Properties

Label 5225.2.a.w.1.1
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, 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("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - x^{14} - 21 x^{13} + 19 x^{12} + 170 x^{11} - 137 x^{10} - 669 x^{9} + 458 x^{8} + 1327 x^{7} + \cdots - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.47407\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.47407 q^{2} -3.11629 q^{3} +4.12105 q^{4} +7.70992 q^{6} -0.568753 q^{7} -5.24763 q^{8} +6.71124 q^{9} +O(q^{10})\) \(q-2.47407 q^{2} -3.11629 q^{3} +4.12105 q^{4} +7.70992 q^{6} -0.568753 q^{7} -5.24763 q^{8} +6.71124 q^{9} -1.00000 q^{11} -12.8424 q^{12} -1.23613 q^{13} +1.40714 q^{14} +4.74093 q^{16} -1.06253 q^{17} -16.6041 q^{18} -1.00000 q^{19} +1.77240 q^{21} +2.47407 q^{22} +5.72002 q^{23} +16.3531 q^{24} +3.05828 q^{26} -11.5653 q^{27} -2.34386 q^{28} +1.16291 q^{29} -0.252612 q^{31} -1.23416 q^{32} +3.11629 q^{33} +2.62878 q^{34} +27.6573 q^{36} +0.974602 q^{37} +2.47407 q^{38} +3.85214 q^{39} -1.90322 q^{41} -4.38504 q^{42} +3.31381 q^{43} -4.12105 q^{44} -14.1518 q^{46} +2.74967 q^{47} -14.7741 q^{48} -6.67652 q^{49} +3.31114 q^{51} -5.09416 q^{52} -4.08290 q^{53} +28.6134 q^{54} +2.98460 q^{56} +3.11629 q^{57} -2.87712 q^{58} -2.54912 q^{59} +7.49069 q^{61} +0.624981 q^{62} -3.81704 q^{63} -6.42846 q^{64} -7.70992 q^{66} -13.5077 q^{67} -4.37873 q^{68} -17.8252 q^{69} -1.42963 q^{71} -35.2181 q^{72} -4.15670 q^{73} -2.41124 q^{74} -4.12105 q^{76} +0.568753 q^{77} -9.53049 q^{78} -9.12259 q^{79} +15.9070 q^{81} +4.70870 q^{82} +10.1749 q^{83} +7.30413 q^{84} -8.19862 q^{86} -3.62396 q^{87} +5.24763 q^{88} -6.03138 q^{89} +0.703054 q^{91} +23.5725 q^{92} +0.787212 q^{93} -6.80289 q^{94} +3.84599 q^{96} +16.9507 q^{97} +16.5182 q^{98} -6.71124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + q^{2} - 4 q^{3} + 13 q^{4} - q^{6} - 11 q^{7} + 3 q^{8} + 19 q^{9} - 15 q^{11} - 11 q^{12} - 3 q^{13} - 11 q^{14} + 13 q^{16} + 5 q^{17} - 12 q^{18} - 15 q^{19} + 10 q^{21} - q^{22} - 26 q^{23} - 11 q^{24} - 5 q^{26} - 19 q^{27} - 18 q^{28} + 7 q^{29} - 10 q^{31} + 12 q^{32} + 4 q^{33} + 17 q^{34} + 24 q^{36} - 31 q^{37} - q^{38} + 4 q^{39} + 2 q^{41} - 22 q^{42} - 26 q^{43} - 13 q^{44} - 23 q^{46} - 26 q^{47} - 46 q^{48} + 12 q^{49} + 12 q^{51} - 16 q^{52} - 21 q^{53} + 5 q^{54} - 10 q^{56} + 4 q^{57} - 34 q^{58} - 11 q^{59} + 20 q^{61} - 25 q^{62} - 27 q^{63} - 3 q^{64} + q^{66} - 41 q^{67} + 6 q^{68} + q^{69} + 25 q^{71} - 54 q^{72} + 6 q^{73} - 9 q^{74} - 13 q^{76} + 11 q^{77} - 28 q^{78} - 6 q^{79} + 43 q^{81} - 18 q^{82} + 20 q^{83} - 14 q^{84} + 35 q^{86} - 29 q^{87} - 3 q^{88} - 3 q^{89} + 30 q^{91} - 54 q^{92} - 2 q^{93} - 28 q^{94} - 61 q^{96} - 28 q^{97} + 2 q^{98} - 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.47407 −1.74944 −0.874718 0.484633i \(-0.838953\pi\)
−0.874718 + 0.484633i \(0.838953\pi\)
\(3\) −3.11629 −1.79919 −0.899594 0.436727i \(-0.856138\pi\)
−0.899594 + 0.436727i \(0.856138\pi\)
\(4\) 4.12105 2.06052
\(5\) 0 0
\(6\) 7.70992 3.14756
\(7\) −0.568753 −0.214968 −0.107484 0.994207i \(-0.534280\pi\)
−0.107484 + 0.994207i \(0.534280\pi\)
\(8\) −5.24763 −1.85532
\(9\) 6.71124 2.23708
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −12.8424 −3.70727
\(13\) −1.23613 −0.342841 −0.171421 0.985198i \(-0.554836\pi\)
−0.171421 + 0.985198i \(0.554836\pi\)
\(14\) 1.40714 0.376073
\(15\) 0 0
\(16\) 4.74093 1.18523
\(17\) −1.06253 −0.257701 −0.128851 0.991664i \(-0.541129\pi\)
−0.128851 + 0.991664i \(0.541129\pi\)
\(18\) −16.6041 −3.91362
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.77240 0.386769
\(22\) 2.47407 0.527475
\(23\) 5.72002 1.19271 0.596354 0.802722i \(-0.296616\pi\)
0.596354 + 0.802722i \(0.296616\pi\)
\(24\) 16.3531 3.33806
\(25\) 0 0
\(26\) 3.05828 0.599779
\(27\) −11.5653 −2.22574
\(28\) −2.34386 −0.442947
\(29\) 1.16291 0.215947 0.107973 0.994154i \(-0.465564\pi\)
0.107973 + 0.994154i \(0.465564\pi\)
\(30\) 0 0
\(31\) −0.252612 −0.0453705 −0.0226852 0.999743i \(-0.507222\pi\)
−0.0226852 + 0.999743i \(0.507222\pi\)
\(32\) −1.23416 −0.218170
\(33\) 3.11629 0.542476
\(34\) 2.62878 0.450831
\(35\) 0 0
\(36\) 27.6573 4.60955
\(37\) 0.974602 0.160224 0.0801118 0.996786i \(-0.474472\pi\)
0.0801118 + 0.996786i \(0.474472\pi\)
\(38\) 2.47407 0.401348
\(39\) 3.85214 0.616836
\(40\) 0 0
\(41\) −1.90322 −0.297232 −0.148616 0.988895i \(-0.547482\pi\)
−0.148616 + 0.988895i \(0.547482\pi\)
\(42\) −4.38504 −0.676627
\(43\) 3.31381 0.505352 0.252676 0.967551i \(-0.418689\pi\)
0.252676 + 0.967551i \(0.418689\pi\)
\(44\) −4.12105 −0.621271
\(45\) 0 0
\(46\) −14.1518 −2.08656
\(47\) 2.74967 0.401081 0.200540 0.979685i \(-0.435730\pi\)
0.200540 + 0.979685i \(0.435730\pi\)
\(48\) −14.7741 −2.13246
\(49\) −6.67652 −0.953789
\(50\) 0 0
\(51\) 3.31114 0.463653
\(52\) −5.09416 −0.706433
\(53\) −4.08290 −0.560829 −0.280415 0.959879i \(-0.590472\pi\)
−0.280415 + 0.959879i \(0.590472\pi\)
\(54\) 28.6134 3.89378
\(55\) 0 0
\(56\) 2.98460 0.398834
\(57\) 3.11629 0.412762
\(58\) −2.87712 −0.377785
\(59\) −2.54912 −0.331868 −0.165934 0.986137i \(-0.553064\pi\)
−0.165934 + 0.986137i \(0.553064\pi\)
\(60\) 0 0
\(61\) 7.49069 0.959085 0.479542 0.877519i \(-0.340803\pi\)
0.479542 + 0.877519i \(0.340803\pi\)
\(62\) 0.624981 0.0793727
\(63\) −3.81704 −0.480901
\(64\) −6.42846 −0.803557
\(65\) 0 0
\(66\) −7.70992 −0.949026
\(67\) −13.5077 −1.65022 −0.825112 0.564969i \(-0.808888\pi\)
−0.825112 + 0.564969i \(0.808888\pi\)
\(68\) −4.37873 −0.530999
\(69\) −17.8252 −2.14591
\(70\) 0 0
\(71\) −1.42963 −0.169665 −0.0848327 0.996395i \(-0.527036\pi\)
−0.0848327 + 0.996395i \(0.527036\pi\)
\(72\) −35.2181 −4.15049
\(73\) −4.15670 −0.486505 −0.243253 0.969963i \(-0.578214\pi\)
−0.243253 + 0.969963i \(0.578214\pi\)
\(74\) −2.41124 −0.280301
\(75\) 0 0
\(76\) −4.12105 −0.472716
\(77\) 0.568753 0.0648154
\(78\) −9.53049 −1.07912
\(79\) −9.12259 −1.02637 −0.513186 0.858278i \(-0.671535\pi\)
−0.513186 + 0.858278i \(0.671535\pi\)
\(80\) 0 0
\(81\) 15.9070 1.76744
\(82\) 4.70870 0.519989
\(83\) 10.1749 1.11684 0.558420 0.829558i \(-0.311408\pi\)
0.558420 + 0.829558i \(0.311408\pi\)
\(84\) 7.30413 0.796946
\(85\) 0 0
\(86\) −8.19862 −0.884080
\(87\) −3.62396 −0.388529
\(88\) 5.24763 0.559399
\(89\) −6.03138 −0.639325 −0.319662 0.947531i \(-0.603570\pi\)
−0.319662 + 0.947531i \(0.603570\pi\)
\(90\) 0 0
\(91\) 0.703054 0.0737001
\(92\) 23.5725 2.45760
\(93\) 0.787212 0.0816301
\(94\) −6.80289 −0.701665
\(95\) 0 0
\(96\) 3.84599 0.392529
\(97\) 16.9507 1.72108 0.860542 0.509380i \(-0.170125\pi\)
0.860542 + 0.509380i \(0.170125\pi\)
\(98\) 16.5182 1.66859
\(99\) −6.71124 −0.674505
\(100\) 0 0
\(101\) −3.69240 −0.367407 −0.183704 0.982982i \(-0.558809\pi\)
−0.183704 + 0.982982i \(0.558809\pi\)
\(102\) −8.19202 −0.811130
\(103\) −5.19662 −0.512038 −0.256019 0.966672i \(-0.582411\pi\)
−0.256019 + 0.966672i \(0.582411\pi\)
\(104\) 6.48676 0.636079
\(105\) 0 0
\(106\) 10.1014 0.981135
\(107\) −1.32425 −0.128020 −0.0640101 0.997949i \(-0.520389\pi\)
−0.0640101 + 0.997949i \(0.520389\pi\)
\(108\) −47.6610 −4.58619
\(109\) 15.6513 1.49912 0.749560 0.661936i \(-0.230265\pi\)
0.749560 + 0.661936i \(0.230265\pi\)
\(110\) 0 0
\(111\) −3.03714 −0.288272
\(112\) −2.69642 −0.254787
\(113\) −16.1223 −1.51666 −0.758329 0.651873i \(-0.773984\pi\)
−0.758329 + 0.651873i \(0.773984\pi\)
\(114\) −7.70992 −0.722101
\(115\) 0 0
\(116\) 4.79240 0.444963
\(117\) −8.29598 −0.766964
\(118\) 6.30672 0.580581
\(119\) 0.604316 0.0553976
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −18.5325 −1.67786
\(123\) 5.93096 0.534777
\(124\) −1.04103 −0.0934869
\(125\) 0 0
\(126\) 9.44363 0.841306
\(127\) −2.42378 −0.215075 −0.107538 0.994201i \(-0.534297\pi\)
−0.107538 + 0.994201i \(0.534297\pi\)
\(128\) 18.3728 1.62394
\(129\) −10.3268 −0.909223
\(130\) 0 0
\(131\) 17.0055 1.48577 0.742887 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(132\) 12.8424 1.11778
\(133\) 0.568753 0.0493171
\(134\) 33.4190 2.88696
\(135\) 0 0
\(136\) 5.57575 0.478117
\(137\) 13.9041 1.18791 0.593955 0.804498i \(-0.297566\pi\)
0.593955 + 0.804498i \(0.297566\pi\)
\(138\) 44.1009 3.75412
\(139\) −5.48558 −0.465281 −0.232641 0.972563i \(-0.574737\pi\)
−0.232641 + 0.972563i \(0.574737\pi\)
\(140\) 0 0
\(141\) −8.56876 −0.721620
\(142\) 3.53700 0.296819
\(143\) 1.23613 0.103371
\(144\) 31.8175 2.65146
\(145\) 0 0
\(146\) 10.2840 0.851109
\(147\) 20.8059 1.71605
\(148\) 4.01638 0.330144
\(149\) 12.5126 1.02508 0.512538 0.858665i \(-0.328705\pi\)
0.512538 + 0.858665i \(0.328705\pi\)
\(150\) 0 0
\(151\) −18.1370 −1.47596 −0.737982 0.674820i \(-0.764221\pi\)
−0.737982 + 0.674820i \(0.764221\pi\)
\(152\) 5.24763 0.425639
\(153\) −7.13088 −0.576498
\(154\) −1.40714 −0.113390
\(155\) 0 0
\(156\) 15.8749 1.27101
\(157\) −0.430792 −0.0343809 −0.0171905 0.999852i \(-0.505472\pi\)
−0.0171905 + 0.999852i \(0.505472\pi\)
\(158\) 22.5700 1.79557
\(159\) 12.7235 1.00904
\(160\) 0 0
\(161\) −3.25328 −0.256394
\(162\) −39.3551 −3.09203
\(163\) −19.8052 −1.55127 −0.775633 0.631184i \(-0.782569\pi\)
−0.775633 + 0.631184i \(0.782569\pi\)
\(164\) −7.84324 −0.612454
\(165\) 0 0
\(166\) −25.1735 −1.95384
\(167\) 4.73947 0.366751 0.183376 0.983043i \(-0.441298\pi\)
0.183376 + 0.983043i \(0.441298\pi\)
\(168\) −9.30088 −0.717578
\(169\) −11.4720 −0.882460
\(170\) 0 0
\(171\) −6.71124 −0.513221
\(172\) 13.6564 1.04129
\(173\) 13.1702 1.00131 0.500654 0.865647i \(-0.333093\pi\)
0.500654 + 0.865647i \(0.333093\pi\)
\(174\) 8.96594 0.679706
\(175\) 0 0
\(176\) −4.74093 −0.357361
\(177\) 7.94380 0.597093
\(178\) 14.9221 1.11846
\(179\) −2.43120 −0.181717 −0.0908583 0.995864i \(-0.528961\pi\)
−0.0908583 + 0.995864i \(0.528961\pi\)
\(180\) 0 0
\(181\) 25.8909 1.92446 0.962229 0.272243i \(-0.0877653\pi\)
0.962229 + 0.272243i \(0.0877653\pi\)
\(182\) −1.73941 −0.128934
\(183\) −23.3431 −1.72557
\(184\) −30.0165 −2.21285
\(185\) 0 0
\(186\) −1.94762 −0.142806
\(187\) 1.06253 0.0776998
\(188\) 11.3315 0.826436
\(189\) 6.57779 0.478463
\(190\) 0 0
\(191\) 18.7924 1.35977 0.679886 0.733318i \(-0.262029\pi\)
0.679886 + 0.733318i \(0.262029\pi\)
\(192\) 20.0329 1.44575
\(193\) 16.0283 1.15374 0.576870 0.816836i \(-0.304274\pi\)
0.576870 + 0.816836i \(0.304274\pi\)
\(194\) −41.9373 −3.01092
\(195\) 0 0
\(196\) −27.5142 −1.96530
\(197\) 19.5771 1.39481 0.697405 0.716678i \(-0.254338\pi\)
0.697405 + 0.716678i \(0.254338\pi\)
\(198\) 16.6041 1.18000
\(199\) −24.9130 −1.76604 −0.883019 0.469338i \(-0.844493\pi\)
−0.883019 + 0.469338i \(0.844493\pi\)
\(200\) 0 0
\(201\) 42.0937 2.96906
\(202\) 9.13526 0.642755
\(203\) −0.661408 −0.0464217
\(204\) 13.6454 0.955367
\(205\) 0 0
\(206\) 12.8568 0.895778
\(207\) 38.3884 2.66818
\(208\) −5.86042 −0.406347
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 15.2896 1.05258 0.526290 0.850305i \(-0.323583\pi\)
0.526290 + 0.850305i \(0.323583\pi\)
\(212\) −16.8258 −1.15560
\(213\) 4.45512 0.305260
\(214\) 3.27629 0.223963
\(215\) 0 0
\(216\) 60.6903 4.12945
\(217\) 0.143674 0.00975322
\(218\) −38.7224 −2.62261
\(219\) 12.9535 0.875314
\(220\) 0 0
\(221\) 1.31343 0.0883506
\(222\) 7.51411 0.504314
\(223\) −26.2223 −1.75598 −0.877989 0.478680i \(-0.841115\pi\)
−0.877989 + 0.478680i \(0.841115\pi\)
\(224\) 0.701931 0.0468997
\(225\) 0 0
\(226\) 39.8877 2.65329
\(227\) 21.0571 1.39761 0.698806 0.715311i \(-0.253715\pi\)
0.698806 + 0.715311i \(0.253715\pi\)
\(228\) 12.8424 0.850506
\(229\) 3.64009 0.240544 0.120272 0.992741i \(-0.461623\pi\)
0.120272 + 0.992741i \(0.461623\pi\)
\(230\) 0 0
\(231\) −1.77240 −0.116615
\(232\) −6.10251 −0.400650
\(233\) 17.0553 1.11733 0.558666 0.829393i \(-0.311313\pi\)
0.558666 + 0.829393i \(0.311313\pi\)
\(234\) 20.5249 1.34175
\(235\) 0 0
\(236\) −10.5051 −0.683821
\(237\) 28.4286 1.84664
\(238\) −1.49512 −0.0969145
\(239\) −17.5075 −1.13247 −0.566233 0.824245i \(-0.691600\pi\)
−0.566233 + 0.824245i \(0.691600\pi\)
\(240\) 0 0
\(241\) 10.6563 0.686431 0.343215 0.939257i \(-0.388484\pi\)
0.343215 + 0.939257i \(0.388484\pi\)
\(242\) −2.47407 −0.159040
\(243\) −14.8749 −0.954226
\(244\) 30.8695 1.97622
\(245\) 0 0
\(246\) −14.6736 −0.935558
\(247\) 1.23613 0.0786532
\(248\) 1.32561 0.0841766
\(249\) −31.7079 −2.00941
\(250\) 0 0
\(251\) 25.5266 1.61123 0.805613 0.592442i \(-0.201836\pi\)
0.805613 + 0.592442i \(0.201836\pi\)
\(252\) −15.7302 −0.990908
\(253\) −5.72002 −0.359615
\(254\) 5.99660 0.376260
\(255\) 0 0
\(256\) −32.5988 −2.03742
\(257\) −0.270665 −0.0168836 −0.00844181 0.999964i \(-0.502687\pi\)
−0.00844181 + 0.999964i \(0.502687\pi\)
\(258\) 25.5493 1.59063
\(259\) −0.554308 −0.0344430
\(260\) 0 0
\(261\) 7.80456 0.483090
\(262\) −42.0728 −2.59926
\(263\) −24.6926 −1.52261 −0.761305 0.648394i \(-0.775441\pi\)
−0.761305 + 0.648394i \(0.775441\pi\)
\(264\) −16.3531 −1.00646
\(265\) 0 0
\(266\) −1.40714 −0.0862771
\(267\) 18.7955 1.15027
\(268\) −55.6657 −3.40032
\(269\) 21.3173 1.29974 0.649869 0.760046i \(-0.274824\pi\)
0.649869 + 0.760046i \(0.274824\pi\)
\(270\) 0 0
\(271\) 12.1163 0.736013 0.368006 0.929823i \(-0.380040\pi\)
0.368006 + 0.929823i \(0.380040\pi\)
\(272\) −5.03737 −0.305436
\(273\) −2.19092 −0.132600
\(274\) −34.3998 −2.07817
\(275\) 0 0
\(276\) −73.4586 −4.42169
\(277\) 6.37705 0.383160 0.191580 0.981477i \(-0.438639\pi\)
0.191580 + 0.981477i \(0.438639\pi\)
\(278\) 13.5717 0.813979
\(279\) −1.69534 −0.101497
\(280\) 0 0
\(281\) −14.1056 −0.841469 −0.420735 0.907184i \(-0.638228\pi\)
−0.420735 + 0.907184i \(0.638228\pi\)
\(282\) 21.1998 1.26243
\(283\) −13.9610 −0.829894 −0.414947 0.909845i \(-0.636200\pi\)
−0.414947 + 0.909845i \(0.636200\pi\)
\(284\) −5.89156 −0.349600
\(285\) 0 0
\(286\) −3.05828 −0.180840
\(287\) 1.08246 0.0638956
\(288\) −8.28272 −0.488064
\(289\) −15.8710 −0.933590
\(290\) 0 0
\(291\) −52.8232 −3.09655
\(292\) −17.1300 −1.00246
\(293\) −0.0735511 −0.00429690 −0.00214845 0.999998i \(-0.500684\pi\)
−0.00214845 + 0.999998i \(0.500684\pi\)
\(294\) −51.4755 −3.00211
\(295\) 0 0
\(296\) −5.11435 −0.297265
\(297\) 11.5653 0.671085
\(298\) −30.9572 −1.79330
\(299\) −7.07071 −0.408910
\(300\) 0 0
\(301\) −1.88474 −0.108635
\(302\) 44.8722 2.58210
\(303\) 11.5066 0.661035
\(304\) −4.74093 −0.271911
\(305\) 0 0
\(306\) 17.6423 1.00855
\(307\) −17.1790 −0.980458 −0.490229 0.871594i \(-0.663087\pi\)
−0.490229 + 0.871594i \(0.663087\pi\)
\(308\) 2.34386 0.133554
\(309\) 16.1942 0.921254
\(310\) 0 0
\(311\) 24.8461 1.40890 0.704448 0.709756i \(-0.251195\pi\)
0.704448 + 0.709756i \(0.251195\pi\)
\(312\) −20.2146 −1.14443
\(313\) 20.2896 1.14684 0.573418 0.819263i \(-0.305617\pi\)
0.573418 + 0.819263i \(0.305617\pi\)
\(314\) 1.06581 0.0601472
\(315\) 0 0
\(316\) −37.5946 −2.11486
\(317\) 0.475340 0.0266977 0.0133489 0.999911i \(-0.495751\pi\)
0.0133489 + 0.999911i \(0.495751\pi\)
\(318\) −31.4789 −1.76525
\(319\) −1.16291 −0.0651104
\(320\) 0 0
\(321\) 4.12674 0.230332
\(322\) 8.04886 0.448545
\(323\) 1.06253 0.0591207
\(324\) 65.5535 3.64186
\(325\) 0 0
\(326\) 48.9996 2.71384
\(327\) −48.7738 −2.69720
\(328\) 9.98736 0.551460
\(329\) −1.56388 −0.0862197
\(330\) 0 0
\(331\) −3.55187 −0.195228 −0.0976141 0.995224i \(-0.531121\pi\)
−0.0976141 + 0.995224i \(0.531121\pi\)
\(332\) 41.9312 2.30128
\(333\) 6.54078 0.358433
\(334\) −11.7258 −0.641607
\(335\) 0 0
\(336\) 8.40281 0.458411
\(337\) −13.0883 −0.712963 −0.356482 0.934302i \(-0.616024\pi\)
−0.356482 + 0.934302i \(0.616024\pi\)
\(338\) 28.3825 1.54381
\(339\) 50.2416 2.72875
\(340\) 0 0
\(341\) 0.252612 0.0136797
\(342\) 16.6041 0.897847
\(343\) 7.77856 0.420003
\(344\) −17.3897 −0.937588
\(345\) 0 0
\(346\) −32.5840 −1.75172
\(347\) 2.64724 0.142111 0.0710557 0.997472i \(-0.477363\pi\)
0.0710557 + 0.997472i \(0.477363\pi\)
\(348\) −14.9345 −0.800573
\(349\) −1.45049 −0.0776427 −0.0388214 0.999246i \(-0.512360\pi\)
−0.0388214 + 0.999246i \(0.512360\pi\)
\(350\) 0 0
\(351\) 14.2962 0.763076
\(352\) 1.23416 0.0657808
\(353\) −0.415398 −0.0221094 −0.0110547 0.999939i \(-0.503519\pi\)
−0.0110547 + 0.999939i \(0.503519\pi\)
\(354\) −19.6536 −1.04457
\(355\) 0 0
\(356\) −24.8556 −1.31734
\(357\) −1.88322 −0.0996707
\(358\) 6.01498 0.317901
\(359\) 8.01993 0.423276 0.211638 0.977348i \(-0.432120\pi\)
0.211638 + 0.977348i \(0.432120\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −64.0561 −3.36671
\(363\) −3.11629 −0.163563
\(364\) 2.89732 0.151861
\(365\) 0 0
\(366\) 57.7527 3.01878
\(367\) −12.1728 −0.635415 −0.317707 0.948189i \(-0.602913\pi\)
−0.317707 + 0.948189i \(0.602913\pi\)
\(368\) 27.1182 1.41363
\(369\) −12.7729 −0.664932
\(370\) 0 0
\(371\) 2.32216 0.120561
\(372\) 3.24414 0.168201
\(373\) 15.3210 0.793291 0.396646 0.917972i \(-0.370174\pi\)
0.396646 + 0.917972i \(0.370174\pi\)
\(374\) −2.62878 −0.135931
\(375\) 0 0
\(376\) −14.4292 −0.744132
\(377\) −1.43751 −0.0740355
\(378\) −16.2739 −0.837041
\(379\) −9.78947 −0.502851 −0.251426 0.967877i \(-0.580899\pi\)
−0.251426 + 0.967877i \(0.580899\pi\)
\(380\) 0 0
\(381\) 7.55318 0.386961
\(382\) −46.4939 −2.37883
\(383\) −1.71875 −0.0878238 −0.0439119 0.999035i \(-0.513982\pi\)
−0.0439119 + 0.999035i \(0.513982\pi\)
\(384\) −57.2549 −2.92178
\(385\) 0 0
\(386\) −39.6551 −2.01839
\(387\) 22.2398 1.13051
\(388\) 69.8546 3.54633
\(389\) −24.8643 −1.26067 −0.630336 0.776323i \(-0.717083\pi\)
−0.630336 + 0.776323i \(0.717083\pi\)
\(390\) 0 0
\(391\) −6.07769 −0.307362
\(392\) 35.0359 1.76958
\(393\) −52.9939 −2.67319
\(394\) −48.4352 −2.44013
\(395\) 0 0
\(396\) −27.6573 −1.38983
\(397\) −12.8294 −0.643887 −0.321943 0.946759i \(-0.604336\pi\)
−0.321943 + 0.946759i \(0.604336\pi\)
\(398\) 61.6367 3.08957
\(399\) −1.77240 −0.0887308
\(400\) 0 0
\(401\) −14.2796 −0.713089 −0.356544 0.934278i \(-0.616045\pi\)
−0.356544 + 0.934278i \(0.616045\pi\)
\(402\) −104.143 −5.19418
\(403\) 0.312262 0.0155549
\(404\) −15.2165 −0.757051
\(405\) 0 0
\(406\) 1.63637 0.0812118
\(407\) −0.974602 −0.0483092
\(408\) −17.3756 −0.860222
\(409\) −14.7884 −0.731237 −0.365619 0.930765i \(-0.619143\pi\)
−0.365619 + 0.930765i \(0.619143\pi\)
\(410\) 0 0
\(411\) −43.3292 −2.13727
\(412\) −21.4155 −1.05507
\(413\) 1.44982 0.0713411
\(414\) −94.9758 −4.66781
\(415\) 0 0
\(416\) 1.52558 0.0747978
\(417\) 17.0946 0.837128
\(418\) −2.47407 −0.121011
\(419\) 11.1996 0.547135 0.273568 0.961853i \(-0.411796\pi\)
0.273568 + 0.961853i \(0.411796\pi\)
\(420\) 0 0
\(421\) 35.4914 1.72974 0.864872 0.501992i \(-0.167399\pi\)
0.864872 + 0.501992i \(0.167399\pi\)
\(422\) −37.8277 −1.84142
\(423\) 18.4537 0.897250
\(424\) 21.4255 1.04052
\(425\) 0 0
\(426\) −11.0223 −0.534033
\(427\) −4.26035 −0.206173
\(428\) −5.45730 −0.263788
\(429\) −3.85214 −0.185983
\(430\) 0 0
\(431\) −37.2681 −1.79514 −0.897571 0.440870i \(-0.854670\pi\)
−0.897571 + 0.440870i \(0.854670\pi\)
\(432\) −54.8301 −2.63802
\(433\) 34.5772 1.66168 0.830838 0.556515i \(-0.187862\pi\)
0.830838 + 0.556515i \(0.187862\pi\)
\(434\) −0.355460 −0.0170626
\(435\) 0 0
\(436\) 64.4996 3.08897
\(437\) −5.72002 −0.273626
\(438\) −32.0479 −1.53131
\(439\) −34.7598 −1.65900 −0.829498 0.558510i \(-0.811373\pi\)
−0.829498 + 0.558510i \(0.811373\pi\)
\(440\) 0 0
\(441\) −44.8077 −2.13370
\(442\) −3.24951 −0.154564
\(443\) −37.8832 −1.79988 −0.899942 0.436010i \(-0.856391\pi\)
−0.899942 + 0.436010i \(0.856391\pi\)
\(444\) −12.5162 −0.593992
\(445\) 0 0
\(446\) 64.8761 3.07197
\(447\) −38.9930 −1.84430
\(448\) 3.65620 0.172739
\(449\) 10.6044 0.500451 0.250226 0.968188i \(-0.419495\pi\)
0.250226 + 0.968188i \(0.419495\pi\)
\(450\) 0 0
\(451\) 1.90322 0.0896189
\(452\) −66.4407 −3.12511
\(453\) 56.5200 2.65554
\(454\) −52.0970 −2.44503
\(455\) 0 0
\(456\) −16.3531 −0.765804
\(457\) −4.25836 −0.199198 −0.0995988 0.995028i \(-0.531756\pi\)
−0.0995988 + 0.995028i \(0.531756\pi\)
\(458\) −9.00585 −0.420816
\(459\) 12.2884 0.573575
\(460\) 0 0
\(461\) 36.0411 1.67860 0.839300 0.543669i \(-0.182965\pi\)
0.839300 + 0.543669i \(0.182965\pi\)
\(462\) 4.38504 0.204011
\(463\) −11.9837 −0.556930 −0.278465 0.960446i \(-0.589826\pi\)
−0.278465 + 0.960446i \(0.589826\pi\)
\(464\) 5.51327 0.255947
\(465\) 0 0
\(466\) −42.1962 −1.95470
\(467\) −11.4583 −0.530228 −0.265114 0.964217i \(-0.585410\pi\)
−0.265114 + 0.964217i \(0.585410\pi\)
\(468\) −34.1881 −1.58035
\(469\) 7.68252 0.354746
\(470\) 0 0
\(471\) 1.34247 0.0618577
\(472\) 13.3769 0.615719
\(473\) −3.31381 −0.152369
\(474\) −70.3345 −3.23057
\(475\) 0 0
\(476\) 2.49042 0.114148
\(477\) −27.4013 −1.25462
\(478\) 43.3149 1.98118
\(479\) 11.8394 0.540956 0.270478 0.962726i \(-0.412818\pi\)
0.270478 + 0.962726i \(0.412818\pi\)
\(480\) 0 0
\(481\) −1.20474 −0.0549313
\(482\) −26.3644 −1.20087
\(483\) 10.1381 0.461302
\(484\) 4.12105 0.187320
\(485\) 0 0
\(486\) 36.8017 1.66936
\(487\) −21.3119 −0.965733 −0.482866 0.875694i \(-0.660404\pi\)
−0.482866 + 0.875694i \(0.660404\pi\)
\(488\) −39.3083 −1.77941
\(489\) 61.7188 2.79102
\(490\) 0 0
\(491\) −30.6885 −1.38495 −0.692477 0.721440i \(-0.743481\pi\)
−0.692477 + 0.721440i \(0.743481\pi\)
\(492\) 24.4418 1.10192
\(493\) −1.23562 −0.0556497
\(494\) −3.05828 −0.137599
\(495\) 0 0
\(496\) −1.19762 −0.0537745
\(497\) 0.813104 0.0364727
\(498\) 78.4477 3.51533
\(499\) 22.7442 1.01817 0.509086 0.860716i \(-0.329984\pi\)
0.509086 + 0.860716i \(0.329984\pi\)
\(500\) 0 0
\(501\) −14.7695 −0.659854
\(502\) −63.1548 −2.81874
\(503\) −1.25443 −0.0559324 −0.0279662 0.999609i \(-0.508903\pi\)
−0.0279662 + 0.999609i \(0.508903\pi\)
\(504\) 20.0304 0.892224
\(505\) 0 0
\(506\) 14.1518 0.629123
\(507\) 35.7500 1.58771
\(508\) −9.98849 −0.443168
\(509\) −31.1519 −1.38078 −0.690392 0.723435i \(-0.742562\pi\)
−0.690392 + 0.723435i \(0.742562\pi\)
\(510\) 0 0
\(511\) 2.36414 0.104583
\(512\) 43.9062 1.94040
\(513\) 11.5653 0.510619
\(514\) 0.669646 0.0295368
\(515\) 0 0
\(516\) −42.5572 −1.87348
\(517\) −2.74967 −0.120930
\(518\) 1.37140 0.0602558
\(519\) −41.0420 −1.80154
\(520\) 0 0
\(521\) −31.6591 −1.38701 −0.693506 0.720451i \(-0.743935\pi\)
−0.693506 + 0.720451i \(0.743935\pi\)
\(522\) −19.3091 −0.845135
\(523\) −12.9670 −0.567007 −0.283504 0.958971i \(-0.591497\pi\)
−0.283504 + 0.958971i \(0.591497\pi\)
\(524\) 70.0803 3.06147
\(525\) 0 0
\(526\) 61.0913 2.66371
\(527\) 0.268408 0.0116920
\(528\) 14.7741 0.642960
\(529\) 9.71866 0.422550
\(530\) 0 0
\(531\) −17.1078 −0.742414
\(532\) 2.34386 0.101619
\(533\) 2.35263 0.101904
\(534\) −46.5015 −2.01232
\(535\) 0 0
\(536\) 70.8832 3.06169
\(537\) 7.57632 0.326942
\(538\) −52.7406 −2.27381
\(539\) 6.67652 0.287578
\(540\) 0 0
\(541\) −2.93440 −0.126160 −0.0630799 0.998008i \(-0.520092\pi\)
−0.0630799 + 0.998008i \(0.520092\pi\)
\(542\) −29.9766 −1.28761
\(543\) −80.6835 −3.46246
\(544\) 1.31133 0.0562227
\(545\) 0 0
\(546\) 5.42049 0.231976
\(547\) −15.5813 −0.666210 −0.333105 0.942890i \(-0.608096\pi\)
−0.333105 + 0.942890i \(0.608096\pi\)
\(548\) 57.2995 2.44771
\(549\) 50.2718 2.14555
\(550\) 0 0
\(551\) −1.16291 −0.0495416
\(552\) 93.5401 3.98133
\(553\) 5.18850 0.220637
\(554\) −15.7773 −0.670313
\(555\) 0 0
\(556\) −22.6063 −0.958722
\(557\) 28.8338 1.22173 0.610864 0.791735i \(-0.290822\pi\)
0.610864 + 0.791735i \(0.290822\pi\)
\(558\) 4.19440 0.177563
\(559\) −4.09631 −0.173256
\(560\) 0 0
\(561\) −3.31114 −0.139797
\(562\) 34.8983 1.47210
\(563\) −3.71838 −0.156711 −0.0783556 0.996925i \(-0.524967\pi\)
−0.0783556 + 0.996925i \(0.524967\pi\)
\(564\) −35.3123 −1.48691
\(565\) 0 0
\(566\) 34.5405 1.45185
\(567\) −9.04715 −0.379945
\(568\) 7.50215 0.314783
\(569\) −12.9927 −0.544683 −0.272341 0.962201i \(-0.587798\pi\)
−0.272341 + 0.962201i \(0.587798\pi\)
\(570\) 0 0
\(571\) 34.2334 1.43262 0.716312 0.697780i \(-0.245829\pi\)
0.716312 + 0.697780i \(0.245829\pi\)
\(572\) 5.09416 0.212997
\(573\) −58.5626 −2.44649
\(574\) −2.67809 −0.111781
\(575\) 0 0
\(576\) −43.1429 −1.79762
\(577\) −5.29472 −0.220422 −0.110211 0.993908i \(-0.535153\pi\)
−0.110211 + 0.993908i \(0.535153\pi\)
\(578\) 39.2661 1.63326
\(579\) −49.9487 −2.07580
\(580\) 0 0
\(581\) −5.78700 −0.240085
\(582\) 130.689 5.41722
\(583\) 4.08290 0.169096
\(584\) 21.8128 0.902621
\(585\) 0 0
\(586\) 0.181971 0.00751715
\(587\) 15.9044 0.656447 0.328223 0.944600i \(-0.393550\pi\)
0.328223 + 0.944600i \(0.393550\pi\)
\(588\) 85.7423 3.53595
\(589\) 0.252612 0.0104087
\(590\) 0 0
\(591\) −61.0078 −2.50952
\(592\) 4.62052 0.189902
\(593\) 15.8487 0.650828 0.325414 0.945572i \(-0.394496\pi\)
0.325414 + 0.945572i \(0.394496\pi\)
\(594\) −28.6134 −1.17402
\(595\) 0 0
\(596\) 51.5652 2.11219
\(597\) 77.6361 3.17743
\(598\) 17.4935 0.715361
\(599\) 29.6191 1.21020 0.605102 0.796148i \(-0.293132\pi\)
0.605102 + 0.796148i \(0.293132\pi\)
\(600\) 0 0
\(601\) −35.1273 −1.43287 −0.716435 0.697654i \(-0.754228\pi\)
−0.716435 + 0.697654i \(0.754228\pi\)
\(602\) 4.66299 0.190049
\(603\) −90.6531 −3.69168
\(604\) −74.7432 −3.04126
\(605\) 0 0
\(606\) −28.4681 −1.15644
\(607\) 46.0823 1.87042 0.935211 0.354090i \(-0.115209\pi\)
0.935211 + 0.354090i \(0.115209\pi\)
\(608\) 1.23416 0.0500517
\(609\) 2.06114 0.0835214
\(610\) 0 0
\(611\) −3.39896 −0.137507
\(612\) −29.3867 −1.18789
\(613\) −8.75866 −0.353759 −0.176880 0.984233i \(-0.556600\pi\)
−0.176880 + 0.984233i \(0.556600\pi\)
\(614\) 42.5022 1.71525
\(615\) 0 0
\(616\) −2.98460 −0.120253
\(617\) −23.4750 −0.945069 −0.472534 0.881312i \(-0.656661\pi\)
−0.472534 + 0.881312i \(0.656661\pi\)
\(618\) −40.0656 −1.61167
\(619\) −18.9663 −0.762319 −0.381160 0.924509i \(-0.624475\pi\)
−0.381160 + 0.924509i \(0.624475\pi\)
\(620\) 0 0
\(621\) −66.1536 −2.65465
\(622\) −61.4712 −2.46477
\(623\) 3.43036 0.137435
\(624\) 18.2627 0.731094
\(625\) 0 0
\(626\) −50.1980 −2.00632
\(627\) −3.11629 −0.124452
\(628\) −1.77531 −0.0708426
\(629\) −1.03554 −0.0412898
\(630\) 0 0
\(631\) 2.89078 0.115080 0.0575401 0.998343i \(-0.481674\pi\)
0.0575401 + 0.998343i \(0.481674\pi\)
\(632\) 47.8719 1.90424
\(633\) −47.6468 −1.89379
\(634\) −1.17603 −0.0467060
\(635\) 0 0
\(636\) 52.4341 2.07915
\(637\) 8.25306 0.326998
\(638\) 2.87712 0.113906
\(639\) −9.59456 −0.379555
\(640\) 0 0
\(641\) −36.0171 −1.42259 −0.711295 0.702894i \(-0.751891\pi\)
−0.711295 + 0.702894i \(0.751891\pi\)
\(642\) −10.2099 −0.402951
\(643\) −37.4426 −1.47659 −0.738296 0.674476i \(-0.764369\pi\)
−0.738296 + 0.674476i \(0.764369\pi\)
\(644\) −13.4069 −0.528306
\(645\) 0 0
\(646\) −2.62878 −0.103428
\(647\) −10.6945 −0.420445 −0.210222 0.977654i \(-0.567419\pi\)
−0.210222 + 0.977654i \(0.567419\pi\)
\(648\) −83.4740 −3.27917
\(649\) 2.54912 0.100062
\(650\) 0 0
\(651\) −0.447729 −0.0175479
\(652\) −81.6183 −3.19642
\(653\) 17.5656 0.687397 0.343698 0.939080i \(-0.388320\pi\)
0.343698 + 0.939080i \(0.388320\pi\)
\(654\) 120.670 4.71857
\(655\) 0 0
\(656\) −9.02301 −0.352289
\(657\) −27.8966 −1.08835
\(658\) 3.86917 0.150836
\(659\) −21.1923 −0.825536 −0.412768 0.910836i \(-0.635438\pi\)
−0.412768 + 0.910836i \(0.635438\pi\)
\(660\) 0 0
\(661\) 37.8506 1.47222 0.736108 0.676864i \(-0.236661\pi\)
0.736108 + 0.676864i \(0.236661\pi\)
\(662\) 8.78759 0.341539
\(663\) −4.09301 −0.158959
\(664\) −53.3941 −2.07209
\(665\) 0 0
\(666\) −16.1824 −0.627055
\(667\) 6.65187 0.257561
\(668\) 19.5316 0.755699
\(669\) 81.7163 3.15934
\(670\) 0 0
\(671\) −7.49069 −0.289175
\(672\) −2.18742 −0.0843814
\(673\) −8.22334 −0.316986 −0.158493 0.987360i \(-0.550664\pi\)
−0.158493 + 0.987360i \(0.550664\pi\)
\(674\) 32.3813 1.24728
\(675\) 0 0
\(676\) −47.2765 −1.81833
\(677\) −38.5319 −1.48090 −0.740450 0.672111i \(-0.765388\pi\)
−0.740450 + 0.672111i \(0.765388\pi\)
\(678\) −124.302 −4.77377
\(679\) −9.64076 −0.369979
\(680\) 0 0
\(681\) −65.6201 −2.51457
\(682\) −0.624981 −0.0239318
\(683\) −8.28374 −0.316968 −0.158484 0.987362i \(-0.550661\pi\)
−0.158484 + 0.987362i \(0.550661\pi\)
\(684\) −27.6573 −1.05750
\(685\) 0 0
\(686\) −19.2447 −0.734768
\(687\) −11.3436 −0.432783
\(688\) 15.7106 0.598959
\(689\) 5.04701 0.192276
\(690\) 0 0
\(691\) −28.9853 −1.10265 −0.551327 0.834289i \(-0.685878\pi\)
−0.551327 + 0.834289i \(0.685878\pi\)
\(692\) 54.2748 2.06322
\(693\) 3.81704 0.144997
\(694\) −6.54947 −0.248615
\(695\) 0 0
\(696\) 19.0172 0.720844
\(697\) 2.02222 0.0765971
\(698\) 3.58861 0.135831
\(699\) −53.1493 −2.01029
\(700\) 0 0
\(701\) 1.63265 0.0616643 0.0308321 0.999525i \(-0.490184\pi\)
0.0308321 + 0.999525i \(0.490184\pi\)
\(702\) −35.3699 −1.33495
\(703\) −0.974602 −0.0367578
\(704\) 6.42846 0.242282
\(705\) 0 0
\(706\) 1.02773 0.0386790
\(707\) 2.10006 0.0789809
\(708\) 32.7368 1.23032
\(709\) −35.8643 −1.34691 −0.673455 0.739228i \(-0.735191\pi\)
−0.673455 + 0.739228i \(0.735191\pi\)
\(710\) 0 0
\(711\) −61.2239 −2.29607
\(712\) 31.6504 1.18615
\(713\) −1.44495 −0.0541137
\(714\) 4.65923 0.174367
\(715\) 0 0
\(716\) −10.0191 −0.374431
\(717\) 54.5584 2.03752
\(718\) −19.8419 −0.740493
\(719\) 15.7314 0.586684 0.293342 0.956008i \(-0.405233\pi\)
0.293342 + 0.956008i \(0.405233\pi\)
\(720\) 0 0
\(721\) 2.95559 0.110072
\(722\) −2.47407 −0.0920755
\(723\) −33.2080 −1.23502
\(724\) 106.698 3.96539
\(725\) 0 0
\(726\) 7.70992 0.286142
\(727\) −2.62799 −0.0974669 −0.0487334 0.998812i \(-0.515518\pi\)
−0.0487334 + 0.998812i \(0.515518\pi\)
\(728\) −3.68936 −0.136737
\(729\) −1.36650 −0.0506111
\(730\) 0 0
\(731\) −3.52102 −0.130230
\(732\) −96.1981 −3.55558
\(733\) −23.1636 −0.855567 −0.427783 0.903881i \(-0.640705\pi\)
−0.427783 + 0.903881i \(0.640705\pi\)
\(734\) 30.1164 1.11162
\(735\) 0 0
\(736\) −7.05941 −0.260213
\(737\) 13.5077 0.497561
\(738\) 31.6012 1.16326
\(739\) −15.0948 −0.555272 −0.277636 0.960686i \(-0.589551\pi\)
−0.277636 + 0.960686i \(0.589551\pi\)
\(740\) 0 0
\(741\) −3.85214 −0.141512
\(742\) −5.74520 −0.210913
\(743\) −5.03276 −0.184634 −0.0923170 0.995730i \(-0.529427\pi\)
−0.0923170 + 0.995730i \(0.529427\pi\)
\(744\) −4.13099 −0.151450
\(745\) 0 0
\(746\) −37.9053 −1.38781
\(747\) 68.2862 2.49846
\(748\) 4.37873 0.160102
\(749\) 0.753171 0.0275203
\(750\) 0 0
\(751\) 28.6187 1.04431 0.522155 0.852850i \(-0.325128\pi\)
0.522155 + 0.852850i \(0.325128\pi\)
\(752\) 13.0360 0.475374
\(753\) −79.5483 −2.89890
\(754\) 3.55651 0.129520
\(755\) 0 0
\(756\) 27.1074 0.985885
\(757\) −47.5711 −1.72900 −0.864500 0.502633i \(-0.832365\pi\)
−0.864500 + 0.502633i \(0.832365\pi\)
\(758\) 24.2199 0.879706
\(759\) 17.8252 0.647015
\(760\) 0 0
\(761\) 40.8932 1.48238 0.741189 0.671296i \(-0.234262\pi\)
0.741189 + 0.671296i \(0.234262\pi\)
\(762\) −18.6871 −0.676963
\(763\) −8.90171 −0.322263
\(764\) 77.4444 2.80184
\(765\) 0 0
\(766\) 4.25230 0.153642
\(767\) 3.15106 0.113778
\(768\) 101.587 3.66571
\(769\) −4.20621 −0.151680 −0.0758399 0.997120i \(-0.524164\pi\)
−0.0758399 + 0.997120i \(0.524164\pi\)
\(770\) 0 0
\(771\) 0.843470 0.0303768
\(772\) 66.0532 2.37731
\(773\) 43.8358 1.57666 0.788331 0.615251i \(-0.210945\pi\)
0.788331 + 0.615251i \(0.210945\pi\)
\(774\) −55.0229 −1.97776
\(775\) 0 0
\(776\) −88.9510 −3.19315
\(777\) 1.72738 0.0619695
\(778\) 61.5162 2.20546
\(779\) 1.90322 0.0681898
\(780\) 0 0
\(781\) 1.42963 0.0511561
\(782\) 15.0367 0.537710
\(783\) −13.4494 −0.480641
\(784\) −31.6529 −1.13046
\(785\) 0 0
\(786\) 131.111 4.67657
\(787\) 16.5893 0.591344 0.295672 0.955290i \(-0.404457\pi\)
0.295672 + 0.955290i \(0.404457\pi\)
\(788\) 80.6781 2.87404
\(789\) 76.9492 2.73946
\(790\) 0 0
\(791\) 9.16960 0.326033
\(792\) 35.2181 1.25142
\(793\) −9.25949 −0.328814
\(794\) 31.7408 1.12644
\(795\) 0 0
\(796\) −102.668 −3.63896
\(797\) −50.2487 −1.77990 −0.889951 0.456056i \(-0.849262\pi\)
−0.889951 + 0.456056i \(0.849262\pi\)
\(798\) 4.38504 0.155229
\(799\) −2.92160 −0.103359
\(800\) 0 0
\(801\) −40.4780 −1.43022
\(802\) 35.3288 1.24750
\(803\) 4.15670 0.146687
\(804\) 173.470 6.11782
\(805\) 0 0
\(806\) −0.772560 −0.0272123
\(807\) −66.4308 −2.33847
\(808\) 19.3763 0.681656
\(809\) −24.3490 −0.856066 −0.428033 0.903763i \(-0.640793\pi\)
−0.428033 + 0.903763i \(0.640793\pi\)
\(810\) 0 0
\(811\) −19.9061 −0.698996 −0.349498 0.936937i \(-0.613648\pi\)
−0.349498 + 0.936937i \(0.613648\pi\)
\(812\) −2.72569 −0.0956530
\(813\) −37.7578 −1.32423
\(814\) 2.41124 0.0845138
\(815\) 0 0
\(816\) 15.6979 0.549536
\(817\) −3.31381 −0.115936
\(818\) 36.5875 1.27925
\(819\) 4.71836 0.164873
\(820\) 0 0
\(821\) −25.2915 −0.882679 −0.441339 0.897340i \(-0.645497\pi\)
−0.441339 + 0.897340i \(0.645497\pi\)
\(822\) 107.200 3.73902
\(823\) 8.61762 0.300391 0.150196 0.988656i \(-0.452010\pi\)
0.150196 + 0.988656i \(0.452010\pi\)
\(824\) 27.2699 0.949993
\(825\) 0 0
\(826\) −3.58697 −0.124807
\(827\) −12.5285 −0.435658 −0.217829 0.975987i \(-0.569898\pi\)
−0.217829 + 0.975987i \(0.569898\pi\)
\(828\) 158.200 5.49785
\(829\) 41.9775 1.45794 0.728970 0.684546i \(-0.239999\pi\)
0.728970 + 0.684546i \(0.239999\pi\)
\(830\) 0 0
\(831\) −19.8727 −0.689376
\(832\) 7.94643 0.275493
\(833\) 7.09399 0.245792
\(834\) −42.2934 −1.46450
\(835\) 0 0
\(836\) 4.12105 0.142529
\(837\) 2.92153 0.100983
\(838\) −27.7086 −0.957177
\(839\) 13.8093 0.476750 0.238375 0.971173i \(-0.423385\pi\)
0.238375 + 0.971173i \(0.423385\pi\)
\(840\) 0 0
\(841\) −27.6476 −0.953367
\(842\) −87.8084 −3.02608
\(843\) 43.9571 1.51396
\(844\) 63.0092 2.16887
\(845\) 0 0
\(846\) −45.6558 −1.56968
\(847\) −0.568753 −0.0195426
\(848\) −19.3567 −0.664713
\(849\) 43.5064 1.49314
\(850\) 0 0
\(851\) 5.57474 0.191100
\(852\) 18.3598 0.628995
\(853\) 35.7647 1.22456 0.612280 0.790641i \(-0.290253\pi\)
0.612280 + 0.790641i \(0.290253\pi\)
\(854\) 10.5404 0.360686
\(855\) 0 0
\(856\) 6.94917 0.237518
\(857\) −6.18572 −0.211300 −0.105650 0.994403i \(-0.533692\pi\)
−0.105650 + 0.994403i \(0.533692\pi\)
\(858\) 9.53049 0.325366
\(859\) 40.5430 1.38331 0.691655 0.722228i \(-0.256882\pi\)
0.691655 + 0.722228i \(0.256882\pi\)
\(860\) 0 0
\(861\) −3.37325 −0.114960
\(862\) 92.2041 3.14048
\(863\) −7.58679 −0.258257 −0.129129 0.991628i \(-0.541218\pi\)
−0.129129 + 0.991628i \(0.541218\pi\)
\(864\) 14.2734 0.485590
\(865\) 0 0
\(866\) −85.5466 −2.90699
\(867\) 49.4587 1.67970
\(868\) 0.592087 0.0200967
\(869\) 9.12259 0.309463
\(870\) 0 0
\(871\) 16.6973 0.565765
\(872\) −82.1320 −2.78134
\(873\) 113.760 3.85020
\(874\) 14.1518 0.478691
\(875\) 0 0
\(876\) 53.3819 1.80361
\(877\) −16.2379 −0.548316 −0.274158 0.961685i \(-0.588399\pi\)
−0.274158 + 0.961685i \(0.588399\pi\)
\(878\) 85.9984 2.90231
\(879\) 0.229206 0.00773094
\(880\) 0 0
\(881\) 0.927391 0.0312446 0.0156223 0.999878i \(-0.495027\pi\)
0.0156223 + 0.999878i \(0.495027\pi\)
\(882\) 110.858 3.73277
\(883\) −52.0058 −1.75013 −0.875067 0.484002i \(-0.839183\pi\)
−0.875067 + 0.484002i \(0.839183\pi\)
\(884\) 5.41269 0.182048
\(885\) 0 0
\(886\) 93.7258 3.14878
\(887\) −0.333406 −0.0111947 −0.00559734 0.999984i \(-0.501782\pi\)
−0.00559734 + 0.999984i \(0.501782\pi\)
\(888\) 15.9378 0.534836
\(889\) 1.37853 0.0462344
\(890\) 0 0
\(891\) −15.9070 −0.532904
\(892\) −108.064 −3.61823
\(893\) −2.74967 −0.0920143
\(894\) 96.4715 3.22649
\(895\) 0 0
\(896\) −10.4496 −0.349096
\(897\) 22.0343 0.735705
\(898\) −26.2360 −0.875507
\(899\) −0.293765 −0.00979761
\(900\) 0 0
\(901\) 4.33820 0.144526
\(902\) −4.70870 −0.156782
\(903\) 5.87339 0.195454
\(904\) 84.6037 2.81388
\(905\) 0 0
\(906\) −139.835 −4.64569
\(907\) 1.23050 0.0408580 0.0204290 0.999791i \(-0.493497\pi\)
0.0204290 + 0.999791i \(0.493497\pi\)
\(908\) 86.7775 2.87981
\(909\) −24.7805 −0.821919
\(910\) 0 0
\(911\) 23.8271 0.789428 0.394714 0.918804i \(-0.370844\pi\)
0.394714 + 0.918804i \(0.370844\pi\)
\(912\) 14.7741 0.489219
\(913\) −10.1749 −0.336740
\(914\) 10.5355 0.348483
\(915\) 0 0
\(916\) 15.0010 0.495646
\(917\) −9.67190 −0.319394
\(918\) −30.4025 −1.00343
\(919\) −23.4692 −0.774176 −0.387088 0.922043i \(-0.626519\pi\)
−0.387088 + 0.922043i \(0.626519\pi\)
\(920\) 0 0
\(921\) 53.5347 1.76403
\(922\) −89.1683 −2.93660
\(923\) 1.76721 0.0581684
\(924\) −7.30413 −0.240288
\(925\) 0 0
\(926\) 29.6486 0.974313
\(927\) −34.8758 −1.14547
\(928\) −1.43521 −0.0471132
\(929\) −14.1081 −0.462872 −0.231436 0.972850i \(-0.574342\pi\)
−0.231436 + 0.972850i \(0.574342\pi\)
\(930\) 0 0
\(931\) 6.67652 0.218814
\(932\) 70.2858 2.30229
\(933\) −77.4277 −2.53487
\(934\) 28.3488 0.927600
\(935\) 0 0
\(936\) 43.5342 1.42296
\(937\) −32.2732 −1.05432 −0.527161 0.849766i \(-0.676743\pi\)
−0.527161 + 0.849766i \(0.676743\pi\)
\(938\) −19.0071 −0.620605
\(939\) −63.2282 −2.06337
\(940\) 0 0
\(941\) 4.62299 0.150705 0.0753526 0.997157i \(-0.475992\pi\)
0.0753526 + 0.997157i \(0.475992\pi\)
\(942\) −3.32137 −0.108216
\(943\) −10.8864 −0.354511
\(944\) −12.0852 −0.393340
\(945\) 0 0
\(946\) 8.19862 0.266560
\(947\) 5.86027 0.190433 0.0952166 0.995457i \(-0.469646\pi\)
0.0952166 + 0.995457i \(0.469646\pi\)
\(948\) 117.156 3.80504
\(949\) 5.13823 0.166794
\(950\) 0 0
\(951\) −1.48129 −0.0480343
\(952\) −3.17123 −0.102780
\(953\) 43.9764 1.42454 0.712268 0.701908i \(-0.247668\pi\)
0.712268 + 0.701908i \(0.247668\pi\)
\(954\) 67.7929 2.19488
\(955\) 0 0
\(956\) −72.1492 −2.33347
\(957\) 3.62396 0.117146
\(958\) −29.2916 −0.946367
\(959\) −7.90801 −0.255363
\(960\) 0 0
\(961\) −30.9362 −0.997942
\(962\) 2.98061 0.0960987
\(963\) −8.88736 −0.286391
\(964\) 43.9150 1.41441
\(965\) 0 0
\(966\) −25.0825 −0.807018
\(967\) 21.7275 0.698710 0.349355 0.936990i \(-0.386401\pi\)
0.349355 + 0.936990i \(0.386401\pi\)
\(968\) −5.24763 −0.168665
\(969\) −3.31114 −0.106369
\(970\) 0 0
\(971\) −10.5833 −0.339634 −0.169817 0.985476i \(-0.554318\pi\)
−0.169817 + 0.985476i \(0.554318\pi\)
\(972\) −61.3002 −1.96621
\(973\) 3.11994 0.100021
\(974\) 52.7271 1.68949
\(975\) 0 0
\(976\) 35.5128 1.13674
\(977\) −35.5223 −1.13646 −0.568230 0.822870i \(-0.692372\pi\)
−0.568230 + 0.822870i \(0.692372\pi\)
\(978\) −152.697 −4.88271
\(979\) 6.03138 0.192764
\(980\) 0 0
\(981\) 105.039 3.35365
\(982\) 75.9257 2.42289
\(983\) 20.4638 0.652695 0.326348 0.945250i \(-0.394182\pi\)
0.326348 + 0.945250i \(0.394182\pi\)
\(984\) −31.1235 −0.992180
\(985\) 0 0
\(986\) 3.05703 0.0973555
\(987\) 4.87351 0.155126
\(988\) 5.09416 0.162067
\(989\) 18.9551 0.602737
\(990\) 0 0
\(991\) −26.0437 −0.827305 −0.413653 0.910435i \(-0.635747\pi\)
−0.413653 + 0.910435i \(0.635747\pi\)
\(992\) 0.311763 0.00989849
\(993\) 11.0686 0.351252
\(994\) −2.01168 −0.0638066
\(995\) 0 0
\(996\) −130.670 −4.14043
\(997\) 4.10364 0.129964 0.0649819 0.997886i \(-0.479301\pi\)
0.0649819 + 0.997886i \(0.479301\pi\)
\(998\) −56.2709 −1.78123
\(999\) −11.2715 −0.356616
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 5225.2.a.w.1.1 yes 15
5.4 even 2 5225.2.a.t.1.15 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.t.1.15 15 5.4 even 2
5225.2.a.w.1.1 yes 15 1.1 even 1 trivial