Properties

Label 145.2.a.d.1.2
Level $145$
Weight $2$
Character 145.1
Self dual yes
Analytic conductor $1.158$
Analytic rank $0$
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] = [145,2,Mod(1,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, 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("145.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.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: \(1.15783082931\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 145.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} +1.70928 q^{3} +0.369102 q^{4} -1.00000 q^{5} +2.63090 q^{6} +0.630898 q^{7} -2.51026 q^{8} -0.0783777 q^{9} +O(q^{10})\) \(q+1.53919 q^{2} +1.70928 q^{3} +0.369102 q^{4} -1.00000 q^{5} +2.63090 q^{6} +0.630898 q^{7} -2.51026 q^{8} -0.0783777 q^{9} -1.53919 q^{10} +0.290725 q^{11} +0.630898 q^{12} -0.921622 q^{13} +0.971071 q^{14} -1.70928 q^{15} -4.60197 q^{16} +4.97107 q^{17} -0.120638 q^{18} -6.04945 q^{19} -0.369102 q^{20} +1.07838 q^{21} +0.447480 q^{22} +2.29072 q^{23} -4.29072 q^{24} +1.00000 q^{25} -1.41855 q^{26} -5.26180 q^{27} +0.232866 q^{28} +1.00000 q^{29} -2.63090 q^{30} +10.0494 q^{31} -2.06278 q^{32} +0.496928 q^{33} +7.65142 q^{34} -0.630898 q^{35} -0.0289294 q^{36} +1.55252 q^{37} -9.31124 q^{38} -1.57531 q^{39} +2.51026 q^{40} +0.340173 q^{41} +1.65983 q^{42} -5.70928 q^{43} +0.107307 q^{44} +0.0783777 q^{45} +3.52586 q^{46} -1.12783 q^{47} -7.86603 q^{48} -6.60197 q^{49} +1.53919 q^{50} +8.49693 q^{51} -0.340173 q^{52} -0.340173 q^{53} -8.09890 q^{54} -0.290725 q^{55} -1.58372 q^{56} -10.3402 q^{57} +1.53919 q^{58} +9.75872 q^{59} -0.630898 q^{60} +3.07838 q^{61} +15.4680 q^{62} -0.0494483 q^{63} +6.02893 q^{64} +0.921622 q^{65} +0.764867 q^{66} -5.70928 q^{67} +1.83483 q^{68} +3.91548 q^{69} -0.971071 q^{70} +9.07838 q^{71} +0.196748 q^{72} -6.94441 q^{73} +2.38962 q^{74} +1.70928 q^{75} -2.23287 q^{76} +0.183417 q^{77} -2.42469 q^{78} +12.3896 q^{79} +4.60197 q^{80} -8.75872 q^{81} +0.523590 q^{82} +2.78765 q^{83} +0.398032 q^{84} -4.97107 q^{85} -8.78765 q^{86} +1.70928 q^{87} -0.729794 q^{88} +4.73820 q^{89} +0.120638 q^{90} -0.581449 q^{91} +0.845512 q^{92} +17.1773 q^{93} -1.73594 q^{94} +6.04945 q^{95} -3.52586 q^{96} -15.8927 q^{97} -10.1617 q^{98} -0.0227863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 4 q^{6} - 2 q^{7} + 9 q^{8} + 3 q^{9} - 3 q^{10} + 8 q^{11} - 2 q^{12} - 6 q^{13} - 12 q^{14} + 2 q^{15} + 5 q^{16} - 13 q^{18} - 5 q^{20} + 2 q^{22} + 14 q^{23} - 20 q^{24} + 3 q^{25} + 10 q^{26} - 8 q^{27} - 22 q^{28} + 3 q^{29} - 4 q^{30} + 12 q^{31} + 11 q^{32} - 16 q^{33} - 14 q^{34} + 2 q^{35} - 15 q^{36} + 4 q^{37} - 2 q^{38} + 16 q^{39} - 9 q^{40} - 10 q^{41} + 16 q^{42} - 10 q^{43} + 12 q^{44} - 3 q^{45} + 8 q^{46} + 18 q^{47} - 10 q^{48} - q^{49} + 3 q^{50} + 8 q^{51} + 10 q^{52} + 10 q^{53} + 12 q^{54} - 8 q^{55} - 32 q^{56} - 20 q^{57} + 3 q^{58} + 4 q^{59} + 2 q^{60} + 6 q^{61} + 14 q^{62} + 18 q^{63} + 33 q^{64} + 6 q^{65} + 12 q^{66} - 10 q^{67} - 36 q^{68} - 20 q^{69} + 12 q^{70} + 24 q^{71} - 3 q^{72} - 4 q^{73} - 22 q^{74} - 2 q^{75} + 16 q^{76} - 4 q^{77} - 28 q^{78} + 8 q^{79} - 5 q^{80} - q^{81} - 14 q^{82} - 2 q^{83} + 20 q^{84} - 16 q^{86} - 2 q^{87} + 38 q^{88} + 22 q^{89} + 13 q^{90} - 16 q^{91} + 22 q^{92} + 12 q^{93} - 8 q^{96} - 36 q^{97} + 23 q^{98} + 20 q^{99}+O(q^{100}) \) 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 \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 1.70928 0.986851 0.493425 0.869788i \(-0.335745\pi\)
0.493425 + 0.869788i \(0.335745\pi\)
\(4\) 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 \(-0.486045\pi\)
0.0438284 + 0.999039i \(0.486045\pi\)
\(12\) 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 \(-0.294039\pi\)
0.602831 + 0.797869i \(0.294039\pi\)
\(18\) −0.120638 −0.0284347
\(19\) −6.04945 −1.38784 −0.693919 0.720053i \(-0.744118\pi\)
−0.693919 + 0.720053i \(0.744118\pi\)
\(20\) −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
\(29\) 1.00000 0.185695
\(30\) −2.63090 −0.480334
\(31\) 10.0494 1.80493 0.902467 0.430759i \(-0.141754\pi\)
0.902467 + 0.430759i \(0.141754\pi\)
\(32\) −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 \(-0.459267\pi\)
0.127616 + 0.991824i \(0.459267\pi\)
\(38\) −9.31124 −1.51048
\(39\) −1.57531 −0.252251
\(40\) 2.51026 0.396907
\(41\) 0.340173 0.0531261 0.0265630 0.999647i \(-0.491544\pi\)
0.0265630 + 0.999647i \(0.491544\pi\)
\(42\) 1.65983 0.256117
\(43\) −5.70928 −0.870656 −0.435328 0.900272i \(-0.643368\pi\)
−0.435328 + 0.900272i \(0.643368\pi\)
\(44\) 0.107307 0.0161772
\(45\) 0.0783777 0.0116839
\(46\) 3.52586 0.519859
\(47\) −1.12783 −0.164510 −0.0822552 0.996611i \(-0.526212\pi\)
−0.0822552 + 0.996611i \(0.526212\pi\)
\(48\) −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
\(58\) 1.53919 0.202105
\(59\) 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 \(-0.436856\pi\)
0.197073 + 0.980389i \(0.436856\pi\)
\(62\) 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 \(-0.633213\pi\)
−0.406391 + 0.913699i \(0.633213\pi\)
\(74\) 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 \(-0.254531\pi\)
0.696971 + 0.717100i \(0.254531\pi\)
\(80\) 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
\(87\) 1.70928 0.183254
\(88\) −0.729794 −0.0777963
\(89\) 4.73820 0.502249 0.251124 0.967955i \(-0.419200\pi\)
0.251124 + 0.967955i \(0.419200\pi\)
\(90\) 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 \(-0.798818\pi\)
−0.806829 + 0.590785i \(0.798818\pi\)
\(98\) −10.1617 −1.02648
\(99\) −0.0227863 −0.00229011
\(100\) 0.369102 0.0369102
\(101\) −12.2557 −1.21948 −0.609741 0.792600i \(-0.708727\pi\)
−0.609741 + 0.792600i \(0.708727\pi\)
\(102\) 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 \(-0.298581\pi\)
0.591386 + 0.806389i \(0.298581\pi\)
\(114\) −15.9155 −1.49062
\(115\) −2.29072 −0.213611
\(116\) 0.369102 0.0342703
\(117\) 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 \(-0.878365\pi\)
−0.927874 + 0.372895i \(0.878365\pi\)
\(128\) 13.4052 1.18487
\(129\) −9.75872 −0.859208
\(130\) 1.41855 0.124415
\(131\) −13.4680 −1.17670 −0.588352 0.808605i \(-0.700223\pi\)
−0.588352 + 0.808605i \(0.700223\pi\)
\(132\) 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 \(-0.696531\pi\)
−0.578935 + 0.815374i \(0.696531\pi\)
\(138\) 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
\(145\) −1.00000 −0.0830455
\(146\) −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 \(-0.962535\pi\)
−0.993081 + 0.117428i \(0.962535\pi\)
\(158\) 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 \(-0.494421\pi\)
0.0175247 + 0.999846i \(0.494421\pi\)
\(164\) 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 \(0.917767\pi\)
\(174\) 2.63090 0.199448
\(175\) 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 \(-0.418156\pi\)
0.254298 + 0.967126i \(0.418156\pi\)
\(192\) 10.3051 0.743707
\(193\) 17.8660 1.28603 0.643013 0.765856i \(-0.277684\pi\)
0.643013 + 0.765856i \(0.277684\pi\)
\(194\) −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
\(203\) 0.630898 0.0442803
\(204\) 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 \(-0.389707\pi\)
0.339603 + 0.940569i \(0.389707\pi\)
\(212\) −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 \(-0.219327\pi\)
0.771859 + 0.635794i \(0.219327\pi\)
\(230\) −3.52586 −0.232488
\(231\) 0.313511 0.0206275
\(232\) −2.51026 −0.164807
\(233\) 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 \(-0.337973\pi\)
0.487324 + 0.873221i \(0.337973\pi\)
\(252\) −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
\(261\) −0.0783777 −0.00485146
\(262\) −20.7298 −1.28069
\(263\) 10.0761 0.621320 0.310660 0.950521i \(-0.399450\pi\)
0.310660 + 0.950521i \(0.399450\pi\)
\(264\) −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 \(-0.171203\pi\)
0.858812 + 0.512291i \(0.171203\pi\)
\(270\) 8.09890 0.492883
\(271\) 28.8020 1.74960 0.874799 0.484485i \(-0.160993\pi\)
0.874799 + 0.484485i \(0.160993\pi\)
\(272\) −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
\(290\) −1.53919 −0.0903843
\(291\) −27.1650 −1.59244
\(292\) −2.56320 −0.150000
\(293\) −15.4101 −0.900270 −0.450135 0.892961i \(-0.648624\pi\)
−0.450135 + 0.892961i \(0.648624\pi\)
\(294\) −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 \(-0.801246\pi\)
−0.811312 + 0.584614i \(0.801246\pi\)
\(308\) 0.0676998 0.00385756
\(309\) −13.4452 −0.764871
\(310\) −15.4680 −0.878523
\(311\) −19.6248 −1.11282 −0.556409 0.830909i \(-0.687821\pi\)
−0.556409 + 0.830909i \(0.687821\pi\)
\(312\) 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 \(-0.278118\pi\)
0.641970 + 0.766730i \(0.278118\pi\)
\(318\) −0.894960 −0.0501869
\(319\) 0.290725 0.0162775
\(320\) −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 \(-0.269680\pi\)
0.662066 + 0.749445i \(0.269680\pi\)
\(332\) 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 \(-0.386761\pi\)
0.348294 + 0.937385i \(0.386761\pi\)
\(338\) −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 \(0.427466\pi\)
\(348\) 0.630898 0.0338197
\(349\) 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 \(-0.457407\pi\)
0.133412 + 0.991061i \(0.457407\pi\)
\(360\) −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 \(-0.219760\pi\)
0.770994 + 0.636843i \(0.219760\pi\)
\(368\) −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
\(377\) −0.921622 −0.0474660
\(378\) −5.10957 −0.262808
\(379\) 14.1340 0.726013 0.363007 0.931787i \(-0.381750\pi\)
0.363007 + 0.931787i \(0.381750\pi\)
\(380\) 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 \(-0.613908\pi\)
−0.350264 + 0.936651i \(0.613908\pi\)
\(390\) 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
\(406\) 0.971071 0.0481934
\(407\) 0.451356 0.0223729
\(408\) −21.3295 −1.05597
\(409\) −1.71769 −0.0849341 −0.0424670 0.999098i \(-0.513522\pi\)
−0.0424670 + 0.999098i \(0.513522\pi\)
\(410\) −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 \(-0.595047\pi\)
−0.294182 + 0.955749i \(0.595047\pi\)
\(422\) 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 \(-0.384152\pi\)
0.355965 + 0.934499i \(0.384152\pi\)
\(434\) 9.75872 0.468434
\(435\) −1.70928 −0.0819535
\(436\) 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 \(-0.205871\pi\)
0.798039 + 0.602606i \(0.205871\pi\)
\(444\) 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 \(-0.446584\pi\)
0.167025 + 0.985953i \(0.446584\pi\)
\(450\) −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 \(-0.771099\pi\)
−0.752390 + 0.658718i \(0.771099\pi\)
\(462\) 0.482553 0.0224504
\(463\) 1.44134 0.0669846 0.0334923 0.999439i \(-0.489337\pi\)
0.0334923 + 0.999439i \(0.489337\pi\)
\(464\) −4.60197 −0.213641
\(465\) −17.1773 −0.796577
\(466\) 19.1941 0.889149
\(467\) −11.7503 −0.543740 −0.271870 0.962334i \(-0.587642\pi\)
−0.271870 + 0.962334i \(0.587642\pi\)
\(468\) 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 \(-0.371703\pi\)
0.392233 + 0.919866i \(0.371703\pi\)
\(480\) 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 \(-0.128204\pi\)
0.919981 + 0.391963i \(0.128204\pi\)
\(492\) 0.214614 0.00967556
\(493\) 4.97107 0.223886
\(494\) 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 \(-0.659073\pi\)
−0.479200 + 0.877706i \(0.659073\pi\)
\(504\) 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 \(0.409494\pi\)
\(522\) −0.120638 −0.00528019
\(523\) −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 \(-0.251502\pi\)
0.703763 + 0.710435i \(0.251502\pi\)
\(542\) 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
\(551\) −6.04945 −0.257715
\(552\) −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 \(-0.431582\pi\)
0.213289 + 0.976989i \(0.431582\pi\)
\(564\) −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 \(-0.671195\pi\)
−0.512270 + 0.858825i \(0.671195\pi\)
\(570\) 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 \(-0.0904015\pi\)
0.959941 + 0.280202i \(0.0904015\pi\)
\(578\) 11.8695 0.493707
\(579\) 30.5380 1.26911
\(580\) −0.369102 −0.0153261
\(581\) 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 \(-0.562822\pi\)
−0.196084 + 0.980587i \(0.562822\pi\)
\(600\) −4.29072 −0.175168
\(601\) 6.81044 0.277804 0.138902 0.990306i \(-0.455643\pi\)
0.138902 + 0.990306i \(0.455643\pi\)
\(602\) −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 \(-0.277577\pi\)
0.643271 + 0.765639i \(0.277577\pi\)
\(608\) 12.4787 0.506077
\(609\) 1.07838 0.0436981
\(610\) −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 \(-0.776559\pi\)
−0.763577 + 0.645716i \(0.776559\pi\)
\(618\) −20.6947 −0.832464
\(619\) −4.60424 −0.185060 −0.0925299 0.995710i \(-0.529495\pi\)
−0.0925299 + 0.995710i \(0.529495\pi\)
\(620\) −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
\(638\) 0.447480 0.0177159
\(639\) −0.711543 −0.0281482
\(640\) −13.4052 −0.529888
\(641\) 32.5380 1.28517 0.642586 0.766213i \(-0.277861\pi\)
0.642586 + 0.766213i \(0.277861\pi\)
\(642\) 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 \(-0.513384\pi\)
−0.0420356 + 0.999116i \(0.513384\pi\)
\(654\) 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 \(-0.841201\pi\)
−0.878118 + 0.478444i \(0.841201\pi\)
\(660\) −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
\(667\) 2.29072 0.0886972
\(668\) 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 \(-0.779433\pi\)
−0.769377 + 0.638796i \(0.779433\pi\)
\(678\) 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
\(696\) −4.29072 −0.162639
\(697\) 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
\(725\) 1.00000 0.0371391
\(726\) −28.7175 −1.06581
\(727\) −26.9711 −1.00030 −0.500151 0.865938i \(-0.666722\pi\)
−0.500151 + 0.865938i \(0.666722\pi\)
\(728\) 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 \(-0.687365\pi\)
−0.555216 + 0.831706i \(0.687365\pi\)
\(734\) 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 \(-0.110218\pi\)
0.940648 + 0.339383i \(0.110218\pi\)
\(740\) −0.573039 −0.0210653
\(741\) 9.52973 0.350084
\(742\) −0.330332 −0.0121269
\(743\) 11.1857 0.410363 0.205181 0.978724i \(-0.434222\pi\)
0.205181 + 0.978724i \(0.434222\pi\)
\(744\) −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 \(-0.608749\pi\)
−0.335037 + 0.942205i \(0.608749\pi\)
\(752\) 5.19022 0.189268
\(753\) 26.3935 0.961832
\(754\) −1.41855 −0.0516606
\(755\) −7.60197 −0.276664
\(756\) −1.22529 −0.0445634
\(757\) 15.8927 0.577630 0.288815 0.957385i \(-0.406739\pi\)
0.288815 + 0.957385i \(0.406739\pi\)
\(758\) 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 \(-0.720400\pi\)
−0.638391 + 0.769712i \(0.720400\pi\)
\(770\) −0.282314 −0.0101739
\(771\) −10.7382 −0.386727
\(772\) 6.59439 0.237337
\(773\) −0.488518 −0.0175708 −0.00878539 0.999961i \(-0.502797\pi\)
−0.00878539 + 0.999961i \(0.502797\pi\)
\(774\) 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
\(783\) −5.26180 −0.188041
\(784\) 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 \(-0.598948\pi\)
−0.305871 + 0.952073i \(0.598948\pi\)
\(798\) −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 \(-0.963232\pi\)
−0.993336 + 0.115254i \(0.963232\pi\)
\(810\) 13.4813 0.473686
\(811\) −8.77924 −0.308281 −0.154140 0.988049i \(-0.549261\pi\)
−0.154140 + 0.988049i \(0.549261\pi\)
\(812\) 0.232866 0.00817199
\(813\) 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 \(-0.382248\pi\)
0.361548 + 0.932353i \(0.382248\pi\)
\(824\) 19.7458 0.687877
\(825\) 0.496928 0.0173008
\(826\) 9.47641 0.329726
\(827\) −9.12783 −0.317406 −0.158703 0.987326i \(-0.550731\pi\)
−0.158703 + 0.987326i \(0.550731\pi\)
\(828\) −0.0662693 −0.00230302
\(829\) 31.8576 1.10646 0.553230 0.833028i \(-0.313395\pi\)
0.553230 + 0.833028i \(0.313395\pi\)
\(830\) −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 \(-0.342922\pi\)
0.473690 + 0.880692i \(0.342922\pi\)
\(840\) 2.70701 0.0934006
\(841\) 1.00000 0.0344828
\(842\) −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 \(-0.909194\pi\)
−0.959584 + 0.281422i \(0.909194\pi\)
\(854\) 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 \(-0.292285\pi\)
0.607218 + 0.794535i \(0.292285\pi\)
\(860\) 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
\(870\) −2.63090 −0.0891958
\(871\) 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 \(-0.370498\pi\)
0.395712 + 0.918375i \(0.370498\pi\)
\(882\) 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 \(-0.603734\pi\)
−0.320153 + 0.947366i \(0.603734\pi\)
\(888\) −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
\(899\) 10.0494 0.335168
\(900\) −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 \(-0.470680\pi\)
0.0919802 + 0.995761i \(0.470680\pi\)
\(908\) −4.63090 −0.153682
\(909\) 0.960570 0.0318601
\(910\) 0.894960 0.0296676
\(911\) 53.2990 1.76587 0.882937 0.469492i \(-0.155563\pi\)
0.882937 + 0.469492i \(0.155563\pi\)
\(912\) 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
\(928\) −2.06278 −0.0677140
\(929\) 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 \(-0.552068\pi\)
−0.162848 + 0.986651i \(0.552068\pi\)
\(948\) 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
\(957\) 0.496928 0.0160634
\(958\) 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 \(-0.418412\pi\)
0.253518 + 0.967331i \(0.418412\pi\)
\(968\) 27.4007 0.880691
\(969\) −51.4017 −1.65126
\(970\) 24.4619 0.785423
\(971\) −17.8804 −0.573810 −0.286905 0.957959i \(-0.592626\pi\)
−0.286905 + 0.957959i \(0.592626\pi\)
\(972\) 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 \(-0.506584\pi\)
−0.0206818 + 0.999786i \(0.506584\pi\)
\(984\) −1.45959 −0.0465300
\(985\) −6.09890 −0.194327
\(986\) 7.65142 0.243671
\(987\) −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 \(-0.341010\pi\)
0.478970 + 0.877831i \(0.341010\pi\)
\(998\) −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 145.2.a.d.1.2 3
3.2 odd 2 1305.2.a.o.1.2 3
4.3 odd 2 2320.2.a.s.1.1 3
5.2 odd 4 725.2.b.d.349.5 6
5.3 odd 4 725.2.b.d.349.2 6
5.4 even 2 725.2.a.d.1.2 3
7.6 odd 2 7105.2.a.p.1.2 3
8.3 odd 2 9280.2.a.bm.1.3 3
8.5 even 2 9280.2.a.bu.1.1 3
15.14 odd 2 6525.2.a.bh.1.2 3
29.28 even 2 4205.2.a.e.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.d.1.2 3 1.1 even 1 trivial
725.2.a.d.1.2 3 5.4 even 2
725.2.b.d.349.2 6 5.3 odd 4
725.2.b.d.349.5 6 5.2 odd 4
1305.2.a.o.1.2 3 3.2 odd 2
2320.2.a.s.1.1 3 4.3 odd 2
4205.2.a.e.1.2 3 29.28 even 2
6525.2.a.bh.1.2 3 15.14 odd 2
7105.2.a.p.1.2 3 7.6 odd 2
9280.2.a.bm.1.3 3 8.3 odd 2
9280.2.a.bu.1.1 3 8.5 even 2