Properties

Label 4598.2.a.bs.1.1
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $4$
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] = [4598,2,Mod(1,4598)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4598, 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("4598.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.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: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.258228.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 12x^{2} + 6x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.19461\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.19461 q^{3} +1.00000 q^{4} -4.19461 q^{5} +3.19461 q^{6} -1.32286 q^{7} -1.00000 q^{8} +7.20555 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.19461 q^{3} +1.00000 q^{4} -4.19461 q^{5} +3.19461 q^{6} -1.32286 q^{7} -1.00000 q^{8} +7.20555 q^{9} +4.19461 q^{10} -3.19461 q^{12} +3.52841 q^{13} +1.32286 q^{14} +13.4002 q^{15} +1.00000 q^{16} +2.32286 q^{17} -7.20555 q^{18} -1.00000 q^{19} -4.19461 q^{20} +4.22603 q^{21} +2.52841 q^{23} +3.19461 q^{24} +12.5948 q^{25} -3.52841 q^{26} -13.4351 q^{27} -1.32286 q^{28} +10.4002 q^{29} -13.4002 q^{30} -6.38923 q^{31} -1.00000 q^{32} -2.32286 q^{34} +5.54889 q^{35} +7.20555 q^{36} +7.84034 q^{37} +1.00000 q^{38} -11.2719 q^{39} +4.19461 q^{40} -2.34334 q^{41} -4.22603 q^{42} +11.4461 q^{43} -30.2245 q^{45} -2.52841 q^{46} -6.20555 q^{47} -3.19461 q^{48} -5.25004 q^{49} -12.5948 q^{50} -7.42064 q^{51} +3.52841 q^{52} +4.40017 q^{53} +13.4351 q^{54} +1.32286 q^{56} +3.19461 q^{57} -10.4002 q^{58} -6.72303 q^{59} +13.4002 q^{60} +8.60572 q^{61} +6.38923 q^{62} -9.53195 q^{63} +1.00000 q^{64} -14.8003 q^{65} +6.21509 q^{67} +2.32286 q^{68} -8.07730 q^{69} -5.54889 q^{70} -10.7326 q^{71} -7.20555 q^{72} +1.67714 q^{73} -7.84034 q^{74} -40.2354 q^{75} -1.00000 q^{76} +11.2719 q^{78} -5.65666 q^{79} -4.19461 q^{80} +21.3033 q^{81} +2.34334 q^{82} +1.80539 q^{83} +4.22603 q^{84} -9.74350 q^{85} -11.4461 q^{86} -33.2245 q^{87} +6.04589 q^{89} +30.2245 q^{90} -4.66760 q^{91} +2.52841 q^{92} +20.4111 q^{93} +6.20555 q^{94} +4.19461 q^{95} +3.19461 q^{96} -10.0000 q^{97} +5.25004 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 4 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 4 q^{8} + 16 q^{9} + 2 q^{10} + 2 q^{12} - q^{13} + 3 q^{14} + 26 q^{15} + 4 q^{16} + 7 q^{17} - 16 q^{18} - 4 q^{19} - 2 q^{20} + 9 q^{21} - 5 q^{23} - 2 q^{24} + 8 q^{25} + q^{26} + 8 q^{27} - 3 q^{28} + 14 q^{29} - 26 q^{30} + 4 q^{31} - 4 q^{32} - 7 q^{34} + 12 q^{35} + 16 q^{36} + 12 q^{37} + 4 q^{38} - 5 q^{39} + 2 q^{40} - 12 q^{41} - 9 q^{42} - 14 q^{43} - 2 q^{45} + 5 q^{46} - 12 q^{47} + 2 q^{48} + 23 q^{49} - 8 q^{50} - 7 q^{51} - q^{52} - 10 q^{53} - 8 q^{54} + 3 q^{56} - 2 q^{57} - 14 q^{58} + 3 q^{59} + 26 q^{60} - 6 q^{61} - 4 q^{62} + 18 q^{63} + 4 q^{64} - 4 q^{65} + 15 q^{67} + 7 q^{68} - 7 q^{69} - 12 q^{70} - 16 q^{71} - 16 q^{72} + 9 q^{73} - 12 q^{74} - 44 q^{75} - 4 q^{76} + 5 q^{78} - 20 q^{79} - 2 q^{80} + 52 q^{81} + 12 q^{82} + 22 q^{83} + 9 q^{84} - 14 q^{85} + 14 q^{86} - 14 q^{87} - 8 q^{89} + 2 q^{90} - 18 q^{91} - 5 q^{92} + 56 q^{93} + 12 q^{94} + 2 q^{95} - 2 q^{96} - 40 q^{97} - 23 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −3.19461 −1.84441 −0.922205 0.386700i \(-0.873615\pi\)
−0.922205 + 0.386700i \(0.873615\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.19461 −1.87589 −0.937944 0.346787i \(-0.887273\pi\)
−0.937944 + 0.346787i \(0.887273\pi\)
\(6\) 3.19461 1.30420
\(7\) −1.32286 −0.499995 −0.249997 0.968247i \(-0.580430\pi\)
−0.249997 + 0.968247i \(0.580430\pi\)
\(8\) −1.00000 −0.353553
\(9\) 7.20555 2.40185
\(10\) 4.19461 1.32645
\(11\) 0 0
\(12\) −3.19461 −0.922205
\(13\) 3.52841 0.978606 0.489303 0.872114i \(-0.337251\pi\)
0.489303 + 0.872114i \(0.337251\pi\)
\(14\) 1.32286 0.353550
\(15\) 13.4002 3.45991
\(16\) 1.00000 0.250000
\(17\) 2.32286 0.563377 0.281688 0.959506i \(-0.409106\pi\)
0.281688 + 0.959506i \(0.409106\pi\)
\(18\) −7.20555 −1.69837
\(19\) −1.00000 −0.229416
\(20\) −4.19461 −0.937944
\(21\) 4.22603 0.922195
\(22\) 0 0
\(23\) 2.52841 0.527211 0.263605 0.964631i \(-0.415088\pi\)
0.263605 + 0.964631i \(0.415088\pi\)
\(24\) 3.19461 0.652098
\(25\) 12.5948 2.51896
\(26\) −3.52841 −0.691979
\(27\) −13.4351 −2.58559
\(28\) −1.32286 −0.249997
\(29\) 10.4002 1.93126 0.965631 0.259916i \(-0.0836949\pi\)
0.965631 + 0.259916i \(0.0836949\pi\)
\(30\) −13.4002 −2.44652
\(31\) −6.38923 −1.14754 −0.573769 0.819017i \(-0.694519\pi\)
−0.573769 + 0.819017i \(0.694519\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.32286 −0.398367
\(35\) 5.54889 0.937934
\(36\) 7.20555 1.20093
\(37\) 7.84034 1.28894 0.644472 0.764628i \(-0.277077\pi\)
0.644472 + 0.764628i \(0.277077\pi\)
\(38\) 1.00000 0.162221
\(39\) −11.2719 −1.80495
\(40\) 4.19461 0.663227
\(41\) −2.34334 −0.365968 −0.182984 0.983116i \(-0.558576\pi\)
−0.182984 + 0.983116i \(0.558576\pi\)
\(42\) −4.22603 −0.652091
\(43\) 11.4461 1.74551 0.872753 0.488161i \(-0.162332\pi\)
0.872753 + 0.488161i \(0.162332\pi\)
\(44\) 0 0
\(45\) −30.2245 −4.50560
\(46\) −2.52841 −0.372794
\(47\) −6.20555 −0.905173 −0.452586 0.891721i \(-0.649499\pi\)
−0.452586 + 0.891721i \(0.649499\pi\)
\(48\) −3.19461 −0.461103
\(49\) −5.25004 −0.750005
\(50\) −12.5948 −1.78117
\(51\) −7.42064 −1.03910
\(52\) 3.52841 0.489303
\(53\) 4.40017 0.604409 0.302205 0.953243i \(-0.402277\pi\)
0.302205 + 0.953243i \(0.402277\pi\)
\(54\) 13.4351 1.82829
\(55\) 0 0
\(56\) 1.32286 0.176775
\(57\) 3.19461 0.423137
\(58\) −10.4002 −1.36561
\(59\) −6.72303 −0.875264 −0.437632 0.899154i \(-0.644183\pi\)
−0.437632 + 0.899154i \(0.644183\pi\)
\(60\) 13.4002 1.72995
\(61\) 8.60572 1.10185 0.550925 0.834555i \(-0.314275\pi\)
0.550925 + 0.834555i \(0.314275\pi\)
\(62\) 6.38923 0.811433
\(63\) −9.53195 −1.20091
\(64\) 1.00000 0.125000
\(65\) −14.8003 −1.83576
\(66\) 0 0
\(67\) 6.21509 0.759294 0.379647 0.925131i \(-0.376045\pi\)
0.379647 + 0.925131i \(0.376045\pi\)
\(68\) 2.32286 0.281688
\(69\) −8.07730 −0.972393
\(70\) −5.54889 −0.663219
\(71\) −10.7326 −1.27372 −0.636861 0.770979i \(-0.719767\pi\)
−0.636861 + 0.770979i \(0.719767\pi\)
\(72\) −7.20555 −0.849183
\(73\) 1.67714 0.196294 0.0981471 0.995172i \(-0.468708\pi\)
0.0981471 + 0.995172i \(0.468708\pi\)
\(74\) −7.84034 −0.911420
\(75\) −40.2354 −4.64599
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 11.2719 1.27629
\(79\) −5.65666 −0.636424 −0.318212 0.948020i \(-0.603082\pi\)
−0.318212 + 0.948020i \(0.603082\pi\)
\(80\) −4.19461 −0.468972
\(81\) 21.3033 2.36704
\(82\) 2.34334 0.258778
\(83\) 1.80539 0.198167 0.0990835 0.995079i \(-0.468409\pi\)
0.0990835 + 0.995079i \(0.468409\pi\)
\(84\) 4.22603 0.461098
\(85\) −9.74350 −1.05683
\(86\) −11.4461 −1.23426
\(87\) −33.2245 −3.56204
\(88\) 0 0
\(89\) 6.04589 0.640863 0.320431 0.947272i \(-0.396172\pi\)
0.320431 + 0.947272i \(0.396172\pi\)
\(90\) 30.2245 3.18594
\(91\) −4.66760 −0.489298
\(92\) 2.52841 0.263605
\(93\) 20.4111 2.11653
\(94\) 6.20555 0.640054
\(95\) 4.19461 0.430358
\(96\) 3.19461 0.326049
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 5.25004 0.530334
\(99\) 0 0
\(100\) 12.5948 1.25948
\(101\) −18.9949 −1.89007 −0.945034 0.326973i \(-0.893972\pi\)
−0.945034 + 0.326973i \(0.893972\pi\)
\(102\) 7.42064 0.734753
\(103\) 20.0459 1.97518 0.987590 0.157054i \(-0.0501996\pi\)
0.987590 + 0.157054i \(0.0501996\pi\)
\(104\) −3.52841 −0.345989
\(105\) −17.7266 −1.72994
\(106\) −4.40017 −0.427382
\(107\) −0.849872 −0.0821603 −0.0410801 0.999156i \(-0.513080\pi\)
−0.0410801 + 0.999156i \(0.513080\pi\)
\(108\) −13.4351 −1.29279
\(109\) −8.72303 −0.835514 −0.417757 0.908559i \(-0.637184\pi\)
−0.417757 + 0.908559i \(0.637184\pi\)
\(110\) 0 0
\(111\) −25.0468 −2.37734
\(112\) −1.32286 −0.124999
\(113\) 3.65666 0.343990 0.171995 0.985098i \(-0.444979\pi\)
0.171995 + 0.985098i \(0.444979\pi\)
\(114\) −3.19461 −0.299203
\(115\) −10.6057 −0.988988
\(116\) 10.4002 0.965631
\(117\) 25.4242 2.35047
\(118\) 6.72303 0.618905
\(119\) −3.07282 −0.281685
\(120\) −13.4002 −1.22326
\(121\) 0 0
\(122\) −8.60572 −0.779125
\(123\) 7.48606 0.674995
\(124\) −6.38923 −0.573769
\(125\) −31.8572 −2.84939
\(126\) 9.53195 0.849173
\(127\) 0.256496 0.0227603 0.0113802 0.999935i \(-0.496377\pi\)
0.0113802 + 0.999935i \(0.496377\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −36.5657 −3.21943
\(130\) 14.8003 1.29807
\(131\) 3.70255 0.323493 0.161747 0.986832i \(-0.448287\pi\)
0.161747 + 0.986832i \(0.448287\pi\)
\(132\) 0 0
\(133\) 1.32286 0.114707
\(134\) −6.21509 −0.536902
\(135\) 56.3551 4.85028
\(136\) −2.32286 −0.199184
\(137\) −5.12319 −0.437704 −0.218852 0.975758i \(-0.570231\pi\)
−0.218852 + 0.975758i \(0.570231\pi\)
\(138\) 8.07730 0.687586
\(139\) −6.36735 −0.540071 −0.270036 0.962850i \(-0.587035\pi\)
−0.270036 + 0.962850i \(0.587035\pi\)
\(140\) 5.54889 0.468967
\(141\) 19.8243 1.66951
\(142\) 10.7326 0.900657
\(143\) 0 0
\(144\) 7.20555 0.600463
\(145\) −43.6247 −3.62283
\(146\) −1.67714 −0.138801
\(147\) 16.7718 1.38332
\(148\) 7.84034 0.644472
\(149\) 13.7244 1.12435 0.562175 0.827019i \(-0.309965\pi\)
0.562175 + 0.827019i \(0.309965\pi\)
\(150\) 40.2354 3.28521
\(151\) 4.98906 0.406004 0.203002 0.979178i \(-0.434930\pi\)
0.203002 + 0.979178i \(0.434930\pi\)
\(152\) 1.00000 0.0811107
\(153\) 16.7375 1.35315
\(154\) 0 0
\(155\) 26.8003 2.15265
\(156\) −11.2719 −0.902476
\(157\) 2.04000 0.162810 0.0814050 0.996681i \(-0.474059\pi\)
0.0814050 + 0.996681i \(0.474059\pi\)
\(158\) 5.65666 0.450020
\(159\) −14.0568 −1.11478
\(160\) 4.19461 0.331613
\(161\) −3.34474 −0.263603
\(162\) −21.3033 −1.67375
\(163\) 6.15461 0.482066 0.241033 0.970517i \(-0.422514\pi\)
0.241033 + 0.970517i \(0.422514\pi\)
\(164\) −2.34334 −0.182984
\(165\) 0 0
\(166\) −1.80539 −0.140125
\(167\) −3.74350 −0.289681 −0.144841 0.989455i \(-0.546267\pi\)
−0.144841 + 0.989455i \(0.546267\pi\)
\(168\) −4.22603 −0.326045
\(169\) −0.550294 −0.0423303
\(170\) 9.74350 0.747293
\(171\) −7.20555 −0.551022
\(172\) 11.4461 0.872753
\(173\) −9.08684 −0.690860 −0.345430 0.938445i \(-0.612267\pi\)
−0.345430 + 0.938445i \(0.612267\pi\)
\(174\) 33.2245 2.51874
\(175\) −16.6611 −1.25946
\(176\) 0 0
\(177\) 21.4775 1.61435
\(178\) −6.04589 −0.453158
\(179\) 8.76751 0.655315 0.327657 0.944797i \(-0.393741\pi\)
0.327657 + 0.944797i \(0.393741\pi\)
\(180\) −30.2245 −2.25280
\(181\) 19.6807 1.46285 0.731426 0.681920i \(-0.238855\pi\)
0.731426 + 0.681920i \(0.238855\pi\)
\(182\) 4.66760 0.345986
\(183\) −27.4919 −2.03226
\(184\) −2.52841 −0.186397
\(185\) −32.8872 −2.41791
\(186\) −20.4111 −1.49661
\(187\) 0 0
\(188\) −6.20555 −0.452586
\(189\) 17.7728 1.29278
\(190\) −4.19461 −0.304309
\(191\) −9.47747 −0.685766 −0.342883 0.939378i \(-0.611403\pi\)
−0.342883 + 0.939378i \(0.611403\pi\)
\(192\) −3.19461 −0.230551
\(193\) −8.00000 −0.575853 −0.287926 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 10.0000 0.717958
\(195\) 47.2813 3.38589
\(196\) −5.25004 −0.375003
\(197\) 24.5438 1.74868 0.874338 0.485318i \(-0.161296\pi\)
0.874338 + 0.485318i \(0.161296\pi\)
\(198\) 0 0
\(199\) 20.1991 1.43188 0.715938 0.698164i \(-0.245999\pi\)
0.715938 + 0.698164i \(0.245999\pi\)
\(200\) −12.5948 −0.890585
\(201\) −19.8548 −1.40045
\(202\) 18.9949 1.33648
\(203\) −13.7580 −0.965621
\(204\) −7.42064 −0.519549
\(205\) 9.82940 0.686515
\(206\) −20.0459 −1.39666
\(207\) 18.2186 1.26628
\(208\) 3.52841 0.244652
\(209\) 0 0
\(210\) 17.7266 1.22325
\(211\) −1.88129 −0.129513 −0.0647566 0.997901i \(-0.520627\pi\)
−0.0647566 + 0.997901i \(0.520627\pi\)
\(212\) 4.40017 0.302205
\(213\) 34.2864 2.34926
\(214\) 0.849872 0.0580961
\(215\) −48.0118 −3.27438
\(216\) 13.4351 0.914144
\(217\) 8.45206 0.573763
\(218\) 8.72303 0.590798
\(219\) −5.35781 −0.362047
\(220\) 0 0
\(221\) 8.19602 0.551324
\(222\) 25.0468 1.68103
\(223\) 20.0868 1.34511 0.672557 0.740045i \(-0.265196\pi\)
0.672557 + 0.740045i \(0.265196\pi\)
\(224\) 1.32286 0.0883874
\(225\) 90.7523 6.05016
\(226\) −3.65666 −0.243238
\(227\) −3.55029 −0.235641 −0.117821 0.993035i \(-0.537591\pi\)
−0.117821 + 0.993035i \(0.537591\pi\)
\(228\) 3.19461 0.211568
\(229\) 7.72162 0.510259 0.255130 0.966907i \(-0.417882\pi\)
0.255130 + 0.966907i \(0.417882\pi\)
\(230\) 10.6057 0.699320
\(231\) 0 0
\(232\) −10.4002 −0.682804
\(233\) 21.9081 1.43525 0.717624 0.696431i \(-0.245230\pi\)
0.717624 + 0.696431i \(0.245230\pi\)
\(234\) −25.4242 −1.66203
\(235\) 26.0299 1.69800
\(236\) −6.72303 −0.437632
\(237\) 18.0708 1.17383
\(238\) 3.07282 0.199182
\(239\) −17.6676 −1.14282 −0.571411 0.820664i \(-0.693604\pi\)
−0.571411 + 0.820664i \(0.693604\pi\)
\(240\) 13.4002 0.864977
\(241\) −17.0999 −1.10150 −0.550751 0.834670i \(-0.685659\pi\)
−0.550751 + 0.834670i \(0.685659\pi\)
\(242\) 0 0
\(243\) −27.7506 −1.78020
\(244\) 8.60572 0.550925
\(245\) 22.0219 1.40693
\(246\) −7.48606 −0.477294
\(247\) −3.52841 −0.224508
\(248\) 6.38923 0.405716
\(249\) −5.76751 −0.365501
\(250\) 31.8572 2.01482
\(251\) 13.4461 0.848707 0.424354 0.905497i \(-0.360501\pi\)
0.424354 + 0.905497i \(0.360501\pi\)
\(252\) −9.53195 −0.600456
\(253\) 0 0
\(254\) −0.256496 −0.0160940
\(255\) 31.1267 1.94923
\(256\) 1.00000 0.0625000
\(257\) 10.1546 0.633427 0.316714 0.948521i \(-0.397421\pi\)
0.316714 + 0.948521i \(0.397421\pi\)
\(258\) 36.5657 2.27648
\(259\) −10.3717 −0.644465
\(260\) −14.8003 −0.917878
\(261\) 74.9389 4.63860
\(262\) −3.70255 −0.228744
\(263\) −24.9081 −1.53590 −0.767950 0.640510i \(-0.778723\pi\)
−0.767950 + 0.640510i \(0.778723\pi\)
\(264\) 0 0
\(265\) −18.4570 −1.13380
\(266\) −1.32286 −0.0811098
\(267\) −19.3143 −1.18201
\(268\) 6.21509 0.379647
\(269\) −29.4761 −1.79719 −0.898594 0.438781i \(-0.855410\pi\)
−0.898594 + 0.438781i \(0.855410\pi\)
\(270\) −56.3551 −3.42966
\(271\) 2.99046 0.181658 0.0908289 0.995867i \(-0.471048\pi\)
0.0908289 + 0.995867i \(0.471048\pi\)
\(272\) 2.32286 0.140844
\(273\) 14.9112 0.902466
\(274\) 5.12319 0.309503
\(275\) 0 0
\(276\) −8.07730 −0.486197
\(277\) 16.8622 1.01315 0.506576 0.862195i \(-0.330911\pi\)
0.506576 + 0.862195i \(0.330911\pi\)
\(278\) 6.36735 0.381888
\(279\) −46.0379 −2.75622
\(280\) −5.54889 −0.331610
\(281\) 0.989060 0.0590024 0.0295012 0.999565i \(-0.490608\pi\)
0.0295012 + 0.999565i \(0.490608\pi\)
\(282\) −19.8243 −1.18052
\(283\) 5.97812 0.355362 0.177681 0.984088i \(-0.443140\pi\)
0.177681 + 0.984088i \(0.443140\pi\)
\(284\) −10.7326 −0.636861
\(285\) −13.4002 −0.793757
\(286\) 0 0
\(287\) 3.09991 0.182982
\(288\) −7.20555 −0.424591
\(289\) −11.6043 −0.682607
\(290\) 43.6247 2.56173
\(291\) 31.9461 1.87272
\(292\) 1.67714 0.0981471
\(293\) −14.5489 −0.849955 −0.424978 0.905204i \(-0.639718\pi\)
−0.424978 + 0.905204i \(0.639718\pi\)
\(294\) −16.7718 −0.978154
\(295\) 28.2005 1.64190
\(296\) −7.84034 −0.455710
\(297\) 0 0
\(298\) −13.7244 −0.795035
\(299\) 8.92129 0.515932
\(300\) −40.2354 −2.32299
\(301\) −15.1415 −0.872744
\(302\) −4.98906 −0.287088
\(303\) 60.6815 3.48606
\(304\) −1.00000 −0.0573539
\(305\) −36.0977 −2.06695
\(306\) −16.7375 −0.956819
\(307\) −9.69161 −0.553129 −0.276565 0.960995i \(-0.589196\pi\)
−0.276565 + 0.960995i \(0.589196\pi\)
\(308\) 0 0
\(309\) −64.0389 −3.64304
\(310\) −26.8003 −1.52216
\(311\) −18.8229 −1.06735 −0.533675 0.845689i \(-0.679190\pi\)
−0.533675 + 0.845689i \(0.679190\pi\)
\(312\) 11.2719 0.638147
\(313\) 27.1800 1.53631 0.768153 0.640267i \(-0.221176\pi\)
0.768153 + 0.640267i \(0.221176\pi\)
\(314\) −2.04000 −0.115124
\(315\) 39.9828 2.25278
\(316\) −5.65666 −0.318212
\(317\) −14.3697 −0.807083 −0.403541 0.914961i \(-0.632221\pi\)
−0.403541 + 0.914961i \(0.632221\pi\)
\(318\) 14.0568 0.788268
\(319\) 0 0
\(320\) −4.19461 −0.234486
\(321\) 2.71501 0.151537
\(322\) 3.34474 0.186395
\(323\) −2.32286 −0.129247
\(324\) 21.3033 1.18352
\(325\) 44.4396 2.46507
\(326\) −6.15461 −0.340872
\(327\) 27.8667 1.54103
\(328\) 2.34334 0.129389
\(329\) 8.20909 0.452582
\(330\) 0 0
\(331\) 18.9635 1.04233 0.521165 0.853456i \(-0.325498\pi\)
0.521165 + 0.853456i \(0.325498\pi\)
\(332\) 1.80539 0.0990835
\(333\) 56.4940 3.09585
\(334\) 3.74350 0.204835
\(335\) −26.0699 −1.42435
\(336\) 4.22603 0.230549
\(337\) −30.1595 −1.64290 −0.821448 0.570284i \(-0.806833\pi\)
−0.821448 + 0.570284i \(0.806833\pi\)
\(338\) 0.550294 0.0299320
\(339\) −11.6816 −0.634459
\(340\) −9.74350 −0.528416
\(341\) 0 0
\(342\) 7.20555 0.389632
\(343\) 16.2051 0.874993
\(344\) −11.4461 −0.617130
\(345\) 33.8812 1.82410
\(346\) 9.08684 0.488512
\(347\) −23.5997 −1.26690 −0.633450 0.773784i \(-0.718362\pi\)
−0.633450 + 0.773784i \(0.718362\pi\)
\(348\) −33.2245 −1.78102
\(349\) 29.2086 1.56350 0.781751 0.623591i \(-0.214327\pi\)
0.781751 + 0.623591i \(0.214327\pi\)
\(350\) 16.6611 0.890576
\(351\) −47.4046 −2.53027
\(352\) 0 0
\(353\) −10.2274 −0.544351 −0.272176 0.962248i \(-0.587743\pi\)
−0.272176 + 0.962248i \(0.587743\pi\)
\(354\) −21.4775 −1.14151
\(355\) 45.0190 2.38936
\(356\) 6.04589 0.320431
\(357\) 9.81648 0.519543
\(358\) −8.76751 −0.463377
\(359\) −0.876807 −0.0462761 −0.0231380 0.999732i \(-0.507366\pi\)
−0.0231380 + 0.999732i \(0.507366\pi\)
\(360\) 30.2245 1.59297
\(361\) 1.00000 0.0526316
\(362\) −19.6807 −1.03439
\(363\) 0 0
\(364\) −4.66760 −0.244649
\(365\) −7.03495 −0.368226
\(366\) 27.4919 1.43703
\(367\) 10.5030 0.548252 0.274126 0.961694i \(-0.411611\pi\)
0.274126 + 0.961694i \(0.411611\pi\)
\(368\) 2.52841 0.131803
\(369\) −16.8850 −0.879000
\(370\) 32.8872 1.70972
\(371\) −5.82081 −0.302201
\(372\) 20.4111 1.05827
\(373\) −34.9249 −1.80834 −0.904172 0.427168i \(-0.859511\pi\)
−0.904172 + 0.427168i \(0.859511\pi\)
\(374\) 0 0
\(375\) 101.771 5.25545
\(376\) 6.20555 0.320027
\(377\) 36.6961 1.88994
\(378\) −17.7728 −0.914134
\(379\) −13.9695 −0.717567 −0.358783 0.933421i \(-0.616808\pi\)
−0.358783 + 0.933421i \(0.616808\pi\)
\(380\) 4.19461 0.215179
\(381\) −0.819406 −0.0419794
\(382\) 9.47747 0.484910
\(383\) 35.3223 1.80488 0.902442 0.430811i \(-0.141772\pi\)
0.902442 + 0.430811i \(0.141772\pi\)
\(384\) 3.19461 0.163024
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 82.4752 4.19245
\(388\) −10.0000 −0.507673
\(389\) 12.5838 0.638026 0.319013 0.947750i \(-0.396649\pi\)
0.319013 + 0.947750i \(0.396649\pi\)
\(390\) −47.2813 −2.39418
\(391\) 5.87315 0.297018
\(392\) 5.25004 0.265167
\(393\) −11.8282 −0.596655
\(394\) −24.5438 −1.23650
\(395\) 23.7275 1.19386
\(396\) 0 0
\(397\) 0.924098 0.0463792 0.0231896 0.999731i \(-0.492618\pi\)
0.0231896 + 0.999731i \(0.492618\pi\)
\(398\) −20.1991 −1.01249
\(399\) −4.22603 −0.211566
\(400\) 12.5948 0.629739
\(401\) −2.30238 −0.114976 −0.0574878 0.998346i \(-0.518309\pi\)
−0.0574878 + 0.998346i \(0.518309\pi\)
\(402\) 19.8548 0.990268
\(403\) −22.5438 −1.12299
\(404\) −18.9949 −0.945034
\(405\) −89.3592 −4.44030
\(406\) 13.7580 0.682797
\(407\) 0 0
\(408\) 7.42064 0.367377
\(409\) 11.4870 0.567996 0.283998 0.958825i \(-0.408339\pi\)
0.283998 + 0.958825i \(0.408339\pi\)
\(410\) −9.82940 −0.485439
\(411\) 16.3666 0.807306
\(412\) 20.0459 0.987590
\(413\) 8.89363 0.437627
\(414\) −18.2186 −0.895396
\(415\) −7.57290 −0.371739
\(416\) −3.52841 −0.172995
\(417\) 20.3412 0.996113
\(418\) 0 0
\(419\) 6.21369 0.303558 0.151779 0.988414i \(-0.451500\pi\)
0.151779 + 0.988414i \(0.451500\pi\)
\(420\) −17.7266 −0.864968
\(421\) 12.5934 0.613764 0.306882 0.951748i \(-0.400714\pi\)
0.306882 + 0.951748i \(0.400714\pi\)
\(422\) 1.88129 0.0915797
\(423\) −44.7144 −2.17409
\(424\) −4.40017 −0.213691
\(425\) 29.2559 1.41912
\(426\) −34.2864 −1.66118
\(427\) −11.3842 −0.550919
\(428\) −0.849872 −0.0410801
\(429\) 0 0
\(430\) 48.0118 2.31533
\(431\) −24.4570 −1.17805 −0.589026 0.808114i \(-0.700488\pi\)
−0.589026 + 0.808114i \(0.700488\pi\)
\(432\) −13.4351 −0.646397
\(433\) 16.3892 0.787616 0.393808 0.919193i \(-0.371157\pi\)
0.393808 + 0.919193i \(0.371157\pi\)
\(434\) −8.45206 −0.405712
\(435\) 139.364 6.68199
\(436\) −8.72303 −0.417757
\(437\) −2.52841 −0.120950
\(438\) 5.35781 0.256006
\(439\) 31.8353 1.51942 0.759708 0.650265i \(-0.225342\pi\)
0.759708 + 0.650265i \(0.225342\pi\)
\(440\) 0 0
\(441\) −37.8294 −1.80140
\(442\) −8.19602 −0.389845
\(443\) −14.7594 −0.701239 −0.350620 0.936518i \(-0.614029\pi\)
−0.350620 + 0.936518i \(0.614029\pi\)
\(444\) −25.0468 −1.18867
\(445\) −25.3602 −1.20219
\(446\) −20.0868 −0.951140
\(447\) −43.8442 −2.07376
\(448\) −1.32286 −0.0624993
\(449\) 1.44605 0.0682435 0.0341218 0.999418i \(-0.489137\pi\)
0.0341218 + 0.999418i \(0.489137\pi\)
\(450\) −90.7523 −4.27811
\(451\) 0 0
\(452\) 3.65666 0.171995
\(453\) −15.9381 −0.748838
\(454\) 3.55029 0.166624
\(455\) 19.5788 0.917868
\(456\) −3.19461 −0.149601
\(457\) −9.58524 −0.448379 −0.224189 0.974546i \(-0.571973\pi\)
−0.224189 + 0.974546i \(0.571973\pi\)
\(458\) −7.72162 −0.360808
\(459\) −31.2079 −1.45666
\(460\) −10.6057 −0.494494
\(461\) 9.91624 0.461845 0.230923 0.972972i \(-0.425826\pi\)
0.230923 + 0.972972i \(0.425826\pi\)
\(462\) 0 0
\(463\) 3.22743 0.149992 0.0749958 0.997184i \(-0.476106\pi\)
0.0749958 + 0.997184i \(0.476106\pi\)
\(464\) 10.4002 0.482816
\(465\) −85.6167 −3.97038
\(466\) −21.9081 −1.01487
\(467\) 36.0118 1.66643 0.833213 0.552952i \(-0.186499\pi\)
0.833213 + 0.552952i \(0.186499\pi\)
\(468\) 25.4242 1.17523
\(469\) −8.22170 −0.379643
\(470\) −26.0299 −1.20067
\(471\) −6.51702 −0.300288
\(472\) 6.72303 0.309452
\(473\) 0 0
\(474\) −18.0708 −0.830021
\(475\) −12.5948 −0.577888
\(476\) −3.07282 −0.140843
\(477\) 31.7056 1.45170
\(478\) 17.6676 0.808097
\(479\) 7.68668 0.351213 0.175607 0.984460i \(-0.443811\pi\)
0.175607 + 0.984460i \(0.443811\pi\)
\(480\) −13.4002 −0.611631
\(481\) 27.6640 1.26137
\(482\) 17.0999 0.778880
\(483\) 10.6852 0.486191
\(484\) 0 0
\(485\) 41.9461 1.90468
\(486\) 27.7506 1.25879
\(487\) −34.6356 −1.56949 −0.784745 0.619819i \(-0.787206\pi\)
−0.784745 + 0.619819i \(0.787206\pi\)
\(488\) −8.60572 −0.389563
\(489\) −19.6616 −0.889128
\(490\) −22.0219 −0.994847
\(491\) −33.1458 −1.49585 −0.747924 0.663784i \(-0.768949\pi\)
−0.747924 + 0.663784i \(0.768949\pi\)
\(492\) 7.48606 0.337498
\(493\) 24.1581 1.08803
\(494\) 3.52841 0.158751
\(495\) 0 0
\(496\) −6.38923 −0.286885
\(497\) 14.1977 0.636854
\(498\) 5.76751 0.258448
\(499\) −22.9949 −1.02940 −0.514698 0.857372i \(-0.672096\pi\)
−0.514698 + 0.857372i \(0.672096\pi\)
\(500\) −31.8572 −1.42470
\(501\) 11.9590 0.534291
\(502\) −13.4461 −0.600127
\(503\) −7.30098 −0.325535 −0.162767 0.986664i \(-0.552042\pi\)
−0.162767 + 0.986664i \(0.552042\pi\)
\(504\) 9.53195 0.424587
\(505\) 79.6764 3.54556
\(506\) 0 0
\(507\) 1.75798 0.0780744
\(508\) 0.256496 0.0113802
\(509\) −38.3213 −1.69856 −0.849282 0.527940i \(-0.822965\pi\)
−0.849282 + 0.527940i \(0.822965\pi\)
\(510\) −31.1267 −1.37831
\(511\) −2.21862 −0.0981461
\(512\) −1.00000 −0.0441942
\(513\) 13.4351 0.593175
\(514\) −10.1546 −0.447901
\(515\) −84.0847 −3.70522
\(516\) −36.5657 −1.60972
\(517\) 0 0
\(518\) 10.3717 0.455705
\(519\) 29.0289 1.27423
\(520\) 14.8003 0.649037
\(521\) 8.17649 0.358219 0.179109 0.983829i \(-0.442678\pi\)
0.179109 + 0.983829i \(0.442678\pi\)
\(522\) −74.9389 −3.27999
\(523\) −24.3806 −1.06609 −0.533045 0.846087i \(-0.678953\pi\)
−0.533045 + 0.846087i \(0.678953\pi\)
\(524\) 3.70255 0.161747
\(525\) 53.2259 2.32297
\(526\) 24.9081 1.08604
\(527\) −14.8413 −0.646497
\(528\) 0 0
\(529\) −16.6071 −0.722049
\(530\) 18.4570 0.801721
\(531\) −48.4431 −2.10225
\(532\) 1.32286 0.0573533
\(533\) −8.26827 −0.358138
\(534\) 19.3143 0.835810
\(535\) 3.56489 0.154123
\(536\) −6.21509 −0.268451
\(537\) −28.0088 −1.20867
\(538\) 29.4761 1.27080
\(539\) 0 0
\(540\) 56.3551 2.42514
\(541\) 41.5788 1.78761 0.893806 0.448454i \(-0.148025\pi\)
0.893806 + 0.448454i \(0.148025\pi\)
\(542\) −2.99046 −0.128451
\(543\) −62.8721 −2.69810
\(544\) −2.32286 −0.0995919
\(545\) 36.5897 1.56733
\(546\) −14.9112 −0.638140
\(547\) 6.05778 0.259012 0.129506 0.991579i \(-0.458661\pi\)
0.129506 + 0.991579i \(0.458661\pi\)
\(548\) −5.12319 −0.218852
\(549\) 62.0090 2.64648
\(550\) 0 0
\(551\) −10.4002 −0.443062
\(552\) 8.07730 0.343793
\(553\) 7.48298 0.318209
\(554\) −16.8622 −0.716407
\(555\) 105.062 4.45962
\(556\) −6.36735 −0.270036
\(557\) 30.0527 1.27337 0.636687 0.771122i \(-0.280304\pi\)
0.636687 + 0.771122i \(0.280304\pi\)
\(558\) 46.0379 1.94894
\(559\) 40.3864 1.70816
\(560\) 5.54889 0.234483
\(561\) 0 0
\(562\) −0.989060 −0.0417210
\(563\) 29.4410 1.24079 0.620395 0.784289i \(-0.286972\pi\)
0.620395 + 0.784289i \(0.286972\pi\)
\(564\) 19.8243 0.834755
\(565\) −15.3383 −0.645286
\(566\) −5.97812 −0.251279
\(567\) −28.1814 −1.18351
\(568\) 10.7326 0.450328
\(569\) 30.5438 1.28046 0.640232 0.768182i \(-0.278838\pi\)
0.640232 + 0.768182i \(0.278838\pi\)
\(570\) 13.4002 0.561271
\(571\) −2.95399 −0.123621 −0.0618104 0.998088i \(-0.519687\pi\)
−0.0618104 + 0.998088i \(0.519687\pi\)
\(572\) 0 0
\(573\) 30.2769 1.26483
\(574\) −3.09991 −0.129388
\(575\) 31.8448 1.32802
\(576\) 7.20555 0.300231
\(577\) −4.69021 −0.195256 −0.0976280 0.995223i \(-0.531126\pi\)
−0.0976280 + 0.995223i \(0.531126\pi\)
\(578\) 11.6043 0.482676
\(579\) 25.5569 1.06211
\(580\) −43.6247 −1.81142
\(581\) −2.38828 −0.0990824
\(582\) −31.9461 −1.32421
\(583\) 0 0
\(584\) −1.67714 −0.0694005
\(585\) −106.645 −4.40921
\(586\) 14.5489 0.601009
\(587\) −11.8273 −0.488164 −0.244082 0.969755i \(-0.578487\pi\)
−0.244082 + 0.969755i \(0.578487\pi\)
\(588\) 16.7718 0.691659
\(589\) 6.38923 0.263263
\(590\) −28.2005 −1.16100
\(591\) −78.4081 −3.22528
\(592\) 7.84034 0.322236
\(593\) 24.9621 1.02507 0.512536 0.858666i \(-0.328706\pi\)
0.512536 + 0.858666i \(0.328706\pi\)
\(594\) 0 0
\(595\) 12.8893 0.528410
\(596\) 13.7244 0.562175
\(597\) −64.5283 −2.64097
\(598\) −8.92129 −0.364819
\(599\) −27.0131 −1.10372 −0.551862 0.833935i \(-0.686083\pi\)
−0.551862 + 0.833935i \(0.686083\pi\)
\(600\) 40.2354 1.64261
\(601\) −7.24343 −0.295466 −0.147733 0.989027i \(-0.547198\pi\)
−0.147733 + 0.989027i \(0.547198\pi\)
\(602\) 15.1415 0.617123
\(603\) 44.7832 1.82371
\(604\) 4.98906 0.203002
\(605\) 0 0
\(606\) −60.6815 −2.46502
\(607\) −27.4411 −1.11380 −0.556900 0.830579i \(-0.688009\pi\)
−0.556900 + 0.830579i \(0.688009\pi\)
\(608\) 1.00000 0.0405554
\(609\) 43.9514 1.78100
\(610\) 36.0977 1.46155
\(611\) −21.8958 −0.885808
\(612\) 16.7375 0.676573
\(613\) 1.97011 0.0795718 0.0397859 0.999208i \(-0.487332\pi\)
0.0397859 + 0.999208i \(0.487332\pi\)
\(614\) 9.69161 0.391122
\(615\) −31.4011 −1.26622
\(616\) 0 0
\(617\) 2.42110 0.0974696 0.0487348 0.998812i \(-0.484481\pi\)
0.0487348 + 0.998812i \(0.484481\pi\)
\(618\) 64.0389 2.57602
\(619\) 38.0118 1.52782 0.763911 0.645322i \(-0.223277\pi\)
0.763911 + 0.645322i \(0.223277\pi\)
\(620\) 26.8003 1.07633
\(621\) −33.9695 −1.36315
\(622\) 18.8229 0.754731
\(623\) −7.99787 −0.320428
\(624\) −11.2719 −0.451238
\(625\) 70.6546 2.82618
\(626\) −27.1800 −1.08633
\(627\) 0 0
\(628\) 2.04000 0.0814050
\(629\) 18.2120 0.726160
\(630\) −39.9828 −1.59295
\(631\) 27.2873 1.08629 0.543146 0.839638i \(-0.317233\pi\)
0.543146 + 0.839638i \(0.317233\pi\)
\(632\) 5.65666 0.225010
\(633\) 6.00999 0.238876
\(634\) 14.3697 0.570694
\(635\) −1.07590 −0.0426959
\(636\) −14.0568 −0.557389
\(637\) −18.5243 −0.733960
\(638\) 0 0
\(639\) −77.3341 −3.05929
\(640\) 4.19461 0.165807
\(641\) 0.589724 0.0232927 0.0116464 0.999932i \(-0.496293\pi\)
0.0116464 + 0.999932i \(0.496293\pi\)
\(642\) −2.71501 −0.107153
\(643\) 11.1197 0.438517 0.219258 0.975667i \(-0.429636\pi\)
0.219258 + 0.975667i \(0.429636\pi\)
\(644\) −3.34474 −0.131801
\(645\) 153.379 6.03929
\(646\) 2.32286 0.0913918
\(647\) 9.19837 0.361625 0.180813 0.983518i \(-0.442127\pi\)
0.180813 + 0.983518i \(0.442127\pi\)
\(648\) −21.3033 −0.836874
\(649\) 0 0
\(650\) −44.4396 −1.74306
\(651\) −27.0011 −1.05825
\(652\) 6.15461 0.241033
\(653\) −12.9940 −0.508494 −0.254247 0.967139i \(-0.581828\pi\)
−0.254247 + 0.967139i \(0.581828\pi\)
\(654\) −27.8667 −1.08967
\(655\) −15.5308 −0.606837
\(656\) −2.34334 −0.0914920
\(657\) 12.0847 0.471470
\(658\) −8.20909 −0.320023
\(659\) 2.11012 0.0821988 0.0410994 0.999155i \(-0.486914\pi\)
0.0410994 + 0.999155i \(0.486914\pi\)
\(660\) 0 0
\(661\) 7.58665 0.295086 0.147543 0.989056i \(-0.452863\pi\)
0.147543 + 0.989056i \(0.452863\pi\)
\(662\) −18.9635 −0.737038
\(663\) −26.1831 −1.01687
\(664\) −1.80539 −0.0700626
\(665\) −5.54889 −0.215177
\(666\) −56.4940 −2.18910
\(667\) 26.2959 1.01818
\(668\) −3.74350 −0.144841
\(669\) −64.1697 −2.48094
\(670\) 26.0699 1.00717
\(671\) 0 0
\(672\) −4.22603 −0.163023
\(673\) −33.2792 −1.28282 −0.641410 0.767199i \(-0.721650\pi\)
−0.641410 + 0.767199i \(0.721650\pi\)
\(674\) 30.1595 1.16170
\(675\) −169.212 −6.51298
\(676\) −0.550294 −0.0211651
\(677\) 0.160391 0.00616431 0.00308216 0.999995i \(-0.499019\pi\)
0.00308216 + 0.999995i \(0.499019\pi\)
\(678\) 11.6816 0.448630
\(679\) 13.2286 0.507668
\(680\) 9.74350 0.373646
\(681\) 11.3418 0.434619
\(682\) 0 0
\(683\) 4.60477 0.176197 0.0880983 0.996112i \(-0.471921\pi\)
0.0880983 + 0.996112i \(0.471921\pi\)
\(684\) −7.20555 −0.275511
\(685\) 21.4898 0.821084
\(686\) −16.2051 −0.618714
\(687\) −24.6676 −0.941128
\(688\) 11.4461 0.436377
\(689\) 15.5256 0.591479
\(690\) −33.8812 −1.28983
\(691\) 6.54008 0.248796 0.124398 0.992232i \(-0.460300\pi\)
0.124398 + 0.992232i \(0.460300\pi\)
\(692\) −9.08684 −0.345430
\(693\) 0 0
\(694\) 23.5997 0.895833
\(695\) 26.7086 1.01311
\(696\) 33.2245 1.25937
\(697\) −5.44325 −0.206178
\(698\) −29.2086 −1.10556
\(699\) −69.9879 −2.64719
\(700\) −16.6611 −0.629732
\(701\) 12.3455 0.466282 0.233141 0.972443i \(-0.425100\pi\)
0.233141 + 0.972443i \(0.425100\pi\)
\(702\) 47.4046 1.78917
\(703\) −7.84034 −0.295704
\(704\) 0 0
\(705\) −83.1554 −3.13182
\(706\) 10.2274 0.384915
\(707\) 25.1277 0.945023
\(708\) 21.4775 0.807173
\(709\) −26.2464 −0.985704 −0.492852 0.870113i \(-0.664046\pi\)
−0.492852 + 0.870113i \(0.664046\pi\)
\(710\) −45.0190 −1.68953
\(711\) −40.7594 −1.52860
\(712\) −6.04589 −0.226579
\(713\) −16.1546 −0.604995
\(714\) −9.81648 −0.367373
\(715\) 0 0
\(716\) 8.76751 0.327657
\(717\) 56.4412 2.10783
\(718\) 0.876807 0.0327221
\(719\) 5.61032 0.209230 0.104615 0.994513i \(-0.466639\pi\)
0.104615 + 0.994513i \(0.466639\pi\)
\(720\) −30.2245 −1.12640
\(721\) −26.5179 −0.987579
\(722\) −1.00000 −0.0372161
\(723\) 54.6276 2.03162
\(724\) 19.6807 0.731426
\(725\) 130.988 4.86476
\(726\) 0 0
\(727\) −9.17060 −0.340119 −0.170059 0.985434i \(-0.554396\pi\)
−0.170059 + 0.985434i \(0.554396\pi\)
\(728\) 4.66760 0.172993
\(729\) 24.7423 0.916382
\(730\) 7.03495 0.260375
\(731\) 26.5876 0.983378
\(732\) −27.4919 −1.01613
\(733\) −26.9140 −0.994091 −0.497045 0.867725i \(-0.665582\pi\)
−0.497045 + 0.867725i \(0.665582\pi\)
\(734\) −10.5030 −0.387673
\(735\) −70.3514 −2.59495
\(736\) −2.52841 −0.0931986
\(737\) 0 0
\(738\) 16.8850 0.621547
\(739\) −43.4032 −1.59661 −0.798307 0.602251i \(-0.794271\pi\)
−0.798307 + 0.602251i \(0.794271\pi\)
\(740\) −32.8872 −1.20896
\(741\) 11.2719 0.414084
\(742\) 5.82081 0.213689
\(743\) −18.0671 −0.662817 −0.331409 0.943487i \(-0.607524\pi\)
−0.331409 + 0.943487i \(0.607524\pi\)
\(744\) −20.4111 −0.748307
\(745\) −57.5687 −2.10915
\(746\) 34.9249 1.27869
\(747\) 13.0088 0.475967
\(748\) 0 0
\(749\) 1.12426 0.0410797
\(750\) −101.771 −3.71616
\(751\) −44.0336 −1.60681 −0.803405 0.595432i \(-0.796981\pi\)
−0.803405 + 0.595432i \(0.796981\pi\)
\(752\) −6.20555 −0.226293
\(753\) −42.9549 −1.56537
\(754\) −36.6961 −1.33639
\(755\) −20.9272 −0.761618
\(756\) 17.7728 0.646390
\(757\) 19.5489 0.710517 0.355258 0.934768i \(-0.384393\pi\)
0.355258 + 0.934768i \(0.384393\pi\)
\(758\) 13.9695 0.507396
\(759\) 0 0
\(760\) −4.19461 −0.152155
\(761\) 36.8548 1.33599 0.667993 0.744168i \(-0.267154\pi\)
0.667993 + 0.744168i \(0.267154\pi\)
\(762\) 0.819406 0.0296839
\(763\) 11.5394 0.417753
\(764\) −9.47747 −0.342883
\(765\) −70.2073 −2.53835
\(766\) −35.3223 −1.27625
\(767\) −23.7216 −0.856538
\(768\) −3.19461 −0.115276
\(769\) 6.09778 0.219892 0.109946 0.993938i \(-0.464932\pi\)
0.109946 + 0.993938i \(0.464932\pi\)
\(770\) 0 0
\(771\) −32.4400 −1.16830
\(772\) −8.00000 −0.287926
\(773\) 3.70407 0.133226 0.0666131 0.997779i \(-0.478781\pi\)
0.0666131 + 0.997779i \(0.478781\pi\)
\(774\) −82.4752 −2.96451
\(775\) −80.4709 −2.89060
\(776\) 10.0000 0.358979
\(777\) 33.1335 1.18866
\(778\) −12.5838 −0.451152
\(779\) 2.34334 0.0839588
\(780\) 47.2813 1.69294
\(781\) 0 0
\(782\) −5.87315 −0.210024
\(783\) −139.727 −4.99345
\(784\) −5.25004 −0.187501
\(785\) −8.55702 −0.305413
\(786\) 11.8282 0.421899
\(787\) 6.14367 0.218998 0.109499 0.993987i \(-0.465075\pi\)
0.109499 + 0.993987i \(0.465075\pi\)
\(788\) 24.5438 0.874338
\(789\) 79.5718 2.83283
\(790\) −23.7275 −0.844187
\(791\) −4.83726 −0.171993
\(792\) 0 0
\(793\) 30.3645 1.07828
\(794\) −0.924098 −0.0327950
\(795\) 58.9630 2.09120
\(796\) 20.1991 0.715938
\(797\) 11.3957 0.403656 0.201828 0.979421i \(-0.435312\pi\)
0.201828 + 0.979421i \(0.435312\pi\)
\(798\) 4.22603 0.149600
\(799\) −14.4146 −0.509953
\(800\) −12.5948 −0.445293
\(801\) 43.5640 1.53926
\(802\) 2.30238 0.0813000
\(803\) 0 0
\(804\) −19.8548 −0.700225
\(805\) 14.0299 0.494489
\(806\) 22.5438 0.794073
\(807\) 94.1646 3.31475
\(808\) 18.9949 0.668240
\(809\) 2.18367 0.0767739 0.0383869 0.999263i \(-0.487778\pi\)
0.0383869 + 0.999263i \(0.487778\pi\)
\(810\) 89.3592 3.13976
\(811\) −41.8142 −1.46830 −0.734148 0.678989i \(-0.762418\pi\)
−0.734148 + 0.678989i \(0.762418\pi\)
\(812\) −13.7580 −0.482810
\(813\) −9.55337 −0.335051
\(814\) 0 0
\(815\) −25.8162 −0.904302
\(816\) −7.42064 −0.259774
\(817\) −11.4461 −0.400447
\(818\) −11.4870 −0.401634
\(819\) −33.6327 −1.17522
\(820\) 9.82940 0.343257
\(821\) 7.65974 0.267327 0.133663 0.991027i \(-0.457326\pi\)
0.133663 + 0.991027i \(0.457326\pi\)
\(822\) −16.3666 −0.570851
\(823\) −13.8893 −0.484151 −0.242075 0.970257i \(-0.577828\pi\)
−0.242075 + 0.970257i \(0.577828\pi\)
\(824\) −20.0459 −0.698332
\(825\) 0 0
\(826\) −8.89363 −0.309449
\(827\) 29.3887 1.02194 0.510972 0.859597i \(-0.329286\pi\)
0.510972 + 0.859597i \(0.329286\pi\)
\(828\) 18.2186 0.633141
\(829\) 8.60432 0.298840 0.149420 0.988774i \(-0.452259\pi\)
0.149420 + 0.988774i \(0.452259\pi\)
\(830\) 7.57290 0.262859
\(831\) −53.8683 −1.86867
\(832\) 3.52841 0.122326
\(833\) −12.1951 −0.422536
\(834\) −20.3412 −0.704358
\(835\) 15.7026 0.543409
\(836\) 0 0
\(837\) 85.8400 2.96706
\(838\) −6.21369 −0.214648
\(839\) 0.815377 0.0281499 0.0140750 0.999901i \(-0.495520\pi\)
0.0140750 + 0.999901i \(0.495520\pi\)
\(840\) 17.7266 0.611624
\(841\) 79.1635 2.72977
\(842\) −12.5934 −0.433996
\(843\) −3.15966 −0.108825
\(844\) −1.88129 −0.0647566
\(845\) 2.30827 0.0794069
\(846\) 44.7144 1.53731
\(847\) 0 0
\(848\) 4.40017 0.151102
\(849\) −19.0978 −0.655434
\(850\) −29.2559 −1.00347
\(851\) 19.8236 0.679545
\(852\) 34.2864 1.17463
\(853\) 6.70856 0.229697 0.114848 0.993383i \(-0.463362\pi\)
0.114848 + 0.993383i \(0.463362\pi\)
\(854\) 11.3842 0.389558
\(855\) 30.2245 1.03366
\(856\) 0.849872 0.0290480
\(857\) −17.6416 −0.602626 −0.301313 0.953525i \(-0.597425\pi\)
−0.301313 + 0.953525i \(0.597425\pi\)
\(858\) 0 0
\(859\) 36.3520 1.24032 0.620158 0.784477i \(-0.287069\pi\)
0.620158 + 0.784477i \(0.287069\pi\)
\(860\) −48.0118 −1.63719
\(861\) −9.90302 −0.337494
\(862\) 24.4570 0.833009
\(863\) 24.1984 0.823722 0.411861 0.911247i \(-0.364879\pi\)
0.411861 + 0.911247i \(0.364879\pi\)
\(864\) 13.4351 0.457072
\(865\) 38.1158 1.29598
\(866\) −16.3892 −0.556929
\(867\) 37.0713 1.25901
\(868\) 8.45206 0.286882
\(869\) 0 0
\(870\) −139.364 −4.72488
\(871\) 21.9294 0.743050
\(872\) 8.72303 0.295399
\(873\) −72.0555 −2.43871
\(874\) 2.52841 0.0855249
\(875\) 42.1426 1.42468
\(876\) −5.35781 −0.181024
\(877\) −43.3477 −1.46375 −0.731874 0.681440i \(-0.761354\pi\)
−0.731874 + 0.681440i \(0.761354\pi\)
\(878\) −31.8353 −1.07439
\(879\) 46.4781 1.56767
\(880\) 0 0
\(881\) 1.34121 0.0451865 0.0225932 0.999745i \(-0.492808\pi\)
0.0225932 + 0.999745i \(0.492808\pi\)
\(882\) 37.8294 1.27378
\(883\) 41.2924 1.38960 0.694800 0.719203i \(-0.255493\pi\)
0.694800 + 0.719203i \(0.255493\pi\)
\(884\) 8.19602 0.275662
\(885\) −90.0897 −3.02833
\(886\) 14.7594 0.495851
\(887\) 8.18142 0.274705 0.137353 0.990522i \(-0.456141\pi\)
0.137353 + 0.990522i \(0.456141\pi\)
\(888\) 25.0468 0.840517
\(889\) −0.339309 −0.0113800
\(890\) 25.3602 0.850075
\(891\) 0 0
\(892\) 20.0868 0.672557
\(893\) 6.20555 0.207661
\(894\) 43.8442 1.46637
\(895\) −36.7763 −1.22930
\(896\) 1.32286 0.0441937
\(897\) −28.5001 −0.951590
\(898\) −1.44605 −0.0482555
\(899\) −66.4490 −2.21620
\(900\) 90.7523 3.02508
\(901\) 10.2210 0.340510
\(902\) 0 0
\(903\) 48.3714 1.60970
\(904\) −3.65666 −0.121619
\(905\) −82.5528 −2.74415
\(906\) 15.9381 0.529508
\(907\) −4.76398 −0.158185 −0.0790927 0.996867i \(-0.525202\pi\)
−0.0790927 + 0.996867i \(0.525202\pi\)
\(908\) −3.55029 −0.117821
\(909\) −136.869 −4.53966
\(910\) −19.5788 −0.649030
\(911\) 10.5589 0.349831 0.174916 0.984583i \(-0.444035\pi\)
0.174916 + 0.984583i \(0.444035\pi\)
\(912\) 3.19461 0.105784
\(913\) 0 0
\(914\) 9.58524 0.317052
\(915\) 115.318 3.81230
\(916\) 7.72162 0.255130
\(917\) −4.89796 −0.161745
\(918\) 31.2079 1.03001
\(919\) 37.5764 1.23953 0.619766 0.784787i \(-0.287227\pi\)
0.619766 + 0.784787i \(0.287227\pi\)
\(920\) 10.6057 0.349660
\(921\) 30.9609 1.02020
\(922\) −9.91624 −0.326574
\(923\) −37.8689 −1.24647
\(924\) 0 0
\(925\) 98.7473 3.24679
\(926\) −3.22743 −0.106060
\(927\) 144.442 4.74409
\(928\) −10.4002 −0.341402
\(929\) −4.07590 −0.133726 −0.0668630 0.997762i \(-0.521299\pi\)
−0.0668630 + 0.997762i \(0.521299\pi\)
\(930\) 85.6167 2.80748
\(931\) 5.25004 0.172063
\(932\) 21.9081 0.717624
\(933\) 60.1320 1.96863
\(934\) −36.0118 −1.17834
\(935\) 0 0
\(936\) −25.4242 −0.831015
\(937\) −54.4273 −1.77806 −0.889030 0.457848i \(-0.848620\pi\)
−0.889030 + 0.457848i \(0.848620\pi\)
\(938\) 8.22170 0.268448
\(939\) −86.8297 −2.83358
\(940\) 26.0299 0.849002
\(941\) −26.7152 −0.870890 −0.435445 0.900215i \(-0.643409\pi\)
−0.435445 + 0.900215i \(0.643409\pi\)
\(942\) 6.51702 0.212336
\(943\) −5.92493 −0.192942
\(944\) −6.72303 −0.218816
\(945\) −74.5500 −2.42511
\(946\) 0 0
\(947\) 56.9448 1.85046 0.925229 0.379409i \(-0.123873\pi\)
0.925229 + 0.379409i \(0.123873\pi\)
\(948\) 18.0708 0.586914
\(949\) 5.91764 0.192095
\(950\) 12.5948 0.408629
\(951\) 45.9056 1.48859
\(952\) 3.07282 0.0995908
\(953\) −0.764553 −0.0247663 −0.0123832 0.999923i \(-0.503942\pi\)
−0.0123832 + 0.999923i \(0.503942\pi\)
\(954\) −31.7056 −1.02651
\(955\) 39.7543 1.28642
\(956\) −17.6676 −0.571411
\(957\) 0 0
\(958\) −7.68668 −0.248345
\(959\) 6.77727 0.218850
\(960\) 13.4002 0.432489
\(961\) 9.82221 0.316846
\(962\) −27.6640 −0.891922
\(963\) −6.12380 −0.197337
\(964\) −17.0999 −0.550751
\(965\) 33.5569 1.08023
\(966\) −10.6852 −0.343789
\(967\) −26.5510 −0.853823 −0.426912 0.904293i \(-0.640398\pi\)
−0.426912 + 0.904293i \(0.640398\pi\)
\(968\) 0 0
\(969\) 7.42064 0.238385
\(970\) −41.9461 −1.34681
\(971\) −35.6335 −1.14353 −0.571766 0.820416i \(-0.693742\pi\)
−0.571766 + 0.820416i \(0.693742\pi\)
\(972\) −27.7506 −0.890100
\(973\) 8.42312 0.270033
\(974\) 34.6356 1.10980
\(975\) −141.967 −4.54659
\(976\) 8.60572 0.275462
\(977\) 23.8544 0.763168 0.381584 0.924334i \(-0.375379\pi\)
0.381584 + 0.924334i \(0.375379\pi\)
\(978\) 19.6616 0.628709
\(979\) 0 0
\(980\) 22.0219 0.703463
\(981\) −62.8542 −2.00678
\(982\) 33.1458 1.05772
\(983\) −21.5520 −0.687401 −0.343701 0.939079i \(-0.611681\pi\)
−0.343701 + 0.939079i \(0.611681\pi\)
\(984\) −7.48606 −0.238647
\(985\) −102.952 −3.28032
\(986\) −24.1581 −0.769352
\(987\) −26.2249 −0.834746
\(988\) −3.52841 −0.112254
\(989\) 28.9404 0.920250
\(990\) 0 0
\(991\) 14.0049 0.444881 0.222441 0.974946i \(-0.428598\pi\)
0.222441 + 0.974946i \(0.428598\pi\)
\(992\) 6.38923 0.202858
\(993\) −60.5811 −1.92248
\(994\) −14.1977 −0.450323
\(995\) −84.7274 −2.68604
\(996\) −5.76751 −0.182751
\(997\) 47.2733 1.49716 0.748581 0.663044i \(-0.230736\pi\)
0.748581 + 0.663044i \(0.230736\pi\)
\(998\) 22.9949 0.727892
\(999\) −105.336 −3.33268
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 4598.2.a.bs.1.1 4
11.10 odd 2 4598.2.a.bv.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.bs.1.1 4 1.1 even 1 trivial
4598.2.a.bv.1.1 yes 4 11.10 odd 2