Properties

Label 8034.2.a.n.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $4$
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] = [8034,2,Mod(1,8034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8034, 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("8034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.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: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.72329.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 9x^{2} + 2x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.11664\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.59680 q^{5} +1.00000 q^{6} +1.51984 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.59680 q^{5} +1.00000 q^{6} +1.51984 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.59680 q^{10} +4.86001 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.51984 q^{14} +2.59680 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -6.23328 q^{19} +2.59680 q^{20} +1.51984 q^{21} +4.86001 q^{22} -3.23328 q^{23} +1.00000 q^{24} +1.74337 q^{25} +1.00000 q^{26} +1.00000 q^{27} +1.51984 q^{28} +4.97665 q^{29} +2.59680 q^{30} +6.23328 q^{31} +1.00000 q^{32} +4.86001 q^{33} +2.00000 q^{34} +3.94672 q^{35} +1.00000 q^{36} -6.86001 q^{37} -6.23328 q^{38} +1.00000 q^{39} +2.59680 q^{40} -0.193601 q^{41} +1.51984 q^{42} +3.13999 q^{43} +4.86001 q^{44} +2.59680 q^{45} -3.23328 q^{46} -7.83666 q^{47} +1.00000 q^{48} -4.69009 q^{49} +1.74337 q^{50} +2.00000 q^{51} +1.00000 q^{52} +3.47358 q^{53} +1.00000 q^{54} +12.6205 q^{55} +1.51984 q^{56} -6.23328 q^{57} +4.97665 q^{58} +2.65983 q^{59} +2.59680 q^{60} +5.09329 q^{61} +6.23328 q^{62} +1.51984 q^{63} +1.00000 q^{64} +2.59680 q^{65} +4.86001 q^{66} +8.83008 q^{67} +2.00000 q^{68} -3.23328 q^{69} +3.94672 q^{70} -12.5404 q^{71} +1.00000 q^{72} +9.93697 q^{73} -6.86001 q^{74} +1.74337 q^{75} -6.23328 q^{76} +7.38643 q^{77} +1.00000 q^{78} +3.42688 q^{79} +2.59680 q^{80} +1.00000 q^{81} -0.193601 q^{82} -6.59680 q^{83} +1.51984 q^{84} +5.19360 q^{85} +3.13999 q^{86} +4.97665 q^{87} +4.86001 q^{88} -12.9136 q^{89} +2.59680 q^{90} +1.51984 q^{91} -3.23328 q^{92} +6.23328 q^{93} -7.83666 q^{94} -16.1866 q^{95} +1.00000 q^{96} +12.9005 q^{97} -4.69009 q^{98} +4.86001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 5 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} + 2 q^{5} + 4 q^{6} + 5 q^{7} + 4 q^{8} + 4 q^{9} + 2 q^{10} + 9 q^{11} + 4 q^{12} + 4 q^{13} + 5 q^{14} + 2 q^{15} + 4 q^{16} + 8 q^{17} + 4 q^{18} - 6 q^{19} + 2 q^{20} + 5 q^{21} + 9 q^{22} + 6 q^{23} + 4 q^{24} + 6 q^{25} + 4 q^{26} + 4 q^{27} + 5 q^{28} + 2 q^{30} + 6 q^{31} + 4 q^{32} + 9 q^{33} + 8 q^{34} - 21 q^{35} + 4 q^{36} - 17 q^{37} - 6 q^{38} + 4 q^{39} + 2 q^{40} + 16 q^{41} + 5 q^{42} + 23 q^{43} + 9 q^{44} + 2 q^{45} + 6 q^{46} - q^{47} + 4 q^{48} + 19 q^{49} + 6 q^{50} + 8 q^{51} + 4 q^{52} + 18 q^{53} + 4 q^{54} - 2 q^{55} + 5 q^{56} - 6 q^{57} + 20 q^{59} + 2 q^{60} - 9 q^{61} + 6 q^{62} + 5 q^{63} + 4 q^{64} + 2 q^{65} + 9 q^{66} + 8 q^{67} + 8 q^{68} + 6 q^{69} - 21 q^{70} - 21 q^{71} + 4 q^{72} + 22 q^{73} - 17 q^{74} + 6 q^{75} - 6 q^{76} + 15 q^{77} + 4 q^{78} - 22 q^{79} + 2 q^{80} + 4 q^{81} + 16 q^{82} - 18 q^{83} + 5 q^{84} + 4 q^{85} + 23 q^{86} + 9 q^{88} - 14 q^{89} + 2 q^{90} + 5 q^{91} + 6 q^{92} + 6 q^{93} - q^{94} - 6 q^{95} + 4 q^{96} + 20 q^{97} + 19 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 2.59680 1.16132 0.580662 0.814145i \(-0.302794\pi\)
0.580662 + 0.814145i \(0.302794\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.51984 0.574445 0.287223 0.957864i \(-0.407268\pi\)
0.287223 + 0.957864i \(0.407268\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.59680 0.821180
\(11\) 4.86001 1.46535 0.732674 0.680579i \(-0.238272\pi\)
0.732674 + 0.680579i \(0.238272\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 1.51984 0.406194
\(15\) 2.59680 0.670491
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.23328 −1.43001 −0.715006 0.699118i \(-0.753576\pi\)
−0.715006 + 0.699118i \(0.753576\pi\)
\(20\) 2.59680 0.580662
\(21\) 1.51984 0.331656
\(22\) 4.86001 1.03616
\(23\) −3.23328 −0.674185 −0.337093 0.941472i \(-0.609444\pi\)
−0.337093 + 0.941472i \(0.609444\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.74337 0.348674
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) 1.51984 0.287223
\(29\) 4.97665 0.924141 0.462070 0.886843i \(-0.347107\pi\)
0.462070 + 0.886843i \(0.347107\pi\)
\(30\) 2.59680 0.474109
\(31\) 6.23328 1.11953 0.559765 0.828651i \(-0.310891\pi\)
0.559765 + 0.828651i \(0.310891\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.86001 0.846019
\(34\) 2.00000 0.342997
\(35\) 3.94672 0.667117
\(36\) 1.00000 0.166667
\(37\) −6.86001 −1.12778 −0.563889 0.825850i \(-0.690696\pi\)
−0.563889 + 0.825850i \(0.690696\pi\)
\(38\) −6.23328 −1.01117
\(39\) 1.00000 0.160128
\(40\) 2.59680 0.410590
\(41\) −0.193601 −0.0302354 −0.0151177 0.999886i \(-0.504812\pi\)
−0.0151177 + 0.999886i \(0.504812\pi\)
\(42\) 1.51984 0.234516
\(43\) 3.13999 0.478844 0.239422 0.970916i \(-0.423042\pi\)
0.239422 + 0.970916i \(0.423042\pi\)
\(44\) 4.86001 0.732674
\(45\) 2.59680 0.387108
\(46\) −3.23328 −0.476721
\(47\) −7.83666 −1.14309 −0.571547 0.820569i \(-0.693657\pi\)
−0.571547 + 0.820569i \(0.693657\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.69009 −0.670013
\(50\) 1.74337 0.246550
\(51\) 2.00000 0.280056
\(52\) 1.00000 0.138675
\(53\) 3.47358 0.477133 0.238566 0.971126i \(-0.423323\pi\)
0.238566 + 0.971126i \(0.423323\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.6205 1.70175
\(56\) 1.51984 0.203097
\(57\) −6.23328 −0.825618
\(58\) 4.97665 0.653466
\(59\) 2.65983 0.346280 0.173140 0.984897i \(-0.444609\pi\)
0.173140 + 0.984897i \(0.444609\pi\)
\(60\) 2.59680 0.335245
\(61\) 5.09329 0.652129 0.326064 0.945348i \(-0.394277\pi\)
0.326064 + 0.945348i \(0.394277\pi\)
\(62\) 6.23328 0.791627
\(63\) 1.51984 0.191482
\(64\) 1.00000 0.125000
\(65\) 2.59680 0.322093
\(66\) 4.86001 0.598226
\(67\) 8.83008 1.07877 0.539383 0.842061i \(-0.318658\pi\)
0.539383 + 0.842061i \(0.318658\pi\)
\(68\) 2.00000 0.242536
\(69\) −3.23328 −0.389241
\(70\) 3.94672 0.471723
\(71\) −12.5404 −1.48827 −0.744133 0.668031i \(-0.767137\pi\)
−0.744133 + 0.668031i \(0.767137\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.93697 1.16303 0.581517 0.813534i \(-0.302459\pi\)
0.581517 + 0.813534i \(0.302459\pi\)
\(74\) −6.86001 −0.797460
\(75\) 1.74337 0.201307
\(76\) −6.23328 −0.715006
\(77\) 7.38643 0.841762
\(78\) 1.00000 0.113228
\(79\) 3.42688 0.385554 0.192777 0.981243i \(-0.438251\pi\)
0.192777 + 0.981243i \(0.438251\pi\)
\(80\) 2.59680 0.290331
\(81\) 1.00000 0.111111
\(82\) −0.193601 −0.0213796
\(83\) −6.59680 −0.724093 −0.362046 0.932160i \(-0.617922\pi\)
−0.362046 + 0.932160i \(0.617922\pi\)
\(84\) 1.51984 0.165828
\(85\) 5.19360 0.563325
\(86\) 3.13999 0.338594
\(87\) 4.97665 0.533553
\(88\) 4.86001 0.518079
\(89\) −12.9136 −1.36884 −0.684421 0.729087i \(-0.739945\pi\)
−0.684421 + 0.729087i \(0.739945\pi\)
\(90\) 2.59680 0.273727
\(91\) 1.51984 0.159322
\(92\) −3.23328 −0.337093
\(93\) 6.23328 0.646361
\(94\) −7.83666 −0.808290
\(95\) −16.1866 −1.66071
\(96\) 1.00000 0.102062
\(97\) 12.9005 1.30984 0.654921 0.755697i \(-0.272702\pi\)
0.654921 + 0.755697i \(0.272702\pi\)
\(98\) −4.69009 −0.473771
\(99\) 4.86001 0.488450
\(100\) 1.74337 0.174337
\(101\) −10.0069 −0.995725 −0.497863 0.867256i \(-0.665882\pi\)
−0.497863 + 0.867256i \(0.665882\pi\)
\(102\) 2.00000 0.198030
\(103\) −1.00000 −0.0985329
\(104\) 1.00000 0.0980581
\(105\) 3.94672 0.385160
\(106\) 3.47358 0.337384
\(107\) 15.7200 1.51971 0.759856 0.650091i \(-0.225269\pi\)
0.759856 + 0.650091i \(0.225269\pi\)
\(108\) 1.00000 0.0962250
\(109\) −17.7138 −1.69667 −0.848336 0.529459i \(-0.822395\pi\)
−0.848336 + 0.529459i \(0.822395\pi\)
\(110\) 12.6205 1.20332
\(111\) −6.86001 −0.651123
\(112\) 1.51984 0.143611
\(113\) 2.79698 0.263118 0.131559 0.991308i \(-0.458002\pi\)
0.131559 + 0.991308i \(0.458002\pi\)
\(114\) −6.23328 −0.583800
\(115\) −8.39618 −0.782948
\(116\) 4.97665 0.462070
\(117\) 1.00000 0.0924500
\(118\) 2.65983 0.244857
\(119\) 3.03968 0.278647
\(120\) 2.59680 0.237054
\(121\) 12.6197 1.14725
\(122\) 5.09329 0.461125
\(123\) −0.193601 −0.0174564
\(124\) 6.23328 0.559765
\(125\) −8.45681 −0.756400
\(126\) 1.51984 0.135398
\(127\) −2.55293 −0.226536 −0.113268 0.993564i \(-0.536132\pi\)
−0.113268 + 0.993564i \(0.536132\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.13999 0.276461
\(130\) 2.59680 0.227754
\(131\) −6.68035 −0.583665 −0.291832 0.956470i \(-0.594265\pi\)
−0.291832 + 0.956470i \(0.594265\pi\)
\(132\) 4.86001 0.423010
\(133\) −9.47358 −0.821463
\(134\) 8.83008 0.762803
\(135\) 2.59680 0.223497
\(136\) 2.00000 0.171499
\(137\) −12.9467 −1.10611 −0.553056 0.833144i \(-0.686539\pi\)
−0.553056 + 0.833144i \(0.686539\pi\)
\(138\) −3.23328 −0.275235
\(139\) −7.80015 −0.661600 −0.330800 0.943701i \(-0.607319\pi\)
−0.330800 + 0.943701i \(0.607319\pi\)
\(140\) 3.94672 0.333559
\(141\) −7.83666 −0.659966
\(142\) −12.5404 −1.05236
\(143\) 4.86001 0.406415
\(144\) 1.00000 0.0833333
\(145\) 12.9234 1.07323
\(146\) 9.93697 0.822390
\(147\) −4.69009 −0.386832
\(148\) −6.86001 −0.563889
\(149\) 6.05361 0.495931 0.247966 0.968769i \(-0.420238\pi\)
0.247966 + 0.968769i \(0.420238\pi\)
\(150\) 1.74337 0.142346
\(151\) −22.5738 −1.83703 −0.918514 0.395388i \(-0.870610\pi\)
−0.918514 + 0.395388i \(0.870610\pi\)
\(152\) −6.23328 −0.505586
\(153\) 2.00000 0.161690
\(154\) 7.38643 0.595216
\(155\) 16.1866 1.30014
\(156\) 1.00000 0.0800641
\(157\) 9.13057 0.728699 0.364350 0.931262i \(-0.381291\pi\)
0.364350 + 0.931262i \(0.381291\pi\)
\(158\) 3.42688 0.272628
\(159\) 3.47358 0.275473
\(160\) 2.59680 0.205295
\(161\) −4.91406 −0.387282
\(162\) 1.00000 0.0785674
\(163\) −5.94639 −0.465757 −0.232879 0.972506i \(-0.574814\pi\)
−0.232879 + 0.972506i \(0.574814\pi\)
\(164\) −0.193601 −0.0151177
\(165\) 12.6205 0.982503
\(166\) −6.59680 −0.512011
\(167\) 10.3405 0.800172 0.400086 0.916478i \(-0.368980\pi\)
0.400086 + 0.916478i \(0.368980\pi\)
\(168\) 1.51984 0.117258
\(169\) 1.00000 0.0769231
\(170\) 5.19360 0.398331
\(171\) −6.23328 −0.476671
\(172\) 3.13999 0.239422
\(173\) −15.0933 −1.14752 −0.573761 0.819023i \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(174\) 4.97665 0.377279
\(175\) 2.64964 0.200294
\(176\) 4.86001 0.366337
\(177\) 2.65983 0.199925
\(178\) −12.9136 −0.967917
\(179\) −2.84684 −0.212783 −0.106392 0.994324i \(-0.533930\pi\)
−0.106392 + 0.994324i \(0.533930\pi\)
\(180\) 2.59680 0.193554
\(181\) 8.38720 0.623416 0.311708 0.950178i \(-0.399099\pi\)
0.311708 + 0.950178i \(0.399099\pi\)
\(182\) 1.51984 0.112658
\(183\) 5.09329 0.376507
\(184\) −3.23328 −0.238360
\(185\) −17.8141 −1.30972
\(186\) 6.23328 0.457046
\(187\) 9.72002 0.710798
\(188\) −7.83666 −0.571547
\(189\) 1.51984 0.110552
\(190\) −16.1866 −1.17430
\(191\) 17.9533 1.29906 0.649528 0.760338i \(-0.274967\pi\)
0.649528 + 0.760338i \(0.274967\pi\)
\(192\) 1.00000 0.0721688
\(193\) 13.9669 1.00536 0.502680 0.864473i \(-0.332347\pi\)
0.502680 + 0.864473i \(0.332347\pi\)
\(194\) 12.9005 0.926199
\(195\) 2.59680 0.185961
\(196\) −4.69009 −0.335006
\(197\) 7.78305 0.554519 0.277260 0.960795i \(-0.410574\pi\)
0.277260 + 0.960795i \(0.410574\pi\)
\(198\) 4.86001 0.345386
\(199\) −4.71344 −0.334127 −0.167063 0.985946i \(-0.553428\pi\)
−0.167063 + 0.985946i \(0.553428\pi\)
\(200\) 1.74337 0.123275
\(201\) 8.83008 0.622826
\(202\) −10.0069 −0.704084
\(203\) 7.56371 0.530868
\(204\) 2.00000 0.140028
\(205\) −0.502743 −0.0351131
\(206\) −1.00000 −0.0696733
\(207\) −3.23328 −0.224728
\(208\) 1.00000 0.0693375
\(209\) −30.2938 −2.09547
\(210\) 3.94672 0.272349
\(211\) −21.4338 −1.47556 −0.737782 0.675040i \(-0.764127\pi\)
−0.737782 + 0.675040i \(0.764127\pi\)
\(212\) 3.47358 0.238566
\(213\) −12.5404 −0.859251
\(214\) 15.7200 1.07460
\(215\) 8.15392 0.556093
\(216\) 1.00000 0.0680414
\(217\) 9.47358 0.643108
\(218\) −17.7138 −1.19973
\(219\) 9.93697 0.671478
\(220\) 12.6205 0.850873
\(221\) 2.00000 0.134535
\(222\) −6.86001 −0.460414
\(223\) 11.9160 0.797956 0.398978 0.916961i \(-0.369365\pi\)
0.398978 + 0.916961i \(0.369365\pi\)
\(224\) 1.51984 0.101548
\(225\) 1.74337 0.116225
\(226\) 2.79698 0.186053
\(227\) −11.1469 −0.739846 −0.369923 0.929062i \(-0.620616\pi\)
−0.369923 + 0.929062i \(0.620616\pi\)
\(228\) −6.23328 −0.412809
\(229\) 10.4805 0.692570 0.346285 0.938129i \(-0.387443\pi\)
0.346285 + 0.938129i \(0.387443\pi\)
\(230\) −8.39618 −0.553628
\(231\) 7.38643 0.485992
\(232\) 4.97665 0.326733
\(233\) 18.8141 1.23255 0.616276 0.787530i \(-0.288641\pi\)
0.616276 + 0.787530i \(0.288641\pi\)
\(234\) 1.00000 0.0653720
\(235\) −20.3502 −1.32750
\(236\) 2.65983 0.173140
\(237\) 3.42688 0.222600
\(238\) 3.03968 0.197033
\(239\) −23.3203 −1.50847 −0.754233 0.656607i \(-0.771991\pi\)
−0.754233 + 0.656607i \(0.771991\pi\)
\(240\) 2.59680 0.167623
\(241\) −9.31263 −0.599879 −0.299940 0.953958i \(-0.596967\pi\)
−0.299940 + 0.953958i \(0.596967\pi\)
\(242\) 12.6197 0.811226
\(243\) 1.00000 0.0641500
\(244\) 5.09329 0.326064
\(245\) −12.1792 −0.778102
\(246\) −0.193601 −0.0123435
\(247\) −6.23328 −0.396614
\(248\) 6.23328 0.395814
\(249\) −6.59680 −0.418055
\(250\) −8.45681 −0.534856
\(251\) −7.04387 −0.444605 −0.222302 0.974978i \(-0.571357\pi\)
−0.222302 + 0.974978i \(0.571357\pi\)
\(252\) 1.51984 0.0957408
\(253\) −15.7138 −0.987916
\(254\) −2.55293 −0.160185
\(255\) 5.19360 0.325236
\(256\) 1.00000 0.0625000
\(257\) −6.04001 −0.376765 −0.188383 0.982096i \(-0.560325\pi\)
−0.188383 + 0.982096i \(0.560325\pi\)
\(258\) 3.13999 0.195487
\(259\) −10.4261 −0.647847
\(260\) 2.59680 0.161047
\(261\) 4.97665 0.308047
\(262\) −6.68035 −0.412713
\(263\) −19.5194 −1.20362 −0.601809 0.798640i \(-0.705553\pi\)
−0.601809 + 0.798640i \(0.705553\pi\)
\(264\) 4.86001 0.299113
\(265\) 9.02019 0.554106
\(266\) −9.47358 −0.580862
\(267\) −12.9136 −0.790301
\(268\) 8.83008 0.539383
\(269\) −5.23045 −0.318906 −0.159453 0.987206i \(-0.550973\pi\)
−0.159453 + 0.987206i \(0.550973\pi\)
\(270\) 2.59680 0.158036
\(271\) −1.08638 −0.0659926 −0.0329963 0.999455i \(-0.510505\pi\)
−0.0329963 + 0.999455i \(0.510505\pi\)
\(272\) 2.00000 0.121268
\(273\) 1.51984 0.0919848
\(274\) −12.9467 −0.782140
\(275\) 8.47281 0.510930
\(276\) −3.23328 −0.194620
\(277\) 7.85343 0.471867 0.235933 0.971769i \(-0.424185\pi\)
0.235933 + 0.971769i \(0.424185\pi\)
\(278\) −7.80015 −0.467822
\(279\) 6.23328 0.373177
\(280\) 3.94672 0.235862
\(281\) 29.0535 1.73319 0.866593 0.499015i \(-0.166305\pi\)
0.866593 + 0.499015i \(0.166305\pi\)
\(282\) −7.83666 −0.466666
\(283\) −10.6594 −0.633635 −0.316817 0.948487i \(-0.602614\pi\)
−0.316817 + 0.948487i \(0.602614\pi\)
\(284\) −12.5404 −0.744133
\(285\) −16.1866 −0.958810
\(286\) 4.86001 0.287379
\(287\) −0.294242 −0.0173686
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 12.9234 0.758886
\(291\) 12.9005 0.756238
\(292\) 9.93697 0.581517
\(293\) −18.4143 −1.07578 −0.537888 0.843017i \(-0.680778\pi\)
−0.537888 + 0.843017i \(0.680778\pi\)
\(294\) −4.69009 −0.273532
\(295\) 6.90704 0.402143
\(296\) −6.86001 −0.398730
\(297\) 4.86001 0.282006
\(298\) 6.05361 0.350676
\(299\) −3.23328 −0.186985
\(300\) 1.74337 0.100654
\(301\) 4.77228 0.275069
\(302\) −22.5738 −1.29898
\(303\) −10.0069 −0.574882
\(304\) −6.23328 −0.357503
\(305\) 13.2263 0.757333
\(306\) 2.00000 0.114332
\(307\) −7.92620 −0.452372 −0.226186 0.974084i \(-0.572626\pi\)
−0.226186 + 0.974084i \(0.572626\pi\)
\(308\) 7.38643 0.420881
\(309\) −1.00000 −0.0568880
\(310\) 16.1866 0.919336
\(311\) 4.06259 0.230368 0.115184 0.993344i \(-0.463254\pi\)
0.115184 + 0.993344i \(0.463254\pi\)
\(312\) 1.00000 0.0566139
\(313\) 4.20018 0.237408 0.118704 0.992930i \(-0.462126\pi\)
0.118704 + 0.992930i \(0.462126\pi\)
\(314\) 9.13057 0.515268
\(315\) 3.94672 0.222372
\(316\) 3.42688 0.192777
\(317\) −6.43390 −0.361364 −0.180682 0.983542i \(-0.557830\pi\)
−0.180682 + 0.983542i \(0.557830\pi\)
\(318\) 3.47358 0.194789
\(319\) 24.1866 1.35419
\(320\) 2.59680 0.145166
\(321\) 15.7200 0.877406
\(322\) −4.91406 −0.273850
\(323\) −12.4666 −0.693658
\(324\) 1.00000 0.0555556
\(325\) 1.74337 0.0967049
\(326\) −5.94639 −0.329340
\(327\) −17.7138 −0.979574
\(328\) −0.193601 −0.0106898
\(329\) −11.9105 −0.656645
\(330\) 12.6205 0.694735
\(331\) −28.9168 −1.58941 −0.794705 0.606996i \(-0.792375\pi\)
−0.794705 + 0.606996i \(0.792375\pi\)
\(332\) −6.59680 −0.362046
\(333\) −6.86001 −0.375926
\(334\) 10.3405 0.565807
\(335\) 22.9300 1.25280
\(336\) 1.51984 0.0829140
\(337\) −6.53344 −0.355899 −0.177950 0.984040i \(-0.556946\pi\)
−0.177950 + 0.984040i \(0.556946\pi\)
\(338\) 1.00000 0.0543928
\(339\) 2.79698 0.151911
\(340\) 5.19360 0.281663
\(341\) 30.2938 1.64050
\(342\) −6.23328 −0.337057
\(343\) −17.7671 −0.959331
\(344\) 3.13999 0.169297
\(345\) −8.39618 −0.452035
\(346\) −15.0933 −0.811420
\(347\) 33.7931 1.81411 0.907055 0.421012i \(-0.138325\pi\)
0.907055 + 0.421012i \(0.138325\pi\)
\(348\) 4.97665 0.266776
\(349\) 21.1748 1.13346 0.566730 0.823904i \(-0.308208\pi\)
0.566730 + 0.823904i \(0.308208\pi\)
\(350\) 2.64964 0.141629
\(351\) 1.00000 0.0533761
\(352\) 4.86001 0.259039
\(353\) 1.87395 0.0997401 0.0498701 0.998756i \(-0.484119\pi\)
0.0498701 + 0.998756i \(0.484119\pi\)
\(354\) 2.65983 0.141368
\(355\) −32.5648 −1.72836
\(356\) −12.9136 −0.684421
\(357\) 3.03968 0.160877
\(358\) −2.84684 −0.150460
\(359\) 17.7395 0.936256 0.468128 0.883661i \(-0.344929\pi\)
0.468128 + 0.883661i \(0.344929\pi\)
\(360\) 2.59680 0.136863
\(361\) 19.8538 1.04493
\(362\) 8.38720 0.440822
\(363\) 12.6197 0.662363
\(364\) 1.51984 0.0796612
\(365\) 25.8043 1.35066
\(366\) 5.09329 0.266231
\(367\) 31.5536 1.64708 0.823542 0.567255i \(-0.191995\pi\)
0.823542 + 0.567255i \(0.191995\pi\)
\(368\) −3.23328 −0.168546
\(369\) −0.193601 −0.0100785
\(370\) −17.8141 −0.926110
\(371\) 5.27928 0.274086
\(372\) 6.23328 0.323180
\(373\) −22.9804 −1.18988 −0.594940 0.803770i \(-0.702824\pi\)
−0.594940 + 0.803770i \(0.702824\pi\)
\(374\) 9.72002 0.502610
\(375\) −8.45681 −0.436708
\(376\) −7.83666 −0.404145
\(377\) 4.97665 0.256311
\(378\) 1.51984 0.0781721
\(379\) 34.0810 1.75063 0.875313 0.483557i \(-0.160655\pi\)
0.875313 + 0.483557i \(0.160655\pi\)
\(380\) −16.1866 −0.830354
\(381\) −2.55293 −0.130791
\(382\) 17.9533 0.918571
\(383\) 11.2462 0.574653 0.287327 0.957833i \(-0.407234\pi\)
0.287327 + 0.957833i \(0.407234\pi\)
\(384\) 1.00000 0.0510310
\(385\) 19.1811 0.977559
\(386\) 13.9669 0.710897
\(387\) 3.13999 0.159615
\(388\) 12.9005 0.654921
\(389\) 5.06063 0.256584 0.128292 0.991736i \(-0.459050\pi\)
0.128292 + 0.991736i \(0.459050\pi\)
\(390\) 2.59680 0.131494
\(391\) −6.46656 −0.327028
\(392\) −4.69009 −0.236885
\(393\) −6.68035 −0.336979
\(394\) 7.78305 0.392104
\(395\) 8.89892 0.447753
\(396\) 4.86001 0.244225
\(397\) 10.7597 0.540014 0.270007 0.962858i \(-0.412974\pi\)
0.270007 + 0.962858i \(0.412974\pi\)
\(398\) −4.71344 −0.236263
\(399\) −9.47358 −0.474272
\(400\) 1.74337 0.0871686
\(401\) 5.44048 0.271685 0.135842 0.990730i \(-0.456626\pi\)
0.135842 + 0.990730i \(0.456626\pi\)
\(402\) 8.83008 0.440404
\(403\) 6.23328 0.310502
\(404\) −10.0069 −0.497863
\(405\) 2.59680 0.129036
\(406\) 7.56371 0.375380
\(407\) −33.3397 −1.65259
\(408\) 2.00000 0.0990148
\(409\) 9.94013 0.491508 0.245754 0.969332i \(-0.420964\pi\)
0.245754 + 0.969332i \(0.420964\pi\)
\(410\) −0.502743 −0.0248287
\(411\) −12.9467 −0.638614
\(412\) −1.00000 −0.0492665
\(413\) 4.04251 0.198919
\(414\) −3.23328 −0.158907
\(415\) −17.1306 −0.840907
\(416\) 1.00000 0.0490290
\(417\) −7.80015 −0.381975
\(418\) −30.2938 −1.48172
\(419\) 29.2137 1.42718 0.713591 0.700563i \(-0.247068\pi\)
0.713591 + 0.700563i \(0.247068\pi\)
\(420\) 3.94672 0.192580
\(421\) −11.9370 −0.581772 −0.290886 0.956758i \(-0.593950\pi\)
−0.290886 + 0.956758i \(0.593950\pi\)
\(422\) −21.4338 −1.04338
\(423\) −7.83666 −0.381032
\(424\) 3.47358 0.168692
\(425\) 3.48674 0.169132
\(426\) −12.5404 −0.607582
\(427\) 7.74098 0.374612
\(428\) 15.7200 0.759856
\(429\) 4.86001 0.234644
\(430\) 8.15392 0.393217
\(431\) 28.4199 1.36894 0.684468 0.729043i \(-0.260034\pi\)
0.684468 + 0.729043i \(0.260034\pi\)
\(432\) 1.00000 0.0481125
\(433\) 23.6266 1.13542 0.567712 0.823228i \(-0.307829\pi\)
0.567712 + 0.823228i \(0.307829\pi\)
\(434\) 9.47358 0.454746
\(435\) 12.9234 0.619628
\(436\) −17.7138 −0.848336
\(437\) 20.1539 0.964093
\(438\) 9.93697 0.474807
\(439\) −14.9934 −0.715597 −0.357798 0.933799i \(-0.616472\pi\)
−0.357798 + 0.933799i \(0.616472\pi\)
\(440\) 12.6205 0.601658
\(441\) −4.69009 −0.223338
\(442\) 2.00000 0.0951303
\(443\) −25.5971 −1.21616 −0.608078 0.793877i \(-0.708059\pi\)
−0.608078 + 0.793877i \(0.708059\pi\)
\(444\) −6.86001 −0.325562
\(445\) −33.5341 −1.58967
\(446\) 11.9160 0.564240
\(447\) 6.05361 0.286326
\(448\) 1.51984 0.0718056
\(449\) −28.8538 −1.36169 −0.680847 0.732426i \(-0.738388\pi\)
−0.680847 + 0.732426i \(0.738388\pi\)
\(450\) 1.74337 0.0821834
\(451\) −0.940902 −0.0443054
\(452\) 2.79698 0.131559
\(453\) −22.5738 −1.06061
\(454\) −11.1469 −0.523150
\(455\) 3.94672 0.185025
\(456\) −6.23328 −0.291900
\(457\) −15.9439 −0.745824 −0.372912 0.927867i \(-0.621641\pi\)
−0.372912 + 0.927867i \(0.621641\pi\)
\(458\) 10.4805 0.489721
\(459\) 2.00000 0.0933520
\(460\) −8.39618 −0.391474
\(461\) 22.6127 1.05318 0.526589 0.850120i \(-0.323471\pi\)
0.526589 + 0.850120i \(0.323471\pi\)
\(462\) 7.38643 0.343648
\(463\) −9.36069 −0.435028 −0.217514 0.976057i \(-0.569795\pi\)
−0.217514 + 0.976057i \(0.569795\pi\)
\(464\) 4.97665 0.231035
\(465\) 16.1866 0.750635
\(466\) 18.8141 0.871545
\(467\) 4.53334 0.209778 0.104889 0.994484i \(-0.466551\pi\)
0.104889 + 0.994484i \(0.466551\pi\)
\(468\) 1.00000 0.0462250
\(469\) 13.4203 0.619692
\(470\) −20.3502 −0.938687
\(471\) 9.13057 0.420715
\(472\) 2.65983 0.122428
\(473\) 15.2604 0.701673
\(474\) 3.42688 0.157402
\(475\) −10.8669 −0.498609
\(476\) 3.03968 0.139323
\(477\) 3.47358 0.159044
\(478\) −23.3203 −1.06665
\(479\) −9.66060 −0.441404 −0.220702 0.975341i \(-0.570835\pi\)
−0.220702 + 0.975341i \(0.570835\pi\)
\(480\) 2.59680 0.118527
\(481\) −6.86001 −0.312790
\(482\) −9.31263 −0.424179
\(483\) −4.91406 −0.223598
\(484\) 12.6197 0.573623
\(485\) 33.4999 1.52115
\(486\) 1.00000 0.0453609
\(487\) 28.2938 1.28211 0.641057 0.767493i \(-0.278496\pi\)
0.641057 + 0.767493i \(0.278496\pi\)
\(488\) 5.09329 0.230562
\(489\) −5.94639 −0.268905
\(490\) −12.1792 −0.550201
\(491\) 18.5591 0.837559 0.418780 0.908088i \(-0.362458\pi\)
0.418780 + 0.908088i \(0.362458\pi\)
\(492\) −0.193601 −0.00872820
\(493\) 9.95330 0.448274
\(494\) −6.23328 −0.280448
\(495\) 12.6205 0.567248
\(496\) 6.23328 0.279882
\(497\) −19.0593 −0.854927
\(498\) −6.59680 −0.295610
\(499\) 25.0863 1.12302 0.561508 0.827471i \(-0.310221\pi\)
0.561508 + 0.827471i \(0.310221\pi\)
\(500\) −8.45681 −0.378200
\(501\) 10.3405 0.461980
\(502\) −7.04387 −0.314383
\(503\) −43.5320 −1.94100 −0.970499 0.241106i \(-0.922490\pi\)
−0.970499 + 0.241106i \(0.922490\pi\)
\(504\) 1.51984 0.0676990
\(505\) −25.9860 −1.15636
\(506\) −15.7138 −0.698562
\(507\) 1.00000 0.0444116
\(508\) −2.55293 −0.113268
\(509\) −2.69292 −0.119362 −0.0596808 0.998218i \(-0.519008\pi\)
−0.0596808 + 0.998218i \(0.519008\pi\)
\(510\) 5.19360 0.229977
\(511\) 15.1026 0.668100
\(512\) 1.00000 0.0441942
\(513\) −6.23328 −0.275206
\(514\) −6.04001 −0.266413
\(515\) −2.59680 −0.114429
\(516\) 3.13999 0.138230
\(517\) −38.0863 −1.67503
\(518\) −10.4261 −0.458097
\(519\) −15.0933 −0.662522
\(520\) 2.59680 0.113877
\(521\) 31.5142 1.38066 0.690331 0.723494i \(-0.257465\pi\)
0.690331 + 0.723494i \(0.257465\pi\)
\(522\) 4.97665 0.217822
\(523\) −2.23012 −0.0975162 −0.0487581 0.998811i \(-0.515526\pi\)
−0.0487581 + 0.998811i \(0.515526\pi\)
\(524\) −6.68035 −0.291832
\(525\) 2.64964 0.115640
\(526\) −19.5194 −0.851086
\(527\) 12.4666 0.543052
\(528\) 4.86001 0.211505
\(529\) −12.5459 −0.545474
\(530\) 9.02019 0.391812
\(531\) 2.65983 0.115427
\(532\) −9.47358 −0.410732
\(533\) −0.193601 −0.00838578
\(534\) −12.9136 −0.558827
\(535\) 40.8218 1.76488
\(536\) 8.83008 0.381401
\(537\) −2.84684 −0.122850
\(538\) −5.23045 −0.225501
\(539\) −22.7939 −0.981802
\(540\) 2.59680 0.111748
\(541\) 0.256297 0.0110191 0.00550954 0.999985i \(-0.498246\pi\)
0.00550954 + 0.999985i \(0.498246\pi\)
\(542\) −1.08638 −0.0466639
\(543\) 8.38720 0.359929
\(544\) 2.00000 0.0857493
\(545\) −45.9991 −1.97039
\(546\) 1.51984 0.0650431
\(547\) −30.3175 −1.29628 −0.648141 0.761520i \(-0.724453\pi\)
−0.648141 + 0.761520i \(0.724453\pi\)
\(548\) −12.9467 −0.553056
\(549\) 5.09329 0.217376
\(550\) 8.47281 0.361282
\(551\) −31.0208 −1.32153
\(552\) −3.23328 −0.137617
\(553\) 5.20830 0.221480
\(554\) 7.85343 0.333660
\(555\) −17.8141 −0.756166
\(556\) −7.80015 −0.330800
\(557\) 13.4400 0.569473 0.284737 0.958606i \(-0.408094\pi\)
0.284737 + 0.958606i \(0.408094\pi\)
\(558\) 6.23328 0.263876
\(559\) 3.13999 0.132807
\(560\) 3.94672 0.166779
\(561\) 9.72002 0.410380
\(562\) 29.0535 1.22555
\(563\) 25.0298 1.05488 0.527441 0.849592i \(-0.323152\pi\)
0.527441 + 0.849592i \(0.323152\pi\)
\(564\) −7.83666 −0.329983
\(565\) 7.26321 0.305566
\(566\) −10.6594 −0.448048
\(567\) 1.51984 0.0638272
\(568\) −12.5404 −0.526182
\(569\) −16.6341 −0.697337 −0.348669 0.937246i \(-0.613366\pi\)
−0.348669 + 0.937246i \(0.613366\pi\)
\(570\) −16.1866 −0.677981
\(571\) −11.2705 −0.471654 −0.235827 0.971795i \(-0.575780\pi\)
−0.235827 + 0.971795i \(0.575780\pi\)
\(572\) 4.86001 0.203207
\(573\) 17.9533 0.750010
\(574\) −0.294242 −0.0122814
\(575\) −5.63681 −0.235071
\(576\) 1.00000 0.0416667
\(577\) −35.5772 −1.48110 −0.740549 0.672002i \(-0.765435\pi\)
−0.740549 + 0.672002i \(0.765435\pi\)
\(578\) −13.0000 −0.540729
\(579\) 13.9669 0.580445
\(580\) 12.9234 0.536614
\(581\) −10.0261 −0.415952
\(582\) 12.9005 0.534741
\(583\) 16.8816 0.699166
\(584\) 9.93697 0.411195
\(585\) 2.59680 0.107364
\(586\) −18.4143 −0.760688
\(587\) −8.87009 −0.366108 −0.183054 0.983103i \(-0.558598\pi\)
−0.183054 + 0.983103i \(0.558598\pi\)
\(588\) −4.69009 −0.193416
\(589\) −38.8538 −1.60094
\(590\) 6.90704 0.284358
\(591\) 7.78305 0.320152
\(592\) −6.86001 −0.281945
\(593\) −45.4874 −1.86794 −0.933972 0.357346i \(-0.883682\pi\)
−0.933972 + 0.357346i \(0.883682\pi\)
\(594\) 4.86001 0.199409
\(595\) 7.89344 0.323599
\(596\) 6.05361 0.247966
\(597\) −4.71344 −0.192908
\(598\) −3.23328 −0.132219
\(599\) −5.69369 −0.232638 −0.116319 0.993212i \(-0.537109\pi\)
−0.116319 + 0.993212i \(0.537109\pi\)
\(600\) 1.74337 0.0711729
\(601\) −28.2938 −1.15413 −0.577064 0.816699i \(-0.695802\pi\)
−0.577064 + 0.816699i \(0.695802\pi\)
\(602\) 4.77228 0.194503
\(603\) 8.83008 0.359589
\(604\) −22.5738 −0.918514
\(605\) 32.7709 1.33233
\(606\) −10.0069 −0.406503
\(607\) −7.30017 −0.296305 −0.148152 0.988965i \(-0.547333\pi\)
−0.148152 + 0.988965i \(0.547333\pi\)
\(608\) −6.23328 −0.252793
\(609\) 7.56371 0.306497
\(610\) 13.2263 0.535515
\(611\) −7.83666 −0.317037
\(612\) 2.00000 0.0808452
\(613\) 23.5070 0.949439 0.474719 0.880137i \(-0.342550\pi\)
0.474719 + 0.880137i \(0.342550\pi\)
\(614\) −7.92620 −0.319875
\(615\) −0.502743 −0.0202725
\(616\) 7.38643 0.297608
\(617\) 9.06755 0.365046 0.182523 0.983202i \(-0.441574\pi\)
0.182523 + 0.983202i \(0.441574\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −2.64769 −0.106420 −0.0532098 0.998583i \(-0.516945\pi\)
−0.0532098 + 0.998583i \(0.516945\pi\)
\(620\) 16.1866 0.650069
\(621\) −3.23328 −0.129747
\(622\) 4.06259 0.162895
\(623\) −19.6266 −0.786324
\(624\) 1.00000 0.0400320
\(625\) −30.6775 −1.22710
\(626\) 4.20018 0.167873
\(627\) −30.2938 −1.20982
\(628\) 9.13057 0.364350
\(629\) −13.7200 −0.547053
\(630\) 3.94672 0.157241
\(631\) 18.0069 0.716844 0.358422 0.933560i \(-0.383315\pi\)
0.358422 + 0.933560i \(0.383315\pi\)
\(632\) 3.42688 0.136314
\(633\) −21.4338 −0.851917
\(634\) −6.43390 −0.255523
\(635\) −6.62946 −0.263082
\(636\) 3.47358 0.137736
\(637\) −4.69009 −0.185828
\(638\) 24.1866 0.957556
\(639\) −12.5404 −0.496089
\(640\) 2.59680 0.102648
\(641\) 47.6740 1.88301 0.941505 0.337000i \(-0.109412\pi\)
0.941505 + 0.337000i \(0.109412\pi\)
\(642\) 15.7200 0.620420
\(643\) −11.6664 −0.460078 −0.230039 0.973181i \(-0.573885\pi\)
−0.230039 + 0.973181i \(0.573885\pi\)
\(644\) −4.91406 −0.193641
\(645\) 8.15392 0.321060
\(646\) −12.4666 −0.490490
\(647\) 5.01291 0.197078 0.0985388 0.995133i \(-0.468583\pi\)
0.0985388 + 0.995133i \(0.468583\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.9268 0.507421
\(650\) 1.74337 0.0683807
\(651\) 9.47358 0.371299
\(652\) −5.94639 −0.232879
\(653\) −19.8997 −0.778735 −0.389368 0.921082i \(-0.627306\pi\)
−0.389368 + 0.921082i \(0.627306\pi\)
\(654\) −17.7138 −0.692663
\(655\) −17.3475 −0.677824
\(656\) −0.193601 −0.00755884
\(657\) 9.93697 0.387678
\(658\) −11.9105 −0.464318
\(659\) −4.32580 −0.168509 −0.0842546 0.996444i \(-0.526851\pi\)
−0.0842546 + 0.996444i \(0.526851\pi\)
\(660\) 12.6205 0.491252
\(661\) 40.2806 1.56674 0.783368 0.621559i \(-0.213500\pi\)
0.783368 + 0.621559i \(0.213500\pi\)
\(662\) −28.9168 −1.12388
\(663\) 2.00000 0.0776736
\(664\) −6.59680 −0.256006
\(665\) −24.6010 −0.953985
\(666\) −6.86001 −0.265820
\(667\) −16.0909 −0.623042
\(668\) 10.3405 0.400086
\(669\) 11.9160 0.460700
\(670\) 22.9300 0.885861
\(671\) 24.7534 0.955596
\(672\) 1.51984 0.0586291
\(673\) −12.3573 −0.476338 −0.238169 0.971224i \(-0.576547\pi\)
−0.238169 + 0.971224i \(0.576547\pi\)
\(674\) −6.53344 −0.251659
\(675\) 1.74337 0.0671024
\(676\) 1.00000 0.0384615
\(677\) −2.16606 −0.0832485 −0.0416242 0.999133i \(-0.513253\pi\)
−0.0416242 + 0.999133i \(0.513253\pi\)
\(678\) 2.79698 0.107418
\(679\) 19.6066 0.752433
\(680\) 5.19360 0.199165
\(681\) −11.1469 −0.427150
\(682\) 30.2938 1.16001
\(683\) 10.8864 0.416557 0.208279 0.978070i \(-0.433214\pi\)
0.208279 + 0.978070i \(0.433214\pi\)
\(684\) −6.23328 −0.238335
\(685\) −33.6200 −1.28456
\(686\) −17.7671 −0.678349
\(687\) 10.4805 0.399856
\(688\) 3.13999 0.119711
\(689\) 3.47358 0.132333
\(690\) −8.39618 −0.319637
\(691\) 3.94432 0.150049 0.0750246 0.997182i \(-0.476096\pi\)
0.0750246 + 0.997182i \(0.476096\pi\)
\(692\) −15.0933 −0.573761
\(693\) 7.38643 0.280587
\(694\) 33.7931 1.28277
\(695\) −20.2554 −0.768332
\(696\) 4.97665 0.188639
\(697\) −0.387202 −0.0146663
\(698\) 21.1748 0.801477
\(699\) 18.8141 0.711614
\(700\) 2.64964 0.100147
\(701\) 26.8906 1.01564 0.507822 0.861462i \(-0.330451\pi\)
0.507822 + 0.861462i \(0.330451\pi\)
\(702\) 1.00000 0.0377426
\(703\) 42.7604 1.61274
\(704\) 4.86001 0.183169
\(705\) −20.3502 −0.766435
\(706\) 1.87395 0.0705269
\(707\) −15.2089 −0.571989
\(708\) 2.65983 0.0999624
\(709\) −24.7367 −0.929006 −0.464503 0.885572i \(-0.653767\pi\)
−0.464503 + 0.885572i \(0.653767\pi\)
\(710\) −32.5648 −1.22214
\(711\) 3.42688 0.128518
\(712\) −12.9136 −0.483959
\(713\) −20.1539 −0.754770
\(714\) 3.03968 0.113757
\(715\) 12.6205 0.471979
\(716\) −2.84684 −0.106392
\(717\) −23.3203 −0.870913
\(718\) 17.7395 0.662033
\(719\) −17.4727 −0.651622 −0.325811 0.945435i \(-0.605637\pi\)
−0.325811 + 0.945435i \(0.605637\pi\)
\(720\) 2.59680 0.0967770
\(721\) −1.51984 −0.0566018
\(722\) 19.8538 0.738880
\(723\) −9.31263 −0.346341
\(724\) 8.38720 0.311708
\(725\) 8.67616 0.322224
\(726\) 12.6197 0.468361
\(727\) −11.2803 −0.418363 −0.209182 0.977877i \(-0.567080\pi\)
−0.209182 + 0.977877i \(0.567080\pi\)
\(728\) 1.51984 0.0563290
\(729\) 1.00000 0.0370370
\(730\) 25.8043 0.955061
\(731\) 6.27998 0.232273
\(732\) 5.09329 0.188253
\(733\) 7.20051 0.265957 0.132979 0.991119i \(-0.457546\pi\)
0.132979 + 0.991119i \(0.457546\pi\)
\(734\) 31.5536 1.16466
\(735\) −12.1792 −0.449238
\(736\) −3.23328 −0.119180
\(737\) 42.9143 1.58077
\(738\) −0.193601 −0.00712654
\(739\) 11.7931 0.433817 0.216909 0.976192i \(-0.430403\pi\)
0.216909 + 0.976192i \(0.430403\pi\)
\(740\) −17.8141 −0.654859
\(741\) −6.23328 −0.228985
\(742\) 5.27928 0.193808
\(743\) −26.8659 −0.985614 −0.492807 0.870139i \(-0.664029\pi\)
−0.492807 + 0.870139i \(0.664029\pi\)
\(744\) 6.23328 0.228523
\(745\) 15.7200 0.575937
\(746\) −22.9804 −0.841372
\(747\) −6.59680 −0.241364
\(748\) 9.72002 0.355399
\(749\) 23.8919 0.872991
\(750\) −8.45681 −0.308799
\(751\) 35.8404 1.30784 0.653918 0.756566i \(-0.273124\pi\)
0.653918 + 0.756566i \(0.273124\pi\)
\(752\) −7.83666 −0.285774
\(753\) −7.04387 −0.256693
\(754\) 4.97665 0.181239
\(755\) −58.6196 −2.13339
\(756\) 1.51984 0.0552760
\(757\) 10.0668 0.365883 0.182942 0.983124i \(-0.441438\pi\)
0.182942 + 0.983124i \(0.441438\pi\)
\(758\) 34.0810 1.23788
\(759\) −15.7138 −0.570374
\(760\) −16.1866 −0.587149
\(761\) 20.5411 0.744615 0.372308 0.928109i \(-0.378567\pi\)
0.372308 + 0.928109i \(0.378567\pi\)
\(762\) −2.55293 −0.0924830
\(763\) −26.9221 −0.974644
\(764\) 17.9533 0.649528
\(765\) 5.19360 0.187775
\(766\) 11.2462 0.406341
\(767\) 2.65983 0.0960408
\(768\) 1.00000 0.0360844
\(769\) −9.25869 −0.333877 −0.166938 0.985967i \(-0.553388\pi\)
−0.166938 + 0.985967i \(0.553388\pi\)
\(770\) 19.1811 0.691239
\(771\) −6.04001 −0.217526
\(772\) 13.9669 0.502680
\(773\) −21.1538 −0.760850 −0.380425 0.924812i \(-0.624222\pi\)
−0.380425 + 0.924812i \(0.624222\pi\)
\(774\) 3.13999 0.112865
\(775\) 10.8669 0.390351
\(776\) 12.9005 0.463099
\(777\) −10.4261 −0.374035
\(778\) 5.06063 0.181433
\(779\) 1.20677 0.0432369
\(780\) 2.59680 0.0929804
\(781\) −60.9463 −2.18083
\(782\) −6.46656 −0.231244
\(783\) 4.97665 0.177851
\(784\) −4.69009 −0.167503
\(785\) 23.7103 0.846256
\(786\) −6.68035 −0.238280
\(787\) −38.9804 −1.38950 −0.694751 0.719251i \(-0.744485\pi\)
−0.694751 + 0.719251i \(0.744485\pi\)
\(788\) 7.78305 0.277260
\(789\) −19.5194 −0.694909
\(790\) 8.89892 0.316609
\(791\) 4.25097 0.151147
\(792\) 4.86001 0.172693
\(793\) 5.09329 0.180868
\(794\) 10.7597 0.381847
\(795\) 9.02019 0.319913
\(796\) −4.71344 −0.167063
\(797\) −35.4685 −1.25636 −0.628180 0.778068i \(-0.716200\pi\)
−0.628180 + 0.778068i \(0.716200\pi\)
\(798\) −9.47358 −0.335361
\(799\) −15.6733 −0.554482
\(800\) 1.74337 0.0616375
\(801\) −12.9136 −0.456280
\(802\) 5.44048 0.192110
\(803\) 48.2938 1.70425
\(804\) 8.83008 0.311413
\(805\) −12.7608 −0.449760
\(806\) 6.23328 0.219558
\(807\) −5.23045 −0.184120
\(808\) −10.0069 −0.352042
\(809\) 43.3440 1.52389 0.761947 0.647640i \(-0.224244\pi\)
0.761947 + 0.647640i \(0.224244\pi\)
\(810\) 2.59680 0.0912423
\(811\) 15.1101 0.530586 0.265293 0.964168i \(-0.414531\pi\)
0.265293 + 0.964168i \(0.414531\pi\)
\(812\) 7.56371 0.265434
\(813\) −1.08638 −0.0381009
\(814\) −33.3397 −1.16856
\(815\) −15.4416 −0.540895
\(816\) 2.00000 0.0700140
\(817\) −19.5724 −0.684752
\(818\) 9.94013 0.347549
\(819\) 1.51984 0.0531075
\(820\) −0.502743 −0.0175565
\(821\) 15.9806 0.557726 0.278863 0.960331i \(-0.410042\pi\)
0.278863 + 0.960331i \(0.410042\pi\)
\(822\) −12.9467 −0.451569
\(823\) 2.85582 0.0995477 0.0497738 0.998761i \(-0.484150\pi\)
0.0497738 + 0.998761i \(0.484150\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 8.47281 0.294985
\(826\) 4.04251 0.140657
\(827\) −4.50634 −0.156701 −0.0783504 0.996926i \(-0.524965\pi\)
−0.0783504 + 0.996926i \(0.524965\pi\)
\(828\) −3.23328 −0.112364
\(829\) −17.7437 −0.616264 −0.308132 0.951344i \(-0.599704\pi\)
−0.308132 + 0.951344i \(0.599704\pi\)
\(830\) −17.1306 −0.594611
\(831\) 7.85343 0.272432
\(832\) 1.00000 0.0346688
\(833\) −9.38018 −0.325004
\(834\) −7.80015 −0.270097
\(835\) 26.8522 0.929259
\(836\) −30.2938 −1.04773
\(837\) 6.23328 0.215454
\(838\) 29.2137 1.00917
\(839\) −39.9665 −1.37980 −0.689898 0.723907i \(-0.742345\pi\)
−0.689898 + 0.723907i \(0.742345\pi\)
\(840\) 3.94672 0.136175
\(841\) −4.23295 −0.145964
\(842\) −11.9370 −0.411375
\(843\) 29.0535 1.00066
\(844\) −21.4338 −0.737782
\(845\) 2.59680 0.0893326
\(846\) −7.83666 −0.269430
\(847\) 19.1799 0.659030
\(848\) 3.47358 0.119283
\(849\) −10.6594 −0.365829
\(850\) 3.48674 0.119594
\(851\) 22.1803 0.760332
\(852\) −12.5404 −0.429626
\(853\) −7.38437 −0.252836 −0.126418 0.991977i \(-0.540348\pi\)
−0.126418 + 0.991977i \(0.540348\pi\)
\(854\) 7.74098 0.264891
\(855\) −16.1866 −0.553569
\(856\) 15.7200 0.537299
\(857\) −23.4727 −0.801812 −0.400906 0.916119i \(-0.631305\pi\)
−0.400906 + 0.916119i \(0.631305\pi\)
\(858\) 4.86001 0.165918
\(859\) −16.3474 −0.557767 −0.278883 0.960325i \(-0.589964\pi\)
−0.278883 + 0.960325i \(0.589964\pi\)
\(860\) 8.15392 0.278046
\(861\) −0.294242 −0.0100277
\(862\) 28.4199 0.967984
\(863\) 0.153332 0.00521948 0.00260974 0.999997i \(-0.499169\pi\)
0.00260974 + 0.999997i \(0.499169\pi\)
\(864\) 1.00000 0.0340207
\(865\) −39.1943 −1.33264
\(866\) 23.6266 0.802865
\(867\) −13.0000 −0.441503
\(868\) 9.47358 0.321554
\(869\) 16.6547 0.564971
\(870\) 12.9234 0.438143
\(871\) 8.83008 0.299196
\(872\) −17.7138 −0.599864
\(873\) 12.9005 0.436614
\(874\) 20.1539 0.681717
\(875\) −12.8530 −0.434510
\(876\) 9.93697 0.335739
\(877\) 21.1071 0.712737 0.356368 0.934346i \(-0.384015\pi\)
0.356368 + 0.934346i \(0.384015\pi\)
\(878\) −14.9934 −0.506003
\(879\) −18.4143 −0.621099
\(880\) 12.6205 0.425436
\(881\) 22.9937 0.774679 0.387340 0.921937i \(-0.373394\pi\)
0.387340 + 0.921937i \(0.373394\pi\)
\(882\) −4.69009 −0.157924
\(883\) −44.2152 −1.48796 −0.743980 0.668202i \(-0.767064\pi\)
−0.743980 + 0.668202i \(0.767064\pi\)
\(884\) 2.00000 0.0672673
\(885\) 6.90704 0.232178
\(886\) −25.5971 −0.859952
\(887\) 34.6988 1.16507 0.582536 0.812805i \(-0.302061\pi\)
0.582536 + 0.812805i \(0.302061\pi\)
\(888\) −6.86001 −0.230207
\(889\) −3.88005 −0.130133
\(890\) −33.5341 −1.12407
\(891\) 4.86001 0.162817
\(892\) 11.9160 0.398978
\(893\) 48.8481 1.63464
\(894\) 6.05361 0.202463
\(895\) −7.39269 −0.247110
\(896\) 1.51984 0.0507742
\(897\) −3.23328 −0.107956
\(898\) −28.8538 −0.962862
\(899\) 31.0208 1.03460
\(900\) 1.74337 0.0581124
\(901\) 6.94716 0.231443
\(902\) −0.940902 −0.0313286
\(903\) 4.77228 0.158811
\(904\) 2.79698 0.0930263
\(905\) 21.7799 0.723988
\(906\) −22.5738 −0.749964
\(907\) −15.4711 −0.513709 −0.256854 0.966450i \(-0.582686\pi\)
−0.256854 + 0.966450i \(0.582686\pi\)
\(908\) −11.1469 −0.369923
\(909\) −10.0069 −0.331908
\(910\) 3.94672 0.130832
\(911\) 13.7137 0.454355 0.227178 0.973853i \(-0.427050\pi\)
0.227178 + 0.973853i \(0.427050\pi\)
\(912\) −6.23328 −0.206404
\(913\) −32.0605 −1.06105
\(914\) −15.9439 −0.527377
\(915\) 13.2263 0.437247
\(916\) 10.4805 0.346285
\(917\) −10.1530 −0.335283
\(918\) 2.00000 0.0660098
\(919\) −9.67060 −0.319004 −0.159502 0.987198i \(-0.550989\pi\)
−0.159502 + 0.987198i \(0.550989\pi\)
\(920\) −8.39618 −0.276814
\(921\) −7.92620 −0.261177
\(922\) 22.6127 0.744709
\(923\) −12.5404 −0.412771
\(924\) 7.38643 0.242996
\(925\) −11.9596 −0.393228
\(926\) −9.36069 −0.307611
\(927\) −1.00000 −0.0328443
\(928\) 4.97665 0.163367
\(929\) −39.0031 −1.27965 −0.639824 0.768521i \(-0.720993\pi\)
−0.639824 + 0.768521i \(0.720993\pi\)
\(930\) 16.1866 0.530779
\(931\) 29.2346 0.958127
\(932\) 18.8141 0.616276
\(933\) 4.06259 0.133003
\(934\) 4.53334 0.148335
\(935\) 25.2410 0.825468
\(936\) 1.00000 0.0326860
\(937\) 50.6538 1.65479 0.827394 0.561622i \(-0.189823\pi\)
0.827394 + 0.561622i \(0.189823\pi\)
\(938\) 13.4203 0.438188
\(939\) 4.20018 0.137068
\(940\) −20.3502 −0.663752
\(941\) −22.0752 −0.719632 −0.359816 0.933023i \(-0.617161\pi\)
−0.359816 + 0.933023i \(0.617161\pi\)
\(942\) 9.13057 0.297490
\(943\) 0.625965 0.0203842
\(944\) 2.65983 0.0865700
\(945\) 3.94672 0.128387
\(946\) 15.2604 0.496158
\(947\) 9.62125 0.312649 0.156324 0.987706i \(-0.450036\pi\)
0.156324 + 0.987706i \(0.450036\pi\)
\(948\) 3.42688 0.111300
\(949\) 9.93697 0.322568
\(950\) −10.8669 −0.352570
\(951\) −6.43390 −0.208634
\(952\) 3.03968 0.0985165
\(953\) −11.3670 −0.368214 −0.184107 0.982906i \(-0.558939\pi\)
−0.184107 + 0.982906i \(0.558939\pi\)
\(954\) 3.47358 0.112461
\(955\) 46.6211 1.50862
\(956\) −23.3203 −0.754233
\(957\) 24.1866 0.781841
\(958\) −9.66060 −0.312120
\(959\) −19.6769 −0.635401
\(960\) 2.59680 0.0838114
\(961\) 7.85376 0.253347
\(962\) −6.86001 −0.221176
\(963\) 15.7200 0.506571
\(964\) −9.31263 −0.299940
\(965\) 36.2693 1.16755
\(966\) −4.91406 −0.158107
\(967\) −13.6143 −0.437807 −0.218904 0.975746i \(-0.570248\pi\)
−0.218904 + 0.975746i \(0.570248\pi\)
\(968\) 12.6197 0.405613
\(969\) −12.4666 −0.400483
\(970\) 33.4999 1.07562
\(971\) 1.45306 0.0466309 0.0233154 0.999728i \(-0.492578\pi\)
0.0233154 + 0.999728i \(0.492578\pi\)
\(972\) 1.00000 0.0320750
\(973\) −11.8550 −0.380053
\(974\) 28.2938 0.906592
\(975\) 1.74337 0.0558326
\(976\) 5.09329 0.163032
\(977\) 6.95264 0.222435 0.111217 0.993796i \(-0.464525\pi\)
0.111217 + 0.993796i \(0.464525\pi\)
\(978\) −5.94639 −0.190145
\(979\) −62.7604 −2.00583
\(980\) −12.1792 −0.389051
\(981\) −17.7138 −0.565557
\(982\) 18.5591 0.592244
\(983\) −36.9930 −1.17989 −0.589946 0.807442i \(-0.700851\pi\)
−0.589946 + 0.807442i \(0.700851\pi\)
\(984\) −0.193601 −0.00617177
\(985\) 20.2110 0.643977
\(986\) 9.95330 0.316978
\(987\) −11.9105 −0.379114
\(988\) −6.23328 −0.198307
\(989\) −10.1525 −0.322829
\(990\) 12.6205 0.401105
\(991\) 50.0465 1.58978 0.794889 0.606754i \(-0.207529\pi\)
0.794889 + 0.606754i \(0.207529\pi\)
\(992\) 6.23328 0.197907
\(993\) −28.9168 −0.917646
\(994\) −19.0593 −0.604525
\(995\) −12.2399 −0.388030
\(996\) −6.59680 −0.209028
\(997\) 29.5167 0.934805 0.467402 0.884045i \(-0.345190\pi\)
0.467402 + 0.884045i \(0.345190\pi\)
\(998\) 25.0863 0.794092
\(999\) −6.86001 −0.217041
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 8034.2.a.n.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.n.1.3 4 1.1 even 1 trivial