Properties

Label 6003.2.a.s.1.16
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $0$
Dimension $20$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-1.94926\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.94926 q^{2} +1.79960 q^{4} -3.69296 q^{5} -2.45473 q^{7} -0.390627 q^{8} +O(q^{10})\) \(q+1.94926 q^{2} +1.79960 q^{4} -3.69296 q^{5} -2.45473 q^{7} -0.390627 q^{8} -7.19852 q^{10} -4.43817 q^{11} -4.28535 q^{13} -4.78490 q^{14} -4.36064 q^{16} +1.49875 q^{17} +1.06731 q^{19} -6.64585 q^{20} -8.65112 q^{22} -1.00000 q^{23} +8.63793 q^{25} -8.35324 q^{26} -4.41754 q^{28} -1.00000 q^{29} +2.48186 q^{31} -7.71875 q^{32} +2.92144 q^{34} +9.06522 q^{35} +3.20872 q^{37} +2.08047 q^{38} +1.44257 q^{40} -4.57310 q^{41} +5.98441 q^{43} -7.98693 q^{44} -1.94926 q^{46} +4.84278 q^{47} -0.974292 q^{49} +16.8375 q^{50} -7.71192 q^{52} -4.65981 q^{53} +16.3900 q^{55} +0.958884 q^{56} -1.94926 q^{58} -0.167254 q^{59} +13.1401 q^{61} +4.83778 q^{62} -6.32455 q^{64} +15.8256 q^{65} +11.9525 q^{67} +2.69715 q^{68} +17.6704 q^{70} -8.22431 q^{71} -10.4723 q^{73} +6.25462 q^{74} +1.92074 q^{76} +10.8945 q^{77} +0.909584 q^{79} +16.1036 q^{80} -8.91415 q^{82} +4.61155 q^{83} -5.53480 q^{85} +11.6652 q^{86} +1.73367 q^{88} -12.8616 q^{89} +10.5194 q^{91} -1.79960 q^{92} +9.43982 q^{94} -3.94154 q^{95} +1.88580 q^{97} -1.89915 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94926 1.37833 0.689166 0.724603i \(-0.257977\pi\)
0.689166 + 0.724603i \(0.257977\pi\)
\(3\) 0 0
\(4\) 1.79960 0.899801
\(5\) −3.69296 −1.65154 −0.825770 0.564007i \(-0.809259\pi\)
−0.825770 + 0.564007i \(0.809259\pi\)
\(6\) 0 0
\(7\) −2.45473 −0.927801 −0.463901 0.885887i \(-0.653551\pi\)
−0.463901 + 0.885887i \(0.653551\pi\)
\(8\) −0.390627 −0.138107
\(9\) 0 0
\(10\) −7.19852 −2.27637
\(11\) −4.43817 −1.33816 −0.669079 0.743192i \(-0.733311\pi\)
−0.669079 + 0.743192i \(0.733311\pi\)
\(12\) 0 0
\(13\) −4.28535 −1.18854 −0.594271 0.804265i \(-0.702559\pi\)
−0.594271 + 0.804265i \(0.702559\pi\)
\(14\) −4.78490 −1.27882
\(15\) 0 0
\(16\) −4.36064 −1.09016
\(17\) 1.49875 0.363499 0.181750 0.983345i \(-0.441824\pi\)
0.181750 + 0.983345i \(0.441824\pi\)
\(18\) 0 0
\(19\) 1.06731 0.244858 0.122429 0.992477i \(-0.460932\pi\)
0.122429 + 0.992477i \(0.460932\pi\)
\(20\) −6.64585 −1.48606
\(21\) 0 0
\(22\) −8.65112 −1.84443
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 8.63793 1.72759
\(26\) −8.35324 −1.63821
\(27\) 0 0
\(28\) −4.41754 −0.834837
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 2.48186 0.445755 0.222878 0.974846i \(-0.428455\pi\)
0.222878 + 0.974846i \(0.428455\pi\)
\(32\) −7.71875 −1.36449
\(33\) 0 0
\(34\) 2.92144 0.501023
\(35\) 9.06522 1.53230
\(36\) 0 0
\(37\) 3.20872 0.527510 0.263755 0.964590i \(-0.415039\pi\)
0.263755 + 0.964590i \(0.415039\pi\)
\(38\) 2.08047 0.337496
\(39\) 0 0
\(40\) 1.44257 0.228090
\(41\) −4.57310 −0.714199 −0.357099 0.934066i \(-0.616234\pi\)
−0.357099 + 0.934066i \(0.616234\pi\)
\(42\) 0 0
\(43\) 5.98441 0.912614 0.456307 0.889822i \(-0.349172\pi\)
0.456307 + 0.889822i \(0.349172\pi\)
\(44\) −7.98693 −1.20408
\(45\) 0 0
\(46\) −1.94926 −0.287402
\(47\) 4.84278 0.706392 0.353196 0.935549i \(-0.385095\pi\)
0.353196 + 0.935549i \(0.385095\pi\)
\(48\) 0 0
\(49\) −0.974292 −0.139185
\(50\) 16.8375 2.38119
\(51\) 0 0
\(52\) −7.71192 −1.06945
\(53\) −4.65981 −0.640074 −0.320037 0.947405i \(-0.603695\pi\)
−0.320037 + 0.947405i \(0.603695\pi\)
\(54\) 0 0
\(55\) 16.3900 2.21002
\(56\) 0.958884 0.128136
\(57\) 0 0
\(58\) −1.94926 −0.255950
\(59\) −0.167254 −0.0217746 −0.0108873 0.999941i \(-0.503466\pi\)
−0.0108873 + 0.999941i \(0.503466\pi\)
\(60\) 0 0
\(61\) 13.1401 1.68242 0.841212 0.540706i \(-0.181843\pi\)
0.841212 + 0.540706i \(0.181843\pi\)
\(62\) 4.83778 0.614399
\(63\) 0 0
\(64\) −6.32455 −0.790568
\(65\) 15.8256 1.96292
\(66\) 0 0
\(67\) 11.9525 1.46023 0.730113 0.683327i \(-0.239467\pi\)
0.730113 + 0.683327i \(0.239467\pi\)
\(68\) 2.69715 0.327077
\(69\) 0 0
\(70\) 17.6704 2.11202
\(71\) −8.22431 −0.976046 −0.488023 0.872831i \(-0.662282\pi\)
−0.488023 + 0.872831i \(0.662282\pi\)
\(72\) 0 0
\(73\) −10.4723 −1.22569 −0.612845 0.790203i \(-0.709975\pi\)
−0.612845 + 0.790203i \(0.709975\pi\)
\(74\) 6.25462 0.727084
\(75\) 0 0
\(76\) 1.92074 0.220324
\(77\) 10.8945 1.24154
\(78\) 0 0
\(79\) 0.909584 0.102336 0.0511681 0.998690i \(-0.483706\pi\)
0.0511681 + 0.998690i \(0.483706\pi\)
\(80\) 16.1036 1.80044
\(81\) 0 0
\(82\) −8.91415 −0.984403
\(83\) 4.61155 0.506183 0.253092 0.967442i \(-0.418553\pi\)
0.253092 + 0.967442i \(0.418553\pi\)
\(84\) 0 0
\(85\) −5.53480 −0.600334
\(86\) 11.6652 1.25789
\(87\) 0 0
\(88\) 1.73367 0.184809
\(89\) −12.8616 −1.36333 −0.681664 0.731665i \(-0.738744\pi\)
−0.681664 + 0.731665i \(0.738744\pi\)
\(90\) 0 0
\(91\) 10.5194 1.10273
\(92\) −1.79960 −0.187622
\(93\) 0 0
\(94\) 9.43982 0.973643
\(95\) −3.94154 −0.404394
\(96\) 0 0
\(97\) 1.88580 0.191474 0.0957372 0.995407i \(-0.469479\pi\)
0.0957372 + 0.995407i \(0.469479\pi\)
\(98\) −1.89915 −0.191843
\(99\) 0 0
\(100\) 15.5448 1.55448
\(101\) −5.40314 −0.537633 −0.268816 0.963191i \(-0.586632\pi\)
−0.268816 + 0.963191i \(0.586632\pi\)
\(102\) 0 0
\(103\) −6.14017 −0.605009 −0.302505 0.953148i \(-0.597823\pi\)
−0.302505 + 0.953148i \(0.597823\pi\)
\(104\) 1.67397 0.164146
\(105\) 0 0
\(106\) −9.08316 −0.882234
\(107\) 3.00788 0.290783 0.145391 0.989374i \(-0.453556\pi\)
0.145391 + 0.989374i \(0.453556\pi\)
\(108\) 0 0
\(109\) −13.3033 −1.27423 −0.637113 0.770771i \(-0.719871\pi\)
−0.637113 + 0.770771i \(0.719871\pi\)
\(110\) 31.9482 3.04614
\(111\) 0 0
\(112\) 10.7042 1.01145
\(113\) 18.0625 1.69918 0.849590 0.527443i \(-0.176849\pi\)
0.849590 + 0.527443i \(0.176849\pi\)
\(114\) 0 0
\(115\) 3.69296 0.344370
\(116\) −1.79960 −0.167089
\(117\) 0 0
\(118\) −0.326020 −0.0300126
\(119\) −3.67902 −0.337255
\(120\) 0 0
\(121\) 8.69731 0.790665
\(122\) 25.6135 2.31894
\(123\) 0 0
\(124\) 4.46636 0.401091
\(125\) −13.4347 −1.20164
\(126\) 0 0
\(127\) −6.61372 −0.586873 −0.293436 0.955979i \(-0.594799\pi\)
−0.293436 + 0.955979i \(0.594799\pi\)
\(128\) 3.10933 0.274828
\(129\) 0 0
\(130\) 30.8482 2.70556
\(131\) 10.8669 0.949448 0.474724 0.880135i \(-0.342548\pi\)
0.474724 + 0.880135i \(0.342548\pi\)
\(132\) 0 0
\(133\) −2.61997 −0.227180
\(134\) 23.2984 2.01268
\(135\) 0 0
\(136\) −0.585450 −0.0502019
\(137\) −19.4409 −1.66095 −0.830474 0.557057i \(-0.811930\pi\)
−0.830474 + 0.557057i \(0.811930\pi\)
\(138\) 0 0
\(139\) −14.9146 −1.26504 −0.632520 0.774544i \(-0.717979\pi\)
−0.632520 + 0.774544i \(0.717979\pi\)
\(140\) 16.3138 1.37877
\(141\) 0 0
\(142\) −16.0313 −1.34532
\(143\) 19.0191 1.59046
\(144\) 0 0
\(145\) 3.69296 0.306683
\(146\) −20.4132 −1.68941
\(147\) 0 0
\(148\) 5.77442 0.474654
\(149\) 7.73831 0.633947 0.316974 0.948434i \(-0.397333\pi\)
0.316974 + 0.948434i \(0.397333\pi\)
\(150\) 0 0
\(151\) 12.8699 1.04733 0.523667 0.851923i \(-0.324564\pi\)
0.523667 + 0.851923i \(0.324564\pi\)
\(152\) −0.416921 −0.0338168
\(153\) 0 0
\(154\) 21.2362 1.71126
\(155\) −9.16540 −0.736183
\(156\) 0 0
\(157\) 12.6332 1.00824 0.504120 0.863634i \(-0.331817\pi\)
0.504120 + 0.863634i \(0.331817\pi\)
\(158\) 1.77301 0.141053
\(159\) 0 0
\(160\) 28.5050 2.25352
\(161\) 2.45473 0.193460
\(162\) 0 0
\(163\) −8.96876 −0.702488 −0.351244 0.936284i \(-0.614241\pi\)
−0.351244 + 0.936284i \(0.614241\pi\)
\(164\) −8.22977 −0.642637
\(165\) 0 0
\(166\) 8.98909 0.697689
\(167\) −12.9614 −1.00298 −0.501492 0.865162i \(-0.667215\pi\)
−0.501492 + 0.865162i \(0.667215\pi\)
\(168\) 0 0
\(169\) 5.36420 0.412631
\(170\) −10.7888 −0.827460
\(171\) 0 0
\(172\) 10.7696 0.821171
\(173\) 4.02670 0.306144 0.153072 0.988215i \(-0.451083\pi\)
0.153072 + 0.988215i \(0.451083\pi\)
\(174\) 0 0
\(175\) −21.2038 −1.60286
\(176\) 19.3532 1.45880
\(177\) 0 0
\(178\) −25.0706 −1.87912
\(179\) 0.565219 0.0422464 0.0211232 0.999777i \(-0.493276\pi\)
0.0211232 + 0.999777i \(0.493276\pi\)
\(180\) 0 0
\(181\) 14.4322 1.07274 0.536368 0.843984i \(-0.319796\pi\)
0.536368 + 0.843984i \(0.319796\pi\)
\(182\) 20.5050 1.51993
\(183\) 0 0
\(184\) 0.390627 0.0287974
\(185\) −11.8497 −0.871204
\(186\) 0 0
\(187\) −6.65168 −0.486419
\(188\) 8.71508 0.635612
\(189\) 0 0
\(190\) −7.68308 −0.557389
\(191\) 11.1284 0.805223 0.402612 0.915371i \(-0.368103\pi\)
0.402612 + 0.915371i \(0.368103\pi\)
\(192\) 0 0
\(193\) −18.0097 −1.29637 −0.648183 0.761485i \(-0.724471\pi\)
−0.648183 + 0.761485i \(0.724471\pi\)
\(194\) 3.67591 0.263915
\(195\) 0 0
\(196\) −1.75334 −0.125238
\(197\) −22.4270 −1.59786 −0.798930 0.601424i \(-0.794600\pi\)
−0.798930 + 0.601424i \(0.794600\pi\)
\(198\) 0 0
\(199\) 26.8224 1.90139 0.950694 0.310129i \(-0.100372\pi\)
0.950694 + 0.310129i \(0.100372\pi\)
\(200\) −3.37421 −0.238592
\(201\) 0 0
\(202\) −10.5321 −0.741037
\(203\) 2.45473 0.172288
\(204\) 0 0
\(205\) 16.8883 1.17953
\(206\) −11.9688 −0.833904
\(207\) 0 0
\(208\) 18.6868 1.29570
\(209\) −4.73691 −0.327659
\(210\) 0 0
\(211\) 9.39698 0.646915 0.323457 0.946243i \(-0.395155\pi\)
0.323457 + 0.946243i \(0.395155\pi\)
\(212\) −8.38580 −0.575939
\(213\) 0 0
\(214\) 5.86313 0.400795
\(215\) −22.1002 −1.50722
\(216\) 0 0
\(217\) −6.09230 −0.413572
\(218\) −25.9316 −1.75631
\(219\) 0 0
\(220\) 29.4954 1.98858
\(221\) −6.42265 −0.432034
\(222\) 0 0
\(223\) 14.7193 0.985681 0.492840 0.870120i \(-0.335959\pi\)
0.492840 + 0.870120i \(0.335959\pi\)
\(224\) 18.9475 1.26598
\(225\) 0 0
\(226\) 35.2085 2.34204
\(227\) −25.3440 −1.68214 −0.841070 0.540927i \(-0.818074\pi\)
−0.841070 + 0.540927i \(0.818074\pi\)
\(228\) 0 0
\(229\) 22.2652 1.47133 0.735664 0.677347i \(-0.236870\pi\)
0.735664 + 0.677347i \(0.236870\pi\)
\(230\) 7.19852 0.474656
\(231\) 0 0
\(232\) 0.390627 0.0256459
\(233\) −5.91632 −0.387591 −0.193796 0.981042i \(-0.562080\pi\)
−0.193796 + 0.981042i \(0.562080\pi\)
\(234\) 0 0
\(235\) −17.8842 −1.16664
\(236\) −0.300990 −0.0195928
\(237\) 0 0
\(238\) −7.17135 −0.464850
\(239\) −15.1516 −0.980072 −0.490036 0.871702i \(-0.663016\pi\)
−0.490036 + 0.871702i \(0.663016\pi\)
\(240\) 0 0
\(241\) −1.07716 −0.0693858 −0.0346929 0.999398i \(-0.511045\pi\)
−0.0346929 + 0.999398i \(0.511045\pi\)
\(242\) 16.9533 1.08980
\(243\) 0 0
\(244\) 23.6470 1.51385
\(245\) 3.59802 0.229869
\(246\) 0 0
\(247\) −4.57381 −0.291024
\(248\) −0.969481 −0.0615621
\(249\) 0 0
\(250\) −26.1877 −1.65626
\(251\) −26.7693 −1.68966 −0.844832 0.535032i \(-0.820299\pi\)
−0.844832 + 0.535032i \(0.820299\pi\)
\(252\) 0 0
\(253\) 4.43817 0.279025
\(254\) −12.8918 −0.808906
\(255\) 0 0
\(256\) 18.7100 1.16937
\(257\) 19.8433 1.23779 0.618897 0.785472i \(-0.287580\pi\)
0.618897 + 0.785472i \(0.287580\pi\)
\(258\) 0 0
\(259\) −7.87654 −0.489425
\(260\) 28.4798 1.76624
\(261\) 0 0
\(262\) 21.1824 1.30866
\(263\) 16.6065 1.02400 0.512002 0.858984i \(-0.328904\pi\)
0.512002 + 0.858984i \(0.328904\pi\)
\(264\) 0 0
\(265\) 17.2085 1.05711
\(266\) −5.10699 −0.313130
\(267\) 0 0
\(268\) 21.5097 1.31391
\(269\) 17.7138 1.08003 0.540015 0.841655i \(-0.318418\pi\)
0.540015 + 0.841655i \(0.318418\pi\)
\(270\) 0 0
\(271\) 14.0182 0.851544 0.425772 0.904830i \(-0.360003\pi\)
0.425772 + 0.904830i \(0.360003\pi\)
\(272\) −6.53548 −0.396272
\(273\) 0 0
\(274\) −37.8953 −2.28934
\(275\) −38.3366 −2.31178
\(276\) 0 0
\(277\) 10.0263 0.602425 0.301212 0.953557i \(-0.402609\pi\)
0.301212 + 0.953557i \(0.402609\pi\)
\(278\) −29.0724 −1.74364
\(279\) 0 0
\(280\) −3.54112 −0.211622
\(281\) 18.4255 1.09917 0.549587 0.835437i \(-0.314785\pi\)
0.549587 + 0.835437i \(0.314785\pi\)
\(282\) 0 0
\(283\) −25.7278 −1.52936 −0.764680 0.644411i \(-0.777103\pi\)
−0.764680 + 0.644411i \(0.777103\pi\)
\(284\) −14.8005 −0.878248
\(285\) 0 0
\(286\) 37.0731 2.19218
\(287\) 11.2257 0.662635
\(288\) 0 0
\(289\) −14.7538 −0.867868
\(290\) 7.19852 0.422712
\(291\) 0 0
\(292\) −18.8460 −1.10288
\(293\) 24.2334 1.41573 0.707865 0.706348i \(-0.249659\pi\)
0.707865 + 0.706348i \(0.249659\pi\)
\(294\) 0 0
\(295\) 0.617660 0.0359616
\(296\) −1.25341 −0.0728531
\(297\) 0 0
\(298\) 15.0840 0.873790
\(299\) 4.28535 0.247828
\(300\) 0 0
\(301\) −14.6901 −0.846725
\(302\) 25.0867 1.44357
\(303\) 0 0
\(304\) −4.65416 −0.266935
\(305\) −48.5260 −2.77859
\(306\) 0 0
\(307\) −7.93632 −0.452950 −0.226475 0.974017i \(-0.572720\pi\)
−0.226475 + 0.974017i \(0.572720\pi\)
\(308\) 19.6058 1.11714
\(309\) 0 0
\(310\) −17.8657 −1.01470
\(311\) 10.8054 0.612717 0.306358 0.951916i \(-0.400889\pi\)
0.306358 + 0.951916i \(0.400889\pi\)
\(312\) 0 0
\(313\) −25.0705 −1.41707 −0.708535 0.705675i \(-0.750644\pi\)
−0.708535 + 0.705675i \(0.750644\pi\)
\(314\) 24.6254 1.38969
\(315\) 0 0
\(316\) 1.63689 0.0920822
\(317\) 14.5292 0.816041 0.408021 0.912973i \(-0.366219\pi\)
0.408021 + 0.912973i \(0.366219\pi\)
\(318\) 0 0
\(319\) 4.43817 0.248490
\(320\) 23.3563 1.30566
\(321\) 0 0
\(322\) 4.78490 0.266652
\(323\) 1.59963 0.0890058
\(324\) 0 0
\(325\) −37.0165 −2.05331
\(326\) −17.4824 −0.968262
\(327\) 0 0
\(328\) 1.78638 0.0986361
\(329\) −11.8877 −0.655392
\(330\) 0 0
\(331\) −2.56900 −0.141205 −0.0706026 0.997505i \(-0.522492\pi\)
−0.0706026 + 0.997505i \(0.522492\pi\)
\(332\) 8.29895 0.455464
\(333\) 0 0
\(334\) −25.2651 −1.38245
\(335\) −44.1399 −2.41162
\(336\) 0 0
\(337\) −27.1666 −1.47986 −0.739930 0.672683i \(-0.765142\pi\)
−0.739930 + 0.672683i \(0.765142\pi\)
\(338\) 10.4562 0.568743
\(339\) 0 0
\(340\) −9.96044 −0.540181
\(341\) −11.0149 −0.596490
\(342\) 0 0
\(343\) 19.5747 1.05694
\(344\) −2.33767 −0.126039
\(345\) 0 0
\(346\) 7.84907 0.421968
\(347\) −24.3805 −1.30881 −0.654406 0.756143i \(-0.727081\pi\)
−0.654406 + 0.756143i \(0.727081\pi\)
\(348\) 0 0
\(349\) −25.6829 −1.37478 −0.687388 0.726290i \(-0.741243\pi\)
−0.687388 + 0.726290i \(0.741243\pi\)
\(350\) −41.3317 −2.20927
\(351\) 0 0
\(352\) 34.2571 1.82591
\(353\) 27.0213 1.43820 0.719100 0.694907i \(-0.244555\pi\)
0.719100 + 0.694907i \(0.244555\pi\)
\(354\) 0 0
\(355\) 30.3720 1.61198
\(356\) −23.1458 −1.22672
\(357\) 0 0
\(358\) 1.10176 0.0582296
\(359\) −14.3872 −0.759326 −0.379663 0.925125i \(-0.623960\pi\)
−0.379663 + 0.925125i \(0.623960\pi\)
\(360\) 0 0
\(361\) −17.8608 −0.940044
\(362\) 28.1320 1.47859
\(363\) 0 0
\(364\) 18.9307 0.992238
\(365\) 38.6738 2.02428
\(366\) 0 0
\(367\) 13.3059 0.694564 0.347282 0.937761i \(-0.387105\pi\)
0.347282 + 0.937761i \(0.387105\pi\)
\(368\) 4.36064 0.227314
\(369\) 0 0
\(370\) −23.0980 −1.20081
\(371\) 11.4386 0.593861
\(372\) 0 0
\(373\) 20.9296 1.08370 0.541848 0.840477i \(-0.317725\pi\)
0.541848 + 0.840477i \(0.317725\pi\)
\(374\) −12.9658 −0.670447
\(375\) 0 0
\(376\) −1.89172 −0.0975580
\(377\) 4.28535 0.220707
\(378\) 0 0
\(379\) 17.6614 0.907206 0.453603 0.891204i \(-0.350138\pi\)
0.453603 + 0.891204i \(0.350138\pi\)
\(380\) −7.09321 −0.363874
\(381\) 0 0
\(382\) 21.6921 1.10987
\(383\) 24.2702 1.24015 0.620076 0.784542i \(-0.287102\pi\)
0.620076 + 0.784542i \(0.287102\pi\)
\(384\) 0 0
\(385\) −40.2329 −2.05046
\(386\) −35.1055 −1.78682
\(387\) 0 0
\(388\) 3.39370 0.172289
\(389\) 17.6760 0.896208 0.448104 0.893981i \(-0.352099\pi\)
0.448104 + 0.893981i \(0.352099\pi\)
\(390\) 0 0
\(391\) −1.49875 −0.0757948
\(392\) 0.380585 0.0192224
\(393\) 0 0
\(394\) −43.7161 −2.20238
\(395\) −3.35905 −0.169012
\(396\) 0 0
\(397\) −1.88374 −0.0945421 −0.0472711 0.998882i \(-0.515052\pi\)
−0.0472711 + 0.998882i \(0.515052\pi\)
\(398\) 52.2837 2.62075
\(399\) 0 0
\(400\) −37.6669 −1.88334
\(401\) −21.3807 −1.06770 −0.533849 0.845580i \(-0.679255\pi\)
−0.533849 + 0.845580i \(0.679255\pi\)
\(402\) 0 0
\(403\) −10.6356 −0.529798
\(404\) −9.72351 −0.483763
\(405\) 0 0
\(406\) 4.78490 0.237471
\(407\) −14.2408 −0.705891
\(408\) 0 0
\(409\) −23.6232 −1.16809 −0.584045 0.811721i \(-0.698531\pi\)
−0.584045 + 0.811721i \(0.698531\pi\)
\(410\) 32.9196 1.62578
\(411\) 0 0
\(412\) −11.0499 −0.544388
\(413\) 0.410563 0.0202025
\(414\) 0 0
\(415\) −17.0302 −0.835982
\(416\) 33.0775 1.62176
\(417\) 0 0
\(418\) −9.23346 −0.451623
\(419\) −8.32610 −0.406757 −0.203378 0.979100i \(-0.565192\pi\)
−0.203378 + 0.979100i \(0.565192\pi\)
\(420\) 0 0
\(421\) 12.2050 0.594835 0.297417 0.954748i \(-0.403875\pi\)
0.297417 + 0.954748i \(0.403875\pi\)
\(422\) 18.3171 0.891663
\(423\) 0 0
\(424\) 1.82024 0.0883989
\(425\) 12.9461 0.627976
\(426\) 0 0
\(427\) −32.2555 −1.56095
\(428\) 5.41299 0.261647
\(429\) 0 0
\(430\) −43.0789 −2.07745
\(431\) 0.126584 0.00609733 0.00304867 0.999995i \(-0.499030\pi\)
0.00304867 + 0.999995i \(0.499030\pi\)
\(432\) 0 0
\(433\) 32.4809 1.56093 0.780467 0.625197i \(-0.214981\pi\)
0.780467 + 0.625197i \(0.214981\pi\)
\(434\) −11.8755 −0.570040
\(435\) 0 0
\(436\) −23.9407 −1.14655
\(437\) −1.06731 −0.0510565
\(438\) 0 0
\(439\) 28.2416 1.34790 0.673950 0.738777i \(-0.264596\pi\)
0.673950 + 0.738777i \(0.264596\pi\)
\(440\) −6.40235 −0.305220
\(441\) 0 0
\(442\) −12.5194 −0.595486
\(443\) 7.98359 0.379312 0.189656 0.981851i \(-0.439263\pi\)
0.189656 + 0.981851i \(0.439263\pi\)
\(444\) 0 0
\(445\) 47.4974 2.25159
\(446\) 28.6918 1.35860
\(447\) 0 0
\(448\) 15.5251 0.733490
\(449\) 21.4781 1.01361 0.506807 0.862059i \(-0.330826\pi\)
0.506807 + 0.862059i \(0.330826\pi\)
\(450\) 0 0
\(451\) 20.2962 0.955710
\(452\) 32.5054 1.52892
\(453\) 0 0
\(454\) −49.4019 −2.31855
\(455\) −38.8476 −1.82120
\(456\) 0 0
\(457\) −15.5058 −0.725332 −0.362666 0.931919i \(-0.618133\pi\)
−0.362666 + 0.931919i \(0.618133\pi\)
\(458\) 43.4006 2.02798
\(459\) 0 0
\(460\) 6.64585 0.309865
\(461\) 39.6350 1.84599 0.922993 0.384817i \(-0.125735\pi\)
0.922993 + 0.384817i \(0.125735\pi\)
\(462\) 0 0
\(463\) −0.779361 −0.0362200 −0.0181100 0.999836i \(-0.505765\pi\)
−0.0181100 + 0.999836i \(0.505765\pi\)
\(464\) 4.36064 0.202437
\(465\) 0 0
\(466\) −11.5324 −0.534230
\(467\) −26.1858 −1.21173 −0.605867 0.795566i \(-0.707174\pi\)
−0.605867 + 0.795566i \(0.707174\pi\)
\(468\) 0 0
\(469\) −29.3401 −1.35480
\(470\) −34.8609 −1.60801
\(471\) 0 0
\(472\) 0.0653337 0.00300723
\(473\) −26.5598 −1.22122
\(474\) 0 0
\(475\) 9.21938 0.423014
\(476\) −6.62077 −0.303462
\(477\) 0 0
\(478\) −29.5343 −1.35087
\(479\) −15.1362 −0.691589 −0.345794 0.938310i \(-0.612390\pi\)
−0.345794 + 0.938310i \(0.612390\pi\)
\(480\) 0 0
\(481\) −13.7505 −0.626968
\(482\) −2.09966 −0.0956367
\(483\) 0 0
\(484\) 15.6517 0.711441
\(485\) −6.96419 −0.316228
\(486\) 0 0
\(487\) 27.2117 1.23308 0.616540 0.787323i \(-0.288534\pi\)
0.616540 + 0.787323i \(0.288534\pi\)
\(488\) −5.13289 −0.232355
\(489\) 0 0
\(490\) 7.01346 0.316836
\(491\) 35.7369 1.61279 0.806393 0.591380i \(-0.201417\pi\)
0.806393 + 0.591380i \(0.201417\pi\)
\(492\) 0 0
\(493\) −1.49875 −0.0675001
\(494\) −8.91552 −0.401128
\(495\) 0 0
\(496\) −10.8225 −0.485944
\(497\) 20.1885 0.905577
\(498\) 0 0
\(499\) −2.79653 −0.125190 −0.0625950 0.998039i \(-0.519938\pi\)
−0.0625950 + 0.998039i \(0.519938\pi\)
\(500\) −24.1772 −1.08124
\(501\) 0 0
\(502\) −52.1802 −2.32892
\(503\) 1.52256 0.0678877 0.0339438 0.999424i \(-0.489193\pi\)
0.0339438 + 0.999424i \(0.489193\pi\)
\(504\) 0 0
\(505\) 19.9536 0.887922
\(506\) 8.65112 0.384589
\(507\) 0 0
\(508\) −11.9021 −0.528069
\(509\) 20.7212 0.918449 0.459225 0.888320i \(-0.348127\pi\)
0.459225 + 0.888320i \(0.348127\pi\)
\(510\) 0 0
\(511\) 25.7067 1.13720
\(512\) 30.2519 1.33696
\(513\) 0 0
\(514\) 38.6798 1.70609
\(515\) 22.6754 0.999197
\(516\) 0 0
\(517\) −21.4931 −0.945264
\(518\) −15.3534 −0.674590
\(519\) 0 0
\(520\) −6.18190 −0.271094
\(521\) 38.9143 1.70486 0.852432 0.522837i \(-0.175127\pi\)
0.852432 + 0.522837i \(0.175127\pi\)
\(522\) 0 0
\(523\) −23.1164 −1.01081 −0.505405 0.862882i \(-0.668657\pi\)
−0.505405 + 0.862882i \(0.668657\pi\)
\(524\) 19.5562 0.854314
\(525\) 0 0
\(526\) 32.3704 1.41142
\(527\) 3.71968 0.162032
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 33.5437 1.45705
\(531\) 0 0
\(532\) −4.71490 −0.204417
\(533\) 19.5973 0.848855
\(534\) 0 0
\(535\) −11.1080 −0.480239
\(536\) −4.66895 −0.201668
\(537\) 0 0
\(538\) 34.5288 1.48864
\(539\) 4.32407 0.186251
\(540\) 0 0
\(541\) −1.05558 −0.0453827 −0.0226914 0.999743i \(-0.507224\pi\)
−0.0226914 + 0.999743i \(0.507224\pi\)
\(542\) 27.3250 1.17371
\(543\) 0 0
\(544\) −11.5684 −0.495993
\(545\) 49.1285 2.10444
\(546\) 0 0
\(547\) 12.3543 0.528231 0.264115 0.964491i \(-0.414920\pi\)
0.264115 + 0.964491i \(0.414920\pi\)
\(548\) −34.9859 −1.49452
\(549\) 0 0
\(550\) −74.7278 −3.18640
\(551\) −1.06731 −0.0454691
\(552\) 0 0
\(553\) −2.23278 −0.0949477
\(554\) 19.5439 0.830342
\(555\) 0 0
\(556\) −26.8403 −1.13828
\(557\) −37.8043 −1.60182 −0.800909 0.598786i \(-0.795650\pi\)
−0.800909 + 0.598786i \(0.795650\pi\)
\(558\) 0 0
\(559\) −25.6453 −1.08468
\(560\) −39.5301 −1.67045
\(561\) 0 0
\(562\) 35.9161 1.51503
\(563\) 5.16163 0.217537 0.108768 0.994067i \(-0.465309\pi\)
0.108768 + 0.994067i \(0.465309\pi\)
\(564\) 0 0
\(565\) −66.7042 −2.80627
\(566\) −50.1501 −2.10797
\(567\) 0 0
\(568\) 3.21264 0.134799
\(569\) 3.91406 0.164086 0.0820429 0.996629i \(-0.473856\pi\)
0.0820429 + 0.996629i \(0.473856\pi\)
\(570\) 0 0
\(571\) −30.6511 −1.28271 −0.641355 0.767245i \(-0.721627\pi\)
−0.641355 + 0.767245i \(0.721627\pi\)
\(572\) 34.2268 1.43109
\(573\) 0 0
\(574\) 21.8819 0.913331
\(575\) −8.63793 −0.360227
\(576\) 0 0
\(577\) −39.4740 −1.64333 −0.821663 0.569974i \(-0.806953\pi\)
−0.821663 + 0.569974i \(0.806953\pi\)
\(578\) −28.7589 −1.19621
\(579\) 0 0
\(580\) 6.64585 0.275954
\(581\) −11.3201 −0.469637
\(582\) 0 0
\(583\) 20.6810 0.856519
\(584\) 4.09076 0.169277
\(585\) 0 0
\(586\) 47.2371 1.95135
\(587\) −3.03610 −0.125313 −0.0626567 0.998035i \(-0.519957\pi\)
−0.0626567 + 0.998035i \(0.519957\pi\)
\(588\) 0 0
\(589\) 2.64892 0.109147
\(590\) 1.20398 0.0495670
\(591\) 0 0
\(592\) −13.9921 −0.575070
\(593\) 18.8795 0.775289 0.387644 0.921809i \(-0.373289\pi\)
0.387644 + 0.921809i \(0.373289\pi\)
\(594\) 0 0
\(595\) 13.5865 0.556990
\(596\) 13.9259 0.570426
\(597\) 0 0
\(598\) 8.35324 0.341589
\(599\) 12.7183 0.519657 0.259828 0.965655i \(-0.416334\pi\)
0.259828 + 0.965655i \(0.416334\pi\)
\(600\) 0 0
\(601\) 13.5436 0.552456 0.276228 0.961092i \(-0.410916\pi\)
0.276228 + 0.961092i \(0.410916\pi\)
\(602\) −28.6348 −1.16707
\(603\) 0 0
\(604\) 23.1606 0.942392
\(605\) −32.1188 −1.30581
\(606\) 0 0
\(607\) −24.2819 −0.985573 −0.492786 0.870150i \(-0.664022\pi\)
−0.492786 + 0.870150i \(0.664022\pi\)
\(608\) −8.23832 −0.334108
\(609\) 0 0
\(610\) −94.5896 −3.82982
\(611\) −20.7530 −0.839576
\(612\) 0 0
\(613\) 39.0334 1.57654 0.788272 0.615327i \(-0.210976\pi\)
0.788272 + 0.615327i \(0.210976\pi\)
\(614\) −15.4699 −0.624315
\(615\) 0 0
\(616\) −4.25569 −0.171466
\(617\) −0.677906 −0.0272915 −0.0136457 0.999907i \(-0.504344\pi\)
−0.0136457 + 0.999907i \(0.504344\pi\)
\(618\) 0 0
\(619\) −14.0270 −0.563794 −0.281897 0.959445i \(-0.590964\pi\)
−0.281897 + 0.959445i \(0.590964\pi\)
\(620\) −16.4941 −0.662418
\(621\) 0 0
\(622\) 21.0625 0.844527
\(623\) 31.5718 1.26490
\(624\) 0 0
\(625\) 6.42419 0.256968
\(626\) −48.8689 −1.95319
\(627\) 0 0
\(628\) 22.7348 0.907216
\(629\) 4.80905 0.191749
\(630\) 0 0
\(631\) −3.52595 −0.140366 −0.0701829 0.997534i \(-0.522358\pi\)
−0.0701829 + 0.997534i \(0.522358\pi\)
\(632\) −0.355308 −0.0141334
\(633\) 0 0
\(634\) 28.3211 1.12478
\(635\) 24.4242 0.969245
\(636\) 0 0
\(637\) 4.17518 0.165427
\(638\) 8.65112 0.342501
\(639\) 0 0
\(640\) −11.4826 −0.453890
\(641\) 32.2414 1.27346 0.636729 0.771088i \(-0.280287\pi\)
0.636729 + 0.771088i \(0.280287\pi\)
\(642\) 0 0
\(643\) 15.2087 0.599772 0.299886 0.953975i \(-0.403051\pi\)
0.299886 + 0.953975i \(0.403051\pi\)
\(644\) 4.41754 0.174075
\(645\) 0 0
\(646\) 3.11809 0.122680
\(647\) 8.38839 0.329781 0.164891 0.986312i \(-0.447273\pi\)
0.164891 + 0.986312i \(0.447273\pi\)
\(648\) 0 0
\(649\) 0.742299 0.0291378
\(650\) −72.1547 −2.83014
\(651\) 0 0
\(652\) −16.1402 −0.632099
\(653\) −3.35936 −0.131462 −0.0657309 0.997837i \(-0.520938\pi\)
−0.0657309 + 0.997837i \(0.520938\pi\)
\(654\) 0 0
\(655\) −40.1311 −1.56805
\(656\) 19.9416 0.778590
\(657\) 0 0
\(658\) −23.1722 −0.903348
\(659\) 21.5018 0.837591 0.418796 0.908080i \(-0.362452\pi\)
0.418796 + 0.908080i \(0.362452\pi\)
\(660\) 0 0
\(661\) 20.8099 0.809412 0.404706 0.914447i \(-0.367374\pi\)
0.404706 + 0.914447i \(0.367374\pi\)
\(662\) −5.00764 −0.194628
\(663\) 0 0
\(664\) −1.80139 −0.0699076
\(665\) 9.67543 0.375197
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) −23.3254 −0.902487
\(669\) 0 0
\(670\) −86.0401 −3.32402
\(671\) −58.3181 −2.25135
\(672\) 0 0
\(673\) −11.9922 −0.462267 −0.231133 0.972922i \(-0.574243\pi\)
−0.231133 + 0.972922i \(0.574243\pi\)
\(674\) −52.9547 −2.03974
\(675\) 0 0
\(676\) 9.65343 0.371286
\(677\) 40.8268 1.56910 0.784551 0.620064i \(-0.212893\pi\)
0.784551 + 0.620064i \(0.212893\pi\)
\(678\) 0 0
\(679\) −4.62914 −0.177650
\(680\) 2.16204 0.0829105
\(681\) 0 0
\(682\) −21.4709 −0.822162
\(683\) −23.2310 −0.888909 −0.444454 0.895801i \(-0.646602\pi\)
−0.444454 + 0.895801i \(0.646602\pi\)
\(684\) 0 0
\(685\) 71.7944 2.74312
\(686\) 38.1562 1.45681
\(687\) 0 0
\(688\) −26.0958 −0.994895
\(689\) 19.9689 0.760754
\(690\) 0 0
\(691\) −5.77244 −0.219594 −0.109797 0.993954i \(-0.535020\pi\)
−0.109797 + 0.993954i \(0.535020\pi\)
\(692\) 7.24645 0.275469
\(693\) 0 0
\(694\) −47.5238 −1.80398
\(695\) 55.0789 2.08926
\(696\) 0 0
\(697\) −6.85392 −0.259611
\(698\) −50.0627 −1.89490
\(699\) 0 0
\(700\) −38.1584 −1.44225
\(701\) −0.668564 −0.0252513 −0.0126256 0.999920i \(-0.504019\pi\)
−0.0126256 + 0.999920i \(0.504019\pi\)
\(702\) 0 0
\(703\) 3.42471 0.129165
\(704\) 28.0694 1.05790
\(705\) 0 0
\(706\) 52.6715 1.98232
\(707\) 13.2633 0.498816
\(708\) 0 0
\(709\) −8.21463 −0.308507 −0.154253 0.988031i \(-0.549297\pi\)
−0.154253 + 0.988031i \(0.549297\pi\)
\(710\) 59.2029 2.22185
\(711\) 0 0
\(712\) 5.02409 0.188286
\(713\) −2.48186 −0.0929464
\(714\) 0 0
\(715\) −70.2366 −2.62670
\(716\) 1.01717 0.0380134
\(717\) 0 0
\(718\) −28.0443 −1.04660
\(719\) −21.6313 −0.806710 −0.403355 0.915044i \(-0.632156\pi\)
−0.403355 + 0.915044i \(0.632156\pi\)
\(720\) 0 0
\(721\) 15.0725 0.561329
\(722\) −34.8154 −1.29569
\(723\) 0 0
\(724\) 25.9722 0.965249
\(725\) −8.63793 −0.320805
\(726\) 0 0
\(727\) −19.4681 −0.722031 −0.361016 0.932560i \(-0.617570\pi\)
−0.361016 + 0.932560i \(0.617570\pi\)
\(728\) −4.10915 −0.152295
\(729\) 0 0
\(730\) 75.3851 2.79013
\(731\) 8.96911 0.331735
\(732\) 0 0
\(733\) −43.4969 −1.60659 −0.803297 0.595578i \(-0.796923\pi\)
−0.803297 + 0.595578i \(0.796923\pi\)
\(734\) 25.9367 0.957340
\(735\) 0 0
\(736\) 7.71875 0.284517
\(737\) −53.0470 −1.95401
\(738\) 0 0
\(739\) 6.06700 0.223178 0.111589 0.993754i \(-0.464406\pi\)
0.111589 + 0.993754i \(0.464406\pi\)
\(740\) −21.3247 −0.783911
\(741\) 0 0
\(742\) 22.2967 0.818538
\(743\) −9.80597 −0.359746 −0.179873 0.983690i \(-0.557569\pi\)
−0.179873 + 0.983690i \(0.557569\pi\)
\(744\) 0 0
\(745\) −28.5773 −1.04699
\(746\) 40.7972 1.49369
\(747\) 0 0
\(748\) −11.9704 −0.437680
\(749\) −7.38354 −0.269789
\(750\) 0 0
\(751\) −17.4182 −0.635599 −0.317799 0.948158i \(-0.602944\pi\)
−0.317799 + 0.948158i \(0.602944\pi\)
\(752\) −21.1176 −0.770080
\(753\) 0 0
\(754\) 8.35324 0.304207
\(755\) −47.5278 −1.72971
\(756\) 0 0
\(757\) 6.99421 0.254209 0.127104 0.991889i \(-0.459432\pi\)
0.127104 + 0.991889i \(0.459432\pi\)
\(758\) 34.4266 1.25043
\(759\) 0 0
\(760\) 1.53967 0.0558498
\(761\) 22.9466 0.831813 0.415906 0.909407i \(-0.363464\pi\)
0.415906 + 0.909407i \(0.363464\pi\)
\(762\) 0 0
\(763\) 32.6560 1.18223
\(764\) 20.0267 0.724541
\(765\) 0 0
\(766\) 47.3089 1.70934
\(767\) 0.716740 0.0258800
\(768\) 0 0
\(769\) −12.9447 −0.466798 −0.233399 0.972381i \(-0.574985\pi\)
−0.233399 + 0.972381i \(0.574985\pi\)
\(770\) −78.4243 −2.82622
\(771\) 0 0
\(772\) −32.4103 −1.16647
\(773\) −25.9962 −0.935020 −0.467510 0.883988i \(-0.654849\pi\)
−0.467510 + 0.883988i \(0.654849\pi\)
\(774\) 0 0
\(775\) 21.4381 0.770080
\(776\) −0.736645 −0.0264440
\(777\) 0 0
\(778\) 34.4550 1.23527
\(779\) −4.88093 −0.174878
\(780\) 0 0
\(781\) 36.5009 1.30610
\(782\) −2.92144 −0.104470
\(783\) 0 0
\(784\) 4.24853 0.151733
\(785\) −46.6539 −1.66515
\(786\) 0 0
\(787\) 55.5880 1.98150 0.990750 0.135701i \(-0.0433286\pi\)
0.990750 + 0.135701i \(0.0433286\pi\)
\(788\) −40.3597 −1.43776
\(789\) 0 0
\(790\) −6.54766 −0.232955
\(791\) −44.3387 −1.57650
\(792\) 0 0
\(793\) −56.3101 −1.99963
\(794\) −3.67189 −0.130311
\(795\) 0 0
\(796\) 48.2696 1.71087
\(797\) 4.13366 0.146422 0.0732110 0.997316i \(-0.476675\pi\)
0.0732110 + 0.997316i \(0.476675\pi\)
\(798\) 0 0
\(799\) 7.25810 0.256773
\(800\) −66.6740 −2.35728
\(801\) 0 0
\(802\) −41.6764 −1.47164
\(803\) 46.4778 1.64017
\(804\) 0 0
\(805\) −9.06522 −0.319507
\(806\) −20.7316 −0.730239
\(807\) 0 0
\(808\) 2.11061 0.0742511
\(809\) −27.6896 −0.973514 −0.486757 0.873537i \(-0.661820\pi\)
−0.486757 + 0.873537i \(0.661820\pi\)
\(810\) 0 0
\(811\) −1.07695 −0.0378168 −0.0189084 0.999821i \(-0.506019\pi\)
−0.0189084 + 0.999821i \(0.506019\pi\)
\(812\) 4.41754 0.155025
\(813\) 0 0
\(814\) −27.7590 −0.972953
\(815\) 33.1213 1.16019
\(816\) 0 0
\(817\) 6.38724 0.223461
\(818\) −46.0476 −1.61002
\(819\) 0 0
\(820\) 30.3922 1.06134
\(821\) −41.8221 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(822\) 0 0
\(823\) −16.1258 −0.562109 −0.281055 0.959692i \(-0.590684\pi\)
−0.281055 + 0.959692i \(0.590684\pi\)
\(824\) 2.39852 0.0835563
\(825\) 0 0
\(826\) 0.800292 0.0278457
\(827\) 19.5277 0.679046 0.339523 0.940598i \(-0.389734\pi\)
0.339523 + 0.940598i \(0.389734\pi\)
\(828\) 0 0
\(829\) −21.6037 −0.750327 −0.375163 0.926959i \(-0.622413\pi\)
−0.375163 + 0.926959i \(0.622413\pi\)
\(830\) −33.1963 −1.15226
\(831\) 0 0
\(832\) 27.1029 0.939623
\(833\) −1.46022 −0.0505935
\(834\) 0 0
\(835\) 47.8660 1.65647
\(836\) −8.52456 −0.294828
\(837\) 0 0
\(838\) −16.2297 −0.560646
\(839\) 41.4413 1.43071 0.715357 0.698759i \(-0.246264\pi\)
0.715357 + 0.698759i \(0.246264\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 23.7907 0.819880
\(843\) 0 0
\(844\) 16.9108 0.582094
\(845\) −19.8098 −0.681477
\(846\) 0 0
\(847\) −21.3496 −0.733580
\(848\) 20.3197 0.697782
\(849\) 0 0
\(850\) 25.2352 0.865560
\(851\) −3.20872 −0.109993
\(852\) 0 0
\(853\) −6.19307 −0.212047 −0.106023 0.994364i \(-0.533812\pi\)
−0.106023 + 0.994364i \(0.533812\pi\)
\(854\) −62.8743 −2.15152
\(855\) 0 0
\(856\) −1.17496 −0.0401592
\(857\) 46.6836 1.59468 0.797340 0.603530i \(-0.206240\pi\)
0.797340 + 0.603530i \(0.206240\pi\)
\(858\) 0 0
\(859\) 34.7061 1.18416 0.592079 0.805880i \(-0.298307\pi\)
0.592079 + 0.805880i \(0.298307\pi\)
\(860\) −39.7715 −1.35620
\(861\) 0 0
\(862\) 0.246745 0.00840416
\(863\) −11.5543 −0.393312 −0.196656 0.980473i \(-0.563008\pi\)
−0.196656 + 0.980473i \(0.563008\pi\)
\(864\) 0 0
\(865\) −14.8704 −0.505609
\(866\) 63.3137 2.15149
\(867\) 0 0
\(868\) −10.9637 −0.372133
\(869\) −4.03688 −0.136942
\(870\) 0 0
\(871\) −51.2205 −1.73554
\(872\) 5.19663 0.175980
\(873\) 0 0
\(874\) −2.08047 −0.0703729
\(875\) 32.9786 1.11488
\(876\) 0 0
\(877\) 29.6949 1.00273 0.501363 0.865237i \(-0.332832\pi\)
0.501363 + 0.865237i \(0.332832\pi\)
\(878\) 55.0502 1.85785
\(879\) 0 0
\(880\) −71.4706 −2.40927
\(881\) 2.14823 0.0723756 0.0361878 0.999345i \(-0.488479\pi\)
0.0361878 + 0.999345i \(0.488479\pi\)
\(882\) 0 0
\(883\) 12.0264 0.404720 0.202360 0.979311i \(-0.435139\pi\)
0.202360 + 0.979311i \(0.435139\pi\)
\(884\) −11.5582 −0.388745
\(885\) 0 0
\(886\) 15.5621 0.522818
\(887\) 47.0278 1.57904 0.789520 0.613725i \(-0.210329\pi\)
0.789520 + 0.613725i \(0.210329\pi\)
\(888\) 0 0
\(889\) 16.2349 0.544502
\(890\) 92.5846 3.10344
\(891\) 0 0
\(892\) 26.4890 0.886916
\(893\) 5.16876 0.172966
\(894\) 0 0
\(895\) −2.08733 −0.0697717
\(896\) −7.63256 −0.254986
\(897\) 0 0
\(898\) 41.8663 1.39710
\(899\) −2.48186 −0.0827747
\(900\) 0 0
\(901\) −6.98386 −0.232666
\(902\) 39.5625 1.31729
\(903\) 0 0
\(904\) −7.05571 −0.234669
\(905\) −53.2974 −1.77167
\(906\) 0 0
\(907\) 35.1405 1.16682 0.583411 0.812177i \(-0.301718\pi\)
0.583411 + 0.812177i \(0.301718\pi\)
\(908\) −45.6091 −1.51359
\(909\) 0 0
\(910\) −75.7240 −2.51023
\(911\) −43.8758 −1.45367 −0.726835 0.686812i \(-0.759010\pi\)
−0.726835 + 0.686812i \(0.759010\pi\)
\(912\) 0 0
\(913\) −20.4668 −0.677353
\(914\) −30.2249 −0.999749
\(915\) 0 0
\(916\) 40.0685 1.32390
\(917\) −26.6754 −0.880899
\(918\) 0 0
\(919\) −45.6219 −1.50493 −0.752463 0.658634i \(-0.771135\pi\)
−0.752463 + 0.658634i \(0.771135\pi\)
\(920\) −1.44257 −0.0475600
\(921\) 0 0
\(922\) 77.2588 2.54438
\(923\) 35.2440 1.16007
\(924\) 0 0
\(925\) 27.7167 0.911319
\(926\) −1.51918 −0.0499232
\(927\) 0 0
\(928\) 7.71875 0.253380
\(929\) −30.1167 −0.988096 −0.494048 0.869435i \(-0.664483\pi\)
−0.494048 + 0.869435i \(0.664483\pi\)
\(930\) 0 0
\(931\) −1.03987 −0.0340805
\(932\) −10.6470 −0.348755
\(933\) 0 0
\(934\) −51.0428 −1.67017
\(935\) 24.5644 0.803341
\(936\) 0 0
\(937\) −57.7787 −1.88755 −0.943773 0.330594i \(-0.892751\pi\)
−0.943773 + 0.330594i \(0.892751\pi\)
\(938\) −57.1914 −1.86736
\(939\) 0 0
\(940\) −32.1844 −1.04974
\(941\) −4.84815 −0.158045 −0.0790226 0.996873i \(-0.525180\pi\)
−0.0790226 + 0.996873i \(0.525180\pi\)
\(942\) 0 0
\(943\) 4.57310 0.148921
\(944\) 0.729332 0.0237377
\(945\) 0 0
\(946\) −51.7719 −1.68325
\(947\) 24.7151 0.803134 0.401567 0.915830i \(-0.368466\pi\)
0.401567 + 0.915830i \(0.368466\pi\)
\(948\) 0 0
\(949\) 44.8774 1.45678
\(950\) 17.9709 0.583054
\(951\) 0 0
\(952\) 1.43712 0.0465774
\(953\) −32.7211 −1.05994 −0.529970 0.848016i \(-0.677797\pi\)
−0.529970 + 0.848016i \(0.677797\pi\)
\(954\) 0 0
\(955\) −41.0967 −1.32986
\(956\) −27.2668 −0.881870
\(957\) 0 0
\(958\) −29.5043 −0.953239
\(959\) 47.7222 1.54103
\(960\) 0 0
\(961\) −24.8404 −0.801302
\(962\) −26.8032 −0.864170
\(963\) 0 0
\(964\) −1.93845 −0.0624334
\(965\) 66.5090 2.14100
\(966\) 0 0
\(967\) −37.3079 −1.19974 −0.599870 0.800097i \(-0.704781\pi\)
−0.599870 + 0.800097i \(0.704781\pi\)
\(968\) −3.39740 −0.109197
\(969\) 0 0
\(970\) −13.5750 −0.435867
\(971\) −27.8228 −0.892876 −0.446438 0.894815i \(-0.647308\pi\)
−0.446438 + 0.894815i \(0.647308\pi\)
\(972\) 0 0
\(973\) 36.6113 1.17371
\(974\) 53.0426 1.69960
\(975\) 0 0
\(976\) −57.2994 −1.83411
\(977\) 54.0284 1.72852 0.864260 0.503045i \(-0.167787\pi\)
0.864260 + 0.503045i \(0.167787\pi\)
\(978\) 0 0
\(979\) 57.0820 1.82435
\(980\) 6.47500 0.206836
\(981\) 0 0
\(982\) 69.6605 2.22296
\(983\) 15.2109 0.485154 0.242577 0.970132i \(-0.422007\pi\)
0.242577 + 0.970132i \(0.422007\pi\)
\(984\) 0 0
\(985\) 82.8221 2.63893
\(986\) −2.92144 −0.0930376
\(987\) 0 0
\(988\) −8.23103 −0.261864
\(989\) −5.98441 −0.190293
\(990\) 0 0
\(991\) 1.74186 0.0553321 0.0276660 0.999617i \(-0.491193\pi\)
0.0276660 + 0.999617i \(0.491193\pi\)
\(992\) −19.1568 −0.608230
\(993\) 0 0
\(994\) 39.3525 1.24819
\(995\) −99.0539 −3.14022
\(996\) 0 0
\(997\) −16.9495 −0.536797 −0.268398 0.963308i \(-0.586494\pi\)
−0.268398 + 0.963308i \(0.586494\pi\)
\(998\) −5.45116 −0.172553
\(999\) 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 6003.2.a.s.1.16 20
3.2 odd 2 2001.2.a.o.1.5 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.o.1.5 20 3.2 odd 2
6003.2.a.s.1.16 20 1.1 even 1 trivial