Properties

Label 1450.2.a.j.1.2
Level $1450$
Weight $2$
Character 1450.1
Self dual yes
Analytic conductor $11.578$
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] = [1450,2,Mod(1,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, 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("1450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 1450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -4.44949 q^{7} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -4.44949 q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +6.44949 q^{19} -10.8990 q^{21} -2.00000 q^{22} +2.44949 q^{23} -2.44949 q^{24} -2.44949 q^{26} -4.44949 q^{28} -1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +3.44949 q^{37} -6.44949 q^{38} +6.00000 q^{39} +11.3485 q^{41} +10.8990 q^{42} +8.89898 q^{43} +2.00000 q^{44} -2.44949 q^{46} -5.89898 q^{47} +2.44949 q^{48} +12.7980 q^{49} +4.89898 q^{51} +2.44949 q^{52} -10.4495 q^{53} +4.44949 q^{56} +15.7980 q^{57} +1.00000 q^{58} +8.55051 q^{59} +3.44949 q^{61} +3.00000 q^{62} -13.3485 q^{63} +1.00000 q^{64} -4.89898 q^{66} +13.4495 q^{67} +2.00000 q^{68} +6.00000 q^{69} +11.3485 q^{71} -3.00000 q^{72} -13.3485 q^{73} -3.44949 q^{74} +6.44949 q^{76} -8.89898 q^{77} -6.00000 q^{78} -2.89898 q^{79} -9.00000 q^{81} -11.3485 q^{82} +6.00000 q^{83} -10.8990 q^{84} -8.89898 q^{86} -2.44949 q^{87} -2.00000 q^{88} +3.55051 q^{89} -10.8990 q^{91} +2.44949 q^{92} -7.34847 q^{93} +5.89898 q^{94} -2.44949 q^{96} -7.34847 q^{97} -12.7980 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 12 q^{21} - 4 q^{22} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36} + 2 q^{37} - 8 q^{38} + 12 q^{39} + 8 q^{41} + 12 q^{42} + 8 q^{43} + 4 q^{44} - 2 q^{47} + 6 q^{49} - 16 q^{53} + 4 q^{56} + 12 q^{57} + 2 q^{58} + 22 q^{59} + 2 q^{61} + 6 q^{62} - 12 q^{63} + 2 q^{64} + 22 q^{67} + 4 q^{68} + 12 q^{69} + 8 q^{71} - 6 q^{72} - 12 q^{73} - 2 q^{74} + 8 q^{76} - 8 q^{77} - 12 q^{78} + 4 q^{79} - 18 q^{81} - 8 q^{82} + 12 q^{83} - 12 q^{84} - 8 q^{86} - 4 q^{88} + 12 q^{89} - 12 q^{91} + 2 q^{94} - 6 q^{98} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 2.44949 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.44949 −1.00000
\(7\) −4.44949 −1.68175 −0.840875 0.541230i \(-0.817959\pi\)
−0.840875 + 0.541230i \(0.817959\pi\)
\(8\) −1.00000 −0.353553
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 2.44949 0.707107
\(13\) 2.44949 0.679366 0.339683 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(14\) 4.44949 1.18918
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −3.00000 −0.707107
\(19\) 6.44949 1.47961 0.739807 0.672819i \(-0.234917\pi\)
0.739807 + 0.672819i \(0.234917\pi\)
\(20\) 0 0
\(21\) −10.8990 −2.37835
\(22\) −2.00000 −0.426401
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) −2.44949 −0.500000
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 0 0
\(28\) −4.44949 −0.840875
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.89898 0.852803
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 3.44949 0.567093 0.283546 0.958959i \(-0.408489\pi\)
0.283546 + 0.958959i \(0.408489\pi\)
\(38\) −6.44949 −1.04625
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 11.3485 1.77233 0.886167 0.463367i \(-0.153359\pi\)
0.886167 + 0.463367i \(0.153359\pi\)
\(42\) 10.8990 1.68175
\(43\) 8.89898 1.35708 0.678541 0.734563i \(-0.262613\pi\)
0.678541 + 0.734563i \(0.262613\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −2.44949 −0.361158
\(47\) −5.89898 −0.860455 −0.430227 0.902721i \(-0.641567\pi\)
−0.430227 + 0.902721i \(0.641567\pi\)
\(48\) 2.44949 0.353553
\(49\) 12.7980 1.82828
\(50\) 0 0
\(51\) 4.89898 0.685994
\(52\) 2.44949 0.339683
\(53\) −10.4495 −1.43535 −0.717674 0.696379i \(-0.754793\pi\)
−0.717674 + 0.696379i \(0.754793\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.44949 0.594588
\(57\) 15.7980 2.09249
\(58\) 1.00000 0.131306
\(59\) 8.55051 1.11318 0.556591 0.830787i \(-0.312109\pi\)
0.556591 + 0.830787i \(0.312109\pi\)
\(60\) 0 0
\(61\) 3.44949 0.441662 0.220831 0.975312i \(-0.429123\pi\)
0.220831 + 0.975312i \(0.429123\pi\)
\(62\) 3.00000 0.381000
\(63\) −13.3485 −1.68175
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.89898 −0.603023
\(67\) 13.4495 1.64312 0.821558 0.570124i \(-0.193105\pi\)
0.821558 + 0.570124i \(0.193105\pi\)
\(68\) 2.00000 0.242536
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) 11.3485 1.34682 0.673408 0.739271i \(-0.264830\pi\)
0.673408 + 0.739271i \(0.264830\pi\)
\(72\) −3.00000 −0.353553
\(73\) −13.3485 −1.56232 −0.781160 0.624331i \(-0.785372\pi\)
−0.781160 + 0.624331i \(0.785372\pi\)
\(74\) −3.44949 −0.400995
\(75\) 0 0
\(76\) 6.44949 0.739807
\(77\) −8.89898 −1.01413
\(78\) −6.00000 −0.679366
\(79\) −2.89898 −0.326161 −0.163080 0.986613i \(-0.552143\pi\)
−0.163080 + 0.986613i \(0.552143\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) −11.3485 −1.25323
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −10.8990 −1.18918
\(85\) 0 0
\(86\) −8.89898 −0.959602
\(87\) −2.44949 −0.262613
\(88\) −2.00000 −0.213201
\(89\) 3.55051 0.376353 0.188177 0.982135i \(-0.439742\pi\)
0.188177 + 0.982135i \(0.439742\pi\)
\(90\) 0 0
\(91\) −10.8990 −1.14252
\(92\) 2.44949 0.255377
\(93\) −7.34847 −0.762001
\(94\) 5.89898 0.608433
\(95\) 0 0
\(96\) −2.44949 −0.250000
\(97\) −7.34847 −0.746124 −0.373062 0.927806i \(-0.621692\pi\)
−0.373062 + 0.927806i \(0.621692\pi\)
\(98\) −12.7980 −1.29279
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 13.4495 1.33827 0.669137 0.743139i \(-0.266664\pi\)
0.669137 + 0.743139i \(0.266664\pi\)
\(102\) −4.89898 −0.485071
\(103\) 2.44949 0.241355 0.120678 0.992692i \(-0.461493\pi\)
0.120678 + 0.992692i \(0.461493\pi\)
\(104\) −2.44949 −0.240192
\(105\) 0 0
\(106\) 10.4495 1.01494
\(107\) −5.24745 −0.507290 −0.253645 0.967297i \(-0.581630\pi\)
−0.253645 + 0.967297i \(0.581630\pi\)
\(108\) 0 0
\(109\) −9.34847 −0.895421 −0.447710 0.894179i \(-0.647760\pi\)
−0.447710 + 0.894179i \(0.647760\pi\)
\(110\) 0 0
\(111\) 8.44949 0.801990
\(112\) −4.44949 −0.420437
\(113\) −6.24745 −0.587711 −0.293855 0.955850i \(-0.594938\pi\)
−0.293855 + 0.955850i \(0.594938\pi\)
\(114\) −15.7980 −1.47961
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 7.34847 0.679366
\(118\) −8.55051 −0.787138
\(119\) −8.89898 −0.815768
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −3.44949 −0.312302
\(123\) 27.7980 2.50646
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) 13.3485 1.18918
\(127\) −8.79796 −0.780693 −0.390346 0.920668i \(-0.627645\pi\)
−0.390346 + 0.920668i \(0.627645\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 21.7980 1.91920
\(130\) 0 0
\(131\) −7.34847 −0.642039 −0.321019 0.947073i \(-0.604025\pi\)
−0.321019 + 0.947073i \(0.604025\pi\)
\(132\) 4.89898 0.426401
\(133\) −28.6969 −2.48834
\(134\) −13.4495 −1.16186
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) −0.898979 −0.0768050 −0.0384025 0.999262i \(-0.512227\pi\)
−0.0384025 + 0.999262i \(0.512227\pi\)
\(138\) −6.00000 −0.510754
\(139\) −11.4495 −0.971133 −0.485567 0.874200i \(-0.661387\pi\)
−0.485567 + 0.874200i \(0.661387\pi\)
\(140\) 0 0
\(141\) −14.4495 −1.21687
\(142\) −11.3485 −0.952342
\(143\) 4.89898 0.409673
\(144\) 3.00000 0.250000
\(145\) 0 0
\(146\) 13.3485 1.10473
\(147\) 31.3485 2.58558
\(148\) 3.44949 0.283546
\(149\) 15.7980 1.29422 0.647110 0.762397i \(-0.275978\pi\)
0.647110 + 0.762397i \(0.275978\pi\)
\(150\) 0 0
\(151\) 11.3485 0.923525 0.461763 0.887004i \(-0.347217\pi\)
0.461763 + 0.887004i \(0.347217\pi\)
\(152\) −6.44949 −0.523123
\(153\) 6.00000 0.485071
\(154\) 8.89898 0.717100
\(155\) 0 0
\(156\) 6.00000 0.480384
\(157\) −12.3485 −0.985515 −0.492758 0.870167i \(-0.664011\pi\)
−0.492758 + 0.870167i \(0.664011\pi\)
\(158\) 2.89898 0.230630
\(159\) −25.5959 −2.02989
\(160\) 0 0
\(161\) −10.8990 −0.858960
\(162\) 9.00000 0.707107
\(163\) 12.4495 0.975119 0.487560 0.873090i \(-0.337887\pi\)
0.487560 + 0.873090i \(0.337887\pi\)
\(164\) 11.3485 0.886167
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 17.7980 1.37725 0.688624 0.725119i \(-0.258215\pi\)
0.688624 + 0.725119i \(0.258215\pi\)
\(168\) 10.8990 0.840875
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 19.3485 1.47961
\(172\) 8.89898 0.678541
\(173\) 16.0000 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(174\) 2.44949 0.185695
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 20.9444 1.57428
\(178\) −3.55051 −0.266122
\(179\) −17.2474 −1.28913 −0.644567 0.764547i \(-0.722962\pi\)
−0.644567 + 0.764547i \(0.722962\pi\)
\(180\) 0 0
\(181\) −26.0454 −1.93594 −0.967970 0.251066i \(-0.919219\pi\)
−0.967970 + 0.251066i \(0.919219\pi\)
\(182\) 10.8990 0.807886
\(183\) 8.44949 0.624604
\(184\) −2.44949 −0.180579
\(185\) 0 0
\(186\) 7.34847 0.538816
\(187\) 4.00000 0.292509
\(188\) −5.89898 −0.430227
\(189\) 0 0
\(190\) 0 0
\(191\) −11.6969 −0.846361 −0.423180 0.906045i \(-0.639086\pi\)
−0.423180 + 0.906045i \(0.639086\pi\)
\(192\) 2.44949 0.176777
\(193\) 5.34847 0.384991 0.192496 0.981298i \(-0.438342\pi\)
0.192496 + 0.981298i \(0.438342\pi\)
\(194\) 7.34847 0.527589
\(195\) 0 0
\(196\) 12.7980 0.914140
\(197\) −5.10102 −0.363433 −0.181716 0.983351i \(-0.558165\pi\)
−0.181716 + 0.983351i \(0.558165\pi\)
\(198\) −6.00000 −0.426401
\(199\) −0.651531 −0.0461858 −0.0230929 0.999733i \(-0.507351\pi\)
−0.0230929 + 0.999733i \(0.507351\pi\)
\(200\) 0 0
\(201\) 32.9444 2.32372
\(202\) −13.4495 −0.946303
\(203\) 4.44949 0.312293
\(204\) 4.89898 0.342997
\(205\) 0 0
\(206\) −2.44949 −0.170664
\(207\) 7.34847 0.510754
\(208\) 2.44949 0.169842
\(209\) 12.8990 0.892241
\(210\) 0 0
\(211\) −22.4949 −1.54861 −0.774306 0.632811i \(-0.781901\pi\)
−0.774306 + 0.632811i \(0.781901\pi\)
\(212\) −10.4495 −0.717674
\(213\) 27.7980 1.90468
\(214\) 5.24745 0.358708
\(215\) 0 0
\(216\) 0 0
\(217\) 13.3485 0.906153
\(218\) 9.34847 0.633158
\(219\) −32.6969 −2.20945
\(220\) 0 0
\(221\) 4.89898 0.329541
\(222\) −8.44949 −0.567093
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 4.44949 0.297294
\(225\) 0 0
\(226\) 6.24745 0.415574
\(227\) −15.2474 −1.01201 −0.506004 0.862531i \(-0.668878\pi\)
−0.506004 + 0.862531i \(0.668878\pi\)
\(228\) 15.7980 1.04625
\(229\) −17.1010 −1.13007 −0.565034 0.825068i \(-0.691137\pi\)
−0.565034 + 0.825068i \(0.691137\pi\)
\(230\) 0 0
\(231\) −21.7980 −1.43420
\(232\) 1.00000 0.0656532
\(233\) 11.0000 0.720634 0.360317 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) −7.34847 −0.480384
\(235\) 0 0
\(236\) 8.55051 0.556591
\(237\) −7.10102 −0.461261
\(238\) 8.89898 0.576835
\(239\) 21.5959 1.39692 0.698462 0.715647i \(-0.253868\pi\)
0.698462 + 0.715647i \(0.253868\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 7.00000 0.449977
\(243\) −22.0454 −1.41421
\(244\) 3.44949 0.220831
\(245\) 0 0
\(246\) −27.7980 −1.77233
\(247\) 15.7980 1.00520
\(248\) 3.00000 0.190500
\(249\) 14.6969 0.931381
\(250\) 0 0
\(251\) −3.14643 −0.198601 −0.0993004 0.995058i \(-0.531660\pi\)
−0.0993004 + 0.995058i \(0.531660\pi\)
\(252\) −13.3485 −0.840875
\(253\) 4.89898 0.307996
\(254\) 8.79796 0.552033
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.6969 −1.66531 −0.832655 0.553793i \(-0.813180\pi\)
−0.832655 + 0.553793i \(0.813180\pi\)
\(258\) −21.7980 −1.35708
\(259\) −15.3485 −0.953707
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 7.34847 0.453990
\(263\) 16.7980 1.03581 0.517903 0.855439i \(-0.326713\pi\)
0.517903 + 0.855439i \(0.326713\pi\)
\(264\) −4.89898 −0.301511
\(265\) 0 0
\(266\) 28.6969 1.75952
\(267\) 8.69694 0.532244
\(268\) 13.4495 0.821558
\(269\) 24.3485 1.48455 0.742276 0.670094i \(-0.233746\pi\)
0.742276 + 0.670094i \(0.233746\pi\)
\(270\) 0 0
\(271\) 15.6969 0.953521 0.476761 0.879033i \(-0.341811\pi\)
0.476761 + 0.879033i \(0.341811\pi\)
\(272\) 2.00000 0.121268
\(273\) −26.6969 −1.61577
\(274\) 0.898979 0.0543093
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) 22.9444 1.37859 0.689297 0.724479i \(-0.257919\pi\)
0.689297 + 0.724479i \(0.257919\pi\)
\(278\) 11.4495 0.686695
\(279\) −9.00000 −0.538816
\(280\) 0 0
\(281\) −28.7980 −1.71794 −0.858971 0.512024i \(-0.828896\pi\)
−0.858971 + 0.512024i \(0.828896\pi\)
\(282\) 14.4495 0.860455
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 11.3485 0.673408
\(285\) 0 0
\(286\) −4.89898 −0.289683
\(287\) −50.4949 −2.98062
\(288\) −3.00000 −0.176777
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −18.0000 −1.05518
\(292\) −13.3485 −0.781160
\(293\) 7.44949 0.435204 0.217602 0.976038i \(-0.430177\pi\)
0.217602 + 0.976038i \(0.430177\pi\)
\(294\) −31.3485 −1.82828
\(295\) 0 0
\(296\) −3.44949 −0.200498
\(297\) 0 0
\(298\) −15.7980 −0.915151
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) −39.5959 −2.28227
\(302\) −11.3485 −0.653031
\(303\) 32.9444 1.89261
\(304\) 6.44949 0.369904
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −25.3939 −1.44931 −0.724653 0.689114i \(-0.758000\pi\)
−0.724653 + 0.689114i \(0.758000\pi\)
\(308\) −8.89898 −0.507066
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) −19.5959 −1.11118 −0.555591 0.831456i \(-0.687508\pi\)
−0.555591 + 0.831456i \(0.687508\pi\)
\(312\) −6.00000 −0.339683
\(313\) 32.5959 1.84243 0.921215 0.389054i \(-0.127198\pi\)
0.921215 + 0.389054i \(0.127198\pi\)
\(314\) 12.3485 0.696864
\(315\) 0 0
\(316\) −2.89898 −0.163080
\(317\) −13.6515 −0.766746 −0.383373 0.923594i \(-0.625238\pi\)
−0.383373 + 0.923594i \(0.625238\pi\)
\(318\) 25.5959 1.43535
\(319\) −2.00000 −0.111979
\(320\) 0 0
\(321\) −12.8536 −0.717416
\(322\) 10.8990 0.607376
\(323\) 12.8990 0.717718
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −12.4495 −0.689513
\(327\) −22.8990 −1.26632
\(328\) −11.3485 −0.626614
\(329\) 26.2474 1.44707
\(330\) 0 0
\(331\) −30.2474 −1.66255 −0.831275 0.555861i \(-0.812389\pi\)
−0.831275 + 0.555861i \(0.812389\pi\)
\(332\) 6.00000 0.329293
\(333\) 10.3485 0.567093
\(334\) −17.7980 −0.973861
\(335\) 0 0
\(336\) −10.8990 −0.594588
\(337\) −23.7980 −1.29636 −0.648179 0.761488i \(-0.724469\pi\)
−0.648179 + 0.761488i \(0.724469\pi\)
\(338\) 7.00000 0.380750
\(339\) −15.3031 −0.831148
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) −19.3485 −1.04625
\(343\) −25.7980 −1.39296
\(344\) −8.89898 −0.479801
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) 25.0454 1.34451 0.672254 0.740321i \(-0.265326\pi\)
0.672254 + 0.740321i \(0.265326\pi\)
\(348\) −2.44949 −0.131306
\(349\) −30.9444 −1.65642 −0.828208 0.560422i \(-0.810639\pi\)
−0.828208 + 0.560422i \(0.810639\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 16.7980 0.894065 0.447033 0.894518i \(-0.352481\pi\)
0.447033 + 0.894518i \(0.352481\pi\)
\(354\) −20.9444 −1.11318
\(355\) 0 0
\(356\) 3.55051 0.188177
\(357\) −21.7980 −1.15367
\(358\) 17.2474 0.911556
\(359\) −16.5959 −0.875899 −0.437950 0.899000i \(-0.644295\pi\)
−0.437950 + 0.899000i \(0.644295\pi\)
\(360\) 0 0
\(361\) 22.5959 1.18926
\(362\) 26.0454 1.36892
\(363\) −17.1464 −0.899954
\(364\) −10.8990 −0.571262
\(365\) 0 0
\(366\) −8.44949 −0.441662
\(367\) 1.20204 0.0627460 0.0313730 0.999508i \(-0.490012\pi\)
0.0313730 + 0.999508i \(0.490012\pi\)
\(368\) 2.44949 0.127688
\(369\) 34.0454 1.77233
\(370\) 0 0
\(371\) 46.4949 2.41389
\(372\) −7.34847 −0.381000
\(373\) −28.4949 −1.47541 −0.737705 0.675123i \(-0.764090\pi\)
−0.737705 + 0.675123i \(0.764090\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 5.89898 0.304217
\(377\) −2.44949 −0.126155
\(378\) 0 0
\(379\) 25.1464 1.29169 0.645843 0.763471i \(-0.276506\pi\)
0.645843 + 0.763471i \(0.276506\pi\)
\(380\) 0 0
\(381\) −21.5505 −1.10407
\(382\) 11.6969 0.598467
\(383\) 11.7980 0.602848 0.301424 0.953490i \(-0.402538\pi\)
0.301424 + 0.953490i \(0.402538\pi\)
\(384\) −2.44949 −0.125000
\(385\) 0 0
\(386\) −5.34847 −0.272230
\(387\) 26.6969 1.35708
\(388\) −7.34847 −0.373062
\(389\) −24.3485 −1.23452 −0.617258 0.786761i \(-0.711757\pi\)
−0.617258 + 0.786761i \(0.711757\pi\)
\(390\) 0 0
\(391\) 4.89898 0.247752
\(392\) −12.7980 −0.646395
\(393\) −18.0000 −0.907980
\(394\) 5.10102 0.256986
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) −13.1464 −0.659800 −0.329900 0.944016i \(-0.607015\pi\)
−0.329900 + 0.944016i \(0.607015\pi\)
\(398\) 0.651531 0.0326583
\(399\) −70.2929 −3.51904
\(400\) 0 0
\(401\) 38.5959 1.92739 0.963694 0.267009i \(-0.0860353\pi\)
0.963694 + 0.267009i \(0.0860353\pi\)
\(402\) −32.9444 −1.64312
\(403\) −7.34847 −0.366053
\(404\) 13.4495 0.669137
\(405\) 0 0
\(406\) −4.44949 −0.220824
\(407\) 6.89898 0.341970
\(408\) −4.89898 −0.242536
\(409\) −5.79796 −0.286691 −0.143345 0.989673i \(-0.545786\pi\)
−0.143345 + 0.989673i \(0.545786\pi\)
\(410\) 0 0
\(411\) −2.20204 −0.108619
\(412\) 2.44949 0.120678
\(413\) −38.0454 −1.87209
\(414\) −7.34847 −0.361158
\(415\) 0 0
\(416\) −2.44949 −0.120096
\(417\) −28.0454 −1.37339
\(418\) −12.8990 −0.630910
\(419\) −10.1464 −0.495685 −0.247843 0.968800i \(-0.579722\pi\)
−0.247843 + 0.968800i \(0.579722\pi\)
\(420\) 0 0
\(421\) 13.4495 0.655488 0.327744 0.944767i \(-0.393712\pi\)
0.327744 + 0.944767i \(0.393712\pi\)
\(422\) 22.4949 1.09503
\(423\) −17.6969 −0.860455
\(424\) 10.4495 0.507472
\(425\) 0 0
\(426\) −27.7980 −1.34682
\(427\) −15.3485 −0.742764
\(428\) −5.24745 −0.253645
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 18.4495 0.888681 0.444340 0.895858i \(-0.353438\pi\)
0.444340 + 0.895858i \(0.353438\pi\)
\(432\) 0 0
\(433\) −30.4495 −1.46331 −0.731655 0.681676i \(-0.761252\pi\)
−0.731655 + 0.681676i \(0.761252\pi\)
\(434\) −13.3485 −0.640747
\(435\) 0 0
\(436\) −9.34847 −0.447710
\(437\) 15.7980 0.755719
\(438\) 32.6969 1.56232
\(439\) 18.6969 0.892356 0.446178 0.894944i \(-0.352785\pi\)
0.446178 + 0.894944i \(0.352785\pi\)
\(440\) 0 0
\(441\) 38.3939 1.82828
\(442\) −4.89898 −0.233021
\(443\) 13.1010 0.622448 0.311224 0.950337i \(-0.399261\pi\)
0.311224 + 0.950337i \(0.399261\pi\)
\(444\) 8.44949 0.400995
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 38.6969 1.83030
\(448\) −4.44949 −0.210219
\(449\) 25.7980 1.21748 0.608740 0.793369i \(-0.291675\pi\)
0.608740 + 0.793369i \(0.291675\pi\)
\(450\) 0 0
\(451\) 22.6969 1.06876
\(452\) −6.24745 −0.293855
\(453\) 27.7980 1.30606
\(454\) 15.2474 0.715598
\(455\) 0 0
\(456\) −15.7980 −0.739807
\(457\) 17.7980 0.832553 0.416277 0.909238i \(-0.363335\pi\)
0.416277 + 0.909238i \(0.363335\pi\)
\(458\) 17.1010 0.799078
\(459\) 0 0
\(460\) 0 0
\(461\) −28.2929 −1.31773 −0.658865 0.752261i \(-0.728963\pi\)
−0.658865 + 0.752261i \(0.728963\pi\)
\(462\) 21.7980 1.01413
\(463\) 3.10102 0.144117 0.0720583 0.997400i \(-0.477043\pi\)
0.0720583 + 0.997400i \(0.477043\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −11.0000 −0.509565
\(467\) 22.9444 1.06174 0.530870 0.847453i \(-0.321865\pi\)
0.530870 + 0.847453i \(0.321865\pi\)
\(468\) 7.34847 0.339683
\(469\) −59.8434 −2.76331
\(470\) 0 0
\(471\) −30.2474 −1.39373
\(472\) −8.55051 −0.393569
\(473\) 17.7980 0.818351
\(474\) 7.10102 0.326161
\(475\) 0 0
\(476\) −8.89898 −0.407884
\(477\) −31.3485 −1.43535
\(478\) −21.5959 −0.987774
\(479\) −4.20204 −0.191996 −0.0959981 0.995382i \(-0.530604\pi\)
−0.0959981 + 0.995382i \(0.530604\pi\)
\(480\) 0 0
\(481\) 8.44949 0.385264
\(482\) −17.0000 −0.774329
\(483\) −26.6969 −1.21475
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 22.0454 1.00000
\(487\) 9.10102 0.412407 0.206203 0.978509i \(-0.433889\pi\)
0.206203 + 0.978509i \(0.433889\pi\)
\(488\) −3.44949 −0.156151
\(489\) 30.4949 1.37903
\(490\) 0 0
\(491\) 26.4949 1.19570 0.597849 0.801609i \(-0.296022\pi\)
0.597849 + 0.801609i \(0.296022\pi\)
\(492\) 27.7980 1.25323
\(493\) −2.00000 −0.0900755
\(494\) −15.7980 −0.710784
\(495\) 0 0
\(496\) −3.00000 −0.134704
\(497\) −50.4949 −2.26501
\(498\) −14.6969 −0.658586
\(499\) −24.3485 −1.08999 −0.544994 0.838440i \(-0.683468\pi\)
−0.544994 + 0.838440i \(0.683468\pi\)
\(500\) 0 0
\(501\) 43.5959 1.94772
\(502\) 3.14643 0.140432
\(503\) −4.79796 −0.213930 −0.106965 0.994263i \(-0.534113\pi\)
−0.106965 + 0.994263i \(0.534113\pi\)
\(504\) 13.3485 0.594588
\(505\) 0 0
\(506\) −4.89898 −0.217786
\(507\) −17.1464 −0.761500
\(508\) −8.79796 −0.390346
\(509\) −17.3939 −0.770970 −0.385485 0.922714i \(-0.625966\pi\)
−0.385485 + 0.922714i \(0.625966\pi\)
\(510\) 0 0
\(511\) 59.3939 2.62743
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.6969 1.17755
\(515\) 0 0
\(516\) 21.7980 0.959602
\(517\) −11.7980 −0.518874
\(518\) 15.3485 0.674373
\(519\) 39.1918 1.72033
\(520\) 0 0
\(521\) −10.1010 −0.442534 −0.221267 0.975213i \(-0.571019\pi\)
−0.221267 + 0.975213i \(0.571019\pi\)
\(522\) 3.00000 0.131306
\(523\) 27.4495 1.20028 0.600141 0.799894i \(-0.295111\pi\)
0.600141 + 0.799894i \(0.295111\pi\)
\(524\) −7.34847 −0.321019
\(525\) 0 0
\(526\) −16.7980 −0.732426
\(527\) −6.00000 −0.261364
\(528\) 4.89898 0.213201
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) 25.6515 1.11318
\(532\) −28.6969 −1.24417
\(533\) 27.7980 1.20406
\(534\) −8.69694 −0.376353
\(535\) 0 0
\(536\) −13.4495 −0.580929
\(537\) −42.2474 −1.82311
\(538\) −24.3485 −1.04974
\(539\) 25.5959 1.10249
\(540\) 0 0
\(541\) −21.0454 −0.904813 −0.452406 0.891812i \(-0.649434\pi\)
−0.452406 + 0.891812i \(0.649434\pi\)
\(542\) −15.6969 −0.674241
\(543\) −63.7980 −2.73783
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 26.6969 1.14252
\(547\) 22.0000 0.940652 0.470326 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −0.898979 −0.0384025
\(549\) 10.3485 0.441662
\(550\) 0 0
\(551\) −6.44949 −0.274758
\(552\) −6.00000 −0.255377
\(553\) 12.8990 0.548520
\(554\) −22.9444 −0.974814
\(555\) 0 0
\(556\) −11.4495 −0.485567
\(557\) 6.49490 0.275198 0.137599 0.990488i \(-0.456062\pi\)
0.137599 + 0.990488i \(0.456062\pi\)
\(558\) 9.00000 0.381000
\(559\) 21.7980 0.921955
\(560\) 0 0
\(561\) 9.79796 0.413670
\(562\) 28.7980 1.21477
\(563\) 21.7980 0.918674 0.459337 0.888262i \(-0.348087\pi\)
0.459337 + 0.888262i \(0.348087\pi\)
\(564\) −14.4495 −0.608433
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) 40.0454 1.68175
\(568\) −11.3485 −0.476171
\(569\) −8.69694 −0.364595 −0.182297 0.983243i \(-0.558353\pi\)
−0.182297 + 0.983243i \(0.558353\pi\)
\(570\) 0 0
\(571\) 25.0454 1.04812 0.524059 0.851682i \(-0.324417\pi\)
0.524059 + 0.851682i \(0.324417\pi\)
\(572\) 4.89898 0.204837
\(573\) −28.6515 −1.19693
\(574\) 50.4949 2.10762
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) −28.9444 −1.20497 −0.602485 0.798130i \(-0.705823\pi\)
−0.602485 + 0.798130i \(0.705823\pi\)
\(578\) 13.0000 0.540729
\(579\) 13.1010 0.544460
\(580\) 0 0
\(581\) −26.6969 −1.10758
\(582\) 18.0000 0.746124
\(583\) −20.8990 −0.865547
\(584\) 13.3485 0.552364
\(585\) 0 0
\(586\) −7.44949 −0.307736
\(587\) 14.7526 0.608903 0.304451 0.952528i \(-0.401527\pi\)
0.304451 + 0.952528i \(0.401527\pi\)
\(588\) 31.3485 1.29279
\(589\) −19.3485 −0.797240
\(590\) 0 0
\(591\) −12.4949 −0.513971
\(592\) 3.44949 0.141773
\(593\) 18.3939 0.755346 0.377673 0.925939i \(-0.376724\pi\)
0.377673 + 0.925939i \(0.376724\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.7980 0.647110
\(597\) −1.59592 −0.0653166
\(598\) −6.00000 −0.245358
\(599\) −12.1010 −0.494434 −0.247217 0.968960i \(-0.579516\pi\)
−0.247217 + 0.968960i \(0.579516\pi\)
\(600\) 0 0
\(601\) −6.69694 −0.273174 −0.136587 0.990628i \(-0.543613\pi\)
−0.136587 + 0.990628i \(0.543613\pi\)
\(602\) 39.5959 1.61381
\(603\) 40.3485 1.64312
\(604\) 11.3485 0.461763
\(605\) 0 0
\(606\) −32.9444 −1.33827
\(607\) 34.3939 1.39600 0.698002 0.716096i \(-0.254073\pi\)
0.698002 + 0.716096i \(0.254073\pi\)
\(608\) −6.44949 −0.261561
\(609\) 10.8990 0.441649
\(610\) 0 0
\(611\) −14.4495 −0.584564
\(612\) 6.00000 0.242536
\(613\) −46.8990 −1.89423 −0.947116 0.320891i \(-0.896018\pi\)
−0.947116 + 0.320891i \(0.896018\pi\)
\(614\) 25.3939 1.02481
\(615\) 0 0
\(616\) 8.89898 0.358550
\(617\) −14.4495 −0.581715 −0.290857 0.956766i \(-0.593940\pi\)
−0.290857 + 0.956766i \(0.593940\pi\)
\(618\) −6.00000 −0.241355
\(619\) 2.89898 0.116520 0.0582599 0.998301i \(-0.481445\pi\)
0.0582599 + 0.998301i \(0.481445\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 19.5959 0.785725
\(623\) −15.7980 −0.632932
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −32.5959 −1.30279
\(627\) 31.5959 1.26182
\(628\) −12.3485 −0.492758
\(629\) 6.89898 0.275080
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 2.89898 0.115315
\(633\) −55.1010 −2.19007
\(634\) 13.6515 0.542172
\(635\) 0 0
\(636\) −25.5959 −1.01494
\(637\) 31.3485 1.24207
\(638\) 2.00000 0.0791808
\(639\) 34.0454 1.34682
\(640\) 0 0
\(641\) 24.2474 0.957717 0.478858 0.877892i \(-0.341051\pi\)
0.478858 + 0.877892i \(0.341051\pi\)
\(642\) 12.8536 0.507290
\(643\) 24.8434 0.979727 0.489863 0.871799i \(-0.337047\pi\)
0.489863 + 0.871799i \(0.337047\pi\)
\(644\) −10.8990 −0.429480
\(645\) 0 0
\(646\) −12.8990 −0.507504
\(647\) −39.5959 −1.55668 −0.778338 0.627845i \(-0.783937\pi\)
−0.778338 + 0.627845i \(0.783937\pi\)
\(648\) 9.00000 0.353553
\(649\) 17.1010 0.671274
\(650\) 0 0
\(651\) 32.6969 1.28149
\(652\) 12.4495 0.487560
\(653\) −11.2474 −0.440147 −0.220073 0.975483i \(-0.570630\pi\)
−0.220073 + 0.975483i \(0.570630\pi\)
\(654\) 22.8990 0.895421
\(655\) 0 0
\(656\) 11.3485 0.443083
\(657\) −40.0454 −1.56232
\(658\) −26.2474 −1.02323
\(659\) −38.6969 −1.50742 −0.753709 0.657208i \(-0.771737\pi\)
−0.753709 + 0.657208i \(0.771737\pi\)
\(660\) 0 0
\(661\) 9.75255 0.379330 0.189665 0.981849i \(-0.439260\pi\)
0.189665 + 0.981849i \(0.439260\pi\)
\(662\) 30.2474 1.17560
\(663\) 12.0000 0.466041
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −10.3485 −0.400995
\(667\) −2.44949 −0.0948446
\(668\) 17.7980 0.688624
\(669\) −9.79796 −0.378811
\(670\) 0 0
\(671\) 6.89898 0.266332
\(672\) 10.8990 0.420437
\(673\) 16.7980 0.647514 0.323757 0.946140i \(-0.395054\pi\)
0.323757 + 0.946140i \(0.395054\pi\)
\(674\) 23.7980 0.916663
\(675\) 0 0
\(676\) −7.00000 −0.269231
\(677\) 25.0454 0.962573 0.481287 0.876563i \(-0.340170\pi\)
0.481287 + 0.876563i \(0.340170\pi\)
\(678\) 15.3031 0.587711
\(679\) 32.6969 1.25479
\(680\) 0 0
\(681\) −37.3485 −1.43120
\(682\) 6.00000 0.229752
\(683\) −11.1010 −0.424769 −0.212384 0.977186i \(-0.568123\pi\)
−0.212384 + 0.977186i \(0.568123\pi\)
\(684\) 19.3485 0.739807
\(685\) 0 0
\(686\) 25.7980 0.984971
\(687\) −41.8888 −1.59816
\(688\) 8.89898 0.339270
\(689\) −25.5959 −0.975127
\(690\) 0 0
\(691\) −43.9444 −1.67172 −0.835862 0.548940i \(-0.815031\pi\)
−0.835862 + 0.548940i \(0.815031\pi\)
\(692\) 16.0000 0.608229
\(693\) −26.6969 −1.01413
\(694\) −25.0454 −0.950711
\(695\) 0 0
\(696\) 2.44949 0.0928477
\(697\) 22.6969 0.859708
\(698\) 30.9444 1.17126
\(699\) 26.9444 1.01913
\(700\) 0 0
\(701\) −25.3939 −0.959113 −0.479557 0.877511i \(-0.659203\pi\)
−0.479557 + 0.877511i \(0.659203\pi\)
\(702\) 0 0
\(703\) 22.2474 0.839078
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −16.7980 −0.632200
\(707\) −59.8434 −2.25064
\(708\) 20.9444 0.787138
\(709\) −27.3939 −1.02880 −0.514399 0.857551i \(-0.671985\pi\)
−0.514399 + 0.857551i \(0.671985\pi\)
\(710\) 0 0
\(711\) −8.69694 −0.326161
\(712\) −3.55051 −0.133061
\(713\) −7.34847 −0.275202
\(714\) 21.7980 0.815768
\(715\) 0 0
\(716\) −17.2474 −0.644567
\(717\) 52.8990 1.97555
\(718\) 16.5959 0.619354
\(719\) −36.7423 −1.37026 −0.685129 0.728422i \(-0.740254\pi\)
−0.685129 + 0.728422i \(0.740254\pi\)
\(720\) 0 0
\(721\) −10.8990 −0.405899
\(722\) −22.5959 −0.840933
\(723\) 41.6413 1.54866
\(724\) −26.0454 −0.967970
\(725\) 0 0
\(726\) 17.1464 0.636364
\(727\) 33.5959 1.24600 0.623002 0.782220i \(-0.285913\pi\)
0.623002 + 0.782220i \(0.285913\pi\)
\(728\) 10.8990 0.403943
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 17.7980 0.658281
\(732\) 8.44949 0.312302
\(733\) −1.10102 −0.0406671 −0.0203336 0.999793i \(-0.506473\pi\)
−0.0203336 + 0.999793i \(0.506473\pi\)
\(734\) −1.20204 −0.0443681
\(735\) 0 0
\(736\) −2.44949 −0.0902894
\(737\) 26.8990 0.990837
\(738\) −34.0454 −1.25323
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) 38.6969 1.42157
\(742\) −46.4949 −1.70688
\(743\) −12.6969 −0.465805 −0.232903 0.972500i \(-0.574822\pi\)
−0.232903 + 0.972500i \(0.574822\pi\)
\(744\) 7.34847 0.269408
\(745\) 0 0
\(746\) 28.4949 1.04327
\(747\) 18.0000 0.658586
\(748\) 4.00000 0.146254
\(749\) 23.3485 0.853134
\(750\) 0 0
\(751\) 24.1010 0.879459 0.439729 0.898130i \(-0.355074\pi\)
0.439729 + 0.898130i \(0.355074\pi\)
\(752\) −5.89898 −0.215114
\(753\) −7.70714 −0.280864
\(754\) 2.44949 0.0892052
\(755\) 0 0
\(756\) 0 0
\(757\) 19.3939 0.704882 0.352441 0.935834i \(-0.385352\pi\)
0.352441 + 0.935834i \(0.385352\pi\)
\(758\) −25.1464 −0.913359
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 14.1010 0.511162 0.255581 0.966788i \(-0.417733\pi\)
0.255581 + 0.966788i \(0.417733\pi\)
\(762\) 21.5505 0.780693
\(763\) 41.5959 1.50587
\(764\) −11.6969 −0.423180
\(765\) 0 0
\(766\) −11.7980 −0.426278
\(767\) 20.9444 0.756258
\(768\) 2.44949 0.0883883
\(769\) −3.55051 −0.128035 −0.0640173 0.997949i \(-0.520391\pi\)
−0.0640173 + 0.997949i \(0.520391\pi\)
\(770\) 0 0
\(771\) −65.3939 −2.35510
\(772\) 5.34847 0.192496
\(773\) 17.4495 0.627615 0.313807 0.949487i \(-0.398395\pi\)
0.313807 + 0.949487i \(0.398395\pi\)
\(774\) −26.6969 −0.959602
\(775\) 0 0
\(776\) 7.34847 0.263795
\(777\) −37.5959 −1.34875
\(778\) 24.3485 0.872935
\(779\) 73.1918 2.62237
\(780\) 0 0
\(781\) 22.6969 0.812160
\(782\) −4.89898 −0.175187
\(783\) 0 0
\(784\) 12.7980 0.457070
\(785\) 0 0
\(786\) 18.0000 0.642039
\(787\) 40.5505 1.44547 0.722735 0.691125i \(-0.242885\pi\)
0.722735 + 0.691125i \(0.242885\pi\)
\(788\) −5.10102 −0.181716
\(789\) 41.1464 1.46485
\(790\) 0 0
\(791\) 27.7980 0.988382
\(792\) −6.00000 −0.213201
\(793\) 8.44949 0.300050
\(794\) 13.1464 0.466549
\(795\) 0 0
\(796\) −0.651531 −0.0230929
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 70.2929 2.48834
\(799\) −11.7980 −0.417382
\(800\) 0 0
\(801\) 10.6515 0.376353
\(802\) −38.5959 −1.36287
\(803\) −26.6969 −0.942114
\(804\) 32.9444 1.16186
\(805\) 0 0
\(806\) 7.34847 0.258839
\(807\) 59.6413 2.09947
\(808\) −13.4495 −0.473151
\(809\) 31.5959 1.11085 0.555427 0.831566i \(-0.312555\pi\)
0.555427 + 0.831566i \(0.312555\pi\)
\(810\) 0 0
\(811\) −29.4495 −1.03411 −0.517056 0.855952i \(-0.672972\pi\)
−0.517056 + 0.855952i \(0.672972\pi\)
\(812\) 4.44949 0.156146
\(813\) 38.4495 1.34848
\(814\) −6.89898 −0.241809
\(815\) 0 0
\(816\) 4.89898 0.171499
\(817\) 57.3939 2.00796
\(818\) 5.79796 0.202721
\(819\) −32.6969 −1.14252
\(820\) 0 0
\(821\) 19.7526 0.689369 0.344684 0.938719i \(-0.387986\pi\)
0.344684 + 0.938719i \(0.387986\pi\)
\(822\) 2.20204 0.0768050
\(823\) 11.7980 0.411251 0.205625 0.978631i \(-0.434077\pi\)
0.205625 + 0.978631i \(0.434077\pi\)
\(824\) −2.44949 −0.0853320
\(825\) 0 0
\(826\) 38.0454 1.32377
\(827\) −19.3031 −0.671233 −0.335617 0.941999i \(-0.608945\pi\)
−0.335617 + 0.941999i \(0.608945\pi\)
\(828\) 7.34847 0.255377
\(829\) 14.4949 0.503429 0.251714 0.967802i \(-0.419006\pi\)
0.251714 + 0.967802i \(0.419006\pi\)
\(830\) 0 0
\(831\) 56.2020 1.94963
\(832\) 2.44949 0.0849208
\(833\) 25.5959 0.886846
\(834\) 28.0454 0.971133
\(835\) 0 0
\(836\) 12.8990 0.446121
\(837\) 0 0
\(838\) 10.1464 0.350503
\(839\) −30.7980 −1.06326 −0.531632 0.846976i \(-0.678421\pi\)
−0.531632 + 0.846976i \(0.678421\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −13.4495 −0.463500
\(843\) −70.5403 −2.42954
\(844\) −22.4949 −0.774306
\(845\) 0 0
\(846\) 17.6969 0.608433
\(847\) 31.1464 1.07020
\(848\) −10.4495 −0.358837
\(849\) −58.7878 −2.01759
\(850\) 0 0
\(851\) 8.44949 0.289645
\(852\) 27.7980 0.952342
\(853\) −8.20204 −0.280833 −0.140416 0.990093i \(-0.544844\pi\)
−0.140416 + 0.990093i \(0.544844\pi\)
\(854\) 15.3485 0.525214
\(855\) 0 0
\(856\) 5.24745 0.179354
\(857\) 39.3939 1.34567 0.672835 0.739793i \(-0.265077\pi\)
0.672835 + 0.739793i \(0.265077\pi\)
\(858\) −12.0000 −0.409673
\(859\) 22.8990 0.781303 0.390652 0.920539i \(-0.372250\pi\)
0.390652 + 0.920539i \(0.372250\pi\)
\(860\) 0 0
\(861\) −123.687 −4.21523
\(862\) −18.4495 −0.628392
\(863\) −11.7526 −0.400061 −0.200031 0.979790i \(-0.564104\pi\)
−0.200031 + 0.979790i \(0.564104\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 30.4495 1.03472
\(867\) −31.8434 −1.08146
\(868\) 13.3485 0.453077
\(869\) −5.79796 −0.196682
\(870\) 0 0
\(871\) 32.9444 1.11628
\(872\) 9.34847 0.316579
\(873\) −22.0454 −0.746124
\(874\) −15.7980 −0.534374
\(875\) 0 0
\(876\) −32.6969 −1.10473
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −18.6969 −0.630991
\(879\) 18.2474 0.615471
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) −38.3939 −1.29279
\(883\) −29.9444 −1.00771 −0.503854 0.863789i \(-0.668085\pi\)
−0.503854 + 0.863789i \(0.668085\pi\)
\(884\) 4.89898 0.164771
\(885\) 0 0
\(886\) −13.1010 −0.440137
\(887\) 47.7980 1.60490 0.802449 0.596720i \(-0.203530\pi\)
0.802449 + 0.596720i \(0.203530\pi\)
\(888\) −8.44949 −0.283546
\(889\) 39.1464 1.31293
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) −4.00000 −0.133930
\(893\) −38.0454 −1.27314
\(894\) −38.6969 −1.29422
\(895\) 0 0
\(896\) 4.44949 0.148647
\(897\) 14.6969 0.490716
\(898\) −25.7980 −0.860889
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −20.8990 −0.696246
\(902\) −22.6969 −0.755725
\(903\) −96.9898 −3.22762
\(904\) 6.24745 0.207787
\(905\) 0 0
\(906\) −27.7980 −0.923525
\(907\) 21.3485 0.708864 0.354432 0.935082i \(-0.384674\pi\)
0.354432 + 0.935082i \(0.384674\pi\)
\(908\) −15.2474 −0.506004
\(909\) 40.3485 1.33827
\(910\) 0 0
\(911\) 51.4949 1.70610 0.853051 0.521827i \(-0.174750\pi\)
0.853051 + 0.521827i \(0.174750\pi\)
\(912\) 15.7980 0.523123
\(913\) 12.0000 0.397142
\(914\) −17.7980 −0.588704
\(915\) 0 0
\(916\) −17.1010 −0.565034
\(917\) 32.6969 1.07975
\(918\) 0 0
\(919\) −24.8536 −0.819844 −0.409922 0.912121i \(-0.634444\pi\)
−0.409922 + 0.912121i \(0.634444\pi\)
\(920\) 0 0
\(921\) −62.2020 −2.04963
\(922\) 28.2929 0.931776
\(923\) 27.7980 0.914981
\(924\) −21.7980 −0.717100
\(925\) 0 0
\(926\) −3.10102 −0.101906
\(927\) 7.34847 0.241355
\(928\) 1.00000 0.0328266
\(929\) −39.4949 −1.29579 −0.647893 0.761732i \(-0.724350\pi\)
−0.647893 + 0.761732i \(0.724350\pi\)
\(930\) 0 0
\(931\) 82.5403 2.70515
\(932\) 11.0000 0.360317
\(933\) −48.0000 −1.57145
\(934\) −22.9444 −0.750763
\(935\) 0 0
\(936\) −7.34847 −0.240192
\(937\) −53.2929 −1.74100 −0.870501 0.492167i \(-0.836205\pi\)
−0.870501 + 0.492167i \(0.836205\pi\)
\(938\) 59.8434 1.95396
\(939\) 79.8434 2.60559
\(940\) 0 0
\(941\) 8.44949 0.275445 0.137723 0.990471i \(-0.456022\pi\)
0.137723 + 0.990471i \(0.456022\pi\)
\(942\) 30.2474 0.985515
\(943\) 27.7980 0.905226
\(944\) 8.55051 0.278295
\(945\) 0 0
\(946\) −17.7980 −0.578662
\(947\) −37.3485 −1.21366 −0.606831 0.794831i \(-0.707560\pi\)
−0.606831 + 0.794831i \(0.707560\pi\)
\(948\) −7.10102 −0.230630
\(949\) −32.6969 −1.06139
\(950\) 0 0
\(951\) −33.4393 −1.08434
\(952\) 8.89898 0.288418
\(953\) −34.2929 −1.11085 −0.555427 0.831565i \(-0.687445\pi\)
−0.555427 + 0.831565i \(0.687445\pi\)
\(954\) 31.3485 1.01494
\(955\) 0 0
\(956\) 21.5959 0.698462
\(957\) −4.89898 −0.158362
\(958\) 4.20204 0.135762
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −8.44949 −0.272422
\(963\) −15.7423 −0.507290
\(964\) 17.0000 0.547533
\(965\) 0 0
\(966\) 26.6969 0.858960
\(967\) −28.0000 −0.900419 −0.450210 0.892923i \(-0.648651\pi\)
−0.450210 + 0.892923i \(0.648651\pi\)
\(968\) 7.00000 0.224989
\(969\) 31.5959 1.01501
\(970\) 0 0
\(971\) 22.9444 0.736320 0.368160 0.929762i \(-0.379988\pi\)
0.368160 + 0.929762i \(0.379988\pi\)
\(972\) −22.0454 −0.707107
\(973\) 50.9444 1.63320
\(974\) −9.10102 −0.291616
\(975\) 0 0
\(976\) 3.44949 0.110415
\(977\) −0.101021 −0.00323193 −0.00161597 0.999999i \(-0.500514\pi\)
−0.00161597 + 0.999999i \(0.500514\pi\)
\(978\) −30.4949 −0.975119
\(979\) 7.10102 0.226950
\(980\) 0 0
\(981\) −28.0454 −0.895421
\(982\) −26.4949 −0.845486
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) −27.7980 −0.886167
\(985\) 0 0
\(986\) 2.00000 0.0636930
\(987\) 64.2929 2.04646
\(988\) 15.7980 0.502600
\(989\) 21.7980 0.693135
\(990\) 0 0
\(991\) −3.50510 −0.111343 −0.0556716 0.998449i \(-0.517730\pi\)
−0.0556716 + 0.998449i \(0.517730\pi\)
\(992\) 3.00000 0.0952501
\(993\) −74.0908 −2.35120
\(994\) 50.4949 1.60160
\(995\) 0 0
\(996\) 14.6969 0.465690
\(997\) −29.7423 −0.941950 −0.470975 0.882147i \(-0.656098\pi\)
−0.470975 + 0.882147i \(0.656098\pi\)
\(998\) 24.3485 0.770737
\(999\) 0 0
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 1450.2.a.j.1.2 2
5.2 odd 4 1450.2.b.i.349.1 4
5.3 odd 4 1450.2.b.i.349.4 4
5.4 even 2 1450.2.a.o.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.j.1.2 2 1.1 even 1 trivial
1450.2.a.o.1.1 yes 2 5.4 even 2
1450.2.b.i.349.1 4 5.2 odd 4
1450.2.b.i.349.4 4 5.3 odd 4