Properties

Label 4205.2.a.e.1.2
Level $4205$
Weight $2$
Character 4205.1
Self dual yes
Analytic conductor $33.577$
Analytic rank $1$
Dimension $3$
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] = [4205,2,Mod(1,4205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4205, 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("4205.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4205 = 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4205.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: \(33.5770940499\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 4205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53919 q^{2} -1.70928 q^{3} +0.369102 q^{4} -1.00000 q^{5} +2.63090 q^{6} +0.630898 q^{7} +2.51026 q^{8} -0.0783777 q^{9} +O(q^{10})\) \(q-1.53919 q^{2} -1.70928 q^{3} +0.369102 q^{4} -1.00000 q^{5} +2.63090 q^{6} +0.630898 q^{7} +2.51026 q^{8} -0.0783777 q^{9} +1.53919 q^{10} -0.290725 q^{11} -0.630898 q^{12} -0.921622 q^{13} -0.971071 q^{14} +1.70928 q^{15} -4.60197 q^{16} -4.97107 q^{17} +0.120638 q^{18} +6.04945 q^{19} -0.369102 q^{20} -1.07838 q^{21} +0.447480 q^{22} +2.29072 q^{23} -4.29072 q^{24} +1.00000 q^{25} +1.41855 q^{26} +5.26180 q^{27} +0.232866 q^{28} -2.63090 q^{30} -10.0494 q^{31} +2.06278 q^{32} +0.496928 q^{33} +7.65142 q^{34} -0.630898 q^{35} -0.0289294 q^{36} -1.55252 q^{37} -9.31124 q^{38} +1.57531 q^{39} -2.51026 q^{40} -0.340173 q^{41} +1.65983 q^{42} +5.70928 q^{43} -0.107307 q^{44} +0.0783777 q^{45} -3.52586 q^{46} +1.12783 q^{47} +7.86603 q^{48} -6.60197 q^{49} -1.53919 q^{50} +8.49693 q^{51} -0.340173 q^{52} -0.340173 q^{53} -8.09890 q^{54} +0.290725 q^{55} +1.58372 q^{56} -10.3402 q^{57} +9.75872 q^{59} +0.630898 q^{60} -3.07838 q^{61} +15.4680 q^{62} -0.0494483 q^{63} +6.02893 q^{64} +0.921622 q^{65} -0.764867 q^{66} -5.70928 q^{67} -1.83483 q^{68} -3.91548 q^{69} +0.971071 q^{70} +9.07838 q^{71} -0.196748 q^{72} +6.94441 q^{73} +2.38962 q^{74} -1.70928 q^{75} +2.23287 q^{76} -0.183417 q^{77} -2.42469 q^{78} -12.3896 q^{79} +4.60197 q^{80} -8.75872 q^{81} +0.523590 q^{82} +2.78765 q^{83} -0.398032 q^{84} +4.97107 q^{85} -8.78765 q^{86} -0.729794 q^{88} -4.73820 q^{89} -0.120638 q^{90} -0.581449 q^{91} +0.845512 q^{92} +17.1773 q^{93} -1.73594 q^{94} -6.04945 q^{95} -3.52586 q^{96} +15.8927 q^{97} +10.1617 q^{98} +0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 3 q^{10} - 8 q^{11} + 2 q^{12} - 6 q^{13} + 12 q^{14} - 2 q^{15} + 5 q^{16} + 13 q^{18} - 5 q^{20} + 2 q^{22} + 14 q^{23} - 20 q^{24} + 3 q^{25} - 10 q^{26} + 8 q^{27} - 22 q^{28} - 4 q^{30} - 12 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 2 q^{35} - 15 q^{36} - 4 q^{37} - 2 q^{38} - 16 q^{39} + 9 q^{40} + 10 q^{41} + 16 q^{42} + 10 q^{43} - 12 q^{44} - 3 q^{45} - 8 q^{46} - 18 q^{47} + 10 q^{48} - q^{49} - 3 q^{50} + 8 q^{51} + 10 q^{52} + 10 q^{53} + 12 q^{54} + 8 q^{55} + 32 q^{56} - 20 q^{57} + 4 q^{59} - 2 q^{60} - 6 q^{61} + 14 q^{62} + 18 q^{63} + 33 q^{64} + 6 q^{65} - 12 q^{66} - 10 q^{67} + 36 q^{68} + 20 q^{69} - 12 q^{70} + 24 q^{71} + 3 q^{72} + 4 q^{73} - 22 q^{74} + 2 q^{75} - 16 q^{76} + 4 q^{77} - 28 q^{78} - 8 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 2 q^{83} - 20 q^{84} - 16 q^{86} + 38 q^{88} - 22 q^{89} - 13 q^{90} - 16 q^{91} + 22 q^{92} + 12 q^{93} - 8 q^{96} + 36 q^{97} - 23 q^{98} - 20 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.53919 −1.08837 −0.544185 0.838965i \(-0.683161\pi\)
−0.544185 + 0.838965i \(0.683161\pi\)
\(3\) −1.70928 −0.986851 −0.493425 0.869788i \(-0.664255\pi\)
−0.493425 + 0.869788i \(0.664255\pi\)
\(4\) 0.369102 0.184551
\(5\) −1.00000 −0.447214
\(6\) 2.63090 1.07406
\(7\) 0.630898 0.238457 0.119228 0.992867i \(-0.461958\pi\)
0.119228 + 0.992867i \(0.461958\pi\)
\(8\) 2.51026 0.887511
\(9\) −0.0783777 −0.0261259
\(10\) 1.53919 0.486734
\(11\) −0.290725 −0.0876568 −0.0438284 0.999039i \(-0.513955\pi\)
−0.0438284 + 0.999039i \(0.513955\pi\)
\(12\) −0.630898 −0.182124
\(13\) −0.921622 −0.255612 −0.127806 0.991799i \(-0.540793\pi\)
−0.127806 + 0.991799i \(0.540793\pi\)
\(14\) −0.971071 −0.259530
\(15\) 1.70928 0.441333
\(16\) −4.60197 −1.15049
\(17\) −4.97107 −1.20566 −0.602831 0.797869i \(-0.705961\pi\)
−0.602831 + 0.797869i \(0.705961\pi\)
\(18\) 0.120638 0.0284347
\(19\) 6.04945 1.38784 0.693919 0.720053i \(-0.255882\pi\)
0.693919 + 0.720053i \(0.255882\pi\)
\(20\) −0.369102 −0.0825338
\(21\) −1.07838 −0.235321
\(22\) 0.447480 0.0954031
\(23\) 2.29072 0.477649 0.238825 0.971063i \(-0.423238\pi\)
0.238825 + 0.971063i \(0.423238\pi\)
\(24\) −4.29072 −0.875840
\(25\) 1.00000 0.200000
\(26\) 1.41855 0.278201
\(27\) 5.26180 1.01263
\(28\) 0.232866 0.0440075
\(29\) 0 0
\(30\) −2.63090 −0.480334
\(31\) −10.0494 −1.80493 −0.902467 0.430759i \(-0.858246\pi\)
−0.902467 + 0.430759i \(0.858246\pi\)
\(32\) 2.06278 0.364651
\(33\) 0.496928 0.0865041
\(34\) 7.65142 1.31221
\(35\) −0.630898 −0.106641
\(36\) −0.0289294 −0.00482157
\(37\) −1.55252 −0.255233 −0.127616 0.991824i \(-0.540733\pi\)
−0.127616 + 0.991824i \(0.540733\pi\)
\(38\) −9.31124 −1.51048
\(39\) 1.57531 0.252251
\(40\) −2.51026 −0.396907
\(41\) −0.340173 −0.0531261 −0.0265630 0.999647i \(-0.508456\pi\)
−0.0265630 + 0.999647i \(0.508456\pi\)
\(42\) 1.65983 0.256117
\(43\) 5.70928 0.870656 0.435328 0.900272i \(-0.356632\pi\)
0.435328 + 0.900272i \(0.356632\pi\)
\(44\) −0.107307 −0.0161772
\(45\) 0.0783777 0.0116839
\(46\) −3.52586 −0.519859
\(47\) 1.12783 0.164510 0.0822552 0.996611i \(-0.473788\pi\)
0.0822552 + 0.996611i \(0.473788\pi\)
\(48\) 7.86603 1.13536
\(49\) −6.60197 −0.943138
\(50\) −1.53919 −0.217674
\(51\) 8.49693 1.18981
\(52\) −0.340173 −0.0471735
\(53\) −0.340173 −0.0467264 −0.0233632 0.999727i \(-0.507437\pi\)
−0.0233632 + 0.999727i \(0.507437\pi\)
\(54\) −8.09890 −1.10212
\(55\) 0.290725 0.0392013
\(56\) 1.58372 0.211633
\(57\) −10.3402 −1.36959
\(58\) 0 0
\(59\) 9.75872 1.27048 0.635239 0.772316i \(-0.280902\pi\)
0.635239 + 0.772316i \(0.280902\pi\)
\(60\) 0.630898 0.0814485
\(61\) −3.07838 −0.394146 −0.197073 0.980389i \(-0.563144\pi\)
−0.197073 + 0.980389i \(0.563144\pi\)
\(62\) 15.4680 1.96444
\(63\) −0.0494483 −0.00622990
\(64\) 6.02893 0.753616
\(65\) 0.921622 0.114313
\(66\) −0.764867 −0.0941486
\(67\) −5.70928 −0.697499 −0.348749 0.937216i \(-0.613394\pi\)
−0.348749 + 0.937216i \(0.613394\pi\)
\(68\) −1.83483 −0.222506
\(69\) −3.91548 −0.471368
\(70\) 0.971071 0.116065
\(71\) 9.07838 1.07741 0.538703 0.842496i \(-0.318915\pi\)
0.538703 + 0.842496i \(0.318915\pi\)
\(72\) −0.196748 −0.0231870
\(73\) 6.94441 0.812782 0.406391 0.913699i \(-0.366787\pi\)
0.406391 + 0.913699i \(0.366787\pi\)
\(74\) 2.38962 0.277788
\(75\) −1.70928 −0.197370
\(76\) 2.23287 0.256127
\(77\) −0.183417 −0.0209024
\(78\) −2.42469 −0.274543
\(79\) −12.3896 −1.39394 −0.696971 0.717100i \(-0.745469\pi\)
−0.696971 + 0.717100i \(0.745469\pi\)
\(80\) 4.60197 0.514516
\(81\) −8.75872 −0.973192
\(82\) 0.523590 0.0578209
\(83\) 2.78765 0.305985 0.152992 0.988227i \(-0.451109\pi\)
0.152992 + 0.988227i \(0.451109\pi\)
\(84\) −0.398032 −0.0434288
\(85\) 4.97107 0.539188
\(86\) −8.78765 −0.947597
\(87\) 0 0
\(88\) −0.729794 −0.0777963
\(89\) −4.73820 −0.502249 −0.251124 0.967955i \(-0.580800\pi\)
−0.251124 + 0.967955i \(0.580800\pi\)
\(90\) −0.120638 −0.0127164
\(91\) −0.581449 −0.0609524
\(92\) 0.845512 0.0881507
\(93\) 17.1773 1.78120
\(94\) −1.73594 −0.179048
\(95\) −6.04945 −0.620660
\(96\) −3.52586 −0.359856
\(97\) 15.8927 1.61366 0.806829 0.590785i \(-0.201182\pi\)
0.806829 + 0.590785i \(0.201182\pi\)
\(98\) 10.1617 1.02648
\(99\) 0.0227863 0.00229011
\(100\) 0.369102 0.0369102
\(101\) 12.2557 1.21948 0.609741 0.792600i \(-0.291273\pi\)
0.609741 + 0.792600i \(0.291273\pi\)
\(102\) −13.0784 −1.29495
\(103\) −7.86603 −0.775063 −0.387532 0.921856i \(-0.626672\pi\)
−0.387532 + 0.921856i \(0.626672\pi\)
\(104\) −2.31351 −0.226858
\(105\) 1.07838 0.105239
\(106\) 0.523590 0.0508556
\(107\) 12.7298 1.23064 0.615318 0.788279i \(-0.289028\pi\)
0.615318 + 0.788279i \(0.289028\pi\)
\(108\) 1.94214 0.186883
\(109\) 12.4391 1.19145 0.595723 0.803190i \(-0.296865\pi\)
0.595723 + 0.803190i \(0.296865\pi\)
\(110\) −0.447480 −0.0426656
\(111\) 2.65368 0.251877
\(112\) −2.90337 −0.274343
\(113\) −12.5730 −1.18277 −0.591386 0.806389i \(-0.701419\pi\)
−0.591386 + 0.806389i \(0.701419\pi\)
\(114\) 15.9155 1.49062
\(115\) −2.29072 −0.213611
\(116\) 0 0
\(117\) 0.0722347 0.00667810
\(118\) −15.0205 −1.38275
\(119\) −3.13624 −0.287498
\(120\) 4.29072 0.391688
\(121\) −10.9155 −0.992316
\(122\) 4.73820 0.428977
\(123\) 0.581449 0.0524275
\(124\) −3.70928 −0.333103
\(125\) −1.00000 −0.0894427
\(126\) 0.0761103 0.00678045
\(127\) 20.9132 1.85575 0.927874 0.372895i \(-0.121635\pi\)
0.927874 + 0.372895i \(0.121635\pi\)
\(128\) −13.4052 −1.18487
\(129\) −9.75872 −0.859208
\(130\) −1.41855 −0.124415
\(131\) 13.4680 1.17670 0.588352 0.808605i \(-0.299777\pi\)
0.588352 + 0.808605i \(0.299777\pi\)
\(132\) 0.183417 0.0159644
\(133\) 3.81658 0.330940
\(134\) 8.78765 0.759138
\(135\) −5.26180 −0.452863
\(136\) −12.4787 −1.07004
\(137\) 13.5525 1.15787 0.578935 0.815374i \(-0.303469\pi\)
0.578935 + 0.815374i \(0.303469\pi\)
\(138\) 6.02666 0.513024
\(139\) −4.89496 −0.415185 −0.207593 0.978215i \(-0.566563\pi\)
−0.207593 + 0.978215i \(0.566563\pi\)
\(140\) −0.232866 −0.0196808
\(141\) −1.92777 −0.162347
\(142\) −13.9733 −1.17262
\(143\) 0.267938 0.0224061
\(144\) 0.360692 0.0300577
\(145\) 0 0
\(146\) −10.6888 −0.884608
\(147\) 11.2846 0.930737
\(148\) −0.573039 −0.0471035
\(149\) −12.5236 −1.02597 −0.512986 0.858397i \(-0.671461\pi\)
−0.512986 + 0.858397i \(0.671461\pi\)
\(150\) 2.63090 0.214812
\(151\) 7.60197 0.618639 0.309320 0.950958i \(-0.399899\pi\)
0.309320 + 0.950958i \(0.399899\pi\)
\(152\) 15.1857 1.23172
\(153\) 0.389621 0.0314990
\(154\) 0.282314 0.0227495
\(155\) 10.0494 0.807191
\(156\) 0.581449 0.0465532
\(157\) 24.8865 1.98616 0.993081 0.117428i \(-0.0374648\pi\)
0.993081 + 0.117428i \(0.0374648\pi\)
\(158\) 19.0700 1.51713
\(159\) 0.581449 0.0461119
\(160\) −2.06278 −0.163077
\(161\) 1.44521 0.113899
\(162\) 13.4813 1.05919
\(163\) −0.447480 −0.0350493 −0.0175247 0.999846i \(-0.505579\pi\)
−0.0175247 + 0.999846i \(0.505579\pi\)
\(164\) −0.125559 −0.00980448
\(165\) −0.496928 −0.0386858
\(166\) −4.29072 −0.333025
\(167\) 19.8660 1.53728 0.768640 0.639682i \(-0.220934\pi\)
0.768640 + 0.639682i \(0.220934\pi\)
\(168\) −2.70701 −0.208850
\(169\) −12.1506 −0.934662
\(170\) −7.65142 −0.586837
\(171\) −0.474142 −0.0362586
\(172\) 2.10731 0.160681
\(173\) −25.4329 −1.93363 −0.966815 0.255478i \(-0.917767\pi\)
−0.966815 + 0.255478i \(0.917767\pi\)
\(174\) 0 0
\(175\) 0.630898 0.0476914
\(176\) 1.33791 0.100848
\(177\) −16.6803 −1.25377
\(178\) 7.29299 0.546633
\(179\) 14.8371 1.10898 0.554489 0.832191i \(-0.312914\pi\)
0.554489 + 0.832191i \(0.312914\pi\)
\(180\) 0.0289294 0.00215627
\(181\) −5.91548 −0.439694 −0.219847 0.975534i \(-0.570556\pi\)
−0.219847 + 0.975534i \(0.570556\pi\)
\(182\) 0.894960 0.0663389
\(183\) 5.26180 0.388963
\(184\) 5.75031 0.423919
\(185\) 1.55252 0.114144
\(186\) −26.4391 −1.93861
\(187\) 1.44521 0.105684
\(188\) 0.416283 0.0303606
\(189\) 3.31965 0.241469
\(190\) 9.31124 0.675509
\(191\) −7.02893 −0.508595 −0.254298 0.967126i \(-0.581844\pi\)
−0.254298 + 0.967126i \(0.581844\pi\)
\(192\) −10.3051 −0.743707
\(193\) −17.8660 −1.28603 −0.643013 0.765856i \(-0.722316\pi\)
−0.643013 + 0.765856i \(0.722316\pi\)
\(194\) −24.4619 −1.75626
\(195\) −1.57531 −0.112810
\(196\) −2.43680 −0.174057
\(197\) 6.09890 0.434528 0.217264 0.976113i \(-0.430287\pi\)
0.217264 + 0.976113i \(0.430287\pi\)
\(198\) −0.0350725 −0.00249249
\(199\) −9.75872 −0.691778 −0.345889 0.938276i \(-0.612423\pi\)
−0.345889 + 0.938276i \(0.612423\pi\)
\(200\) 2.51026 0.177502
\(201\) 9.75872 0.688327
\(202\) −18.8638 −1.32725
\(203\) 0 0
\(204\) 3.13624 0.219580
\(205\) 0.340173 0.0237587
\(206\) 12.1073 0.843556
\(207\) −0.179542 −0.0124790
\(208\) 4.24128 0.294080
\(209\) −1.75872 −0.121653
\(210\) −1.65983 −0.114539
\(211\) −9.86603 −0.679206 −0.339603 0.940569i \(-0.610293\pi\)
−0.339603 + 0.940569i \(0.610293\pi\)
\(212\) −0.125559 −0.00862340
\(213\) −15.5174 −1.06324
\(214\) −19.5936 −1.33939
\(215\) −5.70928 −0.389369
\(216\) 13.2085 0.898723
\(217\) −6.34017 −0.430399
\(218\) −19.1461 −1.29674
\(219\) −11.8699 −0.802094
\(220\) 0.107307 0.00723465
\(221\) 4.58145 0.308182
\(222\) −4.08452 −0.274135
\(223\) 10.9711 0.734677 0.367339 0.930087i \(-0.380269\pi\)
0.367339 + 0.930087i \(0.380269\pi\)
\(224\) 1.30140 0.0869536
\(225\) −0.0783777 −0.00522518
\(226\) 19.3523 1.28729
\(227\) −12.5464 −0.832732 −0.416366 0.909197i \(-0.636697\pi\)
−0.416366 + 0.909197i \(0.636697\pi\)
\(228\) −3.81658 −0.252759
\(229\) −23.3607 −1.54372 −0.771859 0.635794i \(-0.780673\pi\)
−0.771859 + 0.635794i \(0.780673\pi\)
\(230\) 3.52586 0.232488
\(231\) 0.313511 0.0206275
\(232\) 0 0
\(233\) 12.4703 0.816954 0.408477 0.912769i \(-0.366060\pi\)
0.408477 + 0.912769i \(0.366060\pi\)
\(234\) −0.111183 −0.00726825
\(235\) −1.12783 −0.0735713
\(236\) 3.60197 0.234468
\(237\) 21.1773 1.37561
\(238\) 4.82726 0.312905
\(239\) 13.7587 0.889978 0.444989 0.895536i \(-0.353208\pi\)
0.444989 + 0.895536i \(0.353208\pi\)
\(240\) −7.86603 −0.507750
\(241\) −14.6803 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(242\) 16.8010 1.08001
\(243\) −0.814315 −0.0522383
\(244\) −1.13624 −0.0727401
\(245\) 6.60197 0.421784
\(246\) −0.894960 −0.0570606
\(247\) −5.57531 −0.354748
\(248\) −25.2267 −1.60190
\(249\) −4.76487 −0.301961
\(250\) 1.53919 0.0973469
\(251\) −15.4413 −0.974649 −0.487324 0.873221i \(-0.662027\pi\)
−0.487324 + 0.873221i \(0.662027\pi\)
\(252\) −0.0182515 −0.00114974
\(253\) −0.665970 −0.0418692
\(254\) −32.1894 −2.01974
\(255\) −8.49693 −0.532098
\(256\) 8.57531 0.535957
\(257\) −6.28231 −0.391880 −0.195940 0.980616i \(-0.562776\pi\)
−0.195940 + 0.980616i \(0.562776\pi\)
\(258\) 15.0205 0.935137
\(259\) −0.979481 −0.0608620
\(260\) 0.340173 0.0210966
\(261\) 0 0
\(262\) −20.7298 −1.28069
\(263\) −10.0761 −0.621320 −0.310660 0.950521i \(-0.600550\pi\)
−0.310660 + 0.950521i \(0.600550\pi\)
\(264\) 1.24742 0.0767734
\(265\) 0.340173 0.0208967
\(266\) −5.87444 −0.360185
\(267\) 8.09890 0.495644
\(268\) −2.10731 −0.128724
\(269\) −28.1711 −1.71762 −0.858812 0.512291i \(-0.828797\pi\)
−0.858812 + 0.512291i \(0.828797\pi\)
\(270\) 8.09890 0.492883
\(271\) −28.8020 −1.74960 −0.874799 0.484485i \(-0.839007\pi\)
−0.874799 + 0.484485i \(0.839007\pi\)
\(272\) 22.8767 1.38710
\(273\) 0.993857 0.0601510
\(274\) −20.8599 −1.26019
\(275\) −0.290725 −0.0175314
\(276\) −1.44521 −0.0869916
\(277\) 0.0266620 0.00160196 0.000800982 1.00000i \(-0.499745\pi\)
0.000800982 1.00000i \(0.499745\pi\)
\(278\) 7.53427 0.451875
\(279\) 0.787653 0.0471556
\(280\) −1.58372 −0.0946452
\(281\) −28.0722 −1.67465 −0.837325 0.546706i \(-0.815881\pi\)
−0.837325 + 0.546706i \(0.815881\pi\)
\(282\) 2.96719 0.176694
\(283\) −20.8143 −1.23728 −0.618641 0.785674i \(-0.712317\pi\)
−0.618641 + 0.785674i \(0.712317\pi\)
\(284\) 3.35085 0.198836
\(285\) 10.3402 0.612499
\(286\) −0.412408 −0.0243862
\(287\) −0.214614 −0.0126683
\(288\) −0.161676 −0.00952685
\(289\) 7.71154 0.453620
\(290\) 0 0
\(291\) −27.1650 −1.59244
\(292\) 2.56320 0.150000
\(293\) 15.4101 0.900270 0.450135 0.892961i \(-0.351376\pi\)
0.450135 + 0.892961i \(0.351376\pi\)
\(294\) −17.3691 −1.01299
\(295\) −9.75872 −0.568175
\(296\) −3.89723 −0.226522
\(297\) −1.52973 −0.0887641
\(298\) 19.2762 1.11664
\(299\) −2.11118 −0.122093
\(300\) −0.630898 −0.0364249
\(301\) 3.60197 0.207614
\(302\) −11.7009 −0.673309
\(303\) −20.9483 −1.20345
\(304\) −27.8394 −1.59670
\(305\) 3.07838 0.176267
\(306\) −0.599701 −0.0342826
\(307\) 28.4307 1.62262 0.811312 0.584614i \(-0.198754\pi\)
0.811312 + 0.584614i \(0.198754\pi\)
\(308\) −0.0676998 −0.00385756
\(309\) 13.4452 0.764871
\(310\) −15.4680 −0.878523
\(311\) 19.6248 1.11282 0.556409 0.830909i \(-0.312179\pi\)
0.556409 + 0.830909i \(0.312179\pi\)
\(312\) 3.95443 0.223875
\(313\) 22.9093 1.29491 0.647456 0.762103i \(-0.275833\pi\)
0.647456 + 0.762103i \(0.275833\pi\)
\(314\) −38.3051 −2.16168
\(315\) 0.0494483 0.00278610
\(316\) −4.57304 −0.257254
\(317\) −22.8599 −1.28394 −0.641970 0.766730i \(-0.721882\pi\)
−0.641970 + 0.766730i \(0.721882\pi\)
\(318\) −0.894960 −0.0501869
\(319\) 0 0
\(320\) −6.02893 −0.337027
\(321\) −21.7587 −1.21445
\(322\) −2.22446 −0.123964
\(323\) −30.0722 −1.67326
\(324\) −3.23287 −0.179604
\(325\) −0.921622 −0.0511224
\(326\) 0.688756 0.0381467
\(327\) −21.2618 −1.17578
\(328\) −0.853922 −0.0471500
\(329\) 0.711543 0.0392286
\(330\) 0.764867 0.0421045
\(331\) −24.0905 −1.32413 −0.662066 0.749445i \(-0.730320\pi\)
−0.662066 + 0.749445i \(0.730320\pi\)
\(332\) 1.02893 0.0564698
\(333\) 0.121683 0.00666819
\(334\) −30.5776 −1.67313
\(335\) 5.70928 0.311931
\(336\) 4.96266 0.270735
\(337\) −12.7877 −0.696588 −0.348294 0.937385i \(-0.613239\pi\)
−0.348294 + 0.937385i \(0.613239\pi\)
\(338\) 18.7021 1.01726
\(339\) 21.4908 1.16722
\(340\) 1.83483 0.0995078
\(341\) 2.92162 0.158215
\(342\) 0.729794 0.0394628
\(343\) −8.58145 −0.463355
\(344\) 14.3318 0.772717
\(345\) 3.91548 0.210802
\(346\) 39.1461 2.10451
\(347\) 8.41628 0.451810 0.225905 0.974149i \(-0.427466\pi\)
0.225905 + 0.974149i \(0.427466\pi\)
\(348\) 0 0
\(349\) 22.1978 1.18822 0.594110 0.804384i \(-0.297504\pi\)
0.594110 + 0.804384i \(0.297504\pi\)
\(350\) −0.971071 −0.0519059
\(351\) −4.84939 −0.258841
\(352\) −0.599701 −0.0319642
\(353\) −6.18342 −0.329110 −0.164555 0.986368i \(-0.552619\pi\)
−0.164555 + 0.986368i \(0.552619\pi\)
\(354\) 25.6742 1.36457
\(355\) −9.07838 −0.481830
\(356\) −1.74888 −0.0926906
\(357\) 5.36069 0.283718
\(358\) −22.8371 −1.20698
\(359\) −5.05559 −0.266824 −0.133412 0.991061i \(-0.542593\pi\)
−0.133412 + 0.991061i \(0.542593\pi\)
\(360\) 0.196748 0.0103696
\(361\) 17.5958 0.926096
\(362\) 9.10504 0.478550
\(363\) 18.6576 0.979268
\(364\) −0.214614 −0.0112488
\(365\) −6.94441 −0.363487
\(366\) −8.09890 −0.423336
\(367\) −29.5402 −1.54199 −0.770994 0.636843i \(-0.780240\pi\)
−0.770994 + 0.636843i \(0.780240\pi\)
\(368\) −10.5418 −0.549531
\(369\) 0.0266620 0.00138797
\(370\) −2.38962 −0.124230
\(371\) −0.214614 −0.0111422
\(372\) 6.34017 0.328723
\(373\) 14.4124 0.746246 0.373123 0.927782i \(-0.378287\pi\)
0.373123 + 0.927782i \(0.378287\pi\)
\(374\) −2.22446 −0.115024
\(375\) 1.70928 0.0882666
\(376\) 2.83114 0.146005
\(377\) 0 0
\(378\) −5.10957 −0.262808
\(379\) −14.1340 −0.726013 −0.363007 0.931787i \(-0.618250\pi\)
−0.363007 + 0.931787i \(0.618250\pi\)
\(380\) −2.23287 −0.114544
\(381\) −35.7464 −1.83135
\(382\) 10.8188 0.553541
\(383\) −15.7815 −0.806397 −0.403199 0.915112i \(-0.632102\pi\)
−0.403199 + 0.915112i \(0.632102\pi\)
\(384\) 22.9132 1.16928
\(385\) 0.183417 0.00934782
\(386\) 27.4992 1.39967
\(387\) −0.447480 −0.0227467
\(388\) 5.86603 0.297803
\(389\) 13.8166 0.700529 0.350264 0.936651i \(-0.386092\pi\)
0.350264 + 0.936651i \(0.386092\pi\)
\(390\) 2.42469 0.122779
\(391\) −11.3874 −0.575883
\(392\) −16.5727 −0.837045
\(393\) −23.0205 −1.16123
\(394\) −9.38735 −0.472928
\(395\) 12.3896 0.623390
\(396\) 0.00841049 0.000422643 0
\(397\) 9.05172 0.454293 0.227146 0.973861i \(-0.427060\pi\)
0.227146 + 0.973861i \(0.427060\pi\)
\(398\) 15.0205 0.752911
\(399\) −6.52359 −0.326588
\(400\) −4.60197 −0.230098
\(401\) 19.7587 0.986704 0.493352 0.869830i \(-0.335772\pi\)
0.493352 + 0.869830i \(0.335772\pi\)
\(402\) −15.0205 −0.749155
\(403\) 9.26180 0.461363
\(404\) 4.52359 0.225057
\(405\) 8.75872 0.435224
\(406\) 0 0
\(407\) 0.451356 0.0223729
\(408\) 21.3295 1.05597
\(409\) 1.71769 0.0849341 0.0424670 0.999098i \(-0.486478\pi\)
0.0424670 + 0.999098i \(0.486478\pi\)
\(410\) −0.523590 −0.0258583
\(411\) −23.1650 −1.14264
\(412\) −2.90337 −0.143039
\(413\) 6.15676 0.302954
\(414\) 0.276349 0.0135818
\(415\) −2.78765 −0.136841
\(416\) −1.90110 −0.0932093
\(417\) 8.36683 0.409726
\(418\) 2.70701 0.132404
\(419\) −35.5318 −1.73584 −0.867922 0.496701i \(-0.834544\pi\)
−0.867922 + 0.496701i \(0.834544\pi\)
\(420\) 0.398032 0.0194220
\(421\) 12.0722 0.588365 0.294182 0.955749i \(-0.404953\pi\)
0.294182 + 0.955749i \(0.404953\pi\)
\(422\) 15.1857 0.739228
\(423\) −0.0883965 −0.00429798
\(424\) −0.853922 −0.0414701
\(425\) −4.97107 −0.241132
\(426\) 23.8843 1.15720
\(427\) −1.94214 −0.0939868
\(428\) 4.69860 0.227115
\(429\) −0.457980 −0.0221115
\(430\) 8.78765 0.423778
\(431\) 19.8310 0.955224 0.477612 0.878571i \(-0.341503\pi\)
0.477612 + 0.878571i \(0.341503\pi\)
\(432\) −24.2146 −1.16503
\(433\) −14.8143 −0.711931 −0.355965 0.934499i \(-0.615848\pi\)
−0.355965 + 0.934499i \(0.615848\pi\)
\(434\) 9.75872 0.468434
\(435\) 0 0
\(436\) 4.59129 0.219883
\(437\) 13.8576 0.662900
\(438\) 18.2700 0.872976
\(439\) −17.8576 −0.852298 −0.426149 0.904653i \(-0.640130\pi\)
−0.426149 + 0.904653i \(0.640130\pi\)
\(440\) 0.729794 0.0347916
\(441\) 0.517447 0.0246404
\(442\) −7.05172 −0.335416
\(443\) −33.5936 −1.59608 −0.798039 0.602606i \(-0.794129\pi\)
−0.798039 + 0.602606i \(0.794129\pi\)
\(444\) 0.979481 0.0464841
\(445\) 4.73820 0.224612
\(446\) −16.8865 −0.799601
\(447\) 21.4063 1.01248
\(448\) 3.80364 0.179705
\(449\) −7.07838 −0.334049 −0.167025 0.985953i \(-0.553416\pi\)
−0.167025 + 0.985953i \(0.553416\pi\)
\(450\) 0.120638 0.00568694
\(451\) 0.0988967 0.00465686
\(452\) −4.64074 −0.218282
\(453\) −12.9939 −0.610505
\(454\) 19.3112 0.906322
\(455\) 0.581449 0.0272588
\(456\) −25.9565 −1.21553
\(457\) −5.81658 −0.272088 −0.136044 0.990703i \(-0.543439\pi\)
−0.136044 + 0.990703i \(0.543439\pi\)
\(458\) 35.9565 1.68014
\(459\) −26.1568 −1.22089
\(460\) −0.845512 −0.0394222
\(461\) 32.3090 1.50478 0.752390 0.658718i \(-0.228901\pi\)
0.752390 + 0.658718i \(0.228901\pi\)
\(462\) −0.482553 −0.0224504
\(463\) 1.44134 0.0669846 0.0334923 0.999439i \(-0.489337\pi\)
0.0334923 + 0.999439i \(0.489337\pi\)
\(464\) 0 0
\(465\) −17.1773 −0.796577
\(466\) −19.1941 −0.889149
\(467\) 11.7503 0.543740 0.271870 0.962334i \(-0.412358\pi\)
0.271870 + 0.962334i \(0.412358\pi\)
\(468\) 0.0266620 0.00123245
\(469\) −3.60197 −0.166323
\(470\) 1.73594 0.0800728
\(471\) −42.5380 −1.96005
\(472\) 24.4969 1.12756
\(473\) −1.65983 −0.0763189
\(474\) −32.5958 −1.49718
\(475\) 6.04945 0.277568
\(476\) −1.15759 −0.0530582
\(477\) 0.0266620 0.00122077
\(478\) −21.1773 −0.968626
\(479\) −17.1689 −0.784465 −0.392233 0.919866i \(-0.628297\pi\)
−0.392233 + 0.919866i \(0.628297\pi\)
\(480\) 3.52586 0.160933
\(481\) 1.43084 0.0652405
\(482\) 22.5958 1.02921
\(483\) −2.47027 −0.112401
\(484\) −4.02893 −0.183133
\(485\) −15.8927 −0.721650
\(486\) 1.25338 0.0568547
\(487\) −4.10277 −0.185914 −0.0929572 0.995670i \(-0.529632\pi\)
−0.0929572 + 0.995670i \(0.529632\pi\)
\(488\) −7.72753 −0.349809
\(489\) 0.764867 0.0345885
\(490\) −10.1617 −0.459058
\(491\) −40.7708 −1.83996 −0.919981 0.391963i \(-0.871796\pi\)
−0.919981 + 0.391963i \(0.871796\pi\)
\(492\) 0.214614 0.00967556
\(493\) 0 0
\(494\) 8.58145 0.386098
\(495\) −0.0227863 −0.00102417
\(496\) 46.2472 2.07656
\(497\) 5.72753 0.256915
\(498\) 7.33403 0.328646
\(499\) −18.4703 −0.826843 −0.413421 0.910540i \(-0.635666\pi\)
−0.413421 + 0.910540i \(0.635666\pi\)
\(500\) −0.369102 −0.0165068
\(501\) −33.9565 −1.51707
\(502\) 23.7671 1.06078
\(503\) 21.4947 0.958400 0.479200 0.877706i \(-0.340927\pi\)
0.479200 + 0.877706i \(0.340927\pi\)
\(504\) −0.124128 −0.00552911
\(505\) −12.2557 −0.545369
\(506\) 1.02505 0.0455692
\(507\) 20.7687 0.922372
\(508\) 7.71912 0.342480
\(509\) 3.75872 0.166602 0.0833012 0.996524i \(-0.473454\pi\)
0.0833012 + 0.996524i \(0.473454\pi\)
\(510\) 13.0784 0.579120
\(511\) 4.38121 0.193813
\(512\) 13.6114 0.601546
\(513\) 31.8310 1.40537
\(514\) 9.66967 0.426511
\(515\) 7.86603 0.346619
\(516\) −3.60197 −0.158568
\(517\) −0.327887 −0.0144204
\(518\) 1.50761 0.0662404
\(519\) 43.4719 1.90820
\(520\) 2.31351 0.101454
\(521\) 12.8059 0.561037 0.280518 0.959849i \(-0.409494\pi\)
0.280518 + 0.959849i \(0.409494\pi\)
\(522\) 0 0
\(523\) −21.1278 −0.923855 −0.461928 0.886918i \(-0.652842\pi\)
−0.461928 + 0.886918i \(0.652842\pi\)
\(524\) 4.97107 0.217162
\(525\) −1.07838 −0.0470643
\(526\) 15.5090 0.676226
\(527\) 49.9565 2.17614
\(528\) −2.28685 −0.0995223
\(529\) −17.7526 −0.771851
\(530\) −0.523590 −0.0227433
\(531\) −0.764867 −0.0331924
\(532\) 1.40871 0.0610753
\(533\) 0.313511 0.0135797
\(534\) −12.4657 −0.539445
\(535\) −12.7298 −0.550357
\(536\) −14.3318 −0.619038
\(537\) −25.3607 −1.09439
\(538\) 43.3607 1.86941
\(539\) 1.91935 0.0826725
\(540\) −1.94214 −0.0835764
\(541\) −32.7382 −1.40753 −0.703763 0.710435i \(-0.748498\pi\)
−0.703763 + 0.710435i \(0.748498\pi\)
\(542\) 44.3318 1.90421
\(543\) 10.1112 0.433912
\(544\) −10.2542 −0.439646
\(545\) −12.4391 −0.532831
\(546\) −1.52973 −0.0654666
\(547\) −22.1073 −0.945240 −0.472620 0.881266i \(-0.656692\pi\)
−0.472620 + 0.881266i \(0.656692\pi\)
\(548\) 5.00227 0.213686
\(549\) 0.241276 0.0102974
\(550\) 0.447480 0.0190806
\(551\) 0 0
\(552\) −9.82887 −0.418344
\(553\) −7.81658 −0.332395
\(554\) −0.0410378 −0.00174353
\(555\) −2.65368 −0.112643
\(556\) −1.80674 −0.0766229
\(557\) −39.8720 −1.68943 −0.844715 0.535216i \(-0.820230\pi\)
−0.844715 + 0.535216i \(0.820230\pi\)
\(558\) −1.21235 −0.0513227
\(559\) −5.26180 −0.222550
\(560\) 2.90337 0.122690
\(561\) −2.47027 −0.104295
\(562\) 43.2085 1.82264
\(563\) −10.1217 −0.426578 −0.213289 0.976989i \(-0.568418\pi\)
−0.213289 + 0.976989i \(0.568418\pi\)
\(564\) −0.711543 −0.0299614
\(565\) 12.5730 0.528952
\(566\) 32.0372 1.34662
\(567\) −5.52586 −0.232064
\(568\) 22.7891 0.956209
\(569\) 24.4391 1.02454 0.512270 0.858825i \(-0.328805\pi\)
0.512270 + 0.858825i \(0.328805\pi\)
\(570\) −15.9155 −0.666626
\(571\) −28.2511 −1.18227 −0.591136 0.806572i \(-0.701320\pi\)
−0.591136 + 0.806572i \(0.701320\pi\)
\(572\) 0.0988967 0.00413508
\(573\) 12.0144 0.501908
\(574\) 0.330332 0.0137878
\(575\) 2.29072 0.0955298
\(576\) −0.472534 −0.0196889
\(577\) −46.1171 −1.91988 −0.959941 0.280202i \(-0.909598\pi\)
−0.959941 + 0.280202i \(0.909598\pi\)
\(578\) −11.8695 −0.493707
\(579\) 30.5380 1.26911
\(580\) 0 0
\(581\) 1.75872 0.0729642
\(582\) 41.8120 1.73317
\(583\) 0.0988967 0.00409588
\(584\) 17.4323 0.721352
\(585\) −0.0722347 −0.00298654
\(586\) −23.7191 −0.979828
\(587\) −0.715418 −0.0295285 −0.0147642 0.999891i \(-0.504700\pi\)
−0.0147642 + 0.999891i \(0.504700\pi\)
\(588\) 4.16517 0.171769
\(589\) −60.7936 −2.50496
\(590\) 15.0205 0.618385
\(591\) −10.4247 −0.428815
\(592\) 7.14465 0.293643
\(593\) 15.5441 0.638320 0.319160 0.947701i \(-0.396599\pi\)
0.319160 + 0.947701i \(0.396599\pi\)
\(594\) 2.35455 0.0966083
\(595\) 3.13624 0.128573
\(596\) −4.62249 −0.189344
\(597\) 16.6803 0.682681
\(598\) 3.24951 0.132882
\(599\) 9.59809 0.392167 0.196084 0.980587i \(-0.437178\pi\)
0.196084 + 0.980587i \(0.437178\pi\)
\(600\) −4.29072 −0.175168
\(601\) −6.81044 −0.277804 −0.138902 0.990306i \(-0.544357\pi\)
−0.138902 + 0.990306i \(0.544357\pi\)
\(602\) −5.54411 −0.225961
\(603\) 0.447480 0.0182228
\(604\) 2.80590 0.114171
\(605\) 10.9155 0.443777
\(606\) 32.2434 1.30980
\(607\) −31.6970 −1.28654 −0.643271 0.765639i \(-0.722423\pi\)
−0.643271 + 0.765639i \(0.722423\pi\)
\(608\) 12.4787 0.506077
\(609\) 0 0
\(610\) −4.73820 −0.191844
\(611\) −1.03943 −0.0420508
\(612\) 0.143810 0.00581318
\(613\) 1.20394 0.0486265 0.0243133 0.999704i \(-0.492260\pi\)
0.0243133 + 0.999704i \(0.492260\pi\)
\(614\) −43.7602 −1.76602
\(615\) −0.581449 −0.0234463
\(616\) −0.460425 −0.0185511
\(617\) 37.9337 1.52715 0.763577 0.645716i \(-0.223441\pi\)
0.763577 + 0.645716i \(0.223441\pi\)
\(618\) −20.6947 −0.832464
\(619\) 4.60424 0.185060 0.0925299 0.995710i \(-0.470505\pi\)
0.0925299 + 0.995710i \(0.470505\pi\)
\(620\) 3.70928 0.148968
\(621\) 12.0533 0.483683
\(622\) −30.2062 −1.21116
\(623\) −2.98932 −0.119765
\(624\) −7.24951 −0.290213
\(625\) 1.00000 0.0400000
\(626\) −35.2618 −1.40934
\(627\) 3.00614 0.120054
\(628\) 9.18568 0.366549
\(629\) 7.71769 0.307724
\(630\) −0.0761103 −0.00303231
\(631\) −8.41241 −0.334893 −0.167446 0.985881i \(-0.553552\pi\)
−0.167446 + 0.985881i \(0.553552\pi\)
\(632\) −31.1012 −1.23714
\(633\) 16.8638 0.670274
\(634\) 35.1857 1.39740
\(635\) −20.9132 −0.829915
\(636\) 0.214614 0.00851001
\(637\) 6.08452 0.241077
\(638\) 0 0
\(639\) −0.711543 −0.0281482
\(640\) 13.4052 0.529888
\(641\) −32.5380 −1.28517 −0.642586 0.766213i \(-0.722139\pi\)
−0.642586 + 0.766213i \(0.722139\pi\)
\(642\) 33.4908 1.32178
\(643\) −2.09293 −0.0825372 −0.0412686 0.999148i \(-0.513140\pi\)
−0.0412686 + 0.999148i \(0.513140\pi\)
\(644\) 0.533431 0.0210201
\(645\) 9.75872 0.384249
\(646\) 46.2868 1.82113
\(647\) 45.1955 1.77682 0.888410 0.459051i \(-0.151811\pi\)
0.888410 + 0.459051i \(0.151811\pi\)
\(648\) −21.9867 −0.863718
\(649\) −2.83710 −0.111366
\(650\) 1.41855 0.0556401
\(651\) 10.8371 0.424739
\(652\) −0.165166 −0.00646840
\(653\) 2.14834 0.0840712 0.0420356 0.999116i \(-0.486616\pi\)
0.0420356 + 0.999116i \(0.486616\pi\)
\(654\) 32.7259 1.27968
\(655\) −13.4680 −0.526238
\(656\) 1.56547 0.0611211
\(657\) −0.544287 −0.0212347
\(658\) −1.09520 −0.0426953
\(659\) 45.0843 1.75624 0.878118 0.478444i \(-0.158799\pi\)
0.878118 + 0.478444i \(0.158799\pi\)
\(660\) −0.183417 −0.00713952
\(661\) −36.3234 −1.41281 −0.706407 0.707806i \(-0.749685\pi\)
−0.706407 + 0.707806i \(0.749685\pi\)
\(662\) 37.0798 1.44115
\(663\) −7.83096 −0.304129
\(664\) 6.99773 0.271565
\(665\) −3.81658 −0.148001
\(666\) −0.187293 −0.00725746
\(667\) 0 0
\(668\) 7.33260 0.283707
\(669\) −18.7526 −0.725017
\(670\) −8.78765 −0.339497
\(671\) 0.894960 0.0345496
\(672\) −2.22446 −0.0858102
\(673\) −17.4719 −0.673491 −0.336746 0.941596i \(-0.609326\pi\)
−0.336746 + 0.941596i \(0.609326\pi\)
\(674\) 19.6826 0.758146
\(675\) 5.26180 0.202527
\(676\) −4.48482 −0.172493
\(677\) 40.0372 1.53875 0.769377 0.638796i \(-0.220567\pi\)
0.769377 + 0.638796i \(0.220567\pi\)
\(678\) −33.0784 −1.27037
\(679\) 10.0267 0.384788
\(680\) 12.4787 0.478535
\(681\) 21.4452 0.821782
\(682\) −4.49693 −0.172196
\(683\) 2.07611 0.0794402 0.0397201 0.999211i \(-0.487353\pi\)
0.0397201 + 0.999211i \(0.487353\pi\)
\(684\) −0.175007 −0.00669156
\(685\) −13.5525 −0.517815
\(686\) 13.2085 0.504302
\(687\) 39.9299 1.52342
\(688\) −26.2739 −1.00168
\(689\) 0.313511 0.0119438
\(690\) −6.02666 −0.229431
\(691\) 26.7070 1.01598 0.507991 0.861362i \(-0.330388\pi\)
0.507991 + 0.861362i \(0.330388\pi\)
\(692\) −9.38735 −0.356854
\(693\) 0.0143758 0.000546093 0
\(694\) −12.9542 −0.491737
\(695\) 4.89496 0.185676
\(696\) 0 0
\(697\) 1.69102 0.0640521
\(698\) −34.1666 −1.29322
\(699\) −21.3151 −0.806212
\(700\) 0.232866 0.00880150
\(701\) −21.9155 −0.827736 −0.413868 0.910337i \(-0.635823\pi\)
−0.413868 + 0.910337i \(0.635823\pi\)
\(702\) 7.46412 0.281715
\(703\) −9.39189 −0.354222
\(704\) −1.75276 −0.0660596
\(705\) 1.92777 0.0726038
\(706\) 9.51745 0.358194
\(707\) 7.73206 0.290794
\(708\) −6.15676 −0.231385
\(709\) −4.60811 −0.173061 −0.0865306 0.996249i \(-0.527578\pi\)
−0.0865306 + 0.996249i \(0.527578\pi\)
\(710\) 13.9733 0.524410
\(711\) 0.971071 0.0364180
\(712\) −11.8941 −0.445751
\(713\) −23.0205 −0.862125
\(714\) −8.25112 −0.308790
\(715\) −0.267938 −0.0100203
\(716\) 5.47641 0.204663
\(717\) −23.5174 −0.878275
\(718\) 7.78151 0.290403
\(719\) 6.80590 0.253817 0.126909 0.991914i \(-0.459494\pi\)
0.126909 + 0.991914i \(0.459494\pi\)
\(720\) −0.360692 −0.0134422
\(721\) −4.96266 −0.184819
\(722\) −27.0833 −1.00794
\(723\) 25.0928 0.933210
\(724\) −2.18342 −0.0811461
\(725\) 0 0
\(726\) −28.7175 −1.06581
\(727\) 26.9711 1.00030 0.500151 0.865938i \(-0.333278\pi\)
0.500151 + 0.865938i \(0.333278\pi\)
\(728\) −1.45959 −0.0540960
\(729\) 27.6681 1.02474
\(730\) 10.6888 0.395609
\(731\) −28.3812 −1.04972
\(732\) 1.94214 0.0717836
\(733\) 30.0638 1.11043 0.555216 0.831706i \(-0.312635\pi\)
0.555216 + 0.831706i \(0.312635\pi\)
\(734\) 45.4680 1.67825
\(735\) −11.2846 −0.416238
\(736\) 4.72526 0.174175
\(737\) 1.65983 0.0611405
\(738\) −0.0410378 −0.00151062
\(739\) −51.1422 −1.88130 −0.940648 0.339383i \(-0.889782\pi\)
−0.940648 + 0.339383i \(0.889782\pi\)
\(740\) 0.573039 0.0210653
\(741\) 9.52973 0.350084
\(742\) 0.330332 0.0121269
\(743\) −11.1857 −0.410363 −0.205181 0.978724i \(-0.565778\pi\)
−0.205181 + 0.978724i \(0.565778\pi\)
\(744\) 43.1194 1.58083
\(745\) 12.5236 0.458829
\(746\) −22.1834 −0.812193
\(747\) −0.218490 −0.00799413
\(748\) 0.533431 0.0195042
\(749\) 8.03120 0.293454
\(750\) −2.63090 −0.0960668
\(751\) 18.3630 0.670074 0.335037 0.942205i \(-0.391251\pi\)
0.335037 + 0.942205i \(0.391251\pi\)
\(752\) −5.19022 −0.189268
\(753\) 26.3935 0.961832
\(754\) 0 0
\(755\) −7.60197 −0.276664
\(756\) 1.22529 0.0445634
\(757\) −15.8927 −0.577630 −0.288815 0.957385i \(-0.593261\pi\)
−0.288815 + 0.957385i \(0.593261\pi\)
\(758\) 21.7548 0.790172
\(759\) 1.13833 0.0413186
\(760\) −15.1857 −0.550843
\(761\) 13.8843 0.503305 0.251652 0.967818i \(-0.419026\pi\)
0.251652 + 0.967818i \(0.419026\pi\)
\(762\) 55.0205 1.99318
\(763\) 7.84778 0.284109
\(764\) −2.59439 −0.0938619
\(765\) −0.389621 −0.0140868
\(766\) 24.2907 0.877660
\(767\) −8.99386 −0.324749
\(768\) −14.6576 −0.528909
\(769\) 35.4063 1.27678 0.638391 0.769712i \(-0.279600\pi\)
0.638391 + 0.769712i \(0.279600\pi\)
\(770\) −0.282314 −0.0101739
\(771\) 10.7382 0.386727
\(772\) −6.59439 −0.237337
\(773\) 0.488518 0.0175708 0.00878539 0.999961i \(-0.497203\pi\)
0.00878539 + 0.999961i \(0.497203\pi\)
\(774\) 0.688756 0.0247568
\(775\) −10.0494 −0.360987
\(776\) 39.8948 1.43214
\(777\) 1.67420 0.0600617
\(778\) −21.2663 −0.762435
\(779\) −2.05786 −0.0737304
\(780\) −0.581449 −0.0208192
\(781\) −2.63931 −0.0944419
\(782\) 17.5273 0.626775
\(783\) 0 0
\(784\) 30.3820 1.08507
\(785\) −24.8865 −0.888239
\(786\) 35.4329 1.26385
\(787\) −1.99159 −0.0709925 −0.0354962 0.999370i \(-0.511301\pi\)
−0.0354962 + 0.999370i \(0.511301\pi\)
\(788\) 2.25112 0.0801927
\(789\) 17.2228 0.613150
\(790\) −19.0700 −0.678479
\(791\) −7.93230 −0.282040
\(792\) 0.0571996 0.00203250
\(793\) 2.83710 0.100748
\(794\) −13.9323 −0.494439
\(795\) −0.581449 −0.0206219
\(796\) −3.60197 −0.127668
\(797\) 17.2702 0.611742 0.305871 0.952073i \(-0.401052\pi\)
0.305871 + 0.952073i \(0.401052\pi\)
\(798\) 10.0410 0.355449
\(799\) −5.60650 −0.198344
\(800\) 2.06278 0.0729303
\(801\) 0.371370 0.0131217
\(802\) −30.4124 −1.07390
\(803\) −2.01891 −0.0712458
\(804\) 3.60197 0.127032
\(805\) −1.44521 −0.0509371
\(806\) −14.2557 −0.502134
\(807\) 48.1522 1.69504
\(808\) 30.7649 1.08230
\(809\) 56.5068 1.98667 0.993336 0.115254i \(-0.0367680\pi\)
0.993336 + 0.115254i \(0.0367680\pi\)
\(810\) −13.4813 −0.473686
\(811\) −8.77924 −0.308281 −0.154140 0.988049i \(-0.549261\pi\)
−0.154140 + 0.988049i \(0.549261\pi\)
\(812\) 0 0
\(813\) 49.2306 1.72659
\(814\) −0.694722 −0.0243500
\(815\) 0.447480 0.0156745
\(816\) −39.1026 −1.36886
\(817\) 34.5380 1.20833
\(818\) −2.64384 −0.0924398
\(819\) 0.0455727 0.00159244
\(820\) 0.125559 0.00438470
\(821\) −28.1568 −0.982678 −0.491339 0.870969i \(-0.663492\pi\)
−0.491339 + 0.870969i \(0.663492\pi\)
\(822\) 35.6553 1.24362
\(823\) −20.7442 −0.723096 −0.361548 0.932353i \(-0.617752\pi\)
−0.361548 + 0.932353i \(0.617752\pi\)
\(824\) −19.7458 −0.687877
\(825\) 0.496928 0.0173008
\(826\) −9.47641 −0.329726
\(827\) 9.12783 0.317406 0.158703 0.987326i \(-0.449269\pi\)
0.158703 + 0.987326i \(0.449269\pi\)
\(828\) −0.0662693 −0.00230302
\(829\) −31.8576 −1.10646 −0.553230 0.833028i \(-0.686605\pi\)
−0.553230 + 0.833028i \(0.686605\pi\)
\(830\) 4.29072 0.148933
\(831\) −0.0455727 −0.00158090
\(832\) −5.55640 −0.192633
\(833\) 32.8188 1.13711
\(834\) −12.8781 −0.445933
\(835\) −19.8660 −0.687492
\(836\) −0.649149 −0.0224513
\(837\) −52.8781 −1.82774
\(838\) 54.6902 1.88924
\(839\) −27.4413 −0.947380 −0.473690 0.880692i \(-0.657078\pi\)
−0.473690 + 0.880692i \(0.657078\pi\)
\(840\) 2.70701 0.0934006
\(841\) 0 0
\(842\) −18.5814 −0.640359
\(843\) 47.9832 1.65263
\(844\) −3.64158 −0.125348
\(845\) 12.1506 0.417994
\(846\) 0.136059 0.00467780
\(847\) −6.88655 −0.236625
\(848\) 1.56547 0.0537583
\(849\) 35.5774 1.22101
\(850\) 7.65142 0.262441
\(851\) −3.55640 −0.121912
\(852\) −5.72753 −0.196222
\(853\) 56.0515 1.91917 0.959584 0.281422i \(-0.0908061\pi\)
0.959584 + 0.281422i \(0.0908061\pi\)
\(854\) 2.98932 0.102292
\(855\) 0.474142 0.0162153
\(856\) 31.9551 1.09220
\(857\) −6.08452 −0.207843 −0.103922 0.994585i \(-0.533139\pi\)
−0.103922 + 0.994585i \(0.533139\pi\)
\(858\) 0.704918 0.0240655
\(859\) −35.5936 −1.21444 −0.607218 0.794535i \(-0.707715\pi\)
−0.607218 + 0.794535i \(0.707715\pi\)
\(860\) −2.10731 −0.0718586
\(861\) 0.366835 0.0125017
\(862\) −30.5236 −1.03964
\(863\) −12.8287 −0.436694 −0.218347 0.975871i \(-0.570066\pi\)
−0.218347 + 0.975871i \(0.570066\pi\)
\(864\) 10.8539 0.369258
\(865\) 25.4329 0.864745
\(866\) 22.8020 0.774844
\(867\) −13.1812 −0.447655
\(868\) −2.34017 −0.0794306
\(869\) 3.60197 0.122188
\(870\) 0 0
\(871\) 5.26180 0.178289
\(872\) 31.2253 1.05742
\(873\) −1.24563 −0.0421583
\(874\) −21.3295 −0.721481
\(875\) −0.630898 −0.0213282
\(876\) −4.38121 −0.148027
\(877\) −1.50307 −0.0507551 −0.0253776 0.999678i \(-0.508079\pi\)
−0.0253776 + 0.999678i \(0.508079\pi\)
\(878\) 27.4863 0.927616
\(879\) −26.3402 −0.888432
\(880\) −1.33791 −0.0451008
\(881\) −23.4908 −0.791425 −0.395712 0.918375i \(-0.629502\pi\)
−0.395712 + 0.918375i \(0.629502\pi\)
\(882\) −0.796449 −0.0268178
\(883\) −29.0433 −0.977385 −0.488693 0.872456i \(-0.662526\pi\)
−0.488693 + 0.872456i \(0.662526\pi\)
\(884\) 1.69102 0.0568753
\(885\) 16.6803 0.560704
\(886\) 51.7068 1.73712
\(887\) 19.0700 0.640307 0.320153 0.947366i \(-0.396266\pi\)
0.320153 + 0.947366i \(0.396266\pi\)
\(888\) 6.66144 0.223543
\(889\) 13.1941 0.442516
\(890\) −7.29299 −0.244462
\(891\) 2.54638 0.0853068
\(892\) 4.04945 0.135586
\(893\) 6.82273 0.228314
\(894\) −32.9483 −1.10196
\(895\) −14.8371 −0.495950
\(896\) −8.45732 −0.282539
\(897\) 3.60859 0.120487
\(898\) 10.8950 0.363570
\(899\) 0 0
\(900\) −0.0289294 −0.000964314 0
\(901\) 1.69102 0.0563362
\(902\) −0.152221 −0.00506839
\(903\) −6.15676 −0.204884
\(904\) −31.5616 −1.04972
\(905\) 5.91548 0.196637
\(906\) 20.0000 0.664455
\(907\) −5.54023 −0.183960 −0.0919802 0.995761i \(-0.529320\pi\)
−0.0919802 + 0.995761i \(0.529320\pi\)
\(908\) −4.63090 −0.153682
\(909\) −0.960570 −0.0318601
\(910\) −0.894960 −0.0296676
\(911\) −53.2990 −1.76587 −0.882937 0.469492i \(-0.844437\pi\)
−0.882937 + 0.469492i \(0.844437\pi\)
\(912\) 47.5851 1.57570
\(913\) −0.810439 −0.0268216
\(914\) 8.95282 0.296133
\(915\) −5.26180 −0.173950
\(916\) −8.62249 −0.284895
\(917\) 8.49693 0.280593
\(918\) 40.2602 1.32878
\(919\) 37.5897 1.23997 0.619985 0.784614i \(-0.287139\pi\)
0.619985 + 0.784614i \(0.287139\pi\)
\(920\) −5.75031 −0.189582
\(921\) −48.5958 −1.60129
\(922\) −49.7296 −1.63776
\(923\) −8.36683 −0.275398
\(924\) 0.115718 0.00380683
\(925\) −1.55252 −0.0510465
\(926\) −2.21849 −0.0729041
\(927\) 0.616522 0.0202492
\(928\) 0 0
\(929\) 37.3197 1.22442 0.612209 0.790696i \(-0.290281\pi\)
0.612209 + 0.790696i \(0.290281\pi\)
\(930\) 26.4391 0.866971
\(931\) −39.9383 −1.30892
\(932\) 4.60281 0.150770
\(933\) −33.5441 −1.09818
\(934\) −18.0860 −0.591790
\(935\) −1.44521 −0.0472635
\(936\) 0.181328 0.00592688
\(937\) −22.8638 −0.746927 −0.373463 0.927645i \(-0.621830\pi\)
−0.373463 + 0.927645i \(0.621830\pi\)
\(938\) 5.54411 0.181022
\(939\) −39.1584 −1.27788
\(940\) −0.416283 −0.0135777
\(941\) 0.523590 0.0170686 0.00853428 0.999964i \(-0.497283\pi\)
0.00853428 + 0.999964i \(0.497283\pi\)
\(942\) 65.4740 2.13326
\(943\) −0.779243 −0.0253756
\(944\) −44.9093 −1.46167
\(945\) −3.31965 −0.107988
\(946\) 2.55479 0.0830633
\(947\) 10.0228 0.325697 0.162848 0.986651i \(-0.447932\pi\)
0.162848 + 0.986651i \(0.447932\pi\)
\(948\) 7.81658 0.253871
\(949\) −6.40012 −0.207757
\(950\) −9.31124 −0.302097
\(951\) 39.0738 1.26706
\(952\) −7.87277 −0.255158
\(953\) −8.15676 −0.264223 −0.132112 0.991235i \(-0.542176\pi\)
−0.132112 + 0.991235i \(0.542176\pi\)
\(954\) −0.0410378 −0.00132865
\(955\) 7.02893 0.227451
\(956\) 5.07838 0.164246
\(957\) 0 0
\(958\) 26.4261 0.853789
\(959\) 8.55025 0.276102
\(960\) 10.3051 0.332596
\(961\) 69.9914 2.25779
\(962\) −2.20233 −0.0710059
\(963\) −0.997733 −0.0321515
\(964\) −5.41855 −0.174520
\(965\) 17.8660 0.575128
\(966\) 3.80221 0.122334
\(967\) −15.7671 −0.507037 −0.253518 0.967331i \(-0.581588\pi\)
−0.253518 + 0.967331i \(0.581588\pi\)
\(968\) −27.4007 −0.880691
\(969\) 51.4017 1.65126
\(970\) 24.4619 0.785423
\(971\) 17.8804 0.573810 0.286905 0.957959i \(-0.407374\pi\)
0.286905 + 0.957959i \(0.407374\pi\)
\(972\) −0.300566 −0.00964065
\(973\) −3.08822 −0.0990037
\(974\) 6.31494 0.202344
\(975\) 1.57531 0.0504502
\(976\) 14.1666 0.453462
\(977\) −55.1071 −1.76303 −0.881517 0.472153i \(-0.843477\pi\)
−0.881517 + 0.472153i \(0.843477\pi\)
\(978\) −1.17727 −0.0376451
\(979\) 1.37751 0.0440255
\(980\) 2.43680 0.0778408
\(981\) −0.974946 −0.0311276
\(982\) 62.7540 2.00256
\(983\) 1.29687 0.0413637 0.0206818 0.999786i \(-0.493416\pi\)
0.0206818 + 0.999786i \(0.493416\pi\)
\(984\) 1.45959 0.0465300
\(985\) −6.09890 −0.194327
\(986\) 0 0
\(987\) −1.21622 −0.0387128
\(988\) −2.05786 −0.0654692
\(989\) 13.0784 0.415868
\(990\) 0.0350725 0.00111468
\(991\) −3.11942 −0.0990915 −0.0495458 0.998772i \(-0.515777\pi\)
−0.0495458 + 0.998772i \(0.515777\pi\)
\(992\) −20.7298 −0.658172
\(993\) 41.1773 1.30672
\(994\) −8.81575 −0.279618
\(995\) 9.75872 0.309372
\(996\) −1.75872 −0.0557273
\(997\) −30.2472 −0.957940 −0.478970 0.877831i \(-0.658990\pi\)
−0.478970 + 0.877831i \(0.658990\pi\)
\(998\) 28.4292 0.899912
\(999\) −8.16904 −0.258457
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 4205.2.a.e.1.2 3
29.28 even 2 145.2.a.d.1.2 3
87.86 odd 2 1305.2.a.o.1.2 3
116.115 odd 2 2320.2.a.s.1.1 3
145.28 odd 4 725.2.b.d.349.2 6
145.57 odd 4 725.2.b.d.349.5 6
145.144 even 2 725.2.a.d.1.2 3
203.202 odd 2 7105.2.a.p.1.2 3
232.115 odd 2 9280.2.a.bm.1.3 3
232.173 even 2 9280.2.a.bu.1.1 3
435.434 odd 2 6525.2.a.bh.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.2 3 29.28 even 2
725.2.a.d.1.2 3 145.144 even 2
725.2.b.d.349.2 6 145.28 odd 4
725.2.b.d.349.5 6 145.57 odd 4
1305.2.a.o.1.2 3 87.86 odd 2
2320.2.a.s.1.1 3 116.115 odd 2
4205.2.a.e.1.2 3 1.1 even 1 trivial
6525.2.a.bh.1.2 3 435.434 odd 2
7105.2.a.p.1.2 3 203.202 odd 2
9280.2.a.bm.1.3 3 232.115 odd 2
9280.2.a.bu.1.1 3 232.173 even 2