Properties

Label 5225.2.a.x.1.12
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $0$
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: \(0\)
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: \( x^{15} - 5 x^{14} - 11 x^{13} + 85 x^{12} + 6 x^{11} - 537 x^{10} + 327 x^{9} + 1556 x^{8} - 1451 x^{7} + \cdots - 13 \) 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 1.12
Root \(2.12341\) 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)
 
\(f(q)\) \(=\) \(q+2.12341 q^{2} -0.300215 q^{3} +2.50886 q^{4} -0.637479 q^{6} -1.66086 q^{7} +1.08053 q^{8} -2.90987 q^{9} +O(q^{10})\) \(q+2.12341 q^{2} -0.300215 q^{3} +2.50886 q^{4} -0.637479 q^{6} -1.66086 q^{7} +1.08053 q^{8} -2.90987 q^{9} +1.00000 q^{11} -0.753198 q^{12} +0.406405 q^{13} -3.52669 q^{14} -2.72333 q^{16} +4.20766 q^{17} -6.17885 q^{18} -1.00000 q^{19} +0.498615 q^{21} +2.12341 q^{22} +3.54516 q^{23} -0.324390 q^{24} +0.862965 q^{26} +1.77423 q^{27} -4.16688 q^{28} +7.47483 q^{29} +5.10468 q^{31} -7.94379 q^{32} -0.300215 q^{33} +8.93459 q^{34} -7.30047 q^{36} -1.45804 q^{37} -2.12341 q^{38} -0.122009 q^{39} +8.69765 q^{41} +1.05876 q^{42} -0.953442 q^{43} +2.50886 q^{44} +7.52782 q^{46} +5.58315 q^{47} +0.817583 q^{48} -4.24154 q^{49} -1.26320 q^{51} +1.01962 q^{52} -1.77008 q^{53} +3.76742 q^{54} -1.79461 q^{56} +0.300215 q^{57} +15.8721 q^{58} +6.26485 q^{59} -1.18689 q^{61} +10.8393 q^{62} +4.83290 q^{63} -11.4213 q^{64} -0.637479 q^{66} +12.1533 q^{67} +10.5565 q^{68} -1.06431 q^{69} +5.93683 q^{71} -3.14420 q^{72} +2.39287 q^{73} -3.09601 q^{74} -2.50886 q^{76} -1.66086 q^{77} -0.259075 q^{78} -14.2878 q^{79} +8.19696 q^{81} +18.4687 q^{82} +8.85637 q^{83} +1.25096 q^{84} -2.02455 q^{86} -2.24405 q^{87} +1.08053 q^{88} -2.79190 q^{89} -0.674984 q^{91} +8.89432 q^{92} -1.53250 q^{93} +11.8553 q^{94} +2.38484 q^{96} +6.33470 q^{97} -9.00651 q^{98} -2.90987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 5 q^{2} + 4 q^{3} + 17 q^{4} - q^{6} + 15 q^{7} + 15 q^{8} + 19 q^{9} + 15 q^{11} + 9 q^{12} + 13 q^{13} + 9 q^{14} + 21 q^{16} + 11 q^{17} + 16 q^{18} - 15 q^{19} + 5 q^{22} + 10 q^{23} + 17 q^{24} - 17 q^{26} + 13 q^{27} + 30 q^{28} + 5 q^{29} + 6 q^{31} + 40 q^{32} + 4 q^{33} + 17 q^{34} + 28 q^{36} + 13 q^{37} - 5 q^{38} - 22 q^{39} + 30 q^{42} + 36 q^{43} + 17 q^{44} + 13 q^{46} - 6 q^{47} + 14 q^{48} + 16 q^{49} + 4 q^{51} + 50 q^{52} + 9 q^{53} + 9 q^{54} - 18 q^{56} - 4 q^{57} + 2 q^{58} - 7 q^{59} - 2 q^{61} + 11 q^{62} + 39 q^{63} + 17 q^{64} - q^{66} + 35 q^{67} - 18 q^{68} - 9 q^{69} + 13 q^{71} + 68 q^{72} + 2 q^{73} + 13 q^{74} - 17 q^{76} + 15 q^{77} - 10 q^{78} + 6 q^{79} + 11 q^{81} + 14 q^{82} + 30 q^{83} - 6 q^{84} - 25 q^{86} + 19 q^{87} + 15 q^{88} + 55 q^{89} + 26 q^{91} - 18 q^{92} + 14 q^{93} - 22 q^{94} - 17 q^{96} + 28 q^{97} - 22 q^{98} + 19 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12341 1.50148 0.750738 0.660600i \(-0.229698\pi\)
0.750738 + 0.660600i \(0.229698\pi\)
\(3\) −0.300215 −0.173329 −0.0866645 0.996238i \(-0.527621\pi\)
−0.0866645 + 0.996238i \(0.527621\pi\)
\(4\) 2.50886 1.25443
\(5\) 0 0
\(6\) −0.637479 −0.260250
\(7\) −1.66086 −0.627747 −0.313874 0.949465i \(-0.601627\pi\)
−0.313874 + 0.949465i \(0.601627\pi\)
\(8\) 1.08053 0.382024
\(9\) −2.90987 −0.969957
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −0.753198 −0.217430
\(13\) 0.406405 0.112717 0.0563583 0.998411i \(-0.482051\pi\)
0.0563583 + 0.998411i \(0.482051\pi\)
\(14\) −3.52669 −0.942548
\(15\) 0 0
\(16\) −2.72333 −0.680832
\(17\) 4.20766 1.02051 0.510254 0.860024i \(-0.329551\pi\)
0.510254 + 0.860024i \(0.329551\pi\)
\(18\) −6.17885 −1.45637
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0.498615 0.108807
\(22\) 2.12341 0.452712
\(23\) 3.54516 0.739217 0.369608 0.929188i \(-0.379492\pi\)
0.369608 + 0.929188i \(0.379492\pi\)
\(24\) −0.324390 −0.0662159
\(25\) 0 0
\(26\) 0.862965 0.169241
\(27\) 1.77423 0.341451
\(28\) −4.16688 −0.787466
\(29\) 7.47483 1.38804 0.694021 0.719955i \(-0.255838\pi\)
0.694021 + 0.719955i \(0.255838\pi\)
\(30\) 0 0
\(31\) 5.10468 0.916827 0.458414 0.888739i \(-0.348418\pi\)
0.458414 + 0.888739i \(0.348418\pi\)
\(32\) −7.94379 −1.40428
\(33\) −0.300215 −0.0522607
\(34\) 8.93459 1.53227
\(35\) 0 0
\(36\) −7.30047 −1.21675
\(37\) −1.45804 −0.239700 −0.119850 0.992792i \(-0.538241\pi\)
−0.119850 + 0.992792i \(0.538241\pi\)
\(38\) −2.12341 −0.344462
\(39\) −0.122009 −0.0195371
\(40\) 0 0
\(41\) 8.69765 1.35834 0.679172 0.733979i \(-0.262339\pi\)
0.679172 + 0.733979i \(0.262339\pi\)
\(42\) 1.05876 0.163371
\(43\) −0.953442 −0.145399 −0.0726993 0.997354i \(-0.523161\pi\)
−0.0726993 + 0.997354i \(0.523161\pi\)
\(44\) 2.50886 0.378226
\(45\) 0 0
\(46\) 7.52782 1.10992
\(47\) 5.58315 0.814386 0.407193 0.913342i \(-0.366508\pi\)
0.407193 + 0.913342i \(0.366508\pi\)
\(48\) 0.817583 0.118008
\(49\) −4.24154 −0.605934
\(50\) 0 0
\(51\) −1.26320 −0.176884
\(52\) 1.01962 0.141395
\(53\) −1.77008 −0.243139 −0.121570 0.992583i \(-0.538793\pi\)
−0.121570 + 0.992583i \(0.538793\pi\)
\(54\) 3.76742 0.512680
\(55\) 0 0
\(56\) −1.79461 −0.239815
\(57\) 0.300215 0.0397644
\(58\) 15.8721 2.08411
\(59\) 6.26485 0.815614 0.407807 0.913068i \(-0.366294\pi\)
0.407807 + 0.913068i \(0.366294\pi\)
\(60\) 0 0
\(61\) −1.18689 −0.151965 −0.0759827 0.997109i \(-0.524209\pi\)
−0.0759827 + 0.997109i \(0.524209\pi\)
\(62\) 10.8393 1.37660
\(63\) 4.83290 0.608888
\(64\) −11.4213 −1.42766
\(65\) 0 0
\(66\) −0.637479 −0.0784682
\(67\) 12.1533 1.48476 0.742381 0.669978i \(-0.233697\pi\)
0.742381 + 0.669978i \(0.233697\pi\)
\(68\) 10.5565 1.28016
\(69\) −1.06431 −0.128128
\(70\) 0 0
\(71\) 5.93683 0.704572 0.352286 0.935892i \(-0.385405\pi\)
0.352286 + 0.935892i \(0.385405\pi\)
\(72\) −3.14420 −0.370547
\(73\) 2.39287 0.280064 0.140032 0.990147i \(-0.455279\pi\)
0.140032 + 0.990147i \(0.455279\pi\)
\(74\) −3.09601 −0.359904
\(75\) 0 0
\(76\) −2.50886 −0.287787
\(77\) −1.66086 −0.189273
\(78\) −0.259075 −0.0293344
\(79\) −14.2878 −1.60751 −0.803754 0.594962i \(-0.797167\pi\)
−0.803754 + 0.594962i \(0.797167\pi\)
\(80\) 0 0
\(81\) 8.19696 0.910774
\(82\) 18.4687 2.03952
\(83\) 8.85637 0.972113 0.486057 0.873927i \(-0.338435\pi\)
0.486057 + 0.873927i \(0.338435\pi\)
\(84\) 1.25096 0.136491
\(85\) 0 0
\(86\) −2.02455 −0.218313
\(87\) −2.24405 −0.240588
\(88\) 1.08053 0.115185
\(89\) −2.79190 −0.295940 −0.147970 0.988992i \(-0.547274\pi\)
−0.147970 + 0.988992i \(0.547274\pi\)
\(90\) 0 0
\(91\) −0.674984 −0.0707575
\(92\) 8.89432 0.927297
\(93\) −1.53250 −0.158913
\(94\) 11.8553 1.22278
\(95\) 0 0
\(96\) 2.38484 0.243402
\(97\) 6.33470 0.643192 0.321596 0.946877i \(-0.395781\pi\)
0.321596 + 0.946877i \(0.395781\pi\)
\(98\) −9.00651 −0.909795
\(99\) −2.90987 −0.292453
\(100\) 0 0
\(101\) 12.8093 1.27458 0.637289 0.770625i \(-0.280056\pi\)
0.637289 + 0.770625i \(0.280056\pi\)
\(102\) −2.68229 −0.265587
\(103\) −13.6053 −1.34057 −0.670284 0.742105i \(-0.733828\pi\)
−0.670284 + 0.742105i \(0.733828\pi\)
\(104\) 0.439132 0.0430605
\(105\) 0 0
\(106\) −3.75860 −0.365068
\(107\) 10.3908 1.00452 0.502258 0.864718i \(-0.332503\pi\)
0.502258 + 0.864718i \(0.332503\pi\)
\(108\) 4.45130 0.428327
\(109\) 6.56231 0.628555 0.314277 0.949331i \(-0.398238\pi\)
0.314277 + 0.949331i \(0.398238\pi\)
\(110\) 0 0
\(111\) 0.437725 0.0415470
\(112\) 4.52307 0.427390
\(113\) 1.73328 0.163053 0.0815267 0.996671i \(-0.474020\pi\)
0.0815267 + 0.996671i \(0.474020\pi\)
\(114\) 0.637479 0.0597053
\(115\) 0 0
\(116\) 18.7533 1.74120
\(117\) −1.18259 −0.109330
\(118\) 13.3028 1.22463
\(119\) −6.98835 −0.640621
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −2.52025 −0.228172
\(123\) −2.61116 −0.235441
\(124\) 12.8070 1.15010
\(125\) 0 0
\(126\) 10.2622 0.914231
\(127\) 6.49967 0.576753 0.288376 0.957517i \(-0.406885\pi\)
0.288376 + 0.957517i \(0.406885\pi\)
\(128\) −8.36443 −0.739318
\(129\) 0.286237 0.0252018
\(130\) 0 0
\(131\) 1.28315 0.112109 0.0560546 0.998428i \(-0.482148\pi\)
0.0560546 + 0.998428i \(0.482148\pi\)
\(132\) −0.753198 −0.0655575
\(133\) 1.66086 0.144015
\(134\) 25.8064 2.22934
\(135\) 0 0
\(136\) 4.54650 0.389859
\(137\) 6.33302 0.541067 0.270533 0.962711i \(-0.412800\pi\)
0.270533 + 0.962711i \(0.412800\pi\)
\(138\) −2.25996 −0.192381
\(139\) −4.92149 −0.417435 −0.208717 0.977976i \(-0.566929\pi\)
−0.208717 + 0.977976i \(0.566929\pi\)
\(140\) 0 0
\(141\) −1.67614 −0.141157
\(142\) 12.6063 1.05790
\(143\) 0.406405 0.0339853
\(144\) 7.92453 0.660378
\(145\) 0 0
\(146\) 5.08104 0.420510
\(147\) 1.27337 0.105026
\(148\) −3.65802 −0.300688
\(149\) −11.1592 −0.914194 −0.457097 0.889417i \(-0.651111\pi\)
−0.457097 + 0.889417i \(0.651111\pi\)
\(150\) 0 0
\(151\) −9.38884 −0.764053 −0.382027 0.924151i \(-0.624774\pi\)
−0.382027 + 0.924151i \(0.624774\pi\)
\(152\) −1.08053 −0.0876424
\(153\) −12.2438 −0.989849
\(154\) −3.52669 −0.284189
\(155\) 0 0
\(156\) −0.306104 −0.0245079
\(157\) 19.9255 1.59022 0.795112 0.606462i \(-0.207412\pi\)
0.795112 + 0.606462i \(0.207412\pi\)
\(158\) −30.3389 −2.41364
\(159\) 0.531404 0.0421431
\(160\) 0 0
\(161\) −5.88802 −0.464041
\(162\) 17.4055 1.36751
\(163\) −18.6951 −1.46431 −0.732156 0.681137i \(-0.761486\pi\)
−0.732156 + 0.681137i \(0.761486\pi\)
\(164\) 21.8212 1.70395
\(165\) 0 0
\(166\) 18.8057 1.45961
\(167\) 5.92467 0.458465 0.229233 0.973372i \(-0.426378\pi\)
0.229233 + 0.973372i \(0.426378\pi\)
\(168\) 0.538768 0.0415668
\(169\) −12.8348 −0.987295
\(170\) 0 0
\(171\) 2.90987 0.222523
\(172\) −2.39206 −0.182393
\(173\) 4.45145 0.338438 0.169219 0.985579i \(-0.445876\pi\)
0.169219 + 0.985579i \(0.445876\pi\)
\(174\) −4.76504 −0.361237
\(175\) 0 0
\(176\) −2.72333 −0.205279
\(177\) −1.88080 −0.141370
\(178\) −5.92834 −0.444348
\(179\) −5.29409 −0.395699 −0.197850 0.980232i \(-0.563396\pi\)
−0.197850 + 0.980232i \(0.563396\pi\)
\(180\) 0 0
\(181\) −24.6556 −1.83264 −0.916319 0.400450i \(-0.868854\pi\)
−0.916319 + 0.400450i \(0.868854\pi\)
\(182\) −1.43327 −0.106241
\(183\) 0.356321 0.0263400
\(184\) 3.83064 0.282399
\(185\) 0 0
\(186\) −3.25412 −0.238604
\(187\) 4.20766 0.307695
\(188\) 14.0074 1.02159
\(189\) −2.94675 −0.214345
\(190\) 0 0
\(191\) 25.2282 1.82545 0.912724 0.408577i \(-0.133975\pi\)
0.912724 + 0.408577i \(0.133975\pi\)
\(192\) 3.42883 0.247455
\(193\) 0.939382 0.0676182 0.0338091 0.999428i \(-0.489236\pi\)
0.0338091 + 0.999428i \(0.489236\pi\)
\(194\) 13.4512 0.965737
\(195\) 0 0
\(196\) −10.6414 −0.760103
\(197\) −6.91170 −0.492438 −0.246219 0.969214i \(-0.579188\pi\)
−0.246219 + 0.969214i \(0.579188\pi\)
\(198\) −6.17885 −0.439111
\(199\) 22.5252 1.59677 0.798385 0.602147i \(-0.205688\pi\)
0.798385 + 0.602147i \(0.205688\pi\)
\(200\) 0 0
\(201\) −3.64860 −0.257352
\(202\) 27.1995 1.91375
\(203\) −12.4147 −0.871339
\(204\) −3.16920 −0.221889
\(205\) 0 0
\(206\) −28.8896 −2.01283
\(207\) −10.3160 −0.717008
\(208\) −1.10678 −0.0767410
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −13.5534 −0.933054 −0.466527 0.884507i \(-0.654495\pi\)
−0.466527 + 0.884507i \(0.654495\pi\)
\(212\) −4.44089 −0.305002
\(213\) −1.78232 −0.122123
\(214\) 22.0639 1.50826
\(215\) 0 0
\(216\) 1.91711 0.130442
\(217\) −8.47817 −0.575536
\(218\) 13.9345 0.943761
\(219\) −0.718375 −0.0485433
\(220\) 0 0
\(221\) 1.71002 0.115028
\(222\) 0.929469 0.0623819
\(223\) 3.13674 0.210052 0.105026 0.994469i \(-0.466507\pi\)
0.105026 + 0.994469i \(0.466507\pi\)
\(224\) 13.1935 0.881531
\(225\) 0 0
\(226\) 3.68046 0.244821
\(227\) −17.7305 −1.17681 −0.588407 0.808565i \(-0.700245\pi\)
−0.588407 + 0.808565i \(0.700245\pi\)
\(228\) 0.753198 0.0498818
\(229\) −1.58042 −0.104437 −0.0522186 0.998636i \(-0.516629\pi\)
−0.0522186 + 0.998636i \(0.516629\pi\)
\(230\) 0 0
\(231\) 0.498615 0.0328065
\(232\) 8.07676 0.530265
\(233\) 10.1587 0.665519 0.332760 0.943012i \(-0.392020\pi\)
0.332760 + 0.943012i \(0.392020\pi\)
\(234\) −2.51112 −0.164157
\(235\) 0 0
\(236\) 15.7177 1.02313
\(237\) 4.28942 0.278628
\(238\) −14.8391 −0.961878
\(239\) −6.60437 −0.427201 −0.213601 0.976921i \(-0.568519\pi\)
−0.213601 + 0.976921i \(0.568519\pi\)
\(240\) 0 0
\(241\) −27.4790 −1.77008 −0.885041 0.465514i \(-0.845869\pi\)
−0.885041 + 0.465514i \(0.845869\pi\)
\(242\) 2.12341 0.136498
\(243\) −7.78354 −0.499314
\(244\) −2.97774 −0.190630
\(245\) 0 0
\(246\) −5.54457 −0.353509
\(247\) −0.406405 −0.0258590
\(248\) 5.51575 0.350250
\(249\) −2.65881 −0.168495
\(250\) 0 0
\(251\) −6.10901 −0.385597 −0.192799 0.981238i \(-0.561756\pi\)
−0.192799 + 0.981238i \(0.561756\pi\)
\(252\) 12.1251 0.763808
\(253\) 3.54516 0.222882
\(254\) 13.8015 0.865981
\(255\) 0 0
\(256\) 5.08143 0.317589
\(257\) 0.740664 0.0462014 0.0231007 0.999733i \(-0.492646\pi\)
0.0231007 + 0.999733i \(0.492646\pi\)
\(258\) 0.607799 0.0378399
\(259\) 2.42160 0.150471
\(260\) 0 0
\(261\) −21.7508 −1.34634
\(262\) 2.72465 0.168329
\(263\) −0.968172 −0.0597001 −0.0298500 0.999554i \(-0.509503\pi\)
−0.0298500 + 0.999554i \(0.509503\pi\)
\(264\) −0.324390 −0.0199648
\(265\) 0 0
\(266\) 3.52669 0.216235
\(267\) 0.838168 0.0512951
\(268\) 30.4910 1.86253
\(269\) −4.52415 −0.275842 −0.137921 0.990443i \(-0.544042\pi\)
−0.137921 + 0.990443i \(0.544042\pi\)
\(270\) 0 0
\(271\) 3.44651 0.209361 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(272\) −11.4588 −0.694794
\(273\) 0.202640 0.0122643
\(274\) 13.4476 0.812399
\(275\) 0 0
\(276\) −2.67021 −0.160728
\(277\) −5.47688 −0.329074 −0.164537 0.986371i \(-0.552613\pi\)
−0.164537 + 0.986371i \(0.552613\pi\)
\(278\) −10.4503 −0.626769
\(279\) −14.8540 −0.889283
\(280\) 0 0
\(281\) 20.1704 1.20327 0.601633 0.798773i \(-0.294517\pi\)
0.601633 + 0.798773i \(0.294517\pi\)
\(282\) −3.55914 −0.211944
\(283\) −7.06912 −0.420215 −0.210108 0.977678i \(-0.567381\pi\)
−0.210108 + 0.977678i \(0.567381\pi\)
\(284\) 14.8947 0.883837
\(285\) 0 0
\(286\) 0.862965 0.0510282
\(287\) −14.4456 −0.852697
\(288\) 23.1154 1.36209
\(289\) 0.704429 0.0414370
\(290\) 0 0
\(291\) −1.90177 −0.111484
\(292\) 6.00339 0.351322
\(293\) 25.8891 1.51246 0.756228 0.654308i \(-0.227040\pi\)
0.756228 + 0.654308i \(0.227040\pi\)
\(294\) 2.70389 0.157694
\(295\) 0 0
\(296\) −1.57545 −0.0915713
\(297\) 1.77423 0.102951
\(298\) −23.6954 −1.37264
\(299\) 1.44077 0.0833220
\(300\) 0 0
\(301\) 1.58354 0.0912735
\(302\) −19.9364 −1.14721
\(303\) −3.84555 −0.220921
\(304\) 2.72333 0.156194
\(305\) 0 0
\(306\) −25.9985 −1.48624
\(307\) 34.3162 1.95853 0.979264 0.202590i \(-0.0649358\pi\)
0.979264 + 0.202590i \(0.0649358\pi\)
\(308\) −4.16688 −0.237430
\(309\) 4.08450 0.232359
\(310\) 0 0
\(311\) 12.9422 0.733887 0.366944 0.930243i \(-0.380404\pi\)
0.366944 + 0.930243i \(0.380404\pi\)
\(312\) −0.131834 −0.00746363
\(313\) 6.03044 0.340860 0.170430 0.985370i \(-0.445484\pi\)
0.170430 + 0.985370i \(0.445484\pi\)
\(314\) 42.3099 2.38769
\(315\) 0 0
\(316\) −35.8463 −2.01651
\(317\) 7.55456 0.424306 0.212153 0.977236i \(-0.431952\pi\)
0.212153 + 0.977236i \(0.431952\pi\)
\(318\) 1.12839 0.0632768
\(319\) 7.47483 0.418510
\(320\) 0 0
\(321\) −3.11947 −0.174112
\(322\) −12.5027 −0.696747
\(323\) −4.20766 −0.234121
\(324\) 20.5651 1.14250
\(325\) 0 0
\(326\) −39.6973 −2.19863
\(327\) −1.97010 −0.108947
\(328\) 9.39805 0.518921
\(329\) −9.27284 −0.511228
\(330\) 0 0
\(331\) −21.3753 −1.17489 −0.587445 0.809264i \(-0.699866\pi\)
−0.587445 + 0.809264i \(0.699866\pi\)
\(332\) 22.2194 1.21945
\(333\) 4.24271 0.232499
\(334\) 12.5805 0.688375
\(335\) 0 0
\(336\) −1.35789 −0.0740791
\(337\) −20.6132 −1.12287 −0.561436 0.827520i \(-0.689751\pi\)
−0.561436 + 0.827520i \(0.689751\pi\)
\(338\) −27.2536 −1.48240
\(339\) −0.520357 −0.0282619
\(340\) 0 0
\(341\) 5.10468 0.276434
\(342\) 6.17885 0.334114
\(343\) 18.6706 1.00812
\(344\) −1.03022 −0.0555458
\(345\) 0 0
\(346\) 9.45225 0.508156
\(347\) 32.0956 1.72298 0.861490 0.507774i \(-0.169532\pi\)
0.861490 + 0.507774i \(0.169532\pi\)
\(348\) −5.63003 −0.301801
\(349\) −1.86111 −0.0996232 −0.0498116 0.998759i \(-0.515862\pi\)
−0.0498116 + 0.998759i \(0.515862\pi\)
\(350\) 0 0
\(351\) 0.721057 0.0384872
\(352\) −7.94379 −0.423406
\(353\) −11.1724 −0.594648 −0.297324 0.954777i \(-0.596094\pi\)
−0.297324 + 0.954777i \(0.596094\pi\)
\(354\) −3.99371 −0.212263
\(355\) 0 0
\(356\) −7.00449 −0.371237
\(357\) 2.09801 0.111038
\(358\) −11.2415 −0.594133
\(359\) 8.55089 0.451299 0.225649 0.974209i \(-0.427550\pi\)
0.225649 + 0.974209i \(0.427550\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −52.3540 −2.75166
\(363\) −0.300215 −0.0157572
\(364\) −1.69344 −0.0887605
\(365\) 0 0
\(366\) 0.756615 0.0395489
\(367\) −6.53421 −0.341083 −0.170542 0.985350i \(-0.554552\pi\)
−0.170542 + 0.985350i \(0.554552\pi\)
\(368\) −9.65463 −0.503282
\(369\) −25.3090 −1.31754
\(370\) 0 0
\(371\) 2.93986 0.152630
\(372\) −3.84483 −0.199345
\(373\) −29.9850 −1.55257 −0.776283 0.630385i \(-0.782897\pi\)
−0.776283 + 0.630385i \(0.782897\pi\)
\(374\) 8.93459 0.461997
\(375\) 0 0
\(376\) 6.03275 0.311115
\(377\) 3.03781 0.156455
\(378\) −6.25716 −0.321834
\(379\) 5.14929 0.264501 0.132251 0.991216i \(-0.457780\pi\)
0.132251 + 0.991216i \(0.457780\pi\)
\(380\) 0 0
\(381\) −1.95130 −0.0999680
\(382\) 53.5698 2.74087
\(383\) 5.87340 0.300117 0.150058 0.988677i \(-0.452054\pi\)
0.150058 + 0.988677i \(0.452054\pi\)
\(384\) 2.51112 0.128145
\(385\) 0 0
\(386\) 1.99469 0.101527
\(387\) 2.77439 0.141030
\(388\) 15.8929 0.806841
\(389\) 17.3191 0.878113 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(390\) 0 0
\(391\) 14.9168 0.754377
\(392\) −4.58310 −0.231481
\(393\) −0.385220 −0.0194318
\(394\) −14.6764 −0.739385
\(395\) 0 0
\(396\) −7.30047 −0.366863
\(397\) −2.93936 −0.147522 −0.0737610 0.997276i \(-0.523500\pi\)
−0.0737610 + 0.997276i \(0.523500\pi\)
\(398\) 47.8302 2.39751
\(399\) −0.498615 −0.0249620
\(400\) 0 0
\(401\) −39.8344 −1.98924 −0.994618 0.103609i \(-0.966961\pi\)
−0.994618 + 0.103609i \(0.966961\pi\)
\(402\) −7.74747 −0.386409
\(403\) 2.07457 0.103342
\(404\) 32.1369 1.59887
\(405\) 0 0
\(406\) −26.3614 −1.30829
\(407\) −1.45804 −0.0722724
\(408\) −1.36493 −0.0675739
\(409\) −34.7653 −1.71904 −0.859518 0.511106i \(-0.829236\pi\)
−0.859518 + 0.511106i \(0.829236\pi\)
\(410\) 0 0
\(411\) −1.90127 −0.0937826
\(412\) −34.1338 −1.68165
\(413\) −10.4051 −0.511999
\(414\) −21.9050 −1.07657
\(415\) 0 0
\(416\) −3.22840 −0.158285
\(417\) 1.47750 0.0723536
\(418\) −2.12341 −0.103859
\(419\) 4.20222 0.205292 0.102646 0.994718i \(-0.467269\pi\)
0.102646 + 0.994718i \(0.467269\pi\)
\(420\) 0 0
\(421\) 20.9129 1.01923 0.509616 0.860402i \(-0.329788\pi\)
0.509616 + 0.860402i \(0.329788\pi\)
\(422\) −28.7794 −1.40096
\(423\) −16.2462 −0.789919
\(424\) −1.91262 −0.0928850
\(425\) 0 0
\(426\) −3.78460 −0.183364
\(427\) 1.97126 0.0953958
\(428\) 26.0691 1.26010
\(429\) −0.122009 −0.00589065
\(430\) 0 0
\(431\) 10.5831 0.509771 0.254886 0.966971i \(-0.417962\pi\)
0.254886 + 0.966971i \(0.417962\pi\)
\(432\) −4.83181 −0.232471
\(433\) 11.8075 0.567430 0.283715 0.958909i \(-0.408433\pi\)
0.283715 + 0.958909i \(0.408433\pi\)
\(434\) −18.0026 −0.864154
\(435\) 0 0
\(436\) 16.4639 0.788480
\(437\) −3.54516 −0.169588
\(438\) −1.52540 −0.0728866
\(439\) −9.66940 −0.461495 −0.230748 0.973014i \(-0.574117\pi\)
−0.230748 + 0.973014i \(0.574117\pi\)
\(440\) 0 0
\(441\) 12.3423 0.587730
\(442\) 3.63107 0.172712
\(443\) 26.2089 1.24522 0.622610 0.782532i \(-0.286072\pi\)
0.622610 + 0.782532i \(0.286072\pi\)
\(444\) 1.09819 0.0521179
\(445\) 0 0
\(446\) 6.66059 0.315388
\(447\) 3.35014 0.158456
\(448\) 18.9691 0.896208
\(449\) 1.35633 0.0640090 0.0320045 0.999488i \(-0.489811\pi\)
0.0320045 + 0.999488i \(0.489811\pi\)
\(450\) 0 0
\(451\) 8.69765 0.409556
\(452\) 4.34857 0.204539
\(453\) 2.81867 0.132433
\(454\) −37.6491 −1.76696
\(455\) 0 0
\(456\) 0.324390 0.0151910
\(457\) 0.925283 0.0432829 0.0216415 0.999766i \(-0.493111\pi\)
0.0216415 + 0.999766i \(0.493111\pi\)
\(458\) −3.35588 −0.156810
\(459\) 7.46536 0.348453
\(460\) 0 0
\(461\) −17.9298 −0.835073 −0.417536 0.908660i \(-0.637106\pi\)
−0.417536 + 0.908660i \(0.637106\pi\)
\(462\) 1.05876 0.0492582
\(463\) −18.3406 −0.852360 −0.426180 0.904638i \(-0.640141\pi\)
−0.426180 + 0.904638i \(0.640141\pi\)
\(464\) −20.3564 −0.945023
\(465\) 0 0
\(466\) 21.5711 0.999262
\(467\) −9.78809 −0.452939 −0.226469 0.974018i \(-0.572718\pi\)
−0.226469 + 0.974018i \(0.572718\pi\)
\(468\) −2.96695 −0.137147
\(469\) −20.1850 −0.932055
\(470\) 0 0
\(471\) −5.98192 −0.275632
\(472\) 6.76934 0.311584
\(473\) −0.953442 −0.0438393
\(474\) 9.10820 0.418353
\(475\) 0 0
\(476\) −17.5328 −0.803616
\(477\) 5.15070 0.235834
\(478\) −14.0238 −0.641433
\(479\) −23.2528 −1.06245 −0.531223 0.847232i \(-0.678267\pi\)
−0.531223 + 0.847232i \(0.678267\pi\)
\(480\) 0 0
\(481\) −0.592555 −0.0270182
\(482\) −58.3492 −2.65774
\(483\) 1.76767 0.0804318
\(484\) 2.50886 0.114039
\(485\) 0 0
\(486\) −16.5276 −0.749709
\(487\) 20.1880 0.914807 0.457403 0.889259i \(-0.348780\pi\)
0.457403 + 0.889259i \(0.348780\pi\)
\(488\) −1.28247 −0.0580545
\(489\) 5.61254 0.253808
\(490\) 0 0
\(491\) −39.7054 −1.79188 −0.895940 0.444176i \(-0.853497\pi\)
−0.895940 + 0.444176i \(0.853497\pi\)
\(492\) −6.55105 −0.295344
\(493\) 31.4516 1.41651
\(494\) −0.862965 −0.0388266
\(495\) 0 0
\(496\) −13.9017 −0.624205
\(497\) −9.86025 −0.442293
\(498\) −5.64575 −0.252992
\(499\) −3.54572 −0.158728 −0.0793640 0.996846i \(-0.525289\pi\)
−0.0793640 + 0.996846i \(0.525289\pi\)
\(500\) 0 0
\(501\) −1.77867 −0.0794653
\(502\) −12.9719 −0.578966
\(503\) 14.6798 0.654542 0.327271 0.944931i \(-0.393871\pi\)
0.327271 + 0.944931i \(0.393871\pi\)
\(504\) 5.22208 0.232610
\(505\) 0 0
\(506\) 7.52782 0.334652
\(507\) 3.85321 0.171127
\(508\) 16.3068 0.723497
\(509\) −16.2586 −0.720651 −0.360325 0.932827i \(-0.617334\pi\)
−0.360325 + 0.932827i \(0.617334\pi\)
\(510\) 0 0
\(511\) −3.97423 −0.175810
\(512\) 27.5188 1.21617
\(513\) −1.77423 −0.0783342
\(514\) 1.57273 0.0693703
\(515\) 0 0
\(516\) 0.718131 0.0316139
\(517\) 5.58315 0.245547
\(518\) 5.14205 0.225929
\(519\) −1.33639 −0.0586611
\(520\) 0 0
\(521\) 30.2820 1.32668 0.663338 0.748320i \(-0.269139\pi\)
0.663338 + 0.748320i \(0.269139\pi\)
\(522\) −46.1858 −2.02150
\(523\) 22.9391 1.00305 0.501527 0.865142i \(-0.332772\pi\)
0.501527 + 0.865142i \(0.332772\pi\)
\(524\) 3.21925 0.140634
\(525\) 0 0
\(526\) −2.05583 −0.0896383
\(527\) 21.4788 0.935630
\(528\) 0.817583 0.0355807
\(529\) −10.4319 −0.453559
\(530\) 0 0
\(531\) −18.2299 −0.791111
\(532\) 4.16688 0.180657
\(533\) 3.53477 0.153108
\(534\) 1.77977 0.0770184
\(535\) 0 0
\(536\) 13.1320 0.567215
\(537\) 1.58936 0.0685861
\(538\) −9.60661 −0.414170
\(539\) −4.24154 −0.182696
\(540\) 0 0
\(541\) −41.8092 −1.79752 −0.898758 0.438445i \(-0.855529\pi\)
−0.898758 + 0.438445i \(0.855529\pi\)
\(542\) 7.31835 0.314350
\(543\) 7.40198 0.317649
\(544\) −33.4248 −1.43308
\(545\) 0 0
\(546\) 0.430288 0.0184146
\(547\) 18.4227 0.787698 0.393849 0.919175i \(-0.371143\pi\)
0.393849 + 0.919175i \(0.371143\pi\)
\(548\) 15.8887 0.678731
\(549\) 3.45369 0.147400
\(550\) 0 0
\(551\) −7.47483 −0.318438
\(552\) −1.15002 −0.0489479
\(553\) 23.7301 1.00911
\(554\) −11.6296 −0.494096
\(555\) 0 0
\(556\) −12.3473 −0.523644
\(557\) −25.4381 −1.07785 −0.538923 0.842355i \(-0.681168\pi\)
−0.538923 + 0.842355i \(0.681168\pi\)
\(558\) −31.5410 −1.33524
\(559\) −0.387484 −0.0163888
\(560\) 0 0
\(561\) −1.26320 −0.0533324
\(562\) 42.8300 1.80668
\(563\) 17.9758 0.757589 0.378795 0.925481i \(-0.376339\pi\)
0.378795 + 0.925481i \(0.376339\pi\)
\(564\) −4.20522 −0.177072
\(565\) 0 0
\(566\) −15.0106 −0.630943
\(567\) −13.6140 −0.571735
\(568\) 6.41491 0.269163
\(569\) −6.12536 −0.256789 −0.128394 0.991723i \(-0.540982\pi\)
−0.128394 + 0.991723i \(0.540982\pi\)
\(570\) 0 0
\(571\) 16.5581 0.692936 0.346468 0.938062i \(-0.387381\pi\)
0.346468 + 0.938062i \(0.387381\pi\)
\(572\) 1.01962 0.0426323
\(573\) −7.57387 −0.316403
\(574\) −30.6739 −1.28030
\(575\) 0 0
\(576\) 33.2344 1.38477
\(577\) 43.0297 1.79135 0.895676 0.444708i \(-0.146693\pi\)
0.895676 + 0.444708i \(0.146693\pi\)
\(578\) 1.49579 0.0622167
\(579\) −0.282016 −0.0117202
\(580\) 0 0
\(581\) −14.7092 −0.610241
\(582\) −4.03824 −0.167390
\(583\) −1.77008 −0.0733092
\(584\) 2.58556 0.106991
\(585\) 0 0
\(586\) 54.9731 2.27092
\(587\) 5.74599 0.237162 0.118581 0.992944i \(-0.462165\pi\)
0.118581 + 0.992944i \(0.462165\pi\)
\(588\) 3.19472 0.131748
\(589\) −5.10468 −0.210335
\(590\) 0 0
\(591\) 2.07499 0.0853539
\(592\) 3.97072 0.163196
\(593\) −11.8917 −0.488334 −0.244167 0.969733i \(-0.578514\pi\)
−0.244167 + 0.969733i \(0.578514\pi\)
\(594\) 3.76742 0.154579
\(595\) 0 0
\(596\) −27.9968 −1.14679
\(597\) −6.76240 −0.276767
\(598\) 3.05935 0.125106
\(599\) 34.5106 1.41007 0.705033 0.709174i \(-0.250932\pi\)
0.705033 + 0.709174i \(0.250932\pi\)
\(600\) 0 0
\(601\) −28.8714 −1.17769 −0.588845 0.808246i \(-0.700417\pi\)
−0.588845 + 0.808246i \(0.700417\pi\)
\(602\) 3.36249 0.137045
\(603\) −35.3645 −1.44016
\(604\) −23.5553 −0.958453
\(605\) 0 0
\(606\) −8.16568 −0.331708
\(607\) −19.5305 −0.792718 −0.396359 0.918096i \(-0.629726\pi\)
−0.396359 + 0.918096i \(0.629726\pi\)
\(608\) 7.94379 0.322163
\(609\) 3.72706 0.151028
\(610\) 0 0
\(611\) 2.26902 0.0917948
\(612\) −30.7179 −1.24170
\(613\) 34.5585 1.39581 0.697903 0.716193i \(-0.254117\pi\)
0.697903 + 0.716193i \(0.254117\pi\)
\(614\) 72.8673 2.94068
\(615\) 0 0
\(616\) −1.79461 −0.0723068
\(617\) 22.1066 0.889980 0.444990 0.895536i \(-0.353207\pi\)
0.444990 + 0.895536i \(0.353207\pi\)
\(618\) 8.67307 0.348882
\(619\) 45.0191 1.80947 0.904736 0.425973i \(-0.140068\pi\)
0.904736 + 0.425973i \(0.140068\pi\)
\(620\) 0 0
\(621\) 6.28993 0.252406
\(622\) 27.4817 1.10191
\(623\) 4.63696 0.185776
\(624\) 0.332270 0.0133015
\(625\) 0 0
\(626\) 12.8051 0.511794
\(627\) 0.300215 0.0119894
\(628\) 49.9903 1.99483
\(629\) −6.13494 −0.244616
\(630\) 0 0
\(631\) 1.39480 0.0555261 0.0277631 0.999615i \(-0.491162\pi\)
0.0277631 + 0.999615i \(0.491162\pi\)
\(632\) −15.4384 −0.614107
\(633\) 4.06893 0.161725
\(634\) 16.0414 0.637086
\(635\) 0 0
\(636\) 1.33322 0.0528656
\(637\) −1.72378 −0.0682988
\(638\) 15.8721 0.628383
\(639\) −17.2754 −0.683404
\(640\) 0 0
\(641\) 46.7278 1.84564 0.922819 0.385234i \(-0.125879\pi\)
0.922819 + 0.385234i \(0.125879\pi\)
\(642\) −6.62390 −0.261425
\(643\) −26.6503 −1.05098 −0.525492 0.850798i \(-0.676119\pi\)
−0.525492 + 0.850798i \(0.676119\pi\)
\(644\) −14.7722 −0.582108
\(645\) 0 0
\(646\) −8.93459 −0.351527
\(647\) −48.3556 −1.90105 −0.950526 0.310644i \(-0.899455\pi\)
−0.950526 + 0.310644i \(0.899455\pi\)
\(648\) 8.85705 0.347938
\(649\) 6.26485 0.245917
\(650\) 0 0
\(651\) 2.54527 0.0997571
\(652\) −46.9034 −1.83688
\(653\) 11.5581 0.452304 0.226152 0.974092i \(-0.427385\pi\)
0.226152 + 0.974092i \(0.427385\pi\)
\(654\) −4.18333 −0.163581
\(655\) 0 0
\(656\) −23.6865 −0.924804
\(657\) −6.96294 −0.271650
\(658\) −19.6900 −0.767597
\(659\) −12.2772 −0.478251 −0.239125 0.970989i \(-0.576861\pi\)
−0.239125 + 0.970989i \(0.576861\pi\)
\(660\) 0 0
\(661\) 2.16082 0.0840461 0.0420231 0.999117i \(-0.486620\pi\)
0.0420231 + 0.999117i \(0.486620\pi\)
\(662\) −45.3884 −1.76407
\(663\) −0.513372 −0.0199377
\(664\) 9.56956 0.371371
\(665\) 0 0
\(666\) 9.00900 0.349092
\(667\) 26.4995 1.02606
\(668\) 14.8642 0.575113
\(669\) −0.941696 −0.0364081
\(670\) 0 0
\(671\) −1.18689 −0.0458193
\(672\) −3.96090 −0.152795
\(673\) 24.2038 0.932988 0.466494 0.884524i \(-0.345517\pi\)
0.466494 + 0.884524i \(0.345517\pi\)
\(674\) −43.7702 −1.68597
\(675\) 0 0
\(676\) −32.2009 −1.23849
\(677\) −17.0810 −0.656476 −0.328238 0.944595i \(-0.606455\pi\)
−0.328238 + 0.944595i \(0.606455\pi\)
\(678\) −1.10493 −0.0424346
\(679\) −10.5211 −0.403762
\(680\) 0 0
\(681\) 5.32295 0.203976
\(682\) 10.8393 0.415059
\(683\) −15.6456 −0.598663 −0.299332 0.954149i \(-0.596764\pi\)
−0.299332 + 0.954149i \(0.596764\pi\)
\(684\) 7.30047 0.279141
\(685\) 0 0
\(686\) 39.6454 1.51367
\(687\) 0.474465 0.0181020
\(688\) 2.59653 0.0989919
\(689\) −0.719370 −0.0274058
\(690\) 0 0
\(691\) −32.7991 −1.24774 −0.623869 0.781529i \(-0.714440\pi\)
−0.623869 + 0.781529i \(0.714440\pi\)
\(692\) 11.1681 0.424547
\(693\) 4.83290 0.183587
\(694\) 68.1520 2.58702
\(695\) 0 0
\(696\) −2.42476 −0.0919104
\(697\) 36.5968 1.38620
\(698\) −3.95191 −0.149582
\(699\) −3.04979 −0.115354
\(700\) 0 0
\(701\) 45.3838 1.71412 0.857061 0.515216i \(-0.172288\pi\)
0.857061 + 0.515216i \(0.172288\pi\)
\(702\) 1.53110 0.0577876
\(703\) 1.45804 0.0549910
\(704\) −11.4213 −0.430455
\(705\) 0 0
\(706\) −23.7236 −0.892851
\(707\) −21.2746 −0.800112
\(708\) −4.71867 −0.177339
\(709\) −22.2346 −0.835037 −0.417519 0.908668i \(-0.637100\pi\)
−0.417519 + 0.908668i \(0.637100\pi\)
\(710\) 0 0
\(711\) 41.5758 1.55921
\(712\) −3.01672 −0.113056
\(713\) 18.0969 0.677734
\(714\) 4.45492 0.166721
\(715\) 0 0
\(716\) −13.2822 −0.496378
\(717\) 1.98273 0.0740464
\(718\) 18.1570 0.677615
\(719\) −39.4219 −1.47019 −0.735094 0.677965i \(-0.762862\pi\)
−0.735094 + 0.677965i \(0.762862\pi\)
\(720\) 0 0
\(721\) 22.5965 0.841537
\(722\) 2.12341 0.0790251
\(723\) 8.24961 0.306806
\(724\) −61.8576 −2.29892
\(725\) 0 0
\(726\) −0.637479 −0.0236590
\(727\) 14.3222 0.531182 0.265591 0.964086i \(-0.414433\pi\)
0.265591 + 0.964086i \(0.414433\pi\)
\(728\) −0.729339 −0.0270311
\(729\) −22.2542 −0.824228
\(730\) 0 0
\(731\) −4.01176 −0.148380
\(732\) 0.893961 0.0330418
\(733\) 45.2907 1.67285 0.836425 0.548082i \(-0.184642\pi\)
0.836425 + 0.548082i \(0.184642\pi\)
\(734\) −13.8748 −0.512128
\(735\) 0 0
\(736\) −28.1620 −1.03807
\(737\) 12.1533 0.447672
\(738\) −53.7414 −1.97825
\(739\) −38.1634 −1.40386 −0.701931 0.712245i \(-0.747679\pi\)
−0.701931 + 0.712245i \(0.747679\pi\)
\(740\) 0 0
\(741\) 0.122009 0.00448211
\(742\) 6.24252 0.229170
\(743\) −10.5543 −0.387201 −0.193600 0.981080i \(-0.562017\pi\)
−0.193600 + 0.981080i \(0.562017\pi\)
\(744\) −1.65591 −0.0607086
\(745\) 0 0
\(746\) −63.6705 −2.33114
\(747\) −25.7709 −0.942908
\(748\) 10.5565 0.385982
\(749\) −17.2577 −0.630581
\(750\) 0 0
\(751\) −53.4076 −1.94887 −0.974436 0.224665i \(-0.927871\pi\)
−0.974436 + 0.224665i \(0.927871\pi\)
\(752\) −15.2047 −0.554460
\(753\) 1.83402 0.0668352
\(754\) 6.45052 0.234914
\(755\) 0 0
\(756\) −7.39300 −0.268881
\(757\) −20.4970 −0.744977 −0.372488 0.928037i \(-0.621495\pi\)
−0.372488 + 0.928037i \(0.621495\pi\)
\(758\) 10.9340 0.397142
\(759\) −1.06431 −0.0386320
\(760\) 0 0
\(761\) 23.3188 0.845305 0.422653 0.906292i \(-0.361099\pi\)
0.422653 + 0.906292i \(0.361099\pi\)
\(762\) −4.14340 −0.150100
\(763\) −10.8991 −0.394573
\(764\) 63.2941 2.28990
\(765\) 0 0
\(766\) 12.4716 0.450619
\(767\) 2.54607 0.0919332
\(768\) −1.52552 −0.0550475
\(769\) −33.4479 −1.20616 −0.603081 0.797680i \(-0.706060\pi\)
−0.603081 + 0.797680i \(0.706060\pi\)
\(770\) 0 0
\(771\) −0.222358 −0.00800804
\(772\) 2.35678 0.0848225
\(773\) −0.0528628 −0.00190134 −0.000950671 1.00000i \(-0.500303\pi\)
−0.000950671 1.00000i \(0.500303\pi\)
\(774\) 5.89117 0.211754
\(775\) 0 0
\(776\) 6.84482 0.245715
\(777\) −0.727001 −0.0260810
\(778\) 36.7755 1.31847
\(779\) −8.69765 −0.311626
\(780\) 0 0
\(781\) 5.93683 0.212436
\(782\) 31.6745 1.13268
\(783\) 13.2621 0.473948
\(784\) 11.5511 0.412539
\(785\) 0 0
\(786\) −0.817980 −0.0291764
\(787\) 26.7570 0.953785 0.476893 0.878962i \(-0.341763\pi\)
0.476893 + 0.878962i \(0.341763\pi\)
\(788\) −17.3405 −0.617730
\(789\) 0.290660 0.0103478
\(790\) 0 0
\(791\) −2.87874 −0.102356
\(792\) −3.14420 −0.111724
\(793\) −0.482358 −0.0171290
\(794\) −6.24145 −0.221501
\(795\) 0 0
\(796\) 56.5127 2.00304
\(797\) −15.8450 −0.561258 −0.280629 0.959816i \(-0.590543\pi\)
−0.280629 + 0.959816i \(0.590543\pi\)
\(798\) −1.05876 −0.0374798
\(799\) 23.4920 0.831087
\(800\) 0 0
\(801\) 8.12406 0.287049
\(802\) −84.5848 −2.98679
\(803\) 2.39287 0.0844426
\(804\) −9.15384 −0.322831
\(805\) 0 0
\(806\) 4.40516 0.155165
\(807\) 1.35822 0.0478114
\(808\) 13.8409 0.486920
\(809\) 31.2953 1.10028 0.550142 0.835071i \(-0.314573\pi\)
0.550142 + 0.835071i \(0.314573\pi\)
\(810\) 0 0
\(811\) 2.29851 0.0807116 0.0403558 0.999185i \(-0.487151\pi\)
0.0403558 + 0.999185i \(0.487151\pi\)
\(812\) −31.1467 −1.09304
\(813\) −1.03469 −0.0362883
\(814\) −3.09601 −0.108515
\(815\) 0 0
\(816\) 3.44011 0.120428
\(817\) 0.953442 0.0333567
\(818\) −73.8210 −2.58109
\(819\) 1.96412 0.0686317
\(820\) 0 0
\(821\) 3.75192 0.130943 0.0654714 0.997854i \(-0.479145\pi\)
0.0654714 + 0.997854i \(0.479145\pi\)
\(822\) −4.03717 −0.140812
\(823\) 40.2049 1.40145 0.700727 0.713429i \(-0.252859\pi\)
0.700727 + 0.713429i \(0.252859\pi\)
\(824\) −14.7009 −0.512129
\(825\) 0 0
\(826\) −22.0942 −0.768755
\(827\) 36.4082 1.26604 0.633019 0.774136i \(-0.281815\pi\)
0.633019 + 0.774136i \(0.281815\pi\)
\(828\) −25.8813 −0.899438
\(829\) −37.9222 −1.31709 −0.658546 0.752540i \(-0.728828\pi\)
−0.658546 + 0.752540i \(0.728828\pi\)
\(830\) 0 0
\(831\) 1.64424 0.0570380
\(832\) −4.64166 −0.160921
\(833\) −17.8470 −0.618360
\(834\) 3.13734 0.108637
\(835\) 0 0
\(836\) −2.50886 −0.0867709
\(837\) 9.05688 0.313051
\(838\) 8.92304 0.308241
\(839\) 21.3401 0.736743 0.368372 0.929679i \(-0.379915\pi\)
0.368372 + 0.929679i \(0.379915\pi\)
\(840\) 0 0
\(841\) 26.8731 0.926658
\(842\) 44.4066 1.53035
\(843\) −6.05545 −0.208561
\(844\) −34.0036 −1.17045
\(845\) 0 0
\(846\) −34.4974 −1.18605
\(847\) −1.66086 −0.0570679
\(848\) 4.82050 0.165537
\(849\) 2.12225 0.0728355
\(850\) 0 0
\(851\) −5.16898 −0.177190
\(852\) −4.47161 −0.153195
\(853\) −35.0632 −1.20054 −0.600270 0.799798i \(-0.704940\pi\)
−0.600270 + 0.799798i \(0.704940\pi\)
\(854\) 4.18578 0.143235
\(855\) 0 0
\(856\) 11.2275 0.383749
\(857\) 44.9677 1.53607 0.768034 0.640409i \(-0.221235\pi\)
0.768034 + 0.640409i \(0.221235\pi\)
\(858\) −0.259075 −0.00884467
\(859\) 29.2008 0.996319 0.498160 0.867085i \(-0.334009\pi\)
0.498160 + 0.867085i \(0.334009\pi\)
\(860\) 0 0
\(861\) 4.33678 0.147797
\(862\) 22.4723 0.765409
\(863\) 40.8821 1.39164 0.695822 0.718214i \(-0.255040\pi\)
0.695822 + 0.718214i \(0.255040\pi\)
\(864\) −14.0941 −0.479492
\(865\) 0 0
\(866\) 25.0720 0.851983
\(867\) −0.211480 −0.00718224
\(868\) −21.2706 −0.721971
\(869\) −14.2878 −0.484682
\(870\) 0 0
\(871\) 4.93917 0.167357
\(872\) 7.09075 0.240123
\(873\) −18.4332 −0.623868
\(874\) −7.52782 −0.254632
\(875\) 0 0
\(876\) −1.80231 −0.0608942
\(877\) −8.58398 −0.289860 −0.144930 0.989442i \(-0.546296\pi\)
−0.144930 + 0.989442i \(0.546296\pi\)
\(878\) −20.5321 −0.692924
\(879\) −7.77228 −0.262153
\(880\) 0 0
\(881\) −6.82174 −0.229830 −0.114915 0.993375i \(-0.536660\pi\)
−0.114915 + 0.993375i \(0.536660\pi\)
\(882\) 26.2078 0.882462
\(883\) 27.8523 0.937305 0.468653 0.883383i \(-0.344740\pi\)
0.468653 + 0.883383i \(0.344740\pi\)
\(884\) 4.29020 0.144295
\(885\) 0 0
\(886\) 55.6521 1.86967
\(887\) −13.9072 −0.466958 −0.233479 0.972362i \(-0.575011\pi\)
−0.233479 + 0.972362i \(0.575011\pi\)
\(888\) 0.472974 0.0158720
\(889\) −10.7951 −0.362055
\(890\) 0 0
\(891\) 8.19696 0.274609
\(892\) 7.86966 0.263496
\(893\) −5.58315 −0.186833
\(894\) 7.11372 0.237918
\(895\) 0 0
\(896\) 13.8922 0.464105
\(897\) −0.432541 −0.0144421
\(898\) 2.88003 0.0961080
\(899\) 38.1566 1.27259
\(900\) 0 0
\(901\) −7.44790 −0.248125
\(902\) 18.4687 0.614939
\(903\) −0.475401 −0.0158203
\(904\) 1.87286 0.0622904
\(905\) 0 0
\(906\) 5.98519 0.198844
\(907\) 10.8762 0.361139 0.180569 0.983562i \(-0.442206\pi\)
0.180569 + 0.983562i \(0.442206\pi\)
\(908\) −44.4834 −1.47623
\(909\) −37.2735 −1.23629
\(910\) 0 0
\(911\) 13.6506 0.452265 0.226133 0.974097i \(-0.427392\pi\)
0.226133 + 0.974097i \(0.427392\pi\)
\(912\) −0.817583 −0.0270729
\(913\) 8.85637 0.293103
\(914\) 1.96475 0.0649883
\(915\) 0 0
\(916\) −3.96506 −0.131009
\(917\) −2.13113 −0.0703763
\(918\) 15.8520 0.523195
\(919\) 16.0698 0.530093 0.265047 0.964236i \(-0.414613\pi\)
0.265047 + 0.964236i \(0.414613\pi\)
\(920\) 0 0
\(921\) −10.3022 −0.339470
\(922\) −38.0722 −1.25384
\(923\) 2.41276 0.0794169
\(924\) 1.25096 0.0411535
\(925\) 0 0
\(926\) −38.9446 −1.27980
\(927\) 39.5896 1.30029
\(928\) −59.3785 −1.94919
\(929\) −4.90033 −0.160775 −0.0803873 0.996764i \(-0.525616\pi\)
−0.0803873 + 0.996764i \(0.525616\pi\)
\(930\) 0 0
\(931\) 4.24154 0.139011
\(932\) 25.4868 0.834849
\(933\) −3.88545 −0.127204
\(934\) −20.7841 −0.680077
\(935\) 0 0
\(936\) −1.27782 −0.0417668
\(937\) −18.5173 −0.604934 −0.302467 0.953160i \(-0.597810\pi\)
−0.302467 + 0.953160i \(0.597810\pi\)
\(938\) −42.8609 −1.39946
\(939\) −1.81043 −0.0590810
\(940\) 0 0
\(941\) −4.56469 −0.148805 −0.0744024 0.997228i \(-0.523705\pi\)
−0.0744024 + 0.997228i \(0.523705\pi\)
\(942\) −12.7021 −0.413855
\(943\) 30.8345 1.00411
\(944\) −17.0612 −0.555296
\(945\) 0 0
\(946\) −2.02455 −0.0658237
\(947\) −32.7129 −1.06302 −0.531512 0.847050i \(-0.678376\pi\)
−0.531512 + 0.847050i \(0.678376\pi\)
\(948\) 10.7616 0.349520
\(949\) 0.972476 0.0315679
\(950\) 0 0
\(951\) −2.26799 −0.0735446
\(952\) −7.55111 −0.244733
\(953\) 17.7014 0.573403 0.286702 0.958020i \(-0.407441\pi\)
0.286702 + 0.958020i \(0.407441\pi\)
\(954\) 10.9370 0.354100
\(955\) 0 0
\(956\) −16.5695 −0.535895
\(957\) −2.24405 −0.0725400
\(958\) −49.3751 −1.59524
\(959\) −10.5183 −0.339653
\(960\) 0 0
\(961\) −4.94225 −0.159427
\(962\) −1.25824 −0.0405672
\(963\) −30.2358 −0.974337
\(964\) −68.9412 −2.22045
\(965\) 0 0
\(966\) 3.75349 0.120766
\(967\) 1.49038 0.0479273 0.0239637 0.999713i \(-0.492371\pi\)
0.0239637 + 0.999713i \(0.492371\pi\)
\(968\) 1.08053 0.0347295
\(969\) 1.26320 0.0405799
\(970\) 0 0
\(971\) −14.0384 −0.450515 −0.225258 0.974299i \(-0.572322\pi\)
−0.225258 + 0.974299i \(0.572322\pi\)
\(972\) −19.5278 −0.626356
\(973\) 8.17391 0.262044
\(974\) 42.8674 1.37356
\(975\) 0 0
\(976\) 3.23228 0.103463
\(977\) −52.4306 −1.67740 −0.838701 0.544592i \(-0.816685\pi\)
−0.838701 + 0.544592i \(0.816685\pi\)
\(978\) 11.9177 0.381087
\(979\) −2.79190 −0.0892294
\(980\) 0 0
\(981\) −19.0955 −0.609671
\(982\) −84.3108 −2.69046
\(983\) −42.2403 −1.34726 −0.673628 0.739071i \(-0.735265\pi\)
−0.673628 + 0.739071i \(0.735265\pi\)
\(984\) −2.82143 −0.0899440
\(985\) 0 0
\(986\) 66.7845 2.12685
\(987\) 2.78384 0.0886107
\(988\) −1.01962 −0.0324383
\(989\) −3.38010 −0.107481
\(990\) 0 0
\(991\) −11.3461 −0.360420 −0.180210 0.983628i \(-0.557678\pi\)
−0.180210 + 0.983628i \(0.557678\pi\)
\(992\) −40.5505 −1.28748
\(993\) 6.41716 0.203643
\(994\) −20.9373 −0.664092
\(995\) 0 0
\(996\) −6.67060 −0.211366
\(997\) 27.2874 0.864200 0.432100 0.901826i \(-0.357773\pi\)
0.432100 + 0.901826i \(0.357773\pi\)
\(998\) −7.52901 −0.238327
\(999\) −2.58690 −0.0818458
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.x.1.12 yes 15
5.4 even 2 5225.2.a.s.1.4 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.s.1.4 15 5.4 even 2
5225.2.a.x.1.12 yes 15 1.1 even 1 trivial