Properties

Label 2738.2.a.r.1.5
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $6$
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] = [2738,2,Mod(1,2738)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2738, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2738.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.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: \(21.8630400734\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.96962\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.53209 q^{3} +1.00000 q^{4} -0.384754 q^{5} -1.53209 q^{6} -3.26414 q^{7} -1.00000 q^{8} -0.652704 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.53209 q^{3} +1.00000 q^{4} -0.384754 q^{5} -1.53209 q^{6} -3.26414 q^{7} -1.00000 q^{8} -0.652704 q^{9} +0.384754 q^{10} -2.58472 q^{11} +1.53209 q^{12} -1.36492 q^{13} +3.26414 q^{14} -0.589478 q^{15} +1.00000 q^{16} +3.33213 q^{17} +0.652704 q^{18} +5.98578 q^{19} -0.384754 q^{20} -5.00095 q^{21} +2.58472 q^{22} +7.54605 q^{23} -1.53209 q^{24} -4.85196 q^{25} +1.36492 q^{26} -5.59627 q^{27} -3.26414 q^{28} +3.21297 q^{29} +0.589478 q^{30} -2.53737 q^{31} -1.00000 q^{32} -3.96002 q^{33} -3.33213 q^{34} +1.25589 q^{35} -0.652704 q^{36} -5.98578 q^{38} -2.09118 q^{39} +0.384754 q^{40} +8.26915 q^{41} +5.00095 q^{42} -4.33920 q^{43} -2.58472 q^{44} +0.251131 q^{45} -7.54605 q^{46} +5.22909 q^{47} +1.53209 q^{48} +3.65461 q^{49} +4.85196 q^{50} +5.10511 q^{51} -1.36492 q^{52} -8.67122 q^{53} +5.59627 q^{54} +0.994481 q^{55} +3.26414 q^{56} +9.17075 q^{57} -3.21297 q^{58} -12.9674 q^{59} -0.589478 q^{60} +7.01264 q^{61} +2.53737 q^{62} +2.13052 q^{63} +1.00000 q^{64} +0.525160 q^{65} +3.96002 q^{66} +11.2380 q^{67} +3.33213 q^{68} +11.5612 q^{69} -1.25589 q^{70} +14.9995 q^{71} +0.652704 q^{72} +15.0792 q^{73} -7.43364 q^{75} +5.98578 q^{76} +8.43688 q^{77} +2.09118 q^{78} +1.46329 q^{79} -0.384754 q^{80} -6.61587 q^{81} -8.26915 q^{82} -0.0111178 q^{83} -5.00095 q^{84} -1.28205 q^{85} +4.33920 q^{86} +4.92256 q^{87} +2.58472 q^{88} -0.953262 q^{89} -0.251131 q^{90} +4.45530 q^{91} +7.54605 q^{92} -3.88748 q^{93} -5.22909 q^{94} -2.30306 q^{95} -1.53209 q^{96} +14.8185 q^{97} -3.65461 q^{98} +1.68705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} - 6 q^{10} - 6 q^{11} + 12 q^{13} - 6 q^{15} + 6 q^{16} + 12 q^{17} + 6 q^{18} + 12 q^{19} + 6 q^{20} - 12 q^{21} + 6 q^{22} - 6 q^{23} + 6 q^{25} - 12 q^{26} - 6 q^{27} - 6 q^{29} + 6 q^{30} + 18 q^{31} - 6 q^{32} + 6 q^{33} - 12 q^{34} + 24 q^{35} - 6 q^{36} - 12 q^{38} + 6 q^{39} - 6 q^{40} - 12 q^{41} + 12 q^{42} - 6 q^{44} + 6 q^{45} + 6 q^{46} - 6 q^{47} - 12 q^{49} - 6 q^{50} + 24 q^{51} + 12 q^{52} - 24 q^{53} + 6 q^{54} + 36 q^{55} + 24 q^{57} + 6 q^{58} + 12 q^{59} - 6 q^{60} + 12 q^{61} - 18 q^{62} + 6 q^{63} + 6 q^{64} + 18 q^{65} - 6 q^{66} + 18 q^{67} + 12 q^{68} + 24 q^{69} - 24 q^{70} + 24 q^{71} + 6 q^{72} - 18 q^{75} + 12 q^{76} + 30 q^{77} - 6 q^{78} + 12 q^{79} + 6 q^{80} - 18 q^{81} + 12 q^{82} - 6 q^{83} - 12 q^{84} + 18 q^{85} + 6 q^{87} + 6 q^{88} - 6 q^{90} + 12 q^{91} - 6 q^{92} - 36 q^{93} + 6 q^{94} + 18 q^{95} - 12 q^{97} + 12 q^{98} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 1.53209 0.884552 0.442276 0.896879i \(-0.354171\pi\)
0.442276 + 0.896879i \(0.354171\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.384754 −0.172067 −0.0860337 0.996292i \(-0.527419\pi\)
−0.0860337 + 0.996292i \(0.527419\pi\)
\(6\) −1.53209 −0.625473
\(7\) −3.26414 −1.23373 −0.616864 0.787069i \(-0.711597\pi\)
−0.616864 + 0.787069i \(0.711597\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.652704 −0.217568
\(10\) 0.384754 0.121670
\(11\) −2.58472 −0.779321 −0.389661 0.920959i \(-0.627408\pi\)
−0.389661 + 0.920959i \(0.627408\pi\)
\(12\) 1.53209 0.442276
\(13\) −1.36492 −0.378561 −0.189281 0.981923i \(-0.560616\pi\)
−0.189281 + 0.981923i \(0.560616\pi\)
\(14\) 3.26414 0.872378
\(15\) −0.589478 −0.152203
\(16\) 1.00000 0.250000
\(17\) 3.33213 0.808159 0.404080 0.914724i \(-0.367592\pi\)
0.404080 + 0.914724i \(0.367592\pi\)
\(18\) 0.652704 0.153844
\(19\) 5.98578 1.37323 0.686616 0.727020i \(-0.259095\pi\)
0.686616 + 0.727020i \(0.259095\pi\)
\(20\) −0.384754 −0.0860337
\(21\) −5.00095 −1.09130
\(22\) 2.58472 0.551063
\(23\) 7.54605 1.57346 0.786730 0.617297i \(-0.211772\pi\)
0.786730 + 0.617297i \(0.211772\pi\)
\(24\) −1.53209 −0.312736
\(25\) −4.85196 −0.970393
\(26\) 1.36492 0.267683
\(27\) −5.59627 −1.07700
\(28\) −3.26414 −0.616864
\(29\) 3.21297 0.596634 0.298317 0.954467i \(-0.403575\pi\)
0.298317 + 0.954467i \(0.403575\pi\)
\(30\) 0.589478 0.107623
\(31\) −2.53737 −0.455726 −0.227863 0.973693i \(-0.573174\pi\)
−0.227863 + 0.973693i \(0.573174\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.96002 −0.689350
\(34\) −3.33213 −0.571455
\(35\) 1.25589 0.212285
\(36\) −0.652704 −0.108784
\(37\) 0 0
\(38\) −5.98578 −0.971022
\(39\) −2.09118 −0.334857
\(40\) 0.384754 0.0608350
\(41\) 8.26915 1.29142 0.645712 0.763581i \(-0.276561\pi\)
0.645712 + 0.763581i \(0.276561\pi\)
\(42\) 5.00095 0.771664
\(43\) −4.33920 −0.661722 −0.330861 0.943680i \(-0.607339\pi\)
−0.330861 + 0.943680i \(0.607339\pi\)
\(44\) −2.58472 −0.389661
\(45\) 0.251131 0.0374363
\(46\) −7.54605 −1.11260
\(47\) 5.22909 0.762742 0.381371 0.924422i \(-0.375452\pi\)
0.381371 + 0.924422i \(0.375452\pi\)
\(48\) 1.53209 0.221138
\(49\) 3.65461 0.522087
\(50\) 4.85196 0.686171
\(51\) 5.10511 0.714859
\(52\) −1.36492 −0.189281
\(53\) −8.67122 −1.19108 −0.595542 0.803324i \(-0.703063\pi\)
−0.595542 + 0.803324i \(0.703063\pi\)
\(54\) 5.59627 0.761555
\(55\) 0.994481 0.134096
\(56\) 3.26414 0.436189
\(57\) 9.17075 1.21470
\(58\) −3.21297 −0.421884
\(59\) −12.9674 −1.68821 −0.844107 0.536174i \(-0.819869\pi\)
−0.844107 + 0.536174i \(0.819869\pi\)
\(60\) −0.589478 −0.0761013
\(61\) 7.01264 0.897877 0.448938 0.893563i \(-0.351802\pi\)
0.448938 + 0.893563i \(0.351802\pi\)
\(62\) 2.53737 0.322247
\(63\) 2.13052 0.268420
\(64\) 1.00000 0.125000
\(65\) 0.525160 0.0651381
\(66\) 3.96002 0.487444
\(67\) 11.2380 1.37294 0.686470 0.727158i \(-0.259159\pi\)
0.686470 + 0.727158i \(0.259159\pi\)
\(68\) 3.33213 0.404080
\(69\) 11.5612 1.39181
\(70\) −1.25589 −0.150108
\(71\) 14.9995 1.78012 0.890059 0.455845i \(-0.150663\pi\)
0.890059 + 0.455845i \(0.150663\pi\)
\(72\) 0.652704 0.0769219
\(73\) 15.0792 1.76489 0.882445 0.470416i \(-0.155896\pi\)
0.882445 + 0.470416i \(0.155896\pi\)
\(74\) 0 0
\(75\) −7.43364 −0.858363
\(76\) 5.98578 0.686616
\(77\) 8.43688 0.961471
\(78\) 2.09118 0.236780
\(79\) 1.46329 0.164633 0.0823167 0.996606i \(-0.473768\pi\)
0.0823167 + 0.996606i \(0.473768\pi\)
\(80\) −0.384754 −0.0430169
\(81\) −6.61587 −0.735096
\(82\) −8.26915 −0.913175
\(83\) −0.0111178 −0.00122034 −0.000610170 1.00000i \(-0.500194\pi\)
−0.000610170 1.00000i \(0.500194\pi\)
\(84\) −5.00095 −0.545649
\(85\) −1.28205 −0.139058
\(86\) 4.33920 0.467908
\(87\) 4.92256 0.527754
\(88\) 2.58472 0.275532
\(89\) −0.953262 −0.101046 −0.0505228 0.998723i \(-0.516089\pi\)
−0.0505228 + 0.998723i \(0.516089\pi\)
\(90\) −0.251131 −0.0264715
\(91\) 4.45530 0.467042
\(92\) 7.54605 0.786730
\(93\) −3.88748 −0.403113
\(94\) −5.22909 −0.539340
\(95\) −2.30306 −0.236289
\(96\) −1.53209 −0.156368
\(97\) 14.8185 1.50459 0.752297 0.658824i \(-0.228946\pi\)
0.752297 + 0.658824i \(0.228946\pi\)
\(98\) −3.65461 −0.369171
\(99\) 1.68705 0.169555
\(100\) −4.85196 −0.485196
\(101\) −1.27252 −0.126621 −0.0633103 0.997994i \(-0.520166\pi\)
−0.0633103 + 0.997994i \(0.520166\pi\)
\(102\) −5.10511 −0.505482
\(103\) 10.8461 1.06870 0.534350 0.845264i \(-0.320557\pi\)
0.534350 + 0.845264i \(0.320557\pi\)
\(104\) 1.36492 0.133842
\(105\) 1.92414 0.187777
\(106\) 8.67122 0.842224
\(107\) −4.75958 −0.460126 −0.230063 0.973176i \(-0.573893\pi\)
−0.230063 + 0.973176i \(0.573893\pi\)
\(108\) −5.59627 −0.538501
\(109\) 9.63744 0.923099 0.461550 0.887114i \(-0.347294\pi\)
0.461550 + 0.887114i \(0.347294\pi\)
\(110\) −0.994481 −0.0948201
\(111\) 0 0
\(112\) −3.26414 −0.308432
\(113\) 12.9549 1.21870 0.609348 0.792903i \(-0.291431\pi\)
0.609348 + 0.792903i \(0.291431\pi\)
\(114\) −9.17075 −0.858920
\(115\) −2.90338 −0.270741
\(116\) 3.21297 0.298317
\(117\) 0.890890 0.0823628
\(118\) 12.9674 1.19375
\(119\) −10.8765 −0.997050
\(120\) 0.589478 0.0538117
\(121\) −4.31924 −0.392658
\(122\) −7.01264 −0.634895
\(123\) 12.6691 1.14233
\(124\) −2.53737 −0.227863
\(125\) 3.79059 0.339040
\(126\) −2.13052 −0.189801
\(127\) 11.2408 0.997458 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.64804 −0.585327
\(130\) −0.525160 −0.0460596
\(131\) 8.36959 0.731254 0.365627 0.930761i \(-0.380855\pi\)
0.365627 + 0.930761i \(0.380855\pi\)
\(132\) −3.96002 −0.344675
\(133\) −19.5384 −1.69420
\(134\) −11.2380 −0.970815
\(135\) 2.15319 0.185317
\(136\) −3.33213 −0.285728
\(137\) −3.98413 −0.340387 −0.170194 0.985411i \(-0.554439\pi\)
−0.170194 + 0.985411i \(0.554439\pi\)
\(138\) −11.5612 −0.984156
\(139\) −5.11140 −0.433543 −0.216772 0.976222i \(-0.569553\pi\)
−0.216772 + 0.976222i \(0.569553\pi\)
\(140\) 1.25589 0.106142
\(141\) 8.01143 0.674684
\(142\) −14.9995 −1.25873
\(143\) 3.52794 0.295021
\(144\) −0.652704 −0.0543920
\(145\) −1.23620 −0.102661
\(146\) −15.0792 −1.24797
\(147\) 5.59918 0.461813
\(148\) 0 0
\(149\) −15.8700 −1.30012 −0.650060 0.759883i \(-0.725256\pi\)
−0.650060 + 0.759883i \(0.725256\pi\)
\(150\) 7.43364 0.606954
\(151\) 21.7680 1.77146 0.885729 0.464202i \(-0.153659\pi\)
0.885729 + 0.464202i \(0.153659\pi\)
\(152\) −5.98578 −0.485511
\(153\) −2.17489 −0.175830
\(154\) −8.43688 −0.679863
\(155\) 0.976266 0.0784155
\(156\) −2.09118 −0.167429
\(157\) −9.75142 −0.778248 −0.389124 0.921185i \(-0.627222\pi\)
−0.389124 + 0.921185i \(0.627222\pi\)
\(158\) −1.46329 −0.116413
\(159\) −13.2851 −1.05358
\(160\) 0.384754 0.0304175
\(161\) −24.6314 −1.94122
\(162\) 6.61587 0.519792
\(163\) 13.1647 1.03114 0.515571 0.856847i \(-0.327580\pi\)
0.515571 + 0.856847i \(0.327580\pi\)
\(164\) 8.26915 0.645712
\(165\) 1.52363 0.118615
\(166\) 0.0111178 0.000862910 0
\(167\) −5.60228 −0.433518 −0.216759 0.976225i \(-0.569549\pi\)
−0.216759 + 0.976225i \(0.569549\pi\)
\(168\) 5.00095 0.385832
\(169\) −11.1370 −0.856691
\(170\) 1.28205 0.0983288
\(171\) −3.90694 −0.298771
\(172\) −4.33920 −0.330861
\(173\) 10.8354 0.823800 0.411900 0.911229i \(-0.364865\pi\)
0.411900 + 0.911229i \(0.364865\pi\)
\(174\) −4.92256 −0.373178
\(175\) 15.8375 1.19720
\(176\) −2.58472 −0.194830
\(177\) −19.8672 −1.49331
\(178\) 0.953262 0.0714500
\(179\) 6.07192 0.453837 0.226918 0.973914i \(-0.427135\pi\)
0.226918 + 0.973914i \(0.427135\pi\)
\(180\) 0.251131 0.0187182
\(181\) 21.5538 1.60208 0.801042 0.598608i \(-0.204279\pi\)
0.801042 + 0.598608i \(0.204279\pi\)
\(182\) −4.45530 −0.330249
\(183\) 10.7440 0.794219
\(184\) −7.54605 −0.556302
\(185\) 0 0
\(186\) 3.88748 0.285044
\(187\) −8.61260 −0.629816
\(188\) 5.22909 0.381371
\(189\) 18.2670 1.32873
\(190\) 2.30306 0.167081
\(191\) −6.62693 −0.479508 −0.239754 0.970834i \(-0.577067\pi\)
−0.239754 + 0.970834i \(0.577067\pi\)
\(192\) 1.53209 0.110569
\(193\) 22.8544 1.64510 0.822548 0.568696i \(-0.192552\pi\)
0.822548 + 0.568696i \(0.192552\pi\)
\(194\) −14.8185 −1.06391
\(195\) 0.804592 0.0576180
\(196\) 3.65461 0.261043
\(197\) −14.7561 −1.05133 −0.525663 0.850693i \(-0.676183\pi\)
−0.525663 + 0.850693i \(0.676183\pi\)
\(198\) −1.68705 −0.119894
\(199\) 1.47478 0.104544 0.0522721 0.998633i \(-0.483354\pi\)
0.0522721 + 0.998633i \(0.483354\pi\)
\(200\) 4.85196 0.343086
\(201\) 17.2176 1.21444
\(202\) 1.27252 0.0895342
\(203\) −10.4876 −0.736084
\(204\) 5.10511 0.357430
\(205\) −3.18159 −0.222212
\(206\) −10.8461 −0.755685
\(207\) −4.92533 −0.342334
\(208\) −1.36492 −0.0946403
\(209\) −15.4716 −1.07019
\(210\) −1.92414 −0.132778
\(211\) −2.41952 −0.166567 −0.0832835 0.996526i \(-0.526541\pi\)
−0.0832835 + 0.996526i \(0.526541\pi\)
\(212\) −8.67122 −0.595542
\(213\) 22.9806 1.57461
\(214\) 4.75958 0.325358
\(215\) 1.66953 0.113861
\(216\) 5.59627 0.380778
\(217\) 8.28234 0.562242
\(218\) −9.63744 −0.652730
\(219\) 23.1027 1.56114
\(220\) 0.994481 0.0670479
\(221\) −4.54809 −0.305938
\(222\) 0 0
\(223\) −6.94726 −0.465223 −0.232611 0.972570i \(-0.574727\pi\)
−0.232611 + 0.972570i \(0.574727\pi\)
\(224\) 3.26414 0.218095
\(225\) 3.16689 0.211126
\(226\) −12.9549 −0.861749
\(227\) −21.8490 −1.45017 −0.725084 0.688660i \(-0.758199\pi\)
−0.725084 + 0.688660i \(0.758199\pi\)
\(228\) 9.17075 0.607348
\(229\) −4.25010 −0.280854 −0.140427 0.990091i \(-0.544848\pi\)
−0.140427 + 0.990091i \(0.544848\pi\)
\(230\) 2.90338 0.191443
\(231\) 12.9260 0.850471
\(232\) −3.21297 −0.210942
\(233\) −17.3443 −1.13626 −0.568131 0.822938i \(-0.692333\pi\)
−0.568131 + 0.822938i \(0.692333\pi\)
\(234\) −0.890890 −0.0582393
\(235\) −2.01192 −0.131243
\(236\) −12.9674 −0.844107
\(237\) 2.24190 0.145627
\(238\) 10.8765 0.705021
\(239\) 18.3584 1.18750 0.593752 0.804648i \(-0.297646\pi\)
0.593752 + 0.804648i \(0.297646\pi\)
\(240\) −0.589478 −0.0380506
\(241\) −23.4661 −1.51158 −0.755792 0.654811i \(-0.772748\pi\)
−0.755792 + 0.654811i \(0.772748\pi\)
\(242\) 4.31924 0.277651
\(243\) 6.65270 0.426771
\(244\) 7.01264 0.448938
\(245\) −1.40613 −0.0898341
\(246\) −12.6691 −0.807751
\(247\) −8.17013 −0.519853
\(248\) 2.53737 0.161123
\(249\) −0.0170335 −0.00107945
\(250\) −3.79059 −0.239738
\(251\) 6.10149 0.385123 0.192561 0.981285i \(-0.438321\pi\)
0.192561 + 0.981285i \(0.438321\pi\)
\(252\) 2.13052 0.134210
\(253\) −19.5044 −1.22623
\(254\) −11.2408 −0.705309
\(255\) −1.96422 −0.123004
\(256\) 1.00000 0.0625000
\(257\) 4.34710 0.271165 0.135582 0.990766i \(-0.456709\pi\)
0.135582 + 0.990766i \(0.456709\pi\)
\(258\) 6.64804 0.413889
\(259\) 0 0
\(260\) 0.525160 0.0325690
\(261\) −2.09712 −0.129808
\(262\) −8.36959 −0.517075
\(263\) 4.05485 0.250033 0.125016 0.992155i \(-0.460102\pi\)
0.125016 + 0.992155i \(0.460102\pi\)
\(264\) 3.96002 0.243722
\(265\) 3.33629 0.204947
\(266\) 19.5384 1.19798
\(267\) −1.46048 −0.0893801
\(268\) 11.2380 0.686470
\(269\) 8.79337 0.536141 0.268071 0.963399i \(-0.413614\pi\)
0.268071 + 0.963399i \(0.413614\pi\)
\(270\) −2.15319 −0.131039
\(271\) −12.9122 −0.784361 −0.392180 0.919888i \(-0.628279\pi\)
−0.392180 + 0.919888i \(0.628279\pi\)
\(272\) 3.33213 0.202040
\(273\) 6.82591 0.413123
\(274\) 3.98413 0.240690
\(275\) 12.5410 0.756248
\(276\) 11.5612 0.695904
\(277\) 7.77643 0.467240 0.233620 0.972328i \(-0.424943\pi\)
0.233620 + 0.972328i \(0.424943\pi\)
\(278\) 5.11140 0.306561
\(279\) 1.65615 0.0991513
\(280\) −1.25589 −0.0750539
\(281\) −0.232510 −0.0138704 −0.00693518 0.999976i \(-0.502208\pi\)
−0.00693518 + 0.999976i \(0.502208\pi\)
\(282\) −8.01143 −0.477074
\(283\) 12.8330 0.762843 0.381421 0.924401i \(-0.375435\pi\)
0.381421 + 0.924401i \(0.375435\pi\)
\(284\) 14.9995 0.890059
\(285\) −3.52849 −0.209010
\(286\) −3.52794 −0.208611
\(287\) −26.9917 −1.59327
\(288\) 0.652704 0.0384609
\(289\) −5.89693 −0.346878
\(290\) 1.23620 0.0725925
\(291\) 22.7033 1.33089
\(292\) 15.0792 0.882445
\(293\) −24.6443 −1.43974 −0.719868 0.694111i \(-0.755797\pi\)
−0.719868 + 0.694111i \(0.755797\pi\)
\(294\) −5.59918 −0.326551
\(295\) 4.98927 0.290487
\(296\) 0 0
\(297\) 14.4648 0.839331
\(298\) 15.8700 0.919324
\(299\) −10.2998 −0.595651
\(300\) −7.43364 −0.429181
\(301\) 14.1638 0.816385
\(302\) −21.7680 −1.25261
\(303\) −1.94961 −0.112002
\(304\) 5.98578 0.343308
\(305\) −2.69815 −0.154495
\(306\) 2.17489 0.124330
\(307\) −6.91132 −0.394450 −0.197225 0.980358i \(-0.563193\pi\)
−0.197225 + 0.980358i \(0.563193\pi\)
\(308\) 8.43688 0.480736
\(309\) 16.6172 0.945320
\(310\) −0.976266 −0.0554482
\(311\) 1.65859 0.0940501 0.0470251 0.998894i \(-0.485026\pi\)
0.0470251 + 0.998894i \(0.485026\pi\)
\(312\) 2.09118 0.118390
\(313\) −25.9509 −1.46683 −0.733416 0.679780i \(-0.762075\pi\)
−0.733416 + 0.679780i \(0.762075\pi\)
\(314\) 9.75142 0.550304
\(315\) −0.819725 −0.0461863
\(316\) 1.46329 0.0823167
\(317\) 11.3498 0.637470 0.318735 0.947844i \(-0.396742\pi\)
0.318735 + 0.947844i \(0.396742\pi\)
\(318\) 13.2851 0.744991
\(319\) −8.30462 −0.464969
\(320\) −0.384754 −0.0215084
\(321\) −7.29210 −0.407006
\(322\) 24.6314 1.37265
\(323\) 19.9454 1.10979
\(324\) −6.61587 −0.367548
\(325\) 6.62255 0.367353
\(326\) −13.1647 −0.729127
\(327\) 14.7654 0.816529
\(328\) −8.26915 −0.456588
\(329\) −17.0685 −0.941016
\(330\) −1.52363 −0.0838733
\(331\) −12.8377 −0.705623 −0.352811 0.935694i \(-0.614774\pi\)
−0.352811 + 0.935694i \(0.614774\pi\)
\(332\) −0.0111178 −0.000610170 0
\(333\) 0 0
\(334\) 5.60228 0.306543
\(335\) −4.32387 −0.236238
\(336\) −5.00095 −0.272824
\(337\) −5.71183 −0.311143 −0.155572 0.987825i \(-0.549722\pi\)
−0.155572 + 0.987825i \(0.549722\pi\)
\(338\) 11.1370 0.605772
\(339\) 19.8481 1.07800
\(340\) −1.28205 −0.0695290
\(341\) 6.55839 0.355157
\(342\) 3.90694 0.211263
\(343\) 10.9198 0.589615
\(344\) 4.33920 0.233954
\(345\) −4.44823 −0.239485
\(346\) −10.8354 −0.582515
\(347\) −32.3416 −1.73619 −0.868095 0.496398i \(-0.834656\pi\)
−0.868095 + 0.496398i \(0.834656\pi\)
\(348\) 4.92256 0.263877
\(349\) −24.9320 −1.33458 −0.667290 0.744798i \(-0.732546\pi\)
−0.667290 + 0.744798i \(0.732546\pi\)
\(350\) −15.8375 −0.846549
\(351\) 7.63847 0.407711
\(352\) 2.58472 0.137766
\(353\) 8.64114 0.459921 0.229961 0.973200i \(-0.426140\pi\)
0.229961 + 0.973200i \(0.426140\pi\)
\(354\) 19.8672 1.05593
\(355\) −5.77114 −0.306300
\(356\) −0.953262 −0.0505228
\(357\) −16.6638 −0.881942
\(358\) −6.07192 −0.320911
\(359\) 3.85848 0.203643 0.101821 0.994803i \(-0.467533\pi\)
0.101821 + 0.994803i \(0.467533\pi\)
\(360\) −0.251131 −0.0132357
\(361\) 16.8296 0.885768
\(362\) −21.5538 −1.13284
\(363\) −6.61746 −0.347327
\(364\) 4.45530 0.233521
\(365\) −5.80180 −0.303680
\(366\) −10.7440 −0.561597
\(367\) 10.4563 0.545813 0.272906 0.962041i \(-0.412015\pi\)
0.272906 + 0.962041i \(0.412015\pi\)
\(368\) 7.54605 0.393365
\(369\) −5.39731 −0.280973
\(370\) 0 0
\(371\) 28.3041 1.46947
\(372\) −3.88748 −0.201557
\(373\) −9.86566 −0.510825 −0.255412 0.966832i \(-0.582211\pi\)
−0.255412 + 0.966832i \(0.582211\pi\)
\(374\) 8.61260 0.445347
\(375\) 5.80752 0.299899
\(376\) −5.22909 −0.269670
\(377\) −4.38546 −0.225862
\(378\) −18.2670 −0.939553
\(379\) 2.48121 0.127451 0.0637255 0.997967i \(-0.479702\pi\)
0.0637255 + 0.997967i \(0.479702\pi\)
\(380\) −2.30306 −0.118144
\(381\) 17.2219 0.882303
\(382\) 6.62693 0.339063
\(383\) 25.5639 1.30625 0.653127 0.757249i \(-0.273457\pi\)
0.653127 + 0.757249i \(0.273457\pi\)
\(384\) −1.53209 −0.0781841
\(385\) −3.24613 −0.165438
\(386\) −22.8544 −1.16326
\(387\) 2.83221 0.143969
\(388\) 14.8185 0.752297
\(389\) 6.37509 0.323230 0.161615 0.986854i \(-0.448330\pi\)
0.161615 + 0.986854i \(0.448330\pi\)
\(390\) −0.804592 −0.0407421
\(391\) 25.1444 1.27161
\(392\) −3.65461 −0.184586
\(393\) 12.8229 0.646832
\(394\) 14.7561 0.743400
\(395\) −0.563009 −0.0283281
\(396\) 1.68705 0.0847776
\(397\) 37.4977 1.88196 0.940979 0.338465i \(-0.109908\pi\)
0.940979 + 0.338465i \(0.109908\pi\)
\(398\) −1.47478 −0.0739240
\(399\) −29.9346 −1.49861
\(400\) −4.85196 −0.242598
\(401\) 35.1464 1.75513 0.877563 0.479462i \(-0.159168\pi\)
0.877563 + 0.479462i \(0.159168\pi\)
\(402\) −17.2176 −0.858737
\(403\) 3.46332 0.172520
\(404\) −1.27252 −0.0633103
\(405\) 2.54548 0.126486
\(406\) 10.4876 0.520490
\(407\) 0 0
\(408\) −5.10511 −0.252741
\(409\) −0.450741 −0.0222877 −0.0111439 0.999938i \(-0.503547\pi\)
−0.0111439 + 0.999938i \(0.503547\pi\)
\(410\) 3.18159 0.157128
\(411\) −6.10404 −0.301090
\(412\) 10.8461 0.534350
\(413\) 42.3275 2.08280
\(414\) 4.92533 0.242067
\(415\) 0.00427763 0.000209981 0
\(416\) 1.36492 0.0669208
\(417\) −7.83112 −0.383492
\(418\) 15.4716 0.756738
\(419\) −15.3113 −0.748006 −0.374003 0.927428i \(-0.622015\pi\)
−0.374003 + 0.927428i \(0.622015\pi\)
\(420\) 1.92414 0.0938884
\(421\) 22.3965 1.09154 0.545769 0.837936i \(-0.316238\pi\)
0.545769 + 0.837936i \(0.316238\pi\)
\(422\) 2.41952 0.117781
\(423\) −3.41305 −0.165948
\(424\) 8.67122 0.421112
\(425\) −16.1674 −0.784232
\(426\) −22.9806 −1.11342
\(427\) −22.8902 −1.10774
\(428\) −4.75958 −0.230063
\(429\) 5.40511 0.260961
\(430\) −1.66953 −0.0805117
\(431\) 0.102080 0.00491701 0.00245851 0.999997i \(-0.499217\pi\)
0.00245851 + 0.999997i \(0.499217\pi\)
\(432\) −5.59627 −0.269251
\(433\) 7.92714 0.380954 0.190477 0.981692i \(-0.438997\pi\)
0.190477 + 0.981692i \(0.438997\pi\)
\(434\) −8.28234 −0.397565
\(435\) −1.89398 −0.0908092
\(436\) 9.63744 0.461550
\(437\) 45.1690 2.16073
\(438\) −23.1027 −1.10389
\(439\) −7.57063 −0.361326 −0.180663 0.983545i \(-0.557824\pi\)
−0.180663 + 0.983545i \(0.557824\pi\)
\(440\) −0.994481 −0.0474100
\(441\) −2.38538 −0.113589
\(442\) 4.54809 0.216331
\(443\) −15.9856 −0.759499 −0.379749 0.925089i \(-0.623990\pi\)
−0.379749 + 0.925089i \(0.623990\pi\)
\(444\) 0 0
\(445\) 0.366772 0.0173867
\(446\) 6.94726 0.328962
\(447\) −24.3142 −1.15002
\(448\) −3.26414 −0.154216
\(449\) −28.6753 −1.35327 −0.676636 0.736318i \(-0.736563\pi\)
−0.676636 + 0.736318i \(0.736563\pi\)
\(450\) −3.16689 −0.149289
\(451\) −21.3734 −1.00643
\(452\) 12.9549 0.609348
\(453\) 33.3506 1.56695
\(454\) 21.8490 1.02542
\(455\) −1.71420 −0.0803627
\(456\) −9.17075 −0.429460
\(457\) 0.105116 0.00491713 0.00245857 0.999997i \(-0.499217\pi\)
0.00245857 + 0.999997i \(0.499217\pi\)
\(458\) 4.25010 0.198594
\(459\) −18.6475 −0.870389
\(460\) −2.90338 −0.135371
\(461\) −23.9701 −1.11640 −0.558199 0.829707i \(-0.688507\pi\)
−0.558199 + 0.829707i \(0.688507\pi\)
\(462\) −12.9260 −0.601374
\(463\) 20.7514 0.964398 0.482199 0.876062i \(-0.339838\pi\)
0.482199 + 0.876062i \(0.339838\pi\)
\(464\) 3.21297 0.149158
\(465\) 1.49573 0.0693626
\(466\) 17.3443 0.803459
\(467\) 17.4928 0.809472 0.404736 0.914434i \(-0.367363\pi\)
0.404736 + 0.914434i \(0.367363\pi\)
\(468\) 0.890890 0.0411814
\(469\) −36.6824 −1.69384
\(470\) 2.01192 0.0928028
\(471\) −14.9400 −0.688401
\(472\) 12.9674 0.596874
\(473\) 11.2156 0.515694
\(474\) −2.24190 −0.102974
\(475\) −29.0428 −1.33258
\(476\) −10.8765 −0.498525
\(477\) 5.65974 0.259142
\(478\) −18.3584 −0.839692
\(479\) −6.88046 −0.314376 −0.157188 0.987569i \(-0.550243\pi\)
−0.157188 + 0.987569i \(0.550243\pi\)
\(480\) 0.589478 0.0269059
\(481\) 0 0
\(482\) 23.4661 1.06885
\(483\) −37.7374 −1.71711
\(484\) −4.31924 −0.196329
\(485\) −5.70150 −0.258892
\(486\) −6.65270 −0.301773
\(487\) −1.70810 −0.0774014 −0.0387007 0.999251i \(-0.512322\pi\)
−0.0387007 + 0.999251i \(0.512322\pi\)
\(488\) −7.01264 −0.317447
\(489\) 20.1696 0.912099
\(490\) 1.40613 0.0635223
\(491\) 10.6231 0.479412 0.239706 0.970846i \(-0.422949\pi\)
0.239706 + 0.970846i \(0.422949\pi\)
\(492\) 12.6691 0.571166
\(493\) 10.7060 0.482175
\(494\) 8.17013 0.367591
\(495\) −0.649101 −0.0291749
\(496\) −2.53737 −0.113931
\(497\) −48.9606 −2.19618
\(498\) 0.0170335 0.000763289 0
\(499\) 35.4707 1.58789 0.793944 0.607991i \(-0.208024\pi\)
0.793944 + 0.607991i \(0.208024\pi\)
\(500\) 3.79059 0.169520
\(501\) −8.58320 −0.383469
\(502\) −6.10149 −0.272323
\(503\) 30.0880 1.34156 0.670778 0.741658i \(-0.265960\pi\)
0.670778 + 0.741658i \(0.265960\pi\)
\(504\) −2.13052 −0.0949007
\(505\) 0.489608 0.0217873
\(506\) 19.5044 0.867076
\(507\) −17.0629 −0.757788
\(508\) 11.2408 0.498729
\(509\) 24.0909 1.06781 0.533905 0.845544i \(-0.320724\pi\)
0.533905 + 0.845544i \(0.320724\pi\)
\(510\) 1.96422 0.0869769
\(511\) −49.2207 −2.17740
\(512\) −1.00000 −0.0441942
\(513\) −33.4980 −1.47897
\(514\) −4.34710 −0.191742
\(515\) −4.17309 −0.183888
\(516\) −6.64804 −0.292664
\(517\) −13.5157 −0.594421
\(518\) 0 0
\(519\) 16.6008 0.728694
\(520\) −0.525160 −0.0230298
\(521\) 0.749547 0.0328383 0.0164191 0.999865i \(-0.494773\pi\)
0.0164191 + 0.999865i \(0.494773\pi\)
\(522\) 2.09712 0.0917884
\(523\) 2.03859 0.0891413 0.0445706 0.999006i \(-0.485808\pi\)
0.0445706 + 0.999006i \(0.485808\pi\)
\(524\) 8.36959 0.365627
\(525\) 24.2644 1.05899
\(526\) −4.05485 −0.176800
\(527\) −8.45485 −0.368299
\(528\) −3.96002 −0.172338
\(529\) 33.9429 1.47578
\(530\) −3.33629 −0.144919
\(531\) 8.46389 0.367301
\(532\) −19.5384 −0.847098
\(533\) −11.2868 −0.488883
\(534\) 1.46048 0.0632013
\(535\) 1.83127 0.0791727
\(536\) −11.2380 −0.485408
\(537\) 9.30273 0.401442
\(538\) −8.79337 −0.379109
\(539\) −9.44612 −0.406873
\(540\) 2.15319 0.0926585
\(541\) −31.3434 −1.34756 −0.673778 0.738934i \(-0.735330\pi\)
−0.673778 + 0.738934i \(0.735330\pi\)
\(542\) 12.9122 0.554627
\(543\) 33.0224 1.41713
\(544\) −3.33213 −0.142864
\(545\) −3.70805 −0.158835
\(546\) −6.82591 −0.292122
\(547\) 18.1663 0.776737 0.388368 0.921504i \(-0.373039\pi\)
0.388368 + 0.921504i \(0.373039\pi\)
\(548\) −3.98413 −0.170194
\(549\) −4.57718 −0.195349
\(550\) −12.5410 −0.534748
\(551\) 19.2321 0.819317
\(552\) −11.5612 −0.492078
\(553\) −4.77640 −0.203113
\(554\) −7.77643 −0.330389
\(555\) 0 0
\(556\) −5.11140 −0.216772
\(557\) −32.9061 −1.39427 −0.697137 0.716938i \(-0.745543\pi\)
−0.697137 + 0.716938i \(0.745543\pi\)
\(558\) −1.65615 −0.0701105
\(559\) 5.92267 0.250502
\(560\) 1.25589 0.0530711
\(561\) −13.1953 −0.557105
\(562\) 0.232510 0.00980782
\(563\) −34.3008 −1.44561 −0.722804 0.691053i \(-0.757147\pi\)
−0.722804 + 0.691053i \(0.757147\pi\)
\(564\) 8.01143 0.337342
\(565\) −4.98446 −0.209698
\(566\) −12.8330 −0.539411
\(567\) 21.5951 0.906910
\(568\) −14.9995 −0.629367
\(569\) −21.2198 −0.889579 −0.444789 0.895635i \(-0.646722\pi\)
−0.444789 + 0.895635i \(0.646722\pi\)
\(570\) 3.52849 0.147792
\(571\) 16.4467 0.688272 0.344136 0.938920i \(-0.388172\pi\)
0.344136 + 0.938920i \(0.388172\pi\)
\(572\) 3.52794 0.147510
\(573\) −10.1530 −0.424150
\(574\) 26.9917 1.12661
\(575\) −36.6132 −1.52687
\(576\) −0.652704 −0.0271960
\(577\) 36.3216 1.51209 0.756044 0.654521i \(-0.227130\pi\)
0.756044 + 0.654521i \(0.227130\pi\)
\(578\) 5.89693 0.245280
\(579\) 35.0150 1.45517
\(580\) −1.23620 −0.0513306
\(581\) 0.0362901 0.00150557
\(582\) −22.7033 −0.941083
\(583\) 22.4127 0.928237
\(584\) −15.0792 −0.623983
\(585\) −0.342774 −0.0141720
\(586\) 24.6443 1.01805
\(587\) −37.5244 −1.54880 −0.774398 0.632699i \(-0.781947\pi\)
−0.774398 + 0.632699i \(0.781947\pi\)
\(588\) 5.59918 0.230906
\(589\) −15.1882 −0.625817
\(590\) −4.98927 −0.205405
\(591\) −22.6076 −0.929953
\(592\) 0 0
\(593\) −10.0159 −0.411302 −0.205651 0.978625i \(-0.565931\pi\)
−0.205651 + 0.978625i \(0.565931\pi\)
\(594\) −14.4648 −0.593496
\(595\) 4.18479 0.171560
\(596\) −15.8700 −0.650060
\(597\) 2.25949 0.0924748
\(598\) 10.2998 0.421189
\(599\) −33.6544 −1.37508 −0.687541 0.726145i \(-0.741310\pi\)
−0.687541 + 0.726145i \(0.741310\pi\)
\(600\) 7.43364 0.303477
\(601\) −42.8067 −1.74612 −0.873061 0.487610i \(-0.837869\pi\)
−0.873061 + 0.487610i \(0.837869\pi\)
\(602\) −14.1638 −0.577272
\(603\) −7.33508 −0.298708
\(604\) 21.7680 0.885729
\(605\) 1.66185 0.0675637
\(606\) 1.94961 0.0791977
\(607\) 5.37572 0.218194 0.109097 0.994031i \(-0.465204\pi\)
0.109097 + 0.994031i \(0.465204\pi\)
\(608\) −5.98578 −0.242756
\(609\) −16.0679 −0.651105
\(610\) 2.69815 0.109245
\(611\) −7.13731 −0.288744
\(612\) −2.17489 −0.0879148
\(613\) 33.4187 1.34977 0.674884 0.737924i \(-0.264194\pi\)
0.674884 + 0.737924i \(0.264194\pi\)
\(614\) 6.91132 0.278918
\(615\) −4.87448 −0.196558
\(616\) −8.43688 −0.339931
\(617\) 5.05959 0.203691 0.101846 0.994800i \(-0.467525\pi\)
0.101846 + 0.994800i \(0.467525\pi\)
\(618\) −16.6172 −0.668442
\(619\) −35.7818 −1.43819 −0.719096 0.694911i \(-0.755444\pi\)
−0.719096 + 0.694911i \(0.755444\pi\)
\(620\) 0.976266 0.0392078
\(621\) −42.2297 −1.69462
\(622\) −1.65859 −0.0665035
\(623\) 3.11158 0.124663
\(624\) −2.09118 −0.0837143
\(625\) 22.8014 0.912055
\(626\) 25.9509 1.03721
\(627\) −23.7038 −0.946638
\(628\) −9.75142 −0.389124
\(629\) 0 0
\(630\) 0.819725 0.0326586
\(631\) −6.62105 −0.263580 −0.131790 0.991278i \(-0.542072\pi\)
−0.131790 + 0.991278i \(0.542072\pi\)
\(632\) −1.46329 −0.0582067
\(633\) −3.70693 −0.147337
\(634\) −11.3498 −0.450759
\(635\) −4.32494 −0.171630
\(636\) −13.2851 −0.526788
\(637\) −4.98826 −0.197642
\(638\) 8.30462 0.328783
\(639\) −9.79025 −0.387296
\(640\) 0.384754 0.0152088
\(641\) −35.5949 −1.40591 −0.702956 0.711233i \(-0.748137\pi\)
−0.702956 + 0.711233i \(0.748137\pi\)
\(642\) 7.29210 0.287796
\(643\) −5.01240 −0.197670 −0.0988350 0.995104i \(-0.531512\pi\)
−0.0988350 + 0.995104i \(0.531512\pi\)
\(644\) −24.6314 −0.970612
\(645\) 2.55786 0.100716
\(646\) −19.9454 −0.784741
\(647\) −35.2707 −1.38663 −0.693317 0.720633i \(-0.743851\pi\)
−0.693317 + 0.720633i \(0.743851\pi\)
\(648\) 6.61587 0.259896
\(649\) 33.5171 1.31566
\(650\) −6.62255 −0.259758
\(651\) 12.6893 0.497332
\(652\) 13.1647 0.515571
\(653\) −2.70516 −0.105861 −0.0529305 0.998598i \(-0.516856\pi\)
−0.0529305 + 0.998598i \(0.516856\pi\)
\(654\) −14.7654 −0.577373
\(655\) −3.22024 −0.125825
\(656\) 8.26915 0.322856
\(657\) −9.84227 −0.383983
\(658\) 17.0685 0.665399
\(659\) −27.8599 −1.08527 −0.542634 0.839969i \(-0.682573\pi\)
−0.542634 + 0.839969i \(0.682573\pi\)
\(660\) 1.52363 0.0593074
\(661\) 10.6125 0.412777 0.206388 0.978470i \(-0.433829\pi\)
0.206388 + 0.978470i \(0.433829\pi\)
\(662\) 12.8377 0.498951
\(663\) −6.96809 −0.270618
\(664\) 0.0111178 0.000431455 0
\(665\) 7.51750 0.291516
\(666\) 0 0
\(667\) 24.2452 0.938779
\(668\) −5.60228 −0.216759
\(669\) −10.6438 −0.411514
\(670\) 4.32387 0.167046
\(671\) −18.1257 −0.699735
\(672\) 5.00095 0.192916
\(673\) −0.657395 −0.0253407 −0.0126704 0.999920i \(-0.504033\pi\)
−0.0126704 + 0.999920i \(0.504033\pi\)
\(674\) 5.71183 0.220011
\(675\) 27.1529 1.04512
\(676\) −11.1370 −0.428346
\(677\) 32.0071 1.23013 0.615067 0.788475i \(-0.289129\pi\)
0.615067 + 0.788475i \(0.289129\pi\)
\(678\) −19.8481 −0.762261
\(679\) −48.3698 −1.85626
\(680\) 1.28205 0.0491644
\(681\) −33.4746 −1.28275
\(682\) −6.55839 −0.251134
\(683\) 3.99530 0.152876 0.0764379 0.997074i \(-0.475645\pi\)
0.0764379 + 0.997074i \(0.475645\pi\)
\(684\) −3.90694 −0.149386
\(685\) 1.53291 0.0585695
\(686\) −10.9198 −0.416921
\(687\) −6.51153 −0.248430
\(688\) −4.33920 −0.165430
\(689\) 11.8355 0.450898
\(690\) 4.44823 0.169341
\(691\) 4.39613 0.167237 0.0836184 0.996498i \(-0.473352\pi\)
0.0836184 + 0.996498i \(0.473352\pi\)
\(692\) 10.8354 0.411900
\(693\) −5.50678 −0.209185
\(694\) 32.3416 1.22767
\(695\) 1.96663 0.0745987
\(696\) −4.92256 −0.186589
\(697\) 27.5539 1.04368
\(698\) 24.9320 0.943691
\(699\) −26.5730 −1.00508
\(700\) 15.8375 0.598601
\(701\) 14.3530 0.542106 0.271053 0.962564i \(-0.412628\pi\)
0.271053 + 0.962564i \(0.412628\pi\)
\(702\) −7.63847 −0.288295
\(703\) 0 0
\(704\) −2.58472 −0.0974152
\(705\) −3.08244 −0.116091
\(706\) −8.64114 −0.325214
\(707\) 4.15368 0.156215
\(708\) −19.8672 −0.746657
\(709\) −6.87475 −0.258187 −0.129093 0.991632i \(-0.541207\pi\)
−0.129093 + 0.991632i \(0.541207\pi\)
\(710\) 5.77114 0.216587
\(711\) −0.955097 −0.0358190
\(712\) 0.953262 0.0357250
\(713\) −19.1471 −0.717066
\(714\) 16.6638 0.623627
\(715\) −1.35739 −0.0507635
\(716\) 6.07192 0.226918
\(717\) 28.1266 1.05041
\(718\) −3.85848 −0.143997
\(719\) −17.0965 −0.637592 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(720\) 0.251131 0.00935909
\(721\) −35.4032 −1.31849
\(722\) −16.8296 −0.626333
\(723\) −35.9522 −1.33708
\(724\) 21.5538 0.801042
\(725\) −15.5892 −0.578969
\(726\) 6.61746 0.245597
\(727\) 42.6268 1.58094 0.790470 0.612501i \(-0.209837\pi\)
0.790470 + 0.612501i \(0.209837\pi\)
\(728\) −4.45530 −0.165124
\(729\) 30.0401 1.11260
\(730\) 5.80180 0.214734
\(731\) −14.4588 −0.534777
\(732\) 10.7440 0.397109
\(733\) −42.6192 −1.57418 −0.787088 0.616841i \(-0.788412\pi\)
−0.787088 + 0.616841i \(0.788412\pi\)
\(734\) −10.4563 −0.385948
\(735\) −2.15431 −0.0794630
\(736\) −7.54605 −0.278151
\(737\) −29.0470 −1.06996
\(738\) 5.39731 0.198678
\(739\) −1.69175 −0.0622321 −0.0311161 0.999516i \(-0.509906\pi\)
−0.0311161 + 0.999516i \(0.509906\pi\)
\(740\) 0 0
\(741\) −12.5174 −0.459837
\(742\) −28.3041 −1.03908
\(743\) 12.8065 0.469823 0.234912 0.972017i \(-0.424520\pi\)
0.234912 + 0.972017i \(0.424520\pi\)
\(744\) 3.88748 0.142522
\(745\) 6.10605 0.223708
\(746\) 9.86566 0.361208
\(747\) 0.00725664 0.000265507 0
\(748\) −8.61260 −0.314908
\(749\) 15.5359 0.567671
\(750\) −5.80752 −0.212061
\(751\) 48.4720 1.76877 0.884384 0.466759i \(-0.154579\pi\)
0.884384 + 0.466759i \(0.154579\pi\)
\(752\) 5.22909 0.190685
\(753\) 9.34802 0.340661
\(754\) 4.38546 0.159709
\(755\) −8.37535 −0.304810
\(756\) 18.2670 0.664364
\(757\) −32.1365 −1.16802 −0.584010 0.811746i \(-0.698517\pi\)
−0.584010 + 0.811746i \(0.698517\pi\)
\(758\) −2.48121 −0.0901215
\(759\) −29.8825 −1.08467
\(760\) 2.30306 0.0835406
\(761\) −10.3730 −0.376022 −0.188011 0.982167i \(-0.560204\pi\)
−0.188011 + 0.982167i \(0.560204\pi\)
\(762\) −17.2219 −0.623883
\(763\) −31.4579 −1.13885
\(764\) −6.62693 −0.239754
\(765\) 0.836799 0.0302545
\(766\) −25.5639 −0.923661
\(767\) 17.6995 0.639093
\(768\) 1.53209 0.0552845
\(769\) 33.9273 1.22345 0.611725 0.791070i \(-0.290476\pi\)
0.611725 + 0.791070i \(0.290476\pi\)
\(770\) 3.24613 0.116982
\(771\) 6.66015 0.239859
\(772\) 22.8544 0.822548
\(773\) −28.9843 −1.04249 −0.521247 0.853406i \(-0.674533\pi\)
−0.521247 + 0.853406i \(0.674533\pi\)
\(774\) −2.83221 −0.101802
\(775\) 12.3112 0.442233
\(776\) −14.8185 −0.531954
\(777\) 0 0
\(778\) −6.37509 −0.228558
\(779\) 49.4973 1.77343
\(780\) 0.804592 0.0288090
\(781\) −38.7696 −1.38728
\(782\) −25.1444 −0.899162
\(783\) −17.9806 −0.642576
\(784\) 3.65461 0.130522
\(785\) 3.75190 0.133911
\(786\) −12.8229 −0.457379
\(787\) −16.7100 −0.595647 −0.297823 0.954621i \(-0.596261\pi\)
−0.297823 + 0.954621i \(0.596261\pi\)
\(788\) −14.7561 −0.525663
\(789\) 6.21239 0.221167
\(790\) 0.563009 0.0200310
\(791\) −42.2867 −1.50354
\(792\) −1.68705 −0.0599468
\(793\) −9.57171 −0.339901
\(794\) −37.4977 −1.33075
\(795\) 5.11150 0.181286
\(796\) 1.47478 0.0522721
\(797\) 5.21454 0.184709 0.0923543 0.995726i \(-0.470561\pi\)
0.0923543 + 0.995726i \(0.470561\pi\)
\(798\) 29.9346 1.05967
\(799\) 17.4240 0.616417
\(800\) 4.85196 0.171543
\(801\) 0.622198 0.0219843
\(802\) −35.1464 −1.24106
\(803\) −38.9755 −1.37542
\(804\) 17.2176 0.607219
\(805\) 9.47703 0.334021
\(806\) −3.46332 −0.121990
\(807\) 13.4722 0.474245
\(808\) 1.27252 0.0447671
\(809\) −50.5612 −1.77764 −0.888818 0.458260i \(-0.848473\pi\)
−0.888818 + 0.458260i \(0.848473\pi\)
\(810\) −2.54548 −0.0894392
\(811\) 36.0684 1.26653 0.633267 0.773934i \(-0.281714\pi\)
0.633267 + 0.773934i \(0.281714\pi\)
\(812\) −10.4876 −0.368042
\(813\) −19.7826 −0.693808
\(814\) 0 0
\(815\) −5.06519 −0.177426
\(816\) 5.10511 0.178715
\(817\) −25.9735 −0.908698
\(818\) 0.450741 0.0157598
\(819\) −2.90799 −0.101613
\(820\) −3.18159 −0.111106
\(821\) 51.1334 1.78457 0.892285 0.451473i \(-0.149101\pi\)
0.892285 + 0.451473i \(0.149101\pi\)
\(822\) 6.10404 0.212903
\(823\) 8.08213 0.281725 0.140863 0.990029i \(-0.455012\pi\)
0.140863 + 0.990029i \(0.455012\pi\)
\(824\) −10.8461 −0.377842
\(825\) 19.2139 0.668940
\(826\) −42.3275 −1.47276
\(827\) 25.8525 0.898981 0.449490 0.893285i \(-0.351606\pi\)
0.449490 + 0.893285i \(0.351606\pi\)
\(828\) −4.92533 −0.171167
\(829\) 8.19775 0.284720 0.142360 0.989815i \(-0.454531\pi\)
0.142360 + 0.989815i \(0.454531\pi\)
\(830\) −0.00427763 −0.000148479 0
\(831\) 11.9142 0.413298
\(832\) −1.36492 −0.0473202
\(833\) 12.1776 0.421929
\(834\) 7.83112 0.271170
\(835\) 2.15550 0.0745943
\(836\) −15.4716 −0.535095
\(837\) 14.1998 0.490817
\(838\) 15.3113 0.528920
\(839\) 52.8482 1.82452 0.912261 0.409610i \(-0.134335\pi\)
0.912261 + 0.409610i \(0.134335\pi\)
\(840\) −1.92414 −0.0663891
\(841\) −18.6768 −0.644028
\(842\) −22.3965 −0.771833
\(843\) −0.356225 −0.0122691
\(844\) −2.41952 −0.0832835
\(845\) 4.28501 0.147409
\(846\) 3.41305 0.117343
\(847\) 14.0986 0.484434
\(848\) −8.67122 −0.297771
\(849\) 19.6613 0.674774
\(850\) 16.1674 0.554536
\(851\) 0 0
\(852\) 22.9806 0.787303
\(853\) 32.9906 1.12958 0.564788 0.825236i \(-0.308958\pi\)
0.564788 + 0.825236i \(0.308958\pi\)
\(854\) 22.8902 0.783288
\(855\) 1.50321 0.0514088
\(856\) 4.75958 0.162679
\(857\) −38.4078 −1.31199 −0.655993 0.754767i \(-0.727750\pi\)
−0.655993 + 0.754767i \(0.727750\pi\)
\(858\) −5.40511 −0.184528
\(859\) 20.7158 0.706814 0.353407 0.935470i \(-0.385023\pi\)
0.353407 + 0.935470i \(0.385023\pi\)
\(860\) 1.66953 0.0569304
\(861\) −41.3536 −1.40933
\(862\) −0.102080 −0.00347685
\(863\) 35.3661 1.20388 0.601938 0.798543i \(-0.294396\pi\)
0.601938 + 0.798543i \(0.294396\pi\)
\(864\) 5.59627 0.190389
\(865\) −4.16897 −0.141749
\(866\) −7.92714 −0.269375
\(867\) −9.03462 −0.306832
\(868\) 8.28234 0.281121
\(869\) −3.78220 −0.128302
\(870\) 1.89398 0.0642118
\(871\) −15.3390 −0.519742
\(872\) −9.63744 −0.326365
\(873\) −9.67211 −0.327351
\(874\) −45.1690 −1.52786
\(875\) −12.3730 −0.418284
\(876\) 23.1027 0.780568
\(877\) 26.3702 0.890459 0.445230 0.895416i \(-0.353122\pi\)
0.445230 + 0.895416i \(0.353122\pi\)
\(878\) 7.57063 0.255496
\(879\) −37.7573 −1.27352
\(880\) 0.994481 0.0335240
\(881\) −33.7205 −1.13607 −0.568037 0.823003i \(-0.692297\pi\)
−0.568037 + 0.823003i \(0.692297\pi\)
\(882\) 2.38538 0.0803198
\(883\) 57.8576 1.94706 0.973531 0.228555i \(-0.0734001\pi\)
0.973531 + 0.228555i \(0.0734001\pi\)
\(884\) −4.54809 −0.152969
\(885\) 7.64401 0.256951
\(886\) 15.9856 0.537047
\(887\) 44.4679 1.49309 0.746544 0.665336i \(-0.231712\pi\)
0.746544 + 0.665336i \(0.231712\pi\)
\(888\) 0 0
\(889\) −36.6915 −1.23059
\(890\) −0.366772 −0.0122942
\(891\) 17.1001 0.572876
\(892\) −6.94726 −0.232611
\(893\) 31.3002 1.04742
\(894\) 24.3142 0.813190
\(895\) −2.33620 −0.0780905
\(896\) 3.26414 0.109047
\(897\) −15.7802 −0.526884
\(898\) 28.6753 0.956908
\(899\) −8.15251 −0.271901
\(900\) 3.16689 0.105563
\(901\) −28.8936 −0.962586
\(902\) 21.3734 0.711657
\(903\) 21.7001 0.722135
\(904\) −12.9549 −0.430874
\(905\) −8.29294 −0.275667
\(906\) −33.3506 −1.10800
\(907\) 13.0951 0.434815 0.217407 0.976081i \(-0.430240\pi\)
0.217407 + 0.976081i \(0.430240\pi\)
\(908\) −21.8490 −0.725084
\(909\) 0.830579 0.0275486
\(910\) 1.71420 0.0568250
\(911\) −38.4466 −1.27379 −0.636896 0.770950i \(-0.719782\pi\)
−0.636896 + 0.770950i \(0.719782\pi\)
\(912\) 9.17075 0.303674
\(913\) 0.0287364 0.000951036 0
\(914\) −0.105116 −0.00347694
\(915\) −4.13380 −0.136659
\(916\) −4.25010 −0.140427
\(917\) −27.3195 −0.902169
\(918\) 18.6475 0.615458
\(919\) −14.8613 −0.490229 −0.245115 0.969494i \(-0.578826\pi\)
−0.245115 + 0.969494i \(0.578826\pi\)
\(920\) 2.90338 0.0957215
\(921\) −10.5888 −0.348911
\(922\) 23.9701 0.789413
\(923\) −20.4732 −0.673884
\(924\) 12.9260 0.425236
\(925\) 0 0
\(926\) −20.7514 −0.681933
\(927\) −7.07930 −0.232515
\(928\) −3.21297 −0.105471
\(929\) −7.26526 −0.238366 −0.119183 0.992872i \(-0.538027\pi\)
−0.119183 + 0.992872i \(0.538027\pi\)
\(930\) −1.49573 −0.0490468
\(931\) 21.8757 0.716947
\(932\) −17.3443 −0.568131
\(933\) 2.54111 0.0831922
\(934\) −17.4928 −0.572383
\(935\) 3.31374 0.108371
\(936\) −0.890890 −0.0291196
\(937\) −26.3830 −0.861897 −0.430948 0.902377i \(-0.641821\pi\)
−0.430948 + 0.902377i \(0.641821\pi\)
\(938\) 36.6824 1.19772
\(939\) −39.7591 −1.29749
\(940\) −2.01192 −0.0656215
\(941\) 36.1584 1.17873 0.589366 0.807866i \(-0.299378\pi\)
0.589366 + 0.807866i \(0.299378\pi\)
\(942\) 14.9400 0.486773
\(943\) 62.3994 2.03201
\(944\) −12.9674 −0.422054
\(945\) −7.02831 −0.228631
\(946\) −11.2156 −0.364651
\(947\) 47.1144 1.53101 0.765506 0.643428i \(-0.222489\pi\)
0.765506 + 0.643428i \(0.222489\pi\)
\(948\) 2.24190 0.0728134
\(949\) −20.5820 −0.668119
\(950\) 29.0428 0.942273
\(951\) 17.3889 0.563875
\(952\) 10.8765 0.352510
\(953\) −9.35174 −0.302933 −0.151466 0.988462i \(-0.548399\pi\)
−0.151466 + 0.988462i \(0.548399\pi\)
\(954\) −5.65974 −0.183241
\(955\) 2.54974 0.0825077
\(956\) 18.3584 0.593752
\(957\) −12.7234 −0.411290
\(958\) 6.88046 0.222298
\(959\) 13.0048 0.419945
\(960\) −0.589478 −0.0190253
\(961\) −24.5617 −0.792314
\(962\) 0 0
\(963\) 3.10660 0.100109
\(964\) −23.4661 −0.755792
\(965\) −8.79333 −0.283067
\(966\) 37.7374 1.21418
\(967\) 49.9460 1.60615 0.803077 0.595875i \(-0.203195\pi\)
0.803077 + 0.595875i \(0.203195\pi\)
\(968\) 4.31924 0.138826
\(969\) 30.5581 0.981668
\(970\) 5.70150 0.183064
\(971\) 37.6187 1.20724 0.603621 0.797271i \(-0.293724\pi\)
0.603621 + 0.797271i \(0.293724\pi\)
\(972\) 6.65270 0.213386
\(973\) 16.6843 0.534875
\(974\) 1.70810 0.0547311
\(975\) 10.1463 0.324943
\(976\) 7.01264 0.224469
\(977\) 28.1356 0.900136 0.450068 0.892994i \(-0.351400\pi\)
0.450068 + 0.892994i \(0.351400\pi\)
\(978\) −20.1696 −0.644951
\(979\) 2.46391 0.0787470
\(980\) −1.40613 −0.0449171
\(981\) −6.29039 −0.200837
\(982\) −10.6231 −0.338995
\(983\) 32.9418 1.05068 0.525340 0.850893i \(-0.323938\pi\)
0.525340 + 0.850893i \(0.323938\pi\)
\(984\) −12.6691 −0.403875
\(985\) 5.67746 0.180899
\(986\) −10.7060 −0.340949
\(987\) −26.1504 −0.832378
\(988\) −8.17013 −0.259926
\(989\) −32.7438 −1.04119
\(990\) 0.649101 0.0206298
\(991\) 38.9993 1.23885 0.619426 0.785055i \(-0.287365\pi\)
0.619426 + 0.785055i \(0.287365\pi\)
\(992\) 2.53737 0.0805617
\(993\) −19.6685 −0.624160
\(994\) 48.9606 1.55294
\(995\) −0.567428 −0.0179887
\(996\) −0.0170335 −0.000539727 0
\(997\) −7.98046 −0.252744 −0.126372 0.991983i \(-0.540333\pi\)
−0.126372 + 0.991983i \(0.540333\pi\)
\(998\) −35.4707 −1.12281
\(999\) 0 0
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 2738.2.a.r.1.5 6
37.18 odd 36 74.2.h.a.65.2 yes 12
37.35 odd 36 74.2.h.a.41.2 12
37.36 even 2 2738.2.a.s.1.6 6
111.35 even 36 666.2.bj.c.559.1 12
111.92 even 36 666.2.bj.c.361.1 12
148.35 even 36 592.2.bq.b.337.2 12
148.55 even 36 592.2.bq.b.65.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.41.2 12 37.35 odd 36
74.2.h.a.65.2 yes 12 37.18 odd 36
592.2.bq.b.65.2 12 148.55 even 36
592.2.bq.b.337.2 12 148.35 even 36
666.2.bj.c.361.1 12 111.92 even 36
666.2.bj.c.559.1 12 111.35 even 36
2738.2.a.r.1.5 6 1.1 even 1 trivial
2738.2.a.s.1.6 6 37.36 even 2