Properties

Label 5047.2.a.a.1.1
Level $5047$
Weight $2$
Character 5047.1
Self dual yes
Analytic conductor $40.300$
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] = [5047,2,Mod(1,5047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5047, 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("5047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5047 = 7^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5047.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.3004979001\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 103)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 5047.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} +0.381966 q^{5} -2.61803 q^{6} -7.47214 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-2.61803 q^{2} +1.00000 q^{3} +4.85410 q^{4} +0.381966 q^{5} -2.61803 q^{6} -7.47214 q^{8} -2.00000 q^{9} -1.00000 q^{10} -2.61803 q^{11} +4.85410 q^{12} +4.85410 q^{13} +0.381966 q^{15} +9.85410 q^{16} +5.61803 q^{17} +5.23607 q^{18} -5.85410 q^{19} +1.85410 q^{20} +6.85410 q^{22} +4.47214 q^{23} -7.47214 q^{24} -4.85410 q^{25} -12.7082 q^{26} -5.00000 q^{27} -5.23607 q^{29} -1.00000 q^{30} +6.70820 q^{31} -10.8541 q^{32} -2.61803 q^{33} -14.7082 q^{34} -9.70820 q^{36} +6.70820 q^{37} +15.3262 q^{38} +4.85410 q^{39} -2.85410 q^{40} -8.94427 q^{41} -8.70820 q^{43} -12.7082 q^{44} -0.763932 q^{45} -11.7082 q^{46} -4.09017 q^{47} +9.85410 q^{48} +12.7082 q^{50} +5.61803 q^{51} +23.5623 q^{52} +1.09017 q^{53} +13.0902 q^{54} -1.00000 q^{55} -5.85410 q^{57} +13.7082 q^{58} -6.38197 q^{59} +1.85410 q^{60} -4.14590 q^{61} -17.5623 q^{62} +8.70820 q^{64} +1.85410 q^{65} +6.85410 q^{66} +14.4164 q^{67} +27.2705 q^{68} +4.47214 q^{69} -4.09017 q^{71} +14.9443 q^{72} +10.8541 q^{73} -17.5623 q^{74} -4.85410 q^{75} -28.4164 q^{76} -12.7082 q^{78} -6.56231 q^{79} +3.76393 q^{80} +1.00000 q^{81} +23.4164 q^{82} +6.32624 q^{83} +2.14590 q^{85} +22.7984 q^{86} -5.23607 q^{87} +19.5623 q^{88} +2.29180 q^{89} +2.00000 q^{90} +21.7082 q^{92} +6.70820 q^{93} +10.7082 q^{94} -2.23607 q^{95} -10.8541 q^{96} +1.70820 q^{97} +5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 3 q^{6} - 6 q^{8} - 4 q^{9} - 2 q^{10} - 3 q^{11} + 3 q^{12} + 3 q^{13} + 3 q^{15} + 13 q^{16} + 9 q^{17} + 6 q^{18} - 5 q^{19} - 3 q^{20} + 7 q^{22} - 6 q^{24} - 3 q^{25} - 12 q^{26} - 10 q^{27} - 6 q^{29} - 2 q^{30} - 15 q^{32} - 3 q^{33} - 16 q^{34} - 6 q^{36} + 15 q^{38} + 3 q^{39} + q^{40} - 4 q^{43} - 12 q^{44} - 6 q^{45} - 10 q^{46} + 3 q^{47} + 13 q^{48} + 12 q^{50} + 9 q^{51} + 27 q^{52} - 9 q^{53} + 15 q^{54} - 2 q^{55} - 5 q^{57} + 14 q^{58} - 15 q^{59} - 3 q^{60} - 15 q^{61} - 15 q^{62} + 4 q^{64} - 3 q^{65} + 7 q^{66} + 2 q^{67} + 21 q^{68} + 3 q^{71} + 12 q^{72} + 15 q^{73} - 15 q^{74} - 3 q^{75} - 30 q^{76} - 12 q^{78} + 7 q^{79} + 12 q^{80} + 2 q^{81} + 20 q^{82} - 3 q^{83} + 11 q^{85} + 21 q^{86} - 6 q^{87} + 19 q^{88} + 18 q^{89} + 4 q^{90} + 30 q^{92} + 8 q^{94} - 15 q^{96} - 10 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61803 −1.85123 −0.925615 0.378467i \(-0.876451\pi\)
−0.925615 + 0.378467i \(0.876451\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 4.85410 2.42705
\(5\) 0.381966 0.170820 0.0854102 0.996346i \(-0.472780\pi\)
0.0854102 + 0.996346i \(0.472780\pi\)
\(6\) −2.61803 −1.06881
\(7\) 0 0
\(8\) −7.47214 −2.64180
\(9\) −2.00000 −0.666667
\(10\) −1.00000 −0.316228
\(11\) −2.61803 −0.789367 −0.394683 0.918817i \(-0.629146\pi\)
−0.394683 + 0.918817i \(0.629146\pi\)
\(12\) 4.85410 1.40126
\(13\) 4.85410 1.34629 0.673143 0.739512i \(-0.264944\pi\)
0.673143 + 0.739512i \(0.264944\pi\)
\(14\) 0 0
\(15\) 0.381966 0.0986232
\(16\) 9.85410 2.46353
\(17\) 5.61803 1.36257 0.681287 0.732017i \(-0.261421\pi\)
0.681287 + 0.732017i \(0.261421\pi\)
\(18\) 5.23607 1.23415
\(19\) −5.85410 −1.34302 −0.671512 0.740994i \(-0.734355\pi\)
−0.671512 + 0.740994i \(0.734355\pi\)
\(20\) 1.85410 0.414590
\(21\) 0 0
\(22\) 6.85410 1.46130
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) −7.47214 −1.52524
\(25\) −4.85410 −0.970820
\(26\) −12.7082 −2.49228
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −5.23607 −0.972313 −0.486157 0.873872i \(-0.661602\pi\)
−0.486157 + 0.873872i \(0.661602\pi\)
\(30\) −1.00000 −0.182574
\(31\) 6.70820 1.20483 0.602414 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) −10.8541 −1.91875
\(33\) −2.61803 −0.455741
\(34\) −14.7082 −2.52244
\(35\) 0 0
\(36\) −9.70820 −1.61803
\(37\) 6.70820 1.10282 0.551411 0.834234i \(-0.314090\pi\)
0.551411 + 0.834234i \(0.314090\pi\)
\(38\) 15.3262 2.48624
\(39\) 4.85410 0.777278
\(40\) −2.85410 −0.451273
\(41\) −8.94427 −1.39686 −0.698430 0.715678i \(-0.746118\pi\)
−0.698430 + 0.715678i \(0.746118\pi\)
\(42\) 0 0
\(43\) −8.70820 −1.32799 −0.663994 0.747738i \(-0.731140\pi\)
−0.663994 + 0.747738i \(0.731140\pi\)
\(44\) −12.7082 −1.91583
\(45\) −0.763932 −0.113880
\(46\) −11.7082 −1.72628
\(47\) −4.09017 −0.596613 −0.298306 0.954470i \(-0.596422\pi\)
−0.298306 + 0.954470i \(0.596422\pi\)
\(48\) 9.85410 1.42232
\(49\) 0 0
\(50\) 12.7082 1.79721
\(51\) 5.61803 0.786682
\(52\) 23.5623 3.26750
\(53\) 1.09017 0.149746 0.0748732 0.997193i \(-0.476145\pi\)
0.0748732 + 0.997193i \(0.476145\pi\)
\(54\) 13.0902 1.78135
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) −5.85410 −0.775395
\(58\) 13.7082 1.79998
\(59\) −6.38197 −0.830861 −0.415431 0.909625i \(-0.636369\pi\)
−0.415431 + 0.909625i \(0.636369\pi\)
\(60\) 1.85410 0.239364
\(61\) −4.14590 −0.530828 −0.265414 0.964135i \(-0.585509\pi\)
−0.265414 + 0.964135i \(0.585509\pi\)
\(62\) −17.5623 −2.23042
\(63\) 0 0
\(64\) 8.70820 1.08853
\(65\) 1.85410 0.229973
\(66\) 6.85410 0.843682
\(67\) 14.4164 1.76124 0.880622 0.473819i \(-0.157125\pi\)
0.880622 + 0.473819i \(0.157125\pi\)
\(68\) 27.2705 3.30704
\(69\) 4.47214 0.538382
\(70\) 0 0
\(71\) −4.09017 −0.485414 −0.242707 0.970100i \(-0.578035\pi\)
−0.242707 + 0.970100i \(0.578035\pi\)
\(72\) 14.9443 1.76120
\(73\) 10.8541 1.27038 0.635188 0.772357i \(-0.280923\pi\)
0.635188 + 0.772357i \(0.280923\pi\)
\(74\) −17.5623 −2.04158
\(75\) −4.85410 −0.560503
\(76\) −28.4164 −3.25959
\(77\) 0 0
\(78\) −12.7082 −1.43892
\(79\) −6.56231 −0.738317 −0.369159 0.929366i \(-0.620354\pi\)
−0.369159 + 0.929366i \(0.620354\pi\)
\(80\) 3.76393 0.420820
\(81\) 1.00000 0.111111
\(82\) 23.4164 2.58591
\(83\) 6.32624 0.694395 0.347197 0.937792i \(-0.387133\pi\)
0.347197 + 0.937792i \(0.387133\pi\)
\(84\) 0 0
\(85\) 2.14590 0.232755
\(86\) 22.7984 2.45841
\(87\) −5.23607 −0.561365
\(88\) 19.5623 2.08535
\(89\) 2.29180 0.242930 0.121465 0.992596i \(-0.461241\pi\)
0.121465 + 0.992596i \(0.461241\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 21.7082 2.26324
\(93\) 6.70820 0.695608
\(94\) 10.7082 1.10447
\(95\) −2.23607 −0.229416
\(96\) −10.8541 −1.10779
\(97\) 1.70820 0.173442 0.0867209 0.996233i \(-0.472361\pi\)
0.0867209 + 0.996233i \(0.472361\pi\)
\(98\) 0 0
\(99\) 5.23607 0.526245
\(100\) −23.5623 −2.35623
\(101\) −1.90983 −0.190035 −0.0950176 0.995476i \(-0.530291\pi\)
−0.0950176 + 0.995476i \(0.530291\pi\)
\(102\) −14.7082 −1.45633
\(103\) 1.00000 0.0985329
\(104\) −36.2705 −3.55662
\(105\) 0 0
\(106\) −2.85410 −0.277215
\(107\) −7.09017 −0.685433 −0.342716 0.939439i \(-0.611347\pi\)
−0.342716 + 0.939439i \(0.611347\pi\)
\(108\) −24.2705 −2.33543
\(109\) −9.56231 −0.915903 −0.457951 0.888977i \(-0.651417\pi\)
−0.457951 + 0.888977i \(0.651417\pi\)
\(110\) 2.61803 0.249620
\(111\) 6.70820 0.636715
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) 15.3262 1.43543
\(115\) 1.70820 0.159291
\(116\) −25.4164 −2.35985
\(117\) −9.70820 −0.897524
\(118\) 16.7082 1.53811
\(119\) 0 0
\(120\) −2.85410 −0.260543
\(121\) −4.14590 −0.376900
\(122\) 10.8541 0.982684
\(123\) −8.94427 −0.806478
\(124\) 32.5623 2.92418
\(125\) −3.76393 −0.336656
\(126\) 0 0
\(127\) 15.2705 1.35504 0.677519 0.735505i \(-0.263055\pi\)
0.677519 + 0.735505i \(0.263055\pi\)
\(128\) −1.09017 −0.0963583
\(129\) −8.70820 −0.766715
\(130\) −4.85410 −0.425733
\(131\) 2.23607 0.195366 0.0976831 0.995218i \(-0.468857\pi\)
0.0976831 + 0.995218i \(0.468857\pi\)
\(132\) −12.7082 −1.10611
\(133\) 0 0
\(134\) −37.7426 −3.26047
\(135\) −1.90983 −0.164372
\(136\) −41.9787 −3.59965
\(137\) −12.7082 −1.08574 −0.542868 0.839818i \(-0.682661\pi\)
−0.542868 + 0.839818i \(0.682661\pi\)
\(138\) −11.7082 −0.996669
\(139\) 11.1459 0.945383 0.472691 0.881228i \(-0.343283\pi\)
0.472691 + 0.881228i \(0.343283\pi\)
\(140\) 0 0
\(141\) −4.09017 −0.344454
\(142\) 10.7082 0.898613
\(143\) −12.7082 −1.06271
\(144\) −19.7082 −1.64235
\(145\) −2.00000 −0.166091
\(146\) −28.4164 −2.35176
\(147\) 0 0
\(148\) 32.5623 2.67661
\(149\) −7.47214 −0.612141 −0.306071 0.952009i \(-0.599014\pi\)
−0.306071 + 0.952009i \(0.599014\pi\)
\(150\) 12.7082 1.03762
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 43.7426 3.54800
\(153\) −11.2361 −0.908382
\(154\) 0 0
\(155\) 2.56231 0.205809
\(156\) 23.5623 1.88649
\(157\) −16.7082 −1.33346 −0.666730 0.745299i \(-0.732307\pi\)
−0.666730 + 0.745299i \(0.732307\pi\)
\(158\) 17.1803 1.36679
\(159\) 1.09017 0.0864561
\(160\) −4.14590 −0.327762
\(161\) 0 0
\(162\) −2.61803 −0.205692
\(163\) 2.70820 0.212123 0.106061 0.994360i \(-0.466176\pi\)
0.106061 + 0.994360i \(0.466176\pi\)
\(164\) −43.4164 −3.39025
\(165\) −1.00000 −0.0778499
\(166\) −16.5623 −1.28548
\(167\) −9.00000 −0.696441 −0.348220 0.937413i \(-0.613214\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(168\) 0 0
\(169\) 10.5623 0.812485
\(170\) −5.61803 −0.430884
\(171\) 11.7082 0.895349
\(172\) −42.2705 −3.22310
\(173\) −16.0344 −1.21908 −0.609538 0.792757i \(-0.708645\pi\)
−0.609538 + 0.792757i \(0.708645\pi\)
\(174\) 13.7082 1.03922
\(175\) 0 0
\(176\) −25.7984 −1.94463
\(177\) −6.38197 −0.479698
\(178\) −6.00000 −0.449719
\(179\) −7.85410 −0.587043 −0.293522 0.955952i \(-0.594827\pi\)
−0.293522 + 0.955952i \(0.594827\pi\)
\(180\) −3.70820 −0.276393
\(181\) −3.85410 −0.286473 −0.143237 0.989688i \(-0.545751\pi\)
−0.143237 + 0.989688i \(0.545751\pi\)
\(182\) 0 0
\(183\) −4.14590 −0.306474
\(184\) −33.4164 −2.46349
\(185\) 2.56231 0.188384
\(186\) −17.5623 −1.28773
\(187\) −14.7082 −1.07557
\(188\) −19.8541 −1.44801
\(189\) 0 0
\(190\) 5.85410 0.424701
\(191\) 5.61803 0.406507 0.203253 0.979126i \(-0.434848\pi\)
0.203253 + 0.979126i \(0.434848\pi\)
\(192\) 8.70820 0.628460
\(193\) 20.1246 1.44860 0.724301 0.689484i \(-0.242163\pi\)
0.724301 + 0.689484i \(0.242163\pi\)
\(194\) −4.47214 −0.321081
\(195\) 1.85410 0.132775
\(196\) 0 0
\(197\) −16.4164 −1.16962 −0.584810 0.811170i \(-0.698831\pi\)
−0.584810 + 0.811170i \(0.698831\pi\)
\(198\) −13.7082 −0.974200
\(199\) 23.4164 1.65995 0.829973 0.557804i \(-0.188356\pi\)
0.829973 + 0.557804i \(0.188356\pi\)
\(200\) 36.2705 2.56471
\(201\) 14.4164 1.01686
\(202\) 5.00000 0.351799
\(203\) 0 0
\(204\) 27.2705 1.90932
\(205\) −3.41641 −0.238612
\(206\) −2.61803 −0.182407
\(207\) −8.94427 −0.621670
\(208\) 47.8328 3.31661
\(209\) 15.3262 1.06014
\(210\) 0 0
\(211\) 14.8541 1.02260 0.511299 0.859403i \(-0.329164\pi\)
0.511299 + 0.859403i \(0.329164\pi\)
\(212\) 5.29180 0.363442
\(213\) −4.09017 −0.280254
\(214\) 18.5623 1.26889
\(215\) −3.32624 −0.226848
\(216\) 37.3607 2.54207
\(217\) 0 0
\(218\) 25.0344 1.69555
\(219\) 10.8541 0.733452
\(220\) −4.85410 −0.327263
\(221\) 27.2705 1.83441
\(222\) −17.5623 −1.17870
\(223\) −7.70820 −0.516180 −0.258090 0.966121i \(-0.583093\pi\)
−0.258090 + 0.966121i \(0.583093\pi\)
\(224\) 0 0
\(225\) 9.70820 0.647214
\(226\) 39.2705 2.61224
\(227\) −2.94427 −0.195418 −0.0977091 0.995215i \(-0.531151\pi\)
−0.0977091 + 0.995215i \(0.531151\pi\)
\(228\) −28.4164 −1.88192
\(229\) −6.70820 −0.443291 −0.221645 0.975127i \(-0.571143\pi\)
−0.221645 + 0.975127i \(0.571143\pi\)
\(230\) −4.47214 −0.294884
\(231\) 0 0
\(232\) 39.1246 2.56866
\(233\) −26.8885 −1.76153 −0.880764 0.473556i \(-0.842970\pi\)
−0.880764 + 0.473556i \(0.842970\pi\)
\(234\) 25.4164 1.66152
\(235\) −1.56231 −0.101914
\(236\) −30.9787 −2.01654
\(237\) −6.56231 −0.426268
\(238\) 0 0
\(239\) 8.67376 0.561059 0.280530 0.959845i \(-0.409490\pi\)
0.280530 + 0.959845i \(0.409490\pi\)
\(240\) 3.76393 0.242961
\(241\) 8.27051 0.532750 0.266375 0.963869i \(-0.414174\pi\)
0.266375 + 0.963869i \(0.414174\pi\)
\(242\) 10.8541 0.697728
\(243\) 16.0000 1.02640
\(244\) −20.1246 −1.28835
\(245\) 0 0
\(246\) 23.4164 1.49298
\(247\) −28.4164 −1.80809
\(248\) −50.1246 −3.18292
\(249\) 6.32624 0.400909
\(250\) 9.85410 0.623228
\(251\) −11.2361 −0.709214 −0.354607 0.935015i \(-0.615385\pi\)
−0.354607 + 0.935015i \(0.615385\pi\)
\(252\) 0 0
\(253\) −11.7082 −0.736088
\(254\) −39.9787 −2.50849
\(255\) 2.14590 0.134381
\(256\) −14.5623 −0.910144
\(257\) 13.4721 0.840369 0.420184 0.907439i \(-0.361965\pi\)
0.420184 + 0.907439i \(0.361965\pi\)
\(258\) 22.7984 1.41936
\(259\) 0 0
\(260\) 9.00000 0.558156
\(261\) 10.4721 0.648209
\(262\) −5.85410 −0.361668
\(263\) −20.6180 −1.27136 −0.635681 0.771952i \(-0.719281\pi\)
−0.635681 + 0.771952i \(0.719281\pi\)
\(264\) 19.5623 1.20398
\(265\) 0.416408 0.0255797
\(266\) 0 0
\(267\) 2.29180 0.140256
\(268\) 69.9787 4.27463
\(269\) −12.3262 −0.751544 −0.375772 0.926712i \(-0.622622\pi\)
−0.375772 + 0.926712i \(0.622622\pi\)
\(270\) 5.00000 0.304290
\(271\) −1.00000 −0.0607457 −0.0303728 0.999539i \(-0.509669\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 55.3607 3.35673
\(273\) 0 0
\(274\) 33.2705 2.00995
\(275\) 12.7082 0.766334
\(276\) 21.7082 1.30668
\(277\) 4.70820 0.282889 0.141444 0.989946i \(-0.454825\pi\)
0.141444 + 0.989946i \(0.454825\pi\)
\(278\) −29.1803 −1.75012
\(279\) −13.4164 −0.803219
\(280\) 0 0
\(281\) −31.4721 −1.87747 −0.938735 0.344640i \(-0.888001\pi\)
−0.938735 + 0.344640i \(0.888001\pi\)
\(282\) 10.7082 0.637664
\(283\) 19.7082 1.17153 0.585766 0.810481i \(-0.300794\pi\)
0.585766 + 0.810481i \(0.300794\pi\)
\(284\) −19.8541 −1.17812
\(285\) −2.23607 −0.132453
\(286\) 33.2705 1.96733
\(287\) 0 0
\(288\) 21.7082 1.27917
\(289\) 14.5623 0.856606
\(290\) 5.23607 0.307472
\(291\) 1.70820 0.100137
\(292\) 52.6869 3.08327
\(293\) 21.6525 1.26495 0.632476 0.774580i \(-0.282039\pi\)
0.632476 + 0.774580i \(0.282039\pi\)
\(294\) 0 0
\(295\) −2.43769 −0.141928
\(296\) −50.1246 −2.91343
\(297\) 13.0902 0.759569
\(298\) 19.5623 1.13321
\(299\) 21.7082 1.25542
\(300\) −23.5623 −1.36037
\(301\) 0 0
\(302\) 49.7426 2.86237
\(303\) −1.90983 −0.109717
\(304\) −57.6869 −3.30857
\(305\) −1.58359 −0.0906762
\(306\) 29.4164 1.68162
\(307\) 2.85410 0.162892 0.0814461 0.996678i \(-0.474046\pi\)
0.0814461 + 0.996678i \(0.474046\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) −6.70820 −0.381000
\(311\) −2.88854 −0.163794 −0.0818971 0.996641i \(-0.526098\pi\)
−0.0818971 + 0.996641i \(0.526098\pi\)
\(312\) −36.2705 −2.05341
\(313\) −16.2918 −0.920867 −0.460433 0.887694i \(-0.652306\pi\)
−0.460433 + 0.887694i \(0.652306\pi\)
\(314\) 43.7426 2.46854
\(315\) 0 0
\(316\) −31.8541 −1.79193
\(317\) 28.4164 1.59602 0.798012 0.602641i \(-0.205885\pi\)
0.798012 + 0.602641i \(0.205885\pi\)
\(318\) −2.85410 −0.160050
\(319\) 13.7082 0.767512
\(320\) 3.32624 0.185942
\(321\) −7.09017 −0.395735
\(322\) 0 0
\(323\) −32.8885 −1.82997
\(324\) 4.85410 0.269672
\(325\) −23.5623 −1.30700
\(326\) −7.09017 −0.392688
\(327\) −9.56231 −0.528797
\(328\) 66.8328 3.69022
\(329\) 0 0
\(330\) 2.61803 0.144118
\(331\) −16.1459 −0.887459 −0.443729 0.896161i \(-0.646345\pi\)
−0.443729 + 0.896161i \(0.646345\pi\)
\(332\) 30.7082 1.68533
\(333\) −13.4164 −0.735215
\(334\) 23.5623 1.28927
\(335\) 5.50658 0.300856
\(336\) 0 0
\(337\) −2.43769 −0.132790 −0.0663948 0.997793i \(-0.521150\pi\)
−0.0663948 + 0.997793i \(0.521150\pi\)
\(338\) −27.6525 −1.50410
\(339\) −15.0000 −0.814688
\(340\) 10.4164 0.564909
\(341\) −17.5623 −0.951052
\(342\) −30.6525 −1.65750
\(343\) 0 0
\(344\) 65.0689 3.50828
\(345\) 1.70820 0.0919666
\(346\) 41.9787 2.25679
\(347\) −1.47214 −0.0790284 −0.0395142 0.999219i \(-0.512581\pi\)
−0.0395142 + 0.999219i \(0.512581\pi\)
\(348\) −25.4164 −1.36246
\(349\) −15.4164 −0.825221 −0.412611 0.910907i \(-0.635383\pi\)
−0.412611 + 0.910907i \(0.635383\pi\)
\(350\) 0 0
\(351\) −24.2705 −1.29546
\(352\) 28.4164 1.51460
\(353\) 25.0344 1.33245 0.666224 0.745751i \(-0.267909\pi\)
0.666224 + 0.745751i \(0.267909\pi\)
\(354\) 16.7082 0.888031
\(355\) −1.56231 −0.0829186
\(356\) 11.1246 0.589603
\(357\) 0 0
\(358\) 20.5623 1.08675
\(359\) 14.6738 0.774452 0.387226 0.921985i \(-0.373433\pi\)
0.387226 + 0.921985i \(0.373433\pi\)
\(360\) 5.70820 0.300849
\(361\) 15.2705 0.803711
\(362\) 10.0902 0.530328
\(363\) −4.14590 −0.217603
\(364\) 0 0
\(365\) 4.14590 0.217006
\(366\) 10.8541 0.567353
\(367\) −16.4377 −0.858041 −0.429020 0.903295i \(-0.641141\pi\)
−0.429020 + 0.903295i \(0.641141\pi\)
\(368\) 44.0689 2.29725
\(369\) 17.8885 0.931240
\(370\) −6.70820 −0.348743
\(371\) 0 0
\(372\) 32.5623 1.68828
\(373\) −22.6869 −1.17468 −0.587342 0.809339i \(-0.699826\pi\)
−0.587342 + 0.809339i \(0.699826\pi\)
\(374\) 38.5066 1.99113
\(375\) −3.76393 −0.194369
\(376\) 30.5623 1.57613
\(377\) −25.4164 −1.30901
\(378\) 0 0
\(379\) 5.00000 0.256833 0.128416 0.991720i \(-0.459011\pi\)
0.128416 + 0.991720i \(0.459011\pi\)
\(380\) −10.8541 −0.556804
\(381\) 15.2705 0.782332
\(382\) −14.7082 −0.752537
\(383\) −0.819660 −0.0418827 −0.0209413 0.999781i \(-0.506666\pi\)
−0.0209413 + 0.999781i \(0.506666\pi\)
\(384\) −1.09017 −0.0556325
\(385\) 0 0
\(386\) −52.6869 −2.68169
\(387\) 17.4164 0.885326
\(388\) 8.29180 0.420952
\(389\) 7.41641 0.376027 0.188013 0.982166i \(-0.439795\pi\)
0.188013 + 0.982166i \(0.439795\pi\)
\(390\) −4.85410 −0.245797
\(391\) 25.1246 1.27061
\(392\) 0 0
\(393\) 2.23607 0.112795
\(394\) 42.9787 2.16524
\(395\) −2.50658 −0.126120
\(396\) 25.4164 1.27722
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −61.3050 −3.07294
\(399\) 0 0
\(400\) −47.8328 −2.39164
\(401\) −23.8885 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(402\) −37.7426 −1.88243
\(403\) 32.5623 1.62204
\(404\) −9.27051 −0.461225
\(405\) 0.381966 0.0189800
\(406\) 0 0
\(407\) −17.5623 −0.870531
\(408\) −41.9787 −2.07826
\(409\) −36.7082 −1.81510 −0.907552 0.419940i \(-0.862051\pi\)
−0.907552 + 0.419940i \(0.862051\pi\)
\(410\) 8.94427 0.441726
\(411\) −12.7082 −0.626849
\(412\) 4.85410 0.239144
\(413\) 0 0
\(414\) 23.4164 1.15085
\(415\) 2.41641 0.118617
\(416\) −52.6869 −2.58319
\(417\) 11.1459 0.545817
\(418\) −40.1246 −1.96256
\(419\) 4.09017 0.199818 0.0999089 0.994997i \(-0.468145\pi\)
0.0999089 + 0.994997i \(0.468145\pi\)
\(420\) 0 0
\(421\) 3.00000 0.146211 0.0731055 0.997324i \(-0.476709\pi\)
0.0731055 + 0.997324i \(0.476709\pi\)
\(422\) −38.8885 −1.89306
\(423\) 8.18034 0.397742
\(424\) −8.14590 −0.395600
\(425\) −27.2705 −1.32281
\(426\) 10.7082 0.518814
\(427\) 0 0
\(428\) −34.4164 −1.66358
\(429\) −12.7082 −0.613558
\(430\) 8.70820 0.419947
\(431\) 34.3607 1.65510 0.827548 0.561395i \(-0.189735\pi\)
0.827548 + 0.561395i \(0.189735\pi\)
\(432\) −49.2705 −2.37053
\(433\) −14.4164 −0.692808 −0.346404 0.938085i \(-0.612597\pi\)
−0.346404 + 0.938085i \(0.612597\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) −46.4164 −2.22294
\(437\) −26.1803 −1.25238
\(438\) −28.4164 −1.35779
\(439\) −29.5623 −1.41093 −0.705466 0.708744i \(-0.749262\pi\)
−0.705466 + 0.708744i \(0.749262\pi\)
\(440\) 7.47214 0.356220
\(441\) 0 0
\(442\) −71.3951 −3.39592
\(443\) −15.4377 −0.733467 −0.366733 0.930326i \(-0.619524\pi\)
−0.366733 + 0.930326i \(0.619524\pi\)
\(444\) 32.5623 1.54534
\(445\) 0.875388 0.0414974
\(446\) 20.1803 0.955567
\(447\) −7.47214 −0.353420
\(448\) 0 0
\(449\) −13.3607 −0.630529 −0.315265 0.949004i \(-0.602093\pi\)
−0.315265 + 0.949004i \(0.602093\pi\)
\(450\) −25.4164 −1.19814
\(451\) 23.4164 1.10264
\(452\) −72.8115 −3.42477
\(453\) −19.0000 −0.892698
\(454\) 7.70820 0.361764
\(455\) 0 0
\(456\) 43.7426 2.04844
\(457\) −9.85410 −0.460955 −0.230478 0.973078i \(-0.574029\pi\)
−0.230478 + 0.973078i \(0.574029\pi\)
\(458\) 17.5623 0.820633
\(459\) −28.0902 −1.31114
\(460\) 8.29180 0.386607
\(461\) −12.2148 −0.568899 −0.284450 0.958691i \(-0.591811\pi\)
−0.284450 + 0.958691i \(0.591811\pi\)
\(462\) 0 0
\(463\) −21.4164 −0.995305 −0.497652 0.867377i \(-0.665804\pi\)
−0.497652 + 0.867377i \(0.665804\pi\)
\(464\) −51.5967 −2.39532
\(465\) 2.56231 0.118824
\(466\) 70.3951 3.26099
\(467\) 3.65248 0.169016 0.0845082 0.996423i \(-0.473068\pi\)
0.0845082 + 0.996423i \(0.473068\pi\)
\(468\) −47.1246 −2.17834
\(469\) 0 0
\(470\) 4.09017 0.188665
\(471\) −16.7082 −0.769873
\(472\) 47.6869 2.19497
\(473\) 22.7984 1.04827
\(474\) 17.1803 0.789119
\(475\) 28.4164 1.30383
\(476\) 0 0
\(477\) −2.18034 −0.0998309
\(478\) −22.7082 −1.03865
\(479\) −8.18034 −0.373769 −0.186885 0.982382i \(-0.559839\pi\)
−0.186885 + 0.982382i \(0.559839\pi\)
\(480\) −4.14590 −0.189233
\(481\) 32.5623 1.48471
\(482\) −21.6525 −0.986243
\(483\) 0 0
\(484\) −20.1246 −0.914755
\(485\) 0.652476 0.0296274
\(486\) −41.8885 −1.90010
\(487\) −23.0000 −1.04223 −0.521115 0.853487i \(-0.674484\pi\)
−0.521115 + 0.853487i \(0.674484\pi\)
\(488\) 30.9787 1.40234
\(489\) 2.70820 0.122469
\(490\) 0 0
\(491\) 35.7771 1.61460 0.807299 0.590143i \(-0.200929\pi\)
0.807299 + 0.590143i \(0.200929\pi\)
\(492\) −43.4164 −1.95736
\(493\) −29.4164 −1.32485
\(494\) 74.3951 3.34719
\(495\) 2.00000 0.0898933
\(496\) 66.1033 2.96813
\(497\) 0 0
\(498\) −16.5623 −0.742175
\(499\) 13.2705 0.594070 0.297035 0.954867i \(-0.404002\pi\)
0.297035 + 0.954867i \(0.404002\pi\)
\(500\) −18.2705 −0.817082
\(501\) −9.00000 −0.402090
\(502\) 29.4164 1.31292
\(503\) −13.3607 −0.595723 −0.297862 0.954609i \(-0.596273\pi\)
−0.297862 + 0.954609i \(0.596273\pi\)
\(504\) 0 0
\(505\) −0.729490 −0.0324619
\(506\) 30.6525 1.36267
\(507\) 10.5623 0.469088
\(508\) 74.1246 3.28875
\(509\) −26.6180 −1.17982 −0.589912 0.807468i \(-0.700837\pi\)
−0.589912 + 0.807468i \(0.700837\pi\)
\(510\) −5.61803 −0.248771
\(511\) 0 0
\(512\) 40.3050 1.78124
\(513\) 29.2705 1.29232
\(514\) −35.2705 −1.55572
\(515\) 0.381966 0.0168314
\(516\) −42.2705 −1.86086
\(517\) 10.7082 0.470946
\(518\) 0 0
\(519\) −16.0344 −0.703834
\(520\) −13.8541 −0.607543
\(521\) −17.1803 −0.752684 −0.376342 0.926481i \(-0.622818\pi\)
−0.376342 + 0.926481i \(0.622818\pi\)
\(522\) −27.4164 −1.19998
\(523\) 10.5836 0.462788 0.231394 0.972860i \(-0.425671\pi\)
0.231394 + 0.972860i \(0.425671\pi\)
\(524\) 10.8541 0.474164
\(525\) 0 0
\(526\) 53.9787 2.35358
\(527\) 37.6869 1.64167
\(528\) −25.7984 −1.12273
\(529\) −3.00000 −0.130435
\(530\) −1.09017 −0.0473540
\(531\) 12.7639 0.553907
\(532\) 0 0
\(533\) −43.4164 −1.88057
\(534\) −6.00000 −0.259645
\(535\) −2.70820 −0.117086
\(536\) −107.721 −4.65285
\(537\) −7.85410 −0.338930
\(538\) 32.2705 1.39128
\(539\) 0 0
\(540\) −9.27051 −0.398939
\(541\) 9.14590 0.393213 0.196606 0.980482i \(-0.437008\pi\)
0.196606 + 0.980482i \(0.437008\pi\)
\(542\) 2.61803 0.112454
\(543\) −3.85410 −0.165395
\(544\) −60.9787 −2.61444
\(545\) −3.65248 −0.156455
\(546\) 0 0
\(547\) −2.27051 −0.0970800 −0.0485400 0.998821i \(-0.515457\pi\)
−0.0485400 + 0.998821i \(0.515457\pi\)
\(548\) −61.6869 −2.63513
\(549\) 8.29180 0.353885
\(550\) −33.2705 −1.41866
\(551\) 30.6525 1.30584
\(552\) −33.4164 −1.42230
\(553\) 0 0
\(554\) −12.3262 −0.523692
\(555\) 2.56231 0.108764
\(556\) 54.1033 2.29449
\(557\) −31.6869 −1.34262 −0.671309 0.741178i \(-0.734268\pi\)
−0.671309 + 0.741178i \(0.734268\pi\)
\(558\) 35.1246 1.48694
\(559\) −42.2705 −1.78785
\(560\) 0 0
\(561\) −14.7082 −0.620981
\(562\) 82.3951 3.47563
\(563\) 1.20163 0.0506425 0.0253213 0.999679i \(-0.491939\pi\)
0.0253213 + 0.999679i \(0.491939\pi\)
\(564\) −19.8541 −0.836009
\(565\) −5.72949 −0.241041
\(566\) −51.5967 −2.16877
\(567\) 0 0
\(568\) 30.5623 1.28237
\(569\) −27.1591 −1.13857 −0.569283 0.822141i \(-0.692779\pi\)
−0.569283 + 0.822141i \(0.692779\pi\)
\(570\) 5.85410 0.245201
\(571\) −22.5623 −0.944203 −0.472102 0.881544i \(-0.656504\pi\)
−0.472102 + 0.881544i \(0.656504\pi\)
\(572\) −61.6869 −2.57926
\(573\) 5.61803 0.234697
\(574\) 0 0
\(575\) −21.7082 −0.905295
\(576\) −17.4164 −0.725684
\(577\) 36.8328 1.53337 0.766685 0.642023i \(-0.221905\pi\)
0.766685 + 0.642023i \(0.221905\pi\)
\(578\) −38.1246 −1.58577
\(579\) 20.1246 0.836350
\(580\) −9.70820 −0.403111
\(581\) 0 0
\(582\) −4.47214 −0.185376
\(583\) −2.85410 −0.118205
\(584\) −81.1033 −3.35608
\(585\) −3.70820 −0.153315
\(586\) −56.6869 −2.34171
\(587\) 47.0689 1.94274 0.971370 0.237570i \(-0.0763510\pi\)
0.971370 + 0.237570i \(0.0763510\pi\)
\(588\) 0 0
\(589\) −39.2705 −1.61811
\(590\) 6.38197 0.262741
\(591\) −16.4164 −0.675281
\(592\) 66.1033 2.71683
\(593\) −2.18034 −0.0895358 −0.0447679 0.998997i \(-0.514255\pi\)
−0.0447679 + 0.998997i \(0.514255\pi\)
\(594\) −34.2705 −1.40614
\(595\) 0 0
\(596\) −36.2705 −1.48570
\(597\) 23.4164 0.958370
\(598\) −56.8328 −2.32407
\(599\) −35.4508 −1.44848 −0.724241 0.689547i \(-0.757810\pi\)
−0.724241 + 0.689547i \(0.757810\pi\)
\(600\) 36.2705 1.48074
\(601\) −3.56231 −0.145309 −0.0726547 0.997357i \(-0.523147\pi\)
−0.0726547 + 0.997357i \(0.523147\pi\)
\(602\) 0 0
\(603\) −28.8328 −1.17416
\(604\) −92.2279 −3.75270
\(605\) −1.58359 −0.0643822
\(606\) 5.00000 0.203111
\(607\) −5.70820 −0.231689 −0.115844 0.993267i \(-0.536957\pi\)
−0.115844 + 0.993267i \(0.536957\pi\)
\(608\) 63.5410 2.57693
\(609\) 0 0
\(610\) 4.14590 0.167863
\(611\) −19.8541 −0.803211
\(612\) −54.5410 −2.20469
\(613\) 24.4164 0.986169 0.493085 0.869981i \(-0.335869\pi\)
0.493085 + 0.869981i \(0.335869\pi\)
\(614\) −7.47214 −0.301551
\(615\) −3.41641 −0.137763
\(616\) 0 0
\(617\) 3.27051 0.131666 0.0658329 0.997831i \(-0.479030\pi\)
0.0658329 + 0.997831i \(0.479030\pi\)
\(618\) −2.61803 −0.105313
\(619\) 28.6869 1.15302 0.576512 0.817088i \(-0.304413\pi\)
0.576512 + 0.817088i \(0.304413\pi\)
\(620\) 12.4377 0.499510
\(621\) −22.3607 −0.897303
\(622\) 7.56231 0.303221
\(623\) 0 0
\(624\) 47.8328 1.91485
\(625\) 22.8328 0.913313
\(626\) 42.6525 1.70474
\(627\) 15.3262 0.612071
\(628\) −81.1033 −3.23638
\(629\) 37.6869 1.50268
\(630\) 0 0
\(631\) 42.2705 1.68276 0.841381 0.540442i \(-0.181743\pi\)
0.841381 + 0.540442i \(0.181743\pi\)
\(632\) 49.0344 1.95049
\(633\) 14.8541 0.590398
\(634\) −74.3951 −2.95461
\(635\) 5.83282 0.231468
\(636\) 5.29180 0.209833
\(637\) 0 0
\(638\) −35.8885 −1.42084
\(639\) 8.18034 0.323609
\(640\) −0.416408 −0.0164600
\(641\) −15.0000 −0.592464 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 18.5623 0.732596
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −3.32624 −0.130970
\(646\) 86.1033 3.38769
\(647\) −1.25735 −0.0494317 −0.0247158 0.999695i \(-0.507868\pi\)
−0.0247158 + 0.999695i \(0.507868\pi\)
\(648\) −7.47214 −0.293533
\(649\) 16.7082 0.655854
\(650\) 61.6869 2.41956
\(651\) 0 0
\(652\) 13.1459 0.514833
\(653\) −5.23607 −0.204903 −0.102452 0.994738i \(-0.532669\pi\)
−0.102452 + 0.994738i \(0.532669\pi\)
\(654\) 25.0344 0.978924
\(655\) 0.854102 0.0333725
\(656\) −88.1378 −3.44120
\(657\) −21.7082 −0.846918
\(658\) 0 0
\(659\) 36.5967 1.42561 0.712803 0.701364i \(-0.247425\pi\)
0.712803 + 0.701364i \(0.247425\pi\)
\(660\) −4.85410 −0.188946
\(661\) 31.5623 1.22763 0.613816 0.789449i \(-0.289634\pi\)
0.613816 + 0.789449i \(0.289634\pi\)
\(662\) 42.2705 1.64289
\(663\) 27.2705 1.05910
\(664\) −47.2705 −1.83445
\(665\) 0 0
\(666\) 35.1246 1.36105
\(667\) −23.4164 −0.906687
\(668\) −43.6869 −1.69030
\(669\) −7.70820 −0.298016
\(670\) −14.4164 −0.556954
\(671\) 10.8541 0.419018
\(672\) 0 0
\(673\) −24.2918 −0.936380 −0.468190 0.883628i \(-0.655094\pi\)
−0.468190 + 0.883628i \(0.655094\pi\)
\(674\) 6.38197 0.245824
\(675\) 24.2705 0.934172
\(676\) 51.2705 1.97194
\(677\) −40.0344 −1.53865 −0.769324 0.638858i \(-0.779407\pi\)
−0.769324 + 0.638858i \(0.779407\pi\)
\(678\) 39.2705 1.50817
\(679\) 0 0
\(680\) −16.0344 −0.614893
\(681\) −2.94427 −0.112825
\(682\) 45.9787 1.76062
\(683\) 47.2361 1.80744 0.903719 0.428126i \(-0.140826\pi\)
0.903719 + 0.428126i \(0.140826\pi\)
\(684\) 56.8328 2.17306
\(685\) −4.85410 −0.185466
\(686\) 0 0
\(687\) −6.70820 −0.255934
\(688\) −85.8115 −3.27153
\(689\) 5.29180 0.201601
\(690\) −4.47214 −0.170251
\(691\) −7.85410 −0.298784 −0.149392 0.988778i \(-0.547732\pi\)
−0.149392 + 0.988778i \(0.547732\pi\)
\(692\) −77.8328 −2.95876
\(693\) 0 0
\(694\) 3.85410 0.146300
\(695\) 4.25735 0.161491
\(696\) 39.1246 1.48301
\(697\) −50.2492 −1.90333
\(698\) 40.3607 1.52767
\(699\) −26.8885 −1.01702
\(700\) 0 0
\(701\) −25.7984 −0.974391 −0.487196 0.873293i \(-0.661980\pi\)
−0.487196 + 0.873293i \(0.661980\pi\)
\(702\) 63.5410 2.39820
\(703\) −39.2705 −1.48112
\(704\) −22.7984 −0.859246
\(705\) −1.56231 −0.0588398
\(706\) −65.5410 −2.46667
\(707\) 0 0
\(708\) −30.9787 −1.16425
\(709\) −1.02129 −0.0383552 −0.0191776 0.999816i \(-0.506105\pi\)
−0.0191776 + 0.999816i \(0.506105\pi\)
\(710\) 4.09017 0.153501
\(711\) 13.1246 0.492211
\(712\) −17.1246 −0.641772
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) −4.85410 −0.181533
\(716\) −38.1246 −1.42478
\(717\) 8.67376 0.323928
\(718\) −38.4164 −1.43369
\(719\) −39.3262 −1.46662 −0.733311 0.679894i \(-0.762026\pi\)
−0.733311 + 0.679894i \(0.762026\pi\)
\(720\) −7.52786 −0.280547
\(721\) 0 0
\(722\) −39.9787 −1.48785
\(723\) 8.27051 0.307584
\(724\) −18.7082 −0.695285
\(725\) 25.4164 0.943942
\(726\) 10.8541 0.402834
\(727\) 4.72949 0.175407 0.0877035 0.996147i \(-0.472047\pi\)
0.0877035 + 0.996147i \(0.472047\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −10.8541 −0.401728
\(731\) −48.9230 −1.80948
\(732\) −20.1246 −0.743827
\(733\) −14.7082 −0.543260 −0.271630 0.962402i \(-0.587563\pi\)
−0.271630 + 0.962402i \(0.587563\pi\)
\(734\) 43.0344 1.58843
\(735\) 0 0
\(736\) −48.5410 −1.78925
\(737\) −37.7426 −1.39027
\(738\) −46.8328 −1.72394
\(739\) −16.8328 −0.619205 −0.309603 0.950866i \(-0.600196\pi\)
−0.309603 + 0.950866i \(0.600196\pi\)
\(740\) 12.4377 0.457219
\(741\) −28.4164 −1.04390
\(742\) 0 0
\(743\) 30.7082 1.12657 0.563287 0.826261i \(-0.309536\pi\)
0.563287 + 0.826261i \(0.309536\pi\)
\(744\) −50.1246 −1.83766
\(745\) −2.85410 −0.104566
\(746\) 59.3951 2.17461
\(747\) −12.6525 −0.462930
\(748\) −71.3951 −2.61046
\(749\) 0 0
\(750\) 9.85410 0.359821
\(751\) −44.1246 −1.61013 −0.805065 0.593187i \(-0.797870\pi\)
−0.805065 + 0.593187i \(0.797870\pi\)
\(752\) −40.3050 −1.46977
\(753\) −11.2361 −0.409465
\(754\) 66.5410 2.42328
\(755\) −7.25735 −0.264122
\(756\) 0 0
\(757\) −21.2918 −0.773863 −0.386932 0.922108i \(-0.626465\pi\)
−0.386932 + 0.922108i \(0.626465\pi\)
\(758\) −13.0902 −0.475456
\(759\) −11.7082 −0.424981
\(760\) 16.7082 0.606070
\(761\) 7.47214 0.270865 0.135432 0.990787i \(-0.456758\pi\)
0.135432 + 0.990787i \(0.456758\pi\)
\(762\) −39.9787 −1.44828
\(763\) 0 0
\(764\) 27.2705 0.986612
\(765\) −4.29180 −0.155170
\(766\) 2.14590 0.0775344
\(767\) −30.9787 −1.11858
\(768\) −14.5623 −0.525472
\(769\) −31.3951 −1.13214 −0.566069 0.824358i \(-0.691536\pi\)
−0.566069 + 0.824358i \(0.691536\pi\)
\(770\) 0 0
\(771\) 13.4721 0.485187
\(772\) 97.6869 3.51583
\(773\) 16.5279 0.594466 0.297233 0.954805i \(-0.403936\pi\)
0.297233 + 0.954805i \(0.403936\pi\)
\(774\) −45.5967 −1.63894
\(775\) −32.5623 −1.16967
\(776\) −12.7639 −0.458198
\(777\) 0 0
\(778\) −19.4164 −0.696112
\(779\) 52.3607 1.87602
\(780\) 9.00000 0.322252
\(781\) 10.7082 0.383170
\(782\) −65.7771 −2.35218
\(783\) 26.1803 0.935609
\(784\) 0 0
\(785\) −6.38197 −0.227782
\(786\) −5.85410 −0.208809
\(787\) −43.4164 −1.54763 −0.773814 0.633413i \(-0.781653\pi\)
−0.773814 + 0.633413i \(0.781653\pi\)
\(788\) −79.6869 −2.83873
\(789\) −20.6180 −0.734021
\(790\) 6.56231 0.233476
\(791\) 0 0
\(792\) −39.1246 −1.39023
\(793\) −20.1246 −0.714646
\(794\) 52.3607 1.85821
\(795\) 0.416408 0.0147685
\(796\) 113.666 4.02877
\(797\) 50.1246 1.77550 0.887752 0.460321i \(-0.152266\pi\)
0.887752 + 0.460321i \(0.152266\pi\)
\(798\) 0 0
\(799\) −22.9787 −0.812928
\(800\) 52.6869 1.86276
\(801\) −4.58359 −0.161953
\(802\) 62.5410 2.20840
\(803\) −28.4164 −1.00279
\(804\) 69.9787 2.46796
\(805\) 0 0
\(806\) −85.2492 −3.00278
\(807\) −12.3262 −0.433904
\(808\) 14.2705 0.502035
\(809\) 2.94427 0.103515 0.0517575 0.998660i \(-0.483518\pi\)
0.0517575 + 0.998660i \(0.483518\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −21.5410 −0.756408 −0.378204 0.925722i \(-0.623458\pi\)
−0.378204 + 0.925722i \(0.623458\pi\)
\(812\) 0 0
\(813\) −1.00000 −0.0350715
\(814\) 45.9787 1.61155
\(815\) 1.03444 0.0362349
\(816\) 55.3607 1.93801
\(817\) 50.9787 1.78352
\(818\) 96.1033 3.36017
\(819\) 0 0
\(820\) −16.5836 −0.579124
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 33.2705 1.16044
\(823\) 3.43769 0.119830 0.0599152 0.998203i \(-0.480917\pi\)
0.0599152 + 0.998203i \(0.480917\pi\)
\(824\) −7.47214 −0.260304
\(825\) 12.7082 0.442443
\(826\) 0 0
\(827\) 6.70820 0.233267 0.116634 0.993175i \(-0.462790\pi\)
0.116634 + 0.993175i \(0.462790\pi\)
\(828\) −43.4164 −1.50882
\(829\) −14.2705 −0.495635 −0.247818 0.968807i \(-0.579713\pi\)
−0.247818 + 0.968807i \(0.579713\pi\)
\(830\) −6.32624 −0.219587
\(831\) 4.70820 0.163326
\(832\) 42.2705 1.46547
\(833\) 0 0
\(834\) −29.1803 −1.01043
\(835\) −3.43769 −0.118966
\(836\) 74.3951 2.57301
\(837\) −33.5410 −1.15935
\(838\) −10.7082 −0.369909
\(839\) −23.6180 −0.815385 −0.407693 0.913119i \(-0.633666\pi\)
−0.407693 + 0.913119i \(0.633666\pi\)
\(840\) 0 0
\(841\) −1.58359 −0.0546066
\(842\) −7.85410 −0.270670
\(843\) −31.4721 −1.08396
\(844\) 72.1033 2.48190
\(845\) 4.03444 0.138789
\(846\) −21.4164 −0.736311
\(847\) 0 0
\(848\) 10.7426 0.368904
\(849\) 19.7082 0.676384
\(850\) 71.3951 2.44883
\(851\) 30.0000 1.02839
\(852\) −19.8541 −0.680190
\(853\) 44.2705 1.51579 0.757897 0.652375i \(-0.226227\pi\)
0.757897 + 0.652375i \(0.226227\pi\)
\(854\) 0 0
\(855\) 4.47214 0.152944
\(856\) 52.9787 1.81078
\(857\) −8.23607 −0.281339 −0.140669 0.990057i \(-0.544925\pi\)
−0.140669 + 0.990057i \(0.544925\pi\)
\(858\) 33.2705 1.13584
\(859\) −10.5623 −0.360381 −0.180191 0.983632i \(-0.557671\pi\)
−0.180191 + 0.983632i \(0.557671\pi\)
\(860\) −16.1459 −0.550571
\(861\) 0 0
\(862\) −89.9574 −3.06396
\(863\) −21.7082 −0.738956 −0.369478 0.929240i \(-0.620463\pi\)
−0.369478 + 0.929240i \(0.620463\pi\)
\(864\) 54.2705 1.84632
\(865\) −6.12461 −0.208243
\(866\) 37.7426 1.28255
\(867\) 14.5623 0.494562
\(868\) 0 0
\(869\) 17.1803 0.582803
\(870\) 5.23607 0.177519
\(871\) 69.9787 2.37114
\(872\) 71.4508 2.41963
\(873\) −3.41641 −0.115628
\(874\) 68.5410 2.31843
\(875\) 0 0
\(876\) 52.6869 1.78013
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 77.3951 2.61196
\(879\) 21.6525 0.730320
\(880\) −9.85410 −0.332182
\(881\) −29.8885 −1.00697 −0.503485 0.864004i \(-0.667949\pi\)
−0.503485 + 0.864004i \(0.667949\pi\)
\(882\) 0 0
\(883\) 5.87539 0.197723 0.0988613 0.995101i \(-0.468480\pi\)
0.0988613 + 0.995101i \(0.468480\pi\)
\(884\) 132.374 4.45221
\(885\) −2.43769 −0.0819422
\(886\) 40.4164 1.35782
\(887\) −40.1935 −1.34957 −0.674783 0.738016i \(-0.735763\pi\)
−0.674783 + 0.738016i \(0.735763\pi\)
\(888\) −50.1246 −1.68207
\(889\) 0 0
\(890\) −2.29180 −0.0768212
\(891\) −2.61803 −0.0877074
\(892\) −37.4164 −1.25279
\(893\) 23.9443 0.801265
\(894\) 19.5623 0.654261
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 21.7082 0.724816
\(898\) 34.9787 1.16725
\(899\) −35.1246 −1.17147
\(900\) 47.1246 1.57082
\(901\) 6.12461 0.204040
\(902\) −61.3050 −2.04123
\(903\) 0 0
\(904\) 112.082 3.72779
\(905\) −1.47214 −0.0489355
\(906\) 49.7426 1.65259
\(907\) −7.12461 −0.236569 −0.118284 0.992980i \(-0.537739\pi\)
−0.118284 + 0.992980i \(0.537739\pi\)
\(908\) −14.2918 −0.474290
\(909\) 3.81966 0.126690
\(910\) 0 0
\(911\) −10.0344 −0.332456 −0.166228 0.986087i \(-0.553159\pi\)
−0.166228 + 0.986087i \(0.553159\pi\)
\(912\) −57.6869 −1.91020
\(913\) −16.5623 −0.548132
\(914\) 25.7984 0.853334
\(915\) −1.58359 −0.0523519
\(916\) −32.5623 −1.07589
\(917\) 0 0
\(918\) 73.5410 2.42722
\(919\) −27.9787 −0.922933 −0.461466 0.887158i \(-0.652676\pi\)
−0.461466 + 0.887158i \(0.652676\pi\)
\(920\) −12.7639 −0.420814
\(921\) 2.85410 0.0940459
\(922\) 31.9787 1.05316
\(923\) −19.8541 −0.653506
\(924\) 0 0
\(925\) −32.5623 −1.07064
\(926\) 56.0689 1.84254
\(927\) −2.00000 −0.0656886
\(928\) 56.8328 1.86563
\(929\) −14.9443 −0.490306 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(930\) −6.70820 −0.219971
\(931\) 0 0
\(932\) −130.520 −4.27532
\(933\) −2.88854 −0.0945667
\(934\) −9.56231 −0.312888
\(935\) −5.61803 −0.183729
\(936\) 72.5410 2.37108
\(937\) 11.0000 0.359354 0.179677 0.983726i \(-0.442495\pi\)
0.179677 + 0.983726i \(0.442495\pi\)
\(938\) 0 0
\(939\) −16.2918 −0.531663
\(940\) −7.58359 −0.247350
\(941\) 23.3951 0.762659 0.381330 0.924439i \(-0.375466\pi\)
0.381330 + 0.924439i \(0.375466\pi\)
\(942\) 43.7426 1.42521
\(943\) −40.0000 −1.30258
\(944\) −62.8885 −2.04685
\(945\) 0 0
\(946\) −59.6869 −1.94059
\(947\) −41.0132 −1.33275 −0.666374 0.745617i \(-0.732155\pi\)
−0.666374 + 0.745617i \(0.732155\pi\)
\(948\) −31.8541 −1.03457
\(949\) 52.6869 1.71029
\(950\) −74.3951 −2.41370
\(951\) 28.4164 0.921465
\(952\) 0 0
\(953\) 13.3607 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(954\) 5.70820 0.184810
\(955\) 2.14590 0.0694396
\(956\) 42.1033 1.36172
\(957\) 13.7082 0.443123
\(958\) 21.4164 0.691933
\(959\) 0 0
\(960\) 3.32624 0.107354
\(961\) 14.0000 0.451613
\(962\) −85.2492 −2.74855
\(963\) 14.1803 0.456955
\(964\) 40.1459 1.29301
\(965\) 7.68692 0.247451
\(966\) 0 0
\(967\) −14.4164 −0.463600 −0.231800 0.972763i \(-0.574462\pi\)
−0.231800 + 0.972763i \(0.574462\pi\)
\(968\) 30.9787 0.995694
\(969\) −32.8885 −1.05653
\(970\) −1.70820 −0.0548471
\(971\) 53.0132 1.70127 0.850637 0.525754i \(-0.176217\pi\)
0.850637 + 0.525754i \(0.176217\pi\)
\(972\) 77.6656 2.49113
\(973\) 0 0
\(974\) 60.2148 1.92941
\(975\) −23.5623 −0.754598
\(976\) −40.8541 −1.30771
\(977\) 46.7426 1.49543 0.747715 0.664020i \(-0.231151\pi\)
0.747715 + 0.664020i \(0.231151\pi\)
\(978\) −7.09017 −0.226719
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 19.1246 0.610602
\(982\) −93.6656 −2.98899
\(983\) −18.6525 −0.594922 −0.297461 0.954734i \(-0.596140\pi\)
−0.297461 + 0.954734i \(0.596140\pi\)
\(984\) 66.8328 2.13055
\(985\) −6.27051 −0.199795
\(986\) 77.0132 2.45260
\(987\) 0 0
\(988\) −137.936 −4.38833
\(989\) −38.9443 −1.23836
\(990\) −5.23607 −0.166413
\(991\) 24.2705 0.770978 0.385489 0.922712i \(-0.374033\pi\)
0.385489 + 0.922712i \(0.374033\pi\)
\(992\) −72.8115 −2.31177
\(993\) −16.1459 −0.512375
\(994\) 0 0
\(995\) 8.94427 0.283552
\(996\) 30.7082 0.973027
\(997\) 39.2918 1.24438 0.622192 0.782865i \(-0.286242\pi\)
0.622192 + 0.782865i \(0.286242\pi\)
\(998\) −34.7426 −1.09976
\(999\) −33.5410 −1.06119
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 5047.2.a.a.1.1 2
7.6 odd 2 103.2.a.a.1.1 2
21.20 even 2 927.2.a.b.1.2 2
28.27 even 2 1648.2.a.f.1.2 2
35.34 odd 2 2575.2.a.g.1.2 2
56.13 odd 2 6592.2.a.t.1.1 2
56.27 even 2 6592.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.2.a.a.1.1 2 7.6 odd 2
927.2.a.b.1.2 2 21.20 even 2
1648.2.a.f.1.2 2 28.27 even 2
2575.2.a.g.1.2 2 35.34 odd 2
5047.2.a.a.1.1 2 1.1 even 1 trivial
6592.2.a.h.1.1 2 56.27 even 2
6592.2.a.t.1.1 2 56.13 odd 2