Properties

Label 2695.2.a.u.1.10
Level $2695$
Weight $2$
Character 2695.1
Self dual yes
Analytic conductor $21.520$
Analytic rank $1$
Dimension $10$
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] = [2695,2,Mod(1,2695)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, 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("2695.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2695 = 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2695.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: \(21.5196833447\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 13x^{8} + 24x^{7} + 56x^{6} - 92x^{5} - 86x^{4} + 116x^{3} + 31x^{2} - 30x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-2.25931\) of defining polynomial
Character \(\chi\) \(=\) 2695.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.25931 q^{2} -2.89439 q^{3} +3.10446 q^{4} +1.00000 q^{5} -6.53930 q^{6} +2.49532 q^{8} +5.37747 q^{9} +O(q^{10})\) \(q+2.25931 q^{2} -2.89439 q^{3} +3.10446 q^{4} +1.00000 q^{5} -6.53930 q^{6} +2.49532 q^{8} +5.37747 q^{9} +2.25931 q^{10} +1.00000 q^{11} -8.98551 q^{12} -5.18923 q^{13} -2.89439 q^{15} -0.571242 q^{16} -1.23865 q^{17} +12.1493 q^{18} -0.0890109 q^{19} +3.10446 q^{20} +2.25931 q^{22} -6.56566 q^{23} -7.22241 q^{24} +1.00000 q^{25} -11.7241 q^{26} -6.88131 q^{27} -0.308703 q^{29} -6.53930 q^{30} +7.01096 q^{31} -6.28124 q^{32} -2.89439 q^{33} -2.79849 q^{34} +16.6941 q^{36} -0.545621 q^{37} -0.201103 q^{38} +15.0196 q^{39} +2.49532 q^{40} -9.60425 q^{41} +2.61867 q^{43} +3.10446 q^{44} +5.37747 q^{45} -14.8338 q^{46} -10.1126 q^{47} +1.65340 q^{48} +2.25931 q^{50} +3.58513 q^{51} -16.1098 q^{52} -4.66178 q^{53} -15.5470 q^{54} +1.00000 q^{55} +0.257632 q^{57} -0.697455 q^{58} +11.5718 q^{59} -8.98551 q^{60} +3.97047 q^{61} +15.8399 q^{62} -13.0488 q^{64} -5.18923 q^{65} -6.53930 q^{66} -2.15100 q^{67} -3.84534 q^{68} +19.0036 q^{69} -14.8329 q^{71} +13.4185 q^{72} +4.27711 q^{73} -1.23273 q^{74} -2.89439 q^{75} -0.276331 q^{76} +33.9339 q^{78} +5.28131 q^{79} -0.571242 q^{80} +3.78476 q^{81} -21.6989 q^{82} -9.15911 q^{83} -1.23865 q^{85} +5.91637 q^{86} +0.893506 q^{87} +2.49532 q^{88} -12.3013 q^{89} +12.1493 q^{90} -20.3828 q^{92} -20.2924 q^{93} -22.8474 q^{94} -0.0890109 q^{95} +18.1803 q^{96} -19.0137 q^{97} +5.37747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 8 q^{3} + 10 q^{4} + 10 q^{5} - 4 q^{6} - 6 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{2} - 8 q^{3} + 10 q^{4} + 10 q^{5} - 4 q^{6} - 6 q^{8} + 10 q^{9} - 2 q^{10} + 10 q^{11} - 20 q^{12} - 8 q^{13} - 8 q^{15} + 6 q^{16} - 28 q^{17} - 14 q^{18} + 10 q^{20} - 2 q^{22} - 16 q^{23} + 8 q^{24} + 10 q^{25} - 20 q^{26} - 32 q^{27} - 4 q^{30} - 20 q^{31} - 14 q^{32} - 8 q^{33} + 4 q^{34} + 42 q^{36} - 36 q^{37} - 24 q^{38} + 24 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{43} + 10 q^{44} + 10 q^{45} - 4 q^{46} - 12 q^{47} - 40 q^{48} - 2 q^{50} + 20 q^{51} - 4 q^{52} - 16 q^{53} + 48 q^{54} + 10 q^{55} + 4 q^{57} + 16 q^{58} - 32 q^{59} - 20 q^{60} + 16 q^{61} + 4 q^{62} - 34 q^{64} - 8 q^{65} - 4 q^{66} - 20 q^{67} - 32 q^{68} - 28 q^{69} + 12 q^{71} - 2 q^{72} - 20 q^{73} + 32 q^{74} - 8 q^{75} - 12 q^{76} + 20 q^{78} + 12 q^{79} + 6 q^{80} + 42 q^{81} + 40 q^{82} - 8 q^{83} - 28 q^{85} - 4 q^{86} - 28 q^{87} - 6 q^{88} - 68 q^{89} - 14 q^{90} + 32 q^{92} - 32 q^{93} - 16 q^{94} + 80 q^{96} - 36 q^{97} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.25931 1.59757 0.798785 0.601616i \(-0.205476\pi\)
0.798785 + 0.601616i \(0.205476\pi\)
\(3\) −2.89439 −1.67107 −0.835537 0.549434i \(-0.814843\pi\)
−0.835537 + 0.549434i \(0.814843\pi\)
\(4\) 3.10446 1.55223
\(5\) 1.00000 0.447214
\(6\) −6.53930 −2.66966
\(7\) 0 0
\(8\) 2.49532 0.882227
\(9\) 5.37747 1.79249
\(10\) 2.25931 0.714455
\(11\) 1.00000 0.301511
\(12\) −8.98551 −2.59389
\(13\) −5.18923 −1.43923 −0.719617 0.694371i \(-0.755683\pi\)
−0.719617 + 0.694371i \(0.755683\pi\)
\(14\) 0 0
\(15\) −2.89439 −0.747327
\(16\) −0.571242 −0.142811
\(17\) −1.23865 −0.300416 −0.150208 0.988654i \(-0.547994\pi\)
−0.150208 + 0.988654i \(0.547994\pi\)
\(18\) 12.1493 2.86363
\(19\) −0.0890109 −0.0204205 −0.0102102 0.999948i \(-0.503250\pi\)
−0.0102102 + 0.999948i \(0.503250\pi\)
\(20\) 3.10446 0.694179
\(21\) 0 0
\(22\) 2.25931 0.481686
\(23\) −6.56566 −1.36904 −0.684518 0.728996i \(-0.739987\pi\)
−0.684518 + 0.728996i \(0.739987\pi\)
\(24\) −7.22241 −1.47427
\(25\) 1.00000 0.200000
\(26\) −11.7241 −2.29928
\(27\) −6.88131 −1.32431
\(28\) 0 0
\(29\) −0.308703 −0.0573247 −0.0286624 0.999589i \(-0.509125\pi\)
−0.0286624 + 0.999589i \(0.509125\pi\)
\(30\) −6.53930 −1.19391
\(31\) 7.01096 1.25921 0.629603 0.776917i \(-0.283218\pi\)
0.629603 + 0.776917i \(0.283218\pi\)
\(32\) −6.28124 −1.11038
\(33\) −2.89439 −0.503848
\(34\) −2.79849 −0.479936
\(35\) 0 0
\(36\) 16.6941 2.78236
\(37\) −0.545621 −0.0896996 −0.0448498 0.998994i \(-0.514281\pi\)
−0.0448498 + 0.998994i \(0.514281\pi\)
\(38\) −0.201103 −0.0326232
\(39\) 15.0196 2.40507
\(40\) 2.49532 0.394544
\(41\) −9.60425 −1.49993 −0.749966 0.661476i \(-0.769930\pi\)
−0.749966 + 0.661476i \(0.769930\pi\)
\(42\) 0 0
\(43\) 2.61867 0.399343 0.199671 0.979863i \(-0.436013\pi\)
0.199671 + 0.979863i \(0.436013\pi\)
\(44\) 3.10446 0.468015
\(45\) 5.37747 0.801626
\(46\) −14.8338 −2.18713
\(47\) −10.1126 −1.47507 −0.737535 0.675309i \(-0.764010\pi\)
−0.737535 + 0.675309i \(0.764010\pi\)
\(48\) 1.65340 0.238647
\(49\) 0 0
\(50\) 2.25931 0.319514
\(51\) 3.58513 0.502018
\(52\) −16.1098 −2.23402
\(53\) −4.66178 −0.640345 −0.320172 0.947359i \(-0.603741\pi\)
−0.320172 + 0.947359i \(0.603741\pi\)
\(54\) −15.5470 −2.11568
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0.257632 0.0341242
\(58\) −0.697455 −0.0915803
\(59\) 11.5718 1.50652 0.753262 0.657721i \(-0.228479\pi\)
0.753262 + 0.657721i \(0.228479\pi\)
\(60\) −8.98551 −1.16002
\(61\) 3.97047 0.508367 0.254183 0.967156i \(-0.418193\pi\)
0.254183 + 0.967156i \(0.418193\pi\)
\(62\) 15.8399 2.01167
\(63\) 0 0
\(64\) −13.0488 −1.63110
\(65\) −5.18923 −0.643645
\(66\) −6.53930 −0.804932
\(67\) −2.15100 −0.262787 −0.131393 0.991330i \(-0.541945\pi\)
−0.131393 + 0.991330i \(0.541945\pi\)
\(68\) −3.84534 −0.466315
\(69\) 19.0036 2.28776
\(70\) 0 0
\(71\) −14.8329 −1.76034 −0.880170 0.474659i \(-0.842572\pi\)
−0.880170 + 0.474659i \(0.842572\pi\)
\(72\) 13.4185 1.58138
\(73\) 4.27711 0.500598 0.250299 0.968169i \(-0.419471\pi\)
0.250299 + 0.968169i \(0.419471\pi\)
\(74\) −1.23273 −0.143301
\(75\) −2.89439 −0.334215
\(76\) −0.276331 −0.0316973
\(77\) 0 0
\(78\) 33.9339 3.84226
\(79\) 5.28131 0.594194 0.297097 0.954847i \(-0.403981\pi\)
0.297097 + 0.954847i \(0.403981\pi\)
\(80\) −0.571242 −0.0638668
\(81\) 3.78476 0.420529
\(82\) −21.6989 −2.39625
\(83\) −9.15911 −1.00534 −0.502671 0.864478i \(-0.667649\pi\)
−0.502671 + 0.864478i \(0.667649\pi\)
\(84\) 0 0
\(85\) −1.23865 −0.134350
\(86\) 5.91637 0.637978
\(87\) 0.893506 0.0957939
\(88\) 2.49532 0.266002
\(89\) −12.3013 −1.30393 −0.651966 0.758248i \(-0.726056\pi\)
−0.651966 + 0.758248i \(0.726056\pi\)
\(90\) 12.1493 1.28065
\(91\) 0 0
\(92\) −20.3828 −2.12506
\(93\) −20.2924 −2.10423
\(94\) −22.8474 −2.35653
\(95\) −0.0890109 −0.00913232
\(96\) 18.1803 1.85552
\(97\) −19.0137 −1.93055 −0.965277 0.261229i \(-0.915872\pi\)
−0.965277 + 0.261229i \(0.915872\pi\)
\(98\) 0 0
\(99\) 5.37747 0.540456
\(100\) 3.10446 0.310446
\(101\) −15.5111 −1.54341 −0.771706 0.635979i \(-0.780597\pi\)
−0.771706 + 0.635979i \(0.780597\pi\)
\(102\) 8.09989 0.802009
\(103\) 16.8800 1.66323 0.831616 0.555352i \(-0.187416\pi\)
0.831616 + 0.555352i \(0.187416\pi\)
\(104\) −12.9488 −1.26973
\(105\) 0 0
\(106\) −10.5324 −1.02300
\(107\) −20.1474 −1.94772 −0.973860 0.227149i \(-0.927060\pi\)
−0.973860 + 0.227149i \(0.927060\pi\)
\(108\) −21.3628 −2.05563
\(109\) −16.7050 −1.60005 −0.800026 0.599965i \(-0.795181\pi\)
−0.800026 + 0.599965i \(0.795181\pi\)
\(110\) 2.25931 0.215416
\(111\) 1.57924 0.149895
\(112\) 0 0
\(113\) 11.2463 1.05796 0.528980 0.848634i \(-0.322575\pi\)
0.528980 + 0.848634i \(0.322575\pi\)
\(114\) 0.582069 0.0545157
\(115\) −6.56566 −0.612251
\(116\) −0.958357 −0.0889812
\(117\) −27.9049 −2.57981
\(118\) 26.1443 2.40678
\(119\) 0 0
\(120\) −7.22241 −0.659312
\(121\) 1.00000 0.0909091
\(122\) 8.97050 0.812151
\(123\) 27.7984 2.50650
\(124\) 21.7653 1.95458
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.35425 −0.120170 −0.0600849 0.998193i \(-0.519137\pi\)
−0.0600849 + 0.998193i \(0.519137\pi\)
\(128\) −16.9187 −1.49541
\(129\) −7.57943 −0.667332
\(130\) −11.7241 −1.02827
\(131\) 7.97900 0.697129 0.348564 0.937285i \(-0.386669\pi\)
0.348564 + 0.937285i \(0.386669\pi\)
\(132\) −8.98551 −0.782088
\(133\) 0 0
\(134\) −4.85977 −0.419820
\(135\) −6.88131 −0.592249
\(136\) −3.09082 −0.265036
\(137\) 12.6254 1.07866 0.539328 0.842095i \(-0.318678\pi\)
0.539328 + 0.842095i \(0.318678\pi\)
\(138\) 42.9348 3.65486
\(139\) 18.0063 1.52727 0.763635 0.645648i \(-0.223413\pi\)
0.763635 + 0.645648i \(0.223413\pi\)
\(140\) 0 0
\(141\) 29.2697 2.46495
\(142\) −33.5120 −2.81227
\(143\) −5.18923 −0.433945
\(144\) −3.07184 −0.255986
\(145\) −0.308703 −0.0256364
\(146\) 9.66330 0.799741
\(147\) 0 0
\(148\) −1.69386 −0.139234
\(149\) −1.46846 −0.120301 −0.0601506 0.998189i \(-0.519158\pi\)
−0.0601506 + 0.998189i \(0.519158\pi\)
\(150\) −6.53930 −0.533932
\(151\) 3.06680 0.249573 0.124786 0.992184i \(-0.460175\pi\)
0.124786 + 0.992184i \(0.460175\pi\)
\(152\) −0.222110 −0.0180155
\(153\) −6.66079 −0.538493
\(154\) 0 0
\(155\) 7.01096 0.563134
\(156\) 46.6279 3.73322
\(157\) 0.0106622 0.000850937 0 0.000425469 1.00000i \(-0.499865\pi\)
0.000425469 1.00000i \(0.499865\pi\)
\(158\) 11.9321 0.949267
\(159\) 13.4930 1.07006
\(160\) −6.28124 −0.496576
\(161\) 0 0
\(162\) 8.55093 0.671824
\(163\) −24.7147 −1.93580 −0.967902 0.251326i \(-0.919133\pi\)
−0.967902 + 0.251326i \(0.919133\pi\)
\(164\) −29.8160 −2.32824
\(165\) −2.89439 −0.225328
\(166\) −20.6932 −1.60611
\(167\) 8.51313 0.658766 0.329383 0.944196i \(-0.393159\pi\)
0.329383 + 0.944196i \(0.393159\pi\)
\(168\) 0 0
\(169\) 13.9281 1.07139
\(170\) −2.79849 −0.214634
\(171\) −0.478653 −0.0366035
\(172\) 8.12955 0.619872
\(173\) 15.1992 1.15558 0.577788 0.816187i \(-0.303916\pi\)
0.577788 + 0.816187i \(0.303916\pi\)
\(174\) 2.01870 0.153037
\(175\) 0 0
\(176\) −0.571242 −0.0430590
\(177\) −33.4933 −2.51751
\(178\) −27.7923 −2.08312
\(179\) 20.0498 1.49859 0.749297 0.662234i \(-0.230392\pi\)
0.749297 + 0.662234i \(0.230392\pi\)
\(180\) 16.6941 1.24431
\(181\) −18.1470 −1.34886 −0.674429 0.738340i \(-0.735610\pi\)
−0.674429 + 0.738340i \(0.735610\pi\)
\(182\) 0 0
\(183\) −11.4921 −0.849518
\(184\) −16.3834 −1.20780
\(185\) −0.545621 −0.0401149
\(186\) −45.8468 −3.36165
\(187\) −1.23865 −0.0905789
\(188\) −31.3941 −2.28965
\(189\) 0 0
\(190\) −0.201103 −0.0145895
\(191\) 11.9314 0.863326 0.431663 0.902035i \(-0.357927\pi\)
0.431663 + 0.902035i \(0.357927\pi\)
\(192\) 37.7681 2.72568
\(193\) −6.85826 −0.493668 −0.246834 0.969058i \(-0.579390\pi\)
−0.246834 + 0.969058i \(0.579390\pi\)
\(194\) −42.9579 −3.08420
\(195\) 15.0196 1.07558
\(196\) 0 0
\(197\) 4.70609 0.335295 0.167647 0.985847i \(-0.446383\pi\)
0.167647 + 0.985847i \(0.446383\pi\)
\(198\) 12.1493 0.863416
\(199\) 4.68345 0.332001 0.166000 0.986126i \(-0.446915\pi\)
0.166000 + 0.986126i \(0.446915\pi\)
\(200\) 2.49532 0.176445
\(201\) 6.22583 0.439136
\(202\) −35.0443 −2.46571
\(203\) 0 0
\(204\) 11.1299 0.779248
\(205\) −9.60425 −0.670790
\(206\) 38.1370 2.65713
\(207\) −35.3066 −2.45398
\(208\) 2.96431 0.205538
\(209\) −0.0890109 −0.00615701
\(210\) 0 0
\(211\) −6.21950 −0.428168 −0.214084 0.976815i \(-0.568677\pi\)
−0.214084 + 0.976815i \(0.568677\pi\)
\(212\) −14.4723 −0.993963
\(213\) 42.9321 2.94166
\(214\) −45.5191 −3.11162
\(215\) 2.61867 0.178592
\(216\) −17.1710 −1.16834
\(217\) 0 0
\(218\) −37.7418 −2.55620
\(219\) −12.3796 −0.836537
\(220\) 3.10446 0.209303
\(221\) 6.42763 0.432369
\(222\) 3.56798 0.239467
\(223\) −9.98943 −0.668942 −0.334471 0.942406i \(-0.608558\pi\)
−0.334471 + 0.942406i \(0.608558\pi\)
\(224\) 0 0
\(225\) 5.37747 0.358498
\(226\) 25.4088 1.69017
\(227\) 2.41877 0.160539 0.0802697 0.996773i \(-0.474422\pi\)
0.0802697 + 0.996773i \(0.474422\pi\)
\(228\) 0.799808 0.0529686
\(229\) 3.38871 0.223932 0.111966 0.993712i \(-0.464285\pi\)
0.111966 + 0.993712i \(0.464285\pi\)
\(230\) −14.8338 −0.978114
\(231\) 0 0
\(232\) −0.770312 −0.0505735
\(233\) 15.2101 0.996446 0.498223 0.867049i \(-0.333986\pi\)
0.498223 + 0.867049i \(0.333986\pi\)
\(234\) −63.0457 −4.12143
\(235\) −10.1126 −0.659671
\(236\) 35.9243 2.33847
\(237\) −15.2862 −0.992943
\(238\) 0 0
\(239\) −16.6034 −1.07398 −0.536992 0.843587i \(-0.680439\pi\)
−0.536992 + 0.843587i \(0.680439\pi\)
\(240\) 1.65340 0.106726
\(241\) −11.7656 −0.757891 −0.378946 0.925419i \(-0.623713\pi\)
−0.378946 + 0.925419i \(0.623713\pi\)
\(242\) 2.25931 0.145234
\(243\) 9.68938 0.621574
\(244\) 12.3262 0.789102
\(245\) 0 0
\(246\) 62.8051 4.00431
\(247\) 0.461898 0.0293899
\(248\) 17.4946 1.11091
\(249\) 26.5100 1.68000
\(250\) 2.25931 0.142891
\(251\) −4.73206 −0.298685 −0.149342 0.988786i \(-0.547716\pi\)
−0.149342 + 0.988786i \(0.547716\pi\)
\(252\) 0 0
\(253\) −6.56566 −0.412780
\(254\) −3.05965 −0.191980
\(255\) 3.58513 0.224509
\(256\) −12.1269 −0.757930
\(257\) 7.62211 0.475454 0.237727 0.971332i \(-0.423598\pi\)
0.237727 + 0.971332i \(0.423598\pi\)
\(258\) −17.1243 −1.06611
\(259\) 0 0
\(260\) −16.1098 −0.999085
\(261\) −1.66004 −0.102754
\(262\) 18.0270 1.11371
\(263\) 25.1797 1.55265 0.776325 0.630333i \(-0.217082\pi\)
0.776325 + 0.630333i \(0.217082\pi\)
\(264\) −7.22241 −0.444508
\(265\) −4.66178 −0.286371
\(266\) 0 0
\(267\) 35.6046 2.17897
\(268\) −6.67770 −0.407906
\(269\) 10.5440 0.642877 0.321438 0.946930i \(-0.395834\pi\)
0.321438 + 0.946930i \(0.395834\pi\)
\(270\) −15.5470 −0.946159
\(271\) 18.6620 1.13363 0.566817 0.823844i \(-0.308175\pi\)
0.566817 + 0.823844i \(0.308175\pi\)
\(272\) 0.707568 0.0429026
\(273\) 0 0
\(274\) 28.5245 1.72323
\(275\) 1.00000 0.0603023
\(276\) 58.9958 3.55113
\(277\) 2.97955 0.179024 0.0895119 0.995986i \(-0.471469\pi\)
0.0895119 + 0.995986i \(0.471469\pi\)
\(278\) 40.6816 2.43992
\(279\) 37.7012 2.25711
\(280\) 0 0
\(281\) −20.6873 −1.23410 −0.617051 0.786923i \(-0.711673\pi\)
−0.617051 + 0.786923i \(0.711673\pi\)
\(282\) 66.1292 3.93793
\(283\) 31.0609 1.84638 0.923188 0.384349i \(-0.125574\pi\)
0.923188 + 0.384349i \(0.125574\pi\)
\(284\) −46.0481 −2.73245
\(285\) 0.257632 0.0152608
\(286\) −11.7241 −0.693258
\(287\) 0 0
\(288\) −33.7772 −1.99034
\(289\) −15.4658 −0.909750
\(290\) −0.697455 −0.0409560
\(291\) 55.0331 3.22610
\(292\) 13.2781 0.777044
\(293\) 4.89906 0.286206 0.143103 0.989708i \(-0.454292\pi\)
0.143103 + 0.989708i \(0.454292\pi\)
\(294\) 0 0
\(295\) 11.5718 0.673738
\(296\) −1.36150 −0.0791354
\(297\) −6.88131 −0.399294
\(298\) −3.31770 −0.192189
\(299\) 34.0707 1.97036
\(300\) −8.98551 −0.518779
\(301\) 0 0
\(302\) 6.92884 0.398710
\(303\) 44.8951 2.57916
\(304\) 0.0508468 0.00291626
\(305\) 3.97047 0.227348
\(306\) −15.0488 −0.860281
\(307\) 11.2937 0.644564 0.322282 0.946644i \(-0.395550\pi\)
0.322282 + 0.946644i \(0.395550\pi\)
\(308\) 0 0
\(309\) −48.8571 −2.77938
\(310\) 15.8399 0.899646
\(311\) −9.94078 −0.563690 −0.281845 0.959460i \(-0.590946\pi\)
−0.281845 + 0.959460i \(0.590946\pi\)
\(312\) 37.4787 2.12182
\(313\) −23.9712 −1.35493 −0.677465 0.735555i \(-0.736921\pi\)
−0.677465 + 0.735555i \(0.736921\pi\)
\(314\) 0.0240892 0.00135943
\(315\) 0 0
\(316\) 16.3956 0.922327
\(317\) −3.53943 −0.198794 −0.0993971 0.995048i \(-0.531691\pi\)
−0.0993971 + 0.995048i \(0.531691\pi\)
\(318\) 30.4848 1.70950
\(319\) −0.308703 −0.0172841
\(320\) −13.0488 −0.729448
\(321\) 58.3143 3.25478
\(322\) 0 0
\(323\) 0.110253 0.00613465
\(324\) 11.7496 0.652758
\(325\) −5.18923 −0.287847
\(326\) −55.8381 −3.09258
\(327\) 48.3508 2.67381
\(328\) −23.9656 −1.32328
\(329\) 0 0
\(330\) −6.53930 −0.359977
\(331\) 21.7702 1.19660 0.598300 0.801272i \(-0.295843\pi\)
0.598300 + 0.801272i \(0.295843\pi\)
\(332\) −28.4341 −1.56052
\(333\) −2.93406 −0.160786
\(334\) 19.2338 1.05242
\(335\) −2.15100 −0.117522
\(336\) 0 0
\(337\) 15.9441 0.868533 0.434266 0.900784i \(-0.357008\pi\)
0.434266 + 0.900784i \(0.357008\pi\)
\(338\) 31.4679 1.71163
\(339\) −32.5511 −1.76793
\(340\) −3.84534 −0.208543
\(341\) 7.01096 0.379665
\(342\) −1.08142 −0.0584767
\(343\) 0 0
\(344\) 6.53440 0.352311
\(345\) 19.0036 1.02312
\(346\) 34.3397 1.84612
\(347\) 29.9536 1.60799 0.803996 0.594634i \(-0.202703\pi\)
0.803996 + 0.594634i \(0.202703\pi\)
\(348\) 2.77386 0.148694
\(349\) −7.06337 −0.378094 −0.189047 0.981968i \(-0.560540\pi\)
−0.189047 + 0.981968i \(0.560540\pi\)
\(350\) 0 0
\(351\) 35.7087 1.90599
\(352\) −6.28124 −0.334791
\(353\) −10.9105 −0.580705 −0.290353 0.956920i \(-0.593773\pi\)
−0.290353 + 0.956920i \(0.593773\pi\)
\(354\) −75.6717 −4.02190
\(355\) −14.8329 −0.787248
\(356\) −38.1888 −2.02400
\(357\) 0 0
\(358\) 45.2987 2.39411
\(359\) −4.29511 −0.226687 −0.113344 0.993556i \(-0.536156\pi\)
−0.113344 + 0.993556i \(0.536156\pi\)
\(360\) 13.4185 0.707216
\(361\) −18.9921 −0.999583
\(362\) −40.9997 −2.15489
\(363\) −2.89439 −0.151916
\(364\) 0 0
\(365\) 4.27711 0.223874
\(366\) −25.9641 −1.35717
\(367\) −18.9852 −0.991018 −0.495509 0.868603i \(-0.665018\pi\)
−0.495509 + 0.868603i \(0.665018\pi\)
\(368\) 3.75058 0.195513
\(369\) −51.6466 −2.68861
\(370\) −1.23273 −0.0640863
\(371\) 0 0
\(372\) −62.9971 −3.26624
\(373\) 16.7806 0.868869 0.434434 0.900704i \(-0.356948\pi\)
0.434434 + 0.900704i \(0.356948\pi\)
\(374\) −2.79849 −0.144706
\(375\) −2.89439 −0.149465
\(376\) −25.2341 −1.30135
\(377\) 1.60193 0.0825037
\(378\) 0 0
\(379\) 16.4327 0.844089 0.422045 0.906575i \(-0.361313\pi\)
0.422045 + 0.906575i \(0.361313\pi\)
\(380\) −0.276331 −0.0141755
\(381\) 3.91971 0.200813
\(382\) 26.9567 1.37922
\(383\) −22.3089 −1.13993 −0.569966 0.821668i \(-0.693044\pi\)
−0.569966 + 0.821668i \(0.693044\pi\)
\(384\) 48.9691 2.49894
\(385\) 0 0
\(386\) −15.4949 −0.788670
\(387\) 14.0818 0.715818
\(388\) −59.0275 −2.99666
\(389\) 8.54367 0.433181 0.216591 0.976263i \(-0.430506\pi\)
0.216591 + 0.976263i \(0.430506\pi\)
\(390\) 33.9339 1.71831
\(391\) 8.13255 0.411281
\(392\) 0 0
\(393\) −23.0943 −1.16495
\(394\) 10.6325 0.535657
\(395\) 5.28131 0.265732
\(396\) 16.6941 0.838912
\(397\) 24.6326 1.23627 0.618137 0.786070i \(-0.287888\pi\)
0.618137 + 0.786070i \(0.287888\pi\)
\(398\) 10.5813 0.530395
\(399\) 0 0
\(400\) −0.571242 −0.0285621
\(401\) 1.08299 0.0540820 0.0270410 0.999634i \(-0.491392\pi\)
0.0270410 + 0.999634i \(0.491392\pi\)
\(402\) 14.0660 0.701551
\(403\) −36.3815 −1.81229
\(404\) −48.1536 −2.39573
\(405\) 3.78476 0.188066
\(406\) 0 0
\(407\) −0.545621 −0.0270455
\(408\) 8.94602 0.442894
\(409\) 20.8402 1.03048 0.515241 0.857045i \(-0.327702\pi\)
0.515241 + 0.857045i \(0.327702\pi\)
\(410\) −21.6989 −1.07163
\(411\) −36.5426 −1.80252
\(412\) 52.4032 2.58172
\(413\) 0 0
\(414\) −79.7685 −3.92041
\(415\) −9.15911 −0.449603
\(416\) 32.5948 1.59809
\(417\) −52.1170 −2.55218
\(418\) −0.201103 −0.00983626
\(419\) −10.1828 −0.497463 −0.248731 0.968573i \(-0.580014\pi\)
−0.248731 + 0.968573i \(0.580014\pi\)
\(420\) 0 0
\(421\) 27.4719 1.33890 0.669449 0.742858i \(-0.266530\pi\)
0.669449 + 0.742858i \(0.266530\pi\)
\(422\) −14.0518 −0.684029
\(423\) −54.3800 −2.64405
\(424\) −11.6326 −0.564930
\(425\) −1.23865 −0.0600833
\(426\) 96.9967 4.69951
\(427\) 0 0
\(428\) −62.5467 −3.02331
\(429\) 15.0196 0.725155
\(430\) 5.91637 0.285313
\(431\) 29.7975 1.43530 0.717648 0.696406i \(-0.245219\pi\)
0.717648 + 0.696406i \(0.245219\pi\)
\(432\) 3.93090 0.189125
\(433\) −7.88526 −0.378942 −0.189471 0.981886i \(-0.560677\pi\)
−0.189471 + 0.981886i \(0.560677\pi\)
\(434\) 0 0
\(435\) 0.893506 0.0428403
\(436\) −51.8602 −2.48365
\(437\) 0.584415 0.0279564
\(438\) −27.9693 −1.33643
\(439\) 34.7141 1.65682 0.828408 0.560126i \(-0.189247\pi\)
0.828408 + 0.560126i \(0.189247\pi\)
\(440\) 2.49532 0.118960
\(441\) 0 0
\(442\) 14.5220 0.690740
\(443\) −0.195671 −0.00929662 −0.00464831 0.999989i \(-0.501480\pi\)
−0.00464831 + 0.999989i \(0.501480\pi\)
\(444\) 4.90269 0.232671
\(445\) −12.3013 −0.583136
\(446\) −22.5692 −1.06868
\(447\) 4.25030 0.201032
\(448\) 0 0
\(449\) 11.5122 0.543296 0.271648 0.962397i \(-0.412431\pi\)
0.271648 + 0.962397i \(0.412431\pi\)
\(450\) 12.1493 0.572726
\(451\) −9.60425 −0.452247
\(452\) 34.9136 1.64220
\(453\) −8.87650 −0.417054
\(454\) 5.46474 0.256473
\(455\) 0 0
\(456\) 0.642873 0.0301053
\(457\) −5.36664 −0.251041 −0.125521 0.992091i \(-0.540060\pi\)
−0.125521 + 0.992091i \(0.540060\pi\)
\(458\) 7.65614 0.357748
\(459\) 8.52352 0.397844
\(460\) −20.3828 −0.950355
\(461\) −29.5146 −1.37463 −0.687317 0.726358i \(-0.741212\pi\)
−0.687317 + 0.726358i \(0.741212\pi\)
\(462\) 0 0
\(463\) −23.6366 −1.09848 −0.549242 0.835663i \(-0.685084\pi\)
−0.549242 + 0.835663i \(0.685084\pi\)
\(464\) 0.176344 0.00818658
\(465\) −20.2924 −0.941039
\(466\) 34.3642 1.59189
\(467\) −25.7195 −1.19016 −0.595078 0.803668i \(-0.702879\pi\)
−0.595078 + 0.803668i \(0.702879\pi\)
\(468\) −86.6298 −4.00446
\(469\) 0 0
\(470\) −22.8474 −1.05387
\(471\) −0.0308605 −0.00142198
\(472\) 28.8754 1.32910
\(473\) 2.61867 0.120406
\(474\) −34.5361 −1.58630
\(475\) −0.0890109 −0.00408410
\(476\) 0 0
\(477\) −25.0686 −1.14781
\(478\) −37.5121 −1.71576
\(479\) 36.5837 1.67155 0.835776 0.549071i \(-0.185018\pi\)
0.835776 + 0.549071i \(0.185018\pi\)
\(480\) 18.1803 0.829815
\(481\) 2.83136 0.129099
\(482\) −26.5822 −1.21078
\(483\) 0 0
\(484\) 3.10446 0.141112
\(485\) −19.0137 −0.863370
\(486\) 21.8913 0.993008
\(487\) 4.85250 0.219888 0.109944 0.993938i \(-0.464933\pi\)
0.109944 + 0.993938i \(0.464933\pi\)
\(488\) 9.90758 0.448495
\(489\) 71.5339 3.23487
\(490\) 0 0
\(491\) −18.0213 −0.813290 −0.406645 0.913586i \(-0.633301\pi\)
−0.406645 + 0.913586i \(0.633301\pi\)
\(492\) 86.2991 3.89066
\(493\) 0.382375 0.0172213
\(494\) 1.04357 0.0469524
\(495\) 5.37747 0.241699
\(496\) −4.00496 −0.179828
\(497\) 0 0
\(498\) 59.8942 2.68392
\(499\) −0.419359 −0.0187731 −0.00938655 0.999956i \(-0.502988\pi\)
−0.00938655 + 0.999956i \(0.502988\pi\)
\(500\) 3.10446 0.138836
\(501\) −24.6403 −1.10085
\(502\) −10.6912 −0.477170
\(503\) −37.0947 −1.65397 −0.826986 0.562223i \(-0.809946\pi\)
−0.826986 + 0.562223i \(0.809946\pi\)
\(504\) 0 0
\(505\) −15.5111 −0.690235
\(506\) −14.8338 −0.659445
\(507\) −40.3133 −1.79038
\(508\) −4.20420 −0.186531
\(509\) −30.1531 −1.33651 −0.668256 0.743931i \(-0.732959\pi\)
−0.668256 + 0.743931i \(0.732959\pi\)
\(510\) 8.09989 0.358669
\(511\) 0 0
\(512\) 6.43897 0.284565
\(513\) 0.612511 0.0270430
\(514\) 17.2207 0.759571
\(515\) 16.8800 0.743820
\(516\) −23.5301 −1.03585
\(517\) −10.1126 −0.444750
\(518\) 0 0
\(519\) −43.9925 −1.93105
\(520\) −12.9488 −0.567841
\(521\) −13.1706 −0.577012 −0.288506 0.957478i \(-0.593159\pi\)
−0.288506 + 0.957478i \(0.593159\pi\)
\(522\) −3.75054 −0.164157
\(523\) −27.6655 −1.20973 −0.604864 0.796328i \(-0.706773\pi\)
−0.604864 + 0.796328i \(0.706773\pi\)
\(524\) 24.7705 1.08210
\(525\) 0 0
\(526\) 56.8887 2.48047
\(527\) −8.68412 −0.378286
\(528\) 1.65340 0.0719548
\(529\) 20.1079 0.874258
\(530\) −10.5324 −0.457498
\(531\) 62.2271 2.70043
\(532\) 0 0
\(533\) 49.8387 2.15875
\(534\) 80.4417 3.48105
\(535\) −20.1474 −0.871047
\(536\) −5.36743 −0.231838
\(537\) −58.0319 −2.50426
\(538\) 23.8220 1.02704
\(539\) 0 0
\(540\) −21.3628 −0.919307
\(541\) 13.5889 0.584233 0.292117 0.956383i \(-0.405640\pi\)
0.292117 + 0.956383i \(0.405640\pi\)
\(542\) 42.1631 1.81106
\(543\) 52.5245 2.25404
\(544\) 7.78025 0.333575
\(545\) −16.7050 −0.715565
\(546\) 0 0
\(547\) 32.8660 1.40525 0.702624 0.711562i \(-0.252012\pi\)
0.702624 + 0.711562i \(0.252012\pi\)
\(548\) 39.1949 1.67432
\(549\) 21.3511 0.911242
\(550\) 2.25931 0.0963371
\(551\) 0.0274779 0.00117060
\(552\) 47.4199 2.01832
\(553\) 0 0
\(554\) 6.73171 0.286003
\(555\) 1.57924 0.0670350
\(556\) 55.8997 2.37068
\(557\) 35.3570 1.49812 0.749062 0.662500i \(-0.230505\pi\)
0.749062 + 0.662500i \(0.230505\pi\)
\(558\) 85.1786 3.60590
\(559\) −13.5889 −0.574748
\(560\) 0 0
\(561\) 3.58513 0.151364
\(562\) −46.7390 −1.97157
\(563\) −24.3034 −1.02426 −0.512132 0.858907i \(-0.671144\pi\)
−0.512132 + 0.858907i \(0.671144\pi\)
\(564\) 90.8666 3.82617
\(565\) 11.2463 0.473134
\(566\) 70.1760 2.94972
\(567\) 0 0
\(568\) −37.0127 −1.55302
\(569\) −45.3206 −1.89994 −0.949969 0.312345i \(-0.898886\pi\)
−0.949969 + 0.312345i \(0.898886\pi\)
\(570\) 0.582069 0.0243802
\(571\) 19.9418 0.834537 0.417269 0.908783i \(-0.362987\pi\)
0.417269 + 0.908783i \(0.362987\pi\)
\(572\) −16.1098 −0.673583
\(573\) −34.5341 −1.44268
\(574\) 0 0
\(575\) −6.56566 −0.273807
\(576\) −70.1693 −2.92372
\(577\) −11.6234 −0.483887 −0.241944 0.970290i \(-0.577785\pi\)
−0.241944 + 0.970290i \(0.577785\pi\)
\(578\) −34.9419 −1.45339
\(579\) 19.8504 0.824956
\(580\) −0.958357 −0.0397936
\(581\) 0 0
\(582\) 124.337 5.15392
\(583\) −4.66178 −0.193071
\(584\) 10.6727 0.441641
\(585\) −27.9049 −1.15373
\(586\) 11.0685 0.457234
\(587\) 9.03748 0.373017 0.186508 0.982453i \(-0.440283\pi\)
0.186508 + 0.982453i \(0.440283\pi\)
\(588\) 0 0
\(589\) −0.624052 −0.0257136
\(590\) 26.1443 1.07634
\(591\) −13.6212 −0.560303
\(592\) 0.311682 0.0128101
\(593\) −2.98060 −0.122399 −0.0611993 0.998126i \(-0.519493\pi\)
−0.0611993 + 0.998126i \(0.519493\pi\)
\(594\) −15.5470 −0.637900
\(595\) 0 0
\(596\) −4.55878 −0.186735
\(597\) −13.5557 −0.554798
\(598\) 76.9762 3.14779
\(599\) −39.2979 −1.60567 −0.802834 0.596202i \(-0.796676\pi\)
−0.802834 + 0.596202i \(0.796676\pi\)
\(600\) −7.22241 −0.294853
\(601\) −20.4001 −0.832138 −0.416069 0.909333i \(-0.636593\pi\)
−0.416069 + 0.909333i \(0.636593\pi\)
\(602\) 0 0
\(603\) −11.5669 −0.471042
\(604\) 9.52076 0.387394
\(605\) 1.00000 0.0406558
\(606\) 101.432 4.12039
\(607\) 6.70707 0.272231 0.136116 0.990693i \(-0.456538\pi\)
0.136116 + 0.990693i \(0.456538\pi\)
\(608\) 0.559099 0.0226744
\(609\) 0 0
\(610\) 8.97050 0.363205
\(611\) 52.4765 2.12297
\(612\) −20.6782 −0.835866
\(613\) −2.52305 −0.101905 −0.0509524 0.998701i \(-0.516226\pi\)
−0.0509524 + 0.998701i \(0.516226\pi\)
\(614\) 25.5159 1.02974
\(615\) 27.7984 1.12094
\(616\) 0 0
\(617\) −11.2173 −0.451591 −0.225795 0.974175i \(-0.572498\pi\)
−0.225795 + 0.974175i \(0.572498\pi\)
\(618\) −110.383 −4.44026
\(619\) −14.5132 −0.583337 −0.291668 0.956520i \(-0.594210\pi\)
−0.291668 + 0.956520i \(0.594210\pi\)
\(620\) 21.7653 0.874114
\(621\) 45.1804 1.81303
\(622\) −22.4593 −0.900535
\(623\) 0 0
\(624\) −8.57985 −0.343469
\(625\) 1.00000 0.0400000
\(626\) −54.1582 −2.16460
\(627\) 0.257632 0.0102888
\(628\) 0.0331004 0.00132085
\(629\) 0.675833 0.0269472
\(630\) 0 0
\(631\) 46.8572 1.86535 0.932677 0.360713i \(-0.117467\pi\)
0.932677 + 0.360713i \(0.117467\pi\)
\(632\) 13.1785 0.524214
\(633\) 18.0016 0.715501
\(634\) −7.99665 −0.317588
\(635\) −1.35425 −0.0537416
\(636\) 41.8885 1.66099
\(637\) 0 0
\(638\) −0.697455 −0.0276125
\(639\) −79.7634 −3.15539
\(640\) −16.9187 −0.668768
\(641\) 2.35168 0.0928858 0.0464429 0.998921i \(-0.485211\pi\)
0.0464429 + 0.998921i \(0.485211\pi\)
\(642\) 131.750 5.19975
\(643\) −6.05724 −0.238874 −0.119437 0.992842i \(-0.538109\pi\)
−0.119437 + 0.992842i \(0.538109\pi\)
\(644\) 0 0
\(645\) −7.57943 −0.298440
\(646\) 0.249096 0.00980053
\(647\) −26.7941 −1.05338 −0.526692 0.850056i \(-0.676568\pi\)
−0.526692 + 0.850056i \(0.676568\pi\)
\(648\) 9.44417 0.371002
\(649\) 11.5718 0.454234
\(650\) −11.7241 −0.459855
\(651\) 0 0
\(652\) −76.7258 −3.00482
\(653\) 14.2400 0.557254 0.278627 0.960399i \(-0.410121\pi\)
0.278627 + 0.960399i \(0.410121\pi\)
\(654\) 109.239 4.27159
\(655\) 7.97900 0.311766
\(656\) 5.48636 0.214206
\(657\) 23.0000 0.897317
\(658\) 0 0
\(659\) −10.0130 −0.390050 −0.195025 0.980798i \(-0.562479\pi\)
−0.195025 + 0.980798i \(0.562479\pi\)
\(660\) −8.98551 −0.349760
\(661\) −8.61490 −0.335081 −0.167540 0.985865i \(-0.553582\pi\)
−0.167540 + 0.985865i \(0.553582\pi\)
\(662\) 49.1856 1.91165
\(663\) −18.6040 −0.722521
\(664\) −22.8549 −0.886941
\(665\) 0 0
\(666\) −6.62894 −0.256866
\(667\) 2.02684 0.0784796
\(668\) 26.4287 1.02256
\(669\) 28.9133 1.11785
\(670\) −4.85977 −0.187749
\(671\) 3.97047 0.153278
\(672\) 0 0
\(673\) −46.3364 −1.78614 −0.893069 0.449920i \(-0.851452\pi\)
−0.893069 + 0.449920i \(0.851452\pi\)
\(674\) 36.0227 1.38754
\(675\) −6.88131 −0.264862
\(676\) 43.2393 1.66305
\(677\) 5.52562 0.212367 0.106183 0.994347i \(-0.466137\pi\)
0.106183 + 0.994347i \(0.466137\pi\)
\(678\) −73.5428 −2.82439
\(679\) 0 0
\(680\) −3.09082 −0.118527
\(681\) −7.00086 −0.268273
\(682\) 15.8399 0.606541
\(683\) −5.69730 −0.218001 −0.109001 0.994042i \(-0.534765\pi\)
−0.109001 + 0.994042i \(0.534765\pi\)
\(684\) −1.48596 −0.0568171
\(685\) 12.6254 0.482390
\(686\) 0 0
\(687\) −9.80824 −0.374208
\(688\) −1.49589 −0.0570304
\(689\) 24.1911 0.921606
\(690\) 42.9348 1.63450
\(691\) −24.7075 −0.939917 −0.469958 0.882689i \(-0.655731\pi\)
−0.469958 + 0.882689i \(0.655731\pi\)
\(692\) 47.1855 1.79372
\(693\) 0 0
\(694\) 67.6743 2.56888
\(695\) 18.0063 0.683016
\(696\) 2.22958 0.0845120
\(697\) 11.8963 0.450604
\(698\) −15.9583 −0.604031
\(699\) −44.0239 −1.66514
\(700\) 0 0
\(701\) 1.97453 0.0745771 0.0372885 0.999305i \(-0.488128\pi\)
0.0372885 + 0.999305i \(0.488128\pi\)
\(702\) 80.6769 3.04495
\(703\) 0.0485662 0.00183171
\(704\) −13.0488 −0.491794
\(705\) 29.2697 1.10236
\(706\) −24.6501 −0.927717
\(707\) 0 0
\(708\) −103.979 −3.90776
\(709\) 4.47721 0.168145 0.0840725 0.996460i \(-0.473207\pi\)
0.0840725 + 0.996460i \(0.473207\pi\)
\(710\) −33.5120 −1.25768
\(711\) 28.4001 1.06509
\(712\) −30.6956 −1.15036
\(713\) −46.0316 −1.72390
\(714\) 0 0
\(715\) −5.18923 −0.194066
\(716\) 62.2439 2.32616
\(717\) 48.0566 1.79471
\(718\) −9.70396 −0.362149
\(719\) 3.98375 0.148569 0.0742843 0.997237i \(-0.476333\pi\)
0.0742843 + 0.997237i \(0.476333\pi\)
\(720\) −3.07184 −0.114481
\(721\) 0 0
\(722\) −42.9089 −1.59690
\(723\) 34.0543 1.26649
\(724\) −56.3367 −2.09374
\(725\) −0.308703 −0.0114649
\(726\) −6.53930 −0.242696
\(727\) −2.98387 −0.110666 −0.0553329 0.998468i \(-0.517622\pi\)
−0.0553329 + 0.998468i \(0.517622\pi\)
\(728\) 0 0
\(729\) −39.3991 −1.45922
\(730\) 9.66330 0.357655
\(731\) −3.24361 −0.119969
\(732\) −35.6767 −1.31865
\(733\) −0.613761 −0.0226698 −0.0113349 0.999936i \(-0.503608\pi\)
−0.0113349 + 0.999936i \(0.503608\pi\)
\(734\) −42.8933 −1.58322
\(735\) 0 0
\(736\) 41.2405 1.52015
\(737\) −2.15100 −0.0792332
\(738\) −116.685 −4.29525
\(739\) 1.30551 0.0480241 0.0240120 0.999712i \(-0.492356\pi\)
0.0240120 + 0.999712i \(0.492356\pi\)
\(740\) −1.69386 −0.0622676
\(741\) −1.33691 −0.0491126
\(742\) 0 0
\(743\) 18.9550 0.695392 0.347696 0.937607i \(-0.386964\pi\)
0.347696 + 0.937607i \(0.386964\pi\)
\(744\) −50.6360 −1.85641
\(745\) −1.46846 −0.0538003
\(746\) 37.9126 1.38808
\(747\) −49.2528 −1.80207
\(748\) −3.84534 −0.140599
\(749\) 0 0
\(750\) −6.53930 −0.238782
\(751\) −36.1767 −1.32011 −0.660053 0.751219i \(-0.729466\pi\)
−0.660053 + 0.751219i \(0.729466\pi\)
\(752\) 5.77673 0.210656
\(753\) 13.6964 0.499125
\(754\) 3.61925 0.131805
\(755\) 3.06680 0.111612
\(756\) 0 0
\(757\) −9.78039 −0.355474 −0.177737 0.984078i \(-0.556878\pi\)
−0.177737 + 0.984078i \(0.556878\pi\)
\(758\) 37.1264 1.34849
\(759\) 19.0036 0.689786
\(760\) −0.222110 −0.00805678
\(761\) −39.8118 −1.44318 −0.721589 0.692322i \(-0.756588\pi\)
−0.721589 + 0.692322i \(0.756588\pi\)
\(762\) 8.85582 0.320813
\(763\) 0 0
\(764\) 37.0406 1.34008
\(765\) −6.66079 −0.240821
\(766\) −50.4026 −1.82112
\(767\) −60.0489 −2.16824
\(768\) 35.0999 1.26656
\(769\) −10.0809 −0.363527 −0.181764 0.983342i \(-0.558181\pi\)
−0.181764 + 0.983342i \(0.558181\pi\)
\(770\) 0 0
\(771\) −22.0613 −0.794519
\(772\) −21.2912 −0.766287
\(773\) 46.0264 1.65545 0.827727 0.561130i \(-0.189633\pi\)
0.827727 + 0.561130i \(0.189633\pi\)
\(774\) 31.8151 1.14357
\(775\) 7.01096 0.251841
\(776\) −47.4453 −1.70319
\(777\) 0 0
\(778\) 19.3028 0.692037
\(779\) 0.854883 0.0306294
\(780\) 46.6279 1.66955
\(781\) −14.8329 −0.530762
\(782\) 18.3739 0.657050
\(783\) 2.12428 0.0759157
\(784\) 0 0
\(785\) 0.0106622 0.000380551 0
\(786\) −52.1771 −1.86110
\(787\) 11.3150 0.403336 0.201668 0.979454i \(-0.435364\pi\)
0.201668 + 0.979454i \(0.435364\pi\)
\(788\) 14.6099 0.520455
\(789\) −72.8799 −2.59459
\(790\) 11.9321 0.424525
\(791\) 0 0
\(792\) 13.4185 0.476805
\(793\) −20.6037 −0.731658
\(794\) 55.6525 1.97503
\(795\) 13.4930 0.478547
\(796\) 14.5396 0.515342
\(797\) −5.13243 −0.181800 −0.0909001 0.995860i \(-0.528974\pi\)
−0.0909001 + 0.995860i \(0.528974\pi\)
\(798\) 0 0
\(799\) 12.5259 0.443135
\(800\) −6.28124 −0.222075
\(801\) −66.1497 −2.33728
\(802\) 2.44681 0.0863998
\(803\) 4.27711 0.150936
\(804\) 19.3278 0.681640
\(805\) 0 0
\(806\) −82.1969 −2.89526
\(807\) −30.5183 −1.07429
\(808\) −38.7051 −1.36164
\(809\) 21.3222 0.749647 0.374824 0.927096i \(-0.377703\pi\)
0.374824 + 0.927096i \(0.377703\pi\)
\(810\) 8.55093 0.300449
\(811\) −12.5828 −0.441843 −0.220922 0.975292i \(-0.570907\pi\)
−0.220922 + 0.975292i \(0.570907\pi\)
\(812\) 0 0
\(813\) −54.0149 −1.89439
\(814\) −1.23273 −0.0432070
\(815\) −24.7147 −0.865718
\(816\) −2.04798 −0.0716935
\(817\) −0.233090 −0.00815478
\(818\) 47.0844 1.64627
\(819\) 0 0
\(820\) −29.8160 −1.04122
\(821\) 31.0379 1.08323 0.541616 0.840626i \(-0.317813\pi\)
0.541616 + 0.840626i \(0.317813\pi\)
\(822\) −82.5610 −2.87965
\(823\) 17.5694 0.612432 0.306216 0.951962i \(-0.400937\pi\)
0.306216 + 0.951962i \(0.400937\pi\)
\(824\) 42.1208 1.46735
\(825\) −2.89439 −0.100770
\(826\) 0 0
\(827\) 29.9112 1.04012 0.520058 0.854131i \(-0.325910\pi\)
0.520058 + 0.854131i \(0.325910\pi\)
\(828\) −109.608 −3.80915
\(829\) 37.7004 1.30939 0.654694 0.755894i \(-0.272798\pi\)
0.654694 + 0.755894i \(0.272798\pi\)
\(830\) −20.6932 −0.718272
\(831\) −8.62396 −0.299162
\(832\) 67.7130 2.34753
\(833\) 0 0
\(834\) −117.748 −4.07729
\(835\) 8.51313 0.294609
\(836\) −0.276331 −0.00955710
\(837\) −48.2446 −1.66758
\(838\) −23.0061 −0.794732
\(839\) 35.1426 1.21326 0.606628 0.794985i \(-0.292522\pi\)
0.606628 + 0.794985i \(0.292522\pi\)
\(840\) 0 0
\(841\) −28.9047 −0.996714
\(842\) 62.0674 2.13898
\(843\) 59.8771 2.06228
\(844\) −19.3082 −0.664616
\(845\) 13.9281 0.479142
\(846\) −122.861 −4.22405
\(847\) 0 0
\(848\) 2.66301 0.0914480
\(849\) −89.9021 −3.08543
\(850\) −2.79849 −0.0959872
\(851\) 3.58237 0.122802
\(852\) 133.281 4.56613
\(853\) 44.6401 1.52845 0.764223 0.644952i \(-0.223123\pi\)
0.764223 + 0.644952i \(0.223123\pi\)
\(854\) 0 0
\(855\) −0.478653 −0.0163696
\(856\) −50.2741 −1.71833
\(857\) −42.8895 −1.46508 −0.732538 0.680726i \(-0.761664\pi\)
−0.732538 + 0.680726i \(0.761664\pi\)
\(858\) 33.9339 1.15849
\(859\) −10.8330 −0.369616 −0.184808 0.982775i \(-0.559166\pi\)
−0.184808 + 0.982775i \(0.559166\pi\)
\(860\) 8.12955 0.277215
\(861\) 0 0
\(862\) 67.3217 2.29298
\(863\) −11.0789 −0.377130 −0.188565 0.982061i \(-0.560384\pi\)
−0.188565 + 0.982061i \(0.560384\pi\)
\(864\) 43.2232 1.47048
\(865\) 15.1992 0.516790
\(866\) −17.8152 −0.605386
\(867\) 44.7638 1.52026
\(868\) 0 0
\(869\) 5.28131 0.179156
\(870\) 2.01870 0.0684404
\(871\) 11.1620 0.378211
\(872\) −41.6844 −1.41161
\(873\) −102.246 −3.46050
\(874\) 1.32037 0.0446623
\(875\) 0 0
\(876\) −38.4320 −1.29850
\(877\) −37.4882 −1.26589 −0.632943 0.774198i \(-0.718153\pi\)
−0.632943 + 0.774198i \(0.718153\pi\)
\(878\) 78.4298 2.64688
\(879\) −14.1798 −0.478272
\(880\) −0.571242 −0.0192566
\(881\) −35.2942 −1.18909 −0.594546 0.804062i \(-0.702668\pi\)
−0.594546 + 0.804062i \(0.702668\pi\)
\(882\) 0 0
\(883\) 20.3090 0.683451 0.341725 0.939800i \(-0.388989\pi\)
0.341725 + 0.939800i \(0.388989\pi\)
\(884\) 19.9543 0.671137
\(885\) −33.4933 −1.12587
\(886\) −0.442081 −0.0148520
\(887\) −56.0806 −1.88300 −0.941502 0.337008i \(-0.890585\pi\)
−0.941502 + 0.337008i \(0.890585\pi\)
\(888\) 3.94070 0.132241
\(889\) 0 0
\(890\) −27.7923 −0.931601
\(891\) 3.78476 0.126794
\(892\) −31.0118 −1.03835
\(893\) 0.900129 0.0301217
\(894\) 9.60272 0.321163
\(895\) 20.0498 0.670192
\(896\) 0 0
\(897\) −98.6139 −3.29262
\(898\) 26.0097 0.867954
\(899\) −2.16431 −0.0721836
\(900\) 16.6941 0.556471
\(901\) 5.77431 0.192370
\(902\) −21.6989 −0.722496
\(903\) 0 0
\(904\) 28.0630 0.933362
\(905\) −18.1470 −0.603227
\(906\) −20.0547 −0.666274
\(907\) −8.12019 −0.269626 −0.134813 0.990871i \(-0.543043\pi\)
−0.134813 + 0.990871i \(0.543043\pi\)
\(908\) 7.50898 0.249194
\(909\) −83.4105 −2.76655
\(910\) 0 0
\(911\) −49.4761 −1.63922 −0.819608 0.572925i \(-0.805809\pi\)
−0.819608 + 0.572925i \(0.805809\pi\)
\(912\) −0.147170 −0.00487329
\(913\) −9.15911 −0.303122
\(914\) −12.1249 −0.401056
\(915\) −11.4921 −0.379916
\(916\) 10.5201 0.347595
\(917\) 0 0
\(918\) 19.2572 0.635584
\(919\) 8.42669 0.277971 0.138985 0.990294i \(-0.455616\pi\)
0.138985 + 0.990294i \(0.455616\pi\)
\(920\) −16.3834 −0.540145
\(921\) −32.6883 −1.07711
\(922\) −66.6826 −2.19607
\(923\) 76.9713 2.53354
\(924\) 0 0
\(925\) −0.545621 −0.0179399
\(926\) −53.4022 −1.75491
\(927\) 90.7714 2.98132
\(928\) 1.93904 0.0636521
\(929\) 31.8919 1.04634 0.523169 0.852229i \(-0.324749\pi\)
0.523169 + 0.852229i \(0.324749\pi\)
\(930\) −45.8468 −1.50338
\(931\) 0 0
\(932\) 47.2191 1.54671
\(933\) 28.7725 0.941968
\(934\) −58.1081 −1.90136
\(935\) −1.23865 −0.0405081
\(936\) −69.6316 −2.27598
\(937\) −27.4103 −0.895457 −0.447728 0.894170i \(-0.647767\pi\)
−0.447728 + 0.894170i \(0.647767\pi\)
\(938\) 0 0
\(939\) 69.3818 2.26419
\(940\) −31.3941 −1.02396
\(941\) 40.2446 1.31194 0.655968 0.754789i \(-0.272261\pi\)
0.655968 + 0.754789i \(0.272261\pi\)
\(942\) −0.0697234 −0.00227171
\(943\) 63.0583 2.05346
\(944\) −6.61032 −0.215147
\(945\) 0 0
\(946\) 5.91637 0.192358
\(947\) 47.3761 1.53952 0.769758 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(948\) −47.4553 −1.54128
\(949\) −22.1949 −0.720478
\(950\) −0.201103 −0.00652463
\(951\) 10.2445 0.332200
\(952\) 0 0
\(953\) −47.8245 −1.54919 −0.774593 0.632460i \(-0.782045\pi\)
−0.774593 + 0.632460i \(0.782045\pi\)
\(954\) −56.6376 −1.83371
\(955\) 11.9314 0.386091
\(956\) −51.5446 −1.66707
\(957\) 0.893506 0.0288829
\(958\) 82.6537 2.67042
\(959\) 0 0
\(960\) 37.7681 1.21896
\(961\) 18.1536 0.585599
\(962\) 6.39690 0.206244
\(963\) −108.342 −3.49127
\(964\) −36.5260 −1.17642
\(965\) −6.85826 −0.220775
\(966\) 0 0
\(967\) −0.607859 −0.0195474 −0.00977371 0.999952i \(-0.503111\pi\)
−0.00977371 + 0.999952i \(0.503111\pi\)
\(968\) 2.49532 0.0802025
\(969\) −0.319115 −0.0102515
\(970\) −42.9579 −1.37929
\(971\) −23.5739 −0.756523 −0.378262 0.925699i \(-0.623478\pi\)
−0.378262 + 0.925699i \(0.623478\pi\)
\(972\) 30.0803 0.964826
\(973\) 0 0
\(974\) 10.9633 0.351286
\(975\) 15.0196 0.481013
\(976\) −2.26810 −0.0726001
\(977\) −54.3184 −1.73780 −0.868900 0.494988i \(-0.835172\pi\)
−0.868900 + 0.494988i \(0.835172\pi\)
\(978\) 161.617 5.16794
\(979\) −12.3013 −0.393150
\(980\) 0 0
\(981\) −89.8308 −2.86808
\(982\) −40.7156 −1.29929
\(983\) 20.6054 0.657209 0.328604 0.944468i \(-0.393422\pi\)
0.328604 + 0.944468i \(0.393422\pi\)
\(984\) 69.3658 2.21130
\(985\) 4.70609 0.149948
\(986\) 0.863901 0.0275122
\(987\) 0 0
\(988\) 1.43394 0.0456198
\(989\) −17.1933 −0.546715
\(990\) 12.1493 0.386131
\(991\) 34.9827 1.11126 0.555630 0.831429i \(-0.312477\pi\)
0.555630 + 0.831429i \(0.312477\pi\)
\(992\) −44.0375 −1.39819
\(993\) −63.0114 −1.99961
\(994\) 0 0
\(995\) 4.68345 0.148475
\(996\) 82.2992 2.60775
\(997\) −18.0226 −0.570783 −0.285391 0.958411i \(-0.592124\pi\)
−0.285391 + 0.958411i \(0.592124\pi\)
\(998\) −0.947461 −0.0299913
\(999\) 3.75459 0.118790
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 2695.2.a.u.1.10 10
7.6 odd 2 2695.2.a.v.1.10 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2695.2.a.u.1.10 10 1.1 even 1 trivial
2695.2.a.v.1.10 yes 10 7.6 odd 2