Properties

Label 750.2.a.b.1.2
Level $750$
Weight $2$
Character 750.1
Self dual yes
Analytic conductor $5.989$
Analytic rank $1$
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: \(1\)
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 1.2
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.61803 q^{11} -1.00000 q^{12} -4.85410 q^{13} -0.618034 q^{14} +1.00000 q^{16} -0.763932 q^{17} -1.00000 q^{18} +5.85410 q^{19} -0.618034 q^{21} +1.61803 q^{22} -4.85410 q^{23} +1.00000 q^{24} +4.85410 q^{26} -1.00000 q^{27} +0.618034 q^{28} -2.76393 q^{29} -2.47214 q^{31} -1.00000 q^{32} +1.61803 q^{33} +0.763932 q^{34} +1.00000 q^{36} +9.56231 q^{37} -5.85410 q^{38} +4.85410 q^{39} -9.38197 q^{41} +0.618034 q^{42} -5.70820 q^{43} -1.61803 q^{44} +4.85410 q^{46} -1.61803 q^{47} -1.00000 q^{48} -6.61803 q^{49} +0.763932 q^{51} -4.85410 q^{52} -5.38197 q^{53} +1.00000 q^{54} -0.618034 q^{56} -5.85410 q^{57} +2.76393 q^{58} -13.0902 q^{59} -9.70820 q^{61} +2.47214 q^{62} +0.618034 q^{63} +1.00000 q^{64} -1.61803 q^{66} +3.70820 q^{67} -0.763932 q^{68} +4.85410 q^{69} -3.52786 q^{71} -1.00000 q^{72} -12.9443 q^{73} -9.56231 q^{74} +5.85410 q^{76} -1.00000 q^{77} -4.85410 q^{78} +13.4164 q^{79} +1.00000 q^{81} +9.38197 q^{82} +14.9443 q^{83} -0.618034 q^{84} +5.70820 q^{86} +2.76393 q^{87} +1.61803 q^{88} -18.0902 q^{89} -3.00000 q^{91} -4.85410 q^{92} +2.47214 q^{93} +1.61803 q^{94} +1.00000 q^{96} -9.70820 q^{97} +6.61803 q^{98} -1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} - q^{11} - 2 q^{12} - 3 q^{13} + q^{14} + 2 q^{16} - 6 q^{17} - 2 q^{18} + 5 q^{19} + q^{21} + q^{22} - 3 q^{23} + 2 q^{24} + 3 q^{26} - 2 q^{27} - q^{28} - 10 q^{29} + 4 q^{31} - 2 q^{32} + q^{33} + 6 q^{34} + 2 q^{36} - q^{37} - 5 q^{38} + 3 q^{39} - 21 q^{41} - q^{42} + 2 q^{43} - q^{44} + 3 q^{46} - q^{47} - 2 q^{48} - 11 q^{49} + 6 q^{51} - 3 q^{52} - 13 q^{53} + 2 q^{54} + q^{56} - 5 q^{57} + 10 q^{58} - 15 q^{59} - 6 q^{61} - 4 q^{62} - q^{63} + 2 q^{64} - q^{66} - 6 q^{67} - 6 q^{68} + 3 q^{69} - 16 q^{71} - 2 q^{72} - 8 q^{73} + q^{74} + 5 q^{76} - 2 q^{77} - 3 q^{78} + 2 q^{81} + 21 q^{82} + 12 q^{83} + q^{84} - 2 q^{86} + 10 q^{87} + q^{88} - 25 q^{89} - 6 q^{91} - 3 q^{92} - 4 q^{93} + q^{94} + 2 q^{96} - 6 q^{97} + 11 q^{98} - q^{99}+O(q^{100}) \) 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
\(7\) 0.618034 0.233595 0.116797 0.993156i \(-0.462737\pi\)
0.116797 + 0.993156i \(0.462737\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.61803 −0.487856 −0.243928 0.969793i \(-0.578436\pi\)
−0.243928 + 0.969793i \(0.578436\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.85410 −1.34629 −0.673143 0.739512i \(-0.735056\pi\)
−0.673143 + 0.739512i \(0.735056\pi\)
\(14\) −0.618034 −0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.763932 −0.185281 −0.0926404 0.995700i \(-0.529531\pi\)
−0.0926404 + 0.995700i \(0.529531\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.85410 1.34302 0.671512 0.740994i \(-0.265645\pi\)
0.671512 + 0.740994i \(0.265645\pi\)
\(20\) 0 0
\(21\) −0.618034 −0.134866
\(22\) 1.61803 0.344966
\(23\) −4.85410 −1.01215 −0.506075 0.862489i \(-0.668904\pi\)
−0.506075 + 0.862489i \(0.668904\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.85410 0.951968
\(27\) −1.00000 −0.192450
\(28\) 0.618034 0.116797
\(29\) −2.76393 −0.513249 −0.256625 0.966511i \(-0.582610\pi\)
−0.256625 + 0.966511i \(0.582610\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.61803 0.281664
\(34\) 0.763932 0.131013
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 9.56231 1.57203 0.786017 0.618205i \(-0.212140\pi\)
0.786017 + 0.618205i \(0.212140\pi\)
\(38\) −5.85410 −0.949661
\(39\) 4.85410 0.777278
\(40\) 0 0
\(41\) −9.38197 −1.46522 −0.732608 0.680650i \(-0.761697\pi\)
−0.732608 + 0.680650i \(0.761697\pi\)
\(42\) 0.618034 0.0953647
\(43\) −5.70820 −0.870493 −0.435246 0.900311i \(-0.643339\pi\)
−0.435246 + 0.900311i \(0.643339\pi\)
\(44\) −1.61803 −0.243928
\(45\) 0 0
\(46\) 4.85410 0.715698
\(47\) −1.61803 −0.236015 −0.118007 0.993013i \(-0.537651\pi\)
−0.118007 + 0.993013i \(0.537651\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.61803 −0.945433
\(50\) 0 0
\(51\) 0.763932 0.106972
\(52\) −4.85410 −0.673143
\(53\) −5.38197 −0.739270 −0.369635 0.929177i \(-0.620517\pi\)
−0.369635 + 0.929177i \(0.620517\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.618034 −0.0825883
\(57\) −5.85410 −0.775395
\(58\) 2.76393 0.362922
\(59\) −13.0902 −1.70419 −0.852097 0.523383i \(-0.824670\pi\)
−0.852097 + 0.523383i \(0.824670\pi\)
\(60\) 0 0
\(61\) −9.70820 −1.24301 −0.621504 0.783411i \(-0.713478\pi\)
−0.621504 + 0.783411i \(0.713478\pi\)
\(62\) 2.47214 0.313962
\(63\) 0.618034 0.0778650
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.61803 −0.199166
\(67\) 3.70820 0.453029 0.226515 0.974008i \(-0.427267\pi\)
0.226515 + 0.974008i \(0.427267\pi\)
\(68\) −0.763932 −0.0926404
\(69\) 4.85410 0.584365
\(70\) 0 0
\(71\) −3.52786 −0.418680 −0.209340 0.977843i \(-0.567132\pi\)
−0.209340 + 0.977843i \(0.567132\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.9443 −1.51501 −0.757506 0.652828i \(-0.773582\pi\)
−0.757506 + 0.652828i \(0.773582\pi\)
\(74\) −9.56231 −1.11160
\(75\) 0 0
\(76\) 5.85410 0.671512
\(77\) −1.00000 −0.113961
\(78\) −4.85410 −0.549619
\(79\) 13.4164 1.50946 0.754732 0.656033i \(-0.227767\pi\)
0.754732 + 0.656033i \(0.227767\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.38197 1.03606
\(83\) 14.9443 1.64035 0.820173 0.572115i \(-0.193877\pi\)
0.820173 + 0.572115i \(0.193877\pi\)
\(84\) −0.618034 −0.0674330
\(85\) 0 0
\(86\) 5.70820 0.615531
\(87\) 2.76393 0.296325
\(88\) 1.61803 0.172483
\(89\) −18.0902 −1.91755 −0.958777 0.284159i \(-0.908286\pi\)
−0.958777 + 0.284159i \(0.908286\pi\)
\(90\) 0 0
\(91\) −3.00000 −0.314485
\(92\) −4.85410 −0.506075
\(93\) 2.47214 0.256349
\(94\) 1.61803 0.166887
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −9.70820 −0.985719 −0.492859 0.870109i \(-0.664048\pi\)
−0.492859 + 0.870109i \(0.664048\pi\)
\(98\) 6.61803 0.668522
\(99\) −1.61803 −0.162619
\(100\) 0 0
\(101\) 12.6525 1.25897 0.629484 0.777013i \(-0.283266\pi\)
0.629484 + 0.777013i \(0.283266\pi\)
\(102\) −0.763932 −0.0756405
\(103\) 8.56231 0.843669 0.421835 0.906673i \(-0.361386\pi\)
0.421835 + 0.906673i \(0.361386\pi\)
\(104\) 4.85410 0.475984
\(105\) 0 0
\(106\) 5.38197 0.522743
\(107\) −6.94427 −0.671328 −0.335664 0.941982i \(-0.608961\pi\)
−0.335664 + 0.941982i \(0.608961\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.18034 −0.591969 −0.295985 0.955193i \(-0.595648\pi\)
−0.295985 + 0.955193i \(0.595648\pi\)
\(110\) 0 0
\(111\) −9.56231 −0.907614
\(112\) 0.618034 0.0583987
\(113\) 3.23607 0.304424 0.152212 0.988348i \(-0.451360\pi\)
0.152212 + 0.988348i \(0.451360\pi\)
\(114\) 5.85410 0.548287
\(115\) 0 0
\(116\) −2.76393 −0.256625
\(117\) −4.85410 −0.448762
\(118\) 13.0902 1.20505
\(119\) −0.472136 −0.0432806
\(120\) 0 0
\(121\) −8.38197 −0.761997
\(122\) 9.70820 0.878939
\(123\) 9.38197 0.845943
\(124\) −2.47214 −0.222004
\(125\) 0 0
\(126\) −0.618034 −0.0550588
\(127\) −11.4164 −1.01304 −0.506521 0.862228i \(-0.669069\pi\)
−0.506521 + 0.862228i \(0.669069\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.70820 0.502579
\(130\) 0 0
\(131\) 3.90983 0.341603 0.170802 0.985305i \(-0.445364\pi\)
0.170802 + 0.985305i \(0.445364\pi\)
\(132\) 1.61803 0.140832
\(133\) 3.61803 0.313723
\(134\) −3.70820 −0.320340
\(135\) 0 0
\(136\) 0.763932 0.0655066
\(137\) 3.05573 0.261068 0.130534 0.991444i \(-0.458331\pi\)
0.130534 + 0.991444i \(0.458331\pi\)
\(138\) −4.85410 −0.413209
\(139\) 12.5623 1.06552 0.532760 0.846266i \(-0.321155\pi\)
0.532760 + 0.846266i \(0.321155\pi\)
\(140\) 0 0
\(141\) 1.61803 0.136263
\(142\) 3.52786 0.296052
\(143\) 7.85410 0.656793
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 12.9443 1.07128
\(147\) 6.61803 0.545846
\(148\) 9.56231 0.786017
\(149\) −13.4164 −1.09911 −0.549557 0.835456i \(-0.685204\pi\)
−0.549557 + 0.835456i \(0.685204\pi\)
\(150\) 0 0
\(151\) 10.2918 0.837534 0.418767 0.908094i \(-0.362462\pi\)
0.418767 + 0.908094i \(0.362462\pi\)
\(152\) −5.85410 −0.474830
\(153\) −0.763932 −0.0617602
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 4.85410 0.388639
\(157\) 10.9443 0.873448 0.436724 0.899596i \(-0.356139\pi\)
0.436724 + 0.899596i \(0.356139\pi\)
\(158\) −13.4164 −1.06735
\(159\) 5.38197 0.426818
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) 4.94427 0.387265 0.193633 0.981074i \(-0.437973\pi\)
0.193633 + 0.981074i \(0.437973\pi\)
\(164\) −9.38197 −0.732608
\(165\) 0 0
\(166\) −14.9443 −1.15990
\(167\) 7.85410 0.607769 0.303884 0.952709i \(-0.401716\pi\)
0.303884 + 0.952709i \(0.401716\pi\)
\(168\) 0.618034 0.0476824
\(169\) 10.5623 0.812485
\(170\) 0 0
\(171\) 5.85410 0.447674
\(172\) −5.70820 −0.435246
\(173\) 14.6180 1.11139 0.555694 0.831387i \(-0.312453\pi\)
0.555694 + 0.831387i \(0.312453\pi\)
\(174\) −2.76393 −0.209533
\(175\) 0 0
\(176\) −1.61803 −0.121964
\(177\) 13.0902 0.983917
\(178\) 18.0902 1.35592
\(179\) 22.0344 1.64693 0.823466 0.567366i \(-0.192038\pi\)
0.823466 + 0.567366i \(0.192038\pi\)
\(180\) 0 0
\(181\) 25.4164 1.88919 0.944593 0.328243i \(-0.106456\pi\)
0.944593 + 0.328243i \(0.106456\pi\)
\(182\) 3.00000 0.222375
\(183\) 9.70820 0.717651
\(184\) 4.85410 0.357849
\(185\) 0 0
\(186\) −2.47214 −0.181266
\(187\) 1.23607 0.0903902
\(188\) −1.61803 −0.118007
\(189\) −0.618034 −0.0449554
\(190\) 0 0
\(191\) −16.9443 −1.22604 −0.613022 0.790066i \(-0.710046\pi\)
−0.613022 + 0.790066i \(0.710046\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 4.29180 0.308930 0.154465 0.987998i \(-0.450635\pi\)
0.154465 + 0.987998i \(0.450635\pi\)
\(194\) 9.70820 0.697008
\(195\) 0 0
\(196\) −6.61803 −0.472717
\(197\) −6.94427 −0.494759 −0.247379 0.968919i \(-0.579569\pi\)
−0.247379 + 0.968919i \(0.579569\pi\)
\(198\) 1.61803 0.114989
\(199\) −6.18034 −0.438113 −0.219056 0.975712i \(-0.570298\pi\)
−0.219056 + 0.975712i \(0.570298\pi\)
\(200\) 0 0
\(201\) −3.70820 −0.261557
\(202\) −12.6525 −0.890225
\(203\) −1.70820 −0.119892
\(204\) 0.763932 0.0534859
\(205\) 0 0
\(206\) −8.56231 −0.596564
\(207\) −4.85410 −0.337383
\(208\) −4.85410 −0.336571
\(209\) −9.47214 −0.655201
\(210\) 0 0
\(211\) −17.2705 −1.18895 −0.594475 0.804114i \(-0.702640\pi\)
−0.594475 + 0.804114i \(0.702640\pi\)
\(212\) −5.38197 −0.369635
\(213\) 3.52786 0.241725
\(214\) 6.94427 0.474701
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −1.52786 −0.103718
\(218\) 6.18034 0.418585
\(219\) 12.9443 0.874693
\(220\) 0 0
\(221\) 3.70820 0.249441
\(222\) 9.56231 0.641780
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −0.618034 −0.0412941
\(225\) 0 0
\(226\) −3.23607 −0.215260
\(227\) 28.1803 1.87039 0.935197 0.354127i \(-0.115222\pi\)
0.935197 + 0.354127i \(0.115222\pi\)
\(228\) −5.85410 −0.387697
\(229\) 1.70820 0.112881 0.0564406 0.998406i \(-0.482025\pi\)
0.0564406 + 0.998406i \(0.482025\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 2.76393 0.181461
\(233\) 19.4164 1.27201 0.636006 0.771684i \(-0.280586\pi\)
0.636006 + 0.771684i \(0.280586\pi\)
\(234\) 4.85410 0.317323
\(235\) 0 0
\(236\) −13.0902 −0.852097
\(237\) −13.4164 −0.871489
\(238\) 0.472136 0.0306040
\(239\) 8.94427 0.578557 0.289278 0.957245i \(-0.406585\pi\)
0.289278 + 0.957245i \(0.406585\pi\)
\(240\) 0 0
\(241\) −7.14590 −0.460308 −0.230154 0.973154i \(-0.573923\pi\)
−0.230154 + 0.973154i \(0.573923\pi\)
\(242\) 8.38197 0.538813
\(243\) −1.00000 −0.0641500
\(244\) −9.70820 −0.621504
\(245\) 0 0
\(246\) −9.38197 −0.598172
\(247\) −28.4164 −1.80809
\(248\) 2.47214 0.156981
\(249\) −14.9443 −0.947055
\(250\) 0 0
\(251\) 20.9443 1.32199 0.660995 0.750390i \(-0.270134\pi\)
0.660995 + 0.750390i \(0.270134\pi\)
\(252\) 0.618034 0.0389325
\(253\) 7.85410 0.493783
\(254\) 11.4164 0.716329
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −13.5279 −0.843845 −0.421922 0.906632i \(-0.638645\pi\)
−0.421922 + 0.906632i \(0.638645\pi\)
\(258\) −5.70820 −0.355377
\(259\) 5.90983 0.367219
\(260\) 0 0
\(261\) −2.76393 −0.171083
\(262\) −3.90983 −0.241550
\(263\) 14.6180 0.901387 0.450693 0.892679i \(-0.351177\pi\)
0.450693 + 0.892679i \(0.351177\pi\)
\(264\) −1.61803 −0.0995831
\(265\) 0 0
\(266\) −3.61803 −0.221836
\(267\) 18.0902 1.10710
\(268\) 3.70820 0.226515
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) 25.4164 1.54394 0.771968 0.635661i \(-0.219272\pi\)
0.771968 + 0.635661i \(0.219272\pi\)
\(272\) −0.763932 −0.0463202
\(273\) 3.00000 0.181568
\(274\) −3.05573 −0.184603
\(275\) 0 0
\(276\) 4.85410 0.292183
\(277\) 12.8541 0.772328 0.386164 0.922430i \(-0.373800\pi\)
0.386164 + 0.922430i \(0.373800\pi\)
\(278\) −12.5623 −0.753437
\(279\) −2.47214 −0.148003
\(280\) 0 0
\(281\) −6.09017 −0.363309 −0.181655 0.983362i \(-0.558145\pi\)
−0.181655 + 0.983362i \(0.558145\pi\)
\(282\) −1.61803 −0.0963525
\(283\) −30.1803 −1.79403 −0.897017 0.441995i \(-0.854271\pi\)
−0.897017 + 0.441995i \(0.854271\pi\)
\(284\) −3.52786 −0.209340
\(285\) 0 0
\(286\) −7.85410 −0.464423
\(287\) −5.79837 −0.342267
\(288\) −1.00000 −0.0589256
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 9.70820 0.569105
\(292\) −12.9443 −0.757506
\(293\) −22.0902 −1.29052 −0.645261 0.763962i \(-0.723251\pi\)
−0.645261 + 0.763962i \(0.723251\pi\)
\(294\) −6.61803 −0.385972
\(295\) 0 0
\(296\) −9.56231 −0.555798
\(297\) 1.61803 0.0938879
\(298\) 13.4164 0.777192
\(299\) 23.5623 1.36264
\(300\) 0 0
\(301\) −3.52786 −0.203343
\(302\) −10.2918 −0.592226
\(303\) −12.6525 −0.726866
\(304\) 5.85410 0.335756
\(305\) 0 0
\(306\) 0.763932 0.0436711
\(307\) −20.7639 −1.18506 −0.592530 0.805548i \(-0.701871\pi\)
−0.592530 + 0.805548i \(0.701871\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −8.56231 −0.487093
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −4.85410 −0.274809
\(313\) −2.29180 −0.129540 −0.0647700 0.997900i \(-0.520631\pi\)
−0.0647700 + 0.997900i \(0.520631\pi\)
\(314\) −10.9443 −0.617621
\(315\) 0 0
\(316\) 13.4164 0.754732
\(317\) 1.14590 0.0643600 0.0321800 0.999482i \(-0.489755\pi\)
0.0321800 + 0.999482i \(0.489755\pi\)
\(318\) −5.38197 −0.301806
\(319\) 4.47214 0.250392
\(320\) 0 0
\(321\) 6.94427 0.387591
\(322\) 3.00000 0.167183
\(323\) −4.47214 −0.248836
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.94427 −0.273838
\(327\) 6.18034 0.341774
\(328\) 9.38197 0.518032
\(329\) −1.00000 −0.0551318
\(330\) 0 0
\(331\) −2.47214 −0.135881 −0.0679404 0.997689i \(-0.521643\pi\)
−0.0679404 + 0.997689i \(0.521643\pi\)
\(332\) 14.9443 0.820173
\(333\) 9.56231 0.524011
\(334\) −7.85410 −0.429757
\(335\) 0 0
\(336\) −0.618034 −0.0337165
\(337\) 19.8885 1.08340 0.541699 0.840573i \(-0.317781\pi\)
0.541699 + 0.840573i \(0.317781\pi\)
\(338\) −10.5623 −0.574514
\(339\) −3.23607 −0.175759
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −5.85410 −0.316554
\(343\) −8.41641 −0.454443
\(344\) 5.70820 0.307766
\(345\) 0 0
\(346\) −14.6180 −0.785870
\(347\) 26.0689 1.39945 0.699726 0.714412i \(-0.253306\pi\)
0.699726 + 0.714412i \(0.253306\pi\)
\(348\) 2.76393 0.148162
\(349\) −26.8328 −1.43633 −0.718164 0.695874i \(-0.755017\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) 0 0
\(351\) 4.85410 0.259093
\(352\) 1.61803 0.0862415
\(353\) −15.7082 −0.836063 −0.418032 0.908432i \(-0.637280\pi\)
−0.418032 + 0.908432i \(0.637280\pi\)
\(354\) −13.0902 −0.695735
\(355\) 0 0
\(356\) −18.0902 −0.958777
\(357\) 0.472136 0.0249881
\(358\) −22.0344 −1.16456
\(359\) 4.47214 0.236030 0.118015 0.993012i \(-0.462347\pi\)
0.118015 + 0.993012i \(0.462347\pi\)
\(360\) 0 0
\(361\) 15.2705 0.803711
\(362\) −25.4164 −1.33586
\(363\) 8.38197 0.439939
\(364\) −3.00000 −0.157243
\(365\) 0 0
\(366\) −9.70820 −0.507456
\(367\) −4.58359 −0.239262 −0.119631 0.992818i \(-0.538171\pi\)
−0.119631 + 0.992818i \(0.538171\pi\)
\(368\) −4.85410 −0.253038
\(369\) −9.38197 −0.488406
\(370\) 0 0
\(371\) −3.32624 −0.172690
\(372\) 2.47214 0.128174
\(373\) −30.5066 −1.57957 −0.789785 0.613383i \(-0.789808\pi\)
−0.789785 + 0.613383i \(0.789808\pi\)
\(374\) −1.23607 −0.0639156
\(375\) 0 0
\(376\) 1.61803 0.0834437
\(377\) 13.4164 0.690980
\(378\) 0.618034 0.0317882
\(379\) −15.9787 −0.820771 −0.410386 0.911912i \(-0.634606\pi\)
−0.410386 + 0.911912i \(0.634606\pi\)
\(380\) 0 0
\(381\) 11.4164 0.584880
\(382\) 16.9443 0.866944
\(383\) −37.0902 −1.89522 −0.947610 0.319431i \(-0.896508\pi\)
−0.947610 + 0.319431i \(0.896508\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −4.29180 −0.218447
\(387\) −5.70820 −0.290164
\(388\) −9.70820 −0.492859
\(389\) 12.7639 0.647157 0.323579 0.946201i \(-0.395114\pi\)
0.323579 + 0.946201i \(0.395114\pi\)
\(390\) 0 0
\(391\) 3.70820 0.187532
\(392\) 6.61803 0.334261
\(393\) −3.90983 −0.197225
\(394\) 6.94427 0.349847
\(395\) 0 0
\(396\) −1.61803 −0.0813093
\(397\) −17.2705 −0.866782 −0.433391 0.901206i \(-0.642683\pi\)
−0.433391 + 0.901206i \(0.642683\pi\)
\(398\) 6.18034 0.309792
\(399\) −3.61803 −0.181128
\(400\) 0 0
\(401\) 2.32624 0.116167 0.0580834 0.998312i \(-0.481501\pi\)
0.0580834 + 0.998312i \(0.481501\pi\)
\(402\) 3.70820 0.184948
\(403\) 12.0000 0.597763
\(404\) 12.6525 0.629484
\(405\) 0 0
\(406\) 1.70820 0.0847767
\(407\) −15.4721 −0.766925
\(408\) −0.763932 −0.0378203
\(409\) −14.1459 −0.699470 −0.349735 0.936849i \(-0.613728\pi\)
−0.349735 + 0.936849i \(0.613728\pi\)
\(410\) 0 0
\(411\) −3.05573 −0.150728
\(412\) 8.56231 0.421835
\(413\) −8.09017 −0.398091
\(414\) 4.85410 0.238566
\(415\) 0 0
\(416\) 4.85410 0.237992
\(417\) −12.5623 −0.615179
\(418\) 9.47214 0.463297
\(419\) 11.0557 0.540108 0.270054 0.962845i \(-0.412958\pi\)
0.270054 + 0.962845i \(0.412958\pi\)
\(420\) 0 0
\(421\) 18.1803 0.886056 0.443028 0.896508i \(-0.353904\pi\)
0.443028 + 0.896508i \(0.353904\pi\)
\(422\) 17.2705 0.840715
\(423\) −1.61803 −0.0786715
\(424\) 5.38197 0.261371
\(425\) 0 0
\(426\) −3.52786 −0.170926
\(427\) −6.00000 −0.290360
\(428\) −6.94427 −0.335664
\(429\) −7.85410 −0.379200
\(430\) 0 0
\(431\) 15.8197 0.762006 0.381003 0.924574i \(-0.375579\pi\)
0.381003 + 0.924574i \(0.375579\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −17.4164 −0.836979 −0.418490 0.908222i \(-0.637440\pi\)
−0.418490 + 0.908222i \(0.637440\pi\)
\(434\) 1.52786 0.0733398
\(435\) 0 0
\(436\) −6.18034 −0.295985
\(437\) −28.4164 −1.35934
\(438\) −12.9443 −0.618501
\(439\) 1.70820 0.0815281 0.0407641 0.999169i \(-0.487021\pi\)
0.0407641 + 0.999169i \(0.487021\pi\)
\(440\) 0 0
\(441\) −6.61803 −0.315144
\(442\) −3.70820 −0.176381
\(443\) −13.5967 −0.646001 −0.323000 0.946399i \(-0.604692\pi\)
−0.323000 + 0.946399i \(0.604692\pi\)
\(444\) −9.56231 −0.453807
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 13.4164 0.634574
\(448\) 0.618034 0.0291994
\(449\) 28.2148 1.33154 0.665769 0.746158i \(-0.268104\pi\)
0.665769 + 0.746158i \(0.268104\pi\)
\(450\) 0 0
\(451\) 15.1803 0.714814
\(452\) 3.23607 0.152212
\(453\) −10.2918 −0.483551
\(454\) −28.1803 −1.32257
\(455\) 0 0
\(456\) 5.85410 0.274143
\(457\) 38.8328 1.81652 0.908261 0.418404i \(-0.137410\pi\)
0.908261 + 0.418404i \(0.137410\pi\)
\(458\) −1.70820 −0.0798191
\(459\) 0.763932 0.0356573
\(460\) 0 0
\(461\) 38.1803 1.77824 0.889118 0.457678i \(-0.151319\pi\)
0.889118 + 0.457678i \(0.151319\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 31.7771 1.47681 0.738403 0.674359i \(-0.235580\pi\)
0.738403 + 0.674359i \(0.235580\pi\)
\(464\) −2.76393 −0.128312
\(465\) 0 0
\(466\) −19.4164 −0.899448
\(467\) −15.2361 −0.705041 −0.352521 0.935804i \(-0.614675\pi\)
−0.352521 + 0.935804i \(0.614675\pi\)
\(468\) −4.85410 −0.224381
\(469\) 2.29180 0.105825
\(470\) 0 0
\(471\) −10.9443 −0.504285
\(472\) 13.0902 0.602524
\(473\) 9.23607 0.424675
\(474\) 13.4164 0.616236
\(475\) 0 0
\(476\) −0.472136 −0.0216403
\(477\) −5.38197 −0.246423
\(478\) −8.94427 −0.409101
\(479\) 8.29180 0.378862 0.189431 0.981894i \(-0.439336\pi\)
0.189431 + 0.981894i \(0.439336\pi\)
\(480\) 0 0
\(481\) −46.4164 −2.11641
\(482\) 7.14590 0.325487
\(483\) 3.00000 0.136505
\(484\) −8.38197 −0.380998
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 17.3262 0.785127 0.392563 0.919725i \(-0.371588\pi\)
0.392563 + 0.919725i \(0.371588\pi\)
\(488\) 9.70820 0.439470
\(489\) −4.94427 −0.223588
\(490\) 0 0
\(491\) −18.9787 −0.856497 −0.428249 0.903661i \(-0.640869\pi\)
−0.428249 + 0.903661i \(0.640869\pi\)
\(492\) 9.38197 0.422972
\(493\) 2.11146 0.0950952
\(494\) 28.4164 1.27851
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) −2.18034 −0.0978016
\(498\) 14.9443 0.669669
\(499\) −5.72949 −0.256487 −0.128244 0.991743i \(-0.540934\pi\)
−0.128244 + 0.991743i \(0.540934\pi\)
\(500\) 0 0
\(501\) −7.85410 −0.350895
\(502\) −20.9443 −0.934789
\(503\) 13.0344 0.581177 0.290589 0.956848i \(-0.406149\pi\)
0.290589 + 0.956848i \(0.406149\pi\)
\(504\) −0.618034 −0.0275294
\(505\) 0 0
\(506\) −7.85410 −0.349157
\(507\) −10.5623 −0.469088
\(508\) −11.4164 −0.506521
\(509\) 30.6525 1.35865 0.679324 0.733839i \(-0.262273\pi\)
0.679324 + 0.733839i \(0.262273\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) −5.85410 −0.258465
\(514\) 13.5279 0.596689
\(515\) 0 0
\(516\) 5.70820 0.251290
\(517\) 2.61803 0.115141
\(518\) −5.90983 −0.259663
\(519\) −14.6180 −0.641660
\(520\) 0 0
\(521\) −38.4508 −1.68456 −0.842281 0.539038i \(-0.818788\pi\)
−0.842281 + 0.539038i \(0.818788\pi\)
\(522\) 2.76393 0.120974
\(523\) −29.5279 −1.29116 −0.645582 0.763691i \(-0.723385\pi\)
−0.645582 + 0.763691i \(0.723385\pi\)
\(524\) 3.90983 0.170802
\(525\) 0 0
\(526\) −14.6180 −0.637377
\(527\) 1.88854 0.0822663
\(528\) 1.61803 0.0704159
\(529\) 0.562306 0.0244481
\(530\) 0 0
\(531\) −13.0902 −0.568065
\(532\) 3.61803 0.156862
\(533\) 45.5410 1.97260
\(534\) −18.0902 −0.782838
\(535\) 0 0
\(536\) −3.70820 −0.160170
\(537\) −22.0344 −0.950856
\(538\) 13.4164 0.578422
\(539\) 10.7082 0.461235
\(540\) 0 0
\(541\) 45.4164 1.95260 0.976302 0.216413i \(-0.0694357\pi\)
0.976302 + 0.216413i \(0.0694357\pi\)
\(542\) −25.4164 −1.09173
\(543\) −25.4164 −1.09072
\(544\) 0.763932 0.0327533
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) −23.1246 −0.988737 −0.494369 0.869252i \(-0.664601\pi\)
−0.494369 + 0.869252i \(0.664601\pi\)
\(548\) 3.05573 0.130534
\(549\) −9.70820 −0.414336
\(550\) 0 0
\(551\) −16.1803 −0.689306
\(552\) −4.85410 −0.206604
\(553\) 8.29180 0.352603
\(554\) −12.8541 −0.546118
\(555\) 0 0
\(556\) 12.5623 0.532760
\(557\) −11.6180 −0.492272 −0.246136 0.969235i \(-0.579161\pi\)
−0.246136 + 0.969235i \(0.579161\pi\)
\(558\) 2.47214 0.104654
\(559\) 27.7082 1.17193
\(560\) 0 0
\(561\) −1.23607 −0.0521868
\(562\) 6.09017 0.256898
\(563\) −2.94427 −0.124086 −0.0620431 0.998073i \(-0.519762\pi\)
−0.0620431 + 0.998073i \(0.519762\pi\)
\(564\) 1.61803 0.0681315
\(565\) 0 0
\(566\) 30.1803 1.26857
\(567\) 0.618034 0.0259550
\(568\) 3.52786 0.148026
\(569\) −14.6738 −0.615156 −0.307578 0.951523i \(-0.599519\pi\)
−0.307578 + 0.951523i \(0.599519\pi\)
\(570\) 0 0
\(571\) −26.2148 −1.09705 −0.548527 0.836133i \(-0.684811\pi\)
−0.548527 + 0.836133i \(0.684811\pi\)
\(572\) 7.85410 0.328397
\(573\) 16.9443 0.707857
\(574\) 5.79837 0.242019
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 33.7082 1.40329 0.701645 0.712526i \(-0.252449\pi\)
0.701645 + 0.712526i \(0.252449\pi\)
\(578\) 16.4164 0.682833
\(579\) −4.29180 −0.178361
\(580\) 0 0
\(581\) 9.23607 0.383177
\(582\) −9.70820 −0.402418
\(583\) 8.70820 0.360657
\(584\) 12.9443 0.535638
\(585\) 0 0
\(586\) 22.0902 0.912537
\(587\) 15.8197 0.652947 0.326474 0.945206i \(-0.394140\pi\)
0.326474 + 0.945206i \(0.394140\pi\)
\(588\) 6.61803 0.272923
\(589\) −14.4721 −0.596314
\(590\) 0 0
\(591\) 6.94427 0.285649
\(592\) 9.56231 0.393008
\(593\) −1.63932 −0.0673188 −0.0336594 0.999433i \(-0.510716\pi\)
−0.0336594 + 0.999433i \(0.510716\pi\)
\(594\) −1.61803 −0.0663887
\(595\) 0 0
\(596\) −13.4164 −0.549557
\(597\) 6.18034 0.252944
\(598\) −23.5623 −0.963534
\(599\) 48.5410 1.98333 0.991666 0.128834i \(-0.0411235\pi\)
0.991666 + 0.128834i \(0.0411235\pi\)
\(600\) 0 0
\(601\) −32.2705 −1.31634 −0.658171 0.752869i \(-0.728670\pi\)
−0.658171 + 0.752869i \(0.728670\pi\)
\(602\) 3.52786 0.143785
\(603\) 3.70820 0.151010
\(604\) 10.2918 0.418767
\(605\) 0 0
\(606\) 12.6525 0.513972
\(607\) −20.6869 −0.839656 −0.419828 0.907604i \(-0.637910\pi\)
−0.419828 + 0.907604i \(0.637910\pi\)
\(608\) −5.85410 −0.237415
\(609\) 1.70820 0.0692199
\(610\) 0 0
\(611\) 7.85410 0.317743
\(612\) −0.763932 −0.0308801
\(613\) 22.3820 0.903999 0.452000 0.892018i \(-0.350711\pi\)
0.452000 + 0.892018i \(0.350711\pi\)
\(614\) 20.7639 0.837964
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −29.3050 −1.17977 −0.589886 0.807486i \(-0.700828\pi\)
−0.589886 + 0.807486i \(0.700828\pi\)
\(618\) 8.56231 0.344426
\(619\) −18.0902 −0.727105 −0.363553 0.931574i \(-0.618436\pi\)
−0.363553 + 0.931574i \(0.618436\pi\)
\(620\) 0 0
\(621\) 4.85410 0.194788
\(622\) −12.0000 −0.481156
\(623\) −11.1803 −0.447931
\(624\) 4.85410 0.194320
\(625\) 0 0
\(626\) 2.29180 0.0915986
\(627\) 9.47214 0.378281
\(628\) 10.9443 0.436724
\(629\) −7.30495 −0.291267
\(630\) 0 0
\(631\) −1.81966 −0.0724395 −0.0362198 0.999344i \(-0.511532\pi\)
−0.0362198 + 0.999344i \(0.511532\pi\)
\(632\) −13.4164 −0.533676
\(633\) 17.2705 0.686441
\(634\) −1.14590 −0.0455094
\(635\) 0 0
\(636\) 5.38197 0.213409
\(637\) 32.1246 1.27282
\(638\) −4.47214 −0.177054
\(639\) −3.52786 −0.139560
\(640\) 0 0
\(641\) −1.74265 −0.0688304 −0.0344152 0.999408i \(-0.510957\pi\)
−0.0344152 + 0.999408i \(0.510957\pi\)
\(642\) −6.94427 −0.274069
\(643\) 7.05573 0.278251 0.139125 0.990275i \(-0.455571\pi\)
0.139125 + 0.990275i \(0.455571\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) 4.47214 0.175954
\(647\) 5.09017 0.200115 0.100058 0.994982i \(-0.468097\pi\)
0.100058 + 0.994982i \(0.468097\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 21.1803 0.831401
\(650\) 0 0
\(651\) 1.52786 0.0598817
\(652\) 4.94427 0.193633
\(653\) −13.1459 −0.514439 −0.257219 0.966353i \(-0.582806\pi\)
−0.257219 + 0.966353i \(0.582806\pi\)
\(654\) −6.18034 −0.241670
\(655\) 0 0
\(656\) −9.38197 −0.366304
\(657\) −12.9443 −0.505004
\(658\) 1.00000 0.0389841
\(659\) −1.25735 −0.0489796 −0.0244898 0.999700i \(-0.507796\pi\)
−0.0244898 + 0.999700i \(0.507796\pi\)
\(660\) 0 0
\(661\) 2.65248 0.103169 0.0515847 0.998669i \(-0.483573\pi\)
0.0515847 + 0.998669i \(0.483573\pi\)
\(662\) 2.47214 0.0960823
\(663\) −3.70820 −0.144015
\(664\) −14.9443 −0.579950
\(665\) 0 0
\(666\) −9.56231 −0.370532
\(667\) 13.4164 0.519485
\(668\) 7.85410 0.303884
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) 15.7082 0.606408
\(672\) 0.618034 0.0238412
\(673\) 21.5279 0.829838 0.414919 0.909858i \(-0.363810\pi\)
0.414919 + 0.909858i \(0.363810\pi\)
\(674\) −19.8885 −0.766078
\(675\) 0 0
\(676\) 10.5623 0.406243
\(677\) 7.85410 0.301858 0.150929 0.988545i \(-0.451774\pi\)
0.150929 + 0.988545i \(0.451774\pi\)
\(678\) 3.23607 0.124280
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) −28.1803 −1.07987
\(682\) −4.00000 −0.153168
\(683\) 24.9443 0.954466 0.477233 0.878777i \(-0.341640\pi\)
0.477233 + 0.878777i \(0.341640\pi\)
\(684\) 5.85410 0.223837
\(685\) 0 0
\(686\) 8.41641 0.321340
\(687\) −1.70820 −0.0651720
\(688\) −5.70820 −0.217623
\(689\) 26.1246 0.995268
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 14.6180 0.555694
\(693\) −1.00000 −0.0379869
\(694\) −26.0689 −0.989561
\(695\) 0 0
\(696\) −2.76393 −0.104767
\(697\) 7.16718 0.271476
\(698\) 26.8328 1.01564
\(699\) −19.4164 −0.734396
\(700\) 0 0
\(701\) −25.2361 −0.953153 −0.476577 0.879133i \(-0.658122\pi\)
−0.476577 + 0.879133i \(0.658122\pi\)
\(702\) −4.85410 −0.183206
\(703\) 55.9787 2.11128
\(704\) −1.61803 −0.0609820
\(705\) 0 0
\(706\) 15.7082 0.591186
\(707\) 7.81966 0.294089
\(708\) 13.0902 0.491959
\(709\) −2.76393 −0.103802 −0.0519008 0.998652i \(-0.516528\pi\)
−0.0519008 + 0.998652i \(0.516528\pi\)
\(710\) 0 0
\(711\) 13.4164 0.503155
\(712\) 18.0902 0.677958
\(713\) 12.0000 0.449404
\(714\) −0.472136 −0.0176692
\(715\) 0 0
\(716\) 22.0344 0.823466
\(717\) −8.94427 −0.334030
\(718\) −4.47214 −0.166899
\(719\) −27.8885 −1.04007 −0.520034 0.854146i \(-0.674081\pi\)
−0.520034 + 0.854146i \(0.674081\pi\)
\(720\) 0 0
\(721\) 5.29180 0.197077
\(722\) −15.2705 −0.568310
\(723\) 7.14590 0.265759
\(724\) 25.4164 0.944593
\(725\) 0 0
\(726\) −8.38197 −0.311084
\(727\) −44.1033 −1.63570 −0.817851 0.575430i \(-0.804835\pi\)
−0.817851 + 0.575430i \(0.804835\pi\)
\(728\) 3.00000 0.111187
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.36068 0.161286
\(732\) 9.70820 0.358826
\(733\) 31.3262 1.15706 0.578530 0.815661i \(-0.303626\pi\)
0.578530 + 0.815661i \(0.303626\pi\)
\(734\) 4.58359 0.169183
\(735\) 0 0
\(736\) 4.85410 0.178925
\(737\) −6.00000 −0.221013
\(738\) 9.38197 0.345355
\(739\) 10.7295 0.394691 0.197345 0.980334i \(-0.436768\pi\)
0.197345 + 0.980334i \(0.436768\pi\)
\(740\) 0 0
\(741\) 28.4164 1.04390
\(742\) 3.32624 0.122110
\(743\) −4.85410 −0.178080 −0.0890399 0.996028i \(-0.528380\pi\)
−0.0890399 + 0.996028i \(0.528380\pi\)
\(744\) −2.47214 −0.0906329
\(745\) 0 0
\(746\) 30.5066 1.11693
\(747\) 14.9443 0.546782
\(748\) 1.23607 0.0451951
\(749\) −4.29180 −0.156819
\(750\) 0 0
\(751\) −47.5967 −1.73683 −0.868415 0.495838i \(-0.834861\pi\)
−0.868415 + 0.495838i \(0.834861\pi\)
\(752\) −1.61803 −0.0590036
\(753\) −20.9443 −0.763252
\(754\) −13.4164 −0.488597
\(755\) 0 0
\(756\) −0.618034 −0.0224777
\(757\) 21.7984 0.792275 0.396138 0.918191i \(-0.370350\pi\)
0.396138 + 0.918191i \(0.370350\pi\)
\(758\) 15.9787 0.580373
\(759\) −7.85410 −0.285086
\(760\) 0 0
\(761\) −10.0344 −0.363748 −0.181874 0.983322i \(-0.558216\pi\)
−0.181874 + 0.983322i \(0.558216\pi\)
\(762\) −11.4164 −0.413573
\(763\) −3.81966 −0.138281
\(764\) −16.9443 −0.613022
\(765\) 0 0
\(766\) 37.0902 1.34012
\(767\) 63.5410 2.29433
\(768\) −1.00000 −0.0360844
\(769\) −15.7295 −0.567220 −0.283610 0.958940i \(-0.591532\pi\)
−0.283610 + 0.958940i \(0.591532\pi\)
\(770\) 0 0
\(771\) 13.5279 0.487194
\(772\) 4.29180 0.154465
\(773\) −2.94427 −0.105898 −0.0529491 0.998597i \(-0.516862\pi\)
−0.0529491 + 0.998597i \(0.516862\pi\)
\(774\) 5.70820 0.205177
\(775\) 0 0
\(776\) 9.70820 0.348504
\(777\) −5.90983 −0.212014
\(778\) −12.7639 −0.457609
\(779\) −54.9230 −1.96782
\(780\) 0 0
\(781\) 5.70820 0.204256
\(782\) −3.70820 −0.132605
\(783\) 2.76393 0.0987749
\(784\) −6.61803 −0.236358
\(785\) 0 0
\(786\) 3.90983 0.139459
\(787\) 25.4164 0.905997 0.452999 0.891511i \(-0.350354\pi\)
0.452999 + 0.891511i \(0.350354\pi\)
\(788\) −6.94427 −0.247379
\(789\) −14.6180 −0.520416
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 1.61803 0.0574943
\(793\) 47.1246 1.67344
\(794\) 17.2705 0.612907
\(795\) 0 0
\(796\) −6.18034 −0.219056
\(797\) 5.09017 0.180303 0.0901515 0.995928i \(-0.471265\pi\)
0.0901515 + 0.995928i \(0.471265\pi\)
\(798\) 3.61803 0.128077
\(799\) 1.23607 0.0437289
\(800\) 0 0
\(801\) −18.0902 −0.639185
\(802\) −2.32624 −0.0821423
\(803\) 20.9443 0.739107
\(804\) −3.70820 −0.130778
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) 13.4164 0.472280
\(808\) −12.6525 −0.445113
\(809\) 15.9787 0.561782 0.280891 0.959740i \(-0.409370\pi\)
0.280891 + 0.959740i \(0.409370\pi\)
\(810\) 0 0
\(811\) 53.1033 1.86471 0.932355 0.361544i \(-0.117750\pi\)
0.932355 + 0.361544i \(0.117750\pi\)
\(812\) −1.70820 −0.0599462
\(813\) −25.4164 −0.891392
\(814\) 15.4721 0.542298
\(815\) 0 0
\(816\) 0.763932 0.0267430
\(817\) −33.4164 −1.16909
\(818\) 14.1459 0.494600
\(819\) −3.00000 −0.104828
\(820\) 0 0
\(821\) −16.9443 −0.591359 −0.295680 0.955287i \(-0.595546\pi\)
−0.295680 + 0.955287i \(0.595546\pi\)
\(822\) 3.05573 0.106581
\(823\) 8.03444 0.280063 0.140032 0.990147i \(-0.455280\pi\)
0.140032 + 0.990147i \(0.455280\pi\)
\(824\) −8.56231 −0.298282
\(825\) 0 0
\(826\) 8.09017 0.281493
\(827\) −36.5410 −1.27066 −0.635328 0.772243i \(-0.719135\pi\)
−0.635328 + 0.772243i \(0.719135\pi\)
\(828\) −4.85410 −0.168692
\(829\) −40.2492 −1.39791 −0.698957 0.715164i \(-0.746352\pi\)
−0.698957 + 0.715164i \(0.746352\pi\)
\(830\) 0 0
\(831\) −12.8541 −0.445904
\(832\) −4.85410 −0.168286
\(833\) 5.05573 0.175171
\(834\) 12.5623 0.434997
\(835\) 0 0
\(836\) −9.47214 −0.327601
\(837\) 2.47214 0.0854495
\(838\) −11.0557 −0.381914
\(839\) −16.1803 −0.558607 −0.279304 0.960203i \(-0.590104\pi\)
−0.279304 + 0.960203i \(0.590104\pi\)
\(840\) 0 0
\(841\) −21.3607 −0.736575
\(842\) −18.1803 −0.626536
\(843\) 6.09017 0.209757
\(844\) −17.2705 −0.594475
\(845\) 0 0
\(846\) 1.61803 0.0556292
\(847\) −5.18034 −0.177999
\(848\) −5.38197 −0.184817
\(849\) 30.1803 1.03579
\(850\) 0 0
\(851\) −46.4164 −1.59113
\(852\) 3.52786 0.120863
\(853\) −6.43769 −0.220422 −0.110211 0.993908i \(-0.535153\pi\)
−0.110211 + 0.993908i \(0.535153\pi\)
\(854\) 6.00000 0.205316
\(855\) 0 0
\(856\) 6.94427 0.237350
\(857\) 8.18034 0.279435 0.139718 0.990191i \(-0.455381\pi\)
0.139718 + 0.990191i \(0.455381\pi\)
\(858\) 7.85410 0.268135
\(859\) 0.978714 0.0333933 0.0166966 0.999861i \(-0.494685\pi\)
0.0166966 + 0.999861i \(0.494685\pi\)
\(860\) 0 0
\(861\) 5.79837 0.197608
\(862\) −15.8197 −0.538820
\(863\) 2.90983 0.0990518 0.0495259 0.998773i \(-0.484229\pi\)
0.0495259 + 0.998773i \(0.484229\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 17.4164 0.591834
\(867\) 16.4164 0.557530
\(868\) −1.52786 −0.0518591
\(869\) −21.7082 −0.736400
\(870\) 0 0
\(871\) −18.0000 −0.609907
\(872\) 6.18034 0.209293
\(873\) −9.70820 −0.328573
\(874\) 28.4164 0.961199
\(875\) 0 0
\(876\) 12.9443 0.437346
\(877\) 7.72949 0.261006 0.130503 0.991448i \(-0.458341\pi\)
0.130503 + 0.991448i \(0.458341\pi\)
\(878\) −1.70820 −0.0576491
\(879\) 22.0902 0.745083
\(880\) 0 0
\(881\) −8.72949 −0.294104 −0.147052 0.989129i \(-0.546978\pi\)
−0.147052 + 0.989129i \(0.546978\pi\)
\(882\) 6.61803 0.222841
\(883\) −38.4721 −1.29469 −0.647345 0.762197i \(-0.724121\pi\)
−0.647345 + 0.762197i \(0.724121\pi\)
\(884\) 3.70820 0.124720
\(885\) 0 0
\(886\) 13.5967 0.456792
\(887\) −34.9098 −1.17216 −0.586079 0.810254i \(-0.699329\pi\)
−0.586079 + 0.810254i \(0.699329\pi\)
\(888\) 9.56231 0.320890
\(889\) −7.05573 −0.236642
\(890\) 0 0
\(891\) −1.61803 −0.0542062
\(892\) −24.0000 −0.803579
\(893\) −9.47214 −0.316973
\(894\) −13.4164 −0.448712
\(895\) 0 0
\(896\) −0.618034 −0.0206471
\(897\) −23.5623 −0.786722
\(898\) −28.2148 −0.941539
\(899\) 6.83282 0.227887
\(900\) 0 0
\(901\) 4.11146 0.136972
\(902\) −15.1803 −0.505450
\(903\) 3.52786 0.117400
\(904\) −3.23607 −0.107630
\(905\) 0 0
\(906\) 10.2918 0.341922
\(907\) −13.1246 −0.435796 −0.217898 0.975972i \(-0.569920\pi\)
−0.217898 + 0.975972i \(0.569920\pi\)
\(908\) 28.1803 0.935197
\(909\) 12.6525 0.419656
\(910\) 0 0
\(911\) −35.8885 −1.18904 −0.594520 0.804081i \(-0.702658\pi\)
−0.594520 + 0.804081i \(0.702658\pi\)
\(912\) −5.85410 −0.193849
\(913\) −24.1803 −0.800252
\(914\) −38.8328 −1.28448
\(915\) 0 0
\(916\) 1.70820 0.0564406
\(917\) 2.41641 0.0797968
\(918\) −0.763932 −0.0252135
\(919\) −29.5967 −0.976307 −0.488153 0.872758i \(-0.662329\pi\)
−0.488153 + 0.872758i \(0.662329\pi\)
\(920\) 0 0
\(921\) 20.7639 0.684195
\(922\) −38.1803 −1.25740
\(923\) 17.1246 0.563663
\(924\) 1.00000 0.0328976
\(925\) 0 0
\(926\) −31.7771 −1.04426
\(927\) 8.56231 0.281223
\(928\) 2.76393 0.0907305
\(929\) −30.4508 −0.999060 −0.499530 0.866297i \(-0.666494\pi\)
−0.499530 + 0.866297i \(0.666494\pi\)
\(930\) 0 0
\(931\) −38.7426 −1.26974
\(932\) 19.4164 0.636006
\(933\) −12.0000 −0.392862
\(934\) 15.2361 0.498539
\(935\) 0 0
\(936\) 4.85410 0.158661
\(937\) 27.1246 0.886122 0.443061 0.896491i \(-0.353892\pi\)
0.443061 + 0.896491i \(0.353892\pi\)
\(938\) −2.29180 −0.0748298
\(939\) 2.29180 0.0747899
\(940\) 0 0
\(941\) 17.1246 0.558246 0.279123 0.960255i \(-0.409956\pi\)
0.279123 + 0.960255i \(0.409956\pi\)
\(942\) 10.9443 0.356584
\(943\) 45.5410 1.48302
\(944\) −13.0902 −0.426049
\(945\) 0 0
\(946\) −9.23607 −0.300290
\(947\) 29.8885 0.971247 0.485624 0.874168i \(-0.338593\pi\)
0.485624 + 0.874168i \(0.338593\pi\)
\(948\) −13.4164 −0.435745
\(949\) 62.8328 2.03964
\(950\) 0 0
\(951\) −1.14590 −0.0371583
\(952\) 0.472136 0.0153020
\(953\) −60.1803 −1.94943 −0.974716 0.223446i \(-0.928269\pi\)
−0.974716 + 0.223446i \(0.928269\pi\)
\(954\) 5.38197 0.174248
\(955\) 0 0
\(956\) 8.94427 0.289278
\(957\) −4.47214 −0.144564
\(958\) −8.29180 −0.267896
\(959\) 1.88854 0.0609843
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 46.4164 1.49653
\(963\) −6.94427 −0.223776
\(964\) −7.14590 −0.230154
\(965\) 0 0
\(966\) −3.00000 −0.0965234
\(967\) −13.3262 −0.428543 −0.214271 0.976774i \(-0.568738\pi\)
−0.214271 + 0.976774i \(0.568738\pi\)
\(968\) 8.38197 0.269407
\(969\) 4.47214 0.143666
\(970\) 0 0
\(971\) −2.02129 −0.0648662 −0.0324331 0.999474i \(-0.510326\pi\)
−0.0324331 + 0.999474i \(0.510326\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 7.76393 0.248900
\(974\) −17.3262 −0.555168
\(975\) 0 0
\(976\) −9.70820 −0.310752
\(977\) −4.58359 −0.146642 −0.0733211 0.997308i \(-0.523360\pi\)
−0.0733211 + 0.997308i \(0.523360\pi\)
\(978\) 4.94427 0.158100
\(979\) 29.2705 0.935490
\(980\) 0 0
\(981\) −6.18034 −0.197323
\(982\) 18.9787 0.605635
\(983\) −24.4508 −0.779861 −0.389930 0.920844i \(-0.627501\pi\)
−0.389930 + 0.920844i \(0.627501\pi\)
\(984\) −9.38197 −0.299086
\(985\) 0 0
\(986\) −2.11146 −0.0672425
\(987\) 1.00000 0.0318304
\(988\) −28.4164 −0.904046
\(989\) 27.7082 0.881070
\(990\) 0 0
\(991\) 42.0000 1.33417 0.667087 0.744980i \(-0.267541\pi\)
0.667087 + 0.744980i \(0.267541\pi\)
\(992\) 2.47214 0.0784904
\(993\) 2.47214 0.0784509
\(994\) 2.18034 0.0691562
\(995\) 0 0
\(996\) −14.9443 −0.473527
\(997\) −22.1459 −0.701368 −0.350684 0.936494i \(-0.614051\pi\)
−0.350684 + 0.936494i \(0.614051\pi\)
\(998\) 5.72949 0.181364
\(999\) −9.56231 −0.302538
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.b.1.2 2
3.2 odd 2 2250.2.a.l.1.2 2
4.3 odd 2 6000.2.a.w.1.1 2
5.2 odd 4 750.2.c.d.499.2 4
5.3 odd 4 750.2.c.d.499.3 4
5.4 even 2 750.2.a.g.1.1 yes 2
15.2 even 4 2250.2.c.d.1999.4 4
15.8 even 4 2250.2.c.d.1999.1 4
15.14 odd 2 2250.2.a.e.1.1 2
20.3 even 4 6000.2.f.h.1249.4 4
20.7 even 4 6000.2.f.h.1249.1 4
20.19 odd 2 6000.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
750.2.a.b.1.2 2 1.1 even 1 trivial
750.2.a.g.1.1 yes 2 5.4 even 2
750.2.c.d.499.2 4 5.2 odd 4
750.2.c.d.499.3 4 5.3 odd 4
2250.2.a.e.1.1 2 15.14 odd 2
2250.2.a.l.1.2 2 3.2 odd 2
2250.2.c.d.1999.1 4 15.8 even 4
2250.2.c.d.1999.4 4 15.2 even 4
6000.2.a.f.1.2 2 20.19 odd 2
6000.2.a.w.1.1 2 4.3 odd 2
6000.2.f.h.1249.1 4 20.7 even 4
6000.2.f.h.1249.4 4 20.3 even 4