Properties

Label 1045.2.f.a.626.2
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 626.2
Character \(\chi\) \(=\) 1045.626
Dual form 1045.2.f.a.626.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.71965 q^{2} +2.28311i q^{3} +5.39648 q^{4} -1.00000 q^{5} -6.20925i q^{6} -2.59337i q^{7} -9.23723 q^{8} -2.21259 q^{9} +O(q^{10})\) \(q-2.71965 q^{2} +2.28311i q^{3} +5.39648 q^{4} -1.00000 q^{5} -6.20925i q^{6} -2.59337i q^{7} -9.23723 q^{8} -2.21259 q^{9} +2.71965 q^{10} +(-2.91614 + 1.57992i) q^{11} +12.3208i q^{12} -3.84876 q^{13} +7.05304i q^{14} -2.28311i q^{15} +14.3290 q^{16} +1.32421i q^{17} +6.01746 q^{18} +(1.51223 + 4.08817i) q^{19} -5.39648 q^{20} +5.92094 q^{21} +(7.93086 - 4.29683i) q^{22} -4.75248 q^{23} -21.0896i q^{24} +1.00000 q^{25} +10.4673 q^{26} +1.79775i q^{27} -13.9951i q^{28} +5.07960 q^{29} +6.20925i q^{30} -5.94228i q^{31} -20.4955 q^{32} +(-3.60714 - 6.65785i) q^{33} -3.60137i q^{34} +2.59337i q^{35} -11.9402 q^{36} -0.538972i q^{37} +(-4.11273 - 11.1184i) q^{38} -8.78715i q^{39} +9.23723 q^{40} -6.13890 q^{41} -16.1029 q^{42} -10.0967i q^{43} +(-15.7369 + 8.52602i) q^{44} +2.21259 q^{45} +12.9251 q^{46} +9.19886 q^{47} +32.7148i q^{48} +0.274446 q^{49} -2.71965 q^{50} -3.02331 q^{51} -20.7698 q^{52} +1.16719i q^{53} -4.88924i q^{54} +(2.91614 - 1.57992i) q^{55} +23.9555i q^{56} +(-9.33375 + 3.45258i) q^{57} -13.8147 q^{58} -12.3841i q^{59} -12.3208i q^{60} +6.07026i q^{61} +16.1609i q^{62} +5.73805i q^{63} +27.0824 q^{64} +3.84876 q^{65} +(9.81014 + 18.1070i) q^{66} +5.71516i q^{67} +7.14606i q^{68} -10.8504i q^{69} -7.05304i q^{70} -11.0687i q^{71} +20.4382 q^{72} -15.5827i q^{73} +1.46581i q^{74} +2.28311i q^{75} +(8.16071 + 22.0618i) q^{76} +(4.09732 + 7.56261i) q^{77} +23.8979i q^{78} +11.4547 q^{79} -14.3290 q^{80} -10.7422 q^{81} +16.6956 q^{82} +7.45980i q^{83} +31.9522 q^{84} -1.32421i q^{85} +27.4594i q^{86} +11.5973i q^{87} +(26.9370 - 14.5941i) q^{88} -8.00568i q^{89} -6.01746 q^{90} +9.98126i q^{91} -25.6466 q^{92} +13.5669 q^{93} -25.0177 q^{94} +(-1.51223 - 4.08817i) q^{95} -46.7934i q^{96} -17.6549i q^{97} -0.746397 q^{98} +(6.45220 - 3.49572i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + 40 q^{4} - 40 q^{5} - 28 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.71965 −1.92308 −0.961541 0.274663i \(-0.911434\pi\)
−0.961541 + 0.274663i \(0.911434\pi\)
\(3\) 2.28311i 1.31815i 0.752076 + 0.659077i \(0.229053\pi\)
−0.752076 + 0.659077i \(0.770947\pi\)
\(4\) 5.39648 2.69824
\(5\) −1.00000 −0.447214
\(6\) 6.20925i 2.53492i
\(7\) 2.59337i 0.980201i −0.871666 0.490100i \(-0.836960\pi\)
0.871666 0.490100i \(-0.163040\pi\)
\(8\) −9.23723 −3.26585
\(9\) −2.21259 −0.737529
\(10\) 2.71965 0.860028
\(11\) −2.91614 + 1.57992i −0.879248 + 0.476365i
\(12\) 12.3208i 3.55670i
\(13\) −3.84876 −1.06745 −0.533727 0.845657i \(-0.679209\pi\)
−0.533727 + 0.845657i \(0.679209\pi\)
\(14\) 7.05304i 1.88501i
\(15\) 2.28311i 0.589496i
\(16\) 14.3290 3.58226
\(17\) 1.32421i 0.321167i 0.987022 + 0.160584i \(0.0513376\pi\)
−0.987022 + 0.160584i \(0.948662\pi\)
\(18\) 6.01746 1.41833
\(19\) 1.51223 + 4.08817i 0.346929 + 0.937891i
\(20\) −5.39648 −1.20669
\(21\) 5.92094 1.29206
\(22\) 7.93086 4.29683i 1.69086 0.916088i
\(23\) −4.75248 −0.990960 −0.495480 0.868619i \(-0.665008\pi\)
−0.495480 + 0.868619i \(0.665008\pi\)
\(24\) 21.0896i 4.30490i
\(25\) 1.00000 0.200000
\(26\) 10.4673 2.05280
\(27\) 1.79775i 0.345977i
\(28\) 13.9951i 2.64482i
\(29\) 5.07960 0.943258 0.471629 0.881797i \(-0.343666\pi\)
0.471629 + 0.881797i \(0.343666\pi\)
\(30\) 6.20925i 1.13365i
\(31\) 5.94228i 1.06726i −0.845717 0.533632i \(-0.820827\pi\)
0.845717 0.533632i \(-0.179173\pi\)
\(32\) −20.4955 −3.62313
\(33\) −3.60714 6.65785i −0.627922 1.15898i
\(34\) 3.60137i 0.617631i
\(35\) 2.59337i 0.438359i
\(36\) −11.9402 −1.99003
\(37\) 0.538972i 0.0886064i −0.999018 0.0443032i \(-0.985893\pi\)
0.999018 0.0443032i \(-0.0141068\pi\)
\(38\) −4.11273 11.1184i −0.667172 1.80364i
\(39\) 8.78715i 1.40707i
\(40\) 9.23723 1.46053
\(41\) −6.13890 −0.958735 −0.479367 0.877614i \(-0.659134\pi\)
−0.479367 + 0.877614i \(0.659134\pi\)
\(42\) −16.1029 −2.48473
\(43\) 10.0967i 1.53973i −0.638206 0.769866i \(-0.720323\pi\)
0.638206 0.769866i \(-0.279677\pi\)
\(44\) −15.7369 + 8.52602i −2.37242 + 1.28535i
\(45\) 2.21259 0.329833
\(46\) 12.9251 1.90570
\(47\) 9.19886 1.34179 0.670896 0.741552i \(-0.265910\pi\)
0.670896 + 0.741552i \(0.265910\pi\)
\(48\) 32.7148i 4.72197i
\(49\) 0.274446 0.0392066
\(50\) −2.71965 −0.384616
\(51\) −3.02331 −0.423348
\(52\) −20.7698 −2.88025
\(53\) 1.16719i 0.160326i 0.996782 + 0.0801632i \(0.0255441\pi\)
−0.996782 + 0.0801632i \(0.974456\pi\)
\(54\) 4.88924i 0.665342i
\(55\) 2.91614 1.57992i 0.393212 0.213037i
\(56\) 23.9555i 3.20119i
\(57\) −9.33375 + 3.45258i −1.23629 + 0.457306i
\(58\) −13.8147 −1.81396
\(59\) 12.3841i 1.61228i −0.591728 0.806138i \(-0.701554\pi\)
0.591728 0.806138i \(-0.298446\pi\)
\(60\) 12.3208i 1.59060i
\(61\) 6.07026i 0.777218i 0.921403 + 0.388609i \(0.127044\pi\)
−0.921403 + 0.388609i \(0.872956\pi\)
\(62\) 16.1609i 2.05244i
\(63\) 5.73805i 0.722927i
\(64\) 27.0824 3.38530
\(65\) 3.84876 0.477380
\(66\) 9.81014 + 18.1070i 1.20754 + 2.22882i
\(67\) 5.71516i 0.698217i 0.937082 + 0.349109i \(0.113516\pi\)
−0.937082 + 0.349109i \(0.886484\pi\)
\(68\) 7.14606i 0.866586i
\(69\) 10.8504i 1.30624i
\(70\) 7.05304i 0.843000i
\(71\) 11.0687i 1.31361i −0.754061 0.656805i \(-0.771907\pi\)
0.754061 0.656805i \(-0.228093\pi\)
\(72\) 20.4382 2.40866
\(73\) 15.5827i 1.82382i −0.410394 0.911908i \(-0.634609\pi\)
0.410394 0.911908i \(-0.365391\pi\)
\(74\) 1.46581i 0.170397i
\(75\) 2.28311i 0.263631i
\(76\) 8.16071 + 22.0618i 0.936097 + 2.53066i
\(77\) 4.09732 + 7.56261i 0.466933 + 0.861839i
\(78\) 23.8979i 2.70591i
\(79\) 11.4547 1.28876 0.644379 0.764706i \(-0.277116\pi\)
0.644379 + 0.764706i \(0.277116\pi\)
\(80\) −14.3290 −1.60204
\(81\) −10.7422 −1.19358
\(82\) 16.6956 1.84372
\(83\) 7.45980i 0.818819i 0.912351 + 0.409410i \(0.134265\pi\)
−0.912351 + 0.409410i \(0.865735\pi\)
\(84\) 31.9522 3.48628
\(85\) 1.32421i 0.143630i
\(86\) 27.4594i 2.96103i
\(87\) 11.5973i 1.24336i
\(88\) 26.9370 14.5941i 2.87150 1.55574i
\(89\) 8.00568i 0.848600i −0.905522 0.424300i \(-0.860520\pi\)
0.905522 0.424300i \(-0.139480\pi\)
\(90\) −6.01746 −0.634296
\(91\) 9.98126i 1.04632i
\(92\) −25.6466 −2.67385
\(93\) 13.5669 1.40682
\(94\) −25.0177 −2.58037
\(95\) −1.51223 4.08817i −0.155151 0.419438i
\(96\) 46.7934i 4.77584i
\(97\) 17.6549i 1.79259i −0.443462 0.896293i \(-0.646250\pi\)
0.443462 0.896293i \(-0.353750\pi\)
\(98\) −0.746397 −0.0753975
\(99\) 6.45220 3.49572i 0.648471 0.351333i
\(100\) 5.39648 0.539648
\(101\) 12.0320i 1.19723i −0.801037 0.598615i \(-0.795718\pi\)
0.801037 0.598615i \(-0.204282\pi\)
\(102\) 8.22233 0.814132
\(103\) 2.53280i 0.249565i 0.992184 + 0.124782i \(0.0398232\pi\)
−0.992184 + 0.124782i \(0.960177\pi\)
\(104\) 35.5519 3.48615
\(105\) −5.92094 −0.577825
\(106\) 3.17435i 0.308321i
\(107\) 4.40937 0.426270 0.213135 0.977023i \(-0.431633\pi\)
0.213135 + 0.977023i \(0.431633\pi\)
\(108\) 9.70152i 0.933529i
\(109\) −10.1230 −0.969607 −0.484804 0.874623i \(-0.661109\pi\)
−0.484804 + 0.874623i \(0.661109\pi\)
\(110\) −7.93086 + 4.29683i −0.756178 + 0.409687i
\(111\) 1.23053 0.116797
\(112\) 37.1605i 3.51134i
\(113\) 5.82809i 0.548261i −0.961693 0.274130i \(-0.911610\pi\)
0.961693 0.274130i \(-0.0883900\pi\)
\(114\) 25.3845 9.38980i 2.37748 0.879436i
\(115\) 4.75248 0.443171
\(116\) 27.4120 2.54514
\(117\) 8.51572 0.787279
\(118\) 33.6804i 3.10054i
\(119\) 3.43415 0.314808
\(120\) 21.0896i 1.92521i
\(121\) 6.00769 9.21454i 0.546154 0.837685i
\(122\) 16.5090i 1.49465i
\(123\) 14.0158i 1.26376i
\(124\) 32.0674i 2.87974i
\(125\) −1.00000 −0.0894427
\(126\) 15.6055i 1.39025i
\(127\) 0.196466 0.0174335 0.00871677 0.999962i \(-0.497225\pi\)
0.00871677 + 0.999962i \(0.497225\pi\)
\(128\) −32.6636 −2.88708
\(129\) 23.0519 2.02960
\(130\) −10.4673 −0.918041
\(131\) 9.10643i 0.795632i −0.917465 0.397816i \(-0.869768\pi\)
0.917465 0.397816i \(-0.130232\pi\)
\(132\) −19.4658 35.9290i −1.69428 3.12722i
\(133\) 10.6021 3.92176i 0.919322 0.340060i
\(134\) 15.5432i 1.34273i
\(135\) 1.79775i 0.154726i
\(136\) 12.2320i 1.04889i
\(137\) −3.76731 −0.321863 −0.160932 0.986966i \(-0.551450\pi\)
−0.160932 + 0.986966i \(0.551450\pi\)
\(138\) 29.5093i 2.51200i
\(139\) 3.48063i 0.295223i 0.989045 + 0.147612i \(0.0471586\pi\)
−0.989045 + 0.147612i \(0.952841\pi\)
\(140\) 13.9951i 1.18280i
\(141\) 21.0020i 1.76869i
\(142\) 30.1029i 2.52618i
\(143\) 11.2235 6.08075i 0.938557 0.508498i
\(144\) −31.7043 −2.64202
\(145\) −5.07960 −0.421838
\(146\) 42.3794i 3.50735i
\(147\) 0.626591i 0.0516803i
\(148\) 2.90855i 0.239082i
\(149\) 3.22238i 0.263987i −0.991251 0.131994i \(-0.957862\pi\)
0.991251 0.131994i \(-0.0421378\pi\)
\(150\) 6.20925i 0.506983i
\(151\) 0.819171 0.0666632 0.0333316 0.999444i \(-0.489388\pi\)
0.0333316 + 0.999444i \(0.489388\pi\)
\(152\) −13.9688 37.7634i −1.13302 3.06302i
\(153\) 2.92992i 0.236870i
\(154\) −11.1433 20.5676i −0.897950 1.65739i
\(155\) 5.94228i 0.477295i
\(156\) 47.4197i 3.79661i
\(157\) 19.6930 1.57167 0.785835 0.618436i \(-0.212234\pi\)
0.785835 + 0.618436i \(0.212234\pi\)
\(158\) −31.1528 −2.47839
\(159\) −2.66483 −0.211335
\(160\) 20.4955 1.62031
\(161\) 12.3249i 0.971339i
\(162\) 29.2150 2.29535
\(163\) −13.4915 −1.05673 −0.528367 0.849016i \(-0.677195\pi\)
−0.528367 + 0.849016i \(0.677195\pi\)
\(164\) −33.1284 −2.58690
\(165\) 3.60714 + 6.65785i 0.280815 + 0.518313i
\(166\) 20.2880i 1.57466i
\(167\) 19.6312 1.51911 0.759554 0.650444i \(-0.225417\pi\)
0.759554 + 0.650444i \(0.225417\pi\)
\(168\) −54.6931 −4.21966
\(169\) 1.81298 0.139460
\(170\) 3.60137i 0.276213i
\(171\) −3.34594 9.04544i −0.255870 0.691722i
\(172\) 54.4866i 4.15457i
\(173\) −4.76420 −0.362215 −0.181108 0.983463i \(-0.557968\pi\)
−0.181108 + 0.983463i \(0.557968\pi\)
\(174\) 31.5405i 2.39108i
\(175\) 2.59337i 0.196040i
\(176\) −41.7854 + 22.6388i −3.14970 + 1.70646i
\(177\) 28.2743 2.12523
\(178\) 21.7726i 1.63193i
\(179\) 17.4461i 1.30399i 0.758225 + 0.651993i \(0.226067\pi\)
−0.758225 + 0.651993i \(0.773933\pi\)
\(180\) 11.9402 0.889969
\(181\) 12.2191i 0.908237i 0.890941 + 0.454118i \(0.150046\pi\)
−0.890941 + 0.454118i \(0.849954\pi\)
\(182\) 27.1455i 2.01216i
\(183\) −13.8591 −1.02449
\(184\) 43.8997 3.23633
\(185\) 0.538972i 0.0396260i
\(186\) −36.8971 −2.70543
\(187\) −2.09214 3.86156i −0.152993 0.282386i
\(188\) 49.6415 3.62048
\(189\) 4.66222 0.339127
\(190\) 4.11273 + 11.1184i 0.298369 + 0.806613i
\(191\) −3.86569 −0.279711 −0.139856 0.990172i \(-0.544664\pi\)
−0.139856 + 0.990172i \(0.544664\pi\)
\(192\) 61.8321i 4.46235i
\(193\) 11.6387 0.837773 0.418887 0.908038i \(-0.362420\pi\)
0.418887 + 0.908038i \(0.362420\pi\)
\(194\) 48.0152i 3.44729i
\(195\) 8.78715i 0.629261i
\(196\) 1.48104 0.105789
\(197\) 14.4045i 1.02627i 0.858306 + 0.513137i \(0.171517\pi\)
−0.858306 + 0.513137i \(0.828483\pi\)
\(198\) −17.5477 + 9.50712i −1.24706 + 0.675641i
\(199\) −10.2338 −0.725453 −0.362727 0.931896i \(-0.618154\pi\)
−0.362727 + 0.931896i \(0.618154\pi\)
\(200\) −9.23723 −0.653171
\(201\) −13.0483 −0.920358
\(202\) 32.7228i 2.30237i
\(203\) 13.1733i 0.924582i
\(204\) −16.3152 −1.14229
\(205\) 6.13890 0.428759
\(206\) 6.88833i 0.479933i
\(207\) 10.5153 0.730862
\(208\) −55.1491 −3.82390
\(209\) −10.8689 9.53247i −0.751815 0.659374i
\(210\) 16.1029 1.11120
\(211\) 5.84312 0.402257 0.201129 0.979565i \(-0.435539\pi\)
0.201129 + 0.979565i \(0.435539\pi\)
\(212\) 6.29874i 0.432599i
\(213\) 25.2710 1.73154
\(214\) −11.9919 −0.819752
\(215\) 10.0967i 0.688589i
\(216\) 16.6062i 1.12991i
\(217\) −15.4105 −1.04613
\(218\) 27.5310 1.86463
\(219\) 35.5770 2.40407
\(220\) 15.7369 8.52602i 1.06098 0.574824i
\(221\) 5.09656i 0.342832i
\(222\) −3.34661 −0.224610
\(223\) 28.8841i 1.93422i 0.254357 + 0.967110i \(0.418136\pi\)
−0.254357 + 0.967110i \(0.581864\pi\)
\(224\) 53.1523i 3.55139i
\(225\) −2.21259 −0.147506
\(226\) 15.8504i 1.05435i
\(227\) 17.0489 1.13158 0.565788 0.824551i \(-0.308572\pi\)
0.565788 + 0.824551i \(0.308572\pi\)
\(228\) −50.3694 + 18.6318i −3.33579 + 1.23392i
\(229\) −16.6298 −1.09893 −0.549463 0.835518i \(-0.685168\pi\)
−0.549463 + 0.835518i \(0.685168\pi\)
\(230\) −12.9251 −0.852253
\(231\) −17.2663 + 9.35463i −1.13604 + 0.615489i
\(232\) −46.9215 −3.08054
\(233\) 15.5584i 1.01926i −0.860393 0.509632i \(-0.829782\pi\)
0.860393 0.509632i \(-0.170218\pi\)
\(234\) −23.1598 −1.51400
\(235\) −9.19886 −0.600067
\(236\) 66.8307i 4.35031i
\(237\) 26.1524i 1.69878i
\(238\) −9.33969 −0.605402
\(239\) 4.36882i 0.282595i 0.989967 + 0.141298i \(0.0451274\pi\)
−0.989967 + 0.141298i \(0.954873\pi\)
\(240\) 32.7148i 2.11173i
\(241\) −27.0174 −1.74035 −0.870173 0.492746i \(-0.835993\pi\)
−0.870173 + 0.492746i \(0.835993\pi\)
\(242\) −16.3388 + 25.0603i −1.05030 + 1.61094i
\(243\) 19.1324i 1.22734i
\(244\) 32.7581i 2.09712i
\(245\) −0.274446 −0.0175337
\(246\) 38.1180i 2.43031i
\(247\) −5.82021 15.7344i −0.370331 1.00116i
\(248\) 54.8902i 3.48553i
\(249\) −17.0315 −1.07933
\(250\) 2.71965 0.172006
\(251\) −29.3382 −1.85181 −0.925907 0.377752i \(-0.876697\pi\)
−0.925907 + 0.377752i \(0.876697\pi\)
\(252\) 30.9653i 1.95063i
\(253\) 13.8589 7.50854i 0.871299 0.472058i
\(254\) −0.534318 −0.0335261
\(255\) 3.02331 0.189327
\(256\) 34.6687 2.16680
\(257\) 20.4152i 1.27347i −0.771084 0.636734i \(-0.780285\pi\)
0.771084 0.636734i \(-0.219715\pi\)
\(258\) −62.6929 −3.90309
\(259\) −1.39775 −0.0868521
\(260\) 20.7698 1.28809
\(261\) −11.2391 −0.695680
\(262\) 24.7663i 1.53007i
\(263\) 1.95755i 0.120708i 0.998177 + 0.0603538i \(0.0192229\pi\)
−0.998177 + 0.0603538i \(0.980777\pi\)
\(264\) 33.3199 + 61.5001i 2.05070 + 3.78507i
\(265\) 1.16719i 0.0717001i
\(266\) −28.8341 + 10.6658i −1.76793 + 0.653963i
\(267\) 18.2778 1.11859
\(268\) 30.8417i 1.88396i
\(269\) 8.82051i 0.537796i 0.963169 + 0.268898i \(0.0866594\pi\)
−0.963169 + 0.268898i \(0.913341\pi\)
\(270\) 4.88924i 0.297550i
\(271\) 1.56326i 0.0949611i −0.998872 0.0474806i \(-0.984881\pi\)
0.998872 0.0474806i \(-0.0151192\pi\)
\(272\) 18.9746i 1.15051i
\(273\) −22.7883 −1.37921
\(274\) 10.2458 0.618969
\(275\) −2.91614 + 1.57992i −0.175850 + 0.0952729i
\(276\) 58.5541i 3.52454i
\(277\) 0.228901i 0.0137533i −0.999976 0.00687667i \(-0.997811\pi\)
0.999976 0.00687667i \(-0.00218893\pi\)
\(278\) 9.46609i 0.567738i
\(279\) 13.1478i 0.787139i
\(280\) 23.9555i 1.43162i
\(281\) −7.93845 −0.473568 −0.236784 0.971562i \(-0.576093\pi\)
−0.236784 + 0.971562i \(0.576093\pi\)
\(282\) 57.1180i 3.40133i
\(283\) 7.72636i 0.459285i −0.973275 0.229642i \(-0.926244\pi\)
0.973275 0.229642i \(-0.0737556\pi\)
\(284\) 59.7319i 3.54444i
\(285\) 9.33375 3.45258i 0.552883 0.204513i
\(286\) −30.5240 + 16.5375i −1.80492 + 0.977882i
\(287\) 15.9204i 0.939753i
\(288\) 45.3481 2.67216
\(289\) 15.2465 0.896852
\(290\) 13.8147 0.811229
\(291\) 40.3081 2.36290
\(292\) 84.0917i 4.92110i
\(293\) −22.2268 −1.29850 −0.649252 0.760573i \(-0.724918\pi\)
−0.649252 + 0.760573i \(0.724918\pi\)
\(294\) 1.70411i 0.0993855i
\(295\) 12.3841i 0.721032i
\(296\) 4.97861i 0.289376i
\(297\) −2.84030 5.24248i −0.164811 0.304199i
\(298\) 8.76372i 0.507669i
\(299\) 18.2912 1.05780
\(300\) 12.3208i 0.711339i
\(301\) −26.1844 −1.50925
\(302\) −2.22786 −0.128199
\(303\) 27.4704 1.57813
\(304\) 21.6688 + 58.5796i 1.24279 + 3.35977i
\(305\) 6.07026i 0.347582i
\(306\) 7.96836i 0.455521i
\(307\) −1.70905 −0.0975404 −0.0487702 0.998810i \(-0.515530\pi\)
−0.0487702 + 0.998810i \(0.515530\pi\)
\(308\) 22.1111 + 40.8115i 1.25990 + 2.32545i
\(309\) −5.78267 −0.328964
\(310\) 16.1609i 0.917877i
\(311\) 32.7579 1.85753 0.928766 0.370667i \(-0.120871\pi\)
0.928766 + 0.370667i \(0.120871\pi\)
\(312\) 81.1689i 4.59528i
\(313\) 3.61072 0.204090 0.102045 0.994780i \(-0.467461\pi\)
0.102045 + 0.994780i \(0.467461\pi\)
\(314\) −53.5579 −3.02245
\(315\) 5.73805i 0.323303i
\(316\) 61.8152 3.47738
\(317\) 13.3609i 0.750421i −0.926940 0.375211i \(-0.877570\pi\)
0.926940 0.375211i \(-0.122430\pi\)
\(318\) 7.24740 0.406414
\(319\) −14.8128 + 8.02538i −0.829358 + 0.449335i
\(320\) −27.0824 −1.51395
\(321\) 10.0671i 0.561890i
\(322\) 33.5194i 1.86796i
\(323\) −5.41359 + 2.00250i −0.301220 + 0.111422i
\(324\) −57.9702 −3.22057
\(325\) −3.84876 −0.213491
\(326\) 36.6920 2.03218
\(327\) 23.1119i 1.27809i
\(328\) 56.7064 3.13109
\(329\) 23.8560i 1.31523i
\(330\) −9.81014 18.1070i −0.540030 0.996758i
\(331\) 18.0012i 0.989437i −0.869053 0.494718i \(-0.835271\pi\)
0.869053 0.494718i \(-0.164729\pi\)
\(332\) 40.2567i 2.20937i
\(333\) 1.19252i 0.0653498i
\(334\) −53.3900 −2.92137
\(335\) 5.71516i 0.312252i
\(336\) 84.8414 4.62848
\(337\) −17.8934 −0.974717 −0.487359 0.873202i \(-0.662040\pi\)
−0.487359 + 0.873202i \(0.662040\pi\)
\(338\) −4.93066 −0.268192
\(339\) 13.3062 0.722692
\(340\) 7.14606i 0.387549i
\(341\) 9.38834 + 17.3285i 0.508407 + 0.938390i
\(342\) 9.09977 + 24.6004i 0.492059 + 1.33024i
\(343\) 18.8653i 1.01863i
\(344\) 93.2655i 5.02854i
\(345\) 10.8504i 0.584167i
\(346\) 12.9569 0.696569
\(347\) 31.5547i 1.69395i −0.531635 0.846974i \(-0.678422\pi\)
0.531635 0.846974i \(-0.321578\pi\)
\(348\) 62.5845i 3.35488i
\(349\) 10.5028i 0.562205i −0.959678 0.281102i \(-0.909300\pi\)
0.959678 0.281102i \(-0.0907000\pi\)
\(350\) 7.05304i 0.377001i
\(351\) 6.91911i 0.369315i
\(352\) 59.7676 32.3813i 3.18563 1.72593i
\(353\) −23.9603 −1.27528 −0.637640 0.770335i \(-0.720089\pi\)
−0.637640 + 0.770335i \(0.720089\pi\)
\(354\) −76.8961 −4.08698
\(355\) 11.0687i 0.587464i
\(356\) 43.2025i 2.28973i
\(357\) 7.84055i 0.414966i
\(358\) 47.4473i 2.50767i
\(359\) 10.5701i 0.557868i 0.960310 + 0.278934i \(0.0899810\pi\)
−0.960310 + 0.278934i \(0.910019\pi\)
\(360\) −20.4382 −1.07719
\(361\) −14.4263 + 12.3645i −0.759281 + 0.650763i
\(362\) 33.2316i 1.74661i
\(363\) 21.0378 + 13.7162i 1.10420 + 0.719914i
\(364\) 53.8637i 2.82322i
\(365\) 15.5827i 0.815635i
\(366\) 37.6918 1.97018
\(367\) −27.3093 −1.42553 −0.712766 0.701402i \(-0.752558\pi\)
−0.712766 + 0.701402i \(0.752558\pi\)
\(368\) −68.0984 −3.54988
\(369\) 13.5828 0.707095
\(370\) 1.46581i 0.0762040i
\(371\) 3.02696 0.157152
\(372\) 73.2134 3.79594
\(373\) 27.9735 1.44841 0.724206 0.689583i \(-0.242206\pi\)
0.724206 + 0.689583i \(0.242206\pi\)
\(374\) 5.68989 + 10.5021i 0.294217 + 0.543050i
\(375\) 2.28311i 0.117899i
\(376\) −84.9720 −4.38210
\(377\) −19.5502 −1.00689
\(378\) −12.6796 −0.652168
\(379\) 3.00076i 0.154138i −0.997026 0.0770692i \(-0.975444\pi\)
0.997026 0.0770692i \(-0.0245562\pi\)
\(380\) −8.16071 22.0618i −0.418636 1.13174i
\(381\) 0.448553i 0.0229801i
\(382\) 10.5133 0.537908
\(383\) 12.2086i 0.623828i −0.950110 0.311914i \(-0.899030\pi\)
0.950110 0.311914i \(-0.100970\pi\)
\(384\) 74.5746i 3.80562i
\(385\) −4.09732 7.56261i −0.208819 0.385426i
\(386\) −31.6532 −1.61111
\(387\) 22.3398i 1.13560i
\(388\) 95.2745i 4.83683i
\(389\) −32.4838 −1.64700 −0.823498 0.567320i \(-0.807980\pi\)
−0.823498 + 0.567320i \(0.807980\pi\)
\(390\) 23.8979i 1.21012i
\(391\) 6.29326i 0.318264i
\(392\) −2.53512 −0.128043
\(393\) 20.7910 1.04877
\(394\) 39.1750i 1.97361i
\(395\) −11.4547 −0.576350
\(396\) 34.8192 18.8646i 1.74973 0.947980i
\(397\) −21.7253 −1.09036 −0.545180 0.838319i \(-0.683539\pi\)
−0.545180 + 0.838319i \(0.683539\pi\)
\(398\) 27.8323 1.39511
\(399\) 8.95381 + 24.2058i 0.448251 + 1.21181i
\(400\) 14.3290 0.716452
\(401\) 7.68637i 0.383839i −0.981411 0.191919i \(-0.938529\pi\)
0.981411 0.191919i \(-0.0614712\pi\)
\(402\) 35.4868 1.76992
\(403\) 22.8704i 1.13926i
\(404\) 64.9305i 3.23041i
\(405\) 10.7422 0.533785
\(406\) 35.8267i 1.77805i
\(407\) 0.851534 + 1.57172i 0.0422090 + 0.0779070i
\(408\) 27.9270 1.38259
\(409\) −1.84466 −0.0912123 −0.0456062 0.998959i \(-0.514522\pi\)
−0.0456062 + 0.998959i \(0.514522\pi\)
\(410\) −16.6956 −0.824539
\(411\) 8.60119i 0.424265i
\(412\) 13.6682i 0.673385i
\(413\) −32.1166 −1.58035
\(414\) −28.5978 −1.40551
\(415\) 7.45980i 0.366187i
\(416\) 78.8823 3.86752
\(417\) −7.94666 −0.389150
\(418\) 29.5595 + 25.9249i 1.44580 + 1.26803i
\(419\) 5.98752 0.292509 0.146255 0.989247i \(-0.453278\pi\)
0.146255 + 0.989247i \(0.453278\pi\)
\(420\) −31.9522 −1.55911
\(421\) 23.1954i 1.13047i 0.824929 + 0.565237i \(0.191215\pi\)
−0.824929 + 0.565237i \(0.808785\pi\)
\(422\) −15.8912 −0.773573
\(423\) −20.3533 −0.989610
\(424\) 10.7816i 0.523603i
\(425\) 1.32421i 0.0642334i
\(426\) −68.7282 −3.32989
\(427\) 15.7424 0.761829
\(428\) 23.7951 1.15018
\(429\) 13.8830 + 25.6245i 0.670278 + 1.23716i
\(430\) 27.4594i 1.32421i
\(431\) 24.4251 1.17651 0.588257 0.808674i \(-0.299814\pi\)
0.588257 + 0.808674i \(0.299814\pi\)
\(432\) 25.7600i 1.23938i
\(433\) 20.2267i 0.972032i 0.873950 + 0.486016i \(0.161550\pi\)
−0.873950 + 0.486016i \(0.838450\pi\)
\(434\) 41.9111 2.01180
\(435\) 11.5973i 0.556047i
\(436\) −54.6286 −2.61623
\(437\) −7.18682 19.4289i −0.343792 0.929413i
\(438\) −96.7569 −4.62322
\(439\) −6.92575 −0.330548 −0.165274 0.986248i \(-0.552851\pi\)
−0.165274 + 0.986248i \(0.552851\pi\)
\(440\) −26.9370 + 14.5941i −1.28417 + 0.695747i
\(441\) −0.607236 −0.0289160
\(442\) 13.8608i 0.659293i
\(443\) −6.46044 −0.306945 −0.153472 0.988153i \(-0.549046\pi\)
−0.153472 + 0.988153i \(0.549046\pi\)
\(444\) 6.64054 0.315146
\(445\) 8.00568i 0.379506i
\(446\) 78.5545i 3.71966i
\(447\) 7.35703 0.347976
\(448\) 70.2347i 3.31828i
\(449\) 0.954419i 0.0450418i −0.999746 0.0225209i \(-0.992831\pi\)
0.999746 0.0225209i \(-0.00716924\pi\)
\(450\) 6.01746 0.283666
\(451\) 17.9019 9.69898i 0.842966 0.456707i
\(452\) 31.4512i 1.47934i
\(453\) 1.87026i 0.0878723i
\(454\) −46.3670 −2.17611
\(455\) 9.98126i 0.467929i
\(456\) 86.2180 31.8923i 4.03753 1.49349i
\(457\) 27.5121i 1.28696i 0.765462 + 0.643481i \(0.222510\pi\)
−0.765462 + 0.643481i \(0.777490\pi\)
\(458\) 45.2271 2.11332
\(459\) −2.38059 −0.111116
\(460\) 25.6466 1.19578
\(461\) 42.9295i 1.99943i −0.0239396 0.999713i \(-0.507621\pi\)
0.0239396 0.999713i \(-0.492379\pi\)
\(462\) 46.9581 25.4413i 2.18469 1.18364i
\(463\) 27.5331 1.27957 0.639787 0.768553i \(-0.279023\pi\)
0.639787 + 0.768553i \(0.279023\pi\)
\(464\) 72.7859 3.37900
\(465\) −13.5669 −0.629149
\(466\) 42.3133i 1.96013i
\(467\) 7.91119 0.366086 0.183043 0.983105i \(-0.441405\pi\)
0.183043 + 0.983105i \(0.441405\pi\)
\(468\) 45.9549 2.12427
\(469\) 14.8215 0.684393
\(470\) 25.0177 1.15398
\(471\) 44.9612i 2.07170i
\(472\) 114.395i 5.26546i
\(473\) 15.9520 + 29.4433i 0.733474 + 1.35381i
\(474\) 71.1253i 3.26689i
\(475\) 1.51223 + 4.08817i 0.0693858 + 0.187578i
\(476\) 18.5323 0.849429
\(477\) 2.58252i 0.118245i
\(478\) 11.8816i 0.543453i
\(479\) 8.70174i 0.397593i −0.980041 0.198796i \(-0.936297\pi\)
0.980041 0.198796i \(-0.0637032\pi\)
\(480\) 46.7934i 2.13582i
\(481\) 2.07438i 0.0945834i
\(482\) 73.4779 3.34683
\(483\) −28.1391 −1.28037
\(484\) 32.4204 49.7261i 1.47365 2.26028i
\(485\) 17.6549i 0.801669i
\(486\) 52.0334i 2.36028i
\(487\) 12.0708i 0.546980i −0.961875 0.273490i \(-0.911822\pi\)
0.961875 0.273490i \(-0.0881780\pi\)
\(488\) 56.0724i 2.53828i
\(489\) 30.8025i 1.39294i
\(490\) 0.746397 0.0337188
\(491\) 24.0982i 1.08754i −0.839235 0.543769i \(-0.816997\pi\)
0.839235 0.543769i \(-0.183003\pi\)
\(492\) 75.6359i 3.40993i
\(493\) 6.72644i 0.302944i
\(494\) 15.8289 + 42.7921i 0.712176 + 1.92531i
\(495\) −6.45220 + 3.49572i −0.290005 + 0.157121i
\(496\) 85.1472i 3.82322i
\(497\) −28.7051 −1.28760
\(498\) 46.3198 2.07564
\(499\) 29.7475 1.33168 0.665840 0.746095i \(-0.268073\pi\)
0.665840 + 0.746095i \(0.268073\pi\)
\(500\) −5.39648 −0.241338
\(501\) 44.8202i 2.00242i
\(502\) 79.7897 3.56119
\(503\) 3.52627i 0.157229i 0.996905 + 0.0786144i \(0.0250496\pi\)
−0.996905 + 0.0786144i \(0.974950\pi\)
\(504\) 53.0037i 2.36097i
\(505\) 12.0320i 0.535417i
\(506\) −37.6912 + 20.4206i −1.67558 + 0.907806i
\(507\) 4.13922i 0.183829i
\(508\) 1.06022 0.0470399
\(509\) 26.4681i 1.17318i −0.809885 0.586588i \(-0.800471\pi\)
0.809885 0.586588i \(-0.199529\pi\)
\(510\) −8.22233 −0.364091
\(511\) −40.4116 −1.78771
\(512\) −28.9594 −1.27984
\(513\) −7.34951 + 2.71861i −0.324489 + 0.120029i
\(514\) 55.5223i 2.44898i
\(515\) 2.53280i 0.111609i
\(516\) 124.399 5.47636
\(517\) −26.8251 + 14.5335i −1.17977 + 0.639182i
\(518\) 3.80139 0.167024
\(519\) 10.8772i 0.477455i
\(520\) −35.5519 −1.55905
\(521\) 8.09065i 0.354458i −0.984170 0.177229i \(-0.943287\pi\)
0.984170 0.177229i \(-0.0567133\pi\)
\(522\) 30.5663 1.33785
\(523\) −4.77075 −0.208611 −0.104305 0.994545i \(-0.533262\pi\)
−0.104305 + 0.994545i \(0.533262\pi\)
\(524\) 49.1427i 2.14681i
\(525\) 5.92094 0.258411
\(526\) 5.32384i 0.232130i
\(527\) 7.86880 0.342770
\(528\) −51.6868 95.4007i −2.24938 4.15178i
\(529\) −0.413980 −0.0179991
\(530\) 3.17435i 0.137885i
\(531\) 27.4010i 1.18910i
\(532\) 57.2142 21.1637i 2.48055 0.917563i
\(533\) 23.6272 1.02341
\(534\) −49.7093 −2.15113
\(535\) −4.40937 −0.190634
\(536\) 52.7922i 2.28028i
\(537\) −39.8314 −1.71885
\(538\) 23.9887i 1.03422i
\(539\) −0.800322 + 0.433604i −0.0344723 + 0.0186766i
\(540\) 9.70152i 0.417487i
\(541\) 34.6340i 1.48903i 0.667605 + 0.744516i \(0.267320\pi\)
−0.667605 + 0.744516i \(0.732680\pi\)
\(542\) 4.25151i 0.182618i
\(543\) −27.8975 −1.19720
\(544\) 27.1403i 1.16363i
\(545\) 10.1230 0.433622
\(546\) 61.9761 2.65233
\(547\) −7.85791 −0.335980 −0.167990 0.985789i \(-0.553728\pi\)
−0.167990 + 0.985789i \(0.553728\pi\)
\(548\) −20.3302 −0.868465
\(549\) 13.4310i 0.573221i
\(550\) 7.93086 4.29683i 0.338173 0.183218i
\(551\) 7.68151 + 20.7663i 0.327244 + 0.884674i
\(552\) 100.228i 4.26598i
\(553\) 29.7063i 1.26324i
\(554\) 0.622531i 0.0264488i
\(555\) −1.23053 −0.0522332
\(556\) 18.7832i 0.796584i
\(557\) 4.67243i 0.197977i −0.995089 0.0989886i \(-0.968439\pi\)
0.995089 0.0989886i \(-0.0315607\pi\)
\(558\) 35.7574i 1.51373i
\(559\) 38.8598i 1.64359i
\(560\) 37.1605i 1.57032i
\(561\) 8.81637 4.77659i 0.372228 0.201668i
\(562\) 21.5898 0.910710
\(563\) −36.5088 −1.53866 −0.769331 0.638850i \(-0.779410\pi\)
−0.769331 + 0.638850i \(0.779410\pi\)
\(564\) 113.337i 4.77234i
\(565\) 5.82809i 0.245190i
\(566\) 21.0130i 0.883242i
\(567\) 27.8585i 1.16995i
\(568\) 102.244i 4.29006i
\(569\) −35.9943 −1.50896 −0.754480 0.656323i \(-0.772111\pi\)
−0.754480 + 0.656323i \(0.772111\pi\)
\(570\) −25.3845 + 9.38980i −1.06324 + 0.393296i
\(571\) 18.0377i 0.754856i −0.926039 0.377428i \(-0.876809\pi\)
0.926039 0.377428i \(-0.123191\pi\)
\(572\) 60.5675 32.8146i 2.53245 1.37205i
\(573\) 8.82579i 0.368703i
\(574\) 43.2979i 1.80722i
\(575\) −4.75248 −0.198192
\(576\) −59.9222 −2.49676
\(577\) −29.0808 −1.21065 −0.605325 0.795978i \(-0.706957\pi\)
−0.605325 + 0.795978i \(0.706957\pi\)
\(578\) −41.4650 −1.72472
\(579\) 26.5725i 1.10431i
\(580\) −27.4120 −1.13822
\(581\) 19.3460 0.802607
\(582\) −109.624 −4.54406
\(583\) −1.84408 3.40369i −0.0763738 0.140967i
\(584\) 143.941i 5.95632i
\(585\) −8.51572 −0.352082
\(586\) 60.4491 2.49713
\(587\) −0.497758 −0.0205447 −0.0102723 0.999947i \(-0.503270\pi\)
−0.0102723 + 0.999947i \(0.503270\pi\)
\(588\) 3.38139i 0.139446i
\(589\) 24.2931 8.98608i 1.00098 0.370265i
\(590\) 33.6804i 1.38660i
\(591\) −32.8869 −1.35279
\(592\) 7.72296i 0.317411i
\(593\) 12.3964i 0.509058i −0.967065 0.254529i \(-0.918080\pi\)
0.967065 0.254529i \(-0.0819205\pi\)
\(594\) 7.72463 + 14.2577i 0.316945 + 0.585000i
\(595\) −3.43415 −0.140787
\(596\) 17.3895i 0.712301i
\(597\) 23.3648i 0.956259i
\(598\) −49.7455 −2.03424
\(599\) 26.9617i 1.10162i −0.834630 0.550812i \(-0.814318\pi\)
0.834630 0.550812i \(-0.185682\pi\)
\(600\) 21.0896i 0.860980i
\(601\) −47.9125 −1.95439 −0.977197 0.212337i \(-0.931893\pi\)
−0.977197 + 0.212337i \(0.931893\pi\)
\(602\) 71.2124 2.90240
\(603\) 12.6453i 0.514956i
\(604\) 4.42064 0.179873
\(605\) −6.00769 + 9.21454i −0.244247 + 0.374624i
\(606\) −74.7097 −3.03488
\(607\) 17.5422 0.712017 0.356009 0.934483i \(-0.384137\pi\)
0.356009 + 0.934483i \(0.384137\pi\)
\(608\) −30.9939 83.7891i −1.25697 3.39810i
\(609\) 30.0760 1.21874
\(610\) 16.5090i 0.668429i
\(611\) −35.4042 −1.43230
\(612\) 15.8113i 0.639133i
\(613\) 6.93942i 0.280281i −0.990132 0.140140i \(-0.955245\pi\)
0.990132 0.140140i \(-0.0447553\pi\)
\(614\) 4.64800 0.187578
\(615\) 14.0158i 0.565171i
\(616\) −37.8479 69.8576i −1.52493 2.81464i
\(617\) −4.52330 −0.182101 −0.0910506 0.995846i \(-0.529023\pi\)
−0.0910506 + 0.995846i \(0.529023\pi\)
\(618\) 15.7268 0.632625
\(619\) 25.7607 1.03541 0.517705 0.855559i \(-0.326787\pi\)
0.517705 + 0.855559i \(0.326787\pi\)
\(620\) 32.0674i 1.28786i
\(621\) 8.54376i 0.342849i
\(622\) −89.0900 −3.57218
\(623\) −20.7617 −0.831799
\(624\) 125.911i 5.04049i
\(625\) 1.00000 0.0400000
\(626\) −9.81988 −0.392481
\(627\) 21.7637 24.8148i 0.869157 0.991007i
\(628\) 106.273 4.24074
\(629\) 0.713710 0.0284575
\(630\) 15.6055i 0.621737i
\(631\) −7.45186 −0.296654 −0.148327 0.988938i \(-0.547389\pi\)
−0.148327 + 0.988938i \(0.547389\pi\)
\(632\) −105.810 −4.20889
\(633\) 13.3405i 0.530237i
\(634\) 36.3369i 1.44312i
\(635\) −0.196466 −0.00779651
\(636\) −14.3807 −0.570232
\(637\) −1.05628 −0.0418513
\(638\) 40.2856 21.8262i 1.59492 0.864107i
\(639\) 24.4904i 0.968826i
\(640\) 32.6636 1.29114
\(641\) 46.1595i 1.82319i 0.411091 + 0.911594i \(0.365148\pi\)
−0.411091 + 0.911594i \(0.634852\pi\)
\(642\) 27.3789i 1.08056i
\(643\) 29.2285 1.15266 0.576331 0.817217i \(-0.304484\pi\)
0.576331 + 0.817217i \(0.304484\pi\)
\(644\) 66.5112i 2.62091i
\(645\) −23.0519 −0.907666
\(646\) 14.7230 5.44610i 0.579270 0.214274i
\(647\) 11.3553 0.446423 0.223211 0.974770i \(-0.428346\pi\)
0.223211 + 0.974770i \(0.428346\pi\)
\(648\) 99.2284 3.89806
\(649\) 19.5660 + 36.1138i 0.768031 + 1.41759i
\(650\) 10.4673 0.410560
\(651\) 35.1839i 1.37896i
\(652\) −72.8064 −2.85132
\(653\) 19.4499 0.761135 0.380568 0.924753i \(-0.375729\pi\)
0.380568 + 0.924753i \(0.375729\pi\)
\(654\) 62.8562i 2.45787i
\(655\) 9.10643i 0.355817i
\(656\) −87.9646 −3.43444
\(657\) 34.4781i 1.34512i
\(658\) 64.8800i 2.52928i
\(659\) 25.6726 1.00006 0.500031 0.866008i \(-0.333322\pi\)
0.500031 + 0.866008i \(0.333322\pi\)
\(660\) 19.4658 + 35.9290i 0.757707 + 1.39853i
\(661\) 24.8328i 0.965882i −0.875653 0.482941i \(-0.839568\pi\)
0.875653 0.482941i \(-0.160432\pi\)
\(662\) 48.9570i 1.90277i
\(663\) 11.6360 0.451905
\(664\) 68.9079i 2.67415i
\(665\) −10.6021 + 3.92176i −0.411133 + 0.152079i
\(666\) 3.24324i 0.125673i
\(667\) −24.1407 −0.934731
\(668\) 105.939 4.09892
\(669\) −65.9455 −2.54960
\(670\) 15.5432i 0.600487i
\(671\) −9.59055 17.7017i −0.370239 0.683367i
\(672\) −121.353 −4.68128
\(673\) −27.2501 −1.05042 −0.525208 0.850974i \(-0.676012\pi\)
−0.525208 + 0.850974i \(0.676012\pi\)
\(674\) 48.6638 1.87446
\(675\) 1.79775i 0.0691954i
\(676\) 9.78370 0.376296
\(677\) 32.2614 1.23991 0.619953 0.784639i \(-0.287152\pi\)
0.619953 + 0.784639i \(0.287152\pi\)
\(678\) −36.1881 −1.38980
\(679\) −45.7857 −1.75709
\(680\) 12.2320i 0.469076i
\(681\) 38.9245i 1.49159i
\(682\) −25.5330 47.1274i −0.977708 1.80460i
\(683\) 28.1759i 1.07812i 0.842268 + 0.539060i \(0.181220\pi\)
−0.842268 + 0.539060i \(0.818780\pi\)
\(684\) −18.0563 48.8136i −0.690399 1.86643i
\(685\) 3.76731 0.143942
\(686\) 51.3070i 1.95891i
\(687\) 37.9675i 1.44855i
\(688\) 144.676i 5.51572i
\(689\) 4.49225i 0.171141i
\(690\) 29.5093i 1.12340i
\(691\) −15.2544 −0.580306 −0.290153 0.956980i \(-0.593706\pi\)
−0.290153 + 0.956980i \(0.593706\pi\)
\(692\) −25.7099 −0.977344
\(693\) −9.06568 16.7329i −0.344377 0.635632i
\(694\) 85.8178i 3.25760i
\(695\) 3.48063i 0.132028i
\(696\) 107.127i 4.06063i
\(697\) 8.12917i 0.307914i
\(698\) 28.5640i 1.08116i
\(699\) 35.5215 1.34355
\(700\) 13.9951i 0.528963i
\(701\) 10.7433i 0.405767i 0.979203 + 0.202884i \(0.0650313\pi\)
−0.979203 + 0.202884i \(0.934969\pi\)
\(702\) 18.8175i 0.710222i
\(703\) 2.20341 0.815048i 0.0831032 0.0307401i
\(704\) −78.9760 + 42.7881i −2.97652 + 1.61264i
\(705\) 21.0020i 0.790981i
\(706\) 65.1636 2.45247
\(707\) −31.2034 −1.17352
\(708\) 152.582 5.73437
\(709\) 38.5054 1.44610 0.723050 0.690796i \(-0.242740\pi\)
0.723050 + 0.690796i \(0.242740\pi\)
\(710\) 30.1029i 1.12974i
\(711\) −25.3446 −0.950496
\(712\) 73.9503i 2.77141i
\(713\) 28.2405i 1.05762i
\(714\) 21.3235i 0.798013i
\(715\) −11.2235 + 6.08075i −0.419736 + 0.227407i
\(716\) 94.1478i 3.51847i
\(717\) −9.97448 −0.372504
\(718\) 28.7469i 1.07282i
\(719\) 45.5394 1.69833 0.849167 0.528125i \(-0.177105\pi\)
0.849167 + 0.528125i \(0.177105\pi\)
\(720\) 31.7043 1.18155
\(721\) 6.56849 0.244623
\(722\) 39.2345 33.6271i 1.46016 1.25147i
\(723\) 61.6838i 2.29404i
\(724\) 65.9400i 2.45064i
\(725\) 5.07960 0.188652
\(726\) −57.2154 37.3032i −2.12346 1.38445i
\(727\) −3.29804 −0.122317 −0.0611587 0.998128i \(-0.519480\pi\)
−0.0611587 + 0.998128i \(0.519480\pi\)
\(728\) 92.1992i 3.41713i
\(729\) 11.4547 0.424249
\(730\) 42.3794i 1.56853i
\(731\) 13.3701 0.494511
\(732\) −74.7902 −2.76433
\(733\) 18.4626i 0.681930i 0.940076 + 0.340965i \(0.110754\pi\)
−0.940076 + 0.340965i \(0.889246\pi\)
\(734\) 74.2716 2.74141
\(735\) 0.626591i 0.0231121i
\(736\) 97.4043 3.59037
\(737\) −9.02951 16.6662i −0.332606 0.613906i
\(738\) −36.9406 −1.35980
\(739\) 24.7991i 0.912248i −0.889916 0.456124i \(-0.849237\pi\)
0.889916 0.456124i \(-0.150763\pi\)
\(740\) 2.90855i 0.106921i
\(741\) 35.9234 13.2882i 1.31968 0.488153i
\(742\) −8.23227 −0.302216
\(743\) 4.35764 0.159866 0.0799331 0.996800i \(-0.474529\pi\)
0.0799331 + 0.996800i \(0.474529\pi\)
\(744\) −125.320 −4.59447
\(745\) 3.22238i 0.118059i
\(746\) −76.0780 −2.78541
\(747\) 16.5055i 0.603903i
\(748\) −11.2902 20.8389i −0.412811 0.761944i
\(749\) 11.4351i 0.417830i
\(750\) 6.20925i 0.226730i
\(751\) 28.4909i 1.03965i 0.854273 + 0.519824i \(0.174002\pi\)
−0.854273 + 0.519824i \(0.825998\pi\)
\(752\) 131.811 4.80665
\(753\) 66.9824i 2.44098i
\(754\) 53.1696 1.93632
\(755\) −0.819171 −0.0298127
\(756\) 25.1596 0.915046
\(757\) −16.1577 −0.587262 −0.293631 0.955919i \(-0.594864\pi\)
−0.293631 + 0.955919i \(0.594864\pi\)
\(758\) 8.16100i 0.296421i
\(759\) 17.1428 + 31.6413i 0.622245 + 1.14851i
\(760\) 13.9688 + 37.7634i 0.506702 + 1.36982i
\(761\) 40.0708i 1.45257i 0.687396 + 0.726283i \(0.258754\pi\)
−0.687396 + 0.726283i \(0.741246\pi\)
\(762\) 1.21991i 0.0441926i
\(763\) 26.2526i 0.950410i
\(764\) −20.8611 −0.754729
\(765\) 2.92992i 0.105932i
\(766\) 33.2030i 1.19967i
\(767\) 47.6636i 1.72103i
\(768\) 79.1525i 2.85617i
\(769\) 42.8114i 1.54382i −0.635732 0.771910i \(-0.719302\pi\)
0.635732 0.771910i \(-0.280698\pi\)
\(770\) 11.1433 + 20.5676i 0.401575 + 0.741206i
\(771\) 46.6102 1.67863
\(772\) 62.8081 2.26051
\(773\) 35.6628i 1.28270i 0.767248 + 0.641351i \(0.221626\pi\)
−0.767248 + 0.641351i \(0.778374\pi\)
\(774\) 60.7564i 2.18384i
\(775\) 5.94228i 0.213453i
\(776\) 163.083i 5.85433i
\(777\) 3.19122i 0.114484i
\(778\) 88.3445 3.16731
\(779\) −9.28341 25.0969i −0.332613 0.899189i
\(780\) 47.4197i 1.69790i
\(781\) 17.4876 + 32.2777i 0.625757 + 1.15499i
\(782\) 17.1154i 0.612047i
\(783\) 9.13185i 0.326346i
\(784\) 3.93255 0.140448
\(785\) −19.6930 −0.702872
\(786\) −56.5441 −2.01686
\(787\) 10.6336 0.379046 0.189523 0.981876i \(-0.439306\pi\)
0.189523 + 0.981876i \(0.439306\pi\)
\(788\) 77.7334i 2.76914i
\(789\) −4.46929 −0.159111
\(790\) 31.1528 1.10837
\(791\) −15.1144 −0.537406
\(792\) −59.6005 + 32.2907i −2.11781 + 1.14740i
\(793\) 23.3630i 0.829645i
\(794\) 59.0851 2.09685
\(795\) 2.66483 0.0945118
\(796\) −55.2264 −1.95745
\(797\) 50.8767i 1.80215i −0.433668 0.901073i \(-0.642781\pi\)
0.433668 0.901073i \(-0.357219\pi\)
\(798\) −24.3512 65.8313i −0.862023 2.33040i
\(799\) 12.1812i 0.430940i
\(800\) −20.4955 −0.724625
\(801\) 17.7133i 0.625868i
\(802\) 20.9042i 0.738153i
\(803\) 24.6195 + 45.4412i 0.868802 + 1.60359i
\(804\) −70.4150 −2.48335
\(805\) 12.3249i 0.434396i
\(806\) 62.1995i 2.19088i
\(807\) −20.1382 −0.708897
\(808\) 111.142i 3.90998i
\(809\) 43.5683i 1.53178i 0.642973 + 0.765889i \(0.277701\pi\)
−0.642973 + 0.765889i \(0.722299\pi\)
\(810\) −29.2150 −1.02651
\(811\) −7.19373 −0.252606 −0.126303 0.991992i \(-0.540311\pi\)
−0.126303 + 0.991992i \(0.540311\pi\)
\(812\) 71.0893i 2.49475i
\(813\) 3.56909 0.125173
\(814\) −2.31587 4.27451i −0.0811713 0.149822i
\(815\) 13.4915 0.472585
\(816\) −43.3211 −1.51654
\(817\) 41.2770 15.2685i 1.44410 0.534177i
\(818\) 5.01681 0.175409
\(819\) 22.0844i 0.771691i
\(820\) 33.1284 1.15690
\(821\) 15.0312i 0.524593i 0.964987 + 0.262297i \(0.0844798\pi\)
−0.964987 + 0.262297i \(0.915520\pi\)
\(822\) 23.3922i 0.815897i
\(823\) 44.3453 1.54578 0.772889 0.634541i \(-0.218811\pi\)
0.772889 + 0.634541i \(0.218811\pi\)
\(824\) 23.3961i 0.815041i
\(825\) −3.60714 6.65785i −0.125584 0.231797i
\(826\) 87.3458 3.03915
\(827\) −2.11368 −0.0735000 −0.0367500 0.999324i \(-0.511701\pi\)
−0.0367500 + 0.999324i \(0.511701\pi\)
\(828\) 56.7454 1.97204
\(829\) 42.8233i 1.48731i −0.668562 0.743657i \(-0.733090\pi\)
0.668562 0.743657i \(-0.266910\pi\)
\(830\) 20.2880i 0.704208i
\(831\) 0.522607 0.0181290
\(832\) −104.234 −3.61366
\(833\) 0.363423i 0.0125919i
\(834\) 21.6121 0.748366
\(835\) −19.6312 −0.679366
\(836\) −58.6536 51.4418i −2.02858 1.77915i
\(837\) 10.6827 0.369249
\(838\) −16.2839 −0.562519
\(839\) 54.5334i 1.88270i −0.337427 0.941352i \(-0.609557\pi\)
0.337427 0.941352i \(-0.390443\pi\)
\(840\) 54.6931 1.88709
\(841\) −3.19765 −0.110264
\(842\) 63.0833i 2.17399i
\(843\) 18.1243i 0.624236i
\(844\) 31.5323 1.08539
\(845\) −1.81298 −0.0623683
\(846\) 55.3537 1.90310
\(847\) −23.8967 15.5801i −0.821100 0.535340i
\(848\) 16.7248i 0.574331i
\(849\) 17.6401 0.605408
\(850\) 3.60137i 0.123526i
\(851\) 2.56145i 0.0878054i
\(852\) 136.374 4.67211
\(853\) 17.3880i 0.595354i −0.954667 0.297677i \(-0.903788\pi\)
0.954667 0.297677i \(-0.0962118\pi\)
\(854\) −42.8138 −1.46506
\(855\) 3.34594 + 9.04544i 0.114429 + 0.309348i
\(856\) −40.7304 −1.39214
\(857\) 38.1650 1.30369 0.651845 0.758352i \(-0.273995\pi\)
0.651845 + 0.758352i \(0.273995\pi\)
\(858\) −37.7569 69.6896i −1.28900 2.37916i
\(859\) 3.54926 0.121099 0.0605495 0.998165i \(-0.480715\pi\)
0.0605495 + 0.998165i \(0.480715\pi\)
\(860\) 54.4866i 1.85798i
\(861\) −36.3481 −1.23874
\(862\) −66.4275 −2.26253
\(863\) 12.1644i 0.414081i −0.978332 0.207040i \(-0.933617\pi\)
0.978332 0.207040i \(-0.0663832\pi\)
\(864\) 36.8458i 1.25352i
\(865\) 4.76420 0.161988
\(866\) 55.0094i 1.86930i
\(867\) 34.8094i 1.18219i
\(868\) −83.1625 −2.82272
\(869\) −33.4035 + 18.0976i −1.13314 + 0.613918i
\(870\) 31.5405i 1.06932i
\(871\) 21.9963i 0.745316i
\(872\) 93.5085 3.16660
\(873\) 39.0631i 1.32208i
\(874\) 19.5456 + 52.8399i 0.661141 + 1.78734i
\(875\) 2.59337i 0.0876718i
\(876\) 191.991 6.48676
\(877\) −14.3803 −0.485589 −0.242794 0.970078i \(-0.578064\pi\)
−0.242794 + 0.970078i \(0.578064\pi\)
\(878\) 18.8356 0.635671
\(879\) 50.7462i 1.71163i
\(880\) 41.7854 22.6388i 1.40859 0.763153i
\(881\) −27.8657 −0.938820 −0.469410 0.882980i \(-0.655533\pi\)
−0.469410 + 0.882980i \(0.655533\pi\)
\(882\) 1.65147 0.0556078
\(883\) −54.7948 −1.84399 −0.921995 0.387201i \(-0.873442\pi\)
−0.921995 + 0.387201i \(0.873442\pi\)
\(884\) 27.5035i 0.925042i
\(885\) −28.2743 −0.950430
\(886\) 17.5701 0.590280
\(887\) −15.7314 −0.528210 −0.264105 0.964494i \(-0.585077\pi\)
−0.264105 + 0.964494i \(0.585077\pi\)
\(888\) −11.3667 −0.381442
\(889\) 0.509508i 0.0170884i
\(890\) 21.7726i 0.729820i
\(891\) 31.3258 16.9719i 1.04945 0.568579i
\(892\) 155.872i 5.21899i
\(893\) 13.9108 + 37.6065i 0.465506 + 1.25845i
\(894\) −20.0085 −0.669186
\(895\) 17.4461i 0.583160i
\(896\) 84.7088i 2.82992i
\(897\) 41.7607i 1.39435i
\(898\) 2.59568i 0.0866191i
\(899\) 30.1844i 1.00671i
\(900\) −11.9402 −0.398006
\(901\) −1.54561 −0.0514916
\(902\) −48.6867 + 26.3778i −1.62109 + 0.878285i
\(903\) 59.7819i 1.98942i
\(904\) 53.8354i 1.79054i
\(905\) 12.2191i 0.406176i
\(906\) 5.08644i 0.168986i
\(907\) 7.02813i 0.233365i 0.993169 + 0.116683i \(0.0372261\pi\)
−0.993169 + 0.116683i \(0.962774\pi\)
\(908\) 92.0041 3.05326
\(909\) 26.6219i 0.882991i
\(910\) 27.1455i 0.899864i
\(911\) 2.08325i 0.0690213i −0.999404 0.0345106i \(-0.989013\pi\)
0.999404 0.0345106i \(-0.0109873\pi\)
\(912\) −133.744 + 49.4722i −4.42870 + 1.63819i
\(913\) −11.7859 21.7538i −0.390057 0.719945i
\(914\) 74.8232i 2.47493i
\(915\) 13.8591 0.458167
\(916\) −89.7422 −2.96516
\(917\) −23.6163 −0.779879
\(918\) 6.47437 0.213686
\(919\) 14.5525i 0.480042i 0.970768 + 0.240021i \(0.0771544\pi\)
−0.970768 + 0.240021i \(0.922846\pi\)
\(920\) −43.8997 −1.44733
\(921\) 3.90194i 0.128573i
\(922\) 116.753i 3.84506i
\(923\) 42.6007i 1.40222i
\(924\) −93.1771 + 50.4821i −3.06530 + 1.66074i
\(925\) 0.538972i 0.0177213i
\(926\) −74.8804 −2.46072
\(927\) 5.60405i 0.184061i
\(928\) −104.109 −3.41754
\(929\) 26.1795 0.858923 0.429461 0.903085i \(-0.358703\pi\)
0.429461 + 0.903085i \(0.358703\pi\)
\(930\) 36.8971 1.20990
\(931\) 0.415025 + 1.12198i 0.0136019 + 0.0367715i
\(932\) 83.9606i 2.75022i
\(933\) 74.7899i 2.44851i
\(934\) −21.5156 −0.704013
\(935\) 2.09214 + 3.86156i 0.0684204 + 0.126287i
\(936\) −78.6617 −2.57114
\(937\) 6.03224i 0.197065i 0.995134 + 0.0985323i \(0.0314148\pi\)
−0.995134 + 0.0985323i \(0.968585\pi\)
\(938\) −40.3092 −1.31614
\(939\) 8.24366i 0.269022i
\(940\) −49.6415 −1.61913
\(941\) 18.2697 0.595575 0.297787 0.954632i \(-0.403751\pi\)
0.297787 + 0.954632i \(0.403751\pi\)
\(942\) 122.279i 3.98405i
\(943\) 29.1750 0.950067
\(944\) 177.453i 5.77559i
\(945\) −4.66222 −0.151662
\(946\) −43.3838 80.0755i −1.41053 2.60348i
\(947\) 35.9840 1.16932 0.584662 0.811277i \(-0.301227\pi\)
0.584662 + 0.811277i \(0.301227\pi\)
\(948\) 141.131i 4.58372i
\(949\) 59.9741i 1.94684i
\(950\) −4.11273 11.1184i −0.133434 0.360728i
\(951\) 30.5043 0.989171
\(952\) −31.7221 −1.02812
\(953\) 3.75257 0.121558 0.0607788 0.998151i \(-0.480642\pi\)
0.0607788 + 0.998151i \(0.480642\pi\)
\(954\) 7.02354i 0.227395i
\(955\) 3.86569 0.125091
\(956\) 23.5762i 0.762510i
\(957\) −18.3228 33.8192i −0.592292 1.09322i
\(958\) 23.6657i 0.764603i
\(959\) 9.77003i 0.315491i
\(960\) 61.8321i 1.99562i
\(961\) −4.31067 −0.139054
\(962\) 5.64157i 0.181892i
\(963\) −9.75612 −0.314387
\(964\) −145.799 −4.69587
\(965\) −11.6387 −0.374664
\(966\) 76.5285 2.46226
\(967\) 10.7161i 0.344605i 0.985044 + 0.172303i \(0.0551207\pi\)
−0.985044 + 0.172303i \(0.944879\pi\)
\(968\) −55.4944 + 85.1168i −1.78366 + 2.73576i
\(969\) −4.57193 12.3598i −0.146872 0.397054i
\(970\) 48.0152i 1.54167i
\(971\) 0.656184i 0.0210579i −0.999945 0.0105290i \(-0.996648\pi\)
0.999945 0.0105290i \(-0.00335154\pi\)
\(972\) 103.248i 3.31167i
\(973\) 9.02656 0.289378
\(974\) 32.8283i 1.05189i
\(975\) 8.78715i 0.281414i
\(976\) 86.9811i 2.78420i
\(977\) 21.7556i 0.696024i −0.937490 0.348012i \(-0.886857\pi\)
0.937490 0.348012i \(-0.113143\pi\)
\(978\) 83.7719i 2.67873i
\(979\) 12.6484 + 23.3456i 0.404243 + 0.746130i
\(980\) −1.48104 −0.0473102
\(981\) 22.3980 0.715114
\(982\) 65.5387i 2.09142i
\(983\) 50.6245i 1.61467i −0.590093 0.807336i \(-0.700909\pi\)
0.590093 0.807336i \(-0.299091\pi\)
\(984\) 129.467i 4.12726i
\(985\) 14.4045i 0.458964i
\(986\) 18.2935i 0.582585i
\(987\) 54.4659 1.73367
\(988\) −31.4086 84.9105i −0.999242 2.70136i
\(989\) 47.9843i 1.52581i
\(990\) 17.5477 9.50712i 0.557703 0.302156i
\(991\) 31.9998i 1.01651i −0.861208 0.508253i \(-0.830291\pi\)
0.861208 0.508253i \(-0.169709\pi\)
\(992\) 121.790i 3.86683i
\(993\) 41.0988 1.30423
\(994\) 78.0678 2.47616
\(995\) 10.2338 0.324433
\(996\) −91.9104 −2.91229
\(997\) 2.80146i 0.0887230i 0.999016 + 0.0443615i \(0.0141253\pi\)
−0.999016 + 0.0443615i \(0.985875\pi\)
\(998\) −80.9026 −2.56093
\(999\) 0.968936 0.0306558
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.626.2 yes 40
11.10 odd 2 inner 1045.2.f.a.626.40 yes 40
19.18 odd 2 inner 1045.2.f.a.626.39 yes 40
209.208 even 2 inner 1045.2.f.a.626.1 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.f.a.626.1 40 209.208 even 2 inner
1045.2.f.a.626.2 yes 40 1.1 even 1 trivial
1045.2.f.a.626.39 yes 40 19.18 odd 2 inner
1045.2.f.a.626.40 yes 40 11.10 odd 2 inner