Properties

Label 1305.2.a.r.1.3
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
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] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.820249\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.82025 q^{2} +1.31331 q^{4} +1.00000 q^{5} -0.729126 q^{7} -1.24995 q^{8} +O(q^{10})\) \(q+1.82025 q^{2} +1.31331 q^{4} +1.00000 q^{5} -0.729126 q^{7} -1.24995 q^{8} +1.82025 q^{10} -0.729126 q^{11} +3.38351 q^{13} -1.32719 q^{14} -4.90184 q^{16} +5.74301 q^{17} +6.11263 q^{19} +1.31331 q^{20} -1.32719 q^{22} +9.48602 q^{23} +1.00000 q^{25} +6.15883 q^{26} -0.957567 q^{28} +1.00000 q^{29} +5.48602 q^{31} -6.42266 q^{32} +10.4537 q^{34} -0.729126 q^{35} -10.2949 q^{37} +11.1265 q^{38} -1.24995 q^{40} +11.3088 q^{41} -10.1404 q^{43} -0.957567 q^{44} +17.2669 q^{46} +1.89749 q^{47} -6.46838 q^{49} +1.82025 q^{50} +4.44359 q^{52} -8.14040 q^{53} -0.729126 q^{55} +0.911372 q^{56} +1.82025 q^{58} -8.68215 q^{59} -15.5709 q^{61} +9.98592 q^{62} -1.88717 q^{64} +3.38351 q^{65} -2.55187 q^{67} +7.54234 q^{68} -1.32719 q^{70} +4.83164 q^{71} +6.29488 q^{73} -18.7392 q^{74} +8.02777 q^{76} +0.531625 q^{77} -5.39363 q^{79} -4.90184 q^{80} +20.5848 q^{82} -0.0848668 q^{83} +5.74301 q^{85} -18.4581 q^{86} +0.911372 q^{88} -4.63674 q^{89} -2.46700 q^{91} +12.4581 q^{92} +3.45390 q^{94} +6.11263 q^{95} +1.30377 q^{97} -11.7741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 5 q^{4} + 4 q^{5} + 2 q^{7} + 12 q^{8} + 3 q^{10} + 2 q^{11} - 8 q^{13} + 3 q^{14} + 11 q^{16} + 10 q^{17} - 2 q^{19} + 5 q^{20} + 3 q^{22} + 12 q^{23} + 4 q^{25} + 7 q^{26} - 9 q^{28} + 4 q^{29} - 4 q^{31} + 17 q^{32} - q^{34} + 2 q^{35} - 16 q^{37} + 10 q^{38} + 12 q^{40} + 12 q^{41} + 2 q^{43} - 9 q^{44} - 8 q^{46} + 12 q^{47} + 6 q^{49} + 3 q^{50} - 3 q^{52} + 10 q^{53} + 2 q^{55} + 3 q^{58} - 2 q^{59} - 26 q^{61} - 20 q^{62} + 34 q^{64} - 8 q^{65} + 2 q^{67} - 9 q^{68} + 3 q^{70} + 10 q^{71} - 48 q^{74} + 16 q^{76} + 34 q^{77} + 22 q^{79} + 11 q^{80} + 38 q^{82} + 10 q^{83} + 10 q^{85} + 4 q^{86} + 4 q^{89} - 8 q^{91} - 28 q^{92} + 39 q^{94} - 2 q^{95} - 22 q^{97} - 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82025 1.28711 0.643555 0.765400i \(-0.277459\pi\)
0.643555 + 0.765400i \(0.277459\pi\)
\(3\) 0 0
\(4\) 1.31331 0.656654
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.729126 −0.275584 −0.137792 0.990461i \(-0.544001\pi\)
−0.137792 + 0.990461i \(0.544001\pi\)
\(8\) −1.24995 −0.441925
\(9\) 0 0
\(10\) 1.82025 0.575613
\(11\) −0.729126 −0.219840 −0.109920 0.993940i \(-0.535059\pi\)
−0.109920 + 0.993940i \(0.535059\pi\)
\(12\) 0 0
\(13\) 3.38351 0.938416 0.469208 0.883088i \(-0.344539\pi\)
0.469208 + 0.883088i \(0.344539\pi\)
\(14\) −1.32719 −0.354707
\(15\) 0 0
\(16\) −4.90184 −1.22546
\(17\) 5.74301 1.39288 0.696442 0.717613i \(-0.254765\pi\)
0.696442 + 0.717613i \(0.254765\pi\)
\(18\) 0 0
\(19\) 6.11263 1.40233 0.701167 0.712997i \(-0.252663\pi\)
0.701167 + 0.712997i \(0.252663\pi\)
\(20\) 1.31331 0.293664
\(21\) 0 0
\(22\) −1.32719 −0.282958
\(23\) 9.48602 1.97797 0.988986 0.148010i \(-0.0472866\pi\)
0.988986 + 0.148010i \(0.0472866\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.15883 1.20785
\(27\) 0 0
\(28\) −0.957567 −0.180963
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.48602 0.985318 0.492659 0.870222i \(-0.336025\pi\)
0.492659 + 0.870222i \(0.336025\pi\)
\(32\) −6.42266 −1.13538
\(33\) 0 0
\(34\) 10.4537 1.79280
\(35\) −0.729126 −0.123245
\(36\) 0 0
\(37\) −10.2949 −1.69247 −0.846234 0.532811i \(-0.821135\pi\)
−0.846234 + 0.532811i \(0.821135\pi\)
\(38\) 11.1265 1.80496
\(39\) 0 0
\(40\) −1.24995 −0.197635
\(41\) 11.3088 1.76613 0.883066 0.469249i \(-0.155475\pi\)
0.883066 + 0.469249i \(0.155475\pi\)
\(42\) 0 0
\(43\) −10.1404 −1.54640 −0.773198 0.634164i \(-0.781344\pi\)
−0.773198 + 0.634164i \(0.781344\pi\)
\(44\) −0.957567 −0.144359
\(45\) 0 0
\(46\) 17.2669 2.54587
\(47\) 1.89749 0.276777 0.138389 0.990378i \(-0.455808\pi\)
0.138389 + 0.990378i \(0.455808\pi\)
\(48\) 0 0
\(49\) −6.46838 −0.924054
\(50\) 1.82025 0.257422
\(51\) 0 0
\(52\) 4.44359 0.616215
\(53\) −8.14040 −1.11817 −0.559085 0.829110i \(-0.688848\pi\)
−0.559085 + 0.829110i \(0.688848\pi\)
\(54\) 0 0
\(55\) −0.729126 −0.0983153
\(56\) 0.911372 0.121787
\(57\) 0 0
\(58\) 1.82025 0.239010
\(59\) −8.68215 −1.13032 −0.565160 0.824981i \(-0.691186\pi\)
−0.565160 + 0.824981i \(0.691186\pi\)
\(60\) 0 0
\(61\) −15.5709 −1.99365 −0.996824 0.0796378i \(-0.974624\pi\)
−0.996824 + 0.0796378i \(0.974624\pi\)
\(62\) 9.98592 1.26821
\(63\) 0 0
\(64\) −1.88717 −0.235897
\(65\) 3.38351 0.419673
\(66\) 0 0
\(67\) −2.55187 −0.311761 −0.155880 0.987776i \(-0.549821\pi\)
−0.155880 + 0.987776i \(0.549821\pi\)
\(68\) 7.54234 0.914643
\(69\) 0 0
\(70\) −1.32719 −0.158630
\(71\) 4.83164 0.573410 0.286705 0.958019i \(-0.407440\pi\)
0.286705 + 0.958019i \(0.407440\pi\)
\(72\) 0 0
\(73\) 6.29488 0.736760 0.368380 0.929675i \(-0.379913\pi\)
0.368380 + 0.929675i \(0.379913\pi\)
\(74\) −18.7392 −2.17839
\(75\) 0 0
\(76\) 8.02777 0.920848
\(77\) 0.531625 0.0605843
\(78\) 0 0
\(79\) −5.39363 −0.606831 −0.303415 0.952858i \(-0.598127\pi\)
−0.303415 + 0.952858i \(0.598127\pi\)
\(80\) −4.90184 −0.548042
\(81\) 0 0
\(82\) 20.5848 2.27321
\(83\) −0.0848668 −0.00931534 −0.00465767 0.999989i \(-0.501483\pi\)
−0.00465767 + 0.999989i \(0.501483\pi\)
\(84\) 0 0
\(85\) 5.74301 0.622917
\(86\) −18.4581 −1.99038
\(87\) 0 0
\(88\) 0.911372 0.0971526
\(89\) −4.63674 −0.491493 −0.245747 0.969334i \(-0.579033\pi\)
−0.245747 + 0.969334i \(0.579033\pi\)
\(90\) 0 0
\(91\) −2.46700 −0.258612
\(92\) 12.4581 1.29884
\(93\) 0 0
\(94\) 3.45390 0.356243
\(95\) 6.11263 0.627143
\(96\) 0 0
\(97\) 1.30377 0.132378 0.0661891 0.997807i \(-0.478916\pi\)
0.0661891 + 0.997807i \(0.478916\pi\)
\(98\) −11.7741 −1.18936
\(99\) 0 0
\(100\) 1.31331 0.131331
\(101\) 5.35574 0.532916 0.266458 0.963847i \(-0.414147\pi\)
0.266458 + 0.963847i \(0.414147\pi\)
\(102\) 0 0
\(103\) 16.7670 1.65210 0.826052 0.563594i \(-0.190582\pi\)
0.826052 + 0.563594i \(0.190582\pi\)
\(104\) −4.22922 −0.414709
\(105\) 0 0
\(106\) −14.8176 −1.43921
\(107\) −7.30876 −0.706565 −0.353282 0.935517i \(-0.614935\pi\)
−0.353282 + 0.935517i \(0.614935\pi\)
\(108\) 0 0
\(109\) −8.21515 −0.786868 −0.393434 0.919353i \(-0.628713\pi\)
−0.393434 + 0.919353i \(0.628713\pi\)
\(110\) −1.32719 −0.126543
\(111\) 0 0
\(112\) 3.57406 0.337717
\(113\) −8.51003 −0.800556 −0.400278 0.916394i \(-0.631086\pi\)
−0.400278 + 0.916394i \(0.631086\pi\)
\(114\) 0 0
\(115\) 9.48602 0.884576
\(116\) 1.31331 0.121938
\(117\) 0 0
\(118\) −15.8037 −1.45485
\(119\) −4.18738 −0.383856
\(120\) 0 0
\(121\) −10.4684 −0.951670
\(122\) −28.3429 −2.56605
\(123\) 0 0
\(124\) 7.20483 0.647013
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 4.51398 0.400551 0.200275 0.979740i \(-0.435816\pi\)
0.200275 + 0.979740i \(0.435816\pi\)
\(128\) 9.41020 0.831752
\(129\) 0 0
\(130\) 6.15883 0.540165
\(131\) 14.7771 1.29108 0.645542 0.763724i \(-0.276631\pi\)
0.645542 + 0.763724i \(0.276631\pi\)
\(132\) 0 0
\(133\) −4.45688 −0.386461
\(134\) −4.64504 −0.401270
\(135\) 0 0
\(136\) −7.17849 −0.615550
\(137\) 5.19114 0.443509 0.221754 0.975103i \(-0.428822\pi\)
0.221754 + 0.975103i \(0.428822\pi\)
\(138\) 0 0
\(139\) −1.15824 −0.0982406 −0.0491203 0.998793i \(-0.515642\pi\)
−0.0491203 + 0.998793i \(0.515642\pi\)
\(140\) −0.957567 −0.0809291
\(141\) 0 0
\(142\) 8.79478 0.738042
\(143\) −2.46700 −0.206301
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 11.4583 0.948292
\(147\) 0 0
\(148\) −13.5203 −1.11137
\(149\) −6.68215 −0.547423 −0.273712 0.961812i \(-0.588251\pi\)
−0.273712 + 0.961812i \(0.588251\pi\)
\(150\) 0 0
\(151\) 20.2253 1.64591 0.822955 0.568107i \(-0.192324\pi\)
0.822955 + 0.568107i \(0.192324\pi\)
\(152\) −7.64050 −0.619726
\(153\) 0 0
\(154\) 0.967690 0.0779787
\(155\) 5.48602 0.440648
\(156\) 0 0
\(157\) −23.8658 −1.90470 −0.952348 0.305014i \(-0.901339\pi\)
−0.952348 + 0.305014i \(0.901339\pi\)
\(158\) −9.81775 −0.781059
\(159\) 0 0
\(160\) −6.42266 −0.507756
\(161\) −6.91650 −0.545097
\(162\) 0 0
\(163\) −1.93538 −0.151591 −0.0757953 0.997123i \(-0.524150\pi\)
−0.0757953 + 0.997123i \(0.524150\pi\)
\(164\) 14.8519 1.15974
\(165\) 0 0
\(166\) −0.154479 −0.0119899
\(167\) 12.4025 0.959736 0.479868 0.877341i \(-0.340685\pi\)
0.479868 + 0.877341i \(0.340685\pi\)
\(168\) 0 0
\(169\) −1.55187 −0.119375
\(170\) 10.4537 0.801763
\(171\) 0 0
\(172\) −13.3175 −1.01545
\(173\) 21.7113 1.65068 0.825339 0.564637i \(-0.190984\pi\)
0.825339 + 0.564637i \(0.190984\pi\)
\(174\) 0 0
\(175\) −0.729126 −0.0551167
\(176\) 3.57406 0.269405
\(177\) 0 0
\(178\) −8.44002 −0.632606
\(179\) −6.08487 −0.454804 −0.227402 0.973801i \(-0.573023\pi\)
−0.227402 + 0.973801i \(0.573023\pi\)
\(180\) 0 0
\(181\) 6.77714 0.503741 0.251870 0.967761i \(-0.418954\pi\)
0.251870 + 0.967761i \(0.418954\pi\)
\(182\) −4.49056 −0.332863
\(183\) 0 0
\(184\) −11.8571 −0.874115
\(185\) −10.2949 −0.756895
\(186\) 0 0
\(187\) −4.18738 −0.306211
\(188\) 2.49199 0.181747
\(189\) 0 0
\(190\) 11.1265 0.807202
\(191\) 18.9645 1.37222 0.686112 0.727496i \(-0.259316\pi\)
0.686112 + 0.727496i \(0.259316\pi\)
\(192\) 0 0
\(193\) 3.69760 0.266159 0.133079 0.991105i \(-0.457513\pi\)
0.133079 + 0.991105i \(0.457513\pi\)
\(194\) 2.37319 0.170385
\(195\) 0 0
\(196\) −8.49496 −0.606783
\(197\) −20.4783 −1.45902 −0.729509 0.683971i \(-0.760252\pi\)
−0.729509 + 0.683971i \(0.760252\pi\)
\(198\) 0 0
\(199\) −10.8418 −0.768552 −0.384276 0.923218i \(-0.625549\pi\)
−0.384276 + 0.923218i \(0.625549\pi\)
\(200\) −1.24995 −0.0883849
\(201\) 0 0
\(202\) 9.74878 0.685922
\(203\) −0.729126 −0.0511746
\(204\) 0 0
\(205\) 11.3088 0.789838
\(206\) 30.5201 2.12644
\(207\) 0 0
\(208\) −16.5854 −1.14999
\(209\) −4.45688 −0.308289
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −10.6908 −0.734251
\(213\) 0 0
\(214\) −13.3038 −0.909427
\(215\) −10.1404 −0.691570
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −14.9536 −1.01279
\(219\) 0 0
\(220\) −0.957567 −0.0645591
\(221\) 19.4315 1.30711
\(222\) 0 0
\(223\) −4.21515 −0.282267 −0.141134 0.989991i \(-0.545075\pi\)
−0.141134 + 0.989991i \(0.545075\pi\)
\(224\) 4.68293 0.312892
\(225\) 0 0
\(226\) −15.4904 −1.03040
\(227\) 15.4214 1.02355 0.511777 0.859118i \(-0.328987\pi\)
0.511777 + 0.859118i \(0.328987\pi\)
\(228\) 0 0
\(229\) −24.2530 −1.60269 −0.801343 0.598205i \(-0.795881\pi\)
−0.801343 + 0.598205i \(0.795881\pi\)
\(230\) 17.2669 1.13855
\(231\) 0 0
\(232\) −1.24995 −0.0820634
\(233\) 5.33653 0.349608 0.174804 0.984603i \(-0.444071\pi\)
0.174804 + 0.984603i \(0.444071\pi\)
\(234\) 0 0
\(235\) 1.89749 0.123779
\(236\) −11.4023 −0.742229
\(237\) 0 0
\(238\) −7.62207 −0.494066
\(239\) −16.0202 −1.03626 −0.518132 0.855301i \(-0.673372\pi\)
−0.518132 + 0.855301i \(0.673372\pi\)
\(240\) 0 0
\(241\) −6.60741 −0.425620 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(242\) −19.0551 −1.22491
\(243\) 0 0
\(244\) −20.4494 −1.30914
\(245\) −6.46838 −0.413249
\(246\) 0 0
\(247\) 20.6821 1.31597
\(248\) −6.85726 −0.435436
\(249\) 0 0
\(250\) 1.82025 0.115123
\(251\) −25.6834 −1.62112 −0.810560 0.585655i \(-0.800837\pi\)
−0.810560 + 0.585655i \(0.800837\pi\)
\(252\) 0 0
\(253\) −6.91650 −0.434837
\(254\) 8.21657 0.515553
\(255\) 0 0
\(256\) 20.9033 1.30645
\(257\) 7.93538 0.494995 0.247498 0.968888i \(-0.420392\pi\)
0.247498 + 0.968888i \(0.420392\pi\)
\(258\) 0 0
\(259\) 7.50627 0.466417
\(260\) 4.44359 0.275580
\(261\) 0 0
\(262\) 26.8981 1.66177
\(263\) −23.5618 −1.45288 −0.726441 0.687228i \(-0.758827\pi\)
−0.726441 + 0.687228i \(0.758827\pi\)
\(264\) 0 0
\(265\) −8.14040 −0.500061
\(266\) −8.11263 −0.497418
\(267\) 0 0
\(268\) −3.35139 −0.204719
\(269\) 10.6645 0.650226 0.325113 0.945675i \(-0.394598\pi\)
0.325113 + 0.945675i \(0.394598\pi\)
\(270\) 0 0
\(271\) −3.30876 −0.200993 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(272\) −28.1513 −1.70692
\(273\) 0 0
\(274\) 9.44917 0.570845
\(275\) −0.729126 −0.0439680
\(276\) 0 0
\(277\) −0.0469761 −0.00282252 −0.00141126 0.999999i \(-0.500449\pi\)
−0.00141126 + 0.999999i \(0.500449\pi\)
\(278\) −2.10828 −0.126447
\(279\) 0 0
\(280\) 0.911372 0.0544649
\(281\) −29.8037 −1.77794 −0.888969 0.457967i \(-0.848578\pi\)
−0.888969 + 0.457967i \(0.848578\pi\)
\(282\) 0 0
\(283\) 5.79498 0.344476 0.172238 0.985055i \(-0.444900\pi\)
0.172238 + 0.985055i \(0.444900\pi\)
\(284\) 6.34542 0.376532
\(285\) 0 0
\(286\) −4.49056 −0.265533
\(287\) −8.24552 −0.486717
\(288\) 0 0
\(289\) 15.9822 0.940127
\(290\) 1.82025 0.106889
\(291\) 0 0
\(292\) 8.26711 0.483796
\(293\) −2.99624 −0.175042 −0.0875211 0.996163i \(-0.527895\pi\)
−0.0875211 + 0.996163i \(0.527895\pi\)
\(294\) 0 0
\(295\) −8.68215 −0.505494
\(296\) 12.8681 0.747943
\(297\) 0 0
\(298\) −12.1632 −0.704594
\(299\) 32.0960 1.85616
\(300\) 0 0
\(301\) 7.39363 0.426162
\(302\) 36.8150 2.11847
\(303\) 0 0
\(304\) −29.9631 −1.71850
\(305\) −15.5709 −0.891586
\(306\) 0 0
\(307\) 4.05554 0.231462 0.115731 0.993281i \(-0.463079\pi\)
0.115731 + 0.993281i \(0.463079\pi\)
\(308\) 0.698187 0.0397829
\(309\) 0 0
\(310\) 9.98592 0.567162
\(311\) −13.6734 −0.775347 −0.387674 0.921797i \(-0.626721\pi\)
−0.387674 + 0.921797i \(0.626721\pi\)
\(312\) 0 0
\(313\) −4.25200 −0.240337 −0.120169 0.992753i \(-0.538344\pi\)
−0.120169 + 0.992753i \(0.538344\pi\)
\(314\) −43.4416 −2.45155
\(315\) 0 0
\(316\) −7.08350 −0.398478
\(317\) 4.28476 0.240656 0.120328 0.992734i \(-0.461605\pi\)
0.120328 + 0.992734i \(0.461605\pi\)
\(318\) 0 0
\(319\) −0.729126 −0.0408232
\(320\) −1.88717 −0.105496
\(321\) 0 0
\(322\) −12.5898 −0.701600
\(323\) 35.1049 1.95329
\(324\) 0 0
\(325\) 3.38351 0.187683
\(326\) −3.52287 −0.195114
\(327\) 0 0
\(328\) −14.1354 −0.780497
\(329\) −1.38351 −0.0762753
\(330\) 0 0
\(331\) 3.61772 0.198848 0.0994240 0.995045i \(-0.468300\pi\)
0.0994240 + 0.995045i \(0.468300\pi\)
\(332\) −0.111456 −0.00611695
\(333\) 0 0
\(334\) 22.5757 1.23529
\(335\) −2.55187 −0.139424
\(336\) 0 0
\(337\) 23.4442 1.27709 0.638543 0.769586i \(-0.279538\pi\)
0.638543 + 0.769586i \(0.279538\pi\)
\(338\) −2.82479 −0.153648
\(339\) 0 0
\(340\) 7.54234 0.409041
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 9.82014 0.530238
\(344\) 12.6750 0.683391
\(345\) 0 0
\(346\) 39.5200 2.12461
\(347\) −20.8226 −1.11781 −0.558907 0.829231i \(-0.688779\pi\)
−0.558907 + 0.829231i \(0.688779\pi\)
\(348\) 0 0
\(349\) −18.0555 −0.966491 −0.483245 0.875485i \(-0.660542\pi\)
−0.483245 + 0.875485i \(0.660542\pi\)
\(350\) −1.32719 −0.0709414
\(351\) 0 0
\(352\) 4.68293 0.249601
\(353\) 14.3645 0.764545 0.382272 0.924050i \(-0.375142\pi\)
0.382272 + 0.924050i \(0.375142\pi\)
\(354\) 0 0
\(355\) 4.83164 0.256437
\(356\) −6.08946 −0.322741
\(357\) 0 0
\(358\) −11.0760 −0.585383
\(359\) −6.20502 −0.327489 −0.163744 0.986503i \(-0.552357\pi\)
−0.163744 + 0.986503i \(0.552357\pi\)
\(360\) 0 0
\(361\) 18.3643 0.966542
\(362\) 12.3361 0.648370
\(363\) 0 0
\(364\) −3.23993 −0.169819
\(365\) 6.29488 0.329489
\(366\) 0 0
\(367\) 18.1975 0.949902 0.474951 0.880012i \(-0.342466\pi\)
0.474951 + 0.880012i \(0.342466\pi\)
\(368\) −46.4989 −2.42392
\(369\) 0 0
\(370\) −18.7392 −0.974207
\(371\) 5.93538 0.308150
\(372\) 0 0
\(373\) 34.8606 1.80501 0.902506 0.430677i \(-0.141725\pi\)
0.902506 + 0.430677i \(0.141725\pi\)
\(374\) −7.62207 −0.394128
\(375\) 0 0
\(376\) −2.37177 −0.122315
\(377\) 3.38351 0.174260
\(378\) 0 0
\(379\) −6.41005 −0.329262 −0.164631 0.986355i \(-0.552643\pi\)
−0.164631 + 0.986355i \(0.552643\pi\)
\(380\) 8.02777 0.411816
\(381\) 0 0
\(382\) 34.5201 1.76620
\(383\) −20.7202 −1.05875 −0.529376 0.848387i \(-0.677574\pi\)
−0.529376 + 0.848387i \(0.677574\pi\)
\(384\) 0 0
\(385\) 0.531625 0.0270941
\(386\) 6.73055 0.342576
\(387\) 0 0
\(388\) 1.71226 0.0869266
\(389\) 24.7581 1.25528 0.627642 0.778502i \(-0.284020\pi\)
0.627642 + 0.778502i \(0.284020\pi\)
\(390\) 0 0
\(391\) 54.4783 2.75509
\(392\) 8.08516 0.408362
\(393\) 0 0
\(394\) −37.2756 −1.87792
\(395\) −5.39363 −0.271383
\(396\) 0 0
\(397\) −12.7468 −0.639742 −0.319871 0.947461i \(-0.603640\pi\)
−0.319871 + 0.947461i \(0.603640\pi\)
\(398\) −19.7347 −0.989211
\(399\) 0 0
\(400\) −4.90184 −0.245092
\(401\) −28.1959 −1.40804 −0.704019 0.710181i \(-0.748613\pi\)
−0.704019 + 0.710181i \(0.748613\pi\)
\(402\) 0 0
\(403\) 18.5620 0.924639
\(404\) 7.03373 0.349941
\(405\) 0 0
\(406\) −1.32719 −0.0658674
\(407\) 7.50627 0.372072
\(408\) 0 0
\(409\) −24.0112 −1.18728 −0.593638 0.804732i \(-0.702309\pi\)
−0.593638 + 0.804732i \(0.702309\pi\)
\(410\) 20.5848 1.01661
\(411\) 0 0
\(412\) 22.0202 1.08486
\(413\) 6.33038 0.311498
\(414\) 0 0
\(415\) −0.0848668 −0.00416595
\(416\) −21.7311 −1.06546
\(417\) 0 0
\(418\) −8.11263 −0.396802
\(419\) −14.7670 −0.721416 −0.360708 0.932679i \(-0.617465\pi\)
−0.360708 + 0.932679i \(0.617465\pi\)
\(420\) 0 0
\(421\) −12.7302 −0.620430 −0.310215 0.950666i \(-0.600401\pi\)
−0.310215 + 0.950666i \(0.600401\pi\)
\(422\) 3.64050 0.177217
\(423\) 0 0
\(424\) 10.1751 0.494147
\(425\) 5.74301 0.278577
\(426\) 0 0
\(427\) 11.3531 0.549417
\(428\) −9.59865 −0.463968
\(429\) 0 0
\(430\) −18.4581 −0.890127
\(431\) −9.66327 −0.465464 −0.232732 0.972541i \(-0.574766\pi\)
−0.232732 + 0.972541i \(0.574766\pi\)
\(432\) 0 0
\(433\) 13.1190 0.630459 0.315229 0.949016i \(-0.397919\pi\)
0.315229 + 0.949016i \(0.397919\pi\)
\(434\) −7.28100 −0.349499
\(435\) 0 0
\(436\) −10.7890 −0.516700
\(437\) 57.9846 2.77378
\(438\) 0 0
\(439\) −19.9253 −0.950981 −0.475490 0.879721i \(-0.657729\pi\)
−0.475490 + 0.879721i \(0.657729\pi\)
\(440\) 0.911372 0.0434480
\(441\) 0 0
\(442\) 35.3702 1.68239
\(443\) −1.94304 −0.0923167 −0.0461583 0.998934i \(-0.514698\pi\)
−0.0461583 + 0.998934i \(0.514698\pi\)
\(444\) 0 0
\(445\) −4.63674 −0.219802
\(446\) −7.67262 −0.363309
\(447\) 0 0
\(448\) 1.37599 0.0650093
\(449\) −17.0215 −0.803293 −0.401647 0.915795i \(-0.631562\pi\)
−0.401647 + 0.915795i \(0.631562\pi\)
\(450\) 0 0
\(451\) −8.24552 −0.388266
\(452\) −11.1763 −0.525688
\(453\) 0 0
\(454\) 28.0708 1.31743
\(455\) −2.46700 −0.115655
\(456\) 0 0
\(457\) 15.9633 0.746731 0.373366 0.927684i \(-0.378204\pi\)
0.373366 + 0.927684i \(0.378204\pi\)
\(458\) −44.1466 −2.06283
\(459\) 0 0
\(460\) 12.4581 0.580860
\(461\) 17.6910 0.823954 0.411977 0.911194i \(-0.364838\pi\)
0.411977 + 0.911194i \(0.364838\pi\)
\(462\) 0 0
\(463\) −2.60741 −0.121176 −0.0605882 0.998163i \(-0.519298\pi\)
−0.0605882 + 0.998163i \(0.519298\pi\)
\(464\) −4.90184 −0.227562
\(465\) 0 0
\(466\) 9.71382 0.449984
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 1.86064 0.0859162
\(470\) 3.45390 0.159317
\(471\) 0 0
\(472\) 10.8523 0.499516
\(473\) 7.39363 0.339960
\(474\) 0 0
\(475\) 6.11263 0.280467
\(476\) −5.49931 −0.252061
\(477\) 0 0
\(478\) −29.1608 −1.33379
\(479\) −20.6833 −0.945045 −0.472523 0.881319i \(-0.656657\pi\)
−0.472523 + 0.881319i \(0.656657\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) −12.0271 −0.547821
\(483\) 0 0
\(484\) −13.7482 −0.624918
\(485\) 1.30377 0.0592013
\(486\) 0 0
\(487\) −32.1035 −1.45475 −0.727375 0.686240i \(-0.759260\pi\)
−0.727375 + 0.686240i \(0.759260\pi\)
\(488\) 19.4629 0.881042
\(489\) 0 0
\(490\) −11.7741 −0.531898
\(491\) 40.6933 1.83646 0.918232 0.396044i \(-0.129617\pi\)
0.918232 + 0.396044i \(0.129617\pi\)
\(492\) 0 0
\(493\) 5.74301 0.258652
\(494\) 37.6467 1.69380
\(495\) 0 0
\(496\) −26.8916 −1.20747
\(497\) −3.52287 −0.158022
\(498\) 0 0
\(499\) 10.8418 0.485344 0.242672 0.970108i \(-0.421976\pi\)
0.242672 + 0.970108i \(0.421976\pi\)
\(500\) 1.31331 0.0587329
\(501\) 0 0
\(502\) −46.7502 −2.08656
\(503\) 2.47727 0.110456 0.0552280 0.998474i \(-0.482411\pi\)
0.0552280 + 0.998474i \(0.482411\pi\)
\(504\) 0 0
\(505\) 5.35574 0.238327
\(506\) −12.5898 −0.559683
\(507\) 0 0
\(508\) 5.92824 0.263023
\(509\) 24.3101 1.07753 0.538764 0.842457i \(-0.318891\pi\)
0.538764 + 0.842457i \(0.318891\pi\)
\(510\) 0 0
\(511\) −4.58976 −0.203039
\(512\) 19.2287 0.849798
\(513\) 0 0
\(514\) 14.4444 0.637114
\(515\) 16.7670 0.738843
\(516\) 0 0
\(517\) −1.38351 −0.0608466
\(518\) 13.6633 0.600330
\(519\) 0 0
\(520\) −4.22922 −0.185464
\(521\) −25.2997 −1.10840 −0.554200 0.832384i \(-0.686976\pi\)
−0.554200 + 0.832384i \(0.686976\pi\)
\(522\) 0 0
\(523\) 39.2277 1.71531 0.857653 0.514228i \(-0.171922\pi\)
0.857653 + 0.514228i \(0.171922\pi\)
\(524\) 19.4069 0.847796
\(525\) 0 0
\(526\) −42.8884 −1.87002
\(527\) 31.5063 1.37243
\(528\) 0 0
\(529\) 66.9846 2.91237
\(530\) −14.8176 −0.643634
\(531\) 0 0
\(532\) −5.85325 −0.253771
\(533\) 38.2633 1.65737
\(534\) 0 0
\(535\) −7.30876 −0.315985
\(536\) 3.18972 0.137775
\(537\) 0 0
\(538\) 19.4121 0.836913
\(539\) 4.71626 0.203144
\(540\) 0 0
\(541\) 6.28080 0.270033 0.135016 0.990843i \(-0.456891\pi\)
0.135016 + 0.990843i \(0.456891\pi\)
\(542\) −6.02278 −0.258700
\(543\) 0 0
\(544\) −36.8854 −1.58145
\(545\) −8.21515 −0.351898
\(546\) 0 0
\(547\) −20.2809 −0.867151 −0.433575 0.901117i \(-0.642748\pi\)
−0.433575 + 0.901117i \(0.642748\pi\)
\(548\) 6.81756 0.291232
\(549\) 0 0
\(550\) −1.32719 −0.0565916
\(551\) 6.11263 0.260407
\(552\) 0 0
\(553\) 3.93264 0.167233
\(554\) −0.0855081 −0.00363289
\(555\) 0 0
\(556\) −1.52112 −0.0645100
\(557\) −16.3252 −0.691720 −0.345860 0.938286i \(-0.612413\pi\)
−0.345860 + 0.938286i \(0.612413\pi\)
\(558\) 0 0
\(559\) −34.3101 −1.45116
\(560\) 3.57406 0.151032
\(561\) 0 0
\(562\) −54.2501 −2.28840
\(563\) 12.8037 0.539613 0.269806 0.962915i \(-0.413040\pi\)
0.269806 + 0.962915i \(0.413040\pi\)
\(564\) 0 0
\(565\) −8.51003 −0.358020
\(566\) 10.5483 0.443378
\(567\) 0 0
\(568\) −6.03931 −0.253404
\(569\) −9.35574 −0.392213 −0.196107 0.980583i \(-0.562830\pi\)
−0.196107 + 0.980583i \(0.562830\pi\)
\(570\) 0 0
\(571\) −27.0125 −1.13044 −0.565220 0.824940i \(-0.691209\pi\)
−0.565220 + 0.824940i \(0.691209\pi\)
\(572\) −3.23993 −0.135468
\(573\) 0 0
\(574\) −15.0089 −0.626459
\(575\) 9.48602 0.395594
\(576\) 0 0
\(577\) −25.9859 −1.08181 −0.540904 0.841084i \(-0.681918\pi\)
−0.540904 + 0.841084i \(0.681918\pi\)
\(578\) 29.0915 1.21005
\(579\) 0 0
\(580\) 1.31331 0.0545321
\(581\) 0.0618786 0.00256716
\(582\) 0 0
\(583\) 5.93538 0.245818
\(584\) −7.86830 −0.325592
\(585\) 0 0
\(586\) −5.45390 −0.225299
\(587\) −19.6467 −0.810905 −0.405452 0.914116i \(-0.632886\pi\)
−0.405452 + 0.914116i \(0.632886\pi\)
\(588\) 0 0
\(589\) 33.5340 1.38175
\(590\) −15.8037 −0.650627
\(591\) 0 0
\(592\) 50.4638 2.07405
\(593\) −26.1682 −1.07460 −0.537299 0.843392i \(-0.680555\pi\)
−0.537299 + 0.843392i \(0.680555\pi\)
\(594\) 0 0
\(595\) −4.18738 −0.171666
\(596\) −8.77572 −0.359467
\(597\) 0 0
\(598\) 58.4228 2.38908
\(599\) 3.45565 0.141194 0.0705970 0.997505i \(-0.477510\pi\)
0.0705970 + 0.997505i \(0.477510\pi\)
\(600\) 0 0
\(601\) 13.5063 0.550932 0.275466 0.961311i \(-0.411168\pi\)
0.275466 + 0.961311i \(0.411168\pi\)
\(602\) 13.4583 0.548517
\(603\) 0 0
\(604\) 26.5620 1.08079
\(605\) −10.4684 −0.425600
\(606\) 0 0
\(607\) 12.0773 0.490204 0.245102 0.969497i \(-0.421178\pi\)
0.245102 + 0.969497i \(0.421178\pi\)
\(608\) −39.2594 −1.59218
\(609\) 0 0
\(610\) −28.3429 −1.14757
\(611\) 6.42017 0.259732
\(612\) 0 0
\(613\) −19.9935 −0.807531 −0.403765 0.914863i \(-0.632299\pi\)
−0.403765 + 0.914863i \(0.632299\pi\)
\(614\) 7.38209 0.297917
\(615\) 0 0
\(616\) −0.664505 −0.0267737
\(617\) 31.9784 1.28740 0.643701 0.765277i \(-0.277398\pi\)
0.643701 + 0.765277i \(0.277398\pi\)
\(618\) 0 0
\(619\) −33.0862 −1.32985 −0.664924 0.746911i \(-0.731536\pi\)
−0.664924 + 0.746911i \(0.731536\pi\)
\(620\) 7.20483 0.289353
\(621\) 0 0
\(622\) −24.8890 −0.997958
\(623\) 3.38077 0.135448
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −7.73970 −0.309341
\(627\) 0 0
\(628\) −31.3431 −1.25073
\(629\) −59.1236 −2.35741
\(630\) 0 0
\(631\) −3.02654 −0.120485 −0.0602423 0.998184i \(-0.519187\pi\)
−0.0602423 + 0.998184i \(0.519187\pi\)
\(632\) 6.74178 0.268174
\(633\) 0 0
\(634\) 7.79933 0.309751
\(635\) 4.51398 0.179132
\(636\) 0 0
\(637\) −21.8858 −0.867147
\(638\) −1.32719 −0.0525440
\(639\) 0 0
\(640\) 9.41020 0.371971
\(641\) −14.0747 −0.555919 −0.277959 0.960593i \(-0.589658\pi\)
−0.277959 + 0.960593i \(0.589658\pi\)
\(642\) 0 0
\(643\) 4.25044 0.167621 0.0838104 0.996482i \(-0.473291\pi\)
0.0838104 + 0.996482i \(0.473291\pi\)
\(644\) −9.08350 −0.357940
\(645\) 0 0
\(646\) 63.8997 2.51410
\(647\) 43.0185 1.69123 0.845616 0.533792i \(-0.179234\pi\)
0.845616 + 0.533792i \(0.179234\pi\)
\(648\) 0 0
\(649\) 6.33038 0.248489
\(650\) 6.15883 0.241569
\(651\) 0 0
\(652\) −2.54175 −0.0995425
\(653\) −18.5175 −0.724648 −0.362324 0.932052i \(-0.618017\pi\)
−0.362324 + 0.932052i \(0.618017\pi\)
\(654\) 0 0
\(655\) 14.7771 0.577391
\(656\) −55.4337 −2.16432
\(657\) 0 0
\(658\) −2.51833 −0.0981747
\(659\) −6.19736 −0.241415 −0.120707 0.992688i \(-0.538516\pi\)
−0.120707 + 0.992688i \(0.538516\pi\)
\(660\) 0 0
\(661\) −29.0936 −1.13161 −0.565805 0.824539i \(-0.691435\pi\)
−0.565805 + 0.824539i \(0.691435\pi\)
\(662\) 6.58516 0.255939
\(663\) 0 0
\(664\) 0.106079 0.00411668
\(665\) −4.45688 −0.172830
\(666\) 0 0
\(667\) 9.48602 0.367300
\(668\) 16.2883 0.630214
\(669\) 0 0
\(670\) −4.64504 −0.179454
\(671\) 11.3531 0.438283
\(672\) 0 0
\(673\) −36.6288 −1.41194 −0.705969 0.708243i \(-0.749488\pi\)
−0.705969 + 0.708243i \(0.749488\pi\)
\(674\) 42.6742 1.64375
\(675\) 0 0
\(676\) −2.03808 −0.0783878
\(677\) −25.9330 −0.996686 −0.498343 0.866980i \(-0.666058\pi\)
−0.498343 + 0.866980i \(0.666058\pi\)
\(678\) 0 0
\(679\) −0.950615 −0.0364813
\(680\) −7.17849 −0.275282
\(681\) 0 0
\(682\) −7.28100 −0.278804
\(683\) −22.4403 −0.858653 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(684\) 0 0
\(685\) 5.19114 0.198343
\(686\) 17.8751 0.682475
\(687\) 0 0
\(688\) 49.7066 1.89505
\(689\) −27.5431 −1.04931
\(690\) 0 0
\(691\) 16.3000 0.620082 0.310041 0.950723i \(-0.399657\pi\)
0.310041 + 0.950723i \(0.399657\pi\)
\(692\) 28.5136 1.08392
\(693\) 0 0
\(694\) −37.9022 −1.43875
\(695\) −1.15824 −0.0439345
\(696\) 0 0
\(697\) 64.9463 2.46002
\(698\) −32.8656 −1.24398
\(699\) 0 0
\(700\) −0.957567 −0.0361926
\(701\) 10.9811 0.414751 0.207376 0.978261i \(-0.433508\pi\)
0.207376 + 0.978261i \(0.433508\pi\)
\(702\) 0 0
\(703\) −62.9288 −2.37341
\(704\) 1.37599 0.0518595
\(705\) 0 0
\(706\) 26.1470 0.984054
\(707\) −3.90501 −0.146863
\(708\) 0 0
\(709\) −33.0681 −1.24190 −0.620949 0.783851i \(-0.713252\pi\)
−0.620949 + 0.783851i \(0.713252\pi\)
\(710\) 8.79478 0.330062
\(711\) 0 0
\(712\) 5.79570 0.217203
\(713\) 52.0405 1.94893
\(714\) 0 0
\(715\) −2.46700 −0.0922607
\(716\) −7.99130 −0.298649
\(717\) 0 0
\(718\) −11.2947 −0.421514
\(719\) 44.0289 1.64200 0.821001 0.570926i \(-0.193416\pi\)
0.821001 + 0.570926i \(0.193416\pi\)
\(720\) 0 0
\(721\) −12.2253 −0.455293
\(722\) 33.4276 1.24405
\(723\) 0 0
\(724\) 8.90047 0.330783
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) 20.2708 0.751803 0.375902 0.926660i \(-0.377333\pi\)
0.375902 + 0.926660i \(0.377333\pi\)
\(728\) 3.08364 0.114287
\(729\) 0 0
\(730\) 11.4583 0.424089
\(731\) −58.2364 −2.15395
\(732\) 0 0
\(733\) 45.9138 1.69586 0.847932 0.530105i \(-0.177847\pi\)
0.847932 + 0.530105i \(0.177847\pi\)
\(734\) 33.1240 1.22263
\(735\) 0 0
\(736\) −60.9255 −2.24574
\(737\) 1.86064 0.0685374
\(738\) 0 0
\(739\) 38.1884 1.40478 0.702392 0.711791i \(-0.252115\pi\)
0.702392 + 0.711791i \(0.252115\pi\)
\(740\) −13.5203 −0.497018
\(741\) 0 0
\(742\) 10.8039 0.396623
\(743\) 34.4773 1.26485 0.632424 0.774622i \(-0.282060\pi\)
0.632424 + 0.774622i \(0.282060\pi\)
\(744\) 0 0
\(745\) −6.68215 −0.244815
\(746\) 63.4550 2.32325
\(747\) 0 0
\(748\) −5.49931 −0.201075
\(749\) 5.32901 0.194718
\(750\) 0 0
\(751\) 41.7113 1.52207 0.761033 0.648713i \(-0.224692\pi\)
0.761033 + 0.648713i \(0.224692\pi\)
\(752\) −9.30118 −0.339179
\(753\) 0 0
\(754\) 6.15883 0.224291
\(755\) 20.2253 0.736073
\(756\) 0 0
\(757\) −13.5659 −0.493061 −0.246530 0.969135i \(-0.579291\pi\)
−0.246530 + 0.969135i \(0.579291\pi\)
\(758\) −11.6679 −0.423797
\(759\) 0 0
\(760\) −7.64050 −0.277150
\(761\) −3.27982 −0.118893 −0.0594467 0.998231i \(-0.518934\pi\)
−0.0594467 + 0.998231i \(0.518934\pi\)
\(762\) 0 0
\(763\) 5.98988 0.216848
\(764\) 24.9062 0.901076
\(765\) 0 0
\(766\) −37.7159 −1.36273
\(767\) −29.3761 −1.06071
\(768\) 0 0
\(769\) −35.5527 −1.28206 −0.641032 0.767514i \(-0.721493\pi\)
−0.641032 + 0.767514i \(0.721493\pi\)
\(770\) 0.967690 0.0348731
\(771\) 0 0
\(772\) 4.85608 0.174774
\(773\) 0.965870 0.0347399 0.0173700 0.999849i \(-0.494471\pi\)
0.0173700 + 0.999849i \(0.494471\pi\)
\(774\) 0 0
\(775\) 5.48602 0.197064
\(776\) −1.62965 −0.0585012
\(777\) 0 0
\(778\) 45.0659 1.61569
\(779\) 69.1263 2.47671
\(780\) 0 0
\(781\) −3.52287 −0.126058
\(782\) 99.1641 3.54610
\(783\) 0 0
\(784\) 31.7069 1.13239
\(785\) −23.8658 −0.851806
\(786\) 0 0
\(787\) 15.0756 0.537387 0.268693 0.963226i \(-0.413408\pi\)
0.268693 + 0.963226i \(0.413408\pi\)
\(788\) −26.8943 −0.958070
\(789\) 0 0
\(790\) −9.81775 −0.349300
\(791\) 6.20488 0.220620
\(792\) 0 0
\(793\) −52.6842 −1.87087
\(794\) −23.2023 −0.823419
\(795\) 0 0
\(796\) −14.2386 −0.504673
\(797\) −27.9884 −0.991399 −0.495700 0.868494i \(-0.665088\pi\)
−0.495700 + 0.868494i \(0.665088\pi\)
\(798\) 0 0
\(799\) 10.8973 0.385519
\(800\) −6.42266 −0.227075
\(801\) 0 0
\(802\) −51.3236 −1.81230
\(803\) −4.58976 −0.161969
\(804\) 0 0
\(805\) −6.91650 −0.243775
\(806\) 33.7875 1.19011
\(807\) 0 0
\(808\) −6.69442 −0.235509
\(809\) −10.0950 −0.354921 −0.177460 0.984128i \(-0.556788\pi\)
−0.177460 + 0.984128i \(0.556788\pi\)
\(810\) 0 0
\(811\) 16.3555 0.574321 0.287160 0.957882i \(-0.407289\pi\)
0.287160 + 0.957882i \(0.407289\pi\)
\(812\) −0.957567 −0.0336040
\(813\) 0 0
\(814\) 13.6633 0.478898
\(815\) −1.93538 −0.0677934
\(816\) 0 0
\(817\) −61.9846 −2.16857
\(818\) −43.7063 −1.52815
\(819\) 0 0
\(820\) 14.8519 0.518650
\(821\) −20.7468 −0.724067 −0.362034 0.932165i \(-0.617917\pi\)
−0.362034 + 0.932165i \(0.617917\pi\)
\(822\) 0 0
\(823\) 33.4619 1.16641 0.583204 0.812326i \(-0.301799\pi\)
0.583204 + 0.812326i \(0.301799\pi\)
\(824\) −20.9580 −0.730105
\(825\) 0 0
\(826\) 11.5229 0.400932
\(827\) 20.3363 0.707164 0.353582 0.935404i \(-0.384964\pi\)
0.353582 + 0.935404i \(0.384964\pi\)
\(828\) 0 0
\(829\) 18.4858 0.642039 0.321020 0.947073i \(-0.395974\pi\)
0.321020 + 0.947073i \(0.395974\pi\)
\(830\) −0.154479 −0.00536203
\(831\) 0 0
\(832\) −6.38526 −0.221369
\(833\) −37.1479 −1.28710
\(834\) 0 0
\(835\) 12.4025 0.429207
\(836\) −5.85325 −0.202439
\(837\) 0 0
\(838\) −26.8797 −0.928542
\(839\) 22.3571 0.771853 0.385927 0.922529i \(-0.373882\pi\)
0.385927 + 0.922529i \(0.373882\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −23.1721 −0.798562
\(843\) 0 0
\(844\) 2.62661 0.0904118
\(845\) −1.55187 −0.0533860
\(846\) 0 0
\(847\) 7.63277 0.262265
\(848\) 39.9029 1.37027
\(849\) 0 0
\(850\) 10.4537 0.358559
\(851\) −97.6574 −3.34765
\(852\) 0 0
\(853\) 16.1950 0.554505 0.277253 0.960797i \(-0.410576\pi\)
0.277253 + 0.960797i \(0.410576\pi\)
\(854\) 20.6655 0.707160
\(855\) 0 0
\(856\) 9.13560 0.312249
\(857\) −11.4785 −0.392098 −0.196049 0.980594i \(-0.562811\pi\)
−0.196049 + 0.980594i \(0.562811\pi\)
\(858\) 0 0
\(859\) −21.6744 −0.739522 −0.369761 0.929127i \(-0.620560\pi\)
−0.369761 + 0.929127i \(0.620560\pi\)
\(860\) −13.3175 −0.454122
\(861\) 0 0
\(862\) −17.5896 −0.599103
\(863\) 30.7226 1.04581 0.522905 0.852391i \(-0.324848\pi\)
0.522905 + 0.852391i \(0.324848\pi\)
\(864\) 0 0
\(865\) 21.7113 0.738206
\(866\) 23.8798 0.811470
\(867\) 0 0
\(868\) −5.25323 −0.178306
\(869\) 3.93264 0.133406
\(870\) 0 0
\(871\) −8.63428 −0.292561
\(872\) 10.2685 0.347737
\(873\) 0 0
\(874\) 105.546 3.57016
\(875\) −0.729126 −0.0246490
\(876\) 0 0
\(877\) 34.2908 1.15792 0.578959 0.815357i \(-0.303459\pi\)
0.578959 + 0.815357i \(0.303459\pi\)
\(878\) −36.2689 −1.22402
\(879\) 0 0
\(880\) 3.57406 0.120481
\(881\) −12.8620 −0.433332 −0.216666 0.976246i \(-0.569518\pi\)
−0.216666 + 0.976246i \(0.569518\pi\)
\(882\) 0 0
\(883\) 1.37495 0.0462707 0.0231354 0.999732i \(-0.492635\pi\)
0.0231354 + 0.999732i \(0.492635\pi\)
\(884\) 25.5196 0.858316
\(885\) 0 0
\(886\) −3.53682 −0.118822
\(887\) 27.1710 0.912312 0.456156 0.889900i \(-0.349226\pi\)
0.456156 + 0.889900i \(0.349226\pi\)
\(888\) 0 0
\(889\) −3.29126 −0.110385
\(890\) −8.44002 −0.282910
\(891\) 0 0
\(892\) −5.53578 −0.185352
\(893\) 11.5987 0.388134
\(894\) 0 0
\(895\) −6.08487 −0.203395
\(896\) −6.86122 −0.229217
\(897\) 0 0
\(898\) −30.9833 −1.03393
\(899\) 5.48602 0.182969
\(900\) 0 0
\(901\) −46.7504 −1.55748
\(902\) −15.0089 −0.499741
\(903\) 0 0
\(904\) 10.6371 0.353785
\(905\) 6.77714 0.225280
\(906\) 0 0
\(907\) 0.327640 0.0108791 0.00543955 0.999985i \(-0.498269\pi\)
0.00543955 + 0.999985i \(0.498269\pi\)
\(908\) 20.2530 0.672121
\(909\) 0 0
\(910\) −4.49056 −0.148861
\(911\) −43.0377 −1.42590 −0.712951 0.701214i \(-0.752642\pi\)
−0.712951 + 0.701214i \(0.752642\pi\)
\(912\) 0 0
\(913\) 0.0618786 0.00204788
\(914\) 29.0572 0.961125
\(915\) 0 0
\(916\) −31.8517 −1.05241
\(917\) −10.7744 −0.355802
\(918\) 0 0
\(919\) −28.2061 −0.930432 −0.465216 0.885197i \(-0.654023\pi\)
−0.465216 + 0.885197i \(0.654023\pi\)
\(920\) −11.8571 −0.390916
\(921\) 0 0
\(922\) 32.2021 1.06052
\(923\) 16.3479 0.538097
\(924\) 0 0
\(925\) −10.2949 −0.338494
\(926\) −4.74613 −0.155967
\(927\) 0 0
\(928\) −6.42266 −0.210834
\(929\) 14.4505 0.474107 0.237053 0.971497i \(-0.423818\pi\)
0.237053 + 0.971497i \(0.423818\pi\)
\(930\) 0 0
\(931\) −39.5388 −1.29583
\(932\) 7.00851 0.229571
\(933\) 0 0
\(934\) 14.5620 0.476483
\(935\) −4.18738 −0.136942
\(936\) 0 0
\(937\) 13.9657 0.456241 0.228121 0.973633i \(-0.426742\pi\)
0.228121 + 0.973633i \(0.426742\pi\)
\(938\) 3.38682 0.110584
\(939\) 0 0
\(940\) 2.49199 0.0812796
\(941\) 40.6262 1.32438 0.662189 0.749337i \(-0.269628\pi\)
0.662189 + 0.749337i \(0.269628\pi\)
\(942\) 0 0
\(943\) 107.275 3.49336
\(944\) 42.5585 1.38516
\(945\) 0 0
\(946\) 13.4583 0.437566
\(947\) 20.4495 0.664519 0.332260 0.943188i \(-0.392189\pi\)
0.332260 + 0.943188i \(0.392189\pi\)
\(948\) 0 0
\(949\) 21.2988 0.691388
\(950\) 11.1265 0.360992
\(951\) 0 0
\(952\) 5.23402 0.169636
\(953\) 51.5527 1.66996 0.834978 0.550283i \(-0.185480\pi\)
0.834978 + 0.550283i \(0.185480\pi\)
\(954\) 0 0
\(955\) 18.9645 0.613677
\(956\) −21.0395 −0.680466
\(957\) 0 0
\(958\) −37.6488 −1.21638
\(959\) −3.78499 −0.122224
\(960\) 0 0
\(961\) −0.903587 −0.0291480
\(962\) −63.4044 −2.04424
\(963\) 0 0
\(964\) −8.67756 −0.279485
\(965\) 3.69760 0.119030
\(966\) 0 0
\(967\) 16.5429 0.531985 0.265992 0.963975i \(-0.414300\pi\)
0.265992 + 0.963975i \(0.414300\pi\)
\(968\) 13.0850 0.420567
\(969\) 0 0
\(970\) 2.37319 0.0761986
\(971\) −32.5417 −1.04431 −0.522157 0.852849i \(-0.674873\pi\)
−0.522157 + 0.852849i \(0.674873\pi\)
\(972\) 0 0
\(973\) 0.844503 0.0270735
\(974\) −58.4365 −1.87242
\(975\) 0 0
\(976\) 76.3260 2.44313
\(977\) 38.7111 1.23848 0.619239 0.785203i \(-0.287441\pi\)
0.619239 + 0.785203i \(0.287441\pi\)
\(978\) 0 0
\(979\) 3.38077 0.108050
\(980\) −8.49496 −0.271362
\(981\) 0 0
\(982\) 74.0720 2.36373
\(983\) 31.7315 1.01208 0.506039 0.862510i \(-0.331109\pi\)
0.506039 + 0.862510i \(0.331109\pi\)
\(984\) 0 0
\(985\) −20.4783 −0.652493
\(986\) 10.4537 0.332914
\(987\) 0 0
\(988\) 27.1620 0.864139
\(989\) −96.1921 −3.05873
\(990\) 0 0
\(991\) −22.8620 −0.726236 −0.363118 0.931743i \(-0.618288\pi\)
−0.363118 + 0.931743i \(0.618288\pi\)
\(992\) −35.2349 −1.11871
\(993\) 0 0
\(994\) −6.41251 −0.203392
\(995\) −10.8418 −0.343707
\(996\) 0 0
\(997\) 40.9024 1.29539 0.647696 0.761898i \(-0.275733\pi\)
0.647696 + 0.761898i \(0.275733\pi\)
\(998\) 19.7347 0.624691
\(999\) 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 1305.2.a.r.1.3 4
3.2 odd 2 435.2.a.j.1.2 4
5.4 even 2 6525.2.a.bi.1.2 4
12.11 even 2 6960.2.a.co.1.3 4
15.2 even 4 2175.2.c.n.349.2 8
15.8 even 4 2175.2.c.n.349.7 8
15.14 odd 2 2175.2.a.v.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.2 4 3.2 odd 2
1305.2.a.r.1.3 4 1.1 even 1 trivial
2175.2.a.v.1.3 4 15.14 odd 2
2175.2.c.n.349.2 8 15.2 even 4
2175.2.c.n.349.7 8 15.8 even 4
6525.2.a.bi.1.2 4 5.4 even 2
6960.2.a.co.1.3 4 12.11 even 2