Properties

Label 507.2.a.d.1.1
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(1,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, 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("507.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.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: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -0.561553 q^{5} -2.56155 q^{6} +3.56155 q^{7} -6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} -0.561553 q^{5} -2.56155 q^{6} +3.56155 q^{7} -6.56155 q^{8} +1.00000 q^{9} +1.43845 q^{10} +2.00000 q^{11} +4.56155 q^{12} -9.12311 q^{14} -0.561553 q^{15} +7.68466 q^{16} +2.56155 q^{17} -2.56155 q^{18} +1.12311 q^{19} -2.56155 q^{20} +3.56155 q^{21} -5.12311 q^{22} +2.00000 q^{23} -6.56155 q^{24} -4.68466 q^{25} +1.00000 q^{27} +16.2462 q^{28} -5.68466 q^{29} +1.43845 q^{30} +1.56155 q^{31} -6.56155 q^{32} +2.00000 q^{33} -6.56155 q^{34} -2.00000 q^{35} +4.56155 q^{36} -3.43845 q^{37} -2.87689 q^{38} +3.68466 q^{40} -2.56155 q^{41} -9.12311 q^{42} +0.438447 q^{43} +9.12311 q^{44} -0.561553 q^{45} -5.12311 q^{46} +8.24621 q^{47} +7.68466 q^{48} +5.68466 q^{49} +12.0000 q^{50} +2.56155 q^{51} +11.6847 q^{53} -2.56155 q^{54} -1.12311 q^{55} -23.3693 q^{56} +1.12311 q^{57} +14.5616 q^{58} +11.1231 q^{59} -2.56155 q^{60} +12.1231 q^{61} -4.00000 q^{62} +3.56155 q^{63} +1.43845 q^{64} -5.12311 q^{66} -0.438447 q^{67} +11.6847 q^{68} +2.00000 q^{69} +5.12311 q^{70} -14.0000 q^{71} -6.56155 q^{72} +1.87689 q^{73} +8.80776 q^{74} -4.68466 q^{75} +5.12311 q^{76} +7.12311 q^{77} +9.56155 q^{79} -4.31534 q^{80} +1.00000 q^{81} +6.56155 q^{82} +9.12311 q^{83} +16.2462 q^{84} -1.43845 q^{85} -1.12311 q^{86} -5.68466 q^{87} -13.1231 q^{88} -13.1231 q^{89} +1.43845 q^{90} +9.12311 q^{92} +1.56155 q^{93} -21.1231 q^{94} -0.630683 q^{95} -6.56155 q^{96} +4.43845 q^{97} -14.5616 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} + 3 q^{7} - 9 q^{8} + 2 q^{9} + 7 q^{10} + 4 q^{11} + 5 q^{12} - 10 q^{14} + 3 q^{15} + 3 q^{16} + q^{17} - q^{18} - 6 q^{19} - q^{20} + 3 q^{21} - 2 q^{22} + 4 q^{23} - 9 q^{24} + 3 q^{25} + 2 q^{27} + 16 q^{28} + q^{29} + 7 q^{30} - q^{31} - 9 q^{32} + 4 q^{33} - 9 q^{34} - 4 q^{35} + 5 q^{36} - 11 q^{37} - 14 q^{38} - 5 q^{40} - q^{41} - 10 q^{42} + 5 q^{43} + 10 q^{44} + 3 q^{45} - 2 q^{46} + 3 q^{48} - q^{49} + 24 q^{50} + q^{51} + 11 q^{53} - q^{54} + 6 q^{55} - 22 q^{56} - 6 q^{57} + 25 q^{58} + 14 q^{59} - q^{60} + 16 q^{61} - 8 q^{62} + 3 q^{63} + 7 q^{64} - 2 q^{66} - 5 q^{67} + 11 q^{68} + 4 q^{69} + 2 q^{70} - 28 q^{71} - 9 q^{72} + 12 q^{73} - 3 q^{74} + 3 q^{75} + 2 q^{76} + 6 q^{77} + 15 q^{79} - 21 q^{80} + 2 q^{81} + 9 q^{82} + 10 q^{83} + 16 q^{84} - 7 q^{85} + 6 q^{86} + q^{87} - 18 q^{88} - 18 q^{89} + 7 q^{90} + 10 q^{92} - q^{93} - 34 q^{94} - 26 q^{95} - 9 q^{96} + 13 q^{97} - 25 q^{98} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56155 −1.81129 −0.905646 0.424035i \(-0.860613\pi\)
−0.905646 + 0.424035i \(0.860613\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.56155 2.28078
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) −2.56155 −1.04575
\(7\) 3.56155 1.34614 0.673070 0.739579i \(-0.264975\pi\)
0.673070 + 0.739579i \(0.264975\pi\)
\(8\) −6.56155 −2.31986
\(9\) 1.00000 0.333333
\(10\) 1.43845 0.454877
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 4.56155 1.31681
\(13\) 0 0
\(14\) −9.12311 −2.43825
\(15\) −0.561553 −0.144992
\(16\) 7.68466 1.92116
\(17\) 2.56155 0.621268 0.310634 0.950530i \(-0.399459\pi\)
0.310634 + 0.950530i \(0.399459\pi\)
\(18\) −2.56155 −0.603764
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) −2.56155 −0.572781
\(21\) 3.56155 0.777195
\(22\) −5.12311 −1.09225
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −6.56155 −1.33937
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 16.2462 3.07025
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 1.43845 0.262623
\(31\) 1.56155 0.280463 0.140232 0.990119i \(-0.455215\pi\)
0.140232 + 0.990119i \(0.455215\pi\)
\(32\) −6.56155 −1.15993
\(33\) 2.00000 0.348155
\(34\) −6.56155 −1.12530
\(35\) −2.00000 −0.338062
\(36\) 4.56155 0.760259
\(37\) −3.43845 −0.565277 −0.282639 0.959226i \(-0.591210\pi\)
−0.282639 + 0.959226i \(0.591210\pi\)
\(38\) −2.87689 −0.466694
\(39\) 0 0
\(40\) 3.68466 0.582596
\(41\) −2.56155 −0.400047 −0.200024 0.979791i \(-0.564102\pi\)
−0.200024 + 0.979791i \(0.564102\pi\)
\(42\) −9.12311 −1.40773
\(43\) 0.438447 0.0668626 0.0334313 0.999441i \(-0.489357\pi\)
0.0334313 + 0.999441i \(0.489357\pi\)
\(44\) 9.12311 1.37536
\(45\) −0.561553 −0.0837114
\(46\) −5.12311 −0.755361
\(47\) 8.24621 1.20283 0.601417 0.798935i \(-0.294603\pi\)
0.601417 + 0.798935i \(0.294603\pi\)
\(48\) 7.68466 1.10918
\(49\) 5.68466 0.812094
\(50\) 12.0000 1.69706
\(51\) 2.56155 0.358689
\(52\) 0 0
\(53\) 11.6847 1.60501 0.802506 0.596645i \(-0.203500\pi\)
0.802506 + 0.596645i \(0.203500\pi\)
\(54\) −2.56155 −0.348583
\(55\) −1.12311 −0.151440
\(56\) −23.3693 −3.12286
\(57\) 1.12311 0.148759
\(58\) 14.5616 1.91203
\(59\) 11.1231 1.44811 0.724053 0.689745i \(-0.242277\pi\)
0.724053 + 0.689745i \(0.242277\pi\)
\(60\) −2.56155 −0.330695
\(61\) 12.1231 1.55220 0.776102 0.630607i \(-0.217194\pi\)
0.776102 + 0.630607i \(0.217194\pi\)
\(62\) −4.00000 −0.508001
\(63\) 3.56155 0.448713
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) −5.12311 −0.630611
\(67\) −0.438447 −0.0535648 −0.0267824 0.999641i \(-0.508526\pi\)
−0.0267824 + 0.999641i \(0.508526\pi\)
\(68\) 11.6847 1.41697
\(69\) 2.00000 0.240772
\(70\) 5.12311 0.612328
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) −6.56155 −0.773286
\(73\) 1.87689 0.219674 0.109837 0.993950i \(-0.464967\pi\)
0.109837 + 0.993950i \(0.464967\pi\)
\(74\) 8.80776 1.02388
\(75\) −4.68466 −0.540938
\(76\) 5.12311 0.587661
\(77\) 7.12311 0.811753
\(78\) 0 0
\(79\) 9.56155 1.07576 0.537879 0.843022i \(-0.319226\pi\)
0.537879 + 0.843022i \(0.319226\pi\)
\(80\) −4.31534 −0.482470
\(81\) 1.00000 0.111111
\(82\) 6.56155 0.724602
\(83\) 9.12311 1.00139 0.500695 0.865624i \(-0.333078\pi\)
0.500695 + 0.865624i \(0.333078\pi\)
\(84\) 16.2462 1.77261
\(85\) −1.43845 −0.156022
\(86\) −1.12311 −0.121108
\(87\) −5.68466 −0.609459
\(88\) −13.1231 −1.39893
\(89\) −13.1231 −1.39105 −0.695523 0.718504i \(-0.744827\pi\)
−0.695523 + 0.718504i \(0.744827\pi\)
\(90\) 1.43845 0.151626
\(91\) 0 0
\(92\) 9.12311 0.951150
\(93\) 1.56155 0.161925
\(94\) −21.1231 −2.17868
\(95\) −0.630683 −0.0647067
\(96\) −6.56155 −0.669686
\(97\) 4.43845 0.450656 0.225328 0.974283i \(-0.427655\pi\)
0.225328 + 0.974283i \(0.427655\pi\)
\(98\) −14.5616 −1.47094
\(99\) 2.00000 0.201008
\(100\) −21.3693 −2.13693
\(101\) −3.43845 −0.342138 −0.171069 0.985259i \(-0.554722\pi\)
−0.171069 + 0.985259i \(0.554722\pi\)
\(102\) −6.56155 −0.649691
\(103\) −7.56155 −0.745062 −0.372531 0.928020i \(-0.621510\pi\)
−0.372531 + 0.928020i \(0.621510\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) −29.9309 −2.90714
\(107\) 8.24621 0.797191 0.398596 0.917127i \(-0.369498\pi\)
0.398596 + 0.917127i \(0.369498\pi\)
\(108\) 4.56155 0.438936
\(109\) −17.8078 −1.70567 −0.852837 0.522177i \(-0.825120\pi\)
−0.852837 + 0.522177i \(0.825120\pi\)
\(110\) 2.87689 0.274301
\(111\) −3.43845 −0.326363
\(112\) 27.3693 2.58616
\(113\) −14.8078 −1.39300 −0.696499 0.717558i \(-0.745260\pi\)
−0.696499 + 0.717558i \(0.745260\pi\)
\(114\) −2.87689 −0.269446
\(115\) −1.12311 −0.104730
\(116\) −25.9309 −2.40762
\(117\) 0 0
\(118\) −28.4924 −2.62294
\(119\) 9.12311 0.836314
\(120\) 3.68466 0.336362
\(121\) −7.00000 −0.636364
\(122\) −31.0540 −2.81149
\(123\) −2.56155 −0.230967
\(124\) 7.12311 0.639674
\(125\) 5.43845 0.486430
\(126\) −9.12311 −0.812751
\(127\) 9.56155 0.848451 0.424225 0.905557i \(-0.360546\pi\)
0.424225 + 0.905557i \(0.360546\pi\)
\(128\) 9.43845 0.834249
\(129\) 0.438447 0.0386031
\(130\) 0 0
\(131\) −17.3693 −1.51756 −0.758782 0.651345i \(-0.774205\pi\)
−0.758782 + 0.651345i \(0.774205\pi\)
\(132\) 9.12311 0.794064
\(133\) 4.00000 0.346844
\(134\) 1.12311 0.0970215
\(135\) −0.561553 −0.0483308
\(136\) −16.8078 −1.44125
\(137\) 1.43845 0.122895 0.0614474 0.998110i \(-0.480428\pi\)
0.0614474 + 0.998110i \(0.480428\pi\)
\(138\) −5.12311 −0.436108
\(139\) 10.9309 0.927144 0.463572 0.886059i \(-0.346567\pi\)
0.463572 + 0.886059i \(0.346567\pi\)
\(140\) −9.12311 −0.771043
\(141\) 8.24621 0.694456
\(142\) 35.8617 3.00945
\(143\) 0 0
\(144\) 7.68466 0.640388
\(145\) 3.19224 0.265101
\(146\) −4.80776 −0.397893
\(147\) 5.68466 0.468863
\(148\) −15.6847 −1.28927
\(149\) 6.56155 0.537543 0.268772 0.963204i \(-0.413382\pi\)
0.268772 + 0.963204i \(0.413382\pi\)
\(150\) 12.0000 0.979796
\(151\) −15.3693 −1.25074 −0.625369 0.780329i \(-0.715051\pi\)
−0.625369 + 0.780329i \(0.715051\pi\)
\(152\) −7.36932 −0.597731
\(153\) 2.56155 0.207089
\(154\) −18.2462 −1.47032
\(155\) −0.876894 −0.0704339
\(156\) 0 0
\(157\) −4.36932 −0.348709 −0.174355 0.984683i \(-0.555784\pi\)
−0.174355 + 0.984683i \(0.555784\pi\)
\(158\) −24.4924 −1.94851
\(159\) 11.6847 0.926654
\(160\) 3.68466 0.291298
\(161\) 7.12311 0.561379
\(162\) −2.56155 −0.201255
\(163\) −15.8078 −1.23816 −0.619080 0.785328i \(-0.712494\pi\)
−0.619080 + 0.785328i \(0.712494\pi\)
\(164\) −11.6847 −0.912419
\(165\) −1.12311 −0.0874337
\(166\) −23.3693 −1.81381
\(167\) 6.24621 0.483346 0.241673 0.970358i \(-0.422304\pi\)
0.241673 + 0.970358i \(0.422304\pi\)
\(168\) −23.3693 −1.80298
\(169\) 0 0
\(170\) 3.68466 0.282600
\(171\) 1.12311 0.0858860
\(172\) 2.00000 0.152499
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) 14.5616 1.10391
\(175\) −16.6847 −1.26124
\(176\) 15.3693 1.15851
\(177\) 11.1231 0.836064
\(178\) 33.6155 2.51959
\(179\) −13.1231 −0.980867 −0.490433 0.871479i \(-0.663162\pi\)
−0.490433 + 0.871479i \(0.663162\pi\)
\(180\) −2.56155 −0.190927
\(181\) 9.68466 0.719855 0.359927 0.932980i \(-0.382801\pi\)
0.359927 + 0.932980i \(0.382801\pi\)
\(182\) 0 0
\(183\) 12.1231 0.896166
\(184\) −13.1231 −0.967448
\(185\) 1.93087 0.141960
\(186\) −4.00000 −0.293294
\(187\) 5.12311 0.374639
\(188\) 37.6155 2.74339
\(189\) 3.56155 0.259065
\(190\) 1.61553 0.117203
\(191\) −0.876894 −0.0634499 −0.0317249 0.999497i \(-0.510100\pi\)
−0.0317249 + 0.999497i \(0.510100\pi\)
\(192\) 1.43845 0.103811
\(193\) −19.4924 −1.40310 −0.701548 0.712623i \(-0.747507\pi\)
−0.701548 + 0.712623i \(0.747507\pi\)
\(194\) −11.3693 −0.816269
\(195\) 0 0
\(196\) 25.9309 1.85220
\(197\) 11.3693 0.810030 0.405015 0.914310i \(-0.367266\pi\)
0.405015 + 0.914310i \(0.367266\pi\)
\(198\) −5.12311 −0.364083
\(199\) −23.1771 −1.64298 −0.821490 0.570223i \(-0.806857\pi\)
−0.821490 + 0.570223i \(0.806857\pi\)
\(200\) 30.7386 2.17355
\(201\) −0.438447 −0.0309257
\(202\) 8.80776 0.619712
\(203\) −20.2462 −1.42101
\(204\) 11.6847 0.818090
\(205\) 1.43845 0.100466
\(206\) 19.3693 1.34952
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 2.24621 0.155374
\(210\) 5.12311 0.353528
\(211\) 7.31534 0.503609 0.251804 0.967778i \(-0.418976\pi\)
0.251804 + 0.967778i \(0.418976\pi\)
\(212\) 53.3002 3.66067
\(213\) −14.0000 −0.959264
\(214\) −21.1231 −1.44395
\(215\) −0.246211 −0.0167915
\(216\) −6.56155 −0.446457
\(217\) 5.56155 0.377543
\(218\) 45.6155 3.08947
\(219\) 1.87689 0.126829
\(220\) −5.12311 −0.345400
\(221\) 0 0
\(222\) 8.80776 0.591138
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) −23.3693 −1.56143
\(225\) −4.68466 −0.312311
\(226\) 37.9309 2.52312
\(227\) 1.12311 0.0745431 0.0372716 0.999305i \(-0.488133\pi\)
0.0372716 + 0.999305i \(0.488133\pi\)
\(228\) 5.12311 0.339286
\(229\) −0.246211 −0.0162701 −0.00813505 0.999967i \(-0.502589\pi\)
−0.00813505 + 0.999967i \(0.502589\pi\)
\(230\) 2.87689 0.189697
\(231\) 7.12311 0.468666
\(232\) 37.3002 2.44888
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −4.63068 −0.302072
\(236\) 50.7386 3.30280
\(237\) 9.56155 0.621090
\(238\) −23.3693 −1.51481
\(239\) 0.630683 0.0407955 0.0203977 0.999792i \(-0.493507\pi\)
0.0203977 + 0.999792i \(0.493507\pi\)
\(240\) −4.31534 −0.278554
\(241\) −2.80776 −0.180864 −0.0904320 0.995903i \(-0.528825\pi\)
−0.0904320 + 0.995903i \(0.528825\pi\)
\(242\) 17.9309 1.15264
\(243\) 1.00000 0.0641500
\(244\) 55.3002 3.54023
\(245\) −3.19224 −0.203944
\(246\) 6.56155 0.418349
\(247\) 0 0
\(248\) −10.2462 −0.650635
\(249\) 9.12311 0.578153
\(250\) −13.9309 −0.881066
\(251\) −30.7386 −1.94021 −0.970103 0.242695i \(-0.921969\pi\)
−0.970103 + 0.242695i \(0.921969\pi\)
\(252\) 16.2462 1.02342
\(253\) 4.00000 0.251478
\(254\) −24.4924 −1.53679
\(255\) −1.43845 −0.0900791
\(256\) −27.0540 −1.69087
\(257\) −16.1771 −1.00910 −0.504549 0.863383i \(-0.668341\pi\)
−0.504549 + 0.863383i \(0.668341\pi\)
\(258\) −1.12311 −0.0699215
\(259\) −12.2462 −0.760943
\(260\) 0 0
\(261\) −5.68466 −0.351872
\(262\) 44.4924 2.74875
\(263\) −15.3693 −0.947713 −0.473856 0.880602i \(-0.657138\pi\)
−0.473856 + 0.880602i \(0.657138\pi\)
\(264\) −13.1231 −0.807671
\(265\) −6.56155 −0.403073
\(266\) −10.2462 −0.628236
\(267\) −13.1231 −0.803121
\(268\) −2.00000 −0.122169
\(269\) 3.36932 0.205431 0.102715 0.994711i \(-0.467247\pi\)
0.102715 + 0.994711i \(0.467247\pi\)
\(270\) 1.43845 0.0875411
\(271\) 1.06913 0.0649450 0.0324725 0.999473i \(-0.489662\pi\)
0.0324725 + 0.999473i \(0.489662\pi\)
\(272\) 19.6847 1.19356
\(273\) 0 0
\(274\) −3.68466 −0.222598
\(275\) −9.36932 −0.564991
\(276\) 9.12311 0.549146
\(277\) 17.6847 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(278\) −28.0000 −1.67933
\(279\) 1.56155 0.0934877
\(280\) 13.1231 0.784256
\(281\) 2.80776 0.167497 0.0837486 0.996487i \(-0.473311\pi\)
0.0837486 + 0.996487i \(0.473311\pi\)
\(282\) −21.1231 −1.25786
\(283\) −1.31534 −0.0781889 −0.0390945 0.999236i \(-0.512447\pi\)
−0.0390945 + 0.999236i \(0.512447\pi\)
\(284\) −63.8617 −3.78950
\(285\) −0.630683 −0.0373584
\(286\) 0 0
\(287\) −9.12311 −0.538520
\(288\) −6.56155 −0.386643
\(289\) −10.4384 −0.614026
\(290\) −8.17708 −0.480175
\(291\) 4.43845 0.260186
\(292\) 8.56155 0.501027
\(293\) −24.5616 −1.43490 −0.717451 0.696609i \(-0.754691\pi\)
−0.717451 + 0.696609i \(0.754691\pi\)
\(294\) −14.5616 −0.849247
\(295\) −6.24621 −0.363668
\(296\) 22.5616 1.31136
\(297\) 2.00000 0.116052
\(298\) −16.8078 −0.973648
\(299\) 0 0
\(300\) −21.3693 −1.23376
\(301\) 1.56155 0.0900064
\(302\) 39.3693 2.26545
\(303\) −3.43845 −0.197534
\(304\) 8.63068 0.495004
\(305\) −6.80776 −0.389811
\(306\) −6.56155 −0.375099
\(307\) −10.1922 −0.581702 −0.290851 0.956768i \(-0.593938\pi\)
−0.290851 + 0.956768i \(0.593938\pi\)
\(308\) 32.4924 1.85143
\(309\) −7.56155 −0.430162
\(310\) 2.24621 0.127576
\(311\) −10.8769 −0.616772 −0.308386 0.951261i \(-0.599789\pi\)
−0.308386 + 0.951261i \(0.599789\pi\)
\(312\) 0 0
\(313\) −1.31534 −0.0743475 −0.0371738 0.999309i \(-0.511835\pi\)
−0.0371738 + 0.999309i \(0.511835\pi\)
\(314\) 11.1922 0.631614
\(315\) −2.00000 −0.112687
\(316\) 43.6155 2.45357
\(317\) 23.0540 1.29484 0.647420 0.762133i \(-0.275848\pi\)
0.647420 + 0.762133i \(0.275848\pi\)
\(318\) −29.9309 −1.67844
\(319\) −11.3693 −0.636560
\(320\) −0.807764 −0.0451554
\(321\) 8.24621 0.460259
\(322\) −18.2462 −1.01682
\(323\) 2.87689 0.160075
\(324\) 4.56155 0.253420
\(325\) 0 0
\(326\) 40.4924 2.24267
\(327\) −17.8078 −0.984772
\(328\) 16.8078 0.928054
\(329\) 29.3693 1.61918
\(330\) 2.87689 0.158368
\(331\) 23.8078 1.30859 0.654297 0.756238i \(-0.272965\pi\)
0.654297 + 0.756238i \(0.272965\pi\)
\(332\) 41.6155 2.28395
\(333\) −3.43845 −0.188426
\(334\) −16.0000 −0.875481
\(335\) 0.246211 0.0134520
\(336\) 27.3693 1.49312
\(337\) 2.12311 0.115653 0.0578265 0.998327i \(-0.481583\pi\)
0.0578265 + 0.998327i \(0.481583\pi\)
\(338\) 0 0
\(339\) −14.8078 −0.804247
\(340\) −6.56155 −0.355850
\(341\) 3.12311 0.169126
\(342\) −2.87689 −0.155565
\(343\) −4.68466 −0.252948
\(344\) −2.87689 −0.155112
\(345\) −1.12311 −0.0604660
\(346\) −9.61553 −0.516934
\(347\) −13.6155 −0.730920 −0.365460 0.930827i \(-0.619088\pi\)
−0.365460 + 0.930827i \(0.619088\pi\)
\(348\) −25.9309 −1.39004
\(349\) 13.8078 0.739113 0.369556 0.929208i \(-0.379510\pi\)
0.369556 + 0.929208i \(0.379510\pi\)
\(350\) 42.7386 2.28448
\(351\) 0 0
\(352\) −13.1231 −0.699464
\(353\) −17.6847 −0.941259 −0.470630 0.882331i \(-0.655973\pi\)
−0.470630 + 0.882331i \(0.655973\pi\)
\(354\) −28.4924 −1.51436
\(355\) 7.86174 0.417258
\(356\) −59.8617 −3.17267
\(357\) 9.12311 0.482846
\(358\) 33.6155 1.77664
\(359\) −15.3693 −0.811162 −0.405581 0.914059i \(-0.632931\pi\)
−0.405581 + 0.914059i \(0.632931\pi\)
\(360\) 3.68466 0.194199
\(361\) −17.7386 −0.933612
\(362\) −24.8078 −1.30387
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −1.05398 −0.0551676
\(366\) −31.0540 −1.62322
\(367\) 20.0540 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(368\) 15.3693 0.801181
\(369\) −2.56155 −0.133349
\(370\) −4.94602 −0.257132
\(371\) 41.6155 2.16057
\(372\) 7.12311 0.369316
\(373\) 3.63068 0.187990 0.0939948 0.995573i \(-0.470036\pi\)
0.0939948 + 0.995573i \(0.470036\pi\)
\(374\) −13.1231 −0.678580
\(375\) 5.43845 0.280840
\(376\) −54.1080 −2.79040
\(377\) 0 0
\(378\) −9.12311 −0.469242
\(379\) −11.3153 −0.581230 −0.290615 0.956840i \(-0.593860\pi\)
−0.290615 + 0.956840i \(0.593860\pi\)
\(380\) −2.87689 −0.147582
\(381\) 9.56155 0.489853
\(382\) 2.24621 0.114926
\(383\) −26.7386 −1.36628 −0.683140 0.730287i \(-0.739386\pi\)
−0.683140 + 0.730287i \(0.739386\pi\)
\(384\) 9.43845 0.481654
\(385\) −4.00000 −0.203859
\(386\) 49.9309 2.54141
\(387\) 0.438447 0.0222875
\(388\) 20.2462 1.02785
\(389\) −3.05398 −0.154843 −0.0774213 0.996998i \(-0.524669\pi\)
−0.0774213 + 0.996998i \(0.524669\pi\)
\(390\) 0 0
\(391\) 5.12311 0.259087
\(392\) −37.3002 −1.88394
\(393\) −17.3693 −0.876166
\(394\) −29.1231 −1.46720
\(395\) −5.36932 −0.270160
\(396\) 9.12311 0.458453
\(397\) 12.0540 0.604972 0.302486 0.953154i \(-0.402184\pi\)
0.302486 + 0.953154i \(0.402184\pi\)
\(398\) 59.3693 2.97591
\(399\) 4.00000 0.200250
\(400\) −36.0000 −1.80000
\(401\) −18.5616 −0.926920 −0.463460 0.886118i \(-0.653392\pi\)
−0.463460 + 0.886118i \(0.653392\pi\)
\(402\) 1.12311 0.0560154
\(403\) 0 0
\(404\) −15.6847 −0.780341
\(405\) −0.561553 −0.0279038
\(406\) 51.8617 2.57385
\(407\) −6.87689 −0.340875
\(408\) −16.8078 −0.832108
\(409\) 18.3693 0.908304 0.454152 0.890924i \(-0.349942\pi\)
0.454152 + 0.890924i \(0.349942\pi\)
\(410\) −3.68466 −0.181972
\(411\) 1.43845 0.0709534
\(412\) −34.4924 −1.69932
\(413\) 39.6155 1.94935
\(414\) −5.12311 −0.251787
\(415\) −5.12311 −0.251483
\(416\) 0 0
\(417\) 10.9309 0.535287
\(418\) −5.75379 −0.281427
\(419\) −17.7538 −0.867329 −0.433665 0.901074i \(-0.642780\pi\)
−0.433665 + 0.901074i \(0.642780\pi\)
\(420\) −9.12311 −0.445162
\(421\) −14.7538 −0.719056 −0.359528 0.933134i \(-0.617062\pi\)
−0.359528 + 0.933134i \(0.617062\pi\)
\(422\) −18.7386 −0.912182
\(423\) 8.24621 0.400945
\(424\) −76.6695 −3.72340
\(425\) −12.0000 −0.582086
\(426\) 35.8617 1.73751
\(427\) 43.1771 2.08949
\(428\) 37.6155 1.81822
\(429\) 0 0
\(430\) 0.630683 0.0304142
\(431\) 2.87689 0.138575 0.0692876 0.997597i \(-0.477927\pi\)
0.0692876 + 0.997597i \(0.477927\pi\)
\(432\) 7.68466 0.369728
\(433\) 25.2462 1.21326 0.606628 0.794986i \(-0.292522\pi\)
0.606628 + 0.794986i \(0.292522\pi\)
\(434\) −14.2462 −0.683840
\(435\) 3.19224 0.153056
\(436\) −81.2311 −3.89026
\(437\) 2.24621 0.107451
\(438\) −4.80776 −0.229724
\(439\) −1.31534 −0.0627778 −0.0313889 0.999507i \(-0.509993\pi\)
−0.0313889 + 0.999507i \(0.509993\pi\)
\(440\) 7.36932 0.351318
\(441\) 5.68466 0.270698
\(442\) 0 0
\(443\) 14.7386 0.700254 0.350127 0.936702i \(-0.386138\pi\)
0.350127 + 0.936702i \(0.386138\pi\)
\(444\) −15.6847 −0.744361
\(445\) 7.36932 0.349339
\(446\) 20.4924 0.970344
\(447\) 6.56155 0.310351
\(448\) 5.12311 0.242044
\(449\) −8.24621 −0.389163 −0.194581 0.980886i \(-0.562335\pi\)
−0.194581 + 0.980886i \(0.562335\pi\)
\(450\) 12.0000 0.565685
\(451\) −5.12311 −0.241238
\(452\) −67.5464 −3.17712
\(453\) −15.3693 −0.722113
\(454\) −2.87689 −0.135019
\(455\) 0 0
\(456\) −7.36932 −0.345100
\(457\) 28.6155 1.33858 0.669289 0.743002i \(-0.266599\pi\)
0.669289 + 0.743002i \(0.266599\pi\)
\(458\) 0.630683 0.0294699
\(459\) 2.56155 0.119563
\(460\) −5.12311 −0.238866
\(461\) 36.8078 1.71431 0.857154 0.515060i \(-0.172230\pi\)
0.857154 + 0.515060i \(0.172230\pi\)
\(462\) −18.2462 −0.848891
\(463\) 26.6847 1.24014 0.620071 0.784546i \(-0.287104\pi\)
0.620071 + 0.784546i \(0.287104\pi\)
\(464\) −43.6847 −2.02801
\(465\) −0.876894 −0.0406650
\(466\) −66.6004 −3.08520
\(467\) −26.0000 −1.20314 −0.601568 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) 0 0
\(469\) −1.56155 −0.0721058
\(470\) 11.8617 0.547141
\(471\) −4.36932 −0.201327
\(472\) −72.9848 −3.35940
\(473\) 0.876894 0.0403196
\(474\) −24.4924 −1.12497
\(475\) −5.26137 −0.241408
\(476\) 41.6155 1.90744
\(477\) 11.6847 0.535004
\(478\) −1.61553 −0.0738925
\(479\) 6.24621 0.285397 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(480\) 3.68466 0.168181
\(481\) 0 0
\(482\) 7.19224 0.327597
\(483\) 7.12311 0.324113
\(484\) −31.9309 −1.45140
\(485\) −2.49242 −0.113175
\(486\) −2.56155 −0.116194
\(487\) 1.12311 0.0508928 0.0254464 0.999676i \(-0.491899\pi\)
0.0254464 + 0.999676i \(0.491899\pi\)
\(488\) −79.5464 −3.60090
\(489\) −15.8078 −0.714852
\(490\) 8.17708 0.369403
\(491\) −19.7538 −0.891476 −0.445738 0.895163i \(-0.647059\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(492\) −11.6847 −0.526785
\(493\) −14.5616 −0.655819
\(494\) 0 0
\(495\) −1.12311 −0.0504798
\(496\) 12.0000 0.538816
\(497\) −49.8617 −2.23660
\(498\) −23.3693 −1.04720
\(499\) 28.4924 1.27550 0.637748 0.770245i \(-0.279866\pi\)
0.637748 + 0.770245i \(0.279866\pi\)
\(500\) 24.8078 1.10944
\(501\) 6.24621 0.279060
\(502\) 78.7386 3.51428
\(503\) −11.7538 −0.524076 −0.262038 0.965058i \(-0.584395\pi\)
−0.262038 + 0.965058i \(0.584395\pi\)
\(504\) −23.3693 −1.04095
\(505\) 1.93087 0.0859226
\(506\) −10.2462 −0.455500
\(507\) 0 0
\(508\) 43.6155 1.93513
\(509\) −6.80776 −0.301749 −0.150874 0.988553i \(-0.548209\pi\)
−0.150874 + 0.988553i \(0.548209\pi\)
\(510\) 3.68466 0.163159
\(511\) 6.68466 0.295712
\(512\) 50.4233 2.22842
\(513\) 1.12311 0.0495863
\(514\) 41.4384 1.82777
\(515\) 4.24621 0.187110
\(516\) 2.00000 0.0880451
\(517\) 16.4924 0.725336
\(518\) 31.3693 1.37829
\(519\) 3.75379 0.164773
\(520\) 0 0
\(521\) −37.9309 −1.66178 −0.830891 0.556436i \(-0.812169\pi\)
−0.830891 + 0.556436i \(0.812169\pi\)
\(522\) 14.5616 0.637342
\(523\) −23.8617 −1.04340 −0.521701 0.853129i \(-0.674702\pi\)
−0.521701 + 0.853129i \(0.674702\pi\)
\(524\) −79.2311 −3.46122
\(525\) −16.6847 −0.728178
\(526\) 39.3693 1.71658
\(527\) 4.00000 0.174243
\(528\) 15.3693 0.668864
\(529\) −19.0000 −0.826087
\(530\) 16.8078 0.730083
\(531\) 11.1231 0.482702
\(532\) 18.2462 0.791074
\(533\) 0 0
\(534\) 33.6155 1.45469
\(535\) −4.63068 −0.200202
\(536\) 2.87689 0.124263
\(537\) −13.1231 −0.566304
\(538\) −8.63068 −0.372095
\(539\) 11.3693 0.489711
\(540\) −2.56155 −0.110232
\(541\) 29.7386 1.27856 0.639282 0.768972i \(-0.279232\pi\)
0.639282 + 0.768972i \(0.279232\pi\)
\(542\) −2.73863 −0.117634
\(543\) 9.68466 0.415608
\(544\) −16.8078 −0.720627
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −24.9309 −1.06597 −0.532984 0.846126i \(-0.678929\pi\)
−0.532984 + 0.846126i \(0.678929\pi\)
\(548\) 6.56155 0.280296
\(549\) 12.1231 0.517402
\(550\) 24.0000 1.02336
\(551\) −6.38447 −0.271988
\(552\) −13.1231 −0.558556
\(553\) 34.0540 1.44812
\(554\) −45.3002 −1.92462
\(555\) 1.93087 0.0819609
\(556\) 49.8617 2.11461
\(557\) 14.0691 0.596128 0.298064 0.954546i \(-0.403659\pi\)
0.298064 + 0.954546i \(0.403659\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) −15.3693 −0.649472
\(561\) 5.12311 0.216298
\(562\) −7.19224 −0.303386
\(563\) 1.36932 0.0577098 0.0288549 0.999584i \(-0.490814\pi\)
0.0288549 + 0.999584i \(0.490814\pi\)
\(564\) 37.6155 1.58390
\(565\) 8.31534 0.349829
\(566\) 3.36932 0.141623
\(567\) 3.56155 0.149571
\(568\) 91.8617 3.85443
\(569\) 40.7386 1.70785 0.853926 0.520394i \(-0.174215\pi\)
0.853926 + 0.520394i \(0.174215\pi\)
\(570\) 1.61553 0.0676670
\(571\) 19.3693 0.810581 0.405290 0.914188i \(-0.367170\pi\)
0.405290 + 0.914188i \(0.367170\pi\)
\(572\) 0 0
\(573\) −0.876894 −0.0366328
\(574\) 23.3693 0.975416
\(575\) −9.36932 −0.390728
\(576\) 1.43845 0.0599353
\(577\) 29.6847 1.23579 0.617894 0.786261i \(-0.287986\pi\)
0.617894 + 0.786261i \(0.287986\pi\)
\(578\) 26.7386 1.11218
\(579\) −19.4924 −0.810077
\(580\) 14.5616 0.604636
\(581\) 32.4924 1.34801
\(582\) −11.3693 −0.471273
\(583\) 23.3693 0.967858
\(584\) −12.3153 −0.509612
\(585\) 0 0
\(586\) 62.9157 2.59902
\(587\) −14.6307 −0.603873 −0.301936 0.953328i \(-0.597633\pi\)
−0.301936 + 0.953328i \(0.597633\pi\)
\(588\) 25.9309 1.06937
\(589\) 1.75379 0.0722636
\(590\) 16.0000 0.658710
\(591\) 11.3693 0.467671
\(592\) −26.4233 −1.08599
\(593\) −44.4233 −1.82425 −0.912123 0.409917i \(-0.865558\pi\)
−0.912123 + 0.409917i \(0.865558\pi\)
\(594\) −5.12311 −0.210204
\(595\) −5.12311 −0.210027
\(596\) 29.9309 1.22602
\(597\) −23.1771 −0.948575
\(598\) 0 0
\(599\) −0.384472 −0.0157091 −0.00785455 0.999969i \(-0.502500\pi\)
−0.00785455 + 0.999969i \(0.502500\pi\)
\(600\) 30.7386 1.25490
\(601\) −35.9309 −1.46565 −0.732825 0.680417i \(-0.761799\pi\)
−0.732825 + 0.680417i \(0.761799\pi\)
\(602\) −4.00000 −0.163028
\(603\) −0.438447 −0.0178549
\(604\) −70.1080 −2.85265
\(605\) 3.93087 0.159813
\(606\) 8.80776 0.357791
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −7.36932 −0.298865
\(609\) −20.2462 −0.820418
\(610\) 17.4384 0.706062
\(611\) 0 0
\(612\) 11.6847 0.472324
\(613\) −22.8617 −0.923377 −0.461688 0.887042i \(-0.652756\pi\)
−0.461688 + 0.887042i \(0.652756\pi\)
\(614\) 26.1080 1.05363
\(615\) 1.43845 0.0580038
\(616\) −46.7386 −1.88315
\(617\) 10.8078 0.435104 0.217552 0.976049i \(-0.430193\pi\)
0.217552 + 0.976049i \(0.430193\pi\)
\(618\) 19.3693 0.779148
\(619\) 24.3002 0.976707 0.488353 0.872646i \(-0.337598\pi\)
0.488353 + 0.872646i \(0.337598\pi\)
\(620\) −4.00000 −0.160644
\(621\) 2.00000 0.0802572
\(622\) 27.8617 1.11715
\(623\) −46.7386 −1.87254
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 3.36932 0.134665
\(627\) 2.24621 0.0897050
\(628\) −19.9309 −0.795328
\(629\) −8.80776 −0.351189
\(630\) 5.12311 0.204109
\(631\) 14.4384 0.574786 0.287393 0.957813i \(-0.407212\pi\)
0.287393 + 0.957813i \(0.407212\pi\)
\(632\) −62.7386 −2.49561
\(633\) 7.31534 0.290759
\(634\) −59.0540 −2.34533
\(635\) −5.36932 −0.213075
\(636\) 53.3002 2.11349
\(637\) 0 0
\(638\) 29.1231 1.15299
\(639\) −14.0000 −0.553831
\(640\) −5.30019 −0.209508
\(641\) 26.1771 1.03393 0.516966 0.856006i \(-0.327061\pi\)
0.516966 + 0.856006i \(0.327061\pi\)
\(642\) −21.1231 −0.833662
\(643\) −38.5464 −1.52012 −0.760061 0.649852i \(-0.774831\pi\)
−0.760061 + 0.649852i \(0.774831\pi\)
\(644\) 32.4924 1.28038
\(645\) −0.246211 −0.00969456
\(646\) −7.36932 −0.289942
\(647\) −47.6155 −1.87196 −0.935980 0.352054i \(-0.885483\pi\)
−0.935980 + 0.352054i \(0.885483\pi\)
\(648\) −6.56155 −0.257762
\(649\) 22.2462 0.873240
\(650\) 0 0
\(651\) 5.56155 0.217974
\(652\) −72.1080 −2.82397
\(653\) 14.8769 0.582178 0.291089 0.956696i \(-0.405982\pi\)
0.291089 + 0.956696i \(0.405982\pi\)
\(654\) 45.6155 1.78371
\(655\) 9.75379 0.381112
\(656\) −19.6847 −0.768557
\(657\) 1.87689 0.0732246
\(658\) −75.2311 −2.93281
\(659\) 14.2462 0.554954 0.277477 0.960732i \(-0.410502\pi\)
0.277477 + 0.960732i \(0.410502\pi\)
\(660\) −5.12311 −0.199417
\(661\) −30.3693 −1.18123 −0.590615 0.806954i \(-0.701115\pi\)
−0.590615 + 0.806954i \(0.701115\pi\)
\(662\) −60.9848 −2.37024
\(663\) 0 0
\(664\) −59.8617 −2.32309
\(665\) −2.24621 −0.0871043
\(666\) 8.80776 0.341294
\(667\) −11.3693 −0.440222
\(668\) 28.4924 1.10240
\(669\) −8.00000 −0.309298
\(670\) −0.630683 −0.0243654
\(671\) 24.2462 0.936015
\(672\) −23.3693 −0.901491
\(673\) −6.75379 −0.260339 −0.130170 0.991492i \(-0.541552\pi\)
−0.130170 + 0.991492i \(0.541552\pi\)
\(674\) −5.43845 −0.209481
\(675\) −4.68466 −0.180313
\(676\) 0 0
\(677\) 25.6155 0.984485 0.492242 0.870458i \(-0.336177\pi\)
0.492242 + 0.870458i \(0.336177\pi\)
\(678\) 37.9309 1.45673
\(679\) 15.8078 0.606646
\(680\) 9.43845 0.361948
\(681\) 1.12311 0.0430375
\(682\) −8.00000 −0.306336
\(683\) −36.1080 −1.38163 −0.690816 0.723030i \(-0.742749\pi\)
−0.690816 + 0.723030i \(0.742749\pi\)
\(684\) 5.12311 0.195887
\(685\) −0.807764 −0.0308631
\(686\) 12.0000 0.458162
\(687\) −0.246211 −0.00939355
\(688\) 3.36932 0.128454
\(689\) 0 0
\(690\) 2.87689 0.109521
\(691\) −2.30019 −0.0875032 −0.0437516 0.999042i \(-0.513931\pi\)
−0.0437516 + 0.999042i \(0.513931\pi\)
\(692\) 17.1231 0.650923
\(693\) 7.12311 0.270584
\(694\) 34.8769 1.32391
\(695\) −6.13826 −0.232837
\(696\) 37.3002 1.41386
\(697\) −6.56155 −0.248537
\(698\) −35.3693 −1.33875
\(699\) 26.0000 0.983410
\(700\) −76.1080 −2.87661
\(701\) 19.3693 0.731569 0.365785 0.930700i \(-0.380801\pi\)
0.365785 + 0.930700i \(0.380801\pi\)
\(702\) 0 0
\(703\) −3.86174 −0.145648
\(704\) 2.87689 0.108427
\(705\) −4.63068 −0.174402
\(706\) 45.3002 1.70490
\(707\) −12.2462 −0.460566
\(708\) 50.7386 1.90687
\(709\) −25.4924 −0.957388 −0.478694 0.877982i \(-0.658890\pi\)
−0.478694 + 0.877982i \(0.658890\pi\)
\(710\) −20.1383 −0.755775
\(711\) 9.56155 0.358586
\(712\) 86.1080 3.22703
\(713\) 3.12311 0.116961
\(714\) −23.3693 −0.874575
\(715\) 0 0
\(716\) −59.8617 −2.23714
\(717\) 0.630683 0.0235533
\(718\) 39.3693 1.46925
\(719\) 1.36932 0.0510669 0.0255335 0.999674i \(-0.491872\pi\)
0.0255335 + 0.999674i \(0.491872\pi\)
\(720\) −4.31534 −0.160823
\(721\) −26.9309 −1.00296
\(722\) 45.4384 1.69104
\(723\) −2.80776 −0.104422
\(724\) 44.1771 1.64183
\(725\) 26.6307 0.989039
\(726\) 17.9309 0.665477
\(727\) 39.6695 1.47126 0.735630 0.677383i \(-0.236886\pi\)
0.735630 + 0.677383i \(0.236886\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.69981 0.0999246
\(731\) 1.12311 0.0415396
\(732\) 55.3002 2.04395
\(733\) −53.4924 −1.97579 −0.987894 0.155131i \(-0.950420\pi\)
−0.987894 + 0.155131i \(0.950420\pi\)
\(734\) −51.3693 −1.89608
\(735\) −3.19224 −0.117747
\(736\) −13.1231 −0.483724
\(737\) −0.876894 −0.0323008
\(738\) 6.56155 0.241534
\(739\) 6.24621 0.229771 0.114885 0.993379i \(-0.463350\pi\)
0.114885 + 0.993379i \(0.463350\pi\)
\(740\) 8.80776 0.323780
\(741\) 0 0
\(742\) −106.600 −3.91342
\(743\) 37.3693 1.37095 0.685474 0.728097i \(-0.259595\pi\)
0.685474 + 0.728097i \(0.259595\pi\)
\(744\) −10.2462 −0.375644
\(745\) −3.68466 −0.134995
\(746\) −9.30019 −0.340504
\(747\) 9.12311 0.333797
\(748\) 23.3693 0.854467
\(749\) 29.3693 1.07313
\(750\) −13.9309 −0.508683
\(751\) −30.1080 −1.09865 −0.549327 0.835607i \(-0.685116\pi\)
−0.549327 + 0.835607i \(0.685116\pi\)
\(752\) 63.3693 2.31084
\(753\) −30.7386 −1.12018
\(754\) 0 0
\(755\) 8.63068 0.314103
\(756\) 16.2462 0.590869
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 28.9848 1.05278
\(759\) 4.00000 0.145191
\(760\) 4.13826 0.150110
\(761\) 15.3693 0.557137 0.278569 0.960416i \(-0.410140\pi\)
0.278569 + 0.960416i \(0.410140\pi\)
\(762\) −24.4924 −0.887267
\(763\) −63.4233 −2.29608
\(764\) −4.00000 −0.144715
\(765\) −1.43845 −0.0520072
\(766\) 68.4924 2.47473
\(767\) 0 0
\(768\) −27.0540 −0.976226
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 10.2462 0.369248
\(771\) −16.1771 −0.582603
\(772\) −88.9157 −3.20015
\(773\) 7.75379 0.278884 0.139442 0.990230i \(-0.455469\pi\)
0.139442 + 0.990230i \(0.455469\pi\)
\(774\) −1.12311 −0.0403692
\(775\) −7.31534 −0.262775
\(776\) −29.1231 −1.04546
\(777\) −12.2462 −0.439330
\(778\) 7.82292 0.280465
\(779\) −2.87689 −0.103075
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) −13.1231 −0.469281
\(783\) −5.68466 −0.203153
\(784\) 43.6847 1.56017
\(785\) 2.45360 0.0875728
\(786\) 44.4924 1.58699
\(787\) 1.17708 0.0419584 0.0209792 0.999780i \(-0.493322\pi\)
0.0209792 + 0.999780i \(0.493322\pi\)
\(788\) 51.8617 1.84750
\(789\) −15.3693 −0.547162
\(790\) 13.7538 0.489338
\(791\) −52.7386 −1.87517
\(792\) −13.1231 −0.466309
\(793\) 0 0
\(794\) −30.8769 −1.09578
\(795\) −6.56155 −0.232714
\(796\) −105.723 −3.74727
\(797\) −41.6155 −1.47410 −0.737049 0.675840i \(-0.763781\pi\)
−0.737049 + 0.675840i \(0.763781\pi\)
\(798\) −10.2462 −0.362712
\(799\) 21.1231 0.747282
\(800\) 30.7386 1.08677
\(801\) −13.1231 −0.463682
\(802\) 47.5464 1.67892
\(803\) 3.75379 0.132468
\(804\) −2.00000 −0.0705346
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 3.36932 0.118606
\(808\) 22.5616 0.793713
\(809\) 37.3002 1.31140 0.655702 0.755019i \(-0.272373\pi\)
0.655702 + 0.755019i \(0.272373\pi\)
\(810\) 1.43845 0.0505419
\(811\) 1.56155 0.0548335 0.0274168 0.999624i \(-0.491272\pi\)
0.0274168 + 0.999624i \(0.491272\pi\)
\(812\) −92.3542 −3.24100
\(813\) 1.06913 0.0374960
\(814\) 17.6155 0.617424
\(815\) 8.87689 0.310944
\(816\) 19.6847 0.689101
\(817\) 0.492423 0.0172277
\(818\) −47.0540 −1.64520
\(819\) 0 0
\(820\) 6.56155 0.229139
\(821\) 26.4924 0.924592 0.462296 0.886726i \(-0.347026\pi\)
0.462296 + 0.886726i \(0.347026\pi\)
\(822\) −3.68466 −0.128517
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 49.6155 1.72844
\(825\) −9.36932 −0.326198
\(826\) −101.477 −3.53085
\(827\) −34.7386 −1.20798 −0.603990 0.796992i \(-0.706423\pi\)
−0.603990 + 0.796992i \(0.706423\pi\)
\(828\) 9.12311 0.317050
\(829\) −19.4924 −0.677000 −0.338500 0.940966i \(-0.609919\pi\)
−0.338500 + 0.940966i \(0.609919\pi\)
\(830\) 13.1231 0.455510
\(831\) 17.6847 0.613474
\(832\) 0 0
\(833\) 14.5616 0.504528
\(834\) −28.0000 −0.969561
\(835\) −3.50758 −0.121385
\(836\) 10.2462 0.354373
\(837\) 1.56155 0.0539752
\(838\) 45.4773 1.57099
\(839\) 19.6155 0.677203 0.338602 0.940930i \(-0.390046\pi\)
0.338602 + 0.940930i \(0.390046\pi\)
\(840\) 13.1231 0.452790
\(841\) 3.31534 0.114322
\(842\) 37.7926 1.30242
\(843\) 2.80776 0.0967045
\(844\) 33.3693 1.14862
\(845\) 0 0
\(846\) −21.1231 −0.726227
\(847\) −24.9309 −0.856635
\(848\) 89.7926 3.08349
\(849\) −1.31534 −0.0451424
\(850\) 30.7386 1.05433
\(851\) −6.87689 −0.235737
\(852\) −63.8617 −2.18787
\(853\) 6.12311 0.209651 0.104826 0.994491i \(-0.466572\pi\)
0.104826 + 0.994491i \(0.466572\pi\)
\(854\) −110.600 −3.78467
\(855\) −0.630683 −0.0215689
\(856\) −54.1080 −1.84937
\(857\) 31.4384 1.07392 0.536958 0.843609i \(-0.319573\pi\)
0.536958 + 0.843609i \(0.319573\pi\)
\(858\) 0 0
\(859\) 20.4384 0.697351 0.348675 0.937244i \(-0.386632\pi\)
0.348675 + 0.937244i \(0.386632\pi\)
\(860\) −1.12311 −0.0382976
\(861\) −9.12311 −0.310915
\(862\) −7.36932 −0.251000
\(863\) 2.49242 0.0848430 0.0424215 0.999100i \(-0.486493\pi\)
0.0424215 + 0.999100i \(0.486493\pi\)
\(864\) −6.56155 −0.223229
\(865\) −2.10795 −0.0716725
\(866\) −64.6695 −2.19756
\(867\) −10.4384 −0.354508
\(868\) 25.3693 0.861091
\(869\) 19.1231 0.648707
\(870\) −8.17708 −0.277229
\(871\) 0 0
\(872\) 116.847 3.95692
\(873\) 4.43845 0.150219
\(874\) −5.75379 −0.194625
\(875\) 19.3693 0.654802
\(876\) 8.56155 0.289268
\(877\) −19.4384 −0.656390 −0.328195 0.944610i \(-0.606440\pi\)
−0.328195 + 0.944610i \(0.606440\pi\)
\(878\) 3.36932 0.113709
\(879\) −24.5616 −0.828441
\(880\) −8.63068 −0.290940
\(881\) −37.9309 −1.27792 −0.638962 0.769239i \(-0.720636\pi\)
−0.638962 + 0.769239i \(0.720636\pi\)
\(882\) −14.5616 −0.490313
\(883\) −11.8078 −0.397363 −0.198681 0.980064i \(-0.563666\pi\)
−0.198681 + 0.980064i \(0.563666\pi\)
\(884\) 0 0
\(885\) −6.24621 −0.209964
\(886\) −37.7538 −1.26836
\(887\) 49.3693 1.65766 0.828830 0.559501i \(-0.189007\pi\)
0.828830 + 0.559501i \(0.189007\pi\)
\(888\) 22.5616 0.757116
\(889\) 34.0540 1.14213
\(890\) −18.8769 −0.632755
\(891\) 2.00000 0.0670025
\(892\) −36.4924 −1.22186
\(893\) 9.26137 0.309920
\(894\) −16.8078 −0.562136
\(895\) 7.36932 0.246329
\(896\) 33.6155 1.12302
\(897\) 0 0
\(898\) 21.1231 0.704887
\(899\) −8.87689 −0.296061
\(900\) −21.3693 −0.712311
\(901\) 29.9309 0.997142
\(902\) 13.1231 0.436952
\(903\) 1.56155 0.0519652
\(904\) 97.1619 3.23156
\(905\) −5.43845 −0.180780
\(906\) 39.3693 1.30796
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 5.12311 0.170016
\(909\) −3.43845 −0.114046
\(910\) 0 0
\(911\) −10.7386 −0.355787 −0.177893 0.984050i \(-0.556928\pi\)
−0.177893 + 0.984050i \(0.556928\pi\)
\(912\) 8.63068 0.285790
\(913\) 18.2462 0.603861
\(914\) −73.3002 −2.42455
\(915\) −6.80776 −0.225058
\(916\) −1.12311 −0.0371085
\(917\) −61.8617 −2.04285
\(918\) −6.56155 −0.216564
\(919\) 44.4924 1.46767 0.733835 0.679328i \(-0.237729\pi\)
0.733835 + 0.679328i \(0.237729\pi\)
\(920\) 7.36932 0.242959
\(921\) −10.1922 −0.335846
\(922\) −94.2850 −3.10511
\(923\) 0 0
\(924\) 32.4924 1.06892
\(925\) 16.1080 0.529626
\(926\) −68.3542 −2.24626
\(927\) −7.56155 −0.248354
\(928\) 37.3002 1.22444
\(929\) −12.8078 −0.420209 −0.210105 0.977679i \(-0.567380\pi\)
−0.210105 + 0.977679i \(0.567380\pi\)
\(930\) 2.24621 0.0736562
\(931\) 6.38447 0.209243
\(932\) 118.600 3.88488
\(933\) −10.8769 −0.356094
\(934\) 66.6004 2.17923
\(935\) −2.87689 −0.0940845
\(936\) 0 0
\(937\) −3.43845 −0.112329 −0.0561646 0.998422i \(-0.517887\pi\)
−0.0561646 + 0.998422i \(0.517887\pi\)
\(938\) 4.00000 0.130605
\(939\) −1.31534 −0.0429245
\(940\) −21.1231 −0.688960
\(941\) 2.49242 0.0812507 0.0406253 0.999174i \(-0.487065\pi\)
0.0406253 + 0.999174i \(0.487065\pi\)
\(942\) 11.1922 0.364663
\(943\) −5.12311 −0.166831
\(944\) 85.4773 2.78205
\(945\) −2.00000 −0.0650600
\(946\) −2.24621 −0.0730306
\(947\) −10.7386 −0.348959 −0.174479 0.984661i \(-0.555824\pi\)
−0.174479 + 0.984661i \(0.555824\pi\)
\(948\) 43.6155 1.41657
\(949\) 0 0
\(950\) 13.4773 0.437260
\(951\) 23.0540 0.747576
\(952\) −59.8617 −1.94013
\(953\) −34.9848 −1.13327 −0.566635 0.823969i \(-0.691755\pi\)
−0.566635 + 0.823969i \(0.691755\pi\)
\(954\) −29.9309 −0.969048
\(955\) 0.492423 0.0159344
\(956\) 2.87689 0.0930454
\(957\) −11.3693 −0.367518
\(958\) −16.0000 −0.516937
\(959\) 5.12311 0.165434
\(960\) −0.807764 −0.0260705
\(961\) −28.5616 −0.921340
\(962\) 0 0
\(963\) 8.24621 0.265730
\(964\) −12.8078 −0.412510
\(965\) 10.9460 0.352365
\(966\) −18.2462 −0.587062
\(967\) 9.12311 0.293379 0.146690 0.989183i \(-0.453138\pi\)
0.146690 + 0.989183i \(0.453138\pi\)
\(968\) 45.9309 1.47627
\(969\) 2.87689 0.0924192
\(970\) 6.38447 0.204993
\(971\) 52.9848 1.70036 0.850182 0.526488i \(-0.176492\pi\)
0.850182 + 0.526488i \(0.176492\pi\)
\(972\) 4.56155 0.146312
\(973\) 38.9309 1.24807
\(974\) −2.87689 −0.0921816
\(975\) 0 0
\(976\) 93.1619 2.98204
\(977\) −15.8229 −0.506220 −0.253110 0.967438i \(-0.581453\pi\)
−0.253110 + 0.967438i \(0.581453\pi\)
\(978\) 40.4924 1.29480
\(979\) −26.2462 −0.838833
\(980\) −14.5616 −0.465152
\(981\) −17.8078 −0.568558
\(982\) 50.6004 1.61472
\(983\) 27.6155 0.880799 0.440399 0.897802i \(-0.354837\pi\)
0.440399 + 0.897802i \(0.354837\pi\)
\(984\) 16.8078 0.535812
\(985\) −6.38447 −0.203426
\(986\) 37.3002 1.18788
\(987\) 29.3693 0.934836
\(988\) 0 0
\(989\) 0.876894 0.0278836
\(990\) 2.87689 0.0914337
\(991\) 40.3542 1.28189 0.640946 0.767586i \(-0.278542\pi\)
0.640946 + 0.767586i \(0.278542\pi\)
\(992\) −10.2462 −0.325318
\(993\) 23.8078 0.755517
\(994\) 127.723 4.05114
\(995\) 13.0152 0.412608
\(996\) 41.6155 1.31864
\(997\) 20.6155 0.652900 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(998\) −72.9848 −2.31029
\(999\) −3.43845 −0.108788
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 507.2.a.d.1.1 2
3.2 odd 2 1521.2.a.m.1.2 2
4.3 odd 2 8112.2.a.bo.1.1 2
13.2 odd 12 507.2.j.g.316.1 8
13.3 even 3 507.2.e.g.22.2 4
13.4 even 6 39.2.e.b.16.1 4
13.5 odd 4 507.2.b.d.337.4 4
13.6 odd 12 507.2.j.g.361.4 8
13.7 odd 12 507.2.j.g.361.1 8
13.8 odd 4 507.2.b.d.337.1 4
13.9 even 3 507.2.e.g.484.2 4
13.10 even 6 39.2.e.b.22.1 yes 4
13.11 odd 12 507.2.j.g.316.4 8
13.12 even 2 507.2.a.g.1.2 2
39.5 even 4 1521.2.b.h.1351.1 4
39.8 even 4 1521.2.b.h.1351.4 4
39.17 odd 6 117.2.g.c.55.2 4
39.23 odd 6 117.2.g.c.100.2 4
39.38 odd 2 1521.2.a.g.1.1 2
52.23 odd 6 624.2.q.h.529.2 4
52.43 odd 6 624.2.q.h.289.2 4
52.51 odd 2 8112.2.a.bk.1.2 2
65.4 even 6 975.2.i.k.601.2 4
65.17 odd 12 975.2.bb.i.874.4 8
65.23 odd 12 975.2.bb.i.724.4 8
65.43 odd 12 975.2.bb.i.874.1 8
65.49 even 6 975.2.i.k.451.2 4
65.62 odd 12 975.2.bb.i.724.1 8
156.23 even 6 1872.2.t.r.1153.1 4
156.95 even 6 1872.2.t.r.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.1 4 13.4 even 6
39.2.e.b.22.1 yes 4 13.10 even 6
117.2.g.c.55.2 4 39.17 odd 6
117.2.g.c.100.2 4 39.23 odd 6
507.2.a.d.1.1 2 1.1 even 1 trivial
507.2.a.g.1.2 2 13.12 even 2
507.2.b.d.337.1 4 13.8 odd 4
507.2.b.d.337.4 4 13.5 odd 4
507.2.e.g.22.2 4 13.3 even 3
507.2.e.g.484.2 4 13.9 even 3
507.2.j.g.316.1 8 13.2 odd 12
507.2.j.g.316.4 8 13.11 odd 12
507.2.j.g.361.1 8 13.7 odd 12
507.2.j.g.361.4 8 13.6 odd 12
624.2.q.h.289.2 4 52.43 odd 6
624.2.q.h.529.2 4 52.23 odd 6
975.2.i.k.451.2 4 65.49 even 6
975.2.i.k.601.2 4 65.4 even 6
975.2.bb.i.724.1 8 65.62 odd 12
975.2.bb.i.724.4 8 65.23 odd 12
975.2.bb.i.874.1 8 65.43 odd 12
975.2.bb.i.874.4 8 65.17 odd 12
1521.2.a.g.1.1 2 39.38 odd 2
1521.2.a.m.1.2 2 3.2 odd 2
1521.2.b.h.1351.1 4 39.5 even 4
1521.2.b.h.1351.4 4 39.8 even 4
1872.2.t.r.289.1 4 156.95 even 6
1872.2.t.r.1153.1 4 156.23 even 6
8112.2.a.bk.1.2 2 52.51 odd 2
8112.2.a.bo.1.1 2 4.3 odd 2