Properties

Label 435.2.a.j.1.3
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
Analytic rank $1$
Dimension $4$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.13856\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.138564 q^{2} -1.00000 q^{3} -1.98080 q^{4} -1.00000 q^{5} -0.138564 q^{6} +5.07830 q^{7} -0.551597 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.138564 q^{2} -1.00000 q^{3} -1.98080 q^{4} -1.00000 q^{5} -0.138564 q^{6} +5.07830 q^{7} -0.551597 q^{8} +1.00000 q^{9} -0.138564 q^{10} -5.07830 q^{11} +1.98080 q^{12} -3.67096 q^{13} +0.703671 q^{14} +1.00000 q^{15} +3.88517 q^{16} -2.60617 q^{17} +0.138564 q^{18} -6.74926 q^{19} +1.98080 q^{20} -5.07830 q^{21} -0.703671 q^{22} -3.21234 q^{23} +0.551597 q^{24} +1.00000 q^{25} -0.508664 q^{26} -1.00000 q^{27} -10.0591 q^{28} -1.00000 q^{29} +0.138564 q^{30} -0.787665 q^{31} +1.64154 q^{32} +5.07830 q^{33} -0.361122 q^{34} -5.07830 q^{35} -1.98080 q^{36} -5.13021 q^{37} -0.935207 q^{38} +3.67096 q^{39} +0.551597 q^{40} -8.81469 q^{41} -0.703671 q^{42} -2.61968 q^{43} +10.0591 q^{44} -1.00000 q^{45} -0.445115 q^{46} -1.11670 q^{47} -3.88517 q^{48} +18.7892 q^{49} +0.138564 q^{50} +2.60617 q^{51} +7.27144 q^{52} +0.619678 q^{53} -0.138564 q^{54} +5.07830 q^{55} -2.80118 q^{56} +6.74926 q^{57} -0.138564 q^{58} +12.7763 q^{59} -1.98080 q^{60} +8.90587 q^{61} -0.109142 q^{62} +5.07830 q^{63} -7.54288 q^{64} +3.67096 q^{65} +0.703671 q^{66} -0.524047 q^{67} +5.16230 q^{68} +3.21234 q^{69} -0.703671 q^{70} +0.195007 q^{71} -0.551597 q^{72} +1.13021 q^{73} -0.710864 q^{74} -1.00000 q^{75} +13.3689 q^{76} -25.7892 q^{77} +0.508664 q^{78} +15.3035 q^{79} -3.88517 q^{80} +1.00000 q^{81} -1.22140 q^{82} -18.1182 q^{83} +10.0591 q^{84} +2.60617 q^{85} -0.362994 q^{86} +1.00000 q^{87} +2.80118 q^{88} -15.5942 q^{89} -0.138564 q^{90} -18.6423 q^{91} +6.36299 q^{92} +0.787665 q^{93} -0.154735 q^{94} +6.74926 q^{95} -1.64154 q^{96} -12.6671 q^{97} +2.60351 q^{98} -5.07830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 2 q^{11} - 5 q^{12} - 8 q^{13} - 3 q^{14} + 4 q^{15} + 11 q^{16} - 10 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} - 2 q^{21} + 3 q^{22} - 12 q^{23} + 12 q^{24} + 4 q^{25} - 7 q^{26} - 4 q^{27} - 9 q^{28} - 4 q^{29} - 3 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 5 q^{36} - 16 q^{37} - 10 q^{38} + 8 q^{39} + 12 q^{40} - 12 q^{41} + 3 q^{42} + 2 q^{43} + 9 q^{44} - 4 q^{45} - 8 q^{46} - 12 q^{47} - 11 q^{48} + 6 q^{49} - 3 q^{50} + 10 q^{51} - 3 q^{52} - 10 q^{53} + 3 q^{54} + 2 q^{55} + 2 q^{57} + 3 q^{58} + 2 q^{59} + 5 q^{60} - 26 q^{61} + 20 q^{62} + 2 q^{63} + 34 q^{64} + 8 q^{65} - 3 q^{66} + 2 q^{67} + 9 q^{68} + 12 q^{69} + 3 q^{70} - 10 q^{71} - 12 q^{72} + 48 q^{74} - 4 q^{75} + 16 q^{76} - 34 q^{77} + 7 q^{78} + 22 q^{79} - 11 q^{80} + 4 q^{81} + 38 q^{82} - 10 q^{83} + 9 q^{84} + 10 q^{85} - 4 q^{86} + 4 q^{87} - 4 q^{89} + 3 q^{90} - 8 q^{91} + 28 q^{92} + 4 q^{93} + 39 q^{94} + 2 q^{95} + 17 q^{96} - 22 q^{97} + 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.138564 0.0979797 0.0489899 0.998799i \(-0.484400\pi\)
0.0489899 + 0.998799i \(0.484400\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.98080 −0.990400
\(5\) −1.00000 −0.447214
\(6\) −0.138564 −0.0565686
\(7\) 5.07830 1.91942 0.959709 0.280995i \(-0.0906645\pi\)
0.959709 + 0.280995i \(0.0906645\pi\)
\(8\) −0.551597 −0.195019
\(9\) 1.00000 0.333333
\(10\) −0.138564 −0.0438179
\(11\) −5.07830 −1.53117 −0.765583 0.643337i \(-0.777549\pi\)
−0.765583 + 0.643337i \(0.777549\pi\)
\(12\) 1.98080 0.571808
\(13\) −3.67096 −1.01814 −0.509071 0.860725i \(-0.670011\pi\)
−0.509071 + 0.860725i \(0.670011\pi\)
\(14\) 0.703671 0.188064
\(15\) 1.00000 0.258199
\(16\) 3.88517 0.971292
\(17\) −2.60617 −0.632089 −0.316044 0.948744i \(-0.602355\pi\)
−0.316044 + 0.948744i \(0.602355\pi\)
\(18\) 0.138564 0.0326599
\(19\) −6.74926 −1.54839 −0.774194 0.632949i \(-0.781844\pi\)
−0.774194 + 0.632949i \(0.781844\pi\)
\(20\) 1.98080 0.442920
\(21\) −5.07830 −1.10818
\(22\) −0.703671 −0.150023
\(23\) −3.21234 −0.669818 −0.334909 0.942250i \(-0.608706\pi\)
−0.334909 + 0.942250i \(0.608706\pi\)
\(24\) 0.551597 0.112594
\(25\) 1.00000 0.200000
\(26\) −0.508664 −0.0997572
\(27\) −1.00000 −0.192450
\(28\) −10.0591 −1.90099
\(29\) −1.00000 −0.185695
\(30\) 0.138564 0.0252983
\(31\) −0.787665 −0.141469 −0.0707344 0.997495i \(-0.522534\pi\)
−0.0707344 + 0.997495i \(0.522534\pi\)
\(32\) 1.64154 0.290186
\(33\) 5.07830 0.884019
\(34\) −0.361122 −0.0619319
\(35\) −5.07830 −0.858390
\(36\) −1.98080 −0.330133
\(37\) −5.13021 −0.843402 −0.421701 0.906735i \(-0.638567\pi\)
−0.421701 + 0.906735i \(0.638567\pi\)
\(38\) −0.935207 −0.151711
\(39\) 3.67096 0.587824
\(40\) 0.551597 0.0872151
\(41\) −8.81469 −1.37662 −0.688311 0.725415i \(-0.741648\pi\)
−0.688311 + 0.725415i \(0.741648\pi\)
\(42\) −0.703671 −0.108579
\(43\) −2.61968 −0.399497 −0.199749 0.979847i \(-0.564013\pi\)
−0.199749 + 0.979847i \(0.564013\pi\)
\(44\) 10.0591 1.51647
\(45\) −1.00000 −0.149071
\(46\) −0.445115 −0.0656286
\(47\) −1.11670 −0.162888 −0.0814440 0.996678i \(-0.525953\pi\)
−0.0814440 + 0.996678i \(0.525953\pi\)
\(48\) −3.88517 −0.560776
\(49\) 18.7892 2.68417
\(50\) 0.138564 0.0195959
\(51\) 2.60617 0.364936
\(52\) 7.27144 1.00837
\(53\) 0.619678 0.0851194 0.0425597 0.999094i \(-0.486449\pi\)
0.0425597 + 0.999094i \(0.486449\pi\)
\(54\) −0.138564 −0.0188562
\(55\) 5.07830 0.684758
\(56\) −2.80118 −0.374323
\(57\) 6.74926 0.893962
\(58\) −0.138564 −0.0181944
\(59\) 12.7763 1.66333 0.831665 0.555277i \(-0.187388\pi\)
0.831665 + 0.555277i \(0.187388\pi\)
\(60\) −1.98080 −0.255720
\(61\) 8.90587 1.14028 0.570140 0.821548i \(-0.306889\pi\)
0.570140 + 0.821548i \(0.306889\pi\)
\(62\) −0.109142 −0.0138611
\(63\) 5.07830 0.639806
\(64\) −7.54288 −0.942860
\(65\) 3.67096 0.455327
\(66\) 0.703671 0.0866160
\(67\) −0.524047 −0.0640225 −0.0320112 0.999488i \(-0.510191\pi\)
−0.0320112 + 0.999488i \(0.510191\pi\)
\(68\) 5.16230 0.626020
\(69\) 3.21234 0.386720
\(70\) −0.703671 −0.0841048
\(71\) 0.195007 0.0231431 0.0115716 0.999933i \(-0.496317\pi\)
0.0115716 + 0.999933i \(0.496317\pi\)
\(72\) −0.551597 −0.0650063
\(73\) 1.13021 0.132282 0.0661408 0.997810i \(-0.478931\pi\)
0.0661408 + 0.997810i \(0.478931\pi\)
\(74\) −0.710864 −0.0826363
\(75\) −1.00000 −0.115470
\(76\) 13.3689 1.53352
\(77\) −25.7892 −2.93895
\(78\) 0.508664 0.0575949
\(79\) 15.3035 1.72178 0.860890 0.508791i \(-0.169907\pi\)
0.860890 + 0.508791i \(0.169907\pi\)
\(80\) −3.88517 −0.434375
\(81\) 1.00000 0.111111
\(82\) −1.22140 −0.134881
\(83\) −18.1182 −1.98873 −0.994366 0.106003i \(-0.966195\pi\)
−0.994366 + 0.106003i \(0.966195\pi\)
\(84\) 10.0591 1.09754
\(85\) 2.60617 0.282679
\(86\) −0.362994 −0.0391426
\(87\) 1.00000 0.107211
\(88\) 2.80118 0.298606
\(89\) −15.5942 −1.65298 −0.826489 0.562953i \(-0.809665\pi\)
−0.826489 + 0.562953i \(0.809665\pi\)
\(90\) −0.138564 −0.0146060
\(91\) −18.6423 −1.95424
\(92\) 6.36299 0.663388
\(93\) 0.787665 0.0816770
\(94\) −0.154735 −0.0159597
\(95\) 6.74926 0.692460
\(96\) −1.64154 −0.167539
\(97\) −12.6671 −1.28615 −0.643077 0.765802i \(-0.722342\pi\)
−0.643077 + 0.765802i \(0.722342\pi\)
\(98\) 2.60351 0.262994
\(99\) −5.07830 −0.510389
\(100\) −1.98080 −0.198080
\(101\) 7.03990 0.700497 0.350248 0.936657i \(-0.386097\pi\)
0.350248 + 0.936657i \(0.386097\pi\)
\(102\) 0.361122 0.0357564
\(103\) 2.65808 0.261908 0.130954 0.991388i \(-0.458196\pi\)
0.130954 + 0.991388i \(0.458196\pi\)
\(104\) 2.02489 0.198557
\(105\) 5.07830 0.495592
\(106\) 0.0858653 0.00833997
\(107\) 4.81469 0.465453 0.232727 0.972542i \(-0.425235\pi\)
0.232727 + 0.972542i \(0.425235\pi\)
\(108\) 1.98080 0.190603
\(109\) 3.86597 0.370293 0.185146 0.982711i \(-0.440724\pi\)
0.185146 + 0.982711i \(0.440724\pi\)
\(110\) 0.703671 0.0670924
\(111\) 5.13021 0.486938
\(112\) 19.7301 1.86432
\(113\) −8.73575 −0.821791 −0.410895 0.911683i \(-0.634784\pi\)
−0.410895 + 0.911683i \(0.634784\pi\)
\(114\) 0.935207 0.0875901
\(115\) 3.21234 0.299552
\(116\) 1.98080 0.183913
\(117\) −3.67096 −0.339380
\(118\) 1.77034 0.162973
\(119\) −13.2349 −1.21324
\(120\) −0.551597 −0.0503537
\(121\) 14.7892 1.34447
\(122\) 1.23404 0.111724
\(123\) 8.81469 0.794793
\(124\) 1.56021 0.140111
\(125\) −1.00000 −0.0894427
\(126\) 0.703671 0.0626880
\(127\) 10.7877 0.957250 0.478625 0.878019i \(-0.341135\pi\)
0.478625 + 0.878019i \(0.341135\pi\)
\(128\) −4.32825 −0.382567
\(129\) 2.61968 0.230650
\(130\) 0.508664 0.0446128
\(131\) 12.9745 1.13359 0.566793 0.823860i \(-0.308184\pi\)
0.566793 + 0.823860i \(0.308184\pi\)
\(132\) −10.0591 −0.875533
\(133\) −34.2748 −2.97200
\(134\) −0.0726141 −0.00627291
\(135\) 1.00000 0.0860663
\(136\) 1.43755 0.123269
\(137\) −4.08212 −0.348759 −0.174380 0.984679i \(-0.555792\pi\)
−0.174380 + 0.984679i \(0.555792\pi\)
\(138\) 0.445115 0.0378907
\(139\) −19.8276 −1.68175 −0.840876 0.541228i \(-0.817960\pi\)
−0.840876 + 0.541228i \(0.817960\pi\)
\(140\) 10.0591 0.850149
\(141\) 1.11670 0.0940434
\(142\) 0.0270211 0.00226756
\(143\) 18.6423 1.55894
\(144\) 3.88517 0.323764
\(145\) 1.00000 0.0830455
\(146\) 0.156607 0.0129609
\(147\) −18.7892 −1.54970
\(148\) 10.1619 0.835305
\(149\) 10.7763 0.882828 0.441414 0.897304i \(-0.354477\pi\)
0.441414 + 0.897304i \(0.354477\pi\)
\(150\) −0.138564 −0.0113137
\(151\) −5.49853 −0.447464 −0.223732 0.974651i \(-0.571824\pi\)
−0.223732 + 0.974651i \(0.571824\pi\)
\(152\) 3.72287 0.301965
\(153\) −2.60617 −0.210696
\(154\) −3.57346 −0.287957
\(155\) 0.787665 0.0632667
\(156\) −7.27144 −0.582181
\(157\) 5.77566 0.460948 0.230474 0.973079i \(-0.425972\pi\)
0.230474 + 0.973079i \(0.425972\pi\)
\(158\) 2.12052 0.168700
\(159\) −0.619678 −0.0491437
\(160\) −1.64154 −0.129775
\(161\) −16.3132 −1.28566
\(162\) 0.138564 0.0108866
\(163\) 7.14691 0.559790 0.279895 0.960031i \(-0.409700\pi\)
0.279895 + 0.960031i \(0.409700\pi\)
\(164\) 17.4601 1.36341
\(165\) −5.07830 −0.395345
\(166\) −2.51054 −0.194855
\(167\) 17.1009 1.32331 0.661653 0.749810i \(-0.269855\pi\)
0.661653 + 0.749810i \(0.269855\pi\)
\(168\) 2.80118 0.216115
\(169\) 0.475953 0.0366118
\(170\) 0.361122 0.0276968
\(171\) −6.74926 −0.516129
\(172\) 5.18906 0.395662
\(173\) 10.2862 0.782045 0.391022 0.920381i \(-0.372121\pi\)
0.391022 + 0.920381i \(0.372121\pi\)
\(174\) 0.138564 0.0105045
\(175\) 5.07830 0.383884
\(176\) −19.7301 −1.48721
\(177\) −12.7763 −0.960324
\(178\) −2.16079 −0.161958
\(179\) −12.1182 −0.905757 −0.452879 0.891572i \(-0.649603\pi\)
−0.452879 + 0.891572i \(0.649603\pi\)
\(180\) 1.98080 0.147640
\(181\) −20.9745 −1.55902 −0.779510 0.626389i \(-0.784532\pi\)
−0.779510 + 0.626389i \(0.784532\pi\)
\(182\) −2.58315 −0.191476
\(183\) −8.90587 −0.658341
\(184\) 1.77191 0.130627
\(185\) 5.13021 0.377181
\(186\) 0.109142 0.00800269
\(187\) 13.2349 0.967833
\(188\) 2.21197 0.161324
\(189\) −5.07830 −0.369392
\(190\) 0.935207 0.0678470
\(191\) 26.2094 1.89645 0.948223 0.317607i \(-0.102879\pi\)
0.948223 + 0.317607i \(0.102879\pi\)
\(192\) 7.54288 0.544360
\(193\) −23.7643 −1.71059 −0.855295 0.518141i \(-0.826624\pi\)
−0.855295 + 0.518141i \(0.826624\pi\)
\(194\) −1.75521 −0.126017
\(195\) −3.67096 −0.262883
\(196\) −37.2176 −2.65840
\(197\) −25.6281 −1.82593 −0.912964 0.408041i \(-0.866212\pi\)
−0.912964 + 0.408041i \(0.866212\pi\)
\(198\) −0.703671 −0.0500077
\(199\) 7.82757 0.554882 0.277441 0.960743i \(-0.410514\pi\)
0.277441 + 0.960743i \(0.410514\pi\)
\(200\) −0.551597 −0.0390038
\(201\) 0.524047 0.0369634
\(202\) 0.975479 0.0686345
\(203\) −5.07830 −0.356427
\(204\) −5.16230 −0.361433
\(205\) 8.81469 0.615644
\(206\) 0.368315 0.0256617
\(207\) −3.21234 −0.223273
\(208\) −14.2623 −0.988913
\(209\) 34.2748 2.37084
\(210\) 0.703671 0.0485579
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −1.22746 −0.0843022
\(213\) −0.195007 −0.0133617
\(214\) 0.667143 0.0456050
\(215\) 2.61968 0.178661
\(216\) 0.551597 0.0375314
\(217\) −4.00000 −0.271538
\(218\) 0.535685 0.0362812
\(219\) −1.13021 −0.0763728
\(220\) −10.0591 −0.678185
\(221\) 9.56714 0.643555
\(222\) 0.710864 0.0477101
\(223\) 7.86597 0.526744 0.263372 0.964694i \(-0.415165\pi\)
0.263372 + 0.964694i \(0.415165\pi\)
\(224\) 8.33623 0.556988
\(225\) 1.00000 0.0666667
\(226\) −1.21046 −0.0805188
\(227\) −0.0654212 −0.00434216 −0.00217108 0.999998i \(-0.500691\pi\)
−0.00217108 + 0.999998i \(0.500691\pi\)
\(228\) −13.3689 −0.885380
\(229\) −3.87041 −0.255764 −0.127882 0.991789i \(-0.540818\pi\)
−0.127882 + 0.991789i \(0.540818\pi\)
\(230\) 0.445115 0.0293500
\(231\) 25.7892 1.69680
\(232\) 0.551597 0.0362141
\(233\) −8.18363 −0.536127 −0.268064 0.963401i \(-0.586384\pi\)
−0.268064 + 0.963401i \(0.586384\pi\)
\(234\) −0.508664 −0.0332524
\(235\) 1.11670 0.0728457
\(236\) −25.3073 −1.64736
\(237\) −15.3035 −0.994071
\(238\) −1.83389 −0.118873
\(239\) −11.2651 −0.728680 −0.364340 0.931266i \(-0.618705\pi\)
−0.364340 + 0.931266i \(0.618705\pi\)
\(240\) 3.88517 0.250787
\(241\) −15.2619 −0.983107 −0.491554 0.870847i \(-0.663571\pi\)
−0.491554 + 0.870847i \(0.663571\pi\)
\(242\) 2.04925 0.131731
\(243\) −1.00000 −0.0641500
\(244\) −17.6408 −1.12933
\(245\) −18.7892 −1.20040
\(246\) 1.22140 0.0778736
\(247\) 24.7763 1.57648
\(248\) 0.434473 0.0275891
\(249\) 18.1182 1.14819
\(250\) −0.138564 −0.00876357
\(251\) 24.9411 1.57427 0.787134 0.616783i \(-0.211564\pi\)
0.787134 + 0.616783i \(0.211564\pi\)
\(252\) −10.0591 −0.633664
\(253\) 16.3132 1.02560
\(254\) 1.49478 0.0937911
\(255\) −2.60617 −0.163205
\(256\) 14.4860 0.905376
\(257\) 1.14691 0.0715425 0.0357713 0.999360i \(-0.488611\pi\)
0.0357713 + 0.999360i \(0.488611\pi\)
\(258\) 0.362994 0.0225990
\(259\) −26.0528 −1.61884
\(260\) −7.27144 −0.450956
\(261\) −1.00000 −0.0618984
\(262\) 1.79780 0.111068
\(263\) 0.685099 0.0422450 0.0211225 0.999777i \(-0.493276\pi\)
0.0211225 + 0.999777i \(0.493276\pi\)
\(264\) −2.80118 −0.172400
\(265\) −0.619678 −0.0380665
\(266\) −4.74926 −0.291196
\(267\) 15.5942 0.954347
\(268\) 1.03803 0.0634079
\(269\) 4.22522 0.257616 0.128808 0.991670i \(-0.458885\pi\)
0.128808 + 0.991670i \(0.458885\pi\)
\(270\) 0.138564 0.00843275
\(271\) −0.814686 −0.0494886 −0.0247443 0.999694i \(-0.507877\pi\)
−0.0247443 + 0.999694i \(0.507877\pi\)
\(272\) −10.1254 −0.613943
\(273\) 18.6423 1.12828
\(274\) −0.565636 −0.0341713
\(275\) −5.07830 −0.306233
\(276\) −6.36299 −0.383007
\(277\) 9.85459 0.592105 0.296052 0.955172i \(-0.404330\pi\)
0.296052 + 0.955172i \(0.404330\pi\)
\(278\) −2.74739 −0.164778
\(279\) −0.787665 −0.0471562
\(280\) 2.80118 0.167402
\(281\) 12.2297 0.729561 0.364780 0.931094i \(-0.381144\pi\)
0.364780 + 0.931094i \(0.381144\pi\)
\(282\) 0.154735 0.00921435
\(283\) 4.23341 0.251650 0.125825 0.992052i \(-0.459842\pi\)
0.125825 + 0.992052i \(0.459842\pi\)
\(284\) −0.386271 −0.0229209
\(285\) −6.74926 −0.399792
\(286\) 2.58315 0.152745
\(287\) −44.7637 −2.64231
\(288\) 1.64154 0.0967286
\(289\) −10.2079 −0.600464
\(290\) 0.138564 0.00813677
\(291\) 12.6671 0.742561
\(292\) −2.23873 −0.131012
\(293\) −13.3170 −0.777989 −0.388995 0.921240i \(-0.627178\pi\)
−0.388995 + 0.921240i \(0.627178\pi\)
\(294\) −2.60351 −0.151840
\(295\) −12.7763 −0.743864
\(296\) 2.82981 0.164479
\(297\) 5.07830 0.294673
\(298\) 1.49321 0.0864992
\(299\) 11.7924 0.681970
\(300\) 1.98080 0.114362
\(301\) −13.3035 −0.766802
\(302\) −0.761900 −0.0438424
\(303\) −7.03990 −0.404432
\(304\) −26.2220 −1.50394
\(305\) −8.90587 −0.509949
\(306\) −0.361122 −0.0206440
\(307\) 14.7379 0.841136 0.420568 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(308\) 51.0832 2.91073
\(309\) −2.65808 −0.151213
\(310\) 0.109142 0.00619886
\(311\) −10.0226 −0.568328 −0.284164 0.958776i \(-0.591716\pi\)
−0.284164 + 0.958776i \(0.591716\pi\)
\(312\) −2.02489 −0.114637
\(313\) 4.08800 0.231067 0.115534 0.993304i \(-0.463142\pi\)
0.115534 + 0.993304i \(0.463142\pi\)
\(314\) 0.800300 0.0451635
\(315\) −5.07830 −0.286130
\(316\) −30.3132 −1.70525
\(317\) −12.7628 −0.716829 −0.358414 0.933563i \(-0.616683\pi\)
−0.358414 + 0.933563i \(0.616683\pi\)
\(318\) −0.0858653 −0.00481508
\(319\) 5.07830 0.284330
\(320\) 7.54288 0.421660
\(321\) −4.81469 −0.268730
\(322\) −2.26043 −0.125969
\(323\) 17.5897 0.978718
\(324\) −1.98080 −0.110044
\(325\) −3.67096 −0.203628
\(326\) 0.990307 0.0548480
\(327\) −3.86597 −0.213789
\(328\) 4.86215 0.268467
\(329\) −5.67096 −0.312650
\(330\) −0.703671 −0.0387358
\(331\) 5.83576 0.320762 0.160381 0.987055i \(-0.448728\pi\)
0.160381 + 0.987055i \(0.448728\pi\)
\(332\) 35.8885 1.96964
\(333\) −5.13021 −0.281134
\(334\) 2.36957 0.129657
\(335\) 0.524047 0.0286317
\(336\) −19.7301 −1.07636
\(337\) 1.95253 0.106361 0.0531807 0.998585i \(-0.483064\pi\)
0.0531807 + 0.998585i \(0.483064\pi\)
\(338\) 0.0659501 0.00358721
\(339\) 8.73575 0.474461
\(340\) −5.16230 −0.279965
\(341\) 4.00000 0.216612
\(342\) −0.935207 −0.0505702
\(343\) 59.8690 3.23262
\(344\) 1.44501 0.0779095
\(345\) −3.21234 −0.172946
\(346\) 1.42530 0.0766245
\(347\) 17.3960 0.933864 0.466932 0.884293i \(-0.345359\pi\)
0.466932 + 0.884293i \(0.345359\pi\)
\(348\) −1.98080 −0.106182
\(349\) −28.7379 −1.53830 −0.769152 0.639066i \(-0.779321\pi\)
−0.769152 + 0.639066i \(0.779321\pi\)
\(350\) 0.703671 0.0376128
\(351\) 3.67096 0.195941
\(352\) −8.33623 −0.444323
\(353\) −29.7590 −1.58391 −0.791955 0.610580i \(-0.790936\pi\)
−0.791955 + 0.610580i \(0.790936\pi\)
\(354\) −1.77034 −0.0940923
\(355\) −0.195007 −0.0103499
\(356\) 30.8889 1.63711
\(357\) 13.2349 0.700466
\(358\) −1.67915 −0.0887459
\(359\) 7.76659 0.409905 0.204953 0.978772i \(-0.434296\pi\)
0.204953 + 0.978772i \(0.434296\pi\)
\(360\) 0.551597 0.0290717
\(361\) 26.5526 1.39750
\(362\) −2.90631 −0.152752
\(363\) −14.7892 −0.776230
\(364\) 36.9266 1.93548
\(365\) −1.13021 −0.0591581
\(366\) −1.23404 −0.0645041
\(367\) −12.8675 −0.671677 −0.335838 0.941920i \(-0.609020\pi\)
−0.335838 + 0.941920i \(0.609020\pi\)
\(368\) −12.4805 −0.650589
\(369\) −8.81469 −0.458874
\(370\) 0.710864 0.0369561
\(371\) 3.14691 0.163380
\(372\) −1.56021 −0.0808929
\(373\) −13.4639 −0.697133 −0.348566 0.937284i \(-0.613331\pi\)
−0.348566 + 0.937284i \(0.613331\pi\)
\(374\) 1.83389 0.0948280
\(375\) 1.00000 0.0516398
\(376\) 0.615970 0.0317662
\(377\) 3.67096 0.189064
\(378\) −0.703671 −0.0361930
\(379\) −9.53318 −0.489687 −0.244843 0.969563i \(-0.578737\pi\)
−0.244843 + 0.969563i \(0.578737\pi\)
\(380\) −13.3689 −0.685812
\(381\) −10.7877 −0.552669
\(382\) 3.63169 0.185813
\(383\) −20.0836 −1.02622 −0.513111 0.858322i \(-0.671507\pi\)
−0.513111 + 0.858322i \(0.671507\pi\)
\(384\) 4.32825 0.220875
\(385\) 25.7892 1.31434
\(386\) −3.29288 −0.167603
\(387\) −2.61968 −0.133166
\(388\) 25.0911 1.27381
\(389\) 24.3472 1.23445 0.617225 0.786787i \(-0.288257\pi\)
0.617225 + 0.786787i \(0.288257\pi\)
\(390\) −0.508664 −0.0257572
\(391\) 8.37188 0.423384
\(392\) −10.3640 −0.523463
\(393\) −12.9745 −0.654476
\(394\) −3.55114 −0.178904
\(395\) −15.3035 −0.770004
\(396\) 10.0591 0.505489
\(397\) −25.9232 −1.30105 −0.650524 0.759486i \(-0.725451\pi\)
−0.650524 + 0.759486i \(0.725451\pi\)
\(398\) 1.08462 0.0543671
\(399\) 34.2748 1.71589
\(400\) 3.88517 0.194258
\(401\) 31.3576 1.56592 0.782961 0.622071i \(-0.213708\pi\)
0.782961 + 0.622071i \(0.213708\pi\)
\(402\) 0.0726141 0.00362166
\(403\) 2.89149 0.144035
\(404\) −13.9446 −0.693772
\(405\) −1.00000 −0.0496904
\(406\) −0.703671 −0.0349226
\(407\) 26.0528 1.29139
\(408\) −1.43755 −0.0711695
\(409\) 1.67415 0.0827814 0.0413907 0.999143i \(-0.486821\pi\)
0.0413907 + 0.999143i \(0.486821\pi\)
\(410\) 1.22140 0.0603207
\(411\) 4.08212 0.201356
\(412\) −5.26512 −0.259394
\(413\) 64.8819 3.19263
\(414\) −0.445115 −0.0218762
\(415\) 18.1182 0.889388
\(416\) −6.02602 −0.295450
\(417\) 19.8276 0.970960
\(418\) 4.74926 0.232294
\(419\) 0.658078 0.0321492 0.0160746 0.999871i \(-0.494883\pi\)
0.0160746 + 0.999871i \(0.494883\pi\)
\(420\) −10.0591 −0.490834
\(421\) 5.11989 0.249528 0.124764 0.992186i \(-0.460183\pi\)
0.124764 + 0.992186i \(0.460183\pi\)
\(422\) 0.277129 0.0134904
\(423\) −1.11670 −0.0542960
\(424\) −0.341812 −0.0165999
\(425\) −2.60617 −0.126418
\(426\) −0.0270211 −0.00130917
\(427\) 45.2267 2.18867
\(428\) −9.53693 −0.460985
\(429\) −18.6423 −0.900056
\(430\) 0.362994 0.0175051
\(431\) −0.390015 −0.0187864 −0.00939318 0.999956i \(-0.502990\pi\)
−0.00939318 + 0.999956i \(0.502990\pi\)
\(432\) −3.88517 −0.186925
\(433\) −29.6989 −1.42724 −0.713618 0.700535i \(-0.752945\pi\)
−0.713618 + 0.700535i \(0.752945\pi\)
\(434\) −0.554257 −0.0266052
\(435\) −1.00000 −0.0479463
\(436\) −7.65771 −0.366738
\(437\) 21.6809 1.03714
\(438\) −0.156607 −0.00748299
\(439\) −24.4856 −1.16864 −0.584318 0.811525i \(-0.698638\pi\)
−0.584318 + 0.811525i \(0.698638\pi\)
\(440\) −2.80118 −0.133541
\(441\) 18.7892 0.894722
\(442\) 1.32566 0.0630554
\(443\) −11.1091 −0.527808 −0.263904 0.964549i \(-0.585010\pi\)
−0.263904 + 0.964549i \(0.585010\pi\)
\(444\) −10.1619 −0.482264
\(445\) 15.5942 0.739234
\(446\) 1.08994 0.0516103
\(447\) −10.7763 −0.509701
\(448\) −38.3050 −1.80974
\(449\) −15.1003 −0.712628 −0.356314 0.934366i \(-0.615967\pi\)
−0.356314 + 0.934366i \(0.615967\pi\)
\(450\) 0.138564 0.00653198
\(451\) 44.7637 2.10784
\(452\) 17.3038 0.813901
\(453\) 5.49853 0.258343
\(454\) −0.00906504 −0.000425443 0
\(455\) 18.6423 0.873962
\(456\) −3.72287 −0.174339
\(457\) −24.3742 −1.14018 −0.570088 0.821583i \(-0.693091\pi\)
−0.570088 + 0.821583i \(0.693091\pi\)
\(458\) −0.536301 −0.0250597
\(459\) 2.60617 0.121645
\(460\) −6.36299 −0.296676
\(461\) −12.9789 −0.604489 −0.302244 0.953230i \(-0.597736\pi\)
−0.302244 + 0.953230i \(0.597736\pi\)
\(462\) 3.57346 0.166252
\(463\) −11.2619 −0.523386 −0.261693 0.965151i \(-0.584281\pi\)
−0.261693 + 0.965151i \(0.584281\pi\)
\(464\) −3.88517 −0.180364
\(465\) −0.787665 −0.0365271
\(466\) −1.13396 −0.0525296
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 7.27144 0.336122
\(469\) −2.66127 −0.122886
\(470\) 0.154735 0.00713740
\(471\) −5.77566 −0.266128
\(472\) −7.04736 −0.324381
\(473\) 13.3035 0.611697
\(474\) −2.12052 −0.0973988
\(475\) −6.74926 −0.309677
\(476\) 26.2157 1.20160
\(477\) 0.619678 0.0283731
\(478\) −1.56094 −0.0713959
\(479\) −23.8615 −1.09026 −0.545130 0.838351i \(-0.683520\pi\)
−0.545130 + 0.838351i \(0.683520\pi\)
\(480\) 1.64154 0.0749256
\(481\) 18.8328 0.858702
\(482\) −2.11476 −0.0963246
\(483\) 16.3132 0.742277
\(484\) −29.2944 −1.33156
\(485\) 12.6671 0.575185
\(486\) −0.138564 −0.00628540
\(487\) −20.8417 −0.944428 −0.472214 0.881484i \(-0.656545\pi\)
−0.472214 + 0.881484i \(0.656545\pi\)
\(488\) −4.91245 −0.222376
\(489\) −7.14691 −0.323195
\(490\) −2.60351 −0.117614
\(491\) −19.1021 −0.862067 −0.431034 0.902336i \(-0.641851\pi\)
−0.431034 + 0.902336i \(0.641851\pi\)
\(492\) −17.4601 −0.787163
\(493\) 2.60617 0.117376
\(494\) 3.43311 0.154463
\(495\) 5.07830 0.228253
\(496\) −3.06021 −0.137407
\(497\) 0.990307 0.0444213
\(498\) 2.51054 0.112500
\(499\) −7.82757 −0.350410 −0.175205 0.984532i \(-0.556059\pi\)
−0.175205 + 0.984532i \(0.556059\pi\)
\(500\) 1.98080 0.0885841
\(501\) −17.1009 −0.764011
\(502\) 3.45594 0.154246
\(503\) 31.5865 1.40837 0.704187 0.710015i \(-0.251312\pi\)
0.704187 + 0.710015i \(0.251312\pi\)
\(504\) −2.80118 −0.124774
\(505\) −7.03990 −0.313272
\(506\) 2.26043 0.100488
\(507\) −0.475953 −0.0211378
\(508\) −21.3682 −0.948061
\(509\) 19.6167 0.869497 0.434748 0.900552i \(-0.356837\pi\)
0.434748 + 0.900552i \(0.356837\pi\)
\(510\) −0.361122 −0.0159907
\(511\) 5.73957 0.253904
\(512\) 10.6637 0.471275
\(513\) 6.74926 0.297987
\(514\) 0.158921 0.00700972
\(515\) −2.65808 −0.117129
\(516\) −5.18906 −0.228436
\(517\) 5.67096 0.249409
\(518\) −3.60999 −0.158614
\(519\) −10.2862 −0.451514
\(520\) −2.02489 −0.0887973
\(521\) 24.4057 1.06923 0.534616 0.845095i \(-0.320456\pi\)
0.534616 + 0.845095i \(0.320456\pi\)
\(522\) −0.138564 −0.00606479
\(523\) −39.9715 −1.74783 −0.873917 0.486076i \(-0.838428\pi\)
−0.873917 + 0.486076i \(0.838428\pi\)
\(524\) −25.6999 −1.12270
\(525\) −5.07830 −0.221635
\(526\) 0.0949303 0.00413916
\(527\) 2.05279 0.0894208
\(528\) 19.7301 0.858641
\(529\) −12.6809 −0.551344
\(530\) −0.0858653 −0.00372975
\(531\) 12.7763 0.554444
\(532\) 67.8916 2.94347
\(533\) 32.3584 1.40160
\(534\) 2.16079 0.0935067
\(535\) −4.81469 −0.208157
\(536\) 0.289062 0.0124856
\(537\) 12.1182 0.522939
\(538\) 0.585464 0.0252412
\(539\) −95.4171 −4.10991
\(540\) −1.98080 −0.0852401
\(541\) −8.76064 −0.376649 −0.188325 0.982107i \(-0.560306\pi\)
−0.188325 + 0.982107i \(0.560306\pi\)
\(542\) −0.112886 −0.00484888
\(543\) 20.9745 0.900101
\(544\) −4.27813 −0.183423
\(545\) −3.86597 −0.165600
\(546\) 2.58315 0.110549
\(547\) 31.3569 1.34072 0.670361 0.742035i \(-0.266139\pi\)
0.670361 + 0.742035i \(0.266139\pi\)
\(548\) 8.08587 0.345411
\(549\) 8.90587 0.380093
\(550\) −0.703671 −0.0300046
\(551\) 6.74926 0.287528
\(552\) −1.77191 −0.0754176
\(553\) 77.7159 3.30482
\(554\) 1.36549 0.0580143
\(555\) −5.13021 −0.217765
\(556\) 39.2744 1.66561
\(557\) 37.6514 1.59534 0.797670 0.603094i \(-0.206066\pi\)
0.797670 + 0.603094i \(0.206066\pi\)
\(558\) −0.109142 −0.00462036
\(559\) 9.61674 0.406745
\(560\) −19.7301 −0.833747
\(561\) −13.2349 −0.558778
\(562\) 1.69459 0.0714821
\(563\) −39.0323 −1.64501 −0.822507 0.568755i \(-0.807425\pi\)
−0.822507 + 0.568755i \(0.807425\pi\)
\(564\) −2.21197 −0.0931406
\(565\) 8.73575 0.367516
\(566\) 0.586599 0.0246566
\(567\) 5.07830 0.213269
\(568\) −0.107565 −0.00451335
\(569\) −3.03990 −0.127439 −0.0637197 0.997968i \(-0.520296\pi\)
−0.0637197 + 0.997968i \(0.520296\pi\)
\(570\) −0.935207 −0.0391715
\(571\) 40.1056 1.67837 0.839183 0.543849i \(-0.183034\pi\)
0.839183 + 0.543849i \(0.183034\pi\)
\(572\) −36.9266 −1.54398
\(573\) −26.2094 −1.09491
\(574\) −6.20264 −0.258893
\(575\) −3.21234 −0.133964
\(576\) −7.54288 −0.314287
\(577\) −16.1091 −0.670632 −0.335316 0.942106i \(-0.608843\pi\)
−0.335316 + 0.942106i \(0.608843\pi\)
\(578\) −1.41445 −0.0588333
\(579\) 23.7643 0.987610
\(580\) −1.98080 −0.0822482
\(581\) −92.0098 −3.81721
\(582\) 1.75521 0.0727559
\(583\) −3.14691 −0.130332
\(584\) −0.623422 −0.0257974
\(585\) 3.67096 0.151776
\(586\) −1.84526 −0.0762272
\(587\) −21.4331 −0.884639 −0.442320 0.896858i \(-0.645844\pi\)
−0.442320 + 0.896858i \(0.645844\pi\)
\(588\) 37.2176 1.53483
\(589\) 5.31616 0.219048
\(590\) −1.77034 −0.0728836
\(591\) 25.6281 1.05420
\(592\) −19.9317 −0.819190
\(593\) 23.9886 0.985095 0.492547 0.870286i \(-0.336066\pi\)
0.492547 + 0.870286i \(0.336066\pi\)
\(594\) 0.703671 0.0288720
\(595\) 13.2349 0.542578
\(596\) −21.3457 −0.874353
\(597\) −7.82757 −0.320361
\(598\) 1.63400 0.0668192
\(599\) −38.1100 −1.55713 −0.778567 0.627562i \(-0.784053\pi\)
−0.778567 + 0.627562i \(0.784053\pi\)
\(600\) 0.551597 0.0225188
\(601\) −20.0528 −0.817970 −0.408985 0.912541i \(-0.634117\pi\)
−0.408985 + 0.912541i \(0.634117\pi\)
\(602\) −1.84339 −0.0751311
\(603\) −0.524047 −0.0213408
\(604\) 10.8915 0.443168
\(605\) −14.7892 −0.601265
\(606\) −0.975479 −0.0396261
\(607\) −38.7523 −1.57291 −0.786453 0.617650i \(-0.788085\pi\)
−0.786453 + 0.617650i \(0.788085\pi\)
\(608\) −11.0792 −0.449320
\(609\) 5.07830 0.205783
\(610\) −1.23404 −0.0499646
\(611\) 4.09938 0.165843
\(612\) 5.16230 0.208673
\(613\) 24.6757 0.996640 0.498320 0.866993i \(-0.333950\pi\)
0.498320 + 0.866993i \(0.333950\pi\)
\(614\) 2.04214 0.0824142
\(615\) −8.81469 −0.355442
\(616\) 14.2252 0.573150
\(617\) 10.5249 0.423717 0.211859 0.977300i \(-0.432048\pi\)
0.211859 + 0.977300i \(0.432048\pi\)
\(618\) −0.368315 −0.0148158
\(619\) 26.5496 1.06712 0.533560 0.845762i \(-0.320854\pi\)
0.533560 + 0.845762i \(0.320854\pi\)
\(620\) −1.56021 −0.0626594
\(621\) 3.21234 0.128907
\(622\) −1.38877 −0.0556846
\(623\) −79.1919 −3.17276
\(624\) 14.2623 0.570949
\(625\) 1.00000 0.0400000
\(626\) 0.566450 0.0226399
\(627\) −34.2748 −1.36880
\(628\) −11.4404 −0.456523
\(629\) 13.3702 0.533105
\(630\) −0.703671 −0.0280349
\(631\) −13.2041 −0.525649 −0.262824 0.964844i \(-0.584654\pi\)
−0.262824 + 0.964844i \(0.584654\pi\)
\(632\) −8.44137 −0.335780
\(633\) −2.00000 −0.0794929
\(634\) −1.76846 −0.0702347
\(635\) −10.7877 −0.428095
\(636\) 1.22746 0.0486719
\(637\) −68.9743 −2.73286
\(638\) 0.703671 0.0278586
\(639\) 0.195007 0.00771437
\(640\) 4.32825 0.171089
\(641\) 9.51435 0.375794 0.187897 0.982189i \(-0.439833\pi\)
0.187897 + 0.982189i \(0.439833\pi\)
\(642\) −0.667143 −0.0263300
\(643\) 30.1370 1.18849 0.594244 0.804285i \(-0.297451\pi\)
0.594244 + 0.804285i \(0.297451\pi\)
\(644\) 32.3132 1.27332
\(645\) −2.61968 −0.103150
\(646\) 2.43731 0.0958945
\(647\) −42.7535 −1.68081 −0.840407 0.541955i \(-0.817684\pi\)
−0.840407 + 0.541955i \(0.817684\pi\)
\(648\) −0.551597 −0.0216688
\(649\) −64.8819 −2.54684
\(650\) −0.508664 −0.0199514
\(651\) 4.00000 0.156772
\(652\) −14.1566 −0.554416
\(653\) 33.8983 1.32654 0.663272 0.748379i \(-0.269167\pi\)
0.663272 + 0.748379i \(0.269167\pi\)
\(654\) −0.535685 −0.0209469
\(655\) −12.9745 −0.506955
\(656\) −34.2465 −1.33710
\(657\) 1.13021 0.0440939
\(658\) −0.785793 −0.0306334
\(659\) 11.7287 0.456887 0.228444 0.973557i \(-0.426636\pi\)
0.228444 + 0.973557i \(0.426636\pi\)
\(660\) 10.0591 0.391550
\(661\) −38.6807 −1.50450 −0.752252 0.658876i \(-0.771032\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) 0.808627 0.0314282
\(663\) −9.56714 −0.371557
\(664\) 9.99394 0.387840
\(665\) 34.2748 1.32912
\(666\) −0.710864 −0.0275454
\(667\) 3.21234 0.124382
\(668\) −33.8734 −1.31060
\(669\) −7.86597 −0.304116
\(670\) 0.0726141 0.00280533
\(671\) −45.2267 −1.74596
\(672\) −8.33623 −0.321577
\(673\) 30.6410 1.18112 0.590562 0.806992i \(-0.298906\pi\)
0.590562 + 0.806992i \(0.298906\pi\)
\(674\) 0.270552 0.0104213
\(675\) −1.00000 −0.0384900
\(676\) −0.942768 −0.0362603
\(677\) −40.8954 −1.57174 −0.785868 0.618394i \(-0.787784\pi\)
−0.785868 + 0.618394i \(0.787784\pi\)
\(678\) 1.21046 0.0464876
\(679\) −64.3276 −2.46867
\(680\) −1.43755 −0.0551277
\(681\) 0.0654212 0.00250694
\(682\) 0.554257 0.0212236
\(683\) 21.2317 0.812409 0.406205 0.913782i \(-0.366852\pi\)
0.406205 + 0.913782i \(0.366852\pi\)
\(684\) 13.3689 0.511174
\(685\) 4.08212 0.155970
\(686\) 8.29570 0.316731
\(687\) 3.87041 0.147665
\(688\) −10.1779 −0.388028
\(689\) −2.27481 −0.0866635
\(690\) −0.445115 −0.0169452
\(691\) −13.9842 −0.531983 −0.265992 0.963975i \(-0.585699\pi\)
−0.265992 + 0.963975i \(0.585699\pi\)
\(692\) −20.3749 −0.774537
\(693\) −25.7892 −0.979649
\(694\) 2.41046 0.0914998
\(695\) 19.8276 0.752103
\(696\) −0.551597 −0.0209082
\(697\) 22.9725 0.870147
\(698\) −3.98204 −0.150723
\(699\) 8.18363 0.309533
\(700\) −10.0591 −0.380198
\(701\) 3.16630 0.119590 0.0597948 0.998211i \(-0.480955\pi\)
0.0597948 + 0.998211i \(0.480955\pi\)
\(702\) 0.508664 0.0191983
\(703\) 34.6252 1.30591
\(704\) 38.3050 1.44367
\(705\) −1.11670 −0.0420575
\(706\) −4.12353 −0.155191
\(707\) 35.7508 1.34455
\(708\) 25.3073 0.951105
\(709\) 23.3677 0.877592 0.438796 0.898587i \(-0.355405\pi\)
0.438796 + 0.898587i \(0.355405\pi\)
\(710\) −0.0270211 −0.00101408
\(711\) 15.3035 0.573927
\(712\) 8.60169 0.322362
\(713\) 2.53024 0.0947583
\(714\) 1.83389 0.0686314
\(715\) −18.6423 −0.697181
\(716\) 24.0037 0.897062
\(717\) 11.2651 0.420704
\(718\) 1.07617 0.0401624
\(719\) −0.731134 −0.0272667 −0.0136334 0.999907i \(-0.504340\pi\)
−0.0136334 + 0.999907i \(0.504340\pi\)
\(720\) −3.88517 −0.144792
\(721\) 13.4985 0.502711
\(722\) 3.67924 0.136927
\(723\) 15.2619 0.567597
\(724\) 41.5463 1.54405
\(725\) −1.00000 −0.0371391
\(726\) −2.04925 −0.0760548
\(727\) −17.7243 −0.657358 −0.328679 0.944442i \(-0.606603\pi\)
−0.328679 + 0.944442i \(0.606603\pi\)
\(728\) 10.2830 0.381113
\(729\) 1.00000 0.0370370
\(730\) −0.156607 −0.00579630
\(731\) 6.82732 0.252518
\(732\) 17.6408 0.652021
\(733\) −5.67184 −0.209494 −0.104747 0.994499i \(-0.533403\pi\)
−0.104747 + 0.994499i \(0.533403\pi\)
\(734\) −1.78297 −0.0658107
\(735\) 18.7892 0.693049
\(736\) −5.27317 −0.194372
\(737\) 2.66127 0.0980291
\(738\) −1.22140 −0.0449604
\(739\) 8.72350 0.320899 0.160450 0.987044i \(-0.448706\pi\)
0.160450 + 0.987044i \(0.448706\pi\)
\(740\) −10.1619 −0.373560
\(741\) −24.7763 −0.910180
\(742\) 0.436050 0.0160079
\(743\) −0.413474 −0.0151689 −0.00758445 0.999971i \(-0.502414\pi\)
−0.00758445 + 0.999971i \(0.502414\pi\)
\(744\) −0.434473 −0.0159286
\(745\) −10.7763 −0.394813
\(746\) −1.86561 −0.0683049
\(747\) −18.1182 −0.662911
\(748\) −26.2157 −0.958541
\(749\) 24.4504 0.893399
\(750\) 0.138564 0.00505965
\(751\) 9.71381 0.354462 0.177231 0.984169i \(-0.443286\pi\)
0.177231 + 0.984169i \(0.443286\pi\)
\(752\) −4.33858 −0.158212
\(753\) −24.9411 −0.908904
\(754\) 0.508664 0.0185244
\(755\) 5.49853 0.200112
\(756\) 10.0591 0.365846
\(757\) 22.3877 0.813695 0.406847 0.913496i \(-0.366628\pi\)
0.406847 + 0.913496i \(0.366628\pi\)
\(758\) −1.32096 −0.0479794
\(759\) −16.3132 −0.592132
\(760\) −3.72287 −0.135043
\(761\) 44.0836 1.59803 0.799014 0.601313i \(-0.205355\pi\)
0.799014 + 0.601313i \(0.205355\pi\)
\(762\) −1.49478 −0.0541503
\(763\) 19.6326 0.710746
\(764\) −51.9156 −1.87824
\(765\) 2.60617 0.0942262
\(766\) −2.78286 −0.100549
\(767\) −46.9012 −1.69351
\(768\) −14.4860 −0.522719
\(769\) −14.2761 −0.514808 −0.257404 0.966304i \(-0.582867\pi\)
−0.257404 + 0.966304i \(0.582867\pi\)
\(770\) 3.57346 0.128778
\(771\) −1.14691 −0.0413051
\(772\) 47.0723 1.69417
\(773\) −25.5807 −0.920072 −0.460036 0.887900i \(-0.652164\pi\)
−0.460036 + 0.887900i \(0.652164\pi\)
\(774\) −0.362994 −0.0130475
\(775\) −0.787665 −0.0282937
\(776\) 6.98715 0.250824
\(777\) 26.0528 0.934638
\(778\) 3.37365 0.120951
\(779\) 59.4926 2.13155
\(780\) 7.27144 0.260359
\(781\) −0.990307 −0.0354360
\(782\) 1.16004 0.0414831
\(783\) 1.00000 0.0357371
\(784\) 72.9991 2.60711
\(785\) −5.77566 −0.206142
\(786\) −1.79780 −0.0641254
\(787\) −8.73362 −0.311320 −0.155660 0.987811i \(-0.549750\pi\)
−0.155660 + 0.987811i \(0.549750\pi\)
\(788\) 50.7642 1.80840
\(789\) −0.685099 −0.0243902
\(790\) −2.12052 −0.0754448
\(791\) −44.3628 −1.57736
\(792\) 2.80118 0.0995354
\(793\) −32.6931 −1.16097
\(794\) −3.59203 −0.127476
\(795\) 0.619678 0.0219777
\(796\) −15.5048 −0.549555
\(797\) 8.43874 0.298915 0.149458 0.988768i \(-0.452247\pi\)
0.149458 + 0.988768i \(0.452247\pi\)
\(798\) 4.74926 0.168122
\(799\) 2.91032 0.102960
\(800\) 1.64154 0.0580372
\(801\) −15.5942 −0.550993
\(802\) 4.34504 0.153429
\(803\) −5.73957 −0.202545
\(804\) −1.03803 −0.0366085
\(805\) 16.3132 0.574965
\(806\) 0.400657 0.0141125
\(807\) −4.22522 −0.148735
\(808\) −3.88319 −0.136610
\(809\) −21.7508 −0.764716 −0.382358 0.924014i \(-0.624888\pi\)
−0.382358 + 0.924014i \(0.624888\pi\)
\(810\) −0.138564 −0.00486865
\(811\) −3.24629 −0.113993 −0.0569963 0.998374i \(-0.518152\pi\)
−0.0569963 + 0.998374i \(0.518152\pi\)
\(812\) 10.0591 0.353005
\(813\) 0.814686 0.0285723
\(814\) 3.60999 0.126530
\(815\) −7.14691 −0.250345
\(816\) 10.1254 0.354460
\(817\) 17.6809 0.618576
\(818\) 0.231977 0.00811090
\(819\) −18.6423 −0.651413
\(820\) −17.4601 −0.609734
\(821\) 33.9232 1.18393 0.591964 0.805964i \(-0.298353\pi\)
0.591964 + 0.805964i \(0.298353\pi\)
\(822\) 0.565636 0.0197288
\(823\) −36.4648 −1.27108 −0.635542 0.772066i \(-0.719223\pi\)
−0.635542 + 0.772066i \(0.719223\pi\)
\(824\) −1.46619 −0.0510770
\(825\) 5.07830 0.176804
\(826\) 8.99031 0.312813
\(827\) −15.9772 −0.555583 −0.277792 0.960641i \(-0.589602\pi\)
−0.277792 + 0.960641i \(0.589602\pi\)
\(828\) 6.36299 0.221129
\(829\) 5.00595 0.173864 0.0869319 0.996214i \(-0.472294\pi\)
0.0869319 + 0.996214i \(0.472294\pi\)
\(830\) 2.51054 0.0871420
\(831\) −9.85459 −0.341852
\(832\) 27.6896 0.959964
\(833\) −48.9677 −1.69663
\(834\) 2.74739 0.0951344
\(835\) −17.1009 −0.591800
\(836\) −67.8916 −2.34808
\(837\) 0.787665 0.0272257
\(838\) 0.0911861 0.00314997
\(839\) 31.4713 1.08651 0.543255 0.839567i \(-0.317192\pi\)
0.543255 + 0.839567i \(0.317192\pi\)
\(840\) −2.80118 −0.0966497
\(841\) 1.00000 0.0344828
\(842\) 0.709434 0.0244487
\(843\) −12.2297 −0.421212
\(844\) −3.96160 −0.136364
\(845\) −0.475953 −0.0163733
\(846\) −0.154735 −0.00531991
\(847\) 75.1039 2.58060
\(848\) 2.40755 0.0826758
\(849\) −4.23341 −0.145290
\(850\) −0.361122 −0.0123864
\(851\) 16.4800 0.564926
\(852\) 0.386271 0.0132334
\(853\) −36.0197 −1.23329 −0.616646 0.787241i \(-0.711509\pi\)
−0.616646 + 0.787241i \(0.711509\pi\)
\(854\) 6.26681 0.214446
\(855\) 6.74926 0.230820
\(856\) −2.65576 −0.0907722
\(857\) −27.4217 −0.936708 −0.468354 0.883541i \(-0.655153\pi\)
−0.468354 + 0.883541i \(0.655153\pi\)
\(858\) −2.58315 −0.0881873
\(859\) 14.0642 0.479863 0.239932 0.970790i \(-0.422875\pi\)
0.239932 + 0.970790i \(0.422875\pi\)
\(860\) −5.18906 −0.176945
\(861\) 44.7637 1.52554
\(862\) −0.0540421 −0.00184068
\(863\) 19.7540 0.672433 0.336216 0.941785i \(-0.390853\pi\)
0.336216 + 0.941785i \(0.390853\pi\)
\(864\) −1.64154 −0.0558463
\(865\) −10.2862 −0.349741
\(866\) −4.11520 −0.139840
\(867\) 10.2079 0.346678
\(868\) 7.92320 0.268931
\(869\) −77.7159 −2.63633
\(870\) −0.138564 −0.00469777
\(871\) 1.92375 0.0651839
\(872\) −2.13246 −0.0722140
\(873\) −12.6671 −0.428718
\(874\) 3.00420 0.101619
\(875\) −5.07830 −0.171678
\(876\) 2.23873 0.0756396
\(877\) 42.2030 1.42509 0.712547 0.701624i \(-0.247541\pi\)
0.712547 + 0.701624i \(0.247541\pi\)
\(878\) −3.39284 −0.114503
\(879\) 13.3170 0.449172
\(880\) 19.7301 0.665100
\(881\) −33.0927 −1.11492 −0.557461 0.830203i \(-0.688224\pi\)
−0.557461 + 0.830203i \(0.688224\pi\)
\(882\) 2.60351 0.0876646
\(883\) −26.2634 −0.883835 −0.441917 0.897056i \(-0.645702\pi\)
−0.441917 + 0.897056i \(0.645702\pi\)
\(884\) −18.9506 −0.637377
\(885\) 12.7763 0.429470
\(886\) −1.53932 −0.0517145
\(887\) 14.0716 0.472479 0.236239 0.971695i \(-0.424085\pi\)
0.236239 + 0.971695i \(0.424085\pi\)
\(888\) −2.82981 −0.0949622
\(889\) 54.7830 1.83736
\(890\) 2.16079 0.0724300
\(891\) −5.07830 −0.170130
\(892\) −15.5809 −0.521687
\(893\) 7.53693 0.252214
\(894\) −1.49321 −0.0499403
\(895\) 12.1182 0.405067
\(896\) −21.9802 −0.734306
\(897\) −11.7924 −0.393735
\(898\) −2.09237 −0.0698231
\(899\) 0.787665 0.0262701
\(900\) −1.98080 −0.0660267
\(901\) −1.61499 −0.0538030
\(902\) 6.20264 0.206525
\(903\) 13.3035 0.442713
\(904\) 4.81861 0.160265
\(905\) 20.9745 0.697215
\(906\) 0.761900 0.0253124
\(907\) 11.9810 0.397822 0.198911 0.980018i \(-0.436260\pi\)
0.198911 + 0.980018i \(0.436260\pi\)
\(908\) 0.129586 0.00430047
\(909\) 7.03990 0.233499
\(910\) 2.58315 0.0856306
\(911\) 27.5300 0.912109 0.456055 0.889952i \(-0.349262\pi\)
0.456055 + 0.889952i \(0.349262\pi\)
\(912\) 26.2220 0.868298
\(913\) 92.0098 3.04508
\(914\) −3.37739 −0.111714
\(915\) 8.90587 0.294419
\(916\) 7.66652 0.253309
\(917\) 65.8884 2.17583
\(918\) 0.361122 0.0119188
\(919\) −17.7250 −0.584694 −0.292347 0.956312i \(-0.594436\pi\)
−0.292347 + 0.956312i \(0.594436\pi\)
\(920\) −1.77191 −0.0584183
\(921\) −14.7379 −0.485630
\(922\) −1.79842 −0.0592277
\(923\) −0.715865 −0.0235630
\(924\) −51.0832 −1.68051
\(925\) −5.13021 −0.168680
\(926\) −1.56050 −0.0512813
\(927\) 2.65808 0.0873027
\(928\) −1.64154 −0.0538861
\(929\) 36.9971 1.21383 0.606917 0.794765i \(-0.292406\pi\)
0.606917 + 0.794765i \(0.292406\pi\)
\(930\) −0.109142 −0.00357891
\(931\) −126.813 −4.15613
\(932\) 16.2101 0.530980
\(933\) 10.0226 0.328124
\(934\) −1.10851 −0.0362717
\(935\) −13.2349 −0.432828
\(936\) 2.02489 0.0661856
\(937\) −36.0446 −1.17753 −0.588763 0.808306i \(-0.700385\pi\)
−0.588763 + 0.808306i \(0.700385\pi\)
\(938\) −0.368757 −0.0120403
\(939\) −4.08800 −0.133407
\(940\) −2.21197 −0.0721464
\(941\) −19.6256 −0.639777 −0.319889 0.947455i \(-0.603645\pi\)
−0.319889 + 0.947455i \(0.603645\pi\)
\(942\) −0.800300 −0.0260752
\(943\) 28.3157 0.922087
\(944\) 49.6380 1.61558
\(945\) 5.07830 0.165197
\(946\) 1.84339 0.0599339
\(947\) 18.9555 0.615970 0.307985 0.951391i \(-0.400345\pi\)
0.307985 + 0.951391i \(0.400345\pi\)
\(948\) 30.3132 0.984527
\(949\) −4.14897 −0.134681
\(950\) −0.935207 −0.0303421
\(951\) 12.7628 0.413861
\(952\) 7.30033 0.236605
\(953\) −30.2761 −0.980738 −0.490369 0.871515i \(-0.663138\pi\)
−0.490369 + 0.871515i \(0.663138\pi\)
\(954\) 0.0858653 0.00277999
\(955\) −26.2094 −0.848116
\(956\) 22.3140 0.721685
\(957\) −5.07830 −0.164158
\(958\) −3.30635 −0.106823
\(959\) −20.7303 −0.669415
\(960\) −7.54288 −0.243445
\(961\) −30.3796 −0.979987
\(962\) 2.60956 0.0841354
\(963\) 4.81469 0.155151
\(964\) 30.2308 0.973670
\(965\) 23.7643 0.764999
\(966\) 2.26043 0.0727281
\(967\) −20.4812 −0.658631 −0.329316 0.944220i \(-0.606818\pi\)
−0.329316 + 0.944220i \(0.606818\pi\)
\(968\) −8.15766 −0.262197
\(969\) −17.5897 −0.565063
\(970\) 1.75521 0.0563565
\(971\) 44.1566 1.41705 0.708526 0.705684i \(-0.249360\pi\)
0.708526 + 0.705684i \(0.249360\pi\)
\(972\) 1.98080 0.0635342
\(973\) −100.690 −3.22799
\(974\) −2.88792 −0.0925348
\(975\) 3.67096 0.117565
\(976\) 34.6008 1.10754
\(977\) 0.492580 0.0157590 0.00787952 0.999969i \(-0.497492\pi\)
0.00787952 + 0.999969i \(0.497492\pi\)
\(978\) −0.990307 −0.0316665
\(979\) 79.1919 2.53098
\(980\) 37.2176 1.18887
\(981\) 3.86597 0.123431
\(982\) −2.64687 −0.0844651
\(983\) 27.5513 0.878750 0.439375 0.898304i \(-0.355200\pi\)
0.439375 + 0.898304i \(0.355200\pi\)
\(984\) −4.86215 −0.155000
\(985\) 25.6281 0.816580
\(986\) 0.361122 0.0115005
\(987\) 5.67096 0.180509
\(988\) −49.0769 −1.56134
\(989\) 8.41529 0.267590
\(990\) 0.703671 0.0223641
\(991\) 23.0927 0.733563 0.366782 0.930307i \(-0.380460\pi\)
0.366782 + 0.930307i \(0.380460\pi\)
\(992\) −1.29298 −0.0410522
\(993\) −5.83576 −0.185192
\(994\) 0.137221 0.00435239
\(995\) −7.82757 −0.248151
\(996\) −35.8885 −1.13717
\(997\) 7.79593 0.246900 0.123450 0.992351i \(-0.460604\pi\)
0.123450 + 0.992351i \(0.460604\pi\)
\(998\) −1.08462 −0.0343331
\(999\) 5.13021 0.162313
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 435.2.a.j.1.3 4
3.2 odd 2 1305.2.a.r.1.2 4
4.3 odd 2 6960.2.a.co.1.1 4
5.2 odd 4 2175.2.c.n.349.5 8
5.3 odd 4 2175.2.c.n.349.4 8
5.4 even 2 2175.2.a.v.1.2 4
15.14 odd 2 6525.2.a.bi.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.3 4 1.1 even 1 trivial
1305.2.a.r.1.2 4 3.2 odd 2
2175.2.a.v.1.2 4 5.4 even 2
2175.2.c.n.349.4 8 5.3 odd 4
2175.2.c.n.349.5 8 5.2 odd 4
6525.2.a.bi.1.3 4 15.14 odd 2
6960.2.a.co.1.1 4 4.3 odd 2