Properties

Label 4005.2.a.q.1.7
Level $4005$
Weight $2$
Character 4005.1
Self dual yes
Analytic conductor $31.980$
Analytic rank $1$
Dimension $9$
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] = [4005,2,Mod(1,4005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, 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("4005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4005.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: \(31.9800860095\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 6x^{7} + 31x^{6} + 13x^{5} - 75x^{4} - 17x^{3} + 52x^{2} + 11x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1335)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.89144\) of defining polynomial
Character \(\chi\) \(=\) 4005.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.891444 q^{2} -1.20533 q^{4} +1.00000 q^{5} +3.64096 q^{7} -2.85737 q^{8} +O(q^{10})\) \(q+0.891444 q^{2} -1.20533 q^{4} +1.00000 q^{5} +3.64096 q^{7} -2.85737 q^{8} +0.891444 q^{10} -4.33096 q^{11} -1.57936 q^{13} +3.24571 q^{14} -0.136530 q^{16} +4.04140 q^{17} -3.87139 q^{19} -1.20533 q^{20} -3.86080 q^{22} -8.46521 q^{23} +1.00000 q^{25} -1.40791 q^{26} -4.38855 q^{28} +4.60342 q^{29} +4.14525 q^{31} +5.59303 q^{32} +3.60268 q^{34} +3.64096 q^{35} -5.13942 q^{37} -3.45112 q^{38} -2.85737 q^{40} -5.53546 q^{41} -4.03101 q^{43} +5.22022 q^{44} -7.54626 q^{46} -2.01901 q^{47} +6.25657 q^{49} +0.891444 q^{50} +1.90364 q^{52} -7.34755 q^{53} -4.33096 q^{55} -10.4036 q^{56} +4.10369 q^{58} -4.74102 q^{59} -14.7844 q^{61} +3.69525 q^{62} +5.25893 q^{64} -1.57936 q^{65} +2.05578 q^{67} -4.87121 q^{68} +3.24571 q^{70} -9.08588 q^{71} +13.8742 q^{73} -4.58151 q^{74} +4.66629 q^{76} -15.7688 q^{77} +4.23690 q^{79} -0.136530 q^{80} -4.93455 q^{82} -17.2495 q^{83} +4.04140 q^{85} -3.59342 q^{86} +12.3751 q^{88} -1.00000 q^{89} -5.75037 q^{91} +10.2033 q^{92} -1.79984 q^{94} -3.87139 q^{95} +6.13872 q^{97} +5.57739 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} + 11 q^{4} + 9 q^{5} - 3 q^{7} - 18 q^{8} - 5 q^{10} - 4 q^{11} + 5 q^{13} + q^{14} + 15 q^{16} - 21 q^{17} - 18 q^{19} + 11 q^{20} + 6 q^{22} - 16 q^{23} + 9 q^{25} - 8 q^{26} + 6 q^{28} - 3 q^{29} - 6 q^{31} - 46 q^{32} + 12 q^{34} - 3 q^{35} + 11 q^{37} - 20 q^{38} - 18 q^{40} + q^{41} - 3 q^{43} - 38 q^{44} + 16 q^{46} - 27 q^{47} + 24 q^{49} - 5 q^{50} + 17 q^{52} - 43 q^{53} - 4 q^{55} - 5 q^{56} + 34 q^{58} - 3 q^{59} - 30 q^{61} - 36 q^{62} + 50 q^{64} + 5 q^{65} - 12 q^{67} - 64 q^{68} + q^{70} + 4 q^{71} + 26 q^{73} - 2 q^{74} - 12 q^{76} - 34 q^{77} + q^{79} + 15 q^{80} - 51 q^{82} - 24 q^{83} - 21 q^{85} - 18 q^{86} + 64 q^{88} - 9 q^{89} - 50 q^{91} - 10 q^{92} - 11 q^{94} - 18 q^{95} - 4 q^{97} - 75 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.891444 0.630346 0.315173 0.949034i \(-0.397937\pi\)
0.315173 + 0.949034i \(0.397937\pi\)
\(3\) 0 0
\(4\) −1.20533 −0.602664
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.64096 1.37615 0.688076 0.725638i \(-0.258455\pi\)
0.688076 + 0.725638i \(0.258455\pi\)
\(8\) −2.85737 −1.01023
\(9\) 0 0
\(10\) 0.891444 0.281899
\(11\) −4.33096 −1.30583 −0.652916 0.757430i \(-0.726455\pi\)
−0.652916 + 0.757430i \(0.726455\pi\)
\(12\) 0 0
\(13\) −1.57936 −0.438035 −0.219017 0.975721i \(-0.570285\pi\)
−0.219017 + 0.975721i \(0.570285\pi\)
\(14\) 3.24571 0.867452
\(15\) 0 0
\(16\) −0.136530 −0.0341324
\(17\) 4.04140 0.980183 0.490092 0.871671i \(-0.336963\pi\)
0.490092 + 0.871671i \(0.336963\pi\)
\(18\) 0 0
\(19\) −3.87139 −0.888157 −0.444078 0.895988i \(-0.646469\pi\)
−0.444078 + 0.895988i \(0.646469\pi\)
\(20\) −1.20533 −0.269519
\(21\) 0 0
\(22\) −3.86080 −0.823126
\(23\) −8.46521 −1.76512 −0.882559 0.470202i \(-0.844181\pi\)
−0.882559 + 0.470202i \(0.844181\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.40791 −0.276113
\(27\) 0 0
\(28\) −4.38855 −0.829358
\(29\) 4.60342 0.854833 0.427416 0.904055i \(-0.359424\pi\)
0.427416 + 0.904055i \(0.359424\pi\)
\(30\) 0 0
\(31\) 4.14525 0.744508 0.372254 0.928131i \(-0.378585\pi\)
0.372254 + 0.928131i \(0.378585\pi\)
\(32\) 5.59303 0.988718
\(33\) 0 0
\(34\) 3.60268 0.617855
\(35\) 3.64096 0.615434
\(36\) 0 0
\(37\) −5.13942 −0.844916 −0.422458 0.906383i \(-0.638833\pi\)
−0.422458 + 0.906383i \(0.638833\pi\)
\(38\) −3.45112 −0.559846
\(39\) 0 0
\(40\) −2.85737 −0.451790
\(41\) −5.53546 −0.864493 −0.432247 0.901755i \(-0.642279\pi\)
−0.432247 + 0.901755i \(0.642279\pi\)
\(42\) 0 0
\(43\) −4.03101 −0.614724 −0.307362 0.951593i \(-0.599446\pi\)
−0.307362 + 0.951593i \(0.599446\pi\)
\(44\) 5.22022 0.786978
\(45\) 0 0
\(46\) −7.54626 −1.11263
\(47\) −2.01901 −0.294503 −0.147252 0.989099i \(-0.547043\pi\)
−0.147252 + 0.989099i \(0.547043\pi\)
\(48\) 0 0
\(49\) 6.25657 0.893796
\(50\) 0.891444 0.126069
\(51\) 0 0
\(52\) 1.90364 0.263988
\(53\) −7.34755 −1.00926 −0.504632 0.863335i \(-0.668372\pi\)
−0.504632 + 0.863335i \(0.668372\pi\)
\(54\) 0 0
\(55\) −4.33096 −0.583986
\(56\) −10.4036 −1.39023
\(57\) 0 0
\(58\) 4.10369 0.538840
\(59\) −4.74102 −0.617228 −0.308614 0.951187i \(-0.599865\pi\)
−0.308614 + 0.951187i \(0.599865\pi\)
\(60\) 0 0
\(61\) −14.7844 −1.89295 −0.946476 0.322775i \(-0.895384\pi\)
−0.946476 + 0.322775i \(0.895384\pi\)
\(62\) 3.69525 0.469298
\(63\) 0 0
\(64\) 5.25893 0.657367
\(65\) −1.57936 −0.195895
\(66\) 0 0
\(67\) 2.05578 0.251153 0.125577 0.992084i \(-0.459922\pi\)
0.125577 + 0.992084i \(0.459922\pi\)
\(68\) −4.87121 −0.590721
\(69\) 0 0
\(70\) 3.24571 0.387937
\(71\) −9.08588 −1.07830 −0.539148 0.842211i \(-0.681254\pi\)
−0.539148 + 0.842211i \(0.681254\pi\)
\(72\) 0 0
\(73\) 13.8742 1.62385 0.811926 0.583760i \(-0.198419\pi\)
0.811926 + 0.583760i \(0.198419\pi\)
\(74\) −4.58151 −0.532589
\(75\) 0 0
\(76\) 4.66629 0.535260
\(77\) −15.7688 −1.79702
\(78\) 0 0
\(79\) 4.23690 0.476689 0.238344 0.971181i \(-0.423395\pi\)
0.238344 + 0.971181i \(0.423395\pi\)
\(80\) −0.136530 −0.0152645
\(81\) 0 0
\(82\) −4.93455 −0.544930
\(83\) −17.2495 −1.89338 −0.946688 0.322152i \(-0.895594\pi\)
−0.946688 + 0.322152i \(0.895594\pi\)
\(84\) 0 0
\(85\) 4.04140 0.438351
\(86\) −3.59342 −0.387489
\(87\) 0 0
\(88\) 12.3751 1.31919
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −5.75037 −0.602803
\(92\) 10.2033 1.06377
\(93\) 0 0
\(94\) −1.79984 −0.185639
\(95\) −3.87139 −0.397196
\(96\) 0 0
\(97\) 6.13872 0.623292 0.311646 0.950198i \(-0.399120\pi\)
0.311646 + 0.950198i \(0.399120\pi\)
\(98\) 5.57739 0.563401
\(99\) 0 0
\(100\) −1.20533 −0.120533
\(101\) 10.5014 1.04493 0.522464 0.852661i \(-0.325013\pi\)
0.522464 + 0.852661i \(0.325013\pi\)
\(102\) 0 0
\(103\) 13.2616 1.30670 0.653350 0.757056i \(-0.273363\pi\)
0.653350 + 0.757056i \(0.273363\pi\)
\(104\) 4.51281 0.442517
\(105\) 0 0
\(106\) −6.54993 −0.636185
\(107\) −14.6857 −1.41972 −0.709862 0.704341i \(-0.751243\pi\)
−0.709862 + 0.704341i \(0.751243\pi\)
\(108\) 0 0
\(109\) −7.63418 −0.731222 −0.365611 0.930768i \(-0.619140\pi\)
−0.365611 + 0.930768i \(0.619140\pi\)
\(110\) −3.86080 −0.368113
\(111\) 0 0
\(112\) −0.497099 −0.0469714
\(113\) 15.0328 1.41416 0.707082 0.707131i \(-0.250011\pi\)
0.707082 + 0.707131i \(0.250011\pi\)
\(114\) 0 0
\(115\) −8.46521 −0.789384
\(116\) −5.54862 −0.515177
\(117\) 0 0
\(118\) −4.22635 −0.389067
\(119\) 14.7146 1.34888
\(120\) 0 0
\(121\) 7.75718 0.705198
\(122\) −13.1795 −1.19321
\(123\) 0 0
\(124\) −4.99638 −0.448688
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 10.6036 0.940916 0.470458 0.882422i \(-0.344089\pi\)
0.470458 + 0.882422i \(0.344089\pi\)
\(128\) −6.49802 −0.574349
\(129\) 0 0
\(130\) −1.40791 −0.123482
\(131\) −10.0863 −0.881240 −0.440620 0.897694i \(-0.645241\pi\)
−0.440620 + 0.897694i \(0.645241\pi\)
\(132\) 0 0
\(133\) −14.0956 −1.22224
\(134\) 1.83261 0.158313
\(135\) 0 0
\(136\) −11.5478 −0.990213
\(137\) 8.64359 0.738472 0.369236 0.929336i \(-0.379619\pi\)
0.369236 + 0.929336i \(0.379619\pi\)
\(138\) 0 0
\(139\) −17.6476 −1.49685 −0.748423 0.663221i \(-0.769189\pi\)
−0.748423 + 0.663221i \(0.769189\pi\)
\(140\) −4.38855 −0.370900
\(141\) 0 0
\(142\) −8.09955 −0.679699
\(143\) 6.84012 0.572000
\(144\) 0 0
\(145\) 4.60342 0.382293
\(146\) 12.3681 1.02359
\(147\) 0 0
\(148\) 6.19469 0.509200
\(149\) −8.05076 −0.659544 −0.329772 0.944061i \(-0.606972\pi\)
−0.329772 + 0.944061i \(0.606972\pi\)
\(150\) 0 0
\(151\) 8.17384 0.665178 0.332589 0.943072i \(-0.392078\pi\)
0.332589 + 0.943072i \(0.392078\pi\)
\(152\) 11.0620 0.897245
\(153\) 0 0
\(154\) −14.0570 −1.13275
\(155\) 4.14525 0.332954
\(156\) 0 0
\(157\) 19.0147 1.51754 0.758770 0.651359i \(-0.225801\pi\)
0.758770 + 0.651359i \(0.225801\pi\)
\(158\) 3.77696 0.300479
\(159\) 0 0
\(160\) 5.59303 0.442168
\(161\) −30.8215 −2.42907
\(162\) 0 0
\(163\) −7.89684 −0.618529 −0.309264 0.950976i \(-0.600083\pi\)
−0.309264 + 0.950976i \(0.600083\pi\)
\(164\) 6.67204 0.520999
\(165\) 0 0
\(166\) −15.3769 −1.19348
\(167\) 5.95886 0.461110 0.230555 0.973059i \(-0.425946\pi\)
0.230555 + 0.973059i \(0.425946\pi\)
\(168\) 0 0
\(169\) −10.5056 −0.808126
\(170\) 3.60268 0.276313
\(171\) 0 0
\(172\) 4.85869 0.370472
\(173\) −17.0408 −1.29559 −0.647794 0.761816i \(-0.724308\pi\)
−0.647794 + 0.761816i \(0.724308\pi\)
\(174\) 0 0
\(175\) 3.64096 0.275231
\(176\) 0.591304 0.0445712
\(177\) 0 0
\(178\) −0.891444 −0.0668165
\(179\) −15.6647 −1.17084 −0.585419 0.810731i \(-0.699070\pi\)
−0.585419 + 0.810731i \(0.699070\pi\)
\(180\) 0 0
\(181\) −10.9347 −0.812771 −0.406385 0.913702i \(-0.633211\pi\)
−0.406385 + 0.913702i \(0.633211\pi\)
\(182\) −5.12613 −0.379974
\(183\) 0 0
\(184\) 24.1882 1.78318
\(185\) −5.13942 −0.377858
\(186\) 0 0
\(187\) −17.5031 −1.27995
\(188\) 2.43357 0.177487
\(189\) 0 0
\(190\) −3.45112 −0.250371
\(191\) −11.6720 −0.844556 −0.422278 0.906466i \(-0.638769\pi\)
−0.422278 + 0.906466i \(0.638769\pi\)
\(192\) 0 0
\(193\) −4.60334 −0.331356 −0.165678 0.986180i \(-0.552981\pi\)
−0.165678 + 0.986180i \(0.552981\pi\)
\(194\) 5.47232 0.392890
\(195\) 0 0
\(196\) −7.54122 −0.538659
\(197\) −21.3603 −1.52186 −0.760929 0.648835i \(-0.775257\pi\)
−0.760929 + 0.648835i \(0.775257\pi\)
\(198\) 0 0
\(199\) −20.2630 −1.43640 −0.718202 0.695835i \(-0.755035\pi\)
−0.718202 + 0.695835i \(0.755035\pi\)
\(200\) −2.85737 −0.202047
\(201\) 0 0
\(202\) 9.36141 0.658666
\(203\) 16.7608 1.17638
\(204\) 0 0
\(205\) −5.53546 −0.386613
\(206\) 11.8219 0.823674
\(207\) 0 0
\(208\) 0.215629 0.0149512
\(209\) 16.7668 1.15978
\(210\) 0 0
\(211\) −15.2602 −1.05055 −0.525276 0.850932i \(-0.676038\pi\)
−0.525276 + 0.850932i \(0.676038\pi\)
\(212\) 8.85620 0.608247
\(213\) 0 0
\(214\) −13.0915 −0.894917
\(215\) −4.03101 −0.274913
\(216\) 0 0
\(217\) 15.0927 1.02456
\(218\) −6.80545 −0.460923
\(219\) 0 0
\(220\) 5.22022 0.351947
\(221\) −6.38281 −0.429354
\(222\) 0 0
\(223\) −8.24096 −0.551856 −0.275928 0.961178i \(-0.588985\pi\)
−0.275928 + 0.961178i \(0.588985\pi\)
\(224\) 20.3640 1.36063
\(225\) 0 0
\(226\) 13.4009 0.891413
\(227\) 6.21752 0.412672 0.206336 0.978481i \(-0.433846\pi\)
0.206336 + 0.978481i \(0.433846\pi\)
\(228\) 0 0
\(229\) 12.9557 0.856134 0.428067 0.903747i \(-0.359195\pi\)
0.428067 + 0.903747i \(0.359195\pi\)
\(230\) −7.54626 −0.497585
\(231\) 0 0
\(232\) −13.1537 −0.863580
\(233\) −3.25893 −0.213500 −0.106750 0.994286i \(-0.534044\pi\)
−0.106750 + 0.994286i \(0.534044\pi\)
\(234\) 0 0
\(235\) −2.01901 −0.131706
\(236\) 5.71448 0.371981
\(237\) 0 0
\(238\) 13.1172 0.850262
\(239\) −15.9708 −1.03307 −0.516534 0.856267i \(-0.672778\pi\)
−0.516534 + 0.856267i \(0.672778\pi\)
\(240\) 0 0
\(241\) 9.32756 0.600841 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(242\) 6.91509 0.444519
\(243\) 0 0
\(244\) 17.8201 1.14081
\(245\) 6.25657 0.399718
\(246\) 0 0
\(247\) 6.11430 0.389044
\(248\) −11.8445 −0.752127
\(249\) 0 0
\(250\) 0.891444 0.0563799
\(251\) 13.7818 0.869899 0.434949 0.900455i \(-0.356766\pi\)
0.434949 + 0.900455i \(0.356766\pi\)
\(252\) 0 0
\(253\) 36.6624 2.30495
\(254\) 9.45250 0.593103
\(255\) 0 0
\(256\) −16.3105 −1.01941
\(257\) −22.7831 −1.42117 −0.710584 0.703612i \(-0.751569\pi\)
−0.710584 + 0.703612i \(0.751569\pi\)
\(258\) 0 0
\(259\) −18.7124 −1.16273
\(260\) 1.90364 0.118059
\(261\) 0 0
\(262\) −8.99133 −0.555486
\(263\) 0.550629 0.0339532 0.0169766 0.999856i \(-0.494596\pi\)
0.0169766 + 0.999856i \(0.494596\pi\)
\(264\) 0 0
\(265\) −7.34755 −0.451356
\(266\) −12.5654 −0.770434
\(267\) 0 0
\(268\) −2.47789 −0.151361
\(269\) 11.5809 0.706098 0.353049 0.935605i \(-0.385145\pi\)
0.353049 + 0.935605i \(0.385145\pi\)
\(270\) 0 0
\(271\) 21.1026 1.28189 0.640946 0.767586i \(-0.278542\pi\)
0.640946 + 0.767586i \(0.278542\pi\)
\(272\) −0.551771 −0.0334560
\(273\) 0 0
\(274\) 7.70528 0.465493
\(275\) −4.33096 −0.261166
\(276\) 0 0
\(277\) 17.4880 1.05076 0.525378 0.850869i \(-0.323924\pi\)
0.525378 + 0.850869i \(0.323924\pi\)
\(278\) −15.7318 −0.943531
\(279\) 0 0
\(280\) −10.4036 −0.621732
\(281\) 5.55849 0.331592 0.165796 0.986160i \(-0.446981\pi\)
0.165796 + 0.986160i \(0.446981\pi\)
\(282\) 0 0
\(283\) 12.6228 0.750350 0.375175 0.926954i \(-0.377583\pi\)
0.375175 + 0.926954i \(0.377583\pi\)
\(284\) 10.9515 0.649850
\(285\) 0 0
\(286\) 6.09759 0.360558
\(287\) −20.1544 −1.18967
\(288\) 0 0
\(289\) −0.667098 −0.0392411
\(290\) 4.10369 0.240977
\(291\) 0 0
\(292\) −16.7230 −0.978637
\(293\) −2.15302 −0.125781 −0.0628904 0.998020i \(-0.520032\pi\)
−0.0628904 + 0.998020i \(0.520032\pi\)
\(294\) 0 0
\(295\) −4.74102 −0.276033
\(296\) 14.6852 0.853562
\(297\) 0 0
\(298\) −7.17680 −0.415741
\(299\) 13.3696 0.773183
\(300\) 0 0
\(301\) −14.6767 −0.845953
\(302\) 7.28652 0.419292
\(303\) 0 0
\(304\) 0.528559 0.0303149
\(305\) −14.7844 −0.846554
\(306\) 0 0
\(307\) −19.2711 −1.09986 −0.549931 0.835210i \(-0.685346\pi\)
−0.549931 + 0.835210i \(0.685346\pi\)
\(308\) 19.0066 1.08300
\(309\) 0 0
\(310\) 3.69525 0.209876
\(311\) 12.8473 0.728501 0.364250 0.931301i \(-0.381325\pi\)
0.364250 + 0.931301i \(0.381325\pi\)
\(312\) 0 0
\(313\) 4.42924 0.250355 0.125178 0.992134i \(-0.460050\pi\)
0.125178 + 0.992134i \(0.460050\pi\)
\(314\) 16.9506 0.956575
\(315\) 0 0
\(316\) −5.10686 −0.287283
\(317\) −21.0082 −1.17994 −0.589970 0.807425i \(-0.700860\pi\)
−0.589970 + 0.807425i \(0.700860\pi\)
\(318\) 0 0
\(319\) −19.9372 −1.11627
\(320\) 5.25893 0.293983
\(321\) 0 0
\(322\) −27.4756 −1.53116
\(323\) −15.6458 −0.870556
\(324\) 0 0
\(325\) −1.57936 −0.0876069
\(326\) −7.03959 −0.389887
\(327\) 0 0
\(328\) 15.8168 0.873339
\(329\) −7.35114 −0.405282
\(330\) 0 0
\(331\) 20.8474 1.14587 0.572937 0.819599i \(-0.305804\pi\)
0.572937 + 0.819599i \(0.305804\pi\)
\(332\) 20.7913 1.14107
\(333\) 0 0
\(334\) 5.31199 0.290659
\(335\) 2.05578 0.112319
\(336\) 0 0
\(337\) 21.9591 1.19619 0.598094 0.801426i \(-0.295925\pi\)
0.598094 + 0.801426i \(0.295925\pi\)
\(338\) −9.36518 −0.509399
\(339\) 0 0
\(340\) −4.87121 −0.264178
\(341\) −17.9529 −0.972203
\(342\) 0 0
\(343\) −2.70678 −0.146152
\(344\) 11.5181 0.621014
\(345\) 0 0
\(346\) −15.1909 −0.816669
\(347\) −34.5428 −1.85435 −0.927176 0.374626i \(-0.877771\pi\)
−0.927176 + 0.374626i \(0.877771\pi\)
\(348\) 0 0
\(349\) −7.28304 −0.389852 −0.194926 0.980818i \(-0.562447\pi\)
−0.194926 + 0.980818i \(0.562447\pi\)
\(350\) 3.24571 0.173490
\(351\) 0 0
\(352\) −24.2232 −1.29110
\(353\) 13.1365 0.699184 0.349592 0.936902i \(-0.386320\pi\)
0.349592 + 0.936902i \(0.386320\pi\)
\(354\) 0 0
\(355\) −9.08588 −0.482228
\(356\) 1.20533 0.0638822
\(357\) 0 0
\(358\) −13.9642 −0.738033
\(359\) 5.40281 0.285149 0.142575 0.989784i \(-0.454462\pi\)
0.142575 + 0.989784i \(0.454462\pi\)
\(360\) 0 0
\(361\) −4.01237 −0.211177
\(362\) −9.74768 −0.512327
\(363\) 0 0
\(364\) 6.93108 0.363287
\(365\) 13.8742 0.726209
\(366\) 0 0
\(367\) −18.3751 −0.959175 −0.479587 0.877494i \(-0.659214\pi\)
−0.479587 + 0.877494i \(0.659214\pi\)
\(368\) 1.15575 0.0602477
\(369\) 0 0
\(370\) −4.58151 −0.238181
\(371\) −26.7521 −1.38890
\(372\) 0 0
\(373\) 7.80100 0.403921 0.201960 0.979394i \(-0.435269\pi\)
0.201960 + 0.979394i \(0.435269\pi\)
\(374\) −15.6030 −0.806814
\(375\) 0 0
\(376\) 5.76907 0.297517
\(377\) −7.27043 −0.374446
\(378\) 0 0
\(379\) 2.27380 0.116797 0.0583986 0.998293i \(-0.481401\pi\)
0.0583986 + 0.998293i \(0.481401\pi\)
\(380\) 4.66629 0.239376
\(381\) 0 0
\(382\) −10.4049 −0.532362
\(383\) 37.6645 1.92457 0.962284 0.272047i \(-0.0877006\pi\)
0.962284 + 0.272047i \(0.0877006\pi\)
\(384\) 0 0
\(385\) −15.7688 −0.803654
\(386\) −4.10362 −0.208869
\(387\) 0 0
\(388\) −7.39916 −0.375636
\(389\) 17.0462 0.864276 0.432138 0.901807i \(-0.357759\pi\)
0.432138 + 0.901807i \(0.357759\pi\)
\(390\) 0 0
\(391\) −34.2113 −1.73014
\(392\) −17.8773 −0.902942
\(393\) 0 0
\(394\) −19.0415 −0.959298
\(395\) 4.23690 0.213182
\(396\) 0 0
\(397\) 14.9629 0.750968 0.375484 0.926829i \(-0.377476\pi\)
0.375484 + 0.926829i \(0.377476\pi\)
\(398\) −18.0633 −0.905431
\(399\) 0 0
\(400\) −0.136530 −0.00682648
\(401\) −4.52568 −0.226002 −0.113001 0.993595i \(-0.536046\pi\)
−0.113001 + 0.993595i \(0.536046\pi\)
\(402\) 0 0
\(403\) −6.54682 −0.326120
\(404\) −12.6576 −0.629740
\(405\) 0 0
\(406\) 14.9414 0.741527
\(407\) 22.2586 1.10332
\(408\) 0 0
\(409\) 15.5609 0.769435 0.384718 0.923034i \(-0.374299\pi\)
0.384718 + 0.923034i \(0.374299\pi\)
\(410\) −4.93455 −0.243700
\(411\) 0 0
\(412\) −15.9845 −0.787501
\(413\) −17.2618 −0.849400
\(414\) 0 0
\(415\) −17.2495 −0.846743
\(416\) −8.83339 −0.433093
\(417\) 0 0
\(418\) 14.9467 0.731065
\(419\) 23.2741 1.13701 0.568507 0.822678i \(-0.307521\pi\)
0.568507 + 0.822678i \(0.307521\pi\)
\(420\) 0 0
\(421\) 1.37894 0.0672053 0.0336026 0.999435i \(-0.489302\pi\)
0.0336026 + 0.999435i \(0.489302\pi\)
\(422\) −13.6036 −0.662211
\(423\) 0 0
\(424\) 20.9947 1.01959
\(425\) 4.04140 0.196037
\(426\) 0 0
\(427\) −53.8295 −2.60499
\(428\) 17.7011 0.855616
\(429\) 0 0
\(430\) −3.59342 −0.173290
\(431\) −33.3212 −1.60503 −0.802513 0.596635i \(-0.796504\pi\)
−0.802513 + 0.596635i \(0.796504\pi\)
\(432\) 0 0
\(433\) −4.59832 −0.220981 −0.110491 0.993877i \(-0.535242\pi\)
−0.110491 + 0.993877i \(0.535242\pi\)
\(434\) 13.4543 0.645825
\(435\) 0 0
\(436\) 9.20169 0.440681
\(437\) 32.7721 1.56770
\(438\) 0 0
\(439\) 4.93623 0.235593 0.117797 0.993038i \(-0.462417\pi\)
0.117797 + 0.993038i \(0.462417\pi\)
\(440\) 12.3751 0.589962
\(441\) 0 0
\(442\) −5.68992 −0.270642
\(443\) 7.06699 0.335763 0.167881 0.985807i \(-0.446307\pi\)
0.167881 + 0.985807i \(0.446307\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) −7.34636 −0.347860
\(447\) 0 0
\(448\) 19.1476 0.904637
\(449\) −17.4236 −0.822270 −0.411135 0.911574i \(-0.634868\pi\)
−0.411135 + 0.911574i \(0.634868\pi\)
\(450\) 0 0
\(451\) 23.9738 1.12888
\(452\) −18.1194 −0.852266
\(453\) 0 0
\(454\) 5.54257 0.260126
\(455\) −5.75037 −0.269582
\(456\) 0 0
\(457\) −32.3360 −1.51262 −0.756308 0.654216i \(-0.772999\pi\)
−0.756308 + 0.654216i \(0.772999\pi\)
\(458\) 11.5492 0.539661
\(459\) 0 0
\(460\) 10.2033 0.475733
\(461\) 18.8138 0.876247 0.438124 0.898915i \(-0.355643\pi\)
0.438124 + 0.898915i \(0.355643\pi\)
\(462\) 0 0
\(463\) −2.66483 −0.123845 −0.0619225 0.998081i \(-0.519723\pi\)
−0.0619225 + 0.998081i \(0.519723\pi\)
\(464\) −0.628503 −0.0291775
\(465\) 0 0
\(466\) −2.90515 −0.134579
\(467\) 36.0626 1.66878 0.834388 0.551177i \(-0.185821\pi\)
0.834388 + 0.551177i \(0.185821\pi\)
\(468\) 0 0
\(469\) 7.48500 0.345625
\(470\) −1.79984 −0.0830203
\(471\) 0 0
\(472\) 13.5468 0.623544
\(473\) 17.4581 0.802726
\(474\) 0 0
\(475\) −3.87139 −0.177631
\(476\) −17.7359 −0.812922
\(477\) 0 0
\(478\) −14.2371 −0.651190
\(479\) −13.3114 −0.608214 −0.304107 0.952638i \(-0.598358\pi\)
−0.304107 + 0.952638i \(0.598358\pi\)
\(480\) 0 0
\(481\) 8.11698 0.370103
\(482\) 8.31500 0.378738
\(483\) 0 0
\(484\) −9.34994 −0.424997
\(485\) 6.13872 0.278745
\(486\) 0 0
\(487\) 36.8868 1.67150 0.835750 0.549111i \(-0.185033\pi\)
0.835750 + 0.549111i \(0.185033\pi\)
\(488\) 42.2446 1.91232
\(489\) 0 0
\(490\) 5.57739 0.251961
\(491\) 36.6260 1.65291 0.826454 0.563004i \(-0.190354\pi\)
0.826454 + 0.563004i \(0.190354\pi\)
\(492\) 0 0
\(493\) 18.6042 0.837893
\(494\) 5.45055 0.245232
\(495\) 0 0
\(496\) −0.565949 −0.0254119
\(497\) −33.0813 −1.48390
\(498\) 0 0
\(499\) 43.5384 1.94905 0.974523 0.224288i \(-0.0720057\pi\)
0.974523 + 0.224288i \(0.0720057\pi\)
\(500\) −1.20533 −0.0539039
\(501\) 0 0
\(502\) 12.2857 0.548337
\(503\) 33.6446 1.50014 0.750070 0.661358i \(-0.230020\pi\)
0.750070 + 0.661358i \(0.230020\pi\)
\(504\) 0 0
\(505\) 10.5014 0.467306
\(506\) 32.6825 1.45291
\(507\) 0 0
\(508\) −12.7808 −0.567056
\(509\) 25.4064 1.12612 0.563058 0.826417i \(-0.309625\pi\)
0.563058 + 0.826417i \(0.309625\pi\)
\(510\) 0 0
\(511\) 50.5154 2.23467
\(512\) −1.54385 −0.0682290
\(513\) 0 0
\(514\) −20.3098 −0.895828
\(515\) 13.2616 0.584374
\(516\) 0 0
\(517\) 8.74426 0.384572
\(518\) −16.6811 −0.732924
\(519\) 0 0
\(520\) 4.51281 0.197900
\(521\) 39.1230 1.71401 0.857006 0.515307i \(-0.172322\pi\)
0.857006 + 0.515307i \(0.172322\pi\)
\(522\) 0 0
\(523\) −10.5640 −0.461929 −0.230965 0.972962i \(-0.574188\pi\)
−0.230965 + 0.972962i \(0.574188\pi\)
\(524\) 12.1572 0.531092
\(525\) 0 0
\(526\) 0.490855 0.0214023
\(527\) 16.7526 0.729754
\(528\) 0 0
\(529\) 48.6597 2.11564
\(530\) −6.54993 −0.284511
\(531\) 0 0
\(532\) 16.9898 0.736600
\(533\) 8.74246 0.378678
\(534\) 0 0
\(535\) −14.6857 −0.634920
\(536\) −5.87412 −0.253723
\(537\) 0 0
\(538\) 10.3237 0.445086
\(539\) −27.0969 −1.16715
\(540\) 0 0
\(541\) −12.3516 −0.531037 −0.265519 0.964106i \(-0.585543\pi\)
−0.265519 + 0.964106i \(0.585543\pi\)
\(542\) 18.8118 0.808036
\(543\) 0 0
\(544\) 22.6037 0.969124
\(545\) −7.63418 −0.327013
\(546\) 0 0
\(547\) 26.7753 1.14483 0.572414 0.819965i \(-0.306007\pi\)
0.572414 + 0.819965i \(0.306007\pi\)
\(548\) −10.4184 −0.445050
\(549\) 0 0
\(550\) −3.86080 −0.164625
\(551\) −17.8216 −0.759226
\(552\) 0 0
\(553\) 15.4264 0.655997
\(554\) 15.5896 0.662339
\(555\) 0 0
\(556\) 21.2711 0.902095
\(557\) −28.8270 −1.22144 −0.610719 0.791847i \(-0.709119\pi\)
−0.610719 + 0.791847i \(0.709119\pi\)
\(558\) 0 0
\(559\) 6.36640 0.269270
\(560\) −0.497099 −0.0210062
\(561\) 0 0
\(562\) 4.95508 0.209018
\(563\) −4.36594 −0.184002 −0.0920012 0.995759i \(-0.529326\pi\)
−0.0920012 + 0.995759i \(0.529326\pi\)
\(564\) 0 0
\(565\) 15.0328 0.632433
\(566\) 11.2526 0.472980
\(567\) 0 0
\(568\) 25.9617 1.08933
\(569\) −6.39205 −0.267969 −0.133984 0.990983i \(-0.542777\pi\)
−0.133984 + 0.990983i \(0.542777\pi\)
\(570\) 0 0
\(571\) 36.7859 1.53944 0.769722 0.638380i \(-0.220395\pi\)
0.769722 + 0.638380i \(0.220395\pi\)
\(572\) −8.24459 −0.344724
\(573\) 0 0
\(574\) −17.9665 −0.749907
\(575\) −8.46521 −0.353023
\(576\) 0 0
\(577\) −20.3844 −0.848612 −0.424306 0.905519i \(-0.639482\pi\)
−0.424306 + 0.905519i \(0.639482\pi\)
\(578\) −0.594681 −0.0247355
\(579\) 0 0
\(580\) −5.54862 −0.230394
\(581\) −62.8046 −2.60557
\(582\) 0 0
\(583\) 31.8219 1.31793
\(584\) −39.6437 −1.64047
\(585\) 0 0
\(586\) −1.91930 −0.0792854
\(587\) 32.3272 1.33429 0.667143 0.744930i \(-0.267517\pi\)
0.667143 + 0.744930i \(0.267517\pi\)
\(588\) 0 0
\(589\) −16.0478 −0.661240
\(590\) −4.22635 −0.173996
\(591\) 0 0
\(592\) 0.701683 0.0288390
\(593\) 0.138120 0.00567190 0.00283595 0.999996i \(-0.499097\pi\)
0.00283595 + 0.999996i \(0.499097\pi\)
\(594\) 0 0
\(595\) 14.7146 0.603238
\(596\) 9.70380 0.397483
\(597\) 0 0
\(598\) 11.9182 0.487373
\(599\) −1.46178 −0.0597269 −0.0298634 0.999554i \(-0.509507\pi\)
−0.0298634 + 0.999554i \(0.509507\pi\)
\(600\) 0 0
\(601\) −26.7601 −1.09157 −0.545784 0.837926i \(-0.683768\pi\)
−0.545784 + 0.837926i \(0.683768\pi\)
\(602\) −13.0835 −0.533243
\(603\) 0 0
\(604\) −9.85216 −0.400879
\(605\) 7.75718 0.315374
\(606\) 0 0
\(607\) −5.30554 −0.215345 −0.107673 0.994186i \(-0.534340\pi\)
−0.107673 + 0.994186i \(0.534340\pi\)
\(608\) −21.6528 −0.878136
\(609\) 0 0
\(610\) −13.1795 −0.533622
\(611\) 3.18874 0.129003
\(612\) 0 0
\(613\) 30.0385 1.21324 0.606621 0.794991i \(-0.292525\pi\)
0.606621 + 0.794991i \(0.292525\pi\)
\(614\) −17.1791 −0.693294
\(615\) 0 0
\(616\) 45.0574 1.81541
\(617\) −3.19073 −0.128454 −0.0642269 0.997935i \(-0.520458\pi\)
−0.0642269 + 0.997935i \(0.520458\pi\)
\(618\) 0 0
\(619\) −0.490182 −0.0197021 −0.00985105 0.999951i \(-0.503136\pi\)
−0.00985105 + 0.999951i \(0.503136\pi\)
\(620\) −4.99638 −0.200659
\(621\) 0 0
\(622\) 11.4526 0.459208
\(623\) −3.64096 −0.145872
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 3.94842 0.157810
\(627\) 0 0
\(628\) −22.9190 −0.914566
\(629\) −20.7705 −0.828172
\(630\) 0 0
\(631\) −38.7287 −1.54176 −0.770882 0.636977i \(-0.780184\pi\)
−0.770882 + 0.636977i \(0.780184\pi\)
\(632\) −12.1064 −0.481567
\(633\) 0 0
\(634\) −18.7277 −0.743770
\(635\) 10.6036 0.420790
\(636\) 0 0
\(637\) −9.88136 −0.391514
\(638\) −17.7729 −0.703635
\(639\) 0 0
\(640\) −6.49802 −0.256857
\(641\) −48.2249 −1.90477 −0.952385 0.304897i \(-0.901378\pi\)
−0.952385 + 0.304897i \(0.901378\pi\)
\(642\) 0 0
\(643\) 29.9571 1.18139 0.590696 0.806894i \(-0.298853\pi\)
0.590696 + 0.806894i \(0.298853\pi\)
\(644\) 37.1500 1.46391
\(645\) 0 0
\(646\) −13.9474 −0.548752
\(647\) 21.9578 0.863250 0.431625 0.902053i \(-0.357940\pi\)
0.431625 + 0.902053i \(0.357940\pi\)
\(648\) 0 0
\(649\) 20.5331 0.805996
\(650\) −1.40791 −0.0552227
\(651\) 0 0
\(652\) 9.51828 0.372765
\(653\) 45.7267 1.78943 0.894713 0.446642i \(-0.147380\pi\)
0.894713 + 0.446642i \(0.147380\pi\)
\(654\) 0 0
\(655\) −10.0863 −0.394103
\(656\) 0.755754 0.0295072
\(657\) 0 0
\(658\) −6.55313 −0.255468
\(659\) 48.1741 1.87660 0.938298 0.345828i \(-0.112402\pi\)
0.938298 + 0.345828i \(0.112402\pi\)
\(660\) 0 0
\(661\) 7.24751 0.281895 0.140948 0.990017i \(-0.454985\pi\)
0.140948 + 0.990017i \(0.454985\pi\)
\(662\) 18.5842 0.722297
\(663\) 0 0
\(664\) 49.2881 1.91275
\(665\) −14.0956 −0.546602
\(666\) 0 0
\(667\) −38.9689 −1.50888
\(668\) −7.18237 −0.277894
\(669\) 0 0
\(670\) 1.83261 0.0707999
\(671\) 64.0307 2.47188
\(672\) 0 0
\(673\) −17.9965 −0.693715 −0.346858 0.937918i \(-0.612751\pi\)
−0.346858 + 0.937918i \(0.612751\pi\)
\(674\) 19.5753 0.754013
\(675\) 0 0
\(676\) 12.6627 0.487028
\(677\) 9.42242 0.362133 0.181067 0.983471i \(-0.442045\pi\)
0.181067 + 0.983471i \(0.442045\pi\)
\(678\) 0 0
\(679\) 22.3508 0.857745
\(680\) −11.5478 −0.442837
\(681\) 0 0
\(682\) −16.0040 −0.612824
\(683\) −34.3152 −1.31304 −0.656518 0.754310i \(-0.727971\pi\)
−0.656518 + 0.754310i \(0.727971\pi\)
\(684\) 0 0
\(685\) 8.64359 0.330255
\(686\) −2.41294 −0.0921266
\(687\) 0 0
\(688\) 0.550352 0.0209820
\(689\) 11.6044 0.442092
\(690\) 0 0
\(691\) 18.6125 0.708051 0.354026 0.935236i \(-0.384813\pi\)
0.354026 + 0.935236i \(0.384813\pi\)
\(692\) 20.5397 0.780804
\(693\) 0 0
\(694\) −30.7929 −1.16888
\(695\) −17.6476 −0.669410
\(696\) 0 0
\(697\) −22.3710 −0.847362
\(698\) −6.49242 −0.245742
\(699\) 0 0
\(700\) −4.38855 −0.165872
\(701\) −18.4560 −0.697073 −0.348537 0.937295i \(-0.613321\pi\)
−0.348537 + 0.937295i \(0.613321\pi\)
\(702\) 0 0
\(703\) 19.8967 0.750418
\(704\) −22.7762 −0.858411
\(705\) 0 0
\(706\) 11.7104 0.440728
\(707\) 38.2351 1.43798
\(708\) 0 0
\(709\) −8.29517 −0.311532 −0.155766 0.987794i \(-0.549785\pi\)
−0.155766 + 0.987794i \(0.549785\pi\)
\(710\) −8.09955 −0.303971
\(711\) 0 0
\(712\) 2.85737 0.107084
\(713\) −35.0904 −1.31414
\(714\) 0 0
\(715\) 6.84012 0.255806
\(716\) 18.8811 0.705621
\(717\) 0 0
\(718\) 4.81630 0.179743
\(719\) −6.54967 −0.244261 −0.122131 0.992514i \(-0.538973\pi\)
−0.122131 + 0.992514i \(0.538973\pi\)
\(720\) 0 0
\(721\) 48.2848 1.79822
\(722\) −3.57680 −0.133115
\(723\) 0 0
\(724\) 13.1799 0.489828
\(725\) 4.60342 0.170967
\(726\) 0 0
\(727\) −43.9969 −1.63176 −0.815878 0.578224i \(-0.803746\pi\)
−0.815878 + 0.578224i \(0.803746\pi\)
\(728\) 16.4309 0.608971
\(729\) 0 0
\(730\) 12.3681 0.457763
\(731\) −16.2909 −0.602542
\(732\) 0 0
\(733\) 16.3217 0.602857 0.301429 0.953489i \(-0.402536\pi\)
0.301429 + 0.953489i \(0.402536\pi\)
\(734\) −16.3804 −0.604612
\(735\) 0 0
\(736\) −47.3462 −1.74520
\(737\) −8.90348 −0.327964
\(738\) 0 0
\(739\) 7.56278 0.278201 0.139101 0.990278i \(-0.455579\pi\)
0.139101 + 0.990278i \(0.455579\pi\)
\(740\) 6.19469 0.227721
\(741\) 0 0
\(742\) −23.8480 −0.875488
\(743\) 22.5633 0.827767 0.413883 0.910330i \(-0.364172\pi\)
0.413883 + 0.910330i \(0.364172\pi\)
\(744\) 0 0
\(745\) −8.05076 −0.294957
\(746\) 6.95416 0.254610
\(747\) 0 0
\(748\) 21.0970 0.771383
\(749\) −53.4702 −1.95376
\(750\) 0 0
\(751\) −39.6351 −1.44631 −0.723153 0.690688i \(-0.757308\pi\)
−0.723153 + 0.690688i \(0.757308\pi\)
\(752\) 0.275655 0.0100521
\(753\) 0 0
\(754\) −6.48119 −0.236031
\(755\) 8.17384 0.297476
\(756\) 0 0
\(757\) −16.1353 −0.586449 −0.293224 0.956044i \(-0.594728\pi\)
−0.293224 + 0.956044i \(0.594728\pi\)
\(758\) 2.02696 0.0736226
\(759\) 0 0
\(760\) 11.0620 0.401260
\(761\) 11.4845 0.416314 0.208157 0.978095i \(-0.433253\pi\)
0.208157 + 0.978095i \(0.433253\pi\)
\(762\) 0 0
\(763\) −27.7957 −1.00627
\(764\) 14.0686 0.508983
\(765\) 0 0
\(766\) 33.5758 1.21314
\(767\) 7.48776 0.270367
\(768\) 0 0
\(769\) 4.05249 0.146136 0.0730682 0.997327i \(-0.476721\pi\)
0.0730682 + 0.997327i \(0.476721\pi\)
\(770\) −14.0570 −0.506580
\(771\) 0 0
\(772\) 5.54854 0.199696
\(773\) −34.6875 −1.24762 −0.623812 0.781575i \(-0.714417\pi\)
−0.623812 + 0.781575i \(0.714417\pi\)
\(774\) 0 0
\(775\) 4.14525 0.148902
\(776\) −17.5406 −0.629670
\(777\) 0 0
\(778\) 15.1957 0.544793
\(779\) 21.4299 0.767805
\(780\) 0 0
\(781\) 39.3505 1.40807
\(782\) −30.4974 −1.09059
\(783\) 0 0
\(784\) −0.854208 −0.0305074
\(785\) 19.0147 0.678664
\(786\) 0 0
\(787\) −3.48999 −0.124405 −0.0622023 0.998064i \(-0.519812\pi\)
−0.0622023 + 0.998064i \(0.519812\pi\)
\(788\) 25.7462 0.917169
\(789\) 0 0
\(790\) 3.77696 0.134378
\(791\) 54.7337 1.94611
\(792\) 0 0
\(793\) 23.3499 0.829179
\(794\) 13.3386 0.473370
\(795\) 0 0
\(796\) 24.4235 0.865669
\(797\) 39.0614 1.38363 0.691813 0.722076i \(-0.256812\pi\)
0.691813 + 0.722076i \(0.256812\pi\)
\(798\) 0 0
\(799\) −8.15964 −0.288667
\(800\) 5.59303 0.197744
\(801\) 0 0
\(802\) −4.03439 −0.142459
\(803\) −60.0886 −2.12048
\(804\) 0 0
\(805\) −30.8215 −1.08631
\(806\) −5.83612 −0.205569
\(807\) 0 0
\(808\) −30.0064 −1.05562
\(809\) 48.9260 1.72015 0.860073 0.510171i \(-0.170418\pi\)
0.860073 + 0.510171i \(0.170418\pi\)
\(810\) 0 0
\(811\) 44.2201 1.55278 0.776389 0.630254i \(-0.217049\pi\)
0.776389 + 0.630254i \(0.217049\pi\)
\(812\) −20.2023 −0.708962
\(813\) 0 0
\(814\) 19.8423 0.695472
\(815\) −7.89684 −0.276614
\(816\) 0 0
\(817\) 15.6056 0.545971
\(818\) 13.8716 0.485011
\(819\) 0 0
\(820\) 6.67204 0.232998
\(821\) 13.2757 0.463326 0.231663 0.972796i \(-0.425583\pi\)
0.231663 + 0.972796i \(0.425583\pi\)
\(822\) 0 0
\(823\) −52.8359 −1.84175 −0.920873 0.389864i \(-0.872522\pi\)
−0.920873 + 0.389864i \(0.872522\pi\)
\(824\) −37.8932 −1.32007
\(825\) 0 0
\(826\) −15.3880 −0.535416
\(827\) −42.8247 −1.48916 −0.744581 0.667532i \(-0.767351\pi\)
−0.744581 + 0.667532i \(0.767351\pi\)
\(828\) 0 0
\(829\) −5.81578 −0.201990 −0.100995 0.994887i \(-0.532203\pi\)
−0.100995 + 0.994887i \(0.532203\pi\)
\(830\) −15.3769 −0.533741
\(831\) 0 0
\(832\) −8.30573 −0.287949
\(833\) 25.2853 0.876084
\(834\) 0 0
\(835\) 5.95886 0.206215
\(836\) −20.2095 −0.698960
\(837\) 0 0
\(838\) 20.7476 0.716712
\(839\) −27.5758 −0.952023 −0.476011 0.879439i \(-0.657918\pi\)
−0.476011 + 0.879439i \(0.657918\pi\)
\(840\) 0 0
\(841\) −7.80857 −0.269261
\(842\) 1.22924 0.0423626
\(843\) 0 0
\(844\) 18.3935 0.633130
\(845\) −10.5056 −0.361405
\(846\) 0 0
\(847\) 28.2436 0.970460
\(848\) 1.00316 0.0344486
\(849\) 0 0
\(850\) 3.60268 0.123571
\(851\) 43.5063 1.49138
\(852\) 0 0
\(853\) −57.6857 −1.97512 −0.987560 0.157240i \(-0.949740\pi\)
−0.987560 + 0.157240i \(0.949740\pi\)
\(854\) −47.9860 −1.64205
\(855\) 0 0
\(856\) 41.9626 1.43425
\(857\) 38.6275 1.31949 0.659746 0.751489i \(-0.270664\pi\)
0.659746 + 0.751489i \(0.270664\pi\)
\(858\) 0 0
\(859\) −21.7778 −0.743050 −0.371525 0.928423i \(-0.621165\pi\)
−0.371525 + 0.928423i \(0.621165\pi\)
\(860\) 4.85869 0.165680
\(861\) 0 0
\(862\) −29.7040 −1.01172
\(863\) −18.4365 −0.627586 −0.313793 0.949491i \(-0.601600\pi\)
−0.313793 + 0.949491i \(0.601600\pi\)
\(864\) 0 0
\(865\) −17.0408 −0.579405
\(866\) −4.09914 −0.139295
\(867\) 0 0
\(868\) −18.1916 −0.617463
\(869\) −18.3498 −0.622476
\(870\) 0 0
\(871\) −3.24681 −0.110014
\(872\) 21.8137 0.738705
\(873\) 0 0
\(874\) 29.2145 0.988194
\(875\) 3.64096 0.123087
\(876\) 0 0
\(877\) 4.62129 0.156050 0.0780249 0.996951i \(-0.475139\pi\)
0.0780249 + 0.996951i \(0.475139\pi\)
\(878\) 4.40037 0.148505
\(879\) 0 0
\(880\) 0.591304 0.0199328
\(881\) −36.2322 −1.22069 −0.610347 0.792134i \(-0.708970\pi\)
−0.610347 + 0.792134i \(0.708970\pi\)
\(882\) 0 0
\(883\) −19.2717 −0.648545 −0.324273 0.945964i \(-0.605120\pi\)
−0.324273 + 0.945964i \(0.605120\pi\)
\(884\) 7.69338 0.258756
\(885\) 0 0
\(886\) 6.29982 0.211647
\(887\) −48.4068 −1.62534 −0.812671 0.582722i \(-0.801988\pi\)
−0.812671 + 0.582722i \(0.801988\pi\)
\(888\) 0 0
\(889\) 38.6072 1.29484
\(890\) −0.891444 −0.0298813
\(891\) 0 0
\(892\) 9.93306 0.332583
\(893\) 7.81638 0.261565
\(894\) 0 0
\(895\) −15.6647 −0.523614
\(896\) −23.6590 −0.790392
\(897\) 0 0
\(898\) −15.5322 −0.518315
\(899\) 19.0823 0.636430
\(900\) 0 0
\(901\) −29.6944 −0.989263
\(902\) 21.3713 0.711587
\(903\) 0 0
\(904\) −42.9542 −1.42864
\(905\) −10.9347 −0.363482
\(906\) 0 0
\(907\) −32.5739 −1.08160 −0.540800 0.841151i \(-0.681878\pi\)
−0.540800 + 0.841151i \(0.681878\pi\)
\(908\) −7.49415 −0.248702
\(909\) 0 0
\(910\) −5.12613 −0.169930
\(911\) −7.76551 −0.257283 −0.128641 0.991691i \(-0.541062\pi\)
−0.128641 + 0.991691i \(0.541062\pi\)
\(912\) 0 0
\(913\) 74.7067 2.47243
\(914\) −28.8258 −0.953471
\(915\) 0 0
\(916\) −15.6158 −0.515961
\(917\) −36.7236 −1.21272
\(918\) 0 0
\(919\) 15.1765 0.500627 0.250314 0.968165i \(-0.419466\pi\)
0.250314 + 0.968165i \(0.419466\pi\)
\(920\) 24.1882 0.797462
\(921\) 0 0
\(922\) 16.7715 0.552339
\(923\) 14.3498 0.472331
\(924\) 0 0
\(925\) −5.13942 −0.168983
\(926\) −2.37554 −0.0780652
\(927\) 0 0
\(928\) 25.7470 0.845188
\(929\) −6.13515 −0.201288 −0.100644 0.994923i \(-0.532090\pi\)
−0.100644 + 0.994923i \(0.532090\pi\)
\(930\) 0 0
\(931\) −24.2216 −0.793831
\(932\) 3.92808 0.128668
\(933\) 0 0
\(934\) 32.1478 1.05191
\(935\) −17.5031 −0.572413
\(936\) 0 0
\(937\) 48.9924 1.60051 0.800257 0.599658i \(-0.204697\pi\)
0.800257 + 0.599658i \(0.204697\pi\)
\(938\) 6.67246 0.217864
\(939\) 0 0
\(940\) 2.43357 0.0793744
\(941\) −17.2926 −0.563724 −0.281862 0.959455i \(-0.590952\pi\)
−0.281862 + 0.959455i \(0.590952\pi\)
\(942\) 0 0
\(943\) 46.8588 1.52593
\(944\) 0.647289 0.0210675
\(945\) 0 0
\(946\) 15.5629 0.505995
\(947\) 31.6953 1.02996 0.514980 0.857202i \(-0.327799\pi\)
0.514980 + 0.857202i \(0.327799\pi\)
\(948\) 0 0
\(949\) −21.9123 −0.711304
\(950\) −3.45112 −0.111969
\(951\) 0 0
\(952\) −42.0449 −1.36268
\(953\) 31.8797 1.03268 0.516342 0.856382i \(-0.327293\pi\)
0.516342 + 0.856382i \(0.327293\pi\)
\(954\) 0 0
\(955\) −11.6720 −0.377697
\(956\) 19.2501 0.622593
\(957\) 0 0
\(958\) −11.8664 −0.383386
\(959\) 31.4710 1.01625
\(960\) 0 0
\(961\) −13.8169 −0.445708
\(962\) 7.23583 0.233293
\(963\) 0 0
\(964\) −11.2428 −0.362105
\(965\) −4.60334 −0.148187
\(966\) 0 0
\(967\) 34.1950 1.09964 0.549818 0.835284i \(-0.314697\pi\)
0.549818 + 0.835284i \(0.314697\pi\)
\(968\) −22.1651 −0.712414
\(969\) 0 0
\(970\) 5.47232 0.175706
\(971\) 26.2314 0.841807 0.420903 0.907105i \(-0.361713\pi\)
0.420903 + 0.907105i \(0.361713\pi\)
\(972\) 0 0
\(973\) −64.2540 −2.05989
\(974\) 32.8825 1.05362
\(975\) 0 0
\(976\) 2.01851 0.0646110
\(977\) −11.3282 −0.362423 −0.181211 0.983444i \(-0.558002\pi\)
−0.181211 + 0.983444i \(0.558002\pi\)
\(978\) 0 0
\(979\) 4.33096 0.138418
\(980\) −7.54122 −0.240896
\(981\) 0 0
\(982\) 32.6500 1.04190
\(983\) −53.1093 −1.69392 −0.846962 0.531653i \(-0.821571\pi\)
−0.846962 + 0.531653i \(0.821571\pi\)
\(984\) 0 0
\(985\) −21.3603 −0.680596
\(986\) 16.5846 0.528162
\(987\) 0 0
\(988\) −7.36973 −0.234462
\(989\) 34.1233 1.08506
\(990\) 0 0
\(991\) −32.5725 −1.03470 −0.517350 0.855774i \(-0.673081\pi\)
−0.517350 + 0.855774i \(0.673081\pi\)
\(992\) 23.1845 0.736108
\(993\) 0 0
\(994\) −29.4901 −0.935370
\(995\) −20.2630 −0.642379
\(996\) 0 0
\(997\) −4.20259 −0.133098 −0.0665488 0.997783i \(-0.521199\pi\)
−0.0665488 + 0.997783i \(0.521199\pi\)
\(998\) 38.8120 1.22857
\(999\) 0 0
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 4005.2.a.q.1.7 9
3.2 odd 2 1335.2.a.h.1.3 9
15.14 odd 2 6675.2.a.x.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1335.2.a.h.1.3 9 3.2 odd 2
4005.2.a.q.1.7 9 1.1 even 1 trivial
6675.2.a.x.1.7 9 15.14 odd 2