Properties

Label 1075.2.a.n.1.3
Level $1075$
Weight $2$
Character 1075.1
Self dual yes
Analytic conductor $8.584$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1075,2,Mod(1,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.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: \(8.58391821729\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.24217.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 5x^{3} - x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.369680\) of defining polynomial
Character \(\chi\) \(=\) 1075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.369680 q^{2} -1.27684 q^{3} -1.86334 q^{4} -0.472023 q^{6} +1.49366 q^{7} -1.42820 q^{8} -1.36968 q^{9} +O(q^{10})\) \(q+0.369680 q^{2} -1.27684 q^{3} -1.86334 q^{4} -0.472023 q^{6} +1.49366 q^{7} -1.42820 q^{8} -1.36968 q^{9} +3.68692 q^{11} +2.37918 q^{12} +3.61627 q^{13} +0.552175 q^{14} +3.19870 q^{16} -5.00064 q^{17} -0.506344 q^{18} -6.14424 q^{19} -1.90716 q^{21} +1.36298 q^{22} +1.06002 q^{23} +1.82358 q^{24} +1.33686 q^{26} +5.57938 q^{27} -2.78318 q^{28} +0.426517 q^{29} -10.0955 q^{31} +4.03889 q^{32} -4.70760 q^{33} -1.84864 q^{34} +2.55218 q^{36} +8.49430 q^{37} -2.27141 q^{38} -4.61739 q^{39} -10.0225 q^{41} -0.705039 q^{42} -1.00000 q^{43} -6.86997 q^{44} +0.391870 q^{46} -12.7077 q^{47} -4.08422 q^{48} -4.76899 q^{49} +6.38502 q^{51} -6.73832 q^{52} -1.96199 q^{53} +2.06259 q^{54} -2.13324 q^{56} +7.84522 q^{57} +0.157675 q^{58} -8.00909 q^{59} -14.9538 q^{61} -3.73211 q^{62} -2.04583 q^{63} -4.90429 q^{64} -1.74031 q^{66} +3.94582 q^{67} +9.31788 q^{68} -1.35348 q^{69} +1.62426 q^{71} +1.95618 q^{72} -8.77264 q^{73} +3.14018 q^{74} +11.4488 q^{76} +5.50699 q^{77} -1.70696 q^{78} +5.75850 q^{79} -3.01493 q^{81} -3.70513 q^{82} -1.38813 q^{83} +3.55368 q^{84} -0.369680 q^{86} -0.544594 q^{87} -5.26566 q^{88} +10.8191 q^{89} +5.40146 q^{91} -1.97518 q^{92} +12.8903 q^{93} -4.69778 q^{94} -5.15702 q^{96} +4.27085 q^{97} -1.76300 q^{98} -5.04990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{6} - 3 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{6} - 3 q^{8} - 5 q^{9} - 9 q^{11} + 8 q^{12} - q^{13} - 7 q^{14} - 2 q^{16} + 3 q^{17} - 10 q^{18} - 11 q^{19} - 5 q^{21} + 13 q^{22} - 9 q^{24} - 5 q^{26} + 3 q^{27} - 15 q^{28} - 22 q^{29} - 5 q^{31} + 4 q^{32} - 12 q^{33} - 7 q^{34} + 3 q^{36} + 7 q^{37} - 3 q^{38} - 3 q^{39} - 21 q^{41} + 7 q^{42} - 5 q^{43} - 20 q^{44} + 13 q^{46} - 3 q^{47} - 6 q^{48} - 13 q^{49} - 5 q^{51} + 18 q^{52} - 5 q^{53} + 4 q^{54} + 4 q^{56} + 13 q^{57} - 16 q^{58} - 8 q^{59} - 16 q^{61} - 2 q^{62} + 7 q^{63} - 17 q^{64} + 19 q^{66} - 8 q^{67} - 7 q^{68} - 15 q^{69} - 10 q^{71} + 5 q^{72} + q^{73} - 3 q^{76} + 7 q^{77} - 25 q^{78} + 12 q^{79} - 15 q^{81} + 41 q^{82} + 20 q^{83} + 5 q^{84} - 22 q^{87} - 2 q^{88} - 36 q^{89} - 13 q^{91} - q^{92} - 13 q^{93} + 28 q^{96} - 8 q^{97} + 25 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.369680 0.261404 0.130702 0.991422i \(-0.458277\pi\)
0.130702 + 0.991422i \(0.458277\pi\)
\(3\) −1.27684 −0.737184 −0.368592 0.929591i \(-0.620160\pi\)
−0.368592 + 0.929591i \(0.620160\pi\)
\(4\) −1.86334 −0.931668
\(5\) 0 0
\(6\) −0.472023 −0.192702
\(7\) 1.49366 0.564549 0.282274 0.959334i \(-0.408911\pi\)
0.282274 + 0.959334i \(0.408911\pi\)
\(8\) −1.42820 −0.504945
\(9\) −1.36968 −0.456560
\(10\) 0 0
\(11\) 3.68692 1.11165 0.555824 0.831300i \(-0.312403\pi\)
0.555824 + 0.831300i \(0.312403\pi\)
\(12\) 2.37918 0.686811
\(13\) 3.61627 1.00297 0.501486 0.865166i \(-0.332787\pi\)
0.501486 + 0.865166i \(0.332787\pi\)
\(14\) 0.552175 0.147575
\(15\) 0 0
\(16\) 3.19870 0.799674
\(17\) −5.00064 −1.21283 −0.606417 0.795147i \(-0.707394\pi\)
−0.606417 + 0.795147i \(0.707394\pi\)
\(18\) −0.506344 −0.119346
\(19\) −6.14424 −1.40959 −0.704793 0.709413i \(-0.748960\pi\)
−0.704793 + 0.709413i \(0.748960\pi\)
\(20\) 0 0
\(21\) −1.90716 −0.416176
\(22\) 1.36298 0.290589
\(23\) 1.06002 0.221030 0.110515 0.993874i \(-0.464750\pi\)
0.110515 + 0.993874i \(0.464750\pi\)
\(24\) 1.82358 0.372237
\(25\) 0 0
\(26\) 1.33686 0.262180
\(27\) 5.57938 1.07375
\(28\) −2.78318 −0.525972
\(29\) 0.426517 0.0792023 0.0396011 0.999216i \(-0.487391\pi\)
0.0396011 + 0.999216i \(0.487391\pi\)
\(30\) 0 0
\(31\) −10.0955 −1.81320 −0.906602 0.421987i \(-0.861333\pi\)
−0.906602 + 0.421987i \(0.861333\pi\)
\(32\) 4.03889 0.713982
\(33\) −4.70760 −0.819489
\(34\) −1.84864 −0.317039
\(35\) 0 0
\(36\) 2.55218 0.425363
\(37\) 8.49430 1.39645 0.698227 0.715876i \(-0.253973\pi\)
0.698227 + 0.715876i \(0.253973\pi\)
\(38\) −2.27141 −0.368471
\(39\) −4.61739 −0.739375
\(40\) 0 0
\(41\) −10.0225 −1.56525 −0.782627 0.622491i \(-0.786121\pi\)
−0.782627 + 0.622491i \(0.786121\pi\)
\(42\) −0.705039 −0.108790
\(43\) −1.00000 −0.152499
\(44\) −6.86997 −1.03569
\(45\) 0 0
\(46\) 0.391870 0.0577781
\(47\) −12.7077 −1.85361 −0.926804 0.375546i \(-0.877455\pi\)
−0.926804 + 0.375546i \(0.877455\pi\)
\(48\) −4.08422 −0.589507
\(49\) −4.76899 −0.681285
\(50\) 0 0
\(51\) 6.38502 0.894082
\(52\) −6.73832 −0.934437
\(53\) −1.96199 −0.269500 −0.134750 0.990880i \(-0.543023\pi\)
−0.134750 + 0.990880i \(0.543023\pi\)
\(54\) 2.06259 0.280683
\(55\) 0 0
\(56\) −2.13324 −0.285066
\(57\) 7.84522 1.03912
\(58\) 0.157675 0.0207038
\(59\) −8.00909 −1.04269 −0.521347 0.853345i \(-0.674570\pi\)
−0.521347 + 0.853345i \(0.674570\pi\)
\(60\) 0 0
\(61\) −14.9538 −1.91463 −0.957317 0.289039i \(-0.906664\pi\)
−0.957317 + 0.289039i \(0.906664\pi\)
\(62\) −3.73211 −0.473978
\(63\) −2.04583 −0.257751
\(64\) −4.90429 −0.613036
\(65\) 0 0
\(66\) −1.74031 −0.214217
\(67\) 3.94582 0.482058 0.241029 0.970518i \(-0.422515\pi\)
0.241029 + 0.970518i \(0.422515\pi\)
\(68\) 9.31788 1.12996
\(69\) −1.35348 −0.162940
\(70\) 0 0
\(71\) 1.62426 0.192765 0.0963824 0.995344i \(-0.469273\pi\)
0.0963824 + 0.995344i \(0.469273\pi\)
\(72\) 1.95618 0.230538
\(73\) −8.77264 −1.02676 −0.513380 0.858161i \(-0.671607\pi\)
−0.513380 + 0.858161i \(0.671607\pi\)
\(74\) 3.14018 0.365038
\(75\) 0 0
\(76\) 11.4488 1.31327
\(77\) 5.50699 0.627580
\(78\) −1.70696 −0.193275
\(79\) 5.75850 0.647882 0.323941 0.946077i \(-0.394992\pi\)
0.323941 + 0.946077i \(0.394992\pi\)
\(80\) 0 0
\(81\) −3.01493 −0.334993
\(82\) −3.70513 −0.409163
\(83\) −1.38813 −0.152367 −0.0761834 0.997094i \(-0.524273\pi\)
−0.0761834 + 0.997094i \(0.524273\pi\)
\(84\) 3.55368 0.387738
\(85\) 0 0
\(86\) −0.369680 −0.0398637
\(87\) −0.544594 −0.0583866
\(88\) −5.26566 −0.561321
\(89\) 10.8191 1.14682 0.573412 0.819267i \(-0.305619\pi\)
0.573412 + 0.819267i \(0.305619\pi\)
\(90\) 0 0
\(91\) 5.40146 0.566227
\(92\) −1.97518 −0.205927
\(93\) 12.8903 1.33666
\(94\) −4.69778 −0.484539
\(95\) 0 0
\(96\) −5.15702 −0.526336
\(97\) 4.27085 0.433639 0.216820 0.976212i \(-0.430432\pi\)
0.216820 + 0.976212i \(0.430432\pi\)
\(98\) −1.76300 −0.178090
\(99\) −5.04990 −0.507534
\(100\) 0 0
\(101\) −2.92378 −0.290927 −0.145463 0.989364i \(-0.546467\pi\)
−0.145463 + 0.989364i \(0.546467\pi\)
\(102\) 2.36042 0.233716
\(103\) 7.90261 0.778667 0.389334 0.921097i \(-0.372705\pi\)
0.389334 + 0.921097i \(0.372705\pi\)
\(104\) −5.16475 −0.506446
\(105\) 0 0
\(106\) −0.725308 −0.0704482
\(107\) 4.86805 0.470612 0.235306 0.971921i \(-0.424391\pi\)
0.235306 + 0.971921i \(0.424391\pi\)
\(108\) −10.3963 −1.00038
\(109\) 10.1676 0.973878 0.486939 0.873436i \(-0.338113\pi\)
0.486939 + 0.873436i \(0.338113\pi\)
\(110\) 0 0
\(111\) −10.8459 −1.02944
\(112\) 4.77775 0.451455
\(113\) −7.45085 −0.700917 −0.350459 0.936578i \(-0.613974\pi\)
−0.350459 + 0.936578i \(0.613974\pi\)
\(114\) 2.90022 0.271631
\(115\) 0 0
\(116\) −0.794745 −0.0737902
\(117\) −4.95313 −0.457917
\(118\) −2.96080 −0.272564
\(119\) −7.46924 −0.684704
\(120\) 0 0
\(121\) 2.59337 0.235761
\(122\) −5.52812 −0.500492
\(123\) 12.7971 1.15388
\(124\) 18.8113 1.68930
\(125\) 0 0
\(126\) −0.756304 −0.0673769
\(127\) −7.24053 −0.642493 −0.321247 0.946996i \(-0.604102\pi\)
−0.321247 + 0.946996i \(0.604102\pi\)
\(128\) −9.89081 −0.874232
\(129\) 1.27684 0.112419
\(130\) 0 0
\(131\) −2.60582 −0.227671 −0.113836 0.993500i \(-0.536314\pi\)
−0.113836 + 0.993500i \(0.536314\pi\)
\(132\) 8.77185 0.763492
\(133\) −9.17739 −0.795780
\(134\) 1.45869 0.126012
\(135\) 0 0
\(136\) 7.14192 0.612414
\(137\) −20.3626 −1.73969 −0.869847 0.493322i \(-0.835782\pi\)
−0.869847 + 0.493322i \(0.835782\pi\)
\(138\) −0.500355 −0.0425930
\(139\) −9.90995 −0.840551 −0.420276 0.907397i \(-0.638067\pi\)
−0.420276 + 0.907397i \(0.638067\pi\)
\(140\) 0 0
\(141\) 16.2257 1.36645
\(142\) 0.600459 0.0503894
\(143\) 13.3329 1.11495
\(144\) −4.38119 −0.365099
\(145\) 0 0
\(146\) −3.24307 −0.268399
\(147\) 6.08924 0.502232
\(148\) −15.8277 −1.30103
\(149\) −5.62745 −0.461019 −0.230509 0.973070i \(-0.574039\pi\)
−0.230509 + 0.973070i \(0.574039\pi\)
\(150\) 0 0
\(151\) −1.99840 −0.162627 −0.0813136 0.996689i \(-0.525912\pi\)
−0.0813136 + 0.996689i \(0.525912\pi\)
\(152\) 8.77521 0.711763
\(153\) 6.84928 0.553732
\(154\) 2.03583 0.164051
\(155\) 0 0
\(156\) 8.60376 0.688852
\(157\) 7.10674 0.567179 0.283590 0.958946i \(-0.408475\pi\)
0.283590 + 0.958946i \(0.408475\pi\)
\(158\) 2.12881 0.169359
\(159\) 2.50514 0.198671
\(160\) 0 0
\(161\) 1.58331 0.124782
\(162\) −1.11456 −0.0875683
\(163\) −1.80915 −0.141704 −0.0708519 0.997487i \(-0.522572\pi\)
−0.0708519 + 0.997487i \(0.522572\pi\)
\(164\) 18.6753 1.45830
\(165\) 0 0
\(166\) −0.513163 −0.0398292
\(167\) 18.7765 1.45297 0.726484 0.687183i \(-0.241153\pi\)
0.726484 + 0.687183i \(0.241153\pi\)
\(168\) 2.72380 0.210146
\(169\) 0.0773888 0.00595299
\(170\) 0 0
\(171\) 8.41565 0.643561
\(172\) 1.86334 0.142078
\(173\) 22.8361 1.73620 0.868099 0.496391i \(-0.165342\pi\)
0.868099 + 0.496391i \(0.165342\pi\)
\(174\) −0.201326 −0.0152625
\(175\) 0 0
\(176\) 11.7933 0.888956
\(177\) 10.2263 0.768657
\(178\) 3.99962 0.299784
\(179\) 21.0013 1.56971 0.784856 0.619678i \(-0.212737\pi\)
0.784856 + 0.619678i \(0.212737\pi\)
\(180\) 0 0
\(181\) 13.8122 1.02665 0.513326 0.858194i \(-0.328413\pi\)
0.513326 + 0.858194i \(0.328413\pi\)
\(182\) 1.99681 0.148014
\(183\) 19.0936 1.41144
\(184\) −1.51393 −0.111608
\(185\) 0 0
\(186\) 4.76530 0.349409
\(187\) −18.4370 −1.34824
\(188\) 23.6787 1.72695
\(189\) 8.33368 0.606186
\(190\) 0 0
\(191\) −9.23502 −0.668223 −0.334111 0.942534i \(-0.608436\pi\)
−0.334111 + 0.942534i \(0.608436\pi\)
\(192\) 6.26199 0.451920
\(193\) −22.2674 −1.60285 −0.801423 0.598098i \(-0.795923\pi\)
−0.801423 + 0.598098i \(0.795923\pi\)
\(194\) 1.57885 0.113355
\(195\) 0 0
\(196\) 8.88624 0.634731
\(197\) −15.1088 −1.07646 −0.538229 0.842799i \(-0.680906\pi\)
−0.538229 + 0.842799i \(0.680906\pi\)
\(198\) −1.86685 −0.132671
\(199\) −3.09122 −0.219131 −0.109565 0.993980i \(-0.534946\pi\)
−0.109565 + 0.993980i \(0.534946\pi\)
\(200\) 0 0
\(201\) −5.03817 −0.355365
\(202\) −1.08086 −0.0760492
\(203\) 0.637070 0.0447136
\(204\) −11.8974 −0.832988
\(205\) 0 0
\(206\) 2.92144 0.203546
\(207\) −1.45189 −0.100914
\(208\) 11.5673 0.802051
\(209\) −22.6533 −1.56696
\(210\) 0 0
\(211\) −1.82495 −0.125634 −0.0628172 0.998025i \(-0.520009\pi\)
−0.0628172 + 0.998025i \(0.520009\pi\)
\(212\) 3.65584 0.251084
\(213\) −2.07393 −0.142103
\(214\) 1.79962 0.123020
\(215\) 0 0
\(216\) −7.96847 −0.542186
\(217\) −15.0792 −1.02364
\(218\) 3.75876 0.254575
\(219\) 11.2013 0.756911
\(220\) 0 0
\(221\) −18.0837 −1.21644
\(222\) −4.00950 −0.269100
\(223\) 25.9914 1.74051 0.870257 0.492598i \(-0.163953\pi\)
0.870257 + 0.492598i \(0.163953\pi\)
\(224\) 6.03272 0.403078
\(225\) 0 0
\(226\) −2.75443 −0.183222
\(227\) −21.5629 −1.43118 −0.715588 0.698522i \(-0.753841\pi\)
−0.715588 + 0.698522i \(0.753841\pi\)
\(228\) −14.6183 −0.968119
\(229\) −13.9807 −0.923868 −0.461934 0.886914i \(-0.652844\pi\)
−0.461934 + 0.886914i \(0.652844\pi\)
\(230\) 0 0
\(231\) −7.03154 −0.462641
\(232\) −0.609152 −0.0399928
\(233\) 9.27410 0.607566 0.303783 0.952741i \(-0.401750\pi\)
0.303783 + 0.952741i \(0.401750\pi\)
\(234\) −1.83108 −0.119701
\(235\) 0 0
\(236\) 14.9236 0.971445
\(237\) −7.35269 −0.477608
\(238\) −2.76123 −0.178984
\(239\) 19.8255 1.28240 0.641202 0.767372i \(-0.278436\pi\)
0.641202 + 0.767372i \(0.278436\pi\)
\(240\) 0 0
\(241\) 6.12193 0.394348 0.197174 0.980368i \(-0.436824\pi\)
0.197174 + 0.980368i \(0.436824\pi\)
\(242\) 0.958717 0.0616287
\(243\) −12.8886 −0.826801
\(244\) 27.8639 1.78380
\(245\) 0 0
\(246\) 4.73085 0.301628
\(247\) −22.2192 −1.41378
\(248\) 14.4184 0.915568
\(249\) 1.77242 0.112322
\(250\) 0 0
\(251\) −4.80213 −0.303108 −0.151554 0.988449i \(-0.548428\pi\)
−0.151554 + 0.988449i \(0.548428\pi\)
\(252\) 3.81207 0.240138
\(253\) 3.90822 0.245708
\(254\) −2.67668 −0.167950
\(255\) 0 0
\(256\) 6.15214 0.384509
\(257\) −26.2603 −1.63807 −0.819037 0.573740i \(-0.805492\pi\)
−0.819037 + 0.573740i \(0.805492\pi\)
\(258\) 0.472023 0.0293868
\(259\) 12.6876 0.788367
\(260\) 0 0
\(261\) −0.584192 −0.0361606
\(262\) −0.963320 −0.0595141
\(263\) 15.0015 0.925032 0.462516 0.886611i \(-0.346947\pi\)
0.462516 + 0.886611i \(0.346947\pi\)
\(264\) 6.72340 0.413797
\(265\) 0 0
\(266\) −3.39270 −0.208020
\(267\) −13.8143 −0.845420
\(268\) −7.35238 −0.449118
\(269\) −14.5177 −0.885158 −0.442579 0.896729i \(-0.645936\pi\)
−0.442579 + 0.896729i \(0.645936\pi\)
\(270\) 0 0
\(271\) 15.7125 0.954465 0.477233 0.878777i \(-0.341640\pi\)
0.477233 + 0.878777i \(0.341640\pi\)
\(272\) −15.9955 −0.969872
\(273\) −6.89680 −0.417413
\(274\) −7.52765 −0.454762
\(275\) 0 0
\(276\) 2.52199 0.151806
\(277\) 1.55966 0.0937108 0.0468554 0.998902i \(-0.485080\pi\)
0.0468554 + 0.998902i \(0.485080\pi\)
\(278\) −3.66351 −0.219723
\(279\) 13.8276 0.827837
\(280\) 0 0
\(281\) −15.8939 −0.948153 −0.474077 0.880484i \(-0.657218\pi\)
−0.474077 + 0.880484i \(0.657218\pi\)
\(282\) 5.99832 0.357195
\(283\) −2.62323 −0.155935 −0.0779675 0.996956i \(-0.524843\pi\)
−0.0779675 + 0.996956i \(0.524843\pi\)
\(284\) −3.02655 −0.179593
\(285\) 0 0
\(286\) 4.92891 0.291452
\(287\) −14.9702 −0.883662
\(288\) −5.53199 −0.325976
\(289\) 8.00644 0.470967
\(290\) 0 0
\(291\) −5.45319 −0.319672
\(292\) 16.3464 0.956600
\(293\) −8.11264 −0.473946 −0.236973 0.971516i \(-0.576155\pi\)
−0.236973 + 0.971516i \(0.576155\pi\)
\(294\) 2.25107 0.131285
\(295\) 0 0
\(296\) −12.1316 −0.705132
\(297\) 20.5707 1.19363
\(298\) −2.08036 −0.120512
\(299\) 3.83333 0.221687
\(300\) 0 0
\(301\) −1.49366 −0.0860929
\(302\) −0.738768 −0.0425113
\(303\) 3.73319 0.214466
\(304\) −19.6536 −1.12721
\(305\) 0 0
\(306\) 2.53205 0.144747
\(307\) 24.0736 1.37395 0.686975 0.726681i \(-0.258938\pi\)
0.686975 + 0.726681i \(0.258938\pi\)
\(308\) −10.2614 −0.584696
\(309\) −10.0904 −0.574021
\(310\) 0 0
\(311\) 6.39803 0.362799 0.181400 0.983409i \(-0.441937\pi\)
0.181400 + 0.983409i \(0.441937\pi\)
\(312\) 6.59456 0.373343
\(313\) −19.6040 −1.10808 −0.554041 0.832489i \(-0.686915\pi\)
−0.554041 + 0.832489i \(0.686915\pi\)
\(314\) 2.62722 0.148263
\(315\) 0 0
\(316\) −10.7300 −0.603611
\(317\) 5.68164 0.319113 0.159556 0.987189i \(-0.448994\pi\)
0.159556 + 0.987189i \(0.448994\pi\)
\(318\) 0.926103 0.0519333
\(319\) 1.57253 0.0880450
\(320\) 0 0
\(321\) −6.21572 −0.346928
\(322\) 0.585319 0.0326185
\(323\) 30.7252 1.70959
\(324\) 5.61784 0.312102
\(325\) 0 0
\(326\) −0.668808 −0.0370418
\(327\) −12.9824 −0.717927
\(328\) 14.3142 0.790367
\(329\) −18.9809 −1.04645
\(330\) 0 0
\(331\) 11.2711 0.619513 0.309756 0.950816i \(-0.399752\pi\)
0.309756 + 0.950816i \(0.399752\pi\)
\(332\) 2.58655 0.141955
\(333\) −11.6345 −0.637565
\(334\) 6.94130 0.379811
\(335\) 0 0
\(336\) −6.10042 −0.332805
\(337\) 16.7561 0.912765 0.456383 0.889784i \(-0.349145\pi\)
0.456383 + 0.889784i \(0.349145\pi\)
\(338\) 0.0286091 0.00155613
\(339\) 9.51355 0.516705
\(340\) 0 0
\(341\) −37.2213 −2.01564
\(342\) 3.11110 0.168229
\(343\) −17.5788 −0.949167
\(344\) 1.42820 0.0770034
\(345\) 0 0
\(346\) 8.44207 0.453848
\(347\) −34.1337 −1.83239 −0.916197 0.400729i \(-0.868757\pi\)
−0.916197 + 0.400729i \(0.868757\pi\)
\(348\) 1.01476 0.0543970
\(349\) 34.4994 1.84671 0.923354 0.383949i \(-0.125436\pi\)
0.923354 + 0.383949i \(0.125436\pi\)
\(350\) 0 0
\(351\) 20.1765 1.07694
\(352\) 14.8911 0.793697
\(353\) −6.40545 −0.340927 −0.170464 0.985364i \(-0.554527\pi\)
−0.170464 + 0.985364i \(0.554527\pi\)
\(354\) 3.78047 0.200930
\(355\) 0 0
\(356\) −20.1597 −1.06846
\(357\) 9.53702 0.504753
\(358\) 7.76377 0.410328
\(359\) 15.4742 0.816698 0.408349 0.912826i \(-0.366105\pi\)
0.408349 + 0.912826i \(0.366105\pi\)
\(360\) 0 0
\(361\) 18.7517 0.986934
\(362\) 5.10609 0.268370
\(363\) −3.31132 −0.173799
\(364\) −10.0647 −0.527536
\(365\) 0 0
\(366\) 7.05852 0.368955
\(367\) −13.4623 −0.702727 −0.351363 0.936239i \(-0.614282\pi\)
−0.351363 + 0.936239i \(0.614282\pi\)
\(368\) 3.39069 0.176752
\(369\) 13.7276 0.714633
\(370\) 0 0
\(371\) −2.93053 −0.152146
\(372\) −24.0190 −1.24533
\(373\) 16.6957 0.864471 0.432235 0.901761i \(-0.357725\pi\)
0.432235 + 0.901761i \(0.357725\pi\)
\(374\) −6.81579 −0.352436
\(375\) 0 0
\(376\) 18.1491 0.935969
\(377\) 1.54240 0.0794377
\(378\) 3.08080 0.158459
\(379\) 19.2363 0.988102 0.494051 0.869433i \(-0.335516\pi\)
0.494051 + 0.869433i \(0.335516\pi\)
\(380\) 0 0
\(381\) 9.24500 0.473636
\(382\) −3.41401 −0.174676
\(383\) −12.3023 −0.628618 −0.314309 0.949321i \(-0.601773\pi\)
−0.314309 + 0.949321i \(0.601773\pi\)
\(384\) 12.6290 0.644470
\(385\) 0 0
\(386\) −8.23184 −0.418990
\(387\) 1.36968 0.0696248
\(388\) −7.95803 −0.404008
\(389\) −38.1174 −1.93263 −0.966315 0.257363i \(-0.917146\pi\)
−0.966315 + 0.257363i \(0.917146\pi\)
\(390\) 0 0
\(391\) −5.30080 −0.268073
\(392\) 6.81107 0.344011
\(393\) 3.32721 0.167836
\(394\) −5.58543 −0.281390
\(395\) 0 0
\(396\) 9.40966 0.472853
\(397\) 11.9741 0.600963 0.300482 0.953788i \(-0.402853\pi\)
0.300482 + 0.953788i \(0.402853\pi\)
\(398\) −1.14276 −0.0572815
\(399\) 11.7181 0.586636
\(400\) 0 0
\(401\) 5.28174 0.263757 0.131879 0.991266i \(-0.457899\pi\)
0.131879 + 0.991266i \(0.457899\pi\)
\(402\) −1.86251 −0.0928938
\(403\) −36.5080 −1.81859
\(404\) 5.44798 0.271047
\(405\) 0 0
\(406\) 0.235512 0.0116883
\(407\) 31.3178 1.55237
\(408\) −9.11908 −0.451462
\(409\) −13.4050 −0.662835 −0.331418 0.943484i \(-0.607527\pi\)
−0.331418 + 0.943484i \(0.607527\pi\)
\(410\) 0 0
\(411\) 25.9998 1.28247
\(412\) −14.7252 −0.725459
\(413\) −11.9628 −0.588652
\(414\) −0.536736 −0.0263792
\(415\) 0 0
\(416\) 14.6057 0.716104
\(417\) 12.6534 0.619641
\(418\) −8.37449 −0.409610
\(419\) −24.4193 −1.19296 −0.596480 0.802628i \(-0.703435\pi\)
−0.596480 + 0.802628i \(0.703435\pi\)
\(420\) 0 0
\(421\) −20.1650 −0.982782 −0.491391 0.870939i \(-0.663511\pi\)
−0.491391 + 0.870939i \(0.663511\pi\)
\(422\) −0.674647 −0.0328413
\(423\) 17.4055 0.846283
\(424\) 2.80211 0.136083
\(425\) 0 0
\(426\) −0.766690 −0.0371462
\(427\) −22.3358 −1.08090
\(428\) −9.07081 −0.438454
\(429\) −17.0240 −0.821924
\(430\) 0 0
\(431\) 36.3117 1.74907 0.874536 0.484961i \(-0.161166\pi\)
0.874536 + 0.484961i \(0.161166\pi\)
\(432\) 17.8467 0.858652
\(433\) 3.88729 0.186811 0.0934056 0.995628i \(-0.470225\pi\)
0.0934056 + 0.995628i \(0.470225\pi\)
\(434\) −5.57448 −0.267584
\(435\) 0 0
\(436\) −18.9456 −0.907331
\(437\) −6.51304 −0.311561
\(438\) 4.14089 0.197859
\(439\) −0.0901999 −0.00430500 −0.00215250 0.999998i \(-0.500685\pi\)
−0.00215250 + 0.999998i \(0.500685\pi\)
\(440\) 0 0
\(441\) 6.53199 0.311047
\(442\) −6.68518 −0.317981
\(443\) −34.7294 −1.65004 −0.825021 0.565102i \(-0.808837\pi\)
−0.825021 + 0.565102i \(0.808837\pi\)
\(444\) 20.2095 0.959100
\(445\) 0 0
\(446\) 9.60852 0.454976
\(447\) 7.18535 0.339856
\(448\) −7.32532 −0.346089
\(449\) −17.5143 −0.826549 −0.413275 0.910606i \(-0.635615\pi\)
−0.413275 + 0.910606i \(0.635615\pi\)
\(450\) 0 0
\(451\) −36.9522 −1.74001
\(452\) 13.8834 0.653023
\(453\) 2.55163 0.119886
\(454\) −7.97136 −0.374115
\(455\) 0 0
\(456\) −11.2045 −0.524700
\(457\) 16.2351 0.759448 0.379724 0.925100i \(-0.376019\pi\)
0.379724 + 0.925100i \(0.376019\pi\)
\(458\) −5.16838 −0.241502
\(459\) −27.9005 −1.30228
\(460\) 0 0
\(461\) −39.2284 −1.82705 −0.913524 0.406785i \(-0.866650\pi\)
−0.913524 + 0.406785i \(0.866650\pi\)
\(462\) −2.59942 −0.120936
\(463\) 32.1679 1.49497 0.747486 0.664278i \(-0.231261\pi\)
0.747486 + 0.664278i \(0.231261\pi\)
\(464\) 1.36430 0.0633360
\(465\) 0 0
\(466\) 3.42845 0.158820
\(467\) 12.5648 0.581429 0.290715 0.956810i \(-0.406107\pi\)
0.290715 + 0.956810i \(0.406107\pi\)
\(468\) 9.22935 0.426627
\(469\) 5.89369 0.272145
\(470\) 0 0
\(471\) −9.07416 −0.418115
\(472\) 11.4386 0.526503
\(473\) −3.68692 −0.169525
\(474\) −2.71814 −0.124848
\(475\) 0 0
\(476\) 13.9177 0.637917
\(477\) 2.68730 0.123043
\(478\) 7.32909 0.335225
\(479\) −11.1129 −0.507763 −0.253882 0.967235i \(-0.581707\pi\)
−0.253882 + 0.967235i \(0.581707\pi\)
\(480\) 0 0
\(481\) 30.7177 1.40060
\(482\) 2.26316 0.103084
\(483\) −2.02163 −0.0919875
\(484\) −4.83232 −0.219651
\(485\) 0 0
\(486\) −4.76465 −0.216129
\(487\) 26.5032 1.20097 0.600487 0.799634i \(-0.294973\pi\)
0.600487 + 0.799634i \(0.294973\pi\)
\(488\) 21.3570 0.966785
\(489\) 2.31000 0.104462
\(490\) 0 0
\(491\) 29.7631 1.34319 0.671595 0.740919i \(-0.265610\pi\)
0.671595 + 0.740919i \(0.265610\pi\)
\(492\) −23.8454 −1.07503
\(493\) −2.13286 −0.0960592
\(494\) −8.21401 −0.369566
\(495\) 0 0
\(496\) −32.2924 −1.44997
\(497\) 2.42609 0.108825
\(498\) 0.655227 0.0293614
\(499\) 10.2804 0.460215 0.230108 0.973165i \(-0.426092\pi\)
0.230108 + 0.973165i \(0.426092\pi\)
\(500\) 0 0
\(501\) −23.9746 −1.07111
\(502\) −1.77525 −0.0792334
\(503\) 16.6895 0.744149 0.372075 0.928203i \(-0.378647\pi\)
0.372075 + 0.928203i \(0.378647\pi\)
\(504\) 2.92186 0.130150
\(505\) 0 0
\(506\) 1.44479 0.0642288
\(507\) −0.0988131 −0.00438845
\(508\) 13.4915 0.598591
\(509\) −29.5802 −1.31112 −0.655559 0.755144i \(-0.727567\pi\)
−0.655559 + 0.755144i \(0.727567\pi\)
\(510\) 0 0
\(511\) −13.1033 −0.579656
\(512\) 22.0559 0.974744
\(513\) −34.2811 −1.51355
\(514\) −9.70793 −0.428198
\(515\) 0 0
\(516\) −2.37918 −0.104738
\(517\) −46.8522 −2.06056
\(518\) 4.69034 0.206082
\(519\) −29.1581 −1.27990
\(520\) 0 0
\(521\) −36.7760 −1.61119 −0.805593 0.592469i \(-0.798153\pi\)
−0.805593 + 0.592469i \(0.798153\pi\)
\(522\) −0.215964 −0.00945251
\(523\) 2.28308 0.0998322 0.0499161 0.998753i \(-0.484105\pi\)
0.0499161 + 0.998753i \(0.484105\pi\)
\(524\) 4.85551 0.212114
\(525\) 0 0
\(526\) 5.54576 0.241807
\(527\) 50.4840 2.19912
\(528\) −15.0582 −0.655324
\(529\) −21.8764 −0.951146
\(530\) 0 0
\(531\) 10.9699 0.476053
\(532\) 17.1006 0.741403
\(533\) −36.2441 −1.56991
\(534\) −5.10687 −0.220996
\(535\) 0 0
\(536\) −5.63541 −0.243413
\(537\) −26.8153 −1.15717
\(538\) −5.36690 −0.231383
\(539\) −17.5829 −0.757348
\(540\) 0 0
\(541\) −7.42128 −0.319066 −0.159533 0.987193i \(-0.550999\pi\)
−0.159533 + 0.987193i \(0.550999\pi\)
\(542\) 5.80860 0.249501
\(543\) −17.6359 −0.756831
\(544\) −20.1971 −0.865942
\(545\) 0 0
\(546\) −2.54961 −0.109113
\(547\) 9.83182 0.420378 0.210189 0.977661i \(-0.432592\pi\)
0.210189 + 0.977661i \(0.432592\pi\)
\(548\) 37.9424 1.62082
\(549\) 20.4819 0.874146
\(550\) 0 0
\(551\) −2.62063 −0.111642
\(552\) 1.93304 0.0822756
\(553\) 8.60122 0.365761
\(554\) 0.576575 0.0244963
\(555\) 0 0
\(556\) 18.4656 0.783115
\(557\) 15.2177 0.644794 0.322397 0.946605i \(-0.395511\pi\)
0.322397 + 0.946605i \(0.395511\pi\)
\(558\) 5.11179 0.216399
\(559\) −3.61627 −0.152952
\(560\) 0 0
\(561\) 23.5411 0.993904
\(562\) −5.87568 −0.247851
\(563\) 31.8616 1.34281 0.671404 0.741092i \(-0.265692\pi\)
0.671404 + 0.741092i \(0.265692\pi\)
\(564\) −30.2339 −1.27308
\(565\) 0 0
\(566\) −0.969758 −0.0407620
\(567\) −4.50328 −0.189120
\(568\) −2.31977 −0.0973356
\(569\) −8.07208 −0.338399 −0.169200 0.985582i \(-0.554118\pi\)
−0.169200 + 0.985582i \(0.554118\pi\)
\(570\) 0 0
\(571\) −31.1724 −1.30452 −0.652262 0.757993i \(-0.726180\pi\)
−0.652262 + 0.757993i \(0.726180\pi\)
\(572\) −24.8436 −1.03877
\(573\) 11.7916 0.492603
\(574\) −5.53419 −0.230992
\(575\) 0 0
\(576\) 6.71731 0.279888
\(577\) 39.1328 1.62912 0.814560 0.580079i \(-0.196978\pi\)
0.814560 + 0.580079i \(0.196978\pi\)
\(578\) 2.95982 0.123112
\(579\) 28.4320 1.18159
\(580\) 0 0
\(581\) −2.07338 −0.0860185
\(582\) −2.01594 −0.0835633
\(583\) −7.23369 −0.299589
\(584\) 12.5291 0.518457
\(585\) 0 0
\(586\) −2.99908 −0.123891
\(587\) −15.7507 −0.650100 −0.325050 0.945697i \(-0.605381\pi\)
−0.325050 + 0.945697i \(0.605381\pi\)
\(588\) −11.3463 −0.467913
\(589\) 62.0292 2.55587
\(590\) 0 0
\(591\) 19.2915 0.793548
\(592\) 27.1707 1.11671
\(593\) 1.51862 0.0623624 0.0311812 0.999514i \(-0.490073\pi\)
0.0311812 + 0.999514i \(0.490073\pi\)
\(594\) 7.60459 0.312020
\(595\) 0 0
\(596\) 10.4858 0.429517
\(597\) 3.94699 0.161540
\(598\) 1.41711 0.0579498
\(599\) −12.9076 −0.527390 −0.263695 0.964606i \(-0.584941\pi\)
−0.263695 + 0.964606i \(0.584941\pi\)
\(600\) 0 0
\(601\) 6.34026 0.258624 0.129312 0.991604i \(-0.458723\pi\)
0.129312 + 0.991604i \(0.458723\pi\)
\(602\) −0.552175 −0.0225050
\(603\) −5.40451 −0.220089
\(604\) 3.72369 0.151515
\(605\) 0 0
\(606\) 1.38009 0.0560623
\(607\) −22.4914 −0.912897 −0.456449 0.889750i \(-0.650879\pi\)
−0.456449 + 0.889750i \(0.650879\pi\)
\(608\) −24.8160 −1.00642
\(609\) −0.813437 −0.0329621
\(610\) 0 0
\(611\) −45.9544 −1.85912
\(612\) −12.7625 −0.515894
\(613\) 12.9445 0.522824 0.261412 0.965227i \(-0.415812\pi\)
0.261412 + 0.965227i \(0.415812\pi\)
\(614\) 8.89952 0.359156
\(615\) 0 0
\(616\) −7.86508 −0.316893
\(617\) −4.50654 −0.181426 −0.0907132 0.995877i \(-0.528915\pi\)
−0.0907132 + 0.995877i \(0.528915\pi\)
\(618\) −3.73021 −0.150051
\(619\) 22.9420 0.922115 0.461058 0.887370i \(-0.347470\pi\)
0.461058 + 0.887370i \(0.347470\pi\)
\(620\) 0 0
\(621\) 5.91427 0.237332
\(622\) 2.36523 0.0948370
\(623\) 16.1600 0.647438
\(624\) −14.7696 −0.591259
\(625\) 0 0
\(626\) −7.24720 −0.289657
\(627\) 28.9247 1.15514
\(628\) −13.2422 −0.528423
\(629\) −42.4770 −1.69367
\(630\) 0 0
\(631\) 25.5001 1.01514 0.507571 0.861610i \(-0.330543\pi\)
0.507571 + 0.861610i \(0.330543\pi\)
\(632\) −8.22429 −0.327145
\(633\) 2.33016 0.0926157
\(634\) 2.10039 0.0834172
\(635\) 0 0
\(636\) −4.66793 −0.185095
\(637\) −17.2459 −0.683309
\(638\) 0.581335 0.0230153
\(639\) −2.22472 −0.0880087
\(640\) 0 0
\(641\) −41.6768 −1.64613 −0.823067 0.567944i \(-0.807739\pi\)
−0.823067 + 0.567944i \(0.807739\pi\)
\(642\) −2.29783 −0.0906881
\(643\) 12.7866 0.504253 0.252127 0.967694i \(-0.418870\pi\)
0.252127 + 0.967694i \(0.418870\pi\)
\(644\) −2.95024 −0.116256
\(645\) 0 0
\(646\) 11.3585 0.446894
\(647\) 17.9295 0.704881 0.352440 0.935834i \(-0.385352\pi\)
0.352440 + 0.935834i \(0.385352\pi\)
\(648\) 4.30593 0.169153
\(649\) −29.5288 −1.15911
\(650\) 0 0
\(651\) 19.2537 0.754613
\(652\) 3.37106 0.132021
\(653\) −19.8948 −0.778542 −0.389271 0.921123i \(-0.627273\pi\)
−0.389271 + 0.921123i \(0.627273\pi\)
\(654\) −4.79933 −0.187669
\(655\) 0 0
\(656\) −32.0590 −1.25169
\(657\) 12.0157 0.468778
\(658\) −7.01687 −0.273546
\(659\) 15.5607 0.606158 0.303079 0.952965i \(-0.401985\pi\)
0.303079 + 0.952965i \(0.401985\pi\)
\(660\) 0 0
\(661\) −3.83696 −0.149240 −0.0746202 0.997212i \(-0.523774\pi\)
−0.0746202 + 0.997212i \(0.523774\pi\)
\(662\) 4.16669 0.161943
\(663\) 23.0899 0.896739
\(664\) 1.98252 0.0769368
\(665\) 0 0
\(666\) −4.30104 −0.166662
\(667\) 0.452118 0.0175061
\(668\) −34.9869 −1.35368
\(669\) −33.1869 −1.28308
\(670\) 0 0
\(671\) −55.1333 −2.12840
\(672\) −7.70281 −0.297143
\(673\) 10.5613 0.407109 0.203555 0.979064i \(-0.434751\pi\)
0.203555 + 0.979064i \(0.434751\pi\)
\(674\) 6.19442 0.238600
\(675\) 0 0
\(676\) −0.144201 −0.00554621
\(677\) 12.6405 0.485814 0.242907 0.970050i \(-0.421899\pi\)
0.242907 + 0.970050i \(0.421899\pi\)
\(678\) 3.51697 0.135068
\(679\) 6.37918 0.244811
\(680\) 0 0
\(681\) 27.5323 1.05504
\(682\) −13.7600 −0.526896
\(683\) 19.6198 0.750732 0.375366 0.926877i \(-0.377517\pi\)
0.375366 + 0.926877i \(0.377517\pi\)
\(684\) −15.6812 −0.599585
\(685\) 0 0
\(686\) −6.49855 −0.248116
\(687\) 17.8511 0.681061
\(688\) −3.19870 −0.121949
\(689\) −7.09507 −0.270301
\(690\) 0 0
\(691\) 39.6013 1.50651 0.753253 0.657731i \(-0.228484\pi\)
0.753253 + 0.657731i \(0.228484\pi\)
\(692\) −42.5514 −1.61756
\(693\) −7.54281 −0.286528
\(694\) −12.6186 −0.478994
\(695\) 0 0
\(696\) 0.777789 0.0294820
\(697\) 50.1190 1.89839
\(698\) 12.7537 0.482736
\(699\) −11.8415 −0.447888
\(700\) 0 0
\(701\) 25.0682 0.946811 0.473405 0.880845i \(-0.343025\pi\)
0.473405 + 0.880845i \(0.343025\pi\)
\(702\) 7.45887 0.281517
\(703\) −52.1911 −1.96842
\(704\) −18.0817 −0.681480
\(705\) 0 0
\(706\) −2.36797 −0.0891196
\(707\) −4.36712 −0.164242
\(708\) −19.0551 −0.716133
\(709\) −18.6983 −0.702231 −0.351115 0.936332i \(-0.614197\pi\)
−0.351115 + 0.936332i \(0.614197\pi\)
\(710\) 0 0
\(711\) −7.88731 −0.295797
\(712\) −15.4519 −0.579083
\(713\) −10.7015 −0.400773
\(714\) 3.52565 0.131944
\(715\) 0 0
\(716\) −39.1325 −1.46245
\(717\) −25.3140 −0.945367
\(718\) 5.72052 0.213488
\(719\) 39.5258 1.47406 0.737031 0.675859i \(-0.236227\pi\)
0.737031 + 0.675859i \(0.236227\pi\)
\(720\) 0 0
\(721\) 11.8038 0.439596
\(722\) 6.93215 0.257988
\(723\) −7.81672 −0.290707
\(724\) −25.7367 −0.956498
\(725\) 0 0
\(726\) −1.22413 −0.0454317
\(727\) 7.56957 0.280740 0.140370 0.990099i \(-0.455171\pi\)
0.140370 + 0.990099i \(0.455171\pi\)
\(728\) −7.71436 −0.285913
\(729\) 25.5014 0.944497
\(730\) 0 0
\(731\) 5.00064 0.184956
\(732\) −35.5777 −1.31499
\(733\) −7.52401 −0.277906 −0.138953 0.990299i \(-0.544374\pi\)
−0.138953 + 0.990299i \(0.544374\pi\)
\(734\) −4.97675 −0.183695
\(735\) 0 0
\(736\) 4.28132 0.157812
\(737\) 14.5479 0.535879
\(738\) 5.07484 0.186807
\(739\) −45.8820 −1.68780 −0.843898 0.536504i \(-0.819744\pi\)
−0.843898 + 0.536504i \(0.819744\pi\)
\(740\) 0 0
\(741\) 28.3704 1.04221
\(742\) −1.08336 −0.0397714
\(743\) −10.1826 −0.373562 −0.186781 0.982402i \(-0.559806\pi\)
−0.186781 + 0.982402i \(0.559806\pi\)
\(744\) −18.4100 −0.674942
\(745\) 0 0
\(746\) 6.17207 0.225976
\(747\) 1.90129 0.0695646
\(748\) 34.3543 1.25612
\(749\) 7.27119 0.265684
\(750\) 0 0
\(751\) 27.2391 0.993968 0.496984 0.867760i \(-0.334441\pi\)
0.496984 + 0.867760i \(0.334441\pi\)
\(752\) −40.6480 −1.48228
\(753\) 6.13155 0.223446
\(754\) 0.570195 0.0207653
\(755\) 0 0
\(756\) −15.5284 −0.564764
\(757\) 46.8819 1.70395 0.851976 0.523580i \(-0.175404\pi\)
0.851976 + 0.523580i \(0.175404\pi\)
\(758\) 7.11128 0.258293
\(759\) −4.99017 −0.181132
\(760\) 0 0
\(761\) 12.5395 0.454556 0.227278 0.973830i \(-0.427017\pi\)
0.227278 + 0.973830i \(0.427017\pi\)
\(762\) 3.41769 0.123810
\(763\) 15.1869 0.549802
\(764\) 17.2079 0.622562
\(765\) 0 0
\(766\) −4.54792 −0.164323
\(767\) −28.9630 −1.04579
\(768\) −7.85530 −0.283454
\(769\) 36.2542 1.30736 0.653680 0.756771i \(-0.273224\pi\)
0.653680 + 0.756771i \(0.273224\pi\)
\(770\) 0 0
\(771\) 33.5302 1.20756
\(772\) 41.4917 1.49332
\(773\) −41.4306 −1.49016 −0.745078 0.666977i \(-0.767588\pi\)
−0.745078 + 0.666977i \(0.767588\pi\)
\(774\) 0.506344 0.0182002
\(775\) 0 0
\(776\) −6.09963 −0.218964
\(777\) −16.2000 −0.581171
\(778\) −14.0913 −0.505196
\(779\) 61.5808 2.20636
\(780\) 0 0
\(781\) 5.98853 0.214287
\(782\) −1.95960 −0.0700752
\(783\) 2.37970 0.0850436
\(784\) −15.2546 −0.544805
\(785\) 0 0
\(786\) 1.23000 0.0438728
\(787\) −42.8889 −1.52882 −0.764411 0.644729i \(-0.776970\pi\)
−0.764411 + 0.644729i \(0.776970\pi\)
\(788\) 28.1528 1.00290
\(789\) −19.1545 −0.681919
\(790\) 0 0
\(791\) −11.1290 −0.395702
\(792\) 7.21227 0.256277
\(793\) −54.0768 −1.92033
\(794\) 4.42659 0.157094
\(795\) 0 0
\(796\) 5.75998 0.204157
\(797\) 1.50370 0.0532638 0.0266319 0.999645i \(-0.491522\pi\)
0.0266319 + 0.999645i \(0.491522\pi\)
\(798\) 4.33193 0.153349
\(799\) 63.5466 2.24812
\(800\) 0 0
\(801\) −14.8187 −0.523594
\(802\) 1.95256 0.0689471
\(803\) −32.3440 −1.14140
\(804\) 9.38781 0.331083
\(805\) 0 0
\(806\) −13.4963 −0.475387
\(807\) 18.5367 0.652524
\(808\) 4.17574 0.146902
\(809\) 12.9134 0.454012 0.227006 0.973893i \(-0.427106\pi\)
0.227006 + 0.973893i \(0.427106\pi\)
\(810\) 0 0
\(811\) −4.89382 −0.171845 −0.0859227 0.996302i \(-0.527384\pi\)
−0.0859227 + 0.996302i \(0.527384\pi\)
\(812\) −1.18708 −0.0416582
\(813\) −20.0623 −0.703616
\(814\) 11.5776 0.405794
\(815\) 0 0
\(816\) 20.4237 0.714974
\(817\) 6.14424 0.214960
\(818\) −4.95557 −0.173267
\(819\) −7.39827 −0.258517
\(820\) 0 0
\(821\) −4.41317 −0.154021 −0.0770104 0.997030i \(-0.524537\pi\)
−0.0770104 + 0.997030i \(0.524537\pi\)
\(822\) 9.61160 0.335243
\(823\) 17.2753 0.602181 0.301090 0.953596i \(-0.402649\pi\)
0.301090 + 0.953596i \(0.402649\pi\)
\(824\) −11.2865 −0.393184
\(825\) 0 0
\(826\) −4.42242 −0.153876
\(827\) −30.7326 −1.06868 −0.534339 0.845270i \(-0.679439\pi\)
−0.534339 + 0.845270i \(0.679439\pi\)
\(828\) 2.70537 0.0940180
\(829\) 22.8572 0.793864 0.396932 0.917848i \(-0.370075\pi\)
0.396932 + 0.917848i \(0.370075\pi\)
\(830\) 0 0
\(831\) −1.99143 −0.0690821
\(832\) −17.7352 −0.614858
\(833\) 23.8480 0.826285
\(834\) 4.67772 0.161976
\(835\) 0 0
\(836\) 42.2108 1.45989
\(837\) −56.3266 −1.94693
\(838\) −9.02733 −0.311844
\(839\) −4.34672 −0.150065 −0.0750327 0.997181i \(-0.523906\pi\)
−0.0750327 + 0.997181i \(0.523906\pi\)
\(840\) 0 0
\(841\) −28.8181 −0.993727
\(842\) −7.45460 −0.256903
\(843\) 20.2940 0.698963
\(844\) 3.40049 0.117050
\(845\) 0 0
\(846\) 6.43446 0.221221
\(847\) 3.87360 0.133098
\(848\) −6.27580 −0.215512
\(849\) 3.34945 0.114953
\(850\) 0 0
\(851\) 9.00416 0.308659
\(852\) 3.86442 0.132393
\(853\) −15.1263 −0.517913 −0.258957 0.965889i \(-0.583379\pi\)
−0.258957 + 0.965889i \(0.583379\pi\)
\(854\) −8.25711 −0.282552
\(855\) 0 0
\(856\) −6.95255 −0.237633
\(857\) 48.5201 1.65741 0.828707 0.559683i \(-0.189077\pi\)
0.828707 + 0.559683i \(0.189077\pi\)
\(858\) −6.29342 −0.214854
\(859\) 35.1488 1.19926 0.599631 0.800277i \(-0.295314\pi\)
0.599631 + 0.800277i \(0.295314\pi\)
\(860\) 0 0
\(861\) 19.1145 0.651422
\(862\) 13.4237 0.457213
\(863\) −50.2393 −1.71017 −0.855083 0.518492i \(-0.826494\pi\)
−0.855083 + 0.518492i \(0.826494\pi\)
\(864\) 22.5345 0.766640
\(865\) 0 0
\(866\) 1.43705 0.0488331
\(867\) −10.2229 −0.347189
\(868\) 28.0976 0.953695
\(869\) 21.2311 0.720217
\(870\) 0 0
\(871\) 14.2691 0.483491
\(872\) −14.5213 −0.491755
\(873\) −5.84970 −0.197982
\(874\) −2.40774 −0.0814432
\(875\) 0 0
\(876\) −20.8717 −0.705190
\(877\) 19.5625 0.660580 0.330290 0.943880i \(-0.392854\pi\)
0.330290 + 0.943880i \(0.392854\pi\)
\(878\) −0.0333451 −0.00112534
\(879\) 10.3585 0.349385
\(880\) 0 0
\(881\) 20.8014 0.700815 0.350408 0.936597i \(-0.386043\pi\)
0.350408 + 0.936597i \(0.386043\pi\)
\(882\) 2.41475 0.0813089
\(883\) 23.2400 0.782089 0.391045 0.920372i \(-0.372114\pi\)
0.391045 + 0.920372i \(0.372114\pi\)
\(884\) 33.6960 1.13332
\(885\) 0 0
\(886\) −12.8388 −0.431327
\(887\) 10.7445 0.360764 0.180382 0.983597i \(-0.442267\pi\)
0.180382 + 0.983597i \(0.442267\pi\)
\(888\) 15.4901 0.519812
\(889\) −10.8149 −0.362719
\(890\) 0 0
\(891\) −11.1158 −0.372394
\(892\) −48.4308 −1.62158
\(893\) 78.0792 2.61282
\(894\) 2.65628 0.0888394
\(895\) 0 0
\(896\) −14.7735 −0.493547
\(897\) −4.89455 −0.163424
\(898\) −6.47468 −0.216063
\(899\) −4.30590 −0.143610
\(900\) 0 0
\(901\) 9.81120 0.326859
\(902\) −13.6605 −0.454845
\(903\) 1.90716 0.0634663
\(904\) 10.6413 0.353925
\(905\) 0 0
\(906\) 0.943289 0.0313387
\(907\) 2.28878 0.0759976 0.0379988 0.999278i \(-0.487902\pi\)
0.0379988 + 0.999278i \(0.487902\pi\)
\(908\) 40.1788 1.33338
\(909\) 4.00464 0.132825
\(910\) 0 0
\(911\) −18.4981 −0.612871 −0.306435 0.951891i \(-0.599136\pi\)
−0.306435 + 0.951891i \(0.599136\pi\)
\(912\) 25.0945 0.830960
\(913\) −5.11791 −0.169378
\(914\) 6.00181 0.198522
\(915\) 0 0
\(916\) 26.0507 0.860739
\(917\) −3.89220 −0.128532
\(918\) −10.3143 −0.340422
\(919\) −17.9354 −0.591635 −0.295817 0.955245i \(-0.595592\pi\)
−0.295817 + 0.955245i \(0.595592\pi\)
\(920\) 0 0
\(921\) −30.7381 −1.01285
\(922\) −14.5020 −0.477597
\(923\) 5.87378 0.193338
\(924\) 13.1021 0.431028
\(925\) 0 0
\(926\) 11.8919 0.390791
\(927\) −10.8240 −0.355508
\(928\) 1.72266 0.0565490
\(929\) 26.5507 0.871100 0.435550 0.900164i \(-0.356554\pi\)
0.435550 + 0.900164i \(0.356554\pi\)
\(930\) 0 0
\(931\) 29.3019 0.960329
\(932\) −17.2808 −0.566050
\(933\) −8.16926 −0.267450
\(934\) 4.64496 0.151988
\(935\) 0 0
\(936\) 7.07406 0.231223
\(937\) 4.50444 0.147154 0.0735768 0.997290i \(-0.476559\pi\)
0.0735768 + 0.997290i \(0.476559\pi\)
\(938\) 2.17878 0.0711398
\(939\) 25.0311 0.816860
\(940\) 0 0
\(941\) 23.4544 0.764592 0.382296 0.924040i \(-0.375133\pi\)
0.382296 + 0.924040i \(0.375133\pi\)
\(942\) −3.35454 −0.109297
\(943\) −10.6241 −0.345968
\(944\) −25.6186 −0.833815
\(945\) 0 0
\(946\) −1.36298 −0.0443143
\(947\) −43.5192 −1.41418 −0.707092 0.707121i \(-0.749993\pi\)
−0.707092 + 0.707121i \(0.749993\pi\)
\(948\) 13.7005 0.444972
\(949\) −31.7242 −1.02981
\(950\) 0 0
\(951\) −7.25455 −0.235245
\(952\) 10.6676 0.345738
\(953\) −16.4190 −0.531863 −0.265931 0.963992i \(-0.585679\pi\)
−0.265931 + 0.963992i \(0.585679\pi\)
\(954\) 0.993441 0.0321638
\(955\) 0 0
\(956\) −36.9415 −1.19478
\(957\) −2.00787 −0.0649054
\(958\) −4.10824 −0.132731
\(959\) −30.4147 −0.982142
\(960\) 0 0
\(961\) 70.9190 2.28771
\(962\) 11.3557 0.366123
\(963\) −6.66767 −0.214863
\(964\) −11.4072 −0.367402
\(965\) 0 0
\(966\) −0.747358 −0.0240459
\(967\) −24.3957 −0.784513 −0.392257 0.919856i \(-0.628306\pi\)
−0.392257 + 0.919856i \(0.628306\pi\)
\(968\) −3.70385 −0.119046
\(969\) −39.2311 −1.26029
\(970\) 0 0
\(971\) 58.7521 1.88545 0.942723 0.333577i \(-0.108256\pi\)
0.942723 + 0.333577i \(0.108256\pi\)
\(972\) 24.0157 0.770304
\(973\) −14.8021 −0.474532
\(974\) 9.79771 0.313939
\(975\) 0 0
\(976\) −47.8326 −1.53108
\(977\) 16.7134 0.534708 0.267354 0.963598i \(-0.413851\pi\)
0.267354 + 0.963598i \(0.413851\pi\)
\(978\) 0.853961 0.0273066
\(979\) 39.8892 1.27486
\(980\) 0 0
\(981\) −13.9263 −0.444634
\(982\) 11.0028 0.351114
\(983\) 40.7041 1.29826 0.649130 0.760677i \(-0.275133\pi\)
0.649130 + 0.760677i \(0.275133\pi\)
\(984\) −18.2769 −0.582646
\(985\) 0 0
\(986\) −0.788477 −0.0251102
\(987\) 24.2356 0.771427
\(988\) 41.4019 1.31717
\(989\) −1.06002 −0.0337068
\(990\) 0 0
\(991\) −30.2848 −0.962029 −0.481015 0.876713i \(-0.659732\pi\)
−0.481015 + 0.876713i \(0.659732\pi\)
\(992\) −40.7746 −1.29460
\(993\) −14.3913 −0.456695
\(994\) 0.896879 0.0284473
\(995\) 0 0
\(996\) −3.30261 −0.104647
\(997\) −10.3490 −0.327756 −0.163878 0.986481i \(-0.552400\pi\)
−0.163878 + 0.986481i \(0.552400\pi\)
\(998\) 3.80047 0.120302
\(999\) 47.3929 1.49945
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 1075.2.a.n.1.3 5
3.2 odd 2 9675.2.a.cc.1.3 5
5.2 odd 4 1075.2.b.i.474.6 10
5.3 odd 4 1075.2.b.i.474.5 10
5.4 even 2 1075.2.a.o.1.3 yes 5
15.14 odd 2 9675.2.a.cb.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1075.2.a.n.1.3 5 1.1 even 1 trivial
1075.2.a.o.1.3 yes 5 5.4 even 2
1075.2.b.i.474.5 10 5.3 odd 4
1075.2.b.i.474.6 10 5.2 odd 4
9675.2.a.cb.1.3 5 15.14 odd 2
9675.2.a.cc.1.3 5 3.2 odd 2