Properties

Label 1045.2.f.a.626.4
Level $1045$
Weight $2$
Character 1045.626
Analytic conductor $8.344$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,2,Mod(626,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.626");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 626.4
Character \(\chi\) \(=\) 1045.626
Dual form 1045.2.f.a.626.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.52972 q^{2} +0.384887i q^{3} +4.39948 q^{4} -1.00000 q^{5} -0.973657i q^{6} -1.35297i q^{7} -6.07000 q^{8} +2.85186 q^{9} +O(q^{10})\) \(q-2.52972 q^{2} +0.384887i q^{3} +4.39948 q^{4} -1.00000 q^{5} -0.973657i q^{6} -1.35297i q^{7} -6.07000 q^{8} +2.85186 q^{9} +2.52972 q^{10} +(1.95002 - 2.68280i) q^{11} +1.69330i q^{12} +2.33668 q^{13} +3.42264i q^{14} -0.384887i q^{15} +6.55645 q^{16} +5.43006i q^{17} -7.21441 q^{18} +(0.531936 - 4.32632i) q^{19} -4.39948 q^{20} +0.520743 q^{21} +(-4.93299 + 6.78674i) q^{22} +0.981945 q^{23} -2.33627i q^{24} +1.00000 q^{25} -5.91114 q^{26} +2.25231i q^{27} -5.95238i q^{28} -3.04589 q^{29} +0.973657i q^{30} -8.09312i q^{31} -4.44597 q^{32} +(1.03258 + 0.750537i) q^{33} -13.7365i q^{34} +1.35297i q^{35} +12.5467 q^{36} +1.40295i q^{37} +(-1.34565 + 10.9444i) q^{38} +0.899359i q^{39} +6.07000 q^{40} +1.12340 q^{41} -1.31733 q^{42} -2.01890i q^{43} +(8.57905 - 11.8029i) q^{44} -2.85186 q^{45} -2.48404 q^{46} -12.2984 q^{47} +2.52350i q^{48} +5.16946 q^{49} -2.52972 q^{50} -2.08996 q^{51} +10.2802 q^{52} -3.95467i q^{53} -5.69771i q^{54} +(-1.95002 + 2.68280i) q^{55} +8.21255i q^{56} +(1.66515 + 0.204736i) q^{57} +7.70526 q^{58} +1.26044i q^{59} -1.69330i q^{60} +9.55786i q^{61} +20.4733i q^{62} -3.85849i q^{63} -1.86586 q^{64} -2.33668 q^{65} +(-2.61213 - 1.89865i) q^{66} -13.2712i q^{67} +23.8894i q^{68} +0.377938i q^{69} -3.42264i q^{70} +14.3607i q^{71} -17.3108 q^{72} +2.05840i q^{73} -3.54907i q^{74} +0.384887i q^{75} +(2.34024 - 19.0335i) q^{76} +(-3.62976 - 2.63832i) q^{77} -2.27512i q^{78} +9.57041 q^{79} -6.55645 q^{80} +7.68870 q^{81} -2.84188 q^{82} -10.1838i q^{83} +2.29100 q^{84} -5.43006i q^{85} +5.10724i q^{86} -1.17233i q^{87} +(-11.8366 + 16.2846i) q^{88} -12.3375i q^{89} +7.21441 q^{90} -3.16147i q^{91} +4.32004 q^{92} +3.11494 q^{93} +31.1114 q^{94} +(-0.531936 + 4.32632i) q^{95} -1.71120i q^{96} -2.23673i q^{97} -13.0773 q^{98} +(5.56118 - 7.65098i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 40 q^{4} - 40 q^{5} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 40 q^{4} - 40 q^{5} - 28 q^{9} - 4 q^{11} + 32 q^{16} - 40 q^{20} - 16 q^{23} + 40 q^{25} + 8 q^{26} + 8 q^{36} + 28 q^{38} - 84 q^{42} - 48 q^{44} + 28 q^{45} + 32 q^{47} - 20 q^{49} + 4 q^{55} - 20 q^{58} + 72 q^{64} + 36 q^{66} + 16 q^{77} - 32 q^{80} + 16 q^{81} + 16 q^{82} - 20 q^{92} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.52972 −1.78878 −0.894391 0.447287i \(-0.852390\pi\)
−0.894391 + 0.447287i \(0.852390\pi\)
\(3\) 0.384887i 0.222215i 0.993808 + 0.111107i \(0.0354398\pi\)
−0.993808 + 0.111107i \(0.964560\pi\)
\(4\) 4.39948 2.19974
\(5\) −1.00000 −0.447214
\(6\) 0.973657i 0.397494i
\(7\) 1.35297i 0.511376i −0.966759 0.255688i \(-0.917698\pi\)
0.966759 0.255688i \(-0.0823020\pi\)
\(8\) −6.07000 −2.14607
\(9\) 2.85186 0.950621
\(10\) 2.52972 0.799967
\(11\) 1.95002 2.68280i 0.587952 0.808896i
\(12\) 1.69330i 0.488815i
\(13\) 2.33668 0.648078 0.324039 0.946044i \(-0.394959\pi\)
0.324039 + 0.946044i \(0.394959\pi\)
\(14\) 3.42264i 0.914740i
\(15\) 0.384887i 0.0993775i
\(16\) 6.55645 1.63911
\(17\) 5.43006i 1.31698i 0.752588 + 0.658492i \(0.228805\pi\)
−0.752588 + 0.658492i \(0.771195\pi\)
\(18\) −7.21441 −1.70045
\(19\) 0.531936 4.32632i 0.122035 0.992526i
\(20\) −4.39948 −0.983753
\(21\) 0.520743 0.113635
\(22\) −4.93299 + 6.78674i −1.05172 + 1.44694i
\(23\) 0.981945 0.204750 0.102375 0.994746i \(-0.467356\pi\)
0.102375 + 0.994746i \(0.467356\pi\)
\(24\) 2.33627i 0.476889i
\(25\) 1.00000 0.200000
\(26\) −5.91114 −1.15927
\(27\) 2.25231i 0.433457i
\(28\) 5.95238i 1.12489i
\(29\) −3.04589 −0.565608 −0.282804 0.959178i \(-0.591265\pi\)
−0.282804 + 0.959178i \(0.591265\pi\)
\(30\) 0.973657i 0.177765i
\(31\) 8.09312i 1.45357i −0.686866 0.726784i \(-0.741014\pi\)
0.686866 0.726784i \(-0.258986\pi\)
\(32\) −4.44597 −0.785943
\(33\) 1.03258 + 0.750537i 0.179749 + 0.130652i
\(34\) 13.7365i 2.35580i
\(35\) 1.35297i 0.228694i
\(36\) 12.5467 2.09112
\(37\) 1.40295i 0.230644i 0.993328 + 0.115322i \(0.0367899\pi\)
−0.993328 + 0.115322i \(0.963210\pi\)
\(38\) −1.34565 + 10.9444i −0.218293 + 1.77541i
\(39\) 0.899359i 0.144013i
\(40\) 6.07000 0.959752
\(41\) 1.12340 0.175445 0.0877226 0.996145i \(-0.472041\pi\)
0.0877226 + 0.996145i \(0.472041\pi\)
\(42\) −1.31733 −0.203269
\(43\) 2.01890i 0.307879i −0.988080 0.153939i \(-0.950804\pi\)
0.988080 0.153939i \(-0.0491960\pi\)
\(44\) 8.57905 11.8029i 1.29334 1.77936i
\(45\) −2.85186 −0.425130
\(46\) −2.48404 −0.366252
\(47\) −12.2984 −1.79390 −0.896950 0.442131i \(-0.854222\pi\)
−0.896950 + 0.442131i \(0.854222\pi\)
\(48\) 2.52350i 0.364235i
\(49\) 5.16946 0.738495
\(50\) −2.52972 −0.357756
\(51\) −2.08996 −0.292653
\(52\) 10.2802 1.42560
\(53\) 3.95467i 0.543216i −0.962408 0.271608i \(-0.912445\pi\)
0.962408 0.271608i \(-0.0875555\pi\)
\(54\) 5.69771i 0.775360i
\(55\) −1.95002 + 2.68280i −0.262940 + 0.361749i
\(56\) 8.21255i 1.09745i
\(57\) 1.66515 + 0.204736i 0.220554 + 0.0271179i
\(58\) 7.70526 1.01175
\(59\) 1.26044i 0.164095i 0.996628 + 0.0820474i \(0.0261459\pi\)
−0.996628 + 0.0820474i \(0.973854\pi\)
\(60\) 1.69330i 0.218605i
\(61\) 9.55786i 1.22376i 0.790951 + 0.611879i \(0.209586\pi\)
−0.790951 + 0.611879i \(0.790414\pi\)
\(62\) 20.4733i 2.60011i
\(63\) 3.85849i 0.486124i
\(64\) −1.86586 −0.233232
\(65\) −2.33668 −0.289829
\(66\) −2.61213 1.89865i −0.321531 0.233707i
\(67\) 13.2712i 1.62134i −0.585505 0.810669i \(-0.699104\pi\)
0.585505 0.810669i \(-0.300896\pi\)
\(68\) 23.8894i 2.89702i
\(69\) 0.377938i 0.0454984i
\(70\) 3.42264i 0.409084i
\(71\) 14.3607i 1.70431i 0.523292 + 0.852154i \(0.324704\pi\)
−0.523292 + 0.852154i \(0.675296\pi\)
\(72\) −17.3108 −2.04010
\(73\) 2.05840i 0.240918i 0.992718 + 0.120459i \(0.0384366\pi\)
−0.992718 + 0.120459i \(0.961563\pi\)
\(74\) 3.54907i 0.412571i
\(75\) 0.384887i 0.0444430i
\(76\) 2.34024 19.0335i 0.268444 2.18330i
\(77\) −3.62976 2.63832i −0.413650 0.300665i
\(78\) 2.27512i 0.257607i
\(79\) 9.57041 1.07676 0.538378 0.842704i \(-0.319037\pi\)
0.538378 + 0.842704i \(0.319037\pi\)
\(80\) −6.55645 −0.733033
\(81\) 7.68870 0.854300
\(82\) −2.84188 −0.313833
\(83\) 10.1838i 1.11782i −0.829230 0.558908i \(-0.811221\pi\)
0.829230 0.558908i \(-0.188779\pi\)
\(84\) 2.29100 0.249968
\(85\) 5.43006i 0.588973i
\(86\) 5.10724i 0.550728i
\(87\) 1.17233i 0.125687i
\(88\) −11.8366 + 16.2846i −1.26179 + 1.73595i
\(89\) 12.3375i 1.30777i −0.756594 0.653885i \(-0.773138\pi\)
0.756594 0.653885i \(-0.226862\pi\)
\(90\) 7.21441 0.760465
\(91\) 3.16147i 0.331412i
\(92\) 4.32004 0.450396
\(93\) 3.11494 0.323004
\(94\) 31.1114 3.20890
\(95\) −0.531936 + 4.32632i −0.0545755 + 0.443871i
\(96\) 1.71120i 0.174648i
\(97\) 2.23673i 0.227106i −0.993532 0.113553i \(-0.963777\pi\)
0.993532 0.113553i \(-0.0362231\pi\)
\(98\) −13.0773 −1.32101
\(99\) 5.56118 7.65098i 0.558919 0.768953i
\(100\) 4.39948 0.439948
\(101\) 5.08567i 0.506043i 0.967461 + 0.253022i \(0.0814244\pi\)
−0.967461 + 0.253022i \(0.918576\pi\)
\(102\) 5.28702 0.523493
\(103\) 6.81557i 0.671558i −0.941941 0.335779i \(-0.891000\pi\)
0.941941 0.335779i \(-0.109000\pi\)
\(104\) −14.1837 −1.39082
\(105\) −0.520743 −0.0508193
\(106\) 10.0042i 0.971695i
\(107\) 17.4233 1.68438 0.842188 0.539184i \(-0.181267\pi\)
0.842188 + 0.539184i \(0.181267\pi\)
\(108\) 9.90898i 0.953492i
\(109\) 16.9175 1.62040 0.810201 0.586152i \(-0.199358\pi\)
0.810201 + 0.586152i \(0.199358\pi\)
\(110\) 4.93299 6.78674i 0.470342 0.647090i
\(111\) −0.539978 −0.0512524
\(112\) 8.87070i 0.838203i
\(113\) 9.65154i 0.907940i −0.891017 0.453970i \(-0.850007\pi\)
0.891017 0.453970i \(-0.149993\pi\)
\(114\) −4.21235 0.517924i −0.394523 0.0485080i
\(115\) −0.981945 −0.0915668
\(116\) −13.4003 −1.24419
\(117\) 6.66389 0.616077
\(118\) 3.18855i 0.293530i
\(119\) 7.34673 0.673474
\(120\) 2.33627i 0.213271i
\(121\) −3.39487 10.4630i −0.308625 0.951184i
\(122\) 24.1787i 2.18904i
\(123\) 0.432382i 0.0389865i
\(124\) 35.6055i 3.19747i
\(125\) −1.00000 −0.0894427
\(126\) 9.76090i 0.869570i
\(127\) 11.1944 0.993347 0.496673 0.867938i \(-0.334555\pi\)
0.496673 + 0.867938i \(0.334555\pi\)
\(128\) 13.6120 1.20314
\(129\) 0.777048 0.0684152
\(130\) 5.91114 0.518442
\(131\) 11.7600i 1.02748i −0.857947 0.513738i \(-0.828260\pi\)
0.857947 0.513738i \(-0.171740\pi\)
\(132\) 4.54280 + 3.30197i 0.395400 + 0.287400i
\(133\) −5.85340 0.719696i −0.507554 0.0624055i
\(134\) 33.5725i 2.90022i
\(135\) 2.25231i 0.193848i
\(136\) 32.9605i 2.82634i
\(137\) 5.82804 0.497923 0.248962 0.968513i \(-0.419911\pi\)
0.248962 + 0.968513i \(0.419911\pi\)
\(138\) 0.956078i 0.0813867i
\(139\) 1.94056i 0.164596i −0.996608 0.0822980i \(-0.973774\pi\)
0.996608 0.0822980i \(-0.0262259\pi\)
\(140\) 5.95238i 0.503068i
\(141\) 4.73349i 0.398631i
\(142\) 36.3287i 3.04863i
\(143\) 4.55656 6.26885i 0.381039 0.524228i
\(144\) 18.6981 1.55817
\(145\) 3.04589 0.252948
\(146\) 5.20718i 0.430950i
\(147\) 1.98966i 0.164105i
\(148\) 6.17225i 0.507356i
\(149\) 9.35580i 0.766457i −0.923654 0.383228i \(-0.874812\pi\)
0.923654 0.383228i \(-0.125188\pi\)
\(150\) 0.973657i 0.0794988i
\(151\) −9.35512 −0.761309 −0.380655 0.924717i \(-0.624301\pi\)
−0.380655 + 0.924717i \(0.624301\pi\)
\(152\) −3.22886 + 26.2608i −0.261895 + 2.13003i
\(153\) 15.4858i 1.25195i
\(154\) 9.18228 + 6.67421i 0.739929 + 0.537823i
\(155\) 8.09312i 0.650055i
\(156\) 3.95671i 0.316790i
\(157\) 6.63790 0.529762 0.264881 0.964281i \(-0.414667\pi\)
0.264881 + 0.964281i \(0.414667\pi\)
\(158\) −24.2104 −1.92608
\(159\) 1.52210 0.120711
\(160\) 4.44597 0.351484
\(161\) 1.32855i 0.104704i
\(162\) −19.4502 −1.52816
\(163\) 14.1577 1.10892 0.554458 0.832212i \(-0.312926\pi\)
0.554458 + 0.832212i \(0.312926\pi\)
\(164\) 4.94236 0.385934
\(165\) −1.03258 0.750537i −0.0803861 0.0584292i
\(166\) 25.7621i 1.99953i
\(167\) 2.40922 0.186431 0.0932155 0.995646i \(-0.470285\pi\)
0.0932155 + 0.995646i \(0.470285\pi\)
\(168\) −3.16091 −0.243869
\(169\) −7.53993 −0.579994
\(170\) 13.7365i 1.05354i
\(171\) 1.51701 12.3381i 0.116009 0.943515i
\(172\) 8.88209i 0.677253i
\(173\) −10.2515 −0.779410 −0.389705 0.920940i \(-0.627423\pi\)
−0.389705 + 0.920940i \(0.627423\pi\)
\(174\) 2.96566i 0.224826i
\(175\) 1.35297i 0.102275i
\(176\) 12.7852 17.5897i 0.963720 1.32587i
\(177\) −0.485126 −0.0364643
\(178\) 31.2103i 2.33931i
\(179\) 16.5494i 1.23696i −0.785799 0.618482i \(-0.787748\pi\)
0.785799 0.618482i \(-0.212252\pi\)
\(180\) −12.5467 −0.935176
\(181\) 17.2195i 1.27992i 0.768410 + 0.639958i \(0.221048\pi\)
−0.768410 + 0.639958i \(0.778952\pi\)
\(182\) 7.99762i 0.592823i
\(183\) −3.67870 −0.271937
\(184\) −5.96041 −0.439407
\(185\) 1.40295i 0.103147i
\(186\) −7.87993 −0.577784
\(187\) 14.5678 + 10.5887i 1.06530 + 0.774323i
\(188\) −54.1064 −3.94611
\(189\) 3.04731 0.221659
\(190\) 1.34565 10.9444i 0.0976237 0.793988i
\(191\) 10.7378 0.776960 0.388480 0.921457i \(-0.373000\pi\)
0.388480 + 0.921457i \(0.373000\pi\)
\(192\) 0.718145i 0.0518276i
\(193\) −11.7328 −0.844543 −0.422272 0.906469i \(-0.638767\pi\)
−0.422272 + 0.906469i \(0.638767\pi\)
\(194\) 5.65830i 0.406242i
\(195\) 0.899359i 0.0644044i
\(196\) 22.7429 1.62450
\(197\) 10.9577i 0.780704i −0.920666 0.390352i \(-0.872353\pi\)
0.920666 0.390352i \(-0.127647\pi\)
\(198\) −14.0682 + 19.3548i −0.999785 + 1.37549i
\(199\) 15.9815 1.13290 0.566448 0.824098i \(-0.308317\pi\)
0.566448 + 0.824098i \(0.308317\pi\)
\(200\) −6.07000 −0.429214
\(201\) 5.10793 0.360285
\(202\) 12.8653i 0.905201i
\(203\) 4.12101i 0.289238i
\(204\) −9.19475 −0.643761
\(205\) −1.12340 −0.0784615
\(206\) 17.2415i 1.20127i
\(207\) 2.80037 0.194639
\(208\) 15.3203 1.06227
\(209\) −10.5694 9.86348i −0.731099 0.682271i
\(210\) 1.31733 0.0909046
\(211\) −24.8040 −1.70758 −0.853790 0.520617i \(-0.825702\pi\)
−0.853790 + 0.520617i \(0.825702\pi\)
\(212\) 17.3985i 1.19493i
\(213\) −5.52727 −0.378722
\(214\) −44.0761 −3.01298
\(215\) 2.01890i 0.137688i
\(216\) 13.6715i 0.930229i
\(217\) −10.9498 −0.743319
\(218\) −42.7965 −2.89855
\(219\) −0.792254 −0.0535356
\(220\) −8.57905 + 11.8029i −0.578400 + 0.795754i
\(221\) 12.6883i 0.853509i
\(222\) 1.36599 0.0916794
\(223\) 26.5583i 1.77848i 0.457442 + 0.889239i \(0.348766\pi\)
−0.457442 + 0.889239i \(0.651234\pi\)
\(224\) 6.01527i 0.401912i
\(225\) 2.85186 0.190124
\(226\) 24.4157i 1.62411i
\(227\) −3.31971 −0.220337 −0.110168 0.993913i \(-0.535139\pi\)
−0.110168 + 0.993913i \(0.535139\pi\)
\(228\) 7.32577 + 0.900730i 0.485161 + 0.0596523i
\(229\) 2.71143 0.179177 0.0895883 0.995979i \(-0.471445\pi\)
0.0895883 + 0.995979i \(0.471445\pi\)
\(230\) 2.48404 0.163793
\(231\) 1.01546 1.39705i 0.0668121 0.0919191i
\(232\) 18.4886 1.21384
\(233\) 2.91234i 0.190794i −0.995439 0.0953970i \(-0.969588\pi\)
0.995439 0.0953970i \(-0.0304120\pi\)
\(234\) −16.8578 −1.10203
\(235\) 12.2984 0.802257
\(236\) 5.54526i 0.360966i
\(237\) 3.68353i 0.239271i
\(238\) −18.5852 −1.20470
\(239\) 20.7111i 1.33969i 0.742502 + 0.669843i \(0.233639\pi\)
−0.742502 + 0.669843i \(0.766361\pi\)
\(240\) 2.52350i 0.162891i
\(241\) −6.54934 −0.421880 −0.210940 0.977499i \(-0.567652\pi\)
−0.210940 + 0.977499i \(0.567652\pi\)
\(242\) 8.58807 + 26.4685i 0.552062 + 1.70146i
\(243\) 9.71621i 0.623295i
\(244\) 42.0496i 2.69195i
\(245\) −5.16946 −0.330265
\(246\) 1.09380i 0.0697384i
\(247\) 1.24296 10.1092i 0.0790879 0.643234i
\(248\) 49.1253i 3.11946i
\(249\) 3.91961 0.248395
\(250\) 2.52972 0.159993
\(251\) −8.69353 −0.548731 −0.274365 0.961626i \(-0.588468\pi\)
−0.274365 + 0.961626i \(0.588468\pi\)
\(252\) 16.9754i 1.06935i
\(253\) 1.91481 2.63437i 0.120383 0.165621i
\(254\) −28.3188 −1.77688
\(255\) 2.08996 0.130879
\(256\) −30.7029 −1.91893
\(257\) 17.6353i 1.10006i −0.835145 0.550030i \(-0.814616\pi\)
0.835145 0.550030i \(-0.185384\pi\)
\(258\) −1.96571 −0.122380
\(259\) 1.89815 0.117946
\(260\) −10.2802 −0.637549
\(261\) −8.68647 −0.537679
\(262\) 29.7495i 1.83793i
\(263\) 24.9750i 1.54002i 0.638030 + 0.770012i \(0.279750\pi\)
−0.638030 + 0.770012i \(0.720250\pi\)
\(264\) −6.26775 4.55576i −0.385753 0.280388i
\(265\) 3.95467i 0.242934i
\(266\) 14.8074 + 1.82063i 0.907903 + 0.111630i
\(267\) 4.74854 0.290606
\(268\) 58.3864i 3.56652i
\(269\) 14.4347i 0.880096i 0.897974 + 0.440048i \(0.145039\pi\)
−0.897974 + 0.440048i \(0.854961\pi\)
\(270\) 5.69771i 0.346751i
\(271\) 3.81840i 0.231951i −0.993252 0.115976i \(-0.963001\pi\)
0.993252 0.115976i \(-0.0369994\pi\)
\(272\) 35.6019i 2.15868i
\(273\) 1.21681 0.0736446
\(274\) −14.7433 −0.890676
\(275\) 1.95002 2.68280i 0.117590 0.161779i
\(276\) 1.66273i 0.100085i
\(277\) 8.53684i 0.512929i −0.966554 0.256464i \(-0.917442\pi\)
0.966554 0.256464i \(-0.0825577\pi\)
\(278\) 4.90907i 0.294426i
\(279\) 23.0805i 1.38179i
\(280\) 8.21255i 0.490794i
\(281\) 11.5517 0.689114 0.344557 0.938765i \(-0.388029\pi\)
0.344557 + 0.938765i \(0.388029\pi\)
\(282\) 11.9744i 0.713064i
\(283\) 23.1870i 1.37833i −0.724606 0.689163i \(-0.757978\pi\)
0.724606 0.689163i \(-0.242022\pi\)
\(284\) 63.1798i 3.74903i
\(285\) −1.66515 0.204736i −0.0986348 0.0121275i
\(286\) −11.5268 + 15.8584i −0.681596 + 0.937729i
\(287\) 1.51993i 0.0897185i
\(288\) −12.6793 −0.747134
\(289\) −12.4856 −0.734446
\(290\) −7.70526 −0.452468
\(291\) 0.860890 0.0504663
\(292\) 9.05590i 0.529957i
\(293\) 24.5380 1.43353 0.716763 0.697317i \(-0.245623\pi\)
0.716763 + 0.697317i \(0.245623\pi\)
\(294\) 5.03328i 0.293547i
\(295\) 1.26044i 0.0733854i
\(296\) 8.51591i 0.494977i
\(297\) 6.04250 + 4.39204i 0.350621 + 0.254852i
\(298\) 23.6675i 1.37102i
\(299\) 2.29449 0.132694
\(300\) 1.69330i 0.0977629i
\(301\) −2.73151 −0.157442
\(302\) 23.6658 1.36182
\(303\) −1.95741 −0.112450
\(304\) 3.48761 28.3653i 0.200028 1.62686i
\(305\) 9.55786i 0.547281i
\(306\) 39.1747i 2.23947i
\(307\) −21.7400 −1.24077 −0.620384 0.784298i \(-0.713023\pi\)
−0.620384 + 0.784298i \(0.713023\pi\)
\(308\) −15.9691 11.6072i −0.909921 0.661383i
\(309\) 2.62323 0.149230
\(310\) 20.4733i 1.16281i
\(311\) −1.75399 −0.0994594 −0.0497297 0.998763i \(-0.515836\pi\)
−0.0497297 + 0.998763i \(0.515836\pi\)
\(312\) 5.45911i 0.309061i
\(313\) 9.69040 0.547734 0.273867 0.961768i \(-0.411697\pi\)
0.273867 + 0.961768i \(0.411697\pi\)
\(314\) −16.7920 −0.947629
\(315\) 3.85849i 0.217401i
\(316\) 42.1048 2.36858
\(317\) 1.24283i 0.0698042i −0.999391 0.0349021i \(-0.988888\pi\)
0.999391 0.0349021i \(-0.0111119\pi\)
\(318\) −3.85050 −0.215925
\(319\) −5.93954 + 8.17154i −0.332551 + 0.457518i
\(320\) 1.86586 0.104305
\(321\) 6.70602i 0.374293i
\(322\) 3.36085i 0.187293i
\(323\) 23.4922 + 2.88845i 1.30714 + 0.160718i
\(324\) 33.8263 1.87924
\(325\) 2.33668 0.129616
\(326\) −35.8150 −1.98361
\(327\) 6.51133i 0.360078i
\(328\) −6.81903 −0.376518
\(329\) 16.6394i 0.917357i
\(330\) 2.61213 + 1.89865i 0.143793 + 0.104517i
\(331\) 33.0877i 1.81866i 0.416073 + 0.909331i \(0.363406\pi\)
−0.416073 + 0.909331i \(0.636594\pi\)
\(332\) 44.8033i 2.45890i
\(333\) 4.00102i 0.219255i
\(334\) −6.09465 −0.333484
\(335\) 13.2712i 0.725084i
\(336\) 3.41422 0.186261
\(337\) −0.317131 −0.0172752 −0.00863760 0.999963i \(-0.502749\pi\)
−0.00863760 + 0.999963i \(0.502749\pi\)
\(338\) 19.0739 1.03748
\(339\) 3.71476 0.201758
\(340\) 23.8894i 1.29559i
\(341\) −21.7123 15.7817i −1.17578 0.854628i
\(342\) −3.83761 + 31.2118i −0.207514 + 1.68774i
\(343\) 16.4650i 0.889024i
\(344\) 12.2547i 0.660729i
\(345\) 0.377938i 0.0203475i
\(346\) 25.9335 1.39419
\(347\) 10.3945i 0.558008i 0.960290 + 0.279004i \(0.0900043\pi\)
−0.960290 + 0.279004i \(0.909996\pi\)
\(348\) 5.15763i 0.276478i
\(349\) 14.6428i 0.783812i −0.920005 0.391906i \(-0.871816\pi\)
0.920005 0.391906i \(-0.128184\pi\)
\(350\) 3.42264i 0.182948i
\(351\) 5.26292i 0.280914i
\(352\) −8.66971 + 11.9277i −0.462097 + 0.635746i
\(353\) 10.5498 0.561507 0.280753 0.959780i \(-0.409416\pi\)
0.280753 + 0.959780i \(0.409416\pi\)
\(354\) 1.22723 0.0652267
\(355\) 14.3607i 0.762189i
\(356\) 54.2784i 2.87675i
\(357\) 2.82766i 0.149656i
\(358\) 41.8654i 2.21266i
\(359\) 20.9700i 1.10675i −0.832931 0.553376i \(-0.813339\pi\)
0.832931 0.553376i \(-0.186661\pi\)
\(360\) 17.3108 0.912360
\(361\) −18.4341 4.60265i −0.970215 0.242245i
\(362\) 43.5605i 2.28949i
\(363\) 4.02709 1.30664i 0.211367 0.0685810i
\(364\) 13.9088i 0.729019i
\(365\) 2.05840i 0.107742i
\(366\) 9.30608 0.486437
\(367\) −18.1151 −0.945599 −0.472799 0.881170i \(-0.656757\pi\)
−0.472799 + 0.881170i \(0.656757\pi\)
\(368\) 6.43807 0.335608
\(369\) 3.20377 0.166782
\(370\) 3.54907i 0.184507i
\(371\) −5.35057 −0.277788
\(372\) 13.7041 0.710525
\(373\) −22.2513 −1.15213 −0.576064 0.817405i \(-0.695412\pi\)
−0.576064 + 0.817405i \(0.695412\pi\)
\(374\) −36.8524 26.7865i −1.90559 1.38510i
\(375\) 0.384887i 0.0198755i
\(376\) 74.6511 3.84984
\(377\) −7.11728 −0.366559
\(378\) −7.70885 −0.396500
\(379\) 21.0036i 1.07888i 0.842024 + 0.539441i \(0.181364\pi\)
−0.842024 + 0.539441i \(0.818636\pi\)
\(380\) −2.34024 + 19.0335i −0.120052 + 0.976400i
\(381\) 4.30860i 0.220736i
\(382\) −27.1636 −1.38981
\(383\) 17.6182i 0.900249i −0.892966 0.450125i \(-0.851380\pi\)
0.892966 0.450125i \(-0.148620\pi\)
\(384\) 5.23910i 0.267357i
\(385\) 3.62976 + 2.63832i 0.184990 + 0.134461i
\(386\) 29.6806 1.51070
\(387\) 5.75761i 0.292676i
\(388\) 9.84045i 0.499573i
\(389\) −12.7126 −0.644555 −0.322278 0.946645i \(-0.604448\pi\)
−0.322278 + 0.946645i \(0.604448\pi\)
\(390\) 2.27512i 0.115205i
\(391\) 5.33202i 0.269652i
\(392\) −31.3787 −1.58486
\(393\) 4.52628 0.228321
\(394\) 27.7199i 1.39651i
\(395\) −9.57041 −0.481540
\(396\) 24.4663 33.6603i 1.22948 1.69150i
\(397\) −3.16750 −0.158972 −0.0794862 0.996836i \(-0.525328\pi\)
−0.0794862 + 0.996836i \(0.525328\pi\)
\(398\) −40.4286 −2.02650
\(399\) 0.277002 2.25290i 0.0138674 0.112786i
\(400\) 6.55645 0.327822
\(401\) 10.7406i 0.536358i 0.963369 + 0.268179i \(0.0864219\pi\)
−0.963369 + 0.268179i \(0.913578\pi\)
\(402\) −12.9216 −0.644472
\(403\) 18.9110i 0.942026i
\(404\) 22.3743i 1.11316i
\(405\) −7.68870 −0.382055
\(406\) 10.4250i 0.517384i
\(407\) 3.76384 + 2.73578i 0.186567 + 0.135607i
\(408\) 12.6861 0.628055
\(409\) −16.5390 −0.817801 −0.408900 0.912579i \(-0.634088\pi\)
−0.408900 + 0.912579i \(0.634088\pi\)
\(410\) 2.84188 0.140350
\(411\) 2.24314i 0.110646i
\(412\) 29.9849i 1.47725i
\(413\) 1.70534 0.0839141
\(414\) −7.08415 −0.348167
\(415\) 10.1838i 0.499902i
\(416\) −10.3888 −0.509353
\(417\) 0.746897 0.0365757
\(418\) 26.7376 + 24.9518i 1.30778 + 1.22043i
\(419\) 7.71720 0.377010 0.188505 0.982072i \(-0.439636\pi\)
0.188505 + 0.982072i \(0.439636\pi\)
\(420\) −2.29100 −0.111789
\(421\) 25.5946i 1.24740i −0.781662 0.623702i \(-0.785628\pi\)
0.781662 0.623702i \(-0.214372\pi\)
\(422\) 62.7473 3.05449
\(423\) −35.0732 −1.70532
\(424\) 24.0049i 1.16578i
\(425\) 5.43006i 0.263397i
\(426\) 13.9824 0.677452
\(427\) 12.9315 0.625801
\(428\) 76.6535 3.70519
\(429\) 2.41280 + 1.75376i 0.116491 + 0.0846725i
\(430\) 5.10724i 0.246293i
\(431\) −33.3982 −1.60873 −0.804367 0.594133i \(-0.797495\pi\)
−0.804367 + 0.594133i \(0.797495\pi\)
\(432\) 14.7671i 0.710485i
\(433\) 24.2308i 1.16446i 0.813025 + 0.582228i \(0.197819\pi\)
−0.813025 + 0.582228i \(0.802181\pi\)
\(434\) 27.6999 1.32964
\(435\) 1.17233i 0.0562088i
\(436\) 74.4282 3.56446
\(437\) 0.522332 4.24821i 0.0249865 0.203219i
\(438\) 2.00418 0.0957634
\(439\) 30.2441 1.44347 0.721737 0.692167i \(-0.243344\pi\)
0.721737 + 0.692167i \(0.243344\pi\)
\(440\) 11.8366 16.2846i 0.564288 0.776339i
\(441\) 14.7426 0.702028
\(442\) 32.0979i 1.52674i
\(443\) −2.75876 −0.131073 −0.0655363 0.997850i \(-0.520876\pi\)
−0.0655363 + 0.997850i \(0.520876\pi\)
\(444\) −2.37562 −0.112742
\(445\) 12.3375i 0.584852i
\(446\) 67.1852i 3.18131i
\(447\) 3.60093 0.170318
\(448\) 2.52445i 0.119269i
\(449\) 32.7641i 1.54623i 0.634265 + 0.773116i \(0.281303\pi\)
−0.634265 + 0.773116i \(0.718697\pi\)
\(450\) −7.21441 −0.340090
\(451\) 2.19064 3.01386i 0.103153 0.141917i
\(452\) 42.4617i 1.99723i
\(453\) 3.60067i 0.169174i
\(454\) 8.39793 0.394134
\(455\) 3.16147i 0.148212i
\(456\) −10.1074 1.24275i −0.473325 0.0581969i
\(457\) 13.2581i 0.620186i 0.950706 + 0.310093i \(0.100360\pi\)
−0.950706 + 0.310093i \(0.899640\pi\)
\(458\) −6.85916 −0.320508
\(459\) −12.2302 −0.570856
\(460\) −4.32004 −0.201423
\(461\) 23.8716i 1.11181i 0.831246 + 0.555905i \(0.187628\pi\)
−0.831246 + 0.555905i \(0.812372\pi\)
\(462\) −2.56882 + 3.53414i −0.119512 + 0.164423i
\(463\) 7.08380 0.329212 0.164606 0.986359i \(-0.447365\pi\)
0.164606 + 0.986359i \(0.447365\pi\)
\(464\) −19.9703 −0.927096
\(465\) −3.11494 −0.144452
\(466\) 7.36741i 0.341289i
\(467\) 0.535622 0.0247856 0.0123928 0.999923i \(-0.496055\pi\)
0.0123928 + 0.999923i \(0.496055\pi\)
\(468\) 29.3176 1.35521
\(469\) −17.9556 −0.829113
\(470\) −31.1114 −1.43506
\(471\) 2.55485i 0.117721i
\(472\) 7.65085i 0.352159i
\(473\) −5.41630 3.93688i −0.249042 0.181018i
\(474\) 9.31830i 0.428004i
\(475\) 0.531936 4.32632i 0.0244069 0.198505i
\(476\) 32.3218 1.48147
\(477\) 11.2782i 0.516393i
\(478\) 52.3931i 2.39641i
\(479\) 19.4054i 0.886657i −0.896359 0.443329i \(-0.853797\pi\)
0.896359 0.443329i \(-0.146203\pi\)
\(480\) 1.71120i 0.0781051i
\(481\) 3.27824i 0.149475i
\(482\) 16.5680 0.754651
\(483\) 0.511340 0.0232668
\(484\) −14.9357 46.0318i −0.678894 2.09236i
\(485\) 2.23673i 0.101565i
\(486\) 24.5793i 1.11494i
\(487\) 7.55916i 0.342538i 0.985224 + 0.171269i \(0.0547867\pi\)
−0.985224 + 0.171269i \(0.945213\pi\)
\(488\) 58.0163i 2.62627i
\(489\) 5.44912i 0.246418i
\(490\) 13.0773 0.590772
\(491\) 30.9535i 1.39691i −0.715654 0.698455i \(-0.753871\pi\)
0.715654 0.698455i \(-0.246129\pi\)
\(492\) 1.90225i 0.0857602i
\(493\) 16.5394i 0.744897i
\(494\) −3.14435 + 25.5735i −0.141471 + 1.15061i
\(495\) −5.56118 + 7.65098i −0.249956 + 0.343886i
\(496\) 53.0621i 2.38256i
\(497\) 19.4297 0.871542
\(498\) −9.91551 −0.444325
\(499\) 40.9890 1.83492 0.917459 0.397831i \(-0.130237\pi\)
0.917459 + 0.397831i \(0.130237\pi\)
\(500\) −4.39948 −0.196751
\(501\) 0.927278i 0.0414277i
\(502\) 21.9922 0.981559
\(503\) 17.7033i 0.789350i 0.918821 + 0.394675i \(0.129143\pi\)
−0.918821 + 0.394675i \(0.870857\pi\)
\(504\) 23.4211i 1.04326i
\(505\) 5.08567i 0.226310i
\(506\) −4.84393 + 6.66420i −0.215339 + 0.296260i
\(507\) 2.90202i 0.128883i
\(508\) 49.2497 2.18510
\(509\) 40.5717i 1.79831i −0.437630 0.899155i \(-0.644182\pi\)
0.437630 0.899155i \(-0.355818\pi\)
\(510\) −5.28702 −0.234113
\(511\) 2.78497 0.123200
\(512\) 50.4456 2.22940
\(513\) 9.74421 + 1.19808i 0.430217 + 0.0528967i
\(514\) 44.6124i 1.96777i
\(515\) 6.81557i 0.300330i
\(516\) 3.41860 0.150496
\(517\) −23.9820 + 32.9941i −1.05473 + 1.45108i
\(518\) −4.80180 −0.210979
\(519\) 3.94569i 0.173196i
\(520\) 14.1837 0.621994
\(521\) 28.1322i 1.23249i 0.787553 + 0.616247i \(0.211348\pi\)
−0.787553 + 0.616247i \(0.788652\pi\)
\(522\) 21.9743 0.961790
\(523\) −13.7730 −0.602252 −0.301126 0.953584i \(-0.597362\pi\)
−0.301126 + 0.953584i \(0.597362\pi\)
\(524\) 51.7379i 2.26018i
\(525\) 0.520743 0.0227271
\(526\) 63.1797i 2.75477i
\(527\) 43.9462 1.91432
\(528\) 6.77004 + 4.92086i 0.294628 + 0.214153i
\(529\) −22.0358 −0.958078
\(530\) 10.0042i 0.434555i
\(531\) 3.59459i 0.155992i
\(532\) −25.7519 3.16628i −1.11649 0.137276i
\(533\) 2.62502 0.113702
\(534\) −12.0125 −0.519830
\(535\) −17.4233 −0.753276
\(536\) 80.5564i 3.47951i
\(537\) 6.36967 0.274872
\(538\) 36.5156i 1.57430i
\(539\) 10.0805 13.8687i 0.434200 0.597365i
\(540\) 9.90898i 0.426415i
\(541\) 19.7937i 0.850996i 0.904959 + 0.425498i \(0.139901\pi\)
−0.904959 + 0.425498i \(0.860099\pi\)
\(542\) 9.65947i 0.414910i
\(543\) −6.62757 −0.284416
\(544\) 24.1419i 1.03507i
\(545\) −16.9175 −0.724666
\(546\) −3.07818 −0.131734
\(547\) 38.3756 1.64082 0.820411 0.571774i \(-0.193744\pi\)
0.820411 + 0.571774i \(0.193744\pi\)
\(548\) 25.6403 1.09530
\(549\) 27.2577i 1.16333i
\(550\) −4.93299 + 6.78674i −0.210344 + 0.289388i
\(551\) −1.62022 + 13.1775i −0.0690238 + 0.561381i
\(552\) 2.29409i 0.0976428i
\(553\) 12.9485i 0.550627i
\(554\) 21.5958i 0.917518i
\(555\) 0.539978 0.0229208
\(556\) 8.53744i 0.362068i
\(557\) 17.2554i 0.731136i 0.930785 + 0.365568i \(0.119125\pi\)
−0.930785 + 0.365568i \(0.880875\pi\)
\(558\) 58.3871i 2.47172i
\(559\) 4.71751i 0.199529i
\(560\) 8.87070i 0.374856i
\(561\) −4.07546 + 5.60696i −0.172066 + 0.236726i
\(562\) −29.2224 −1.23267
\(563\) 27.2632 1.14901 0.574503 0.818502i \(-0.305195\pi\)
0.574503 + 0.818502i \(0.305195\pi\)
\(564\) 20.8249i 0.876885i
\(565\) 9.65154i 0.406043i
\(566\) 58.6567i 2.46553i
\(567\) 10.4026i 0.436868i
\(568\) 87.1698i 3.65756i
\(569\) 28.1885 1.18172 0.590862 0.806772i \(-0.298788\pi\)
0.590862 + 0.806772i \(0.298788\pi\)
\(570\) 4.21235 + 0.517924i 0.176436 + 0.0216934i
\(571\) 20.8031i 0.870581i 0.900290 + 0.435290i \(0.143354\pi\)
−0.900290 + 0.435290i \(0.856646\pi\)
\(572\) 20.0465 27.5797i 0.838186 1.15316i
\(573\) 4.13284i 0.172652i
\(574\) 3.84499i 0.160487i
\(575\) 0.981945 0.0409499
\(576\) −5.32116 −0.221715
\(577\) −43.6723 −1.81810 −0.909051 0.416685i \(-0.863192\pi\)
−0.909051 + 0.416685i \(0.863192\pi\)
\(578\) 31.5850 1.31376
\(579\) 4.51580i 0.187670i
\(580\) 13.4003 0.556419
\(581\) −13.7784 −0.571624
\(582\) −2.17781 −0.0902731
\(583\) −10.6096 7.71168i −0.439405 0.319385i
\(584\) 12.4945i 0.517027i
\(585\) −6.66389 −0.275518
\(586\) −62.0743 −2.56426
\(587\) −27.5563 −1.13737 −0.568685 0.822555i \(-0.692548\pi\)
−0.568685 + 0.822555i \(0.692548\pi\)
\(588\) 8.75347i 0.360987i
\(589\) −35.0134 4.30502i −1.44270 0.177385i
\(590\) 3.18855i 0.131270i
\(591\) 4.21748 0.173484
\(592\) 9.19837i 0.378051i
\(593\) 17.1694i 0.705063i −0.935800 0.352531i \(-0.885321\pi\)
0.935800 0.352531i \(-0.114679\pi\)
\(594\) −15.2858 11.1106i −0.627185 0.455874i
\(595\) −7.34673 −0.301187
\(596\) 41.1606i 1.68600i
\(597\) 6.15106i 0.251746i
\(598\) −5.80442 −0.237360
\(599\) 6.49020i 0.265182i 0.991171 + 0.132591i \(0.0423297\pi\)
−0.991171 + 0.132591i \(0.957670\pi\)
\(600\) 2.33627i 0.0953778i
\(601\) −44.8595 −1.82986 −0.914928 0.403618i \(-0.867753\pi\)
−0.914928 + 0.403618i \(0.867753\pi\)
\(602\) 6.90996 0.281629
\(603\) 37.8477i 1.54128i
\(604\) −41.1577 −1.67468
\(605\) 3.39487 + 10.4630i 0.138021 + 0.425382i
\(606\) 4.95170 0.201149
\(607\) −30.0795 −1.22089 −0.610446 0.792058i \(-0.709010\pi\)
−0.610446 + 0.792058i \(0.709010\pi\)
\(608\) −2.36497 + 19.2347i −0.0959122 + 0.780069i
\(609\) −1.58613 −0.0642731
\(610\) 24.1787i 0.978967i
\(611\) −28.7373 −1.16259
\(612\) 68.1294i 2.75397i
\(613\) 13.4630i 0.543764i 0.962331 + 0.271882i \(0.0876460\pi\)
−0.962331 + 0.271882i \(0.912354\pi\)
\(614\) 54.9962 2.21946
\(615\) 0.432382i 0.0174353i
\(616\) 22.0327 + 16.0146i 0.887722 + 0.645247i
\(617\) 9.32634 0.375464 0.187732 0.982220i \(-0.439886\pi\)
0.187732 + 0.982220i \(0.439886\pi\)
\(618\) −6.63603 −0.266940
\(619\) 17.9855 0.722899 0.361449 0.932392i \(-0.382282\pi\)
0.361449 + 0.932392i \(0.382282\pi\)
\(620\) 35.6055i 1.42995i
\(621\) 2.21164i 0.0887502i
\(622\) 4.43709 0.177911
\(623\) −16.6923 −0.668762
\(624\) 5.89660i 0.236053i
\(625\) 1.00000 0.0400000
\(626\) −24.5140 −0.979776
\(627\) 3.79633 4.06802i 0.151611 0.162461i
\(628\) 29.2033 1.16534
\(629\) −7.61811 −0.303754
\(630\) 9.76090i 0.388884i
\(631\) 10.4925 0.417701 0.208850 0.977948i \(-0.433028\pi\)
0.208850 + 0.977948i \(0.433028\pi\)
\(632\) −58.0924 −2.31079
\(633\) 9.54677i 0.379450i
\(634\) 3.14401i 0.124864i
\(635\) −11.1944 −0.444238
\(636\) 6.69647 0.265532
\(637\) 12.0794 0.478602
\(638\) 15.0254 20.6717i 0.594860 0.818400i
\(639\) 40.9549i 1.62015i
\(640\) −13.6120 −0.538062
\(641\) 6.59173i 0.260358i 0.991491 + 0.130179i \(0.0415551\pi\)
−0.991491 + 0.130179i \(0.958445\pi\)
\(642\) 16.9643i 0.669529i
\(643\) 42.3494 1.67010 0.835049 0.550176i \(-0.185439\pi\)
0.835049 + 0.550176i \(0.185439\pi\)
\(644\) 5.84491i 0.230322i
\(645\) −0.777048 −0.0305962
\(646\) −59.4286 7.30696i −2.33819 0.287488i
\(647\) −37.6226 −1.47910 −0.739549 0.673103i \(-0.764961\pi\)
−0.739549 + 0.673103i \(0.764961\pi\)
\(648\) −46.6704 −1.83339
\(649\) 3.38150 + 2.45787i 0.132736 + 0.0964798i
\(650\) −5.91114 −0.231854
\(651\) 4.21443i 0.165177i
\(652\) 62.2864 2.43932
\(653\) 2.79930 0.109545 0.0547726 0.998499i \(-0.482557\pi\)
0.0547726 + 0.998499i \(0.482557\pi\)
\(654\) 16.4718i 0.644100i
\(655\) 11.7600i 0.459501i
\(656\) 7.36550 0.287574
\(657\) 5.87028i 0.229022i
\(658\) 42.0929i 1.64095i
\(659\) −28.1649 −1.09715 −0.548574 0.836102i \(-0.684829\pi\)
−0.548574 + 0.836102i \(0.684829\pi\)
\(660\) −4.54280 3.30197i −0.176828 0.128529i
\(661\) 23.2863i 0.905731i 0.891579 + 0.452865i \(0.149598\pi\)
−0.891579 + 0.452865i \(0.850402\pi\)
\(662\) 83.7025i 3.25319i
\(663\) −4.88357 −0.189662
\(664\) 61.8156i 2.39891i
\(665\) 5.85340 + 0.719696i 0.226985 + 0.0279086i
\(666\) 10.1215i 0.392198i
\(667\) −2.99090 −0.115808
\(668\) 10.5993 0.410099
\(669\) −10.2220 −0.395204
\(670\) 33.5725i 1.29702i
\(671\) 25.6419 + 18.6380i 0.989893 + 0.719511i
\(672\) −2.31520 −0.0893109
\(673\) 21.5800 0.831847 0.415924 0.909400i \(-0.363458\pi\)
0.415924 + 0.909400i \(0.363458\pi\)
\(674\) 0.802251 0.0309016
\(675\) 2.25231i 0.0866914i
\(676\) −33.1717 −1.27584
\(677\) 40.3794 1.55190 0.775952 0.630791i \(-0.217270\pi\)
0.775952 + 0.630791i \(0.217270\pi\)
\(678\) −9.39729 −0.360901
\(679\) −3.02624 −0.116136
\(680\) 32.9605i 1.26398i
\(681\) 1.27771i 0.0489621i
\(682\) 54.9259 + 39.9233i 2.10322 + 1.52874i
\(683\) 36.7147i 1.40485i −0.711758 0.702424i \(-0.752101\pi\)
0.711758 0.702424i \(-0.247899\pi\)
\(684\) 6.67405 54.2810i 0.255189 2.07549i
\(685\) −5.82804 −0.222678
\(686\) 41.6517i 1.59027i
\(687\) 1.04360i 0.0398157i
\(688\) 13.2368i 0.504648i
\(689\) 9.24081i 0.352047i
\(690\) 0.956078i 0.0363973i
\(691\) 13.6882 0.520724 0.260362 0.965511i \(-0.416158\pi\)
0.260362 + 0.965511i \(0.416158\pi\)
\(692\) −45.1014 −1.71450
\(693\) −10.3516 7.52412i −0.393224 0.285818i
\(694\) 26.2953i 0.998155i
\(695\) 1.94056i 0.0736096i
\(696\) 7.11603i 0.269732i
\(697\) 6.10012i 0.231059i
\(698\) 37.0422i 1.40207i
\(699\) 1.12092 0.0423972
\(700\) 5.95238i 0.224979i
\(701\) 28.9116i 1.09197i 0.837793 + 0.545987i \(0.183845\pi\)
−0.837793 + 0.545987i \(0.816155\pi\)
\(702\) 13.3137i 0.502494i
\(703\) 6.06961 + 0.746280i 0.228920 + 0.0281465i
\(704\) −3.63845 + 5.00573i −0.137129 + 0.188660i
\(705\) 4.73349i 0.178273i
\(706\) −26.6879 −1.00441
\(707\) 6.88078 0.258778
\(708\) −2.13430 −0.0802119
\(709\) 16.2679 0.610954 0.305477 0.952199i \(-0.401184\pi\)
0.305477 + 0.952199i \(0.401184\pi\)
\(710\) 36.3287i 1.36339i
\(711\) 27.2935 1.02359
\(712\) 74.8885i 2.80657i
\(713\) 7.94700i 0.297617i
\(714\) 7.15320i 0.267702i
\(715\) −4.55656 + 6.26885i −0.170406 + 0.234442i
\(716\) 72.8089i 2.72100i
\(717\) −7.97143 −0.297698
\(718\) 53.0481i 1.97974i
\(719\) −9.82583 −0.366442 −0.183221 0.983072i \(-0.558652\pi\)
−0.183221 + 0.983072i \(0.558652\pi\)
\(720\) −18.6981 −0.696837
\(721\) −9.22128 −0.343418
\(722\) 46.6331 + 11.6434i 1.73550 + 0.433323i
\(723\) 2.52076i 0.0937480i
\(724\) 75.7568i 2.81548i
\(725\) −3.04589 −0.113122
\(726\) −10.1874 + 3.30544i −0.378090 + 0.122676i
\(727\) −29.2149 −1.08352 −0.541760 0.840533i \(-0.682242\pi\)
−0.541760 + 0.840533i \(0.682242\pi\)
\(728\) 19.1901i 0.711233i
\(729\) 19.3265 0.715795
\(730\) 5.20718i 0.192727i
\(731\) 10.9627 0.405471
\(732\) −16.1844 −0.598191
\(733\) 27.2427i 1.00623i −0.864219 0.503116i \(-0.832187\pi\)
0.864219 0.503116i \(-0.167813\pi\)
\(734\) 45.8260 1.69147
\(735\) 1.98966i 0.0733898i
\(736\) −4.36569 −0.160922
\(737\) −35.6041 25.8791i −1.31149 0.953269i
\(738\) −8.10465 −0.298336
\(739\) 12.3404i 0.453950i 0.973901 + 0.226975i \(0.0728835\pi\)
−0.973901 + 0.226975i \(0.927116\pi\)
\(740\) 6.17225i 0.226896i
\(741\) 3.89091 + 0.478402i 0.142936 + 0.0175745i
\(742\) 13.5354 0.496901
\(743\) −16.2242 −0.595207 −0.297603 0.954690i \(-0.596187\pi\)
−0.297603 + 0.954690i \(0.596187\pi\)
\(744\) −18.9077 −0.693190
\(745\) 9.35580i 0.342770i
\(746\) 56.2895 2.06090
\(747\) 29.0427i 1.06262i
\(748\) 64.0907 + 46.5848i 2.34339 + 1.70331i
\(749\) 23.5733i 0.861349i
\(750\) 0.973657i 0.0355529i
\(751\) 5.85396i 0.213614i −0.994280 0.106807i \(-0.965937\pi\)
0.994280 0.106807i \(-0.0340627\pi\)
\(752\) −80.6336 −2.94040
\(753\) 3.34603i 0.121936i
\(754\) 18.0047 0.655693
\(755\) 9.35512 0.340468
\(756\) 13.4066 0.487593
\(757\) −40.7510 −1.48112 −0.740561 0.671989i \(-0.765440\pi\)
−0.740561 + 0.671989i \(0.765440\pi\)
\(758\) 53.1331i 1.92988i
\(759\) 1.01393 + 0.736986i 0.0368035 + 0.0267509i
\(760\) 3.22886 26.2608i 0.117123 0.952579i
\(761\) 12.3774i 0.448681i −0.974511 0.224340i \(-0.927977\pi\)
0.974511 0.224340i \(-0.0720227\pi\)
\(762\) 10.8996i 0.394849i
\(763\) 22.8889i 0.828635i
\(764\) 47.2407 1.70911
\(765\) 15.4858i 0.559890i
\(766\) 44.5691i 1.61035i
\(767\) 2.94523i 0.106346i
\(768\) 11.8172i 0.426415i
\(769\) 27.3673i 0.986889i 0.869777 + 0.493444i \(0.164262\pi\)
−0.869777 + 0.493444i \(0.835738\pi\)
\(770\) −9.18228 6.67421i −0.330906 0.240522i
\(771\) 6.78761 0.244450
\(772\) −51.6181 −1.85778
\(773\) 44.5435i 1.60212i 0.598586 + 0.801059i \(0.295729\pi\)
−0.598586 + 0.801059i \(0.704271\pi\)
\(774\) 14.5651i 0.523533i
\(775\) 8.09312i 0.290714i
\(776\) 13.5770i 0.487385i
\(777\) 0.730576i 0.0262093i
\(778\) 32.1593 1.15297
\(779\) 0.597576 4.86018i 0.0214104 0.174134i
\(780\) 3.95671i 0.141673i
\(781\) 38.5271 + 28.0037i 1.37861 + 1.00205i
\(782\) 13.4885i 0.482348i
\(783\) 6.86029i 0.245167i
\(784\) 33.8933 1.21048
\(785\) −6.63790 −0.236917
\(786\) −11.4502 −0.408416
\(787\) −14.5204 −0.517596 −0.258798 0.965932i \(-0.583326\pi\)
−0.258798 + 0.965932i \(0.583326\pi\)
\(788\) 48.2082i 1.71734i
\(789\) −9.61256 −0.342216
\(790\) 24.2104 0.861369
\(791\) −13.0583 −0.464299
\(792\) −33.7564 + 46.4415i −1.19948 + 1.65023i
\(793\) 22.3337i 0.793091i
\(794\) 8.01289 0.284367
\(795\) −1.52210 −0.0539835
\(796\) 70.3101 2.49207
\(797\) 4.45183i 0.157692i 0.996887 + 0.0788459i \(0.0251235\pi\)
−0.996887 + 0.0788459i \(0.974876\pi\)
\(798\) −0.700737 + 5.69920i −0.0248058 + 0.201750i
\(799\) 66.7809i 2.36254i
\(800\) −4.44597 −0.157189
\(801\) 35.1848i 1.24319i
\(802\) 27.1706i 0.959428i
\(803\) 5.52229 + 4.01392i 0.194878 + 0.141648i
\(804\) 22.4722 0.792534
\(805\) 1.32855i 0.0468251i
\(806\) 47.8396i 1.68508i
\(807\) −5.55572 −0.195571
\(808\) 30.8701i 1.08600i
\(809\) 44.2999i 1.55750i −0.627334 0.778750i \(-0.715854\pi\)
0.627334 0.778750i \(-0.284146\pi\)
\(810\) 19.4502 0.683412
\(811\) 23.0013 0.807683 0.403842 0.914829i \(-0.367675\pi\)
0.403842 + 0.914829i \(0.367675\pi\)
\(812\) 18.1303i 0.636249i
\(813\) 1.46965 0.0515430
\(814\) −9.52146 6.92074i −0.333727 0.242572i
\(815\) −14.1577 −0.495922
\(816\) −13.7027 −0.479692
\(817\) −8.73439 1.07392i −0.305578 0.0375718i
\(818\) 41.8390 1.46287
\(819\) 9.01606i 0.315047i
\(820\) −4.94236 −0.172595
\(821\) 27.6565i 0.965218i 0.875836 + 0.482609i \(0.160311\pi\)
−0.875836 + 0.482609i \(0.839689\pi\)
\(822\) 5.67451i 0.197921i
\(823\) 11.9083 0.415098 0.207549 0.978225i \(-0.433451\pi\)
0.207549 + 0.978225i \(0.433451\pi\)
\(824\) 41.3705i 1.44121i
\(825\) 1.03258 + 0.750537i 0.0359497 + 0.0261303i
\(826\) −4.31402 −0.150104
\(827\) 11.4708 0.398879 0.199440 0.979910i \(-0.436088\pi\)
0.199440 + 0.979910i \(0.436088\pi\)
\(828\) 12.3202 0.428156
\(829\) 39.9028i 1.38588i 0.720995 + 0.692941i \(0.243685\pi\)
−0.720995 + 0.692941i \(0.756315\pi\)
\(830\) 25.7621i 0.894216i
\(831\) 3.28572 0.113980
\(832\) −4.35991 −0.151153
\(833\) 28.0705i 0.972586i
\(834\) −1.88944 −0.0654259
\(835\) −2.40922 −0.0833745
\(836\) −46.4998 43.3941i −1.60823 1.50082i
\(837\) 18.2282 0.630059
\(838\) −19.5223 −0.674388
\(839\) 31.4430i 1.08553i −0.839884 0.542766i \(-0.817377\pi\)
0.839884 0.542766i \(-0.182623\pi\)
\(840\) 3.16091 0.109062
\(841\) −19.7225 −0.680087
\(842\) 64.7471i 2.23133i
\(843\) 4.44609i 0.153131i
\(844\) −109.125 −3.75623
\(845\) 7.53993 0.259381
\(846\) 88.7254 3.05044
\(847\) −14.1562 + 4.59317i −0.486412 + 0.157823i
\(848\) 25.9286i 0.890393i
\(849\) 8.92440 0.306285
\(850\) 13.7365i 0.471159i
\(851\) 1.37762i 0.0472242i
\(852\) −24.3171 −0.833090
\(853\) 29.2514i 1.00155i −0.865578 0.500775i \(-0.833049\pi\)
0.865578 0.500775i \(-0.166951\pi\)
\(854\) −32.7131 −1.11942
\(855\) −1.51701 + 12.3381i −0.0518806 + 0.421953i
\(856\) −105.760 −3.61479
\(857\) −3.26967 −0.111690 −0.0558449 0.998439i \(-0.517785\pi\)
−0.0558449 + 0.998439i \(0.517785\pi\)
\(858\) −6.10371 4.43653i −0.208377 0.151461i
\(859\) 44.4634 1.51707 0.758536 0.651631i \(-0.225915\pi\)
0.758536 + 0.651631i \(0.225915\pi\)
\(860\) 8.88209i 0.302877i
\(861\) 0.585001 0.0199368
\(862\) 84.4880 2.87767
\(863\) 24.8514i 0.845951i 0.906141 + 0.422975i \(0.139014\pi\)
−0.906141 + 0.422975i \(0.860986\pi\)
\(864\) 10.0137i 0.340672i
\(865\) 10.2515 0.348563
\(866\) 61.2971i 2.08296i
\(867\) 4.80555i 0.163205i
\(868\) −48.1733 −1.63511
\(869\) 18.6625 25.6755i 0.633081 0.870983i
\(870\) 2.96566i 0.100545i
\(871\) 31.0106i 1.05075i
\(872\) −102.689 −3.47750
\(873\) 6.37885i 0.215891i
\(874\) −1.32135 + 10.7468i −0.0446954 + 0.363515i
\(875\) 1.35297i 0.0457388i
\(876\) −3.48550 −0.117764
\(877\) 1.81191 0.0611839 0.0305919 0.999532i \(-0.490261\pi\)
0.0305919 + 0.999532i \(0.490261\pi\)
\(878\) −76.5092 −2.58206
\(879\) 9.44437i 0.318551i
\(880\) −12.7852 + 17.5897i −0.430989 + 0.592948i
\(881\) −45.3998 −1.52956 −0.764779 0.644293i \(-0.777152\pi\)
−0.764779 + 0.644293i \(0.777152\pi\)
\(882\) −37.2946 −1.25578
\(883\) 31.4299 1.05770 0.528851 0.848715i \(-0.322623\pi\)
0.528851 + 0.848715i \(0.322623\pi\)
\(884\) 55.8220i 1.87750i
\(885\) 0.485126 0.0163073
\(886\) 6.97889 0.234460
\(887\) 12.8799 0.432464 0.216232 0.976342i \(-0.430623\pi\)
0.216232 + 0.976342i \(0.430623\pi\)
\(888\) 3.27767 0.109991
\(889\) 15.1458i 0.507974i
\(890\) 31.2103i 1.04617i
\(891\) 14.9931 20.6273i 0.502287 0.691040i
\(892\) 116.843i 3.91219i
\(893\) −6.54195 + 53.2067i −0.218918 + 1.78049i
\(894\) −9.10934 −0.304662
\(895\) 16.5494i 0.553187i
\(896\) 18.4167i 0.615259i
\(897\) 0.883121i 0.0294865i
\(898\) 82.8839i 2.76587i
\(899\) 24.6508i 0.822150i
\(900\) 12.5467 0.418223
\(901\) 21.4741 0.715407
\(902\) −5.54171 + 7.62421i −0.184519 + 0.253858i
\(903\) 1.05132i 0.0349859i
\(904\) 58.5849i 1.94850i
\(905\) 17.2195i 0.572395i
\(906\) 9.10868i 0.302616i
\(907\) 45.1329i 1.49861i 0.662222 + 0.749307i \(0.269613\pi\)
−0.662222 + 0.749307i \(0.730387\pi\)
\(908\) −14.6050 −0.484683
\(909\) 14.5036i 0.481055i
\(910\) 7.99762i 0.265118i
\(911\) 45.2270i 1.49844i −0.662323 0.749219i \(-0.730429\pi\)
0.662323 0.749219i \(-0.269571\pi\)
\(912\) 10.9174 + 1.34234i 0.361513 + 0.0444493i
\(913\) −27.3211 19.8585i −0.904196 0.657222i
\(914\) 33.5392i 1.10938i
\(915\) 3.67870 0.121614
\(916\) 11.9289 0.394142
\(917\) −15.9110 −0.525427
\(918\) 30.9389 1.02114
\(919\) 50.4139i 1.66300i 0.555525 + 0.831500i \(0.312517\pi\)
−0.555525 + 0.831500i \(0.687483\pi\)
\(920\) 5.96041 0.196509
\(921\) 8.36746i 0.275717i
\(922\) 60.3884i 1.98879i
\(923\) 33.5565i 1.10452i
\(924\) 4.46748 6.14629i 0.146969 0.202198i
\(925\) 1.40295i 0.0461287i
\(926\) −17.9200 −0.588889
\(927\) 19.4371i 0.638397i
\(928\) 13.5419 0.444536
\(929\) −18.1263 −0.594704 −0.297352 0.954768i \(-0.596103\pi\)
−0.297352 + 0.954768i \(0.596103\pi\)
\(930\) 7.87993 0.258393
\(931\) 2.74982 22.3648i 0.0901219 0.732975i
\(932\) 12.8128i 0.419697i
\(933\) 0.675087i 0.0221014i
\(934\) −1.35497 −0.0443361
\(935\) −14.5678 10.5887i −0.476418 0.346288i
\(936\) −40.4498 −1.32214
\(937\) 39.2538i 1.28237i 0.767388 + 0.641183i \(0.221556\pi\)
−0.767388 + 0.641183i \(0.778444\pi\)
\(938\) 45.4226 1.48310
\(939\) 3.72971i 0.121715i
\(940\) 54.1064 1.76476
\(941\) −19.7778 −0.644739 −0.322370 0.946614i \(-0.604479\pi\)
−0.322370 + 0.946614i \(0.604479\pi\)
\(942\) 6.46304i 0.210577i
\(943\) 1.10311 0.0359224
\(944\) 8.26398i 0.268970i
\(945\) −3.04731 −0.0991291
\(946\) 13.7017 + 9.95920i 0.445481 + 0.323801i
\(947\) 14.7874 0.480527 0.240263 0.970708i \(-0.422766\pi\)
0.240263 + 0.970708i \(0.422766\pi\)
\(948\) 16.2056i 0.526334i
\(949\) 4.80983i 0.156134i
\(950\) −1.34565 + 10.9444i −0.0436586 + 0.355082i
\(951\) 0.478349 0.0155115
\(952\) −44.5947 −1.44532
\(953\) 24.2111 0.784274 0.392137 0.919907i \(-0.371736\pi\)
0.392137 + 0.919907i \(0.371736\pi\)
\(954\) 28.5306i 0.923713i
\(955\) −10.7378 −0.347467
\(956\) 91.1178i 2.94696i
\(957\) −3.14512 2.28606i −0.101667 0.0738977i
\(958\) 49.0903i 1.58604i
\(959\) 7.88518i 0.254626i
\(960\) 0.718145i 0.0231780i
\(961\) −34.4986 −1.11286
\(962\) 8.29304i 0.267378i
\(963\) 49.6889 1.60120
\(964\) −28.8137 −0.928026
\(965\) 11.7328 0.377691
\(966\) −1.29355 −0.0416192
\(967\) 0.150839i 0.00485064i 0.999997 + 0.00242532i \(0.000772005\pi\)
−0.999997 + 0.00242532i \(0.999228\pi\)
\(968\) 20.6069 + 63.5106i 0.662330 + 2.04131i
\(969\) −1.11173 + 9.04185i −0.0357138 + 0.290466i
\(970\) 5.65830i 0.181677i
\(971\) 1.13243i 0.0363415i −0.999835 0.0181708i \(-0.994216\pi\)
0.999835 0.0181708i \(-0.00578425\pi\)
\(972\) 42.7462i 1.37109i
\(973\) −2.62552 −0.0841704
\(974\) 19.1225i 0.612726i
\(975\) 0.899359i 0.0288025i
\(976\) 62.6656i 2.00588i
\(977\) 14.0511i 0.449534i −0.974413 0.224767i \(-0.927838\pi\)
0.974413 0.224767i \(-0.0721621\pi\)
\(978\) 13.7847i 0.440787i
\(979\) −33.0990 24.0583i −1.05785 0.768906i
\(980\) −22.7429 −0.726497
\(981\) 48.2464 1.54039
\(982\) 78.3035i 2.49877i
\(983\) 29.6116i 0.944464i 0.881474 + 0.472232i \(0.156551\pi\)
−0.881474 + 0.472232i \(0.843449\pi\)
\(984\) 2.62456i 0.0836679i
\(985\) 10.9577i 0.349141i
\(986\) 41.8400i 1.33246i
\(987\) −6.40428 −0.203850
\(988\) 5.46840 44.4753i 0.173973 1.41495i
\(989\) 1.98244i 0.0630381i
\(990\) 14.0682 19.3548i 0.447117 0.615137i
\(991\) 1.64752i 0.0523351i 0.999658 + 0.0261675i \(0.00833034\pi\)
−0.999658 + 0.0261675i \(0.991670\pi\)
\(992\) 35.9817i 1.14242i
\(993\) −12.7350 −0.404134
\(994\) −49.1517 −1.55900
\(995\) −15.9815 −0.506646
\(996\) 17.2442 0.546405
\(997\) 23.4405i 0.742369i 0.928559 + 0.371185i \(0.121048\pi\)
−0.928559 + 0.371185i \(0.878952\pi\)
\(998\) −103.691 −3.28227
\(999\) −3.15988 −0.0999741
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 1045.2.f.a.626.4 yes 40
11.10 odd 2 inner 1045.2.f.a.626.38 yes 40
19.18 odd 2 inner 1045.2.f.a.626.37 yes 40
209.208 even 2 inner 1045.2.f.a.626.3 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.f.a.626.3 40 209.208 even 2 inner
1045.2.f.a.626.4 yes 40 1.1 even 1 trivial
1045.2.f.a.626.37 yes 40 19.18 odd 2 inner
1045.2.f.a.626.38 yes 40 11.10 odd 2 inner