Properties

Label 1425.2.a.k.1.2
Level $1425$
Weight $2$
Character 1425.1
Self dual yes
Analytic conductor $11.379$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,2,Mod(1,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, 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("1425.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.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: \(11.3786822880\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1425.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} -1.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -0.414214 q^{6} -1.41421 q^{7} -1.58579 q^{8} +1.00000 q^{9} +6.24264 q^{11} +1.82843 q^{12} +0.585786 q^{13} -0.585786 q^{14} +3.00000 q^{16} -6.82843 q^{17} +0.414214 q^{18} -1.00000 q^{19} +1.41421 q^{21} +2.58579 q^{22} +3.65685 q^{23} +1.58579 q^{24} +0.242641 q^{26} -1.00000 q^{27} +2.58579 q^{28} -1.41421 q^{29} -8.82843 q^{31} +4.41421 q^{32} -6.24264 q^{33} -2.82843 q^{34} -1.82843 q^{36} +0.585786 q^{37} -0.414214 q^{38} -0.585786 q^{39} +8.24264 q^{41} +0.585786 q^{42} -3.75736 q^{43} -11.4142 q^{44} +1.51472 q^{46} -3.65685 q^{47} -3.00000 q^{48} -5.00000 q^{49} +6.82843 q^{51} -1.07107 q^{52} -8.00000 q^{53} -0.414214 q^{54} +2.24264 q^{56} +1.00000 q^{57} -0.585786 q^{58} -4.48528 q^{59} -15.3137 q^{61} -3.65685 q^{62} -1.41421 q^{63} -4.17157 q^{64} -2.58579 q^{66} -1.65685 q^{67} +12.4853 q^{68} -3.65685 q^{69} -5.17157 q^{71} -1.58579 q^{72} -3.65685 q^{73} +0.242641 q^{74} +1.82843 q^{76} -8.82843 q^{77} -0.242641 q^{78} +1.00000 q^{81} +3.41421 q^{82} -7.17157 q^{83} -2.58579 q^{84} -1.55635 q^{86} +1.41421 q^{87} -9.89949 q^{88} -13.8995 q^{89} -0.828427 q^{91} -6.68629 q^{92} +8.82843 q^{93} -1.51472 q^{94} -4.41421 q^{96} +18.2426 q^{97} -2.07107 q^{98} +6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 2 q^{9} + 4 q^{11} - 2 q^{12} + 4 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 2 q^{18} - 2 q^{19} + 8 q^{22} - 4 q^{23} + 6 q^{24} - 8 q^{26} - 2 q^{27} + 8 q^{28} - 12 q^{31} + 6 q^{32} - 4 q^{33} + 2 q^{36} + 4 q^{37} + 2 q^{38} - 4 q^{39} + 8 q^{41} + 4 q^{42} - 16 q^{43} - 20 q^{44} + 20 q^{46} + 4 q^{47} - 6 q^{48} - 10 q^{49} + 8 q^{51} + 12 q^{52} - 16 q^{53} + 2 q^{54} - 4 q^{56} + 2 q^{57} - 4 q^{58} + 8 q^{59} - 8 q^{61} + 4 q^{62} - 14 q^{64} - 8 q^{66} + 8 q^{67} + 8 q^{68} + 4 q^{69} - 16 q^{71} - 6 q^{72} + 4 q^{73} - 8 q^{74} - 2 q^{76} - 12 q^{77} + 8 q^{78} + 2 q^{81} + 4 q^{82} - 20 q^{83} - 8 q^{84} + 28 q^{86} - 8 q^{89} + 4 q^{91} - 36 q^{92} + 12 q^{93} - 20 q^{94} - 6 q^{96} + 28 q^{97} + 10 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) −0.414214 −0.169102
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.24264 1.88223 0.941113 0.338091i \(-0.109781\pi\)
0.941113 + 0.338091i \(0.109781\pi\)
\(12\) 1.82843 0.527821
\(13\) 0.585786 0.162468 0.0812340 0.996695i \(-0.474114\pi\)
0.0812340 + 0.996695i \(0.474114\pi\)
\(14\) −0.585786 −0.156558
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0.414214 0.0976311
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 2.58579 0.551292
\(23\) 3.65685 0.762507 0.381253 0.924471i \(-0.375493\pi\)
0.381253 + 0.924471i \(0.375493\pi\)
\(24\) 1.58579 0.323697
\(25\) 0 0
\(26\) 0.242641 0.0475858
\(27\) −1.00000 −0.192450
\(28\) 2.58579 0.488668
\(29\) −1.41421 −0.262613 −0.131306 0.991342i \(-0.541917\pi\)
−0.131306 + 0.991342i \(0.541917\pi\)
\(30\) 0 0
\(31\) −8.82843 −1.58563 −0.792816 0.609461i \(-0.791386\pi\)
−0.792816 + 0.609461i \(0.791386\pi\)
\(32\) 4.41421 0.780330
\(33\) −6.24264 −1.08670
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) 0.585786 0.0963027 0.0481513 0.998840i \(-0.484667\pi\)
0.0481513 + 0.998840i \(0.484667\pi\)
\(38\) −0.414214 −0.0671943
\(39\) −0.585786 −0.0938009
\(40\) 0 0
\(41\) 8.24264 1.28728 0.643642 0.765327i \(-0.277423\pi\)
0.643642 + 0.765327i \(0.277423\pi\)
\(42\) 0.585786 0.0903888
\(43\) −3.75736 −0.572992 −0.286496 0.958081i \(-0.592491\pi\)
−0.286496 + 0.958081i \(0.592491\pi\)
\(44\) −11.4142 −1.72076
\(45\) 0 0
\(46\) 1.51472 0.223333
\(47\) −3.65685 −0.533407 −0.266704 0.963779i \(-0.585934\pi\)
−0.266704 + 0.963779i \(0.585934\pi\)
\(48\) −3.00000 −0.433013
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 6.82843 0.956171
\(52\) −1.07107 −0.148530
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0 0
\(56\) 2.24264 0.299685
\(57\) 1.00000 0.132453
\(58\) −0.585786 −0.0769175
\(59\) −4.48528 −0.583934 −0.291967 0.956428i \(-0.594310\pi\)
−0.291967 + 0.956428i \(0.594310\pi\)
\(60\) 0 0
\(61\) −15.3137 −1.96072 −0.980360 0.197218i \(-0.936809\pi\)
−0.980360 + 0.197218i \(0.936809\pi\)
\(62\) −3.65685 −0.464421
\(63\) −1.41421 −0.178174
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) −2.58579 −0.318288
\(67\) −1.65685 −0.202417 −0.101208 0.994865i \(-0.532271\pi\)
−0.101208 + 0.994865i \(0.532271\pi\)
\(68\) 12.4853 1.51406
\(69\) −3.65685 −0.440234
\(70\) 0 0
\(71\) −5.17157 −0.613753 −0.306876 0.951749i \(-0.599284\pi\)
−0.306876 + 0.951749i \(0.599284\pi\)
\(72\) −1.58579 −0.186887
\(73\) −3.65685 −0.428002 −0.214001 0.976833i \(-0.568650\pi\)
−0.214001 + 0.976833i \(0.568650\pi\)
\(74\) 0.242641 0.0282064
\(75\) 0 0
\(76\) 1.82843 0.209735
\(77\) −8.82843 −1.00609
\(78\) −0.242641 −0.0274736
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.41421 0.377037
\(83\) −7.17157 −0.787182 −0.393591 0.919286i \(-0.628767\pi\)
−0.393591 + 0.919286i \(0.628767\pi\)
\(84\) −2.58579 −0.282132
\(85\) 0 0
\(86\) −1.55635 −0.167825
\(87\) 1.41421 0.151620
\(88\) −9.89949 −1.05529
\(89\) −13.8995 −1.47334 −0.736672 0.676250i \(-0.763604\pi\)
−0.736672 + 0.676250i \(0.763604\pi\)
\(90\) 0 0
\(91\) −0.828427 −0.0868428
\(92\) −6.68629 −0.697094
\(93\) 8.82843 0.915465
\(94\) −1.51472 −0.156231
\(95\) 0 0
\(96\) −4.41421 −0.450524
\(97\) 18.2426 1.85226 0.926130 0.377205i \(-0.123115\pi\)
0.926130 + 0.377205i \(0.123115\pi\)
\(98\) −2.07107 −0.209209
\(99\) 6.24264 0.627409
\(100\) 0 0
\(101\) −8.82843 −0.878461 −0.439231 0.898374i \(-0.644749\pi\)
−0.439231 + 0.898374i \(0.644749\pi\)
\(102\) 2.82843 0.280056
\(103\) −15.3137 −1.50890 −0.754452 0.656355i \(-0.772097\pi\)
−0.754452 + 0.656355i \(0.772097\pi\)
\(104\) −0.928932 −0.0910893
\(105\) 0 0
\(106\) −3.31371 −0.321856
\(107\) 3.31371 0.320348 0.160174 0.987089i \(-0.448794\pi\)
0.160174 + 0.987089i \(0.448794\pi\)
\(108\) 1.82843 0.175940
\(109\) 10.4853 1.00431 0.502154 0.864778i \(-0.332541\pi\)
0.502154 + 0.864778i \(0.332541\pi\)
\(110\) 0 0
\(111\) −0.585786 −0.0556004
\(112\) −4.24264 −0.400892
\(113\) 18.1421 1.70667 0.853334 0.521364i \(-0.174577\pi\)
0.853334 + 0.521364i \(0.174577\pi\)
\(114\) 0.414214 0.0387947
\(115\) 0 0
\(116\) 2.58579 0.240084
\(117\) 0.585786 0.0541560
\(118\) −1.85786 −0.171030
\(119\) 9.65685 0.885242
\(120\) 0 0
\(121\) 27.9706 2.54278
\(122\) −6.34315 −0.574281
\(123\) −8.24264 −0.743214
\(124\) 16.1421 1.44961
\(125\) 0 0
\(126\) −0.585786 −0.0521860
\(127\) 3.31371 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(128\) −10.5563 −0.933058
\(129\) 3.75736 0.330817
\(130\) 0 0
\(131\) −11.4142 −0.997264 −0.498632 0.866814i \(-0.666164\pi\)
−0.498632 + 0.866814i \(0.666164\pi\)
\(132\) 11.4142 0.993480
\(133\) 1.41421 0.122628
\(134\) −0.686292 −0.0592866
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) −1.51472 −0.128941
\(139\) 14.1421 1.19952 0.599760 0.800180i \(-0.295263\pi\)
0.599760 + 0.800180i \(0.295263\pi\)
\(140\) 0 0
\(141\) 3.65685 0.307963
\(142\) −2.14214 −0.179764
\(143\) 3.65685 0.305802
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) −1.51472 −0.125359
\(147\) 5.00000 0.412393
\(148\) −1.07107 −0.0880412
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 10.4853 0.853280 0.426640 0.904422i \(-0.359697\pi\)
0.426640 + 0.904422i \(0.359697\pi\)
\(152\) 1.58579 0.128624
\(153\) −6.82843 −0.552046
\(154\) −3.65685 −0.294678
\(155\) 0 0
\(156\) 1.07107 0.0857541
\(157\) 6.48528 0.517582 0.258791 0.965933i \(-0.416676\pi\)
0.258791 + 0.965933i \(0.416676\pi\)
\(158\) 0 0
\(159\) 8.00000 0.634441
\(160\) 0 0
\(161\) −5.17157 −0.407577
\(162\) 0.414214 0.0325437
\(163\) −2.10051 −0.164524 −0.0822621 0.996611i \(-0.526214\pi\)
−0.0822621 + 0.996611i \(0.526214\pi\)
\(164\) −15.0711 −1.17685
\(165\) 0 0
\(166\) −2.97056 −0.230560
\(167\) −5.31371 −0.411187 −0.205594 0.978637i \(-0.565912\pi\)
−0.205594 + 0.978637i \(0.565912\pi\)
\(168\) −2.24264 −0.173023
\(169\) −12.6569 −0.973604
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 6.87006 0.523837
\(173\) −19.7990 −1.50529 −0.752645 0.658427i \(-0.771222\pi\)
−0.752645 + 0.658427i \(0.771222\pi\)
\(174\) 0.585786 0.0444084
\(175\) 0 0
\(176\) 18.7279 1.41167
\(177\) 4.48528 0.337134
\(178\) −5.75736 −0.431532
\(179\) −0.485281 −0.0362716 −0.0181358 0.999836i \(-0.505773\pi\)
−0.0181358 + 0.999836i \(0.505773\pi\)
\(180\) 0 0
\(181\) −15.1716 −1.12769 −0.563847 0.825879i \(-0.690679\pi\)
−0.563847 + 0.825879i \(0.690679\pi\)
\(182\) −0.343146 −0.0254357
\(183\) 15.3137 1.13202
\(184\) −5.79899 −0.427507
\(185\) 0 0
\(186\) 3.65685 0.268134
\(187\) −42.6274 −3.11723
\(188\) 6.68629 0.487648
\(189\) 1.41421 0.102869
\(190\) 0 0
\(191\) 1.75736 0.127158 0.0635790 0.997977i \(-0.479749\pi\)
0.0635790 + 0.997977i \(0.479749\pi\)
\(192\) 4.17157 0.301057
\(193\) 9.07107 0.652950 0.326475 0.945206i \(-0.394139\pi\)
0.326475 + 0.945206i \(0.394139\pi\)
\(194\) 7.55635 0.542514
\(195\) 0 0
\(196\) 9.14214 0.653010
\(197\) −1.17157 −0.0834711 −0.0417356 0.999129i \(-0.513289\pi\)
−0.0417356 + 0.999129i \(0.513289\pi\)
\(198\) 2.58579 0.183764
\(199\) −10.1421 −0.718957 −0.359478 0.933153i \(-0.617045\pi\)
−0.359478 + 0.933153i \(0.617045\pi\)
\(200\) 0 0
\(201\) 1.65685 0.116865
\(202\) −3.65685 −0.257295
\(203\) 2.00000 0.140372
\(204\) −12.4853 −0.874145
\(205\) 0 0
\(206\) −6.34315 −0.441948
\(207\) 3.65685 0.254169
\(208\) 1.75736 0.121851
\(209\) −6.24264 −0.431812
\(210\) 0 0
\(211\) −15.3137 −1.05424 −0.527120 0.849791i \(-0.676728\pi\)
−0.527120 + 0.849791i \(0.676728\pi\)
\(212\) 14.6274 1.00462
\(213\) 5.17157 0.354350
\(214\) 1.37258 0.0938278
\(215\) 0 0
\(216\) 1.58579 0.107899
\(217\) 12.4853 0.847556
\(218\) 4.34315 0.294155
\(219\) 3.65685 0.247107
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) −0.242641 −0.0162850
\(223\) 26.6274 1.78310 0.891552 0.452919i \(-0.149617\pi\)
0.891552 + 0.452919i \(0.149617\pi\)
\(224\) −6.24264 −0.417104
\(225\) 0 0
\(226\) 7.51472 0.499872
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) −1.82843 −0.121091
\(229\) 22.6274 1.49526 0.747631 0.664114i \(-0.231191\pi\)
0.747631 + 0.664114i \(0.231191\pi\)
\(230\) 0 0
\(231\) 8.82843 0.580868
\(232\) 2.24264 0.147237
\(233\) 11.6569 0.763666 0.381833 0.924231i \(-0.375293\pi\)
0.381833 + 0.924231i \(0.375293\pi\)
\(234\) 0.242641 0.0158619
\(235\) 0 0
\(236\) 8.20101 0.533840
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) −1.27208 −0.0822839 −0.0411419 0.999153i \(-0.513100\pi\)
−0.0411419 + 0.999153i \(0.513100\pi\)
\(240\) 0 0
\(241\) −8.34315 −0.537429 −0.268715 0.963220i \(-0.586599\pi\)
−0.268715 + 0.963220i \(0.586599\pi\)
\(242\) 11.5858 0.744763
\(243\) −1.00000 −0.0641500
\(244\) 28.0000 1.79252
\(245\) 0 0
\(246\) −3.41421 −0.217682
\(247\) −0.585786 −0.0372727
\(248\) 14.0000 0.889001
\(249\) 7.17157 0.454480
\(250\) 0 0
\(251\) −10.2426 −0.646510 −0.323255 0.946312i \(-0.604777\pi\)
−0.323255 + 0.946312i \(0.604777\pi\)
\(252\) 2.58579 0.162889
\(253\) 22.8284 1.43521
\(254\) 1.37258 0.0861235
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −12.4853 −0.778810 −0.389405 0.921067i \(-0.627319\pi\)
−0.389405 + 0.921067i \(0.627319\pi\)
\(258\) 1.55635 0.0968941
\(259\) −0.828427 −0.0514760
\(260\) 0 0
\(261\) −1.41421 −0.0875376
\(262\) −4.72792 −0.292092
\(263\) −27.4558 −1.69300 −0.846500 0.532389i \(-0.821294\pi\)
−0.846500 + 0.532389i \(0.821294\pi\)
\(264\) 9.89949 0.609272
\(265\) 0 0
\(266\) 0.585786 0.0359169
\(267\) 13.8995 0.850635
\(268\) 3.02944 0.185052
\(269\) 11.0711 0.675015 0.337507 0.941323i \(-0.390416\pi\)
0.337507 + 0.941323i \(0.390416\pi\)
\(270\) 0 0
\(271\) −5.17157 −0.314151 −0.157075 0.987587i \(-0.550207\pi\)
−0.157075 + 0.987587i \(0.550207\pi\)
\(272\) −20.4853 −1.24210
\(273\) 0.828427 0.0501387
\(274\) −4.14214 −0.250236
\(275\) 0 0
\(276\) 6.68629 0.402467
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 5.85786 0.351331
\(279\) −8.82843 −0.528544
\(280\) 0 0
\(281\) −17.4142 −1.03884 −0.519422 0.854518i \(-0.673853\pi\)
−0.519422 + 0.854518i \(0.673853\pi\)
\(282\) 1.51472 0.0902002
\(283\) −26.3848 −1.56841 −0.784206 0.620500i \(-0.786929\pi\)
−0.784206 + 0.620500i \(0.786929\pi\)
\(284\) 9.45584 0.561101
\(285\) 0 0
\(286\) 1.51472 0.0895672
\(287\) −11.6569 −0.688082
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) −18.2426 −1.06940
\(292\) 6.68629 0.391286
\(293\) 28.4853 1.66413 0.832064 0.554680i \(-0.187159\pi\)
0.832064 + 0.554680i \(0.187159\pi\)
\(294\) 2.07107 0.120787
\(295\) 0 0
\(296\) −0.928932 −0.0539931
\(297\) −6.24264 −0.362235
\(298\) −2.48528 −0.143968
\(299\) 2.14214 0.123883
\(300\) 0 0
\(301\) 5.31371 0.306277
\(302\) 4.34315 0.249920
\(303\) 8.82843 0.507180
\(304\) −3.00000 −0.172062
\(305\) 0 0
\(306\) −2.82843 −0.161690
\(307\) −26.8284 −1.53118 −0.765590 0.643329i \(-0.777553\pi\)
−0.765590 + 0.643329i \(0.777553\pi\)
\(308\) 16.1421 0.919784
\(309\) 15.3137 0.871166
\(310\) 0 0
\(311\) 2.24264 0.127168 0.0635842 0.997976i \(-0.479747\pi\)
0.0635842 + 0.997976i \(0.479747\pi\)
\(312\) 0.928932 0.0525904
\(313\) 33.7990 1.91043 0.955216 0.295910i \(-0.0956227\pi\)
0.955216 + 0.295910i \(0.0956227\pi\)
\(314\) 2.68629 0.151596
\(315\) 0 0
\(316\) 0 0
\(317\) −18.6274 −1.04622 −0.523110 0.852265i \(-0.675228\pi\)
−0.523110 + 0.852265i \(0.675228\pi\)
\(318\) 3.31371 0.185824
\(319\) −8.82843 −0.494297
\(320\) 0 0
\(321\) −3.31371 −0.184953
\(322\) −2.14214 −0.119377
\(323\) 6.82843 0.379944
\(324\) −1.82843 −0.101579
\(325\) 0 0
\(326\) −0.870058 −0.0481880
\(327\) −10.4853 −0.579837
\(328\) −13.0711 −0.721729
\(329\) 5.17157 0.285118
\(330\) 0 0
\(331\) −0.142136 −0.00781248 −0.00390624 0.999992i \(-0.501243\pi\)
−0.00390624 + 0.999992i \(0.501243\pi\)
\(332\) 13.1127 0.719653
\(333\) 0.585786 0.0321009
\(334\) −2.20101 −0.120434
\(335\) 0 0
\(336\) 4.24264 0.231455
\(337\) 27.4142 1.49335 0.746674 0.665191i \(-0.231650\pi\)
0.746674 + 0.665191i \(0.231650\pi\)
\(338\) −5.24264 −0.285162
\(339\) −18.1421 −0.985346
\(340\) 0 0
\(341\) −55.1127 −2.98452
\(342\) −0.414214 −0.0223981
\(343\) 16.9706 0.916324
\(344\) 5.95837 0.321254
\(345\) 0 0
\(346\) −8.20101 −0.440889
\(347\) −23.4558 −1.25918 −0.629588 0.776929i \(-0.716776\pi\)
−0.629588 + 0.776929i \(0.716776\pi\)
\(348\) −2.58579 −0.138613
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) −0.585786 −0.0312670
\(352\) 27.5563 1.46876
\(353\) 7.65685 0.407533 0.203767 0.979019i \(-0.434682\pi\)
0.203767 + 0.979019i \(0.434682\pi\)
\(354\) 1.85786 0.0987444
\(355\) 0 0
\(356\) 25.4142 1.34695
\(357\) −9.65685 −0.511095
\(358\) −0.201010 −0.0106237
\(359\) −28.8701 −1.52370 −0.761852 0.647752i \(-0.775710\pi\)
−0.761852 + 0.647752i \(0.775710\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −6.28427 −0.330294
\(363\) −27.9706 −1.46807
\(364\) 1.51472 0.0793928
\(365\) 0 0
\(366\) 6.34315 0.331562
\(367\) −22.3848 −1.16848 −0.584238 0.811582i \(-0.698606\pi\)
−0.584238 + 0.811582i \(0.698606\pi\)
\(368\) 10.9706 0.571880
\(369\) 8.24264 0.429095
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) −16.1421 −0.836931
\(373\) −3.41421 −0.176781 −0.0883906 0.996086i \(-0.528172\pi\)
−0.0883906 + 0.996086i \(0.528172\pi\)
\(374\) −17.6569 −0.913014
\(375\) 0 0
\(376\) 5.79899 0.299060
\(377\) −0.828427 −0.0426662
\(378\) 0.585786 0.0301296
\(379\) 8.82843 0.453486 0.226743 0.973955i \(-0.427192\pi\)
0.226743 + 0.973955i \(0.427192\pi\)
\(380\) 0 0
\(381\) −3.31371 −0.169766
\(382\) 0.727922 0.0372437
\(383\) −28.0000 −1.43073 −0.715367 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) 10.5563 0.538701
\(385\) 0 0
\(386\) 3.75736 0.191245
\(387\) −3.75736 −0.190997
\(388\) −33.3553 −1.69336
\(389\) −2.97056 −0.150614 −0.0753068 0.997160i \(-0.523994\pi\)
−0.0753068 + 0.997160i \(0.523994\pi\)
\(390\) 0 0
\(391\) −24.9706 −1.26282
\(392\) 7.92893 0.400472
\(393\) 11.4142 0.575771
\(394\) −0.485281 −0.0244481
\(395\) 0 0
\(396\) −11.4142 −0.573586
\(397\) 16.6274 0.834506 0.417253 0.908790i \(-0.362993\pi\)
0.417253 + 0.908790i \(0.362993\pi\)
\(398\) −4.20101 −0.210578
\(399\) −1.41421 −0.0707992
\(400\) 0 0
\(401\) −16.2426 −0.811119 −0.405559 0.914069i \(-0.632923\pi\)
−0.405559 + 0.914069i \(0.632923\pi\)
\(402\) 0.686292 0.0342291
\(403\) −5.17157 −0.257614
\(404\) 16.1421 0.803101
\(405\) 0 0
\(406\) 0.828427 0.0411141
\(407\) 3.65685 0.181264
\(408\) −10.8284 −0.536087
\(409\) −7.17157 −0.354611 −0.177306 0.984156i \(-0.556738\pi\)
−0.177306 + 0.984156i \(0.556738\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 28.0000 1.37946
\(413\) 6.34315 0.312126
\(414\) 1.51472 0.0744444
\(415\) 0 0
\(416\) 2.58579 0.126779
\(417\) −14.1421 −0.692543
\(418\) −2.58579 −0.126475
\(419\) 0.585786 0.0286175 0.0143088 0.999898i \(-0.495445\pi\)
0.0143088 + 0.999898i \(0.495445\pi\)
\(420\) 0 0
\(421\) −13.3137 −0.648870 −0.324435 0.945908i \(-0.605174\pi\)
−0.324435 + 0.945908i \(0.605174\pi\)
\(422\) −6.34315 −0.308780
\(423\) −3.65685 −0.177802
\(424\) 12.6863 0.616101
\(425\) 0 0
\(426\) 2.14214 0.103787
\(427\) 21.6569 1.04805
\(428\) −6.05887 −0.292867
\(429\) −3.65685 −0.176555
\(430\) 0 0
\(431\) 31.1127 1.49865 0.749323 0.662205i \(-0.230379\pi\)
0.749323 + 0.662205i \(0.230379\pi\)
\(432\) −3.00000 −0.144338
\(433\) 25.0711 1.20484 0.602419 0.798180i \(-0.294204\pi\)
0.602419 + 0.798180i \(0.294204\pi\)
\(434\) 5.17157 0.248243
\(435\) 0 0
\(436\) −19.1716 −0.918152
\(437\) −3.65685 −0.174931
\(438\) 1.51472 0.0723761
\(439\) −10.3431 −0.493651 −0.246826 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) −1.65685 −0.0788085
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 1.07107 0.0508306
\(445\) 0 0
\(446\) 11.0294 0.522259
\(447\) 6.00000 0.283790
\(448\) 5.89949 0.278725
\(449\) 26.8701 1.26808 0.634038 0.773302i \(-0.281396\pi\)
0.634038 + 0.773302i \(0.281396\pi\)
\(450\) 0 0
\(451\) 51.4558 2.42296
\(452\) −33.1716 −1.56026
\(453\) −10.4853 −0.492641
\(454\) 7.85786 0.368788
\(455\) 0 0
\(456\) −1.58579 −0.0742613
\(457\) −27.1716 −1.27103 −0.635516 0.772087i \(-0.719213\pi\)
−0.635516 + 0.772087i \(0.719213\pi\)
\(458\) 9.37258 0.437952
\(459\) 6.82843 0.318724
\(460\) 0 0
\(461\) 8.34315 0.388579 0.194290 0.980944i \(-0.437760\pi\)
0.194290 + 0.980944i \(0.437760\pi\)
\(462\) 3.65685 0.170132
\(463\) −15.7574 −0.732307 −0.366153 0.930555i \(-0.619325\pi\)
−0.366153 + 0.930555i \(0.619325\pi\)
\(464\) −4.24264 −0.196960
\(465\) 0 0
\(466\) 4.82843 0.223673
\(467\) −24.3431 −1.12647 −0.563233 0.826298i \(-0.690443\pi\)
−0.563233 + 0.826298i \(0.690443\pi\)
\(468\) −1.07107 −0.0495101
\(469\) 2.34315 0.108196
\(470\) 0 0
\(471\) −6.48528 −0.298826
\(472\) 7.11270 0.327388
\(473\) −23.4558 −1.07850
\(474\) 0 0
\(475\) 0 0
\(476\) −17.6569 −0.809301
\(477\) −8.00000 −0.366295
\(478\) −0.526912 −0.0241004
\(479\) 2.92893 0.133826 0.0669132 0.997759i \(-0.478685\pi\)
0.0669132 + 0.997759i \(0.478685\pi\)
\(480\) 0 0
\(481\) 0.343146 0.0156461
\(482\) −3.45584 −0.157409
\(483\) 5.17157 0.235315
\(484\) −51.1421 −2.32464
\(485\) 0 0
\(486\) −0.414214 −0.0187891
\(487\) −3.51472 −0.159267 −0.0796336 0.996824i \(-0.525375\pi\)
−0.0796336 + 0.996824i \(0.525375\pi\)
\(488\) 24.2843 1.09930
\(489\) 2.10051 0.0949881
\(490\) 0 0
\(491\) −10.2426 −0.462244 −0.231122 0.972925i \(-0.574240\pi\)
−0.231122 + 0.972925i \(0.574240\pi\)
\(492\) 15.0711 0.679456
\(493\) 9.65685 0.434923
\(494\) −0.242641 −0.0109169
\(495\) 0 0
\(496\) −26.4853 −1.18922
\(497\) 7.31371 0.328065
\(498\) 2.97056 0.133114
\(499\) −10.8284 −0.484747 −0.242373 0.970183i \(-0.577926\pi\)
−0.242373 + 0.970183i \(0.577926\pi\)
\(500\) 0 0
\(501\) 5.31371 0.237399
\(502\) −4.24264 −0.189358
\(503\) 0.828427 0.0369377 0.0184689 0.999829i \(-0.494121\pi\)
0.0184689 + 0.999829i \(0.494121\pi\)
\(504\) 2.24264 0.0998952
\(505\) 0 0
\(506\) 9.45584 0.420364
\(507\) 12.6569 0.562111
\(508\) −6.05887 −0.268819
\(509\) 24.7279 1.09605 0.548023 0.836463i \(-0.315381\pi\)
0.548023 + 0.836463i \(0.315381\pi\)
\(510\) 0 0
\(511\) 5.17157 0.228777
\(512\) 22.7574 1.00574
\(513\) 1.00000 0.0441511
\(514\) −5.17157 −0.228108
\(515\) 0 0
\(516\) −6.87006 −0.302437
\(517\) −22.8284 −1.00399
\(518\) −0.343146 −0.0150770
\(519\) 19.7990 0.869079
\(520\) 0 0
\(521\) 16.2426 0.711603 0.355802 0.934562i \(-0.384208\pi\)
0.355802 + 0.934562i \(0.384208\pi\)
\(522\) −0.585786 −0.0256392
\(523\) 32.4853 1.42048 0.710241 0.703959i \(-0.248586\pi\)
0.710241 + 0.703959i \(0.248586\pi\)
\(524\) 20.8701 0.911713
\(525\) 0 0
\(526\) −11.3726 −0.495868
\(527\) 60.2843 2.62602
\(528\) −18.7279 −0.815028
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) −4.48528 −0.194645
\(532\) −2.58579 −0.112108
\(533\) 4.82843 0.209142
\(534\) 5.75736 0.249145
\(535\) 0 0
\(536\) 2.62742 0.113487
\(537\) 0.485281 0.0209414
\(538\) 4.58579 0.197707
\(539\) −31.2132 −1.34445
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) −2.14214 −0.0920126
\(543\) 15.1716 0.651075
\(544\) −30.1421 −1.29233
\(545\) 0 0
\(546\) 0.343146 0.0146853
\(547\) −7.51472 −0.321306 −0.160653 0.987011i \(-0.551360\pi\)
−0.160653 + 0.987011i \(0.551360\pi\)
\(548\) 18.2843 0.781065
\(549\) −15.3137 −0.653573
\(550\) 0 0
\(551\) 1.41421 0.0602475
\(552\) 5.79899 0.246821
\(553\) 0 0
\(554\) 9.11270 0.387161
\(555\) 0 0
\(556\) −25.8579 −1.09662
\(557\) 2.68629 0.113822 0.0569109 0.998379i \(-0.481875\pi\)
0.0569109 + 0.998379i \(0.481875\pi\)
\(558\) −3.65685 −0.154807
\(559\) −2.20101 −0.0930928
\(560\) 0 0
\(561\) 42.6274 1.79973
\(562\) −7.21320 −0.304271
\(563\) −26.2843 −1.10775 −0.553875 0.832600i \(-0.686851\pi\)
−0.553875 + 0.832600i \(0.686851\pi\)
\(564\) −6.68629 −0.281544
\(565\) 0 0
\(566\) −10.9289 −0.459377
\(567\) −1.41421 −0.0593914
\(568\) 8.20101 0.344107
\(569\) 16.9289 0.709698 0.354849 0.934924i \(-0.384532\pi\)
0.354849 + 0.934924i \(0.384532\pi\)
\(570\) 0 0
\(571\) 14.8284 0.620550 0.310275 0.950647i \(-0.399579\pi\)
0.310275 + 0.950647i \(0.399579\pi\)
\(572\) −6.68629 −0.279568
\(573\) −1.75736 −0.0734147
\(574\) −4.82843 −0.201535
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) 31.4558 1.30952 0.654762 0.755835i \(-0.272769\pi\)
0.654762 + 0.755835i \(0.272769\pi\)
\(578\) 12.2721 0.510451
\(579\) −9.07107 −0.376981
\(580\) 0 0
\(581\) 10.1421 0.420767
\(582\) −7.55635 −0.313221
\(583\) −49.9411 −2.06835
\(584\) 5.79899 0.239964
\(585\) 0 0
\(586\) 11.7990 0.487412
\(587\) 23.6569 0.976423 0.488211 0.872725i \(-0.337649\pi\)
0.488211 + 0.872725i \(0.337649\pi\)
\(588\) −9.14214 −0.377015
\(589\) 8.82843 0.363769
\(590\) 0 0
\(591\) 1.17157 0.0481921
\(592\) 1.75736 0.0722270
\(593\) 8.62742 0.354286 0.177143 0.984185i \(-0.443315\pi\)
0.177143 + 0.984185i \(0.443315\pi\)
\(594\) −2.58579 −0.106096
\(595\) 0 0
\(596\) 10.9706 0.449372
\(597\) 10.1421 0.415090
\(598\) 0.887302 0.0362845
\(599\) −36.9706 −1.51058 −0.755288 0.655393i \(-0.772503\pi\)
−0.755288 + 0.655393i \(0.772503\pi\)
\(600\) 0 0
\(601\) −20.1421 −0.821615 −0.410807 0.911722i \(-0.634753\pi\)
−0.410807 + 0.911722i \(0.634753\pi\)
\(602\) 2.20101 0.0897065
\(603\) −1.65685 −0.0674723
\(604\) −19.1716 −0.780080
\(605\) 0 0
\(606\) 3.65685 0.148550
\(607\) 0.485281 0.0196970 0.00984848 0.999952i \(-0.496865\pi\)
0.00984848 + 0.999952i \(0.496865\pi\)
\(608\) −4.41421 −0.179020
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −2.14214 −0.0866615
\(612\) 12.4853 0.504688
\(613\) −6.48528 −0.261938 −0.130969 0.991386i \(-0.541809\pi\)
−0.130969 + 0.991386i \(0.541809\pi\)
\(614\) −11.1127 −0.448472
\(615\) 0 0
\(616\) 14.0000 0.564076
\(617\) −11.5147 −0.463565 −0.231783 0.972768i \(-0.574456\pi\)
−0.231783 + 0.972768i \(0.574456\pi\)
\(618\) 6.34315 0.255159
\(619\) −3.51472 −0.141268 −0.0706342 0.997502i \(-0.522502\pi\)
−0.0706342 + 0.997502i \(0.522502\pi\)
\(620\) 0 0
\(621\) −3.65685 −0.146745
\(622\) 0.928932 0.0372468
\(623\) 19.6569 0.787535
\(624\) −1.75736 −0.0703507
\(625\) 0 0
\(626\) 14.0000 0.559553
\(627\) 6.24264 0.249307
\(628\) −11.8579 −0.473180
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 15.3137 0.608665
\(634\) −7.71573 −0.306431
\(635\) 0 0
\(636\) −14.6274 −0.580015
\(637\) −2.92893 −0.116049
\(638\) −3.65685 −0.144776
\(639\) −5.17157 −0.204584
\(640\) 0 0
\(641\) 39.3553 1.55444 0.777221 0.629227i \(-0.216629\pi\)
0.777221 + 0.629227i \(0.216629\pi\)
\(642\) −1.37258 −0.0541715
\(643\) −5.61522 −0.221443 −0.110721 0.993851i \(-0.535316\pi\)
−0.110721 + 0.993851i \(0.535316\pi\)
\(644\) 9.45584 0.372612
\(645\) 0 0
\(646\) 2.82843 0.111283
\(647\) 7.85786 0.308925 0.154462 0.987999i \(-0.450635\pi\)
0.154462 + 0.987999i \(0.450635\pi\)
\(648\) −1.58579 −0.0622956
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) −12.4853 −0.489337
\(652\) 3.84062 0.150410
\(653\) −37.1716 −1.45464 −0.727318 0.686301i \(-0.759233\pi\)
−0.727318 + 0.686301i \(0.759233\pi\)
\(654\) −4.34315 −0.169830
\(655\) 0 0
\(656\) 24.7279 0.965463
\(657\) −3.65685 −0.142667
\(658\) 2.14214 0.0835091
\(659\) 26.6274 1.03726 0.518628 0.855000i \(-0.326443\pi\)
0.518628 + 0.855000i \(0.326443\pi\)
\(660\) 0 0
\(661\) −43.4558 −1.69024 −0.845118 0.534579i \(-0.820470\pi\)
−0.845118 + 0.534579i \(0.820470\pi\)
\(662\) −0.0588745 −0.00228822
\(663\) 4.00000 0.155347
\(664\) 11.3726 0.441342
\(665\) 0 0
\(666\) 0.242641 0.00940214
\(667\) −5.17157 −0.200244
\(668\) 9.71573 0.375913
\(669\) −26.6274 −1.02948
\(670\) 0 0
\(671\) −95.5980 −3.69052
\(672\) 6.24264 0.240815
\(673\) −32.1838 −1.24059 −0.620297 0.784367i \(-0.712988\pi\)
−0.620297 + 0.784367i \(0.712988\pi\)
\(674\) 11.3553 0.437391
\(675\) 0 0
\(676\) 23.1421 0.890082
\(677\) 16.9706 0.652232 0.326116 0.945330i \(-0.394260\pi\)
0.326116 + 0.945330i \(0.394260\pi\)
\(678\) −7.51472 −0.288601
\(679\) −25.7990 −0.990074
\(680\) 0 0
\(681\) −18.9706 −0.726954
\(682\) −22.8284 −0.874146
\(683\) −29.6569 −1.13479 −0.567394 0.823446i \(-0.692048\pi\)
−0.567394 + 0.823446i \(0.692048\pi\)
\(684\) 1.82843 0.0699117
\(685\) 0 0
\(686\) 7.02944 0.268385
\(687\) −22.6274 −0.863290
\(688\) −11.2721 −0.429744
\(689\) −4.68629 −0.178533
\(690\) 0 0
\(691\) 41.1716 1.56624 0.783120 0.621870i \(-0.213627\pi\)
0.783120 + 0.621870i \(0.213627\pi\)
\(692\) 36.2010 1.37616
\(693\) −8.82843 −0.335364
\(694\) −9.71573 −0.368804
\(695\) 0 0
\(696\) −2.24264 −0.0850071
\(697\) −56.2843 −2.13192
\(698\) 7.45584 0.282208
\(699\) −11.6569 −0.440903
\(700\) 0 0
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) −0.242641 −0.00915788
\(703\) −0.585786 −0.0220934
\(704\) −26.0416 −0.981481
\(705\) 0 0
\(706\) 3.17157 0.119364
\(707\) 12.4853 0.469557
\(708\) −8.20101 −0.308213
\(709\) −4.97056 −0.186673 −0.0933367 0.995635i \(-0.529753\pi\)
−0.0933367 + 0.995635i \(0.529753\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 22.0416 0.826045
\(713\) −32.2843 −1.20906
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 0.887302 0.0331600
\(717\) 1.27208 0.0475066
\(718\) −11.9584 −0.446282
\(719\) 13.0711 0.487469 0.243734 0.969842i \(-0.421628\pi\)
0.243734 + 0.969842i \(0.421628\pi\)
\(720\) 0 0
\(721\) 21.6569 0.806543
\(722\) 0.414214 0.0154154
\(723\) 8.34315 0.310285
\(724\) 27.7401 1.03095
\(725\) 0 0
\(726\) −11.5858 −0.429989
\(727\) 23.3553 0.866202 0.433101 0.901345i \(-0.357419\pi\)
0.433101 + 0.901345i \(0.357419\pi\)
\(728\) 1.31371 0.0486893
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 25.6569 0.948953
\(732\) −28.0000 −1.03491
\(733\) 22.0000 0.812589 0.406294 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) −9.27208 −0.342239
\(735\) 0 0
\(736\) 16.1421 0.595007
\(737\) −10.3431 −0.380995
\(738\) 3.41421 0.125679
\(739\) 25.6569 0.943803 0.471901 0.881651i \(-0.343568\pi\)
0.471901 + 0.881651i \(0.343568\pi\)
\(740\) 0 0
\(741\) 0.585786 0.0215194
\(742\) 4.68629 0.172039
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) −14.0000 −0.513265
\(745\) 0 0
\(746\) −1.41421 −0.0517780
\(747\) −7.17157 −0.262394
\(748\) 77.9411 2.84981
\(749\) −4.68629 −0.171233
\(750\) 0 0
\(751\) 4.14214 0.151149 0.0755743 0.997140i \(-0.475921\pi\)
0.0755743 + 0.997140i \(0.475921\pi\)
\(752\) −10.9706 −0.400055
\(753\) 10.2426 0.373263
\(754\) −0.343146 −0.0124966
\(755\) 0 0
\(756\) −2.58579 −0.0940441
\(757\) −52.4264 −1.90547 −0.952735 0.303802i \(-0.901744\pi\)
−0.952735 + 0.303802i \(0.901744\pi\)
\(758\) 3.65685 0.132823
\(759\) −22.8284 −0.828619
\(760\) 0 0
\(761\) −23.9411 −0.867865 −0.433933 0.900945i \(-0.642874\pi\)
−0.433933 + 0.900945i \(0.642874\pi\)
\(762\) −1.37258 −0.0497234
\(763\) −14.8284 −0.536825
\(764\) −3.21320 −0.116250
\(765\) 0 0
\(766\) −11.5980 −0.419052
\(767\) −2.62742 −0.0948705
\(768\) −3.97056 −0.143275
\(769\) 9.02944 0.325610 0.162805 0.986658i \(-0.447946\pi\)
0.162805 + 0.986658i \(0.447946\pi\)
\(770\) 0 0
\(771\) 12.4853 0.449646
\(772\) −16.5858 −0.596936
\(773\) 14.3431 0.515887 0.257944 0.966160i \(-0.416955\pi\)
0.257944 + 0.966160i \(0.416955\pi\)
\(774\) −1.55635 −0.0559418
\(775\) 0 0
\(776\) −28.9289 −1.03849
\(777\) 0.828427 0.0297197
\(778\) −1.23045 −0.0441137
\(779\) −8.24264 −0.295323
\(780\) 0 0
\(781\) −32.2843 −1.15522
\(782\) −10.3431 −0.369870
\(783\) 1.41421 0.0505399
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 4.72792 0.168639
\(787\) 37.1716 1.32502 0.662512 0.749052i \(-0.269490\pi\)
0.662512 + 0.749052i \(0.269490\pi\)
\(788\) 2.14214 0.0763104
\(789\) 27.4558 0.977454
\(790\) 0 0
\(791\) −25.6569 −0.912253
\(792\) −9.89949 −0.351763
\(793\) −8.97056 −0.318554
\(794\) 6.88730 0.244421
\(795\) 0 0
\(796\) 18.5442 0.657280
\(797\) −14.1421 −0.500940 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(798\) −0.585786 −0.0207366
\(799\) 24.9706 0.883395
\(800\) 0 0
\(801\) −13.8995 −0.491115
\(802\) −6.72792 −0.237571
\(803\) −22.8284 −0.805598
\(804\) −3.02944 −0.106840
\(805\) 0 0
\(806\) −2.14214 −0.0754535
\(807\) −11.0711 −0.389720
\(808\) 14.0000 0.492518
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) 31.3137 1.09957 0.549787 0.835305i \(-0.314709\pi\)
0.549787 + 0.835305i \(0.314709\pi\)
\(812\) −3.65685 −0.128330
\(813\) 5.17157 0.181375
\(814\) 1.51472 0.0530909
\(815\) 0 0
\(816\) 20.4853 0.717128
\(817\) 3.75736 0.131453
\(818\) −2.97056 −0.103863
\(819\) −0.828427 −0.0289476
\(820\) 0 0
\(821\) −2.48528 −0.0867369 −0.0433685 0.999059i \(-0.513809\pi\)
−0.0433685 + 0.999059i \(0.513809\pi\)
\(822\) 4.14214 0.144474
\(823\) −57.0122 −1.98732 −0.993660 0.112426i \(-0.964138\pi\)
−0.993660 + 0.112426i \(0.964138\pi\)
\(824\) 24.2843 0.845983
\(825\) 0 0
\(826\) 2.62742 0.0914195
\(827\) 25.3137 0.880244 0.440122 0.897938i \(-0.354935\pi\)
0.440122 + 0.897938i \(0.354935\pi\)
\(828\) −6.68629 −0.232365
\(829\) 29.5147 1.02509 0.512544 0.858661i \(-0.328703\pi\)
0.512544 + 0.858661i \(0.328703\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) −2.44365 −0.0847183
\(833\) 34.1421 1.18295
\(834\) −5.85786 −0.202841
\(835\) 0 0
\(836\) 11.4142 0.394769
\(837\) 8.82843 0.305155
\(838\) 0.242641 0.00838188
\(839\) −38.1421 −1.31681 −0.658406 0.752663i \(-0.728769\pi\)
−0.658406 + 0.752663i \(0.728769\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) −5.51472 −0.190050
\(843\) 17.4142 0.599777
\(844\) 28.0000 0.963800
\(845\) 0 0
\(846\) −1.51472 −0.0520771
\(847\) −39.5563 −1.35917
\(848\) −24.0000 −0.824163
\(849\) 26.3848 0.905523
\(850\) 0 0
\(851\) 2.14214 0.0734315
\(852\) −9.45584 −0.323952
\(853\) 11.1716 0.382507 0.191254 0.981541i \(-0.438745\pi\)
0.191254 + 0.981541i \(0.438745\pi\)
\(854\) 8.97056 0.306966
\(855\) 0 0
\(856\) −5.25483 −0.179607
\(857\) −35.3137 −1.20629 −0.603147 0.797630i \(-0.706087\pi\)
−0.603147 + 0.797630i \(0.706087\pi\)
\(858\) −1.51472 −0.0517116
\(859\) 17.9411 0.612143 0.306072 0.952008i \(-0.400985\pi\)
0.306072 + 0.952008i \(0.400985\pi\)
\(860\) 0 0
\(861\) 11.6569 0.397265
\(862\) 12.8873 0.438943
\(863\) −31.3137 −1.06593 −0.532966 0.846137i \(-0.678922\pi\)
−0.532966 + 0.846137i \(0.678922\pi\)
\(864\) −4.41421 −0.150175
\(865\) 0 0
\(866\) 10.3848 0.352889
\(867\) −29.6274 −1.00620
\(868\) −22.8284 −0.774847
\(869\) 0 0
\(870\) 0 0
\(871\) −0.970563 −0.0328863
\(872\) −16.6274 −0.563075
\(873\) 18.2426 0.617420
\(874\) −1.51472 −0.0512361
\(875\) 0 0
\(876\) −6.68629 −0.225909
\(877\) 49.0711 1.65701 0.828506 0.559980i \(-0.189191\pi\)
0.828506 + 0.559980i \(0.189191\pi\)
\(878\) −4.28427 −0.144587
\(879\) −28.4853 −0.960785
\(880\) 0 0
\(881\) 33.7990 1.13872 0.569358 0.822089i \(-0.307192\pi\)
0.569358 + 0.822089i \(0.307192\pi\)
\(882\) −2.07107 −0.0697365
\(883\) −36.0416 −1.21290 −0.606449 0.795123i \(-0.707406\pi\)
−0.606449 + 0.795123i \(0.707406\pi\)
\(884\) 7.31371 0.245987
\(885\) 0 0
\(886\) −7.45584 −0.250484
\(887\) 41.9411 1.40825 0.704123 0.710078i \(-0.251341\pi\)
0.704123 + 0.710078i \(0.251341\pi\)
\(888\) 0.928932 0.0311729
\(889\) −4.68629 −0.157173
\(890\) 0 0
\(891\) 6.24264 0.209136
\(892\) −48.6863 −1.63014
\(893\) 3.65685 0.122372
\(894\) 2.48528 0.0831202
\(895\) 0 0
\(896\) 14.9289 0.498741
\(897\) −2.14214 −0.0715238
\(898\) 11.1299 0.371411
\(899\) 12.4853 0.416407
\(900\) 0 0
\(901\) 54.6274 1.81990
\(902\) 21.3137 0.709669
\(903\) −5.31371 −0.176829
\(904\) −28.7696 −0.956861
\(905\) 0 0
\(906\) −4.34315 −0.144291
\(907\) −10.1421 −0.336764 −0.168382 0.985722i \(-0.553854\pi\)
−0.168382 + 0.985722i \(0.553854\pi\)
\(908\) −34.6863 −1.15111
\(909\) −8.82843 −0.292820
\(910\) 0 0
\(911\) 31.3137 1.03747 0.518735 0.854935i \(-0.326403\pi\)
0.518735 + 0.854935i \(0.326403\pi\)
\(912\) 3.00000 0.0993399
\(913\) −44.7696 −1.48166
\(914\) −11.2548 −0.372277
\(915\) 0 0
\(916\) −41.3726 −1.36699
\(917\) 16.1421 0.533060
\(918\) 2.82843 0.0933520
\(919\) −12.0000 −0.395843 −0.197922 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 0 0
\(921\) 26.8284 0.884027
\(922\) 3.45584 0.113812
\(923\) −3.02944 −0.0997151
\(924\) −16.1421 −0.531037
\(925\) 0 0
\(926\) −6.52691 −0.214488
\(927\) −15.3137 −0.502968
\(928\) −6.24264 −0.204925
\(929\) −29.1127 −0.955157 −0.477578 0.878589i \(-0.658485\pi\)
−0.477578 + 0.878589i \(0.658485\pi\)
\(930\) 0 0
\(931\) 5.00000 0.163868
\(932\) −21.3137 −0.698154
\(933\) −2.24264 −0.0734208
\(934\) −10.0833 −0.329934
\(935\) 0 0
\(936\) −0.928932 −0.0303631
\(937\) −9.79899 −0.320119 −0.160060 0.987107i \(-0.551169\pi\)
−0.160060 + 0.987107i \(0.551169\pi\)
\(938\) 0.970563 0.0316900
\(939\) −33.7990 −1.10299
\(940\) 0 0
\(941\) −30.8701 −1.00634 −0.503168 0.864189i \(-0.667832\pi\)
−0.503168 + 0.864189i \(0.667832\pi\)
\(942\) −2.68629 −0.0875241
\(943\) 30.1421 0.981563
\(944\) −13.4558 −0.437950
\(945\) 0 0
\(946\) −9.71573 −0.315886
\(947\) 52.8284 1.71669 0.858347 0.513070i \(-0.171492\pi\)
0.858347 + 0.513070i \(0.171492\pi\)
\(948\) 0 0
\(949\) −2.14214 −0.0695367
\(950\) 0 0
\(951\) 18.6274 0.604035
\(952\) −15.3137 −0.496320
\(953\) 2.54416 0.0824133 0.0412066 0.999151i \(-0.486880\pi\)
0.0412066 + 0.999151i \(0.486880\pi\)
\(954\) −3.31371 −0.107285
\(955\) 0 0
\(956\) 2.32590 0.0752250
\(957\) 8.82843 0.285383
\(958\) 1.21320 0.0391968
\(959\) 14.1421 0.456673
\(960\) 0 0
\(961\) 46.9411 1.51423
\(962\) 0.142136 0.00458264
\(963\) 3.31371 0.106783
\(964\) 15.2548 0.491325
\(965\) 0 0
\(966\) 2.14214 0.0689221
\(967\) 40.0416 1.28765 0.643826 0.765172i \(-0.277346\pi\)
0.643826 + 0.765172i \(0.277346\pi\)
\(968\) −44.3553 −1.42563
\(969\) −6.82843 −0.219361
\(970\) 0 0
\(971\) 33.6569 1.08010 0.540050 0.841633i \(-0.318405\pi\)
0.540050 + 0.841633i \(0.318405\pi\)
\(972\) 1.82843 0.0586468
\(973\) −20.0000 −0.641171
\(974\) −1.45584 −0.0466483
\(975\) 0 0
\(976\) −45.9411 −1.47054
\(977\) 24.2843 0.776923 0.388461 0.921465i \(-0.373007\pi\)
0.388461 + 0.921465i \(0.373007\pi\)
\(978\) 0.870058 0.0278214
\(979\) −86.7696 −2.77317
\(980\) 0 0
\(981\) 10.4853 0.334769
\(982\) −4.24264 −0.135388
\(983\) −20.6274 −0.657912 −0.328956 0.944345i \(-0.606697\pi\)
−0.328956 + 0.944345i \(0.606697\pi\)
\(984\) 13.0711 0.416690
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) −5.17157 −0.164613
\(988\) 1.07107 0.0340752
\(989\) −13.7401 −0.436910
\(990\) 0 0
\(991\) 13.6569 0.433824 0.216912 0.976191i \(-0.430401\pi\)
0.216912 + 0.976191i \(0.430401\pi\)
\(992\) −38.9706 −1.23732
\(993\) 0.142136 0.00451054
\(994\) 3.02944 0.0960879
\(995\) 0 0
\(996\) −13.1127 −0.415492
\(997\) −58.0833 −1.83952 −0.919758 0.392487i \(-0.871615\pi\)
−0.919758 + 0.392487i \(0.871615\pi\)
\(998\) −4.48528 −0.141979
\(999\) −0.585786 −0.0185335
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 1425.2.a.k.1.2 2
3.2 odd 2 4275.2.a.y.1.1 2
5.2 odd 4 1425.2.c.l.799.3 4
5.3 odd 4 1425.2.c.l.799.2 4
5.4 even 2 285.2.a.g.1.1 2
15.14 odd 2 855.2.a.d.1.2 2
20.19 odd 2 4560.2.a.bf.1.1 2
95.94 odd 2 5415.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.1 2 5.4 even 2
855.2.a.d.1.2 2 15.14 odd 2
1425.2.a.k.1.2 2 1.1 even 1 trivial
1425.2.c.l.799.2 4 5.3 odd 4
1425.2.c.l.799.3 4 5.2 odd 4
4275.2.a.y.1.1 2 3.2 odd 2
4560.2.a.bf.1.1 2 20.19 odd 2
5415.2.a.n.1.2 2 95.94 odd 2