Properties

Label 5550.2.a.cb.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -0.414214 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -0.414214 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.58579 q^{11} +1.00000 q^{12} +3.24264 q^{13} -0.414214 q^{14} +1.00000 q^{16} -6.41421 q^{17} +1.00000 q^{18} -7.82843 q^{19} -0.414214 q^{21} -3.58579 q^{22} -3.00000 q^{23} +1.00000 q^{24} +3.24264 q^{26} +1.00000 q^{27} -0.414214 q^{28} -7.07107 q^{29} +7.65685 q^{31} +1.00000 q^{32} -3.58579 q^{33} -6.41421 q^{34} +1.00000 q^{36} +1.00000 q^{37} -7.82843 q^{38} +3.24264 q^{39} -9.41421 q^{41} -0.414214 q^{42} +6.48528 q^{43} -3.58579 q^{44} -3.00000 q^{46} -7.89949 q^{47} +1.00000 q^{48} -6.82843 q^{49} -6.41421 q^{51} +3.24264 q^{52} -1.82843 q^{53} +1.00000 q^{54} -0.414214 q^{56} -7.82843 q^{57} -7.07107 q^{58} +1.07107 q^{59} +10.4853 q^{61} +7.65685 q^{62} -0.414214 q^{63} +1.00000 q^{64} -3.58579 q^{66} -4.82843 q^{67} -6.41421 q^{68} -3.00000 q^{69} -6.58579 q^{71} +1.00000 q^{72} -7.82843 q^{73} +1.00000 q^{74} -7.82843 q^{76} +1.48528 q^{77} +3.24264 q^{78} +3.75736 q^{79} +1.00000 q^{81} -9.41421 q^{82} +12.8995 q^{83} -0.414214 q^{84} +6.48528 q^{86} -7.07107 q^{87} -3.58579 q^{88} -16.8995 q^{89} -1.34315 q^{91} -3.00000 q^{92} +7.65685 q^{93} -7.89949 q^{94} +1.00000 q^{96} -18.7279 q^{97} -6.82843 q^{98} -3.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 10 q^{11} + 2 q^{12} - 2 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} + 2 q^{18} - 10 q^{19} + 2 q^{21} - 10 q^{22} - 6 q^{23} + 2 q^{24} - 2 q^{26} + 2 q^{27} + 2 q^{28} + 4 q^{31} + 2 q^{32} - 10 q^{33} - 10 q^{34} + 2 q^{36} + 2 q^{37} - 10 q^{38} - 2 q^{39} - 16 q^{41} + 2 q^{42} - 4 q^{43} - 10 q^{44} - 6 q^{46} + 4 q^{47} + 2 q^{48} - 8 q^{49} - 10 q^{51} - 2 q^{52} + 2 q^{53} + 2 q^{54} + 2 q^{56} - 10 q^{57} - 12 q^{59} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} - 10 q^{66} - 4 q^{67} - 10 q^{68} - 6 q^{69} - 16 q^{71} + 2 q^{72} - 10 q^{73} + 2 q^{74} - 10 q^{76} - 14 q^{77} - 2 q^{78} + 16 q^{79} + 2 q^{81} - 16 q^{82} + 6 q^{83} + 2 q^{84} - 4 q^{86} - 10 q^{88} - 14 q^{89} - 14 q^{91} - 6 q^{92} + 4 q^{93} + 4 q^{94} + 2 q^{96} - 12 q^{97} - 8 q^{98} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −0.414214 −0.156558 −0.0782790 0.996931i \(-0.524942\pi\)
−0.0782790 + 0.996931i \(0.524942\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.58579 −1.08116 −0.540578 0.841294i \(-0.681794\pi\)
−0.540578 + 0.841294i \(0.681794\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.24264 0.899347 0.449673 0.893193i \(-0.351540\pi\)
0.449673 + 0.893193i \(0.351540\pi\)
\(14\) −0.414214 −0.110703
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.41421 −1.55568 −0.777838 0.628465i \(-0.783683\pi\)
−0.777838 + 0.628465i \(0.783683\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.82843 −1.79596 −0.897982 0.440032i \(-0.854967\pi\)
−0.897982 + 0.440032i \(0.854967\pi\)
\(20\) 0 0
\(21\) −0.414214 −0.0903888
\(22\) −3.58579 −0.764492
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 3.24264 0.635934
\(27\) 1.00000 0.192450
\(28\) −0.414214 −0.0782790
\(29\) −7.07107 −1.31306 −0.656532 0.754298i \(-0.727977\pi\)
−0.656532 + 0.754298i \(0.727977\pi\)
\(30\) 0 0
\(31\) 7.65685 1.37521 0.687606 0.726084i \(-0.258662\pi\)
0.687606 + 0.726084i \(0.258662\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.58579 −0.624205
\(34\) −6.41421 −1.10003
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) −7.82843 −1.26994
\(39\) 3.24264 0.519238
\(40\) 0 0
\(41\) −9.41421 −1.47025 −0.735127 0.677930i \(-0.762877\pi\)
−0.735127 + 0.677930i \(0.762877\pi\)
\(42\) −0.414214 −0.0639145
\(43\) 6.48528 0.988996 0.494498 0.869179i \(-0.335352\pi\)
0.494498 + 0.869179i \(0.335352\pi\)
\(44\) −3.58579 −0.540578
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) −7.89949 −1.15226 −0.576130 0.817358i \(-0.695438\pi\)
−0.576130 + 0.817358i \(0.695438\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.82843 −0.975490
\(50\) 0 0
\(51\) −6.41421 −0.898170
\(52\) 3.24264 0.449673
\(53\) −1.82843 −0.251154 −0.125577 0.992084i \(-0.540078\pi\)
−0.125577 + 0.992084i \(0.540078\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.414214 −0.0553516
\(57\) −7.82843 −1.03690
\(58\) −7.07107 −0.928477
\(59\) 1.07107 0.139441 0.0697206 0.997567i \(-0.477789\pi\)
0.0697206 + 0.997567i \(0.477789\pi\)
\(60\) 0 0
\(61\) 10.4853 1.34250 0.671251 0.741230i \(-0.265757\pi\)
0.671251 + 0.741230i \(0.265757\pi\)
\(62\) 7.65685 0.972421
\(63\) −0.414214 −0.0521860
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.58579 −0.441380
\(67\) −4.82843 −0.589886 −0.294943 0.955515i \(-0.595301\pi\)
−0.294943 + 0.955515i \(0.595301\pi\)
\(68\) −6.41421 −0.777838
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −6.58579 −0.781589 −0.390795 0.920478i \(-0.627800\pi\)
−0.390795 + 0.920478i \(0.627800\pi\)
\(72\) 1.00000 0.117851
\(73\) −7.82843 −0.916248 −0.458124 0.888888i \(-0.651478\pi\)
−0.458124 + 0.888888i \(0.651478\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −7.82843 −0.897982
\(77\) 1.48528 0.169264
\(78\) 3.24264 0.367157
\(79\) 3.75736 0.422736 0.211368 0.977407i \(-0.432208\pi\)
0.211368 + 0.977407i \(0.432208\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −9.41421 −1.03963
\(83\) 12.8995 1.41590 0.707952 0.706261i \(-0.249619\pi\)
0.707952 + 0.706261i \(0.249619\pi\)
\(84\) −0.414214 −0.0451944
\(85\) 0 0
\(86\) 6.48528 0.699326
\(87\) −7.07107 −0.758098
\(88\) −3.58579 −0.382246
\(89\) −16.8995 −1.79134 −0.895671 0.444716i \(-0.853304\pi\)
−0.895671 + 0.444716i \(0.853304\pi\)
\(90\) 0 0
\(91\) −1.34315 −0.140800
\(92\) −3.00000 −0.312772
\(93\) 7.65685 0.793979
\(94\) −7.89949 −0.814771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −18.7279 −1.90153 −0.950766 0.309909i \(-0.899701\pi\)
−0.950766 + 0.309909i \(0.899701\pi\)
\(98\) −6.82843 −0.689775
\(99\) −3.58579 −0.360385
\(100\) 0 0
\(101\) 17.3137 1.72278 0.861389 0.507946i \(-0.169595\pi\)
0.861389 + 0.507946i \(0.169595\pi\)
\(102\) −6.41421 −0.635102
\(103\) 0.928932 0.0915304 0.0457652 0.998952i \(-0.485427\pi\)
0.0457652 + 0.998952i \(0.485427\pi\)
\(104\) 3.24264 0.317967
\(105\) 0 0
\(106\) −1.82843 −0.177593
\(107\) −1.92893 −0.186477 −0.0932385 0.995644i \(-0.529722\pi\)
−0.0932385 + 0.995644i \(0.529722\pi\)
\(108\) 1.00000 0.0962250
\(109\) 13.2426 1.26841 0.634207 0.773163i \(-0.281327\pi\)
0.634207 + 0.773163i \(0.281327\pi\)
\(110\) 0 0
\(111\) 1.00000 0.0949158
\(112\) −0.414214 −0.0391395
\(113\) −2.82843 −0.266076 −0.133038 0.991111i \(-0.542473\pi\)
−0.133038 + 0.991111i \(0.542473\pi\)
\(114\) −7.82843 −0.733199
\(115\) 0 0
\(116\) −7.07107 −0.656532
\(117\) 3.24264 0.299782
\(118\) 1.07107 0.0985998
\(119\) 2.65685 0.243553
\(120\) 0 0
\(121\) 1.85786 0.168897
\(122\) 10.4853 0.949293
\(123\) −9.41421 −0.848851
\(124\) 7.65685 0.687606
\(125\) 0 0
\(126\) −0.414214 −0.0369011
\(127\) −12.4142 −1.10158 −0.550792 0.834643i \(-0.685674\pi\)
−0.550792 + 0.834643i \(0.685674\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.48528 0.570997
\(130\) 0 0
\(131\) −17.8995 −1.56389 −0.781943 0.623350i \(-0.785771\pi\)
−0.781943 + 0.623350i \(0.785771\pi\)
\(132\) −3.58579 −0.312103
\(133\) 3.24264 0.281173
\(134\) −4.82843 −0.417113
\(135\) 0 0
\(136\) −6.41421 −0.550014
\(137\) 13.5563 1.15820 0.579099 0.815258i \(-0.303405\pi\)
0.579099 + 0.815258i \(0.303405\pi\)
\(138\) −3.00000 −0.255377
\(139\) 14.4853 1.22863 0.614313 0.789063i \(-0.289433\pi\)
0.614313 + 0.789063i \(0.289433\pi\)
\(140\) 0 0
\(141\) −7.89949 −0.665257
\(142\) −6.58579 −0.552667
\(143\) −11.6274 −0.972333
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −7.82843 −0.647885
\(147\) −6.82843 −0.563199
\(148\) 1.00000 0.0821995
\(149\) 20.4853 1.67822 0.839110 0.543962i \(-0.183076\pi\)
0.839110 + 0.543962i \(0.183076\pi\)
\(150\) 0 0
\(151\) −0.414214 −0.0337082 −0.0168541 0.999858i \(-0.505365\pi\)
−0.0168541 + 0.999858i \(0.505365\pi\)
\(152\) −7.82843 −0.634969
\(153\) −6.41421 −0.518558
\(154\) 1.48528 0.119687
\(155\) 0 0
\(156\) 3.24264 0.259619
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 3.75736 0.298919
\(159\) −1.82843 −0.145004
\(160\) 0 0
\(161\) 1.24264 0.0979338
\(162\) 1.00000 0.0785674
\(163\) 13.1421 1.02937 0.514686 0.857379i \(-0.327909\pi\)
0.514686 + 0.857379i \(0.327909\pi\)
\(164\) −9.41421 −0.735127
\(165\) 0 0
\(166\) 12.8995 1.00119
\(167\) 7.34315 0.568230 0.284115 0.958790i \(-0.408300\pi\)
0.284115 + 0.958790i \(0.408300\pi\)
\(168\) −0.414214 −0.0319573
\(169\) −2.48528 −0.191175
\(170\) 0 0
\(171\) −7.82843 −0.598655
\(172\) 6.48528 0.494498
\(173\) 10.6569 0.810226 0.405113 0.914267i \(-0.367232\pi\)
0.405113 + 0.914267i \(0.367232\pi\)
\(174\) −7.07107 −0.536056
\(175\) 0 0
\(176\) −3.58579 −0.270289
\(177\) 1.07107 0.0805064
\(178\) −16.8995 −1.26667
\(179\) −13.1716 −0.984490 −0.492245 0.870457i \(-0.663824\pi\)
−0.492245 + 0.870457i \(0.663824\pi\)
\(180\) 0 0
\(181\) 23.4142 1.74036 0.870182 0.492730i \(-0.164001\pi\)
0.870182 + 0.492730i \(0.164001\pi\)
\(182\) −1.34315 −0.0995606
\(183\) 10.4853 0.775094
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 7.65685 0.561428
\(187\) 23.0000 1.68193
\(188\) −7.89949 −0.576130
\(189\) −0.414214 −0.0301296
\(190\) 0 0
\(191\) −2.65685 −0.192243 −0.0961216 0.995370i \(-0.530644\pi\)
−0.0961216 + 0.995370i \(0.530644\pi\)
\(192\) 1.00000 0.0721688
\(193\) −13.3137 −0.958342 −0.479171 0.877722i \(-0.659063\pi\)
−0.479171 + 0.877722i \(0.659063\pi\)
\(194\) −18.7279 −1.34459
\(195\) 0 0
\(196\) −6.82843 −0.487745
\(197\) 13.1421 0.936338 0.468169 0.883639i \(-0.344914\pi\)
0.468169 + 0.883639i \(0.344914\pi\)
\(198\) −3.58579 −0.254831
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) −4.82843 −0.340571
\(202\) 17.3137 1.21819
\(203\) 2.92893 0.205571
\(204\) −6.41421 −0.449085
\(205\) 0 0
\(206\) 0.928932 0.0647218
\(207\) −3.00000 −0.208514
\(208\) 3.24264 0.224837
\(209\) 28.0711 1.94172
\(210\) 0 0
\(211\) −26.3848 −1.81640 −0.908201 0.418533i \(-0.862544\pi\)
−0.908201 + 0.418533i \(0.862544\pi\)
\(212\) −1.82843 −0.125577
\(213\) −6.58579 −0.451251
\(214\) −1.92893 −0.131859
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −3.17157 −0.215300
\(218\) 13.2426 0.896905
\(219\) −7.82843 −0.528996
\(220\) 0 0
\(221\) −20.7990 −1.39909
\(222\) 1.00000 0.0671156
\(223\) 1.65685 0.110951 0.0554756 0.998460i \(-0.482333\pi\)
0.0554756 + 0.998460i \(0.482333\pi\)
\(224\) −0.414214 −0.0276758
\(225\) 0 0
\(226\) −2.82843 −0.188144
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −7.82843 −0.518450
\(229\) −6.72792 −0.444594 −0.222297 0.974979i \(-0.571355\pi\)
−0.222297 + 0.974979i \(0.571355\pi\)
\(230\) 0 0
\(231\) 1.48528 0.0977243
\(232\) −7.07107 −0.464238
\(233\) −14.4853 −0.948962 −0.474481 0.880266i \(-0.657364\pi\)
−0.474481 + 0.880266i \(0.657364\pi\)
\(234\) 3.24264 0.211978
\(235\) 0 0
\(236\) 1.07107 0.0697206
\(237\) 3.75736 0.244067
\(238\) 2.65685 0.172218
\(239\) −24.6274 −1.59302 −0.796508 0.604629i \(-0.793322\pi\)
−0.796508 + 0.604629i \(0.793322\pi\)
\(240\) 0 0
\(241\) 5.41421 0.348760 0.174380 0.984678i \(-0.444208\pi\)
0.174380 + 0.984678i \(0.444208\pi\)
\(242\) 1.85786 0.119428
\(243\) 1.00000 0.0641500
\(244\) 10.4853 0.671251
\(245\) 0 0
\(246\) −9.41421 −0.600228
\(247\) −25.3848 −1.61519
\(248\) 7.65685 0.486211
\(249\) 12.8995 0.817472
\(250\) 0 0
\(251\) −0.828427 −0.0522899 −0.0261449 0.999658i \(-0.508323\pi\)
−0.0261449 + 0.999658i \(0.508323\pi\)
\(252\) −0.414214 −0.0260930
\(253\) 10.7574 0.676309
\(254\) −12.4142 −0.778937
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.4142 0.774377 0.387189 0.922001i \(-0.373446\pi\)
0.387189 + 0.922001i \(0.373446\pi\)
\(258\) 6.48528 0.403756
\(259\) −0.414214 −0.0257380
\(260\) 0 0
\(261\) −7.07107 −0.437688
\(262\) −17.8995 −1.10583
\(263\) −28.5858 −1.76268 −0.881338 0.472487i \(-0.843356\pi\)
−0.881338 + 0.472487i \(0.843356\pi\)
\(264\) −3.58579 −0.220690
\(265\) 0 0
\(266\) 3.24264 0.198819
\(267\) −16.8995 −1.03423
\(268\) −4.82843 −0.294943
\(269\) 19.6274 1.19670 0.598352 0.801233i \(-0.295822\pi\)
0.598352 + 0.801233i \(0.295822\pi\)
\(270\) 0 0
\(271\) −1.17157 −0.0711680 −0.0355840 0.999367i \(-0.511329\pi\)
−0.0355840 + 0.999367i \(0.511329\pi\)
\(272\) −6.41421 −0.388919
\(273\) −1.34315 −0.0812909
\(274\) 13.5563 0.818969
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −16.8995 −1.01539 −0.507696 0.861536i \(-0.669503\pi\)
−0.507696 + 0.861536i \(0.669503\pi\)
\(278\) 14.4853 0.868769
\(279\) 7.65685 0.458404
\(280\) 0 0
\(281\) 23.3848 1.39502 0.697509 0.716576i \(-0.254292\pi\)
0.697509 + 0.716576i \(0.254292\pi\)
\(282\) −7.89949 −0.470408
\(283\) −20.7990 −1.23637 −0.618186 0.786032i \(-0.712132\pi\)
−0.618186 + 0.786032i \(0.712132\pi\)
\(284\) −6.58579 −0.390795
\(285\) 0 0
\(286\) −11.6274 −0.687544
\(287\) 3.89949 0.230180
\(288\) 1.00000 0.0589256
\(289\) 24.1421 1.42013
\(290\) 0 0
\(291\) −18.7279 −1.09785
\(292\) −7.82843 −0.458124
\(293\) −2.65685 −0.155215 −0.0776075 0.996984i \(-0.524728\pi\)
−0.0776075 + 0.996984i \(0.524728\pi\)
\(294\) −6.82843 −0.398242
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) −3.58579 −0.208068
\(298\) 20.4853 1.18668
\(299\) −9.72792 −0.562580
\(300\) 0 0
\(301\) −2.68629 −0.154835
\(302\) −0.414214 −0.0238353
\(303\) 17.3137 0.994647
\(304\) −7.82843 −0.448991
\(305\) 0 0
\(306\) −6.41421 −0.366676
\(307\) −5.89949 −0.336702 −0.168351 0.985727i \(-0.553844\pi\)
−0.168351 + 0.985727i \(0.553844\pi\)
\(308\) 1.48528 0.0846318
\(309\) 0.928932 0.0528451
\(310\) 0 0
\(311\) −7.17157 −0.406663 −0.203331 0.979110i \(-0.565177\pi\)
−0.203331 + 0.979110i \(0.565177\pi\)
\(312\) 3.24264 0.183578
\(313\) −22.7279 −1.28466 −0.642329 0.766429i \(-0.722032\pi\)
−0.642329 + 0.766429i \(0.722032\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 3.75736 0.211368
\(317\) −26.8284 −1.50683 −0.753417 0.657543i \(-0.771596\pi\)
−0.753417 + 0.657543i \(0.771596\pi\)
\(318\) −1.82843 −0.102533
\(319\) 25.3553 1.41963
\(320\) 0 0
\(321\) −1.92893 −0.107662
\(322\) 1.24264 0.0692497
\(323\) 50.2132 2.79394
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 13.1421 0.727876
\(327\) 13.2426 0.732320
\(328\) −9.41421 −0.519813
\(329\) 3.27208 0.180395
\(330\) 0 0
\(331\) −27.4558 −1.50911 −0.754555 0.656237i \(-0.772147\pi\)
−0.754555 + 0.656237i \(0.772147\pi\)
\(332\) 12.8995 0.707952
\(333\) 1.00000 0.0547997
\(334\) 7.34315 0.401799
\(335\) 0 0
\(336\) −0.414214 −0.0225972
\(337\) 5.97056 0.325237 0.162619 0.986689i \(-0.448006\pi\)
0.162619 + 0.986689i \(0.448006\pi\)
\(338\) −2.48528 −0.135181
\(339\) −2.82843 −0.153619
\(340\) 0 0
\(341\) −27.4558 −1.48682
\(342\) −7.82843 −0.423313
\(343\) 5.72792 0.309279
\(344\) 6.48528 0.349663
\(345\) 0 0
\(346\) 10.6569 0.572916
\(347\) −7.55635 −0.405646 −0.202823 0.979215i \(-0.565012\pi\)
−0.202823 + 0.979215i \(0.565012\pi\)
\(348\) −7.07107 −0.379049
\(349\) −20.0416 −1.07280 −0.536402 0.843963i \(-0.680217\pi\)
−0.536402 + 0.843963i \(0.680217\pi\)
\(350\) 0 0
\(351\) 3.24264 0.173079
\(352\) −3.58579 −0.191123
\(353\) 12.1421 0.646261 0.323130 0.946354i \(-0.395265\pi\)
0.323130 + 0.946354i \(0.395265\pi\)
\(354\) 1.07107 0.0569266
\(355\) 0 0
\(356\) −16.8995 −0.895671
\(357\) 2.65685 0.140616
\(358\) −13.1716 −0.696139
\(359\) 14.9289 0.787919 0.393959 0.919128i \(-0.371105\pi\)
0.393959 + 0.919128i \(0.371105\pi\)
\(360\) 0 0
\(361\) 42.2843 2.22549
\(362\) 23.4142 1.23062
\(363\) 1.85786 0.0975126
\(364\) −1.34315 −0.0704000
\(365\) 0 0
\(366\) 10.4853 0.548074
\(367\) −6.75736 −0.352731 −0.176366 0.984325i \(-0.556434\pi\)
−0.176366 + 0.984325i \(0.556434\pi\)
\(368\) −3.00000 −0.156386
\(369\) −9.41421 −0.490084
\(370\) 0 0
\(371\) 0.757359 0.0393201
\(372\) 7.65685 0.396989
\(373\) −10.2426 −0.530344 −0.265172 0.964201i \(-0.585429\pi\)
−0.265172 + 0.964201i \(0.585429\pi\)
\(374\) 23.0000 1.18930
\(375\) 0 0
\(376\) −7.89949 −0.407385
\(377\) −22.9289 −1.18090
\(378\) −0.414214 −0.0213048
\(379\) −3.27208 −0.168075 −0.0840377 0.996463i \(-0.526782\pi\)
−0.0840377 + 0.996463i \(0.526782\pi\)
\(380\) 0 0
\(381\) −12.4142 −0.635999
\(382\) −2.65685 −0.135936
\(383\) 3.34315 0.170827 0.0854134 0.996346i \(-0.472779\pi\)
0.0854134 + 0.996346i \(0.472779\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −13.3137 −0.677650
\(387\) 6.48528 0.329665
\(388\) −18.7279 −0.950766
\(389\) 10.9706 0.556230 0.278115 0.960548i \(-0.410290\pi\)
0.278115 + 0.960548i \(0.410290\pi\)
\(390\) 0 0
\(391\) 19.2426 0.973142
\(392\) −6.82843 −0.344888
\(393\) −17.8995 −0.902910
\(394\) 13.1421 0.662091
\(395\) 0 0
\(396\) −3.58579 −0.180193
\(397\) −8.48528 −0.425864 −0.212932 0.977067i \(-0.568301\pi\)
−0.212932 + 0.977067i \(0.568301\pi\)
\(398\) 20.0000 1.00251
\(399\) 3.24264 0.162335
\(400\) 0 0
\(401\) 33.0416 1.65002 0.825010 0.565118i \(-0.191169\pi\)
0.825010 + 0.565118i \(0.191169\pi\)
\(402\) −4.82843 −0.240820
\(403\) 24.8284 1.23679
\(404\) 17.3137 0.861389
\(405\) 0 0
\(406\) 2.92893 0.145360
\(407\) −3.58579 −0.177741
\(408\) −6.41421 −0.317551
\(409\) −4.38478 −0.216813 −0.108407 0.994107i \(-0.534575\pi\)
−0.108407 + 0.994107i \(0.534575\pi\)
\(410\) 0 0
\(411\) 13.5563 0.668685
\(412\) 0.928932 0.0457652
\(413\) −0.443651 −0.0218306
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) 3.24264 0.158984
\(417\) 14.4853 0.709347
\(418\) 28.0711 1.37300
\(419\) −2.27208 −0.110998 −0.0554991 0.998459i \(-0.517675\pi\)
−0.0554991 + 0.998459i \(0.517675\pi\)
\(420\) 0 0
\(421\) 37.4558 1.82549 0.912743 0.408534i \(-0.133960\pi\)
0.912743 + 0.408534i \(0.133960\pi\)
\(422\) −26.3848 −1.28439
\(423\) −7.89949 −0.384087
\(424\) −1.82843 −0.0887963
\(425\) 0 0
\(426\) −6.58579 −0.319082
\(427\) −4.34315 −0.210180
\(428\) −1.92893 −0.0932385
\(429\) −11.6274 −0.561377
\(430\) 0 0
\(431\) 24.1716 1.16430 0.582152 0.813080i \(-0.302211\pi\)
0.582152 + 0.813080i \(0.302211\pi\)
\(432\) 1.00000 0.0481125
\(433\) −6.51472 −0.313077 −0.156539 0.987672i \(-0.550034\pi\)
−0.156539 + 0.987672i \(0.550034\pi\)
\(434\) −3.17157 −0.152240
\(435\) 0 0
\(436\) 13.2426 0.634207
\(437\) 23.4853 1.12345
\(438\) −7.82843 −0.374057
\(439\) −0.727922 −0.0347418 −0.0173709 0.999849i \(-0.505530\pi\)
−0.0173709 + 0.999849i \(0.505530\pi\)
\(440\) 0 0
\(441\) −6.82843 −0.325163
\(442\) −20.7990 −0.989307
\(443\) −12.3431 −0.586441 −0.293220 0.956045i \(-0.594727\pi\)
−0.293220 + 0.956045i \(0.594727\pi\)
\(444\) 1.00000 0.0474579
\(445\) 0 0
\(446\) 1.65685 0.0784543
\(447\) 20.4853 0.968921
\(448\) −0.414214 −0.0195698
\(449\) 25.7990 1.21753 0.608765 0.793351i \(-0.291665\pi\)
0.608765 + 0.793351i \(0.291665\pi\)
\(450\) 0 0
\(451\) 33.7574 1.58957
\(452\) −2.82843 −0.133038
\(453\) −0.414214 −0.0194615
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −7.82843 −0.366600
\(457\) 32.9706 1.54230 0.771149 0.636655i \(-0.219682\pi\)
0.771149 + 0.636655i \(0.219682\pi\)
\(458\) −6.72792 −0.314375
\(459\) −6.41421 −0.299390
\(460\) 0 0
\(461\) 15.1716 0.706611 0.353305 0.935508i \(-0.385058\pi\)
0.353305 + 0.935508i \(0.385058\pi\)
\(462\) 1.48528 0.0691015
\(463\) −10.1005 −0.469410 −0.234705 0.972067i \(-0.575412\pi\)
−0.234705 + 0.972067i \(0.575412\pi\)
\(464\) −7.07107 −0.328266
\(465\) 0 0
\(466\) −14.4853 −0.671018
\(467\) −31.7574 −1.46956 −0.734778 0.678308i \(-0.762714\pi\)
−0.734778 + 0.678308i \(0.762714\pi\)
\(468\) 3.24264 0.149891
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 0 0
\(472\) 1.07107 0.0492999
\(473\) −23.2548 −1.06926
\(474\) 3.75736 0.172581
\(475\) 0 0
\(476\) 2.65685 0.121777
\(477\) −1.82843 −0.0837179
\(478\) −24.6274 −1.12643
\(479\) 11.8284 0.540455 0.270227 0.962797i \(-0.412901\pi\)
0.270227 + 0.962797i \(0.412901\pi\)
\(480\) 0 0
\(481\) 3.24264 0.147852
\(482\) 5.41421 0.246611
\(483\) 1.24264 0.0565421
\(484\) 1.85786 0.0844484
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −30.0000 −1.35943 −0.679715 0.733476i \(-0.737896\pi\)
−0.679715 + 0.733476i \(0.737896\pi\)
\(488\) 10.4853 0.474646
\(489\) 13.1421 0.594308
\(490\) 0 0
\(491\) −13.5858 −0.613118 −0.306559 0.951852i \(-0.599178\pi\)
−0.306559 + 0.951852i \(0.599178\pi\)
\(492\) −9.41421 −0.424426
\(493\) 45.3553 2.04270
\(494\) −25.3848 −1.14212
\(495\) 0 0
\(496\) 7.65685 0.343803
\(497\) 2.72792 0.122364
\(498\) 12.8995 0.578040
\(499\) −9.48528 −0.424620 −0.212310 0.977202i \(-0.568099\pi\)
−0.212310 + 0.977202i \(0.568099\pi\)
\(500\) 0 0
\(501\) 7.34315 0.328068
\(502\) −0.828427 −0.0369745
\(503\) −12.3431 −0.550354 −0.275177 0.961394i \(-0.588736\pi\)
−0.275177 + 0.961394i \(0.588736\pi\)
\(504\) −0.414214 −0.0184505
\(505\) 0 0
\(506\) 10.7574 0.478223
\(507\) −2.48528 −0.110375
\(508\) −12.4142 −0.550792
\(509\) −7.68629 −0.340689 −0.170344 0.985385i \(-0.554488\pi\)
−0.170344 + 0.985385i \(0.554488\pi\)
\(510\) 0 0
\(511\) 3.24264 0.143446
\(512\) 1.00000 0.0441942
\(513\) −7.82843 −0.345634
\(514\) 12.4142 0.547567
\(515\) 0 0
\(516\) 6.48528 0.285499
\(517\) 28.3259 1.24577
\(518\) −0.414214 −0.0181995
\(519\) 10.6569 0.467784
\(520\) 0 0
\(521\) 32.0416 1.40377 0.701885 0.712291i \(-0.252342\pi\)
0.701885 + 0.712291i \(0.252342\pi\)
\(522\) −7.07107 −0.309492
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) −17.8995 −0.781943
\(525\) 0 0
\(526\) −28.5858 −1.24640
\(527\) −49.1127 −2.13938
\(528\) −3.58579 −0.156051
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 1.07107 0.0464804
\(532\) 3.24264 0.140586
\(533\) −30.5269 −1.32227
\(534\) −16.8995 −0.731313
\(535\) 0 0
\(536\) −4.82843 −0.208556
\(537\) −13.1716 −0.568395
\(538\) 19.6274 0.846198
\(539\) 24.4853 1.05466
\(540\) 0 0
\(541\) −6.21320 −0.267126 −0.133563 0.991040i \(-0.542642\pi\)
−0.133563 + 0.991040i \(0.542642\pi\)
\(542\) −1.17157 −0.0503234
\(543\) 23.4142 1.00480
\(544\) −6.41421 −0.275007
\(545\) 0 0
\(546\) −1.34315 −0.0574813
\(547\) 45.6274 1.95089 0.975444 0.220249i \(-0.0706869\pi\)
0.975444 + 0.220249i \(0.0706869\pi\)
\(548\) 13.5563 0.579099
\(549\) 10.4853 0.447501
\(550\) 0 0
\(551\) 55.3553 2.35822
\(552\) −3.00000 −0.127688
\(553\) −1.55635 −0.0661827
\(554\) −16.8995 −0.717991
\(555\) 0 0
\(556\) 14.4853 0.614313
\(557\) 6.44365 0.273026 0.136513 0.990638i \(-0.456410\pi\)
0.136513 + 0.990638i \(0.456410\pi\)
\(558\) 7.65685 0.324140
\(559\) 21.0294 0.889450
\(560\) 0 0
\(561\) 23.0000 0.971061
\(562\) 23.3848 0.986427
\(563\) −19.4142 −0.818212 −0.409106 0.912487i \(-0.634159\pi\)
−0.409106 + 0.912487i \(0.634159\pi\)
\(564\) −7.89949 −0.332629
\(565\) 0 0
\(566\) −20.7990 −0.874247
\(567\) −0.414214 −0.0173953
\(568\) −6.58579 −0.276333
\(569\) −11.0416 −0.462889 −0.231445 0.972848i \(-0.574345\pi\)
−0.231445 + 0.972848i \(0.574345\pi\)
\(570\) 0 0
\(571\) 20.0416 0.838716 0.419358 0.907821i \(-0.362255\pi\)
0.419358 + 0.907821i \(0.362255\pi\)
\(572\) −11.6274 −0.486167
\(573\) −2.65685 −0.110992
\(574\) 3.89949 0.162762
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −26.4853 −1.10260 −0.551298 0.834308i \(-0.685867\pi\)
−0.551298 + 0.834308i \(0.685867\pi\)
\(578\) 24.1421 1.00418
\(579\) −13.3137 −0.553299
\(580\) 0 0
\(581\) −5.34315 −0.221671
\(582\) −18.7279 −0.776297
\(583\) 6.55635 0.271536
\(584\) −7.82843 −0.323943
\(585\) 0 0
\(586\) −2.65685 −0.109754
\(587\) −2.82843 −0.116742 −0.0583708 0.998295i \(-0.518591\pi\)
−0.0583708 + 0.998295i \(0.518591\pi\)
\(588\) −6.82843 −0.281600
\(589\) −59.9411 −2.46983
\(590\) 0 0
\(591\) 13.1421 0.540595
\(592\) 1.00000 0.0410997
\(593\) 28.9706 1.18968 0.594839 0.803845i \(-0.297216\pi\)
0.594839 + 0.803845i \(0.297216\pi\)
\(594\) −3.58579 −0.147127
\(595\) 0 0
\(596\) 20.4853 0.839110
\(597\) 20.0000 0.818546
\(598\) −9.72792 −0.397804
\(599\) −20.8701 −0.852727 −0.426364 0.904552i \(-0.640206\pi\)
−0.426364 + 0.904552i \(0.640206\pi\)
\(600\) 0 0
\(601\) −23.9706 −0.977780 −0.488890 0.872346i \(-0.662598\pi\)
−0.488890 + 0.872346i \(0.662598\pi\)
\(602\) −2.68629 −0.109485
\(603\) −4.82843 −0.196629
\(604\) −0.414214 −0.0168541
\(605\) 0 0
\(606\) 17.3137 0.703321
\(607\) 40.1421 1.62932 0.814660 0.579940i \(-0.196924\pi\)
0.814660 + 0.579940i \(0.196924\pi\)
\(608\) −7.82843 −0.317485
\(609\) 2.92893 0.118686
\(610\) 0 0
\(611\) −25.6152 −1.03628
\(612\) −6.41421 −0.259279
\(613\) 21.3137 0.860853 0.430426 0.902626i \(-0.358363\pi\)
0.430426 + 0.902626i \(0.358363\pi\)
\(614\) −5.89949 −0.238084
\(615\) 0 0
\(616\) 1.48528 0.0598437
\(617\) −23.6985 −0.954065 −0.477033 0.878886i \(-0.658288\pi\)
−0.477033 + 0.878886i \(0.658288\pi\)
\(618\) 0.928932 0.0373671
\(619\) 35.2132 1.41534 0.707669 0.706544i \(-0.249747\pi\)
0.707669 + 0.706544i \(0.249747\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) −7.17157 −0.287554
\(623\) 7.00000 0.280449
\(624\) 3.24264 0.129810
\(625\) 0 0
\(626\) −22.7279 −0.908390
\(627\) 28.0711 1.12105
\(628\) 0 0
\(629\) −6.41421 −0.255751
\(630\) 0 0
\(631\) 24.4853 0.974744 0.487372 0.873195i \(-0.337956\pi\)
0.487372 + 0.873195i \(0.337956\pi\)
\(632\) 3.75736 0.149460
\(633\) −26.3848 −1.04870
\(634\) −26.8284 −1.06549
\(635\) 0 0
\(636\) −1.82843 −0.0725019
\(637\) −22.1421 −0.877303
\(638\) 25.3553 1.00383
\(639\) −6.58579 −0.260530
\(640\) 0 0
\(641\) −21.1716 −0.836227 −0.418113 0.908395i \(-0.637309\pi\)
−0.418113 + 0.908395i \(0.637309\pi\)
\(642\) −1.92893 −0.0761289
\(643\) −32.4558 −1.27993 −0.639967 0.768403i \(-0.721052\pi\)
−0.639967 + 0.768403i \(0.721052\pi\)
\(644\) 1.24264 0.0489669
\(645\) 0 0
\(646\) 50.2132 1.97561
\(647\) 21.0000 0.825595 0.412798 0.910823i \(-0.364552\pi\)
0.412798 + 0.910823i \(0.364552\pi\)
\(648\) 1.00000 0.0392837
\(649\) −3.84062 −0.150758
\(650\) 0 0
\(651\) −3.17157 −0.124304
\(652\) 13.1421 0.514686
\(653\) −3.51472 −0.137542 −0.0687708 0.997632i \(-0.521908\pi\)
−0.0687708 + 0.997632i \(0.521908\pi\)
\(654\) 13.2426 0.517828
\(655\) 0 0
\(656\) −9.41421 −0.367563
\(657\) −7.82843 −0.305416
\(658\) 3.27208 0.127559
\(659\) −11.1127 −0.432889 −0.216445 0.976295i \(-0.569446\pi\)
−0.216445 + 0.976295i \(0.569446\pi\)
\(660\) 0 0
\(661\) 16.7574 0.651786 0.325893 0.945407i \(-0.394335\pi\)
0.325893 + 0.945407i \(0.394335\pi\)
\(662\) −27.4558 −1.06710
\(663\) −20.7990 −0.807766
\(664\) 12.8995 0.500597
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 21.2132 0.821379
\(668\) 7.34315 0.284115
\(669\) 1.65685 0.0640577
\(670\) 0 0
\(671\) −37.5980 −1.45145
\(672\) −0.414214 −0.0159786
\(673\) −33.9706 −1.30947 −0.654734 0.755859i \(-0.727220\pi\)
−0.654734 + 0.755859i \(0.727220\pi\)
\(674\) 5.97056 0.229977
\(675\) 0 0
\(676\) −2.48528 −0.0955877
\(677\) −40.7990 −1.56803 −0.784016 0.620740i \(-0.786832\pi\)
−0.784016 + 0.620740i \(0.786832\pi\)
\(678\) −2.82843 −0.108625
\(679\) 7.75736 0.297700
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) −27.4558 −1.05134
\(683\) 31.0711 1.18890 0.594451 0.804132i \(-0.297370\pi\)
0.594451 + 0.804132i \(0.297370\pi\)
\(684\) −7.82843 −0.299327
\(685\) 0 0
\(686\) 5.72792 0.218693
\(687\) −6.72792 −0.256686
\(688\) 6.48528 0.247249
\(689\) −5.92893 −0.225874
\(690\) 0 0
\(691\) 33.4142 1.27114 0.635568 0.772045i \(-0.280766\pi\)
0.635568 + 0.772045i \(0.280766\pi\)
\(692\) 10.6569 0.405113
\(693\) 1.48528 0.0564212
\(694\) −7.55635 −0.286835
\(695\) 0 0
\(696\) −7.07107 −0.268028
\(697\) 60.3848 2.28724
\(698\) −20.0416 −0.758587
\(699\) −14.4853 −0.547884
\(700\) 0 0
\(701\) −30.6274 −1.15678 −0.578391 0.815760i \(-0.696319\pi\)
−0.578391 + 0.815760i \(0.696319\pi\)
\(702\) 3.24264 0.122386
\(703\) −7.82843 −0.295255
\(704\) −3.58579 −0.135144
\(705\) 0 0
\(706\) 12.1421 0.456975
\(707\) −7.17157 −0.269715
\(708\) 1.07107 0.0402532
\(709\) 29.9289 1.12400 0.562002 0.827136i \(-0.310031\pi\)
0.562002 + 0.827136i \(0.310031\pi\)
\(710\) 0 0
\(711\) 3.75736 0.140912
\(712\) −16.8995 −0.633335
\(713\) −22.9706 −0.860254
\(714\) 2.65685 0.0994303
\(715\) 0 0
\(716\) −13.1716 −0.492245
\(717\) −24.6274 −0.919728
\(718\) 14.9289 0.557143
\(719\) −1.41421 −0.0527413 −0.0263706 0.999652i \(-0.508395\pi\)
−0.0263706 + 0.999652i \(0.508395\pi\)
\(720\) 0 0
\(721\) −0.384776 −0.0143298
\(722\) 42.2843 1.57366
\(723\) 5.41421 0.201357
\(724\) 23.4142 0.870182
\(725\) 0 0
\(726\) 1.85786 0.0689518
\(727\) 16.3848 0.607678 0.303839 0.952723i \(-0.401732\pi\)
0.303839 + 0.952723i \(0.401732\pi\)
\(728\) −1.34315 −0.0497803
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.5980 −1.53856
\(732\) 10.4853 0.387547
\(733\) −22.9706 −0.848437 −0.424219 0.905560i \(-0.639451\pi\)
−0.424219 + 0.905560i \(0.639451\pi\)
\(734\) −6.75736 −0.249419
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 17.3137 0.637759
\(738\) −9.41421 −0.346542
\(739\) −10.0000 −0.367856 −0.183928 0.982940i \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 0 0
\(741\) −25.3848 −0.932533
\(742\) 0.757359 0.0278035
\(743\) 19.3553 0.710079 0.355039 0.934851i \(-0.384467\pi\)
0.355039 + 0.934851i \(0.384467\pi\)
\(744\) 7.65685 0.280714
\(745\) 0 0
\(746\) −10.2426 −0.375010
\(747\) 12.8995 0.471968
\(748\) 23.0000 0.840963
\(749\) 0.798990 0.0291945
\(750\) 0 0
\(751\) 27.1127 0.989356 0.494678 0.869076i \(-0.335286\pi\)
0.494678 + 0.869076i \(0.335286\pi\)
\(752\) −7.89949 −0.288065
\(753\) −0.828427 −0.0301896
\(754\) −22.9289 −0.835022
\(755\) 0 0
\(756\) −0.414214 −0.0150648
\(757\) −26.2132 −0.952735 −0.476368 0.879246i \(-0.658047\pi\)
−0.476368 + 0.879246i \(0.658047\pi\)
\(758\) −3.27208 −0.118847
\(759\) 10.7574 0.390467
\(760\) 0 0
\(761\) 2.68629 0.0973780 0.0486890 0.998814i \(-0.484496\pi\)
0.0486890 + 0.998814i \(0.484496\pi\)
\(762\) −12.4142 −0.449720
\(763\) −5.48528 −0.198581
\(764\) −2.65685 −0.0961216
\(765\) 0 0
\(766\) 3.34315 0.120793
\(767\) 3.47309 0.125406
\(768\) 1.00000 0.0360844
\(769\) 11.5563 0.416733 0.208366 0.978051i \(-0.433185\pi\)
0.208366 + 0.978051i \(0.433185\pi\)
\(770\) 0 0
\(771\) 12.4142 0.447087
\(772\) −13.3137 −0.479171
\(773\) −3.34315 −0.120245 −0.0601223 0.998191i \(-0.519149\pi\)
−0.0601223 + 0.998191i \(0.519149\pi\)
\(774\) 6.48528 0.233109
\(775\) 0 0
\(776\) −18.7279 −0.672293
\(777\) −0.414214 −0.0148598
\(778\) 10.9706 0.393314
\(779\) 73.6985 2.64052
\(780\) 0 0
\(781\) 23.6152 0.845019
\(782\) 19.2426 0.688115
\(783\) −7.07107 −0.252699
\(784\) −6.82843 −0.243872
\(785\) 0 0
\(786\) −17.8995 −0.638454
\(787\) 21.5147 0.766917 0.383458 0.923558i \(-0.374733\pi\)
0.383458 + 0.923558i \(0.374733\pi\)
\(788\) 13.1421 0.468169
\(789\) −28.5858 −1.01768
\(790\) 0 0
\(791\) 1.17157 0.0416563
\(792\) −3.58579 −0.127415
\(793\) 34.0000 1.20738
\(794\) −8.48528 −0.301131
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −2.87006 −0.101663 −0.0508313 0.998707i \(-0.516187\pi\)
−0.0508313 + 0.998707i \(0.516187\pi\)
\(798\) 3.24264 0.114788
\(799\) 50.6690 1.79254
\(800\) 0 0
\(801\) −16.8995 −0.597114
\(802\) 33.0416 1.16674
\(803\) 28.0711 0.990606
\(804\) −4.82843 −0.170285
\(805\) 0 0
\(806\) 24.8284 0.874544
\(807\) 19.6274 0.690918
\(808\) 17.3137 0.609094
\(809\) 54.2132 1.90603 0.953017 0.302916i \(-0.0979601\pi\)
0.953017 + 0.302916i \(0.0979601\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 2.92893 0.102785
\(813\) −1.17157 −0.0410889
\(814\) −3.58579 −0.125682
\(815\) 0 0
\(816\) −6.41421 −0.224542
\(817\) −50.7696 −1.77620
\(818\) −4.38478 −0.153310
\(819\) −1.34315 −0.0469333
\(820\) 0 0
\(821\) 5.28427 0.184422 0.0922112 0.995739i \(-0.470607\pi\)
0.0922112 + 0.995739i \(0.470607\pi\)
\(822\) 13.5563 0.472832
\(823\) 17.7279 0.617957 0.308978 0.951069i \(-0.400013\pi\)
0.308978 + 0.951069i \(0.400013\pi\)
\(824\) 0.928932 0.0323609
\(825\) 0 0
\(826\) −0.443651 −0.0154366
\(827\) −7.07107 −0.245885 −0.122943 0.992414i \(-0.539233\pi\)
−0.122943 + 0.992414i \(0.539233\pi\)
\(828\) −3.00000 −0.104257
\(829\) −10.7574 −0.373619 −0.186809 0.982396i \(-0.559815\pi\)
−0.186809 + 0.982396i \(0.559815\pi\)
\(830\) 0 0
\(831\) −16.8995 −0.586237
\(832\) 3.24264 0.112418
\(833\) 43.7990 1.51755
\(834\) 14.4853 0.501584
\(835\) 0 0
\(836\) 28.0711 0.970858
\(837\) 7.65685 0.264660
\(838\) −2.27208 −0.0784876
\(839\) −21.2132 −0.732361 −0.366181 0.930544i \(-0.619335\pi\)
−0.366181 + 0.930544i \(0.619335\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 37.4558 1.29081
\(843\) 23.3848 0.805414
\(844\) −26.3848 −0.908201
\(845\) 0 0
\(846\) −7.89949 −0.271590
\(847\) −0.769553 −0.0264421
\(848\) −1.82843 −0.0627884
\(849\) −20.7990 −0.713819
\(850\) 0 0
\(851\) −3.00000 −0.102839
\(852\) −6.58579 −0.225625
\(853\) 44.8406 1.53531 0.767657 0.640861i \(-0.221423\pi\)
0.767657 + 0.640861i \(0.221423\pi\)
\(854\) −4.34315 −0.148619
\(855\) 0 0
\(856\) −1.92893 −0.0659295
\(857\) 42.4975 1.45169 0.725843 0.687860i \(-0.241450\pi\)
0.725843 + 0.687860i \(0.241450\pi\)
\(858\) −11.6274 −0.396953
\(859\) −6.51472 −0.222279 −0.111140 0.993805i \(-0.535450\pi\)
−0.111140 + 0.993805i \(0.535450\pi\)
\(860\) 0 0
\(861\) 3.89949 0.132894
\(862\) 24.1716 0.823287
\(863\) 5.55635 0.189140 0.0945702 0.995518i \(-0.469852\pi\)
0.0945702 + 0.995518i \(0.469852\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −6.51472 −0.221379
\(867\) 24.1421 0.819910
\(868\) −3.17157 −0.107650
\(869\) −13.4731 −0.457043
\(870\) 0 0
\(871\) −15.6569 −0.530512
\(872\) 13.2426 0.448452
\(873\) −18.7279 −0.633844
\(874\) 23.4853 0.794401
\(875\) 0 0
\(876\) −7.82843 −0.264498
\(877\) 31.7574 1.07237 0.536185 0.844101i \(-0.319865\pi\)
0.536185 + 0.844101i \(0.319865\pi\)
\(878\) −0.727922 −0.0245662
\(879\) −2.65685 −0.0896135
\(880\) 0 0
\(881\) −26.6274 −0.897100 −0.448550 0.893758i \(-0.648059\pi\)
−0.448550 + 0.893758i \(0.648059\pi\)
\(882\) −6.82843 −0.229925
\(883\) −55.9706 −1.88356 −0.941780 0.336231i \(-0.890848\pi\)
−0.941780 + 0.336231i \(0.890848\pi\)
\(884\) −20.7990 −0.699546
\(885\) 0 0
\(886\) −12.3431 −0.414676
\(887\) −8.14214 −0.273386 −0.136693 0.990613i \(-0.543647\pi\)
−0.136693 + 0.990613i \(0.543647\pi\)
\(888\) 1.00000 0.0335578
\(889\) 5.14214 0.172462
\(890\) 0 0
\(891\) −3.58579 −0.120128
\(892\) 1.65685 0.0554756
\(893\) 61.8406 2.06942
\(894\) 20.4853 0.685130
\(895\) 0 0
\(896\) −0.414214 −0.0138379
\(897\) −9.72792 −0.324806
\(898\) 25.7990 0.860923
\(899\) −54.1421 −1.80574
\(900\) 0 0
\(901\) 11.7279 0.390714
\(902\) 33.7574 1.12400
\(903\) −2.68629 −0.0893942
\(904\) −2.82843 −0.0940721
\(905\) 0 0
\(906\) −0.414214 −0.0137613
\(907\) −36.7990 −1.22189 −0.610945 0.791673i \(-0.709210\pi\)
−0.610945 + 0.791673i \(0.709210\pi\)
\(908\) −12.0000 −0.398234
\(909\) 17.3137 0.574259
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −7.82843 −0.259225
\(913\) −46.2548 −1.53081
\(914\) 32.9706 1.09057
\(915\) 0 0
\(916\) −6.72792 −0.222297
\(917\) 7.41421 0.244839
\(918\) −6.41421 −0.211701
\(919\) 7.75736 0.255892 0.127946 0.991781i \(-0.459162\pi\)
0.127946 + 0.991781i \(0.459162\pi\)
\(920\) 0 0
\(921\) −5.89949 −0.194395
\(922\) 15.1716 0.499649
\(923\) −21.3553 −0.702920
\(924\) 1.48528 0.0488622
\(925\) 0 0
\(926\) −10.1005 −0.331923
\(927\) 0.928932 0.0305101
\(928\) −7.07107 −0.232119
\(929\) −14.4853 −0.475247 −0.237623 0.971357i \(-0.576368\pi\)
−0.237623 + 0.971357i \(0.576368\pi\)
\(930\) 0 0
\(931\) 53.4558 1.75194
\(932\) −14.4853 −0.474481
\(933\) −7.17157 −0.234787
\(934\) −31.7574 −1.03913
\(935\) 0 0
\(936\) 3.24264 0.105989
\(937\) −17.9411 −0.586111 −0.293056 0.956095i \(-0.594672\pi\)
−0.293056 + 0.956095i \(0.594672\pi\)
\(938\) 2.00000 0.0653023
\(939\) −22.7279 −0.741698
\(940\) 0 0
\(941\) −26.3431 −0.858762 −0.429381 0.903123i \(-0.641268\pi\)
−0.429381 + 0.903123i \(0.641268\pi\)
\(942\) 0 0
\(943\) 28.2426 0.919707
\(944\) 1.07107 0.0348603
\(945\) 0 0
\(946\) −23.2548 −0.756080
\(947\) −20.0416 −0.651265 −0.325633 0.945496i \(-0.605577\pi\)
−0.325633 + 0.945496i \(0.605577\pi\)
\(948\) 3.75736 0.122033
\(949\) −25.3848 −0.824025
\(950\) 0 0
\(951\) −26.8284 −0.869971
\(952\) 2.65685 0.0861091
\(953\) 3.41421 0.110597 0.0552986 0.998470i \(-0.482389\pi\)
0.0552986 + 0.998470i \(0.482389\pi\)
\(954\) −1.82843 −0.0591975
\(955\) 0 0
\(956\) −24.6274 −0.796508
\(957\) 25.3553 0.819622
\(958\) 11.8284 0.382159
\(959\) −5.61522 −0.181325
\(960\) 0 0
\(961\) 27.6274 0.891207
\(962\) 3.24264 0.104547
\(963\) −1.92893 −0.0621590
\(964\) 5.41421 0.174380
\(965\) 0 0
\(966\) 1.24264 0.0399813
\(967\) −40.1005 −1.28955 −0.644773 0.764374i \(-0.723048\pi\)
−0.644773 + 0.764374i \(0.723048\pi\)
\(968\) 1.85786 0.0597140
\(969\) 50.2132 1.61308
\(970\) 0 0
\(971\) −22.9706 −0.737160 −0.368580 0.929596i \(-0.620156\pi\)
−0.368580 + 0.929596i \(0.620156\pi\)
\(972\) 1.00000 0.0320750
\(973\) −6.00000 −0.192351
\(974\) −30.0000 −0.961262
\(975\) 0 0
\(976\) 10.4853 0.335626
\(977\) 20.6985 0.662203 0.331102 0.943595i \(-0.392580\pi\)
0.331102 + 0.943595i \(0.392580\pi\)
\(978\) 13.1421 0.420239
\(979\) 60.5980 1.93672
\(980\) 0 0
\(981\) 13.2426 0.422805
\(982\) −13.5858 −0.433540
\(983\) 37.1127 1.18371 0.591856 0.806044i \(-0.298396\pi\)
0.591856 + 0.806044i \(0.298396\pi\)
\(984\) −9.41421 −0.300114
\(985\) 0 0
\(986\) 45.3553 1.44441
\(987\) 3.27208 0.104151
\(988\) −25.3848 −0.807597
\(989\) −19.4558 −0.618660
\(990\) 0 0
\(991\) −30.6274 −0.972912 −0.486456 0.873705i \(-0.661711\pi\)
−0.486456 + 0.873705i \(0.661711\pi\)
\(992\) 7.65685 0.243105
\(993\) −27.4558 −0.871285
\(994\) 2.72792 0.0865244
\(995\) 0 0
\(996\) 12.8995 0.408736
\(997\) −9.78680 −0.309951 −0.154975 0.987918i \(-0.549530\pi\)
−0.154975 + 0.987918i \(0.549530\pi\)
\(998\) −9.48528 −0.300251
\(999\) 1.00000 0.0316386
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 5550.2.a.cb.1.1 2
5.2 odd 4 1110.2.d.g.889.3 yes 4
5.3 odd 4 1110.2.d.g.889.1 4
5.4 even 2 5550.2.a.bs.1.2 2
15.2 even 4 3330.2.d.k.1999.2 4
15.8 even 4 3330.2.d.k.1999.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.d.g.889.1 4 5.3 odd 4
1110.2.d.g.889.3 yes 4 5.2 odd 4
3330.2.d.k.1999.2 4 15.2 even 4
3330.2.d.k.1999.4 4 15.8 even 4
5550.2.a.bs.1.2 2 5.4 even 2
5550.2.a.cb.1.1 2 1.1 even 1 trivial