Properties

Label 1045.2.f.b.626.9
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 626.9
Character \(\chi\) \(=\) 1045.626
Dual form 1045.2.f.b.626.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.83534 q^{2} +0.556027i q^{3} +1.36849 q^{4} +1.00000 q^{5} -1.02050i q^{6} -4.08954i q^{7} +1.15904 q^{8} +2.69083 q^{9} +O(q^{10})\) \(q-1.83534 q^{2} +0.556027i q^{3} +1.36849 q^{4} +1.00000 q^{5} -1.02050i q^{6} -4.08954i q^{7} +1.15904 q^{8} +2.69083 q^{9} -1.83534 q^{10} +(2.99061 - 1.43396i) q^{11} +0.760918i q^{12} +5.36672 q^{13} +7.50571i q^{14} +0.556027i q^{15} -4.86421 q^{16} -5.36655i q^{17} -4.93861 q^{18} +(0.797318 + 4.28536i) q^{19} +1.36849 q^{20} +2.27389 q^{21} +(-5.48881 + 2.63180i) q^{22} -1.90493 q^{23} +0.644456i q^{24} +1.00000 q^{25} -9.84979 q^{26} +3.16426i q^{27} -5.59649i q^{28} -8.05184 q^{29} -1.02050i q^{30} -5.60442i q^{31} +6.60944 q^{32} +(0.797318 + 1.66286i) q^{33} +9.84947i q^{34} -4.08954i q^{35} +3.68238 q^{36} -3.42346i q^{37} +(-1.46335 - 7.86511i) q^{38} +2.98404i q^{39} +1.15904 q^{40} -9.21852 q^{41} -4.17338 q^{42} -6.62567i q^{43} +(4.09263 - 1.96235i) q^{44} +2.69083 q^{45} +3.49620 q^{46} +7.01016 q^{47} -2.70464i q^{48} -9.72432 q^{49} -1.83534 q^{50} +2.98395 q^{51} +7.34431 q^{52} +7.74220i q^{53} -5.80751i q^{54} +(2.99061 - 1.43396i) q^{55} -4.73993i q^{56} +(-2.38278 + 0.443331i) q^{57} +14.7779 q^{58} -3.01488i q^{59} +0.760918i q^{60} +7.25535i q^{61} +10.2860i q^{62} -11.0043i q^{63} -2.40217 q^{64} +5.36672 q^{65} +(-1.46335 - 3.05193i) q^{66} +6.53890i q^{67} -7.34407i q^{68} -1.05919i q^{69} +7.50571i q^{70} +0.908694i q^{71} +3.11878 q^{72} +9.46787i q^{73} +6.28323i q^{74} +0.556027i q^{75} +(1.09112 + 5.86447i) q^{76} +(-5.86421 - 12.2302i) q^{77} -5.47675i q^{78} +9.42033 q^{79} -4.86421 q^{80} +6.31309 q^{81} +16.9192 q^{82} +3.44016i q^{83} +3.11180 q^{84} -5.36655i q^{85} +12.1604i q^{86} -4.47704i q^{87} +(3.46623 - 1.66201i) q^{88} +6.38503i q^{89} -4.93861 q^{90} -21.9474i q^{91} -2.60688 q^{92} +3.11621 q^{93} -12.8661 q^{94} +(0.797318 + 4.28536i) q^{95} +3.67503i q^{96} -12.5270i q^{97} +17.8475 q^{98} +(8.04724 - 3.85854i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 40 q^{4} + 40 q^{5} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 40 q^{4} + 40 q^{5} - 28 q^{9} - 4 q^{11} + 48 q^{16} + 40 q^{20} + 24 q^{23} + 40 q^{25} - 24 q^{26} + 24 q^{36} - 28 q^{38} - 60 q^{42} - 48 q^{44} - 28 q^{45} - 36 q^{49} - 4 q^{55} - 36 q^{58} - 40 q^{64} - 28 q^{66} + 8 q^{77} + 48 q^{80} + 80 q^{81} + 8 q^{82} + 4 q^{92} + 56 q^{93} - 16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83534 −1.29778 −0.648892 0.760880i \(-0.724767\pi\)
−0.648892 + 0.760880i \(0.724767\pi\)
\(3\) 0.556027i 0.321023i 0.987034 + 0.160511i \(0.0513143\pi\)
−0.987034 + 0.160511i \(0.948686\pi\)
\(4\) 1.36849 0.684245
\(5\) 1.00000 0.447214
\(6\) 1.02050i 0.416618i
\(7\) 4.08954i 1.54570i −0.634589 0.772850i \(-0.718830\pi\)
0.634589 0.772850i \(-0.281170\pi\)
\(8\) 1.15904 0.409782
\(9\) 2.69083 0.896945
\(10\) −1.83534 −0.580387
\(11\) 2.99061 1.43396i 0.901704 0.432354i
\(12\) 0.760918i 0.219658i
\(13\) 5.36672 1.48846 0.744231 0.667923i \(-0.232816\pi\)
0.744231 + 0.667923i \(0.232816\pi\)
\(14\) 7.50571i 2.00599i
\(15\) 0.556027i 0.143566i
\(16\) −4.86421 −1.21605
\(17\) 5.36655i 1.30158i −0.759258 0.650790i \(-0.774438\pi\)
0.759258 0.650790i \(-0.225562\pi\)
\(18\) −4.93861 −1.16404
\(19\) 0.797318 + 4.28536i 0.182917 + 0.983128i
\(20\) 1.36849 0.306004
\(21\) 2.27389 0.496205
\(22\) −5.48881 + 2.63180i −1.17022 + 0.561102i
\(23\) −1.90493 −0.397205 −0.198603 0.980080i \(-0.563640\pi\)
−0.198603 + 0.980080i \(0.563640\pi\)
\(24\) 0.644456i 0.131549i
\(25\) 1.00000 0.200000
\(26\) −9.84979 −1.93170
\(27\) 3.16426i 0.608962i
\(28\) 5.59649i 1.05764i
\(29\) −8.05184 −1.49519 −0.747595 0.664155i \(-0.768792\pi\)
−0.747595 + 0.664155i \(0.768792\pi\)
\(30\) 1.02050i 0.186317i
\(31\) 5.60442i 1.00658i −0.864117 0.503292i \(-0.832122\pi\)
0.864117 0.503292i \(-0.167878\pi\)
\(32\) 6.60944 1.16839
\(33\) 0.797318 + 1.66286i 0.138795 + 0.289467i
\(34\) 9.84947i 1.68917i
\(35\) 4.08954i 0.691258i
\(36\) 3.68238 0.613730
\(37\) 3.42346i 0.562813i −0.959589 0.281407i \(-0.909199\pi\)
0.959589 0.281407i \(-0.0908010\pi\)
\(38\) −1.46335 7.86511i −0.237387 1.27589i
\(39\) 2.98404i 0.477830i
\(40\) 1.15904 0.183260
\(41\) −9.21852 −1.43969 −0.719846 0.694134i \(-0.755787\pi\)
−0.719846 + 0.694134i \(0.755787\pi\)
\(42\) −4.17338 −0.643967
\(43\) 6.62567i 1.01041i −0.863001 0.505203i \(-0.831418\pi\)
0.863001 0.505203i \(-0.168582\pi\)
\(44\) 4.09263 1.96235i 0.616987 0.295836i
\(45\) 2.69083 0.401126
\(46\) 3.49620 0.515487
\(47\) 7.01016 1.02254 0.511268 0.859421i \(-0.329176\pi\)
0.511268 + 0.859421i \(0.329176\pi\)
\(48\) 2.70464i 0.390381i
\(49\) −9.72432 −1.38919
\(50\) −1.83534 −0.259557
\(51\) 2.98395 0.417836
\(52\) 7.34431 1.01847
\(53\) 7.74220i 1.06347i 0.846910 + 0.531736i \(0.178460\pi\)
−0.846910 + 0.531736i \(0.821540\pi\)
\(54\) 5.80751i 0.790301i
\(55\) 2.99061 1.43396i 0.403254 0.193354i
\(56\) 4.73993i 0.633399i
\(57\) −2.38278 + 0.443331i −0.315606 + 0.0587206i
\(58\) 14.7779 1.94043
\(59\) 3.01488i 0.392504i −0.980554 0.196252i \(-0.937123\pi\)
0.980554 0.196252i \(-0.0628770\pi\)
\(60\) 0.760918i 0.0982341i
\(61\) 7.25535i 0.928953i 0.885586 + 0.464476i \(0.153757\pi\)
−0.885586 + 0.464476i \(0.846243\pi\)
\(62\) 10.2860i 1.30633i
\(63\) 11.0043i 1.38641i
\(64\) −2.40217 −0.300271
\(65\) 5.36672 0.665660
\(66\) −1.46335 3.05193i −0.180126 0.375666i
\(67\) 6.53890i 0.798854i 0.916765 + 0.399427i \(0.130791\pi\)
−0.916765 + 0.399427i \(0.869209\pi\)
\(68\) 7.34407i 0.890600i
\(69\) 1.05919i 0.127512i
\(70\) 7.50571i 0.897104i
\(71\) 0.908694i 0.107842i 0.998545 + 0.0539211i \(0.0171719\pi\)
−0.998545 + 0.0539211i \(0.982828\pi\)
\(72\) 3.11878 0.367551
\(73\) 9.46787i 1.10813i 0.832473 + 0.554065i \(0.186924\pi\)
−0.832473 + 0.554065i \(0.813076\pi\)
\(74\) 6.28323i 0.730411i
\(75\) 0.556027i 0.0642045i
\(76\) 1.09112 + 5.86447i 0.125160 + 0.672701i
\(77\) −5.86421 12.2302i −0.668289 1.39376i
\(78\) 5.47675i 0.620120i
\(79\) 9.42033 1.05987 0.529935 0.848038i \(-0.322216\pi\)
0.529935 + 0.848038i \(0.322216\pi\)
\(80\) −4.86421 −0.543836
\(81\) 6.31309 0.701454
\(82\) 16.9192 1.86841
\(83\) 3.44016i 0.377606i 0.982015 + 0.188803i \(0.0604608\pi\)
−0.982015 + 0.188803i \(0.939539\pi\)
\(84\) 3.11180 0.339526
\(85\) 5.36655i 0.582084i
\(86\) 12.1604i 1.31129i
\(87\) 4.47704i 0.479990i
\(88\) 3.46623 1.66201i 0.369502 0.177171i
\(89\) 6.38503i 0.676812i 0.941000 + 0.338406i \(0.109888\pi\)
−0.941000 + 0.338406i \(0.890112\pi\)
\(90\) −4.93861 −0.520575
\(91\) 21.9474i 2.30071i
\(92\) −2.60688 −0.271786
\(93\) 3.11621 0.323136
\(94\) −12.8661 −1.32703
\(95\) 0.797318 + 4.28536i 0.0818031 + 0.439668i
\(96\) 3.67503i 0.375081i
\(97\) 12.5270i 1.27192i −0.771720 0.635962i \(-0.780604\pi\)
0.771720 0.635962i \(-0.219396\pi\)
\(98\) 17.8475 1.80287
\(99\) 8.04724 3.85854i 0.808779 0.387797i
\(100\) 1.36849 0.136849
\(101\) 19.3531i 1.92571i 0.270022 + 0.962854i \(0.412969\pi\)
−0.270022 + 0.962854i \(0.587031\pi\)
\(102\) −5.47657 −0.542262
\(103\) 19.1440i 1.88632i −0.332343 0.943159i \(-0.607839\pi\)
0.332343 0.943159i \(-0.392161\pi\)
\(104\) 6.22023 0.609944
\(105\) 2.27389 0.221909
\(106\) 14.2096i 1.38016i
\(107\) 2.16802 0.209590 0.104795 0.994494i \(-0.466581\pi\)
0.104795 + 0.994494i \(0.466581\pi\)
\(108\) 4.33026i 0.416679i
\(109\) −15.5057 −1.48518 −0.742589 0.669747i \(-0.766403\pi\)
−0.742589 + 0.669747i \(0.766403\pi\)
\(110\) −5.48881 + 2.63180i −0.523337 + 0.250933i
\(111\) 1.90354 0.180676
\(112\) 19.8924i 1.87965i
\(113\) 14.5570i 1.36941i −0.728820 0.684706i \(-0.759931\pi\)
0.728820 0.684706i \(-0.240069\pi\)
\(114\) 4.37321 0.813665i 0.409589 0.0762067i
\(115\) −1.90493 −0.177635
\(116\) −11.0189 −1.02308
\(117\) 14.4410 1.33507
\(118\) 5.53334i 0.509385i
\(119\) −21.9467 −2.01185
\(120\) 0.644456i 0.0588306i
\(121\) 6.88754 8.57681i 0.626140 0.779710i
\(122\) 13.3161i 1.20558i
\(123\) 5.12575i 0.462173i
\(124\) 7.66960i 0.688750i
\(125\) 1.00000 0.0894427
\(126\) 20.1966i 1.79926i
\(127\) 10.9929 0.975462 0.487731 0.872994i \(-0.337825\pi\)
0.487731 + 0.872994i \(0.337825\pi\)
\(128\) −8.81007 −0.778708
\(129\) 3.68406 0.324363
\(130\) −9.84979 −0.863884
\(131\) 7.82540i 0.683708i −0.939753 0.341854i \(-0.888945\pi\)
0.939753 0.341854i \(-0.111055\pi\)
\(132\) 1.09112 + 2.27561i 0.0949700 + 0.198067i
\(133\) 17.5251 3.26066i 1.51962 0.282735i
\(134\) 12.0011i 1.03674i
\(135\) 3.16426i 0.272336i
\(136\) 6.22003i 0.533363i
\(137\) −16.0753 −1.37341 −0.686705 0.726937i \(-0.740943\pi\)
−0.686705 + 0.726937i \(0.740943\pi\)
\(138\) 1.94398i 0.165483i
\(139\) 14.9215i 1.26562i −0.774306 0.632811i \(-0.781901\pi\)
0.774306 0.632811i \(-0.218099\pi\)
\(140\) 5.59649i 0.472990i
\(141\) 3.89784i 0.328257i
\(142\) 1.66777i 0.139956i
\(143\) 16.0498 7.69564i 1.34215 0.643542i
\(144\) −13.0888 −1.09073
\(145\) −8.05184 −0.668669
\(146\) 17.3768i 1.43812i
\(147\) 5.40699i 0.445961i
\(148\) 4.68497i 0.385102i
\(149\) 6.23473i 0.510769i 0.966840 + 0.255384i \(0.0822020\pi\)
−0.966840 + 0.255384i \(0.917798\pi\)
\(150\) 1.02050i 0.0833236i
\(151\) 7.01213 0.570639 0.285319 0.958432i \(-0.407900\pi\)
0.285319 + 0.958432i \(0.407900\pi\)
\(152\) 0.924122 + 4.96689i 0.0749562 + 0.402868i
\(153\) 14.4405i 1.16744i
\(154\) 10.7629 + 22.4467i 0.867296 + 1.80881i
\(155\) 5.60442i 0.450158i
\(156\) 4.08364i 0.326953i
\(157\) 9.47616 0.756280 0.378140 0.925749i \(-0.376564\pi\)
0.378140 + 0.925749i \(0.376564\pi\)
\(158\) −17.2895 −1.37548
\(159\) −4.30487 −0.341399
\(160\) 6.60944 0.522522
\(161\) 7.79028i 0.613960i
\(162\) −11.5867 −0.910336
\(163\) 1.79960 0.140956 0.0704778 0.997513i \(-0.477548\pi\)
0.0704778 + 0.997513i \(0.477548\pi\)
\(164\) −12.6155 −0.985102
\(165\) 0.797318 + 1.66286i 0.0620711 + 0.129454i
\(166\) 6.31387i 0.490051i
\(167\) −16.2612 −1.25833 −0.629166 0.777271i \(-0.716603\pi\)
−0.629166 + 0.777271i \(0.716603\pi\)
\(168\) 2.63553 0.203335
\(169\) 15.8017 1.21552
\(170\) 9.84947i 0.755420i
\(171\) 2.14545 + 11.5312i 0.164067 + 0.881812i
\(172\) 9.06717i 0.691365i
\(173\) 17.5756 1.33625 0.668123 0.744051i \(-0.267098\pi\)
0.668123 + 0.744051i \(0.267098\pi\)
\(174\) 8.21692i 0.622923i
\(175\) 4.08954i 0.309140i
\(176\) −14.5470 + 6.97507i −1.09652 + 0.525765i
\(177\) 1.67635 0.126003
\(178\) 11.7187i 0.878356i
\(179\) 24.1500i 1.80506i 0.430632 + 0.902528i \(0.358291\pi\)
−0.430632 + 0.902528i \(0.641709\pi\)
\(180\) 3.68238 0.274468
\(181\) 12.2267i 0.908804i 0.890797 + 0.454402i \(0.150147\pi\)
−0.890797 + 0.454402i \(0.849853\pi\)
\(182\) 40.2811i 2.98583i
\(183\) −4.03417 −0.298215
\(184\) −2.20788 −0.162767
\(185\) 3.42346i 0.251698i
\(186\) −5.71932 −0.419361
\(187\) −7.69539 16.0493i −0.562743 1.17364i
\(188\) 9.59334 0.699666
\(189\) 12.9404 0.941272
\(190\) −1.46335 7.86511i −0.106163 0.570595i
\(191\) −2.74212 −0.198413 −0.0992065 0.995067i \(-0.531630\pi\)
−0.0992065 + 0.995067i \(0.531630\pi\)
\(192\) 1.33567i 0.0963937i
\(193\) −2.80307 −0.201770 −0.100885 0.994898i \(-0.532167\pi\)
−0.100885 + 0.994898i \(0.532167\pi\)
\(194\) 22.9914i 1.65068i
\(195\) 2.98404i 0.213692i
\(196\) −13.3076 −0.950546
\(197\) 27.3376i 1.94772i 0.227140 + 0.973862i \(0.427062\pi\)
−0.227140 + 0.973862i \(0.572938\pi\)
\(198\) −14.7695 + 7.08174i −1.04962 + 0.503278i
\(199\) −10.9591 −0.776872 −0.388436 0.921476i \(-0.626985\pi\)
−0.388436 + 0.921476i \(0.626985\pi\)
\(200\) 1.15904 0.0819563
\(201\) −3.63581 −0.256450
\(202\) 35.5197i 2.49916i
\(203\) 32.9283i 2.31112i
\(204\) 4.08350 0.285903
\(205\) −9.21852 −0.643849
\(206\) 35.1359i 2.44803i
\(207\) −5.12585 −0.356271
\(208\) −26.1049 −1.81005
\(209\) 8.52948 + 11.6725i 0.589997 + 0.807406i
\(210\) −4.17338 −0.287991
\(211\) 6.28757 0.432854 0.216427 0.976299i \(-0.430560\pi\)
0.216427 + 0.976299i \(0.430560\pi\)
\(212\) 10.5951i 0.727676i
\(213\) −0.505258 −0.0346197
\(214\) −3.97906 −0.272003
\(215\) 6.62567i 0.451867i
\(216\) 3.66749i 0.249541i
\(217\) −22.9195 −1.55588
\(218\) 28.4583 1.92744
\(219\) −5.26440 −0.355735
\(220\) 4.09263 1.96235i 0.275925 0.132302i
\(221\) 28.8008i 1.93735i
\(222\) −3.49365 −0.234478
\(223\) 22.8453i 1.52983i −0.644130 0.764916i \(-0.722780\pi\)
0.644130 0.764916i \(-0.277220\pi\)
\(224\) 27.0295i 1.80599i
\(225\) 2.69083 0.179389
\(226\) 26.7172i 1.77720i
\(227\) −20.2532 −1.34425 −0.672126 0.740437i \(-0.734619\pi\)
−0.672126 + 0.740437i \(0.734619\pi\)
\(228\) −3.26081 + 0.606694i −0.215952 + 0.0401793i
\(229\) 8.42267 0.556586 0.278293 0.960496i \(-0.410231\pi\)
0.278293 + 0.960496i \(0.410231\pi\)
\(230\) 3.49620 0.230533
\(231\) 6.80034 3.26066i 0.447430 0.214536i
\(232\) −9.33239 −0.612701
\(233\) 12.1491i 0.795913i −0.917404 0.397957i \(-0.869719\pi\)
0.917404 0.397957i \(-0.130281\pi\)
\(234\) −26.5041 −1.73263
\(235\) 7.01016 0.457292
\(236\) 4.12583i 0.268569i
\(237\) 5.23796i 0.340242i
\(238\) 40.2798 2.61095
\(239\) 9.71140i 0.628178i −0.949394 0.314089i \(-0.898301\pi\)
0.949394 0.314089i \(-0.101699\pi\)
\(240\) 2.70464i 0.174584i
\(241\) 10.4015 0.670022 0.335011 0.942214i \(-0.391260\pi\)
0.335011 + 0.942214i \(0.391260\pi\)
\(242\) −12.6410 + 15.7414i −0.812596 + 1.01190i
\(243\) 13.0030i 0.834144i
\(244\) 9.92888i 0.635632i
\(245\) −9.72432 −0.621264
\(246\) 9.40752i 0.599801i
\(247\) 4.27899 + 22.9983i 0.272265 + 1.46335i
\(248\) 6.49573i 0.412479i
\(249\) −1.91282 −0.121220
\(250\) −1.83534 −0.116077
\(251\) 14.3271 0.904316 0.452158 0.891938i \(-0.350654\pi\)
0.452158 + 0.891938i \(0.350654\pi\)
\(252\) 15.0592i 0.948643i
\(253\) −5.69691 + 2.73158i −0.358161 + 0.171733i
\(254\) −20.1758 −1.26594
\(255\) 2.98395 0.186862
\(256\) 20.9738 1.31087
\(257\) 7.82897i 0.488358i −0.969730 0.244179i \(-0.921482\pi\)
0.969730 0.244179i \(-0.0785184\pi\)
\(258\) −6.76151 −0.420953
\(259\) −14.0004 −0.869941
\(260\) 7.34431 0.455475
\(261\) −21.6662 −1.34110
\(262\) 14.3623i 0.887306i
\(263\) 4.23088i 0.260887i 0.991456 + 0.130444i \(0.0416402\pi\)
−0.991456 + 0.130444i \(0.958360\pi\)
\(264\) 0.924122 + 1.92732i 0.0568758 + 0.118618i
\(265\) 7.74220i 0.475599i
\(266\) −32.1647 + 5.98444i −1.97214 + 0.366930i
\(267\) −3.55025 −0.217272
\(268\) 8.94842i 0.546612i
\(269\) 22.3455i 1.36243i 0.732083 + 0.681215i \(0.238548\pi\)
−0.732083 + 0.681215i \(0.761452\pi\)
\(270\) 5.80751i 0.353434i
\(271\) 14.8331i 0.901044i −0.892765 0.450522i \(-0.851238\pi\)
0.892765 0.450522i \(-0.148762\pi\)
\(272\) 26.1040i 1.58279i
\(273\) 12.2034 0.738581
\(274\) 29.5038 1.78239
\(275\) 2.99061 1.43396i 0.180341 0.0864708i
\(276\) 1.44949i 0.0872493i
\(277\) 19.4975i 1.17149i 0.810494 + 0.585747i \(0.199199\pi\)
−0.810494 + 0.585747i \(0.800801\pi\)
\(278\) 27.3860i 1.64251i
\(279\) 15.0806i 0.902850i
\(280\) 4.73993i 0.283265i
\(281\) 20.7464 1.23763 0.618813 0.785538i \(-0.287614\pi\)
0.618813 + 0.785538i \(0.287614\pi\)
\(282\) 7.15388i 0.426007i
\(283\) 3.16295i 0.188018i −0.995571 0.0940088i \(-0.970032\pi\)
0.995571 0.0940088i \(-0.0299682\pi\)
\(284\) 1.24354i 0.0737905i
\(285\) −2.38278 + 0.443331i −0.141143 + 0.0262606i
\(286\) −29.4569 + 14.1242i −1.74182 + 0.835179i
\(287\) 37.6995i 2.22533i
\(288\) 17.7849 1.04798
\(289\) −11.7999 −0.694109
\(290\) 14.7779 0.867789
\(291\) 6.96535 0.408316
\(292\) 12.9567i 0.758233i
\(293\) −13.1728 −0.769561 −0.384780 0.923008i \(-0.625723\pi\)
−0.384780 + 0.923008i \(0.625723\pi\)
\(294\) 9.92369i 0.578761i
\(295\) 3.01488i 0.175533i
\(296\) 3.96792i 0.230631i
\(297\) 4.53741 + 9.46308i 0.263287 + 0.549103i
\(298\) 11.4429i 0.662868i
\(299\) −10.2232 −0.591224
\(300\) 0.760918i 0.0439316i
\(301\) −27.0959 −1.56178
\(302\) −12.8697 −0.740567
\(303\) −10.7609 −0.618196
\(304\) −3.87833 20.8449i −0.222437 1.19554i
\(305\) 7.25535i 0.415440i
\(306\) 26.5033i 1.51509i
\(307\) 25.5176 1.45637 0.728184 0.685382i \(-0.240365\pi\)
0.728184 + 0.685382i \(0.240365\pi\)
\(308\) −8.02512 16.7370i −0.457274 0.953677i
\(309\) 10.6446 0.605550
\(310\) 10.2860i 0.584208i
\(311\) 2.64178 0.149802 0.0749009 0.997191i \(-0.476136\pi\)
0.0749009 + 0.997191i \(0.476136\pi\)
\(312\) 3.45862i 0.195806i
\(313\) −22.5891 −1.27681 −0.638406 0.769700i \(-0.720406\pi\)
−0.638406 + 0.769700i \(0.720406\pi\)
\(314\) −17.3920 −0.981488
\(315\) 11.0043i 0.620020i
\(316\) 12.8916 0.725211
\(317\) 9.60819i 0.539650i 0.962909 + 0.269825i \(0.0869658\pi\)
−0.962909 + 0.269825i \(0.913034\pi\)
\(318\) 7.90093 0.443062
\(319\) −24.0800 + 11.5460i −1.34822 + 0.646451i
\(320\) −2.40217 −0.134285
\(321\) 1.20548i 0.0672831i
\(322\) 14.2978i 0.796788i
\(323\) 22.9976 4.27885i 1.27962 0.238081i
\(324\) 8.63940 0.479967
\(325\) 5.36672 0.297692
\(326\) −3.30289 −0.182930
\(327\) 8.62160i 0.476776i
\(328\) −10.6846 −0.589959
\(329\) 28.6683i 1.58053i
\(330\) −1.46335 3.05193i −0.0805550 0.168003i
\(331\) 18.3796i 1.01023i 0.863051 + 0.505116i \(0.168550\pi\)
−0.863051 + 0.505116i \(0.831450\pi\)
\(332\) 4.70782i 0.258375i
\(333\) 9.21196i 0.504812i
\(334\) 29.8450 1.63304
\(335\) 6.53890i 0.357258i
\(336\) −11.0607 −0.603411
\(337\) 16.8472 0.917727 0.458863 0.888507i \(-0.348257\pi\)
0.458863 + 0.888507i \(0.348257\pi\)
\(338\) −29.0016 −1.57748
\(339\) 8.09411 0.439612
\(340\) 7.34407i 0.398288i
\(341\) −8.03649 16.7607i −0.435200 0.907640i
\(342\) −3.93764 21.1637i −0.212923 1.14440i
\(343\) 11.1412i 0.601569i
\(344\) 7.67940i 0.414046i
\(345\) 1.05919i 0.0570250i
\(346\) −32.2572 −1.73416
\(347\) 13.4799i 0.723641i −0.932248 0.361820i \(-0.882155\pi\)
0.932248 0.361820i \(-0.117845\pi\)
\(348\) 6.12679i 0.328431i
\(349\) 17.8479i 0.955376i 0.878530 + 0.477688i \(0.158525\pi\)
−0.878530 + 0.477688i \(0.841475\pi\)
\(350\) 7.50571i 0.401197i
\(351\) 16.9817i 0.906416i
\(352\) 19.7663 9.47764i 1.05355 0.505160i
\(353\) 8.97800 0.477851 0.238925 0.971038i \(-0.423205\pi\)
0.238925 + 0.971038i \(0.423205\pi\)
\(354\) −3.07669 −0.163524
\(355\) 0.908694i 0.0482285i
\(356\) 8.73786i 0.463105i
\(357\) 12.2030i 0.645850i
\(358\) 44.3236i 2.34257i
\(359\) 6.55754i 0.346094i 0.984914 + 0.173047i \(0.0553612\pi\)
−0.984914 + 0.173047i \(0.944639\pi\)
\(360\) 3.11878 0.164374
\(361\) −17.7286 + 6.83359i −0.933082 + 0.359662i
\(362\) 22.4402i 1.17943i
\(363\) 4.76894 + 3.82966i 0.250305 + 0.201005i
\(364\) 30.0348i 1.57425i
\(365\) 9.46787i 0.495571i
\(366\) 7.40410 0.387019
\(367\) 7.89395 0.412061 0.206030 0.978546i \(-0.433945\pi\)
0.206030 + 0.978546i \(0.433945\pi\)
\(368\) 9.26598 0.483023
\(369\) −24.8055 −1.29132
\(370\) 6.28323i 0.326650i
\(371\) 31.6620 1.64381
\(372\) 4.26450 0.221104
\(373\) −8.28038 −0.428742 −0.214371 0.976752i \(-0.568770\pi\)
−0.214371 + 0.976752i \(0.568770\pi\)
\(374\) 14.1237 + 29.4560i 0.730319 + 1.52313i
\(375\) 0.556027i 0.0287131i
\(376\) 8.12503 0.419017
\(377\) −43.2120 −2.22553
\(378\) −23.7500 −1.22157
\(379\) 2.52457i 0.129679i 0.997896 + 0.0648393i \(0.0206535\pi\)
−0.997896 + 0.0648393i \(0.979347\pi\)
\(380\) 1.09112 + 5.86447i 0.0559734 + 0.300841i
\(381\) 6.11235i 0.313145i
\(382\) 5.03274 0.257497
\(383\) 1.62674i 0.0831225i 0.999136 + 0.0415613i \(0.0132332\pi\)
−0.999136 + 0.0415613i \(0.986767\pi\)
\(384\) 4.89864i 0.249983i
\(385\) −5.86421 12.2302i −0.298868 0.623310i
\(386\) 5.14461 0.261854
\(387\) 17.8286i 0.906278i
\(388\) 17.1431i 0.870308i
\(389\) 1.52255 0.0771964 0.0385982 0.999255i \(-0.487711\pi\)
0.0385982 + 0.999255i \(0.487711\pi\)
\(390\) 5.47675i 0.277326i
\(391\) 10.2229i 0.516994i
\(392\) −11.2709 −0.569264
\(393\) 4.35113 0.219486
\(394\) 50.1739i 2.52773i
\(395\) 9.42033 0.473988
\(396\) 11.0126 5.28037i 0.553403 0.265349i
\(397\) 26.5187 1.33093 0.665467 0.746427i \(-0.268232\pi\)
0.665467 + 0.746427i \(0.268232\pi\)
\(398\) 20.1138 1.00821
\(399\) 1.81302 + 9.74445i 0.0907644 + 0.487833i
\(400\) −4.86421 −0.243211
\(401\) 0.583132i 0.0291202i 0.999894 + 0.0145601i \(0.00463479\pi\)
−0.999894 + 0.0145601i \(0.995365\pi\)
\(402\) 6.67296 0.332817
\(403\) 30.0774i 1.49826i
\(404\) 26.4846i 1.31766i
\(405\) 6.31309 0.313700
\(406\) 60.4348i 2.99933i
\(407\) −4.90909 10.2382i −0.243335 0.507491i
\(408\) 3.45851 0.171222
\(409\) −7.96701 −0.393943 −0.196972 0.980409i \(-0.563111\pi\)
−0.196972 + 0.980409i \(0.563111\pi\)
\(410\) 16.9192 0.835578
\(411\) 8.93833i 0.440895i
\(412\) 26.1984i 1.29070i
\(413\) −12.3295 −0.606693
\(414\) 9.40769 0.462363
\(415\) 3.44016i 0.168871i
\(416\) 35.4710 1.73911
\(417\) 8.29674 0.406293
\(418\) −15.6545 21.4231i −0.765689 1.04784i
\(419\) −2.93792 −0.143527 −0.0717634 0.997422i \(-0.522863\pi\)
−0.0717634 + 0.997422i \(0.522863\pi\)
\(420\) 3.11180 0.151840
\(421\) 3.30643i 0.161146i −0.996749 0.0805728i \(-0.974325\pi\)
0.996749 0.0805728i \(-0.0256749\pi\)
\(422\) −11.5399 −0.561752
\(423\) 18.8632 0.917159
\(424\) 8.97350i 0.435792i
\(425\) 5.36655i 0.260316i
\(426\) 0.927324 0.0449290
\(427\) 29.6710 1.43588
\(428\) 2.96691 0.143411
\(429\) 4.27899 + 8.92413i 0.206591 + 0.430861i
\(430\) 12.1604i 0.586426i
\(431\) 9.10067 0.438364 0.219182 0.975684i \(-0.429661\pi\)
0.219182 + 0.975684i \(0.429661\pi\)
\(432\) 15.3916i 0.740530i
\(433\) 36.6982i 1.76360i 0.471621 + 0.881801i \(0.343669\pi\)
−0.471621 + 0.881801i \(0.656331\pi\)
\(434\) 42.0652 2.01919
\(435\) 4.47704i 0.214658i
\(436\) −21.2194 −1.01623
\(437\) −1.51883 8.16330i −0.0726557 0.390503i
\(438\) 9.66198 0.461667
\(439\) 32.3779 1.54531 0.772657 0.634824i \(-0.218927\pi\)
0.772657 + 0.634824i \(0.218927\pi\)
\(440\) 3.46623 1.66201i 0.165246 0.0792331i
\(441\) −26.1665 −1.24603
\(442\) 52.8594i 2.51426i
\(443\) −29.7753 −1.41467 −0.707333 0.706880i \(-0.750102\pi\)
−0.707333 + 0.706880i \(0.750102\pi\)
\(444\) 2.60497 0.123627
\(445\) 6.38503i 0.302680i
\(446\) 41.9290i 1.98539i
\(447\) −3.46668 −0.163968
\(448\) 9.82375i 0.464128i
\(449\) 21.3178i 1.00605i −0.864273 0.503023i \(-0.832221\pi\)
0.864273 0.503023i \(-0.167779\pi\)
\(450\) −4.93861 −0.232808
\(451\) −27.5690 + 13.2189i −1.29818 + 0.622456i
\(452\) 19.9212i 0.937014i
\(453\) 3.89894i 0.183188i
\(454\) 37.1716 1.74455
\(455\) 21.9474i 1.02891i
\(456\) −2.76173 + 0.513837i −0.129330 + 0.0240626i
\(457\) 2.19241i 0.102557i −0.998684 0.0512783i \(-0.983670\pi\)
0.998684 0.0512783i \(-0.0163296\pi\)
\(458\) −15.4585 −0.722329
\(459\) 16.9812 0.792612
\(460\) −2.60688 −0.121546
\(461\) 0.666820i 0.0310569i −0.999879 0.0155285i \(-0.995057\pi\)
0.999879 0.0155285i \(-0.00494306\pi\)
\(462\) −12.4810 + 5.98444i −0.580667 + 0.278421i
\(463\) 7.79469 0.362250 0.181125 0.983460i \(-0.442026\pi\)
0.181125 + 0.983460i \(0.442026\pi\)
\(464\) 39.1659 1.81823
\(465\) 3.11621 0.144511
\(466\) 22.2978i 1.03292i
\(467\) 12.3550 0.571720 0.285860 0.958271i \(-0.407721\pi\)
0.285860 + 0.958271i \(0.407721\pi\)
\(468\) 19.7623 0.913513
\(469\) 26.7411 1.23479
\(470\) −12.8661 −0.593467
\(471\) 5.26900i 0.242783i
\(472\) 3.49436i 0.160841i
\(473\) −9.50092 19.8148i −0.436853 0.911087i
\(474\) 9.61346i 0.441561i
\(475\) 0.797318 + 4.28536i 0.0365835 + 0.196626i
\(476\) −30.0339 −1.37660
\(477\) 20.8330i 0.953876i
\(478\) 17.8238i 0.815240i
\(479\) 12.7066i 0.580581i 0.956939 + 0.290290i \(0.0937519\pi\)
−0.956939 + 0.290290i \(0.906248\pi\)
\(480\) 3.67503i 0.167741i
\(481\) 18.3728i 0.837726i
\(482\) −19.0904 −0.869545
\(483\) −4.33161 −0.197095
\(484\) 9.42554 11.7373i 0.428434 0.533513i
\(485\) 12.5270i 0.568822i
\(486\) 23.8650i 1.08254i
\(487\) 1.56912i 0.0711038i 0.999368 + 0.0355519i \(0.0113189\pi\)
−0.999368 + 0.0355519i \(0.988681\pi\)
\(488\) 8.40923i 0.380668i
\(489\) 1.00063i 0.0452499i
\(490\) 17.8475 0.806267
\(491\) 13.6611i 0.616517i −0.951303 0.308258i \(-0.900254\pi\)
0.951303 0.308258i \(-0.0997461\pi\)
\(492\) 7.01454i 0.316240i
\(493\) 43.2106i 1.94611i
\(494\) −7.85342 42.2099i −0.353342 1.89911i
\(495\) 8.04724 3.85854i 0.361697 0.173428i
\(496\) 27.2611i 1.22406i
\(497\) 3.71614 0.166692
\(498\) 3.51068 0.157318
\(499\) 36.6887 1.64241 0.821205 0.570633i \(-0.193302\pi\)
0.821205 + 0.570633i \(0.193302\pi\)
\(500\) 1.36849 0.0612008
\(501\) 9.04169i 0.403953i
\(502\) −26.2951 −1.17361
\(503\) 39.8719i 1.77780i 0.458102 + 0.888899i \(0.348529\pi\)
−0.458102 + 0.888899i \(0.651471\pi\)
\(504\) 12.7544i 0.568124i
\(505\) 19.3531i 0.861203i
\(506\) 10.4558 5.01339i 0.464816 0.222873i
\(507\) 8.78619i 0.390208i
\(508\) 15.0437 0.667456
\(509\) 11.2708i 0.499571i −0.968301 0.249786i \(-0.919640\pi\)
0.968301 0.249786i \(-0.0803602\pi\)
\(510\) −5.47657 −0.242507
\(511\) 38.7192 1.71284
\(512\) −20.8741 −0.922514
\(513\) −13.5600 + 2.52292i −0.598688 + 0.111390i
\(514\) 14.3689i 0.633783i
\(515\) 19.1440i 0.843587i
\(516\) 5.04160 0.221944
\(517\) 20.9647 10.0523i 0.922025 0.442098i
\(518\) 25.6955 1.12900
\(519\) 9.77249i 0.428965i
\(520\) 6.22023 0.272775
\(521\) 10.8381i 0.474825i −0.971409 0.237413i \(-0.923701\pi\)
0.971409 0.237413i \(-0.0762994\pi\)
\(522\) 39.7649 1.74046
\(523\) 8.09479 0.353960 0.176980 0.984214i \(-0.443367\pi\)
0.176980 + 0.984214i \(0.443367\pi\)
\(524\) 10.7090i 0.467824i
\(525\) 2.27389 0.0992409
\(526\) 7.76513i 0.338576i
\(527\) −30.0764 −1.31015
\(528\) −3.87833 8.08852i −0.168783 0.352008i
\(529\) −19.3712 −0.842228
\(530\) 14.2096i 0.617226i
\(531\) 8.11253i 0.352054i
\(532\) 23.9830 4.46219i 1.03979 0.193460i
\(533\) −49.4733 −2.14292
\(534\) 6.51594 0.281972
\(535\) 2.16802 0.0937315
\(536\) 7.57883i 0.327356i
\(537\) −13.4281 −0.579463
\(538\) 41.0117i 1.76814i
\(539\) −29.0817 + 13.9442i −1.25264 + 0.600621i
\(540\) 4.33026i 0.186345i
\(541\) 29.3769i 1.26301i −0.775371 0.631506i \(-0.782437\pi\)
0.775371 0.631506i \(-0.217563\pi\)
\(542\) 27.2238i 1.16936i
\(543\) −6.79838 −0.291747
\(544\) 35.4699i 1.52076i
\(545\) −15.5057 −0.664192
\(546\) −22.3974 −0.958519
\(547\) −14.2016 −0.607215 −0.303608 0.952797i \(-0.598191\pi\)
−0.303608 + 0.952797i \(0.598191\pi\)
\(548\) −21.9990 −0.939749
\(549\) 19.5229i 0.833219i
\(550\) −5.48881 + 2.63180i −0.234044 + 0.112220i
\(551\) −6.41988 34.5050i −0.273496 1.46996i
\(552\) 1.22764i 0.0522520i
\(553\) 38.5248i 1.63824i
\(554\) 35.7847i 1.52035i
\(555\) 1.90354 0.0808007
\(556\) 20.4199i 0.865996i
\(557\) 38.6125i 1.63606i −0.575173 0.818032i \(-0.695065\pi\)
0.575173 0.818032i \(-0.304935\pi\)
\(558\) 27.6780i 1.17170i
\(559\) 35.5582i 1.50395i
\(560\) 19.8924i 0.840607i
\(561\) 8.92384 4.27885i 0.376765 0.180653i
\(562\) −38.0768 −1.60617
\(563\) −44.7052 −1.88410 −0.942049 0.335474i \(-0.891104\pi\)
−0.942049 + 0.335474i \(0.891104\pi\)
\(564\) 5.33416i 0.224609i
\(565\) 14.5570i 0.612420i
\(566\) 5.80510i 0.244006i
\(567\) 25.8176i 1.08424i
\(568\) 1.05321i 0.0441917i
\(569\) 12.2282 0.512633 0.256316 0.966593i \(-0.417491\pi\)
0.256316 + 0.966593i \(0.417491\pi\)
\(570\) 4.37321 0.813665i 0.183174 0.0340807i
\(571\) 12.7770i 0.534699i 0.963600 + 0.267350i \(0.0861479\pi\)
−0.963600 + 0.267350i \(0.913852\pi\)
\(572\) 21.9640 10.5314i 0.918361 0.440340i
\(573\) 1.52469i 0.0636950i
\(574\) 69.1916i 2.88800i
\(575\) −1.90493 −0.0794410
\(576\) −6.46383 −0.269326
\(577\) −44.8565 −1.86740 −0.933700 0.358057i \(-0.883439\pi\)
−0.933700 + 0.358057i \(0.883439\pi\)
\(578\) 21.6568 0.900804
\(579\) 1.55859i 0.0647726i
\(580\) −11.0189 −0.457534
\(581\) 14.0686 0.583666
\(582\) −12.7838 −0.529907
\(583\) 11.1020 + 23.1539i 0.459796 + 0.958938i
\(584\) 10.9736i 0.454092i
\(585\) 14.4410 0.597060
\(586\) 24.1766 0.998724
\(587\) 3.53027 0.145710 0.0728550 0.997343i \(-0.476789\pi\)
0.0728550 + 0.997343i \(0.476789\pi\)
\(588\) 7.39941i 0.305147i
\(589\) 24.0169 4.46851i 0.989601 0.184122i
\(590\) 5.53334i 0.227804i
\(591\) −15.2005 −0.625263
\(592\) 16.6524i 0.684411i
\(593\) 3.72182i 0.152837i −0.997076 0.0764185i \(-0.975652\pi\)
0.997076 0.0764185i \(-0.0243485\pi\)
\(594\) −8.32770 17.3680i −0.341690 0.712618i
\(595\) −21.9467 −0.899727
\(596\) 8.53217i 0.349491i
\(597\) 6.09358i 0.249393i
\(598\) 18.7631 0.767282
\(599\) 12.0519i 0.492428i 0.969216 + 0.246214i \(0.0791866\pi\)
−0.969216 + 0.246214i \(0.920813\pi\)
\(600\) 0.644456i 0.0263098i
\(601\) 24.0531 0.981148 0.490574 0.871400i \(-0.336787\pi\)
0.490574 + 0.871400i \(0.336787\pi\)
\(602\) 49.7304 2.02686
\(603\) 17.5951i 0.716527i
\(604\) 9.59603 0.390457
\(605\) 6.88754 8.57681i 0.280019 0.348697i
\(606\) 19.7499 0.802285
\(607\) 17.0902 0.693672 0.346836 0.937926i \(-0.387256\pi\)
0.346836 + 0.937926i \(0.387256\pi\)
\(608\) 5.26982 + 28.3238i 0.213720 + 1.14868i
\(609\) −18.3090 −0.741920
\(610\) 13.3161i 0.539152i
\(611\) 37.6216 1.52201
\(612\) 19.7617i 0.798818i
\(613\) 45.1575i 1.82389i 0.410307 + 0.911947i \(0.365421\pi\)
−0.410307 + 0.911947i \(0.634579\pi\)
\(614\) −46.8337 −1.89005
\(615\) 5.12575i 0.206690i
\(616\) −6.79684 14.1753i −0.273853 0.571139i
\(617\) 15.4937 0.623753 0.311877 0.950123i \(-0.399042\pi\)
0.311877 + 0.950123i \(0.399042\pi\)
\(618\) −19.5365 −0.785874
\(619\) −23.4826 −0.943847 −0.471924 0.881639i \(-0.656440\pi\)
−0.471924 + 0.881639i \(0.656440\pi\)
\(620\) 7.66960i 0.308018i
\(621\) 6.02769i 0.241883i
\(622\) −4.84859 −0.194411
\(623\) 26.1118 1.04615
\(624\) 14.5150i 0.581066i
\(625\) 1.00000 0.0400000
\(626\) 41.4588 1.65703
\(627\) −6.49024 + 4.74262i −0.259195 + 0.189402i
\(628\) 12.9680 0.517481
\(629\) −18.3722 −0.732546
\(630\) 20.1966i 0.804653i
\(631\) 27.0277 1.07596 0.537979 0.842958i \(-0.319188\pi\)
0.537979 + 0.842958i \(0.319188\pi\)
\(632\) 10.9185 0.434315
\(633\) 3.49606i 0.138956i
\(634\) 17.6343i 0.700349i
\(635\) 10.9929 0.436240
\(636\) −5.89118 −0.233600
\(637\) −52.1877 −2.06775
\(638\) 44.1950 21.1909i 1.74970 0.838954i
\(639\) 2.44514i 0.0967284i
\(640\) −8.81007 −0.348249
\(641\) 10.3415i 0.408465i −0.978922 0.204233i \(-0.934530\pi\)
0.978922 0.204233i \(-0.0654699\pi\)
\(642\) 2.21246i 0.0873190i
\(643\) 32.8621 1.29596 0.647978 0.761659i \(-0.275615\pi\)
0.647978 + 0.761659i \(0.275615\pi\)
\(644\) 10.6609i 0.420099i
\(645\) 3.68406 0.145060
\(646\) −42.2085 + 7.85316i −1.66067 + 0.308978i
\(647\) −2.08256 −0.0818737 −0.0409369 0.999162i \(-0.513034\pi\)
−0.0409369 + 0.999162i \(0.513034\pi\)
\(648\) 7.31710 0.287443
\(649\) −4.32320 9.01634i −0.169700 0.353922i
\(650\) −9.84979 −0.386340
\(651\) 12.7439i 0.499471i
\(652\) 2.46274 0.0964482
\(653\) −25.5955 −1.00163 −0.500815 0.865555i \(-0.666966\pi\)
−0.500815 + 0.865555i \(0.666966\pi\)
\(654\) 15.8236i 0.618752i
\(655\) 7.82540i 0.305764i
\(656\) 44.8409 1.75074
\(657\) 25.4765i 0.993932i
\(658\) 52.6162i 2.05119i
\(659\) −6.55507 −0.255349 −0.127675 0.991816i \(-0.540751\pi\)
−0.127675 + 0.991816i \(0.540751\pi\)
\(660\) 1.09112 + 2.27561i 0.0424719 + 0.0885781i
\(661\) 44.6883i 1.73817i 0.494660 + 0.869086i \(0.335293\pi\)
−0.494660 + 0.869086i \(0.664707\pi\)
\(662\) 33.7328i 1.31106i
\(663\) 16.0140 0.621933
\(664\) 3.98727i 0.154736i
\(665\) 17.5251 3.26066i 0.679595 0.126443i
\(666\) 16.9071i 0.655138i
\(667\) 15.3382 0.593897
\(668\) −22.2533 −0.861008
\(669\) 12.7026 0.491111
\(670\) 12.0011i 0.463644i
\(671\) 10.4039 + 21.6980i 0.401636 + 0.837640i
\(672\) 15.0292 0.579763
\(673\) −21.4395 −0.826432 −0.413216 0.910633i \(-0.635594\pi\)
−0.413216 + 0.910633i \(0.635594\pi\)
\(674\) −30.9205 −1.19101
\(675\) 3.16426i 0.121792i
\(676\) 21.6245 0.831712
\(677\) −17.0149 −0.653935 −0.326968 0.945036i \(-0.606027\pi\)
−0.326968 + 0.945036i \(0.606027\pi\)
\(678\) −14.8555 −0.570522
\(679\) −51.2296 −1.96601
\(680\) 6.22003i 0.238527i
\(681\) 11.2613i 0.431535i
\(682\) 14.7497 + 30.7616i 0.564796 + 1.17792i
\(683\) 6.01886i 0.230305i −0.993348 0.115153i \(-0.963264\pi\)
0.993348 0.115153i \(-0.0367357\pi\)
\(684\) 2.93603 + 15.7803i 0.112262 + 0.603375i
\(685\) −16.0753 −0.614207
\(686\) 20.4480i 0.780707i
\(687\) 4.68324i 0.178677i
\(688\) 32.2287i 1.22871i
\(689\) 41.5502i 1.58294i
\(690\) 1.94398i 0.0740062i
\(691\) −51.1113 −1.94436 −0.972182 0.234226i \(-0.924744\pi\)
−0.972182 + 0.234226i \(0.924744\pi\)
\(692\) 24.0520 0.914320
\(693\) −15.7796 32.9095i −0.599418 1.25013i
\(694\) 24.7403i 0.939130i
\(695\) 14.9215i 0.566004i
\(696\) 5.18906i 0.196691i
\(697\) 49.4717i 1.87387i
\(698\) 32.7570i 1.23987i
\(699\) 6.75522 0.255506
\(700\) 5.59649i 0.211528i
\(701\) 6.49530i 0.245324i 0.992449 + 0.122662i \(0.0391431\pi\)
−0.992449 + 0.122662i \(0.960857\pi\)
\(702\) 31.1673i 1.17633i
\(703\) 14.6708 2.72959i 0.553318 0.102948i
\(704\) −7.18395 + 3.44460i −0.270755 + 0.129823i
\(705\) 3.89784i 0.146801i
\(706\) −16.4777 −0.620148
\(707\) 79.1454 2.97657
\(708\) 2.29408 0.0862166
\(709\) 4.65020 0.174642 0.0873209 0.996180i \(-0.472169\pi\)
0.0873209 + 0.996180i \(0.472169\pi\)
\(710\) 1.66777i 0.0625902i
\(711\) 25.3485 0.950644
\(712\) 7.40049i 0.277345i
\(713\) 10.6760i 0.399820i
\(714\) 22.3967i 0.838174i
\(715\) 16.0498 7.69564i 0.600228 0.287801i
\(716\) 33.0490i 1.23510i
\(717\) 5.39980 0.201659
\(718\) 12.0353i 0.449155i
\(719\) −14.4010 −0.537066 −0.268533 0.963271i \(-0.586539\pi\)
−0.268533 + 0.963271i \(0.586539\pi\)
\(720\) −13.0888 −0.487790
\(721\) −78.2902 −2.91568
\(722\) 32.5380 12.5420i 1.21094 0.466764i
\(723\) 5.78354i 0.215092i
\(724\) 16.7321i 0.621845i
\(725\) −8.05184 −0.299038
\(726\) −8.75265 7.02875i −0.324841 0.260861i
\(727\) −6.68089 −0.247781 −0.123890 0.992296i \(-0.539537\pi\)
−0.123890 + 0.992296i \(0.539537\pi\)
\(728\) 25.4379i 0.942791i
\(729\) 11.7092 0.433675
\(730\) 17.3768i 0.643145i
\(731\) −35.5570 −1.31512
\(732\) −5.52073 −0.204052
\(733\) 16.2583i 0.600514i 0.953858 + 0.300257i \(0.0970724\pi\)
−0.953858 + 0.300257i \(0.902928\pi\)
\(734\) −14.4881 −0.534766
\(735\) 5.40699i 0.199440i
\(736\) −12.5905 −0.464092
\(737\) 9.37649 + 19.5553i 0.345387 + 0.720330i
\(738\) 45.5267 1.67586
\(739\) 20.5139i 0.754616i −0.926088 0.377308i \(-0.876850\pi\)
0.926088 0.377308i \(-0.123150\pi\)
\(740\) 4.68497i 0.172223i
\(741\) −12.7877 + 2.37923i −0.469768 + 0.0874033i
\(742\) −58.1107 −2.13331
\(743\) −6.96289 −0.255444 −0.127722 0.991810i \(-0.540766\pi\)
−0.127722 + 0.991810i \(0.540766\pi\)
\(744\) 3.61180 0.132415
\(745\) 6.23473i 0.228423i
\(746\) 15.1974 0.556415
\(747\) 9.25689i 0.338692i
\(748\) −10.5311 21.9633i −0.385054 0.803057i
\(749\) 8.86618i 0.323963i
\(750\) 1.02050i 0.0372635i
\(751\) 24.7078i 0.901599i 0.892625 + 0.450800i \(0.148861\pi\)
−0.892625 + 0.450800i \(0.851139\pi\)
\(752\) −34.0989 −1.24346
\(753\) 7.96624i 0.290306i
\(754\) 79.3090 2.88826
\(755\) 7.01213 0.255198
\(756\) 17.7088 0.644061
\(757\) 33.3132 1.21079 0.605395 0.795925i \(-0.293015\pi\)
0.605395 + 0.795925i \(0.293015\pi\)
\(758\) 4.63346i 0.168295i
\(759\) −1.51883 3.16763i −0.0551302 0.114978i
\(760\) 0.924122 + 4.96689i 0.0335214 + 0.180168i
\(761\) 28.7548i 1.04236i 0.853447 + 0.521180i \(0.174508\pi\)
−0.853447 + 0.521180i \(0.825492\pi\)
\(762\) 11.2183i 0.406395i
\(763\) 63.4112i 2.29564i
\(764\) −3.75257 −0.135763
\(765\) 14.4405i 0.522097i
\(766\) 2.98563i 0.107875i
\(767\) 16.1800i 0.584226i
\(768\) 11.6620i 0.420817i
\(769\) 15.9549i 0.575349i −0.957728 0.287675i \(-0.907118\pi\)
0.957728 0.287675i \(-0.0928822\pi\)
\(770\) 10.7629 + 22.4467i 0.387866 + 0.808923i
\(771\) 4.35312 0.156774
\(772\) −3.83598 −0.138060
\(773\) 47.0295i 1.69153i −0.533553 0.845767i \(-0.679144\pi\)
0.533553 0.845767i \(-0.320856\pi\)
\(774\) 32.7216i 1.17615i
\(775\) 5.60442i 0.201317i
\(776\) 14.5193i 0.521211i
\(777\) 7.78459i 0.279271i
\(778\) −2.79441 −0.100184
\(779\) −7.35010 39.5047i −0.263345 1.41540i
\(780\) 4.08364i 0.146218i
\(781\) 1.30303 + 2.71755i 0.0466259 + 0.0972417i
\(782\) 18.7625i 0.670947i
\(783\) 25.4781i 0.910514i
\(784\) 47.3012 1.68933
\(785\) 9.47616 0.338219
\(786\) −7.98583 −0.284845
\(787\) −26.7602 −0.953899 −0.476950 0.878931i \(-0.658258\pi\)
−0.476950 + 0.878931i \(0.658258\pi\)
\(788\) 37.4113i 1.33272i
\(789\) −2.35249 −0.0837507
\(790\) −17.2895 −0.615135
\(791\) −59.5316 −2.11670
\(792\) 9.32706 4.47219i 0.331423 0.158912i
\(793\) 38.9375i 1.38271i
\(794\) −48.6709 −1.72727
\(795\) −4.30487 −0.152678
\(796\) −14.9975 −0.531571
\(797\) 47.7999i 1.69316i 0.532262 + 0.846579i \(0.321342\pi\)
−0.532262 + 0.846579i \(0.678658\pi\)
\(798\) −3.32751 17.8844i −0.117793 0.633102i
\(799\) 37.6204i 1.33091i
\(800\) 6.60944 0.233679
\(801\) 17.1811i 0.607063i
\(802\) 1.07025i 0.0377918i
\(803\) 13.5765 + 28.3148i 0.479105 + 0.999206i
\(804\) −4.97557 −0.175475
\(805\) 7.79028i 0.274571i
\(806\) 55.2023i 1.94442i
\(807\) −12.4247 −0.437371
\(808\) 22.4310i 0.789120i
\(809\) 34.4405i 1.21086i 0.795897 + 0.605432i \(0.207000\pi\)
−0.795897 + 0.605432i \(0.793000\pi\)
\(810\) −11.5867 −0.407115
\(811\) −28.9085 −1.01511 −0.507557 0.861618i \(-0.669451\pi\)
−0.507557 + 0.861618i \(0.669451\pi\)
\(812\) 45.0621i 1.58137i
\(813\) 8.24759 0.289256
\(814\) 9.00987 + 18.7907i 0.315796 + 0.658614i
\(815\) 1.79960 0.0630372
\(816\) −14.5146 −0.508111
\(817\) 28.3934 5.28277i 0.993358 0.184821i
\(818\) 14.6222 0.511254
\(819\) 59.0569i 2.06361i
\(820\) −12.6155 −0.440551
\(821\) 2.02869i 0.0708019i 0.999373 + 0.0354009i \(0.0112708\pi\)
−0.999373 + 0.0354009i \(0.988729\pi\)
\(822\) 16.4049i 0.572187i
\(823\) −21.2219 −0.739750 −0.369875 0.929081i \(-0.620600\pi\)
−0.369875 + 0.929081i \(0.620600\pi\)
\(824\) 22.1886i 0.772978i
\(825\) 0.797318 + 1.66286i 0.0277591 + 0.0578935i
\(826\) 22.6288 0.787357
\(827\) 18.2806 0.635679 0.317840 0.948144i \(-0.397043\pi\)
0.317840 + 0.948144i \(0.397043\pi\)
\(828\) −7.01467 −0.243777
\(829\) 9.24614i 0.321132i −0.987025 0.160566i \(-0.948668\pi\)
0.987025 0.160566i \(-0.0513319\pi\)
\(830\) 6.31387i 0.219158i
\(831\) −10.8412 −0.376076
\(832\) −12.8918 −0.446941
\(833\) 52.1861i 1.80814i
\(834\) −15.2274 −0.527281
\(835\) −16.2612 −0.562743
\(836\) 11.6725 + 15.9737i 0.403702 + 0.552464i
\(837\) 17.7338 0.612971
\(838\) 5.39210 0.186267
\(839\) 51.4071i 1.77477i 0.461028 + 0.887386i \(0.347481\pi\)
−0.461028 + 0.887386i \(0.652519\pi\)
\(840\) 2.63553 0.0909344
\(841\) 35.8322 1.23559
\(842\) 6.06844i 0.209132i
\(843\) 11.5356i 0.397306i
\(844\) 8.60448 0.296178
\(845\) 15.8017 0.543596
\(846\) −34.6204 −1.19027
\(847\) −35.0752 28.1669i −1.20520 0.967825i
\(848\) 37.6597i 1.29324i
\(849\) 1.75868 0.0603579
\(850\) 9.84947i 0.337834i
\(851\) 6.52145i 0.223552i
\(852\) −0.691442 −0.0236884
\(853\) 27.7719i 0.950890i 0.879745 + 0.475445i \(0.157713\pi\)
−0.879745 + 0.475445i \(0.842287\pi\)
\(854\) −54.4566 −1.86347
\(855\) 2.14545 + 11.5312i 0.0733729 + 0.394358i
\(856\) 2.51281 0.0858861
\(857\) −11.5029 −0.392931 −0.196465 0.980511i \(-0.562946\pi\)
−0.196465 + 0.980511i \(0.562946\pi\)
\(858\) −7.85342 16.3788i −0.268111 0.559165i
\(859\) −43.7925 −1.49418 −0.747090 0.664723i \(-0.768550\pi\)
−0.747090 + 0.664723i \(0.768550\pi\)
\(860\) 9.06717i 0.309188i
\(861\) −20.9619 −0.714381
\(862\) −16.7029 −0.568902
\(863\) 18.5796i 0.632457i 0.948683 + 0.316228i \(0.102417\pi\)
−0.948683 + 0.316228i \(0.897583\pi\)
\(864\) 20.9140i 0.711508i
\(865\) 17.5756 0.597587
\(866\) 67.3539i 2.28878i
\(867\) 6.56104i 0.222825i
\(868\) −31.3651 −1.06460
\(869\) 28.1726 13.5083i 0.955689 0.458239i
\(870\) 8.21692i 0.278580i
\(871\) 35.0925i 1.18906i
\(872\) −17.9717 −0.608599
\(873\) 33.7081i 1.14085i
\(874\) 2.78758 + 14.9825i 0.0942914 + 0.506789i
\(875\) 4.08954i 0.138252i
\(876\) −7.20428 −0.243410
\(877\) 41.3037 1.39473 0.697364 0.716717i \(-0.254356\pi\)
0.697364 + 0.716717i \(0.254356\pi\)
\(878\) −59.4246 −2.00548
\(879\) 7.32441i 0.247046i
\(880\) −14.5470 + 6.97507i −0.490379 + 0.235129i
\(881\) 11.1721 0.376399 0.188199 0.982131i \(-0.439735\pi\)
0.188199 + 0.982131i \(0.439735\pi\)
\(882\) 48.0246 1.61707
\(883\) −28.6532 −0.964255 −0.482128 0.876101i \(-0.660136\pi\)
−0.482128 + 0.876101i \(0.660136\pi\)
\(884\) 39.4136i 1.32562i
\(885\) 1.67635 0.0563500
\(886\) 54.6479 1.83593
\(887\) 49.6519 1.66715 0.833575 0.552407i \(-0.186291\pi\)
0.833575 + 0.552407i \(0.186291\pi\)
\(888\) 2.20627 0.0740376
\(889\) 44.9559i 1.50777i
\(890\) 11.7187i 0.392813i
\(891\) 18.8800 9.05268i 0.632504 0.303276i
\(892\) 31.2635i 1.04678i
\(893\) 5.58933 + 30.0410i 0.187040 + 1.00528i
\(894\) 6.36255 0.212796
\(895\) 24.1500i 0.807245i
\(896\) 36.0291i 1.20365i
\(897\) 5.68439i 0.189796i
\(898\) 39.1254i 1.30563i
\(899\) 45.1259i 1.50503i
\(900\) 3.68238 0.122746
\(901\) 41.5489 1.38419
\(902\) 50.5987 24.2613i 1.68475 0.807814i
\(903\) 15.0661i 0.501368i
\(904\) 16.8722i 0.561160i
\(905\) 12.2267i 0.406429i
\(906\) 7.15589i 0.237739i
\(907\) 20.0790i 0.666714i −0.942801 0.333357i \(-0.891819\pi\)
0.942801 0.333357i \(-0.108181\pi\)
\(908\) −27.7163 −0.919798
\(909\) 52.0761i 1.72725i
\(910\) 40.2811i 1.33530i
\(911\) 40.9514i 1.35678i −0.734702 0.678390i \(-0.762678\pi\)
0.734702 0.678390i \(-0.237322\pi\)
\(912\) 11.5903 2.15646i 0.383794 0.0714074i
\(913\) 4.93303 + 10.2882i 0.163259 + 0.340489i
\(914\) 4.02383i 0.133096i
\(915\) −4.03417 −0.133366
\(916\) 11.5264 0.380841
\(917\) −32.0023 −1.05681
\(918\) −31.1663 −1.02864
\(919\) 12.5429i 0.413754i 0.978367 + 0.206877i \(0.0663299\pi\)
−0.978367 + 0.206877i \(0.933670\pi\)
\(920\) −2.20788 −0.0727917
\(921\) 14.1885i 0.467527i
\(922\) 1.22385i 0.0403052i
\(923\) 4.87671i 0.160519i
\(924\) 9.30620 4.46219i 0.306152 0.146795i
\(925\) 3.42346i 0.112563i
\(926\) −14.3059 −0.470123
\(927\) 51.5134i 1.69192i
\(928\) −53.2181 −1.74697
\(929\) 30.1376 0.988782 0.494391 0.869240i \(-0.335391\pi\)
0.494391 + 0.869240i \(0.335391\pi\)
\(930\) −5.71932 −0.187544
\(931\) −7.75338 41.6722i −0.254107 1.36575i
\(932\) 16.6259i 0.544600i
\(933\) 1.46890i 0.0480898i
\(934\) −22.6756 −0.741970
\(935\) −7.69539 16.0493i −0.251666 0.524868i
\(936\) 16.7376 0.547086
\(937\) 27.6531i 0.903386i 0.892173 + 0.451693i \(0.149180\pi\)
−0.892173 + 0.451693i \(0.850820\pi\)
\(938\) −49.0791 −1.60249
\(939\) 12.5602i 0.409885i
\(940\) 9.59334 0.312900
\(941\) 45.3025 1.47682 0.738410 0.674352i \(-0.235577\pi\)
0.738410 + 0.674352i \(0.235577\pi\)
\(942\) 9.67043i 0.315080i
\(943\) 17.5606 0.571853
\(944\) 14.6650i 0.477305i
\(945\) 12.9404 0.420950
\(946\) 17.4375 + 36.3671i 0.566941 + 1.18239i
\(947\) −9.89509 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(948\) 7.16810i 0.232809i
\(949\) 50.8115i 1.64941i
\(950\) −1.46335 7.86511i −0.0474775 0.255178i
\(951\) −5.34242 −0.173240
\(952\) −25.4371 −0.824420
\(953\) −8.39491 −0.271938 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(954\) 38.2357i 1.23793i
\(955\) −2.74212 −0.0887330
\(956\) 13.2900i 0.429828i
\(957\) −6.41988 13.3891i −0.207525 0.432809i
\(958\) 23.3211i 0.753469i
\(959\) 65.7407i 2.12288i
\(960\) 1.33567i 0.0431086i
\(961\) −0.409517 −0.0132102
\(962\) 33.7204i 1.08719i
\(963\) 5.83377 0.187991
\(964\) 14.2344 0.458460
\(965\) −2.80307 −0.0902341
\(966\) 7.94999 0.255787
\(967\) 22.4769i 0.722809i 0.932409 + 0.361405i \(0.117703\pi\)
−0.932409 + 0.361405i \(0.882297\pi\)
\(968\) 7.98292 9.94085i 0.256581 0.319511i
\(969\) 2.37916 + 12.7873i 0.0764295 + 0.410787i
\(970\) 22.9914i 0.738208i
\(971\) 17.9797i 0.576997i −0.957480 0.288498i \(-0.906844\pi\)
0.957480 0.288498i \(-0.0931560\pi\)
\(972\) 17.7945i 0.570759i
\(973\) −61.0219 −1.95627
\(974\) 2.87989i 0.0922775i
\(975\) 2.98404i 0.0955659i
\(976\) 35.2916i 1.12966i
\(977\) 13.1960i 0.422179i 0.977467 + 0.211089i \(0.0677011\pi\)
−0.977467 + 0.211089i \(0.932299\pi\)
\(978\) 1.83650i 0.0587246i
\(979\) 9.15585 + 19.0952i 0.292622 + 0.610284i
\(980\) −13.3076 −0.425097
\(981\) −41.7233 −1.33212
\(982\) 25.0728i 0.800106i
\(983\) 9.75240i 0.311053i −0.987832 0.155527i \(-0.950293\pi\)
0.987832 0.155527i \(-0.0497074\pi\)
\(984\) 5.94094i 0.189390i
\(985\) 27.3376i 0.871049i
\(986\) 79.3064i 2.52563i
\(987\) 15.9404 0.507387
\(988\) 5.85575 + 31.4730i 0.186296 + 1.00129i
\(989\) 12.6214i 0.401338i
\(990\) −14.7695 + 7.08174i −0.469405 + 0.225073i
\(991\) 15.9412i 0.506390i 0.967415 + 0.253195i \(0.0814814\pi\)
−0.967415 + 0.253195i \(0.918519\pi\)
\(992\) 37.0421i 1.17609i
\(993\) −10.2195 −0.324307
\(994\) −6.82039 −0.216330
\(995\) −10.9591 −0.347428
\(996\) −2.61768 −0.0829443
\(997\) 11.8297i 0.374650i −0.982298 0.187325i \(-0.940018\pi\)
0.982298 0.187325i \(-0.0599818\pi\)
\(998\) −67.3364 −2.13150
\(999\) 10.8327 0.342732
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.b.626.9 40
11.10 odd 2 inner 1045.2.f.b.626.31 yes 40
19.18 odd 2 inner 1045.2.f.b.626.32 yes 40
209.208 even 2 inner 1045.2.f.b.626.10 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.f.b.626.9 40 1.1 even 1 trivial
1045.2.f.b.626.10 yes 40 209.208 even 2 inner
1045.2.f.b.626.31 yes 40 11.10 odd 2 inner
1045.2.f.b.626.32 yes 40 19.18 odd 2 inner