Properties

Label 750.2.a.d.1.1
Level $750$
Weight $2$
Character 750.1
Self dual yes
Analytic conductor $5.989$
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] = [750,2,Mod(1,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("750.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 750.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.38197 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.38197 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.85410 q^{11} +1.00000 q^{12} +3.38197 q^{13} -2.38197 q^{14} +1.00000 q^{16} +7.70820 q^{17} -1.00000 q^{18} +4.09017 q^{19} +2.38197 q^{21} +5.85410 q^{22} -3.09017 q^{23} -1.00000 q^{24} -3.38197 q^{26} +1.00000 q^{27} +2.38197 q^{28} +5.70820 q^{29} -6.47214 q^{31} -1.00000 q^{32} -5.85410 q^{33} -7.70820 q^{34} +1.00000 q^{36} -1.61803 q^{37} -4.09017 q^{38} +3.38197 q^{39} +0.381966 q^{41} -2.38197 q^{42} +7.70820 q^{43} -5.85410 q^{44} +3.09017 q^{46} +8.61803 q^{47} +1.00000 q^{48} -1.32624 q^{49} +7.70820 q^{51} +3.38197 q^{52} +0.381966 q^{53} -1.00000 q^{54} -2.38197 q^{56} +4.09017 q^{57} -5.70820 q^{58} -10.8541 q^{59} +11.7082 q^{61} +6.47214 q^{62} +2.38197 q^{63} +1.00000 q^{64} +5.85410 q^{66} +3.23607 q^{67} +7.70820 q^{68} -3.09017 q^{69} -4.47214 q^{71} -1.00000 q^{72} -8.00000 q^{73} +1.61803 q^{74} +4.09017 q^{76} -13.9443 q^{77} -3.38197 q^{78} +12.4721 q^{79} +1.00000 q^{81} -0.381966 q^{82} +2.00000 q^{83} +2.38197 q^{84} -7.70820 q^{86} +5.70820 q^{87} +5.85410 q^{88} +15.5623 q^{89} +8.05573 q^{91} -3.09017 q^{92} -6.47214 q^{93} -8.61803 q^{94} -1.00000 q^{96} -14.1803 q^{97} +1.32624 q^{98} -5.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 7 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 7 q^{7} - 2 q^{8} + 2 q^{9} - 5 q^{11} + 2 q^{12} + 9 q^{13} - 7 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 3 q^{19} + 7 q^{21} + 5 q^{22} + 5 q^{23} - 2 q^{24} - 9 q^{26} + 2 q^{27} + 7 q^{28} - 2 q^{29} - 4 q^{31} - 2 q^{32} - 5 q^{33} - 2 q^{34} + 2 q^{36} - q^{37} + 3 q^{38} + 9 q^{39} + 3 q^{41} - 7 q^{42} + 2 q^{43} - 5 q^{44} - 5 q^{46} + 15 q^{47} + 2 q^{48} + 13 q^{49} + 2 q^{51} + 9 q^{52} + 3 q^{53} - 2 q^{54} - 7 q^{56} - 3 q^{57} + 2 q^{58} - 15 q^{59} + 10 q^{61} + 4 q^{62} + 7 q^{63} + 2 q^{64} + 5 q^{66} + 2 q^{67} + 2 q^{68} + 5 q^{69} - 2 q^{72} - 16 q^{73} + q^{74} - 3 q^{76} - 10 q^{77} - 9 q^{78} + 16 q^{79} + 2 q^{81} - 3 q^{82} + 4 q^{83} + 7 q^{84} - 2 q^{86} - 2 q^{87} + 5 q^{88} + 11 q^{89} + 34 q^{91} + 5 q^{92} - 4 q^{93} - 15 q^{94} - 2 q^{96} - 6 q^{97} - 13 q^{98} - 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 2.38197 0.900299 0.450149 0.892953i \(-0.351371\pi\)
0.450149 + 0.892953i \(0.351371\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.85410 −1.76508 −0.882539 0.470239i \(-0.844168\pi\)
−0.882539 + 0.470239i \(0.844168\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.38197 0.937989 0.468994 0.883201i \(-0.344616\pi\)
0.468994 + 0.883201i \(0.344616\pi\)
\(14\) −2.38197 −0.636607
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.70820 1.86951 0.934757 0.355288i \(-0.115617\pi\)
0.934757 + 0.355288i \(0.115617\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.09017 0.938349 0.469175 0.883105i \(-0.344551\pi\)
0.469175 + 0.883105i \(0.344551\pi\)
\(20\) 0 0
\(21\) 2.38197 0.519788
\(22\) 5.85410 1.24810
\(23\) −3.09017 −0.644345 −0.322172 0.946681i \(-0.604413\pi\)
−0.322172 + 0.946681i \(0.604413\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −3.38197 −0.663258
\(27\) 1.00000 0.192450
\(28\) 2.38197 0.450149
\(29\) 5.70820 1.05999 0.529993 0.848002i \(-0.322194\pi\)
0.529993 + 0.848002i \(0.322194\pi\)
\(30\) 0 0
\(31\) −6.47214 −1.16243 −0.581215 0.813750i \(-0.697422\pi\)
−0.581215 + 0.813750i \(0.697422\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.85410 −1.01907
\(34\) −7.70820 −1.32195
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.61803 −0.266003 −0.133002 0.991116i \(-0.542462\pi\)
−0.133002 + 0.991116i \(0.542462\pi\)
\(38\) −4.09017 −0.663513
\(39\) 3.38197 0.541548
\(40\) 0 0
\(41\) 0.381966 0.0596531 0.0298265 0.999555i \(-0.490505\pi\)
0.0298265 + 0.999555i \(0.490505\pi\)
\(42\) −2.38197 −0.367545
\(43\) 7.70820 1.17549 0.587745 0.809046i \(-0.300016\pi\)
0.587745 + 0.809046i \(0.300016\pi\)
\(44\) −5.85410 −0.882539
\(45\) 0 0
\(46\) 3.09017 0.455621
\(47\) 8.61803 1.25707 0.628535 0.777782i \(-0.283655\pi\)
0.628535 + 0.777782i \(0.283655\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.32624 −0.189463
\(50\) 0 0
\(51\) 7.70820 1.07936
\(52\) 3.38197 0.468994
\(53\) 0.381966 0.0524671 0.0262335 0.999656i \(-0.491649\pi\)
0.0262335 + 0.999656i \(0.491649\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.38197 −0.318304
\(57\) 4.09017 0.541756
\(58\) −5.70820 −0.749524
\(59\) −10.8541 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(60\) 0 0
\(61\) 11.7082 1.49908 0.749541 0.661958i \(-0.230274\pi\)
0.749541 + 0.661958i \(0.230274\pi\)
\(62\) 6.47214 0.821962
\(63\) 2.38197 0.300100
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.85410 0.720590
\(67\) 3.23607 0.395349 0.197674 0.980268i \(-0.436661\pi\)
0.197674 + 0.980268i \(0.436661\pi\)
\(68\) 7.70820 0.934757
\(69\) −3.09017 −0.372013
\(70\) 0 0
\(71\) −4.47214 −0.530745 −0.265372 0.964146i \(-0.585495\pi\)
−0.265372 + 0.964146i \(0.585495\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 1.61803 0.188093
\(75\) 0 0
\(76\) 4.09017 0.469175
\(77\) −13.9443 −1.58910
\(78\) −3.38197 −0.382932
\(79\) 12.4721 1.40322 0.701612 0.712559i \(-0.252464\pi\)
0.701612 + 0.712559i \(0.252464\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.381966 −0.0421811
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 2.38197 0.259894
\(85\) 0 0
\(86\) −7.70820 −0.831197
\(87\) 5.70820 0.611984
\(88\) 5.85410 0.624049
\(89\) 15.5623 1.64960 0.824801 0.565424i \(-0.191287\pi\)
0.824801 + 0.565424i \(0.191287\pi\)
\(90\) 0 0
\(91\) 8.05573 0.844470
\(92\) −3.09017 −0.322172
\(93\) −6.47214 −0.671129
\(94\) −8.61803 −0.888882
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −14.1803 −1.43980 −0.719898 0.694080i \(-0.755811\pi\)
−0.719898 + 0.694080i \(0.755811\pi\)
\(98\) 1.32624 0.133970
\(99\) −5.85410 −0.588359
\(100\) 0 0
\(101\) −13.7082 −1.36402 −0.682009 0.731344i \(-0.738893\pi\)
−0.682009 + 0.731344i \(0.738893\pi\)
\(102\) −7.70820 −0.763226
\(103\) 2.90983 0.286714 0.143357 0.989671i \(-0.454210\pi\)
0.143357 + 0.989671i \(0.454210\pi\)
\(104\) −3.38197 −0.331629
\(105\) 0 0
\(106\) −0.381966 −0.0370998
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 1.00000 0.0962250
\(109\) −1.70820 −0.163616 −0.0818081 0.996648i \(-0.526069\pi\)
−0.0818081 + 0.996648i \(0.526069\pi\)
\(110\) 0 0
\(111\) −1.61803 −0.153577
\(112\) 2.38197 0.225075
\(113\) 3.70820 0.348838 0.174419 0.984671i \(-0.444195\pi\)
0.174419 + 0.984671i \(0.444195\pi\)
\(114\) −4.09017 −0.383080
\(115\) 0 0
\(116\) 5.70820 0.529993
\(117\) 3.38197 0.312663
\(118\) 10.8541 0.999201
\(119\) 18.3607 1.68312
\(120\) 0 0
\(121\) 23.2705 2.11550
\(122\) −11.7082 −1.06001
\(123\) 0.381966 0.0344407
\(124\) −6.47214 −0.581215
\(125\) 0 0
\(126\) −2.38197 −0.212202
\(127\) −5.52786 −0.490519 −0.245259 0.969458i \(-0.578873\pi\)
−0.245259 + 0.969458i \(0.578873\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.70820 0.678670
\(130\) 0 0
\(131\) −20.3262 −1.77591 −0.887956 0.459929i \(-0.847875\pi\)
−0.887956 + 0.459929i \(0.847875\pi\)
\(132\) −5.85410 −0.509534
\(133\) 9.74265 0.844795
\(134\) −3.23607 −0.279554
\(135\) 0 0
\(136\) −7.70820 −0.660973
\(137\) 4.94427 0.422418 0.211209 0.977441i \(-0.432260\pi\)
0.211209 + 0.977441i \(0.432260\pi\)
\(138\) 3.09017 0.263053
\(139\) −2.61803 −0.222059 −0.111029 0.993817i \(-0.535415\pi\)
−0.111029 + 0.993817i \(0.535415\pi\)
\(140\) 0 0
\(141\) 8.61803 0.725769
\(142\) 4.47214 0.375293
\(143\) −19.7984 −1.65562
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 8.00000 0.662085
\(147\) −1.32624 −0.109386
\(148\) −1.61803 −0.133002
\(149\) −9.41641 −0.771422 −0.385711 0.922620i \(-0.626044\pi\)
−0.385711 + 0.922620i \(0.626044\pi\)
\(150\) 0 0
\(151\) −0.291796 −0.0237460 −0.0118730 0.999930i \(-0.503779\pi\)
−0.0118730 + 0.999930i \(0.503779\pi\)
\(152\) −4.09017 −0.331757
\(153\) 7.70820 0.623171
\(154\) 13.9443 1.12366
\(155\) 0 0
\(156\) 3.38197 0.270774
\(157\) −6.94427 −0.554213 −0.277107 0.960839i \(-0.589376\pi\)
−0.277107 + 0.960839i \(0.589376\pi\)
\(158\) −12.4721 −0.992230
\(159\) 0.381966 0.0302919
\(160\) 0 0
\(161\) −7.36068 −0.580103
\(162\) −1.00000 −0.0785674
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0.381966 0.0298265
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −1.90983 −0.147787 −0.0738935 0.997266i \(-0.523543\pi\)
−0.0738935 + 0.997266i \(0.523543\pi\)
\(168\) −2.38197 −0.183773
\(169\) −1.56231 −0.120177
\(170\) 0 0
\(171\) 4.09017 0.312783
\(172\) 7.70820 0.587745
\(173\) 4.38197 0.333155 0.166577 0.986028i \(-0.446728\pi\)
0.166577 + 0.986028i \(0.446728\pi\)
\(174\) −5.70820 −0.432738
\(175\) 0 0
\(176\) −5.85410 −0.441270
\(177\) −10.8541 −0.815844
\(178\) −15.5623 −1.16644
\(179\) 3.79837 0.283904 0.141952 0.989874i \(-0.454662\pi\)
0.141952 + 0.989874i \(0.454662\pi\)
\(180\) 0 0
\(181\) −20.4721 −1.52168 −0.760841 0.648938i \(-0.775213\pi\)
−0.760841 + 0.648938i \(0.775213\pi\)
\(182\) −8.05573 −0.597130
\(183\) 11.7082 0.865495
\(184\) 3.09017 0.227810
\(185\) 0 0
\(186\) 6.47214 0.474560
\(187\) −45.1246 −3.29984
\(188\) 8.61803 0.628535
\(189\) 2.38197 0.173263
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.70820 0.698812 0.349406 0.936971i \(-0.386383\pi\)
0.349406 + 0.936971i \(0.386383\pi\)
\(194\) 14.1803 1.01809
\(195\) 0 0
\(196\) −1.32624 −0.0947313
\(197\) −14.9443 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(198\) 5.85410 0.416033
\(199\) −4.76393 −0.337706 −0.168853 0.985641i \(-0.554006\pi\)
−0.168853 + 0.985641i \(0.554006\pi\)
\(200\) 0 0
\(201\) 3.23607 0.228255
\(202\) 13.7082 0.964506
\(203\) 13.5967 0.954305
\(204\) 7.70820 0.539682
\(205\) 0 0
\(206\) −2.90983 −0.202737
\(207\) −3.09017 −0.214782
\(208\) 3.38197 0.234497
\(209\) −23.9443 −1.65626
\(210\) 0 0
\(211\) −22.5623 −1.55325 −0.776627 0.629961i \(-0.783071\pi\)
−0.776627 + 0.629961i \(0.783071\pi\)
\(212\) 0.381966 0.0262335
\(213\) −4.47214 −0.306426
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −15.4164 −1.04653
\(218\) 1.70820 0.115694
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 26.0689 1.75358
\(222\) 1.61803 0.108595
\(223\) −28.9443 −1.93825 −0.969126 0.246566i \(-0.920698\pi\)
−0.969126 + 0.246566i \(0.920698\pi\)
\(224\) −2.38197 −0.159152
\(225\) 0 0
\(226\) −3.70820 −0.246666
\(227\) 10.7639 0.714427 0.357214 0.934023i \(-0.383727\pi\)
0.357214 + 0.934023i \(0.383727\pi\)
\(228\) 4.09017 0.270878
\(229\) −22.7639 −1.50428 −0.752141 0.659002i \(-0.770979\pi\)
−0.752141 + 0.659002i \(0.770979\pi\)
\(230\) 0 0
\(231\) −13.9443 −0.917466
\(232\) −5.70820 −0.374762
\(233\) −2.47214 −0.161955 −0.0809775 0.996716i \(-0.525804\pi\)
−0.0809775 + 0.996716i \(0.525804\pi\)
\(234\) −3.38197 −0.221086
\(235\) 0 0
\(236\) −10.8541 −0.706542
\(237\) 12.4721 0.810152
\(238\) −18.3607 −1.19015
\(239\) −7.05573 −0.456397 −0.228199 0.973615i \(-0.573284\pi\)
−0.228199 + 0.973615i \(0.573284\pi\)
\(240\) 0 0
\(241\) 15.0902 0.972043 0.486022 0.873947i \(-0.338448\pi\)
0.486022 + 0.873947i \(0.338448\pi\)
\(242\) −23.2705 −1.49589
\(243\) 1.00000 0.0641500
\(244\) 11.7082 0.749541
\(245\) 0 0
\(246\) −0.381966 −0.0243533
\(247\) 13.8328 0.880161
\(248\) 6.47214 0.410981
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 2.38197 0.150050
\(253\) 18.0902 1.13732
\(254\) 5.52786 0.346849
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 17.5279 1.09336 0.546679 0.837342i \(-0.315892\pi\)
0.546679 + 0.837342i \(0.315892\pi\)
\(258\) −7.70820 −0.479892
\(259\) −3.85410 −0.239482
\(260\) 0 0
\(261\) 5.70820 0.353329
\(262\) 20.3262 1.25576
\(263\) 25.3262 1.56168 0.780841 0.624729i \(-0.214791\pi\)
0.780841 + 0.624729i \(0.214791\pi\)
\(264\) 5.85410 0.360295
\(265\) 0 0
\(266\) −9.74265 −0.597360
\(267\) 15.5623 0.952398
\(268\) 3.23607 0.197674
\(269\) −9.41641 −0.574129 −0.287064 0.957911i \(-0.592679\pi\)
−0.287064 + 0.957911i \(0.592679\pi\)
\(270\) 0 0
\(271\) −9.41641 −0.572006 −0.286003 0.958229i \(-0.592327\pi\)
−0.286003 + 0.958229i \(0.592327\pi\)
\(272\) 7.70820 0.467379
\(273\) 8.05573 0.487555
\(274\) −4.94427 −0.298694
\(275\) 0 0
\(276\) −3.09017 −0.186006
\(277\) 27.4508 1.64936 0.824681 0.565598i \(-0.191355\pi\)
0.824681 + 0.565598i \(0.191355\pi\)
\(278\) 2.61803 0.157019
\(279\) −6.47214 −0.387477
\(280\) 0 0
\(281\) 24.5066 1.46194 0.730970 0.682410i \(-0.239068\pi\)
0.730970 + 0.682410i \(0.239068\pi\)
\(282\) −8.61803 −0.513196
\(283\) 8.18034 0.486271 0.243135 0.969992i \(-0.421824\pi\)
0.243135 + 0.969992i \(0.421824\pi\)
\(284\) −4.47214 −0.265372
\(285\) 0 0
\(286\) 19.7984 1.17070
\(287\) 0.909830 0.0537056
\(288\) −1.00000 −0.0589256
\(289\) 42.4164 2.49508
\(290\) 0 0
\(291\) −14.1803 −0.831266
\(292\) −8.00000 −0.468165
\(293\) −20.7984 −1.21505 −0.607527 0.794299i \(-0.707838\pi\)
−0.607527 + 0.794299i \(0.707838\pi\)
\(294\) 1.32624 0.0773478
\(295\) 0 0
\(296\) 1.61803 0.0940463
\(297\) −5.85410 −0.339689
\(298\) 9.41641 0.545478
\(299\) −10.4508 −0.604388
\(300\) 0 0
\(301\) 18.3607 1.05829
\(302\) 0.291796 0.0167910
\(303\) −13.7082 −0.787516
\(304\) 4.09017 0.234587
\(305\) 0 0
\(306\) −7.70820 −0.440649
\(307\) −6.18034 −0.352731 −0.176365 0.984325i \(-0.556434\pi\)
−0.176365 + 0.984325i \(0.556434\pi\)
\(308\) −13.9443 −0.794549
\(309\) 2.90983 0.165534
\(310\) 0 0
\(311\) 8.94427 0.507183 0.253592 0.967311i \(-0.418388\pi\)
0.253592 + 0.967311i \(0.418388\pi\)
\(312\) −3.38197 −0.191466
\(313\) 0.291796 0.0164933 0.00824664 0.999966i \(-0.497375\pi\)
0.00824664 + 0.999966i \(0.497375\pi\)
\(314\) 6.94427 0.391888
\(315\) 0 0
\(316\) 12.4721 0.701612
\(317\) 23.8541 1.33978 0.669890 0.742460i \(-0.266341\pi\)
0.669890 + 0.742460i \(0.266341\pi\)
\(318\) −0.381966 −0.0214196
\(319\) −33.4164 −1.87096
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 7.36068 0.410195
\(323\) 31.5279 1.75426
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) −1.70820 −0.0944639
\(328\) −0.381966 −0.0210905
\(329\) 20.5279 1.13174
\(330\) 0 0
\(331\) 6.47214 0.355741 0.177870 0.984054i \(-0.443079\pi\)
0.177870 + 0.984054i \(0.443079\pi\)
\(332\) 2.00000 0.109764
\(333\) −1.61803 −0.0886677
\(334\) 1.90983 0.104501
\(335\) 0 0
\(336\) 2.38197 0.129947
\(337\) −20.8328 −1.13484 −0.567418 0.823430i \(-0.692058\pi\)
−0.567418 + 0.823430i \(0.692058\pi\)
\(338\) 1.56231 0.0849782
\(339\) 3.70820 0.201402
\(340\) 0 0
\(341\) 37.8885 2.05178
\(342\) −4.09017 −0.221171
\(343\) −19.8328 −1.07087
\(344\) −7.70820 −0.415599
\(345\) 0 0
\(346\) −4.38197 −0.235576
\(347\) −14.1803 −0.761241 −0.380620 0.924731i \(-0.624289\pi\)
−0.380620 + 0.924731i \(0.624289\pi\)
\(348\) 5.70820 0.305992
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 3.38197 0.180516
\(352\) 5.85410 0.312025
\(353\) −9.12461 −0.485654 −0.242827 0.970070i \(-0.578075\pi\)
−0.242827 + 0.970070i \(0.578075\pi\)
\(354\) 10.8541 0.576889
\(355\) 0 0
\(356\) 15.5623 0.824801
\(357\) 18.3607 0.971750
\(358\) −3.79837 −0.200750
\(359\) −17.4164 −0.919203 −0.459601 0.888125i \(-0.652008\pi\)
−0.459601 + 0.888125i \(0.652008\pi\)
\(360\) 0 0
\(361\) −2.27051 −0.119501
\(362\) 20.4721 1.07599
\(363\) 23.2705 1.22139
\(364\) 8.05573 0.422235
\(365\) 0 0
\(366\) −11.7082 −0.611998
\(367\) 15.4164 0.804730 0.402365 0.915479i \(-0.368188\pi\)
0.402365 + 0.915479i \(0.368188\pi\)
\(368\) −3.09017 −0.161086
\(369\) 0.381966 0.0198844
\(370\) 0 0
\(371\) 0.909830 0.0472360
\(372\) −6.47214 −0.335565
\(373\) −19.3262 −1.00067 −0.500337 0.865831i \(-0.666791\pi\)
−0.500337 + 0.865831i \(0.666791\pi\)
\(374\) 45.1246 2.33334
\(375\) 0 0
\(376\) −8.61803 −0.444441
\(377\) 19.3050 0.994256
\(378\) −2.38197 −0.122515
\(379\) 35.9230 1.84524 0.922620 0.385710i \(-0.126044\pi\)
0.922620 + 0.385710i \(0.126044\pi\)
\(380\) 0 0
\(381\) −5.52786 −0.283201
\(382\) 4.00000 0.204658
\(383\) 27.0344 1.38140 0.690698 0.723144i \(-0.257304\pi\)
0.690698 + 0.723144i \(0.257304\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −9.70820 −0.494135
\(387\) 7.70820 0.391830
\(388\) −14.1803 −0.719898
\(389\) −34.5410 −1.75130 −0.875650 0.482947i \(-0.839566\pi\)
−0.875650 + 0.482947i \(0.839566\pi\)
\(390\) 0 0
\(391\) −23.8197 −1.20461
\(392\) 1.32624 0.0669851
\(393\) −20.3262 −1.02532
\(394\) 14.9443 0.752882
\(395\) 0 0
\(396\) −5.85410 −0.294180
\(397\) 23.3262 1.17071 0.585355 0.810777i \(-0.300955\pi\)
0.585355 + 0.810777i \(0.300955\pi\)
\(398\) 4.76393 0.238794
\(399\) 9.74265 0.487742
\(400\) 0 0
\(401\) −1.32624 −0.0662292 −0.0331146 0.999452i \(-0.510543\pi\)
−0.0331146 + 0.999452i \(0.510543\pi\)
\(402\) −3.23607 −0.161400
\(403\) −21.8885 −1.09035
\(404\) −13.7082 −0.682009
\(405\) 0 0
\(406\) −13.5967 −0.674795
\(407\) 9.47214 0.469516
\(408\) −7.70820 −0.381613
\(409\) −4.85410 −0.240020 −0.120010 0.992773i \(-0.538293\pi\)
−0.120010 + 0.992773i \(0.538293\pi\)
\(410\) 0 0
\(411\) 4.94427 0.243883
\(412\) 2.90983 0.143357
\(413\) −25.8541 −1.27220
\(414\) 3.09017 0.151874
\(415\) 0 0
\(416\) −3.38197 −0.165815
\(417\) −2.61803 −0.128206
\(418\) 23.9443 1.17115
\(419\) 17.8885 0.873913 0.436956 0.899483i \(-0.356056\pi\)
0.436956 + 0.899483i \(0.356056\pi\)
\(420\) 0 0
\(421\) −25.1246 −1.22450 −0.612249 0.790665i \(-0.709735\pi\)
−0.612249 + 0.790665i \(0.709735\pi\)
\(422\) 22.5623 1.09832
\(423\) 8.61803 0.419023
\(424\) −0.381966 −0.0185499
\(425\) 0 0
\(426\) 4.47214 0.216676
\(427\) 27.8885 1.34962
\(428\) −2.00000 −0.0966736
\(429\) −19.7984 −0.955874
\(430\) 0 0
\(431\) 17.2361 0.830232 0.415116 0.909768i \(-0.363741\pi\)
0.415116 + 0.909768i \(0.363741\pi\)
\(432\) 1.00000 0.0481125
\(433\) 38.3607 1.84350 0.921748 0.387789i \(-0.126761\pi\)
0.921748 + 0.387789i \(0.126761\pi\)
\(434\) 15.4164 0.740011
\(435\) 0 0
\(436\) −1.70820 −0.0818081
\(437\) −12.6393 −0.604621
\(438\) 8.00000 0.382255
\(439\) −32.6525 −1.55842 −0.779209 0.626764i \(-0.784379\pi\)
−0.779209 + 0.626764i \(0.784379\pi\)
\(440\) 0 0
\(441\) −1.32624 −0.0631542
\(442\) −26.0689 −1.23997
\(443\) −13.1246 −0.623569 −0.311785 0.950153i \(-0.600927\pi\)
−0.311785 + 0.950153i \(0.600927\pi\)
\(444\) −1.61803 −0.0767885
\(445\) 0 0
\(446\) 28.9443 1.37055
\(447\) −9.41641 −0.445381
\(448\) 2.38197 0.112537
\(449\) 16.5623 0.781624 0.390812 0.920471i \(-0.372194\pi\)
0.390812 + 0.920471i \(0.372194\pi\)
\(450\) 0 0
\(451\) −2.23607 −0.105292
\(452\) 3.70820 0.174419
\(453\) −0.291796 −0.0137098
\(454\) −10.7639 −0.505176
\(455\) 0 0
\(456\) −4.09017 −0.191540
\(457\) 33.8885 1.58524 0.792620 0.609716i \(-0.208717\pi\)
0.792620 + 0.609716i \(0.208717\pi\)
\(458\) 22.7639 1.06369
\(459\) 7.70820 0.359788
\(460\) 0 0
\(461\) 19.5967 0.912712 0.456356 0.889797i \(-0.349154\pi\)
0.456356 + 0.889797i \(0.349154\pi\)
\(462\) 13.9443 0.648746
\(463\) −15.0557 −0.699699 −0.349850 0.936806i \(-0.613767\pi\)
−0.349850 + 0.936806i \(0.613767\pi\)
\(464\) 5.70820 0.264997
\(465\) 0 0
\(466\) 2.47214 0.114519
\(467\) 21.2361 0.982688 0.491344 0.870966i \(-0.336506\pi\)
0.491344 + 0.870966i \(0.336506\pi\)
\(468\) 3.38197 0.156331
\(469\) 7.70820 0.355932
\(470\) 0 0
\(471\) −6.94427 −0.319975
\(472\) 10.8541 0.499601
\(473\) −45.1246 −2.07483
\(474\) −12.4721 −0.572864
\(475\) 0 0
\(476\) 18.3607 0.841560
\(477\) 0.381966 0.0174890
\(478\) 7.05573 0.322721
\(479\) −9.12461 −0.416914 −0.208457 0.978032i \(-0.566844\pi\)
−0.208457 + 0.978032i \(0.566844\pi\)
\(480\) 0 0
\(481\) −5.47214 −0.249508
\(482\) −15.0902 −0.687338
\(483\) −7.36068 −0.334923
\(484\) 23.2705 1.05775
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −12.7984 −0.579950 −0.289975 0.957034i \(-0.593647\pi\)
−0.289975 + 0.957034i \(0.593647\pi\)
\(488\) −11.7082 −0.530005
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 11.0344 0.497977 0.248989 0.968506i \(-0.419902\pi\)
0.248989 + 0.968506i \(0.419902\pi\)
\(492\) 0.381966 0.0172204
\(493\) 44.0000 1.98166
\(494\) −13.8328 −0.622368
\(495\) 0 0
\(496\) −6.47214 −0.290607
\(497\) −10.6525 −0.477829
\(498\) −2.00000 −0.0896221
\(499\) −5.38197 −0.240930 −0.120465 0.992718i \(-0.538439\pi\)
−0.120465 + 0.992718i \(0.538439\pi\)
\(500\) 0 0
\(501\) −1.90983 −0.0853249
\(502\) 24.0000 1.07117
\(503\) −30.1459 −1.34414 −0.672070 0.740488i \(-0.734594\pi\)
−0.672070 + 0.740488i \(0.734594\pi\)
\(504\) −2.38197 −0.106101
\(505\) 0 0
\(506\) −18.0902 −0.804206
\(507\) −1.56231 −0.0693844
\(508\) −5.52786 −0.245259
\(509\) 12.2918 0.544824 0.272412 0.962181i \(-0.412179\pi\)
0.272412 + 0.962181i \(0.412179\pi\)
\(510\) 0 0
\(511\) −19.0557 −0.842976
\(512\) −1.00000 −0.0441942
\(513\) 4.09017 0.180585
\(514\) −17.5279 −0.773121
\(515\) 0 0
\(516\) 7.70820 0.339335
\(517\) −50.4508 −2.21883
\(518\) 3.85410 0.169340
\(519\) 4.38197 0.192347
\(520\) 0 0
\(521\) 1.09017 0.0477612 0.0238806 0.999715i \(-0.492398\pi\)
0.0238806 + 0.999715i \(0.492398\pi\)
\(522\) −5.70820 −0.249841
\(523\) 32.3607 1.41503 0.707517 0.706696i \(-0.249815\pi\)
0.707517 + 0.706696i \(0.249815\pi\)
\(524\) −20.3262 −0.887956
\(525\) 0 0
\(526\) −25.3262 −1.10428
\(527\) −49.8885 −2.17318
\(528\) −5.85410 −0.254767
\(529\) −13.4508 −0.584820
\(530\) 0 0
\(531\) −10.8541 −0.471028
\(532\) 9.74265 0.422397
\(533\) 1.29180 0.0559539
\(534\) −15.5623 −0.673447
\(535\) 0 0
\(536\) −3.23607 −0.139777
\(537\) 3.79837 0.163912
\(538\) 9.41641 0.405970
\(539\) 7.76393 0.334416
\(540\) 0 0
\(541\) −13.4164 −0.576816 −0.288408 0.957508i \(-0.593126\pi\)
−0.288408 + 0.957508i \(0.593126\pi\)
\(542\) 9.41641 0.404469
\(543\) −20.4721 −0.878543
\(544\) −7.70820 −0.330487
\(545\) 0 0
\(546\) −8.05573 −0.344753
\(547\) −0.763932 −0.0326634 −0.0163317 0.999867i \(-0.505199\pi\)
−0.0163317 + 0.999867i \(0.505199\pi\)
\(548\) 4.94427 0.211209
\(549\) 11.7082 0.499694
\(550\) 0 0
\(551\) 23.3475 0.994638
\(552\) 3.09017 0.131526
\(553\) 29.7082 1.26332
\(554\) −27.4508 −1.16627
\(555\) 0 0
\(556\) −2.61803 −0.111029
\(557\) 13.4508 0.569931 0.284965 0.958538i \(-0.408018\pi\)
0.284965 + 0.958538i \(0.408018\pi\)
\(558\) 6.47214 0.273987
\(559\) 26.0689 1.10260
\(560\) 0 0
\(561\) −45.1246 −1.90516
\(562\) −24.5066 −1.03375
\(563\) 6.94427 0.292666 0.146333 0.989235i \(-0.453253\pi\)
0.146333 + 0.989235i \(0.453253\pi\)
\(564\) 8.61803 0.362885
\(565\) 0 0
\(566\) −8.18034 −0.343845
\(567\) 2.38197 0.100033
\(568\) 4.47214 0.187647
\(569\) 27.9230 1.17059 0.585296 0.810820i \(-0.300978\pi\)
0.585296 + 0.810820i \(0.300978\pi\)
\(570\) 0 0
\(571\) 4.27051 0.178715 0.0893576 0.996000i \(-0.471519\pi\)
0.0893576 + 0.996000i \(0.471519\pi\)
\(572\) −19.7984 −0.827812
\(573\) −4.00000 −0.167102
\(574\) −0.909830 −0.0379756
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 9.23607 0.384503 0.192251 0.981346i \(-0.438421\pi\)
0.192251 + 0.981346i \(0.438421\pi\)
\(578\) −42.4164 −1.76429
\(579\) 9.70820 0.403459
\(580\) 0 0
\(581\) 4.76393 0.197641
\(582\) 14.1803 0.587794
\(583\) −2.23607 −0.0926085
\(584\) 8.00000 0.331042
\(585\) 0 0
\(586\) 20.7984 0.859173
\(587\) 25.2361 1.04160 0.520802 0.853678i \(-0.325633\pi\)
0.520802 + 0.853678i \(0.325633\pi\)
\(588\) −1.32624 −0.0546931
\(589\) −26.4721 −1.09077
\(590\) 0 0
\(591\) −14.9443 −0.614725
\(592\) −1.61803 −0.0665008
\(593\) 31.3050 1.28554 0.642770 0.766059i \(-0.277785\pi\)
0.642770 + 0.766059i \(0.277785\pi\)
\(594\) 5.85410 0.240197
\(595\) 0 0
\(596\) −9.41641 −0.385711
\(597\) −4.76393 −0.194975
\(598\) 10.4508 0.427367
\(599\) −6.76393 −0.276367 −0.138183 0.990407i \(-0.544126\pi\)
−0.138183 + 0.990407i \(0.544126\pi\)
\(600\) 0 0
\(601\) 10.4377 0.425762 0.212881 0.977078i \(-0.431715\pi\)
0.212881 + 0.977078i \(0.431715\pi\)
\(602\) −18.3607 −0.748325
\(603\) 3.23607 0.131783
\(604\) −0.291796 −0.0118730
\(605\) 0 0
\(606\) 13.7082 0.556858
\(607\) −9.97871 −0.405023 −0.202512 0.979280i \(-0.564910\pi\)
−0.202512 + 0.979280i \(0.564910\pi\)
\(608\) −4.09017 −0.165878
\(609\) 13.5967 0.550968
\(610\) 0 0
\(611\) 29.1459 1.17912
\(612\) 7.70820 0.311586
\(613\) −31.7426 −1.28207 −0.641037 0.767510i \(-0.721495\pi\)
−0.641037 + 0.767510i \(0.721495\pi\)
\(614\) 6.18034 0.249418
\(615\) 0 0
\(616\) 13.9443 0.561831
\(617\) 5.52786 0.222543 0.111272 0.993790i \(-0.464508\pi\)
0.111272 + 0.993790i \(0.464508\pi\)
\(618\) −2.90983 −0.117051
\(619\) −11.8541 −0.476457 −0.238228 0.971209i \(-0.576567\pi\)
−0.238228 + 0.971209i \(0.576567\pi\)
\(620\) 0 0
\(621\) −3.09017 −0.124004
\(622\) −8.94427 −0.358633
\(623\) 37.0689 1.48513
\(624\) 3.38197 0.135387
\(625\) 0 0
\(626\) −0.291796 −0.0116625
\(627\) −23.9443 −0.956242
\(628\) −6.94427 −0.277107
\(629\) −12.4721 −0.497297
\(630\) 0 0
\(631\) 33.7082 1.34190 0.670951 0.741502i \(-0.265886\pi\)
0.670951 + 0.741502i \(0.265886\pi\)
\(632\) −12.4721 −0.496115
\(633\) −22.5623 −0.896771
\(634\) −23.8541 −0.947367
\(635\) 0 0
\(636\) 0.381966 0.0151459
\(637\) −4.48529 −0.177714
\(638\) 33.4164 1.32297
\(639\) −4.47214 −0.176915
\(640\) 0 0
\(641\) −29.1459 −1.15119 −0.575597 0.817734i \(-0.695230\pi\)
−0.575597 + 0.817734i \(0.695230\pi\)
\(642\) 2.00000 0.0789337
\(643\) −15.0557 −0.593740 −0.296870 0.954918i \(-0.595943\pi\)
−0.296870 + 0.954918i \(0.595943\pi\)
\(644\) −7.36068 −0.290051
\(645\) 0 0
\(646\) −31.5279 −1.24045
\(647\) 14.8541 0.583975 0.291988 0.956422i \(-0.405683\pi\)
0.291988 + 0.956422i \(0.405683\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 63.5410 2.49420
\(650\) 0 0
\(651\) −15.4164 −0.604217
\(652\) −24.0000 −0.939913
\(653\) 12.1459 0.475306 0.237653 0.971350i \(-0.423622\pi\)
0.237653 + 0.971350i \(0.423622\pi\)
\(654\) 1.70820 0.0667961
\(655\) 0 0
\(656\) 0.381966 0.0149133
\(657\) −8.00000 −0.312110
\(658\) −20.5279 −0.800259
\(659\) 14.5066 0.565096 0.282548 0.959253i \(-0.408820\pi\)
0.282548 + 0.959253i \(0.408820\pi\)
\(660\) 0 0
\(661\) −0.875388 −0.0340487 −0.0170243 0.999855i \(-0.505419\pi\)
−0.0170243 + 0.999855i \(0.505419\pi\)
\(662\) −6.47214 −0.251547
\(663\) 26.0689 1.01243
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 1.61803 0.0626975
\(667\) −17.6393 −0.682997
\(668\) −1.90983 −0.0738935
\(669\) −28.9443 −1.11905
\(670\) 0 0
\(671\) −68.5410 −2.64600
\(672\) −2.38197 −0.0918863
\(673\) −9.52786 −0.367272 −0.183636 0.982994i \(-0.558787\pi\)
−0.183636 + 0.982994i \(0.558787\pi\)
\(674\) 20.8328 0.802450
\(675\) 0 0
\(676\) −1.56231 −0.0600887
\(677\) −24.7426 −0.950937 −0.475469 0.879733i \(-0.657721\pi\)
−0.475469 + 0.879733i \(0.657721\pi\)
\(678\) −3.70820 −0.142413
\(679\) −33.7771 −1.29625
\(680\) 0 0
\(681\) 10.7639 0.412475
\(682\) −37.8885 −1.45083
\(683\) −29.8885 −1.14365 −0.571827 0.820374i \(-0.693765\pi\)
−0.571827 + 0.820374i \(0.693765\pi\)
\(684\) 4.09017 0.156392
\(685\) 0 0
\(686\) 19.8328 0.757220
\(687\) −22.7639 −0.868498
\(688\) 7.70820 0.293873
\(689\) 1.29180 0.0492135
\(690\) 0 0
\(691\) −33.8885 −1.28918 −0.644590 0.764528i \(-0.722972\pi\)
−0.644590 + 0.764528i \(0.722972\pi\)
\(692\) 4.38197 0.166577
\(693\) −13.9443 −0.529699
\(694\) 14.1803 0.538278
\(695\) 0 0
\(696\) −5.70820 −0.216369
\(697\) 2.94427 0.111522
\(698\) 20.0000 0.757011
\(699\) −2.47214 −0.0935048
\(700\) 0 0
\(701\) −17.7082 −0.668830 −0.334415 0.942426i \(-0.608539\pi\)
−0.334415 + 0.942426i \(0.608539\pi\)
\(702\) −3.38197 −0.127644
\(703\) −6.61803 −0.249604
\(704\) −5.85410 −0.220635
\(705\) 0 0
\(706\) 9.12461 0.343409
\(707\) −32.6525 −1.22802
\(708\) −10.8541 −0.407922
\(709\) 18.6525 0.700508 0.350254 0.936655i \(-0.386095\pi\)
0.350254 + 0.936655i \(0.386095\pi\)
\(710\) 0 0
\(711\) 12.4721 0.467742
\(712\) −15.5623 −0.583222
\(713\) 20.0000 0.749006
\(714\) −18.3607 −0.687131
\(715\) 0 0
\(716\) 3.79837 0.141952
\(717\) −7.05573 −0.263501
\(718\) 17.4164 0.649975
\(719\) 12.1115 0.451681 0.225841 0.974164i \(-0.427487\pi\)
0.225841 + 0.974164i \(0.427487\pi\)
\(720\) 0 0
\(721\) 6.93112 0.258128
\(722\) 2.27051 0.0844996
\(723\) 15.0902 0.561209
\(724\) −20.4721 −0.760841
\(725\) 0 0
\(726\) −23.2705 −0.863650
\(727\) −12.6738 −0.470044 −0.235022 0.971990i \(-0.575516\pi\)
−0.235022 + 0.971990i \(0.575516\pi\)
\(728\) −8.05573 −0.298565
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 59.4164 2.19760
\(732\) 11.7082 0.432748
\(733\) 17.9230 0.662001 0.331000 0.943631i \(-0.392614\pi\)
0.331000 + 0.943631i \(0.392614\pi\)
\(734\) −15.4164 −0.569030
\(735\) 0 0
\(736\) 3.09017 0.113905
\(737\) −18.9443 −0.697821
\(738\) −0.381966 −0.0140604
\(739\) 23.3262 0.858070 0.429035 0.903288i \(-0.358854\pi\)
0.429035 + 0.903288i \(0.358854\pi\)
\(740\) 0 0
\(741\) 13.8328 0.508161
\(742\) −0.909830 −0.0334009
\(743\) −38.1459 −1.39944 −0.699719 0.714419i \(-0.746691\pi\)
−0.699719 + 0.714419i \(0.746691\pi\)
\(744\) 6.47214 0.237280
\(745\) 0 0
\(746\) 19.3262 0.707584
\(747\) 2.00000 0.0731762
\(748\) −45.1246 −1.64992
\(749\) −4.76393 −0.174070
\(750\) 0 0
\(751\) −4.06888 −0.148476 −0.0742378 0.997241i \(-0.523652\pi\)
−0.0742378 + 0.997241i \(0.523652\pi\)
\(752\) 8.61803 0.314267
\(753\) −24.0000 −0.874609
\(754\) −19.3050 −0.703045
\(755\) 0 0
\(756\) 2.38197 0.0866313
\(757\) 11.6738 0.424290 0.212145 0.977238i \(-0.431955\pi\)
0.212145 + 0.977238i \(0.431955\pi\)
\(758\) −35.9230 −1.30478
\(759\) 18.0902 0.656632
\(760\) 0 0
\(761\) 55.0344 1.99500 0.997498 0.0706879i \(-0.0225194\pi\)
0.997498 + 0.0706879i \(0.0225194\pi\)
\(762\) 5.52786 0.200253
\(763\) −4.06888 −0.147303
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −27.0344 −0.976794
\(767\) −36.7082 −1.32546
\(768\) 1.00000 0.0360844
\(769\) 11.6738 0.420967 0.210483 0.977597i \(-0.432496\pi\)
0.210483 + 0.977597i \(0.432496\pi\)
\(770\) 0 0
\(771\) 17.5279 0.631251
\(772\) 9.70820 0.349406
\(773\) −26.9443 −0.969118 −0.484559 0.874759i \(-0.661020\pi\)
−0.484559 + 0.874759i \(0.661020\pi\)
\(774\) −7.70820 −0.277066
\(775\) 0 0
\(776\) 14.1803 0.509045
\(777\) −3.85410 −0.138265
\(778\) 34.5410 1.23836
\(779\) 1.56231 0.0559754
\(780\) 0 0
\(781\) 26.1803 0.936806
\(782\) 23.8197 0.851789
\(783\) 5.70820 0.203995
\(784\) −1.32624 −0.0473656
\(785\) 0 0
\(786\) 20.3262 0.725013
\(787\) −10.5836 −0.377264 −0.188632 0.982048i \(-0.560405\pi\)
−0.188632 + 0.982048i \(0.560405\pi\)
\(788\) −14.9443 −0.532368
\(789\) 25.3262 0.901638
\(790\) 0 0
\(791\) 8.83282 0.314059
\(792\) 5.85410 0.208016
\(793\) 39.5967 1.40612
\(794\) −23.3262 −0.827817
\(795\) 0 0
\(796\) −4.76393 −0.168853
\(797\) −33.8673 −1.19964 −0.599820 0.800135i \(-0.704761\pi\)
−0.599820 + 0.800135i \(0.704761\pi\)
\(798\) −9.74265 −0.344886
\(799\) 66.4296 2.35011
\(800\) 0 0
\(801\) 15.5623 0.549867
\(802\) 1.32624 0.0468311
\(803\) 46.8328 1.65269
\(804\) 3.23607 0.114127
\(805\) 0 0
\(806\) 21.8885 0.770991
\(807\) −9.41641 −0.331473
\(808\) 13.7082 0.482253
\(809\) 46.3262 1.62874 0.814372 0.580343i \(-0.197082\pi\)
0.814372 + 0.580343i \(0.197082\pi\)
\(810\) 0 0
\(811\) −3.27051 −0.114843 −0.0574216 0.998350i \(-0.518288\pi\)
−0.0574216 + 0.998350i \(0.518288\pi\)
\(812\) 13.5967 0.477152
\(813\) −9.41641 −0.330248
\(814\) −9.47214 −0.331998
\(815\) 0 0
\(816\) 7.70820 0.269841
\(817\) 31.5279 1.10302
\(818\) 4.85410 0.169720
\(819\) 8.05573 0.281490
\(820\) 0 0
\(821\) −29.8885 −1.04312 −0.521559 0.853215i \(-0.674649\pi\)
−0.521559 + 0.853215i \(0.674649\pi\)
\(822\) −4.94427 −0.172451
\(823\) −34.9230 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(824\) −2.90983 −0.101369
\(825\) 0 0
\(826\) 25.8541 0.899579
\(827\) 5.81966 0.202369 0.101185 0.994868i \(-0.467737\pi\)
0.101185 + 0.994868i \(0.467737\pi\)
\(828\) −3.09017 −0.107391
\(829\) 5.63932 0.195862 0.0979308 0.995193i \(-0.468778\pi\)
0.0979308 + 0.995193i \(0.468778\pi\)
\(830\) 0 0
\(831\) 27.4508 0.952259
\(832\) 3.38197 0.117249
\(833\) −10.2229 −0.354203
\(834\) 2.61803 0.0906551
\(835\) 0 0
\(836\) −23.9443 −0.828130
\(837\) −6.47214 −0.223710
\(838\) −17.8885 −0.617949
\(839\) 23.3475 0.806046 0.403023 0.915190i \(-0.367959\pi\)
0.403023 + 0.915190i \(0.367959\pi\)
\(840\) 0 0
\(841\) 3.58359 0.123572
\(842\) 25.1246 0.865851
\(843\) 24.5066 0.844051
\(844\) −22.5623 −0.776627
\(845\) 0 0
\(846\) −8.61803 −0.296294
\(847\) 55.4296 1.90458
\(848\) 0.381966 0.0131168
\(849\) 8.18034 0.280749
\(850\) 0 0
\(851\) 5.00000 0.171398
\(852\) −4.47214 −0.153213
\(853\) 48.2705 1.65275 0.826375 0.563120i \(-0.190399\pi\)
0.826375 + 0.563120i \(0.190399\pi\)
\(854\) −27.8885 −0.954326
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 3.70820 0.126670 0.0633349 0.997992i \(-0.479826\pi\)
0.0633349 + 0.997992i \(0.479826\pi\)
\(858\) 19.7984 0.675905
\(859\) −1.03444 −0.0352947 −0.0176474 0.999844i \(-0.505618\pi\)
−0.0176474 + 0.999844i \(0.505618\pi\)
\(860\) 0 0
\(861\) 0.909830 0.0310069
\(862\) −17.2361 −0.587063
\(863\) 11.0344 0.375617 0.187808 0.982206i \(-0.439862\pi\)
0.187808 + 0.982206i \(0.439862\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −38.3607 −1.30355
\(867\) 42.4164 1.44054
\(868\) −15.4164 −0.523267
\(869\) −73.0132 −2.47680
\(870\) 0 0
\(871\) 10.9443 0.370833
\(872\) 1.70820 0.0578471
\(873\) −14.1803 −0.479932
\(874\) 12.6393 0.427531
\(875\) 0 0
\(876\) −8.00000 −0.270295
\(877\) −8.97871 −0.303190 −0.151595 0.988443i \(-0.548441\pi\)
−0.151595 + 0.988443i \(0.548441\pi\)
\(878\) 32.6525 1.10197
\(879\) −20.7984 −0.701512
\(880\) 0 0
\(881\) −16.1591 −0.544412 −0.272206 0.962239i \(-0.587753\pi\)
−0.272206 + 0.962239i \(0.587753\pi\)
\(882\) 1.32624 0.0446568
\(883\) −7.41641 −0.249582 −0.124791 0.992183i \(-0.539826\pi\)
−0.124791 + 0.992183i \(0.539826\pi\)
\(884\) 26.0689 0.876791
\(885\) 0 0
\(886\) 13.1246 0.440930
\(887\) −27.0344 −0.907728 −0.453864 0.891071i \(-0.649955\pi\)
−0.453864 + 0.891071i \(0.649955\pi\)
\(888\) 1.61803 0.0542977
\(889\) −13.1672 −0.441613
\(890\) 0 0
\(891\) −5.85410 −0.196120
\(892\) −28.9443 −0.969126
\(893\) 35.2492 1.17957
\(894\) 9.41641 0.314932
\(895\) 0 0
\(896\) −2.38197 −0.0795759
\(897\) −10.4508 −0.348944
\(898\) −16.5623 −0.552691
\(899\) −36.9443 −1.23216
\(900\) 0 0
\(901\) 2.94427 0.0980879
\(902\) 2.23607 0.0744529
\(903\) 18.3607 0.611005
\(904\) −3.70820 −0.123333
\(905\) 0 0
\(906\) 0.291796 0.00969428
\(907\) −16.6525 −0.552936 −0.276468 0.961023i \(-0.589164\pi\)
−0.276468 + 0.961023i \(0.589164\pi\)
\(908\) 10.7639 0.357214
\(909\) −13.7082 −0.454672
\(910\) 0 0
\(911\) −29.0557 −0.962659 −0.481330 0.876540i \(-0.659846\pi\)
−0.481330 + 0.876540i \(0.659846\pi\)
\(912\) 4.09017 0.135439
\(913\) −11.7082 −0.387485
\(914\) −33.8885 −1.12093
\(915\) 0 0
\(916\) −22.7639 −0.752141
\(917\) −48.4164 −1.59885
\(918\) −7.70820 −0.254409
\(919\) 36.7639 1.21273 0.606365 0.795186i \(-0.292627\pi\)
0.606365 + 0.795186i \(0.292627\pi\)
\(920\) 0 0
\(921\) −6.18034 −0.203649
\(922\) −19.5967 −0.645385
\(923\) −15.1246 −0.497833
\(924\) −13.9443 −0.458733
\(925\) 0 0
\(926\) 15.0557 0.494762
\(927\) 2.90983 0.0955714
\(928\) −5.70820 −0.187381
\(929\) 41.8115 1.37179 0.685896 0.727700i \(-0.259411\pi\)
0.685896 + 0.727700i \(0.259411\pi\)
\(930\) 0 0
\(931\) −5.42454 −0.177782
\(932\) −2.47214 −0.0809775
\(933\) 8.94427 0.292822
\(934\) −21.2361 −0.694865
\(935\) 0 0
\(936\) −3.38197 −0.110543
\(937\) 6.65248 0.217327 0.108663 0.994079i \(-0.465343\pi\)
0.108663 + 0.994079i \(0.465343\pi\)
\(938\) −7.70820 −0.251682
\(939\) 0.291796 0.00952240
\(940\) 0 0
\(941\) −18.1803 −0.592662 −0.296331 0.955085i \(-0.595763\pi\)
−0.296331 + 0.955085i \(0.595763\pi\)
\(942\) 6.94427 0.226257
\(943\) −1.18034 −0.0384372
\(944\) −10.8541 −0.353271
\(945\) 0 0
\(946\) 45.1246 1.46713
\(947\) 13.8885 0.451317 0.225659 0.974206i \(-0.427547\pi\)
0.225659 + 0.974206i \(0.427547\pi\)
\(948\) 12.4721 0.405076
\(949\) −27.0557 −0.878266
\(950\) 0 0
\(951\) 23.8541 0.773522
\(952\) −18.3607 −0.595073
\(953\) −22.5410 −0.730175 −0.365088 0.930973i \(-0.618961\pi\)
−0.365088 + 0.930973i \(0.618961\pi\)
\(954\) −0.381966 −0.0123666
\(955\) 0 0
\(956\) −7.05573 −0.228199
\(957\) −33.4164 −1.08020
\(958\) 9.12461 0.294803
\(959\) 11.7771 0.380302
\(960\) 0 0
\(961\) 10.8885 0.351243
\(962\) 5.47214 0.176429
\(963\) −2.00000 −0.0644491
\(964\) 15.0902 0.486022
\(965\) 0 0
\(966\) 7.36068 0.236826
\(967\) 58.6869 1.88724 0.943622 0.331025i \(-0.107394\pi\)
0.943622 + 0.331025i \(0.107394\pi\)
\(968\) −23.2705 −0.747943
\(969\) 31.5279 1.01282
\(970\) 0 0
\(971\) −6.72949 −0.215960 −0.107980 0.994153i \(-0.534438\pi\)
−0.107980 + 0.994153i \(0.534438\pi\)
\(972\) 1.00000 0.0320750
\(973\) −6.23607 −0.199919
\(974\) 12.7984 0.410086
\(975\) 0 0
\(976\) 11.7082 0.374770
\(977\) 9.30495 0.297692 0.148846 0.988860i \(-0.452444\pi\)
0.148846 + 0.988860i \(0.452444\pi\)
\(978\) 24.0000 0.767435
\(979\) −91.1033 −2.91167
\(980\) 0 0
\(981\) −1.70820 −0.0545388
\(982\) −11.0344 −0.352123
\(983\) −22.2148 −0.708541 −0.354271 0.935143i \(-0.615271\pi\)
−0.354271 + 0.935143i \(0.615271\pi\)
\(984\) −0.381966 −0.0121766
\(985\) 0 0
\(986\) −44.0000 −1.40125
\(987\) 20.5279 0.653409
\(988\) 13.8328 0.440080
\(989\) −23.8197 −0.757421
\(990\) 0 0
\(991\) 16.1115 0.511797 0.255899 0.966704i \(-0.417629\pi\)
0.255899 + 0.966704i \(0.417629\pi\)
\(992\) 6.47214 0.205491
\(993\) 6.47214 0.205387
\(994\) 10.6525 0.337876
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) 4.09017 0.129537 0.0647685 0.997900i \(-0.479369\pi\)
0.0647685 + 0.997900i \(0.479369\pi\)
\(998\) 5.38197 0.170363
\(999\) −1.61803 −0.0511923
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 750.2.a.d.1.1 2
3.2 odd 2 2250.2.a.p.1.1 2
4.3 odd 2 6000.2.a.a.1.2 2
5.2 odd 4 750.2.c.a.499.1 4
5.3 odd 4 750.2.c.a.499.4 4
5.4 even 2 750.2.a.e.1.2 yes 2
15.2 even 4 2250.2.c.g.1999.3 4
15.8 even 4 2250.2.c.g.1999.2 4
15.14 odd 2 2250.2.a.a.1.2 2
20.3 even 4 6000.2.f.k.1249.1 4
20.7 even 4 6000.2.f.k.1249.4 4
20.19 odd 2 6000.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
750.2.a.d.1.1 2 1.1 even 1 trivial
750.2.a.e.1.2 yes 2 5.4 even 2
750.2.c.a.499.1 4 5.2 odd 4
750.2.c.a.499.4 4 5.3 odd 4
2250.2.a.a.1.2 2 15.14 odd 2
2250.2.a.p.1.1 2 3.2 odd 2
2250.2.c.g.1999.2 4 15.8 even 4
2250.2.c.g.1999.3 4 15.2 even 4
6000.2.a.a.1.2 2 4.3 odd 2
6000.2.a.bb.1.1 2 20.19 odd 2
6000.2.f.k.1249.1 4 20.3 even 4
6000.2.f.k.1249.4 4 20.7 even 4