Properties

Label 4031.2.a.a.1.1
Level $4031$
Weight $2$
Character 4031.1
Self dual yes
Analytic conductor $32.188$
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] = [4031,2,Mod(1,4031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, 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("4031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4031 = 29 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4031.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: \(32.1876970548\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.414214 q^{2} -0.414214 q^{3} -1.82843 q^{4} +2.00000 q^{5} +0.171573 q^{6} +1.82843 q^{7} +1.58579 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q-0.414214 q^{2} -0.414214 q^{3} -1.82843 q^{4} +2.00000 q^{5} +0.171573 q^{6} +1.82843 q^{7} +1.58579 q^{8} -2.82843 q^{9} -0.828427 q^{10} +0.757359 q^{12} +5.00000 q^{13} -0.757359 q^{14} -0.828427 q^{15} +3.00000 q^{16} +1.58579 q^{17} +1.17157 q^{18} +3.24264 q^{19} -3.65685 q^{20} -0.757359 q^{21} +8.82843 q^{23} -0.656854 q^{24} -1.00000 q^{25} -2.07107 q^{26} +2.41421 q^{27} -3.34315 q^{28} +1.00000 q^{29} +0.343146 q^{30} -0.828427 q^{31} -4.41421 q^{32} -0.656854 q^{34} +3.65685 q^{35} +5.17157 q^{36} -8.48528 q^{37} -1.34315 q^{38} -2.07107 q^{39} +3.17157 q^{40} +2.00000 q^{41} +0.313708 q^{42} +0.414214 q^{43} -5.65685 q^{45} -3.65685 q^{46} -0.828427 q^{47} -1.24264 q^{48} -3.65685 q^{49} +0.414214 q^{50} -0.656854 q^{51} -9.14214 q^{52} +1.17157 q^{53} -1.00000 q^{54} +2.89949 q^{56} -1.34315 q^{57} -0.414214 q^{58} -0.343146 q^{59} +1.51472 q^{60} -8.89949 q^{61} +0.343146 q^{62} -5.17157 q^{63} -4.17157 q^{64} +10.0000 q^{65} +10.3137 q^{67} -2.89949 q^{68} -3.65685 q^{69} -1.51472 q^{70} +8.31371 q^{71} -4.48528 q^{72} +10.0711 q^{73} +3.51472 q^{74} +0.414214 q^{75} -5.92893 q^{76} +0.857864 q^{78} +14.4853 q^{79} +6.00000 q^{80} +7.48528 q^{81} -0.828427 q^{82} -7.00000 q^{83} +1.38478 q^{84} +3.17157 q^{85} -0.171573 q^{86} -0.414214 q^{87} -11.1716 q^{89} +2.34315 q^{90} +9.14214 q^{91} -16.1421 q^{92} +0.343146 q^{93} +0.343146 q^{94} +6.48528 q^{95} +1.82843 q^{96} +0.757359 q^{97} +1.51472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 4 q^{5} + 6 q^{6} - 2 q^{7} + 6 q^{8} + 4 q^{10} + 10 q^{12} + 10 q^{13} - 10 q^{14} + 4 q^{15} + 6 q^{16} + 6 q^{17} + 8 q^{18} - 2 q^{19} + 4 q^{20} - 10 q^{21} + 12 q^{23} + 10 q^{24} - 2 q^{25} + 10 q^{26} + 2 q^{27} - 18 q^{28} + 2 q^{29} + 12 q^{30} + 4 q^{31} - 6 q^{32} + 10 q^{34} - 4 q^{35} + 16 q^{36} - 14 q^{38} + 10 q^{39} + 12 q^{40} + 4 q^{41} - 22 q^{42} - 2 q^{43} + 4 q^{46} + 4 q^{47} + 6 q^{48} + 4 q^{49} - 2 q^{50} + 10 q^{51} + 10 q^{52} + 8 q^{53} - 2 q^{54} - 14 q^{56} - 14 q^{57} + 2 q^{58} - 12 q^{59} + 20 q^{60} + 2 q^{61} + 12 q^{62} - 16 q^{63} - 14 q^{64} + 20 q^{65} - 2 q^{67} + 14 q^{68} + 4 q^{69} - 20 q^{70} - 6 q^{71} + 8 q^{72} + 6 q^{73} + 24 q^{74} - 2 q^{75} - 26 q^{76} + 30 q^{78} + 12 q^{79} + 12 q^{80} - 2 q^{81} + 4 q^{82} - 14 q^{83} - 34 q^{84} + 12 q^{85} - 6 q^{86} + 2 q^{87} - 28 q^{89} + 16 q^{90} - 10 q^{91} - 4 q^{92} + 12 q^{93} + 12 q^{94} - 4 q^{95} - 2 q^{96} + 10 q^{97} + 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.414214 −0.292893 −0.146447 0.989219i \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) −0.414214 −0.239146 −0.119573 0.992825i \(-0.538153\pi\)
−0.119573 + 0.992825i \(0.538153\pi\)
\(4\) −1.82843 −0.914214
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0.171573 0.0700443
\(7\) 1.82843 0.691080 0.345540 0.938404i \(-0.387696\pi\)
0.345540 + 0.938404i \(0.387696\pi\)
\(8\) 1.58579 0.560660
\(9\) −2.82843 −0.942809
\(10\) −0.828427 −0.261972
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0.757359 0.218631
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −0.757359 −0.202413
\(15\) −0.828427 −0.213899
\(16\) 3.00000 0.750000
\(17\) 1.58579 0.384610 0.192305 0.981335i \(-0.438404\pi\)
0.192305 + 0.981335i \(0.438404\pi\)
\(18\) 1.17157 0.276142
\(19\) 3.24264 0.743913 0.371956 0.928250i \(-0.378687\pi\)
0.371956 + 0.928250i \(0.378687\pi\)
\(20\) −3.65685 −0.817697
\(21\) −0.757359 −0.165269
\(22\) 0 0
\(23\) 8.82843 1.84085 0.920427 0.390914i \(-0.127841\pi\)
0.920427 + 0.390914i \(0.127841\pi\)
\(24\) −0.656854 −0.134080
\(25\) −1.00000 −0.200000
\(26\) −2.07107 −0.406170
\(27\) 2.41421 0.464616
\(28\) −3.34315 −0.631795
\(29\) 1.00000 0.185695
\(30\) 0.343146 0.0626496
\(31\) −0.828427 −0.148790 −0.0743950 0.997229i \(-0.523703\pi\)
−0.0743950 + 0.997229i \(0.523703\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) −0.656854 −0.112650
\(35\) 3.65685 0.618121
\(36\) 5.17157 0.861929
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −1.34315 −0.217887
\(39\) −2.07107 −0.331636
\(40\) 3.17157 0.501470
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0.313708 0.0484063
\(43\) 0.414214 0.0631670 0.0315835 0.999501i \(-0.489945\pi\)
0.0315835 + 0.999501i \(0.489945\pi\)
\(44\) 0 0
\(45\) −5.65685 −0.843274
\(46\) −3.65685 −0.539174
\(47\) −0.828427 −0.120839 −0.0604193 0.998173i \(-0.519244\pi\)
−0.0604193 + 0.998173i \(0.519244\pi\)
\(48\) −1.24264 −0.179360
\(49\) −3.65685 −0.522408
\(50\) 0.414214 0.0585786
\(51\) −0.656854 −0.0919780
\(52\) −9.14214 −1.26779
\(53\) 1.17157 0.160928 0.0804640 0.996758i \(-0.474360\pi\)
0.0804640 + 0.996758i \(0.474360\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.89949 0.387461
\(57\) −1.34315 −0.177904
\(58\) −0.414214 −0.0543889
\(59\) −0.343146 −0.0446738 −0.0223369 0.999751i \(-0.507111\pi\)
−0.0223369 + 0.999751i \(0.507111\pi\)
\(60\) 1.51472 0.195549
\(61\) −8.89949 −1.13946 −0.569732 0.821831i \(-0.692953\pi\)
−0.569732 + 0.821831i \(0.692953\pi\)
\(62\) 0.343146 0.0435796
\(63\) −5.17157 −0.651557
\(64\) −4.17157 −0.521447
\(65\) 10.0000 1.24035
\(66\) 0 0
\(67\) 10.3137 1.26002 0.630010 0.776587i \(-0.283051\pi\)
0.630010 + 0.776587i \(0.283051\pi\)
\(68\) −2.89949 −0.351615
\(69\) −3.65685 −0.440234
\(70\) −1.51472 −0.181044
\(71\) 8.31371 0.986656 0.493328 0.869843i \(-0.335780\pi\)
0.493328 + 0.869843i \(0.335780\pi\)
\(72\) −4.48528 −0.528595
\(73\) 10.0711 1.17873 0.589365 0.807867i \(-0.299378\pi\)
0.589365 + 0.807867i \(0.299378\pi\)
\(74\) 3.51472 0.408578
\(75\) 0.414214 0.0478293
\(76\) −5.92893 −0.680095
\(77\) 0 0
\(78\) 0.857864 0.0971340
\(79\) 14.4853 1.62972 0.814861 0.579657i \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 6.00000 0.670820
\(81\) 7.48528 0.831698
\(82\) −0.828427 −0.0914845
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 1.38478 0.151091
\(85\) 3.17157 0.344005
\(86\) −0.171573 −0.0185012
\(87\) −0.414214 −0.0444084
\(88\) 0 0
\(89\) −11.1716 −1.18418 −0.592092 0.805870i \(-0.701698\pi\)
−0.592092 + 0.805870i \(0.701698\pi\)
\(90\) 2.34315 0.246989
\(91\) 9.14214 0.958356
\(92\) −16.1421 −1.68293
\(93\) 0.343146 0.0355826
\(94\) 0.343146 0.0353928
\(95\) 6.48528 0.665376
\(96\) 1.82843 0.186613
\(97\) 0.757359 0.0768982 0.0384491 0.999261i \(-0.487758\pi\)
0.0384491 + 0.999261i \(0.487758\pi\)
\(98\) 1.51472 0.153010
\(99\) 0 0
\(100\) 1.82843 0.182843
\(101\) 10.1421 1.00918 0.504590 0.863359i \(-0.331644\pi\)
0.504590 + 0.863359i \(0.331644\pi\)
\(102\) 0.272078 0.0269397
\(103\) −6.82843 −0.672825 −0.336412 0.941715i \(-0.609214\pi\)
−0.336412 + 0.941715i \(0.609214\pi\)
\(104\) 7.92893 0.777496
\(105\) −1.51472 −0.147821
\(106\) −0.485281 −0.0471347
\(107\) −19.4853 −1.88371 −0.941857 0.336015i \(-0.890921\pi\)
−0.941857 + 0.336015i \(0.890921\pi\)
\(108\) −4.41421 −0.424758
\(109\) −1.17157 −0.112216 −0.0561082 0.998425i \(-0.517869\pi\)
−0.0561082 + 0.998425i \(0.517869\pi\)
\(110\) 0 0
\(111\) 3.51472 0.333602
\(112\) 5.48528 0.518310
\(113\) −2.48528 −0.233796 −0.116898 0.993144i \(-0.537295\pi\)
−0.116898 + 0.993144i \(0.537295\pi\)
\(114\) 0.556349 0.0521069
\(115\) 17.6569 1.64651
\(116\) −1.82843 −0.169765
\(117\) −14.1421 −1.30744
\(118\) 0.142136 0.0130846
\(119\) 2.89949 0.265796
\(120\) −1.31371 −0.119925
\(121\) −11.0000 −1.00000
\(122\) 3.68629 0.333741
\(123\) −0.828427 −0.0746968
\(124\) 1.51472 0.136026
\(125\) −12.0000 −1.07331
\(126\) 2.14214 0.190837
\(127\) 6.48528 0.575476 0.287738 0.957709i \(-0.407097\pi\)
0.287738 + 0.957709i \(0.407097\pi\)
\(128\) 10.5563 0.933058
\(129\) −0.171573 −0.0151061
\(130\) −4.14214 −0.363289
\(131\) 4.34315 0.379462 0.189731 0.981836i \(-0.439238\pi\)
0.189731 + 0.981836i \(0.439238\pi\)
\(132\) 0 0
\(133\) 5.92893 0.514104
\(134\) −4.27208 −0.369051
\(135\) 4.82843 0.415565
\(136\) 2.51472 0.215635
\(137\) 22.4853 1.92105 0.960524 0.278198i \(-0.0897373\pi\)
0.960524 + 0.278198i \(0.0897373\pi\)
\(138\) 1.51472 0.128941
\(139\) −1.00000 −0.0848189
\(140\) −6.68629 −0.565095
\(141\) 0.343146 0.0288981
\(142\) −3.44365 −0.288985
\(143\) 0 0
\(144\) −8.48528 −0.707107
\(145\) 2.00000 0.166091
\(146\) −4.17157 −0.345242
\(147\) 1.51472 0.124932
\(148\) 15.5147 1.27530
\(149\) −0.828427 −0.0678674 −0.0339337 0.999424i \(-0.510804\pi\)
−0.0339337 + 0.999424i \(0.510804\pi\)
\(150\) −0.171573 −0.0140089
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 5.14214 0.417082
\(153\) −4.48528 −0.362614
\(154\) 0 0
\(155\) −1.65685 −0.133082
\(156\) 3.78680 0.303186
\(157\) 18.2132 1.45357 0.726786 0.686864i \(-0.241013\pi\)
0.726786 + 0.686864i \(0.241013\pi\)
\(158\) −6.00000 −0.477334
\(159\) −0.485281 −0.0384853
\(160\) −8.82843 −0.697948
\(161\) 16.1421 1.27218
\(162\) −3.10051 −0.243599
\(163\) 24.6274 1.92897 0.964484 0.264141i \(-0.0850884\pi\)
0.964484 + 0.264141i \(0.0850884\pi\)
\(164\) −3.65685 −0.285552
\(165\) 0 0
\(166\) 2.89949 0.225044
\(167\) 2.34315 0.181318 0.0906590 0.995882i \(-0.471103\pi\)
0.0906590 + 0.995882i \(0.471103\pi\)
\(168\) −1.20101 −0.0926599
\(169\) 12.0000 0.923077
\(170\) −1.31371 −0.100757
\(171\) −9.17157 −0.701368
\(172\) −0.757359 −0.0577481
\(173\) 4.17157 0.317159 0.158579 0.987346i \(-0.449309\pi\)
0.158579 + 0.987346i \(0.449309\pi\)
\(174\) 0.171573 0.0130069
\(175\) −1.82843 −0.138216
\(176\) 0 0
\(177\) 0.142136 0.0106836
\(178\) 4.62742 0.346840
\(179\) −17.7990 −1.33036 −0.665179 0.746684i \(-0.731645\pi\)
−0.665179 + 0.746684i \(0.731645\pi\)
\(180\) 10.3431 0.770933
\(181\) −17.8284 −1.32518 −0.662588 0.748984i \(-0.730542\pi\)
−0.662588 + 0.748984i \(0.730542\pi\)
\(182\) −3.78680 −0.280696
\(183\) 3.68629 0.272499
\(184\) 14.0000 1.03209
\(185\) −16.9706 −1.24770
\(186\) −0.142136 −0.0104219
\(187\) 0 0
\(188\) 1.51472 0.110472
\(189\) 4.41421 0.321087
\(190\) −2.68629 −0.194884
\(191\) −11.6569 −0.843460 −0.421730 0.906721i \(-0.638577\pi\)
−0.421730 + 0.906721i \(0.638577\pi\)
\(192\) 1.72792 0.124702
\(193\) 19.1716 1.38000 0.690000 0.723809i \(-0.257611\pi\)
0.690000 + 0.723809i \(0.257611\pi\)
\(194\) −0.313708 −0.0225230
\(195\) −4.14214 −0.296624
\(196\) 6.68629 0.477592
\(197\) 24.1421 1.72006 0.860028 0.510247i \(-0.170446\pi\)
0.860028 + 0.510247i \(0.170446\pi\)
\(198\) 0 0
\(199\) −15.7990 −1.11996 −0.559980 0.828506i \(-0.689191\pi\)
−0.559980 + 0.828506i \(0.689191\pi\)
\(200\) −1.58579 −0.112132
\(201\) −4.27208 −0.301329
\(202\) −4.20101 −0.295582
\(203\) 1.82843 0.128330
\(204\) 1.20101 0.0840875
\(205\) 4.00000 0.279372
\(206\) 2.82843 0.197066
\(207\) −24.9706 −1.73557
\(208\) 15.0000 1.04006
\(209\) 0 0
\(210\) 0.627417 0.0432959
\(211\) −9.92893 −0.683536 −0.341768 0.939784i \(-0.611026\pi\)
−0.341768 + 0.939784i \(0.611026\pi\)
\(212\) −2.14214 −0.147122
\(213\) −3.44365 −0.235955
\(214\) 8.07107 0.551727
\(215\) 0.828427 0.0564983
\(216\) 3.82843 0.260491
\(217\) −1.51472 −0.102826
\(218\) 0.485281 0.0328674
\(219\) −4.17157 −0.281889
\(220\) 0 0
\(221\) 7.92893 0.533358
\(222\) −1.45584 −0.0977099
\(223\) 16.8284 1.12691 0.563457 0.826145i \(-0.309471\pi\)
0.563457 + 0.826145i \(0.309471\pi\)
\(224\) −8.07107 −0.539271
\(225\) 2.82843 0.188562
\(226\) 1.02944 0.0684771
\(227\) −24.6274 −1.63458 −0.817290 0.576227i \(-0.804524\pi\)
−0.817290 + 0.576227i \(0.804524\pi\)
\(228\) 2.45584 0.162642
\(229\) −20.4853 −1.35371 −0.676853 0.736118i \(-0.736657\pi\)
−0.676853 + 0.736118i \(0.736657\pi\)
\(230\) −7.31371 −0.482252
\(231\) 0 0
\(232\) 1.58579 0.104112
\(233\) 23.7990 1.55912 0.779562 0.626325i \(-0.215442\pi\)
0.779562 + 0.626325i \(0.215442\pi\)
\(234\) 5.85786 0.382941
\(235\) −1.65685 −0.108081
\(236\) 0.627417 0.0408414
\(237\) −6.00000 −0.389742
\(238\) −1.20101 −0.0778499
\(239\) 15.8284 1.02386 0.511928 0.859028i \(-0.328931\pi\)
0.511928 + 0.859028i \(0.328931\pi\)
\(240\) −2.48528 −0.160424
\(241\) −20.9706 −1.35083 −0.675416 0.737437i \(-0.736036\pi\)
−0.675416 + 0.737437i \(0.736036\pi\)
\(242\) 4.55635 0.292893
\(243\) −10.3431 −0.663513
\(244\) 16.2721 1.04171
\(245\) −7.31371 −0.467256
\(246\) 0.343146 0.0218782
\(247\) 16.2132 1.03162
\(248\) −1.31371 −0.0834206
\(249\) 2.89949 0.183748
\(250\) 4.97056 0.314366
\(251\) 28.9706 1.82861 0.914303 0.405031i \(-0.132739\pi\)
0.914303 + 0.405031i \(0.132739\pi\)
\(252\) 9.45584 0.595662
\(253\) 0 0
\(254\) −2.68629 −0.168553
\(255\) −1.31371 −0.0822676
\(256\) 3.97056 0.248160
\(257\) 29.3137 1.82854 0.914269 0.405107i \(-0.132766\pi\)
0.914269 + 0.405107i \(0.132766\pi\)
\(258\) 0.0710678 0.00442449
\(259\) −15.5147 −0.964038
\(260\) −18.2843 −1.13394
\(261\) −2.82843 −0.175075
\(262\) −1.79899 −0.111142
\(263\) −1.51472 −0.0934016 −0.0467008 0.998909i \(-0.514871\pi\)
−0.0467008 + 0.998909i \(0.514871\pi\)
\(264\) 0 0
\(265\) 2.34315 0.143938
\(266\) −2.45584 −0.150577
\(267\) 4.62742 0.283193
\(268\) −18.8579 −1.15193
\(269\) −7.72792 −0.471180 −0.235590 0.971853i \(-0.575702\pi\)
−0.235590 + 0.971853i \(0.575702\pi\)
\(270\) −2.00000 −0.121716
\(271\) 18.4142 1.11858 0.559292 0.828971i \(-0.311073\pi\)
0.559292 + 0.828971i \(0.311073\pi\)
\(272\) 4.75736 0.288457
\(273\) −3.78680 −0.229187
\(274\) −9.31371 −0.562662
\(275\) 0 0
\(276\) 6.68629 0.402467
\(277\) 21.1716 1.27208 0.636038 0.771658i \(-0.280572\pi\)
0.636038 + 0.771658i \(0.280572\pi\)
\(278\) 0.414214 0.0248429
\(279\) 2.34315 0.140280
\(280\) 5.79899 0.346556
\(281\) 13.6569 0.814700 0.407350 0.913272i \(-0.366453\pi\)
0.407350 + 0.913272i \(0.366453\pi\)
\(282\) −0.142136 −0.00846405
\(283\) 27.9706 1.66268 0.831339 0.555766i \(-0.187575\pi\)
0.831339 + 0.555766i \(0.187575\pi\)
\(284\) −15.2010 −0.902014
\(285\) −2.68629 −0.159122
\(286\) 0 0
\(287\) 3.65685 0.215857
\(288\) 12.4853 0.735702
\(289\) −14.4853 −0.852075
\(290\) −0.828427 −0.0486469
\(291\) −0.313708 −0.0183899
\(292\) −18.4142 −1.07761
\(293\) 20.4853 1.19676 0.598381 0.801211i \(-0.295811\pi\)
0.598381 + 0.801211i \(0.295811\pi\)
\(294\) −0.627417 −0.0365917
\(295\) −0.686292 −0.0399574
\(296\) −13.4558 −0.782105
\(297\) 0 0
\(298\) 0.343146 0.0198779
\(299\) 44.1421 2.55281
\(300\) −0.757359 −0.0437262
\(301\) 0.757359 0.0436535
\(302\) 4.97056 0.286024
\(303\) −4.20101 −0.241342
\(304\) 9.72792 0.557935
\(305\) −17.7990 −1.01917
\(306\) 1.85786 0.106207
\(307\) 27.3137 1.55888 0.779438 0.626480i \(-0.215505\pi\)
0.779438 + 0.626480i \(0.215505\pi\)
\(308\) 0 0
\(309\) 2.82843 0.160904
\(310\) 0.686292 0.0389787
\(311\) 3.65685 0.207361 0.103681 0.994611i \(-0.466938\pi\)
0.103681 + 0.994611i \(0.466938\pi\)
\(312\) −3.28427 −0.185935
\(313\) 29.1421 1.64721 0.823605 0.567163i \(-0.191959\pi\)
0.823605 + 0.567163i \(0.191959\pi\)
\(314\) −7.54416 −0.425741
\(315\) −10.3431 −0.582770
\(316\) −26.4853 −1.48991
\(317\) 10.8284 0.608185 0.304093 0.952643i \(-0.401647\pi\)
0.304093 + 0.952643i \(0.401647\pi\)
\(318\) 0.201010 0.0112721
\(319\) 0 0
\(320\) −8.34315 −0.466396
\(321\) 8.07107 0.450483
\(322\) −6.68629 −0.372612
\(323\) 5.14214 0.286116
\(324\) −13.6863 −0.760350
\(325\) −5.00000 −0.277350
\(326\) −10.2010 −0.564982
\(327\) 0.485281 0.0268361
\(328\) 3.17157 0.175121
\(329\) −1.51472 −0.0835091
\(330\) 0 0
\(331\) −1.02944 −0.0565830 −0.0282915 0.999600i \(-0.509007\pi\)
−0.0282915 + 0.999600i \(0.509007\pi\)
\(332\) 12.7990 0.702436
\(333\) 24.0000 1.31519
\(334\) −0.970563 −0.0531068
\(335\) 20.6274 1.12700
\(336\) −2.27208 −0.123952
\(337\) 10.7574 0.585991 0.292995 0.956114i \(-0.405348\pi\)
0.292995 + 0.956114i \(0.405348\pi\)
\(338\) −4.97056 −0.270363
\(339\) 1.02944 0.0559114
\(340\) −5.79899 −0.314494
\(341\) 0 0
\(342\) 3.79899 0.205426
\(343\) −19.4853 −1.05211
\(344\) 0.656854 0.0354152
\(345\) −7.31371 −0.393757
\(346\) −1.72792 −0.0928937
\(347\) −12.9706 −0.696296 −0.348148 0.937440i \(-0.613189\pi\)
−0.348148 + 0.937440i \(0.613189\pi\)
\(348\) 0.757359 0.0405987
\(349\) 5.97056 0.319597 0.159798 0.987150i \(-0.448916\pi\)
0.159798 + 0.987150i \(0.448916\pi\)
\(350\) 0.757359 0.0404826
\(351\) 12.0711 0.644306
\(352\) 0 0
\(353\) −15.4558 −0.822632 −0.411316 0.911493i \(-0.634931\pi\)
−0.411316 + 0.911493i \(0.634931\pi\)
\(354\) −0.0588745 −0.00312914
\(355\) 16.6274 0.882492
\(356\) 20.4264 1.08260
\(357\) −1.20101 −0.0635642
\(358\) 7.37258 0.389653
\(359\) 8.48528 0.447836 0.223918 0.974608i \(-0.428115\pi\)
0.223918 + 0.974608i \(0.428115\pi\)
\(360\) −8.97056 −0.472790
\(361\) −8.48528 −0.446594
\(362\) 7.38478 0.388135
\(363\) 4.55635 0.239146
\(364\) −16.7157 −0.876142
\(365\) 20.1421 1.05429
\(366\) −1.52691 −0.0798130
\(367\) −10.1421 −0.529415 −0.264708 0.964329i \(-0.585275\pi\)
−0.264708 + 0.964329i \(0.585275\pi\)
\(368\) 26.4853 1.38064
\(369\) −5.65685 −0.294484
\(370\) 7.02944 0.365443
\(371\) 2.14214 0.111214
\(372\) −0.627417 −0.0325301
\(373\) 20.0000 1.03556 0.517780 0.855514i \(-0.326758\pi\)
0.517780 + 0.855514i \(0.326758\pi\)
\(374\) 0 0
\(375\) 4.97056 0.256679
\(376\) −1.31371 −0.0677493
\(377\) 5.00000 0.257513
\(378\) −1.82843 −0.0940441
\(379\) −33.7279 −1.73249 −0.866243 0.499622i \(-0.833472\pi\)
−0.866243 + 0.499622i \(0.833472\pi\)
\(380\) −11.8579 −0.608296
\(381\) −2.68629 −0.137623
\(382\) 4.82843 0.247044
\(383\) −15.7990 −0.807291 −0.403645 0.914916i \(-0.632257\pi\)
−0.403645 + 0.914916i \(0.632257\pi\)
\(384\) −4.37258 −0.223137
\(385\) 0 0
\(386\) −7.94113 −0.404193
\(387\) −1.17157 −0.0595544
\(388\) −1.38478 −0.0703014
\(389\) 29.0416 1.47247 0.736235 0.676726i \(-0.236602\pi\)
0.736235 + 0.676726i \(0.236602\pi\)
\(390\) 1.71573 0.0868793
\(391\) 14.0000 0.708010
\(392\) −5.79899 −0.292893
\(393\) −1.79899 −0.0907470
\(394\) −10.0000 −0.503793
\(395\) 28.9706 1.45767
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 6.54416 0.328029
\(399\) −2.45584 −0.122946
\(400\) −3.00000 −0.150000
\(401\) −10.9706 −0.547844 −0.273922 0.961752i \(-0.588321\pi\)
−0.273922 + 0.961752i \(0.588321\pi\)
\(402\) 1.76955 0.0882573
\(403\) −4.14214 −0.206334
\(404\) −18.5442 −0.922606
\(405\) 14.9706 0.743893
\(406\) −0.757359 −0.0375871
\(407\) 0 0
\(408\) −1.04163 −0.0515684
\(409\) −20.1421 −0.995965 −0.497982 0.867187i \(-0.665926\pi\)
−0.497982 + 0.867187i \(0.665926\pi\)
\(410\) −1.65685 −0.0818262
\(411\) −9.31371 −0.459411
\(412\) 12.4853 0.615106
\(413\) −0.627417 −0.0308732
\(414\) 10.3431 0.508338
\(415\) −14.0000 −0.687233
\(416\) −22.0711 −1.08212
\(417\) 0.414214 0.0202841
\(418\) 0 0
\(419\) 31.7990 1.55348 0.776741 0.629820i \(-0.216871\pi\)
0.776741 + 0.629820i \(0.216871\pi\)
\(420\) 2.76955 0.135140
\(421\) 12.6863 0.618292 0.309146 0.951015i \(-0.399957\pi\)
0.309146 + 0.951015i \(0.399957\pi\)
\(422\) 4.11270 0.200203
\(423\) 2.34315 0.113928
\(424\) 1.85786 0.0902259
\(425\) −1.58579 −0.0769219
\(426\) 1.42641 0.0691096
\(427\) −16.2721 −0.787461
\(428\) 35.6274 1.72212
\(429\) 0 0
\(430\) −0.343146 −0.0165480
\(431\) 5.17157 0.249106 0.124553 0.992213i \(-0.460250\pi\)
0.124553 + 0.992213i \(0.460250\pi\)
\(432\) 7.24264 0.348462
\(433\) 15.6569 0.752420 0.376210 0.926534i \(-0.377227\pi\)
0.376210 + 0.926534i \(0.377227\pi\)
\(434\) 0.627417 0.0301170
\(435\) −0.828427 −0.0397200
\(436\) 2.14214 0.102590
\(437\) 28.6274 1.36944
\(438\) 1.72792 0.0825633
\(439\) −17.4558 −0.833122 −0.416561 0.909108i \(-0.636765\pi\)
−0.416561 + 0.909108i \(0.636765\pi\)
\(440\) 0 0
\(441\) 10.3431 0.492531
\(442\) −3.28427 −0.156217
\(443\) −38.9706 −1.85155 −0.925774 0.378078i \(-0.876585\pi\)
−0.925774 + 0.378078i \(0.876585\pi\)
\(444\) −6.42641 −0.304984
\(445\) −22.3431 −1.05917
\(446\) −6.97056 −0.330066
\(447\) 0.343146 0.0162302
\(448\) −7.62742 −0.360362
\(449\) 37.5269 1.77100 0.885502 0.464635i \(-0.153814\pi\)
0.885502 + 0.464635i \(0.153814\pi\)
\(450\) −1.17157 −0.0552285
\(451\) 0 0
\(452\) 4.54416 0.213739
\(453\) 4.97056 0.233537
\(454\) 10.2010 0.478757
\(455\) 18.2843 0.857180
\(456\) −2.12994 −0.0997437
\(457\) −21.6569 −1.01306 −0.506532 0.862221i \(-0.669073\pi\)
−0.506532 + 0.862221i \(0.669073\pi\)
\(458\) 8.48528 0.396491
\(459\) 3.82843 0.178696
\(460\) −32.2843 −1.50526
\(461\) −15.1716 −0.706611 −0.353305 0.935508i \(-0.614942\pi\)
−0.353305 + 0.935508i \(0.614942\pi\)
\(462\) 0 0
\(463\) 30.1716 1.40219 0.701096 0.713067i \(-0.252694\pi\)
0.701096 + 0.713067i \(0.252694\pi\)
\(464\) 3.00000 0.139272
\(465\) 0.686292 0.0318260
\(466\) −9.85786 −0.456657
\(467\) −35.6569 −1.65000 −0.825001 0.565131i \(-0.808826\pi\)
−0.825001 + 0.565131i \(0.808826\pi\)
\(468\) 25.8579 1.19528
\(469\) 18.8579 0.870775
\(470\) 0.686292 0.0316563
\(471\) −7.54416 −0.347616
\(472\) −0.544156 −0.0250468
\(473\) 0 0
\(474\) 2.48528 0.114153
\(475\) −3.24264 −0.148783
\(476\) −5.30152 −0.242995
\(477\) −3.31371 −0.151724
\(478\) −6.55635 −0.299880
\(479\) 2.68629 0.122740 0.0613699 0.998115i \(-0.480453\pi\)
0.0613699 + 0.998115i \(0.480453\pi\)
\(480\) 3.65685 0.166912
\(481\) −42.4264 −1.93448
\(482\) 8.68629 0.395650
\(483\) −6.68629 −0.304237
\(484\) 20.1127 0.914214
\(485\) 1.51472 0.0687798
\(486\) 4.28427 0.194338
\(487\) −38.1421 −1.72839 −0.864193 0.503161i \(-0.832170\pi\)
−0.864193 + 0.503161i \(0.832170\pi\)
\(488\) −14.1127 −0.638852
\(489\) −10.2010 −0.461306
\(490\) 3.02944 0.136856
\(491\) −34.2843 −1.54723 −0.773614 0.633657i \(-0.781553\pi\)
−0.773614 + 0.633657i \(0.781553\pi\)
\(492\) 1.51472 0.0682888
\(493\) 1.58579 0.0714202
\(494\) −6.71573 −0.302155
\(495\) 0 0
\(496\) −2.48528 −0.111592
\(497\) 15.2010 0.681858
\(498\) −1.20101 −0.0538186
\(499\) 10.6274 0.475749 0.237874 0.971296i \(-0.423549\pi\)
0.237874 + 0.971296i \(0.423549\pi\)
\(500\) 21.9411 0.981237
\(501\) −0.970563 −0.0433615
\(502\) −12.0000 −0.535586
\(503\) −15.3137 −0.682805 −0.341402 0.939917i \(-0.610902\pi\)
−0.341402 + 0.939917i \(0.610902\pi\)
\(504\) −8.20101 −0.365302
\(505\) 20.2843 0.902638
\(506\) 0 0
\(507\) −4.97056 −0.220750
\(508\) −11.8579 −0.526108
\(509\) 4.48528 0.198807 0.0994033 0.995047i \(-0.468307\pi\)
0.0994033 + 0.995047i \(0.468307\pi\)
\(510\) 0.544156 0.0240956
\(511\) 18.4142 0.814597
\(512\) −22.7574 −1.00574
\(513\) 7.82843 0.345634
\(514\) −12.1421 −0.535567
\(515\) −13.6569 −0.601793
\(516\) 0.313708 0.0138102
\(517\) 0 0
\(518\) 6.42641 0.282360
\(519\) −1.72792 −0.0758474
\(520\) 15.8579 0.695413
\(521\) −2.82843 −0.123916 −0.0619578 0.998079i \(-0.519734\pi\)
−0.0619578 + 0.998079i \(0.519734\pi\)
\(522\) 1.17157 0.0512784
\(523\) 10.5147 0.459777 0.229888 0.973217i \(-0.426164\pi\)
0.229888 + 0.973217i \(0.426164\pi\)
\(524\) −7.94113 −0.346910
\(525\) 0.757359 0.0330539
\(526\) 0.627417 0.0273567
\(527\) −1.31371 −0.0572260
\(528\) 0 0
\(529\) 54.9411 2.38874
\(530\) −0.970563 −0.0421586
\(531\) 0.970563 0.0421188
\(532\) −10.8406 −0.470001
\(533\) 10.0000 0.433148
\(534\) −1.91674 −0.0829454
\(535\) −38.9706 −1.68484
\(536\) 16.3553 0.706443
\(537\) 7.37258 0.318150
\(538\) 3.20101 0.138005
\(539\) 0 0
\(540\) −8.82843 −0.379915
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −7.62742 −0.327626
\(543\) 7.38478 0.316911
\(544\) −7.00000 −0.300123
\(545\) −2.34315 −0.100369
\(546\) 1.56854 0.0671274
\(547\) 10.1421 0.433646 0.216823 0.976211i \(-0.430430\pi\)
0.216823 + 0.976211i \(0.430430\pi\)
\(548\) −41.1127 −1.75625
\(549\) 25.1716 1.07430
\(550\) 0 0
\(551\) 3.24264 0.138141
\(552\) −5.79899 −0.246821
\(553\) 26.4853 1.12627
\(554\) −8.76955 −0.372583
\(555\) 7.02944 0.298383
\(556\) 1.82843 0.0775426
\(557\) 11.6569 0.493917 0.246958 0.969026i \(-0.420569\pi\)
0.246958 + 0.969026i \(0.420569\pi\)
\(558\) −0.970563 −0.0410872
\(559\) 2.07107 0.0875968
\(560\) 10.9706 0.463591
\(561\) 0 0
\(562\) −5.65685 −0.238620
\(563\) −17.1127 −0.721214 −0.360607 0.932718i \(-0.617430\pi\)
−0.360607 + 0.932718i \(0.617430\pi\)
\(564\) −0.627417 −0.0264190
\(565\) −4.97056 −0.209113
\(566\) −11.5858 −0.486987
\(567\) 13.6863 0.574770
\(568\) 13.1838 0.553179
\(569\) −11.7990 −0.494639 −0.247320 0.968934i \(-0.579550\pi\)
−0.247320 + 0.968934i \(0.579550\pi\)
\(570\) 1.11270 0.0466058
\(571\) −20.6274 −0.863231 −0.431615 0.902058i \(-0.642056\pi\)
−0.431615 + 0.902058i \(0.642056\pi\)
\(572\) 0 0
\(573\) 4.82843 0.201710
\(574\) −1.51472 −0.0632231
\(575\) −8.82843 −0.368171
\(576\) 11.7990 0.491625
\(577\) −25.2426 −1.05086 −0.525432 0.850835i \(-0.676096\pi\)
−0.525432 + 0.850835i \(0.676096\pi\)
\(578\) 6.00000 0.249567
\(579\) −7.94113 −0.330022
\(580\) −3.65685 −0.151843
\(581\) −12.7990 −0.530992
\(582\) 0.129942 0.00538628
\(583\) 0 0
\(584\) 15.9706 0.660867
\(585\) −28.2843 −1.16941
\(586\) −8.48528 −0.350524
\(587\) 2.65685 0.109660 0.0548301 0.998496i \(-0.482538\pi\)
0.0548301 + 0.998496i \(0.482538\pi\)
\(588\) −2.76955 −0.114214
\(589\) −2.68629 −0.110687
\(590\) 0.284271 0.0117033
\(591\) −10.0000 −0.411345
\(592\) −25.4558 −1.04623
\(593\) −30.9411 −1.27060 −0.635300 0.772266i \(-0.719123\pi\)
−0.635300 + 0.772266i \(0.719123\pi\)
\(594\) 0 0
\(595\) 5.79899 0.237735
\(596\) 1.51472 0.0620453
\(597\) 6.54416 0.267834
\(598\) −18.2843 −0.747699
\(599\) −34.2843 −1.40082 −0.700409 0.713742i \(-0.746999\pi\)
−0.700409 + 0.713742i \(0.746999\pi\)
\(600\) 0.656854 0.0268160
\(601\) 44.4853 1.81459 0.907296 0.420492i \(-0.138143\pi\)
0.907296 + 0.420492i \(0.138143\pi\)
\(602\) −0.313708 −0.0127858
\(603\) −29.1716 −1.18796
\(604\) 21.9411 0.892772
\(605\) −22.0000 −0.894427
\(606\) 1.74012 0.0706874
\(607\) −25.1716 −1.02168 −0.510841 0.859675i \(-0.670666\pi\)
−0.510841 + 0.859675i \(0.670666\pi\)
\(608\) −14.3137 −0.580498
\(609\) −0.757359 −0.0306897
\(610\) 7.37258 0.298507
\(611\) −4.14214 −0.167573
\(612\) 8.20101 0.331506
\(613\) −24.7990 −1.00162 −0.500811 0.865557i \(-0.666965\pi\)
−0.500811 + 0.865557i \(0.666965\pi\)
\(614\) −11.3137 −0.456584
\(615\) −1.65685 −0.0668108
\(616\) 0 0
\(617\) −17.3848 −0.699885 −0.349942 0.936771i \(-0.613799\pi\)
−0.349942 + 0.936771i \(0.613799\pi\)
\(618\) −1.17157 −0.0471276
\(619\) 1.65685 0.0665946 0.0332973 0.999445i \(-0.489399\pi\)
0.0332973 + 0.999445i \(0.489399\pi\)
\(620\) 3.02944 0.121665
\(621\) 21.3137 0.855290
\(622\) −1.51472 −0.0607347
\(623\) −20.4264 −0.818367
\(624\) −6.21320 −0.248727
\(625\) −19.0000 −0.760000
\(626\) −12.0711 −0.482457
\(627\) 0 0
\(628\) −33.3015 −1.32888
\(629\) −13.4558 −0.536520
\(630\) 4.28427 0.170689
\(631\) −10.4853 −0.417412 −0.208706 0.977978i \(-0.566925\pi\)
−0.208706 + 0.977978i \(0.566925\pi\)
\(632\) 22.9706 0.913720
\(633\) 4.11270 0.163465
\(634\) −4.48528 −0.178133
\(635\) 12.9706 0.514721
\(636\) 0.887302 0.0351838
\(637\) −18.2843 −0.724449
\(638\) 0 0
\(639\) −23.5147 −0.930228
\(640\) 21.1127 0.834553
\(641\) 1.17157 0.0462743 0.0231372 0.999732i \(-0.492635\pi\)
0.0231372 + 0.999732i \(0.492635\pi\)
\(642\) −3.34315 −0.131943
\(643\) 35.6569 1.40617 0.703085 0.711106i \(-0.251806\pi\)
0.703085 + 0.711106i \(0.251806\pi\)
\(644\) −29.5147 −1.16304
\(645\) −0.343146 −0.0135114
\(646\) −2.12994 −0.0838015
\(647\) −21.3431 −0.839086 −0.419543 0.907736i \(-0.637810\pi\)
−0.419543 + 0.907736i \(0.637810\pi\)
\(648\) 11.8701 0.466300
\(649\) 0 0
\(650\) 2.07107 0.0812340
\(651\) 0.627417 0.0245904
\(652\) −45.0294 −1.76349
\(653\) 27.7279 1.08508 0.542539 0.840031i \(-0.317463\pi\)
0.542539 + 0.840031i \(0.317463\pi\)
\(654\) −0.201010 −0.00786012
\(655\) 8.68629 0.339401
\(656\) 6.00000 0.234261
\(657\) −28.4853 −1.11132
\(658\) 0.627417 0.0244593
\(659\) −10.4142 −0.405680 −0.202840 0.979212i \(-0.565017\pi\)
−0.202840 + 0.979212i \(0.565017\pi\)
\(660\) 0 0
\(661\) −22.6863 −0.882394 −0.441197 0.897410i \(-0.645446\pi\)
−0.441197 + 0.897410i \(0.645446\pi\)
\(662\) 0.426407 0.0165728
\(663\) −3.28427 −0.127551
\(664\) −11.1005 −0.430783
\(665\) 11.8579 0.459828
\(666\) −9.94113 −0.385211
\(667\) 8.82843 0.341838
\(668\) −4.28427 −0.165763
\(669\) −6.97056 −0.269498
\(670\) −8.54416 −0.330090
\(671\) 0 0
\(672\) 3.34315 0.128965
\(673\) 31.4853 1.21367 0.606834 0.794828i \(-0.292439\pi\)
0.606834 + 0.794828i \(0.292439\pi\)
\(674\) −4.45584 −0.171633
\(675\) −2.41421 −0.0929231
\(676\) −21.9411 −0.843889
\(677\) −45.9411 −1.76566 −0.882830 0.469692i \(-0.844365\pi\)
−0.882830 + 0.469692i \(0.844365\pi\)
\(678\) −0.426407 −0.0163761
\(679\) 1.38478 0.0531428
\(680\) 5.02944 0.192870
\(681\) 10.2010 0.390904
\(682\) 0 0
\(683\) −15.1421 −0.579398 −0.289699 0.957118i \(-0.593555\pi\)
−0.289699 + 0.957118i \(0.593555\pi\)
\(684\) 16.7696 0.641200
\(685\) 44.9706 1.71824
\(686\) 8.07107 0.308155
\(687\) 8.48528 0.323734
\(688\) 1.24264 0.0473752
\(689\) 5.85786 0.223167
\(690\) 3.02944 0.115329
\(691\) 18.4853 0.703213 0.351607 0.936148i \(-0.385635\pi\)
0.351607 + 0.936148i \(0.385635\pi\)
\(692\) −7.62742 −0.289951
\(693\) 0 0
\(694\) 5.37258 0.203940
\(695\) −2.00000 −0.0758643
\(696\) −0.656854 −0.0248980
\(697\) 3.17157 0.120132
\(698\) −2.47309 −0.0936078
\(699\) −9.85786 −0.372859
\(700\) 3.34315 0.126359
\(701\) −52.3137 −1.97586 −0.987931 0.154896i \(-0.950496\pi\)
−0.987931 + 0.154896i \(0.950496\pi\)
\(702\) −5.00000 −0.188713
\(703\) −27.5147 −1.03774
\(704\) 0 0
\(705\) 0.686292 0.0258472
\(706\) 6.40202 0.240943
\(707\) 18.5442 0.697425
\(708\) −0.259885 −0.00976706
\(709\) −21.1716 −0.795115 −0.397558 0.917577i \(-0.630142\pi\)
−0.397558 + 0.917577i \(0.630142\pi\)
\(710\) −6.88730 −0.258476
\(711\) −40.9706 −1.53652
\(712\) −17.7157 −0.663925
\(713\) −7.31371 −0.273901
\(714\) 0.497475 0.0186175
\(715\) 0 0
\(716\) 32.5442 1.21623
\(717\) −6.55635 −0.244851
\(718\) −3.51472 −0.131168
\(719\) 32.3137 1.20510 0.602549 0.798082i \(-0.294152\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(720\) −16.9706 −0.632456
\(721\) −12.4853 −0.464976
\(722\) 3.51472 0.130804
\(723\) 8.68629 0.323047
\(724\) 32.5980 1.21149
\(725\) −1.00000 −0.0371391
\(726\) −1.88730 −0.0700443
\(727\) 27.8701 1.03364 0.516822 0.856093i \(-0.327115\pi\)
0.516822 + 0.856093i \(0.327115\pi\)
\(728\) 14.4975 0.537312
\(729\) −18.1716 −0.673021
\(730\) −8.34315 −0.308794
\(731\) 0.656854 0.0242946
\(732\) −6.74012 −0.249122
\(733\) 6.97056 0.257464 0.128732 0.991679i \(-0.458909\pi\)
0.128732 + 0.991679i \(0.458909\pi\)
\(734\) 4.20101 0.155062
\(735\) 3.02944 0.111742
\(736\) −38.9706 −1.43647
\(737\) 0 0
\(738\) 2.34315 0.0862524
\(739\) 13.1127 0.482358 0.241179 0.970481i \(-0.422466\pi\)
0.241179 + 0.970481i \(0.422466\pi\)
\(740\) 31.0294 1.14066
\(741\) −6.71573 −0.246708
\(742\) −0.887302 −0.0325739
\(743\) 15.5858 0.571787 0.285894 0.958261i \(-0.407710\pi\)
0.285894 + 0.958261i \(0.407710\pi\)
\(744\) 0.544156 0.0199497
\(745\) −1.65685 −0.0607024
\(746\) −8.28427 −0.303309
\(747\) 19.7990 0.724407
\(748\) 0 0
\(749\) −35.6274 −1.30180
\(750\) −2.05887 −0.0751795
\(751\) 4.07107 0.148555 0.0742777 0.997238i \(-0.476335\pi\)
0.0742777 + 0.997238i \(0.476335\pi\)
\(752\) −2.48528 −0.0906289
\(753\) −12.0000 −0.437304
\(754\) −2.07107 −0.0754238
\(755\) −24.0000 −0.873449
\(756\) −8.07107 −0.293542
\(757\) 32.7696 1.19103 0.595515 0.803344i \(-0.296948\pi\)
0.595515 + 0.803344i \(0.296948\pi\)
\(758\) 13.9706 0.507434
\(759\) 0 0
\(760\) 10.2843 0.373050
\(761\) −47.4853 −1.72134 −0.860670 0.509163i \(-0.829955\pi\)
−0.860670 + 0.509163i \(0.829955\pi\)
\(762\) 1.11270 0.0403088
\(763\) −2.14214 −0.0775505
\(764\) 21.3137 0.771103
\(765\) −8.97056 −0.324331
\(766\) 6.54416 0.236450
\(767\) −1.71573 −0.0619514
\(768\) −1.64466 −0.0593466
\(769\) −26.8284 −0.967458 −0.483729 0.875218i \(-0.660718\pi\)
−0.483729 + 0.875218i \(0.660718\pi\)
\(770\) 0 0
\(771\) −12.1421 −0.437288
\(772\) −35.0538 −1.26161
\(773\) 37.1716 1.33697 0.668484 0.743726i \(-0.266943\pi\)
0.668484 + 0.743726i \(0.266943\pi\)
\(774\) 0.485281 0.0174431
\(775\) 0.828427 0.0297580
\(776\) 1.20101 0.0431138
\(777\) 6.42641 0.230546
\(778\) −12.0294 −0.431276
\(779\) 6.48528 0.232359
\(780\) 7.57359 0.271178
\(781\) 0 0
\(782\) −5.79899 −0.207371
\(783\) 2.41421 0.0862770
\(784\) −10.9706 −0.391806
\(785\) 36.4264 1.30011
\(786\) 0.745166 0.0265792
\(787\) 11.6569 0.415522 0.207761 0.978180i \(-0.433382\pi\)
0.207761 + 0.978180i \(0.433382\pi\)
\(788\) −44.1421 −1.57250
\(789\) 0.627417 0.0223366
\(790\) −12.0000 −0.426941
\(791\) −4.54416 −0.161572
\(792\) 0 0
\(793\) −44.4975 −1.58015
\(794\) 1.65685 0.0587996
\(795\) −0.970563 −0.0344223
\(796\) 28.8873 1.02388
\(797\) 17.0416 0.603646 0.301823 0.953364i \(-0.402405\pi\)
0.301823 + 0.953364i \(0.402405\pi\)
\(798\) 1.01724 0.0360100
\(799\) −1.31371 −0.0464757
\(800\) 4.41421 0.156066
\(801\) 31.5980 1.11646
\(802\) 4.54416 0.160460
\(803\) 0 0
\(804\) 7.81118 0.275479
\(805\) 32.2843 1.13787
\(806\) 1.71573 0.0604340
\(807\) 3.20101 0.112681
\(808\) 16.0833 0.565807
\(809\) −25.3848 −0.892481 −0.446241 0.894913i \(-0.647237\pi\)
−0.446241 + 0.894913i \(0.647237\pi\)
\(810\) −6.20101 −0.217881
\(811\) −25.6569 −0.900934 −0.450467 0.892793i \(-0.648743\pi\)
−0.450467 + 0.892793i \(0.648743\pi\)
\(812\) −3.34315 −0.117321
\(813\) −7.62742 −0.267505
\(814\) 0 0
\(815\) 49.2548 1.72532
\(816\) −1.97056 −0.0689835
\(817\) 1.34315 0.0469907
\(818\) 8.34315 0.291711
\(819\) −25.8579 −0.903547
\(820\) −7.31371 −0.255406
\(821\) 30.4264 1.06189 0.530944 0.847407i \(-0.321837\pi\)
0.530944 + 0.847407i \(0.321837\pi\)
\(822\) 3.85786 0.134558
\(823\) −2.41421 −0.0841542 −0.0420771 0.999114i \(-0.513398\pi\)
−0.0420771 + 0.999114i \(0.513398\pi\)
\(824\) −10.8284 −0.377226
\(825\) 0 0
\(826\) 0.259885 0.00904254
\(827\) −24.7574 −0.860898 −0.430449 0.902615i \(-0.641645\pi\)
−0.430449 + 0.902615i \(0.641645\pi\)
\(828\) 45.6569 1.58669
\(829\) 40.7696 1.41599 0.707993 0.706220i \(-0.249601\pi\)
0.707993 + 0.706220i \(0.249601\pi\)
\(830\) 5.79899 0.201286
\(831\) −8.76955 −0.304212
\(832\) −20.8579 −0.723116
\(833\) −5.79899 −0.200923
\(834\) −0.171573 −0.00594108
\(835\) 4.68629 0.162176
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) −13.1716 −0.455004
\(839\) 22.0000 0.759524 0.379762 0.925084i \(-0.376006\pi\)
0.379762 + 0.925084i \(0.376006\pi\)
\(840\) −2.40202 −0.0828776
\(841\) 1.00000 0.0344828
\(842\) −5.25483 −0.181094
\(843\) −5.65685 −0.194832
\(844\) 18.1543 0.624898
\(845\) 24.0000 0.825625
\(846\) −0.970563 −0.0333686
\(847\) −20.1127 −0.691080
\(848\) 3.51472 0.120696
\(849\) −11.5858 −0.397623
\(850\) 0.656854 0.0225299
\(851\) −74.9117 −2.56794
\(852\) 6.29646 0.215713
\(853\) 50.8995 1.74277 0.871383 0.490604i \(-0.163224\pi\)
0.871383 + 0.490604i \(0.163224\pi\)
\(854\) 6.74012 0.230642
\(855\) −18.3431 −0.627322
\(856\) −30.8995 −1.05612
\(857\) −47.9411 −1.63764 −0.818819 0.574052i \(-0.805371\pi\)
−0.818819 + 0.574052i \(0.805371\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) −1.51472 −0.0516515
\(861\) −1.51472 −0.0516215
\(862\) −2.14214 −0.0729614
\(863\) 48.9706 1.66698 0.833489 0.552537i \(-0.186340\pi\)
0.833489 + 0.552537i \(0.186340\pi\)
\(864\) −10.6569 −0.362554
\(865\) 8.34315 0.283675
\(866\) −6.48528 −0.220379
\(867\) 6.00000 0.203771
\(868\) 2.76955 0.0940047
\(869\) 0 0
\(870\) 0.343146 0.0116337
\(871\) 51.5685 1.74733
\(872\) −1.85786 −0.0629152
\(873\) −2.14214 −0.0725003
\(874\) −11.8579 −0.401098
\(875\) −21.9411 −0.741745
\(876\) 7.62742 0.257707
\(877\) −6.68629 −0.225780 −0.112890 0.993607i \(-0.536011\pi\)
−0.112890 + 0.993607i \(0.536011\pi\)
\(878\) 7.23045 0.244016
\(879\) −8.48528 −0.286201
\(880\) 0 0
\(881\) 22.6274 0.762337 0.381169 0.924506i \(-0.375522\pi\)
0.381169 + 0.924506i \(0.375522\pi\)
\(882\) −4.28427 −0.144259
\(883\) −35.4264 −1.19219 −0.596097 0.802913i \(-0.703283\pi\)
−0.596097 + 0.802913i \(0.703283\pi\)
\(884\) −14.4975 −0.487603
\(885\) 0.284271 0.00955567
\(886\) 16.1421 0.542306
\(887\) 38.4142 1.28982 0.644912 0.764257i \(-0.276894\pi\)
0.644912 + 0.764257i \(0.276894\pi\)
\(888\) 5.57359 0.187038
\(889\) 11.8579 0.397700
\(890\) 9.25483 0.310223
\(891\) 0 0
\(892\) −30.7696 −1.03024
\(893\) −2.68629 −0.0898933
\(894\) −0.142136 −0.00475373
\(895\) −35.5980 −1.18991
\(896\) 19.3015 0.644818
\(897\) −18.2843 −0.610494
\(898\) −15.5442 −0.518715
\(899\) −0.828427 −0.0276296
\(900\) −5.17157 −0.172386
\(901\) 1.85786 0.0618944
\(902\) 0 0
\(903\) −0.313708 −0.0104396
\(904\) −3.94113 −0.131080
\(905\) −35.6569 −1.18527
\(906\) −2.05887 −0.0684015
\(907\) 21.3137 0.707710 0.353855 0.935300i \(-0.384871\pi\)
0.353855 + 0.935300i \(0.384871\pi\)
\(908\) 45.0294 1.49435
\(909\) −28.6863 −0.951464
\(910\) −7.57359 −0.251062
\(911\) −19.1716 −0.635183 −0.317591 0.948228i \(-0.602874\pi\)
−0.317591 + 0.948228i \(0.602874\pi\)
\(912\) −4.02944 −0.133428
\(913\) 0 0
\(914\) 8.97056 0.296720
\(915\) 7.37258 0.243730
\(916\) 37.4558 1.23758
\(917\) 7.94113 0.262239
\(918\) −1.58579 −0.0523388
\(919\) 13.6569 0.450498 0.225249 0.974301i \(-0.427680\pi\)
0.225249 + 0.974301i \(0.427680\pi\)
\(920\) 28.0000 0.923133
\(921\) −11.3137 −0.372799
\(922\) 6.28427 0.206961
\(923\) 41.5685 1.36825
\(924\) 0 0
\(925\) 8.48528 0.278994
\(926\) −12.4975 −0.410693
\(927\) 19.3137 0.634345
\(928\) −4.41421 −0.144904
\(929\) −26.9706 −0.884875 −0.442438 0.896799i \(-0.645886\pi\)
−0.442438 + 0.896799i \(0.645886\pi\)
\(930\) −0.284271 −0.00932162
\(931\) −11.8579 −0.388626
\(932\) −43.5147 −1.42537
\(933\) −1.51472 −0.0495897
\(934\) 14.7696 0.483275
\(935\) 0 0
\(936\) −22.4264 −0.733030
\(937\) 24.8284 0.811109 0.405555 0.914071i \(-0.367078\pi\)
0.405555 + 0.914071i \(0.367078\pi\)
\(938\) −7.81118 −0.255044
\(939\) −12.0711 −0.393924
\(940\) 3.02944 0.0988093
\(941\) −17.0294 −0.555144 −0.277572 0.960705i \(-0.589530\pi\)
−0.277572 + 0.960705i \(0.589530\pi\)
\(942\) 3.12489 0.101814
\(943\) 17.6569 0.574986
\(944\) −1.02944 −0.0335053
\(945\) 8.82843 0.287189
\(946\) 0 0
\(947\) −25.1716 −0.817966 −0.408983 0.912542i \(-0.634117\pi\)
−0.408983 + 0.912542i \(0.634117\pi\)
\(948\) 10.9706 0.356307
\(949\) 50.3553 1.63460
\(950\) 1.34315 0.0435774
\(951\) −4.48528 −0.145445
\(952\) 4.59798 0.149021
\(953\) −0.343146 −0.0111156 −0.00555779 0.999985i \(-0.501769\pi\)
−0.00555779 + 0.999985i \(0.501769\pi\)
\(954\) 1.37258 0.0444390
\(955\) −23.3137 −0.754414
\(956\) −28.9411 −0.936023
\(957\) 0 0
\(958\) −1.11270 −0.0359497
\(959\) 41.1127 1.32760
\(960\) 3.45584 0.111537
\(961\) −30.3137 −0.977862
\(962\) 17.5736 0.566595
\(963\) 55.1127 1.77598
\(964\) 38.3431 1.23495
\(965\) 38.3431 1.23431
\(966\) 2.76955 0.0891089
\(967\) 34.2843 1.10251 0.551254 0.834338i \(-0.314150\pi\)
0.551254 + 0.834338i \(0.314150\pi\)
\(968\) −17.4437 −0.560660
\(969\) −2.12994 −0.0684236
\(970\) −0.627417 −0.0201451
\(971\) −39.5980 −1.27076 −0.635380 0.772200i \(-0.719156\pi\)
−0.635380 + 0.772200i \(0.719156\pi\)
\(972\) 18.9117 0.606593
\(973\) −1.82843 −0.0586167
\(974\) 15.7990 0.506232
\(975\) 2.07107 0.0663273
\(976\) −26.6985 −0.854598
\(977\) 14.0294 0.448841 0.224421 0.974492i \(-0.427951\pi\)
0.224421 + 0.974492i \(0.427951\pi\)
\(978\) 4.22540 0.135113
\(979\) 0 0
\(980\) 13.3726 0.427171
\(981\) 3.31371 0.105799
\(982\) 14.2010 0.453172
\(983\) −48.2132 −1.53776 −0.768881 0.639392i \(-0.779186\pi\)
−0.768881 + 0.639392i \(0.779186\pi\)
\(984\) −1.31371 −0.0418795
\(985\) 48.2843 1.53846
\(986\) −0.656854 −0.0209185
\(987\) 0.627417 0.0199709
\(988\) −29.6447 −0.943122
\(989\) 3.65685 0.116281
\(990\) 0 0
\(991\) 18.9706 0.602620 0.301310 0.953526i \(-0.402576\pi\)
0.301310 + 0.953526i \(0.402576\pi\)
\(992\) 3.65685 0.116105
\(993\) 0.426407 0.0135316
\(994\) −6.29646 −0.199712
\(995\) −31.5980 −1.00172
\(996\) −5.30152 −0.167985
\(997\) 21.7990 0.690381 0.345190 0.938533i \(-0.387814\pi\)
0.345190 + 0.938533i \(0.387814\pi\)
\(998\) −4.40202 −0.139344
\(999\) −20.4853 −0.648126
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 4031.2.a.a.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4031.2.a.a.1.1 2 1.1 even 1 trivial