Properties

Label 7942.2.a.z.1.2
Level $7942$
Weight $2$
Character 7942.1
Self dual yes
Analytic conductor $63.417$
Analytic rank $0$
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: \(0\)
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 \(1.61803\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.85410 q^{5} +1.61803 q^{6} +3.23607 q^{7} +1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.61803 q^{3} +1.00000 q^{4} -2.85410 q^{5} +1.61803 q^{6} +3.23607 q^{7} +1.00000 q^{8} -0.381966 q^{9} -2.85410 q^{10} +1.00000 q^{11} +1.61803 q^{12} +5.23607 q^{13} +3.23607 q^{14} -4.61803 q^{15} +1.00000 q^{16} +6.47214 q^{17} -0.381966 q^{18} -2.85410 q^{20} +5.23607 q^{21} +1.00000 q^{22} -2.85410 q^{23} +1.61803 q^{24} +3.14590 q^{25} +5.23607 q^{26} -5.47214 q^{27} +3.23607 q^{28} +10.4721 q^{29} -4.61803 q^{30} -6.09017 q^{31} +1.00000 q^{32} +1.61803 q^{33} +6.47214 q^{34} -9.23607 q^{35} -0.381966 q^{36} +5.38197 q^{37} +8.47214 q^{39} -2.85410 q^{40} -10.9443 q^{41} +5.23607 q^{42} -3.23607 q^{43} +1.00000 q^{44} +1.09017 q^{45} -2.85410 q^{46} +3.09017 q^{47} +1.61803 q^{48} +3.47214 q^{49} +3.14590 q^{50} +10.4721 q^{51} +5.23607 q^{52} -1.14590 q^{53} -5.47214 q^{54} -2.85410 q^{55} +3.23607 q^{56} +10.4721 q^{58} +7.85410 q^{59} -4.61803 q^{60} +6.94427 q^{61} -6.09017 q^{62} -1.23607 q^{63} +1.00000 q^{64} -14.9443 q^{65} +1.61803 q^{66} -4.14590 q^{67} +6.47214 q^{68} -4.61803 q^{69} -9.23607 q^{70} +1.52786 q^{71} -0.381966 q^{72} -5.70820 q^{73} +5.38197 q^{74} +5.09017 q^{75} +3.23607 q^{77} +8.47214 q^{78} +16.4721 q^{79} -2.85410 q^{80} -7.70820 q^{81} -10.9443 q^{82} +6.00000 q^{83} +5.23607 q^{84} -18.4721 q^{85} -3.23607 q^{86} +16.9443 q^{87} +1.00000 q^{88} -1.85410 q^{89} +1.09017 q^{90} +16.9443 q^{91} -2.85410 q^{92} -9.85410 q^{93} +3.09017 q^{94} +1.61803 q^{96} -16.4721 q^{97} +3.47214 q^{98} -0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{5} + q^{6} + 2 q^{7} + 2 q^{8} - 3 q^{9} + q^{10} + 2 q^{11} + q^{12} + 6 q^{13} + 2 q^{14} - 7 q^{15} + 2 q^{16} + 4 q^{17} - 3 q^{18} + q^{20} + 6 q^{21} + 2 q^{22} + q^{23} + q^{24} + 13 q^{25} + 6 q^{26} - 2 q^{27} + 2 q^{28} + 12 q^{29} - 7 q^{30} - q^{31} + 2 q^{32} + q^{33} + 4 q^{34} - 14 q^{35} - 3 q^{36} + 13 q^{37} + 8 q^{39} + q^{40} - 4 q^{41} + 6 q^{42} - 2 q^{43} + 2 q^{44} - 9 q^{45} + q^{46} - 5 q^{47} + q^{48} - 2 q^{49} + 13 q^{50} + 12 q^{51} + 6 q^{52} - 9 q^{53} - 2 q^{54} + q^{55} + 2 q^{56} + 12 q^{58} + 9 q^{59} - 7 q^{60} - 4 q^{61} - q^{62} + 2 q^{63} + 2 q^{64} - 12 q^{65} + q^{66} - 15 q^{67} + 4 q^{68} - 7 q^{69} - 14 q^{70} + 12 q^{71} - 3 q^{72} + 2 q^{73} + 13 q^{74} - q^{75} + 2 q^{77} + 8 q^{78} + 24 q^{79} + q^{80} - 2 q^{81} - 4 q^{82} + 12 q^{83} + 6 q^{84} - 28 q^{85} - 2 q^{86} + 16 q^{87} + 2 q^{88} + 3 q^{89} - 9 q^{90} + 16 q^{91} + q^{92} - 13 q^{93} - 5 q^{94} + q^{96} - 24 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100}) \) 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
\(3\) 1.61803 0.934172 0.467086 0.884212i \(-0.345304\pi\)
0.467086 + 0.884212i \(0.345304\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.85410 −1.27639 −0.638197 0.769873i \(-0.720319\pi\)
−0.638197 + 0.769873i \(0.720319\pi\)
\(6\) 1.61803 0.660560
\(7\) 3.23607 1.22312 0.611559 0.791199i \(-0.290543\pi\)
0.611559 + 0.791199i \(0.290543\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.381966 −0.127322
\(10\) −2.85410 −0.902546
\(11\) 1.00000 0.301511
\(12\) 1.61803 0.467086
\(13\) 5.23607 1.45222 0.726112 0.687576i \(-0.241325\pi\)
0.726112 + 0.687576i \(0.241325\pi\)
\(14\) 3.23607 0.864876
\(15\) −4.61803 −1.19237
\(16\) 1.00000 0.250000
\(17\) 6.47214 1.56972 0.784862 0.619671i \(-0.212734\pi\)
0.784862 + 0.619671i \(0.212734\pi\)
\(18\) −0.381966 −0.0900303
\(19\) 0 0
\(20\) −2.85410 −0.638197
\(21\) 5.23607 1.14260
\(22\) 1.00000 0.213201
\(23\) −2.85410 −0.595121 −0.297561 0.954703i \(-0.596173\pi\)
−0.297561 + 0.954703i \(0.596173\pi\)
\(24\) 1.61803 0.330280
\(25\) 3.14590 0.629180
\(26\) 5.23607 1.02688
\(27\) −5.47214 −1.05311
\(28\) 3.23607 0.611559
\(29\) 10.4721 1.94463 0.972313 0.233681i \(-0.0750770\pi\)
0.972313 + 0.233681i \(0.0750770\pi\)
\(30\) −4.61803 −0.843134
\(31\) −6.09017 −1.09383 −0.546913 0.837189i \(-0.684197\pi\)
−0.546913 + 0.837189i \(0.684197\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.61803 0.281664
\(34\) 6.47214 1.10996
\(35\) −9.23607 −1.56118
\(36\) −0.381966 −0.0636610
\(37\) 5.38197 0.884790 0.442395 0.896820i \(-0.354129\pi\)
0.442395 + 0.896820i \(0.354129\pi\)
\(38\) 0 0
\(39\) 8.47214 1.35663
\(40\) −2.85410 −0.451273
\(41\) −10.9443 −1.70921 −0.854604 0.519280i \(-0.826200\pi\)
−0.854604 + 0.519280i \(0.826200\pi\)
\(42\) 5.23607 0.807943
\(43\) −3.23607 −0.493496 −0.246748 0.969080i \(-0.579362\pi\)
−0.246748 + 0.969080i \(0.579362\pi\)
\(44\) 1.00000 0.150756
\(45\) 1.09017 0.162513
\(46\) −2.85410 −0.420814
\(47\) 3.09017 0.450748 0.225374 0.974272i \(-0.427640\pi\)
0.225374 + 0.974272i \(0.427640\pi\)
\(48\) 1.61803 0.233543
\(49\) 3.47214 0.496019
\(50\) 3.14590 0.444897
\(51\) 10.4721 1.46639
\(52\) 5.23607 0.726112
\(53\) −1.14590 −0.157401 −0.0787006 0.996898i \(-0.525077\pi\)
−0.0787006 + 0.996898i \(0.525077\pi\)
\(54\) −5.47214 −0.744663
\(55\) −2.85410 −0.384847
\(56\) 3.23607 0.432438
\(57\) 0 0
\(58\) 10.4721 1.37506
\(59\) 7.85410 1.02252 0.511258 0.859427i \(-0.329179\pi\)
0.511258 + 0.859427i \(0.329179\pi\)
\(60\) −4.61803 −0.596186
\(61\) 6.94427 0.889123 0.444561 0.895748i \(-0.353360\pi\)
0.444561 + 0.895748i \(0.353360\pi\)
\(62\) −6.09017 −0.773452
\(63\) −1.23607 −0.155730
\(64\) 1.00000 0.125000
\(65\) −14.9443 −1.85361
\(66\) 1.61803 0.199166
\(67\) −4.14590 −0.506502 −0.253251 0.967401i \(-0.581500\pi\)
−0.253251 + 0.967401i \(0.581500\pi\)
\(68\) 6.47214 0.784862
\(69\) −4.61803 −0.555946
\(70\) −9.23607 −1.10392
\(71\) 1.52786 0.181324 0.0906621 0.995882i \(-0.471102\pi\)
0.0906621 + 0.995882i \(0.471102\pi\)
\(72\) −0.381966 −0.0450151
\(73\) −5.70820 −0.668095 −0.334047 0.942556i \(-0.608415\pi\)
−0.334047 + 0.942556i \(0.608415\pi\)
\(74\) 5.38197 0.625641
\(75\) 5.09017 0.587762
\(76\) 0 0
\(77\) 3.23607 0.368784
\(78\) 8.47214 0.959280
\(79\) 16.4721 1.85326 0.926630 0.375974i \(-0.122692\pi\)
0.926630 + 0.375974i \(0.122692\pi\)
\(80\) −2.85410 −0.319098
\(81\) −7.70820 −0.856467
\(82\) −10.9443 −1.20859
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 5.23607 0.571302
\(85\) −18.4721 −2.00358
\(86\) −3.23607 −0.348954
\(87\) 16.9443 1.81662
\(88\) 1.00000 0.106600
\(89\) −1.85410 −0.196534 −0.0982672 0.995160i \(-0.531330\pi\)
−0.0982672 + 0.995160i \(0.531330\pi\)
\(90\) 1.09017 0.114914
\(91\) 16.9443 1.77624
\(92\) −2.85410 −0.297561
\(93\) −9.85410 −1.02182
\(94\) 3.09017 0.318727
\(95\) 0 0
\(96\) 1.61803 0.165140
\(97\) −16.4721 −1.67249 −0.836246 0.548354i \(-0.815254\pi\)
−0.836246 + 0.548354i \(0.815254\pi\)
\(98\) 3.47214 0.350739
\(99\) −0.381966 −0.0383890
\(100\) 3.14590 0.314590
\(101\) 10.7639 1.07105 0.535526 0.844519i \(-0.320114\pi\)
0.535526 + 0.844519i \(0.320114\pi\)
\(102\) 10.4721 1.03690
\(103\) −0.944272 −0.0930419 −0.0465209 0.998917i \(-0.514813\pi\)
−0.0465209 + 0.998917i \(0.514813\pi\)
\(104\) 5.23607 0.513439
\(105\) −14.9443 −1.45841
\(106\) −1.14590 −0.111299
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) −5.47214 −0.526557
\(109\) −10.1803 −0.975100 −0.487550 0.873095i \(-0.662109\pi\)
−0.487550 + 0.873095i \(0.662109\pi\)
\(110\) −2.85410 −0.272128
\(111\) 8.70820 0.826546
\(112\) 3.23607 0.305780
\(113\) 14.7984 1.39211 0.696057 0.717987i \(-0.254936\pi\)
0.696057 + 0.717987i \(0.254936\pi\)
\(114\) 0 0
\(115\) 8.14590 0.759609
\(116\) 10.4721 0.972313
\(117\) −2.00000 −0.184900
\(118\) 7.85410 0.723029
\(119\) 20.9443 1.91996
\(120\) −4.61803 −0.421567
\(121\) 1.00000 0.0909091
\(122\) 6.94427 0.628705
\(123\) −17.7082 −1.59669
\(124\) −6.09017 −0.546913
\(125\) 5.29180 0.473313
\(126\) −1.23607 −0.110118
\(127\) −9.41641 −0.835571 −0.417786 0.908546i \(-0.637194\pi\)
−0.417786 + 0.908546i \(0.637194\pi\)
\(128\) 1.00000 0.0883883
\(129\) −5.23607 −0.461010
\(130\) −14.9443 −1.31070
\(131\) −17.4164 −1.52168 −0.760839 0.648940i \(-0.775212\pi\)
−0.760839 + 0.648940i \(0.775212\pi\)
\(132\) 1.61803 0.140832
\(133\) 0 0
\(134\) −4.14590 −0.358151
\(135\) 15.6180 1.34419
\(136\) 6.47214 0.554981
\(137\) 13.8541 1.18364 0.591818 0.806072i \(-0.298410\pi\)
0.591818 + 0.806072i \(0.298410\pi\)
\(138\) −4.61803 −0.393113
\(139\) 13.4164 1.13796 0.568982 0.822350i \(-0.307337\pi\)
0.568982 + 0.822350i \(0.307337\pi\)
\(140\) −9.23607 −0.780590
\(141\) 5.00000 0.421076
\(142\) 1.52786 0.128216
\(143\) 5.23607 0.437862
\(144\) −0.381966 −0.0318305
\(145\) −29.8885 −2.48211
\(146\) −5.70820 −0.472414
\(147\) 5.61803 0.463368
\(148\) 5.38197 0.442395
\(149\) −18.7639 −1.53720 −0.768601 0.639729i \(-0.779047\pi\)
−0.768601 + 0.639729i \(0.779047\pi\)
\(150\) 5.09017 0.415611
\(151\) −16.4721 −1.34048 −0.670242 0.742143i \(-0.733810\pi\)
−0.670242 + 0.742143i \(0.733810\pi\)
\(152\) 0 0
\(153\) −2.47214 −0.199860
\(154\) 3.23607 0.260770
\(155\) 17.3820 1.39615
\(156\) 8.47214 0.678314
\(157\) 15.5623 1.24201 0.621004 0.783808i \(-0.286725\pi\)
0.621004 + 0.783808i \(0.286725\pi\)
\(158\) 16.4721 1.31045
\(159\) −1.85410 −0.147040
\(160\) −2.85410 −0.225637
\(161\) −9.23607 −0.727904
\(162\) −7.70820 −0.605614
\(163\) 3.14590 0.246406 0.123203 0.992382i \(-0.460683\pi\)
0.123203 + 0.992382i \(0.460683\pi\)
\(164\) −10.9443 −0.854604
\(165\) −4.61803 −0.359513
\(166\) 6.00000 0.465690
\(167\) 25.2361 1.95283 0.976413 0.215912i \(-0.0692725\pi\)
0.976413 + 0.215912i \(0.0692725\pi\)
\(168\) 5.23607 0.403971
\(169\) 14.4164 1.10895
\(170\) −18.4721 −1.41675
\(171\) 0 0
\(172\) −3.23607 −0.246748
\(173\) 15.2361 1.15838 0.579188 0.815194i \(-0.303370\pi\)
0.579188 + 0.815194i \(0.303370\pi\)
\(174\) 16.9443 1.28454
\(175\) 10.1803 0.769561
\(176\) 1.00000 0.0753778
\(177\) 12.7082 0.955207
\(178\) −1.85410 −0.138971
\(179\) −12.0902 −0.903662 −0.451831 0.892104i \(-0.649229\pi\)
−0.451831 + 0.892104i \(0.649229\pi\)
\(180\) 1.09017 0.0812565
\(181\) −1.67376 −0.124410 −0.0622049 0.998063i \(-0.519813\pi\)
−0.0622049 + 0.998063i \(0.519813\pi\)
\(182\) 16.9443 1.25599
\(183\) 11.2361 0.830594
\(184\) −2.85410 −0.210407
\(185\) −15.3607 −1.12934
\(186\) −9.85410 −0.722538
\(187\) 6.47214 0.473289
\(188\) 3.09017 0.225374
\(189\) −17.7082 −1.28808
\(190\) 0 0
\(191\) −3.38197 −0.244710 −0.122355 0.992486i \(-0.539045\pi\)
−0.122355 + 0.992486i \(0.539045\pi\)
\(192\) 1.61803 0.116772
\(193\) −23.8885 −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(194\) −16.4721 −1.18263
\(195\) −24.1803 −1.73159
\(196\) 3.47214 0.248010
\(197\) −13.5279 −0.963820 −0.481910 0.876221i \(-0.660057\pi\)
−0.481910 + 0.876221i \(0.660057\pi\)
\(198\) −0.381966 −0.0271451
\(199\) 6.67376 0.473090 0.236545 0.971620i \(-0.423985\pi\)
0.236545 + 0.971620i \(0.423985\pi\)
\(200\) 3.14590 0.222449
\(201\) −6.70820 −0.473160
\(202\) 10.7639 0.757348
\(203\) 33.8885 2.37851
\(204\) 10.4721 0.733196
\(205\) 31.2361 2.18162
\(206\) −0.944272 −0.0657905
\(207\) 1.09017 0.0757720
\(208\) 5.23607 0.363056
\(209\) 0 0
\(210\) −14.9443 −1.03125
\(211\) −6.94427 −0.478063 −0.239032 0.971012i \(-0.576830\pi\)
−0.239032 + 0.971012i \(0.576830\pi\)
\(212\) −1.14590 −0.0787006
\(213\) 2.47214 0.169388
\(214\) 6.00000 0.410152
\(215\) 9.23607 0.629895
\(216\) −5.47214 −0.372332
\(217\) −19.7082 −1.33788
\(218\) −10.1803 −0.689500
\(219\) −9.23607 −0.624116
\(220\) −2.85410 −0.192424
\(221\) 33.8885 2.27959
\(222\) 8.70820 0.584456
\(223\) 9.56231 0.640339 0.320170 0.947360i \(-0.396260\pi\)
0.320170 + 0.947360i \(0.396260\pi\)
\(224\) 3.23607 0.216219
\(225\) −1.20163 −0.0801084
\(226\) 14.7984 0.984373
\(227\) 22.3607 1.48413 0.742065 0.670328i \(-0.233846\pi\)
0.742065 + 0.670328i \(0.233846\pi\)
\(228\) 0 0
\(229\) −3.61803 −0.239086 −0.119543 0.992829i \(-0.538143\pi\)
−0.119543 + 0.992829i \(0.538143\pi\)
\(230\) 8.14590 0.537125
\(231\) 5.23607 0.344508
\(232\) 10.4721 0.687529
\(233\) 1.52786 0.100094 0.0500469 0.998747i \(-0.484063\pi\)
0.0500469 + 0.998747i \(0.484063\pi\)
\(234\) −2.00000 −0.130744
\(235\) −8.81966 −0.575331
\(236\) 7.85410 0.511258
\(237\) 26.6525 1.73126
\(238\) 20.9443 1.35762
\(239\) 19.7082 1.27482 0.637409 0.770526i \(-0.280006\pi\)
0.637409 + 0.770526i \(0.280006\pi\)
\(240\) −4.61803 −0.298093
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) 3.94427 0.253025
\(244\) 6.94427 0.444561
\(245\) −9.90983 −0.633116
\(246\) −17.7082 −1.12903
\(247\) 0 0
\(248\) −6.09017 −0.386726
\(249\) 9.70820 0.615232
\(250\) 5.29180 0.334683
\(251\) 4.14590 0.261687 0.130843 0.991403i \(-0.458231\pi\)
0.130843 + 0.991403i \(0.458231\pi\)
\(252\) −1.23607 −0.0778650
\(253\) −2.85410 −0.179436
\(254\) −9.41641 −0.590838
\(255\) −29.8885 −1.87169
\(256\) 1.00000 0.0625000
\(257\) 26.9443 1.68074 0.840369 0.542015i \(-0.182338\pi\)
0.840369 + 0.542015i \(0.182338\pi\)
\(258\) −5.23607 −0.325983
\(259\) 17.4164 1.08220
\(260\) −14.9443 −0.926804
\(261\) −4.00000 −0.247594
\(262\) −17.4164 −1.07599
\(263\) 29.2361 1.80277 0.901387 0.433015i \(-0.142550\pi\)
0.901387 + 0.433015i \(0.142550\pi\)
\(264\) 1.61803 0.0995831
\(265\) 3.27051 0.200906
\(266\) 0 0
\(267\) −3.00000 −0.183597
\(268\) −4.14590 −0.253251
\(269\) −19.5623 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(270\) 15.6180 0.950483
\(271\) −20.7639 −1.26132 −0.630660 0.776060i \(-0.717216\pi\)
−0.630660 + 0.776060i \(0.717216\pi\)
\(272\) 6.47214 0.392431
\(273\) 27.4164 1.65932
\(274\) 13.8541 0.836957
\(275\) 3.14590 0.189705
\(276\) −4.61803 −0.277973
\(277\) 13.7082 0.823646 0.411823 0.911264i \(-0.364892\pi\)
0.411823 + 0.911264i \(0.364892\pi\)
\(278\) 13.4164 0.804663
\(279\) 2.32624 0.139268
\(280\) −9.23607 −0.551961
\(281\) 11.2361 0.670288 0.335144 0.942167i \(-0.391215\pi\)
0.335144 + 0.942167i \(0.391215\pi\)
\(282\) 5.00000 0.297746
\(283\) 16.9443 1.00723 0.503616 0.863927i \(-0.332003\pi\)
0.503616 + 0.863927i \(0.332003\pi\)
\(284\) 1.52786 0.0906621
\(285\) 0 0
\(286\) 5.23607 0.309615
\(287\) −35.4164 −2.09056
\(288\) −0.381966 −0.0225076
\(289\) 24.8885 1.46403
\(290\) −29.8885 −1.75512
\(291\) −26.6525 −1.56240
\(292\) −5.70820 −0.334047
\(293\) −23.4164 −1.36800 −0.684001 0.729481i \(-0.739761\pi\)
−0.684001 + 0.729481i \(0.739761\pi\)
\(294\) 5.61803 0.327650
\(295\) −22.4164 −1.30513
\(296\) 5.38197 0.312820
\(297\) −5.47214 −0.317526
\(298\) −18.7639 −1.08697
\(299\) −14.9443 −0.864250
\(300\) 5.09017 0.293881
\(301\) −10.4721 −0.603604
\(302\) −16.4721 −0.947865
\(303\) 17.4164 1.00055
\(304\) 0 0
\(305\) −19.8197 −1.13487
\(306\) −2.47214 −0.141323
\(307\) 2.58359 0.147453 0.0737267 0.997278i \(-0.476511\pi\)
0.0737267 + 0.997278i \(0.476511\pi\)
\(308\) 3.23607 0.184392
\(309\) −1.52786 −0.0869171
\(310\) 17.3820 0.987229
\(311\) −30.5066 −1.72987 −0.864935 0.501884i \(-0.832640\pi\)
−0.864935 + 0.501884i \(0.832640\pi\)
\(312\) 8.47214 0.479640
\(313\) 3.52786 0.199407 0.0997033 0.995017i \(-0.468211\pi\)
0.0997033 + 0.995017i \(0.468211\pi\)
\(314\) 15.5623 0.878232
\(315\) 3.52786 0.198773
\(316\) 16.4721 0.926630
\(317\) 9.41641 0.528878 0.264439 0.964402i \(-0.414813\pi\)
0.264439 + 0.964402i \(0.414813\pi\)
\(318\) −1.85410 −0.103973
\(319\) 10.4721 0.586327
\(320\) −2.85410 −0.159549
\(321\) 9.70820 0.541859
\(322\) −9.23607 −0.514706
\(323\) 0 0
\(324\) −7.70820 −0.428234
\(325\) 16.4721 0.913710
\(326\) 3.14590 0.174235
\(327\) −16.4721 −0.910911
\(328\) −10.9443 −0.604296
\(329\) 10.0000 0.551318
\(330\) −4.61803 −0.254214
\(331\) −7.14590 −0.392774 −0.196387 0.980526i \(-0.562921\pi\)
−0.196387 + 0.980526i \(0.562921\pi\)
\(332\) 6.00000 0.329293
\(333\) −2.05573 −0.112653
\(334\) 25.2361 1.38086
\(335\) 11.8328 0.646496
\(336\) 5.23607 0.285651
\(337\) −3.88854 −0.211822 −0.105911 0.994376i \(-0.533776\pi\)
−0.105911 + 0.994376i \(0.533776\pi\)
\(338\) 14.4164 0.784149
\(339\) 23.9443 1.30047
\(340\) −18.4721 −1.00179
\(341\) −6.09017 −0.329801
\(342\) 0 0
\(343\) −11.4164 −0.616428
\(344\) −3.23607 −0.174477
\(345\) 13.1803 0.709606
\(346\) 15.2361 0.819096
\(347\) −12.9443 −0.694885 −0.347442 0.937701i \(-0.612950\pi\)
−0.347442 + 0.937701i \(0.612950\pi\)
\(348\) 16.9443 0.908308
\(349\) 17.8885 0.957552 0.478776 0.877937i \(-0.341081\pi\)
0.478776 + 0.877937i \(0.341081\pi\)
\(350\) 10.1803 0.544162
\(351\) −28.6525 −1.52936
\(352\) 1.00000 0.0533002
\(353\) −21.8541 −1.16318 −0.581588 0.813483i \(-0.697568\pi\)
−0.581588 + 0.813483i \(0.697568\pi\)
\(354\) 12.7082 0.675433
\(355\) −4.36068 −0.231441
\(356\) −1.85410 −0.0982672
\(357\) 33.8885 1.79357
\(358\) −12.0902 −0.638985
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 1.09017 0.0574570
\(361\) 0 0
\(362\) −1.67376 −0.0879710
\(363\) 1.61803 0.0849248
\(364\) 16.9443 0.888121
\(365\) 16.2918 0.852752
\(366\) 11.2361 0.587319
\(367\) −9.20163 −0.480321 −0.240160 0.970733i \(-0.577200\pi\)
−0.240160 + 0.970733i \(0.577200\pi\)
\(368\) −2.85410 −0.148780
\(369\) 4.18034 0.217620
\(370\) −15.3607 −0.798564
\(371\) −3.70820 −0.192520
\(372\) −9.85410 −0.510911
\(373\) 1.34752 0.0697722 0.0348861 0.999391i \(-0.488893\pi\)
0.0348861 + 0.999391i \(0.488893\pi\)
\(374\) 6.47214 0.334666
\(375\) 8.56231 0.442156
\(376\) 3.09017 0.159363
\(377\) 54.8328 2.82403
\(378\) −17.7082 −0.910812
\(379\) 30.8328 1.58378 0.791888 0.610667i \(-0.209099\pi\)
0.791888 + 0.610667i \(0.209099\pi\)
\(380\) 0 0
\(381\) −15.2361 −0.780567
\(382\) −3.38197 −0.173036
\(383\) 6.85410 0.350228 0.175114 0.984548i \(-0.443971\pi\)
0.175114 + 0.984548i \(0.443971\pi\)
\(384\) 1.61803 0.0825700
\(385\) −9.23607 −0.470714
\(386\) −23.8885 −1.21589
\(387\) 1.23607 0.0628329
\(388\) −16.4721 −0.836246
\(389\) −3.79837 −0.192585 −0.0962926 0.995353i \(-0.530698\pi\)
−0.0962926 + 0.995353i \(0.530698\pi\)
\(390\) −24.1803 −1.22442
\(391\) −18.4721 −0.934176
\(392\) 3.47214 0.175369
\(393\) −28.1803 −1.42151
\(394\) −13.5279 −0.681524
\(395\) −47.0132 −2.36549
\(396\) −0.381966 −0.0191945
\(397\) −16.3820 −0.822187 −0.411094 0.911593i \(-0.634853\pi\)
−0.411094 + 0.911593i \(0.634853\pi\)
\(398\) 6.67376 0.334525
\(399\) 0 0
\(400\) 3.14590 0.157295
\(401\) 13.9787 0.698064 0.349032 0.937111i \(-0.386511\pi\)
0.349032 + 0.937111i \(0.386511\pi\)
\(402\) −6.70820 −0.334575
\(403\) −31.8885 −1.58848
\(404\) 10.7639 0.535526
\(405\) 22.0000 1.09319
\(406\) 33.8885 1.68186
\(407\) 5.38197 0.266774
\(408\) 10.4721 0.518448
\(409\) −15.7082 −0.776721 −0.388360 0.921508i \(-0.626958\pi\)
−0.388360 + 0.921508i \(0.626958\pi\)
\(410\) 31.2361 1.54264
\(411\) 22.4164 1.10572
\(412\) −0.944272 −0.0465209
\(413\) 25.4164 1.25066
\(414\) 1.09017 0.0535789
\(415\) −17.1246 −0.840614
\(416\) 5.23607 0.256719
\(417\) 21.7082 1.06306
\(418\) 0 0
\(419\) −0.673762 −0.0329154 −0.0164577 0.999865i \(-0.505239\pi\)
−0.0164577 + 0.999865i \(0.505239\pi\)
\(420\) −14.9443 −0.729206
\(421\) 33.4508 1.63029 0.815147 0.579254i \(-0.196656\pi\)
0.815147 + 0.579254i \(0.196656\pi\)
\(422\) −6.94427 −0.338042
\(423\) −1.18034 −0.0573901
\(424\) −1.14590 −0.0556497
\(425\) 20.3607 0.987638
\(426\) 2.47214 0.119775
\(427\) 22.4721 1.08750
\(428\) 6.00000 0.290021
\(429\) 8.47214 0.409039
\(430\) 9.23607 0.445403
\(431\) −14.7639 −0.711154 −0.355577 0.934647i \(-0.615716\pi\)
−0.355577 + 0.934647i \(0.615716\pi\)
\(432\) −5.47214 −0.263278
\(433\) −9.79837 −0.470880 −0.235440 0.971889i \(-0.575653\pi\)
−0.235440 + 0.971889i \(0.575653\pi\)
\(434\) −19.7082 −0.946024
\(435\) −48.3607 −2.31872
\(436\) −10.1803 −0.487550
\(437\) 0 0
\(438\) −9.23607 −0.441316
\(439\) 7.34752 0.350678 0.175339 0.984508i \(-0.443898\pi\)
0.175339 + 0.984508i \(0.443898\pi\)
\(440\) −2.85410 −0.136064
\(441\) −1.32624 −0.0631542
\(442\) 33.8885 1.61191
\(443\) 15.2705 0.725524 0.362762 0.931882i \(-0.381834\pi\)
0.362762 + 0.931882i \(0.381834\pi\)
\(444\) 8.70820 0.413273
\(445\) 5.29180 0.250855
\(446\) 9.56231 0.452788
\(447\) −30.3607 −1.43601
\(448\) 3.23607 0.152890
\(449\) 5.56231 0.262501 0.131251 0.991349i \(-0.458101\pi\)
0.131251 + 0.991349i \(0.458101\pi\)
\(450\) −1.20163 −0.0566452
\(451\) −10.9443 −0.515346
\(452\) 14.7984 0.696057
\(453\) −26.6525 −1.25224
\(454\) 22.3607 1.04944
\(455\) −48.3607 −2.26718
\(456\) 0 0
\(457\) −4.76393 −0.222847 −0.111424 0.993773i \(-0.535541\pi\)
−0.111424 + 0.993773i \(0.535541\pi\)
\(458\) −3.61803 −0.169060
\(459\) −35.4164 −1.65310
\(460\) 8.14590 0.379804
\(461\) −20.6525 −0.961882 −0.480941 0.876753i \(-0.659705\pi\)
−0.480941 + 0.876753i \(0.659705\pi\)
\(462\) 5.23607 0.243604
\(463\) −39.1591 −1.81988 −0.909938 0.414745i \(-0.863871\pi\)
−0.909938 + 0.414745i \(0.863871\pi\)
\(464\) 10.4721 0.486157
\(465\) 28.1246 1.30425
\(466\) 1.52786 0.0707769
\(467\) 1.61803 0.0748737 0.0374368 0.999299i \(-0.488081\pi\)
0.0374368 + 0.999299i \(0.488081\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −13.4164 −0.619512
\(470\) −8.81966 −0.406821
\(471\) 25.1803 1.16025
\(472\) 7.85410 0.361514
\(473\) −3.23607 −0.148795
\(474\) 26.6525 1.22419
\(475\) 0 0
\(476\) 20.9443 0.959979
\(477\) 0.437694 0.0200406
\(478\) 19.7082 0.901432
\(479\) −17.1246 −0.782443 −0.391222 0.920296i \(-0.627947\pi\)
−0.391222 + 0.920296i \(0.627947\pi\)
\(480\) −4.61803 −0.210783
\(481\) 28.1803 1.28491
\(482\) −10.0000 −0.455488
\(483\) −14.9443 −0.679988
\(484\) 1.00000 0.0454545
\(485\) 47.0132 2.13476
\(486\) 3.94427 0.178916
\(487\) −10.8541 −0.491846 −0.245923 0.969289i \(-0.579091\pi\)
−0.245923 + 0.969289i \(0.579091\pi\)
\(488\) 6.94427 0.314352
\(489\) 5.09017 0.230185
\(490\) −9.90983 −0.447680
\(491\) 22.9443 1.03546 0.517730 0.855544i \(-0.326777\pi\)
0.517730 + 0.855544i \(0.326777\pi\)
\(492\) −17.7082 −0.798347
\(493\) 67.7771 3.05253
\(494\) 0 0
\(495\) 1.09017 0.0489995
\(496\) −6.09017 −0.273457
\(497\) 4.94427 0.221781
\(498\) 9.70820 0.435035
\(499\) −2.03444 −0.0910741 −0.0455371 0.998963i \(-0.514500\pi\)
−0.0455371 + 0.998963i \(0.514500\pi\)
\(500\) 5.29180 0.236656
\(501\) 40.8328 1.82428
\(502\) 4.14590 0.185040
\(503\) 4.76393 0.212413 0.106207 0.994344i \(-0.466130\pi\)
0.106207 + 0.994344i \(0.466130\pi\)
\(504\) −1.23607 −0.0550588
\(505\) −30.7214 −1.36708
\(506\) −2.85410 −0.126880
\(507\) 23.3262 1.03595
\(508\) −9.41641 −0.417786
\(509\) 16.4721 0.730115 0.365057 0.930985i \(-0.381049\pi\)
0.365057 + 0.930985i \(0.381049\pi\)
\(510\) −29.8885 −1.32349
\(511\) −18.4721 −0.817159
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 26.9443 1.18846
\(515\) 2.69505 0.118758
\(516\) −5.23607 −0.230505
\(517\) 3.09017 0.135906
\(518\) 17.4164 0.765233
\(519\) 24.6525 1.08212
\(520\) −14.9443 −0.655350
\(521\) 26.0000 1.13908 0.569540 0.821963i \(-0.307121\pi\)
0.569540 + 0.821963i \(0.307121\pi\)
\(522\) −4.00000 −0.175075
\(523\) −1.23607 −0.0540495 −0.0270247 0.999635i \(-0.508603\pi\)
−0.0270247 + 0.999635i \(0.508603\pi\)
\(524\) −17.4164 −0.760839
\(525\) 16.4721 0.718903
\(526\) 29.2361 1.27475
\(527\) −39.4164 −1.71701
\(528\) 1.61803 0.0704159
\(529\) −14.8541 −0.645831
\(530\) 3.27051 0.142062
\(531\) −3.00000 −0.130189
\(532\) 0 0
\(533\) −57.3050 −2.48215
\(534\) −3.00000 −0.129823
\(535\) −17.1246 −0.740362
\(536\) −4.14590 −0.179076
\(537\) −19.5623 −0.844176
\(538\) −19.5623 −0.843391
\(539\) 3.47214 0.149555
\(540\) 15.6180 0.672093
\(541\) −8.83282 −0.379752 −0.189876 0.981808i \(-0.560809\pi\)
−0.189876 + 0.981808i \(0.560809\pi\)
\(542\) −20.7639 −0.891887
\(543\) −2.70820 −0.116220
\(544\) 6.47214 0.277491
\(545\) 29.0557 1.24461
\(546\) 27.4164 1.17331
\(547\) −14.7639 −0.631260 −0.315630 0.948882i \(-0.602216\pi\)
−0.315630 + 0.948882i \(0.602216\pi\)
\(548\) 13.8541 0.591818
\(549\) −2.65248 −0.113205
\(550\) 3.14590 0.134142
\(551\) 0 0
\(552\) −4.61803 −0.196557
\(553\) 53.3050 2.26676
\(554\) 13.7082 0.582406
\(555\) −24.8541 −1.05500
\(556\) 13.4164 0.568982
\(557\) −46.1803 −1.95672 −0.978362 0.206901i \(-0.933662\pi\)
−0.978362 + 0.206901i \(0.933662\pi\)
\(558\) 2.32624 0.0984775
\(559\) −16.9443 −0.716666
\(560\) −9.23607 −0.390295
\(561\) 10.4721 0.442134
\(562\) 11.2361 0.473965
\(563\) 4.36068 0.183781 0.0918904 0.995769i \(-0.470709\pi\)
0.0918904 + 0.995769i \(0.470709\pi\)
\(564\) 5.00000 0.210538
\(565\) −42.2361 −1.77688
\(566\) 16.9443 0.712221
\(567\) −24.9443 −1.04756
\(568\) 1.52786 0.0641078
\(569\) −7.05573 −0.295792 −0.147896 0.989003i \(-0.547250\pi\)
−0.147896 + 0.989003i \(0.547250\pi\)
\(570\) 0 0
\(571\) −14.8328 −0.620734 −0.310367 0.950617i \(-0.600452\pi\)
−0.310367 + 0.950617i \(0.600452\pi\)
\(572\) 5.23607 0.218931
\(573\) −5.47214 −0.228602
\(574\) −35.4164 −1.47825
\(575\) −8.97871 −0.374438
\(576\) −0.381966 −0.0159153
\(577\) −19.6869 −0.819577 −0.409789 0.912181i \(-0.634398\pi\)
−0.409789 + 0.912181i \(0.634398\pi\)
\(578\) 24.8885 1.03523
\(579\) −38.6525 −1.60634
\(580\) −29.8885 −1.24105
\(581\) 19.4164 0.805528
\(582\) −26.6525 −1.10478
\(583\) −1.14590 −0.0474582
\(584\) −5.70820 −0.236207
\(585\) 5.70820 0.236005
\(586\) −23.4164 −0.967323
\(587\) −45.4508 −1.87596 −0.937979 0.346693i \(-0.887305\pi\)
−0.937979 + 0.346693i \(0.887305\pi\)
\(588\) 5.61803 0.231684
\(589\) 0 0
\(590\) −22.4164 −0.922869
\(591\) −21.8885 −0.900374
\(592\) 5.38197 0.221197
\(593\) 5.59675 0.229831 0.114915 0.993375i \(-0.463340\pi\)
0.114915 + 0.993375i \(0.463340\pi\)
\(594\) −5.47214 −0.224524
\(595\) −59.7771 −2.45062
\(596\) −18.7639 −0.768601
\(597\) 10.7984 0.441948
\(598\) −14.9443 −0.611117
\(599\) 8.03444 0.328278 0.164139 0.986437i \(-0.447515\pi\)
0.164139 + 0.986437i \(0.447515\pi\)
\(600\) 5.09017 0.207805
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −10.4721 −0.426812
\(603\) 1.58359 0.0644889
\(604\) −16.4721 −0.670242
\(605\) −2.85410 −0.116036
\(606\) 17.4164 0.707493
\(607\) 41.7771 1.69568 0.847840 0.530252i \(-0.177903\pi\)
0.847840 + 0.530252i \(0.177903\pi\)
\(608\) 0 0
\(609\) 54.8328 2.22194
\(610\) −19.8197 −0.802475
\(611\) 16.1803 0.654586
\(612\) −2.47214 −0.0999302
\(613\) −39.2361 −1.58473 −0.792365 0.610047i \(-0.791150\pi\)
−0.792365 + 0.610047i \(0.791150\pi\)
\(614\) 2.58359 0.104265
\(615\) 50.5410 2.03801
\(616\) 3.23607 0.130385
\(617\) −14.9656 −0.602491 −0.301245 0.953547i \(-0.597402\pi\)
−0.301245 + 0.953547i \(0.597402\pi\)
\(618\) −1.52786 −0.0614597
\(619\) −17.9656 −0.722097 −0.361048 0.932547i \(-0.617581\pi\)
−0.361048 + 0.932547i \(0.617581\pi\)
\(620\) 17.3820 0.698077
\(621\) 15.6180 0.626730
\(622\) −30.5066 −1.22320
\(623\) −6.00000 −0.240385
\(624\) 8.47214 0.339157
\(625\) −30.8328 −1.23331
\(626\) 3.52786 0.141002
\(627\) 0 0
\(628\) 15.5623 0.621004
\(629\) 34.8328 1.38888
\(630\) 3.52786 0.140553
\(631\) 23.9787 0.954578 0.477289 0.878747i \(-0.341620\pi\)
0.477289 + 0.878747i \(0.341620\pi\)
\(632\) 16.4721 0.655226
\(633\) −11.2361 −0.446594
\(634\) 9.41641 0.373973
\(635\) 26.8754 1.06652
\(636\) −1.85410 −0.0735199
\(637\) 18.1803 0.720331
\(638\) 10.4721 0.414596
\(639\) −0.583592 −0.0230865
\(640\) −2.85410 −0.112818
\(641\) −5.97871 −0.236145 −0.118072 0.993005i \(-0.537672\pi\)
−0.118072 + 0.993005i \(0.537672\pi\)
\(642\) 9.70820 0.383152
\(643\) −28.1459 −1.10997 −0.554983 0.831862i \(-0.687275\pi\)
−0.554983 + 0.831862i \(0.687275\pi\)
\(644\) −9.23607 −0.363952
\(645\) 14.9443 0.588430
\(646\) 0 0
\(647\) 11.7426 0.461651 0.230826 0.972995i \(-0.425857\pi\)
0.230826 + 0.972995i \(0.425857\pi\)
\(648\) −7.70820 −0.302807
\(649\) 7.85410 0.308300
\(650\) 16.4721 0.646090
\(651\) −31.8885 −1.24981
\(652\) 3.14590 0.123203
\(653\) −46.6180 −1.82430 −0.912152 0.409851i \(-0.865581\pi\)
−0.912152 + 0.409851i \(0.865581\pi\)
\(654\) −16.4721 −0.644111
\(655\) 49.7082 1.94226
\(656\) −10.9443 −0.427302
\(657\) 2.18034 0.0850632
\(658\) 10.0000 0.389841
\(659\) 9.88854 0.385203 0.192601 0.981277i \(-0.438308\pi\)
0.192601 + 0.981277i \(0.438308\pi\)
\(660\) −4.61803 −0.179757
\(661\) −19.7984 −0.770067 −0.385034 0.922902i \(-0.625810\pi\)
−0.385034 + 0.922902i \(0.625810\pi\)
\(662\) −7.14590 −0.277733
\(663\) 54.8328 2.12953
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −2.05573 −0.0796578
\(667\) −29.8885 −1.15729
\(668\) 25.2361 0.976413
\(669\) 15.4721 0.598187
\(670\) 11.8328 0.457142
\(671\) 6.94427 0.268081
\(672\) 5.23607 0.201986
\(673\) −29.8885 −1.15212 −0.576059 0.817408i \(-0.695410\pi\)
−0.576059 + 0.817408i \(0.695410\pi\)
\(674\) −3.88854 −0.149781
\(675\) −17.2148 −0.662597
\(676\) 14.4164 0.554477
\(677\) −24.8328 −0.954403 −0.477201 0.878794i \(-0.658349\pi\)
−0.477201 + 0.878794i \(0.658349\pi\)
\(678\) 23.9443 0.919574
\(679\) −53.3050 −2.04566
\(680\) −18.4721 −0.708374
\(681\) 36.1803 1.38643
\(682\) −6.09017 −0.233205
\(683\) 38.9230 1.48935 0.744673 0.667429i \(-0.232605\pi\)
0.744673 + 0.667429i \(0.232605\pi\)
\(684\) 0 0
\(685\) −39.5410 −1.51078
\(686\) −11.4164 −0.435880
\(687\) −5.85410 −0.223348
\(688\) −3.23607 −0.123374
\(689\) −6.00000 −0.228582
\(690\) 13.1803 0.501767
\(691\) 26.3820 1.00362 0.501809 0.864979i \(-0.332668\pi\)
0.501809 + 0.864979i \(0.332668\pi\)
\(692\) 15.2361 0.579188
\(693\) −1.23607 −0.0469543
\(694\) −12.9443 −0.491358
\(695\) −38.2918 −1.45249
\(696\) 16.9443 0.642271
\(697\) −70.8328 −2.68298
\(698\) 17.8885 0.677091
\(699\) 2.47214 0.0935048
\(700\) 10.1803 0.384781
\(701\) 36.6525 1.38434 0.692172 0.721732i \(-0.256654\pi\)
0.692172 + 0.721732i \(0.256654\pi\)
\(702\) −28.6525 −1.08142
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) −14.2705 −0.537458
\(706\) −21.8541 −0.822490
\(707\) 34.8328 1.31002
\(708\) 12.7082 0.477604
\(709\) −16.8328 −0.632170 −0.316085 0.948731i \(-0.602368\pi\)
−0.316085 + 0.948731i \(0.602368\pi\)
\(710\) −4.36068 −0.163653
\(711\) −6.29180 −0.235961
\(712\) −1.85410 −0.0694854
\(713\) 17.3820 0.650960
\(714\) 33.8885 1.26825
\(715\) −14.9443 −0.558884
\(716\) −12.0902 −0.451831
\(717\) 31.8885 1.19090
\(718\) −18.0000 −0.671754
\(719\) −20.9787 −0.782374 −0.391187 0.920311i \(-0.627935\pi\)
−0.391187 + 0.920311i \(0.627935\pi\)
\(720\) 1.09017 0.0406282
\(721\) −3.05573 −0.113801
\(722\) 0 0
\(723\) −16.1803 −0.601753
\(724\) −1.67376 −0.0622049
\(725\) 32.9443 1.22352
\(726\) 1.61803 0.0600509
\(727\) −19.1591 −0.710570 −0.355285 0.934758i \(-0.615616\pi\)
−0.355285 + 0.934758i \(0.615616\pi\)
\(728\) 16.9443 0.627996
\(729\) 29.5066 1.09284
\(730\) 16.2918 0.602986
\(731\) −20.9443 −0.774652
\(732\) 11.2361 0.415297
\(733\) 18.0689 0.667389 0.333695 0.942681i \(-0.391705\pi\)
0.333695 + 0.942681i \(0.391705\pi\)
\(734\) −9.20163 −0.339638
\(735\) −16.0344 −0.591439
\(736\) −2.85410 −0.105204
\(737\) −4.14590 −0.152716
\(738\) 4.18034 0.153880
\(739\) 15.8197 0.581936 0.290968 0.956733i \(-0.406023\pi\)
0.290968 + 0.956733i \(0.406023\pi\)
\(740\) −15.3607 −0.564670
\(741\) 0 0
\(742\) −3.70820 −0.136132
\(743\) 24.1803 0.887091 0.443545 0.896252i \(-0.353721\pi\)
0.443545 + 0.896252i \(0.353721\pi\)
\(744\) −9.85410 −0.361269
\(745\) 53.5542 1.96207
\(746\) 1.34752 0.0493364
\(747\) −2.29180 −0.0838524
\(748\) 6.47214 0.236645
\(749\) 19.4164 0.709460
\(750\) 8.56231 0.312651
\(751\) −29.3262 −1.07013 −0.535065 0.844811i \(-0.679713\pi\)
−0.535065 + 0.844811i \(0.679713\pi\)
\(752\) 3.09017 0.112687
\(753\) 6.70820 0.244461
\(754\) 54.8328 1.99689
\(755\) 47.0132 1.71098
\(756\) −17.7082 −0.644041
\(757\) −33.0557 −1.20143 −0.600715 0.799463i \(-0.705118\pi\)
−0.600715 + 0.799463i \(0.705118\pi\)
\(758\) 30.8328 1.11990
\(759\) −4.61803 −0.167624
\(760\) 0 0
\(761\) −28.9443 −1.04923 −0.524615 0.851340i \(-0.675791\pi\)
−0.524615 + 0.851340i \(0.675791\pi\)
\(762\) −15.2361 −0.551945
\(763\) −32.9443 −1.19266
\(764\) −3.38197 −0.122355
\(765\) 7.05573 0.255100
\(766\) 6.85410 0.247649
\(767\) 41.1246 1.48492
\(768\) 1.61803 0.0583858
\(769\) −44.3607 −1.59969 −0.799844 0.600209i \(-0.795084\pi\)
−0.799844 + 0.600209i \(0.795084\pi\)
\(770\) −9.23607 −0.332845
\(771\) 43.5967 1.57010
\(772\) −23.8885 −0.859768
\(773\) 46.3262 1.66624 0.833119 0.553093i \(-0.186553\pi\)
0.833119 + 0.553093i \(0.186553\pi\)
\(774\) 1.23607 0.0444295
\(775\) −19.1591 −0.688214
\(776\) −16.4721 −0.591315
\(777\) 28.1803 1.01096
\(778\) −3.79837 −0.136178
\(779\) 0 0
\(780\) −24.1803 −0.865795
\(781\) 1.52786 0.0546713
\(782\) −18.4721 −0.660562
\(783\) −57.3050 −2.04791
\(784\) 3.47214 0.124005
\(785\) −44.4164 −1.58529
\(786\) −28.1803 −1.00516
\(787\) 15.5967 0.555964 0.277982 0.960586i \(-0.410334\pi\)
0.277982 + 0.960586i \(0.410334\pi\)
\(788\) −13.5279 −0.481910
\(789\) 47.3050 1.68410
\(790\) −47.0132 −1.67265
\(791\) 47.8885 1.70272
\(792\) −0.381966 −0.0135726
\(793\) 36.3607 1.29121
\(794\) −16.3820 −0.581374
\(795\) 5.29180 0.187681
\(796\) 6.67376 0.236545
\(797\) −9.90983 −0.351024 −0.175512 0.984477i \(-0.556158\pi\)
−0.175512 + 0.984477i \(0.556158\pi\)
\(798\) 0 0
\(799\) 20.0000 0.707549
\(800\) 3.14590 0.111224
\(801\) 0.708204 0.0250232
\(802\) 13.9787 0.493606
\(803\) −5.70820 −0.201438
\(804\) −6.70820 −0.236580
\(805\) 26.3607 0.929092
\(806\) −31.8885 −1.12323
\(807\) −31.6525 −1.11422
\(808\) 10.7639 0.378674
\(809\) −23.0557 −0.810596 −0.405298 0.914185i \(-0.632832\pi\)
−0.405298 + 0.914185i \(0.632832\pi\)
\(810\) 22.0000 0.773001
\(811\) 17.8885 0.628152 0.314076 0.949398i \(-0.398305\pi\)
0.314076 + 0.949398i \(0.398305\pi\)
\(812\) 33.8885 1.18925
\(813\) −33.5967 −1.17829
\(814\) 5.38197 0.188638
\(815\) −8.97871 −0.314511
\(816\) 10.4721 0.366598
\(817\) 0 0
\(818\) −15.7082 −0.549224
\(819\) −6.47214 −0.226155
\(820\) 31.2361 1.09081
\(821\) −5.23607 −0.182740 −0.0913700 0.995817i \(-0.529125\pi\)
−0.0913700 + 0.995817i \(0.529125\pi\)
\(822\) 22.4164 0.781862
\(823\) −21.1459 −0.737100 −0.368550 0.929608i \(-0.620146\pi\)
−0.368550 + 0.929608i \(0.620146\pi\)
\(824\) −0.944272 −0.0328953
\(825\) 5.09017 0.177217
\(826\) 25.4164 0.884350
\(827\) −52.6525 −1.83091 −0.915453 0.402425i \(-0.868167\pi\)
−0.915453 + 0.402425i \(0.868167\pi\)
\(828\) 1.09017 0.0378860
\(829\) −23.3050 −0.809414 −0.404707 0.914446i \(-0.632627\pi\)
−0.404707 + 0.914446i \(0.632627\pi\)
\(830\) −17.1246 −0.594404
\(831\) 22.1803 0.769427
\(832\) 5.23607 0.181528
\(833\) 22.4721 0.778613
\(834\) 21.7082 0.751694
\(835\) −72.0263 −2.49257
\(836\) 0 0
\(837\) 33.3262 1.15192
\(838\) −0.673762 −0.0232747
\(839\) 22.4508 0.775089 0.387545 0.921851i \(-0.373323\pi\)
0.387545 + 0.921851i \(0.373323\pi\)
\(840\) −14.9443 −0.515626
\(841\) 80.6656 2.78157
\(842\) 33.4508 1.15279
\(843\) 18.1803 0.626164
\(844\) −6.94427 −0.239032
\(845\) −41.1459 −1.41546
\(846\) −1.18034 −0.0405809
\(847\) 3.23607 0.111193
\(848\) −1.14590 −0.0393503
\(849\) 27.4164 0.940929
\(850\) 20.3607 0.698366
\(851\) −15.3607 −0.526557
\(852\) 2.47214 0.0846940
\(853\) −36.8328 −1.26113 −0.630566 0.776136i \(-0.717177\pi\)
−0.630566 + 0.776136i \(0.717177\pi\)
\(854\) 22.4721 0.768981
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −54.5410 −1.86309 −0.931543 0.363632i \(-0.881537\pi\)
−0.931543 + 0.363632i \(0.881537\pi\)
\(858\) 8.47214 0.289234
\(859\) 13.0344 0.444729 0.222365 0.974964i \(-0.428622\pi\)
0.222365 + 0.974964i \(0.428622\pi\)
\(860\) 9.23607 0.314947
\(861\) −57.3050 −1.95295
\(862\) −14.7639 −0.502862
\(863\) 10.4934 0.357200 0.178600 0.983922i \(-0.442843\pi\)
0.178600 + 0.983922i \(0.442843\pi\)
\(864\) −5.47214 −0.186166
\(865\) −43.4853 −1.47854
\(866\) −9.79837 −0.332962
\(867\) 40.2705 1.36766
\(868\) −19.7082 −0.668940
\(869\) 16.4721 0.558779
\(870\) −48.3607 −1.63958
\(871\) −21.7082 −0.735554
\(872\) −10.1803 −0.344750
\(873\) 6.29180 0.212945
\(874\) 0 0
\(875\) 17.1246 0.578918
\(876\) −9.23607 −0.312058
\(877\) 13.3050 0.449276 0.224638 0.974442i \(-0.427880\pi\)
0.224638 + 0.974442i \(0.427880\pi\)
\(878\) 7.34752 0.247967
\(879\) −37.8885 −1.27795
\(880\) −2.85410 −0.0962118
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −1.32624 −0.0446568
\(883\) 33.6180 1.13134 0.565668 0.824633i \(-0.308618\pi\)
0.565668 + 0.824633i \(0.308618\pi\)
\(884\) 33.8885 1.13980
\(885\) −36.2705 −1.21922
\(886\) 15.2705 0.513023
\(887\) −1.70820 −0.0573559 −0.0286779 0.999589i \(-0.509130\pi\)
−0.0286779 + 0.999589i \(0.509130\pi\)
\(888\) 8.70820 0.292228
\(889\) −30.4721 −1.02200
\(890\) 5.29180 0.177381
\(891\) −7.70820 −0.258235
\(892\) 9.56231 0.320170
\(893\) 0 0
\(894\) −30.3607 −1.01541
\(895\) 34.5066 1.15343
\(896\) 3.23607 0.108109
\(897\) −24.1803 −0.807358
\(898\) 5.56231 0.185617
\(899\) −63.7771 −2.12708
\(900\) −1.20163 −0.0400542
\(901\) −7.41641 −0.247076
\(902\) −10.9443 −0.364404
\(903\) −16.9443 −0.563870
\(904\) 14.7984 0.492187
\(905\) 4.77709 0.158796
\(906\) −26.6525 −0.885469
\(907\) 13.4377 0.446191 0.223096 0.974797i \(-0.428384\pi\)
0.223096 + 0.974797i \(0.428384\pi\)
\(908\) 22.3607 0.742065
\(909\) −4.11146 −0.136368
\(910\) −48.3607 −1.60314
\(911\) 16.3820 0.542759 0.271379 0.962472i \(-0.412520\pi\)
0.271379 + 0.962472i \(0.412520\pi\)
\(912\) 0 0
\(913\) 6.00000 0.198571
\(914\) −4.76393 −0.157577
\(915\) −32.0689 −1.06016
\(916\) −3.61803 −0.119543
\(917\) −56.3607 −1.86119
\(918\) −35.4164 −1.16892
\(919\) −49.0132 −1.61679 −0.808397 0.588637i \(-0.799665\pi\)
−0.808397 + 0.588637i \(0.799665\pi\)
\(920\) 8.14590 0.268562
\(921\) 4.18034 0.137747
\(922\) −20.6525 −0.680153
\(923\) 8.00000 0.263323
\(924\) 5.23607 0.172254
\(925\) 16.9311 0.556692
\(926\) −39.1591 −1.28685
\(927\) 0.360680 0.0118463
\(928\) 10.4721 0.343765
\(929\) 42.3394 1.38911 0.694555 0.719439i \(-0.255601\pi\)
0.694555 + 0.719439i \(0.255601\pi\)
\(930\) 28.1246 0.922242
\(931\) 0 0
\(932\) 1.52786 0.0500469
\(933\) −49.3607 −1.61600
\(934\) 1.61803 0.0529437
\(935\) −18.4721 −0.604103
\(936\) −2.00000 −0.0653720
\(937\) 10.1803 0.332577 0.166289 0.986077i \(-0.446822\pi\)
0.166289 + 0.986077i \(0.446822\pi\)
\(938\) −13.4164 −0.438061
\(939\) 5.70820 0.186280
\(940\) −8.81966 −0.287666
\(941\) −6.58359 −0.214619 −0.107309 0.994226i \(-0.534224\pi\)
−0.107309 + 0.994226i \(0.534224\pi\)
\(942\) 25.1803 0.820420
\(943\) 31.2361 1.01719
\(944\) 7.85410 0.255629
\(945\) 50.5410 1.64410
\(946\) −3.23607 −0.105214
\(947\) −54.8328 −1.78183 −0.890914 0.454173i \(-0.849935\pi\)
−0.890914 + 0.454173i \(0.849935\pi\)
\(948\) 26.6525 0.865632
\(949\) −29.8885 −0.970223
\(950\) 0 0
\(951\) 15.2361 0.494063
\(952\) 20.9443 0.678808
\(953\) 0.583592 0.0189044 0.00945220 0.999955i \(-0.496991\pi\)
0.00945220 + 0.999955i \(0.496991\pi\)
\(954\) 0.437694 0.0141709
\(955\) 9.65248 0.312347
\(956\) 19.7082 0.637409
\(957\) 16.9443 0.547731
\(958\) −17.1246 −0.553271
\(959\) 44.8328 1.44773
\(960\) −4.61803 −0.149046
\(961\) 6.09017 0.196457
\(962\) 28.1803 0.908571
\(963\) −2.29180 −0.0738521
\(964\) −10.0000 −0.322078
\(965\) 68.1803 2.19480
\(966\) −14.9443 −0.480824
\(967\) −55.9574 −1.79947 −0.899735 0.436437i \(-0.856240\pi\)
−0.899735 + 0.436437i \(0.856240\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 47.0132 1.50950
\(971\) −31.2705 −1.00352 −0.501759 0.865007i \(-0.667314\pi\)
−0.501759 + 0.865007i \(0.667314\pi\)
\(972\) 3.94427 0.126513
\(973\) 43.4164 1.39187
\(974\) −10.8541 −0.347788
\(975\) 26.6525 0.853562
\(976\) 6.94427 0.222281
\(977\) −2.14590 −0.0686534 −0.0343267 0.999411i \(-0.510929\pi\)
−0.0343267 + 0.999411i \(0.510929\pi\)
\(978\) 5.09017 0.162766
\(979\) −1.85410 −0.0592574
\(980\) −9.90983 −0.316558
\(981\) 3.88854 0.124152
\(982\) 22.9443 0.732181
\(983\) −61.6312 −1.96573 −0.982865 0.184328i \(-0.940989\pi\)
−0.982865 + 0.184328i \(0.940989\pi\)
\(984\) −17.7082 −0.564517
\(985\) 38.6099 1.23021
\(986\) 67.7771 2.15846
\(987\) 16.1803 0.515026
\(988\) 0 0
\(989\) 9.23607 0.293690
\(990\) 1.09017 0.0346479
\(991\) −18.6738 −0.593192 −0.296596 0.955003i \(-0.595851\pi\)
−0.296596 + 0.955003i \(0.595851\pi\)
\(992\) −6.09017 −0.193363
\(993\) −11.5623 −0.366919
\(994\) 4.94427 0.156823
\(995\) −19.0476 −0.603849
\(996\) 9.70820 0.307616
\(997\) −26.0689 −0.825610 −0.412805 0.910819i \(-0.635451\pi\)
−0.412805 + 0.910819i \(0.635451\pi\)
\(998\) −2.03444 −0.0643991
\(999\) −29.4508 −0.931784
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.z.1.2 yes 2
19.18 odd 2 7942.2.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7942.2.a.u.1.1 2 19.18 odd 2
7942.2.a.z.1.2 yes 2 1.1 even 1 trivial