Properties

Label 3549.2.a.o.1.1
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
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] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.87939 q^{2} +1.00000 q^{3} +1.53209 q^{4} +2.34730 q^{5} -1.87939 q^{6} -1.00000 q^{7} +0.879385 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.87939 q^{2} +1.00000 q^{3} +1.53209 q^{4} +2.34730 q^{5} -1.87939 q^{6} -1.00000 q^{7} +0.879385 q^{8} +1.00000 q^{9} -4.41147 q^{10} +4.71688 q^{11} +1.53209 q^{12} +1.87939 q^{14} +2.34730 q^{15} -4.71688 q^{16} +6.24897 q^{17} -1.87939 q^{18} +3.69459 q^{19} +3.59627 q^{20} -1.00000 q^{21} -8.86484 q^{22} +6.98545 q^{23} +0.879385 q^{24} +0.509800 q^{25} +1.00000 q^{27} -1.53209 q^{28} -6.39693 q^{29} -4.41147 q^{30} +1.95811 q^{31} +7.10607 q^{32} +4.71688 q^{33} -11.7442 q^{34} -2.34730 q^{35} +1.53209 q^{36} +8.80066 q^{37} -6.94356 q^{38} +2.06418 q^{40} +6.73917 q^{41} +1.87939 q^{42} -4.66044 q^{43} +7.22668 q^{44} +2.34730 q^{45} -13.1284 q^{46} -3.10607 q^{47} -4.71688 q^{48} +1.00000 q^{49} -0.958111 q^{50} +6.24897 q^{51} -13.3405 q^{53} -1.87939 q^{54} +11.0719 q^{55} -0.879385 q^{56} +3.69459 q^{57} +12.0223 q^{58} +12.3746 q^{59} +3.59627 q^{60} -7.36959 q^{61} -3.68004 q^{62} -1.00000 q^{63} -3.92127 q^{64} -8.86484 q^{66} -13.2199 q^{67} +9.57398 q^{68} +6.98545 q^{69} +4.41147 q^{70} -9.98545 q^{71} +0.879385 q^{72} +9.16250 q^{73} -16.5398 q^{74} +0.509800 q^{75} +5.66044 q^{76} -4.71688 q^{77} -3.58853 q^{79} -11.0719 q^{80} +1.00000 q^{81} -12.6655 q^{82} +0.829755 q^{83} -1.53209 q^{84} +14.6682 q^{85} +8.75877 q^{86} -6.39693 q^{87} +4.14796 q^{88} -2.30541 q^{89} -4.41147 q^{90} +10.7023 q^{92} +1.95811 q^{93} +5.83750 q^{94} +8.67230 q^{95} +7.10607 q^{96} +12.6236 q^{97} -1.87939 q^{98} +4.71688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{5} - 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{5} - 3 q^{7} - 3 q^{8} + 3 q^{9} - 3 q^{10} + 6 q^{11} + 6 q^{15} - 6 q^{16} + 6 q^{17} + 9 q^{19} - 3 q^{20} - 3 q^{21} - 3 q^{22} + 3 q^{23} - 3 q^{24} + 3 q^{25} + 3 q^{27} + 9 q^{29} - 3 q^{30} + 9 q^{31} + 9 q^{32} + 6 q^{33} - 6 q^{34} - 6 q^{35} + 12 q^{37} - 6 q^{38} - 3 q^{40} + 6 q^{41} + 9 q^{43} + 15 q^{44} + 6 q^{45} - 21 q^{46} + 3 q^{47} - 6 q^{48} + 3 q^{49} - 6 q^{50} + 6 q^{51} + 3 q^{53} + 3 q^{56} + 9 q^{57} + 30 q^{58} + 15 q^{59} - 3 q^{60} - 15 q^{61} + 9 q^{62} - 3 q^{63} - 3 q^{64} - 3 q^{66} + 9 q^{67} + 21 q^{68} + 3 q^{69} + 3 q^{70} - 12 q^{71} - 3 q^{72} + 30 q^{73} - 21 q^{74} + 3 q^{75} - 6 q^{76} - 6 q^{77} - 21 q^{79} + 3 q^{81} + 24 q^{83} - 3 q^{85} + 15 q^{86} + 9 q^{87} - 3 q^{88} - 9 q^{89} - 3 q^{90} + 6 q^{92} + 9 q^{93} + 15 q^{94} + 30 q^{95} + 9 q^{96} + 3 q^{97} + 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87939 −1.32893 −0.664463 0.747321i \(-0.731340\pi\)
−0.664463 + 0.747321i \(0.731340\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.53209 0.766044
\(5\) 2.34730 1.04974 0.524871 0.851182i \(-0.324113\pi\)
0.524871 + 0.851182i \(0.324113\pi\)
\(6\) −1.87939 −0.767256
\(7\) −1.00000 −0.377964
\(8\) 0.879385 0.310910
\(9\) 1.00000 0.333333
\(10\) −4.41147 −1.39503
\(11\) 4.71688 1.42219 0.711097 0.703094i \(-0.248199\pi\)
0.711097 + 0.703094i \(0.248199\pi\)
\(12\) 1.53209 0.442276
\(13\) 0 0
\(14\) 1.87939 0.502287
\(15\) 2.34730 0.606069
\(16\) −4.71688 −1.17922
\(17\) 6.24897 1.51560 0.757799 0.652488i \(-0.226275\pi\)
0.757799 + 0.652488i \(0.226275\pi\)
\(18\) −1.87939 −0.442975
\(19\) 3.69459 0.847598 0.423799 0.905756i \(-0.360696\pi\)
0.423799 + 0.905756i \(0.360696\pi\)
\(20\) 3.59627 0.804150
\(21\) −1.00000 −0.218218
\(22\) −8.86484 −1.88999
\(23\) 6.98545 1.45657 0.728284 0.685276i \(-0.240318\pi\)
0.728284 + 0.685276i \(0.240318\pi\)
\(24\) 0.879385 0.179504
\(25\) 0.509800 0.101960
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.53209 −0.289538
\(29\) −6.39693 −1.18788 −0.593940 0.804510i \(-0.702428\pi\)
−0.593940 + 0.804510i \(0.702428\pi\)
\(30\) −4.41147 −0.805421
\(31\) 1.95811 0.351687 0.175844 0.984418i \(-0.443735\pi\)
0.175844 + 0.984418i \(0.443735\pi\)
\(32\) 7.10607 1.25619
\(33\) 4.71688 0.821104
\(34\) −11.7442 −2.01412
\(35\) −2.34730 −0.396766
\(36\) 1.53209 0.255348
\(37\) 8.80066 1.44682 0.723410 0.690419i \(-0.242574\pi\)
0.723410 + 0.690419i \(0.242574\pi\)
\(38\) −6.94356 −1.12639
\(39\) 0 0
\(40\) 2.06418 0.326375
\(41\) 6.73917 1.05248 0.526241 0.850336i \(-0.323601\pi\)
0.526241 + 0.850336i \(0.323601\pi\)
\(42\) 1.87939 0.289995
\(43\) −4.66044 −0.710711 −0.355356 0.934731i \(-0.615640\pi\)
−0.355356 + 0.934731i \(0.615640\pi\)
\(44\) 7.22668 1.08946
\(45\) 2.34730 0.349914
\(46\) −13.1284 −1.93567
\(47\) −3.10607 −0.453066 −0.226533 0.974003i \(-0.572739\pi\)
−0.226533 + 0.974003i \(0.572739\pi\)
\(48\) −4.71688 −0.680823
\(49\) 1.00000 0.142857
\(50\) −0.958111 −0.135497
\(51\) 6.24897 0.875031
\(52\) 0 0
\(53\) −13.3405 −1.83246 −0.916229 0.400656i \(-0.868782\pi\)
−0.916229 + 0.400656i \(0.868782\pi\)
\(54\) −1.87939 −0.255752
\(55\) 11.0719 1.49294
\(56\) −0.879385 −0.117513
\(57\) 3.69459 0.489361
\(58\) 12.0223 1.57860
\(59\) 12.3746 1.61104 0.805520 0.592568i \(-0.201886\pi\)
0.805520 + 0.592568i \(0.201886\pi\)
\(60\) 3.59627 0.464276
\(61\) −7.36959 −0.943579 −0.471789 0.881711i \(-0.656392\pi\)
−0.471789 + 0.881711i \(0.656392\pi\)
\(62\) −3.68004 −0.467366
\(63\) −1.00000 −0.125988
\(64\) −3.92127 −0.490159
\(65\) 0 0
\(66\) −8.86484 −1.09119
\(67\) −13.2199 −1.61506 −0.807532 0.589823i \(-0.799197\pi\)
−0.807532 + 0.589823i \(0.799197\pi\)
\(68\) 9.57398 1.16102
\(69\) 6.98545 0.840950
\(70\) 4.41147 0.527272
\(71\) −9.98545 −1.18506 −0.592528 0.805550i \(-0.701870\pi\)
−0.592528 + 0.805550i \(0.701870\pi\)
\(72\) 0.879385 0.103637
\(73\) 9.16250 1.07239 0.536195 0.844094i \(-0.319861\pi\)
0.536195 + 0.844094i \(0.319861\pi\)
\(74\) −16.5398 −1.92272
\(75\) 0.509800 0.0588667
\(76\) 5.66044 0.649298
\(77\) −4.71688 −0.537539
\(78\) 0 0
\(79\) −3.58853 −0.403741 −0.201870 0.979412i \(-0.564702\pi\)
−0.201870 + 0.979412i \(0.564702\pi\)
\(80\) −11.0719 −1.23788
\(81\) 1.00000 0.111111
\(82\) −12.6655 −1.39867
\(83\) 0.829755 0.0910775 0.0455387 0.998963i \(-0.485500\pi\)
0.0455387 + 0.998963i \(0.485500\pi\)
\(84\) −1.53209 −0.167165
\(85\) 14.6682 1.59099
\(86\) 8.75877 0.944483
\(87\) −6.39693 −0.685822
\(88\) 4.14796 0.442174
\(89\) −2.30541 −0.244373 −0.122186 0.992507i \(-0.538991\pi\)
−0.122186 + 0.992507i \(0.538991\pi\)
\(90\) −4.41147 −0.465010
\(91\) 0 0
\(92\) 10.7023 1.11580
\(93\) 1.95811 0.203047
\(94\) 5.83750 0.602092
\(95\) 8.67230 0.889760
\(96\) 7.10607 0.725260
\(97\) 12.6236 1.28173 0.640867 0.767652i \(-0.278575\pi\)
0.640867 + 0.767652i \(0.278575\pi\)
\(98\) −1.87939 −0.189847
\(99\) 4.71688 0.474064
\(100\) 0.781059 0.0781059
\(101\) −2.40373 −0.239180 −0.119590 0.992823i \(-0.538158\pi\)
−0.119590 + 0.992823i \(0.538158\pi\)
\(102\) −11.7442 −1.16285
\(103\) 1.69459 0.166973 0.0834866 0.996509i \(-0.473394\pi\)
0.0834866 + 0.996509i \(0.473394\pi\)
\(104\) 0 0
\(105\) −2.34730 −0.229073
\(106\) 25.0719 2.43520
\(107\) 3.85710 0.372880 0.186440 0.982466i \(-0.440305\pi\)
0.186440 + 0.982466i \(0.440305\pi\)
\(108\) 1.53209 0.147425
\(109\) 0.268571 0.0257244 0.0128622 0.999917i \(-0.495906\pi\)
0.0128622 + 0.999917i \(0.495906\pi\)
\(110\) −20.8084 −1.98400
\(111\) 8.80066 0.835322
\(112\) 4.71688 0.445703
\(113\) 7.71688 0.725943 0.362972 0.931800i \(-0.381762\pi\)
0.362972 + 0.931800i \(0.381762\pi\)
\(114\) −6.94356 −0.650324
\(115\) 16.3969 1.52902
\(116\) −9.80066 −0.909968
\(117\) 0 0
\(118\) −23.2567 −2.14095
\(119\) −6.24897 −0.572842
\(120\) 2.06418 0.188433
\(121\) 11.2490 1.02263
\(122\) 13.8503 1.25395
\(123\) 6.73917 0.607651
\(124\) 3.00000 0.269408
\(125\) −10.5398 −0.942711
\(126\) 1.87939 0.167429
\(127\) −7.65270 −0.679068 −0.339534 0.940594i \(-0.610269\pi\)
−0.339534 + 0.940594i \(0.610269\pi\)
\(128\) −6.84255 −0.604802
\(129\) −4.66044 −0.410329
\(130\) 0 0
\(131\) −9.25402 −0.808528 −0.404264 0.914642i \(-0.632472\pi\)
−0.404264 + 0.914642i \(0.632472\pi\)
\(132\) 7.22668 0.629002
\(133\) −3.69459 −0.320362
\(134\) 24.8452 2.14630
\(135\) 2.34730 0.202023
\(136\) 5.49525 0.471214
\(137\) −17.5817 −1.50211 −0.751054 0.660241i \(-0.770454\pi\)
−0.751054 + 0.660241i \(0.770454\pi\)
\(138\) −13.1284 −1.11756
\(139\) −3.89899 −0.330708 −0.165354 0.986234i \(-0.552877\pi\)
−0.165354 + 0.986234i \(0.552877\pi\)
\(140\) −3.59627 −0.303940
\(141\) −3.10607 −0.261578
\(142\) 18.7665 1.57485
\(143\) 0 0
\(144\) −4.71688 −0.393073
\(145\) −15.0155 −1.24697
\(146\) −17.2199 −1.42513
\(147\) 1.00000 0.0824786
\(148\) 13.4834 1.10833
\(149\) −11.1702 −0.915102 −0.457551 0.889183i \(-0.651273\pi\)
−0.457551 + 0.889183i \(0.651273\pi\)
\(150\) −0.958111 −0.0782294
\(151\) −5.97090 −0.485905 −0.242953 0.970038i \(-0.578116\pi\)
−0.242953 + 0.970038i \(0.578116\pi\)
\(152\) 3.24897 0.263526
\(153\) 6.24897 0.505199
\(154\) 8.86484 0.714349
\(155\) 4.59627 0.369181
\(156\) 0 0
\(157\) 8.75196 0.698483 0.349241 0.937033i \(-0.386439\pi\)
0.349241 + 0.937033i \(0.386439\pi\)
\(158\) 6.74422 0.536542
\(159\) −13.3405 −1.05797
\(160\) 16.6800 1.31867
\(161\) −6.98545 −0.550531
\(162\) −1.87939 −0.147658
\(163\) 2.22163 0.174011 0.0870057 0.996208i \(-0.472270\pi\)
0.0870057 + 0.996208i \(0.472270\pi\)
\(164\) 10.3250 0.806248
\(165\) 11.0719 0.861948
\(166\) −1.55943 −0.121035
\(167\) −15.9513 −1.23435 −0.617174 0.786826i \(-0.711723\pi\)
−0.617174 + 0.786826i \(0.711723\pi\)
\(168\) −0.879385 −0.0678460
\(169\) 0 0
\(170\) −27.5672 −2.11431
\(171\) 3.69459 0.282533
\(172\) −7.14022 −0.544436
\(173\) 3.09833 0.235561 0.117781 0.993040i \(-0.462422\pi\)
0.117781 + 0.993040i \(0.462422\pi\)
\(174\) 12.0223 0.911407
\(175\) −0.509800 −0.0385373
\(176\) −22.2490 −1.67708
\(177\) 12.3746 0.930135
\(178\) 4.33275 0.324753
\(179\) 15.5895 1.16521 0.582605 0.812755i \(-0.302033\pi\)
0.582605 + 0.812755i \(0.302033\pi\)
\(180\) 3.59627 0.268050
\(181\) 18.0942 1.34493 0.672466 0.740128i \(-0.265235\pi\)
0.672466 + 0.740128i \(0.265235\pi\)
\(182\) 0 0
\(183\) −7.36959 −0.544775
\(184\) 6.14290 0.452861
\(185\) 20.6578 1.51879
\(186\) −3.68004 −0.269834
\(187\) 29.4757 2.15547
\(188\) −4.75877 −0.347069
\(189\) −1.00000 −0.0727393
\(190\) −16.2986 −1.18242
\(191\) −12.9017 −0.933532 −0.466766 0.884381i \(-0.654581\pi\)
−0.466766 + 0.884381i \(0.654581\pi\)
\(192\) −3.92127 −0.282994
\(193\) −12.6313 −0.909224 −0.454612 0.890689i \(-0.650222\pi\)
−0.454612 + 0.890689i \(0.650222\pi\)
\(194\) −23.7246 −1.70333
\(195\) 0 0
\(196\) 1.53209 0.109435
\(197\) −4.72462 −0.336615 −0.168308 0.985735i \(-0.553830\pi\)
−0.168308 + 0.985735i \(0.553830\pi\)
\(198\) −8.86484 −0.629997
\(199\) −15.6604 −1.11014 −0.555070 0.831804i \(-0.687308\pi\)
−0.555070 + 0.831804i \(0.687308\pi\)
\(200\) 0.448311 0.0317004
\(201\) −13.2199 −0.932458
\(202\) 4.51754 0.317853
\(203\) 6.39693 0.448976
\(204\) 9.57398 0.670313
\(205\) 15.8188 1.10483
\(206\) −3.18479 −0.221895
\(207\) 6.98545 0.485522
\(208\) 0 0
\(209\) 17.4270 1.20545
\(210\) 4.41147 0.304421
\(211\) 6.13341 0.422241 0.211121 0.977460i \(-0.432289\pi\)
0.211121 + 0.977460i \(0.432289\pi\)
\(212\) −20.4388 −1.40374
\(213\) −9.98545 −0.684192
\(214\) −7.24897 −0.495529
\(215\) −10.9394 −0.746064
\(216\) 0.879385 0.0598346
\(217\) −1.95811 −0.132925
\(218\) −0.504748 −0.0341858
\(219\) 9.16250 0.619145
\(220\) 16.9632 1.14366
\(221\) 0 0
\(222\) −16.5398 −1.11008
\(223\) −1.36959 −0.0917142 −0.0458571 0.998948i \(-0.514602\pi\)
−0.0458571 + 0.998948i \(0.514602\pi\)
\(224\) −7.10607 −0.474794
\(225\) 0.509800 0.0339867
\(226\) −14.5030 −0.964725
\(227\) −7.34730 −0.487657 −0.243829 0.969818i \(-0.578403\pi\)
−0.243829 + 0.969818i \(0.578403\pi\)
\(228\) 5.66044 0.374872
\(229\) 15.1976 1.00428 0.502142 0.864785i \(-0.332545\pi\)
0.502142 + 0.864785i \(0.332545\pi\)
\(230\) −30.8161 −2.03196
\(231\) −4.71688 −0.310348
\(232\) −5.62536 −0.369323
\(233\) 13.6263 0.892688 0.446344 0.894861i \(-0.352726\pi\)
0.446344 + 0.894861i \(0.352726\pi\)
\(234\) 0 0
\(235\) −7.29086 −0.475603
\(236\) 18.9590 1.23413
\(237\) −3.58853 −0.233100
\(238\) 11.7442 0.761265
\(239\) −2.36959 −0.153276 −0.0766379 0.997059i \(-0.524419\pi\)
−0.0766379 + 0.997059i \(0.524419\pi\)
\(240\) −11.0719 −0.714689
\(241\) −21.0378 −1.35516 −0.677581 0.735448i \(-0.736972\pi\)
−0.677581 + 0.735448i \(0.736972\pi\)
\(242\) −21.1411 −1.35900
\(243\) 1.00000 0.0641500
\(244\) −11.2909 −0.722823
\(245\) 2.34730 0.149963
\(246\) −12.6655 −0.807523
\(247\) 0 0
\(248\) 1.72193 0.109343
\(249\) 0.829755 0.0525836
\(250\) 19.8084 1.25279
\(251\) 7.10607 0.448531 0.224265 0.974528i \(-0.428002\pi\)
0.224265 + 0.974528i \(0.428002\pi\)
\(252\) −1.53209 −0.0965125
\(253\) 32.9495 2.07152
\(254\) 14.3824 0.902431
\(255\) 14.6682 0.918557
\(256\) 20.7023 1.29390
\(257\) −15.2071 −0.948592 −0.474296 0.880366i \(-0.657297\pi\)
−0.474296 + 0.880366i \(0.657297\pi\)
\(258\) 8.75877 0.545297
\(259\) −8.80066 −0.546846
\(260\) 0 0
\(261\) −6.39693 −0.395960
\(262\) 17.3919 1.07447
\(263\) 12.8375 0.791594 0.395797 0.918338i \(-0.370468\pi\)
0.395797 + 0.918338i \(0.370468\pi\)
\(264\) 4.14796 0.255289
\(265\) −31.3141 −1.92361
\(266\) 6.94356 0.425737
\(267\) −2.30541 −0.141089
\(268\) −20.2540 −1.23721
\(269\) −17.8007 −1.08533 −0.542663 0.839951i \(-0.682584\pi\)
−0.542663 + 0.839951i \(0.682584\pi\)
\(270\) −4.41147 −0.268474
\(271\) −0.702333 −0.0426637 −0.0213319 0.999772i \(-0.506791\pi\)
−0.0213319 + 0.999772i \(0.506791\pi\)
\(272\) −29.4757 −1.78722
\(273\) 0 0
\(274\) 33.0428 1.99619
\(275\) 2.40467 0.145007
\(276\) 10.7023 0.644205
\(277\) −0.453363 −0.0272400 −0.0136200 0.999907i \(-0.504336\pi\)
−0.0136200 + 0.999907i \(0.504336\pi\)
\(278\) 7.32770 0.439486
\(279\) 1.95811 0.117229
\(280\) −2.06418 −0.123358
\(281\) 25.3628 1.51302 0.756508 0.653984i \(-0.226904\pi\)
0.756508 + 0.653984i \(0.226904\pi\)
\(282\) 5.83750 0.347618
\(283\) −12.7956 −0.760620 −0.380310 0.924859i \(-0.624183\pi\)
−0.380310 + 0.924859i \(0.624183\pi\)
\(284\) −15.2986 −0.907805
\(285\) 8.67230 0.513703
\(286\) 0 0
\(287\) −6.73917 −0.397801
\(288\) 7.10607 0.418729
\(289\) 22.0496 1.29704
\(290\) 28.2199 1.65713
\(291\) 12.6236 0.740009
\(292\) 14.0378 0.821498
\(293\) 8.06687 0.471271 0.235636 0.971841i \(-0.424283\pi\)
0.235636 + 0.971841i \(0.424283\pi\)
\(294\) −1.87939 −0.109608
\(295\) 29.0469 1.69118
\(296\) 7.73917 0.449830
\(297\) 4.71688 0.273701
\(298\) 20.9932 1.21610
\(299\) 0 0
\(300\) 0.781059 0.0450945
\(301\) 4.66044 0.268624
\(302\) 11.2216 0.645732
\(303\) −2.40373 −0.138091
\(304\) −17.4270 −0.999504
\(305\) −17.2986 −0.990515
\(306\) −11.7442 −0.671373
\(307\) 10.6536 0.608035 0.304018 0.952666i \(-0.401672\pi\)
0.304018 + 0.952666i \(0.401672\pi\)
\(308\) −7.22668 −0.411778
\(309\) 1.69459 0.0964020
\(310\) −8.63816 −0.490614
\(311\) 1.83481 0.104042 0.0520212 0.998646i \(-0.483434\pi\)
0.0520212 + 0.998646i \(0.483434\pi\)
\(312\) 0 0
\(313\) −25.9094 −1.46449 −0.732243 0.681043i \(-0.761527\pi\)
−0.732243 + 0.681043i \(0.761527\pi\)
\(314\) −16.4483 −0.928232
\(315\) −2.34730 −0.132255
\(316\) −5.49794 −0.309283
\(317\) −14.6186 −0.821060 −0.410530 0.911847i \(-0.634656\pi\)
−0.410530 + 0.911847i \(0.634656\pi\)
\(318\) 25.0719 1.40596
\(319\) −30.1735 −1.68939
\(320\) −9.20439 −0.514541
\(321\) 3.85710 0.215282
\(322\) 13.1284 0.731615
\(323\) 23.0874 1.28462
\(324\) 1.53209 0.0851160
\(325\) 0 0
\(326\) −4.17530 −0.231248
\(327\) 0.268571 0.0148520
\(328\) 5.92633 0.327227
\(329\) 3.10607 0.171243
\(330\) −20.8084 −1.14546
\(331\) 9.52940 0.523783 0.261892 0.965097i \(-0.415654\pi\)
0.261892 + 0.965097i \(0.415654\pi\)
\(332\) 1.27126 0.0697694
\(333\) 8.80066 0.482273
\(334\) 29.9786 1.64036
\(335\) −31.0310 −1.69540
\(336\) 4.71688 0.257327
\(337\) −17.2790 −0.941247 −0.470624 0.882334i \(-0.655971\pi\)
−0.470624 + 0.882334i \(0.655971\pi\)
\(338\) 0 0
\(339\) 7.71688 0.419123
\(340\) 22.4730 1.21877
\(341\) 9.23618 0.500167
\(342\) −6.94356 −0.375465
\(343\) −1.00000 −0.0539949
\(344\) −4.09833 −0.220967
\(345\) 16.3969 0.882781
\(346\) −5.82295 −0.313044
\(347\) 19.4320 1.04316 0.521582 0.853201i \(-0.325342\pi\)
0.521582 + 0.853201i \(0.325342\pi\)
\(348\) −9.80066 −0.525370
\(349\) 28.9786 1.55119 0.775596 0.631230i \(-0.217450\pi\)
0.775596 + 0.631230i \(0.217450\pi\)
\(350\) 0.958111 0.0512132
\(351\) 0 0
\(352\) 33.5185 1.78654
\(353\) 2.53983 0.135181 0.0675907 0.997713i \(-0.478469\pi\)
0.0675907 + 0.997713i \(0.478469\pi\)
\(354\) −23.2567 −1.23608
\(355\) −23.4388 −1.24400
\(356\) −3.53209 −0.187200
\(357\) −6.24897 −0.330731
\(358\) −29.2986 −1.54848
\(359\) 26.4243 1.39462 0.697310 0.716770i \(-0.254380\pi\)
0.697310 + 0.716770i \(0.254380\pi\)
\(360\) 2.06418 0.108792
\(361\) −5.34998 −0.281578
\(362\) −34.0060 −1.78731
\(363\) 11.2490 0.590418
\(364\) 0 0
\(365\) 21.5071 1.12573
\(366\) 13.8503 0.723966
\(367\) −2.70502 −0.141201 −0.0706005 0.997505i \(-0.522492\pi\)
−0.0706005 + 0.997505i \(0.522492\pi\)
\(368\) −32.9495 −1.71761
\(369\) 6.73917 0.350827
\(370\) −38.8239 −2.01836
\(371\) 13.3405 0.692604
\(372\) 3.00000 0.155543
\(373\) −30.2071 −1.56406 −0.782032 0.623239i \(-0.785817\pi\)
−0.782032 + 0.623239i \(0.785817\pi\)
\(374\) −55.3961 −2.86446
\(375\) −10.5398 −0.544274
\(376\) −2.73143 −0.140863
\(377\) 0 0
\(378\) 1.87939 0.0966651
\(379\) 23.8871 1.22700 0.613500 0.789695i \(-0.289761\pi\)
0.613500 + 0.789695i \(0.289761\pi\)
\(380\) 13.2867 0.681595
\(381\) −7.65270 −0.392060
\(382\) 24.2472 1.24060
\(383\) 34.5526 1.76556 0.882778 0.469790i \(-0.155670\pi\)
0.882778 + 0.469790i \(0.155670\pi\)
\(384\) −6.84255 −0.349182
\(385\) −11.0719 −0.564277
\(386\) 23.7392 1.20829
\(387\) −4.66044 −0.236904
\(388\) 19.3405 0.981865
\(389\) 24.3114 1.23264 0.616318 0.787497i \(-0.288623\pi\)
0.616318 + 0.787497i \(0.288623\pi\)
\(390\) 0 0
\(391\) 43.6519 2.20757
\(392\) 0.879385 0.0444157
\(393\) −9.25402 −0.466804
\(394\) 8.87939 0.447337
\(395\) −8.42333 −0.423824
\(396\) 7.22668 0.363154
\(397\) 17.2831 0.867415 0.433707 0.901054i \(-0.357205\pi\)
0.433707 + 0.901054i \(0.357205\pi\)
\(398\) 29.4320 1.47529
\(399\) −3.69459 −0.184961
\(400\) −2.40467 −0.120233
\(401\) −4.35773 −0.217614 −0.108807 0.994063i \(-0.534703\pi\)
−0.108807 + 0.994063i \(0.534703\pi\)
\(402\) 24.8452 1.23917
\(403\) 0 0
\(404\) −3.68273 −0.183223
\(405\) 2.34730 0.116638
\(406\) −12.0223 −0.596656
\(407\) 41.5117 2.05766
\(408\) 5.49525 0.272056
\(409\) 31.8307 1.57393 0.786963 0.617000i \(-0.211652\pi\)
0.786963 + 0.617000i \(0.211652\pi\)
\(410\) −29.7297 −1.46824
\(411\) −17.5817 −0.867242
\(412\) 2.59627 0.127909
\(413\) −12.3746 −0.608916
\(414\) −13.1284 −0.645223
\(415\) 1.94768 0.0956079
\(416\) 0 0
\(417\) −3.89899 −0.190934
\(418\) −32.7520 −1.60195
\(419\) 23.4953 1.14782 0.573909 0.818919i \(-0.305426\pi\)
0.573909 + 0.818919i \(0.305426\pi\)
\(420\) −3.59627 −0.175480
\(421\) −9.07697 −0.442384 −0.221192 0.975230i \(-0.570995\pi\)
−0.221192 + 0.975230i \(0.570995\pi\)
\(422\) −11.5270 −0.561127
\(423\) −3.10607 −0.151022
\(424\) −11.7314 −0.569729
\(425\) 3.18573 0.154530
\(426\) 18.7665 0.909240
\(427\) 7.36959 0.356639
\(428\) 5.90941 0.285642
\(429\) 0 0
\(430\) 20.5594 0.991464
\(431\) −23.4020 −1.12723 −0.563617 0.826036i \(-0.690591\pi\)
−0.563617 + 0.826036i \(0.690591\pi\)
\(432\) −4.71688 −0.226941
\(433\) 9.14384 0.439425 0.219712 0.975565i \(-0.429488\pi\)
0.219712 + 0.975565i \(0.429488\pi\)
\(434\) 3.68004 0.176648
\(435\) −15.0155 −0.719937
\(436\) 0.411474 0.0197060
\(437\) 25.8084 1.23458
\(438\) −17.2199 −0.822797
\(439\) 40.8212 1.94829 0.974145 0.225925i \(-0.0725405\pi\)
0.974145 + 0.225925i \(0.0725405\pi\)
\(440\) 9.73648 0.464169
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −11.0838 −0.526606 −0.263303 0.964713i \(-0.584812\pi\)
−0.263303 + 0.964713i \(0.584812\pi\)
\(444\) 13.4834 0.639893
\(445\) −5.41147 −0.256528
\(446\) 2.57398 0.121881
\(447\) −11.1702 −0.528334
\(448\) 3.92127 0.185263
\(449\) −1.47977 −0.0698347 −0.0349173 0.999390i \(-0.511117\pi\)
−0.0349173 + 0.999390i \(0.511117\pi\)
\(450\) −0.958111 −0.0451658
\(451\) 31.7879 1.49683
\(452\) 11.8229 0.556105
\(453\) −5.97090 −0.280538
\(454\) 13.8084 0.648060
\(455\) 0 0
\(456\) 3.24897 0.152147
\(457\) −10.0341 −0.469378 −0.234689 0.972071i \(-0.575407\pi\)
−0.234689 + 0.972071i \(0.575407\pi\)
\(458\) −28.5621 −1.33462
\(459\) 6.24897 0.291677
\(460\) 25.1215 1.17130
\(461\) 4.45605 0.207539 0.103770 0.994601i \(-0.466910\pi\)
0.103770 + 0.994601i \(0.466910\pi\)
\(462\) 8.86484 0.412430
\(463\) 26.3901 1.22645 0.613226 0.789907i \(-0.289871\pi\)
0.613226 + 0.789907i \(0.289871\pi\)
\(464\) 30.1735 1.40077
\(465\) 4.59627 0.213147
\(466\) −25.6091 −1.18632
\(467\) −19.3037 −0.893267 −0.446633 0.894717i \(-0.647377\pi\)
−0.446633 + 0.894717i \(0.647377\pi\)
\(468\) 0 0
\(469\) 13.2199 0.610437
\(470\) 13.7023 0.632042
\(471\) 8.75196 0.403269
\(472\) 10.8821 0.500888
\(473\) −21.9828 −1.01077
\(474\) 6.74422 0.309772
\(475\) 1.88350 0.0864211
\(476\) −9.57398 −0.438823
\(477\) −13.3405 −0.610819
\(478\) 4.45336 0.203692
\(479\) 1.30634 0.0596882 0.0298441 0.999555i \(-0.490499\pi\)
0.0298441 + 0.999555i \(0.490499\pi\)
\(480\) 16.6800 0.761336
\(481\) 0 0
\(482\) 39.5381 1.80091
\(483\) −6.98545 −0.317849
\(484\) 17.2344 0.783383
\(485\) 29.6313 1.34549
\(486\) −1.87939 −0.0852506
\(487\) 5.16519 0.234057 0.117029 0.993129i \(-0.462663\pi\)
0.117029 + 0.993129i \(0.462663\pi\)
\(488\) −6.48070 −0.293368
\(489\) 2.22163 0.100466
\(490\) −4.41147 −0.199290
\(491\) 22.2739 1.00521 0.502605 0.864516i \(-0.332375\pi\)
0.502605 + 0.864516i \(0.332375\pi\)
\(492\) 10.3250 0.465487
\(493\) −39.9742 −1.80035
\(494\) 0 0
\(495\) 11.0719 0.497646
\(496\) −9.23618 −0.414717
\(497\) 9.98545 0.447909
\(498\) −1.55943 −0.0698797
\(499\) 29.6091 1.32548 0.662742 0.748848i \(-0.269393\pi\)
0.662742 + 0.748848i \(0.269393\pi\)
\(500\) −16.1480 −0.722159
\(501\) −15.9513 −0.712652
\(502\) −13.3550 −0.596064
\(503\) −18.3054 −0.816198 −0.408099 0.912938i \(-0.633808\pi\)
−0.408099 + 0.912938i \(0.633808\pi\)
\(504\) −0.879385 −0.0391709
\(505\) −5.64227 −0.251078
\(506\) −61.9249 −2.75290
\(507\) 0 0
\(508\) −11.7246 −0.520196
\(509\) −38.9513 −1.72649 −0.863243 0.504788i \(-0.831571\pi\)
−0.863243 + 0.504788i \(0.831571\pi\)
\(510\) −27.5672 −1.22069
\(511\) −9.16250 −0.405325
\(512\) −25.2226 −1.11469
\(513\) 3.69459 0.163120
\(514\) 28.5800 1.26061
\(515\) 3.97771 0.175279
\(516\) −7.14022 −0.314330
\(517\) −14.6509 −0.644348
\(518\) 16.5398 0.726718
\(519\) 3.09833 0.136001
\(520\) 0 0
\(521\) 18.4415 0.807937 0.403968 0.914773i \(-0.367631\pi\)
0.403968 + 0.914773i \(0.367631\pi\)
\(522\) 12.0223 0.526201
\(523\) 19.7425 0.863278 0.431639 0.902046i \(-0.357935\pi\)
0.431639 + 0.902046i \(0.357935\pi\)
\(524\) −14.1780 −0.619368
\(525\) −0.509800 −0.0222495
\(526\) −24.1266 −1.05197
\(527\) 12.2362 0.533016
\(528\) −22.2490 −0.968262
\(529\) 25.7965 1.12159
\(530\) 58.8512 2.55633
\(531\) 12.3746 0.537014
\(532\) −5.66044 −0.245411
\(533\) 0 0
\(534\) 4.33275 0.187496
\(535\) 9.05375 0.391428
\(536\) −11.6254 −0.502139
\(537\) 15.5895 0.672735
\(538\) 33.4543 1.44232
\(539\) 4.71688 0.203170
\(540\) 3.59627 0.154759
\(541\) −30.7766 −1.32319 −0.661595 0.749861i \(-0.730120\pi\)
−0.661595 + 0.749861i \(0.730120\pi\)
\(542\) 1.31996 0.0566969
\(543\) 18.0942 0.776497
\(544\) 44.4056 1.90387
\(545\) 0.630415 0.0270040
\(546\) 0 0
\(547\) −14.3874 −0.615162 −0.307581 0.951522i \(-0.599520\pi\)
−0.307581 + 0.951522i \(0.599520\pi\)
\(548\) −26.9368 −1.15068
\(549\) −7.36959 −0.314526
\(550\) −4.51930 −0.192703
\(551\) −23.6340 −1.00684
\(552\) 6.14290 0.261459
\(553\) 3.58853 0.152600
\(554\) 0.852044 0.0361999
\(555\) 20.6578 0.876873
\(556\) −5.97359 −0.253337
\(557\) −13.5841 −0.575576 −0.287788 0.957694i \(-0.592920\pi\)
−0.287788 + 0.957694i \(0.592920\pi\)
\(558\) −3.68004 −0.155789
\(559\) 0 0
\(560\) 11.0719 0.467874
\(561\) 29.4757 1.24446
\(562\) −47.6664 −2.01069
\(563\) −26.7469 −1.12725 −0.563624 0.826032i \(-0.690593\pi\)
−0.563624 + 0.826032i \(0.690593\pi\)
\(564\) −4.75877 −0.200380
\(565\) 18.1138 0.762054
\(566\) 24.0479 1.01081
\(567\) −1.00000 −0.0419961
\(568\) −8.78106 −0.368445
\(569\) −2.56118 −0.107370 −0.0536852 0.998558i \(-0.517097\pi\)
−0.0536852 + 0.998558i \(0.517097\pi\)
\(570\) −16.2986 −0.682673
\(571\) −20.7415 −0.868006 −0.434003 0.900911i \(-0.642899\pi\)
−0.434003 + 0.900911i \(0.642899\pi\)
\(572\) 0 0
\(573\) −12.9017 −0.538975
\(574\) 12.6655 0.528648
\(575\) 3.56118 0.148512
\(576\) −3.92127 −0.163386
\(577\) −0.615867 −0.0256389 −0.0128194 0.999918i \(-0.504081\pi\)
−0.0128194 + 0.999918i \(0.504081\pi\)
\(578\) −41.4397 −1.72367
\(579\) −12.6313 −0.524941
\(580\) −23.0051 −0.955233
\(581\) −0.829755 −0.0344241
\(582\) −23.7246 −0.983417
\(583\) −62.9255 −2.60611
\(584\) 8.05737 0.333416
\(585\) 0 0
\(586\) −15.1607 −0.626285
\(587\) 13.4175 0.553798 0.276899 0.960899i \(-0.410693\pi\)
0.276899 + 0.960899i \(0.410693\pi\)
\(588\) 1.53209 0.0631823
\(589\) 7.23442 0.298089
\(590\) −54.5904 −2.24745
\(591\) −4.72462 −0.194345
\(592\) −41.5117 −1.70612
\(593\) 12.7665 0.524258 0.262129 0.965033i \(-0.415575\pi\)
0.262129 + 0.965033i \(0.415575\pi\)
\(594\) −8.86484 −0.363729
\(595\) −14.6682 −0.601337
\(596\) −17.1138 −0.701009
\(597\) −15.6604 −0.640939
\(598\) 0 0
\(599\) −1.22762 −0.0501590 −0.0250795 0.999685i \(-0.507984\pi\)
−0.0250795 + 0.999685i \(0.507984\pi\)
\(600\) 0.448311 0.0183022
\(601\) 22.0823 0.900758 0.450379 0.892838i \(-0.351289\pi\)
0.450379 + 0.892838i \(0.351289\pi\)
\(602\) −8.75877 −0.356981
\(603\) −13.2199 −0.538355
\(604\) −9.14796 −0.372225
\(605\) 26.4047 1.07350
\(606\) 4.51754 0.183513
\(607\) 35.8057 1.45331 0.726655 0.687003i \(-0.241074\pi\)
0.726655 + 0.687003i \(0.241074\pi\)
\(608\) 26.2540 1.06474
\(609\) 6.39693 0.259217
\(610\) 32.5107 1.31632
\(611\) 0 0
\(612\) 9.57398 0.387005
\(613\) 10.4911 0.423733 0.211866 0.977299i \(-0.432046\pi\)
0.211866 + 0.977299i \(0.432046\pi\)
\(614\) −20.0223 −0.808034
\(615\) 15.8188 0.637877
\(616\) −4.14796 −0.167126
\(617\) −7.10195 −0.285914 −0.142957 0.989729i \(-0.545661\pi\)
−0.142957 + 0.989729i \(0.545661\pi\)
\(618\) −3.18479 −0.128111
\(619\) −10.1438 −0.407715 −0.203858 0.979001i \(-0.565348\pi\)
−0.203858 + 0.979001i \(0.565348\pi\)
\(620\) 7.04189 0.282809
\(621\) 6.98545 0.280317
\(622\) −3.44831 −0.138265
\(623\) 2.30541 0.0923642
\(624\) 0 0
\(625\) −27.2891 −1.09156
\(626\) 48.6938 1.94619
\(627\) 17.4270 0.695966
\(628\) 13.4088 0.535069
\(629\) 54.9951 2.19280
\(630\) 4.41147 0.175757
\(631\) 20.0624 0.798672 0.399336 0.916805i \(-0.369241\pi\)
0.399336 + 0.916805i \(0.369241\pi\)
\(632\) −3.15570 −0.125527
\(633\) 6.13341 0.243781
\(634\) 27.4739 1.09113
\(635\) −17.9632 −0.712846
\(636\) −20.4388 −0.810452
\(637\) 0 0
\(638\) 56.7077 2.24508
\(639\) −9.98545 −0.395018
\(640\) −16.0615 −0.634886
\(641\) −44.4056 −1.75392 −0.876958 0.480567i \(-0.840431\pi\)
−0.876958 + 0.480567i \(0.840431\pi\)
\(642\) −7.24897 −0.286094
\(643\) 30.1735 1.18993 0.594964 0.803752i \(-0.297166\pi\)
0.594964 + 0.803752i \(0.297166\pi\)
\(644\) −10.7023 −0.421731
\(645\) −10.9394 −0.430740
\(646\) −43.3901 −1.70716
\(647\) −6.59896 −0.259432 −0.129716 0.991551i \(-0.541407\pi\)
−0.129716 + 0.991551i \(0.541407\pi\)
\(648\) 0.879385 0.0345455
\(649\) 58.3697 2.29121
\(650\) 0 0
\(651\) −1.95811 −0.0767444
\(652\) 3.40373 0.133300
\(653\) 8.56212 0.335062 0.167531 0.985867i \(-0.446421\pi\)
0.167531 + 0.985867i \(0.446421\pi\)
\(654\) −0.504748 −0.0197372
\(655\) −21.7219 −0.848746
\(656\) −31.7879 −1.24111
\(657\) 9.16250 0.357463
\(658\) −5.83750 −0.227569
\(659\) 32.1702 1.25317 0.626587 0.779351i \(-0.284451\pi\)
0.626587 + 0.779351i \(0.284451\pi\)
\(660\) 16.9632 0.660290
\(661\) −7.43645 −0.289244 −0.144622 0.989487i \(-0.546197\pi\)
−0.144622 + 0.989487i \(0.546197\pi\)
\(662\) −17.9094 −0.696069
\(663\) 0 0
\(664\) 0.729675 0.0283169
\(665\) −8.67230 −0.336298
\(666\) −16.5398 −0.640905
\(667\) −44.6854 −1.73023
\(668\) −24.4388 −0.945566
\(669\) −1.36959 −0.0529512
\(670\) 58.3191 2.25306
\(671\) −34.7615 −1.34195
\(672\) −7.10607 −0.274122
\(673\) −46.8827 −1.80719 −0.903597 0.428383i \(-0.859083\pi\)
−0.903597 + 0.428383i \(0.859083\pi\)
\(674\) 32.4739 1.25085
\(675\) 0.509800 0.0196222
\(676\) 0 0
\(677\) −35.1206 −1.34979 −0.674897 0.737912i \(-0.735812\pi\)
−0.674897 + 0.737912i \(0.735812\pi\)
\(678\) −14.5030 −0.556984
\(679\) −12.6236 −0.484450
\(680\) 12.8990 0.494654
\(681\) −7.34730 −0.281549
\(682\) −17.3583 −0.664685
\(683\) 32.2327 1.23335 0.616674 0.787218i \(-0.288480\pi\)
0.616674 + 0.787218i \(0.288480\pi\)
\(684\) 5.66044 0.216433
\(685\) −41.2695 −1.57683
\(686\) 1.87939 0.0717553
\(687\) 15.1976 0.579824
\(688\) 21.9828 0.838085
\(689\) 0 0
\(690\) −30.8161 −1.17315
\(691\) 27.4442 1.04403 0.522013 0.852937i \(-0.325181\pi\)
0.522013 + 0.852937i \(0.325181\pi\)
\(692\) 4.74691 0.180450
\(693\) −4.71688 −0.179180
\(694\) −36.5202 −1.38629
\(695\) −9.15207 −0.347158
\(696\) −5.62536 −0.213229
\(697\) 42.1129 1.59514
\(698\) −54.4620 −2.06142
\(699\) 13.6263 0.515394
\(700\) −0.781059 −0.0295213
\(701\) −21.4611 −0.810575 −0.405287 0.914189i \(-0.632829\pi\)
−0.405287 + 0.914189i \(0.632829\pi\)
\(702\) 0 0
\(703\) 32.5149 1.22632
\(704\) −18.4962 −0.697101
\(705\) −7.29086 −0.274590
\(706\) −4.77332 −0.179646
\(707\) 2.40373 0.0904017
\(708\) 18.9590 0.712525
\(709\) −36.2422 −1.36110 −0.680552 0.732700i \(-0.738260\pi\)
−0.680552 + 0.732700i \(0.738260\pi\)
\(710\) 44.0506 1.65319
\(711\) −3.58853 −0.134580
\(712\) −2.02734 −0.0759778
\(713\) 13.6783 0.512256
\(714\) 11.7442 0.439516
\(715\) 0 0
\(716\) 23.8844 0.892603
\(717\) −2.36959 −0.0884938
\(718\) −49.6614 −1.85335
\(719\) −27.6996 −1.03302 −0.516511 0.856280i \(-0.672770\pi\)
−0.516511 + 0.856280i \(0.672770\pi\)
\(720\) −11.0719 −0.412626
\(721\) −1.69459 −0.0631099
\(722\) 10.0547 0.374197
\(723\) −21.0378 −0.782403
\(724\) 27.7219 1.03028
\(725\) −3.26115 −0.121116
\(726\) −21.1411 −0.784622
\(727\) −11.1516 −0.413589 −0.206795 0.978384i \(-0.566303\pi\)
−0.206795 + 0.978384i \(0.566303\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −40.4201 −1.49602
\(731\) −29.1230 −1.07715
\(732\) −11.2909 −0.417322
\(733\) −18.4219 −0.680429 −0.340214 0.940348i \(-0.610500\pi\)
−0.340214 + 0.940348i \(0.610500\pi\)
\(734\) 5.08378 0.187646
\(735\) 2.34730 0.0865813
\(736\) 49.6391 1.82972
\(737\) −62.3566 −2.29693
\(738\) −12.6655 −0.466223
\(739\) −26.8435 −0.987453 −0.493727 0.869617i \(-0.664366\pi\)
−0.493727 + 0.869617i \(0.664366\pi\)
\(740\) 31.6495 1.16346
\(741\) 0 0
\(742\) −25.0719 −0.920419
\(743\) −15.9590 −0.585481 −0.292740 0.956192i \(-0.594567\pi\)
−0.292740 + 0.956192i \(0.594567\pi\)
\(744\) 1.72193 0.0631292
\(745\) −26.2199 −0.960622
\(746\) 56.7707 2.07852
\(747\) 0.829755 0.0303592
\(748\) 45.1593 1.65119
\(749\) −3.85710 −0.140935
\(750\) 19.8084 0.723301
\(751\) −21.4766 −0.783692 −0.391846 0.920031i \(-0.628163\pi\)
−0.391846 + 0.920031i \(0.628163\pi\)
\(752\) 14.6509 0.534265
\(753\) 7.10607 0.258959
\(754\) 0 0
\(755\) −14.0155 −0.510076
\(756\) −1.53209 −0.0557215
\(757\) 17.5672 0.638490 0.319245 0.947672i \(-0.396571\pi\)
0.319245 + 0.947672i \(0.396571\pi\)
\(758\) −44.8931 −1.63059
\(759\) 32.9495 1.19599
\(760\) 7.62630 0.276635
\(761\) −27.4567 −0.995303 −0.497652 0.867377i \(-0.665804\pi\)
−0.497652 + 0.867377i \(0.665804\pi\)
\(762\) 14.3824 0.521019
\(763\) −0.268571 −0.00972291
\(764\) −19.7665 −0.715127
\(765\) 14.6682 0.530329
\(766\) −64.9377 −2.34629
\(767\) 0 0
\(768\) 20.7023 0.747031
\(769\) −26.9905 −0.973302 −0.486651 0.873596i \(-0.661782\pi\)
−0.486651 + 0.873596i \(0.661782\pi\)
\(770\) 20.8084 0.749883
\(771\) −15.2071 −0.547670
\(772\) −19.3523 −0.696506
\(773\) −7.56799 −0.272202 −0.136101 0.990695i \(-0.543457\pi\)
−0.136101 + 0.990695i \(0.543457\pi\)
\(774\) 8.75877 0.314828
\(775\) 0.998245 0.0358580
\(776\) 11.1010 0.398503
\(777\) −8.80066 −0.315722
\(778\) −45.6905 −1.63808
\(779\) 24.8985 0.892081
\(780\) 0 0
\(781\) −47.1002 −1.68538
\(782\) −82.0387 −2.93370
\(783\) −6.39693 −0.228607
\(784\) −4.71688 −0.168460
\(785\) 20.5435 0.733227
\(786\) 17.3919 0.620348
\(787\) −3.64227 −0.129833 −0.0649165 0.997891i \(-0.520678\pi\)
−0.0649165 + 0.997891i \(0.520678\pi\)
\(788\) −7.23854 −0.257862
\(789\) 12.8375 0.457027
\(790\) 15.8307 0.563231
\(791\) −7.71688 −0.274381
\(792\) 4.14796 0.147391
\(793\) 0 0
\(794\) −32.4816 −1.15273
\(795\) −31.3141 −1.11060
\(796\) −23.9932 −0.850416
\(797\) −24.9949 −0.885366 −0.442683 0.896678i \(-0.645973\pi\)
−0.442683 + 0.896678i \(0.645973\pi\)
\(798\) 6.94356 0.245799
\(799\) −19.4097 −0.686667
\(800\) 3.62267 0.128081
\(801\) −2.30541 −0.0814576
\(802\) 8.18984 0.289193
\(803\) 43.2184 1.52515
\(804\) −20.2540 −0.714304
\(805\) −16.3969 −0.577916
\(806\) 0 0
\(807\) −17.8007 −0.626613
\(808\) −2.11381 −0.0743635
\(809\) 31.2240 1.09778 0.548889 0.835896i \(-0.315051\pi\)
0.548889 + 0.835896i \(0.315051\pi\)
\(810\) −4.41147 −0.155003
\(811\) −23.1780 −0.813889 −0.406945 0.913453i \(-0.633406\pi\)
−0.406945 + 0.913453i \(0.633406\pi\)
\(812\) 9.80066 0.343936
\(813\) −0.702333 −0.0246319
\(814\) −78.0164 −2.73447
\(815\) 5.21482 0.182667
\(816\) −29.4757 −1.03185
\(817\) −17.2184 −0.602397
\(818\) −59.8221 −2.09163
\(819\) 0 0
\(820\) 24.2359 0.846353
\(821\) −53.1688 −1.85560 −0.927802 0.373072i \(-0.878304\pi\)
−0.927802 + 0.373072i \(0.878304\pi\)
\(822\) 33.0428 1.15250
\(823\) −13.7047 −0.477716 −0.238858 0.971055i \(-0.576773\pi\)
−0.238858 + 0.971055i \(0.576773\pi\)
\(824\) 1.49020 0.0519136
\(825\) 2.40467 0.0837198
\(826\) 23.2567 0.809204
\(827\) −29.0137 −1.00891 −0.504453 0.863439i \(-0.668306\pi\)
−0.504453 + 0.863439i \(0.668306\pi\)
\(828\) 10.7023 0.371932
\(829\) 53.2526 1.84954 0.924769 0.380528i \(-0.124258\pi\)
0.924769 + 0.380528i \(0.124258\pi\)
\(830\) −3.66044 −0.127056
\(831\) −0.453363 −0.0157270
\(832\) 0 0
\(833\) 6.24897 0.216514
\(834\) 7.32770 0.253737
\(835\) −37.4424 −1.29575
\(836\) 26.6996 0.923427
\(837\) 1.95811 0.0676822
\(838\) −44.1566 −1.52537
\(839\) 50.1111 1.73003 0.865014 0.501748i \(-0.167309\pi\)
0.865014 + 0.501748i \(0.167309\pi\)
\(840\) −2.06418 −0.0712209
\(841\) 11.9207 0.411057
\(842\) 17.0591 0.587896
\(843\) 25.3628 0.873540
\(844\) 9.39693 0.323456
\(845\) 0 0
\(846\) 5.83750 0.200697
\(847\) −11.2490 −0.386519
\(848\) 62.9255 2.16087
\(849\) −12.7956 −0.439144
\(850\) −5.98721 −0.205359
\(851\) 61.4766 2.10739
\(852\) −15.2986 −0.524121
\(853\) 30.5090 1.04461 0.522304 0.852760i \(-0.325073\pi\)
0.522304 + 0.852760i \(0.325073\pi\)
\(854\) −13.8503 −0.473947
\(855\) 8.67230 0.296587
\(856\) 3.39187 0.115932
\(857\) −4.86390 −0.166148 −0.0830739 0.996543i \(-0.526474\pi\)
−0.0830739 + 0.996543i \(0.526474\pi\)
\(858\) 0 0
\(859\) −36.8066 −1.25583 −0.627913 0.778283i \(-0.716091\pi\)
−0.627913 + 0.778283i \(0.716091\pi\)
\(860\) −16.7602 −0.571518
\(861\) −6.73917 −0.229670
\(862\) 43.9813 1.49801
\(863\) 39.3414 1.33920 0.669599 0.742723i \(-0.266466\pi\)
0.669599 + 0.742723i \(0.266466\pi\)
\(864\) 7.10607 0.241753
\(865\) 7.27269 0.247279
\(866\) −17.1848 −0.583963
\(867\) 22.0496 0.748845
\(868\) −3.00000 −0.101827
\(869\) −16.9267 −0.574197
\(870\) 28.2199 0.956743
\(871\) 0 0
\(872\) 0.236177 0.00799797
\(873\) 12.6236 0.427244
\(874\) −48.5039 −1.64067
\(875\) 10.5398 0.356311
\(876\) 14.0378 0.474292
\(877\) −30.4570 −1.02846 −0.514230 0.857653i \(-0.671922\pi\)
−0.514230 + 0.857653i \(0.671922\pi\)
\(878\) −76.7187 −2.58913
\(879\) 8.06687 0.272089
\(880\) −52.2249 −1.76050
\(881\) 1.20439 0.0405770 0.0202885 0.999794i \(-0.493542\pi\)
0.0202885 + 0.999794i \(0.493542\pi\)
\(882\) −1.87939 −0.0632822
\(883\) 36.8648 1.24060 0.620300 0.784364i \(-0.287011\pi\)
0.620300 + 0.784364i \(0.287011\pi\)
\(884\) 0 0
\(885\) 29.0469 0.976402
\(886\) 20.8307 0.699821
\(887\) −22.4243 −0.752933 −0.376467 0.926430i \(-0.622861\pi\)
−0.376467 + 0.926430i \(0.622861\pi\)
\(888\) 7.73917 0.259710
\(889\) 7.65270 0.256663
\(890\) 10.1702 0.340907
\(891\) 4.71688 0.158021
\(892\) −2.09833 −0.0702572
\(893\) −11.4757 −0.384018
\(894\) 20.9932 0.702117
\(895\) 36.5931 1.22317
\(896\) 6.84255 0.228594
\(897\) 0 0
\(898\) 2.78106 0.0928051
\(899\) −12.5259 −0.417762
\(900\) 0.781059 0.0260353
\(901\) −83.3643 −2.77727
\(902\) −59.7416 −1.98918
\(903\) 4.66044 0.155090
\(904\) 6.78611 0.225703
\(905\) 42.4725 1.41183
\(906\) 11.2216 0.372814
\(907\) 38.3391 1.27303 0.636514 0.771265i \(-0.280376\pi\)
0.636514 + 0.771265i \(0.280376\pi\)
\(908\) −11.2567 −0.373567
\(909\) −2.40373 −0.0797268
\(910\) 0 0
\(911\) 10.3000 0.341255 0.170628 0.985336i \(-0.445420\pi\)
0.170628 + 0.985336i \(0.445420\pi\)
\(912\) −17.4270 −0.577064
\(913\) 3.91386 0.129530
\(914\) 18.8580 0.623768
\(915\) −17.2986 −0.571874
\(916\) 23.2841 0.769327
\(917\) 9.25402 0.305595
\(918\) −11.7442 −0.387617
\(919\) 9.59390 0.316474 0.158237 0.987401i \(-0.449419\pi\)
0.158237 + 0.987401i \(0.449419\pi\)
\(920\) 14.4192 0.475387
\(921\) 10.6536 0.351049
\(922\) −8.37464 −0.275804
\(923\) 0 0
\(924\) −7.22668 −0.237740
\(925\) 4.48658 0.147518
\(926\) −49.5972 −1.62987
\(927\) 1.69459 0.0556577
\(928\) −45.4570 −1.49220
\(929\) −11.3301 −0.371727 −0.185864 0.982576i \(-0.559508\pi\)
−0.185864 + 0.982576i \(0.559508\pi\)
\(930\) −8.63816 −0.283256
\(931\) 3.69459 0.121085
\(932\) 20.8767 0.683839
\(933\) 1.83481 0.0600689
\(934\) 36.2790 1.18709
\(935\) 69.1881 2.26269
\(936\) 0 0
\(937\) −48.7749 −1.59341 −0.796703 0.604372i \(-0.793424\pi\)
−0.796703 + 0.604372i \(0.793424\pi\)
\(938\) −24.8452 −0.811226
\(939\) −25.9094 −0.845522
\(940\) −11.1702 −0.364333
\(941\) −8.55943 −0.279029 −0.139515 0.990220i \(-0.544554\pi\)
−0.139515 + 0.990220i \(0.544554\pi\)
\(942\) −16.4483 −0.535915
\(943\) 47.0761 1.53301
\(944\) −58.3697 −1.89977
\(945\) −2.34730 −0.0763576
\(946\) 41.3141 1.34324
\(947\) −2.53983 −0.0825334 −0.0412667 0.999148i \(-0.513139\pi\)
−0.0412667 + 0.999148i \(0.513139\pi\)
\(948\) −5.49794 −0.178565
\(949\) 0 0
\(950\) −3.53983 −0.114847
\(951\) −14.6186 −0.474039
\(952\) −5.49525 −0.178102
\(953\) −53.5954 −1.73613 −0.868063 0.496453i \(-0.834635\pi\)
−0.868063 + 0.496453i \(0.834635\pi\)
\(954\) 25.0719 0.811733
\(955\) −30.2841 −0.979969
\(956\) −3.63041 −0.117416
\(957\) −30.1735 −0.975372
\(958\) −2.45512 −0.0793213
\(959\) 17.5817 0.567743
\(960\) −9.20439 −0.297071
\(961\) −27.1658 −0.876316
\(962\) 0 0
\(963\) 3.85710 0.124293
\(964\) −32.2317 −1.03811
\(965\) −29.6495 −0.954452
\(966\) 13.1284 0.422398
\(967\) −13.5953 −0.437196 −0.218598 0.975815i \(-0.570148\pi\)
−0.218598 + 0.975815i \(0.570148\pi\)
\(968\) 9.89218 0.317947
\(969\) 23.0874 0.741674
\(970\) −55.6887 −1.78806
\(971\) −48.1397 −1.54488 −0.772439 0.635090i \(-0.780963\pi\)
−0.772439 + 0.635090i \(0.780963\pi\)
\(972\) 1.53209 0.0491418
\(973\) 3.89899 0.124996
\(974\) −9.70739 −0.311045
\(975\) 0 0
\(976\) 34.7615 1.11269
\(977\) −34.9017 −1.11660 −0.558302 0.829638i \(-0.688547\pi\)
−0.558302 + 0.829638i \(0.688547\pi\)
\(978\) −4.17530 −0.133511
\(979\) −10.8743 −0.347545
\(980\) 3.59627 0.114879
\(981\) 0.268571 0.00857480
\(982\) −41.8613 −1.33585
\(983\) −58.3465 −1.86096 −0.930482 0.366338i \(-0.880611\pi\)
−0.930482 + 0.366338i \(0.880611\pi\)
\(984\) 5.92633 0.188924
\(985\) −11.0901 −0.353360
\(986\) 75.1269 2.39253
\(987\) 3.10607 0.0988672
\(988\) 0 0
\(989\) −32.5553 −1.03520
\(990\) −20.8084 −0.661334
\(991\) −60.3988 −1.91863 −0.959315 0.282336i \(-0.908891\pi\)
−0.959315 + 0.282336i \(0.908891\pi\)
\(992\) 13.9145 0.441785
\(993\) 9.52940 0.302406
\(994\) −18.7665 −0.595238
\(995\) −36.7597 −1.16536
\(996\) 1.27126 0.0402814
\(997\) −24.5134 −0.776348 −0.388174 0.921586i \(-0.626894\pi\)
−0.388174 + 0.921586i \(0.626894\pi\)
\(998\) −55.6468 −1.76147
\(999\) 8.80066 0.278441
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 3549.2.a.o.1.1 3
13.3 even 3 273.2.k.a.22.3 6
13.9 even 3 273.2.k.a.211.3 yes 6
13.12 even 2 3549.2.a.n.1.3 3
39.29 odd 6 819.2.o.g.568.1 6
39.35 odd 6 819.2.o.g.757.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.k.a.22.3 6 13.3 even 3
273.2.k.a.211.3 yes 6 13.9 even 3
819.2.o.g.568.1 6 39.29 odd 6
819.2.o.g.757.1 6 39.35 odd 6
3549.2.a.n.1.3 3 13.12 even 2
3549.2.a.o.1.1 3 1.1 even 1 trivial