Properties

Label 418.2.a.d.1.2
Level $418$
Weight $2$
Character 418.1
Self dual yes
Analytic conductor $3.338$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [418,2,Mod(1,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.33774680449\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 418.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -2.30278 q^{5} -0.302776 q^{6} +1.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} +2.30278 q^{10} +1.00000 q^{11} +0.302776 q^{12} -3.30278 q^{13} -1.30278 q^{14} -0.697224 q^{15} +1.00000 q^{16} -4.60555 q^{17} +2.90833 q^{18} +1.00000 q^{19} -2.30278 q^{20} +0.394449 q^{21} -1.00000 q^{22} -1.39445 q^{23} -0.302776 q^{24} +0.302776 q^{25} +3.30278 q^{26} -1.78890 q^{27} +1.30278 q^{28} -8.30278 q^{29} +0.697224 q^{30} -3.30278 q^{31} -1.00000 q^{32} +0.302776 q^{33} +4.60555 q^{34} -3.00000 q^{35} -2.90833 q^{36} +5.21110 q^{37} -1.00000 q^{38} -1.00000 q^{39} +2.30278 q^{40} +3.90833 q^{41} -0.394449 q^{42} +1.09167 q^{43} +1.00000 q^{44} +6.69722 q^{45} +1.39445 q^{46} -6.00000 q^{47} +0.302776 q^{48} -5.30278 q^{49} -0.302776 q^{50} -1.39445 q^{51} -3.30278 q^{52} -10.6056 q^{53} +1.78890 q^{54} -2.30278 q^{55} -1.30278 q^{56} +0.302776 q^{57} +8.30278 q^{58} -0.697224 q^{60} +11.2111 q^{61} +3.30278 q^{62} -3.78890 q^{63} +1.00000 q^{64} +7.60555 q^{65} -0.302776 q^{66} -6.51388 q^{67} -4.60555 q^{68} -0.422205 q^{69} +3.00000 q^{70} -9.69722 q^{71} +2.90833 q^{72} -2.60555 q^{73} -5.21110 q^{74} +0.0916731 q^{75} +1.00000 q^{76} +1.30278 q^{77} +1.00000 q^{78} +15.8167 q^{79} -2.30278 q^{80} +8.18335 q^{81} -3.90833 q^{82} +14.3028 q^{83} +0.394449 q^{84} +10.6056 q^{85} -1.09167 q^{86} -2.51388 q^{87} -1.00000 q^{88} +4.60555 q^{89} -6.69722 q^{90} -4.30278 q^{91} -1.39445 q^{92} -1.00000 q^{93} +6.00000 q^{94} -2.30278 q^{95} -0.302776 q^{96} +8.00000 q^{97} +5.30278 q^{98} -2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - q^{5} + 3 q^{6} - q^{7} - 2 q^{8} + 5 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} - 3 q^{13} + q^{14} - 5 q^{15} + 2 q^{16} - 2 q^{17} - 5 q^{18} + 2 q^{19} - q^{20} + 8 q^{21} - 2 q^{22} - 10 q^{23} + 3 q^{24} - 3 q^{25} + 3 q^{26} - 18 q^{27} - q^{28} - 13 q^{29} + 5 q^{30} - 3 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} - 6 q^{35} + 5 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{39} + q^{40} - 3 q^{41} - 8 q^{42} + 13 q^{43} + 2 q^{44} + 17 q^{45} + 10 q^{46} - 12 q^{47} - 3 q^{48} - 7 q^{49} + 3 q^{50} - 10 q^{51} - 3 q^{52} - 14 q^{53} + 18 q^{54} - q^{55} + q^{56} - 3 q^{57} + 13 q^{58} - 5 q^{60} + 8 q^{61} + 3 q^{62} - 22 q^{63} + 2 q^{64} + 8 q^{65} + 3 q^{66} + 5 q^{67} - 2 q^{68} + 28 q^{69} + 6 q^{70} - 23 q^{71} - 5 q^{72} + 2 q^{73} + 4 q^{74} + 11 q^{75} + 2 q^{76} - q^{77} + 2 q^{78} + 10 q^{79} - q^{80} + 38 q^{81} + 3 q^{82} + 25 q^{83} + 8 q^{84} + 14 q^{85} - 13 q^{86} + 13 q^{87} - 2 q^{88} + 2 q^{89} - 17 q^{90} - 5 q^{91} - 10 q^{92} - 2 q^{93} + 12 q^{94} - q^{95} + 3 q^{96} + 16 q^{97} + 7 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.302776 0.174808 0.0874038 0.996173i \(-0.472143\pi\)
0.0874038 + 0.996173i \(0.472143\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.30278 −1.02983 −0.514916 0.857240i \(-0.672177\pi\)
−0.514916 + 0.857240i \(0.672177\pi\)
\(6\) −0.302776 −0.123608
\(7\) 1.30278 0.492403 0.246201 0.969219i \(-0.420818\pi\)
0.246201 + 0.969219i \(0.420818\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.90833 −0.969442
\(10\) 2.30278 0.728202
\(11\) 1.00000 0.301511
\(12\) 0.302776 0.0874038
\(13\) −3.30278 −0.916025 −0.458013 0.888946i \(-0.651439\pi\)
−0.458013 + 0.888946i \(0.651439\pi\)
\(14\) −1.30278 −0.348181
\(15\) −0.697224 −0.180023
\(16\) 1.00000 0.250000
\(17\) −4.60555 −1.11701 −0.558505 0.829501i \(-0.688625\pi\)
−0.558505 + 0.829501i \(0.688625\pi\)
\(18\) 2.90833 0.685499
\(19\) 1.00000 0.229416
\(20\) −2.30278 −0.514916
\(21\) 0.394449 0.0860758
\(22\) −1.00000 −0.213201
\(23\) −1.39445 −0.290763 −0.145381 0.989376i \(-0.546441\pi\)
−0.145381 + 0.989376i \(0.546441\pi\)
\(24\) −0.302776 −0.0618038
\(25\) 0.302776 0.0605551
\(26\) 3.30278 0.647728
\(27\) −1.78890 −0.344273
\(28\) 1.30278 0.246201
\(29\) −8.30278 −1.54179 −0.770893 0.636964i \(-0.780190\pi\)
−0.770893 + 0.636964i \(0.780190\pi\)
\(30\) 0.697224 0.127295
\(31\) −3.30278 −0.593196 −0.296598 0.955002i \(-0.595852\pi\)
−0.296598 + 0.955002i \(0.595852\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.302776 0.0527065
\(34\) 4.60555 0.789846
\(35\) −3.00000 −0.507093
\(36\) −2.90833 −0.484721
\(37\) 5.21110 0.856700 0.428350 0.903613i \(-0.359095\pi\)
0.428350 + 0.903613i \(0.359095\pi\)
\(38\) −1.00000 −0.162221
\(39\) −1.00000 −0.160128
\(40\) 2.30278 0.364101
\(41\) 3.90833 0.610378 0.305189 0.952292i \(-0.401280\pi\)
0.305189 + 0.952292i \(0.401280\pi\)
\(42\) −0.394449 −0.0608648
\(43\) 1.09167 0.166479 0.0832393 0.996530i \(-0.473473\pi\)
0.0832393 + 0.996530i \(0.473473\pi\)
\(44\) 1.00000 0.150756
\(45\) 6.69722 0.998363
\(46\) 1.39445 0.205600
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0.302776 0.0437019
\(49\) −5.30278 −0.757539
\(50\) −0.302776 −0.0428189
\(51\) −1.39445 −0.195262
\(52\) −3.30278 −0.458013
\(53\) −10.6056 −1.45678 −0.728392 0.685160i \(-0.759732\pi\)
−0.728392 + 0.685160i \(0.759732\pi\)
\(54\) 1.78890 0.243438
\(55\) −2.30278 −0.310506
\(56\) −1.30278 −0.174091
\(57\) 0.302776 0.0401036
\(58\) 8.30278 1.09021
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −0.697224 −0.0900113
\(61\) 11.2111 1.43543 0.717717 0.696335i \(-0.245187\pi\)
0.717717 + 0.696335i \(0.245187\pi\)
\(62\) 3.30278 0.419453
\(63\) −3.78890 −0.477356
\(64\) 1.00000 0.125000
\(65\) 7.60555 0.943353
\(66\) −0.302776 −0.0372691
\(67\) −6.51388 −0.795797 −0.397898 0.917429i \(-0.630260\pi\)
−0.397898 + 0.917429i \(0.630260\pi\)
\(68\) −4.60555 −0.558505
\(69\) −0.422205 −0.0508275
\(70\) 3.00000 0.358569
\(71\) −9.69722 −1.15085 −0.575424 0.817855i \(-0.695163\pi\)
−0.575424 + 0.817855i \(0.695163\pi\)
\(72\) 2.90833 0.342750
\(73\) −2.60555 −0.304957 −0.152478 0.988307i \(-0.548725\pi\)
−0.152478 + 0.988307i \(0.548725\pi\)
\(74\) −5.21110 −0.605778
\(75\) 0.0916731 0.0105855
\(76\) 1.00000 0.114708
\(77\) 1.30278 0.148465
\(78\) 1.00000 0.113228
\(79\) 15.8167 1.77951 0.889756 0.456436i \(-0.150874\pi\)
0.889756 + 0.456436i \(0.150874\pi\)
\(80\) −2.30278 −0.257458
\(81\) 8.18335 0.909261
\(82\) −3.90833 −0.431603
\(83\) 14.3028 1.56993 0.784967 0.619538i \(-0.212680\pi\)
0.784967 + 0.619538i \(0.212680\pi\)
\(84\) 0.394449 0.0430379
\(85\) 10.6056 1.15033
\(86\) −1.09167 −0.117718
\(87\) −2.51388 −0.269516
\(88\) −1.00000 −0.106600
\(89\) 4.60555 0.488187 0.244094 0.969752i \(-0.421510\pi\)
0.244094 + 0.969752i \(0.421510\pi\)
\(90\) −6.69722 −0.705949
\(91\) −4.30278 −0.451053
\(92\) −1.39445 −0.145381
\(93\) −1.00000 −0.103695
\(94\) 6.00000 0.618853
\(95\) −2.30278 −0.236260
\(96\) −0.302776 −0.0309019
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 5.30278 0.535661
\(99\) −2.90833 −0.292298
\(100\) 0.302776 0.0302776
\(101\) 7.81665 0.777786 0.388893 0.921283i \(-0.372858\pi\)
0.388893 + 0.921283i \(0.372858\pi\)
\(102\) 1.39445 0.138071
\(103\) −1.69722 −0.167232 −0.0836162 0.996498i \(-0.526647\pi\)
−0.0836162 + 0.996498i \(0.526647\pi\)
\(104\) 3.30278 0.323864
\(105\) −0.908327 −0.0886436
\(106\) 10.6056 1.03010
\(107\) 3.21110 0.310429 0.155215 0.987881i \(-0.450393\pi\)
0.155215 + 0.987881i \(0.450393\pi\)
\(108\) −1.78890 −0.172137
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 2.30278 0.219561
\(111\) 1.57779 0.149758
\(112\) 1.30278 0.123101
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −0.302776 −0.0283575
\(115\) 3.21110 0.299437
\(116\) −8.30278 −0.770893
\(117\) 9.60555 0.888034
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0.697224 0.0636476
\(121\) 1.00000 0.0909091
\(122\) −11.2111 −1.01501
\(123\) 1.18335 0.106699
\(124\) −3.30278 −0.296598
\(125\) 10.8167 0.967471
\(126\) 3.78890 0.337542
\(127\) −5.39445 −0.478680 −0.239340 0.970936i \(-0.576931\pi\)
−0.239340 + 0.970936i \(0.576931\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.330532 0.0291017
\(130\) −7.60555 −0.667051
\(131\) 19.1194 1.67047 0.835236 0.549891i \(-0.185331\pi\)
0.835236 + 0.549891i \(0.185331\pi\)
\(132\) 0.302776 0.0263532
\(133\) 1.30278 0.112965
\(134\) 6.51388 0.562713
\(135\) 4.11943 0.354544
\(136\) 4.60555 0.394923
\(137\) −3.48612 −0.297839 −0.148920 0.988849i \(-0.547580\pi\)
−0.148920 + 0.988849i \(0.547580\pi\)
\(138\) 0.422205 0.0359405
\(139\) −6.30278 −0.534594 −0.267297 0.963614i \(-0.586131\pi\)
−0.267297 + 0.963614i \(0.586131\pi\)
\(140\) −3.00000 −0.253546
\(141\) −1.81665 −0.152990
\(142\) 9.69722 0.813773
\(143\) −3.30278 −0.276192
\(144\) −2.90833 −0.242361
\(145\) 19.1194 1.58778
\(146\) 2.60555 0.215637
\(147\) −1.60555 −0.132424
\(148\) 5.21110 0.428350
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) −0.0916731 −0.00748508
\(151\) −16.4222 −1.33642 −0.668210 0.743973i \(-0.732939\pi\)
−0.668210 + 0.743973i \(0.732939\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 13.3944 1.08288
\(154\) −1.30278 −0.104981
\(155\) 7.60555 0.610893
\(156\) −1.00000 −0.0800641
\(157\) 1.09167 0.0871250 0.0435625 0.999051i \(-0.486129\pi\)
0.0435625 + 0.999051i \(0.486129\pi\)
\(158\) −15.8167 −1.25831
\(159\) −3.21110 −0.254657
\(160\) 2.30278 0.182050
\(161\) −1.81665 −0.143172
\(162\) −8.18335 −0.642944
\(163\) −11.3944 −0.892482 −0.446241 0.894913i \(-0.647238\pi\)
−0.446241 + 0.894913i \(0.647238\pi\)
\(164\) 3.90833 0.305189
\(165\) −0.697224 −0.0542788
\(166\) −14.3028 −1.11011
\(167\) −12.4222 −0.961259 −0.480630 0.876924i \(-0.659592\pi\)
−0.480630 + 0.876924i \(0.659592\pi\)
\(168\) −0.394449 −0.0304324
\(169\) −2.09167 −0.160898
\(170\) −10.6056 −0.813409
\(171\) −2.90833 −0.222405
\(172\) 1.09167 0.0832393
\(173\) −19.3305 −1.46967 −0.734837 0.678244i \(-0.762741\pi\)
−0.734837 + 0.678244i \(0.762741\pi\)
\(174\) 2.51388 0.190577
\(175\) 0.394449 0.0298175
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −4.60555 −0.345201
\(179\) −13.1194 −0.980592 −0.490296 0.871556i \(-0.663111\pi\)
−0.490296 + 0.871556i \(0.663111\pi\)
\(180\) 6.69722 0.499182
\(181\) −15.0278 −1.11700 −0.558502 0.829503i \(-0.688624\pi\)
−0.558502 + 0.829503i \(0.688624\pi\)
\(182\) 4.30278 0.318943
\(183\) 3.39445 0.250925
\(184\) 1.39445 0.102800
\(185\) −12.0000 −0.882258
\(186\) 1.00000 0.0733236
\(187\) −4.60555 −0.336791
\(188\) −6.00000 −0.437595
\(189\) −2.33053 −0.169521
\(190\) 2.30278 0.167061
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 0.302776 0.0218509
\(193\) −24.7250 −1.77974 −0.889872 0.456211i \(-0.849206\pi\)
−0.889872 + 0.456211i \(0.849206\pi\)
\(194\) −8.00000 −0.574367
\(195\) 2.30278 0.164905
\(196\) −5.30278 −0.378770
\(197\) 21.6333 1.54131 0.770655 0.637253i \(-0.219929\pi\)
0.770655 + 0.637253i \(0.219929\pi\)
\(198\) 2.90833 0.206686
\(199\) −2.18335 −0.154773 −0.0773867 0.997001i \(-0.524658\pi\)
−0.0773867 + 0.997001i \(0.524658\pi\)
\(200\) −0.302776 −0.0214095
\(201\) −1.97224 −0.139111
\(202\) −7.81665 −0.549978
\(203\) −10.8167 −0.759180
\(204\) −1.39445 −0.0976309
\(205\) −9.00000 −0.628587
\(206\) 1.69722 0.118251
\(207\) 4.05551 0.281878
\(208\) −3.30278 −0.229006
\(209\) 1.00000 0.0691714
\(210\) 0.908327 0.0626805
\(211\) 24.6056 1.69392 0.846958 0.531660i \(-0.178431\pi\)
0.846958 + 0.531660i \(0.178431\pi\)
\(212\) −10.6056 −0.728392
\(213\) −2.93608 −0.201177
\(214\) −3.21110 −0.219506
\(215\) −2.51388 −0.171445
\(216\) 1.78890 0.121719
\(217\) −4.30278 −0.292091
\(218\) −2.00000 −0.135457
\(219\) −0.788897 −0.0533087
\(220\) −2.30278 −0.155253
\(221\) 15.2111 1.02321
\(222\) −1.57779 −0.105895
\(223\) −25.2111 −1.68826 −0.844130 0.536138i \(-0.819883\pi\)
−0.844130 + 0.536138i \(0.819883\pi\)
\(224\) −1.30278 −0.0870454
\(225\) −0.880571 −0.0587047
\(226\) 0 0
\(227\) 1.81665 0.120576 0.0602878 0.998181i \(-0.480798\pi\)
0.0602878 + 0.998181i \(0.480798\pi\)
\(228\) 0.302776 0.0200518
\(229\) 11.9083 0.786924 0.393462 0.919341i \(-0.371277\pi\)
0.393462 + 0.919341i \(0.371277\pi\)
\(230\) −3.21110 −0.211734
\(231\) 0.394449 0.0259528
\(232\) 8.30278 0.545104
\(233\) 9.21110 0.603439 0.301720 0.953397i \(-0.402439\pi\)
0.301720 + 0.953397i \(0.402439\pi\)
\(234\) −9.60555 −0.627935
\(235\) 13.8167 0.901299
\(236\) 0 0
\(237\) 4.78890 0.311072
\(238\) 6.00000 0.388922
\(239\) −23.9361 −1.54830 −0.774148 0.633004i \(-0.781822\pi\)
−0.774148 + 0.633004i \(0.781822\pi\)
\(240\) −0.697224 −0.0450056
\(241\) 13.0917 0.843309 0.421654 0.906757i \(-0.361450\pi\)
0.421654 + 0.906757i \(0.361450\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 7.84441 0.503219
\(244\) 11.2111 0.717717
\(245\) 12.2111 0.780139
\(246\) −1.18335 −0.0754474
\(247\) −3.30278 −0.210151
\(248\) 3.30278 0.209726
\(249\) 4.33053 0.274436
\(250\) −10.8167 −0.684105
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) −3.78890 −0.238678
\(253\) −1.39445 −0.0876682
\(254\) 5.39445 0.338478
\(255\) 3.21110 0.201087
\(256\) 1.00000 0.0625000
\(257\) −4.60555 −0.287286 −0.143643 0.989630i \(-0.545882\pi\)
−0.143643 + 0.989630i \(0.545882\pi\)
\(258\) −0.330532 −0.0205780
\(259\) 6.78890 0.421842
\(260\) 7.60555 0.471676
\(261\) 24.1472 1.49467
\(262\) −19.1194 −1.18120
\(263\) −20.3028 −1.25192 −0.625961 0.779854i \(-0.715293\pi\)
−0.625961 + 0.779854i \(0.715293\pi\)
\(264\) −0.302776 −0.0186346
\(265\) 24.4222 1.50024
\(266\) −1.30278 −0.0798783
\(267\) 1.39445 0.0853389
\(268\) −6.51388 −0.397898
\(269\) −19.8167 −1.20824 −0.604121 0.796892i \(-0.706476\pi\)
−0.604121 + 0.796892i \(0.706476\pi\)
\(270\) −4.11943 −0.250700
\(271\) −6.51388 −0.395690 −0.197845 0.980233i \(-0.563394\pi\)
−0.197845 + 0.980233i \(0.563394\pi\)
\(272\) −4.60555 −0.279253
\(273\) −1.30278 −0.0788476
\(274\) 3.48612 0.210604
\(275\) 0.302776 0.0182581
\(276\) −0.422205 −0.0254138
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 6.30278 0.378015
\(279\) 9.60555 0.575069
\(280\) 3.00000 0.179284
\(281\) −0.486122 −0.0289996 −0.0144998 0.999895i \(-0.504616\pi\)
−0.0144998 + 0.999895i \(0.504616\pi\)
\(282\) 1.81665 0.108180
\(283\) −18.9361 −1.12563 −0.562817 0.826582i \(-0.690282\pi\)
−0.562817 + 0.826582i \(0.690282\pi\)
\(284\) −9.69722 −0.575424
\(285\) −0.697224 −0.0413000
\(286\) 3.30278 0.195297
\(287\) 5.09167 0.300552
\(288\) 2.90833 0.171375
\(289\) 4.21110 0.247712
\(290\) −19.1194 −1.12273
\(291\) 2.42221 0.141992
\(292\) −2.60555 −0.152478
\(293\) −25.5416 −1.49216 −0.746079 0.665857i \(-0.768066\pi\)
−0.746079 + 0.665857i \(0.768066\pi\)
\(294\) 1.60555 0.0936377
\(295\) 0 0
\(296\) −5.21110 −0.302889
\(297\) −1.78890 −0.103802
\(298\) 0 0
\(299\) 4.60555 0.266346
\(300\) 0.0916731 0.00529275
\(301\) 1.42221 0.0819745
\(302\) 16.4222 0.944992
\(303\) 2.36669 0.135963
\(304\) 1.00000 0.0573539
\(305\) −25.8167 −1.47826
\(306\) −13.3944 −0.765710
\(307\) −2.18335 −0.124610 −0.0623051 0.998057i \(-0.519845\pi\)
−0.0623051 + 0.998057i \(0.519845\pi\)
\(308\) 1.30278 0.0742325
\(309\) −0.513878 −0.0292335
\(310\) −7.60555 −0.431966
\(311\) 25.8167 1.46393 0.731964 0.681343i \(-0.238604\pi\)
0.731964 + 0.681343i \(0.238604\pi\)
\(312\) 1.00000 0.0566139
\(313\) 18.1194 1.02417 0.512085 0.858935i \(-0.328873\pi\)
0.512085 + 0.858935i \(0.328873\pi\)
\(314\) −1.09167 −0.0616067
\(315\) 8.72498 0.491597
\(316\) 15.8167 0.889756
\(317\) −13.8167 −0.776021 −0.388010 0.921655i \(-0.626838\pi\)
−0.388010 + 0.921655i \(0.626838\pi\)
\(318\) 3.21110 0.180070
\(319\) −8.30278 −0.464866
\(320\) −2.30278 −0.128729
\(321\) 0.972244 0.0542653
\(322\) 1.81665 0.101238
\(323\) −4.60555 −0.256260
\(324\) 8.18335 0.454630
\(325\) −1.00000 −0.0554700
\(326\) 11.3944 0.631080
\(327\) 0.605551 0.0334871
\(328\) −3.90833 −0.215801
\(329\) −7.81665 −0.430946
\(330\) 0.697224 0.0383809
\(331\) 22.7250 1.24908 0.624539 0.780994i \(-0.285287\pi\)
0.624539 + 0.780994i \(0.285287\pi\)
\(332\) 14.3028 0.784967
\(333\) −15.1556 −0.830521
\(334\) 12.4222 0.679713
\(335\) 15.0000 0.819538
\(336\) 0.394449 0.0215189
\(337\) −9.30278 −0.506754 −0.253377 0.967368i \(-0.581541\pi\)
−0.253377 + 0.967368i \(0.581541\pi\)
\(338\) 2.09167 0.113772
\(339\) 0 0
\(340\) 10.6056 0.575167
\(341\) −3.30278 −0.178855
\(342\) 2.90833 0.157264
\(343\) −16.0278 −0.865417
\(344\) −1.09167 −0.0588591
\(345\) 0.972244 0.0523438
\(346\) 19.3305 1.03922
\(347\) 2.78890 0.149716 0.0748579 0.997194i \(-0.476150\pi\)
0.0748579 + 0.997194i \(0.476150\pi\)
\(348\) −2.51388 −0.134758
\(349\) 32.4222 1.73552 0.867760 0.496983i \(-0.165559\pi\)
0.867760 + 0.496983i \(0.165559\pi\)
\(350\) −0.394449 −0.0210842
\(351\) 5.90833 0.315363
\(352\) −1.00000 −0.0533002
\(353\) −15.2111 −0.809605 −0.404803 0.914404i \(-0.632660\pi\)
−0.404803 + 0.914404i \(0.632660\pi\)
\(354\) 0 0
\(355\) 22.3305 1.18518
\(356\) 4.60555 0.244094
\(357\) −1.81665 −0.0961475
\(358\) 13.1194 0.693383
\(359\) −9.48612 −0.500658 −0.250329 0.968161i \(-0.580539\pi\)
−0.250329 + 0.968161i \(0.580539\pi\)
\(360\) −6.69722 −0.352975
\(361\) 1.00000 0.0526316
\(362\) 15.0278 0.789841
\(363\) 0.302776 0.0158916
\(364\) −4.30278 −0.225527
\(365\) 6.00000 0.314054
\(366\) −3.39445 −0.177431
\(367\) −7.21110 −0.376416 −0.188208 0.982129i \(-0.560268\pi\)
−0.188208 + 0.982129i \(0.560268\pi\)
\(368\) −1.39445 −0.0726907
\(369\) −11.3667 −0.591726
\(370\) 12.0000 0.623850
\(371\) −13.8167 −0.717325
\(372\) −1.00000 −0.0518476
\(373\) 7.09167 0.367193 0.183596 0.983002i \(-0.441226\pi\)
0.183596 + 0.983002i \(0.441226\pi\)
\(374\) 4.60555 0.238147
\(375\) 3.27502 0.169121
\(376\) 6.00000 0.309426
\(377\) 27.4222 1.41232
\(378\) 2.33053 0.119870
\(379\) −13.4861 −0.692736 −0.346368 0.938099i \(-0.612585\pi\)
−0.346368 + 0.938099i \(0.612585\pi\)
\(380\) −2.30278 −0.118130
\(381\) −1.63331 −0.0836769
\(382\) 6.00000 0.306987
\(383\) 3.48612 0.178133 0.0890663 0.996026i \(-0.471612\pi\)
0.0890663 + 0.996026i \(0.471612\pi\)
\(384\) −0.302776 −0.0154510
\(385\) −3.00000 −0.152894
\(386\) 24.7250 1.25847
\(387\) −3.17494 −0.161391
\(388\) 8.00000 0.406138
\(389\) 34.5416 1.75133 0.875665 0.482919i \(-0.160423\pi\)
0.875665 + 0.482919i \(0.160423\pi\)
\(390\) −2.30278 −0.116606
\(391\) 6.42221 0.324785
\(392\) 5.30278 0.267831
\(393\) 5.78890 0.292011
\(394\) −21.6333 −1.08987
\(395\) −36.4222 −1.83260
\(396\) −2.90833 −0.146149
\(397\) −32.7527 −1.64381 −0.821906 0.569623i \(-0.807089\pi\)
−0.821906 + 0.569623i \(0.807089\pi\)
\(398\) 2.18335 0.109441
\(399\) 0.394449 0.0197471
\(400\) 0.302776 0.0151388
\(401\) 22.6056 1.12887 0.564434 0.825478i \(-0.309095\pi\)
0.564434 + 0.825478i \(0.309095\pi\)
\(402\) 1.97224 0.0983666
\(403\) 10.9083 0.543382
\(404\) 7.81665 0.388893
\(405\) −18.8444 −0.936386
\(406\) 10.8167 0.536822
\(407\) 5.21110 0.258305
\(408\) 1.39445 0.0690355
\(409\) 23.1472 1.14455 0.572277 0.820060i \(-0.306060\pi\)
0.572277 + 0.820060i \(0.306060\pi\)
\(410\) 9.00000 0.444478
\(411\) −1.05551 −0.0520646
\(412\) −1.69722 −0.0836162
\(413\) 0 0
\(414\) −4.05551 −0.199318
\(415\) −32.9361 −1.61677
\(416\) 3.30278 0.161932
\(417\) −1.90833 −0.0934512
\(418\) −1.00000 −0.0489116
\(419\) 8.78890 0.429366 0.214683 0.976684i \(-0.431128\pi\)
0.214683 + 0.976684i \(0.431128\pi\)
\(420\) −0.908327 −0.0443218
\(421\) −30.2389 −1.47375 −0.736876 0.676028i \(-0.763700\pi\)
−0.736876 + 0.676028i \(0.763700\pi\)
\(422\) −24.6056 −1.19778
\(423\) 17.4500 0.848446
\(424\) 10.6056 0.515051
\(425\) −1.39445 −0.0676407
\(426\) 2.93608 0.142254
\(427\) 14.6056 0.706812
\(428\) 3.21110 0.155215
\(429\) −1.00000 −0.0482805
\(430\) 2.51388 0.121230
\(431\) 2.78890 0.134336 0.0671682 0.997742i \(-0.478604\pi\)
0.0671682 + 0.997742i \(0.478604\pi\)
\(432\) −1.78890 −0.0860684
\(433\) −24.2389 −1.16485 −0.582423 0.812886i \(-0.697895\pi\)
−0.582423 + 0.812886i \(0.697895\pi\)
\(434\) 4.30278 0.206540
\(435\) 5.78890 0.277556
\(436\) 2.00000 0.0957826
\(437\) −1.39445 −0.0667055
\(438\) 0.788897 0.0376950
\(439\) −25.2111 −1.20326 −0.601630 0.798775i \(-0.705482\pi\)
−0.601630 + 0.798775i \(0.705482\pi\)
\(440\) 2.30278 0.109781
\(441\) 15.4222 0.734391
\(442\) −15.2111 −0.723518
\(443\) 39.6333 1.88304 0.941518 0.336964i \(-0.109400\pi\)
0.941518 + 0.336964i \(0.109400\pi\)
\(444\) 1.57779 0.0748788
\(445\) −10.6056 −0.502751
\(446\) 25.2111 1.19378
\(447\) 0 0
\(448\) 1.30278 0.0615504
\(449\) −31.8167 −1.50152 −0.750760 0.660575i \(-0.770313\pi\)
−0.750760 + 0.660575i \(0.770313\pi\)
\(450\) 0.880571 0.0415105
\(451\) 3.90833 0.184036
\(452\) 0 0
\(453\) −4.97224 −0.233616
\(454\) −1.81665 −0.0852598
\(455\) 9.90833 0.464510
\(456\) −0.302776 −0.0141788
\(457\) 21.8167 1.02054 0.510270 0.860014i \(-0.329545\pi\)
0.510270 + 0.860014i \(0.329545\pi\)
\(458\) −11.9083 −0.556440
\(459\) 8.23886 0.384557
\(460\) 3.21110 0.149718
\(461\) 6.42221 0.299112 0.149556 0.988753i \(-0.452216\pi\)
0.149556 + 0.988753i \(0.452216\pi\)
\(462\) −0.394449 −0.0183514
\(463\) −1.21110 −0.0562847 −0.0281424 0.999604i \(-0.508959\pi\)
−0.0281424 + 0.999604i \(0.508959\pi\)
\(464\) −8.30278 −0.385447
\(465\) 2.30278 0.106789
\(466\) −9.21110 −0.426696
\(467\) −22.6056 −1.04606 −0.523030 0.852314i \(-0.675198\pi\)
−0.523030 + 0.852314i \(0.675198\pi\)
\(468\) 9.60555 0.444017
\(469\) −8.48612 −0.391853
\(470\) −13.8167 −0.637315
\(471\) 0.330532 0.0152301
\(472\) 0 0
\(473\) 1.09167 0.0501952
\(474\) −4.78890 −0.219961
\(475\) 0.302776 0.0138923
\(476\) −6.00000 −0.275010
\(477\) 30.8444 1.41227
\(478\) 23.9361 1.09481
\(479\) 35.7250 1.63232 0.816158 0.577829i \(-0.196100\pi\)
0.816158 + 0.577829i \(0.196100\pi\)
\(480\) 0.697224 0.0318238
\(481\) −17.2111 −0.784759
\(482\) −13.0917 −0.596309
\(483\) −0.550039 −0.0250276
\(484\) 1.00000 0.0454545
\(485\) −18.4222 −0.836509
\(486\) −7.84441 −0.355830
\(487\) −29.3305 −1.32909 −0.664547 0.747247i \(-0.731375\pi\)
−0.664547 + 0.747247i \(0.731375\pi\)
\(488\) −11.2111 −0.507503
\(489\) −3.44996 −0.156013
\(490\) −12.2111 −0.551641
\(491\) −14.9361 −0.674056 −0.337028 0.941495i \(-0.609422\pi\)
−0.337028 + 0.941495i \(0.609422\pi\)
\(492\) 1.18335 0.0533494
\(493\) 38.2389 1.72219
\(494\) 3.30278 0.148599
\(495\) 6.69722 0.301018
\(496\) −3.30278 −0.148299
\(497\) −12.6333 −0.566681
\(498\) −4.33053 −0.194056
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 10.8167 0.483735
\(501\) −3.76114 −0.168035
\(502\) −24.0000 −1.07117
\(503\) 22.1194 0.986257 0.493128 0.869957i \(-0.335853\pi\)
0.493128 + 0.869957i \(0.335853\pi\)
\(504\) 3.78890 0.168771
\(505\) −18.0000 −0.800989
\(506\) 1.39445 0.0619908
\(507\) −0.633308 −0.0281262
\(508\) −5.39445 −0.239340
\(509\) −18.4222 −0.816550 −0.408275 0.912859i \(-0.633870\pi\)
−0.408275 + 0.912859i \(0.633870\pi\)
\(510\) −3.21110 −0.142190
\(511\) −3.39445 −0.150162
\(512\) −1.00000 −0.0441942
\(513\) −1.78890 −0.0789818
\(514\) 4.60555 0.203142
\(515\) 3.90833 0.172221
\(516\) 0.330532 0.0145509
\(517\) −6.00000 −0.263880
\(518\) −6.78890 −0.298287
\(519\) −5.85281 −0.256910
\(520\) −7.60555 −0.333525
\(521\) 36.4222 1.59569 0.797843 0.602865i \(-0.205974\pi\)
0.797843 + 0.602865i \(0.205974\pi\)
\(522\) −24.1472 −1.05689
\(523\) 19.0278 0.832026 0.416013 0.909359i \(-0.363427\pi\)
0.416013 + 0.909359i \(0.363427\pi\)
\(524\) 19.1194 0.835236
\(525\) 0.119429 0.00521233
\(526\) 20.3028 0.885243
\(527\) 15.2111 0.662606
\(528\) 0.302776 0.0131766
\(529\) −21.0555 −0.915457
\(530\) −24.4222 −1.06083
\(531\) 0 0
\(532\) 1.30278 0.0564825
\(533\) −12.9083 −0.559122
\(534\) −1.39445 −0.0603437
\(535\) −7.39445 −0.319690
\(536\) 6.51388 0.281357
\(537\) −3.97224 −0.171415
\(538\) 19.8167 0.854357
\(539\) −5.30278 −0.228407
\(540\) 4.11943 0.177272
\(541\) 3.81665 0.164091 0.0820454 0.996629i \(-0.473855\pi\)
0.0820454 + 0.996629i \(0.473855\pi\)
\(542\) 6.51388 0.279795
\(543\) −4.55004 −0.195261
\(544\) 4.60555 0.197461
\(545\) −4.60555 −0.197280
\(546\) 1.30278 0.0557537
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) −3.48612 −0.148920
\(549\) −32.6056 −1.39157
\(550\) −0.302776 −0.0129104
\(551\) −8.30278 −0.353710
\(552\) 0.422205 0.0179702
\(553\) 20.6056 0.876237
\(554\) 10.0000 0.424859
\(555\) −3.63331 −0.154225
\(556\) −6.30278 −0.267297
\(557\) −27.6333 −1.17086 −0.585430 0.810723i \(-0.699074\pi\)
−0.585430 + 0.810723i \(0.699074\pi\)
\(558\) −9.60555 −0.406635
\(559\) −3.60555 −0.152499
\(560\) −3.00000 −0.126773
\(561\) −1.39445 −0.0588737
\(562\) 0.486122 0.0205058
\(563\) 21.6333 0.911735 0.455868 0.890048i \(-0.349329\pi\)
0.455868 + 0.890048i \(0.349329\pi\)
\(564\) −1.81665 −0.0764949
\(565\) 0 0
\(566\) 18.9361 0.795943
\(567\) 10.6611 0.447723
\(568\) 9.69722 0.406886
\(569\) 12.2750 0.514596 0.257298 0.966332i \(-0.417168\pi\)
0.257298 + 0.966332i \(0.417168\pi\)
\(570\) 0.697224 0.0292035
\(571\) −6.93608 −0.290266 −0.145133 0.989412i \(-0.546361\pi\)
−0.145133 + 0.989412i \(0.546361\pi\)
\(572\) −3.30278 −0.138096
\(573\) −1.81665 −0.0758918
\(574\) −5.09167 −0.212522
\(575\) −0.422205 −0.0176072
\(576\) −2.90833 −0.121180
\(577\) −18.0917 −0.753166 −0.376583 0.926383i \(-0.622901\pi\)
−0.376583 + 0.926383i \(0.622901\pi\)
\(578\) −4.21110 −0.175159
\(579\) −7.48612 −0.311113
\(580\) 19.1194 0.793891
\(581\) 18.6333 0.773040
\(582\) −2.42221 −0.100404
\(583\) −10.6056 −0.439237
\(584\) 2.60555 0.107818
\(585\) −22.1194 −0.914526
\(586\) 25.5416 1.05512
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.60555 −0.0662118
\(589\) −3.30278 −0.136088
\(590\) 0 0
\(591\) 6.55004 0.269433
\(592\) 5.21110 0.214175
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 1.78890 0.0733994
\(595\) 13.8167 0.566428
\(596\) 0 0
\(597\) −0.661064 −0.0270555
\(598\) −4.60555 −0.188335
\(599\) −19.8806 −0.812298 −0.406149 0.913807i \(-0.633129\pi\)
−0.406149 + 0.913807i \(0.633129\pi\)
\(600\) −0.0916731 −0.00374254
\(601\) 4.09167 0.166903 0.0834514 0.996512i \(-0.473406\pi\)
0.0834514 + 0.996512i \(0.473406\pi\)
\(602\) −1.42221 −0.0579648
\(603\) 18.9445 0.771479
\(604\) −16.4222 −0.668210
\(605\) −2.30278 −0.0936211
\(606\) −2.36669 −0.0961403
\(607\) 3.81665 0.154913 0.0774566 0.996996i \(-0.475320\pi\)
0.0774566 + 0.996996i \(0.475320\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −3.27502 −0.132710
\(610\) 25.8167 1.04529
\(611\) 19.8167 0.801696
\(612\) 13.3944 0.541438
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 2.18335 0.0881127
\(615\) −2.72498 −0.109882
\(616\) −1.30278 −0.0524903
\(617\) 2.09167 0.0842076 0.0421038 0.999113i \(-0.486594\pi\)
0.0421038 + 0.999113i \(0.486594\pi\)
\(618\) 0.513878 0.0206712
\(619\) −34.8444 −1.40052 −0.700258 0.713890i \(-0.746932\pi\)
−0.700258 + 0.713890i \(0.746932\pi\)
\(620\) 7.60555 0.305446
\(621\) 2.49453 0.100102
\(622\) −25.8167 −1.03515
\(623\) 6.00000 0.240385
\(624\) −1.00000 −0.0400320
\(625\) −26.4222 −1.05689
\(626\) −18.1194 −0.724198
\(627\) 0.302776 0.0120917
\(628\) 1.09167 0.0435625
\(629\) −24.0000 −0.956943
\(630\) −8.72498 −0.347612
\(631\) −45.0278 −1.79253 −0.896263 0.443522i \(-0.853729\pi\)
−0.896263 + 0.443522i \(0.853729\pi\)
\(632\) −15.8167 −0.629153
\(633\) 7.44996 0.296109
\(634\) 13.8167 0.548729
\(635\) 12.4222 0.492960
\(636\) −3.21110 −0.127328
\(637\) 17.5139 0.693925
\(638\) 8.30278 0.328710
\(639\) 28.2027 1.11568
\(640\) 2.30278 0.0910252
\(641\) 20.2389 0.799387 0.399693 0.916649i \(-0.369117\pi\)
0.399693 + 0.916649i \(0.369117\pi\)
\(642\) −0.972244 −0.0383714
\(643\) 26.4222 1.04199 0.520995 0.853560i \(-0.325561\pi\)
0.520995 + 0.853560i \(0.325561\pi\)
\(644\) −1.81665 −0.0715862
\(645\) −0.761141 −0.0299699
\(646\) 4.60555 0.181203
\(647\) 25.8167 1.01496 0.507479 0.861664i \(-0.330578\pi\)
0.507479 + 0.861664i \(0.330578\pi\)
\(648\) −8.18335 −0.321472
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) −1.30278 −0.0510598
\(652\) −11.3944 −0.446241
\(653\) 22.3305 0.873861 0.436931 0.899495i \(-0.356066\pi\)
0.436931 + 0.899495i \(0.356066\pi\)
\(654\) −0.605551 −0.0236789
\(655\) −44.0278 −1.72031
\(656\) 3.90833 0.152595
\(657\) 7.57779 0.295638
\(658\) 7.81665 0.304725
\(659\) 29.0278 1.13076 0.565380 0.824830i \(-0.308729\pi\)
0.565380 + 0.824830i \(0.308729\pi\)
\(660\) −0.697224 −0.0271394
\(661\) 23.6333 0.919229 0.459615 0.888118i \(-0.347988\pi\)
0.459615 + 0.888118i \(0.347988\pi\)
\(662\) −22.7250 −0.883231
\(663\) 4.60555 0.178865
\(664\) −14.3028 −0.555055
\(665\) −3.00000 −0.116335
\(666\) 15.1556 0.587267
\(667\) 11.5778 0.448294
\(668\) −12.4222 −0.480630
\(669\) −7.63331 −0.295121
\(670\) −15.0000 −0.579501
\(671\) 11.2111 0.432800
\(672\) −0.394449 −0.0152162
\(673\) 1.30278 0.0502183 0.0251092 0.999685i \(-0.492007\pi\)
0.0251092 + 0.999685i \(0.492007\pi\)
\(674\) 9.30278 0.358330
\(675\) −0.541635 −0.0208475
\(676\) −2.09167 −0.0804490
\(677\) −8.30278 −0.319102 −0.159551 0.987190i \(-0.551005\pi\)
−0.159551 + 0.987190i \(0.551005\pi\)
\(678\) 0 0
\(679\) 10.4222 0.399968
\(680\) −10.6056 −0.406704
\(681\) 0.550039 0.0210775
\(682\) 3.30278 0.126470
\(683\) −17.5778 −0.672596 −0.336298 0.941756i \(-0.609175\pi\)
−0.336298 + 0.941756i \(0.609175\pi\)
\(684\) −2.90833 −0.111203
\(685\) 8.02776 0.306725
\(686\) 16.0278 0.611943
\(687\) 3.60555 0.137560
\(688\) 1.09167 0.0416196
\(689\) 35.0278 1.33445
\(690\) −0.972244 −0.0370127
\(691\) 32.4222 1.23340 0.616699 0.787199i \(-0.288469\pi\)
0.616699 + 0.787199i \(0.288469\pi\)
\(692\) −19.3305 −0.734837
\(693\) −3.78890 −0.143928
\(694\) −2.78890 −0.105865
\(695\) 14.5139 0.550543
\(696\) 2.51388 0.0952883
\(697\) −18.0000 −0.681799
\(698\) −32.4222 −1.22720
\(699\) 2.78890 0.105486
\(700\) 0.394449 0.0149088
\(701\) −27.6333 −1.04370 −0.521848 0.853039i \(-0.674757\pi\)
−0.521848 + 0.853039i \(0.674757\pi\)
\(702\) −5.90833 −0.222995
\(703\) 5.21110 0.196540
\(704\) 1.00000 0.0376889
\(705\) 4.18335 0.157554
\(706\) 15.2111 0.572477
\(707\) 10.1833 0.382984
\(708\) 0 0
\(709\) 14.6972 0.551966 0.275983 0.961163i \(-0.410997\pi\)
0.275983 + 0.961163i \(0.410997\pi\)
\(710\) −22.3305 −0.838050
\(711\) −46.0000 −1.72513
\(712\) −4.60555 −0.172600
\(713\) 4.60555 0.172479
\(714\) 1.81665 0.0679866
\(715\) 7.60555 0.284431
\(716\) −13.1194 −0.490296
\(717\) −7.24726 −0.270654
\(718\) 9.48612 0.354019
\(719\) −25.8167 −0.962799 −0.481399 0.876501i \(-0.659871\pi\)
−0.481399 + 0.876501i \(0.659871\pi\)
\(720\) 6.69722 0.249591
\(721\) −2.21110 −0.0823458
\(722\) −1.00000 −0.0372161
\(723\) 3.96384 0.147417
\(724\) −15.0278 −0.558502
\(725\) −2.51388 −0.0933631
\(726\) −0.302776 −0.0112371
\(727\) −0.366692 −0.0135999 −0.00679993 0.999977i \(-0.502165\pi\)
−0.00679993 + 0.999977i \(0.502165\pi\)
\(728\) 4.30278 0.159471
\(729\) −22.1749 −0.821294
\(730\) −6.00000 −0.222070
\(731\) −5.02776 −0.185958
\(732\) 3.39445 0.125462
\(733\) −7.21110 −0.266348 −0.133174 0.991093i \(-0.542517\pi\)
−0.133174 + 0.991093i \(0.542517\pi\)
\(734\) 7.21110 0.266167
\(735\) 3.69722 0.136374
\(736\) 1.39445 0.0514001
\(737\) −6.51388 −0.239942
\(738\) 11.3667 0.418414
\(739\) −40.3583 −1.48460 −0.742302 0.670066i \(-0.766266\pi\)
−0.742302 + 0.670066i \(0.766266\pi\)
\(740\) −12.0000 −0.441129
\(741\) −1.00000 −0.0367359
\(742\) 13.8167 0.507225
\(743\) 2.36669 0.0868255 0.0434128 0.999057i \(-0.486177\pi\)
0.0434128 + 0.999057i \(0.486177\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −7.09167 −0.259645
\(747\) −41.5971 −1.52196
\(748\) −4.60555 −0.168396
\(749\) 4.18335 0.152856
\(750\) −3.27502 −0.119587
\(751\) −40.8444 −1.49043 −0.745217 0.666822i \(-0.767654\pi\)
−0.745217 + 0.666822i \(0.767654\pi\)
\(752\) −6.00000 −0.218797
\(753\) 7.26662 0.264810
\(754\) −27.4222 −0.998658
\(755\) 37.8167 1.37629
\(756\) −2.33053 −0.0847606
\(757\) 30.5416 1.11005 0.555027 0.831832i \(-0.312708\pi\)
0.555027 + 0.831832i \(0.312708\pi\)
\(758\) 13.4861 0.489838
\(759\) −0.422205 −0.0153251
\(760\) 2.30278 0.0835305
\(761\) 20.2389 0.733658 0.366829 0.930288i \(-0.380443\pi\)
0.366829 + 0.930288i \(0.380443\pi\)
\(762\) 1.63331 0.0591685
\(763\) 2.60555 0.0943273
\(764\) −6.00000 −0.217072
\(765\) −30.8444 −1.11518
\(766\) −3.48612 −0.125959
\(767\) 0 0
\(768\) 0.302776 0.0109255
\(769\) −37.6333 −1.35709 −0.678546 0.734558i \(-0.737390\pi\)
−0.678546 + 0.734558i \(0.737390\pi\)
\(770\) 3.00000 0.108112
\(771\) −1.39445 −0.0502198
\(772\) −24.7250 −0.889872
\(773\) 48.4222 1.74163 0.870813 0.491615i \(-0.163593\pi\)
0.870813 + 0.491615i \(0.163593\pi\)
\(774\) 3.17494 0.114121
\(775\) −1.00000 −0.0359211
\(776\) −8.00000 −0.287183
\(777\) 2.05551 0.0737411
\(778\) −34.5416 −1.23838
\(779\) 3.90833 0.140030
\(780\) 2.30278 0.0824526
\(781\) −9.69722 −0.346994
\(782\) −6.42221 −0.229658
\(783\) 14.8528 0.530796
\(784\) −5.30278 −0.189385
\(785\) −2.51388 −0.0897242
\(786\) −5.78890 −0.206483
\(787\) −15.0278 −0.535682 −0.267841 0.963463i \(-0.586310\pi\)
−0.267841 + 0.963463i \(0.586310\pi\)
\(788\) 21.6333 0.770655
\(789\) −6.14719 −0.218846
\(790\) 36.4222 1.29584
\(791\) 0 0
\(792\) 2.90833 0.103343
\(793\) −37.0278 −1.31489
\(794\) 32.7527 1.16235
\(795\) 7.39445 0.262254
\(796\) −2.18335 −0.0773867
\(797\) 16.6056 0.588199 0.294099 0.955775i \(-0.404980\pi\)
0.294099 + 0.955775i \(0.404980\pi\)
\(798\) −0.394449 −0.0139633
\(799\) 27.6333 0.977596
\(800\) −0.302776 −0.0107047
\(801\) −13.3944 −0.473270
\(802\) −22.6056 −0.798230
\(803\) −2.60555 −0.0919479
\(804\) −1.97224 −0.0695557
\(805\) 4.18335 0.147444
\(806\) −10.9083 −0.384229
\(807\) −6.00000 −0.211210
\(808\) −7.81665 −0.274989
\(809\) 33.2111 1.16764 0.583820 0.811883i \(-0.301557\pi\)
0.583820 + 0.811883i \(0.301557\pi\)
\(810\) 18.8444 0.662125
\(811\) 29.6333 1.04057 0.520283 0.853994i \(-0.325826\pi\)
0.520283 + 0.853994i \(0.325826\pi\)
\(812\) −10.8167 −0.379590
\(813\) −1.97224 −0.0691696
\(814\) −5.21110 −0.182649
\(815\) 26.2389 0.919107
\(816\) −1.39445 −0.0488155
\(817\) 1.09167 0.0381928
\(818\) −23.1472 −0.809322
\(819\) 12.5139 0.437270
\(820\) −9.00000 −0.314294
\(821\) −34.6056 −1.20774 −0.603871 0.797082i \(-0.706376\pi\)
−0.603871 + 0.797082i \(0.706376\pi\)
\(822\) 1.05551 0.0368152
\(823\) 32.0000 1.11545 0.557725 0.830026i \(-0.311674\pi\)
0.557725 + 0.830026i \(0.311674\pi\)
\(824\) 1.69722 0.0591256
\(825\) 0.0916731 0.00319165
\(826\) 0 0
\(827\) 22.0555 0.766945 0.383473 0.923552i \(-0.374728\pi\)
0.383473 + 0.923552i \(0.374728\pi\)
\(828\) 4.05551 0.140939
\(829\) −27.5778 −0.957816 −0.478908 0.877865i \(-0.658967\pi\)
−0.478908 + 0.877865i \(0.658967\pi\)
\(830\) 32.9361 1.14323
\(831\) −3.02776 −0.105032
\(832\) −3.30278 −0.114503
\(833\) 24.4222 0.846179
\(834\) 1.90833 0.0660800
\(835\) 28.6056 0.989936
\(836\) 1.00000 0.0345857
\(837\) 5.90833 0.204222
\(838\) −8.78890 −0.303607
\(839\) 19.8806 0.686354 0.343177 0.939271i \(-0.388497\pi\)
0.343177 + 0.939271i \(0.388497\pi\)
\(840\) 0.908327 0.0313403
\(841\) 39.9361 1.37711
\(842\) 30.2389 1.04210
\(843\) −0.147186 −0.00506935
\(844\) 24.6056 0.846958
\(845\) 4.81665 0.165698
\(846\) −17.4500 −0.599942
\(847\) 1.30278 0.0447639
\(848\) −10.6056 −0.364196
\(849\) −5.73338 −0.196769
\(850\) 1.39445 0.0478292
\(851\) −7.26662 −0.249096
\(852\) −2.93608 −0.100589
\(853\) −46.4222 −1.58947 −0.794733 0.606959i \(-0.792389\pi\)
−0.794733 + 0.606959i \(0.792389\pi\)
\(854\) −14.6056 −0.499792
\(855\) 6.69722 0.229040
\(856\) −3.21110 −0.109753
\(857\) −43.5416 −1.48735 −0.743677 0.668539i \(-0.766920\pi\)
−0.743677 + 0.668539i \(0.766920\pi\)
\(858\) 1.00000 0.0341394
\(859\) 8.84441 0.301767 0.150884 0.988552i \(-0.451788\pi\)
0.150884 + 0.988552i \(0.451788\pi\)
\(860\) −2.51388 −0.0857225
\(861\) 1.54163 0.0525388
\(862\) −2.78890 −0.0949902
\(863\) −17.7250 −0.603365 −0.301683 0.953408i \(-0.597548\pi\)
−0.301683 + 0.953408i \(0.597548\pi\)
\(864\) 1.78890 0.0608595
\(865\) 44.5139 1.51352
\(866\) 24.2389 0.823670
\(867\) 1.27502 0.0433019
\(868\) −4.30278 −0.146046
\(869\) 15.8167 0.536543
\(870\) −5.78890 −0.196262
\(871\) 21.5139 0.728970
\(872\) −2.00000 −0.0677285
\(873\) −23.2666 −0.787456
\(874\) 1.39445 0.0471679
\(875\) 14.0917 0.476385
\(876\) −0.788897 −0.0266544
\(877\) 47.5694 1.60630 0.803152 0.595774i \(-0.203155\pi\)
0.803152 + 0.595774i \(0.203155\pi\)
\(878\) 25.2111 0.850833
\(879\) −7.73338 −0.260841
\(880\) −2.30278 −0.0776266
\(881\) −33.9083 −1.14240 −0.571200 0.820811i \(-0.693522\pi\)
−0.571200 + 0.820811i \(0.693522\pi\)
\(882\) −15.4222 −0.519293
\(883\) −39.0278 −1.31339 −0.656694 0.754157i \(-0.728046\pi\)
−0.656694 + 0.754157i \(0.728046\pi\)
\(884\) 15.2111 0.511605
\(885\) 0 0
\(886\) −39.6333 −1.33151
\(887\) 24.8444 0.834194 0.417097 0.908862i \(-0.363048\pi\)
0.417097 + 0.908862i \(0.363048\pi\)
\(888\) −1.57779 −0.0529473
\(889\) −7.02776 −0.235703
\(890\) 10.6056 0.355499
\(891\) 8.18335 0.274152
\(892\) −25.2111 −0.844130
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 30.2111 1.00985
\(896\) −1.30278 −0.0435227
\(897\) 1.39445 0.0465593
\(898\) 31.8167 1.06174
\(899\) 27.4222 0.914582
\(900\) −0.880571 −0.0293524
\(901\) 48.8444 1.62724
\(902\) −3.90833 −0.130133
\(903\) 0.430609 0.0143298
\(904\) 0 0
\(905\) 34.6056 1.15033
\(906\) 4.97224 0.165192
\(907\) −4.84441 −0.160856 −0.0804280 0.996760i \(-0.525629\pi\)
−0.0804280 + 0.996760i \(0.525629\pi\)
\(908\) 1.81665 0.0602878
\(909\) −22.7334 −0.754019
\(910\) −9.90833 −0.328458
\(911\) 27.6333 0.915532 0.457766 0.889073i \(-0.348650\pi\)
0.457766 + 0.889073i \(0.348650\pi\)
\(912\) 0.302776 0.0100259
\(913\) 14.3028 0.473353
\(914\) −21.8167 −0.721631
\(915\) −7.81665 −0.258411
\(916\) 11.9083 0.393462
\(917\) 24.9083 0.822545
\(918\) −8.23886 −0.271923
\(919\) 6.11943 0.201861 0.100931 0.994893i \(-0.467818\pi\)
0.100931 + 0.994893i \(0.467818\pi\)
\(920\) −3.21110 −0.105867
\(921\) −0.661064 −0.0217828
\(922\) −6.42221 −0.211504
\(923\) 32.0278 1.05421
\(924\) 0.394449 0.0129764
\(925\) 1.57779 0.0518776
\(926\) 1.21110 0.0397993
\(927\) 4.93608 0.162122
\(928\) 8.30278 0.272552
\(929\) −8.51388 −0.279331 −0.139666 0.990199i \(-0.544603\pi\)
−0.139666 + 0.990199i \(0.544603\pi\)
\(930\) −2.30278 −0.0755110
\(931\) −5.30278 −0.173791
\(932\) 9.21110 0.301720
\(933\) 7.81665 0.255906
\(934\) 22.6056 0.739676
\(935\) 10.6056 0.346839
\(936\) −9.60555 −0.313967
\(937\) 1.15559 0.0377515 0.0188757 0.999822i \(-0.493991\pi\)
0.0188757 + 0.999822i \(0.493991\pi\)
\(938\) 8.48612 0.277082
\(939\) 5.48612 0.179033
\(940\) 13.8167 0.450650
\(941\) −57.6333 −1.87879 −0.939396 0.342834i \(-0.888613\pi\)
−0.939396 + 0.342834i \(0.888613\pi\)
\(942\) −0.330532 −0.0107693
\(943\) −5.44996 −0.177475
\(944\) 0 0
\(945\) 5.36669 0.174579
\(946\) −1.09167 −0.0354934
\(947\) 25.8167 0.838929 0.419464 0.907772i \(-0.362218\pi\)
0.419464 + 0.907772i \(0.362218\pi\)
\(948\) 4.78890 0.155536
\(949\) 8.60555 0.279348
\(950\) −0.302776 −0.00982334
\(951\) −4.18335 −0.135654
\(952\) 6.00000 0.194461
\(953\) −21.6333 −0.700772 −0.350386 0.936605i \(-0.613950\pi\)
−0.350386 + 0.936605i \(0.613950\pi\)
\(954\) −30.8444 −0.998625
\(955\) 13.8167 0.447096
\(956\) −23.9361 −0.774148
\(957\) −2.51388 −0.0812621
\(958\) −35.7250 −1.15422
\(959\) −4.54163 −0.146657
\(960\) −0.697224 −0.0225028
\(961\) −20.0917 −0.648118
\(962\) 17.2111 0.554908
\(963\) −9.33894 −0.300943
\(964\) 13.0917 0.421654
\(965\) 56.9361 1.83284
\(966\) 0.550039 0.0176972
\(967\) 23.6333 0.759996 0.379998 0.924987i \(-0.375925\pi\)
0.379998 + 0.924987i \(0.375925\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −1.39445 −0.0447961
\(970\) 18.4222 0.591501
\(971\) −23.9361 −0.768145 −0.384073 0.923303i \(-0.625479\pi\)
−0.384073 + 0.923303i \(0.625479\pi\)
\(972\) 7.84441 0.251610
\(973\) −8.21110 −0.263236
\(974\) 29.3305 0.939811
\(975\) −0.302776 −0.00969658
\(976\) 11.2111 0.358859
\(977\) 27.6333 0.884068 0.442034 0.896998i \(-0.354257\pi\)
0.442034 + 0.896998i \(0.354257\pi\)
\(978\) 3.44996 0.110318
\(979\) 4.60555 0.147194
\(980\) 12.2111 0.390069
\(981\) −5.81665 −0.185711
\(982\) 14.9361 0.476630
\(983\) 23.3028 0.743243 0.371622 0.928384i \(-0.378802\pi\)
0.371622 + 0.928384i \(0.378802\pi\)
\(984\) −1.18335 −0.0377237
\(985\) −49.8167 −1.58729
\(986\) −38.2389 −1.21777
\(987\) −2.36669 −0.0753326
\(988\) −3.30278 −0.105075
\(989\) −1.52228 −0.0484058
\(990\) −6.69722 −0.212852
\(991\) −25.9083 −0.823005 −0.411503 0.911409i \(-0.634996\pi\)
−0.411503 + 0.911409i \(0.634996\pi\)
\(992\) 3.30278 0.104863
\(993\) 6.88057 0.218348
\(994\) 12.6333 0.400704
\(995\) 5.02776 0.159391
\(996\) 4.33053 0.137218
\(997\) 0.183346 0.00580663 0.00290332 0.999996i \(-0.499076\pi\)
0.00290332 + 0.999996i \(0.499076\pi\)
\(998\) 34.0000 1.07625
\(999\) −9.32213 −0.294939
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 418.2.a.d.1.2 2
3.2 odd 2 3762.2.a.bb.1.2 2
4.3 odd 2 3344.2.a.o.1.1 2
11.10 odd 2 4598.2.a.bc.1.2 2
19.18 odd 2 7942.2.a.bb.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
418.2.a.d.1.2 2 1.1 even 1 trivial
3344.2.a.o.1.1 2 4.3 odd 2
3762.2.a.bb.1.2 2 3.2 odd 2
4598.2.a.bc.1.2 2 11.10 odd 2
7942.2.a.bb.1.1 2 19.18 odd 2