Properties

Label 2070.2.a.u.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2070,2,Mod(1,2070)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2070, 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("2070.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.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: \(16.5290332184\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 2070.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.618034 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.85410 q^{11} -7.09017 q^{13} +0.618034 q^{14} +1.00000 q^{16} -6.09017 q^{17} +1.85410 q^{19} -1.00000 q^{20} -2.85410 q^{22} -1.00000 q^{23} +1.00000 q^{25} +7.09017 q^{26} -0.618034 q^{28} +9.23607 q^{29} +9.09017 q^{31} -1.00000 q^{32} +6.09017 q^{34} +0.618034 q^{35} +6.47214 q^{37} -1.85410 q^{38} +1.00000 q^{40} -3.32624 q^{41} +2.85410 q^{44} +1.00000 q^{46} +3.70820 q^{47} -6.61803 q^{49} -1.00000 q^{50} -7.09017 q^{52} -0.472136 q^{53} -2.85410 q^{55} +0.618034 q^{56} -9.23607 q^{58} -1.70820 q^{59} -9.32624 q^{61} -9.09017 q^{62} +1.00000 q^{64} +7.09017 q^{65} +14.4721 q^{67} -6.09017 q^{68} -0.618034 q^{70} +4.09017 q^{71} +3.23607 q^{73} -6.47214 q^{74} +1.85410 q^{76} -1.76393 q^{77} +1.52786 q^{79} -1.00000 q^{80} +3.32624 q^{82} +6.94427 q^{83} +6.09017 q^{85} -2.85410 q^{88} +10.4721 q^{89} +4.38197 q^{91} -1.00000 q^{92} -3.70820 q^{94} -1.85410 q^{95} +12.3820 q^{97} +6.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + q^{7} - 2 q^{8} + 2 q^{10} - q^{11} - 3 q^{13} - q^{14} + 2 q^{16} - q^{17} - 3 q^{19} - 2 q^{20} + q^{22} - 2 q^{23} + 2 q^{25} + 3 q^{26} + q^{28} + 14 q^{29} + 7 q^{31} - 2 q^{32} + q^{34} - q^{35} + 4 q^{37} + 3 q^{38} + 2 q^{40} + 9 q^{41} - q^{44} + 2 q^{46} - 6 q^{47} - 11 q^{49} - 2 q^{50} - 3 q^{52} + 8 q^{53} + q^{55} - q^{56} - 14 q^{58} + 10 q^{59} - 3 q^{61} - 7 q^{62} + 2 q^{64} + 3 q^{65} + 20 q^{67} - q^{68} + q^{70} - 3 q^{71} + 2 q^{73} - 4 q^{74} - 3 q^{76} - 8 q^{77} + 12 q^{79} - 2 q^{80} - 9 q^{82} - 4 q^{83} + q^{85} + q^{88} + 12 q^{89} + 11 q^{91} - 2 q^{92} + 6 q^{94} + 3 q^{95} + 27 q^{97} + 11 q^{98}+O(q^{100}) \) 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
\(7\) −0.618034 −0.233595 −0.116797 0.993156i \(-0.537263\pi\)
−0.116797 + 0.993156i \(0.537263\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.85410 0.860544 0.430272 0.902699i \(-0.358418\pi\)
0.430272 + 0.902699i \(0.358418\pi\)
\(12\) 0 0
\(13\) −7.09017 −1.96646 −0.983230 0.182372i \(-0.941623\pi\)
−0.983230 + 0.182372i \(0.941623\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.09017 −1.47708 −0.738542 0.674208i \(-0.764485\pi\)
−0.738542 + 0.674208i \(0.764485\pi\)
\(18\) 0 0
\(19\) 1.85410 0.425360 0.212680 0.977122i \(-0.431781\pi\)
0.212680 + 0.977122i \(0.431781\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −2.85410 −0.608497
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 7.09017 1.39050
\(27\) 0 0
\(28\) −0.618034 −0.116797
\(29\) 9.23607 1.71509 0.857547 0.514405i \(-0.171987\pi\)
0.857547 + 0.514405i \(0.171987\pi\)
\(30\) 0 0
\(31\) 9.09017 1.63264 0.816321 0.577598i \(-0.196010\pi\)
0.816321 + 0.577598i \(0.196010\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.09017 1.04446
\(35\) 0.618034 0.104467
\(36\) 0 0
\(37\) 6.47214 1.06401 0.532006 0.846740i \(-0.321438\pi\)
0.532006 + 0.846740i \(0.321438\pi\)
\(38\) −1.85410 −0.300775
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.32624 −0.519471 −0.259736 0.965680i \(-0.583635\pi\)
−0.259736 + 0.965680i \(0.583635\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 2.85410 0.430272
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.70820 0.540897 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) 0 0
\(49\) −6.61803 −0.945433
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −7.09017 −0.983230
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) −2.85410 −0.384847
\(56\) 0.618034 0.0825883
\(57\) 0 0
\(58\) −9.23607 −1.21276
\(59\) −1.70820 −0.222389 −0.111195 0.993799i \(-0.535468\pi\)
−0.111195 + 0.993799i \(0.535468\pi\)
\(60\) 0 0
\(61\) −9.32624 −1.19410 −0.597051 0.802203i \(-0.703661\pi\)
−0.597051 + 0.802203i \(0.703661\pi\)
\(62\) −9.09017 −1.15445
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.09017 0.879427
\(66\) 0 0
\(67\) 14.4721 1.76805 0.884026 0.467437i \(-0.154823\pi\)
0.884026 + 0.467437i \(0.154823\pi\)
\(68\) −6.09017 −0.738542
\(69\) 0 0
\(70\) −0.618034 −0.0738692
\(71\) 4.09017 0.485414 0.242707 0.970100i \(-0.421965\pi\)
0.242707 + 0.970100i \(0.421965\pi\)
\(72\) 0 0
\(73\) 3.23607 0.378753 0.189377 0.981905i \(-0.439353\pi\)
0.189377 + 0.981905i \(0.439353\pi\)
\(74\) −6.47214 −0.752371
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) −1.76393 −0.201019
\(78\) 0 0
\(79\) 1.52786 0.171898 0.0859491 0.996300i \(-0.472608\pi\)
0.0859491 + 0.996300i \(0.472608\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 3.32624 0.367322
\(83\) 6.94427 0.762233 0.381116 0.924527i \(-0.375540\pi\)
0.381116 + 0.924527i \(0.375540\pi\)
\(84\) 0 0
\(85\) 6.09017 0.660572
\(86\) 0 0
\(87\) 0 0
\(88\) −2.85410 −0.304248
\(89\) 10.4721 1.11004 0.555022 0.831836i \(-0.312710\pi\)
0.555022 + 0.831836i \(0.312710\pi\)
\(90\) 0 0
\(91\) 4.38197 0.459355
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) −3.70820 −0.382472
\(95\) −1.85410 −0.190227
\(96\) 0 0
\(97\) 12.3820 1.25720 0.628599 0.777730i \(-0.283629\pi\)
0.628599 + 0.777730i \(0.283629\pi\)
\(98\) 6.61803 0.668522
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 0.291796 0.0290348 0.0145174 0.999895i \(-0.495379\pi\)
0.0145174 + 0.999895i \(0.495379\pi\)
\(102\) 0 0
\(103\) 16.5623 1.63193 0.815966 0.578100i \(-0.196205\pi\)
0.815966 + 0.578100i \(0.196205\pi\)
\(104\) 7.09017 0.695248
\(105\) 0 0
\(106\) 0.472136 0.0458579
\(107\) −18.1803 −1.75756 −0.878780 0.477227i \(-0.841642\pi\)
−0.878780 + 0.477227i \(0.841642\pi\)
\(108\) 0 0
\(109\) −11.5623 −1.10747 −0.553734 0.832694i \(-0.686798\pi\)
−0.553734 + 0.832694i \(0.686798\pi\)
\(110\) 2.85410 0.272128
\(111\) 0 0
\(112\) −0.618034 −0.0583987
\(113\) −1.05573 −0.0993145 −0.0496573 0.998766i \(-0.515813\pi\)
−0.0496573 + 0.998766i \(0.515813\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 9.23607 0.857547
\(117\) 0 0
\(118\) 1.70820 0.157253
\(119\) 3.76393 0.345039
\(120\) 0 0
\(121\) −2.85410 −0.259464
\(122\) 9.32624 0.844358
\(123\) 0 0
\(124\) 9.09017 0.816321
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.1803 1.43577 0.717886 0.696160i \(-0.245110\pi\)
0.717886 + 0.696160i \(0.245110\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −7.09017 −0.621849
\(131\) −2.94427 −0.257242 −0.128621 0.991694i \(-0.541055\pi\)
−0.128621 + 0.991694i \(0.541055\pi\)
\(132\) 0 0
\(133\) −1.14590 −0.0993620
\(134\) −14.4721 −1.25020
\(135\) 0 0
\(136\) 6.09017 0.522228
\(137\) 10.3262 0.882230 0.441115 0.897451i \(-0.354583\pi\)
0.441115 + 0.897451i \(0.354583\pi\)
\(138\) 0 0
\(139\) 12.7639 1.08262 0.541311 0.840822i \(-0.317928\pi\)
0.541311 + 0.840822i \(0.317928\pi\)
\(140\) 0.618034 0.0522334
\(141\) 0 0
\(142\) −4.09017 −0.343239
\(143\) −20.2361 −1.69223
\(144\) 0 0
\(145\) −9.23607 −0.767014
\(146\) −3.23607 −0.267819
\(147\) 0 0
\(148\) 6.47214 0.532006
\(149\) 7.85410 0.643433 0.321717 0.946836i \(-0.395740\pi\)
0.321717 + 0.946836i \(0.395740\pi\)
\(150\) 0 0
\(151\) −2.56231 −0.208517 −0.104259 0.994550i \(-0.533247\pi\)
−0.104259 + 0.994550i \(0.533247\pi\)
\(152\) −1.85410 −0.150388
\(153\) 0 0
\(154\) 1.76393 0.142142
\(155\) −9.09017 −0.730140
\(156\) 0 0
\(157\) −3.70820 −0.295947 −0.147973 0.988991i \(-0.547275\pi\)
−0.147973 + 0.988991i \(0.547275\pi\)
\(158\) −1.52786 −0.121550
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0.618034 0.0487079
\(162\) 0 0
\(163\) 1.38197 0.108244 0.0541220 0.998534i \(-0.482764\pi\)
0.0541220 + 0.998534i \(0.482764\pi\)
\(164\) −3.32624 −0.259736
\(165\) 0 0
\(166\) −6.94427 −0.538980
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 37.2705 2.86696
\(170\) −6.09017 −0.467095
\(171\) 0 0
\(172\) 0 0
\(173\) 1.43769 0.109306 0.0546529 0.998505i \(-0.482595\pi\)
0.0546529 + 0.998505i \(0.482595\pi\)
\(174\) 0 0
\(175\) −0.618034 −0.0467190
\(176\) 2.85410 0.215136
\(177\) 0 0
\(178\) −10.4721 −0.784920
\(179\) −2.18034 −0.162966 −0.0814831 0.996675i \(-0.525966\pi\)
−0.0814831 + 0.996675i \(0.525966\pi\)
\(180\) 0 0
\(181\) −12.1459 −0.902797 −0.451399 0.892322i \(-0.649075\pi\)
−0.451399 + 0.892322i \(0.649075\pi\)
\(182\) −4.38197 −0.324813
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −6.47214 −0.475841
\(186\) 0 0
\(187\) −17.3820 −1.27110
\(188\) 3.70820 0.270449
\(189\) 0 0
\(190\) 1.85410 0.134511
\(191\) 13.7082 0.991891 0.495945 0.868354i \(-0.334822\pi\)
0.495945 + 0.868354i \(0.334822\pi\)
\(192\) 0 0
\(193\) 0.763932 0.0549890 0.0274945 0.999622i \(-0.491247\pi\)
0.0274945 + 0.999622i \(0.491247\pi\)
\(194\) −12.3820 −0.888973
\(195\) 0 0
\(196\) −6.61803 −0.472717
\(197\) 22.5623 1.60750 0.803749 0.594969i \(-0.202836\pi\)
0.803749 + 0.594969i \(0.202836\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −0.291796 −0.0205307
\(203\) −5.70820 −0.400637
\(204\) 0 0
\(205\) 3.32624 0.232315
\(206\) −16.5623 −1.15395
\(207\) 0 0
\(208\) −7.09017 −0.491615
\(209\) 5.29180 0.366041
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) −0.472136 −0.0324264
\(213\) 0 0
\(214\) 18.1803 1.24278
\(215\) 0 0
\(216\) 0 0
\(217\) −5.61803 −0.381377
\(218\) 11.5623 0.783098
\(219\) 0 0
\(220\) −2.85410 −0.192424
\(221\) 43.1803 2.90462
\(222\) 0 0
\(223\) −20.9443 −1.40253 −0.701266 0.712900i \(-0.747382\pi\)
−0.701266 + 0.712900i \(0.747382\pi\)
\(224\) 0.618034 0.0412941
\(225\) 0 0
\(226\) 1.05573 0.0702260
\(227\) 18.7639 1.24541 0.622703 0.782458i \(-0.286035\pi\)
0.622703 + 0.782458i \(0.286035\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −9.23607 −0.606378
\(233\) −6.29180 −0.412189 −0.206095 0.978532i \(-0.566075\pi\)
−0.206095 + 0.978532i \(0.566075\pi\)
\(234\) 0 0
\(235\) −3.70820 −0.241897
\(236\) −1.70820 −0.111195
\(237\) 0 0
\(238\) −3.76393 −0.243979
\(239\) 20.3607 1.31702 0.658511 0.752571i \(-0.271186\pi\)
0.658511 + 0.752571i \(0.271186\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 2.85410 0.183469
\(243\) 0 0
\(244\) −9.32624 −0.597051
\(245\) 6.61803 0.422811
\(246\) 0 0
\(247\) −13.1459 −0.836453
\(248\) −9.09017 −0.577226
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −6.14590 −0.387926 −0.193963 0.981009i \(-0.562134\pi\)
−0.193963 + 0.981009i \(0.562134\pi\)
\(252\) 0 0
\(253\) −2.85410 −0.179436
\(254\) −16.1803 −1.01524
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.81966 0.487777 0.243888 0.969803i \(-0.421577\pi\)
0.243888 + 0.969803i \(0.421577\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 7.09017 0.439714
\(261\) 0 0
\(262\) 2.94427 0.181898
\(263\) −20.7426 −1.27905 −0.639523 0.768772i \(-0.720868\pi\)
−0.639523 + 0.768772i \(0.720868\pi\)
\(264\) 0 0
\(265\) 0.472136 0.0290031
\(266\) 1.14590 0.0702595
\(267\) 0 0
\(268\) 14.4721 0.884026
\(269\) 14.1803 0.864591 0.432295 0.901732i \(-0.357704\pi\)
0.432295 + 0.901732i \(0.357704\pi\)
\(270\) 0 0
\(271\) −30.3262 −1.84219 −0.921094 0.389341i \(-0.872703\pi\)
−0.921094 + 0.389341i \(0.872703\pi\)
\(272\) −6.09017 −0.369271
\(273\) 0 0
\(274\) −10.3262 −0.623831
\(275\) 2.85410 0.172109
\(276\) 0 0
\(277\) 29.4164 1.76746 0.883730 0.467996i \(-0.155024\pi\)
0.883730 + 0.467996i \(0.155024\pi\)
\(278\) −12.7639 −0.765530
\(279\) 0 0
\(280\) −0.618034 −0.0369346
\(281\) −22.7639 −1.35798 −0.678991 0.734146i \(-0.737583\pi\)
−0.678991 + 0.734146i \(0.737583\pi\)
\(282\) 0 0
\(283\) −26.9443 −1.60167 −0.800835 0.598885i \(-0.795611\pi\)
−0.800835 + 0.598885i \(0.795611\pi\)
\(284\) 4.09017 0.242707
\(285\) 0 0
\(286\) 20.2361 1.19658
\(287\) 2.05573 0.121346
\(288\) 0 0
\(289\) 20.0902 1.18177
\(290\) 9.23607 0.542361
\(291\) 0 0
\(292\) 3.23607 0.189377
\(293\) 19.8885 1.16190 0.580951 0.813939i \(-0.302681\pi\)
0.580951 + 0.813939i \(0.302681\pi\)
\(294\) 0 0
\(295\) 1.70820 0.0994555
\(296\) −6.47214 −0.376185
\(297\) 0 0
\(298\) −7.85410 −0.454976
\(299\) 7.09017 0.410035
\(300\) 0 0
\(301\) 0 0
\(302\) 2.56231 0.147444
\(303\) 0 0
\(304\) 1.85410 0.106340
\(305\) 9.32624 0.534019
\(306\) 0 0
\(307\) −28.4508 −1.62378 −0.811888 0.583813i \(-0.801560\pi\)
−0.811888 + 0.583813i \(0.801560\pi\)
\(308\) −1.76393 −0.100509
\(309\) 0 0
\(310\) 9.09017 0.516287
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) 0 0
\(313\) 12.7984 0.723407 0.361703 0.932293i \(-0.382195\pi\)
0.361703 + 0.932293i \(0.382195\pi\)
\(314\) 3.70820 0.209266
\(315\) 0 0
\(316\) 1.52786 0.0859491
\(317\) 11.0902 0.622886 0.311443 0.950265i \(-0.399188\pi\)
0.311443 + 0.950265i \(0.399188\pi\)
\(318\) 0 0
\(319\) 26.3607 1.47591
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −0.618034 −0.0344417
\(323\) −11.2918 −0.628292
\(324\) 0 0
\(325\) −7.09017 −0.393292
\(326\) −1.38197 −0.0765400
\(327\) 0 0
\(328\) 3.32624 0.183661
\(329\) −2.29180 −0.126351
\(330\) 0 0
\(331\) 19.2361 1.05731 0.528655 0.848837i \(-0.322697\pi\)
0.528655 + 0.848837i \(0.322697\pi\)
\(332\) 6.94427 0.381116
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) −14.4721 −0.790697
\(336\) 0 0
\(337\) 13.6738 0.744857 0.372429 0.928061i \(-0.378525\pi\)
0.372429 + 0.928061i \(0.378525\pi\)
\(338\) −37.2705 −2.02725
\(339\) 0 0
\(340\) 6.09017 0.330286
\(341\) 25.9443 1.40496
\(342\) 0 0
\(343\) 8.41641 0.454443
\(344\) 0 0
\(345\) 0 0
\(346\) −1.43769 −0.0772909
\(347\) 6.38197 0.342602 0.171301 0.985219i \(-0.445203\pi\)
0.171301 + 0.985219i \(0.445203\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0.618034 0.0330353
\(351\) 0 0
\(352\) −2.85410 −0.152124
\(353\) −24.0000 −1.27739 −0.638696 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 0 0
\(355\) −4.09017 −0.217084
\(356\) 10.4721 0.555022
\(357\) 0 0
\(358\) 2.18034 0.115235
\(359\) −26.3607 −1.39126 −0.695632 0.718399i \(-0.744875\pi\)
−0.695632 + 0.718399i \(0.744875\pi\)
\(360\) 0 0
\(361\) −15.5623 −0.819069
\(362\) 12.1459 0.638374
\(363\) 0 0
\(364\) 4.38197 0.229677
\(365\) −3.23607 −0.169384
\(366\) 0 0
\(367\) 6.47214 0.337843 0.168921 0.985630i \(-0.445972\pi\)
0.168921 + 0.985630i \(0.445972\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 6.47214 0.336470
\(371\) 0.291796 0.0151493
\(372\) 0 0
\(373\) 20.1803 1.04490 0.522449 0.852670i \(-0.325018\pi\)
0.522449 + 0.852670i \(0.325018\pi\)
\(374\) 17.3820 0.898800
\(375\) 0 0
\(376\) −3.70820 −0.191236
\(377\) −65.4853 −3.37266
\(378\) 0 0
\(379\) −22.4508 −1.15322 −0.576611 0.817019i \(-0.695625\pi\)
−0.576611 + 0.817019i \(0.695625\pi\)
\(380\) −1.85410 −0.0951134
\(381\) 0 0
\(382\) −13.7082 −0.701373
\(383\) 17.8885 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(384\) 0 0
\(385\) 1.76393 0.0898983
\(386\) −0.763932 −0.0388831
\(387\) 0 0
\(388\) 12.3820 0.628599
\(389\) −21.3262 −1.08128 −0.540642 0.841253i \(-0.681818\pi\)
−0.540642 + 0.841253i \(0.681818\pi\)
\(390\) 0 0
\(391\) 6.09017 0.307993
\(392\) 6.61803 0.334261
\(393\) 0 0
\(394\) −22.5623 −1.13667
\(395\) −1.52786 −0.0768752
\(396\) 0 0
\(397\) 7.32624 0.367693 0.183847 0.982955i \(-0.441145\pi\)
0.183847 + 0.982955i \(0.441145\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 1.70820 0.0853036 0.0426518 0.999090i \(-0.486419\pi\)
0.0426518 + 0.999090i \(0.486419\pi\)
\(402\) 0 0
\(403\) −64.4508 −3.21053
\(404\) 0.291796 0.0145174
\(405\) 0 0
\(406\) 5.70820 0.283293
\(407\) 18.4721 0.915630
\(408\) 0 0
\(409\) −30.2148 −1.49402 −0.747012 0.664810i \(-0.768512\pi\)
−0.747012 + 0.664810i \(0.768512\pi\)
\(410\) −3.32624 −0.164271
\(411\) 0 0
\(412\) 16.5623 0.815966
\(413\) 1.05573 0.0519490
\(414\) 0 0
\(415\) −6.94427 −0.340881
\(416\) 7.09017 0.347624
\(417\) 0 0
\(418\) −5.29180 −0.258830
\(419\) −14.4721 −0.707010 −0.353505 0.935433i \(-0.615010\pi\)
−0.353505 + 0.935433i \(0.615010\pi\)
\(420\) 0 0
\(421\) 13.7426 0.669776 0.334888 0.942258i \(-0.391302\pi\)
0.334888 + 0.942258i \(0.391302\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) 0.472136 0.0229289
\(425\) −6.09017 −0.295417
\(426\) 0 0
\(427\) 5.76393 0.278936
\(428\) −18.1803 −0.878780
\(429\) 0 0
\(430\) 0 0
\(431\) −3.34752 −0.161245 −0.0806223 0.996745i \(-0.525691\pi\)
−0.0806223 + 0.996745i \(0.525691\pi\)
\(432\) 0 0
\(433\) 8.50658 0.408800 0.204400 0.978887i \(-0.434476\pi\)
0.204400 + 0.978887i \(0.434476\pi\)
\(434\) 5.61803 0.269674
\(435\) 0 0
\(436\) −11.5623 −0.553734
\(437\) −1.85410 −0.0886937
\(438\) 0 0
\(439\) −13.3820 −0.638686 −0.319343 0.947639i \(-0.603462\pi\)
−0.319343 + 0.947639i \(0.603462\pi\)
\(440\) 2.85410 0.136064
\(441\) 0 0
\(442\) −43.1803 −2.05388
\(443\) −25.0902 −1.19207 −0.596035 0.802958i \(-0.703258\pi\)
−0.596035 + 0.802958i \(0.703258\pi\)
\(444\) 0 0
\(445\) −10.4721 −0.496427
\(446\) 20.9443 0.991740
\(447\) 0 0
\(448\) −0.618034 −0.0291994
\(449\) 1.56231 0.0737298 0.0368649 0.999320i \(-0.488263\pi\)
0.0368649 + 0.999320i \(0.488263\pi\)
\(450\) 0 0
\(451\) −9.49342 −0.447028
\(452\) −1.05573 −0.0496573
\(453\) 0 0
\(454\) −18.7639 −0.880635
\(455\) −4.38197 −0.205430
\(456\) 0 0
\(457\) −37.7771 −1.76714 −0.883569 0.468301i \(-0.844866\pi\)
−0.883569 + 0.468301i \(0.844866\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) 39.2361 1.82741 0.913703 0.406383i \(-0.133210\pi\)
0.913703 + 0.406383i \(0.133210\pi\)
\(462\) 0 0
\(463\) 2.00000 0.0929479 0.0464739 0.998920i \(-0.485202\pi\)
0.0464739 + 0.998920i \(0.485202\pi\)
\(464\) 9.23607 0.428774
\(465\) 0 0
\(466\) 6.29180 0.291462
\(467\) 17.1246 0.792433 0.396216 0.918157i \(-0.370323\pi\)
0.396216 + 0.918157i \(0.370323\pi\)
\(468\) 0 0
\(469\) −8.94427 −0.413008
\(470\) 3.70820 0.171047
\(471\) 0 0
\(472\) 1.70820 0.0786265
\(473\) 0 0
\(474\) 0 0
\(475\) 1.85410 0.0850720
\(476\) 3.76393 0.172520
\(477\) 0 0
\(478\) −20.3607 −0.931276
\(479\) 31.8885 1.45702 0.728512 0.685033i \(-0.240212\pi\)
0.728512 + 0.685033i \(0.240212\pi\)
\(480\) 0 0
\(481\) −45.8885 −2.09234
\(482\) 0 0
\(483\) 0 0
\(484\) −2.85410 −0.129732
\(485\) −12.3820 −0.562236
\(486\) 0 0
\(487\) −19.8197 −0.898115 −0.449057 0.893503i \(-0.648240\pi\)
−0.449057 + 0.893503i \(0.648240\pi\)
\(488\) 9.32624 0.422179
\(489\) 0 0
\(490\) −6.61803 −0.298972
\(491\) −6.18034 −0.278915 −0.139457 0.990228i \(-0.544536\pi\)
−0.139457 + 0.990228i \(0.544536\pi\)
\(492\) 0 0
\(493\) −56.2492 −2.53334
\(494\) 13.1459 0.591462
\(495\) 0 0
\(496\) 9.09017 0.408161
\(497\) −2.52786 −0.113390
\(498\) 0 0
\(499\) 12.3607 0.553340 0.276670 0.960965i \(-0.410769\pi\)
0.276670 + 0.960965i \(0.410769\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 6.14590 0.274305
\(503\) −36.3262 −1.61971 −0.809853 0.586632i \(-0.800453\pi\)
−0.809853 + 0.586632i \(0.800453\pi\)
\(504\) 0 0
\(505\) −0.291796 −0.0129848
\(506\) 2.85410 0.126880
\(507\) 0 0
\(508\) 16.1803 0.717886
\(509\) −36.6525 −1.62459 −0.812296 0.583245i \(-0.801783\pi\)
−0.812296 + 0.583245i \(0.801783\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −7.81966 −0.344910
\(515\) −16.5623 −0.729822
\(516\) 0 0
\(517\) 10.5836 0.465466
\(518\) 4.00000 0.175750
\(519\) 0 0
\(520\) −7.09017 −0.310925
\(521\) 15.5279 0.680288 0.340144 0.940373i \(-0.389524\pi\)
0.340144 + 0.940373i \(0.389524\pi\)
\(522\) 0 0
\(523\) −26.0000 −1.13690 −0.568450 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −2.94427 −0.128621
\(525\) 0 0
\(526\) 20.7426 0.904422
\(527\) −55.3607 −2.41155
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −0.472136 −0.0205083
\(531\) 0 0
\(532\) −1.14590 −0.0496810
\(533\) 23.5836 1.02152
\(534\) 0 0
\(535\) 18.1803 0.786005
\(536\) −14.4721 −0.625101
\(537\) 0 0
\(538\) −14.1803 −0.611358
\(539\) −18.8885 −0.813587
\(540\) 0 0
\(541\) 22.8328 0.981659 0.490830 0.871256i \(-0.336694\pi\)
0.490830 + 0.871256i \(0.336694\pi\)
\(542\) 30.3262 1.30262
\(543\) 0 0
\(544\) 6.09017 0.261114
\(545\) 11.5623 0.495275
\(546\) 0 0
\(547\) 27.9230 1.19390 0.596950 0.802278i \(-0.296379\pi\)
0.596950 + 0.802278i \(0.296379\pi\)
\(548\) 10.3262 0.441115
\(549\) 0 0
\(550\) −2.85410 −0.121699
\(551\) 17.1246 0.729533
\(552\) 0 0
\(553\) −0.944272 −0.0401545
\(554\) −29.4164 −1.24978
\(555\) 0 0
\(556\) 12.7639 0.541311
\(557\) 22.8328 0.967457 0.483729 0.875218i \(-0.339282\pi\)
0.483729 + 0.875218i \(0.339282\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0.618034 0.0261167
\(561\) 0 0
\(562\) 22.7639 0.960239
\(563\) 13.8885 0.585332 0.292666 0.956215i \(-0.405458\pi\)
0.292666 + 0.956215i \(0.405458\pi\)
\(564\) 0 0
\(565\) 1.05573 0.0444148
\(566\) 26.9443 1.13255
\(567\) 0 0
\(568\) −4.09017 −0.171620
\(569\) 2.00000 0.0838444 0.0419222 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) 15.9787 0.668688 0.334344 0.942451i \(-0.391485\pi\)
0.334344 + 0.942451i \(0.391485\pi\)
\(572\) −20.2361 −0.846113
\(573\) 0 0
\(574\) −2.05573 −0.0858044
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −3.52786 −0.146867 −0.0734335 0.997300i \(-0.523396\pi\)
−0.0734335 + 0.997300i \(0.523396\pi\)
\(578\) −20.0902 −0.835641
\(579\) 0 0
\(580\) −9.23607 −0.383507
\(581\) −4.29180 −0.178054
\(582\) 0 0
\(583\) −1.34752 −0.0558087
\(584\) −3.23607 −0.133909
\(585\) 0 0
\(586\) −19.8885 −0.821588
\(587\) −13.6180 −0.562076 −0.281038 0.959697i \(-0.590679\pi\)
−0.281038 + 0.959697i \(0.590679\pi\)
\(588\) 0 0
\(589\) 16.8541 0.694461
\(590\) −1.70820 −0.0703256
\(591\) 0 0
\(592\) 6.47214 0.266003
\(593\) −39.2361 −1.61123 −0.805616 0.592438i \(-0.798166\pi\)
−0.805616 + 0.592438i \(0.798166\pi\)
\(594\) 0 0
\(595\) −3.76393 −0.154306
\(596\) 7.85410 0.321717
\(597\) 0 0
\(598\) −7.09017 −0.289939
\(599\) 18.3820 0.751067 0.375533 0.926809i \(-0.377460\pi\)
0.375533 + 0.926809i \(0.377460\pi\)
\(600\) 0 0
\(601\) −33.2705 −1.35713 −0.678566 0.734539i \(-0.737398\pi\)
−0.678566 + 0.734539i \(0.737398\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −2.56231 −0.104259
\(605\) 2.85410 0.116036
\(606\) 0 0
\(607\) 26.4721 1.07447 0.537235 0.843432i \(-0.319469\pi\)
0.537235 + 0.843432i \(0.319469\pi\)
\(608\) −1.85410 −0.0751938
\(609\) 0 0
\(610\) −9.32624 −0.377608
\(611\) −26.2918 −1.06365
\(612\) 0 0
\(613\) 19.3050 0.779720 0.389860 0.920874i \(-0.372523\pi\)
0.389860 + 0.920874i \(0.372523\pi\)
\(614\) 28.4508 1.14818
\(615\) 0 0
\(616\) 1.76393 0.0710708
\(617\) −34.0902 −1.37242 −0.686209 0.727404i \(-0.740727\pi\)
−0.686209 + 0.727404i \(0.740727\pi\)
\(618\) 0 0
\(619\) −2.79837 −0.112476 −0.0562381 0.998417i \(-0.517911\pi\)
−0.0562381 + 0.998417i \(0.517911\pi\)
\(620\) −9.09017 −0.365070
\(621\) 0 0
\(622\) 4.00000 0.160385
\(623\) −6.47214 −0.259301
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.7984 −0.511526
\(627\) 0 0
\(628\) −3.70820 −0.147973
\(629\) −39.4164 −1.57164
\(630\) 0 0
\(631\) 42.0689 1.67474 0.837368 0.546640i \(-0.184093\pi\)
0.837368 + 0.546640i \(0.184093\pi\)
\(632\) −1.52786 −0.0607752
\(633\) 0 0
\(634\) −11.0902 −0.440447
\(635\) −16.1803 −0.642097
\(636\) 0 0
\(637\) 46.9230 1.85916
\(638\) −26.3607 −1.04363
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −0.360680 −0.0142460 −0.00712300 0.999975i \(-0.502267\pi\)
−0.00712300 + 0.999975i \(0.502267\pi\)
\(642\) 0 0
\(643\) −8.29180 −0.326997 −0.163498 0.986544i \(-0.552278\pi\)
−0.163498 + 0.986544i \(0.552278\pi\)
\(644\) 0.618034 0.0243540
\(645\) 0 0
\(646\) 11.2918 0.444270
\(647\) 36.2492 1.42510 0.712552 0.701619i \(-0.247539\pi\)
0.712552 + 0.701619i \(0.247539\pi\)
\(648\) 0 0
\(649\) −4.87539 −0.191376
\(650\) 7.09017 0.278099
\(651\) 0 0
\(652\) 1.38197 0.0541220
\(653\) 8.03444 0.314412 0.157206 0.987566i \(-0.449751\pi\)
0.157206 + 0.987566i \(0.449751\pi\)
\(654\) 0 0
\(655\) 2.94427 0.115042
\(656\) −3.32624 −0.129868
\(657\) 0 0
\(658\) 2.29180 0.0893435
\(659\) 46.2492 1.80161 0.900807 0.434220i \(-0.142976\pi\)
0.900807 + 0.434220i \(0.142976\pi\)
\(660\) 0 0
\(661\) 18.6738 0.726325 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(662\) −19.2361 −0.747631
\(663\) 0 0
\(664\) −6.94427 −0.269490
\(665\) 1.14590 0.0444360
\(666\) 0 0
\(667\) −9.23607 −0.357622
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 14.4721 0.559107
\(671\) −26.6180 −1.02758
\(672\) 0 0
\(673\) −10.9443 −0.421871 −0.210935 0.977500i \(-0.567651\pi\)
−0.210935 + 0.977500i \(0.567651\pi\)
\(674\) −13.6738 −0.526694
\(675\) 0 0
\(676\) 37.2705 1.43348
\(677\) 50.9443 1.95795 0.978974 0.203986i \(-0.0653899\pi\)
0.978974 + 0.203986i \(0.0653899\pi\)
\(678\) 0 0
\(679\) −7.65248 −0.293675
\(680\) −6.09017 −0.233547
\(681\) 0 0
\(682\) −25.9443 −0.993458
\(683\) 31.5623 1.20770 0.603849 0.797099i \(-0.293633\pi\)
0.603849 + 0.797099i \(0.293633\pi\)
\(684\) 0 0
\(685\) −10.3262 −0.394545
\(686\) −8.41641 −0.321340
\(687\) 0 0
\(688\) 0 0
\(689\) 3.34752 0.127531
\(690\) 0 0
\(691\) −29.2361 −1.11219 −0.556096 0.831118i \(-0.687701\pi\)
−0.556096 + 0.831118i \(0.687701\pi\)
\(692\) 1.43769 0.0546529
\(693\) 0 0
\(694\) −6.38197 −0.242256
\(695\) −12.7639 −0.484164
\(696\) 0 0
\(697\) 20.2574 0.767302
\(698\) 2.00000 0.0757011
\(699\) 0 0
\(700\) −0.618034 −0.0233595
\(701\) −43.3394 −1.63691 −0.818453 0.574573i \(-0.805168\pi\)
−0.818453 + 0.574573i \(0.805168\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 2.85410 0.107568
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) −0.180340 −0.00678238
\(708\) 0 0
\(709\) −26.0902 −0.979837 −0.489918 0.871768i \(-0.662973\pi\)
−0.489918 + 0.871768i \(0.662973\pi\)
\(710\) 4.09017 0.153501
\(711\) 0 0
\(712\) −10.4721 −0.392460
\(713\) −9.09017 −0.340430
\(714\) 0 0
\(715\) 20.2361 0.756786
\(716\) −2.18034 −0.0814831
\(717\) 0 0
\(718\) 26.3607 0.983772
\(719\) −35.2705 −1.31537 −0.657684 0.753294i \(-0.728464\pi\)
−0.657684 + 0.753294i \(0.728464\pi\)
\(720\) 0 0
\(721\) −10.2361 −0.381211
\(722\) 15.5623 0.579169
\(723\) 0 0
\(724\) −12.1459 −0.451399
\(725\) 9.23607 0.343019
\(726\) 0 0
\(727\) 28.2016 1.04594 0.522970 0.852351i \(-0.324824\pi\)
0.522970 + 0.852351i \(0.324824\pi\)
\(728\) −4.38197 −0.162406
\(729\) 0 0
\(730\) 3.23607 0.119772
\(731\) 0 0
\(732\) 0 0
\(733\) −29.4164 −1.08652 −0.543260 0.839565i \(-0.682810\pi\)
−0.543260 + 0.839565i \(0.682810\pi\)
\(734\) −6.47214 −0.238891
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 41.3050 1.52149
\(738\) 0 0
\(739\) −13.8885 −0.510898 −0.255449 0.966822i \(-0.582223\pi\)
−0.255449 + 0.966822i \(0.582223\pi\)
\(740\) −6.47214 −0.237920
\(741\) 0 0
\(742\) −0.291796 −0.0107122
\(743\) −33.6312 −1.23381 −0.616904 0.787038i \(-0.711613\pi\)
−0.616904 + 0.787038i \(0.711613\pi\)
\(744\) 0 0
\(745\) −7.85410 −0.287752
\(746\) −20.1803 −0.738855
\(747\) 0 0
\(748\) −17.3820 −0.635548
\(749\) 11.2361 0.410557
\(750\) 0 0
\(751\) −47.0132 −1.71553 −0.857767 0.514038i \(-0.828149\pi\)
−0.857767 + 0.514038i \(0.828149\pi\)
\(752\) 3.70820 0.135224
\(753\) 0 0
\(754\) 65.4853 2.38483
\(755\) 2.56231 0.0932519
\(756\) 0 0
\(757\) −17.8885 −0.650170 −0.325085 0.945685i \(-0.605393\pi\)
−0.325085 + 0.945685i \(0.605393\pi\)
\(758\) 22.4508 0.815452
\(759\) 0 0
\(760\) 1.85410 0.0672553
\(761\) −46.8673 −1.69894 −0.849468 0.527640i \(-0.823077\pi\)
−0.849468 + 0.527640i \(0.823077\pi\)
\(762\) 0 0
\(763\) 7.14590 0.258699
\(764\) 13.7082 0.495945
\(765\) 0 0
\(766\) −17.8885 −0.646339
\(767\) 12.1115 0.437319
\(768\) 0 0
\(769\) −6.58359 −0.237410 −0.118705 0.992930i \(-0.537874\pi\)
−0.118705 + 0.992930i \(0.537874\pi\)
\(770\) −1.76393 −0.0635677
\(771\) 0 0
\(772\) 0.763932 0.0274945
\(773\) −28.9443 −1.04105 −0.520527 0.853845i \(-0.674264\pi\)
−0.520527 + 0.853845i \(0.674264\pi\)
\(774\) 0 0
\(775\) 9.09017 0.326529
\(776\) −12.3820 −0.444487
\(777\) 0 0
\(778\) 21.3262 0.764583
\(779\) −6.16718 −0.220962
\(780\) 0 0
\(781\) 11.6738 0.417720
\(782\) −6.09017 −0.217784
\(783\) 0 0
\(784\) −6.61803 −0.236358
\(785\) 3.70820 0.132351
\(786\) 0 0
\(787\) −2.87539 −0.102497 −0.0512483 0.998686i \(-0.516320\pi\)
−0.0512483 + 0.998686i \(0.516320\pi\)
\(788\) 22.5623 0.803749
\(789\) 0 0
\(790\) 1.52786 0.0543590
\(791\) 0.652476 0.0231994
\(792\) 0 0
\(793\) 66.1246 2.34815
\(794\) −7.32624 −0.259998
\(795\) 0 0
\(796\) 2.00000 0.0708881
\(797\) −13.7082 −0.485569 −0.242785 0.970080i \(-0.578061\pi\)
−0.242785 + 0.970080i \(0.578061\pi\)
\(798\) 0 0
\(799\) −22.5836 −0.798950
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −1.70820 −0.0603188
\(803\) 9.23607 0.325934
\(804\) 0 0
\(805\) −0.618034 −0.0217828
\(806\) 64.4508 2.27018
\(807\) 0 0
\(808\) −0.291796 −0.0102653
\(809\) 46.7426 1.64338 0.821692 0.569932i \(-0.193030\pi\)
0.821692 + 0.569932i \(0.193030\pi\)
\(810\) 0 0
\(811\) 21.8197 0.766192 0.383096 0.923709i \(-0.374858\pi\)
0.383096 + 0.923709i \(0.374858\pi\)
\(812\) −5.70820 −0.200319
\(813\) 0 0
\(814\) −18.4721 −0.647448
\(815\) −1.38197 −0.0484082
\(816\) 0 0
\(817\) 0 0
\(818\) 30.2148 1.05644
\(819\) 0 0
\(820\) 3.32624 0.116157
\(821\) 33.0557 1.15365 0.576826 0.816867i \(-0.304291\pi\)
0.576826 + 0.816867i \(0.304291\pi\)
\(822\) 0 0
\(823\) 25.4164 0.885960 0.442980 0.896531i \(-0.353921\pi\)
0.442980 + 0.896531i \(0.353921\pi\)
\(824\) −16.5623 −0.576975
\(825\) 0 0
\(826\) −1.05573 −0.0367335
\(827\) −21.7082 −0.754868 −0.377434 0.926036i \(-0.623194\pi\)
−0.377434 + 0.926036i \(0.623194\pi\)
\(828\) 0 0
\(829\) 18.9443 0.657962 0.328981 0.944337i \(-0.393295\pi\)
0.328981 + 0.944337i \(0.393295\pi\)
\(830\) 6.94427 0.241039
\(831\) 0 0
\(832\) −7.09017 −0.245807
\(833\) 40.3050 1.39648
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 5.29180 0.183021
\(837\) 0 0
\(838\) 14.4721 0.499932
\(839\) −33.0132 −1.13974 −0.569870 0.821735i \(-0.693007\pi\)
−0.569870 + 0.821735i \(0.693007\pi\)
\(840\) 0 0
\(841\) 56.3050 1.94155
\(842\) −13.7426 −0.473603
\(843\) 0 0
\(844\) 14.0000 0.481900
\(845\) −37.2705 −1.28214
\(846\) 0 0
\(847\) 1.76393 0.0606094
\(848\) −0.472136 −0.0162132
\(849\) 0 0
\(850\) 6.09017 0.208891
\(851\) −6.47214 −0.221862
\(852\) 0 0
\(853\) −10.7984 −0.369729 −0.184865 0.982764i \(-0.559185\pi\)
−0.184865 + 0.982764i \(0.559185\pi\)
\(854\) −5.76393 −0.197238
\(855\) 0 0
\(856\) 18.1803 0.621391
\(857\) 6.58359 0.224891 0.112446 0.993658i \(-0.464132\pi\)
0.112446 + 0.993658i \(0.464132\pi\)
\(858\) 0 0
\(859\) −24.0689 −0.821220 −0.410610 0.911811i \(-0.634684\pi\)
−0.410610 + 0.911811i \(0.634684\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.34752 0.114017
\(863\) −32.7639 −1.11530 −0.557649 0.830077i \(-0.688296\pi\)
−0.557649 + 0.830077i \(0.688296\pi\)
\(864\) 0 0
\(865\) −1.43769 −0.0488831
\(866\) −8.50658 −0.289065
\(867\) 0 0
\(868\) −5.61803 −0.190688
\(869\) 4.36068 0.147926
\(870\) 0 0
\(871\) −102.610 −3.47680
\(872\) 11.5623 0.391549
\(873\) 0 0
\(874\) 1.85410 0.0627159
\(875\) 0.618034 0.0208934
\(876\) 0 0
\(877\) 18.7426 0.632894 0.316447 0.948610i \(-0.397510\pi\)
0.316447 + 0.948610i \(0.397510\pi\)
\(878\) 13.3820 0.451619
\(879\) 0 0
\(880\) −2.85410 −0.0962118
\(881\) −8.58359 −0.289189 −0.144594 0.989491i \(-0.546188\pi\)
−0.144594 + 0.989491i \(0.546188\pi\)
\(882\) 0 0
\(883\) 15.5623 0.523713 0.261857 0.965107i \(-0.415665\pi\)
0.261857 + 0.965107i \(0.415665\pi\)
\(884\) 43.1803 1.45231
\(885\) 0 0
\(886\) 25.0902 0.842921
\(887\) 5.16718 0.173497 0.0867485 0.996230i \(-0.472352\pi\)
0.0867485 + 0.996230i \(0.472352\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 10.4721 0.351027
\(891\) 0 0
\(892\) −20.9443 −0.701266
\(893\) 6.87539 0.230076
\(894\) 0 0
\(895\) 2.18034 0.0728807
\(896\) 0.618034 0.0206471
\(897\) 0 0
\(898\) −1.56231 −0.0521348
\(899\) 83.9574 2.80014
\(900\) 0 0
\(901\) 2.87539 0.0957931
\(902\) 9.49342 0.316096
\(903\) 0 0
\(904\) 1.05573 0.0351130
\(905\) 12.1459 0.403743
\(906\) 0 0
\(907\) 7.12461 0.236569 0.118284 0.992980i \(-0.462261\pi\)
0.118284 + 0.992980i \(0.462261\pi\)
\(908\) 18.7639 0.622703
\(909\) 0 0
\(910\) 4.38197 0.145261
\(911\) −36.0689 −1.19502 −0.597508 0.801863i \(-0.703842\pi\)
−0.597508 + 0.801863i \(0.703842\pi\)
\(912\) 0 0
\(913\) 19.8197 0.655935
\(914\) 37.7771 1.24955
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 1.81966 0.0600905
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 0 0
\(922\) −39.2361 −1.29217
\(923\) −29.0000 −0.954547
\(924\) 0 0
\(925\) 6.47214 0.212803
\(926\) −2.00000 −0.0657241
\(927\) 0 0
\(928\) −9.23607 −0.303189
\(929\) 3.52786 0.115745 0.0578727 0.998324i \(-0.481568\pi\)
0.0578727 + 0.998324i \(0.481568\pi\)
\(930\) 0 0
\(931\) −12.2705 −0.402150
\(932\) −6.29180 −0.206095
\(933\) 0 0
\(934\) −17.1246 −0.560334
\(935\) 17.3820 0.568451
\(936\) 0 0
\(937\) −36.7984 −1.20215 −0.601075 0.799192i \(-0.705261\pi\)
−0.601075 + 0.799192i \(0.705261\pi\)
\(938\) 8.94427 0.292041
\(939\) 0 0
\(940\) −3.70820 −0.120948
\(941\) 22.4934 0.733265 0.366632 0.930366i \(-0.380511\pi\)
0.366632 + 0.930366i \(0.380511\pi\)
\(942\) 0 0
\(943\) 3.32624 0.108317
\(944\) −1.70820 −0.0555973
\(945\) 0 0
\(946\) 0 0
\(947\) 54.6869 1.77709 0.888543 0.458793i \(-0.151718\pi\)
0.888543 + 0.458793i \(0.151718\pi\)
\(948\) 0 0
\(949\) −22.9443 −0.744803
\(950\) −1.85410 −0.0601550
\(951\) 0 0
\(952\) −3.76393 −0.121990
\(953\) −3.79837 −0.123041 −0.0615207 0.998106i \(-0.519595\pi\)
−0.0615207 + 0.998106i \(0.519595\pi\)
\(954\) 0 0
\(955\) −13.7082 −0.443587
\(956\) 20.3607 0.658511
\(957\) 0 0
\(958\) −31.8885 −1.03027
\(959\) −6.38197 −0.206084
\(960\) 0 0
\(961\) 51.6312 1.66552
\(962\) 45.8885 1.47951
\(963\) 0 0
\(964\) 0 0
\(965\) −0.763932 −0.0245918
\(966\) 0 0
\(967\) −16.5410 −0.531923 −0.265962 0.963984i \(-0.585689\pi\)
−0.265962 + 0.963984i \(0.585689\pi\)
\(968\) 2.85410 0.0917343
\(969\) 0 0
\(970\) 12.3820 0.397561
\(971\) −34.2705 −1.09979 −0.549896 0.835233i \(-0.685333\pi\)
−0.549896 + 0.835233i \(0.685333\pi\)
\(972\) 0 0
\(973\) −7.88854 −0.252895
\(974\) 19.8197 0.635063
\(975\) 0 0
\(976\) −9.32624 −0.298526
\(977\) 23.5623 0.753825 0.376912 0.926249i \(-0.376986\pi\)
0.376912 + 0.926249i \(0.376986\pi\)
\(978\) 0 0
\(979\) 29.8885 0.955242
\(980\) 6.61803 0.211405
\(981\) 0 0
\(982\) 6.18034 0.197223
\(983\) 14.2705 0.455159 0.227579 0.973760i \(-0.426919\pi\)
0.227579 + 0.973760i \(0.426919\pi\)
\(984\) 0 0
\(985\) −22.5623 −0.718895
\(986\) 56.2492 1.79134
\(987\) 0 0
\(988\) −13.1459 −0.418227
\(989\) 0 0
\(990\) 0 0
\(991\) 27.5066 0.873775 0.436888 0.899516i \(-0.356081\pi\)
0.436888 + 0.899516i \(0.356081\pi\)
\(992\) −9.09017 −0.288613
\(993\) 0 0
\(994\) 2.52786 0.0801790
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) 57.1935 1.81134 0.905668 0.423987i \(-0.139370\pi\)
0.905668 + 0.423987i \(0.139370\pi\)
\(998\) −12.3607 −0.391270
\(999\) 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 2070.2.a.u.1.1 2
3.2 odd 2 230.2.a.c.1.2 2
12.11 even 2 1840.2.a.l.1.1 2
15.2 even 4 1150.2.b.i.599.3 4
15.8 even 4 1150.2.b.i.599.2 4
15.14 odd 2 1150.2.a.j.1.1 2
24.5 odd 2 7360.2.a.bh.1.1 2
24.11 even 2 7360.2.a.bn.1.2 2
60.59 even 2 9200.2.a.bu.1.2 2
69.68 even 2 5290.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.c.1.2 2 3.2 odd 2
1150.2.a.j.1.1 2 15.14 odd 2
1150.2.b.i.599.2 4 15.8 even 4
1150.2.b.i.599.3 4 15.2 even 4
1840.2.a.l.1.1 2 12.11 even 2
2070.2.a.u.1.1 2 1.1 even 1 trivial
5290.2.a.o.1.2 2 69.68 even 2
7360.2.a.bh.1.1 2 24.5 odd 2
7360.2.a.bn.1.2 2 24.11 even 2
9200.2.a.bu.1.2 2 60.59 even 2