Properties

Label 6042.2.a.bf.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $12$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + 5237 x^{4} - 12206 x^{3} - 7876 x^{2} + 9264 x + 4048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.22349\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.22349 q^{5} -1.00000 q^{6} +3.89357 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -4.22349 q^{5} -1.00000 q^{6} +3.89357 q^{7} -1.00000 q^{8} +1.00000 q^{9} +4.22349 q^{10} +1.39763 q^{11} +1.00000 q^{12} +6.39181 q^{13} -3.89357 q^{14} -4.22349 q^{15} +1.00000 q^{16} +6.62509 q^{17} -1.00000 q^{18} +1.00000 q^{19} -4.22349 q^{20} +3.89357 q^{21} -1.39763 q^{22} +4.91438 q^{23} -1.00000 q^{24} +12.8379 q^{25} -6.39181 q^{26} +1.00000 q^{27} +3.89357 q^{28} +5.17021 q^{29} +4.22349 q^{30} +9.08792 q^{31} -1.00000 q^{32} +1.39763 q^{33} -6.62509 q^{34} -16.4445 q^{35} +1.00000 q^{36} +3.19935 q^{37} -1.00000 q^{38} +6.39181 q^{39} +4.22349 q^{40} -8.33052 q^{41} -3.89357 q^{42} -2.91438 q^{43} +1.39763 q^{44} -4.22349 q^{45} -4.91438 q^{46} -3.31449 q^{47} +1.00000 q^{48} +8.15991 q^{49} -12.8379 q^{50} +6.62509 q^{51} +6.39181 q^{52} +1.00000 q^{53} -1.00000 q^{54} -5.90289 q^{55} -3.89357 q^{56} +1.00000 q^{57} -5.17021 q^{58} +1.28144 q^{59} -4.22349 q^{60} -14.2334 q^{61} -9.08792 q^{62} +3.89357 q^{63} +1.00000 q^{64} -26.9957 q^{65} -1.39763 q^{66} +1.72928 q^{67} +6.62509 q^{68} +4.91438 q^{69} +16.4445 q^{70} -14.6340 q^{71} -1.00000 q^{72} -8.59442 q^{73} -3.19935 q^{74} +12.8379 q^{75} +1.00000 q^{76} +5.44178 q^{77} -6.39181 q^{78} +0.401082 q^{79} -4.22349 q^{80} +1.00000 q^{81} +8.33052 q^{82} +17.6808 q^{83} +3.89357 q^{84} -27.9810 q^{85} +2.91438 q^{86} +5.17021 q^{87} -1.39763 q^{88} -3.45446 q^{89} +4.22349 q^{90} +24.8870 q^{91} +4.91438 q^{92} +9.08792 q^{93} +3.31449 q^{94} -4.22349 q^{95} -1.00000 q^{96} -12.2819 q^{97} -8.15991 q^{98} +1.39763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 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
\(5\) −4.22349 −1.88880 −0.944401 0.328795i \(-0.893358\pi\)
−0.944401 + 0.328795i \(0.893358\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.89357 1.47163 0.735816 0.677181i \(-0.236799\pi\)
0.735816 + 0.677181i \(0.236799\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 4.22349 1.33559
\(11\) 1.39763 0.421402 0.210701 0.977551i \(-0.432425\pi\)
0.210701 + 0.977551i \(0.432425\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.39181 1.77277 0.886384 0.462950i \(-0.153209\pi\)
0.886384 + 0.462950i \(0.153209\pi\)
\(14\) −3.89357 −1.04060
\(15\) −4.22349 −1.09050
\(16\) 1.00000 0.250000
\(17\) 6.62509 1.60682 0.803410 0.595426i \(-0.203017\pi\)
0.803410 + 0.595426i \(0.203017\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) −4.22349 −0.944401
\(21\) 3.89357 0.849647
\(22\) −1.39763 −0.297976
\(23\) 4.91438 1.02472 0.512359 0.858771i \(-0.328772\pi\)
0.512359 + 0.858771i \(0.328772\pi\)
\(24\) −1.00000 −0.204124
\(25\) 12.8379 2.56758
\(26\) −6.39181 −1.25354
\(27\) 1.00000 0.192450
\(28\) 3.89357 0.735816
\(29\) 5.17021 0.960084 0.480042 0.877246i \(-0.340621\pi\)
0.480042 + 0.877246i \(0.340621\pi\)
\(30\) 4.22349 0.771101
\(31\) 9.08792 1.63224 0.816119 0.577884i \(-0.196121\pi\)
0.816119 + 0.577884i \(0.196121\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.39763 0.243297
\(34\) −6.62509 −1.13619
\(35\) −16.4445 −2.77962
\(36\) 1.00000 0.166667
\(37\) 3.19935 0.525970 0.262985 0.964800i \(-0.415293\pi\)
0.262985 + 0.964800i \(0.415293\pi\)
\(38\) −1.00000 −0.162221
\(39\) 6.39181 1.02351
\(40\) 4.22349 0.667793
\(41\) −8.33052 −1.30101 −0.650504 0.759503i \(-0.725442\pi\)
−0.650504 + 0.759503i \(0.725442\pi\)
\(42\) −3.89357 −0.600791
\(43\) −2.91438 −0.444439 −0.222219 0.974997i \(-0.571330\pi\)
−0.222219 + 0.974997i \(0.571330\pi\)
\(44\) 1.39763 0.210701
\(45\) −4.22349 −0.629601
\(46\) −4.91438 −0.724586
\(47\) −3.31449 −0.483468 −0.241734 0.970343i \(-0.577716\pi\)
−0.241734 + 0.970343i \(0.577716\pi\)
\(48\) 1.00000 0.144338
\(49\) 8.15991 1.16570
\(50\) −12.8379 −1.81555
\(51\) 6.62509 0.927698
\(52\) 6.39181 0.886384
\(53\) 1.00000 0.137361
\(54\) −1.00000 −0.136083
\(55\) −5.90289 −0.795945
\(56\) −3.89357 −0.520301
\(57\) 1.00000 0.132453
\(58\) −5.17021 −0.678882
\(59\) 1.28144 0.166830 0.0834149 0.996515i \(-0.473417\pi\)
0.0834149 + 0.996515i \(0.473417\pi\)
\(60\) −4.22349 −0.545250
\(61\) −14.2334 −1.82240 −0.911199 0.411967i \(-0.864842\pi\)
−0.911199 + 0.411967i \(0.864842\pi\)
\(62\) −9.08792 −1.15417
\(63\) 3.89357 0.490544
\(64\) 1.00000 0.125000
\(65\) −26.9957 −3.34841
\(66\) −1.39763 −0.172037
\(67\) 1.72928 0.211265 0.105633 0.994405i \(-0.466313\pi\)
0.105633 + 0.994405i \(0.466313\pi\)
\(68\) 6.62509 0.803410
\(69\) 4.91438 0.591622
\(70\) 16.4445 1.96549
\(71\) −14.6340 −1.73673 −0.868367 0.495923i \(-0.834830\pi\)
−0.868367 + 0.495923i \(0.834830\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.59442 −1.00590 −0.502951 0.864315i \(-0.667752\pi\)
−0.502951 + 0.864315i \(0.667752\pi\)
\(74\) −3.19935 −0.371917
\(75\) 12.8379 1.48239
\(76\) 1.00000 0.114708
\(77\) 5.44178 0.620149
\(78\) −6.39181 −0.723730
\(79\) 0.401082 0.0451253 0.0225626 0.999745i \(-0.492817\pi\)
0.0225626 + 0.999745i \(0.492817\pi\)
\(80\) −4.22349 −0.472201
\(81\) 1.00000 0.111111
\(82\) 8.33052 0.919952
\(83\) 17.6808 1.94072 0.970360 0.241663i \(-0.0776927\pi\)
0.970360 + 0.241663i \(0.0776927\pi\)
\(84\) 3.89357 0.424824
\(85\) −27.9810 −3.03497
\(86\) 2.91438 0.314266
\(87\) 5.17021 0.554305
\(88\) −1.39763 −0.148988
\(89\) −3.45446 −0.366172 −0.183086 0.983097i \(-0.558609\pi\)
−0.183086 + 0.983097i \(0.558609\pi\)
\(90\) 4.22349 0.445195
\(91\) 24.8870 2.60886
\(92\) 4.91438 0.512359
\(93\) 9.08792 0.942373
\(94\) 3.31449 0.341863
\(95\) −4.22349 −0.433321
\(96\) −1.00000 −0.102062
\(97\) −12.2819 −1.24703 −0.623517 0.781809i \(-0.714297\pi\)
−0.623517 + 0.781809i \(0.714297\pi\)
\(98\) −8.15991 −0.824275
\(99\) 1.39763 0.140467
\(100\) 12.8379 1.28379
\(101\) 16.3423 1.62611 0.813057 0.582183i \(-0.197801\pi\)
0.813057 + 0.582183i \(0.197801\pi\)
\(102\) −6.62509 −0.655982
\(103\) −12.1122 −1.19345 −0.596724 0.802446i \(-0.703531\pi\)
−0.596724 + 0.802446i \(0.703531\pi\)
\(104\) −6.39181 −0.626768
\(105\) −16.4445 −1.60482
\(106\) −1.00000 −0.0971286
\(107\) −7.11873 −0.688194 −0.344097 0.938934i \(-0.611815\pi\)
−0.344097 + 0.938934i \(0.611815\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.5533 1.48973 0.744867 0.667213i \(-0.232513\pi\)
0.744867 + 0.667213i \(0.232513\pi\)
\(110\) 5.90289 0.562818
\(111\) 3.19935 0.303669
\(112\) 3.89357 0.367908
\(113\) 13.3002 1.25118 0.625588 0.780154i \(-0.284859\pi\)
0.625588 + 0.780154i \(0.284859\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −20.7558 −1.93549
\(116\) 5.17021 0.480042
\(117\) 6.39181 0.590923
\(118\) −1.28144 −0.117966
\(119\) 25.7953 2.36465
\(120\) 4.22349 0.385550
\(121\) −9.04662 −0.822420
\(122\) 14.2334 1.28863
\(123\) −8.33052 −0.751138
\(124\) 9.08792 0.816119
\(125\) −33.1032 −2.96084
\(126\) −3.89357 −0.346867
\(127\) −0.284449 −0.0252408 −0.0126204 0.999920i \(-0.504017\pi\)
−0.0126204 + 0.999920i \(0.504017\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.91438 −0.256597
\(130\) 26.9957 2.36768
\(131\) 5.09021 0.444734 0.222367 0.974963i \(-0.428622\pi\)
0.222367 + 0.974963i \(0.428622\pi\)
\(132\) 1.39763 0.121648
\(133\) 3.89357 0.337616
\(134\) −1.72928 −0.149387
\(135\) −4.22349 −0.363500
\(136\) −6.62509 −0.568097
\(137\) −22.5889 −1.92990 −0.964948 0.262441i \(-0.915473\pi\)
−0.964948 + 0.262441i \(0.915473\pi\)
\(138\) −4.91438 −0.418340
\(139\) 20.7843 1.76290 0.881452 0.472273i \(-0.156566\pi\)
0.881452 + 0.472273i \(0.156566\pi\)
\(140\) −16.4445 −1.38981
\(141\) −3.31449 −0.279130
\(142\) 14.6340 1.22806
\(143\) 8.93340 0.747048
\(144\) 1.00000 0.0833333
\(145\) −21.8363 −1.81341
\(146\) 8.59442 0.711280
\(147\) 8.15991 0.673018
\(148\) 3.19935 0.262985
\(149\) −12.9791 −1.06329 −0.531644 0.846968i \(-0.678426\pi\)
−0.531644 + 0.846968i \(0.678426\pi\)
\(150\) −12.8379 −1.04821
\(151\) 14.2437 1.15913 0.579567 0.814924i \(-0.303222\pi\)
0.579567 + 0.814924i \(0.303222\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 6.62509 0.535607
\(154\) −5.44178 −0.438511
\(155\) −38.3827 −3.08298
\(156\) 6.39181 0.511754
\(157\) 14.8382 1.18422 0.592109 0.805858i \(-0.298296\pi\)
0.592109 + 0.805858i \(0.298296\pi\)
\(158\) −0.401082 −0.0319084
\(159\) 1.00000 0.0793052
\(160\) 4.22349 0.333896
\(161\) 19.1345 1.50801
\(162\) −1.00000 −0.0785674
\(163\) −15.5059 −1.21452 −0.607258 0.794505i \(-0.707731\pi\)
−0.607258 + 0.794505i \(0.707731\pi\)
\(164\) −8.33052 −0.650504
\(165\) −5.90289 −0.459539
\(166\) −17.6808 −1.37230
\(167\) 7.97057 0.616781 0.308391 0.951260i \(-0.400210\pi\)
0.308391 + 0.951260i \(0.400210\pi\)
\(168\) −3.89357 −0.300396
\(169\) 27.8552 2.14271
\(170\) 27.9810 2.14605
\(171\) 1.00000 0.0764719
\(172\) −2.91438 −0.222219
\(173\) −21.5813 −1.64080 −0.820398 0.571793i \(-0.806248\pi\)
−0.820398 + 0.571793i \(0.806248\pi\)
\(174\) −5.17021 −0.391953
\(175\) 49.9852 3.77853
\(176\) 1.39763 0.105351
\(177\) 1.28144 0.0963192
\(178\) 3.45446 0.258923
\(179\) −8.43180 −0.630222 −0.315111 0.949055i \(-0.602042\pi\)
−0.315111 + 0.949055i \(0.602042\pi\)
\(180\) −4.22349 −0.314800
\(181\) 8.13253 0.604486 0.302243 0.953231i \(-0.402265\pi\)
0.302243 + 0.953231i \(0.402265\pi\)
\(182\) −24.8870 −1.84475
\(183\) −14.2334 −1.05216
\(184\) −4.91438 −0.362293
\(185\) −13.5124 −0.993454
\(186\) −9.08792 −0.666359
\(187\) 9.25944 0.677117
\(188\) −3.31449 −0.241734
\(189\) 3.89357 0.283216
\(190\) 4.22349 0.306404
\(191\) −13.8455 −1.00183 −0.500914 0.865497i \(-0.667003\pi\)
−0.500914 + 0.865497i \(0.667003\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.356400 0.0256542 0.0128271 0.999918i \(-0.495917\pi\)
0.0128271 + 0.999918i \(0.495917\pi\)
\(194\) 12.2819 0.881787
\(195\) −26.9957 −1.93321
\(196\) 8.15991 0.582851
\(197\) −3.72524 −0.265413 −0.132706 0.991155i \(-0.542367\pi\)
−0.132706 + 0.991155i \(0.542367\pi\)
\(198\) −1.39763 −0.0993254
\(199\) −7.09049 −0.502632 −0.251316 0.967905i \(-0.580863\pi\)
−0.251316 + 0.967905i \(0.580863\pi\)
\(200\) −12.8379 −0.907775
\(201\) 1.72928 0.121974
\(202\) −16.3423 −1.14984
\(203\) 20.1306 1.41289
\(204\) 6.62509 0.463849
\(205\) 35.1839 2.45735
\(206\) 12.1122 0.843895
\(207\) 4.91438 0.341573
\(208\) 6.39181 0.443192
\(209\) 1.39763 0.0966763
\(210\) 16.4445 1.13478
\(211\) −6.34960 −0.437124 −0.218562 0.975823i \(-0.570137\pi\)
−0.218562 + 0.975823i \(0.570137\pi\)
\(212\) 1.00000 0.0686803
\(213\) −14.6340 −1.00270
\(214\) 7.11873 0.486626
\(215\) 12.3089 0.839457
\(216\) −1.00000 −0.0680414
\(217\) 35.3845 2.40205
\(218\) −15.5533 −1.05340
\(219\) −8.59442 −0.580757
\(220\) −5.90289 −0.397973
\(221\) 42.3463 2.84852
\(222\) −3.19935 −0.214726
\(223\) −15.3317 −1.02669 −0.513343 0.858183i \(-0.671593\pi\)
−0.513343 + 0.858183i \(0.671593\pi\)
\(224\) −3.89357 −0.260150
\(225\) 12.8379 0.855859
\(226\) −13.3002 −0.884715
\(227\) 10.2980 0.683503 0.341751 0.939790i \(-0.388980\pi\)
0.341751 + 0.939790i \(0.388980\pi\)
\(228\) 1.00000 0.0662266
\(229\) −21.0001 −1.38772 −0.693862 0.720108i \(-0.744092\pi\)
−0.693862 + 0.720108i \(0.744092\pi\)
\(230\) 20.7558 1.36860
\(231\) 5.44178 0.358043
\(232\) −5.17021 −0.339441
\(233\) 10.4476 0.684442 0.342221 0.939620i \(-0.388821\pi\)
0.342221 + 0.939620i \(0.388821\pi\)
\(234\) −6.39181 −0.417846
\(235\) 13.9987 0.913175
\(236\) 1.28144 0.0834149
\(237\) 0.401082 0.0260531
\(238\) −25.7953 −1.67206
\(239\) −22.3004 −1.44249 −0.721246 0.692679i \(-0.756430\pi\)
−0.721246 + 0.692679i \(0.756430\pi\)
\(240\) −4.22349 −0.272625
\(241\) −3.10563 −0.200051 −0.100026 0.994985i \(-0.531893\pi\)
−0.100026 + 0.994985i \(0.531893\pi\)
\(242\) 9.04662 0.581539
\(243\) 1.00000 0.0641500
\(244\) −14.2334 −0.911199
\(245\) −34.4633 −2.20178
\(246\) 8.33052 0.531135
\(247\) 6.39181 0.406701
\(248\) −9.08792 −0.577083
\(249\) 17.6808 1.12048
\(250\) 33.1032 2.09363
\(251\) 3.24901 0.205076 0.102538 0.994729i \(-0.467304\pi\)
0.102538 + 0.994729i \(0.467304\pi\)
\(252\) 3.89357 0.245272
\(253\) 6.86850 0.431819
\(254\) 0.284449 0.0178479
\(255\) −27.9810 −1.75224
\(256\) 1.00000 0.0625000
\(257\) −2.35501 −0.146902 −0.0734508 0.997299i \(-0.523401\pi\)
−0.0734508 + 0.997299i \(0.523401\pi\)
\(258\) 2.91438 0.181441
\(259\) 12.4569 0.774035
\(260\) −26.9957 −1.67421
\(261\) 5.17021 0.320028
\(262\) −5.09021 −0.314474
\(263\) −2.60931 −0.160897 −0.0804485 0.996759i \(-0.525635\pi\)
−0.0804485 + 0.996759i \(0.525635\pi\)
\(264\) −1.39763 −0.0860183
\(265\) −4.22349 −0.259447
\(266\) −3.89357 −0.238730
\(267\) −3.45446 −0.211409
\(268\) 1.72928 0.105633
\(269\) 22.2422 1.35613 0.678065 0.735002i \(-0.262819\pi\)
0.678065 + 0.735002i \(0.262819\pi\)
\(270\) 4.22349 0.257034
\(271\) −3.51603 −0.213584 −0.106792 0.994281i \(-0.534058\pi\)
−0.106792 + 0.994281i \(0.534058\pi\)
\(272\) 6.62509 0.401705
\(273\) 24.8870 1.50623
\(274\) 22.5889 1.36464
\(275\) 17.9426 1.08198
\(276\) 4.91438 0.295811
\(277\) 0.273929 0.0164588 0.00822939 0.999966i \(-0.497380\pi\)
0.00822939 + 0.999966i \(0.497380\pi\)
\(278\) −20.7843 −1.24656
\(279\) 9.08792 0.544079
\(280\) 16.4445 0.982745
\(281\) 8.62051 0.514257 0.257128 0.966377i \(-0.417224\pi\)
0.257128 + 0.966377i \(0.417224\pi\)
\(282\) 3.31449 0.197375
\(283\) −6.97293 −0.414498 −0.207249 0.978288i \(-0.566451\pi\)
−0.207249 + 0.978288i \(0.566451\pi\)
\(284\) −14.6340 −0.868367
\(285\) −4.22349 −0.250178
\(286\) −8.93340 −0.528243
\(287\) −32.4355 −1.91461
\(288\) −1.00000 −0.0589256
\(289\) 26.8918 1.58187
\(290\) 21.8363 1.28227
\(291\) −12.2819 −0.719976
\(292\) −8.59442 −0.502951
\(293\) 14.1738 0.828044 0.414022 0.910267i \(-0.364124\pi\)
0.414022 + 0.910267i \(0.364124\pi\)
\(294\) −8.15991 −0.475896
\(295\) −5.41217 −0.315109
\(296\) −3.19935 −0.185959
\(297\) 1.39763 0.0810989
\(298\) 12.9791 0.751859
\(299\) 31.4118 1.81659
\(300\) 12.8379 0.741195
\(301\) −11.3473 −0.654050
\(302\) −14.2437 −0.819632
\(303\) 16.3423 0.938838
\(304\) 1.00000 0.0573539
\(305\) 60.1146 3.44215
\(306\) −6.62509 −0.378731
\(307\) −24.9195 −1.42223 −0.711116 0.703075i \(-0.751810\pi\)
−0.711116 + 0.703075i \(0.751810\pi\)
\(308\) 5.44178 0.310074
\(309\) −12.1122 −0.689038
\(310\) 38.3827 2.17999
\(311\) −1.66374 −0.0943418 −0.0471709 0.998887i \(-0.515021\pi\)
−0.0471709 + 0.998887i \(0.515021\pi\)
\(312\) −6.39181 −0.361865
\(313\) −23.7002 −1.33962 −0.669808 0.742535i \(-0.733623\pi\)
−0.669808 + 0.742535i \(0.733623\pi\)
\(314\) −14.8382 −0.837368
\(315\) −16.4445 −0.926541
\(316\) 0.401082 0.0225626
\(317\) −12.5172 −0.703036 −0.351518 0.936181i \(-0.614334\pi\)
−0.351518 + 0.936181i \(0.614334\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 7.22605 0.404581
\(320\) −4.22349 −0.236100
\(321\) −7.11873 −0.397329
\(322\) −19.1345 −1.06632
\(323\) 6.62509 0.368630
\(324\) 1.00000 0.0555556
\(325\) 82.0573 4.55172
\(326\) 15.5059 0.858792
\(327\) 15.5533 0.860098
\(328\) 8.33052 0.459976
\(329\) −12.9052 −0.711487
\(330\) 5.90289 0.324943
\(331\) 12.0207 0.660719 0.330359 0.943855i \(-0.392830\pi\)
0.330359 + 0.943855i \(0.392830\pi\)
\(332\) 17.6808 0.970360
\(333\) 3.19935 0.175323
\(334\) −7.97057 −0.436130
\(335\) −7.30360 −0.399038
\(336\) 3.89357 0.212412
\(337\) −27.6560 −1.50652 −0.753258 0.657725i \(-0.771519\pi\)
−0.753258 + 0.657725i \(0.771519\pi\)
\(338\) −27.8552 −1.51512
\(339\) 13.3002 0.722367
\(340\) −27.9810 −1.51748
\(341\) 12.7016 0.687829
\(342\) −1.00000 −0.0540738
\(343\) 4.51620 0.243852
\(344\) 2.91438 0.157133
\(345\) −20.7558 −1.11746
\(346\) 21.5813 1.16022
\(347\) 19.4805 1.04577 0.522883 0.852404i \(-0.324856\pi\)
0.522883 + 0.852404i \(0.324856\pi\)
\(348\) 5.17021 0.277152
\(349\) 0.935636 0.0500834 0.0250417 0.999686i \(-0.492028\pi\)
0.0250417 + 0.999686i \(0.492028\pi\)
\(350\) −49.9852 −2.67182
\(351\) 6.39181 0.341169
\(352\) −1.39763 −0.0744941
\(353\) 27.2562 1.45070 0.725349 0.688381i \(-0.241678\pi\)
0.725349 + 0.688381i \(0.241678\pi\)
\(354\) −1.28144 −0.0681080
\(355\) 61.8065 3.28035
\(356\) −3.45446 −0.183086
\(357\) 25.7953 1.36523
\(358\) 8.43180 0.445634
\(359\) −14.3368 −0.756665 −0.378332 0.925670i \(-0.623502\pi\)
−0.378332 + 0.925670i \(0.623502\pi\)
\(360\) 4.22349 0.222598
\(361\) 1.00000 0.0526316
\(362\) −8.13253 −0.427436
\(363\) −9.04662 −0.474825
\(364\) 24.8870 1.30443
\(365\) 36.2985 1.89995
\(366\) 14.2334 0.743991
\(367\) 5.07573 0.264951 0.132476 0.991186i \(-0.457707\pi\)
0.132476 + 0.991186i \(0.457707\pi\)
\(368\) 4.91438 0.256180
\(369\) −8.33052 −0.433670
\(370\) 13.5124 0.702478
\(371\) 3.89357 0.202144
\(372\) 9.08792 0.471187
\(373\) 5.06710 0.262365 0.131182 0.991358i \(-0.458123\pi\)
0.131182 + 0.991358i \(0.458123\pi\)
\(374\) −9.25944 −0.478794
\(375\) −33.1032 −1.70944
\(376\) 3.31449 0.170932
\(377\) 33.0470 1.70201
\(378\) −3.89357 −0.200264
\(379\) 21.3462 1.09648 0.548241 0.836321i \(-0.315298\pi\)
0.548241 + 0.836321i \(0.315298\pi\)
\(380\) −4.22349 −0.216661
\(381\) −0.284449 −0.0145728
\(382\) 13.8455 0.708399
\(383\) −2.17408 −0.111090 −0.0555452 0.998456i \(-0.517690\pi\)
−0.0555452 + 0.998456i \(0.517690\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −22.9833 −1.17134
\(386\) −0.356400 −0.0181403
\(387\) −2.91438 −0.148146
\(388\) −12.2819 −0.623517
\(389\) 24.0139 1.21755 0.608777 0.793341i \(-0.291660\pi\)
0.608777 + 0.793341i \(0.291660\pi\)
\(390\) 26.9957 1.36698
\(391\) 32.5582 1.64654
\(392\) −8.15991 −0.412138
\(393\) 5.09021 0.256767
\(394\) 3.72524 0.187675
\(395\) −1.69397 −0.0852328
\(396\) 1.39763 0.0702337
\(397\) −12.3957 −0.622120 −0.311060 0.950390i \(-0.600684\pi\)
−0.311060 + 0.950390i \(0.600684\pi\)
\(398\) 7.09049 0.355414
\(399\) 3.89357 0.194922
\(400\) 12.8379 0.641894
\(401\) −2.39611 −0.119656 −0.0598281 0.998209i \(-0.519055\pi\)
−0.0598281 + 0.998209i \(0.519055\pi\)
\(402\) −1.72928 −0.0862487
\(403\) 58.0882 2.89358
\(404\) 16.3423 0.813057
\(405\) −4.22349 −0.209867
\(406\) −20.1306 −0.999064
\(407\) 4.47152 0.221645
\(408\) −6.62509 −0.327991
\(409\) −12.5246 −0.619302 −0.309651 0.950850i \(-0.600212\pi\)
−0.309651 + 0.950850i \(0.600212\pi\)
\(410\) −35.1839 −1.73761
\(411\) −22.5889 −1.11423
\(412\) −12.1122 −0.596724
\(413\) 4.98940 0.245512
\(414\) −4.91438 −0.241529
\(415\) −74.6747 −3.66564
\(416\) −6.39181 −0.313384
\(417\) 20.7843 1.01781
\(418\) −1.39763 −0.0683604
\(419\) −28.2150 −1.37839 −0.689196 0.724575i \(-0.742036\pi\)
−0.689196 + 0.724575i \(0.742036\pi\)
\(420\) −16.4445 −0.802408
\(421\) 16.4246 0.800486 0.400243 0.916409i \(-0.368926\pi\)
0.400243 + 0.916409i \(0.368926\pi\)
\(422\) 6.34960 0.309094
\(423\) −3.31449 −0.161156
\(424\) −1.00000 −0.0485643
\(425\) 85.0521 4.12563
\(426\) 14.6340 0.709018
\(427\) −55.4187 −2.68190
\(428\) −7.11873 −0.344097
\(429\) 8.93340 0.431309
\(430\) −12.3089 −0.593586
\(431\) −2.01343 −0.0969833 −0.0484917 0.998824i \(-0.515441\pi\)
−0.0484917 + 0.998824i \(0.515441\pi\)
\(432\) 1.00000 0.0481125
\(433\) −17.3549 −0.834025 −0.417013 0.908901i \(-0.636923\pi\)
−0.417013 + 0.908901i \(0.636923\pi\)
\(434\) −35.3845 −1.69851
\(435\) −21.8363 −1.04697
\(436\) 15.5533 0.744867
\(437\) 4.91438 0.235087
\(438\) 8.59442 0.410657
\(439\) −24.5865 −1.17345 −0.586724 0.809787i \(-0.699583\pi\)
−0.586724 + 0.809787i \(0.699583\pi\)
\(440\) 5.90289 0.281409
\(441\) 8.15991 0.388567
\(442\) −42.3463 −2.01421
\(443\) −14.1705 −0.673263 −0.336631 0.941637i \(-0.609288\pi\)
−0.336631 + 0.941637i \(0.609288\pi\)
\(444\) 3.19935 0.151835
\(445\) 14.5899 0.691626
\(446\) 15.3317 0.725977
\(447\) −12.9791 −0.613890
\(448\) 3.89357 0.183954
\(449\) 40.1610 1.89532 0.947658 0.319287i \(-0.103444\pi\)
0.947658 + 0.319287i \(0.103444\pi\)
\(450\) −12.8379 −0.605183
\(451\) −11.6430 −0.548248
\(452\) 13.3002 0.625588
\(453\) 14.2437 0.669227
\(454\) −10.2980 −0.483309
\(455\) −105.110 −4.92763
\(456\) −1.00000 −0.0468293
\(457\) −14.6789 −0.686651 −0.343326 0.939216i \(-0.611553\pi\)
−0.343326 + 0.939216i \(0.611553\pi\)
\(458\) 21.0001 0.981269
\(459\) 6.62509 0.309233
\(460\) −20.7558 −0.967746
\(461\) −37.4126 −1.74248 −0.871240 0.490857i \(-0.836684\pi\)
−0.871240 + 0.490857i \(0.836684\pi\)
\(462\) −5.44178 −0.253175
\(463\) −39.7374 −1.84675 −0.923377 0.383895i \(-0.874582\pi\)
−0.923377 + 0.383895i \(0.874582\pi\)
\(464\) 5.17021 0.240021
\(465\) −38.3827 −1.77996
\(466\) −10.4476 −0.483973
\(467\) 1.57966 0.0730979 0.0365490 0.999332i \(-0.488364\pi\)
0.0365490 + 0.999332i \(0.488364\pi\)
\(468\) 6.39181 0.295461
\(469\) 6.73308 0.310905
\(470\) −13.9987 −0.645712
\(471\) 14.8382 0.683708
\(472\) −1.28144 −0.0589832
\(473\) −4.07323 −0.187287
\(474\) −0.401082 −0.0184223
\(475\) 12.8379 0.589042
\(476\) 25.7953 1.18232
\(477\) 1.00000 0.0457869
\(478\) 22.3004 1.02000
\(479\) −39.8886 −1.82256 −0.911278 0.411792i \(-0.864903\pi\)
−0.911278 + 0.411792i \(0.864903\pi\)
\(480\) 4.22349 0.192775
\(481\) 20.4496 0.932424
\(482\) 3.10563 0.141458
\(483\) 19.1345 0.870650
\(484\) −9.04662 −0.411210
\(485\) 51.8724 2.35540
\(486\) −1.00000 −0.0453609
\(487\) −8.41305 −0.381232 −0.190616 0.981665i \(-0.561048\pi\)
−0.190616 + 0.981665i \(0.561048\pi\)
\(488\) 14.2334 0.644315
\(489\) −15.5059 −0.701201
\(490\) 34.4633 1.55689
\(491\) −17.4356 −0.786858 −0.393429 0.919355i \(-0.628711\pi\)
−0.393429 + 0.919355i \(0.628711\pi\)
\(492\) −8.33052 −0.375569
\(493\) 34.2531 1.54268
\(494\) −6.39181 −0.287581
\(495\) −5.90289 −0.265315
\(496\) 9.08792 0.408060
\(497\) −56.9784 −2.55583
\(498\) −17.6808 −0.792296
\(499\) −13.6058 −0.609080 −0.304540 0.952500i \(-0.598503\pi\)
−0.304540 + 0.952500i \(0.598503\pi\)
\(500\) −33.1032 −1.48042
\(501\) 7.97057 0.356099
\(502\) −3.24901 −0.145010
\(503\) 15.7280 0.701276 0.350638 0.936511i \(-0.385965\pi\)
0.350638 + 0.936511i \(0.385965\pi\)
\(504\) −3.89357 −0.173434
\(505\) −69.0214 −3.07141
\(506\) −6.86850 −0.305342
\(507\) 27.8552 1.23709
\(508\) −0.284449 −0.0126204
\(509\) −1.79321 −0.0794825 −0.0397413 0.999210i \(-0.512653\pi\)
−0.0397413 + 0.999210i \(0.512653\pi\)
\(510\) 27.9810 1.23902
\(511\) −33.4630 −1.48032
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 2.35501 0.103875
\(515\) 51.1557 2.25419
\(516\) −2.91438 −0.128298
\(517\) −4.63243 −0.203734
\(518\) −12.4569 −0.547325
\(519\) −21.5813 −0.947314
\(520\) 26.9957 1.18384
\(521\) −8.06785 −0.353459 −0.176729 0.984259i \(-0.556552\pi\)
−0.176729 + 0.984259i \(0.556552\pi\)
\(522\) −5.17021 −0.226294
\(523\) 26.7063 1.16778 0.583892 0.811831i \(-0.301529\pi\)
0.583892 + 0.811831i \(0.301529\pi\)
\(524\) 5.09021 0.222367
\(525\) 49.9852 2.18153
\(526\) 2.60931 0.113771
\(527\) 60.2083 2.62271
\(528\) 1.39763 0.0608241
\(529\) 1.15112 0.0500489
\(530\) 4.22349 0.183457
\(531\) 1.28144 0.0556099
\(532\) 3.89357 0.168808
\(533\) −53.2471 −2.30639
\(534\) 3.45446 0.149489
\(535\) 30.0659 1.29986
\(536\) −1.72928 −0.0746935
\(537\) −8.43180 −0.363859
\(538\) −22.2422 −0.958929
\(539\) 11.4046 0.491229
\(540\) −4.22349 −0.181750
\(541\) 21.8595 0.939812 0.469906 0.882716i \(-0.344288\pi\)
0.469906 + 0.882716i \(0.344288\pi\)
\(542\) 3.51603 0.151026
\(543\) 8.13253 0.349000
\(544\) −6.62509 −0.284048
\(545\) −65.6891 −2.81381
\(546\) −24.8870 −1.06506
\(547\) −41.1697 −1.76029 −0.880145 0.474706i \(-0.842554\pi\)
−0.880145 + 0.474706i \(0.842554\pi\)
\(548\) −22.5889 −0.964948
\(549\) −14.2334 −0.607466
\(550\) −17.9426 −0.765077
\(551\) 5.17021 0.220258
\(552\) −4.91438 −0.209170
\(553\) 1.56164 0.0664078
\(554\) −0.273929 −0.0116381
\(555\) −13.5124 −0.573571
\(556\) 20.7843 0.881452
\(557\) 1.61913 0.0686047 0.0343024 0.999412i \(-0.489079\pi\)
0.0343024 + 0.999412i \(0.489079\pi\)
\(558\) −9.08792 −0.384722
\(559\) −18.6282 −0.787887
\(560\) −16.4445 −0.694906
\(561\) 9.25944 0.390934
\(562\) −8.62051 −0.363634
\(563\) −3.83903 −0.161796 −0.0808979 0.996722i \(-0.525779\pi\)
−0.0808979 + 0.996722i \(0.525779\pi\)
\(564\) −3.31449 −0.139565
\(565\) −56.1732 −2.36322
\(566\) 6.97293 0.293094
\(567\) 3.89357 0.163515
\(568\) 14.6340 0.614028
\(569\) 21.2492 0.890811 0.445405 0.895329i \(-0.353060\pi\)
0.445405 + 0.895329i \(0.353060\pi\)
\(570\) 4.22349 0.176903
\(571\) 25.3748 1.06190 0.530951 0.847403i \(-0.321835\pi\)
0.530951 + 0.847403i \(0.321835\pi\)
\(572\) 8.93340 0.373524
\(573\) −13.8455 −0.578405
\(574\) 32.4355 1.35383
\(575\) 63.0902 2.63104
\(576\) 1.00000 0.0416667
\(577\) −37.7782 −1.57273 −0.786363 0.617765i \(-0.788038\pi\)
−0.786363 + 0.617765i \(0.788038\pi\)
\(578\) −26.8918 −1.11855
\(579\) 0.356400 0.0148115
\(580\) −21.8363 −0.906704
\(581\) 68.8415 2.85603
\(582\) 12.2819 0.509100
\(583\) 1.39763 0.0578840
\(584\) 8.59442 0.355640
\(585\) −26.9957 −1.11614
\(586\) −14.1738 −0.585515
\(587\) 40.0748 1.65406 0.827032 0.562155i \(-0.190028\pi\)
0.827032 + 0.562155i \(0.190028\pi\)
\(588\) 8.15991 0.336509
\(589\) 9.08792 0.374461
\(590\) 5.41217 0.222815
\(591\) −3.72524 −0.153236
\(592\) 3.19935 0.131493
\(593\) 2.24843 0.0923318 0.0461659 0.998934i \(-0.485300\pi\)
0.0461659 + 0.998934i \(0.485300\pi\)
\(594\) −1.39763 −0.0573456
\(595\) −108.946 −4.46635
\(596\) −12.9791 −0.531644
\(597\) −7.09049 −0.290195
\(598\) −31.4118 −1.28452
\(599\) −39.8670 −1.62892 −0.814460 0.580219i \(-0.802967\pi\)
−0.814460 + 0.580219i \(0.802967\pi\)
\(600\) −12.8379 −0.524104
\(601\) 19.9169 0.812425 0.406213 0.913779i \(-0.366849\pi\)
0.406213 + 0.913779i \(0.366849\pi\)
\(602\) 11.3473 0.462483
\(603\) 1.72928 0.0704217
\(604\) 14.2437 0.579567
\(605\) 38.2083 1.55339
\(606\) −16.3423 −0.663859
\(607\) 6.34974 0.257728 0.128864 0.991662i \(-0.458867\pi\)
0.128864 + 0.991662i \(0.458867\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 20.1306 0.815733
\(610\) −60.1146 −2.43397
\(611\) −21.1856 −0.857076
\(612\) 6.62509 0.267803
\(613\) −1.12114 −0.0452823 −0.0226411 0.999744i \(-0.507208\pi\)
−0.0226411 + 0.999744i \(0.507208\pi\)
\(614\) 24.9195 1.00567
\(615\) 35.1839 1.41875
\(616\) −5.44178 −0.219256
\(617\) 22.3742 0.900750 0.450375 0.892840i \(-0.351290\pi\)
0.450375 + 0.892840i \(0.351290\pi\)
\(618\) 12.1122 0.487223
\(619\) 11.0226 0.443036 0.221518 0.975156i \(-0.428899\pi\)
0.221518 + 0.975156i \(0.428899\pi\)
\(620\) −38.3827 −1.54149
\(621\) 4.91438 0.197207
\(622\) 1.66374 0.0667097
\(623\) −13.4502 −0.538870
\(624\) 6.39181 0.255877
\(625\) 75.6218 3.02487
\(626\) 23.7002 0.947251
\(627\) 1.39763 0.0558161
\(628\) 14.8382 0.592109
\(629\) 21.1960 0.845140
\(630\) 16.4445 0.655163
\(631\) 9.27017 0.369040 0.184520 0.982829i \(-0.440927\pi\)
0.184520 + 0.982829i \(0.440927\pi\)
\(632\) −0.401082 −0.0159542
\(633\) −6.34960 −0.252374
\(634\) 12.5172 0.497121
\(635\) 1.20137 0.0476748
\(636\) 1.00000 0.0396526
\(637\) 52.1566 2.06652
\(638\) −7.22605 −0.286082
\(639\) −14.6340 −0.578911
\(640\) 4.22349 0.166948
\(641\) −3.04109 −0.120116 −0.0600578 0.998195i \(-0.519129\pi\)
−0.0600578 + 0.998195i \(0.519129\pi\)
\(642\) 7.11873 0.280954
\(643\) 21.8414 0.861340 0.430670 0.902509i \(-0.358277\pi\)
0.430670 + 0.902509i \(0.358277\pi\)
\(644\) 19.1345 0.754005
\(645\) 12.3089 0.484661
\(646\) −6.62509 −0.260661
\(647\) 0.490380 0.0192788 0.00963941 0.999954i \(-0.496932\pi\)
0.00963941 + 0.999954i \(0.496932\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 1.79099 0.0703024
\(650\) −82.0573 −3.21855
\(651\) 35.3845 1.38683
\(652\) −15.5059 −0.607258
\(653\) −12.8838 −0.504182 −0.252091 0.967704i \(-0.581118\pi\)
−0.252091 + 0.967704i \(0.581118\pi\)
\(654\) −15.5533 −0.608181
\(655\) −21.4985 −0.840014
\(656\) −8.33052 −0.325252
\(657\) −8.59442 −0.335300
\(658\) 12.9052 0.503097
\(659\) −21.2815 −0.829010 −0.414505 0.910047i \(-0.636045\pi\)
−0.414505 + 0.910047i \(0.636045\pi\)
\(660\) −5.90289 −0.229770
\(661\) −27.7790 −1.08048 −0.540238 0.841512i \(-0.681666\pi\)
−0.540238 + 0.841512i \(0.681666\pi\)
\(662\) −12.0207 −0.467199
\(663\) 42.3463 1.64459
\(664\) −17.6808 −0.686148
\(665\) −16.4445 −0.637689
\(666\) −3.19935 −0.123972
\(667\) 25.4084 0.983816
\(668\) 7.97057 0.308391
\(669\) −15.3317 −0.592758
\(670\) 7.30360 0.282163
\(671\) −19.8930 −0.767962
\(672\) −3.89357 −0.150198
\(673\) −36.5990 −1.41079 −0.705394 0.708815i \(-0.749230\pi\)
−0.705394 + 0.708815i \(0.749230\pi\)
\(674\) 27.6560 1.06527
\(675\) 12.8379 0.494130
\(676\) 27.8552 1.07135
\(677\) −18.3026 −0.703425 −0.351712 0.936108i \(-0.614401\pi\)
−0.351712 + 0.936108i \(0.614401\pi\)
\(678\) −13.3002 −0.510790
\(679\) −47.8204 −1.83518
\(680\) 27.9810 1.07302
\(681\) 10.2980 0.394621
\(682\) −12.7016 −0.486368
\(683\) 27.7691 1.06255 0.531277 0.847198i \(-0.321712\pi\)
0.531277 + 0.847198i \(0.321712\pi\)
\(684\) 1.00000 0.0382360
\(685\) 95.4038 3.64519
\(686\) −4.51620 −0.172429
\(687\) −21.0001 −0.801203
\(688\) −2.91438 −0.111110
\(689\) 6.39181 0.243509
\(690\) 20.7558 0.790161
\(691\) 20.3886 0.775619 0.387810 0.921740i \(-0.373232\pi\)
0.387810 + 0.921740i \(0.373232\pi\)
\(692\) −21.5813 −0.820398
\(693\) 5.44178 0.206716
\(694\) −19.4805 −0.739468
\(695\) −87.7825 −3.32978
\(696\) −5.17021 −0.195976
\(697\) −55.1904 −2.09049
\(698\) −0.935636 −0.0354143
\(699\) 10.4476 0.395163
\(700\) 49.9852 1.88926
\(701\) −0.0916529 −0.00346168 −0.00173084 0.999999i \(-0.500551\pi\)
−0.00173084 + 0.999999i \(0.500551\pi\)
\(702\) −6.39181 −0.241243
\(703\) 3.19935 0.120666
\(704\) 1.39763 0.0526753
\(705\) 13.9987 0.527222
\(706\) −27.2562 −1.02580
\(707\) 63.6298 2.39304
\(708\) 1.28144 0.0481596
\(709\) −2.51581 −0.0944832 −0.0472416 0.998883i \(-0.515043\pi\)
−0.0472416 + 0.998883i \(0.515043\pi\)
\(710\) −61.8065 −2.31956
\(711\) 0.401082 0.0150418
\(712\) 3.45446 0.129461
\(713\) 44.6615 1.67259
\(714\) −25.7953 −0.965364
\(715\) −37.7301 −1.41103
\(716\) −8.43180 −0.315111
\(717\) −22.3004 −0.832823
\(718\) 14.3368 0.535043
\(719\) 7.92057 0.295388 0.147694 0.989033i \(-0.452815\pi\)
0.147694 + 0.989033i \(0.452815\pi\)
\(720\) −4.22349 −0.157400
\(721\) −47.1597 −1.75632
\(722\) −1.00000 −0.0372161
\(723\) −3.10563 −0.115500
\(724\) 8.13253 0.302243
\(725\) 66.3745 2.46509
\(726\) 9.04662 0.335752
\(727\) 37.1761 1.37879 0.689393 0.724388i \(-0.257878\pi\)
0.689393 + 0.724388i \(0.257878\pi\)
\(728\) −24.8870 −0.922373
\(729\) 1.00000 0.0370370
\(730\) −36.2985 −1.34347
\(731\) −19.3080 −0.714133
\(732\) −14.2334 −0.526081
\(733\) −8.20806 −0.303172 −0.151586 0.988444i \(-0.548438\pi\)
−0.151586 + 0.988444i \(0.548438\pi\)
\(734\) −5.07573 −0.187349
\(735\) −34.4633 −1.27120
\(736\) −4.91438 −0.181146
\(737\) 2.41690 0.0890276
\(738\) 8.33052 0.306651
\(739\) 28.4210 1.04548 0.522742 0.852491i \(-0.324909\pi\)
0.522742 + 0.852491i \(0.324909\pi\)
\(740\) −13.5124 −0.496727
\(741\) 6.39181 0.234809
\(742\) −3.89357 −0.142938
\(743\) 10.3261 0.378828 0.189414 0.981897i \(-0.439341\pi\)
0.189414 + 0.981897i \(0.439341\pi\)
\(744\) −9.08792 −0.333179
\(745\) 54.8171 2.00834
\(746\) −5.06710 −0.185520
\(747\) 17.6808 0.646907
\(748\) 9.25944 0.338559
\(749\) −27.7173 −1.01277
\(750\) 33.1032 1.20876
\(751\) 22.9979 0.839207 0.419603 0.907708i \(-0.362169\pi\)
0.419603 + 0.907708i \(0.362169\pi\)
\(752\) −3.31449 −0.120867
\(753\) 3.24901 0.118401
\(754\) −33.0470 −1.20350
\(755\) −60.1581 −2.18938
\(756\) 3.89357 0.141608
\(757\) 20.4806 0.744379 0.372189 0.928157i \(-0.378607\pi\)
0.372189 + 0.928157i \(0.378607\pi\)
\(758\) −21.3462 −0.775330
\(759\) 6.86850 0.249311
\(760\) 4.22349 0.153202
\(761\) −3.61334 −0.130983 −0.0654917 0.997853i \(-0.520862\pi\)
−0.0654917 + 0.997853i \(0.520862\pi\)
\(762\) 0.284449 0.0103045
\(763\) 60.5578 2.19234
\(764\) −13.8455 −0.500914
\(765\) −27.9810 −1.01166
\(766\) 2.17408 0.0785528
\(767\) 8.19075 0.295751
\(768\) 1.00000 0.0360844
\(769\) 28.1348 1.01457 0.507283 0.861780i \(-0.330650\pi\)
0.507283 + 0.861780i \(0.330650\pi\)
\(770\) 22.9833 0.828262
\(771\) −2.35501 −0.0848136
\(772\) 0.356400 0.0128271
\(773\) 28.8510 1.03770 0.518849 0.854866i \(-0.326361\pi\)
0.518849 + 0.854866i \(0.326361\pi\)
\(774\) 2.91438 0.104755
\(775\) 116.670 4.19090
\(776\) 12.2819 0.440893
\(777\) 12.4569 0.446889
\(778\) −24.0139 −0.860941
\(779\) −8.33052 −0.298472
\(780\) −26.9957 −0.966603
\(781\) −20.4529 −0.731863
\(782\) −32.5582 −1.16428
\(783\) 5.17021 0.184768
\(784\) 8.15991 0.291425
\(785\) −62.6690 −2.23675
\(786\) −5.09021 −0.181562
\(787\) −10.8334 −0.386171 −0.193085 0.981182i \(-0.561849\pi\)
−0.193085 + 0.981182i \(0.561849\pi\)
\(788\) −3.72524 −0.132706
\(789\) −2.60931 −0.0928939
\(790\) 1.69397 0.0602687
\(791\) 51.7852 1.84127
\(792\) −1.39763 −0.0496627
\(793\) −90.9770 −3.23069
\(794\) 12.3957 0.439906
\(795\) −4.22349 −0.149792
\(796\) −7.09049 −0.251316
\(797\) 35.7948 1.26792 0.633959 0.773367i \(-0.281429\pi\)
0.633959 + 0.773367i \(0.281429\pi\)
\(798\) −3.89357 −0.137831
\(799\) −21.9588 −0.776845
\(800\) −12.8379 −0.453888
\(801\) −3.45446 −0.122057
\(802\) 2.39611 0.0846097
\(803\) −12.0118 −0.423889
\(804\) 1.72928 0.0609870
\(805\) −80.8144 −2.84833
\(806\) −58.0882 −2.04607
\(807\) 22.2422 0.782963
\(808\) −16.3423 −0.574918
\(809\) −7.72351 −0.271544 −0.135772 0.990740i \(-0.543352\pi\)
−0.135772 + 0.990740i \(0.543352\pi\)
\(810\) 4.22349 0.148398
\(811\) 13.8613 0.486736 0.243368 0.969934i \(-0.421748\pi\)
0.243368 + 0.969934i \(0.421748\pi\)
\(812\) 20.1306 0.706445
\(813\) −3.51603 −0.123313
\(814\) −4.47152 −0.156727
\(815\) 65.4890 2.29398
\(816\) 6.62509 0.231925
\(817\) −2.91438 −0.101961
\(818\) 12.5246 0.437913
\(819\) 24.8870 0.869621
\(820\) 35.1839 1.22867
\(821\) 4.91887 0.171670 0.0858349 0.996309i \(-0.472644\pi\)
0.0858349 + 0.996309i \(0.472644\pi\)
\(822\) 22.5889 0.787877
\(823\) −27.5695 −0.961014 −0.480507 0.876991i \(-0.659547\pi\)
−0.480507 + 0.876991i \(0.659547\pi\)
\(824\) 12.1122 0.421948
\(825\) 17.9426 0.624682
\(826\) −4.98940 −0.173603
\(827\) −0.146357 −0.00508935 −0.00254467 0.999997i \(-0.500810\pi\)
−0.00254467 + 0.999997i \(0.500810\pi\)
\(828\) 4.91438 0.170786
\(829\) 8.69894 0.302127 0.151063 0.988524i \(-0.451730\pi\)
0.151063 + 0.988524i \(0.451730\pi\)
\(830\) 74.6747 2.59200
\(831\) 0.273929 0.00950249
\(832\) 6.39181 0.221596
\(833\) 54.0601 1.87307
\(834\) −20.7843 −0.719703
\(835\) −33.6636 −1.16498
\(836\) 1.39763 0.0483381
\(837\) 9.08792 0.314124
\(838\) 28.2150 0.974670
\(839\) −26.2455 −0.906097 −0.453048 0.891486i \(-0.649664\pi\)
−0.453048 + 0.891486i \(0.649664\pi\)
\(840\) 16.4445 0.567388
\(841\) −2.26894 −0.0782393
\(842\) −16.4246 −0.566029
\(843\) 8.62051 0.296906
\(844\) −6.34960 −0.218562
\(845\) −117.646 −4.04715
\(846\) 3.31449 0.113954
\(847\) −35.2237 −1.21030
\(848\) 1.00000 0.0343401
\(849\) −6.97293 −0.239310
\(850\) −85.0521 −2.91726
\(851\) 15.7228 0.538972
\(852\) −14.6340 −0.501352
\(853\) 51.1742 1.75217 0.876086 0.482156i \(-0.160146\pi\)
0.876086 + 0.482156i \(0.160146\pi\)
\(854\) 55.4187 1.89639
\(855\) −4.22349 −0.144440
\(856\) 7.11873 0.243313
\(857\) −30.5716 −1.04430 −0.522152 0.852852i \(-0.674871\pi\)
−0.522152 + 0.852852i \(0.674871\pi\)
\(858\) −8.93340 −0.304981
\(859\) −19.8443 −0.677078 −0.338539 0.940952i \(-0.609933\pi\)
−0.338539 + 0.940952i \(0.609933\pi\)
\(860\) 12.3089 0.419729
\(861\) −32.4355 −1.10540
\(862\) 2.01343 0.0685776
\(863\) −7.54342 −0.256781 −0.128390 0.991724i \(-0.540981\pi\)
−0.128390 + 0.991724i \(0.540981\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 91.1484 3.09914
\(866\) 17.3549 0.589745
\(867\) 26.8918 0.913294
\(868\) 35.3845 1.20103
\(869\) 0.560566 0.0190159
\(870\) 21.8363 0.740321
\(871\) 11.0532 0.374524
\(872\) −15.5533 −0.526700
\(873\) −12.2819 −0.415678
\(874\) −4.91438 −0.166231
\(875\) −128.890 −4.35727
\(876\) −8.59442 −0.290379
\(877\) −45.2366 −1.52753 −0.763766 0.645494i \(-0.776652\pi\)
−0.763766 + 0.645494i \(0.776652\pi\)
\(878\) 24.5865 0.829753
\(879\) 14.1738 0.478071
\(880\) −5.90289 −0.198986
\(881\) 43.7496 1.47396 0.736981 0.675914i \(-0.236251\pi\)
0.736981 + 0.675914i \(0.236251\pi\)
\(882\) −8.15991 −0.274758
\(883\) −33.1294 −1.11489 −0.557446 0.830213i \(-0.688219\pi\)
−0.557446 + 0.830213i \(0.688219\pi\)
\(884\) 42.3463 1.42426
\(885\) −5.41217 −0.181928
\(886\) 14.1705 0.476068
\(887\) −24.2227 −0.813318 −0.406659 0.913580i \(-0.633306\pi\)
−0.406659 + 0.913580i \(0.633306\pi\)
\(888\) −3.19935 −0.107363
\(889\) −1.10752 −0.0371451
\(890\) −14.5899 −0.489054
\(891\) 1.39763 0.0468224
\(892\) −15.3317 −0.513343
\(893\) −3.31449 −0.110915
\(894\) 12.9791 0.434086
\(895\) 35.6116 1.19037
\(896\) −3.89357 −0.130075
\(897\) 31.4118 1.04881
\(898\) −40.1610 −1.34019
\(899\) 46.9864 1.56709
\(900\) 12.8379 0.427929
\(901\) 6.62509 0.220714
\(902\) 11.6430 0.387670
\(903\) −11.3473 −0.377616
\(904\) −13.3002 −0.442357
\(905\) −34.3477 −1.14176
\(906\) −14.2437 −0.473215
\(907\) 10.2660 0.340878 0.170439 0.985368i \(-0.445481\pi\)
0.170439 + 0.985368i \(0.445481\pi\)
\(908\) 10.2980 0.341751
\(909\) 16.3423 0.542038
\(910\) 105.110 3.48436
\(911\) 8.39506 0.278141 0.139070 0.990283i \(-0.455589\pi\)
0.139070 + 0.990283i \(0.455589\pi\)
\(912\) 1.00000 0.0331133
\(913\) 24.7113 0.817824
\(914\) 14.6789 0.485536
\(915\) 60.1146 1.98733
\(916\) −21.0001 −0.693862
\(917\) 19.8191 0.654484
\(918\) −6.62509 −0.218661
\(919\) 46.9164 1.54763 0.773815 0.633412i \(-0.218346\pi\)
0.773815 + 0.633412i \(0.218346\pi\)
\(920\) 20.7558 0.684300
\(921\) −24.9195 −0.821126
\(922\) 37.4126 1.23212
\(923\) −93.5376 −3.07883
\(924\) 5.44178 0.179022
\(925\) 41.0729 1.35047
\(926\) 39.7374 1.30585
\(927\) −12.1122 −0.397816
\(928\) −5.17021 −0.169720
\(929\) −40.8766 −1.34112 −0.670559 0.741857i \(-0.733946\pi\)
−0.670559 + 0.741857i \(0.733946\pi\)
\(930\) 38.3827 1.25862
\(931\) 8.15991 0.267430
\(932\) 10.4476 0.342221
\(933\) −1.66374 −0.0544682
\(934\) −1.57966 −0.0516880
\(935\) −39.1072 −1.27894
\(936\) −6.39181 −0.208923
\(937\) 54.6283 1.78463 0.892314 0.451416i \(-0.149081\pi\)
0.892314 + 0.451416i \(0.149081\pi\)
\(938\) −6.73308 −0.219843
\(939\) −23.7002 −0.773427
\(940\) 13.9987 0.456587
\(941\) 40.8345 1.33117 0.665584 0.746323i \(-0.268183\pi\)
0.665584 + 0.746323i \(0.268183\pi\)
\(942\) −14.8382 −0.483455
\(943\) −40.9393 −1.33317
\(944\) 1.28144 0.0417075
\(945\) −16.4445 −0.534939
\(946\) 4.07323 0.132432
\(947\) −11.3667 −0.369368 −0.184684 0.982798i \(-0.559126\pi\)
−0.184684 + 0.982798i \(0.559126\pi\)
\(948\) 0.401082 0.0130265
\(949\) −54.9339 −1.78323
\(950\) −12.8379 −0.416516
\(951\) −12.5172 −0.405898
\(952\) −25.7953 −0.836029
\(953\) 3.25471 0.105430 0.0527152 0.998610i \(-0.483212\pi\)
0.0527152 + 0.998610i \(0.483212\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 58.4765 1.89225
\(956\) −22.3004 −0.721246
\(957\) 7.22605 0.233585
\(958\) 39.8886 1.28874
\(959\) −87.9514 −2.84010
\(960\) −4.22349 −0.136313
\(961\) 51.5903 1.66420
\(962\) −20.4496 −0.659323
\(963\) −7.11873 −0.229398
\(964\) −3.10563 −0.100026
\(965\) −1.50525 −0.0484558
\(966\) −19.1345 −0.615642
\(967\) −51.7706 −1.66483 −0.832414 0.554154i \(-0.813042\pi\)
−0.832414 + 0.554154i \(0.813042\pi\)
\(968\) 9.04662 0.290769
\(969\) 6.62509 0.212829
\(970\) −51.8724 −1.66552
\(971\) 55.3421 1.77601 0.888006 0.459832i \(-0.152090\pi\)
0.888006 + 0.459832i \(0.152090\pi\)
\(972\) 1.00000 0.0320750
\(973\) 80.9254 2.59435
\(974\) 8.41305 0.269571
\(975\) 82.0573 2.62794
\(976\) −14.2334 −0.455599
\(977\) 48.8817 1.56386 0.781932 0.623364i \(-0.214234\pi\)
0.781932 + 0.623364i \(0.214234\pi\)
\(978\) 15.5059 0.495824
\(979\) −4.82806 −0.154306
\(980\) −34.4633 −1.10089
\(981\) 15.5533 0.496578
\(982\) 17.4356 0.556393
\(983\) −23.9367 −0.763463 −0.381732 0.924273i \(-0.624672\pi\)
−0.381732 + 0.924273i \(0.624672\pi\)
\(984\) 8.33052 0.265567
\(985\) 15.7335 0.501312
\(986\) −34.2531 −1.09084
\(987\) −12.9052 −0.410777
\(988\) 6.39181 0.203351
\(989\) −14.3224 −0.455425
\(990\) 5.90289 0.187606
\(991\) −15.9344 −0.506174 −0.253087 0.967444i \(-0.581446\pi\)
−0.253087 + 0.967444i \(0.581446\pi\)
\(992\) −9.08792 −0.288542
\(993\) 12.0207 0.381466
\(994\) 56.9784 1.80725
\(995\) 29.9466 0.949372
\(996\) 17.6808 0.560238
\(997\) −41.1437 −1.30303 −0.651517 0.758634i \(-0.725867\pi\)
−0.651517 + 0.758634i \(0.725867\pi\)
\(998\) 13.6058 0.430684
\(999\) 3.19935 0.101223
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 6042.2.a.bf.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.bf.1.1 12 1.1 even 1 trivial