Properties

Label 3330.2.a.bk.1.1
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
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] = [3330,2,Mod(1,3330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3330, 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("3330.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.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: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23544108.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.43959\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.23800 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.23800 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.01784 q^{11} -3.86299 q^{13} +4.23800 q^{14} +1.00000 q^{16} -6.23800 q^{17} -7.49220 q^{19} -1.00000 q^{20} +2.01784 q^{22} -5.86299 q^{23} +1.00000 q^{25} +3.86299 q^{26} -4.23800 q^{28} -0.375011 q^{29} +9.47436 q^{31} -1.00000 q^{32} +6.23800 q^{34} +4.23800 q^{35} +1.00000 q^{37} +7.49220 q^{38} +1.00000 q^{40} -6.72433 q^{41} +8.72433 q^{43} -2.01784 q^{44} +5.86299 q^{46} -11.3195 q^{47} +10.9607 q^{49} -1.00000 q^{50} -3.86299 q^{52} +0.861339 q^{53} +2.01784 q^{55} +4.23800 q^{56} +0.375011 q^{58} +4.03569 q^{59} +4.12247 q^{61} -9.47436 q^{62} +1.00000 q^{64} +3.86299 q^{65} +12.5117 q^{67} -6.23800 q^{68} -4.23800 q^{70} +14.1987 q^{71} -10.3714 q^{73} -1.00000 q^{74} -7.49220 q^{76} +8.55164 q^{77} -12.1987 q^{79} -1.00000 q^{80} +6.72433 q^{82} -1.82730 q^{83} +6.23800 q^{85} -8.72433 q^{86} +2.01784 q^{88} -13.0890 q^{89} +16.3714 q^{91} -5.86299 q^{92} +11.3195 q^{94} +7.49220 q^{95} +6.59517 q^{97} -10.9607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29} + 16 q^{31} - 5 q^{32} + 7 q^{34} - 3 q^{35} + 5 q^{37} + q^{38} + 5 q^{40} - 2 q^{41} + 12 q^{43} - 5 q^{44} + 5 q^{46} - 6 q^{47} + 16 q^{49} - 5 q^{50} + 5 q^{52} - 3 q^{53} + 5 q^{55} - 3 q^{56} + 2 q^{58} + 10 q^{59} + 16 q^{61} - 16 q^{62} + 5 q^{64} - 5 q^{65} + 4 q^{67} - 7 q^{68} + 3 q^{70} + 8 q^{71} - 3 q^{73} - 5 q^{74} - q^{76} - 3 q^{77} + 2 q^{79} - 5 q^{80} + 2 q^{82} + 5 q^{83} + 7 q^{85} - 12 q^{86} + 5 q^{88} + 7 q^{89} + 33 q^{91} - 5 q^{92} + 6 q^{94} + q^{95} + 14 q^{97} - 16 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.23800 −1.60182 −0.800908 0.598788i \(-0.795649\pi\)
−0.800908 + 0.598788i \(0.795649\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −2.01784 −0.608403 −0.304202 0.952608i \(-0.598390\pi\)
−0.304202 + 0.952608i \(0.598390\pi\)
\(12\) 0 0
\(13\) −3.86299 −1.07140 −0.535701 0.844408i \(-0.679953\pi\)
−0.535701 + 0.844408i \(0.679953\pi\)
\(14\) 4.23800 1.13265
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.23800 −1.51294 −0.756469 0.654029i \(-0.773077\pi\)
−0.756469 + 0.654029i \(0.773077\pi\)
\(18\) 0 0
\(19\) −7.49220 −1.71883 −0.859414 0.511280i \(-0.829172\pi\)
−0.859414 + 0.511280i \(0.829172\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 2.01784 0.430206
\(23\) −5.86299 −1.22252 −0.611259 0.791430i \(-0.709337\pi\)
−0.611259 + 0.791430i \(0.709337\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.86299 0.757595
\(27\) 0 0
\(28\) −4.23800 −0.800908
\(29\) −0.375011 −0.0696378 −0.0348189 0.999394i \(-0.511085\pi\)
−0.0348189 + 0.999394i \(0.511085\pi\)
\(30\) 0 0
\(31\) 9.47436 1.70164 0.850822 0.525454i \(-0.176104\pi\)
0.850822 + 0.525454i \(0.176104\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.23800 1.06981
\(35\) 4.23800 0.716354
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 7.49220 1.21540
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −6.72433 −1.05016 −0.525082 0.851052i \(-0.675965\pi\)
−0.525082 + 0.851052i \(0.675965\pi\)
\(42\) 0 0
\(43\) 8.72433 1.33045 0.665224 0.746644i \(-0.268336\pi\)
0.665224 + 0.746644i \(0.268336\pi\)
\(44\) −2.01784 −0.304202
\(45\) 0 0
\(46\) 5.86299 0.864451
\(47\) −11.3195 −1.65112 −0.825560 0.564315i \(-0.809140\pi\)
−0.825560 + 0.564315i \(0.809140\pi\)
\(48\) 0 0
\(49\) 10.9607 1.56581
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.86299 −0.535701
\(53\) 0.861339 0.118314 0.0591570 0.998249i \(-0.481159\pi\)
0.0591570 + 0.998249i \(0.481159\pi\)
\(54\) 0 0
\(55\) 2.01784 0.272086
\(56\) 4.23800 0.566327
\(57\) 0 0
\(58\) 0.375011 0.0492414
\(59\) 4.03569 0.525402 0.262701 0.964877i \(-0.415387\pi\)
0.262701 + 0.964877i \(0.415387\pi\)
\(60\) 0 0
\(61\) 4.12247 0.527828 0.263914 0.964546i \(-0.414986\pi\)
0.263914 + 0.964546i \(0.414986\pi\)
\(62\) −9.47436 −1.20324
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.86299 0.479145
\(66\) 0 0
\(67\) 12.5117 1.52855 0.764274 0.644892i \(-0.223098\pi\)
0.764274 + 0.644892i \(0.223098\pi\)
\(68\) −6.23800 −0.756469
\(69\) 0 0
\(70\) −4.23800 −0.506538
\(71\) 14.1987 1.68507 0.842537 0.538638i \(-0.181061\pi\)
0.842537 + 0.538638i \(0.181061\pi\)
\(72\) 0 0
\(73\) −10.3714 −1.21388 −0.606939 0.794748i \(-0.707603\pi\)
−0.606939 + 0.794748i \(0.707603\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −7.49220 −0.859414
\(77\) 8.55164 0.974549
\(78\) 0 0
\(79\) −12.1987 −1.37246 −0.686230 0.727385i \(-0.740736\pi\)
−0.686230 + 0.727385i \(0.740736\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 6.72433 0.742578
\(83\) −1.82730 −0.200573 −0.100286 0.994959i \(-0.531976\pi\)
−0.100286 + 0.994959i \(0.531976\pi\)
\(84\) 0 0
\(85\) 6.23800 0.676607
\(86\) −8.72433 −0.940769
\(87\) 0 0
\(88\) 2.01784 0.215103
\(89\) −13.0890 −1.38743 −0.693717 0.720248i \(-0.744028\pi\)
−0.693717 + 0.720248i \(0.744028\pi\)
\(90\) 0 0
\(91\) 16.3714 1.71619
\(92\) −5.86299 −0.611259
\(93\) 0 0
\(94\) 11.3195 1.16752
\(95\) 7.49220 0.768684
\(96\) 0 0
\(97\) 6.59517 0.669638 0.334819 0.942282i \(-0.391325\pi\)
0.334819 + 0.942282i \(0.391325\pi\)
\(98\) −10.9607 −1.10720
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.0422 −1.29775 −0.648873 0.760897i \(-0.724759\pi\)
−0.648873 + 0.760897i \(0.724759\pi\)
\(102\) 0 0
\(103\) −9.31950 −0.918278 −0.459139 0.888364i \(-0.651842\pi\)
−0.459139 + 0.888364i \(0.651842\pi\)
\(104\) 3.86299 0.378798
\(105\) 0 0
\(106\) −0.861339 −0.0836607
\(107\) −5.11297 −0.494290 −0.247145 0.968979i \(-0.579492\pi\)
−0.247145 + 0.968979i \(0.579492\pi\)
\(108\) 0 0
\(109\) 14.1808 1.35828 0.679139 0.734009i \(-0.262353\pi\)
0.679139 + 0.734009i \(0.262353\pi\)
\(110\) −2.01784 −0.192394
\(111\) 0 0
\(112\) −4.23800 −0.400454
\(113\) −3.66068 −0.344368 −0.172184 0.985065i \(-0.555082\pi\)
−0.172184 + 0.985065i \(0.555082\pi\)
\(114\) 0 0
\(115\) 5.86299 0.546727
\(116\) −0.375011 −0.0348189
\(117\) 0 0
\(118\) −4.03569 −0.371515
\(119\) 26.4367 2.42345
\(120\) 0 0
\(121\) −6.92830 −0.629846
\(122\) −4.12247 −0.373231
\(123\) 0 0
\(124\) 9.47436 0.850822
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −0.609707 −0.0541028 −0.0270514 0.999634i \(-0.508612\pi\)
−0.0270514 + 0.999634i \(0.508612\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.86299 −0.338807
\(131\) 20.9163 1.82747 0.913734 0.406313i \(-0.133186\pi\)
0.913734 + 0.406313i \(0.133186\pi\)
\(132\) 0 0
\(133\) 31.7520 2.75325
\(134\) −12.5117 −1.08085
\(135\) 0 0
\(136\) 6.23800 0.534904
\(137\) 2.87918 0.245985 0.122993 0.992408i \(-0.460751\pi\)
0.122993 + 0.992408i \(0.460751\pi\)
\(138\) 0 0
\(139\) −15.8594 −1.34517 −0.672587 0.740018i \(-0.734817\pi\)
−0.672587 + 0.740018i \(0.734817\pi\)
\(140\) 4.23800 0.358177
\(141\) 0 0
\(142\) −14.1987 −1.19153
\(143\) 7.79492 0.651844
\(144\) 0 0
\(145\) 0.375011 0.0311430
\(146\) 10.3714 0.858342
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −5.35189 −0.438444 −0.219222 0.975675i \(-0.570352\pi\)
−0.219222 + 0.975675i \(0.570352\pi\)
\(150\) 0 0
\(151\) 17.0857 1.39042 0.695208 0.718809i \(-0.255312\pi\)
0.695208 + 0.718809i \(0.255312\pi\)
\(152\) 7.49220 0.607698
\(153\) 0 0
\(154\) −8.55164 −0.689110
\(155\) −9.47436 −0.760999
\(156\) 0 0
\(157\) 1.16072 0.0926358 0.0463179 0.998927i \(-0.485251\pi\)
0.0463179 + 0.998927i \(0.485251\pi\)
\(158\) 12.1987 0.970475
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 24.8474 1.95825
\(162\) 0 0
\(163\) 9.36973 0.733894 0.366947 0.930242i \(-0.380403\pi\)
0.366947 + 0.930242i \(0.380403\pi\)
\(164\) −6.72433 −0.525082
\(165\) 0 0
\(166\) 1.82730 0.141826
\(167\) 5.86299 0.453692 0.226846 0.973931i \(-0.427159\pi\)
0.226846 + 0.973931i \(0.427159\pi\)
\(168\) 0 0
\(169\) 1.92272 0.147901
\(170\) −6.23800 −0.478433
\(171\) 0 0
\(172\) 8.72433 0.665224
\(173\) −2.36138 −0.179533 −0.0897663 0.995963i \(-0.528612\pi\)
−0.0897663 + 0.995963i \(0.528612\pi\)
\(174\) 0 0
\(175\) −4.23800 −0.320363
\(176\) −2.01784 −0.152101
\(177\) 0 0
\(178\) 13.0890 0.981064
\(179\) −13.9247 −1.04078 −0.520390 0.853929i \(-0.674213\pi\)
−0.520390 + 0.853929i \(0.674213\pi\)
\(180\) 0 0
\(181\) −2.51170 −0.186693 −0.0933466 0.995634i \(-0.529756\pi\)
−0.0933466 + 0.995634i \(0.529756\pi\)
\(182\) −16.3714 −1.21353
\(183\) 0 0
\(184\) 5.86299 0.432226
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 12.5873 0.920476
\(188\) −11.3195 −0.825560
\(189\) 0 0
\(190\) −7.49220 −0.543541
\(191\) 25.6867 1.85862 0.929311 0.369298i \(-0.120402\pi\)
0.929311 + 0.369298i \(0.120402\pi\)
\(192\) 0 0
\(193\) −6.03384 −0.434325 −0.217163 0.976135i \(-0.569680\pi\)
−0.217163 + 0.976135i \(0.569680\pi\)
\(194\) −6.59517 −0.473506
\(195\) 0 0
\(196\) 10.9607 0.782906
\(197\) −22.3747 −1.59413 −0.797065 0.603893i \(-0.793615\pi\)
−0.797065 + 0.603893i \(0.793615\pi\)
\(198\) 0 0
\(199\) −18.1663 −1.28778 −0.643888 0.765120i \(-0.722680\pi\)
−0.643888 + 0.765120i \(0.722680\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 13.0422 0.917645
\(203\) 1.58930 0.111547
\(204\) 0 0
\(205\) 6.72433 0.469648
\(206\) 9.31950 0.649321
\(207\) 0 0
\(208\) −3.86299 −0.267850
\(209\) 15.1181 1.04574
\(210\) 0 0
\(211\) −17.3594 −1.19507 −0.597536 0.801842i \(-0.703853\pi\)
−0.597536 + 0.801842i \(0.703853\pi\)
\(212\) 0.861339 0.0591570
\(213\) 0 0
\(214\) 5.11297 0.349516
\(215\) −8.72433 −0.594995
\(216\) 0 0
\(217\) −40.1524 −2.72572
\(218\) −14.1808 −0.960448
\(219\) 0 0
\(220\) 2.01784 0.136043
\(221\) 24.0974 1.62096
\(222\) 0 0
\(223\) 0.652333 0.0436834 0.0218417 0.999761i \(-0.493047\pi\)
0.0218417 + 0.999761i \(0.493047\pi\)
\(224\) 4.23800 0.283164
\(225\) 0 0
\(226\) 3.66068 0.243505
\(227\) −10.1367 −0.672796 −0.336398 0.941720i \(-0.609209\pi\)
−0.336398 + 0.941720i \(0.609209\pi\)
\(228\) 0 0
\(229\) −26.1747 −1.72968 −0.864838 0.502051i \(-0.832579\pi\)
−0.864838 + 0.502051i \(0.832579\pi\)
\(230\) −5.86299 −0.386594
\(231\) 0 0
\(232\) 0.375011 0.0246207
\(233\) −27.0812 −1.77415 −0.887074 0.461628i \(-0.847265\pi\)
−0.887074 + 0.461628i \(0.847265\pi\)
\(234\) 0 0
\(235\) 11.3195 0.738403
\(236\) 4.03569 0.262701
\(237\) 0 0
\(238\) −26.4367 −1.71364
\(239\) 12.3426 0.798378 0.399189 0.916869i \(-0.369292\pi\)
0.399189 + 0.916869i \(0.369292\pi\)
\(240\) 0 0
\(241\) 12.5000 0.805193 0.402596 0.915378i \(-0.368108\pi\)
0.402596 + 0.915378i \(0.368108\pi\)
\(242\) 6.92830 0.445368
\(243\) 0 0
\(244\) 4.12247 0.263914
\(245\) −10.9607 −0.700252
\(246\) 0 0
\(247\) 28.9423 1.84156
\(248\) −9.47436 −0.601622
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −9.41298 −0.594142 −0.297071 0.954855i \(-0.596010\pi\)
−0.297071 + 0.954855i \(0.596010\pi\)
\(252\) 0 0
\(253\) 11.8306 0.743784
\(254\) 0.609707 0.0382564
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.26431 −0.390757 −0.195379 0.980728i \(-0.562594\pi\)
−0.195379 + 0.980728i \(0.562594\pi\)
\(258\) 0 0
\(259\) −4.23800 −0.263337
\(260\) 3.86299 0.239573
\(261\) 0 0
\(262\) −20.9163 −1.29222
\(263\) 25.6392 1.58098 0.790491 0.612474i \(-0.209826\pi\)
0.790491 + 0.612474i \(0.209826\pi\)
\(264\) 0 0
\(265\) −0.861339 −0.0529117
\(266\) −31.7520 −1.94684
\(267\) 0 0
\(268\) 12.5117 0.764274
\(269\) −6.74549 −0.411280 −0.205640 0.978628i \(-0.565928\pi\)
−0.205640 + 0.978628i \(0.565928\pi\)
\(270\) 0 0
\(271\) −14.6390 −0.889256 −0.444628 0.895715i \(-0.646664\pi\)
−0.444628 + 0.895715i \(0.646664\pi\)
\(272\) −6.23800 −0.378235
\(273\) 0 0
\(274\) −2.87918 −0.173938
\(275\) −2.01784 −0.121681
\(276\) 0 0
\(277\) 17.0533 1.02464 0.512318 0.858796i \(-0.328787\pi\)
0.512318 + 0.858796i \(0.328787\pi\)
\(278\) 15.8594 0.951182
\(279\) 0 0
\(280\) −4.23800 −0.253269
\(281\) 26.0617 1.55471 0.777355 0.629062i \(-0.216561\pi\)
0.777355 + 0.629062i \(0.216561\pi\)
\(282\) 0 0
\(283\) 15.1214 0.898874 0.449437 0.893312i \(-0.351625\pi\)
0.449437 + 0.893312i \(0.351625\pi\)
\(284\) 14.1987 0.842537
\(285\) 0 0
\(286\) −7.79492 −0.460923
\(287\) 28.4978 1.68217
\(288\) 0 0
\(289\) 21.9127 1.28898
\(290\) −0.375011 −0.0220214
\(291\) 0 0
\(292\) −10.3714 −0.606939
\(293\) −6.82896 −0.398952 −0.199476 0.979903i \(-0.563924\pi\)
−0.199476 + 0.979903i \(0.563924\pi\)
\(294\) 0 0
\(295\) −4.03569 −0.234967
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) 5.35189 0.310026
\(299\) 22.6487 1.30981
\(300\) 0 0
\(301\) −36.9738 −2.13113
\(302\) −17.0857 −0.983173
\(303\) 0 0
\(304\) −7.49220 −0.429707
\(305\) −4.12247 −0.236052
\(306\) 0 0
\(307\) −8.03569 −0.458621 −0.229311 0.973353i \(-0.573647\pi\)
−0.229311 + 0.973353i \(0.573647\pi\)
\(308\) 8.55164 0.487275
\(309\) 0 0
\(310\) 9.47436 0.538107
\(311\) −25.5614 −1.44945 −0.724727 0.689036i \(-0.758034\pi\)
−0.724727 + 0.689036i \(0.758034\pi\)
\(312\) 0 0
\(313\) 20.4182 1.15411 0.577053 0.816707i \(-0.304203\pi\)
0.577053 + 0.816707i \(0.304203\pi\)
\(314\) −1.16072 −0.0655034
\(315\) 0 0
\(316\) −12.1987 −0.686230
\(317\) −0.280705 −0.0157660 −0.00788298 0.999969i \(-0.502509\pi\)
−0.00788298 + 0.999969i \(0.502509\pi\)
\(318\) 0 0
\(319\) 0.756714 0.0423679
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −24.8474 −1.38469
\(323\) 46.7364 2.60048
\(324\) 0 0
\(325\) −3.86299 −0.214280
\(326\) −9.36973 −0.518941
\(327\) 0 0
\(328\) 6.72433 0.371289
\(329\) 47.9721 2.64479
\(330\) 0 0
\(331\) −1.85184 −0.101786 −0.0508932 0.998704i \(-0.516207\pi\)
−0.0508932 + 0.998704i \(0.516207\pi\)
\(332\) −1.82730 −0.100286
\(333\) 0 0
\(334\) −5.86299 −0.320809
\(335\) −12.5117 −0.683587
\(336\) 0 0
\(337\) −13.4344 −0.731819 −0.365910 0.930650i \(-0.619242\pi\)
−0.365910 + 0.930650i \(0.619242\pi\)
\(338\) −1.92272 −0.104582
\(339\) 0 0
\(340\) 6.23800 0.338303
\(341\) −19.1178 −1.03529
\(342\) 0 0
\(343\) −16.7854 −0.906326
\(344\) −8.72433 −0.470384
\(345\) 0 0
\(346\) 2.36138 0.126949
\(347\) −13.2260 −0.710011 −0.355005 0.934864i \(-0.615521\pi\)
−0.355005 + 0.934864i \(0.615521\pi\)
\(348\) 0 0
\(349\) 34.1591 1.82849 0.914246 0.405159i \(-0.132784\pi\)
0.914246 + 0.405159i \(0.132784\pi\)
\(350\) 4.23800 0.226531
\(351\) 0 0
\(352\) 2.01784 0.107551
\(353\) 9.08934 0.483777 0.241888 0.970304i \(-0.422233\pi\)
0.241888 + 0.970304i \(0.422233\pi\)
\(354\) 0 0
\(355\) −14.1987 −0.753588
\(356\) −13.0890 −0.693717
\(357\) 0 0
\(358\) 13.9247 0.735942
\(359\) 19.3214 1.01974 0.509871 0.860251i \(-0.329693\pi\)
0.509871 + 0.860251i \(0.329693\pi\)
\(360\) 0 0
\(361\) 37.1331 1.95437
\(362\) 2.51170 0.132012
\(363\) 0 0
\(364\) 16.3714 0.858094
\(365\) 10.3714 0.542863
\(366\) 0 0
\(367\) 3.77043 0.196815 0.0984074 0.995146i \(-0.468625\pi\)
0.0984074 + 0.995146i \(0.468625\pi\)
\(368\) −5.86299 −0.305630
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −3.65036 −0.189517
\(372\) 0 0
\(373\) −12.0390 −0.623356 −0.311678 0.950188i \(-0.600891\pi\)
−0.311678 + 0.950188i \(0.600891\pi\)
\(374\) −12.5873 −0.650875
\(375\) 0 0
\(376\) 11.3195 0.583759
\(377\) 1.44867 0.0746101
\(378\) 0 0
\(379\) 38.1624 1.96027 0.980134 0.198334i \(-0.0635532\pi\)
0.980134 + 0.198334i \(0.0635532\pi\)
\(380\) 7.49220 0.384342
\(381\) 0 0
\(382\) −25.6867 −1.31424
\(383\) 33.1280 1.69276 0.846381 0.532577i \(-0.178776\pi\)
0.846381 + 0.532577i \(0.178776\pi\)
\(384\) 0 0
\(385\) −8.55164 −0.435832
\(386\) 6.03384 0.307114
\(387\) 0 0
\(388\) 6.59517 0.334819
\(389\) −13.1178 −0.665098 −0.332549 0.943086i \(-0.607909\pi\)
−0.332549 + 0.943086i \(0.607909\pi\)
\(390\) 0 0
\(391\) 36.5734 1.84960
\(392\) −10.9607 −0.553598
\(393\) 0 0
\(394\) 22.3747 1.12722
\(395\) 12.1987 0.613783
\(396\) 0 0
\(397\) −2.74159 −0.137596 −0.0687982 0.997631i \(-0.521916\pi\)
−0.0687982 + 0.997631i \(0.521916\pi\)
\(398\) 18.1663 0.910595
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −10.3423 −0.516470 −0.258235 0.966082i \(-0.583141\pi\)
−0.258235 + 0.966082i \(0.583141\pi\)
\(402\) 0 0
\(403\) −36.5994 −1.82314
\(404\) −13.0422 −0.648873
\(405\) 0 0
\(406\) −1.58930 −0.0788756
\(407\) −2.01784 −0.100021
\(408\) 0 0
\(409\) −7.68699 −0.380097 −0.190049 0.981775i \(-0.560865\pi\)
−0.190049 + 0.981775i \(0.560865\pi\)
\(410\) −6.72433 −0.332091
\(411\) 0 0
\(412\) −9.31950 −0.459139
\(413\) −17.1033 −0.841597
\(414\) 0 0
\(415\) 1.82730 0.0896988
\(416\) 3.86299 0.189399
\(417\) 0 0
\(418\) −15.1181 −0.739450
\(419\) 6.70318 0.327472 0.163736 0.986504i \(-0.447646\pi\)
0.163736 + 0.986504i \(0.447646\pi\)
\(420\) 0 0
\(421\) 11.1208 0.541995 0.270998 0.962580i \(-0.412646\pi\)
0.270998 + 0.962580i \(0.412646\pi\)
\(422\) 17.3594 0.845043
\(423\) 0 0
\(424\) −0.861339 −0.0418303
\(425\) −6.23800 −0.302588
\(426\) 0 0
\(427\) −17.4710 −0.845483
\(428\) −5.11297 −0.247145
\(429\) 0 0
\(430\) 8.72433 0.420725
\(431\) −26.7035 −1.28626 −0.643130 0.765757i \(-0.722365\pi\)
−0.643130 + 0.765757i \(0.722365\pi\)
\(432\) 0 0
\(433\) −3.38698 −0.162768 −0.0813840 0.996683i \(-0.525934\pi\)
−0.0813840 + 0.996683i \(0.525934\pi\)
\(434\) 40.1524 1.92737
\(435\) 0 0
\(436\) 14.1808 0.679139
\(437\) 43.9267 2.10130
\(438\) 0 0
\(439\) −27.5847 −1.31655 −0.658274 0.752779i \(-0.728713\pi\)
−0.658274 + 0.752779i \(0.728713\pi\)
\(440\) −2.01784 −0.0961970
\(441\) 0 0
\(442\) −24.0974 −1.14620
\(443\) −15.6474 −0.743428 −0.371714 0.928347i \(-0.621230\pi\)
−0.371714 + 0.928347i \(0.621230\pi\)
\(444\) 0 0
\(445\) 13.0890 0.620479
\(446\) −0.652333 −0.0308888
\(447\) 0 0
\(448\) −4.23800 −0.200227
\(449\) 13.6903 0.646085 0.323043 0.946384i \(-0.395294\pi\)
0.323043 + 0.946384i \(0.395294\pi\)
\(450\) 0 0
\(451\) 13.5687 0.638923
\(452\) −3.66068 −0.172184
\(453\) 0 0
\(454\) 10.1367 0.475738
\(455\) −16.3714 −0.767502
\(456\) 0 0
\(457\) 4.66324 0.218137 0.109069 0.994034i \(-0.465213\pi\)
0.109069 + 0.994034i \(0.465213\pi\)
\(458\) 26.1747 1.22307
\(459\) 0 0
\(460\) 5.86299 0.273363
\(461\) 22.8048 1.06212 0.531062 0.847333i \(-0.321793\pi\)
0.531062 + 0.847333i \(0.321793\pi\)
\(462\) 0 0
\(463\) 15.8551 0.736851 0.368426 0.929657i \(-0.379897\pi\)
0.368426 + 0.929657i \(0.379897\pi\)
\(464\) −0.375011 −0.0174095
\(465\) 0 0
\(466\) 27.0812 1.25451
\(467\) 27.2841 1.26256 0.631278 0.775556i \(-0.282531\pi\)
0.631278 + 0.775556i \(0.282531\pi\)
\(468\) 0 0
\(469\) −53.0246 −2.44845
\(470\) −11.3195 −0.522130
\(471\) 0 0
\(472\) −4.03569 −0.185758
\(473\) −17.6044 −0.809449
\(474\) 0 0
\(475\) −7.49220 −0.343766
\(476\) 26.4367 1.21172
\(477\) 0 0
\(478\) −12.3426 −0.564539
\(479\) 38.3033 1.75012 0.875062 0.484012i \(-0.160821\pi\)
0.875062 + 0.484012i \(0.160821\pi\)
\(480\) 0 0
\(481\) −3.86299 −0.176137
\(482\) −12.5000 −0.569357
\(483\) 0 0
\(484\) −6.92830 −0.314923
\(485\) −6.59517 −0.299471
\(486\) 0 0
\(487\) 7.66820 0.347480 0.173740 0.984792i \(-0.444415\pi\)
0.173740 + 0.984792i \(0.444415\pi\)
\(488\) −4.12247 −0.186615
\(489\) 0 0
\(490\) 10.9607 0.495153
\(491\) −42.1786 −1.90350 −0.951748 0.306882i \(-0.900714\pi\)
−0.951748 + 0.306882i \(0.900714\pi\)
\(492\) 0 0
\(493\) 2.33932 0.105358
\(494\) −28.9423 −1.30218
\(495\) 0 0
\(496\) 9.47436 0.425411
\(497\) −60.1741 −2.69918
\(498\) 0 0
\(499\) −4.70318 −0.210543 −0.105272 0.994444i \(-0.533571\pi\)
−0.105272 + 0.994444i \(0.533571\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 9.41298 0.420122
\(503\) 25.7650 1.14880 0.574402 0.818573i \(-0.305235\pi\)
0.574402 + 0.818573i \(0.305235\pi\)
\(504\) 0 0
\(505\) 13.0422 0.580369
\(506\) −11.8306 −0.525935
\(507\) 0 0
\(508\) −0.609707 −0.0270514
\(509\) 9.86114 0.437087 0.218544 0.975827i \(-0.429869\pi\)
0.218544 + 0.975827i \(0.429869\pi\)
\(510\) 0 0
\(511\) 43.9540 1.94441
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.26431 0.276307
\(515\) 9.31950 0.410666
\(516\) 0 0
\(517\) 22.8410 1.00455
\(518\) 4.23800 0.186207
\(519\) 0 0
\(520\) −3.86299 −0.169403
\(521\) −19.1374 −0.838425 −0.419212 0.907888i \(-0.637694\pi\)
−0.419212 + 0.907888i \(0.637694\pi\)
\(522\) 0 0
\(523\) −31.9313 −1.39626 −0.698129 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(524\) 20.9163 0.913734
\(525\) 0 0
\(526\) −25.6392 −1.11792
\(527\) −59.1011 −2.57448
\(528\) 0 0
\(529\) 11.3747 0.494552
\(530\) 0.861339 0.0374142
\(531\) 0 0
\(532\) 31.7520 1.37662
\(533\) 25.9761 1.12515
\(534\) 0 0
\(535\) 5.11297 0.221053
\(536\) −12.5117 −0.540423
\(537\) 0 0
\(538\) 6.74549 0.290819
\(539\) −22.1170 −0.952645
\(540\) 0 0
\(541\) −16.6825 −0.717238 −0.358619 0.933484i \(-0.616752\pi\)
−0.358619 + 0.933484i \(0.616752\pi\)
\(542\) 14.6390 0.628799
\(543\) 0 0
\(544\) 6.23800 0.267452
\(545\) −14.1808 −0.607441
\(546\) 0 0
\(547\) 15.5600 0.665297 0.332648 0.943051i \(-0.392058\pi\)
0.332648 + 0.943051i \(0.392058\pi\)
\(548\) 2.87918 0.122993
\(549\) 0 0
\(550\) 2.01784 0.0860412
\(551\) 2.80966 0.119695
\(552\) 0 0
\(553\) 51.6981 2.19843
\(554\) −17.0533 −0.724527
\(555\) 0 0
\(556\) −15.8594 −0.672587
\(557\) −30.6986 −1.30074 −0.650372 0.759616i \(-0.725387\pi\)
−0.650372 + 0.759616i \(0.725387\pi\)
\(558\) 0 0
\(559\) −33.7020 −1.42544
\(560\) 4.23800 0.179088
\(561\) 0 0
\(562\) −26.0617 −1.09935
\(563\) −15.0173 −0.632905 −0.316452 0.948608i \(-0.602492\pi\)
−0.316452 + 0.948608i \(0.602492\pi\)
\(564\) 0 0
\(565\) 3.66068 0.154006
\(566\) −15.1214 −0.635600
\(567\) 0 0
\(568\) −14.1987 −0.595764
\(569\) −16.8084 −0.704645 −0.352322 0.935879i \(-0.614608\pi\)
−0.352322 + 0.935879i \(0.614608\pi\)
\(570\) 0 0
\(571\) 40.0050 1.67416 0.837079 0.547082i \(-0.184261\pi\)
0.837079 + 0.547082i \(0.184261\pi\)
\(572\) 7.79492 0.325922
\(573\) 0 0
\(574\) −28.4978 −1.18947
\(575\) −5.86299 −0.244504
\(576\) 0 0
\(577\) −36.7442 −1.52968 −0.764841 0.644219i \(-0.777182\pi\)
−0.764841 + 0.644219i \(0.777182\pi\)
\(578\) −21.9127 −0.911448
\(579\) 0 0
\(580\) 0.375011 0.0155715
\(581\) 7.74412 0.321280
\(582\) 0 0
\(583\) −1.73805 −0.0719826
\(584\) 10.3714 0.429171
\(585\) 0 0
\(586\) 6.82896 0.282102
\(587\) −7.28472 −0.300673 −0.150336 0.988635i \(-0.548036\pi\)
−0.150336 + 0.988635i \(0.548036\pi\)
\(588\) 0 0
\(589\) −70.9838 −2.92484
\(590\) 4.03569 0.166147
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) 31.0025 1.27312 0.636561 0.771226i \(-0.280356\pi\)
0.636561 + 0.771226i \(0.280356\pi\)
\(594\) 0 0
\(595\) −26.4367 −1.08380
\(596\) −5.35189 −0.219222
\(597\) 0 0
\(598\) −22.6487 −0.926174
\(599\) 16.5117 0.674650 0.337325 0.941388i \(-0.390478\pi\)
0.337325 + 0.941388i \(0.390478\pi\)
\(600\) 0 0
\(601\) −13.9997 −0.571059 −0.285529 0.958370i \(-0.592169\pi\)
−0.285529 + 0.958370i \(0.592169\pi\)
\(602\) 36.9738 1.50694
\(603\) 0 0
\(604\) 17.0857 0.695208
\(605\) 6.92830 0.281676
\(606\) 0 0
\(607\) −36.5422 −1.48320 −0.741602 0.670841i \(-0.765933\pi\)
−0.741602 + 0.670841i \(0.765933\pi\)
\(608\) 7.49220 0.303849
\(609\) 0 0
\(610\) 4.12247 0.166914
\(611\) 43.7272 1.76901
\(612\) 0 0
\(613\) 5.90231 0.238392 0.119196 0.992871i \(-0.461968\pi\)
0.119196 + 0.992871i \(0.461968\pi\)
\(614\) 8.03569 0.324294
\(615\) 0 0
\(616\) −8.55164 −0.344555
\(617\) −19.4115 −0.781478 −0.390739 0.920501i \(-0.627781\pi\)
−0.390739 + 0.920501i \(0.627781\pi\)
\(618\) 0 0
\(619\) −28.6723 −1.15244 −0.576219 0.817295i \(-0.695473\pi\)
−0.576219 + 0.817295i \(0.695473\pi\)
\(620\) −9.47436 −0.380499
\(621\) 0 0
\(622\) 25.5614 1.02492
\(623\) 55.4713 2.22241
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −20.4182 −0.816076
\(627\) 0 0
\(628\) 1.16072 0.0463179
\(629\) −6.23800 −0.248726
\(630\) 0 0
\(631\) −17.5167 −0.697327 −0.348664 0.937248i \(-0.613364\pi\)
−0.348664 + 0.937248i \(0.613364\pi\)
\(632\) 12.1987 0.485238
\(633\) 0 0
\(634\) 0.280705 0.0111482
\(635\) 0.609707 0.0241955
\(636\) 0 0
\(637\) −42.3410 −1.67761
\(638\) −0.756714 −0.0299586
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −29.7444 −1.17483 −0.587417 0.809284i \(-0.699855\pi\)
−0.587417 + 0.809284i \(0.699855\pi\)
\(642\) 0 0
\(643\) 16.3541 0.644944 0.322472 0.946579i \(-0.395486\pi\)
0.322472 + 0.946579i \(0.395486\pi\)
\(644\) 24.8474 0.979124
\(645\) 0 0
\(646\) −46.7364 −1.83882
\(647\) −9.67605 −0.380405 −0.190202 0.981745i \(-0.560914\pi\)
−0.190202 + 0.981745i \(0.560914\pi\)
\(648\) 0 0
\(649\) −8.14340 −0.319656
\(650\) 3.86299 0.151519
\(651\) 0 0
\(652\) 9.36973 0.366947
\(653\) 30.9006 1.20924 0.604618 0.796516i \(-0.293326\pi\)
0.604618 + 0.796516i \(0.293326\pi\)
\(654\) 0 0
\(655\) −20.9163 −0.817269
\(656\) −6.72433 −0.262541
\(657\) 0 0
\(658\) −47.9721 −1.87015
\(659\) 30.5779 1.19115 0.595573 0.803301i \(-0.296925\pi\)
0.595573 + 0.803301i \(0.296925\pi\)
\(660\) 0 0
\(661\) 13.4749 0.524115 0.262057 0.965052i \(-0.415599\pi\)
0.262057 + 0.965052i \(0.415599\pi\)
\(662\) 1.85184 0.0719738
\(663\) 0 0
\(664\) 1.82730 0.0709131
\(665\) −31.7520 −1.23129
\(666\) 0 0
\(667\) 2.19869 0.0851335
\(668\) 5.86299 0.226846
\(669\) 0 0
\(670\) 12.5117 0.483369
\(671\) −8.31851 −0.321132
\(672\) 0 0
\(673\) −21.4937 −0.828519 −0.414260 0.910159i \(-0.635959\pi\)
−0.414260 + 0.910159i \(0.635959\pi\)
\(674\) 13.4344 0.517474
\(675\) 0 0
\(676\) 1.92272 0.0739507
\(677\) 27.9981 1.07605 0.538027 0.842927i \(-0.319170\pi\)
0.538027 + 0.842927i \(0.319170\pi\)
\(678\) 0 0
\(679\) −27.9504 −1.07264
\(680\) −6.23800 −0.239217
\(681\) 0 0
\(682\) 19.1178 0.732058
\(683\) −20.6366 −0.789639 −0.394820 0.918759i \(-0.629193\pi\)
−0.394820 + 0.918759i \(0.629193\pi\)
\(684\) 0 0
\(685\) −2.87918 −0.110008
\(686\) 16.7854 0.640869
\(687\) 0 0
\(688\) 8.72433 0.332612
\(689\) −3.32735 −0.126762
\(690\) 0 0
\(691\) 33.2874 1.26631 0.633156 0.774024i \(-0.281759\pi\)
0.633156 + 0.774024i \(0.281759\pi\)
\(692\) −2.36138 −0.0897663
\(693\) 0 0
\(694\) 13.2260 0.502053
\(695\) 15.8594 0.601580
\(696\) 0 0
\(697\) 41.9464 1.58883
\(698\) −34.1591 −1.29294
\(699\) 0 0
\(700\) −4.23800 −0.160182
\(701\) 3.29741 0.124542 0.0622708 0.998059i \(-0.480166\pi\)
0.0622708 + 0.998059i \(0.480166\pi\)
\(702\) 0 0
\(703\) −7.49220 −0.282574
\(704\) −2.01784 −0.0760504
\(705\) 0 0
\(706\) −9.08934 −0.342082
\(707\) 55.2728 2.07875
\(708\) 0 0
\(709\) −27.8744 −1.04685 −0.523423 0.852073i \(-0.675345\pi\)
−0.523423 + 0.852073i \(0.675345\pi\)
\(710\) 14.1987 0.532867
\(711\) 0 0
\(712\) 13.0890 0.490532
\(713\) −55.5481 −2.08029
\(714\) 0 0
\(715\) −7.79492 −0.291514
\(716\) −13.9247 −0.520390
\(717\) 0 0
\(718\) −19.3214 −0.721067
\(719\) 34.0982 1.27165 0.635825 0.771833i \(-0.280660\pi\)
0.635825 + 0.771833i \(0.280660\pi\)
\(720\) 0 0
\(721\) 39.4961 1.47091
\(722\) −37.1331 −1.38195
\(723\) 0 0
\(724\) −2.51170 −0.0933466
\(725\) −0.375011 −0.0139276
\(726\) 0 0
\(727\) −33.3122 −1.23548 −0.617741 0.786381i \(-0.711952\pi\)
−0.617741 + 0.786381i \(0.711952\pi\)
\(728\) −16.3714 −0.606764
\(729\) 0 0
\(730\) −10.3714 −0.383862
\(731\) −54.4224 −2.01289
\(732\) 0 0
\(733\) −17.5569 −0.648480 −0.324240 0.945975i \(-0.605108\pi\)
−0.324240 + 0.945975i \(0.605108\pi\)
\(734\) −3.77043 −0.139169
\(735\) 0 0
\(736\) 5.86299 0.216113
\(737\) −25.2467 −0.929973
\(738\) 0 0
\(739\) −18.4464 −0.678561 −0.339281 0.940685i \(-0.610184\pi\)
−0.339281 + 0.940685i \(0.610184\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 3.65036 0.134009
\(743\) 19.0679 0.699532 0.349766 0.936837i \(-0.386261\pi\)
0.349766 + 0.936837i \(0.386261\pi\)
\(744\) 0 0
\(745\) 5.35189 0.196078
\(746\) 12.0390 0.440779
\(747\) 0 0
\(748\) 12.5873 0.460238
\(749\) 21.6688 0.791761
\(750\) 0 0
\(751\) −35.9571 −1.31209 −0.656046 0.754721i \(-0.727772\pi\)
−0.656046 + 0.754721i \(0.727772\pi\)
\(752\) −11.3195 −0.412780
\(753\) 0 0
\(754\) −1.44867 −0.0527573
\(755\) −17.0857 −0.621813
\(756\) 0 0
\(757\) −22.5940 −0.821194 −0.410597 0.911817i \(-0.634680\pi\)
−0.410597 + 0.911817i \(0.634680\pi\)
\(758\) −38.1624 −1.38612
\(759\) 0 0
\(760\) −7.49220 −0.271771
\(761\) 25.4347 0.922008 0.461004 0.887398i \(-0.347489\pi\)
0.461004 + 0.887398i \(0.347489\pi\)
\(762\) 0 0
\(763\) −60.0985 −2.17571
\(764\) 25.6867 0.929311
\(765\) 0 0
\(766\) −33.1280 −1.19696
\(767\) −15.5898 −0.562917
\(768\) 0 0
\(769\) −2.20263 −0.0794290 −0.0397145 0.999211i \(-0.512645\pi\)
−0.0397145 + 0.999211i \(0.512645\pi\)
\(770\) 8.55164 0.308180
\(771\) 0 0
\(772\) −6.03384 −0.217163
\(773\) 20.5939 0.740713 0.370356 0.928890i \(-0.379236\pi\)
0.370356 + 0.928890i \(0.379236\pi\)
\(774\) 0 0
\(775\) 9.47436 0.340329
\(776\) −6.59517 −0.236753
\(777\) 0 0
\(778\) 13.1178 0.470295
\(779\) 50.3800 1.80505
\(780\) 0 0
\(781\) −28.6508 −1.02520
\(782\) −36.5734 −1.30786
\(783\) 0 0
\(784\) 10.9607 0.391453
\(785\) −1.16072 −0.0414280
\(786\) 0 0
\(787\) 5.94036 0.211751 0.105876 0.994379i \(-0.466235\pi\)
0.105876 + 0.994379i \(0.466235\pi\)
\(788\) −22.3747 −0.797065
\(789\) 0 0
\(790\) −12.1987 −0.434010
\(791\) 15.5140 0.551613
\(792\) 0 0
\(793\) −15.9251 −0.565516
\(794\) 2.74159 0.0972953
\(795\) 0 0
\(796\) −18.1663 −0.643888
\(797\) −13.3214 −0.471867 −0.235933 0.971769i \(-0.575815\pi\)
−0.235933 + 0.971769i \(0.575815\pi\)
\(798\) 0 0
\(799\) 70.6111 2.49804
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 10.3423 0.365200
\(803\) 20.9278 0.738528
\(804\) 0 0
\(805\) −24.8474 −0.875756
\(806\) 36.5994 1.28916
\(807\) 0 0
\(808\) 13.0422 0.458822
\(809\) −3.52974 −0.124099 −0.0620496 0.998073i \(-0.519764\pi\)
−0.0620496 + 0.998073i \(0.519764\pi\)
\(810\) 0 0
\(811\) −3.46105 −0.121534 −0.0607669 0.998152i \(-0.519355\pi\)
−0.0607669 + 0.998152i \(0.519355\pi\)
\(812\) 1.58930 0.0557735
\(813\) 0 0
\(814\) 2.01784 0.0707254
\(815\) −9.36973 −0.328207
\(816\) 0 0
\(817\) −65.3644 −2.28681
\(818\) 7.68699 0.268769
\(819\) 0 0
\(820\) 6.72433 0.234824
\(821\) 6.13452 0.214096 0.107048 0.994254i \(-0.465860\pi\)
0.107048 + 0.994254i \(0.465860\pi\)
\(822\) 0 0
\(823\) −7.18104 −0.250315 −0.125158 0.992137i \(-0.539944\pi\)
−0.125158 + 0.992137i \(0.539944\pi\)
\(824\) 9.31950 0.324660
\(825\) 0 0
\(826\) 17.1033 0.595099
\(827\) −34.6127 −1.20360 −0.601801 0.798646i \(-0.705550\pi\)
−0.601801 + 0.798646i \(0.705550\pi\)
\(828\) 0 0
\(829\) −21.3225 −0.740561 −0.370281 0.928920i \(-0.620738\pi\)
−0.370281 + 0.928920i \(0.620738\pi\)
\(830\) −1.82730 −0.0634266
\(831\) 0 0
\(832\) −3.86299 −0.133925
\(833\) −68.3728 −2.36898
\(834\) 0 0
\(835\) −5.86299 −0.202897
\(836\) 15.1181 0.522870
\(837\) 0 0
\(838\) −6.70318 −0.231557
\(839\) −38.1267 −1.31628 −0.658139 0.752896i \(-0.728656\pi\)
−0.658139 + 0.752896i \(0.728656\pi\)
\(840\) 0 0
\(841\) −28.8594 −0.995151
\(842\) −11.1208 −0.383249
\(843\) 0 0
\(844\) −17.3594 −0.597536
\(845\) −1.92272 −0.0661435
\(846\) 0 0
\(847\) 29.3622 1.00890
\(848\) 0.861339 0.0295785
\(849\) 0 0
\(850\) 6.23800 0.213962
\(851\) −5.86299 −0.200981
\(852\) 0 0
\(853\) 25.8306 0.884423 0.442212 0.896911i \(-0.354194\pi\)
0.442212 + 0.896911i \(0.354194\pi\)
\(854\) 17.4710 0.597847
\(855\) 0 0
\(856\) 5.11297 0.174758
\(857\) −39.5354 −1.35050 −0.675252 0.737587i \(-0.735965\pi\)
−0.675252 + 0.737587i \(0.735965\pi\)
\(858\) 0 0
\(859\) −48.9118 −1.66885 −0.834424 0.551122i \(-0.814200\pi\)
−0.834424 + 0.551122i \(0.814200\pi\)
\(860\) −8.72433 −0.297497
\(861\) 0 0
\(862\) 26.7035 0.909523
\(863\) −33.7426 −1.14861 −0.574305 0.818641i \(-0.694728\pi\)
−0.574305 + 0.818641i \(0.694728\pi\)
\(864\) 0 0
\(865\) 2.36138 0.0802895
\(866\) 3.38698 0.115094
\(867\) 0 0
\(868\) −40.1524 −1.36286
\(869\) 24.6151 0.835009
\(870\) 0 0
\(871\) −48.3326 −1.63769
\(872\) −14.1808 −0.480224
\(873\) 0 0
\(874\) −43.9267 −1.48584
\(875\) 4.23800 0.143271
\(876\) 0 0
\(877\) −41.2829 −1.39402 −0.697012 0.717059i \(-0.745488\pi\)
−0.697012 + 0.717059i \(0.745488\pi\)
\(878\) 27.5847 0.930940
\(879\) 0 0
\(880\) 2.01784 0.0680215
\(881\) −53.0893 −1.78862 −0.894312 0.447443i \(-0.852335\pi\)
−0.894312 + 0.447443i \(0.852335\pi\)
\(882\) 0 0
\(883\) 11.6180 0.390976 0.195488 0.980706i \(-0.437371\pi\)
0.195488 + 0.980706i \(0.437371\pi\)
\(884\) 24.0974 0.810482
\(885\) 0 0
\(886\) 15.6474 0.525683
\(887\) 28.7458 0.965189 0.482595 0.875844i \(-0.339694\pi\)
0.482595 + 0.875844i \(0.339694\pi\)
\(888\) 0 0
\(889\) 2.58394 0.0866627
\(890\) −13.0890 −0.438745
\(891\) 0 0
\(892\) 0.652333 0.0218417
\(893\) 84.8080 2.83799
\(894\) 0 0
\(895\) 13.9247 0.465451
\(896\) 4.23800 0.141582
\(897\) 0 0
\(898\) −13.6903 −0.456851
\(899\) −3.55299 −0.118499
\(900\) 0 0
\(901\) −5.37304 −0.179002
\(902\) −13.5687 −0.451787
\(903\) 0 0
\(904\) 3.66068 0.121752
\(905\) 2.51170 0.0834917
\(906\) 0 0
\(907\) 24.6046 0.816982 0.408491 0.912762i \(-0.366055\pi\)
0.408491 + 0.912762i \(0.366055\pi\)
\(908\) −10.1367 −0.336398
\(909\) 0 0
\(910\) 16.3714 0.542706
\(911\) 41.3494 1.36997 0.684983 0.728559i \(-0.259809\pi\)
0.684983 + 0.728559i \(0.259809\pi\)
\(912\) 0 0
\(913\) 3.68722 0.122029
\(914\) −4.66324 −0.154246
\(915\) 0 0
\(916\) −26.1747 −0.864838
\(917\) −88.6435 −2.92727
\(918\) 0 0
\(919\) 49.8444 1.64422 0.822108 0.569332i \(-0.192798\pi\)
0.822108 + 0.569332i \(0.192798\pi\)
\(920\) −5.86299 −0.193297
\(921\) 0 0
\(922\) −22.8048 −0.751035
\(923\) −54.8494 −1.80539
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −15.8551 −0.521033
\(927\) 0 0
\(928\) 0.375011 0.0123103
\(929\) −15.1440 −0.496859 −0.248429 0.968650i \(-0.579914\pi\)
−0.248429 + 0.968650i \(0.579914\pi\)
\(930\) 0 0
\(931\) −82.1196 −2.69136
\(932\) −27.0812 −0.887074
\(933\) 0 0
\(934\) −27.2841 −0.892763
\(935\) −12.5873 −0.411650
\(936\) 0 0
\(937\) −33.2735 −1.08700 −0.543498 0.839410i \(-0.682901\pi\)
−0.543498 + 0.839410i \(0.682901\pi\)
\(938\) 53.0246 1.73132
\(939\) 0 0
\(940\) 11.3195 0.369201
\(941\) 47.6286 1.55265 0.776323 0.630335i \(-0.217082\pi\)
0.776323 + 0.630335i \(0.217082\pi\)
\(942\) 0 0
\(943\) 39.4247 1.28385
\(944\) 4.03569 0.131351
\(945\) 0 0
\(946\) 17.6044 0.572367
\(947\) −20.6366 −0.670601 −0.335300 0.942111i \(-0.608838\pi\)
−0.335300 + 0.942111i \(0.608838\pi\)
\(948\) 0 0
\(949\) 40.0646 1.30055
\(950\) 7.49220 0.243079
\(951\) 0 0
\(952\) −26.4367 −0.856818
\(953\) −22.8435 −0.739973 −0.369987 0.929037i \(-0.620638\pi\)
−0.369987 + 0.929037i \(0.620638\pi\)
\(954\) 0 0
\(955\) −25.6867 −0.831201
\(956\) 12.3426 0.399189
\(957\) 0 0
\(958\) −38.3033 −1.23752
\(959\) −12.2020 −0.394023
\(960\) 0 0
\(961\) 58.7634 1.89559
\(962\) 3.86299 0.124548
\(963\) 0 0
\(964\) 12.5000 0.402596
\(965\) 6.03384 0.194236
\(966\) 0 0
\(967\) 0.0934721 0.00300586 0.00150293 0.999999i \(-0.499522\pi\)
0.00150293 + 0.999999i \(0.499522\pi\)
\(968\) 6.92830 0.222684
\(969\) 0 0
\(970\) 6.59517 0.211758
\(971\) −34.5628 −1.10917 −0.554586 0.832126i \(-0.687123\pi\)
−0.554586 + 0.832126i \(0.687123\pi\)
\(972\) 0 0
\(973\) 67.2121 2.15472
\(974\) −7.66820 −0.245705
\(975\) 0 0
\(976\) 4.12247 0.131957
\(977\) 61.6264 1.97160 0.985802 0.167915i \(-0.0537033\pi\)
0.985802 + 0.167915i \(0.0537033\pi\)
\(978\) 0 0
\(979\) 26.4116 0.844119
\(980\) −10.9607 −0.350126
\(981\) 0 0
\(982\) 42.1786 1.34597
\(983\) 6.38987 0.203805 0.101903 0.994794i \(-0.467507\pi\)
0.101903 + 0.994794i \(0.467507\pi\)
\(984\) 0 0
\(985\) 22.3747 0.712917
\(986\) −2.33932 −0.0744992
\(987\) 0 0
\(988\) 28.9423 0.920778
\(989\) −51.1507 −1.62650
\(990\) 0 0
\(991\) −24.0777 −0.764852 −0.382426 0.923986i \(-0.624911\pi\)
−0.382426 + 0.923986i \(0.624911\pi\)
\(992\) −9.47436 −0.300811
\(993\) 0 0
\(994\) 60.1741 1.90861
\(995\) 18.1663 0.575911
\(996\) 0 0
\(997\) 50.1397 1.58794 0.793970 0.607957i \(-0.208011\pi\)
0.793970 + 0.607957i \(0.208011\pi\)
\(998\) 4.70318 0.148877
\(999\) 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 3330.2.a.bk.1.1 5
3.2 odd 2 3330.2.a.bl.1.1 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.bk.1.1 5 1.1 even 1 trivial
3330.2.a.bl.1.1 yes 5 3.2 odd 2