Properties

Label 7942.2.a.t.1.2
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $2$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7942.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -0.381966 q^{5} +0.381966 q^{6} -1.85410 q^{7} -1.00000 q^{8} -2.85410 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.381966 q^{3} +1.00000 q^{4} -0.381966 q^{5} +0.381966 q^{6} -1.85410 q^{7} -1.00000 q^{8} -2.85410 q^{9} +0.381966 q^{10} -1.00000 q^{11} -0.381966 q^{12} -5.85410 q^{13} +1.85410 q^{14} +0.145898 q^{15} +1.00000 q^{16} +6.47214 q^{17} +2.85410 q^{18} -0.381966 q^{20} +0.708204 q^{21} +1.00000 q^{22} +4.76393 q^{23} +0.381966 q^{24} -4.85410 q^{25} +5.85410 q^{26} +2.23607 q^{27} -1.85410 q^{28} +8.09017 q^{29} -0.145898 q^{30} -0.145898 q^{31} -1.00000 q^{32} +0.381966 q^{33} -6.47214 q^{34} +0.708204 q^{35} -2.85410 q^{36} -1.23607 q^{37} +2.23607 q^{39} +0.381966 q^{40} -1.09017 q^{41} -0.708204 q^{42} -2.38197 q^{43} -1.00000 q^{44} +1.09017 q^{45} -4.76393 q^{46} +9.23607 q^{47} -0.381966 q^{48} -3.56231 q^{49} +4.85410 q^{50} -2.47214 q^{51} -5.85410 q^{52} -6.76393 q^{53} -2.23607 q^{54} +0.381966 q^{55} +1.85410 q^{56} -8.09017 q^{58} +4.94427 q^{59} +0.145898 q^{60} +7.70820 q^{61} +0.145898 q^{62} +5.29180 q^{63} +1.00000 q^{64} +2.23607 q^{65} -0.381966 q^{66} +11.0902 q^{67} +6.47214 q^{68} -1.81966 q^{69} -0.708204 q^{70} -7.14590 q^{71} +2.85410 q^{72} +9.23607 q^{73} +1.23607 q^{74} +1.85410 q^{75} +1.85410 q^{77} -2.23607 q^{78} -3.23607 q^{79} -0.381966 q^{80} +7.70820 q^{81} +1.09017 q^{82} +0.0901699 q^{83} +0.708204 q^{84} -2.47214 q^{85} +2.38197 q^{86} -3.09017 q^{87} +1.00000 q^{88} +16.6525 q^{89} -1.09017 q^{90} +10.8541 q^{91} +4.76393 q^{92} +0.0557281 q^{93} -9.23607 q^{94} +0.381966 q^{96} +8.00000 q^{97} +3.56231 q^{98} +2.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{5} + 3 q^{6} + 3 q^{7} - 2 q^{8} + q^{9} + 3 q^{10} - 2 q^{11} - 3 q^{12} - 5 q^{13} - 3 q^{14} + 7 q^{15} + 2 q^{16} + 4 q^{17} - q^{18} - 3 q^{20} - 12 q^{21} + 2 q^{22} + 14 q^{23} + 3 q^{24} - 3 q^{25} + 5 q^{26} + 3 q^{28} + 5 q^{29} - 7 q^{30} - 7 q^{31} - 2 q^{32} + 3 q^{33} - 4 q^{34} - 12 q^{35} + q^{36} + 2 q^{37} + 3 q^{40} + 9 q^{41} + 12 q^{42} - 7 q^{43} - 2 q^{44} - 9 q^{45} - 14 q^{46} + 14 q^{47} - 3 q^{48} + 13 q^{49} + 3 q^{50} + 4 q^{51} - 5 q^{52} - 18 q^{53} + 3 q^{55} - 3 q^{56} - 5 q^{58} - 8 q^{59} + 7 q^{60} + 2 q^{61} + 7 q^{62} + 24 q^{63} + 2 q^{64} - 3 q^{66} + 11 q^{67} + 4 q^{68} - 26 q^{69} + 12 q^{70} - 21 q^{71} - q^{72} + 14 q^{73} - 2 q^{74} - 3 q^{75} - 3 q^{77} - 2 q^{79} - 3 q^{80} + 2 q^{81} - 9 q^{82} - 11 q^{83} - 12 q^{84} + 4 q^{85} + 7 q^{86} + 5 q^{87} + 2 q^{88} + 2 q^{89} + 9 q^{90} + 15 q^{91} + 14 q^{92} + 18 q^{93} - 14 q^{94} + 3 q^{96} + 16 q^{97} - 13 q^{98} - q^{99}+O(q^{100}) \) 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
\(3\) −0.381966 −0.220528 −0.110264 0.993902i \(-0.535170\pi\)
−0.110264 + 0.993902i \(0.535170\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 0.381966 0.155937
\(7\) −1.85410 −0.700785 −0.350392 0.936603i \(-0.613952\pi\)
−0.350392 + 0.936603i \(0.613952\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.85410 −0.951367
\(10\) 0.381966 0.120788
\(11\) −1.00000 −0.301511
\(12\) −0.381966 −0.110264
\(13\) −5.85410 −1.62364 −0.811818 0.583911i \(-0.801522\pi\)
−0.811818 + 0.583911i \(0.801522\pi\)
\(14\) 1.85410 0.495530
\(15\) 0.145898 0.0376707
\(16\) 1.00000 0.250000
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) 2.85410 0.672718
\(19\) 0 0
\(20\) −0.381966 −0.0854102
\(21\) 0.708204 0.154543
\(22\) 1.00000 0.213201
\(23\) 4.76393 0.993348 0.496674 0.867937i \(-0.334554\pi\)
0.496674 + 0.867937i \(0.334554\pi\)
\(24\) 0.381966 0.0779685
\(25\) −4.85410 −0.970820
\(26\) 5.85410 1.14808
\(27\) 2.23607 0.430331
\(28\) −1.85410 −0.350392
\(29\) 8.09017 1.50231 0.751153 0.660128i \(-0.229498\pi\)
0.751153 + 0.660128i \(0.229498\pi\)
\(30\) −0.145898 −0.0266372
\(31\) −0.145898 −0.0262041 −0.0131020 0.999914i \(-0.504171\pi\)
−0.0131020 + 0.999914i \(0.504171\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.381966 0.0664917
\(34\) −6.47214 −1.10996
\(35\) 0.708204 0.119708
\(36\) −2.85410 −0.475684
\(37\) −1.23607 −0.203208 −0.101604 0.994825i \(-0.532398\pi\)
−0.101604 + 0.994825i \(0.532398\pi\)
\(38\) 0 0
\(39\) 2.23607 0.358057
\(40\) 0.381966 0.0603941
\(41\) −1.09017 −0.170256 −0.0851280 0.996370i \(-0.527130\pi\)
−0.0851280 + 0.996370i \(0.527130\pi\)
\(42\) −0.708204 −0.109278
\(43\) −2.38197 −0.363246 −0.181623 0.983368i \(-0.558135\pi\)
−0.181623 + 0.983368i \(0.558135\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.09017 0.162513
\(46\) −4.76393 −0.702403
\(47\) 9.23607 1.34722 0.673609 0.739087i \(-0.264743\pi\)
0.673609 + 0.739087i \(0.264743\pi\)
\(48\) −0.381966 −0.0551320
\(49\) −3.56231 −0.508901
\(50\) 4.85410 0.686474
\(51\) −2.47214 −0.346168
\(52\) −5.85410 −0.811818
\(53\) −6.76393 −0.929098 −0.464549 0.885548i \(-0.653783\pi\)
−0.464549 + 0.885548i \(0.653783\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0.381966 0.0515043
\(56\) 1.85410 0.247765
\(57\) 0 0
\(58\) −8.09017 −1.06229
\(59\) 4.94427 0.643689 0.321845 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(60\) 0.145898 0.0188354
\(61\) 7.70820 0.986934 0.493467 0.869764i \(-0.335729\pi\)
0.493467 + 0.869764i \(0.335729\pi\)
\(62\) 0.145898 0.0185291
\(63\) 5.29180 0.666704
\(64\) 1.00000 0.125000
\(65\) 2.23607 0.277350
\(66\) −0.381966 −0.0470168
\(67\) 11.0902 1.35488 0.677440 0.735578i \(-0.263089\pi\)
0.677440 + 0.735578i \(0.263089\pi\)
\(68\) 6.47214 0.784862
\(69\) −1.81966 −0.219061
\(70\) −0.708204 −0.0846466
\(71\) −7.14590 −0.848062 −0.424031 0.905648i \(-0.639385\pi\)
−0.424031 + 0.905648i \(0.639385\pi\)
\(72\) 2.85410 0.336359
\(73\) 9.23607 1.08100 0.540500 0.841344i \(-0.318235\pi\)
0.540500 + 0.841344i \(0.318235\pi\)
\(74\) 1.23607 0.143690
\(75\) 1.85410 0.214093
\(76\) 0 0
\(77\) 1.85410 0.211295
\(78\) −2.23607 −0.253185
\(79\) −3.23607 −0.364086 −0.182043 0.983291i \(-0.558271\pi\)
−0.182043 + 0.983291i \(0.558271\pi\)
\(80\) −0.381966 −0.0427051
\(81\) 7.70820 0.856467
\(82\) 1.09017 0.120389
\(83\) 0.0901699 0.00989744 0.00494872 0.999988i \(-0.498425\pi\)
0.00494872 + 0.999988i \(0.498425\pi\)
\(84\) 0.708204 0.0772714
\(85\) −2.47214 −0.268141
\(86\) 2.38197 0.256854
\(87\) −3.09017 −0.331301
\(88\) 1.00000 0.106600
\(89\) 16.6525 1.76516 0.882579 0.470163i \(-0.155805\pi\)
0.882579 + 0.470163i \(0.155805\pi\)
\(90\) −1.09017 −0.114914
\(91\) 10.8541 1.13782
\(92\) 4.76393 0.496674
\(93\) 0.0557281 0.00577873
\(94\) −9.23607 −0.952628
\(95\) 0 0
\(96\) 0.381966 0.0389842
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 3.56231 0.359847
\(99\) 2.85410 0.286848
\(100\) −4.85410 −0.485410
\(101\) 13.4164 1.33498 0.667491 0.744618i \(-0.267368\pi\)
0.667491 + 0.744618i \(0.267368\pi\)
\(102\) 2.47214 0.244778
\(103\) −18.5623 −1.82900 −0.914499 0.404588i \(-0.867415\pi\)
−0.914499 + 0.404588i \(0.867415\pi\)
\(104\) 5.85410 0.574042
\(105\) −0.270510 −0.0263991
\(106\) 6.76393 0.656971
\(107\) −7.23607 −0.699537 −0.349769 0.936836i \(-0.613740\pi\)
−0.349769 + 0.936836i \(0.613740\pi\)
\(108\) 2.23607 0.215166
\(109\) −13.4164 −1.28506 −0.642529 0.766261i \(-0.722115\pi\)
−0.642529 + 0.766261i \(0.722115\pi\)
\(110\) −0.381966 −0.0364190
\(111\) 0.472136 0.0448132
\(112\) −1.85410 −0.175196
\(113\) 2.29180 0.215594 0.107797 0.994173i \(-0.465620\pi\)
0.107797 + 0.994173i \(0.465620\pi\)
\(114\) 0 0
\(115\) −1.81966 −0.169684
\(116\) 8.09017 0.751153
\(117\) 16.7082 1.54467
\(118\) −4.94427 −0.455157
\(119\) −12.0000 −1.10004
\(120\) −0.145898 −0.0133186
\(121\) 1.00000 0.0909091
\(122\) −7.70820 −0.697868
\(123\) 0.416408 0.0375462
\(124\) −0.145898 −0.0131020
\(125\) 3.76393 0.336656
\(126\) −5.29180 −0.471431
\(127\) −21.7082 −1.92629 −0.963146 0.268980i \(-0.913313\pi\)
−0.963146 + 0.268980i \(0.913313\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.909830 0.0801061
\(130\) −2.23607 −0.196116
\(131\) −15.2705 −1.33419 −0.667095 0.744972i \(-0.732463\pi\)
−0.667095 + 0.744972i \(0.732463\pi\)
\(132\) 0.381966 0.0332459
\(133\) 0 0
\(134\) −11.0902 −0.958045
\(135\) −0.854102 −0.0735094
\(136\) −6.47214 −0.554981
\(137\) −0.326238 −0.0278724 −0.0139362 0.999903i \(-0.504436\pi\)
−0.0139362 + 0.999903i \(0.504436\pi\)
\(138\) 1.81966 0.154900
\(139\) 23.3262 1.97851 0.989253 0.146215i \(-0.0467091\pi\)
0.989253 + 0.146215i \(0.0467091\pi\)
\(140\) 0.708204 0.0598542
\(141\) −3.52786 −0.297100
\(142\) 7.14590 0.599670
\(143\) 5.85410 0.489545
\(144\) −2.85410 −0.237842
\(145\) −3.09017 −0.256625
\(146\) −9.23607 −0.764382
\(147\) 1.36068 0.112227
\(148\) −1.23607 −0.101604
\(149\) −12.9443 −1.06044 −0.530218 0.847861i \(-0.677890\pi\)
−0.530218 + 0.847861i \(0.677890\pi\)
\(150\) −1.85410 −0.151387
\(151\) −16.9443 −1.37891 −0.689453 0.724331i \(-0.742149\pi\)
−0.689453 + 0.724331i \(0.742149\pi\)
\(152\) 0 0
\(153\) −18.4721 −1.49338
\(154\) −1.85410 −0.149408
\(155\) 0.0557281 0.00447619
\(156\) 2.23607 0.179029
\(157\) −7.61803 −0.607985 −0.303993 0.952674i \(-0.598320\pi\)
−0.303993 + 0.952674i \(0.598320\pi\)
\(158\) 3.23607 0.257448
\(159\) 2.58359 0.204892
\(160\) 0.381966 0.0301971
\(161\) −8.83282 −0.696123
\(162\) −7.70820 −0.605614
\(163\) −13.2361 −1.03673 −0.518364 0.855160i \(-0.673459\pi\)
−0.518364 + 0.855160i \(0.673459\pi\)
\(164\) −1.09017 −0.0851280
\(165\) −0.145898 −0.0113581
\(166\) −0.0901699 −0.00699854
\(167\) 4.47214 0.346064 0.173032 0.984916i \(-0.444644\pi\)
0.173032 + 0.984916i \(0.444644\pi\)
\(168\) −0.708204 −0.0546391
\(169\) 21.2705 1.63619
\(170\) 2.47214 0.189604
\(171\) 0 0
\(172\) −2.38197 −0.181623
\(173\) −1.61803 −0.123017 −0.0615084 0.998107i \(-0.519591\pi\)
−0.0615084 + 0.998107i \(0.519591\pi\)
\(174\) 3.09017 0.234265
\(175\) 9.00000 0.680336
\(176\) −1.00000 −0.0753778
\(177\) −1.88854 −0.141952
\(178\) −16.6525 −1.24816
\(179\) −20.0344 −1.49744 −0.748722 0.662884i \(-0.769332\pi\)
−0.748722 + 0.662884i \(0.769332\pi\)
\(180\) 1.09017 0.0812565
\(181\) −8.65248 −0.643133 −0.321567 0.946887i \(-0.604209\pi\)
−0.321567 + 0.946887i \(0.604209\pi\)
\(182\) −10.8541 −0.804560
\(183\) −2.94427 −0.217647
\(184\) −4.76393 −0.351202
\(185\) 0.472136 0.0347121
\(186\) −0.0557281 −0.00408618
\(187\) −6.47214 −0.473289
\(188\) 9.23607 0.673609
\(189\) −4.14590 −0.301570
\(190\) 0 0
\(191\) 19.4164 1.40492 0.702461 0.711722i \(-0.252085\pi\)
0.702461 + 0.711722i \(0.252085\pi\)
\(192\) −0.381966 −0.0275660
\(193\) −8.56231 −0.616328 −0.308164 0.951333i \(-0.599715\pi\)
−0.308164 + 0.951333i \(0.599715\pi\)
\(194\) −8.00000 −0.574367
\(195\) −0.854102 −0.0611635
\(196\) −3.56231 −0.254450
\(197\) 16.4721 1.17359 0.586796 0.809735i \(-0.300389\pi\)
0.586796 + 0.809735i \(0.300389\pi\)
\(198\) −2.85410 −0.202832
\(199\) −2.47214 −0.175245 −0.0876225 0.996154i \(-0.527927\pi\)
−0.0876225 + 0.996154i \(0.527927\pi\)
\(200\) 4.85410 0.343237
\(201\) −4.23607 −0.298789
\(202\) −13.4164 −0.943975
\(203\) −15.0000 −1.05279
\(204\) −2.47214 −0.173084
\(205\) 0.416408 0.0290832
\(206\) 18.5623 1.29330
\(207\) −13.5967 −0.945039
\(208\) −5.85410 −0.405909
\(209\) 0 0
\(210\) 0.270510 0.0186670
\(211\) −3.41641 −0.235195 −0.117598 0.993061i \(-0.537519\pi\)
−0.117598 + 0.993061i \(0.537519\pi\)
\(212\) −6.76393 −0.464549
\(213\) 2.72949 0.187022
\(214\) 7.23607 0.494647
\(215\) 0.909830 0.0620499
\(216\) −2.23607 −0.152145
\(217\) 0.270510 0.0183634
\(218\) 13.4164 0.908674
\(219\) −3.52786 −0.238391
\(220\) 0.381966 0.0257521
\(221\) −37.8885 −2.54866
\(222\) −0.472136 −0.0316877
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 1.85410 0.123882
\(225\) 13.8541 0.923607
\(226\) −2.29180 −0.152448
\(227\) 29.5967 1.96441 0.982203 0.187825i \(-0.0601437\pi\)
0.982203 + 0.187825i \(0.0601437\pi\)
\(228\) 0 0
\(229\) −17.3820 −1.14863 −0.574316 0.818633i \(-0.694732\pi\)
−0.574316 + 0.818633i \(0.694732\pi\)
\(230\) 1.81966 0.119985
\(231\) −0.708204 −0.0465964
\(232\) −8.09017 −0.531146
\(233\) −3.05573 −0.200187 −0.100094 0.994978i \(-0.531914\pi\)
−0.100094 + 0.994978i \(0.531914\pi\)
\(234\) −16.7082 −1.09225
\(235\) −3.52786 −0.230132
\(236\) 4.94427 0.321845
\(237\) 1.23607 0.0802912
\(238\) 12.0000 0.777844
\(239\) 8.56231 0.553850 0.276925 0.960892i \(-0.410685\pi\)
0.276925 + 0.960892i \(0.410685\pi\)
\(240\) 0.145898 0.00941768
\(241\) −10.8541 −0.699174 −0.349587 0.936904i \(-0.613678\pi\)
−0.349587 + 0.936904i \(0.613678\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −9.65248 −0.619207
\(244\) 7.70820 0.493467
\(245\) 1.36068 0.0869306
\(246\) −0.416408 −0.0265492
\(247\) 0 0
\(248\) 0.145898 0.00926453
\(249\) −0.0344419 −0.00218266
\(250\) −3.76393 −0.238052
\(251\) 7.52786 0.475155 0.237577 0.971369i \(-0.423647\pi\)
0.237577 + 0.971369i \(0.423647\pi\)
\(252\) 5.29180 0.333352
\(253\) −4.76393 −0.299506
\(254\) 21.7082 1.36209
\(255\) 0.944272 0.0591326
\(256\) 1.00000 0.0625000
\(257\) −21.7082 −1.35412 −0.677060 0.735928i \(-0.736746\pi\)
−0.677060 + 0.735928i \(0.736746\pi\)
\(258\) −0.909830 −0.0566435
\(259\) 2.29180 0.142405
\(260\) 2.23607 0.138675
\(261\) −23.0902 −1.42925
\(262\) 15.2705 0.943415
\(263\) −10.8541 −0.669293 −0.334646 0.942344i \(-0.608617\pi\)
−0.334646 + 0.942344i \(0.608617\pi\)
\(264\) −0.381966 −0.0235084
\(265\) 2.58359 0.158709
\(266\) 0 0
\(267\) −6.36068 −0.389267
\(268\) 11.0902 0.677440
\(269\) 17.5967 1.07289 0.536446 0.843934i \(-0.319766\pi\)
0.536446 + 0.843934i \(0.319766\pi\)
\(270\) 0.854102 0.0519790
\(271\) 2.43769 0.148079 0.0740397 0.997255i \(-0.476411\pi\)
0.0740397 + 0.997255i \(0.476411\pi\)
\(272\) 6.47214 0.392431
\(273\) −4.14590 −0.250921
\(274\) 0.326238 0.0197088
\(275\) 4.85410 0.292713
\(276\) −1.81966 −0.109531
\(277\) −0.944272 −0.0567358 −0.0283679 0.999598i \(-0.509031\pi\)
−0.0283679 + 0.999598i \(0.509031\pi\)
\(278\) −23.3262 −1.39901
\(279\) 0.416408 0.0249297
\(280\) −0.708204 −0.0423233
\(281\) −30.2705 −1.80579 −0.902894 0.429864i \(-0.858561\pi\)
−0.902894 + 0.429864i \(0.858561\pi\)
\(282\) 3.52786 0.210081
\(283\) 18.0344 1.07204 0.536018 0.844206i \(-0.319928\pi\)
0.536018 + 0.844206i \(0.319928\pi\)
\(284\) −7.14590 −0.424031
\(285\) 0 0
\(286\) −5.85410 −0.346160
\(287\) 2.02129 0.119313
\(288\) 2.85410 0.168180
\(289\) 24.8885 1.46403
\(290\) 3.09017 0.181461
\(291\) −3.05573 −0.179130
\(292\) 9.23607 0.540500
\(293\) −8.03444 −0.469377 −0.234689 0.972071i \(-0.575407\pi\)
−0.234689 + 0.972071i \(0.575407\pi\)
\(294\) −1.36068 −0.0793565
\(295\) −1.88854 −0.109955
\(296\) 1.23607 0.0718450
\(297\) −2.23607 −0.129750
\(298\) 12.9443 0.749842
\(299\) −27.8885 −1.61284
\(300\) 1.85410 0.107047
\(301\) 4.41641 0.254558
\(302\) 16.9443 0.975033
\(303\) −5.12461 −0.294401
\(304\) 0 0
\(305\) −2.94427 −0.168589
\(306\) 18.4721 1.05598
\(307\) 25.2361 1.44030 0.720149 0.693819i \(-0.244073\pi\)
0.720149 + 0.693819i \(0.244073\pi\)
\(308\) 1.85410 0.105647
\(309\) 7.09017 0.403346
\(310\) −0.0557281 −0.00316514
\(311\) 14.4721 0.820640 0.410320 0.911942i \(-0.365417\pi\)
0.410320 + 0.911942i \(0.365417\pi\)
\(312\) −2.23607 −0.126592
\(313\) −18.5623 −1.04920 −0.524602 0.851348i \(-0.675786\pi\)
−0.524602 + 0.851348i \(0.675786\pi\)
\(314\) 7.61803 0.429911
\(315\) −2.02129 −0.113887
\(316\) −3.23607 −0.182043
\(317\) −24.1803 −1.35810 −0.679052 0.734091i \(-0.737609\pi\)
−0.679052 + 0.734091i \(0.737609\pi\)
\(318\) −2.58359 −0.144881
\(319\) −8.09017 −0.452963
\(320\) −0.381966 −0.0213525
\(321\) 2.76393 0.154268
\(322\) 8.83282 0.492234
\(323\) 0 0
\(324\) 7.70820 0.428234
\(325\) 28.4164 1.57626
\(326\) 13.2361 0.733078
\(327\) 5.12461 0.283392
\(328\) 1.09017 0.0601946
\(329\) −17.1246 −0.944110
\(330\) 0.145898 0.00803142
\(331\) −12.7426 −0.700399 −0.350200 0.936675i \(-0.613886\pi\)
−0.350200 + 0.936675i \(0.613886\pi\)
\(332\) 0.0901699 0.00494872
\(333\) 3.52786 0.193326
\(334\) −4.47214 −0.244704
\(335\) −4.23607 −0.231441
\(336\) 0.708204 0.0386357
\(337\) 7.85410 0.427840 0.213920 0.976851i \(-0.431377\pi\)
0.213920 + 0.976851i \(0.431377\pi\)
\(338\) −21.2705 −1.15696
\(339\) −0.875388 −0.0475446
\(340\) −2.47214 −0.134070
\(341\) 0.145898 0.00790082
\(342\) 0 0
\(343\) 19.5836 1.05741
\(344\) 2.38197 0.128427
\(345\) 0.695048 0.0374201
\(346\) 1.61803 0.0869860
\(347\) −13.5279 −0.726214 −0.363107 0.931747i \(-0.618284\pi\)
−0.363107 + 0.931747i \(0.618284\pi\)
\(348\) −3.09017 −0.165650
\(349\) −16.7639 −0.897353 −0.448676 0.893694i \(-0.648104\pi\)
−0.448676 + 0.893694i \(0.648104\pi\)
\(350\) −9.00000 −0.481070
\(351\) −13.0902 −0.698702
\(352\) 1.00000 0.0533002
\(353\) −20.4721 −1.08962 −0.544811 0.838559i \(-0.683399\pi\)
−0.544811 + 0.838559i \(0.683399\pi\)
\(354\) 1.88854 0.100375
\(355\) 2.72949 0.144866
\(356\) 16.6525 0.882579
\(357\) 4.58359 0.242589
\(358\) 20.0344 1.05885
\(359\) 16.0344 0.846265 0.423133 0.906068i \(-0.360930\pi\)
0.423133 + 0.906068i \(0.360930\pi\)
\(360\) −1.09017 −0.0574570
\(361\) 0 0
\(362\) 8.65248 0.454764
\(363\) −0.381966 −0.0200480
\(364\) 10.8541 0.568910
\(365\) −3.52786 −0.184657
\(366\) 2.94427 0.153900
\(367\) 29.1246 1.52029 0.760146 0.649752i \(-0.225127\pi\)
0.760146 + 0.649752i \(0.225127\pi\)
\(368\) 4.76393 0.248337
\(369\) 3.11146 0.161976
\(370\) −0.472136 −0.0245452
\(371\) 12.5410 0.651097
\(372\) 0.0557281 0.00288937
\(373\) 5.79837 0.300228 0.150114 0.988669i \(-0.452036\pi\)
0.150114 + 0.988669i \(0.452036\pi\)
\(374\) 6.47214 0.334666
\(375\) −1.43769 −0.0742422
\(376\) −9.23607 −0.476314
\(377\) −47.3607 −2.43920
\(378\) 4.14590 0.213242
\(379\) −34.5066 −1.77248 −0.886242 0.463223i \(-0.846693\pi\)
−0.886242 + 0.463223i \(0.846693\pi\)
\(380\) 0 0
\(381\) 8.29180 0.424802
\(382\) −19.4164 −0.993430
\(383\) 0.798374 0.0407950 0.0203975 0.999792i \(-0.493507\pi\)
0.0203975 + 0.999792i \(0.493507\pi\)
\(384\) 0.381966 0.0194921
\(385\) −0.708204 −0.0360934
\(386\) 8.56231 0.435810
\(387\) 6.79837 0.345581
\(388\) 8.00000 0.406138
\(389\) −16.3820 −0.830599 −0.415299 0.909685i \(-0.636323\pi\)
−0.415299 + 0.909685i \(0.636323\pi\)
\(390\) 0.854102 0.0432491
\(391\) 30.8328 1.55928
\(392\) 3.56231 0.179924
\(393\) 5.83282 0.294227
\(394\) −16.4721 −0.829854
\(395\) 1.23607 0.0621933
\(396\) 2.85410 0.143424
\(397\) −26.9787 −1.35402 −0.677011 0.735973i \(-0.736725\pi\)
−0.677011 + 0.735973i \(0.736725\pi\)
\(398\) 2.47214 0.123917
\(399\) 0 0
\(400\) −4.85410 −0.242705
\(401\) 18.4721 0.922454 0.461227 0.887282i \(-0.347409\pi\)
0.461227 + 0.887282i \(0.347409\pi\)
\(402\) 4.23607 0.211276
\(403\) 0.854102 0.0425458
\(404\) 13.4164 0.667491
\(405\) −2.94427 −0.146302
\(406\) 15.0000 0.744438
\(407\) 1.23607 0.0612696
\(408\) 2.47214 0.122389
\(409\) 22.4508 1.11012 0.555062 0.831809i \(-0.312695\pi\)
0.555062 + 0.831809i \(0.312695\pi\)
\(410\) −0.416408 −0.0205649
\(411\) 0.124612 0.00614665
\(412\) −18.5623 −0.914499
\(413\) −9.16718 −0.451088
\(414\) 13.5967 0.668244
\(415\) −0.0344419 −0.00169068
\(416\) 5.85410 0.287021
\(417\) −8.90983 −0.436316
\(418\) 0 0
\(419\) −3.23607 −0.158092 −0.0790461 0.996871i \(-0.525187\pi\)
−0.0790461 + 0.996871i \(0.525187\pi\)
\(420\) −0.270510 −0.0131995
\(421\) 37.4164 1.82356 0.911782 0.410674i \(-0.134707\pi\)
0.911782 + 0.410674i \(0.134707\pi\)
\(422\) 3.41641 0.166308
\(423\) −26.3607 −1.28170
\(424\) 6.76393 0.328486
\(425\) −31.4164 −1.52392
\(426\) −2.72949 −0.132244
\(427\) −14.2918 −0.691628
\(428\) −7.23607 −0.349769
\(429\) −2.23607 −0.107958
\(430\) −0.909830 −0.0438759
\(431\) 13.4164 0.646246 0.323123 0.946357i \(-0.395267\pi\)
0.323123 + 0.946357i \(0.395267\pi\)
\(432\) 2.23607 0.107583
\(433\) 14.3607 0.690130 0.345065 0.938579i \(-0.387857\pi\)
0.345065 + 0.938579i \(0.387857\pi\)
\(434\) −0.270510 −0.0129849
\(435\) 1.18034 0.0565930
\(436\) −13.4164 −0.642529
\(437\) 0 0
\(438\) 3.52786 0.168568
\(439\) 11.4164 0.544875 0.272438 0.962173i \(-0.412170\pi\)
0.272438 + 0.962173i \(0.412170\pi\)
\(440\) −0.381966 −0.0182095
\(441\) 10.1672 0.484152
\(442\) 37.8885 1.80217
\(443\) −16.7639 −0.796478 −0.398239 0.917282i \(-0.630379\pi\)
−0.398239 + 0.917282i \(0.630379\pi\)
\(444\) 0.472136 0.0224066
\(445\) −6.36068 −0.301525
\(446\) 8.00000 0.378811
\(447\) 4.94427 0.233856
\(448\) −1.85410 −0.0875981
\(449\) −4.47214 −0.211053 −0.105527 0.994416i \(-0.533653\pi\)
−0.105527 + 0.994416i \(0.533653\pi\)
\(450\) −13.8541 −0.653089
\(451\) 1.09017 0.0513341
\(452\) 2.29180 0.107797
\(453\) 6.47214 0.304087
\(454\) −29.5967 −1.38904
\(455\) −4.14590 −0.194363
\(456\) 0 0
\(457\) −3.52786 −0.165027 −0.0825133 0.996590i \(-0.526295\pi\)
−0.0825133 + 0.996590i \(0.526295\pi\)
\(458\) 17.3820 0.812206
\(459\) 14.4721 0.675501
\(460\) −1.81966 −0.0848421
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0.708204 0.0329486
\(463\) −7.12461 −0.331109 −0.165554 0.986201i \(-0.552941\pi\)
−0.165554 + 0.986201i \(0.552941\pi\)
\(464\) 8.09017 0.375577
\(465\) −0.0212862 −0.000987126 0
\(466\) 3.05573 0.141554
\(467\) −14.1803 −0.656188 −0.328094 0.944645i \(-0.606406\pi\)
−0.328094 + 0.944645i \(0.606406\pi\)
\(468\) 16.7082 0.772337
\(469\) −20.5623 −0.949479
\(470\) 3.52786 0.162728
\(471\) 2.90983 0.134078
\(472\) −4.94427 −0.227579
\(473\) 2.38197 0.109523
\(474\) −1.23607 −0.0567745
\(475\) 0 0
\(476\) −12.0000 −0.550019
\(477\) 19.3050 0.883913
\(478\) −8.56231 −0.391631
\(479\) −27.3820 −1.25111 −0.625557 0.780178i \(-0.715128\pi\)
−0.625557 + 0.780178i \(0.715128\pi\)
\(480\) −0.145898 −0.00665930
\(481\) 7.23607 0.329936
\(482\) 10.8541 0.494391
\(483\) 3.37384 0.153515
\(484\) 1.00000 0.0454545
\(485\) −3.05573 −0.138753
\(486\) 9.65248 0.437845
\(487\) −10.5623 −0.478624 −0.239312 0.970943i \(-0.576922\pi\)
−0.239312 + 0.970943i \(0.576922\pi\)
\(488\) −7.70820 −0.348934
\(489\) 5.05573 0.228628
\(490\) −1.36068 −0.0614692
\(491\) 23.5623 1.06335 0.531676 0.846948i \(-0.321562\pi\)
0.531676 + 0.846948i \(0.321562\pi\)
\(492\) 0.416408 0.0187731
\(493\) 52.3607 2.35821
\(494\) 0 0
\(495\) −1.09017 −0.0489995
\(496\) −0.145898 −0.00655102
\(497\) 13.2492 0.594309
\(498\) 0.0344419 0.00154338
\(499\) −23.7082 −1.06132 −0.530662 0.847583i \(-0.678057\pi\)
−0.530662 + 0.847583i \(0.678057\pi\)
\(500\) 3.76393 0.168328
\(501\) −1.70820 −0.0763169
\(502\) −7.52786 −0.335985
\(503\) 24.9230 1.11126 0.555631 0.831429i \(-0.312477\pi\)
0.555631 + 0.831429i \(0.312477\pi\)
\(504\) −5.29180 −0.235715
\(505\) −5.12461 −0.228042
\(506\) 4.76393 0.211783
\(507\) −8.12461 −0.360827
\(508\) −21.7082 −0.963146
\(509\) 0.180340 0.00799342 0.00399671 0.999992i \(-0.498728\pi\)
0.00399671 + 0.999992i \(0.498728\pi\)
\(510\) −0.944272 −0.0418131
\(511\) −17.1246 −0.757548
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 21.7082 0.957508
\(515\) 7.09017 0.312430
\(516\) 0.909830 0.0400530
\(517\) −9.23607 −0.406202
\(518\) −2.29180 −0.100696
\(519\) 0.618034 0.0271287
\(520\) −2.23607 −0.0980581
\(521\) −38.0689 −1.66783 −0.833914 0.551894i \(-0.813905\pi\)
−0.833914 + 0.551894i \(0.813905\pi\)
\(522\) 23.0902 1.01063
\(523\) 9.34752 0.408739 0.204369 0.978894i \(-0.434486\pi\)
0.204369 + 0.978894i \(0.434486\pi\)
\(524\) −15.2705 −0.667095
\(525\) −3.43769 −0.150033
\(526\) 10.8541 0.473261
\(527\) −0.944272 −0.0411331
\(528\) 0.381966 0.0166229
\(529\) −0.304952 −0.0132588
\(530\) −2.58359 −0.112224
\(531\) −14.1115 −0.612385
\(532\) 0 0
\(533\) 6.38197 0.276434
\(534\) 6.36068 0.275254
\(535\) 2.76393 0.119495
\(536\) −11.0902 −0.479022
\(537\) 7.65248 0.330229
\(538\) −17.5967 −0.758650
\(539\) 3.56231 0.153439
\(540\) −0.854102 −0.0367547
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −2.43769 −0.104708
\(543\) 3.30495 0.141829
\(544\) −6.47214 −0.277491
\(545\) 5.12461 0.219514
\(546\) 4.14590 0.177428
\(547\) −4.18034 −0.178738 −0.0893692 0.995999i \(-0.528485\pi\)
−0.0893692 + 0.995999i \(0.528485\pi\)
\(548\) −0.326238 −0.0139362
\(549\) −22.0000 −0.938937
\(550\) −4.85410 −0.206980
\(551\) 0 0
\(552\) 1.81966 0.0774499
\(553\) 6.00000 0.255146
\(554\) 0.944272 0.0401183
\(555\) −0.180340 −0.00765500
\(556\) 23.3262 0.989253
\(557\) 14.9443 0.633209 0.316605 0.948558i \(-0.397457\pi\)
0.316605 + 0.948558i \(0.397457\pi\)
\(558\) −0.416408 −0.0176280
\(559\) 13.9443 0.589780
\(560\) 0.708204 0.0299271
\(561\) 2.47214 0.104374
\(562\) 30.2705 1.27688
\(563\) −33.2361 −1.40073 −0.700367 0.713783i \(-0.746980\pi\)
−0.700367 + 0.713783i \(0.746980\pi\)
\(564\) −3.52786 −0.148550
\(565\) −0.875388 −0.0368279
\(566\) −18.0344 −0.758044
\(567\) −14.2918 −0.600199
\(568\) 7.14590 0.299835
\(569\) −30.9787 −1.29870 −0.649348 0.760492i \(-0.724958\pi\)
−0.649348 + 0.760492i \(0.724958\pi\)
\(570\) 0 0
\(571\) −5.38197 −0.225228 −0.112614 0.993639i \(-0.535922\pi\)
−0.112614 + 0.993639i \(0.535922\pi\)
\(572\) 5.85410 0.244772
\(573\) −7.41641 −0.309825
\(574\) −2.02129 −0.0843669
\(575\) −23.1246 −0.964363
\(576\) −2.85410 −0.118921
\(577\) −10.7295 −0.446675 −0.223337 0.974741i \(-0.571695\pi\)
−0.223337 + 0.974741i \(0.571695\pi\)
\(578\) −24.8885 −1.03523
\(579\) 3.27051 0.135918
\(580\) −3.09017 −0.128312
\(581\) −0.167184 −0.00693597
\(582\) 3.05573 0.126664
\(583\) 6.76393 0.280133
\(584\) −9.23607 −0.382191
\(585\) −6.38197 −0.263862
\(586\) 8.03444 0.331900
\(587\) −42.6525 −1.76046 −0.880228 0.474551i \(-0.842610\pi\)
−0.880228 + 0.474551i \(0.842610\pi\)
\(588\) 1.36068 0.0561135
\(589\) 0 0
\(590\) 1.88854 0.0777501
\(591\) −6.29180 −0.258810
\(592\) −1.23607 −0.0508021
\(593\) 14.2918 0.586894 0.293447 0.955975i \(-0.405198\pi\)
0.293447 + 0.955975i \(0.405198\pi\)
\(594\) 2.23607 0.0917470
\(595\) 4.58359 0.187909
\(596\) −12.9443 −0.530218
\(597\) 0.944272 0.0386465
\(598\) 27.8885 1.14045
\(599\) 41.6180 1.70047 0.850233 0.526406i \(-0.176461\pi\)
0.850233 + 0.526406i \(0.176461\pi\)
\(600\) −1.85410 −0.0756934
\(601\) 6.03444 0.246150 0.123075 0.992397i \(-0.460724\pi\)
0.123075 + 0.992397i \(0.460724\pi\)
\(602\) −4.41641 −0.179999
\(603\) −31.6525 −1.28899
\(604\) −16.9443 −0.689453
\(605\) −0.381966 −0.0155291
\(606\) 5.12461 0.208173
\(607\) 35.3050 1.43298 0.716492 0.697595i \(-0.245747\pi\)
0.716492 + 0.697595i \(0.245747\pi\)
\(608\) 0 0
\(609\) 5.72949 0.232171
\(610\) 2.94427 0.119210
\(611\) −54.0689 −2.18739
\(612\) −18.4721 −0.746692
\(613\) −36.0000 −1.45403 −0.727013 0.686624i \(-0.759092\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) −25.2361 −1.01844
\(615\) −0.159054 −0.00641366
\(616\) −1.85410 −0.0747039
\(617\) −13.8541 −0.557745 −0.278873 0.960328i \(-0.589961\pi\)
−0.278873 + 0.960328i \(0.589961\pi\)
\(618\) −7.09017 −0.285208
\(619\) 25.7082 1.03330 0.516650 0.856197i \(-0.327179\pi\)
0.516650 + 0.856197i \(0.327179\pi\)
\(620\) 0.0557281 0.00223809
\(621\) 10.6525 0.427469
\(622\) −14.4721 −0.580280
\(623\) −30.8754 −1.23700
\(624\) 2.23607 0.0895144
\(625\) 22.8328 0.913313
\(626\) 18.5623 0.741899
\(627\) 0 0
\(628\) −7.61803 −0.303993
\(629\) −8.00000 −0.318981
\(630\) 2.02129 0.0805300
\(631\) −42.3607 −1.68635 −0.843176 0.537638i \(-0.819317\pi\)
−0.843176 + 0.537638i \(0.819317\pi\)
\(632\) 3.23607 0.128724
\(633\) 1.30495 0.0518672
\(634\) 24.1803 0.960324
\(635\) 8.29180 0.329050
\(636\) 2.58359 0.102446
\(637\) 20.8541 0.826270
\(638\) 8.09017 0.320293
\(639\) 20.3951 0.806819
\(640\) 0.381966 0.0150985
\(641\) 29.2361 1.15476 0.577378 0.816477i \(-0.304076\pi\)
0.577378 + 0.816477i \(0.304076\pi\)
\(642\) −2.76393 −0.109084
\(643\) −21.4164 −0.844581 −0.422290 0.906461i \(-0.638774\pi\)
−0.422290 + 0.906461i \(0.638774\pi\)
\(644\) −8.83282 −0.348062
\(645\) −0.347524 −0.0136838
\(646\) 0 0
\(647\) −2.29180 −0.0900998 −0.0450499 0.998985i \(-0.514345\pi\)
−0.0450499 + 0.998985i \(0.514345\pi\)
\(648\) −7.70820 −0.302807
\(649\) −4.94427 −0.194080
\(650\) −28.4164 −1.11458
\(651\) −0.103326 −0.00404965
\(652\) −13.2361 −0.518364
\(653\) −37.1591 −1.45415 −0.727073 0.686560i \(-0.759120\pi\)
−0.727073 + 0.686560i \(0.759120\pi\)
\(654\) −5.12461 −0.200388
\(655\) 5.83282 0.227907
\(656\) −1.09017 −0.0425640
\(657\) −26.3607 −1.02843
\(658\) 17.1246 0.667587
\(659\) 2.18034 0.0849340 0.0424670 0.999098i \(-0.486478\pi\)
0.0424670 + 0.999098i \(0.486478\pi\)
\(660\) −0.145898 −0.00567907
\(661\) 1.23607 0.0480775 0.0240387 0.999711i \(-0.492347\pi\)
0.0240387 + 0.999711i \(0.492347\pi\)
\(662\) 12.7426 0.495257
\(663\) 14.4721 0.562051
\(664\) −0.0901699 −0.00349927
\(665\) 0 0
\(666\) −3.52786 −0.136702
\(667\) 38.5410 1.49231
\(668\) 4.47214 0.173032
\(669\) 3.05573 0.118141
\(670\) 4.23607 0.163654
\(671\) −7.70820 −0.297572
\(672\) −0.708204 −0.0273196
\(673\) 26.9787 1.03995 0.519976 0.854181i \(-0.325941\pi\)
0.519976 + 0.854181i \(0.325941\pi\)
\(674\) −7.85410 −0.302529
\(675\) −10.8541 −0.417775
\(676\) 21.2705 0.818097
\(677\) −26.3820 −1.01394 −0.506971 0.861963i \(-0.669235\pi\)
−0.506971 + 0.861963i \(0.669235\pi\)
\(678\) 0.875388 0.0336191
\(679\) −14.8328 −0.569231
\(680\) 2.47214 0.0948021
\(681\) −11.3050 −0.433207
\(682\) −0.145898 −0.00558672
\(683\) −21.8885 −0.837542 −0.418771 0.908092i \(-0.637539\pi\)
−0.418771 + 0.908092i \(0.637539\pi\)
\(684\) 0 0
\(685\) 0.124612 0.00476117
\(686\) −19.5836 −0.747705
\(687\) 6.63932 0.253306
\(688\) −2.38197 −0.0908116
\(689\) 39.5967 1.50852
\(690\) −0.695048 −0.0264600
\(691\) 8.29180 0.315435 0.157717 0.987484i \(-0.449586\pi\)
0.157717 + 0.987484i \(0.449586\pi\)
\(692\) −1.61803 −0.0615084
\(693\) −5.29180 −0.201019
\(694\) 13.5279 0.513511
\(695\) −8.90983 −0.337969
\(696\) 3.09017 0.117133
\(697\) −7.05573 −0.267255
\(698\) 16.7639 0.634524
\(699\) 1.16718 0.0441470
\(700\) 9.00000 0.340168
\(701\) −32.6525 −1.23327 −0.616633 0.787250i \(-0.711504\pi\)
−0.616633 + 0.787250i \(0.711504\pi\)
\(702\) 13.0902 0.494057
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 1.34752 0.0507507
\(706\) 20.4721 0.770479
\(707\) −24.8754 −0.935535
\(708\) −1.88854 −0.0709758
\(709\) 24.1459 0.906818 0.453409 0.891303i \(-0.350208\pi\)
0.453409 + 0.891303i \(0.350208\pi\)
\(710\) −2.72949 −0.102436
\(711\) 9.23607 0.346380
\(712\) −16.6525 −0.624078
\(713\) −0.695048 −0.0260298
\(714\) −4.58359 −0.171537
\(715\) −2.23607 −0.0836242
\(716\) −20.0344 −0.748722
\(717\) −3.27051 −0.122139
\(718\) −16.0344 −0.598400
\(719\) 8.47214 0.315957 0.157979 0.987443i \(-0.449502\pi\)
0.157979 + 0.987443i \(0.449502\pi\)
\(720\) 1.09017 0.0406282
\(721\) 34.4164 1.28173
\(722\) 0 0
\(723\) 4.14590 0.154188
\(724\) −8.65248 −0.321567
\(725\) −39.2705 −1.45847
\(726\) 0.381966 0.0141761
\(727\) −3.63932 −0.134975 −0.0674875 0.997720i \(-0.521498\pi\)
−0.0674875 + 0.997720i \(0.521498\pi\)
\(728\) −10.8541 −0.402280
\(729\) −19.4377 −0.719915
\(730\) 3.52786 0.130572
\(731\) −15.4164 −0.570196
\(732\) −2.94427 −0.108823
\(733\) −0.472136 −0.0174387 −0.00871937 0.999962i \(-0.502775\pi\)
−0.00871937 + 0.999962i \(0.502775\pi\)
\(734\) −29.1246 −1.07501
\(735\) −0.519733 −0.0191707
\(736\) −4.76393 −0.175601
\(737\) −11.0902 −0.408512
\(738\) −3.11146 −0.114534
\(739\) 32.6312 1.20036 0.600179 0.799866i \(-0.295096\pi\)
0.600179 + 0.799866i \(0.295096\pi\)
\(740\) 0.472136 0.0173561
\(741\) 0 0
\(742\) −12.5410 −0.460395
\(743\) 18.1115 0.664445 0.332222 0.943201i \(-0.392202\pi\)
0.332222 + 0.943201i \(0.392202\pi\)
\(744\) −0.0557281 −0.00204309
\(745\) 4.94427 0.181144
\(746\) −5.79837 −0.212294
\(747\) −0.257354 −0.00941610
\(748\) −6.47214 −0.236645
\(749\) 13.4164 0.490225
\(750\) 1.43769 0.0524972
\(751\) −38.2492 −1.39573 −0.697867 0.716227i \(-0.745867\pi\)
−0.697867 + 0.716227i \(0.745867\pi\)
\(752\) 9.23607 0.336805
\(753\) −2.87539 −0.104785
\(754\) 47.3607 1.72477
\(755\) 6.47214 0.235545
\(756\) −4.14590 −0.150785
\(757\) 24.5623 0.892732 0.446366 0.894850i \(-0.352718\pi\)
0.446366 + 0.894850i \(0.352718\pi\)
\(758\) 34.5066 1.25334
\(759\) 1.81966 0.0660495
\(760\) 0 0
\(761\) 32.7639 1.18769 0.593846 0.804579i \(-0.297609\pi\)
0.593846 + 0.804579i \(0.297609\pi\)
\(762\) −8.29180 −0.300380
\(763\) 24.8754 0.900550
\(764\) 19.4164 0.702461
\(765\) 7.05573 0.255100
\(766\) −0.798374 −0.0288464
\(767\) −28.9443 −1.04512
\(768\) −0.381966 −0.0137830
\(769\) −20.2918 −0.731741 −0.365870 0.930666i \(-0.619229\pi\)
−0.365870 + 0.930666i \(0.619229\pi\)
\(770\) 0.708204 0.0255219
\(771\) 8.29180 0.298622
\(772\) −8.56231 −0.308164
\(773\) −18.6525 −0.670883 −0.335441 0.942061i \(-0.608885\pi\)
−0.335441 + 0.942061i \(0.608885\pi\)
\(774\) −6.79837 −0.244363
\(775\) 0.708204 0.0254394
\(776\) −8.00000 −0.287183
\(777\) −0.875388 −0.0314044
\(778\) 16.3820 0.587322
\(779\) 0 0
\(780\) −0.854102 −0.0305818
\(781\) 7.14590 0.255700
\(782\) −30.8328 −1.10258
\(783\) 18.0902 0.646490
\(784\) −3.56231 −0.127225
\(785\) 2.90983 0.103856
\(786\) −5.83282 −0.208050
\(787\) −16.1115 −0.574311 −0.287156 0.957884i \(-0.592710\pi\)
−0.287156 + 0.957884i \(0.592710\pi\)
\(788\) 16.4721 0.586796
\(789\) 4.14590 0.147598
\(790\) −1.23607 −0.0439773
\(791\) −4.24922 −0.151085
\(792\) −2.85410 −0.101416
\(793\) −45.1246 −1.60242
\(794\) 26.9787 0.957439
\(795\) −0.986844 −0.0349998
\(796\) −2.47214 −0.0876225
\(797\) −15.0557 −0.533301 −0.266651 0.963793i \(-0.585917\pi\)
−0.266651 + 0.963793i \(0.585917\pi\)
\(798\) 0 0
\(799\) 59.7771 2.11476
\(800\) 4.85410 0.171618
\(801\) −47.5279 −1.67931
\(802\) −18.4721 −0.652274
\(803\) −9.23607 −0.325934
\(804\) −4.23607 −0.149395
\(805\) 3.37384 0.118912
\(806\) −0.854102 −0.0300845
\(807\) −6.72136 −0.236603
\(808\) −13.4164 −0.471988
\(809\) −8.06888 −0.283687 −0.141843 0.989889i \(-0.545303\pi\)
−0.141843 + 0.989889i \(0.545303\pi\)
\(810\) 2.94427 0.103451
\(811\) 25.7771 0.905156 0.452578 0.891725i \(-0.350504\pi\)
0.452578 + 0.891725i \(0.350504\pi\)
\(812\) −15.0000 −0.526397
\(813\) −0.931116 −0.0326557
\(814\) −1.23607 −0.0433242
\(815\) 5.05573 0.177094
\(816\) −2.47214 −0.0865421
\(817\) 0 0
\(818\) −22.4508 −0.784976
\(819\) −30.9787 −1.08248
\(820\) 0.416408 0.0145416
\(821\) 38.0689 1.32861 0.664307 0.747460i \(-0.268727\pi\)
0.664307 + 0.747460i \(0.268727\pi\)
\(822\) −0.124612 −0.00434634
\(823\) −25.5967 −0.892247 −0.446123 0.894972i \(-0.647196\pi\)
−0.446123 + 0.894972i \(0.647196\pi\)
\(824\) 18.5623 0.646649
\(825\) −1.85410 −0.0645515
\(826\) 9.16718 0.318967
\(827\) −36.5410 −1.27066 −0.635328 0.772243i \(-0.719135\pi\)
−0.635328 + 0.772243i \(0.719135\pi\)
\(828\) −13.5967 −0.472520
\(829\) −21.4164 −0.743823 −0.371911 0.928268i \(-0.621297\pi\)
−0.371911 + 0.928268i \(0.621297\pi\)
\(830\) 0.0344419 0.00119549
\(831\) 0.360680 0.0125118
\(832\) −5.85410 −0.202954
\(833\) −23.0557 −0.798834
\(834\) 8.90983 0.308522
\(835\) −1.70820 −0.0591148
\(836\) 0 0
\(837\) −0.326238 −0.0112764
\(838\) 3.23607 0.111788
\(839\) 34.3820 1.18700 0.593499 0.804835i \(-0.297746\pi\)
0.593499 + 0.804835i \(0.297746\pi\)
\(840\) 0.270510 0.00933348
\(841\) 36.4508 1.25693
\(842\) −37.4164 −1.28945
\(843\) 11.5623 0.398227
\(844\) −3.41641 −0.117598
\(845\) −8.12461 −0.279495
\(846\) 26.3607 0.906299
\(847\) −1.85410 −0.0637077
\(848\) −6.76393 −0.232274
\(849\) −6.88854 −0.236414
\(850\) 31.4164 1.07757
\(851\) −5.88854 −0.201857
\(852\) 2.72949 0.0935108
\(853\) −35.7082 −1.22263 −0.611313 0.791389i \(-0.709358\pi\)
−0.611313 + 0.791389i \(0.709358\pi\)
\(854\) 14.2918 0.489055
\(855\) 0 0
\(856\) 7.23607 0.247324
\(857\) −53.3820 −1.82349 −0.911747 0.410753i \(-0.865266\pi\)
−0.911747 + 0.410753i \(0.865266\pi\)
\(858\) 2.23607 0.0763381
\(859\) 37.0132 1.26287 0.631436 0.775428i \(-0.282466\pi\)
0.631436 + 0.775428i \(0.282466\pi\)
\(860\) 0.909830 0.0310249
\(861\) −0.772063 −0.0263118
\(862\) −13.4164 −0.456965
\(863\) −27.2016 −0.925954 −0.462977 0.886370i \(-0.653219\pi\)
−0.462977 + 0.886370i \(0.653219\pi\)
\(864\) −2.23607 −0.0760726
\(865\) 0.618034 0.0210138
\(866\) −14.3607 −0.487996
\(867\) −9.50658 −0.322860
\(868\) 0.270510 0.00918170
\(869\) 3.23607 0.109776
\(870\) −1.18034 −0.0400173
\(871\) −64.9230 −2.19983
\(872\) 13.4164 0.454337
\(873\) −22.8328 −0.772774
\(874\) 0 0
\(875\) −6.97871 −0.235924
\(876\) −3.52786 −0.119195
\(877\) 23.3262 0.787671 0.393836 0.919181i \(-0.371148\pi\)
0.393836 + 0.919181i \(0.371148\pi\)
\(878\) −11.4164 −0.385285
\(879\) 3.06888 0.103511
\(880\) 0.381966 0.0128761
\(881\) 3.67376 0.123772 0.0618861 0.998083i \(-0.480288\pi\)
0.0618861 + 0.998083i \(0.480288\pi\)
\(882\) −10.1672 −0.342347
\(883\) −49.1246 −1.65317 −0.826587 0.562808i \(-0.809721\pi\)
−0.826587 + 0.562808i \(0.809721\pi\)
\(884\) −37.8885 −1.27433
\(885\) 0.721360 0.0242482
\(886\) 16.7639 0.563195
\(887\) −20.0689 −0.673847 −0.336924 0.941532i \(-0.609386\pi\)
−0.336924 + 0.941532i \(0.609386\pi\)
\(888\) −0.472136 −0.0158438
\(889\) 40.2492 1.34992
\(890\) 6.36068 0.213210
\(891\) −7.70820 −0.258235
\(892\) −8.00000 −0.267860
\(893\) 0 0
\(894\) −4.94427 −0.165361
\(895\) 7.65248 0.255794
\(896\) 1.85410 0.0619412
\(897\) 10.6525 0.355676
\(898\) 4.47214 0.149237
\(899\) −1.18034 −0.0393665
\(900\) 13.8541 0.461803
\(901\) −43.7771 −1.45843
\(902\) −1.09017 −0.0362987
\(903\) −1.68692 −0.0561371
\(904\) −2.29180 −0.0762240
\(905\) 3.30495 0.109860
\(906\) −6.47214 −0.215022
\(907\) 47.4164 1.57444 0.787218 0.616675i \(-0.211521\pi\)
0.787218 + 0.616675i \(0.211521\pi\)
\(908\) 29.5967 0.982203
\(909\) −38.2918 −1.27006
\(910\) 4.14590 0.137435
\(911\) 31.4164 1.04087 0.520436 0.853901i \(-0.325769\pi\)
0.520436 + 0.853901i \(0.325769\pi\)
\(912\) 0 0
\(913\) −0.0901699 −0.00298419
\(914\) 3.52786 0.116691
\(915\) 1.12461 0.0371785
\(916\) −17.3820 −0.574316
\(917\) 28.3131 0.934980
\(918\) −14.4721 −0.477652
\(919\) −17.3262 −0.571540 −0.285770 0.958298i \(-0.592249\pi\)
−0.285770 + 0.958298i \(0.592249\pi\)
\(920\) 1.81966 0.0599924
\(921\) −9.63932 −0.317626
\(922\) 18.0000 0.592798
\(923\) 41.8328 1.37694
\(924\) −0.708204 −0.0232982
\(925\) 6.00000 0.197279
\(926\) 7.12461 0.234129
\(927\) 52.9787 1.74005
\(928\) −8.09017 −0.265573
\(929\) −20.6180 −0.676456 −0.338228 0.941064i \(-0.609827\pi\)
−0.338228 + 0.941064i \(0.609827\pi\)
\(930\) 0.0212862 0.000698003 0
\(931\) 0 0
\(932\) −3.05573 −0.100094
\(933\) −5.52786 −0.180974
\(934\) 14.1803 0.463995
\(935\) 2.47214 0.0808475
\(936\) −16.7082 −0.546125
\(937\) −10.7639 −0.351642 −0.175821 0.984422i \(-0.556258\pi\)
−0.175821 + 0.984422i \(0.556258\pi\)
\(938\) 20.5623 0.671383
\(939\) 7.09017 0.231379
\(940\) −3.52786 −0.115066
\(941\) 21.0557 0.686397 0.343199 0.939263i \(-0.388490\pi\)
0.343199 + 0.939263i \(0.388490\pi\)
\(942\) −2.90983 −0.0948074
\(943\) −5.19350 −0.169123
\(944\) 4.94427 0.160922
\(945\) 1.58359 0.0515143
\(946\) −2.38197 −0.0774444
\(947\) −13.3050 −0.432353 −0.216176 0.976354i \(-0.569359\pi\)
−0.216176 + 0.976354i \(0.569359\pi\)
\(948\) 1.23607 0.0401456
\(949\) −54.0689 −1.75515
\(950\) 0 0
\(951\) 9.23607 0.299500
\(952\) 12.0000 0.388922
\(953\) −29.0557 −0.941207 −0.470604 0.882345i \(-0.655964\pi\)
−0.470604 + 0.882345i \(0.655964\pi\)
\(954\) −19.3050 −0.625021
\(955\) −7.41641 −0.239989
\(956\) 8.56231 0.276925
\(957\) 3.09017 0.0998910
\(958\) 27.3820 0.884671
\(959\) 0.604878 0.0195325
\(960\) 0.145898 0.00470884
\(961\) −30.9787 −0.999313
\(962\) −7.23607 −0.233300
\(963\) 20.6525 0.665517
\(964\) −10.8541 −0.349587
\(965\) 3.27051 0.105281
\(966\) −3.37384 −0.108551
\(967\) 58.8328 1.89194 0.945968 0.324260i \(-0.105115\pi\)
0.945968 + 0.324260i \(0.105115\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 3.05573 0.0981135
\(971\) −14.6738 −0.470903 −0.235452 0.971886i \(-0.575657\pi\)
−0.235452 + 0.971886i \(0.575657\pi\)
\(972\) −9.65248 −0.309603
\(973\) −43.2492 −1.38651
\(974\) 10.5623 0.338438
\(975\) −10.8541 −0.347609
\(976\) 7.70820 0.246734
\(977\) 8.83282 0.282587 0.141293 0.989968i \(-0.454874\pi\)
0.141293 + 0.989968i \(0.454874\pi\)
\(978\) −5.05573 −0.161664
\(979\) −16.6525 −0.532215
\(980\) 1.36068 0.0434653
\(981\) 38.2918 1.22256
\(982\) −23.5623 −0.751903
\(983\) 26.0344 0.830370 0.415185 0.909737i \(-0.363717\pi\)
0.415185 + 0.909737i \(0.363717\pi\)
\(984\) −0.416408 −0.0132746
\(985\) −6.29180 −0.200473
\(986\) −52.3607 −1.66750
\(987\) 6.54102 0.208203
\(988\) 0 0
\(989\) −11.3475 −0.360830
\(990\) 1.09017 0.0346479
\(991\) −10.3262 −0.328024 −0.164012 0.986458i \(-0.552444\pi\)
−0.164012 + 0.986458i \(0.552444\pi\)
\(992\) 0.145898 0.00463227
\(993\) 4.86726 0.154458
\(994\) −13.2492 −0.420240
\(995\) 0.944272 0.0299354
\(996\) −0.0344419 −0.00109133
\(997\) 41.4853 1.31385 0.656926 0.753955i \(-0.271856\pi\)
0.656926 + 0.753955i \(0.271856\pi\)
\(998\) 23.7082 0.750470
\(999\) −2.76393 −0.0874469
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 7942.2.a.t.1.2 2
19.18 odd 2 7942.2.a.ba.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7942.2.a.t.1.2 2 1.1 even 1 trivial
7942.2.a.ba.1.1 yes 2 19.18 odd 2