2.41421 −0.140800
\(295\) 0 0
\(296\) −16.1421 −0.938243
\(297\) 2.00000 0.116052
\(298\) −35.7990 −2.07378
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −27.3137 −1.57434
\(302\) −49.4558 −2.84586
\(303\) 3.65685 0.210081
\(304\) 8.48528 0.486664
\(305\) 0 0
\(306\) 8.82843 0.504688
\(307\) 17.1716 0.980033 0.490017 0.871713i \(-0.336991\pi\)
0.490017 + 0.871713i \(0.336991\pi\)
\(308\) −21.6569 −1.23401
\(309\) 13.6569 0.776911
\(310\) 0 0
\(311\) 34.6274 1.96354 0.981770 0.190071i \(-0.0608718\pi\)
0.981770 + 0.190071i \(0.0608718\pi\)
\(312\) −4.41421 −0.249906
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 24.1421 1.36242
\(315\) 0 0
\(316\) 43.3137 2.43659
\(317\) 8.48528 0.476581 0.238290 0.971194i \(-0.423413\pi\)
0.238290 + 0.971194i \(0.423413\pi\)
\(318\) −4.82843 −0.270765
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) 11.3137 0.631470
\(322\) 27.3137 1.52213
\(323\) 10.3431 0.575508
\(324\) 3.82843 0.212690
\(325\) 0 0
\(326\) −31.7990 −1.76118
\(327\) 17.3137 0.957450
\(328\) 47.7990 2.63926
\(329\) 0.970563 0.0535089
\(330\) 0 0
\(331\) −2.14214 −0.117742 −0.0588712 0.998266i \(-0.518750\pi\)
−0.0588712 + 0.998266i \(0.518750\pi\)
\(332\) 29.3137 1.60880
\(333\) −3.65685 −0.200394
\(334\) −18.4853 −1.01147
\(335\) 0 0
\(336\) −8.48528 −0.462910
\(337\) 13.3137 0.725244 0.362622 0.931936i \(-0.381882\pi\)
0.362622 + 0.931936i \(0.381882\pi\)
\(338\) 2.41421 0.131316
\(339\) 17.3137 0.940352
\(340\) 0 0
\(341\) 13.6569 0.739560
\(342\) 6.82843 0.369239
\(343\) −16.9706 −0.916324
\(344\) −42.6274 −2.29832
\(345\) 0 0
\(346\) 0.828427 0.0445365
\(347\) 31.3137 1.68101 0.840504 0.541805i \(-0.182259\pi\)
0.840504 + 0.541805i \(0.182259\pi\)
\(348\) −7.65685 −0.410450
\(349\) −7.65685 −0.409862 −0.204931 0.978776i \(-0.565697\pi\)
−0.204931 + 0.978776i \(0.565697\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 3.17157 0.169045
\(353\) −17.4558 −0.929081 −0.464540 0.885552i \(-0.653780\pi\)
−0.464540 + 0.885552i \(0.653780\pi\)
\(354\) 8.82843 0.469226
\(355\) 0 0
\(356\) 35.1127 1.86097
\(357\) −10.3431 −0.547417
\(358\) −1.65685 −0.0875675
\(359\) 1.02944 0.0543316 0.0271658 0.999631i \(-0.491352\pi\)
0.0271658 + 0.999631i \(0.491352\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 33.7990 1.77644
\(363\) 7.00000 0.367405
\(364\) 10.8284 0.567564
\(365\) 0 0
\(366\) 22.4853 1.17532
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 12.0000 0.625543
\(369\) 10.8284 0.563705
\(370\) 0 0
\(371\) 5.65685 0.293689
\(372\) 26.1421 1.35541
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −17.6569 −0.913014
\(375\) 0 0
\(376\) 1.51472 0.0781156
\(377\) 2.00000 0.103005
\(378\) −6.82843 −0.351216
\(379\) −16.4853 −0.846792 −0.423396 0.905945i \(-0.639162\pi\)
−0.423396 + 0.905945i \(0.639162\pi\)
\(380\) 0 0
\(381\) −5.65685 −0.289809
\(382\) −46.6274 −2.38567
\(383\) 2.97056 0.151789 0.0758943 0.997116i \(-0.475819\pi\)
0.0758943 + 0.997116i \(0.475819\pi\)
\(384\) 20.5563 1.04901
\(385\) 0 0
\(386\) 41.7990 2.12751
\(387\) −9.65685 −0.490885
\(388\) 29.3137 1.48818
\(389\) 6.97056 0.353422 0.176711 0.984263i \(-0.443454\pi\)
0.176711 + 0.984263i \(0.443454\pi\)
\(390\) 0 0
\(391\) 14.6274 0.739740
\(392\) 4.41421 0.222951
\(393\) 8.00000 0.403547
\(394\) 39.7990 2.00504
\(395\) 0 0
\(396\) −7.65685 −0.384771
\(397\) 2.97056 0.149088 0.0745441 0.997218i \(-0.476250\pi\)
0.0745441 + 0.997218i \(0.476250\pi\)
\(398\) 24.9706 1.25166
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −2.14214 −0.106973 −0.0534866 0.998569i \(-0.517033\pi\)
−0.0534866 + 0.998569i \(0.517033\pi\)
\(402\) 2.82843 0.141069
\(403\) −6.82843 −0.340148
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 13.6569 0.677778
\(407\) 7.31371 0.362527
\(408\) −16.1421 −0.799155
\(409\) −1.02944 −0.0509024 −0.0254512 0.999676i \(-0.508102\pi\)
−0.0254512 + 0.999676i \(0.508102\pi\)
\(410\) 0 0
\(411\) −5.17157 −0.255095
\(412\) −52.2843 −2.57586
\(413\) −10.3431 −0.508953
\(414\) 9.65685 0.474608
\(415\) 0 0
\(416\) −1.58579 −0.0777496
\(417\) −15.3137 −0.749916
\(418\) −13.6569 −0.667979
\(419\) −30.6274 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(420\) 0 0
\(421\) 14.6863 0.715766 0.357883 0.933766i \(-0.383499\pi\)
0.357883 + 0.933766i \(0.383499\pi\)
\(422\) −28.9706 −1.41026
\(423\) 0.343146 0.0166843
\(424\) 8.82843 0.428746
\(425\) 0 0
\(426\) −4.82843 −0.233938
\(427\) −26.3431 −1.27483
\(428\) −43.3137 −2.09365
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 19.6569 0.946837 0.473419 0.880838i \(-0.343020\pi\)
0.473419 + 0.880838i \(0.343020\pi\)
\(432\) −3.00000 −0.144338
\(433\) −1.31371 −0.0631328 −0.0315664 0.999502i \(-0.510050\pi\)
−0.0315664 + 0.999502i \(0.510050\pi\)
\(434\) −46.6274 −2.23819
\(435\) 0 0
\(436\) −66.2843 −3.17444
\(437\) 11.3137 0.541208
\(438\) 28.1421 1.34468
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 8.82843 0.419925
\(443\) −41.9411 −1.99268 −0.996342 0.0854611i \(-0.972764\pi\)
−0.996342 + 0.0854611i \(0.972764\pi\)
\(444\) 14.0000 0.664411
\(445\) 0 0
\(446\) −10.8284 −0.512741
\(447\) 14.8284 0.701361
\(448\) −27.7990 −1.31338
\(449\) 7.79899 0.368057 0.184029 0.982921i \(-0.441086\pi\)
0.184029 + 0.982921i \(0.441086\pi\)
\(450\) 0 0
\(451\) −21.6569 −1.01978
\(452\) −66.2843 −3.11775
\(453\) 20.4853 0.962482
\(454\) −12.8284 −0.602068
\(455\) 0 0
\(456\) −12.4853 −0.584677
\(457\) −3.65685 −0.171060 −0.0855302 0.996336i \(-0.527258\pi\)
−0.0855302 + 0.996336i \(0.527258\pi\)
\(458\) 51.4558 2.40437
\(459\) −3.65685 −0.170687
\(460\) 0 0
\(461\) 10.8284 0.504330 0.252165 0.967684i \(-0.418857\pi\)
0.252165 + 0.967684i \(0.418857\pi\)
\(462\) 13.6569 0.635374
\(463\) 7.51472 0.349239 0.174619 0.984636i \(-0.444131\pi\)
0.174619 + 0.984636i \(0.444131\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 65.1127 3.01629
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 3.82843 0.176969
\(469\) −3.31371 −0.153013
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −16.1421 −0.743002
\(473\) 19.3137 0.888045
\(474\) −27.3137 −1.25456
\(475\) 0 0
\(476\) 39.5980 1.81497
\(477\) 2.00000 0.0915737
\(478\) 4.82843 0.220847
\(479\) 2.68629 0.122740 0.0613699 0.998115i \(-0.480453\pi\)
0.0613699 + 0.998115i \(0.480453\pi\)
\(480\) 0 0
\(481\) −3.65685 −0.166738
\(482\) 28.1421 1.28184
\(483\) −11.3137 −0.514792
\(484\) −26.7990 −1.21814
\(485\) 0 0
\(486\) −2.41421 −0.109511
\(487\) −31.7990 −1.44095 −0.720475 0.693481i \(-0.756076\pi\)
−0.720475 + 0.693481i \(0.756076\pi\)
\(488\) −41.1127 −1.86108
\(489\) 13.1716 0.595639
\(490\) 0 0
\(491\) −14.6274 −0.660126 −0.330063 0.943959i \(-0.607070\pi\)
−0.330063 + 0.943959i \(0.607070\pi\)
\(492\) −41.4558 −1.86897
\(493\) 7.31371 0.329393
\(494\) 6.82843 0.307225
\(495\) 0 0
\(496\) −20.4853 −0.919816
\(497\) 5.65685 0.253745
\(498\) −18.4853 −0.828345
\(499\) −2.14214 −0.0958952 −0.0479476 0.998850i \(-0.515268\pi\)
−0.0479476 + 0.998850i \(0.515268\pi\)
\(500\) 0 0
\(501\) 7.65685 0.342083
\(502\) 0 0
\(503\) −15.3137 −0.682805 −0.341402 0.939917i \(-0.610902\pi\)
−0.341402 + 0.939917i \(0.610902\pi\)
\(504\) 12.4853 0.556139
\(505\) 0 0
\(506\) −19.3137 −0.858599
\(507\) −1.00000 −0.0444116
\(508\) 21.6569 0.960868
\(509\) −27.7990 −1.23217 −0.616084 0.787680i \(-0.711282\pi\)
−0.616084 + 0.787680i \(0.711282\pi\)
\(510\) 0 0
\(511\) −32.9706 −1.45853
\(512\) −31.2426 −1.38074
\(513\) −2.82843 −0.124878
\(514\) 37.7990 1.66724
\(515\) 0 0
\(516\) 36.9706 1.62754
\(517\) −0.686292 −0.0301831
\(518\) −24.9706 −1.09714
\(519\) −0.343146 −0.0150624
\(520\) 0 0
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) 4.82843 0.211335
\(523\) −7.31371 −0.319806 −0.159903 0.987133i \(-0.551118\pi\)
−0.159903 + 0.987133i \(0.551118\pi\)
\(524\) −30.6274 −1.33796
\(525\) 0 0
\(526\) −28.9706 −1.26318
\(527\) −24.9706 −1.08773
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −3.65685 −0.158694
\(532\) 30.6274 1.32787
\(533\) 10.8284 0.469031
\(534\) −22.1421 −0.958184
\(535\) 0 0
\(536\) −5.17157 −0.223378
\(537\) 0.686292 0.0296157
\(538\) 43.4558 1.87351
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 28.4853 1.22355
\(543\) −14.0000 −0.600798
\(544\) −5.79899 −0.248630
\(545\) 0 0
\(546\) −6.82843 −0.292230
\(547\) −0.686292 −0.0293437 −0.0146719 0.999892i \(-0.504670\pi\)
−0.0146719 + 0.999892i \(0.504670\pi\)
\(548\) 19.7990 0.845771
\(549\) −9.31371 −0.397499
\(550\) 0 0
\(551\) 5.65685 0.240990
\(552\) −17.6569 −0.751526
\(553\) 32.0000 1.36078
\(554\) 4.82843 0.205140
\(555\) 0 0
\(556\) 58.6274 2.48636
\(557\) −31.7990 −1.34737 −0.673683 0.739020i \(-0.735289\pi\)
−0.673683 + 0.739020i \(0.735289\pi\)
\(558\) −16.4853 −0.697878
\(559\) −9.65685 −0.408441
\(560\) 0 0
\(561\) 7.31371 0.308785
\(562\) 64.7696 2.73214
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −1.31371 −0.0553171
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 2.82843 0.118783
\(568\) 8.82843 0.370433
\(569\) −9.02944 −0.378534 −0.189267 0.981926i \(-0.560611\pi\)
−0.189267 + 0.981926i \(0.560611\pi\)
\(570\) 0 0
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) −7.65685 −0.320149
\(573\) 19.3137 0.806842
\(574\) 73.9411 3.08624
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) −35.9411 −1.49625 −0.748124 0.663559i \(-0.769045\pi\)
−0.748124 + 0.663559i \(0.769045\pi\)
\(578\) −8.75736 −0.364258
\(579\) −17.3137 −0.719533
\(580\) 0 0
\(581\) 21.6569 0.898478
\(582\) −18.4853 −0.766240
\(583\) −4.00000 −0.165663
\(584\) −51.4558 −2.12926
\(585\) 0 0
\(586\) 63.1127 2.60716
\(587\) −22.9706 −0.948097 −0.474048 0.880499i \(-0.657208\pi\)
−0.474048 + 0.880499i \(0.657208\pi\)
\(588\) −3.82843 −0.157882
\(589\) −19.3137 −0.795807
\(590\) 0 0
\(591\) −16.4853 −0.678114
\(592\) −10.9706 −0.450887
\(593\) 3.51472 0.144332 0.0721661 0.997393i \(-0.477009\pi\)
0.0721661 + 0.997393i \(0.477009\pi\)
\(594\) 4.82843 0.198113
\(595\) 0 0
\(596\) −56.7696 −2.32537
\(597\) −10.3431 −0.423317
\(598\) 9.65685 0.394898
\(599\) −0.686292 −0.0280411 −0.0140206 0.999902i \(-0.504463\pi\)
−0.0140206 + 0.999902i \(0.504463\pi\)
\(600\) 0 0
\(601\) 44.6274 1.82039 0.910195 0.414180i \(-0.135931\pi\)
0.910195 + 0.414180i \(0.135931\pi\)
\(602\) −65.9411 −2.68756
\(603\) −1.17157 −0.0477101
\(604\) −78.4264 −3.19113
\(605\) 0 0
\(606\) 8.82843 0.358630
\(607\) 25.9411 1.05292 0.526459 0.850201i \(-0.323519\pi\)
0.526459 + 0.850201i \(0.323519\pi\)
\(608\) −4.48528 −0.181902
\(609\) −5.65685 −0.229227
\(610\) 0 0
\(611\) 0.343146 0.0138822
\(612\) 14.0000 0.565916
\(613\) 36.3431 1.46789 0.733943 0.679211i \(-0.237678\pi\)
0.733943 + 0.679211i \(0.237678\pi\)
\(614\) 41.4558 1.67302
\(615\) 0 0
\(616\) −24.9706 −1.00609
\(617\) 29.1716 1.17440 0.587202 0.809441i \(-0.300230\pi\)
0.587202 + 0.809441i \(0.300230\pi\)
\(618\) 32.9706 1.32627
\(619\) −15.7990 −0.635015 −0.317508 0.948256i \(-0.602846\pi\)
−0.317508 + 0.948256i \(0.602846\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 83.5980 3.35197
\(623\) 25.9411 1.03931
\(624\) −3.00000 −0.120096
\(625\) 0 0
\(626\) −14.4853 −0.578948
\(627\) 5.65685 0.225913
\(628\) 38.2843 1.52771
\(629\) −13.3726 −0.533200
\(630\) 0 0
\(631\) −19.1127 −0.760865 −0.380432 0.924809i \(-0.624225\pi\)
−0.380432 + 0.924809i \(0.624225\pi\)
\(632\) 49.9411 1.98655
\(633\) 12.0000 0.476957
\(634\) 20.4853 0.813574
\(635\) 0 0
\(636\) −7.65685 −0.303614
\(637\) 1.00000 0.0396214
\(638\) −9.65685 −0.382319
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −26.2843 −1.03817 −0.519083 0.854724i \(-0.673727\pi\)
−0.519083 + 0.854724i \(0.673727\pi\)
\(642\) 27.3137 1.07799
\(643\) −17.1716 −0.677181 −0.338590 0.940934i \(-0.609950\pi\)
−0.338590 + 0.940934i \(0.609950\pi\)
\(644\) 43.3137 1.70680
\(645\) 0 0
\(646\) 24.9706 0.982454
\(647\) 11.3137 0.444788 0.222394 0.974957i \(-0.428613\pi\)
0.222394 + 0.974957i \(0.428613\pi\)
\(648\) 4.41421 0.173407
\(649\) 7.31371 0.287088
\(650\) 0 0
\(651\) 19.3137 0.756964
\(652\) −50.4264 −1.97485
\(653\) 2.68629 0.105123 0.0525614 0.998618i \(-0.483261\pi\)
0.0525614 + 0.998618i \(0.483261\pi\)
\(654\) 41.7990 1.63447
\(655\) 0 0
\(656\) 32.4853 1.26834
\(657\) −11.6569 −0.454777
\(658\) 2.34315 0.0913453
\(659\) −24.6863 −0.961641 −0.480821 0.876819i \(-0.659661\pi\)
−0.480821 + 0.876819i \(0.659661\pi\)
\(660\) 0 0
\(661\) −1.02944 −0.0400405 −0.0200202 0.999800i \(-0.506373\pi\)
−0.0200202 + 0.999800i \(0.506373\pi\)
\(662\) −5.17157 −0.200999
\(663\) −3.65685 −0.142020
\(664\) 33.7990 1.31166
\(665\) 0 0
\(666\) −8.82843 −0.342095
\(667\) 8.00000 0.309761
\(668\) −29.3137 −1.13418
\(669\) 4.48528 0.173411
\(670\) 0 0
\(671\) 18.6274 0.719103
\(672\) 4.48528 0.173023
\(673\) 28.6274 1.10351 0.551753 0.834008i \(-0.313959\pi\)
0.551753 + 0.834008i \(0.313959\pi\)
\(674\) 32.1421 1.23807
\(675\) 0 0
\(676\) 3.82843 0.147247
\(677\) −49.3137 −1.89528 −0.947640 0.319341i \(-0.896538\pi\)
−0.947640 + 0.319341i \(0.896538\pi\)
\(678\) 41.7990 1.60528
\(679\) 21.6569 0.831114
\(680\) 0 0
\(681\) 5.31371 0.203622
\(682\) 32.9706 1.26251
\(683\) 19.9411 0.763026 0.381513 0.924363i \(-0.375403\pi\)
0.381513 + 0.924363i \(0.375403\pi\)
\(684\) 10.8284 0.414035
\(685\) 0 0
\(686\) −40.9706 −1.56426
\(687\) −21.3137 −0.813169
\(688\) −28.9706 −1.10449
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −34.1421 −1.29883 −0.649414 0.760435i \(-0.724986\pi\)
−0.649414 + 0.760435i \(0.724986\pi\)
\(692\) 1.31371 0.0499397
\(693\) −5.65685 −0.214886
\(694\) 75.5980 2.86966
\(695\) 0 0
\(696\) −8.82843 −0.334641
\(697\) 39.5980 1.49988
\(698\) −18.4853 −0.699678
\(699\) −26.9706 −1.02012
\(700\) 0 0
\(701\) 38.9706 1.47190 0.735949 0.677037i \(-0.236736\pi\)
0.735949 + 0.677037i \(0.236736\pi\)
\(702\) −2.41421 −0.0911186
\(703\) −10.3431 −0.390099
\(704\) 19.6569 0.740846
\(705\) 0 0
\(706\) −42.1421 −1.58604
\(707\) −10.3431 −0.388994
\(708\) 14.0000 0.526152
\(709\) 40.6274 1.52579 0.762897 0.646520i \(-0.223776\pi\)
0.762897 + 0.646520i \(0.223776\pi\)
\(710\) 0 0
\(711\) 11.3137 0.424297
\(712\) 40.4853 1.51725
\(713\) −27.3137 −1.02291
\(714\) −24.9706 −0.934500
\(715\) 0 0
\(716\) −2.62742 −0.0981912
\(717\) −2.00000 −0.0746914
\(718\) 2.48528 0.0927499
\(719\) 37.9411 1.41497 0.707483 0.706731i \(-0.249831\pi\)
0.707483 + 0.706731i \(0.249831\pi\)
\(720\) 0 0
\(721\) −38.6274 −1.43856
\(722\) −26.5563 −0.988325
\(723\) −11.6569 −0.433523
\(724\) 53.5980 1.99195
\(725\) 0 0
\(726\) 16.8995 0.627199
\(727\) 21.6569 0.803208 0.401604 0.915813i \(-0.368453\pi\)
0.401604 + 0.915813i \(0.368453\pi\)
\(728\) 12.4853 0.462735
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −35.3137 −1.30612
\(732\) 35.6569 1.31792
\(733\) −8.62742 −0.318661 −0.159330 0.987225i \(-0.550934\pi\)
−0.159330 + 0.987225i \(0.550934\pi\)
\(734\) 57.9411 2.13865
\(735\) 0 0
\(736\) −6.34315 −0.233811
\(737\) 2.34315 0.0863109
\(738\) 26.1421 0.962305
\(739\) 10.1421 0.373084 0.186542 0.982447i \(-0.440272\pi\)
0.186542 + 0.982447i \(0.440272\pi\)
\(740\) 0 0
\(741\) −2.82843 −0.103905
\(742\) 13.6569 0.501359
\(743\) −2.00000 −0.0733729 −0.0366864 0.999327i \(-0.511680\pi\)
−0.0366864 + 0.999327i \(0.511680\pi\)
\(744\) 30.1421 1.10506
\(745\) 0 0
\(746\) −24.1421 −0.883906
\(747\) 7.65685 0.280150
\(748\) −28.0000 −1.02378
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) 32.9706 1.20311 0.601556 0.798830i \(-0.294547\pi\)
0.601556 + 0.798830i \(0.294547\pi\)
\(752\) 1.02944 0.0375397
\(753\) 0 0
\(754\) 4.82843 0.175841
\(755\) 0 0
\(756\) −10.8284 −0.393826
\(757\) 15.9411 0.579390 0.289695 0.957119i \(-0.406446\pi\)
0.289695 + 0.957119i \(0.406446\pi\)
\(758\) −39.7990 −1.44556
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 15.5147 0.562408 0.281204 0.959648i \(-0.409266\pi\)
0.281204 + 0.959648i \(0.409266\pi\)
\(762\) −13.6569 −0.494736
\(763\) −48.9706 −1.77285
\(764\) −73.9411 −2.67510
\(765\) 0 0
\(766\) 7.17157 0.259119
\(767\) −3.65685 −0.132041
\(768\) 29.9706 1.08147
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −15.6569 −0.563868
\(772\) 66.2843 2.38562
\(773\) −5.85786 −0.210693 −0.105346 0.994436i \(-0.533595\pi\)
−0.105346 + 0.994436i \(0.533595\pi\)
\(774\) −23.3137 −0.837994
\(775\) 0 0
\(776\) 33.7990 1.21331
\(777\) 10.3431 0.371058
\(778\) 16.8284 0.603328
\(779\) 30.6274 1.09734
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 35.3137 1.26282
\(783\) −2.00000 −0.0714742
\(784\) 3.00000 0.107143
\(785\) 0 0
\(786\) 19.3137 0.688897
\(787\) −32.7696 −1.16811 −0.584054 0.811715i \(-0.698534\pi\)
−0.584054 + 0.811715i \(0.698534\pi\)
\(788\) 63.1127 2.24830
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −48.9706 −1.74119
\(792\) −8.82843 −0.313704
\(793\) −9.31371 −0.330739
\(794\) 7.17157 0.254510
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) −35.6569 −1.26303 −0.631515 0.775363i \(-0.717567\pi\)
−0.631515 + 0.775363i \(0.717567\pi\)
\(798\) −19.3137 −0.683698
\(799\) 1.25483 0.0443928
\(800\) 0 0
\(801\) 9.17157 0.324062
\(802\) −5.17157 −0.182615
\(803\) 23.3137 0.822723
\(804\) 4.48528 0.158184
\(805\) 0 0
\(806\) −16.4853 −0.580669
\(807\) −18.0000 −0.633630
\(808\) −16.1421 −0.567878
\(809\) 41.3137 1.45251 0.726256 0.687424i \(-0.241259\pi\)
0.726256 + 0.687424i \(0.241259\pi\)
\(810\) 0 0
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) 21.6569 0.760007
\(813\) −11.7990 −0.413809
\(814\) 17.6569 0.618872
\(815\) 0 0
\(816\) −10.9706 −0.384047
\(817\) −27.3137 −0.955586
\(818\) −2.48528 −0.0868958
\(819\) 2.82843 0.0988332
\(820\) 0 0
\(821\) 15.7990 0.551389 0.275694 0.961245i \(-0.411092\pi\)
0.275694 + 0.961245i \(0.411092\pi\)
\(822\) −12.4853 −0.435474
\(823\) −48.9706 −1.70701 −0.853503 0.521088i \(-0.825527\pi\)
−0.853503 + 0.521088i \(0.825527\pi\)
\(824\) −60.2843 −2.10010
\(825\) 0 0
\(826\) −24.9706 −0.868837
\(827\) 26.0000 0.904109 0.452054 0.891990i \(-0.350691\pi\)
0.452054 + 0.891990i \(0.350691\pi\)
\(828\) 15.3137 0.532188
\(829\) 5.31371 0.184553 0.0922764 0.995733i \(-0.470586\pi\)
0.0922764 + 0.995733i \(0.470586\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −9.82843 −0.340739
\(833\) 3.65685 0.126702
\(834\) −36.9706 −1.28019
\(835\) 0 0
\(836\) −21.6569 −0.749018
\(837\) 6.82843 0.236025
\(838\) −73.9411 −2.55425
\(839\) −47.2548 −1.63142 −0.815709 0.578462i \(-0.803653\pi\)
−0.815709 + 0.578462i \(0.803653\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 35.4558 1.22189
\(843\) −26.8284 −0.924020
\(844\) −45.9411 −1.58136
\(845\) 0 0
\(846\) 0.828427 0.0284819
\(847\) −19.7990 −0.680301
\(848\) 6.00000 0.206041
\(849\) −4.97056 −0.170589
\(850\) 0 0
\(851\) −14.6274 −0.501421
\(852\) −7.65685 −0.262320
\(853\) 7.65685 0.262166 0.131083 0.991371i \(-0.458155\pi\)
0.131083 + 0.991371i \(0.458155\pi\)
\(854\) −63.5980 −2.17628
\(855\) 0 0
\(856\) −49.9411 −1.70695
\(857\) −29.5980 −1.01105 −0.505524 0.862813i \(-0.668701\pi\)
−0.505524 + 0.862813i \(0.668701\pi\)
\(858\) 4.82843 0.164840
\(859\) −23.3137 −0.795453 −0.397727 0.917504i \(-0.630201\pi\)
−0.397727 + 0.917504i \(0.630201\pi\)
\(860\) 0 0
\(861\) −30.6274 −1.04378
\(862\) 47.4558 1.61635
\(863\) 39.6569 1.34994 0.674968 0.737847i \(-0.264158\pi\)
0.674968 + 0.737847i \(0.264158\pi\)
\(864\) 1.58579 0.0539496
\(865\) 0 0
\(866\) −3.17157 −0.107774
\(867\) 3.62742 0.123194
\(868\) −73.9411 −2.50973
\(869\) −22.6274 −0.767583
\(870\) 0 0
\(871\) −1.17157 −0.0396972
\(872\) −76.4264 −2.58812
\(873\) 7.65685 0.259145
\(874\) 27.3137 0.923900
\(875\) 0 0
\(876\) 44.6274 1.50782
\(877\) 14.2843 0.482346 0.241173 0.970482i \(-0.422468\pi\)
0.241173 + 0.970482i \(0.422468\pi\)
\(878\) −40.9706 −1.38269
\(879\) −26.1421 −0.881752
\(880\) 0 0
\(881\) 53.5980 1.80576 0.902881 0.429891i \(-0.141448\pi\)
0.902881 + 0.429891i \(0.141448\pi\)
\(882\) 2.41421 0.0812908
\(883\) 51.5980 1.73641 0.868205 0.496205i \(-0.165274\pi\)
0.868205 + 0.496205i \(0.165274\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) −101.255 −3.40172
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 16.1421 0.541695
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −17.1716 −0.574947
\(893\) 0.970563 0.0324786
\(894\) 35.7990 1.19730
\(895\) 0 0
\(896\) −58.1421 −1.94239
\(897\) −4.00000 −0.133556
\(898\) 18.8284 0.628313
\(899\) −13.6569 −0.455482
\(900\) 0 0
\(901\) 7.31371 0.243655
\(902\) −52.2843 −1.74088
\(903\) 27.3137 0.908943
\(904\) −76.4264 −2.54190
\(905\) 0 0
\(906\) 49.4558 1.64306
\(907\) −20.9706 −0.696316 −0.348158 0.937436i \(-0.613193\pi\)
−0.348158 + 0.937436i \(0.613193\pi\)
\(908\) −20.3431 −0.675111
\(909\) −3.65685 −0.121290
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) −8.48528 −0.280976
\(913\) −15.3137 −0.506810
\(914\) −8.82843 −0.292018
\(915\) 0 0
\(916\) 81.5980 2.69607
\(917\) −22.6274 −0.747223
\(918\) −8.82843 −0.291382
\(919\) −19.3137 −0.637100 −0.318550 0.947906i \(-0.603196\pi\)
−0.318550 + 0.947906i \(0.603196\pi\)
\(920\) 0 0
\(921\) −17.1716 −0.565823
\(922\) 26.1421 0.860945
\(923\) 2.00000 0.0658308
\(924\) 21.6569 0.712458
\(925\) 0 0
\(926\) 18.1421 0.596188
\(927\) −13.6569 −0.448550
\(928\) −3.17157 −0.104112
\(929\) −27.7990 −0.912055 −0.456028 0.889966i \(-0.650728\pi\)
−0.456028 + 0.889966i \(0.650728\pi\)
\(930\) 0 0
\(931\) 2.82843 0.0926980
\(932\) 103.255 3.38222
\(933\) −34.6274 −1.13365
\(934\) 19.3137 0.631964
\(935\) 0 0
\(936\) 4.41421 0.144283
\(937\) −1.31371 −0.0429170 −0.0214585 0.999770i \(-0.506831\pi\)
−0.0214585 + 0.999770i \(0.506831\pi\)
\(938\) −8.00000 −0.261209
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 5.85786 0.190961 0.0954805 0.995431i \(-0.469561\pi\)
0.0954805 + 0.995431i \(0.469561\pi\)
\(942\) −24.1421 −0.786593
\(943\) 43.3137 1.41049
\(944\) −10.9706 −0.357061
\(945\) 0 0
\(946\) 46.6274 1.51599
\(947\) 54.9706 1.78630 0.893152 0.449756i \(-0.148489\pi\)
0.893152 + 0.449756i \(0.148489\pi\)
\(948\) −43.3137 −1.40676
\(949\) −11.6569 −0.378398
\(950\) 0 0
\(951\) −8.48528 −0.275154
\(952\) 45.6569 1.47975
\(953\) −51.6569 −1.67333 −0.836665 0.547715i \(-0.815498\pi\)
−0.836665 + 0.547715i \(0.815498\pi\)
\(954\) 4.82843 0.156326
\(955\) 0 0
\(956\) 7.65685 0.247640
\(957\) 4.00000 0.129302
\(958\) 6.48528 0.209530
\(959\) 14.6274 0.472344
\(960\) 0 0
\(961\) 15.6274 0.504110
\(962\) −8.82843 −0.284640
\(963\) −11.3137 −0.364579
\(964\) 44.6274 1.43735
\(965\) 0 0
\(966\) −27.3137 −0.878804
\(967\) 10.1421 0.326149 0.163075 0.986614i \(-0.447859\pi\)
0.163075 + 0.986614i \(0.447859\pi\)
\(968\) −30.8995 −0.993147
\(969\) −10.3431 −0.332270
\(970\) 0 0
\(971\) 7.31371 0.234708 0.117354 0.993090i \(-0.462559\pi\)
0.117354 + 0.993090i \(0.462559\pi\)
\(972\) −3.82843 −0.122797
\(973\) 43.3137 1.38857
\(974\) −76.7696 −2.45986
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) −13.8579 −0.443352 −0.221676 0.975120i \(-0.571153\pi\)
−0.221676 + 0.975120i \(0.571153\pi\)
\(978\) 31.7990 1.01682
\(979\) −18.3431 −0.586249
\(980\) 0 0
\(981\) −17.3137 −0.552784
\(982\) −35.3137 −1.12691
\(983\) −2.68629 −0.0856794 −0.0428397 0.999082i \(-0.513640\pi\)
−0.0428397 + 0.999082i \(0.513640\pi\)
\(984\) −47.7990 −1.52378
\(985\) 0 0
\(986\) 17.6569 0.562309
\(987\) −0.970563 −0.0308934
\(988\) 10.8284 0.344498
\(989\) −38.6274 −1.22828
\(990\) 0 0
\(991\) 27.3137 0.867649 0.433824 0.900998i \(-0.357164\pi\)
0.433824 + 0.900998i \(0.357164\pi\)
\(992\) 10.8284 0.343803
\(993\) 2.14214 0.0679786
\(994\) 13.6569 0.433169
\(995\) 0 0
\(996\) −29.3137 −0.928840
\(997\) −51.2548 −1.62326 −0.811628 0.584174i \(-0.801419\pi\)
−0.811628 + 0.584174i \(0.801419\pi\)
\(998\) −5.17157 −0.163703
\(999\) 3.65685 0.115698
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 975.2.a.l.1.2 2
3.2 odd 2 2925.2.a.v.1.1 2
5.2 odd 4 975.2.c.h.274.4 4
5.3 odd 4 975.2.c.h.274.1 4
5.4 even 2 39.2.a.b.1.1 2
15.2 even 4 2925.2.c.u.2224.1 4
15.8 even 4 2925.2.c.u.2224.4 4
15.14 odd 2 117.2.a.c.1.2 2
20.19 odd 2 624.2.a.k.1.2 2
35.34 odd 2 1911.2.a.h.1.1 2
40.19 odd 2 2496.2.a.bi.1.1 2
40.29 even 2 2496.2.a.bf.1.1 2
45.4 even 6 1053.2.e.m.703.2 4
45.14 odd 6 1053.2.e.e.703.1 4
45.29 odd 6 1053.2.e.e.352.1 4
45.34 even 6 1053.2.e.m.352.2 4
55.54 odd 2 4719.2.a.p.1.2 2
60.59 even 2 1872.2.a.w.1.1 2
65.4 even 6 507.2.e.d.484.1 4
65.9 even 6 507.2.e.h.484.2 4
65.19 odd 12 507.2.j.f.361.4 8
65.24 odd 12 507.2.j.f.316.4 8
65.29 even 6 507.2.e.h.22.2 4
65.34 odd 4 507.2.b.e.337.1 4
65.44 odd 4 507.2.b.e.337.4 4
65.49 even 6 507.2.e.d.22.1 4
65.54 odd 12 507.2.j.f.316.1 8
65.59 odd 12 507.2.j.f.361.1 8
65.64 even 2 507.2.a.h.1.2 2
105.104 even 2 5733.2.a.u.1.2 2
120.29 odd 2 7488.2.a.cl.1.2 2
120.59 even 2 7488.2.a.co.1.2 2
195.44 even 4 1521.2.b.j.1351.1 4
195.164 even 4 1521.2.b.j.1351.4 4
195.194 odd 2 1521.2.a.f.1.1 2
260.259 odd 2 8112.2.a.bm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.1 2 5.4 even 2
117.2.a.c.1.2 2 15.14 odd 2
507.2.a.h.1.2 2 65.64 even 2
507.2.b.e.337.1 4 65.34 odd 4
507.2.b.e.337.4 4 65.44 odd 4
507.2.e.d.22.1 4 65.49 even 6
507.2.e.d.484.1 4 65.4 even 6
507.2.e.h.22.2 4 65.29 even 6
507.2.e.h.484.2 4 65.9 even 6
507.2.j.f.316.1 8 65.54 odd 12
507.2.j.f.316.4 8 65.24 odd 12
507.2.j.f.361.1 8 65.59 odd 12
507.2.j.f.361.4 8 65.19 odd 12
624.2.a.k.1.2 2 20.19 odd 2
975.2.a.l.1.2 2 1.1 even 1 trivial
975.2.c.h.274.1 4 5.3 odd 4
975.2.c.h.274.4 4 5.2 odd 4
1053.2.e.e.352.1 4 45.29 odd 6
1053.2.e.e.703.1 4 45.14 odd 6
1053.2.e.m.352.2 4 45.34 even 6
1053.2.e.m.703.2 4 45.4 even 6
1521.2.a.f.1.1 2 195.194 odd 2
1521.2.b.j.1351.1 4 195.44 even 4
1521.2.b.j.1351.4 4 195.164 even 4
1872.2.a.w.1.1 2 60.59 even 2
1911.2.a.h.1.1 2 35.34 odd 2
2496.2.a.bf.1.1 2 40.29 even 2
2496.2.a.bi.1.1 2 40.19 odd 2
2925.2.a.v.1.1 2 3.2 odd 2
2925.2.c.u.2224.1 4 15.2 even 4
2925.2.c.u.2224.4 4 15.8 even 4
4719.2.a.p.1.2 2 55.54 odd 2
5733.2.a.u.1.2 2 105.104 even 2
7488.2.a.cl.1.2 2 120.29 odd 2
7488.2.a.co.1.2 2 120.59 even 2
8112.2.a.bm.1.1 2 260.259 odd 2