Properties

Label 1450.2.a.s.1.2
Level $1450$
Weight $2$
Character 1450.1
Self dual yes
Analytic conductor $11.578$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.470683 q^{3} +1.00000 q^{4} -0.470683 q^{6} +3.30777 q^{7} +1.00000 q^{8} -2.77846 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.470683 q^{3} +1.00000 q^{4} -0.470683 q^{6} +3.30777 q^{7} +1.00000 q^{8} -2.77846 q^{9} +4.71982 q^{11} -0.470683 q^{12} -2.24914 q^{13} +3.30777 q^{14} +1.00000 q^{16} -0.719824 q^{17} -2.77846 q^{18} -0.470683 q^{19} -1.55691 q^{21} +4.71982 q^{22} +5.80605 q^{23} -0.470683 q^{24} -2.24914 q^{26} +2.71982 q^{27} +3.30777 q^{28} +1.00000 q^{29} +10.2181 q^{31} +1.00000 q^{32} -2.22154 q^{33} -0.719824 q^{34} -2.77846 q^{36} -1.02760 q^{37} -0.470683 q^{38} +1.05863 q^{39} -0.0275977 q^{41} -1.55691 q^{42} -6.61555 q^{43} +4.71982 q^{44} +5.80605 q^{46} -4.33537 q^{47} -0.470683 q^{48} +3.94137 q^{49} +0.338809 q^{51} -2.24914 q^{52} -10.7474 q^{53} +2.71982 q^{54} +3.30777 q^{56} +0.221543 q^{57} +1.00000 q^{58} +11.0276 q^{59} +1.35342 q^{61} +10.2181 q^{62} -9.19051 q^{63} +1.00000 q^{64} -2.22154 q^{66} +8.80605 q^{67} -0.719824 q^{68} -2.73281 q^{69} -5.80605 q^{71} -2.77846 q^{72} -3.64658 q^{73} -1.02760 q^{74} -0.470683 q^{76} +15.6121 q^{77} +1.05863 q^{78} +16.6707 q^{79} +7.05520 q^{81} -0.0275977 q^{82} +5.21811 q^{83} -1.55691 q^{84} -6.61555 q^{86} -0.470683 q^{87} +4.71982 q^{88} +9.08623 q^{89} -7.43965 q^{91} +5.80605 q^{92} -4.80949 q^{93} -4.33537 q^{94} -0.470683 q^{96} +11.1905 q^{97} +3.94137 q^{98} -13.1138 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} + 2 q^{7} + 3 q^{8} + 5 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - q^{19} + 12 q^{21} + 5 q^{22} - 8 q^{23} - q^{24} + 2 q^{26} - q^{27} + 2 q^{28} + 3 q^{29} + 4 q^{31} + 3 q^{32} - 15 q^{33} + 7 q^{34} + 14 q^{37} - q^{38} + 4 q^{39} + 17 q^{41} + 12 q^{42} - 4 q^{43} + 5 q^{44} - 8 q^{46} + 12 q^{47} - q^{48} + 11 q^{49} + 11 q^{51} + 2 q^{52} - 6 q^{53} - q^{54} + 2 q^{56} + 9 q^{57} + 3 q^{58} + 16 q^{59} + 2 q^{61} + 4 q^{62} - 18 q^{63} + 3 q^{64} - 15 q^{66} + q^{67} + 7 q^{68} + 6 q^{69} + 8 q^{71} - 13 q^{73} + 14 q^{74} - q^{76} - 4 q^{77} + 4 q^{78} - 13 q^{81} + 17 q^{82} - 11 q^{83} + 12 q^{84} - 4 q^{86} - q^{87} + 5 q^{88} + 11 q^{89} - 4 q^{91} - 8 q^{92} - 24 q^{93} + 12 q^{94} - q^{96} + 24 q^{97} + 11 q^{98} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.470683 −0.271749 −0.135875 0.990726i \(-0.543384\pi\)
−0.135875 + 0.990726i \(0.543384\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −0.470683 −0.192156
\(7\) 3.30777 1.25022 0.625110 0.780536i \(-0.285054\pi\)
0.625110 + 0.780536i \(0.285054\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.77846 −0.926152
\(10\) 0 0
\(11\) 4.71982 1.42308 0.711540 0.702645i \(-0.247998\pi\)
0.711540 + 0.702645i \(0.247998\pi\)
\(12\) −0.470683 −0.135875
\(13\) −2.24914 −0.623799 −0.311900 0.950115i \(-0.600965\pi\)
−0.311900 + 0.950115i \(0.600965\pi\)
\(14\) 3.30777 0.884040
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.719824 −0.174583 −0.0872915 0.996183i \(-0.527821\pi\)
−0.0872915 + 0.996183i \(0.527821\pi\)
\(18\) −2.77846 −0.654889
\(19\) −0.470683 −0.107982 −0.0539911 0.998541i \(-0.517194\pi\)
−0.0539911 + 0.998541i \(0.517194\pi\)
\(20\) 0 0
\(21\) −1.55691 −0.339747
\(22\) 4.71982 1.00627
\(23\) 5.80605 1.21065 0.605323 0.795980i \(-0.293044\pi\)
0.605323 + 0.795980i \(0.293044\pi\)
\(24\) −0.470683 −0.0960779
\(25\) 0 0
\(26\) −2.24914 −0.441093
\(27\) 2.71982 0.523430
\(28\) 3.30777 0.625110
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 10.2181 1.83523 0.917613 0.397475i \(-0.130114\pi\)
0.917613 + 0.397475i \(0.130114\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.22154 −0.386721
\(34\) −0.719824 −0.123449
\(35\) 0 0
\(36\) −2.77846 −0.463076
\(37\) −1.02760 −0.168936 −0.0844680 0.996426i \(-0.526919\pi\)
−0.0844680 + 0.996426i \(0.526919\pi\)
\(38\) −0.470683 −0.0763549
\(39\) 1.05863 0.169517
\(40\) 0 0
\(41\) −0.0275977 −0.00431003 −0.00215501 0.999998i \(-0.500686\pi\)
−0.00215501 + 0.999998i \(0.500686\pi\)
\(42\) −1.55691 −0.240237
\(43\) −6.61555 −1.00886 −0.504431 0.863452i \(-0.668298\pi\)
−0.504431 + 0.863452i \(0.668298\pi\)
\(44\) 4.71982 0.711540
\(45\) 0 0
\(46\) 5.80605 0.856056
\(47\) −4.33537 −0.632379 −0.316189 0.948696i \(-0.602404\pi\)
−0.316189 + 0.948696i \(0.602404\pi\)
\(48\) −0.470683 −0.0679373
\(49\) 3.94137 0.563052
\(50\) 0 0
\(51\) 0.338809 0.0474428
\(52\) −2.24914 −0.311900
\(53\) −10.7474 −1.47627 −0.738136 0.674652i \(-0.764294\pi\)
−0.738136 + 0.674652i \(0.764294\pi\)
\(54\) 2.71982 0.370121
\(55\) 0 0
\(56\) 3.30777 0.442020
\(57\) 0.221543 0.0293441
\(58\) 1.00000 0.131306
\(59\) 11.0276 1.43567 0.717835 0.696213i \(-0.245133\pi\)
0.717835 + 0.696213i \(0.245133\pi\)
\(60\) 0 0
\(61\) 1.35342 0.173287 0.0866437 0.996239i \(-0.472386\pi\)
0.0866437 + 0.996239i \(0.472386\pi\)
\(62\) 10.2181 1.29770
\(63\) −9.19051 −1.15790
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.22154 −0.273453
\(67\) 8.80605 1.07583 0.537915 0.842999i \(-0.319212\pi\)
0.537915 + 0.842999i \(0.319212\pi\)
\(68\) −0.719824 −0.0872915
\(69\) −2.73281 −0.328992
\(70\) 0 0
\(71\) −5.80605 −0.689052 −0.344526 0.938777i \(-0.611960\pi\)
−0.344526 + 0.938777i \(0.611960\pi\)
\(72\) −2.77846 −0.327444
\(73\) −3.64658 −0.426800 −0.213400 0.976965i \(-0.568454\pi\)
−0.213400 + 0.976965i \(0.568454\pi\)
\(74\) −1.02760 −0.119456
\(75\) 0 0
\(76\) −0.470683 −0.0539911
\(77\) 15.6121 1.77917
\(78\) 1.05863 0.119867
\(79\) 16.6707 1.87561 0.937803 0.347169i \(-0.112857\pi\)
0.937803 + 0.347169i \(0.112857\pi\)
\(80\) 0 0
\(81\) 7.05520 0.783911
\(82\) −0.0275977 −0.00304765
\(83\) 5.21811 0.572761 0.286381 0.958116i \(-0.407548\pi\)
0.286381 + 0.958116i \(0.407548\pi\)
\(84\) −1.55691 −0.169873
\(85\) 0 0
\(86\) −6.61555 −0.713373
\(87\) −0.470683 −0.0504626
\(88\) 4.71982 0.503135
\(89\) 9.08623 0.963139 0.481569 0.876408i \(-0.340067\pi\)
0.481569 + 0.876408i \(0.340067\pi\)
\(90\) 0 0
\(91\) −7.43965 −0.779887
\(92\) 5.80605 0.605323
\(93\) −4.80949 −0.498721
\(94\) −4.33537 −0.447159
\(95\) 0 0
\(96\) −0.470683 −0.0480389
\(97\) 11.1905 1.13622 0.568112 0.822951i \(-0.307674\pi\)
0.568112 + 0.822951i \(0.307674\pi\)
\(98\) 3.94137 0.398138
\(99\) −13.1138 −1.31799
\(100\) 0 0
\(101\) −3.02760 −0.301257 −0.150629 0.988590i \(-0.548130\pi\)
−0.150629 + 0.988590i \(0.548130\pi\)
\(102\) 0.338809 0.0335471
\(103\) 6.01461 0.592637 0.296318 0.955089i \(-0.404241\pi\)
0.296318 + 0.955089i \(0.404241\pi\)
\(104\) −2.24914 −0.220546
\(105\) 0 0
\(106\) −10.7474 −1.04388
\(107\) −11.7474 −1.13567 −0.567833 0.823144i \(-0.692218\pi\)
−0.567833 + 0.823144i \(0.692218\pi\)
\(108\) 2.71982 0.261715
\(109\) −9.92332 −0.950482 −0.475241 0.879856i \(-0.657639\pi\)
−0.475241 + 0.879856i \(0.657639\pi\)
\(110\) 0 0
\(111\) 0.483673 0.0459082
\(112\) 3.30777 0.312555
\(113\) −14.0276 −1.31961 −0.659803 0.751439i \(-0.729360\pi\)
−0.659803 + 0.751439i \(0.729360\pi\)
\(114\) 0.221543 0.0207494
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 6.24914 0.577733
\(118\) 11.0276 1.01517
\(119\) −2.38101 −0.218267
\(120\) 0 0
\(121\) 11.2767 1.02516
\(122\) 1.35342 0.122533
\(123\) 0.0129898 0.00117125
\(124\) 10.2181 0.917613
\(125\) 0 0
\(126\) −9.19051 −0.818755
\(127\) −16.2181 −1.43912 −0.719562 0.694428i \(-0.755657\pi\)
−0.719562 + 0.694428i \(0.755657\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.11383 0.274157
\(130\) 0 0
\(131\) 21.3630 1.86649 0.933246 0.359239i \(-0.116964\pi\)
0.933246 + 0.359239i \(0.116964\pi\)
\(132\) −2.22154 −0.193360
\(133\) −1.55691 −0.135002
\(134\) 8.80605 0.760727
\(135\) 0 0
\(136\) −0.719824 −0.0617244
\(137\) −9.54392 −0.815392 −0.407696 0.913118i \(-0.633668\pi\)
−0.407696 + 0.913118i \(0.633668\pi\)
\(138\) −2.73281 −0.232633
\(139\) −3.74742 −0.317852 −0.158926 0.987290i \(-0.550803\pi\)
−0.158926 + 0.987290i \(0.550803\pi\)
\(140\) 0 0
\(141\) 2.04059 0.171848
\(142\) −5.80605 −0.487233
\(143\) −10.6155 −0.887717
\(144\) −2.77846 −0.231538
\(145\) 0 0
\(146\) −3.64658 −0.301793
\(147\) −1.85514 −0.153009
\(148\) −1.02760 −0.0844680
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) −14.2491 −1.15958 −0.579789 0.814766i \(-0.696865\pi\)
−0.579789 + 0.814766i \(0.696865\pi\)
\(152\) −0.470683 −0.0381775
\(153\) 2.00000 0.161690
\(154\) 15.6121 1.25806
\(155\) 0 0
\(156\) 1.05863 0.0847585
\(157\) −12.2035 −0.973945 −0.486973 0.873417i \(-0.661899\pi\)
−0.486973 + 0.873417i \(0.661899\pi\)
\(158\) 16.6707 1.32625
\(159\) 5.05863 0.401176
\(160\) 0 0
\(161\) 19.2051 1.51358
\(162\) 7.05520 0.554308
\(163\) −17.5845 −1.37733 −0.688663 0.725082i \(-0.741802\pi\)
−0.688663 + 0.725082i \(0.741802\pi\)
\(164\) −0.0275977 −0.00215501
\(165\) 0 0
\(166\) 5.21811 0.405003
\(167\) −11.7914 −0.912450 −0.456225 0.889865i \(-0.650799\pi\)
−0.456225 + 0.889865i \(0.650799\pi\)
\(168\) −1.55691 −0.120119
\(169\) −7.94137 −0.610874
\(170\) 0 0
\(171\) 1.30777 0.100008
\(172\) −6.61555 −0.504431
\(173\) 20.7880 1.58048 0.790242 0.612795i \(-0.209955\pi\)
0.790242 + 0.612795i \(0.209955\pi\)
\(174\) −0.470683 −0.0356824
\(175\) 0 0
\(176\) 4.71982 0.355770
\(177\) −5.19051 −0.390142
\(178\) 9.08623 0.681042
\(179\) −7.74742 −0.579069 −0.289535 0.957168i \(-0.593501\pi\)
−0.289535 + 0.957168i \(0.593501\pi\)
\(180\) 0 0
\(181\) −19.7440 −1.46756 −0.733779 0.679388i \(-0.762245\pi\)
−0.733779 + 0.679388i \(0.762245\pi\)
\(182\) −7.43965 −0.551463
\(183\) −0.637031 −0.0470907
\(184\) 5.80605 0.428028
\(185\) 0 0
\(186\) −4.80949 −0.352649
\(187\) −3.39744 −0.248446
\(188\) −4.33537 −0.316189
\(189\) 8.99656 0.654404
\(190\) 0 0
\(191\) 0.397442 0.0287579 0.0143790 0.999897i \(-0.495423\pi\)
0.0143790 + 0.999897i \(0.495423\pi\)
\(192\) −0.470683 −0.0339686
\(193\) −4.35342 −0.313366 −0.156683 0.987649i \(-0.550080\pi\)
−0.156683 + 0.987649i \(0.550080\pi\)
\(194\) 11.1905 0.803432
\(195\) 0 0
\(196\) 3.94137 0.281526
\(197\) 6.70683 0.477842 0.238921 0.971039i \(-0.423206\pi\)
0.238921 + 0.971039i \(0.423206\pi\)
\(198\) −13.1138 −0.931959
\(199\) −9.36297 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(200\) 0 0
\(201\) −4.14486 −0.292356
\(202\) −3.02760 −0.213021
\(203\) 3.30777 0.232160
\(204\) 0.338809 0.0237214
\(205\) 0 0
\(206\) 6.01461 0.419058
\(207\) −16.1319 −1.12124
\(208\) −2.24914 −0.155950
\(209\) −2.22154 −0.153667
\(210\) 0 0
\(211\) −24.5665 −1.69123 −0.845613 0.533797i \(-0.820765\pi\)
−0.845613 + 0.533797i \(0.820765\pi\)
\(212\) −10.7474 −0.738136
\(213\) 2.73281 0.187249
\(214\) −11.7474 −0.803037
\(215\) 0 0
\(216\) 2.71982 0.185061
\(217\) 33.7992 2.29444
\(218\) −9.92332 −0.672092
\(219\) 1.71639 0.115983
\(220\) 0 0
\(221\) 1.61899 0.108905
\(222\) 0.483673 0.0324620
\(223\) 2.11727 0.141783 0.0708913 0.997484i \(-0.477416\pi\)
0.0708913 + 0.997484i \(0.477416\pi\)
\(224\) 3.30777 0.221010
\(225\) 0 0
\(226\) −14.0276 −0.933102
\(227\) −15.9138 −1.05623 −0.528117 0.849172i \(-0.677102\pi\)
−0.528117 + 0.849172i \(0.677102\pi\)
\(228\) 0.221543 0.0146720
\(229\) −22.2897 −1.47295 −0.736473 0.676467i \(-0.763510\pi\)
−0.736473 + 0.676467i \(0.763510\pi\)
\(230\) 0 0
\(231\) −7.34836 −0.483487
\(232\) 1.00000 0.0656532
\(233\) −23.3871 −1.53214 −0.766071 0.642756i \(-0.777791\pi\)
−0.766071 + 0.642756i \(0.777791\pi\)
\(234\) 6.24914 0.408519
\(235\) 0 0
\(236\) 11.0276 0.717835
\(237\) −7.84664 −0.509694
\(238\) −2.38101 −0.154338
\(239\) 20.2277 1.30842 0.654209 0.756314i \(-0.273002\pi\)
0.654209 + 0.756314i \(0.273002\pi\)
\(240\) 0 0
\(241\) 1.26375 0.0814052 0.0407026 0.999171i \(-0.487040\pi\)
0.0407026 + 0.999171i \(0.487040\pi\)
\(242\) 11.2767 0.724896
\(243\) −11.4802 −0.736457
\(244\) 1.35342 0.0866437
\(245\) 0 0
\(246\) 0.0129898 0.000828197 0
\(247\) 1.05863 0.0673592
\(248\) 10.2181 0.648850
\(249\) −2.45608 −0.155647
\(250\) 0 0
\(251\) 9.63971 0.608453 0.304226 0.952600i \(-0.401602\pi\)
0.304226 + 0.952600i \(0.401602\pi\)
\(252\) −9.19051 −0.578948
\(253\) 27.4036 1.72285
\(254\) −16.2181 −1.01761
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.43965 −0.588829 −0.294415 0.955678i \(-0.595125\pi\)
−0.294415 + 0.955678i \(0.595125\pi\)
\(258\) 3.11383 0.193858
\(259\) −3.39906 −0.211207
\(260\) 0 0
\(261\) −2.77846 −0.171982
\(262\) 21.3630 1.31981
\(263\) 11.6646 0.719272 0.359636 0.933093i \(-0.382901\pi\)
0.359636 + 0.933093i \(0.382901\pi\)
\(264\) −2.22154 −0.136727
\(265\) 0 0
\(266\) −1.55691 −0.0954605
\(267\) −4.27674 −0.261732
\(268\) 8.80605 0.537915
\(269\) −20.1414 −1.22804 −0.614022 0.789289i \(-0.710449\pi\)
−0.614022 + 0.789289i \(0.710449\pi\)
\(270\) 0 0
\(271\) −27.0422 −1.64270 −0.821349 0.570427i \(-0.806778\pi\)
−0.821349 + 0.570427i \(0.806778\pi\)
\(272\) −0.719824 −0.0436457
\(273\) 3.50172 0.211934
\(274\) −9.54392 −0.576570
\(275\) 0 0
\(276\) −2.73281 −0.164496
\(277\) −12.4216 −0.746342 −0.373171 0.927763i \(-0.621729\pi\)
−0.373171 + 0.927763i \(0.621729\pi\)
\(278\) −3.74742 −0.224755
\(279\) −28.3906 −1.69970
\(280\) 0 0
\(281\) 24.8957 1.48515 0.742577 0.669761i \(-0.233603\pi\)
0.742577 + 0.669761i \(0.233603\pi\)
\(282\) 2.04059 0.121515
\(283\) −4.83709 −0.287535 −0.143768 0.989611i \(-0.545922\pi\)
−0.143768 + 0.989611i \(0.545922\pi\)
\(284\) −5.80605 −0.344526
\(285\) 0 0
\(286\) −10.6155 −0.627710
\(287\) −0.0912868 −0.00538849
\(288\) −2.77846 −0.163722
\(289\) −16.4819 −0.969521
\(290\) 0 0
\(291\) −5.26719 −0.308768
\(292\) −3.64658 −0.213400
\(293\) 33.5190 1.95820 0.979101 0.203377i \(-0.0651917\pi\)
0.979101 + 0.203377i \(0.0651917\pi\)
\(294\) −1.85514 −0.108194
\(295\) 0 0
\(296\) −1.02760 −0.0597279
\(297\) 12.8371 0.744884
\(298\) 2.00000 0.115857
\(299\) −13.0586 −0.755200
\(300\) 0 0
\(301\) −21.8827 −1.26130
\(302\) −14.2491 −0.819946
\(303\) 1.42504 0.0818664
\(304\) −0.470683 −0.0269955
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) −12.3680 −0.705880 −0.352940 0.935646i \(-0.614818\pi\)
−0.352940 + 0.935646i \(0.614818\pi\)
\(308\) 15.6121 0.889583
\(309\) −2.83098 −0.161049
\(310\) 0 0
\(311\) 15.3224 0.868853 0.434426 0.900707i \(-0.356951\pi\)
0.434426 + 0.900707i \(0.356951\pi\)
\(312\) 1.05863 0.0599333
\(313\) 22.1268 1.25068 0.625341 0.780352i \(-0.284960\pi\)
0.625341 + 0.780352i \(0.284960\pi\)
\(314\) −12.2035 −0.688683
\(315\) 0 0
\(316\) 16.6707 0.937803
\(317\) 10.1775 0.571626 0.285813 0.958285i \(-0.407736\pi\)
0.285813 + 0.958285i \(0.407736\pi\)
\(318\) 5.05863 0.283674
\(319\) 4.71982 0.264259
\(320\) 0 0
\(321\) 5.52932 0.308616
\(322\) 19.2051 1.07026
\(323\) 0.338809 0.0188518
\(324\) 7.05520 0.391955
\(325\) 0 0
\(326\) −17.5845 −0.973916
\(327\) 4.67074 0.258293
\(328\) −0.0275977 −0.00152383
\(329\) −14.3404 −0.790613
\(330\) 0 0
\(331\) 8.69834 0.478104 0.239052 0.971007i \(-0.423163\pi\)
0.239052 + 0.971007i \(0.423163\pi\)
\(332\) 5.21811 0.286381
\(333\) 2.85514 0.156460
\(334\) −11.7914 −0.645199
\(335\) 0 0
\(336\) −1.55691 −0.0849366
\(337\) 16.7198 0.910787 0.455393 0.890290i \(-0.349499\pi\)
0.455393 + 0.890290i \(0.349499\pi\)
\(338\) −7.94137 −0.431953
\(339\) 6.60256 0.358602
\(340\) 0 0
\(341\) 48.2277 2.61167
\(342\) 1.30777 0.0707163
\(343\) −10.1173 −0.546281
\(344\) −6.61555 −0.356686
\(345\) 0 0
\(346\) 20.7880 1.11757
\(347\) 6.74398 0.362036 0.181018 0.983480i \(-0.442061\pi\)
0.181018 + 0.983480i \(0.442061\pi\)
\(348\) −0.470683 −0.0252313
\(349\) 7.42504 0.397453 0.198727 0.980055i \(-0.436319\pi\)
0.198727 + 0.980055i \(0.436319\pi\)
\(350\) 0 0
\(351\) −6.11727 −0.326516
\(352\) 4.71982 0.251567
\(353\) 3.36802 0.179262 0.0896309 0.995975i \(-0.471431\pi\)
0.0896309 + 0.995975i \(0.471431\pi\)
\(354\) −5.19051 −0.275872
\(355\) 0 0
\(356\) 9.08623 0.481569
\(357\) 1.12070 0.0593140
\(358\) −7.74742 −0.409464
\(359\) 1.81111 0.0955868 0.0477934 0.998857i \(-0.484781\pi\)
0.0477934 + 0.998857i \(0.484781\pi\)
\(360\) 0 0
\(361\) −18.7785 −0.988340
\(362\) −19.7440 −1.03772
\(363\) −5.30777 −0.278586
\(364\) −7.43965 −0.389944
\(365\) 0 0
\(366\) −0.637031 −0.0332981
\(367\) −30.8268 −1.60914 −0.804572 0.593855i \(-0.797605\pi\)
−0.804572 + 0.593855i \(0.797605\pi\)
\(368\) 5.80605 0.302662
\(369\) 0.0766789 0.00399174
\(370\) 0 0
\(371\) −35.5500 −1.84567
\(372\) −4.80949 −0.249361
\(373\) 2.06207 0.106770 0.0533850 0.998574i \(-0.482999\pi\)
0.0533850 + 0.998574i \(0.482999\pi\)
\(374\) −3.39744 −0.175678
\(375\) 0 0
\(376\) −4.33537 −0.223580
\(377\) −2.24914 −0.115837
\(378\) 8.99656 0.462733
\(379\) −7.29478 −0.374708 −0.187354 0.982292i \(-0.559991\pi\)
−0.187354 + 0.982292i \(0.559991\pi\)
\(380\) 0 0
\(381\) 7.63359 0.391081
\(382\) 0.397442 0.0203349
\(383\) −15.7846 −0.806554 −0.403277 0.915078i \(-0.632129\pi\)
−0.403277 + 0.915078i \(0.632129\pi\)
\(384\) −0.470683 −0.0240195
\(385\) 0 0
\(386\) −4.35342 −0.221583
\(387\) 18.3810 0.934359
\(388\) 11.1905 0.568112
\(389\) −13.3534 −0.677045 −0.338523 0.940958i \(-0.609927\pi\)
−0.338523 + 0.940958i \(0.609927\pi\)
\(390\) 0 0
\(391\) −4.17934 −0.211358
\(392\) 3.94137 0.199069
\(393\) −10.0552 −0.507218
\(394\) 6.70683 0.337885
\(395\) 0 0
\(396\) −13.1138 −0.658995
\(397\) 20.2130 1.01446 0.507232 0.861810i \(-0.330669\pi\)
0.507232 + 0.861810i \(0.330669\pi\)
\(398\) −9.36297 −0.469323
\(399\) 0.732814 0.0366866
\(400\) 0 0
\(401\) 15.2897 0.763533 0.381766 0.924259i \(-0.375316\pi\)
0.381766 + 0.924259i \(0.375316\pi\)
\(402\) −4.14486 −0.206727
\(403\) −22.9820 −1.14481
\(404\) −3.02760 −0.150629
\(405\) 0 0
\(406\) 3.30777 0.164162
\(407\) −4.85008 −0.240410
\(408\) 0.338809 0.0167736
\(409\) 14.8302 0.733307 0.366653 0.930358i \(-0.380503\pi\)
0.366653 + 0.930358i \(0.380503\pi\)
\(410\) 0 0
\(411\) 4.49217 0.221582
\(412\) 6.01461 0.296318
\(413\) 36.4768 1.79491
\(414\) −16.1319 −0.792838
\(415\) 0 0
\(416\) −2.24914 −0.110273
\(417\) 1.76385 0.0863761
\(418\) −2.22154 −0.108659
\(419\) −13.8647 −0.677334 −0.338667 0.940906i \(-0.609976\pi\)
−0.338667 + 0.940906i \(0.609976\pi\)
\(420\) 0 0
\(421\) 2.70178 0.131677 0.0658383 0.997830i \(-0.479028\pi\)
0.0658383 + 0.997830i \(0.479028\pi\)
\(422\) −24.5665 −1.19588
\(423\) 12.0456 0.585679
\(424\) −10.7474 −0.521941
\(425\) 0 0
\(426\) 2.73281 0.132405
\(427\) 4.47680 0.216647
\(428\) −11.7474 −0.567833
\(429\) 4.99656 0.241236
\(430\) 0 0
\(431\) −27.3009 −1.31504 −0.657519 0.753438i \(-0.728394\pi\)
−0.657519 + 0.753438i \(0.728394\pi\)
\(432\) 2.71982 0.130858
\(433\) −0.262130 −0.0125972 −0.00629859 0.999980i \(-0.502005\pi\)
−0.00629859 + 0.999980i \(0.502005\pi\)
\(434\) 33.7992 1.62241
\(435\) 0 0
\(436\) −9.92332 −0.475241
\(437\) −2.73281 −0.130728
\(438\) 1.71639 0.0820121
\(439\) 20.4983 0.978330 0.489165 0.872191i \(-0.337302\pi\)
0.489165 + 0.872191i \(0.337302\pi\)
\(440\) 0 0
\(441\) −10.9509 −0.521472
\(442\) 1.61899 0.0770073
\(443\) −0.456076 −0.0216688 −0.0108344 0.999941i \(-0.503449\pi\)
−0.0108344 + 0.999941i \(0.503449\pi\)
\(444\) 0.483673 0.0229541
\(445\) 0 0
\(446\) 2.11727 0.100255
\(447\) −0.941367 −0.0445251
\(448\) 3.30777 0.156278
\(449\) −8.62854 −0.407206 −0.203603 0.979054i \(-0.565265\pi\)
−0.203603 + 0.979054i \(0.565265\pi\)
\(450\) 0 0
\(451\) −0.130256 −0.00613352
\(452\) −14.0276 −0.659803
\(453\) 6.70683 0.315115
\(454\) −15.9138 −0.746870
\(455\) 0 0
\(456\) 0.221543 0.0103747
\(457\) −11.6872 −0.546703 −0.273351 0.961914i \(-0.588132\pi\)
−0.273351 + 0.961914i \(0.588132\pi\)
\(458\) −22.2897 −1.04153
\(459\) −1.95779 −0.0913820
\(460\) 0 0
\(461\) −1.82066 −0.0847967 −0.0423984 0.999101i \(-0.513500\pi\)
−0.0423984 + 0.999101i \(0.513500\pi\)
\(462\) −7.34836 −0.341877
\(463\) 26.8241 1.24662 0.623311 0.781974i \(-0.285787\pi\)
0.623311 + 0.781974i \(0.285787\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −23.3871 −1.08339
\(467\) −0.366407 −0.0169553 −0.00847764 0.999964i \(-0.502699\pi\)
−0.00847764 + 0.999964i \(0.502699\pi\)
\(468\) 6.24914 0.288867
\(469\) 29.1284 1.34503
\(470\) 0 0
\(471\) 5.74398 0.264669
\(472\) 11.0276 0.507586
\(473\) −31.2242 −1.43569
\(474\) −7.84664 −0.360408
\(475\) 0 0
\(476\) −2.38101 −0.109134
\(477\) 29.8613 1.36725
\(478\) 20.2277 0.925191
\(479\) 13.7363 0.627625 0.313813 0.949485i \(-0.398394\pi\)
0.313813 + 0.949485i \(0.398394\pi\)
\(480\) 0 0
\(481\) 2.31121 0.105382
\(482\) 1.26375 0.0575622
\(483\) −9.03953 −0.411313
\(484\) 11.2767 0.512579
\(485\) 0 0
\(486\) −11.4802 −0.520754
\(487\) 22.9414 1.03957 0.519786 0.854296i \(-0.326012\pi\)
0.519786 + 0.854296i \(0.326012\pi\)
\(488\) 1.35342 0.0612663
\(489\) 8.27674 0.374287
\(490\) 0 0
\(491\) 12.8241 0.578744 0.289372 0.957217i \(-0.406554\pi\)
0.289372 + 0.957217i \(0.406554\pi\)
\(492\) 0.0129898 0.000585624 0
\(493\) −0.719824 −0.0324192
\(494\) 1.05863 0.0476302
\(495\) 0 0
\(496\) 10.2181 0.458806
\(497\) −19.2051 −0.861467
\(498\) −2.45608 −0.110059
\(499\) −0.349979 −0.0156672 −0.00783361 0.999969i \(-0.502494\pi\)
−0.00783361 + 0.999969i \(0.502494\pi\)
\(500\) 0 0
\(501\) 5.55004 0.247957
\(502\) 9.63971 0.430241
\(503\) 34.1629 1.52325 0.761624 0.648019i \(-0.224402\pi\)
0.761624 + 0.648019i \(0.224402\pi\)
\(504\) −9.19051 −0.409378
\(505\) 0 0
\(506\) 27.4036 1.21824
\(507\) 3.73787 0.166005
\(508\) −16.2181 −0.719562
\(509\) −33.7586 −1.49632 −0.748162 0.663517i \(-0.769063\pi\)
−0.748162 + 0.663517i \(0.769063\pi\)
\(510\) 0 0
\(511\) −12.0621 −0.533595
\(512\) 1.00000 0.0441942
\(513\) −1.28018 −0.0565212
\(514\) −9.43965 −0.416365
\(515\) 0 0
\(516\) 3.11383 0.137079
\(517\) −20.4622 −0.899926
\(518\) −3.39906 −0.149346
\(519\) −9.78457 −0.429495
\(520\) 0 0
\(521\) 31.3810 1.37483 0.687414 0.726266i \(-0.258746\pi\)
0.687414 + 0.726266i \(0.258746\pi\)
\(522\) −2.77846 −0.121610
\(523\) 38.0303 1.66295 0.831474 0.555564i \(-0.187498\pi\)
0.831474 + 0.555564i \(0.187498\pi\)
\(524\) 21.3630 0.933246
\(525\) 0 0
\(526\) 11.6646 0.508602
\(527\) −7.35524 −0.320399
\(528\) −2.22154 −0.0966802
\(529\) 10.7103 0.465664
\(530\) 0 0
\(531\) −30.6397 −1.32965
\(532\) −1.55691 −0.0675008
\(533\) 0.0620710 0.00268859
\(534\) −4.27674 −0.185073
\(535\) 0 0
\(536\) 8.80605 0.380364
\(537\) 3.64658 0.157362
\(538\) −20.1414 −0.868359
\(539\) 18.6026 0.801269
\(540\) 0 0
\(541\) −42.7862 −1.83952 −0.919761 0.392479i \(-0.871618\pi\)
−0.919761 + 0.392479i \(0.871618\pi\)
\(542\) −27.0422 −1.16156
\(543\) 9.29317 0.398808
\(544\) −0.719824 −0.0308622
\(545\) 0 0
\(546\) 3.50172 0.149860
\(547\) −28.0974 −1.20136 −0.600679 0.799490i \(-0.705103\pi\)
−0.600679 + 0.799490i \(0.705103\pi\)
\(548\) −9.54392 −0.407696
\(549\) −3.76041 −0.160490
\(550\) 0 0
\(551\) −0.470683 −0.0200518
\(552\) −2.73281 −0.116316
\(553\) 55.1430 2.34492
\(554\) −12.4216 −0.527743
\(555\) 0 0
\(556\) −3.74742 −0.158926
\(557\) 12.6155 0.534538 0.267269 0.963622i \(-0.413879\pi\)
0.267269 + 0.963622i \(0.413879\pi\)
\(558\) −28.3906 −1.20187
\(559\) 14.8793 0.629327
\(560\) 0 0
\(561\) 1.59912 0.0675149
\(562\) 24.8957 1.05016
\(563\) 33.7586 1.42276 0.711378 0.702810i \(-0.248072\pi\)
0.711378 + 0.702810i \(0.248072\pi\)
\(564\) 2.04059 0.0859242
\(565\) 0 0
\(566\) −4.83709 −0.203318
\(567\) 23.3370 0.980061
\(568\) −5.80605 −0.243617
\(569\) 5.66119 0.237329 0.118665 0.992934i \(-0.462139\pi\)
0.118665 + 0.992934i \(0.462139\pi\)
\(570\) 0 0
\(571\) 30.6105 1.28101 0.640505 0.767954i \(-0.278725\pi\)
0.640505 + 0.767954i \(0.278725\pi\)
\(572\) −10.6155 −0.443858
\(573\) −0.187070 −0.00781494
\(574\) −0.0912868 −0.00381024
\(575\) 0 0
\(576\) −2.77846 −0.115769
\(577\) −2.94298 −0.122518 −0.0612590 0.998122i \(-0.519512\pi\)
−0.0612590 + 0.998122i \(0.519512\pi\)
\(578\) −16.4819 −0.685555
\(579\) 2.04908 0.0851569
\(580\) 0 0
\(581\) 17.2603 0.716078
\(582\) −5.26719 −0.218332
\(583\) −50.7259 −2.10085
\(584\) −3.64658 −0.150897
\(585\) 0 0
\(586\) 33.5190 1.38466
\(587\) −14.5715 −0.601431 −0.300716 0.953714i \(-0.597225\pi\)
−0.300716 + 0.953714i \(0.597225\pi\)
\(588\) −1.85514 −0.0765045
\(589\) −4.80949 −0.198172
\(590\) 0 0
\(591\) −3.15680 −0.129853
\(592\) −1.02760 −0.0422340
\(593\) −19.0552 −0.782503 −0.391252 0.920284i \(-0.627958\pi\)
−0.391252 + 0.920284i \(0.627958\pi\)
\(594\) 12.8371 0.526712
\(595\) 0 0
\(596\) 2.00000 0.0819232
\(597\) 4.40699 0.180366
\(598\) −13.0586 −0.534007
\(599\) 5.86631 0.239691 0.119845 0.992793i \(-0.461760\pi\)
0.119845 + 0.992793i \(0.461760\pi\)
\(600\) 0 0
\(601\) −31.3285 −1.27792 −0.638958 0.769242i \(-0.720634\pi\)
−0.638958 + 0.769242i \(0.720634\pi\)
\(602\) −21.8827 −0.891874
\(603\) −24.4672 −0.996383
\(604\) −14.2491 −0.579789
\(605\) 0 0
\(606\) 1.42504 0.0578883
\(607\) 11.0061 0.446724 0.223362 0.974736i \(-0.428297\pi\)
0.223362 + 0.974736i \(0.428297\pi\)
\(608\) −0.470683 −0.0190887
\(609\) −1.55691 −0.0630893
\(610\) 0 0
\(611\) 9.75086 0.394478
\(612\) 2.00000 0.0808452
\(613\) −14.7068 −0.594003 −0.297002 0.954877i \(-0.595987\pi\)
−0.297002 + 0.954877i \(0.595987\pi\)
\(614\) −12.3680 −0.499133
\(615\) 0 0
\(616\) 15.6121 0.629030
\(617\) 20.1579 0.811525 0.405762 0.913979i \(-0.367006\pi\)
0.405762 + 0.913979i \(0.367006\pi\)
\(618\) −2.83098 −0.113879
\(619\) 39.9379 1.60524 0.802620 0.596490i \(-0.203438\pi\)
0.802620 + 0.596490i \(0.203438\pi\)
\(620\) 0 0
\(621\) 15.7914 0.633689
\(622\) 15.3224 0.614372
\(623\) 30.0552 1.20414
\(624\) 1.05863 0.0423792
\(625\) 0 0
\(626\) 22.1268 0.884366
\(627\) 1.04564 0.0417590
\(628\) −12.2035 −0.486973
\(629\) 0.739689 0.0294933
\(630\) 0 0
\(631\) 40.3189 1.60507 0.802536 0.596604i \(-0.203484\pi\)
0.802536 + 0.596604i \(0.203484\pi\)
\(632\) 16.6707 0.663127
\(633\) 11.5630 0.459589
\(634\) 10.1775 0.404201
\(635\) 0 0
\(636\) 5.05863 0.200588
\(637\) −8.86469 −0.351232
\(638\) 4.71982 0.186860
\(639\) 16.1319 0.638167
\(640\) 0 0
\(641\) −35.7992 −1.41398 −0.706991 0.707222i \(-0.749948\pi\)
−0.706991 + 0.707222i \(0.749948\pi\)
\(642\) 5.52932 0.218225
\(643\) −21.1741 −0.835024 −0.417512 0.908671i \(-0.637098\pi\)
−0.417512 + 0.908671i \(0.637098\pi\)
\(644\) 19.2051 0.756788
\(645\) 0 0
\(646\) 0.338809 0.0133303
\(647\) 33.6742 1.32387 0.661934 0.749562i \(-0.269736\pi\)
0.661934 + 0.749562i \(0.269736\pi\)
\(648\) 7.05520 0.277154
\(649\) 52.0483 2.04308
\(650\) 0 0
\(651\) −15.9087 −0.623512
\(652\) −17.5845 −0.688663
\(653\) −28.3500 −1.10942 −0.554710 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(654\) 4.67074 0.182640
\(655\) 0 0
\(656\) −0.0275977 −0.00107751
\(657\) 10.1319 0.395282
\(658\) −14.3404 −0.559048
\(659\) −13.1268 −0.511348 −0.255674 0.966763i \(-0.582297\pi\)
−0.255674 + 0.966763i \(0.582297\pi\)
\(660\) 0 0
\(661\) −23.3078 −0.906567 −0.453284 0.891366i \(-0.649748\pi\)
−0.453284 + 0.891366i \(0.649748\pi\)
\(662\) 8.69834 0.338071
\(663\) −0.762030 −0.0295948
\(664\) 5.21811 0.202502
\(665\) 0 0
\(666\) 2.85514 0.110634
\(667\) 5.80605 0.224811
\(668\) −11.7914 −0.456225
\(669\) −0.996562 −0.0385293
\(670\) 0 0
\(671\) 6.38789 0.246602
\(672\) −1.55691 −0.0600593
\(673\) 27.4784 1.05922 0.529608 0.848243i \(-0.322339\pi\)
0.529608 + 0.848243i \(0.322339\pi\)
\(674\) 16.7198 0.644024
\(675\) 0 0
\(676\) −7.94137 −0.305437
\(677\) −35.6363 −1.36961 −0.684807 0.728725i \(-0.740113\pi\)
−0.684807 + 0.728725i \(0.740113\pi\)
\(678\) 6.60256 0.253570
\(679\) 37.0157 1.42053
\(680\) 0 0
\(681\) 7.49035 0.287031
\(682\) 48.2277 1.84673
\(683\) 10.3319 0.395340 0.197670 0.980269i \(-0.436662\pi\)
0.197670 + 0.980269i \(0.436662\pi\)
\(684\) 1.30777 0.0500040
\(685\) 0 0
\(686\) −10.1173 −0.386279
\(687\) 10.4914 0.400272
\(688\) −6.61555 −0.252215
\(689\) 24.1725 0.920897
\(690\) 0 0
\(691\) 3.02148 0.114943 0.0574713 0.998347i \(-0.481696\pi\)
0.0574713 + 0.998347i \(0.481696\pi\)
\(692\) 20.7880 0.790242
\(693\) −43.3776 −1.64778
\(694\) 6.74398 0.255998
\(695\) 0 0
\(696\) −0.470683 −0.0178412
\(697\) 0.0198655 0.000752458 0
\(698\) 7.42504 0.281042
\(699\) 11.0079 0.416358
\(700\) 0 0
\(701\) 37.5500 1.41825 0.709123 0.705085i \(-0.249091\pi\)
0.709123 + 0.705085i \(0.249091\pi\)
\(702\) −6.11727 −0.230881
\(703\) 0.483673 0.0182421
\(704\) 4.71982 0.177885
\(705\) 0 0
\(706\) 3.36802 0.126757
\(707\) −10.0146 −0.376638
\(708\) −5.19051 −0.195071
\(709\) −23.7586 −0.892273 −0.446136 0.894965i \(-0.647200\pi\)
−0.446136 + 0.894965i \(0.647200\pi\)
\(710\) 0 0
\(711\) −46.3189 −1.73710
\(712\) 9.08623 0.340521
\(713\) 59.3269 2.22181
\(714\) 1.12070 0.0419413
\(715\) 0 0
\(716\) −7.74742 −0.289535
\(717\) −9.52082 −0.355562
\(718\) 1.81111 0.0675901
\(719\) 48.4768 1.80788 0.903940 0.427660i \(-0.140662\pi\)
0.903940 + 0.427660i \(0.140662\pi\)
\(720\) 0 0
\(721\) 19.8950 0.740927
\(722\) −18.7785 −0.698862
\(723\) −0.594825 −0.0221218
\(724\) −19.7440 −0.733779
\(725\) 0 0
\(726\) −5.30777 −0.196990
\(727\) −31.5208 −1.16904 −0.584521 0.811378i \(-0.698718\pi\)
−0.584521 + 0.811378i \(0.698718\pi\)
\(728\) −7.43965 −0.275732
\(729\) −15.7620 −0.583779
\(730\) 0 0
\(731\) 4.76203 0.176130
\(732\) −0.637031 −0.0235453
\(733\) −21.5017 −0.794184 −0.397092 0.917779i \(-0.629981\pi\)
−0.397092 + 0.917779i \(0.629981\pi\)
\(734\) −30.8268 −1.13784
\(735\) 0 0
\(736\) 5.80605 0.214014
\(737\) 41.5630 1.53099
\(738\) 0.0766789 0.00282259
\(739\) 12.5604 0.462040 0.231020 0.972949i \(-0.425794\pi\)
0.231020 + 0.972949i \(0.425794\pi\)
\(740\) 0 0
\(741\) −0.498281 −0.0183048
\(742\) −35.5500 −1.30508
\(743\) −26.5275 −0.973199 −0.486600 0.873625i \(-0.661763\pi\)
−0.486600 + 0.873625i \(0.661763\pi\)
\(744\) −4.80949 −0.176325
\(745\) 0 0
\(746\) 2.06207 0.0754978
\(747\) −14.4983 −0.530464
\(748\) −3.39744 −0.124223
\(749\) −38.8578 −1.41983
\(750\) 0 0
\(751\) −37.4232 −1.36559 −0.682796 0.730609i \(-0.739236\pi\)
−0.682796 + 0.730609i \(0.739236\pi\)
\(752\) −4.33537 −0.158095
\(753\) −4.53725 −0.165347
\(754\) −2.24914 −0.0819089
\(755\) 0 0
\(756\) 8.99656 0.327202
\(757\) 30.0844 1.09344 0.546718 0.837317i \(-0.315877\pi\)
0.546718 + 0.837317i \(0.315877\pi\)
\(758\) −7.29478 −0.264958
\(759\) −12.8984 −0.468182
\(760\) 0 0
\(761\) 42.1690 1.52863 0.764313 0.644845i \(-0.223078\pi\)
0.764313 + 0.644845i \(0.223078\pi\)
\(762\) 7.63359 0.276536
\(763\) −32.8241 −1.18831
\(764\) 0.397442 0.0143790
\(765\) 0 0
\(766\) −15.7846 −0.570320
\(767\) −24.8026 −0.895571
\(768\) −0.470683 −0.0169843
\(769\) −29.9587 −1.08034 −0.540168 0.841557i \(-0.681639\pi\)
−0.540168 + 0.841557i \(0.681639\pi\)
\(770\) 0 0
\(771\) 4.44309 0.160014
\(772\) −4.35342 −0.156683
\(773\) −50.1966 −1.80545 −0.902723 0.430221i \(-0.858436\pi\)
−0.902723 + 0.430221i \(0.858436\pi\)
\(774\) 18.3810 0.660692
\(775\) 0 0
\(776\) 11.1905 0.401716
\(777\) 1.59988 0.0573954
\(778\) −13.3534 −0.478743
\(779\) 0.0129898 0.000465406 0
\(780\) 0 0
\(781\) −27.4036 −0.980576
\(782\) −4.17934 −0.149453
\(783\) 2.71982 0.0971986
\(784\) 3.94137 0.140763
\(785\) 0 0
\(786\) −10.0552 −0.358657
\(787\) −44.5845 −1.58927 −0.794633 0.607090i \(-0.792337\pi\)
−0.794633 + 0.607090i \(0.792337\pi\)
\(788\) 6.70683 0.238921
\(789\) −5.49035 −0.195462
\(790\) 0 0
\(791\) −46.4001 −1.64980
\(792\) −13.1138 −0.465980
\(793\) −3.04403 −0.108097
\(794\) 20.2130 0.717334
\(795\) 0 0
\(796\) −9.36297 −0.331862
\(797\) −23.3155 −0.825878 −0.412939 0.910759i \(-0.635498\pi\)
−0.412939 + 0.910759i \(0.635498\pi\)
\(798\) 0.732814 0.0259413
\(799\) 3.12070 0.110403
\(800\) 0 0
\(801\) −25.2457 −0.892013
\(802\) 15.2897 0.539899
\(803\) −17.2112 −0.607371
\(804\) −4.14486 −0.146178
\(805\) 0 0
\(806\) −22.9820 −0.809505
\(807\) 9.48024 0.333720
\(808\) −3.02760 −0.106511
\(809\) 50.6570 1.78100 0.890502 0.454978i \(-0.150353\pi\)
0.890502 + 0.454978i \(0.150353\pi\)
\(810\) 0 0
\(811\) 40.6949 1.42899 0.714496 0.699640i \(-0.246656\pi\)
0.714496 + 0.699640i \(0.246656\pi\)
\(812\) 3.30777 0.116080
\(813\) 12.7283 0.446402
\(814\) −4.85008 −0.169995
\(815\) 0 0
\(816\) 0.338809 0.0118607
\(817\) 3.11383 0.108939
\(818\) 14.8302 0.518526
\(819\) 20.6707 0.722294
\(820\) 0 0
\(821\) −11.8421 −0.413294 −0.206647 0.978416i \(-0.566255\pi\)
−0.206647 + 0.978416i \(0.566255\pi\)
\(822\) 4.49217 0.156682
\(823\) 14.5243 0.506284 0.253142 0.967429i \(-0.418536\pi\)
0.253142 + 0.967429i \(0.418536\pi\)
\(824\) 6.01461 0.209529
\(825\) 0 0
\(826\) 36.4768 1.26919
\(827\) −30.9183 −1.07513 −0.537567 0.843221i \(-0.680656\pi\)
−0.537567 + 0.843221i \(0.680656\pi\)
\(828\) −16.1319 −0.560621
\(829\) 1.26719 0.0440112 0.0220056 0.999758i \(-0.492995\pi\)
0.0220056 + 0.999758i \(0.492995\pi\)
\(830\) 0 0
\(831\) 5.84664 0.202818
\(832\) −2.24914 −0.0779749
\(833\) −2.83709 −0.0982994
\(834\) 1.76385 0.0610771
\(835\) 0 0
\(836\) −2.22154 −0.0768337
\(837\) 27.7914 0.960613
\(838\) −13.8647 −0.478948
\(839\) 33.1855 1.14569 0.572845 0.819664i \(-0.305840\pi\)
0.572845 + 0.819664i \(0.305840\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 2.70178 0.0931094
\(843\) −11.7180 −0.403590
\(844\) −24.5665 −0.845613
\(845\) 0 0
\(846\) 12.0456 0.414138
\(847\) 37.3009 1.28167
\(848\) −10.7474 −0.369068
\(849\) 2.27674 0.0781375
\(850\) 0 0
\(851\) −5.96629 −0.204522
\(852\) 2.73281 0.0936247
\(853\) −42.7743 −1.46456 −0.732281 0.681002i \(-0.761544\pi\)
−0.732281 + 0.681002i \(0.761544\pi\)
\(854\) 4.47680 0.153193
\(855\) 0 0
\(856\) −11.7474 −0.401519
\(857\) 44.7061 1.52713 0.763565 0.645731i \(-0.223447\pi\)
0.763565 + 0.645731i \(0.223447\pi\)
\(858\) 4.99656 0.170580
\(859\) −7.01643 −0.239397 −0.119699 0.992810i \(-0.538193\pi\)
−0.119699 + 0.992810i \(0.538193\pi\)
\(860\) 0 0
\(861\) 0.0429672 0.00146432
\(862\) −27.3009 −0.929872
\(863\) 40.4768 1.37785 0.688923 0.724834i \(-0.258084\pi\)
0.688923 + 0.724834i \(0.258084\pi\)
\(864\) 2.71982 0.0925303
\(865\) 0 0
\(866\) −0.262130 −0.00890755
\(867\) 7.75774 0.263466
\(868\) 33.7992 1.14722
\(869\) 78.6830 2.66914
\(870\) 0 0
\(871\) −19.8061 −0.671103
\(872\) −9.92332 −0.336046
\(873\) −31.0923 −1.05232
\(874\) −2.73281 −0.0924388
\(875\) 0 0
\(876\) 1.71639 0.0579913
\(877\) 6.14325 0.207443 0.103721 0.994606i \(-0.466925\pi\)
0.103721 + 0.994606i \(0.466925\pi\)
\(878\) 20.4983 0.691783
\(879\) −15.7768 −0.532140
\(880\) 0 0
\(881\) −56.6639 −1.90905 −0.954527 0.298124i \(-0.903639\pi\)
−0.954527 + 0.298124i \(0.903639\pi\)
\(882\) −10.9509 −0.368737
\(883\) −49.5389 −1.66712 −0.833558 0.552432i \(-0.813700\pi\)
−0.833558 + 0.552432i \(0.813700\pi\)
\(884\) 1.61899 0.0544524
\(885\) 0 0
\(886\) −0.456076 −0.0153222
\(887\) −1.03265 −0.0346731 −0.0173366 0.999850i \(-0.505519\pi\)
−0.0173366 + 0.999850i \(0.505519\pi\)
\(888\) 0.483673 0.0162310
\(889\) −53.6458 −1.79922
\(890\) 0 0
\(891\) 33.2993 1.11557
\(892\) 2.11727 0.0708913
\(893\) 2.04059 0.0682857
\(894\) −0.941367 −0.0314840
\(895\) 0 0
\(896\) 3.30777 0.110505
\(897\) 6.14648 0.205225
\(898\) −8.62854 −0.287938
\(899\) 10.2181 0.340793
\(900\) 0 0
\(901\) 7.73625 0.257732
\(902\) −0.130256 −0.00433705
\(903\) 10.2998 0.342757
\(904\) −14.0276 −0.466551
\(905\) 0 0
\(906\) 6.70683 0.222820
\(907\) −8.04059 −0.266983 −0.133492 0.991050i \(-0.542619\pi\)
−0.133492 + 0.991050i \(0.542619\pi\)
\(908\) −15.9138 −0.528117
\(909\) 8.41205 0.279010
\(910\) 0 0
\(911\) 14.0456 0.465353 0.232676 0.972554i \(-0.425252\pi\)
0.232676 + 0.972554i \(0.425252\pi\)
\(912\) 0.221543 0.00733602
\(913\) 24.6285 0.815086
\(914\) −11.6872 −0.386577
\(915\) 0 0
\(916\) −22.2897 −0.736473
\(917\) 70.6639 2.33353
\(918\) −1.95779 −0.0646168
\(919\) 15.5163 0.511836 0.255918 0.966698i \(-0.417622\pi\)
0.255918 + 0.966698i \(0.417622\pi\)
\(920\) 0 0
\(921\) 5.82142 0.191822
\(922\) −1.82066 −0.0599603
\(923\) 13.0586 0.429830
\(924\) −7.34836 −0.241743
\(925\) 0 0
\(926\) 26.8241 0.881495
\(927\) −16.7113 −0.548872
\(928\) 1.00000 0.0328266
\(929\) 6.93181 0.227425 0.113713 0.993514i \(-0.463726\pi\)
0.113713 + 0.993514i \(0.463726\pi\)
\(930\) 0 0
\(931\) −1.85514 −0.0607996
\(932\) −23.3871 −0.766071
\(933\) −7.21199 −0.236110
\(934\) −0.366407 −0.0119892
\(935\) 0 0
\(936\) 6.24914 0.204260
\(937\) 45.3189 1.48051 0.740253 0.672329i \(-0.234706\pi\)
0.740253 + 0.672329i \(0.234706\pi\)
\(938\) 29.1284 0.951077
\(939\) −10.4147 −0.339872
\(940\) 0 0
\(941\) 7.75086 0.252671 0.126335 0.991988i \(-0.459678\pi\)
0.126335 + 0.991988i \(0.459678\pi\)
\(942\) 5.74398 0.187149
\(943\) −0.160234 −0.00521792
\(944\) 11.0276 0.358918
\(945\) 0 0
\(946\) −31.2242 −1.01519
\(947\) 11.1544 0.362470 0.181235 0.983440i \(-0.441991\pi\)
0.181235 + 0.983440i \(0.441991\pi\)
\(948\) −7.84664 −0.254847
\(949\) 8.20168 0.266238
\(950\) 0 0
\(951\) −4.79039 −0.155339
\(952\) −2.38101 −0.0771691
\(953\) 55.1492 1.78646 0.893228 0.449604i \(-0.148435\pi\)
0.893228 + 0.449604i \(0.148435\pi\)
\(954\) 29.8613 0.966794
\(955\) 0 0
\(956\) 20.2277 0.654209
\(957\) −2.22154 −0.0718123
\(958\) 13.7363 0.443798
\(959\) −31.5691 −1.01942
\(960\) 0 0
\(961\) 73.4097 2.36805
\(962\) 2.31121 0.0745165
\(963\) 32.6397 1.05180
\(964\) 1.26375 0.0407026
\(965\) 0 0
\(966\) −9.03953 −0.290842
\(967\) 28.8984 0.929310 0.464655 0.885492i \(-0.346178\pi\)
0.464655 + 0.885492i \(0.346178\pi\)
\(968\) 11.2767 0.362448
\(969\) −0.159472 −0.00512297
\(970\) 0 0
\(971\) −12.1932 −0.391298 −0.195649 0.980674i \(-0.562681\pi\)
−0.195649 + 0.980674i \(0.562681\pi\)
\(972\) −11.4802 −0.368229
\(973\) −12.3956 −0.397385
\(974\) 22.9414 0.735089
\(975\) 0 0
\(976\) 1.35342 0.0433218
\(977\) −20.5861 −0.658609 −0.329304 0.944224i \(-0.606814\pi\)
−0.329304 + 0.944224i \(0.606814\pi\)
\(978\) 8.27674 0.264661
\(979\) 42.8854 1.37062
\(980\) 0 0
\(981\) 27.5715 0.880291
\(982\) 12.8241 0.409233
\(983\) 19.7681 0.630506 0.315253 0.949008i \(-0.397911\pi\)
0.315253 + 0.949008i \(0.397911\pi\)
\(984\) 0.0129898 0.000414098 0
\(985\) 0 0
\(986\) −0.719824 −0.0229239
\(987\) 6.74980 0.214849
\(988\) 1.05863 0.0336796
\(989\) −38.4102 −1.22137
\(990\) 0 0
\(991\) 47.5139 1.50933 0.754665 0.656110i \(-0.227799\pi\)
0.754665 + 0.656110i \(0.227799\pi\)
\(992\) 10.2181 0.324425
\(993\) −4.09416 −0.129924
\(994\) −19.2051 −0.609149
\(995\) 0 0
\(996\) −2.45608 −0.0778237
\(997\) 31.7276 1.00482 0.502411 0.864629i \(-0.332446\pi\)
0.502411 + 0.864629i \(0.332446\pi\)
\(998\) −0.349979 −0.0110784
\(999\) −2.79488 −0.0884262
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.s.1.2 yes 3
5.2 odd 4 1450.2.b.k.349.5 6
5.3 odd 4 1450.2.b.k.349.2 6
5.4 even 2 1450.2.a.q.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1450.2.a.q.1.2 3 5.4 even 2
1450.2.a.s.1.2 yes 3 1.1 even 1 trivial
1450.2.b.k.349.2 6 5.3 odd 4
1450.2.b.k.349.5 6 5.2 odd 4