Properties

Label 3075.2.a.t.1.3
Level $3075$
Weight $2$
Character 3075.1
Self dual yes
Analytic conductor $24.554$
Analytic rank $1$
Dimension $3$
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] = [3075,2,Mod(1,3075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3075, 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("3075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3075 = 3 \cdot 5^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3075.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: \(24.5539986215\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 3075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.81361 q^{6} -2.52444 q^{7} -1.28917 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.81361 q^{2} +1.00000 q^{3} +1.28917 q^{4} +1.81361 q^{6} -2.52444 q^{7} -1.28917 q^{8} +1.00000 q^{9} +0.813607 q^{11} +1.28917 q^{12} -5.10278 q^{13} -4.57834 q^{14} -4.91638 q^{16} -3.39194 q^{17} +1.81361 q^{18} +3.10278 q^{19} -2.52444 q^{21} +1.47556 q^{22} +0.897225 q^{23} -1.28917 q^{24} -9.25443 q^{26} +1.00000 q^{27} -3.25443 q^{28} +4.44082 q^{29} -8.96526 q^{31} -6.33804 q^{32} +0.813607 q^{33} -6.15165 q^{34} +1.28917 q^{36} -2.08362 q^{37} +5.62721 q^{38} -5.10278 q^{39} +1.00000 q^{41} -4.57834 q^{42} -9.07306 q^{43} +1.04888 q^{44} +1.62721 q^{46} +0.235269 q^{47} -4.91638 q^{48} -0.627213 q^{49} -3.39194 q^{51} -6.57834 q^{52} -13.8328 q^{53} +1.81361 q^{54} +3.25443 q^{56} +3.10278 q^{57} +8.05390 q^{58} +1.04888 q^{59} +1.91638 q^{61} -16.2594 q^{62} -2.52444 q^{63} -1.66196 q^{64} +1.47556 q^{66} +10.0383 q^{67} -4.37279 q^{68} +0.897225 q^{69} -12.8136 q^{71} -1.28917 q^{72} -4.75971 q^{73} -3.77886 q^{74} +4.00000 q^{76} -2.05390 q^{77} -9.25443 q^{78} -15.8328 q^{79} +1.00000 q^{81} +1.81361 q^{82} +7.68111 q^{83} -3.25443 q^{84} -16.4550 q^{86} +4.44082 q^{87} -1.04888 q^{88} +13.2544 q^{89} +12.8816 q^{91} +1.15667 q^{92} -8.96526 q^{93} +0.426686 q^{94} -6.33804 q^{96} -2.72999 q^{97} -1.13752 q^{98} +0.813607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{11} + 3 q^{12} - 8 q^{13} - 12 q^{14} - q^{16} - 2 q^{17} - q^{18} + 2 q^{19} - 2 q^{21} + 10 q^{22} + 10 q^{23} - 3 q^{24} - 2 q^{26} + 3 q^{27} + 16 q^{28} - 6 q^{29} - 2 q^{31} - 7 q^{32} - 4 q^{33} + 3 q^{36} - 20 q^{37} + 4 q^{38} - 8 q^{39} + 3 q^{41} - 12 q^{42} - 10 q^{43} - 8 q^{44} - 8 q^{46} - 4 q^{47} - q^{48} + 11 q^{49} - 2 q^{51} - 18 q^{52} - 14 q^{53} - q^{54} - 16 q^{56} + 2 q^{57} + 28 q^{58} - 8 q^{59} - 8 q^{61} - 38 q^{62} - 2 q^{63} - 17 q^{64} + 10 q^{66} - 12 q^{67} - 26 q^{68} + 10 q^{69} - 32 q^{71} - 3 q^{72} - 4 q^{73} + 20 q^{74} + 12 q^{76} - 10 q^{77} - 2 q^{78} - 20 q^{79} + 3 q^{81} - q^{82} + 14 q^{83} + 16 q^{84} + 6 q^{86} - 6 q^{87} + 8 q^{88} + 14 q^{89} - 2 q^{93} + 18 q^{94} - 7 q^{96} + 12 q^{97} - 21 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.81361 1.28241 0.641207 0.767368i \(-0.278434\pi\)
0.641207 + 0.767368i \(0.278434\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.28917 0.644584
\(5\) 0 0
\(6\) 1.81361 0.740402
\(7\) −2.52444 −0.954148 −0.477074 0.878863i \(-0.658303\pi\)
−0.477074 + 0.878863i \(0.658303\pi\)
\(8\) −1.28917 −0.455790
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.813607 0.245312 0.122656 0.992449i \(-0.460859\pi\)
0.122656 + 0.992449i \(0.460859\pi\)
\(12\) 1.28917 0.372151
\(13\) −5.10278 −1.41526 −0.707628 0.706586i \(-0.750235\pi\)
−0.707628 + 0.706586i \(0.750235\pi\)
\(14\) −4.57834 −1.22361
\(15\) 0 0
\(16\) −4.91638 −1.22910
\(17\) −3.39194 −0.822667 −0.411334 0.911485i \(-0.634937\pi\)
−0.411334 + 0.911485i \(0.634937\pi\)
\(18\) 1.81361 0.427471
\(19\) 3.10278 0.711825 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(20\) 0 0
\(21\) −2.52444 −0.550878
\(22\) 1.47556 0.314591
\(23\) 0.897225 0.187084 0.0935422 0.995615i \(-0.470181\pi\)
0.0935422 + 0.995615i \(0.470181\pi\)
\(24\) −1.28917 −0.263150
\(25\) 0 0
\(26\) −9.25443 −1.81494
\(27\) 1.00000 0.192450
\(28\) −3.25443 −0.615029
\(29\) 4.44082 0.824639 0.412320 0.911039i \(-0.364719\pi\)
0.412320 + 0.911039i \(0.364719\pi\)
\(30\) 0 0
\(31\) −8.96526 −1.61021 −0.805104 0.593134i \(-0.797890\pi\)
−0.805104 + 0.593134i \(0.797890\pi\)
\(32\) −6.33804 −1.12042
\(33\) 0.813607 0.141631
\(34\) −6.15165 −1.05500
\(35\) 0 0
\(36\) 1.28917 0.214861
\(37\) −2.08362 −0.342545 −0.171272 0.985224i \(-0.554788\pi\)
−0.171272 + 0.985224i \(0.554788\pi\)
\(38\) 5.62721 0.912854
\(39\) −5.10278 −0.817098
\(40\) 0 0
\(41\) 1.00000 0.156174
\(42\) −4.57834 −0.706453
\(43\) −9.07306 −1.38363 −0.691814 0.722076i \(-0.743188\pi\)
−0.691814 + 0.722076i \(0.743188\pi\)
\(44\) 1.04888 0.158124
\(45\) 0 0
\(46\) 1.62721 0.239919
\(47\) 0.235269 0.0343176 0.0171588 0.999853i \(-0.494538\pi\)
0.0171588 + 0.999853i \(0.494538\pi\)
\(48\) −4.91638 −0.709619
\(49\) −0.627213 −0.0896019
\(50\) 0 0
\(51\) −3.39194 −0.474967
\(52\) −6.57834 −0.912251
\(53\) −13.8328 −1.90008 −0.950038 0.312134i \(-0.898956\pi\)
−0.950038 + 0.312134i \(0.898956\pi\)
\(54\) 1.81361 0.246801
\(55\) 0 0
\(56\) 3.25443 0.434891
\(57\) 3.10278 0.410973
\(58\) 8.05390 1.05753
\(59\) 1.04888 0.136552 0.0682760 0.997666i \(-0.478250\pi\)
0.0682760 + 0.997666i \(0.478250\pi\)
\(60\) 0 0
\(61\) 1.91638 0.245368 0.122684 0.992446i \(-0.460850\pi\)
0.122684 + 0.992446i \(0.460850\pi\)
\(62\) −16.2594 −2.06495
\(63\) −2.52444 −0.318049
\(64\) −1.66196 −0.207744
\(65\) 0 0
\(66\) 1.47556 0.181629
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) −4.37279 −0.530278
\(69\) 0.897225 0.108013
\(70\) 0 0
\(71\) −12.8136 −1.52070 −0.760348 0.649516i \(-0.774971\pi\)
−0.760348 + 0.649516i \(0.774971\pi\)
\(72\) −1.28917 −0.151930
\(73\) −4.75971 −0.557082 −0.278541 0.960424i \(-0.589851\pi\)
−0.278541 + 0.960424i \(0.589851\pi\)
\(74\) −3.77886 −0.439284
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −2.05390 −0.234064
\(78\) −9.25443 −1.04786
\(79\) −15.8328 −1.78133 −0.890663 0.454665i \(-0.849759\pi\)
−0.890663 + 0.454665i \(0.849759\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.81361 0.200279
\(83\) 7.68111 0.843112 0.421556 0.906802i \(-0.361484\pi\)
0.421556 + 0.906802i \(0.361484\pi\)
\(84\) −3.25443 −0.355087
\(85\) 0 0
\(86\) −16.4550 −1.77438
\(87\) 4.44082 0.476106
\(88\) −1.04888 −0.111811
\(89\) 13.2544 1.40497 0.702483 0.711700i \(-0.252075\pi\)
0.702483 + 0.711700i \(0.252075\pi\)
\(90\) 0 0
\(91\) 12.8816 1.35036
\(92\) 1.15667 0.120592
\(93\) −8.96526 −0.929654
\(94\) 0.426686 0.0440093
\(95\) 0 0
\(96\) −6.33804 −0.646874
\(97\) −2.72999 −0.277188 −0.138594 0.990349i \(-0.544258\pi\)
−0.138594 + 0.990349i \(0.544258\pi\)
\(98\) −1.13752 −0.114907
\(99\) 0.813607 0.0817705
\(100\) 0 0
\(101\) −11.8625 −1.18036 −0.590181 0.807271i \(-0.700943\pi\)
−0.590181 + 0.807271i \(0.700943\pi\)
\(102\) −6.15165 −0.609104
\(103\) 0.494719 0.0487461 0.0243730 0.999703i \(-0.492241\pi\)
0.0243730 + 0.999703i \(0.492241\pi\)
\(104\) 6.57834 0.645059
\(105\) 0 0
\(106\) −25.0872 −2.43668
\(107\) 15.0872 1.45853 0.729267 0.684229i \(-0.239861\pi\)
0.729267 + 0.684229i \(0.239861\pi\)
\(108\) 1.28917 0.124050
\(109\) 5.10278 0.488757 0.244379 0.969680i \(-0.421416\pi\)
0.244379 + 0.969680i \(0.421416\pi\)
\(110\) 0 0
\(111\) −2.08362 −0.197768
\(112\) 12.4111 1.17274
\(113\) 17.5139 1.64757 0.823783 0.566905i \(-0.191859\pi\)
0.823783 + 0.566905i \(0.191859\pi\)
\(114\) 5.62721 0.527037
\(115\) 0 0
\(116\) 5.72496 0.531550
\(117\) −5.10278 −0.471752
\(118\) 1.90225 0.175116
\(119\) 8.56275 0.784946
\(120\) 0 0
\(121\) −10.3380 −0.939822
\(122\) 3.47556 0.314663
\(123\) 1.00000 0.0901670
\(124\) −11.5577 −1.03791
\(125\) 0 0
\(126\) −4.57834 −0.407871
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) 9.66196 0.854004
\(129\) −9.07306 −0.798838
\(130\) 0 0
\(131\) −6.35720 −0.555431 −0.277716 0.960663i \(-0.589577\pi\)
−0.277716 + 0.960663i \(0.589577\pi\)
\(132\) 1.04888 0.0912929
\(133\) −7.83276 −0.679187
\(134\) 18.2056 1.57272
\(135\) 0 0
\(136\) 4.37279 0.374963
\(137\) 17.4897 1.49425 0.747123 0.664686i \(-0.231435\pi\)
0.747123 + 0.664686i \(0.231435\pi\)
\(138\) 1.62721 0.138518
\(139\) 15.7250 1.33377 0.666887 0.745159i \(-0.267626\pi\)
0.666887 + 0.745159i \(0.267626\pi\)
\(140\) 0 0
\(141\) 0.235269 0.0198133
\(142\) −23.2388 −1.95016
\(143\) −4.15165 −0.347178
\(144\) −4.91638 −0.409698
\(145\) 0 0
\(146\) −8.63224 −0.714409
\(147\) −0.627213 −0.0517317
\(148\) −2.68614 −0.220799
\(149\) 11.5678 0.947669 0.473835 0.880614i \(-0.342869\pi\)
0.473835 + 0.880614i \(0.342869\pi\)
\(150\) 0 0
\(151\) 19.2544 1.56690 0.783451 0.621453i \(-0.213457\pi\)
0.783451 + 0.621453i \(0.213457\pi\)
\(152\) −4.00000 −0.324443
\(153\) −3.39194 −0.274222
\(154\) −3.72496 −0.300166
\(155\) 0 0
\(156\) −6.57834 −0.526688
\(157\) 20.9200 1.66959 0.834797 0.550558i \(-0.185585\pi\)
0.834797 + 0.550558i \(0.185585\pi\)
\(158\) −28.7144 −2.28440
\(159\) −13.8328 −1.09701
\(160\) 0 0
\(161\) −2.26499 −0.178506
\(162\) 1.81361 0.142490
\(163\) 12.7980 1.00242 0.501209 0.865326i \(-0.332889\pi\)
0.501209 + 0.865326i \(0.332889\pi\)
\(164\) 1.28917 0.100667
\(165\) 0 0
\(166\) 13.9305 1.08122
\(167\) −9.62721 −0.744976 −0.372488 0.928037i \(-0.621495\pi\)
−0.372488 + 0.928037i \(0.621495\pi\)
\(168\) 3.25443 0.251084
\(169\) 13.0383 1.00295
\(170\) 0 0
\(171\) 3.10278 0.237275
\(172\) −11.6967 −0.891865
\(173\) −11.9461 −0.908245 −0.454123 0.890939i \(-0.650047\pi\)
−0.454123 + 0.890939i \(0.650047\pi\)
\(174\) 8.05390 0.610565
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 1.04888 0.0788383
\(178\) 24.0383 1.80175
\(179\) −2.13752 −0.159766 −0.0798828 0.996804i \(-0.525455\pi\)
−0.0798828 + 0.996804i \(0.525455\pi\)
\(180\) 0 0
\(181\) −1.83276 −0.136228 −0.0681141 0.997678i \(-0.521698\pi\)
−0.0681141 + 0.997678i \(0.521698\pi\)
\(182\) 23.3622 1.73172
\(183\) 1.91638 0.141663
\(184\) −1.15667 −0.0852712
\(185\) 0 0
\(186\) −16.2594 −1.19220
\(187\) −2.75971 −0.201810
\(188\) 0.303302 0.0221206
\(189\) −2.52444 −0.183626
\(190\) 0 0
\(191\) −18.6167 −1.34705 −0.673527 0.739163i \(-0.735221\pi\)
−0.673527 + 0.739163i \(0.735221\pi\)
\(192\) −1.66196 −0.119941
\(193\) −15.6116 −1.12375 −0.561875 0.827222i \(-0.689920\pi\)
−0.561875 + 0.827222i \(0.689920\pi\)
\(194\) −4.95112 −0.355470
\(195\) 0 0
\(196\) −0.808583 −0.0577559
\(197\) 5.21057 0.371238 0.185619 0.982622i \(-0.440571\pi\)
0.185619 + 0.982622i \(0.440571\pi\)
\(198\) 1.47556 0.104864
\(199\) 2.05390 0.145597 0.0727985 0.997347i \(-0.476807\pi\)
0.0727985 + 0.997347i \(0.476807\pi\)
\(200\) 0 0
\(201\) 10.0383 0.708048
\(202\) −21.5139 −1.51371
\(203\) −11.2106 −0.786828
\(204\) −4.37279 −0.306156
\(205\) 0 0
\(206\) 0.897225 0.0625126
\(207\) 0.897225 0.0623614
\(208\) 25.0872 1.73948
\(209\) 2.52444 0.174619
\(210\) 0 0
\(211\) −9.04888 −0.622950 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(212\) −17.8328 −1.22476
\(213\) −12.8136 −0.877974
\(214\) 27.3622 1.87044
\(215\) 0 0
\(216\) −1.28917 −0.0877168
\(217\) 22.6322 1.53638
\(218\) 9.25443 0.626789
\(219\) −4.75971 −0.321631
\(220\) 0 0
\(221\) 17.3083 1.16428
\(222\) −3.77886 −0.253621
\(223\) −24.8222 −1.66222 −0.831109 0.556110i \(-0.812293\pi\)
−0.831109 + 0.556110i \(0.812293\pi\)
\(224\) 16.0000 1.06904
\(225\) 0 0
\(226\) 31.7633 2.11286
\(227\) −16.6464 −1.10486 −0.552429 0.833560i \(-0.686299\pi\)
−0.552429 + 0.833560i \(0.686299\pi\)
\(228\) 4.00000 0.264906
\(229\) −15.7789 −1.04270 −0.521348 0.853344i \(-0.674571\pi\)
−0.521348 + 0.853344i \(0.674571\pi\)
\(230\) 0 0
\(231\) −2.05390 −0.135137
\(232\) −5.72496 −0.375862
\(233\) 24.9200 1.63256 0.816280 0.577656i \(-0.196033\pi\)
0.816280 + 0.577656i \(0.196033\pi\)
\(234\) −9.25443 −0.604981
\(235\) 0 0
\(236\) 1.35218 0.0880193
\(237\) −15.8328 −1.02845
\(238\) 15.5295 1.00663
\(239\) −2.95112 −0.190892 −0.0954462 0.995435i \(-0.530428\pi\)
−0.0954462 + 0.995435i \(0.530428\pi\)
\(240\) 0 0
\(241\) −16.5925 −1.06881 −0.534407 0.845227i \(-0.679465\pi\)
−0.534407 + 0.845227i \(0.679465\pi\)
\(242\) −18.7491 −1.20524
\(243\) 1.00000 0.0641500
\(244\) 2.47054 0.158160
\(245\) 0 0
\(246\) 1.81361 0.115631
\(247\) −15.8328 −1.00741
\(248\) 11.5577 0.733916
\(249\) 7.68111 0.486771
\(250\) 0 0
\(251\) −20.1361 −1.27098 −0.635489 0.772110i \(-0.719201\pi\)
−0.635489 + 0.772110i \(0.719201\pi\)
\(252\) −3.25443 −0.205010
\(253\) 0.729988 0.0458940
\(254\) −23.3622 −1.46588
\(255\) 0 0
\(256\) 20.8469 1.30293
\(257\) −25.9008 −1.61565 −0.807824 0.589424i \(-0.799355\pi\)
−0.807824 + 0.589424i \(0.799355\pi\)
\(258\) −16.4550 −1.02444
\(259\) 5.25997 0.326838
\(260\) 0 0
\(261\) 4.44082 0.274880
\(262\) −11.5295 −0.712292
\(263\) 22.4408 1.38376 0.691880 0.722012i \(-0.256783\pi\)
0.691880 + 0.722012i \(0.256783\pi\)
\(264\) −1.04888 −0.0645538
\(265\) 0 0
\(266\) −14.2056 −0.870998
\(267\) 13.2544 0.811158
\(268\) 12.9411 0.790502
\(269\) 7.62721 0.465039 0.232520 0.972592i \(-0.425303\pi\)
0.232520 + 0.972592i \(0.425303\pi\)
\(270\) 0 0
\(271\) 19.9164 1.20983 0.604917 0.796289i \(-0.293206\pi\)
0.604917 + 0.796289i \(0.293206\pi\)
\(272\) 16.6761 1.01114
\(273\) 12.8816 0.779632
\(274\) 31.7194 1.91624
\(275\) 0 0
\(276\) 1.15667 0.0696236
\(277\) 17.6413 1.05997 0.529983 0.848008i \(-0.322198\pi\)
0.529983 + 0.848008i \(0.322198\pi\)
\(278\) 28.5189 1.71045
\(279\) −8.96526 −0.536736
\(280\) 0 0
\(281\) −0.0297193 −0.00177291 −0.000886453 1.00000i \(-0.500282\pi\)
−0.000886453 1.00000i \(0.500282\pi\)
\(282\) 0.426686 0.0254088
\(283\) 10.1814 0.605220 0.302610 0.953115i \(-0.402142\pi\)
0.302610 + 0.953115i \(0.402142\pi\)
\(284\) −16.5189 −0.980216
\(285\) 0 0
\(286\) −7.52946 −0.445226
\(287\) −2.52444 −0.149013
\(288\) −6.33804 −0.373473
\(289\) −5.49472 −0.323219
\(290\) 0 0
\(291\) −2.72999 −0.160035
\(292\) −6.13607 −0.359086
\(293\) −3.14808 −0.183913 −0.0919564 0.995763i \(-0.529312\pi\)
−0.0919564 + 0.995763i \(0.529312\pi\)
\(294\) −1.13752 −0.0663414
\(295\) 0 0
\(296\) 2.68614 0.156128
\(297\) 0.813607 0.0472102
\(298\) 20.9794 1.21530
\(299\) −4.57834 −0.264772
\(300\) 0 0
\(301\) 22.9044 1.32019
\(302\) 34.9200 2.00942
\(303\) −11.8625 −0.681482
\(304\) −15.2544 −0.874901
\(305\) 0 0
\(306\) −6.15165 −0.351666
\(307\) 12.7980 0.730422 0.365211 0.930925i \(-0.380997\pi\)
0.365211 + 0.930925i \(0.380997\pi\)
\(308\) −2.64782 −0.150874
\(309\) 0.494719 0.0281436
\(310\) 0 0
\(311\) 26.6167 1.50929 0.754646 0.656132i \(-0.227809\pi\)
0.754646 + 0.656132i \(0.227809\pi\)
\(312\) 6.57834 0.372425
\(313\) 3.00502 0.169854 0.0849270 0.996387i \(-0.472934\pi\)
0.0849270 + 0.996387i \(0.472934\pi\)
\(314\) 37.9406 2.14111
\(315\) 0 0
\(316\) −20.4111 −1.14821
\(317\) 4.33302 0.243367 0.121683 0.992569i \(-0.461171\pi\)
0.121683 + 0.992569i \(0.461171\pi\)
\(318\) −25.0872 −1.40682
\(319\) 3.61308 0.202294
\(320\) 0 0
\(321\) 15.0872 0.842085
\(322\) −4.10780 −0.228919
\(323\) −10.5244 −0.585595
\(324\) 1.28917 0.0716205
\(325\) 0 0
\(326\) 23.2106 1.28551
\(327\) 5.10278 0.282184
\(328\) −1.28917 −0.0711824
\(329\) −0.593923 −0.0327440
\(330\) 0 0
\(331\) 5.53500 0.304231 0.152116 0.988363i \(-0.451391\pi\)
0.152116 + 0.988363i \(0.451391\pi\)
\(332\) 9.90225 0.543456
\(333\) −2.08362 −0.114182
\(334\) −17.4600 −0.955367
\(335\) 0 0
\(336\) 12.4111 0.677081
\(337\) 3.71083 0.202142 0.101071 0.994879i \(-0.467773\pi\)
0.101071 + 0.994879i \(0.467773\pi\)
\(338\) 23.6464 1.28619
\(339\) 17.5139 0.951223
\(340\) 0 0
\(341\) −7.29419 −0.395003
\(342\) 5.62721 0.304285
\(343\) 19.2544 1.03964
\(344\) 11.6967 0.630644
\(345\) 0 0
\(346\) −21.6655 −1.16475
\(347\) −3.07860 −0.165268 −0.0826338 0.996580i \(-0.526333\pi\)
−0.0826338 + 0.996580i \(0.526333\pi\)
\(348\) 5.72496 0.306890
\(349\) 14.7980 0.792120 0.396060 0.918225i \(-0.370377\pi\)
0.396060 + 0.918225i \(0.370377\pi\)
\(350\) 0 0
\(351\) −5.10278 −0.272366
\(352\) −5.15667 −0.274852
\(353\) −26.8222 −1.42760 −0.713801 0.700349i \(-0.753028\pi\)
−0.713801 + 0.700349i \(0.753028\pi\)
\(354\) 1.90225 0.101103
\(355\) 0 0
\(356\) 17.0872 0.905619
\(357\) 8.56275 0.453189
\(358\) −3.87662 −0.204886
\(359\) 2.35720 0.124408 0.0622042 0.998063i \(-0.480187\pi\)
0.0622042 + 0.998063i \(0.480187\pi\)
\(360\) 0 0
\(361\) −9.37279 −0.493305
\(362\) −3.32391 −0.174701
\(363\) −10.3380 −0.542607
\(364\) 16.6066 0.870423
\(365\) 0 0
\(366\) 3.47556 0.181671
\(367\) −23.1708 −1.20951 −0.604753 0.796413i \(-0.706728\pi\)
−0.604753 + 0.796413i \(0.706728\pi\)
\(368\) −4.41110 −0.229944
\(369\) 1.00000 0.0520579
\(370\) 0 0
\(371\) 34.9200 1.81295
\(372\) −11.5577 −0.599240
\(373\) −35.2091 −1.82306 −0.911530 0.411235i \(-0.865098\pi\)
−0.911530 + 0.411235i \(0.865098\pi\)
\(374\) −5.00502 −0.258804
\(375\) 0 0
\(376\) −0.303302 −0.0156416
\(377\) −22.6605 −1.16708
\(378\) −4.57834 −0.235484
\(379\) −26.0383 −1.33750 −0.668749 0.743488i \(-0.733170\pi\)
−0.668749 + 0.743488i \(0.733170\pi\)
\(380\) 0 0
\(381\) −12.8816 −0.659946
\(382\) −33.7633 −1.72748
\(383\) −15.3819 −0.785978 −0.392989 0.919543i \(-0.628559\pi\)
−0.392989 + 0.919543i \(0.628559\pi\)
\(384\) 9.66196 0.493060
\(385\) 0 0
\(386\) −28.3133 −1.44111
\(387\) −9.07306 −0.461209
\(388\) −3.51941 −0.178671
\(389\) 4.46500 0.226384 0.113192 0.993573i \(-0.463892\pi\)
0.113192 + 0.993573i \(0.463892\pi\)
\(390\) 0 0
\(391\) −3.04334 −0.153908
\(392\) 0.808583 0.0408396
\(393\) −6.35720 −0.320678
\(394\) 9.44993 0.476081
\(395\) 0 0
\(396\) 1.04888 0.0527080
\(397\) 10.5628 0.530129 0.265065 0.964231i \(-0.414607\pi\)
0.265065 + 0.964231i \(0.414607\pi\)
\(398\) 3.72496 0.186716
\(399\) −7.83276 −0.392129
\(400\) 0 0
\(401\) 13.0872 0.653543 0.326772 0.945103i \(-0.394039\pi\)
0.326772 + 0.945103i \(0.394039\pi\)
\(402\) 18.2056 0.908010
\(403\) 45.7477 2.27885
\(404\) −15.2927 −0.760842
\(405\) 0 0
\(406\) −20.3316 −1.00904
\(407\) −1.69525 −0.0840302
\(408\) 4.37279 0.216485
\(409\) −18.5542 −0.917444 −0.458722 0.888580i \(-0.651693\pi\)
−0.458722 + 0.888580i \(0.651693\pi\)
\(410\) 0 0
\(411\) 17.4897 0.862703
\(412\) 0.637776 0.0314210
\(413\) −2.64782 −0.130291
\(414\) 1.62721 0.0799732
\(415\) 0 0
\(416\) 32.3416 1.58568
\(417\) 15.7250 0.770055
\(418\) 4.57834 0.223934
\(419\) −25.6116 −1.25121 −0.625605 0.780140i \(-0.715148\pi\)
−0.625605 + 0.780140i \(0.715148\pi\)
\(420\) 0 0
\(421\) −8.67609 −0.422847 −0.211423 0.977395i \(-0.567810\pi\)
−0.211423 + 0.977395i \(0.567810\pi\)
\(422\) −16.4111 −0.798880
\(423\) 0.235269 0.0114392
\(424\) 17.8328 0.866036
\(425\) 0 0
\(426\) −23.2388 −1.12593
\(427\) −4.83779 −0.234117
\(428\) 19.4499 0.940148
\(429\) −4.15165 −0.200444
\(430\) 0 0
\(431\) 7.10278 0.342129 0.171064 0.985260i \(-0.445279\pi\)
0.171064 + 0.985260i \(0.445279\pi\)
\(432\) −4.91638 −0.236540
\(433\) 11.0247 0.529813 0.264907 0.964274i \(-0.414659\pi\)
0.264907 + 0.964274i \(0.414659\pi\)
\(434\) 41.0460 1.97027
\(435\) 0 0
\(436\) 6.57834 0.315045
\(437\) 2.78389 0.133171
\(438\) −8.63224 −0.412464
\(439\) −1.32391 −0.0631868 −0.0315934 0.999501i \(-0.510058\pi\)
−0.0315934 + 0.999501i \(0.510058\pi\)
\(440\) 0 0
\(441\) −0.627213 −0.0298673
\(442\) 31.3905 1.49309
\(443\) −15.5889 −0.740651 −0.370325 0.928902i \(-0.620754\pi\)
−0.370325 + 0.928902i \(0.620754\pi\)
\(444\) −2.68614 −0.127478
\(445\) 0 0
\(446\) −45.0177 −2.13165
\(447\) 11.5678 0.547137
\(448\) 4.19550 0.198219
\(449\) −32.7839 −1.54717 −0.773584 0.633694i \(-0.781538\pi\)
−0.773584 + 0.633694i \(0.781538\pi\)
\(450\) 0 0
\(451\) 0.813607 0.0383112
\(452\) 22.5783 1.06200
\(453\) 19.2544 0.904652
\(454\) −30.1900 −1.41689
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 14.1672 0.662715 0.331358 0.943505i \(-0.392493\pi\)
0.331358 + 0.943505i \(0.392493\pi\)
\(458\) −28.6167 −1.33717
\(459\) −3.39194 −0.158322
\(460\) 0 0
\(461\) −30.0766 −1.40081 −0.700404 0.713747i \(-0.746997\pi\)
−0.700404 + 0.713747i \(0.746997\pi\)
\(462\) −3.72496 −0.173301
\(463\) 23.8766 1.10964 0.554820 0.831970i \(-0.312787\pi\)
0.554820 + 0.831970i \(0.312787\pi\)
\(464\) −21.8328 −1.01356
\(465\) 0 0
\(466\) 45.1950 2.09362
\(467\) 9.73501 0.450483 0.225241 0.974303i \(-0.427683\pi\)
0.225241 + 0.974303i \(0.427683\pi\)
\(468\) −6.57834 −0.304084
\(469\) −25.3411 −1.17014
\(470\) 0 0
\(471\) 20.9200 0.963941
\(472\) −1.35218 −0.0622390
\(473\) −7.38190 −0.339420
\(474\) −28.7144 −1.31890
\(475\) 0 0
\(476\) 11.0388 0.505964
\(477\) −13.8328 −0.633359
\(478\) −5.35218 −0.244803
\(479\) −1.17635 −0.0537487 −0.0268743 0.999639i \(-0.508555\pi\)
−0.0268743 + 0.999639i \(0.508555\pi\)
\(480\) 0 0
\(481\) 10.6322 0.484788
\(482\) −30.0922 −1.37066
\(483\) −2.26499 −0.103061
\(484\) −13.3275 −0.605795
\(485\) 0 0
\(486\) 1.81361 0.0822669
\(487\) −20.6025 −0.933589 −0.466795 0.884366i \(-0.654591\pi\)
−0.466795 + 0.884366i \(0.654591\pi\)
\(488\) −2.47054 −0.111836
\(489\) 12.7980 0.578746
\(490\) 0 0
\(491\) 14.1900 0.640384 0.320192 0.947353i \(-0.396253\pi\)
0.320192 + 0.947353i \(0.396253\pi\)
\(492\) 1.28917 0.0581202
\(493\) −15.0630 −0.678404
\(494\) −28.7144 −1.29192
\(495\) 0 0
\(496\) 44.0766 1.97910
\(497\) 32.3472 1.45097
\(498\) 13.9305 0.624241
\(499\) −21.4600 −0.960680 −0.480340 0.877082i \(-0.659487\pi\)
−0.480340 + 0.877082i \(0.659487\pi\)
\(500\) 0 0
\(501\) −9.62721 −0.430112
\(502\) −36.5189 −1.62992
\(503\) 15.5491 0.693302 0.346651 0.937994i \(-0.387319\pi\)
0.346651 + 0.937994i \(0.387319\pi\)
\(504\) 3.25443 0.144964
\(505\) 0 0
\(506\) 1.32391 0.0588550
\(507\) 13.0383 0.579052
\(508\) −16.6066 −0.736799
\(509\) 22.7542 1.00856 0.504280 0.863540i \(-0.331758\pi\)
0.504280 + 0.863540i \(0.331758\pi\)
\(510\) 0 0
\(511\) 12.0156 0.531538
\(512\) 18.4842 0.816892
\(513\) 3.10278 0.136991
\(514\) −46.9739 −2.07193
\(515\) 0 0
\(516\) −11.6967 −0.514918
\(517\) 0.191417 0.00841850
\(518\) 9.53951 0.419142
\(519\) −11.9461 −0.524376
\(520\) 0 0
\(521\) −38.7230 −1.69649 −0.848243 0.529608i \(-0.822339\pi\)
−0.848243 + 0.529608i \(0.822339\pi\)
\(522\) 8.05390 0.352510
\(523\) −6.56829 −0.287211 −0.143606 0.989635i \(-0.545870\pi\)
−0.143606 + 0.989635i \(0.545870\pi\)
\(524\) −8.19550 −0.358022
\(525\) 0 0
\(526\) 40.6988 1.77455
\(527\) 30.4096 1.32467
\(528\) −4.00000 −0.174078
\(529\) −22.1950 −0.964999
\(530\) 0 0
\(531\) 1.04888 0.0455173
\(532\) −10.0978 −0.437793
\(533\) −5.10278 −0.221026
\(534\) 24.0383 1.04024
\(535\) 0 0
\(536\) −12.9411 −0.558969
\(537\) −2.13752 −0.0922407
\(538\) 13.8328 0.596373
\(539\) −0.510305 −0.0219804
\(540\) 0 0
\(541\) 42.0766 1.80902 0.904508 0.426457i \(-0.140239\pi\)
0.904508 + 0.426457i \(0.140239\pi\)
\(542\) 36.1205 1.55151
\(543\) −1.83276 −0.0786514
\(544\) 21.4983 0.921732
\(545\) 0 0
\(546\) 23.3622 0.999811
\(547\) −19.9688 −0.853805 −0.426903 0.904298i \(-0.640395\pi\)
−0.426903 + 0.904298i \(0.640395\pi\)
\(548\) 22.5472 0.963167
\(549\) 1.91638 0.0817892
\(550\) 0 0
\(551\) 13.7789 0.586999
\(552\) −1.15667 −0.0492313
\(553\) 39.9688 1.69965
\(554\) 31.9945 1.35931
\(555\) 0 0
\(556\) 20.2721 0.859730
\(557\) −13.5491 −0.574095 −0.287048 0.957916i \(-0.592674\pi\)
−0.287048 + 0.957916i \(0.592674\pi\)
\(558\) −16.2594 −0.688317
\(559\) 46.2978 1.95819
\(560\) 0 0
\(561\) −2.75971 −0.116515
\(562\) −0.0538991 −0.00227360
\(563\) −9.75468 −0.411111 −0.205555 0.978645i \(-0.565900\pi\)
−0.205555 + 0.978645i \(0.565900\pi\)
\(564\) 0.303302 0.0127713
\(565\) 0 0
\(566\) 18.4650 0.776142
\(567\) −2.52444 −0.106016
\(568\) 16.5189 0.693118
\(569\) −21.4544 −0.899417 −0.449708 0.893175i \(-0.648472\pi\)
−0.449708 + 0.893175i \(0.648472\pi\)
\(570\) 0 0
\(571\) 20.9411 0.876357 0.438178 0.898888i \(-0.355624\pi\)
0.438178 + 0.898888i \(0.355624\pi\)
\(572\) −5.35218 −0.223786
\(573\) −18.6167 −0.777722
\(574\) −4.57834 −0.191096
\(575\) 0 0
\(576\) −1.66196 −0.0692481
\(577\) −18.4705 −0.768939 −0.384469 0.923138i \(-0.625616\pi\)
−0.384469 + 0.923138i \(0.625616\pi\)
\(578\) −9.96526 −0.414500
\(579\) −15.6116 −0.648797
\(580\) 0 0
\(581\) −19.3905 −0.804453
\(582\) −4.95112 −0.205231
\(583\) −11.2544 −0.466111
\(584\) 6.13607 0.253912
\(585\) 0 0
\(586\) −5.70938 −0.235852
\(587\) 0.127471 0.00526130 0.00263065 0.999997i \(-0.499163\pi\)
0.00263065 + 0.999997i \(0.499163\pi\)
\(588\) −0.808583 −0.0333454
\(589\) −27.8172 −1.14619
\(590\) 0 0
\(591\) 5.21057 0.214334
\(592\) 10.2439 0.421020
\(593\) 27.2841 1.12043 0.560213 0.828349i \(-0.310719\pi\)
0.560213 + 0.828349i \(0.310719\pi\)
\(594\) 1.47556 0.0605430
\(595\) 0 0
\(596\) 14.9128 0.610853
\(597\) 2.05390 0.0840605
\(598\) −8.30330 −0.339547
\(599\) −42.3260 −1.72939 −0.864697 0.502293i \(-0.832490\pi\)
−0.864697 + 0.502293i \(0.832490\pi\)
\(600\) 0 0
\(601\) −15.6756 −0.639420 −0.319710 0.947515i \(-0.603585\pi\)
−0.319710 + 0.947515i \(0.603585\pi\)
\(602\) 41.5395 1.69302
\(603\) 10.0383 0.408792
\(604\) 24.8222 1.01000
\(605\) 0 0
\(606\) −21.5139 −0.873941
\(607\) 1.93051 0.0783572 0.0391786 0.999232i \(-0.487526\pi\)
0.0391786 + 0.999232i \(0.487526\pi\)
\(608\) −19.6655 −0.797542
\(609\) −11.2106 −0.454275
\(610\) 0 0
\(611\) −1.20053 −0.0485681
\(612\) −4.37279 −0.176759
\(613\) 0.372787 0.0150567 0.00752836 0.999972i \(-0.497604\pi\)
0.00752836 + 0.999972i \(0.497604\pi\)
\(614\) 23.2106 0.936703
\(615\) 0 0
\(616\) 2.64782 0.106684
\(617\) −31.3083 −1.26043 −0.630213 0.776422i \(-0.717032\pi\)
−0.630213 + 0.776422i \(0.717032\pi\)
\(618\) 0.897225 0.0360917
\(619\) 34.3658 1.38128 0.690639 0.723200i \(-0.257329\pi\)
0.690639 + 0.723200i \(0.257329\pi\)
\(620\) 0 0
\(621\) 0.897225 0.0360044
\(622\) 48.2721 1.93554
\(623\) −33.4600 −1.34055
\(624\) 25.0872 1.00429
\(625\) 0 0
\(626\) 5.44993 0.217823
\(627\) 2.52444 0.100816
\(628\) 26.9693 1.07619
\(629\) 7.06752 0.281800
\(630\) 0 0
\(631\) 7.33804 0.292123 0.146061 0.989276i \(-0.453340\pi\)
0.146061 + 0.989276i \(0.453340\pi\)
\(632\) 20.4111 0.811910
\(633\) −9.04888 −0.359661
\(634\) 7.85840 0.312097
\(635\) 0 0
\(636\) −17.8328 −0.707115
\(637\) 3.20053 0.126809
\(638\) 6.55270 0.259424
\(639\) −12.8136 −0.506898
\(640\) 0 0
\(641\) −31.5764 −1.24719 −0.623596 0.781747i \(-0.714329\pi\)
−0.623596 + 0.781747i \(0.714329\pi\)
\(642\) 27.3622 1.07990
\(643\) 4.51890 0.178208 0.0891040 0.996022i \(-0.471600\pi\)
0.0891040 + 0.996022i \(0.471600\pi\)
\(644\) −2.91995 −0.115062
\(645\) 0 0
\(646\) −19.0872 −0.750975
\(647\) 20.6705 0.812643 0.406322 0.913730i \(-0.366811\pi\)
0.406322 + 0.913730i \(0.366811\pi\)
\(648\) −1.28917 −0.0506433
\(649\) 0.853372 0.0334978
\(650\) 0 0
\(651\) 22.6322 0.887027
\(652\) 16.4988 0.646143
\(653\) −0.381381 −0.0149246 −0.00746229 0.999972i \(-0.502375\pi\)
−0.00746229 + 0.999972i \(0.502375\pi\)
\(654\) 9.25443 0.361877
\(655\) 0 0
\(656\) −4.91638 −0.191952
\(657\) −4.75971 −0.185694
\(658\) −1.07714 −0.0419914
\(659\) 13.4600 0.524326 0.262163 0.965024i \(-0.415564\pi\)
0.262163 + 0.965024i \(0.415564\pi\)
\(660\) 0 0
\(661\) 30.7738 1.19696 0.598482 0.801136i \(-0.295771\pi\)
0.598482 + 0.801136i \(0.295771\pi\)
\(662\) 10.0383 0.390150
\(663\) 17.3083 0.672200
\(664\) −9.90225 −0.384282
\(665\) 0 0
\(666\) −3.77886 −0.146428
\(667\) 3.98441 0.154277
\(668\) −12.4111 −0.480200
\(669\) −24.8222 −0.959682
\(670\) 0 0
\(671\) 1.55918 0.0601915
\(672\) 16.0000 0.617213
\(673\) −1.48110 −0.0570923 −0.0285461 0.999592i \(-0.509088\pi\)
−0.0285461 + 0.999592i \(0.509088\pi\)
\(674\) 6.72999 0.259229
\(675\) 0 0
\(676\) 16.8086 0.646484
\(677\) −45.3749 −1.74390 −0.871950 0.489596i \(-0.837144\pi\)
−0.871950 + 0.489596i \(0.837144\pi\)
\(678\) 31.7633 1.21986
\(679\) 6.89169 0.264479
\(680\) 0 0
\(681\) −16.6464 −0.637890
\(682\) −13.2288 −0.506557
\(683\) 0.0594386 0.00227436 0.00113718 0.999999i \(-0.499638\pi\)
0.00113718 + 0.999999i \(0.499638\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 34.9200 1.33325
\(687\) −15.7789 −0.602001
\(688\) 44.6066 1.70061
\(689\) 70.5855 2.68909
\(690\) 0 0
\(691\) −18.9355 −0.720342 −0.360171 0.932886i \(-0.617282\pi\)
−0.360171 + 0.932886i \(0.617282\pi\)
\(692\) −15.4005 −0.585441
\(693\) −2.05390 −0.0780212
\(694\) −5.58336 −0.211941
\(695\) 0 0
\(696\) −5.72496 −0.217004
\(697\) −3.39194 −0.128479
\(698\) 26.8378 1.01583
\(699\) 24.9200 0.942559
\(700\) 0 0
\(701\) −0.540024 −0.0203964 −0.0101982 0.999948i \(-0.503246\pi\)
−0.0101982 + 0.999948i \(0.503246\pi\)
\(702\) −9.25443 −0.349286
\(703\) −6.46500 −0.243832
\(704\) −1.35218 −0.0509621
\(705\) 0 0
\(706\) −48.6449 −1.83078
\(707\) 29.9461 1.12624
\(708\) 1.35218 0.0508180
\(709\) 28.0867 1.05482 0.527409 0.849612i \(-0.323164\pi\)
0.527409 + 0.849612i \(0.323164\pi\)
\(710\) 0 0
\(711\) −15.8328 −0.593775
\(712\) −17.0872 −0.640369
\(713\) −8.04385 −0.301245
\(714\) 15.5295 0.581175
\(715\) 0 0
\(716\) −2.75562 −0.102982
\(717\) −2.95112 −0.110212
\(718\) 4.27504 0.159543
\(719\) 9.86248 0.367809 0.183904 0.982944i \(-0.441126\pi\)
0.183904 + 0.982944i \(0.441126\pi\)
\(720\) 0 0
\(721\) −1.24889 −0.0465110
\(722\) −16.9985 −0.632620
\(723\) −16.5925 −0.617081
\(724\) −2.36274 −0.0878106
\(725\) 0 0
\(726\) −18.7491 −0.695846
\(727\) 30.4650 1.12988 0.564942 0.825131i \(-0.308898\pi\)
0.564942 + 0.825131i \(0.308898\pi\)
\(728\) −16.6066 −0.615482
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.7753 1.13827
\(732\) 2.47054 0.0913137
\(733\) −27.5436 −1.01735 −0.508673 0.860960i \(-0.669864\pi\)
−0.508673 + 0.860960i \(0.669864\pi\)
\(734\) −42.0227 −1.55109
\(735\) 0 0
\(736\) −5.68665 −0.209613
\(737\) 8.16724 0.300844
\(738\) 1.81361 0.0667598
\(739\) 13.1013 0.481940 0.240970 0.970533i \(-0.422534\pi\)
0.240970 + 0.970533i \(0.422534\pi\)
\(740\) 0 0
\(741\) −15.8328 −0.581631
\(742\) 63.3311 2.32496
\(743\) −34.7527 −1.27495 −0.637477 0.770470i \(-0.720022\pi\)
−0.637477 + 0.770470i \(0.720022\pi\)
\(744\) 11.5577 0.423727
\(745\) 0 0
\(746\) −63.8555 −2.33792
\(747\) 7.68111 0.281037
\(748\) −3.55773 −0.130083
\(749\) −38.0867 −1.39166
\(750\) 0 0
\(751\) 32.7738 1.19593 0.597967 0.801521i \(-0.295975\pi\)
0.597967 + 0.801521i \(0.295975\pi\)
\(752\) −1.15667 −0.0421796
\(753\) −20.1361 −0.733799
\(754\) −41.0972 −1.49667
\(755\) 0 0
\(756\) −3.25443 −0.118362
\(757\) 32.2978 1.17388 0.586941 0.809630i \(-0.300332\pi\)
0.586941 + 0.809630i \(0.300332\pi\)
\(758\) −47.2233 −1.71523
\(759\) 0.729988 0.0264969
\(760\) 0 0
\(761\) 44.4494 1.61129 0.805645 0.592399i \(-0.201819\pi\)
0.805645 + 0.592399i \(0.201819\pi\)
\(762\) −23.3622 −0.846324
\(763\) −12.8816 −0.466347
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) −27.8967 −1.00795
\(767\) −5.35218 −0.193256
\(768\) 20.8469 0.752248
\(769\) −19.7108 −0.710791 −0.355395 0.934716i \(-0.615654\pi\)
−0.355395 + 0.934716i \(0.615654\pi\)
\(770\) 0 0
\(771\) −25.9008 −0.932794
\(772\) −20.1260 −0.724351
\(773\) −43.2530 −1.55570 −0.777851 0.628449i \(-0.783690\pi\)
−0.777851 + 0.628449i \(0.783690\pi\)
\(774\) −16.4550 −0.591461
\(775\) 0 0
\(776\) 3.51941 0.126340
\(777\) 5.25997 0.188700
\(778\) 8.09775 0.290318
\(779\) 3.10278 0.111168
\(780\) 0 0
\(781\) −10.4252 −0.373044
\(782\) −5.51941 −0.197374
\(783\) 4.44082 0.158702
\(784\) 3.08362 0.110129
\(785\) 0 0
\(786\) −11.5295 −0.411242
\(787\) 21.4358 0.764104 0.382052 0.924141i \(-0.375218\pi\)
0.382052 + 0.924141i \(0.375218\pi\)
\(788\) 6.71731 0.239294
\(789\) 22.4408 0.798914
\(790\) 0 0
\(791\) −44.2127 −1.57202
\(792\) −1.04888 −0.0372702
\(793\) −9.77886 −0.347258
\(794\) 19.1567 0.679845
\(795\) 0 0
\(796\) 2.64782 0.0938496
\(797\) 4.40105 0.155893 0.0779467 0.996958i \(-0.475164\pi\)
0.0779467 + 0.996958i \(0.475164\pi\)
\(798\) −14.2056 −0.502871
\(799\) −0.798021 −0.0282319
\(800\) 0 0
\(801\) 13.2544 0.468322
\(802\) 23.7350 0.838112
\(803\) −3.87253 −0.136659
\(804\) 12.9411 0.456397
\(805\) 0 0
\(806\) 82.9683 2.92243
\(807\) 7.62721 0.268491
\(808\) 15.2927 0.537997
\(809\) −26.3799 −0.927469 −0.463734 0.885974i \(-0.653491\pi\)
−0.463734 + 0.885974i \(0.653491\pi\)
\(810\) 0 0
\(811\) 4.96526 0.174354 0.0871769 0.996193i \(-0.472215\pi\)
0.0871769 + 0.996193i \(0.472215\pi\)
\(812\) −14.4523 −0.507177
\(813\) 19.9164 0.698498
\(814\) −3.07451 −0.107761
\(815\) 0 0
\(816\) 16.6761 0.583780
\(817\) −28.1517 −0.984902
\(818\) −33.6499 −1.17654
\(819\) 12.8816 0.450121
\(820\) 0 0
\(821\) −15.7789 −0.550686 −0.275343 0.961346i \(-0.588791\pi\)
−0.275343 + 0.961346i \(0.588791\pi\)
\(822\) 31.7194 1.10634
\(823\) 8.35166 0.291121 0.145560 0.989349i \(-0.453502\pi\)
0.145560 + 0.989349i \(0.453502\pi\)
\(824\) −0.637776 −0.0222180
\(825\) 0 0
\(826\) −4.80211 −0.167087
\(827\) 18.3925 0.639568 0.319784 0.947490i \(-0.396390\pi\)
0.319784 + 0.947490i \(0.396390\pi\)
\(828\) 1.15667 0.0401972
\(829\) −29.4147 −1.02161 −0.510807 0.859695i \(-0.670653\pi\)
−0.510807 + 0.859695i \(0.670653\pi\)
\(830\) 0 0
\(831\) 17.6413 0.611972
\(832\) 8.48059 0.294011
\(833\) 2.12747 0.0737125
\(834\) 28.5189 0.987529
\(835\) 0 0
\(836\) 3.25443 0.112557
\(837\) −8.96526 −0.309885
\(838\) −46.4494 −1.60457
\(839\) −33.4600 −1.15517 −0.577583 0.816332i \(-0.696004\pi\)
−0.577583 + 0.816332i \(0.696004\pi\)
\(840\) 0 0
\(841\) −9.27912 −0.319970
\(842\) −15.7350 −0.542264
\(843\) −0.0297193 −0.00102359
\(844\) −11.6655 −0.401544
\(845\) 0 0
\(846\) 0.426686 0.0146698
\(847\) 26.0978 0.896729
\(848\) 68.0071 2.33537
\(849\) 10.1814 0.349424
\(850\) 0 0
\(851\) −1.86947 −0.0640848
\(852\) −16.5189 −0.565928
\(853\) −40.4494 −1.38496 −0.692481 0.721436i \(-0.743482\pi\)
−0.692481 + 0.721436i \(0.743482\pi\)
\(854\) −8.77384 −0.300235
\(855\) 0 0
\(856\) −19.4499 −0.664785
\(857\) −12.4494 −0.425264 −0.212632 0.977132i \(-0.568204\pi\)
−0.212632 + 0.977132i \(0.568204\pi\)
\(858\) −7.52946 −0.257052
\(859\) −21.6514 −0.738736 −0.369368 0.929283i \(-0.620426\pi\)
−0.369368 + 0.929283i \(0.620426\pi\)
\(860\) 0 0
\(861\) −2.52444 −0.0860326
\(862\) 12.8816 0.438750
\(863\) −19.3778 −0.659628 −0.329814 0.944046i \(-0.606986\pi\)
−0.329814 + 0.944046i \(0.606986\pi\)
\(864\) −6.33804 −0.215625
\(865\) 0 0
\(866\) 19.9945 0.679439
\(867\) −5.49472 −0.186610
\(868\) 29.1768 0.990324
\(869\) −12.8816 −0.436980
\(870\) 0 0
\(871\) −51.2233 −1.73563
\(872\) −6.57834 −0.222771
\(873\) −2.72999 −0.0923961
\(874\) 5.04888 0.170781
\(875\) 0 0
\(876\) −6.13607 −0.207318
\(877\) 3.08413 0.104144 0.0520719 0.998643i \(-0.483417\pi\)
0.0520719 + 0.998643i \(0.483417\pi\)
\(878\) −2.40105 −0.0810316
\(879\) −3.14808 −0.106182
\(880\) 0 0
\(881\) −34.5783 −1.16497 −0.582487 0.812840i \(-0.697920\pi\)
−0.582487 + 0.812840i \(0.697920\pi\)
\(882\) −1.13752 −0.0383022
\(883\) 9.64280 0.324506 0.162253 0.986749i \(-0.448124\pi\)
0.162253 + 0.986749i \(0.448124\pi\)
\(884\) 22.3133 0.750479
\(885\) 0 0
\(886\) −28.2721 −0.949821
\(887\) −32.2041 −1.08131 −0.540654 0.841245i \(-0.681823\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(888\) 2.68614 0.0901408
\(889\) 32.5189 1.09065
\(890\) 0 0
\(891\) 0.813607 0.0272568
\(892\) −32.0000 −1.07144
\(893\) 0.729988 0.0244281
\(894\) 20.9794 0.701656
\(895\) 0 0
\(896\) −24.3910 −0.814846
\(897\) −4.57834 −0.152866
\(898\) −59.4571 −1.98411
\(899\) −39.8131 −1.32784
\(900\) 0 0
\(901\) 46.9200 1.56313
\(902\) 1.47556 0.0491308
\(903\) 22.9044 0.762210
\(904\) −22.5783 −0.750944
\(905\) 0 0
\(906\) 34.9200 1.16014
\(907\) 17.8227 0.591794 0.295897 0.955220i \(-0.404382\pi\)
0.295897 + 0.955220i \(0.404382\pi\)
\(908\) −21.4600 −0.712174
\(909\) −11.8625 −0.393454
\(910\) 0 0
\(911\) −20.7894 −0.688784 −0.344392 0.938826i \(-0.611915\pi\)
−0.344392 + 0.938826i \(0.611915\pi\)
\(912\) −15.2544 −0.505125
\(913\) 6.24940 0.206825
\(914\) 25.6938 0.849875
\(915\) 0 0
\(916\) −20.3416 −0.672106
\(917\) 16.0484 0.529964
\(918\) −6.15165 −0.203035
\(919\) 33.8555 1.11679 0.558395 0.829575i \(-0.311417\pi\)
0.558395 + 0.829575i \(0.311417\pi\)
\(920\) 0 0
\(921\) 12.7980 0.421709
\(922\) −54.5472 −1.79642
\(923\) 65.3850 2.15217
\(924\) −2.64782 −0.0871070
\(925\) 0 0
\(926\) 43.3028 1.42302
\(927\) 0.494719 0.0162487
\(928\) −28.1461 −0.923941
\(929\) −4.10635 −0.134725 −0.0673624 0.997729i \(-0.521458\pi\)
−0.0673624 + 0.997729i \(0.521458\pi\)
\(930\) 0 0
\(931\) −1.94610 −0.0637809
\(932\) 32.1260 1.05232
\(933\) 26.6167 0.871390
\(934\) 17.6555 0.577705
\(935\) 0 0
\(936\) 6.57834 0.215020
\(937\) −13.3028 −0.434583 −0.217292 0.976107i \(-0.569722\pi\)
−0.217292 + 0.976107i \(0.569722\pi\)
\(938\) −45.9588 −1.50061
\(939\) 3.00502 0.0980652
\(940\) 0 0
\(941\) −11.7633 −0.383472 −0.191736 0.981447i \(-0.561412\pi\)
−0.191736 + 0.981447i \(0.561412\pi\)
\(942\) 37.9406 1.23617
\(943\) 0.897225 0.0292177
\(944\) −5.15667 −0.167835
\(945\) 0 0
\(946\) −13.3879 −0.435277
\(947\) −1.40054 −0.0455114 −0.0227557 0.999741i \(-0.507244\pi\)
−0.0227557 + 0.999741i \(0.507244\pi\)
\(948\) −20.4111 −0.662922
\(949\) 24.2877 0.788413
\(950\) 0 0
\(951\) 4.33302 0.140508
\(952\) −11.0388 −0.357771
\(953\) −33.7577 −1.09352 −0.546760 0.837289i \(-0.684139\pi\)
−0.546760 + 0.837289i \(0.684139\pi\)
\(954\) −25.0872 −0.812228
\(955\) 0 0
\(956\) −3.80450 −0.123046
\(957\) 3.61308 0.116794
\(958\) −2.13343 −0.0689280
\(959\) −44.1517 −1.42573
\(960\) 0 0
\(961\) 49.3758 1.59277
\(962\) 19.2827 0.621699
\(963\) 15.0872 0.486178
\(964\) −21.3905 −0.688941
\(965\) 0 0
\(966\) −4.10780 −0.132166
\(967\) 10.4806 0.337033 0.168516 0.985699i \(-0.446102\pi\)
0.168516 + 0.985699i \(0.446102\pi\)
\(968\) 13.3275 0.428361
\(969\) −10.5244 −0.338094
\(970\) 0 0
\(971\) 57.6358 1.84962 0.924811 0.380428i \(-0.124223\pi\)
0.924811 + 0.380428i \(0.124223\pi\)
\(972\) 1.28917 0.0413501
\(973\) −39.6967 −1.27262
\(974\) −37.3649 −1.19725
\(975\) 0 0
\(976\) −9.42166 −0.301580
\(977\) −41.9008 −1.34053 −0.670263 0.742124i \(-0.733819\pi\)
−0.670263 + 0.742124i \(0.733819\pi\)
\(978\) 23.2106 0.742192
\(979\) 10.7839 0.344655
\(980\) 0 0
\(981\) 5.10278 0.162919
\(982\) 25.7350 0.821237
\(983\) 18.4056 0.587046 0.293523 0.955952i \(-0.405172\pi\)
0.293523 + 0.955952i \(0.405172\pi\)
\(984\) −1.28917 −0.0410972
\(985\) 0 0
\(986\) −27.3184 −0.869994
\(987\) −0.593923 −0.0189048
\(988\) −20.4111 −0.649364
\(989\) −8.14057 −0.258855
\(990\) 0 0
\(991\) 26.0666 0.828032 0.414016 0.910270i \(-0.364126\pi\)
0.414016 + 0.910270i \(0.364126\pi\)
\(992\) 56.8222 1.80411
\(993\) 5.53500 0.175648
\(994\) 58.6650 1.86074
\(995\) 0 0
\(996\) 9.90225 0.313765
\(997\) 29.8483 0.945307 0.472653 0.881248i \(-0.343296\pi\)
0.472653 + 0.881248i \(0.343296\pi\)
\(998\) −38.9200 −1.23199
\(999\) −2.08362 −0.0659228
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 3075.2.a.t.1.3 3
3.2 odd 2 9225.2.a.bx.1.1 3
5.4 even 2 123.2.a.d.1.1 3
15.14 odd 2 369.2.a.e.1.3 3
20.19 odd 2 1968.2.a.w.1.1 3
35.34 odd 2 6027.2.a.s.1.1 3
40.19 odd 2 7872.2.a.bs.1.3 3
40.29 even 2 7872.2.a.bx.1.3 3
60.59 even 2 5904.2.a.bd.1.3 3
205.204 even 2 5043.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.2.a.d.1.1 3 5.4 even 2
369.2.a.e.1.3 3 15.14 odd 2
1968.2.a.w.1.1 3 20.19 odd 2
3075.2.a.t.1.3 3 1.1 even 1 trivial
5043.2.a.n.1.1 3 205.204 even 2
5904.2.a.bd.1.3 3 60.59 even 2
6027.2.a.s.1.1 3 35.34 odd 2
7872.2.a.bs.1.3 3 40.19 odd 2
7872.2.a.bx.1.3 3 40.29 even 2
9225.2.a.bx.1.1 3 3.2 odd 2