Properties

Label 9075.2.a.df.1.4
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
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] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.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: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.33826\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.82709 q^{2} +1.00000 q^{3} +1.33826 q^{4} +1.82709 q^{6} +1.74724 q^{7} -1.20906 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.82709 q^{2} +1.00000 q^{3} +1.33826 q^{4} +1.82709 q^{6} +1.74724 q^{7} -1.20906 q^{8} +1.00000 q^{9} +1.33826 q^{12} -5.35772 q^{13} +3.19236 q^{14} -4.88558 q^{16} +7.33070 q^{17} +1.82709 q^{18} -6.01478 q^{19} +1.74724 q^{21} -8.35772 q^{23} -1.20906 q^{24} -9.78903 q^{26} +1.00000 q^{27} +2.33826 q^{28} -5.53818 q^{29} -0.0213642 q^{31} -6.50828 q^{32} +13.3939 q^{34} +1.33826 q^{36} +1.12165 q^{37} -10.9896 q^{38} -5.35772 q^{39} +3.65983 q^{41} +3.19236 q^{42} -1.96950 q^{43} -15.2703 q^{46} -3.81953 q^{47} -4.88558 q^{48} -3.94716 q^{49} +7.33070 q^{51} -7.17002 q^{52} -1.67088 q^{53} +1.82709 q^{54} -2.11251 q^{56} -6.01478 q^{57} -10.1188 q^{58} +5.66332 q^{59} -9.27977 q^{61} -0.0390343 q^{62} +1.74724 q^{63} -2.12007 q^{64} -0.350489 q^{67} +9.81040 q^{68} -8.35772 q^{69} +4.53818 q^{71} -1.20906 q^{72} +10.1395 q^{73} +2.04935 q^{74} -8.04935 q^{76} -9.78903 q^{78} -2.56401 q^{79} +1.00000 q^{81} +6.68684 q^{82} -16.0805 q^{83} +2.33826 q^{84} -3.59845 q^{86} -5.53818 q^{87} -14.5062 q^{89} -9.36121 q^{91} -11.1848 q^{92} -0.0213642 q^{93} -6.97864 q^{94} -6.50828 q^{96} +1.89972 q^{97} -7.21182 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + 4 q^{3} + q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + q^{12} + 7 q^{13} - 5 q^{14} - 9 q^{16} + 8 q^{17} + q^{18} - 11 q^{19} - 5 q^{23} - 3 q^{24} - 12 q^{26} + 4 q^{27} + 5 q^{28} - 17 q^{29} - 5 q^{31} + 17 q^{34} + q^{36} - 15 q^{37} + q^{38} + 7 q^{39} - 10 q^{41} - 5 q^{42} - 4 q^{43} - 15 q^{46} + 8 q^{47} - 9 q^{48} - 8 q^{49} + 8 q^{51} - 7 q^{52} - 10 q^{53} + q^{54} + 10 q^{56} - 11 q^{57} + 7 q^{58} + 9 q^{59} - 37 q^{61} - 20 q^{62} - 7 q^{64} - 3 q^{67} + 17 q^{68} - 5 q^{69} + 13 q^{71} - 3 q^{72} + 15 q^{73} + 5 q^{74} - 29 q^{76} - 12 q^{78} - 20 q^{79} + 4 q^{81} - 5 q^{82} - 17 q^{83} + 5 q^{84} + 34 q^{86} - 17 q^{87} - 24 q^{89} - 20 q^{91} - 10 q^{92} - 5 q^{93} - 23 q^{94} + 32 q^{97} + 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82709 1.29195 0.645974 0.763359i \(-0.276451\pi\)
0.645974 + 0.763359i \(0.276451\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.33826 0.669131
\(5\) 0 0
\(6\) 1.82709 0.745907
\(7\) 1.74724 0.660394 0.330197 0.943912i \(-0.392885\pi\)
0.330197 + 0.943912i \(0.392885\pi\)
\(8\) −1.20906 −0.427466
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 1.33826 0.386323
\(13\) −5.35772 −1.48596 −0.742981 0.669312i \(-0.766589\pi\)
−0.742981 + 0.669312i \(0.766589\pi\)
\(14\) 3.19236 0.853195
\(15\) 0 0
\(16\) −4.88558 −1.22139
\(17\) 7.33070 1.77796 0.888978 0.457949i \(-0.151416\pi\)
0.888978 + 0.457949i \(0.151416\pi\)
\(18\) 1.82709 0.430649
\(19\) −6.01478 −1.37989 −0.689943 0.723864i \(-0.742364\pi\)
−0.689943 + 0.723864i \(0.742364\pi\)
\(20\) 0 0
\(21\) 1.74724 0.381279
\(22\) 0 0
\(23\) −8.35772 −1.74270 −0.871352 0.490658i \(-0.836756\pi\)
−0.871352 + 0.490658i \(0.836756\pi\)
\(24\) −1.20906 −0.246798
\(25\) 0 0
\(26\) −9.78903 −1.91979
\(27\) 1.00000 0.192450
\(28\) 2.33826 0.441890
\(29\) −5.53818 −1.02841 −0.514207 0.857666i \(-0.671914\pi\)
−0.514207 + 0.857666i \(0.671914\pi\)
\(30\) 0 0
\(31\) −0.0213642 −0.00383712 −0.00191856 0.999998i \(-0.500611\pi\)
−0.00191856 + 0.999998i \(0.500611\pi\)
\(32\) −6.50828 −1.15051
\(33\) 0 0
\(34\) 13.3939 2.29703
\(35\) 0 0
\(36\) 1.33826 0.223044
\(37\) 1.12165 0.184398 0.0921988 0.995741i \(-0.470610\pi\)
0.0921988 + 0.995741i \(0.470610\pi\)
\(38\) −10.9896 −1.78274
\(39\) −5.35772 −0.857921
\(40\) 0 0
\(41\) 3.65983 0.571569 0.285785 0.958294i \(-0.407746\pi\)
0.285785 + 0.958294i \(0.407746\pi\)
\(42\) 3.19236 0.492592
\(43\) −1.96950 −0.300346 −0.150173 0.988660i \(-0.547983\pi\)
−0.150173 + 0.988660i \(0.547983\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −15.2703 −2.25148
\(47\) −3.81953 −0.557136 −0.278568 0.960416i \(-0.589860\pi\)
−0.278568 + 0.960416i \(0.589860\pi\)
\(48\) −4.88558 −0.705173
\(49\) −3.94716 −0.563880
\(50\) 0 0
\(51\) 7.33070 1.02650
\(52\) −7.17002 −0.994303
\(53\) −1.67088 −0.229512 −0.114756 0.993394i \(-0.536609\pi\)
−0.114756 + 0.993394i \(0.536609\pi\)
\(54\) 1.82709 0.248636
\(55\) 0 0
\(56\) −2.11251 −0.282296
\(57\) −6.01478 −0.796678
\(58\) −10.1188 −1.32866
\(59\) 5.66332 0.737301 0.368651 0.929568i \(-0.379820\pi\)
0.368651 + 0.929568i \(0.379820\pi\)
\(60\) 0 0
\(61\) −9.27977 −1.18815 −0.594077 0.804408i \(-0.702482\pi\)
−0.594077 + 0.804408i \(0.702482\pi\)
\(62\) −0.0390343 −0.00495737
\(63\) 1.74724 0.220131
\(64\) −2.12007 −0.265008
\(65\) 0 0
\(66\) 0 0
\(67\) −0.350489 −0.0428190 −0.0214095 0.999771i \(-0.506815\pi\)
−0.0214095 + 0.999771i \(0.506815\pi\)
\(68\) 9.81040 1.18969
\(69\) −8.35772 −1.00615
\(70\) 0 0
\(71\) 4.53818 0.538583 0.269292 0.963059i \(-0.413210\pi\)
0.269292 + 0.963059i \(0.413210\pi\)
\(72\) −1.20906 −0.142489
\(73\) 10.1395 1.18674 0.593371 0.804929i \(-0.297797\pi\)
0.593371 + 0.804929i \(0.297797\pi\)
\(74\) 2.04935 0.238232
\(75\) 0 0
\(76\) −8.04935 −0.923324
\(77\) 0 0
\(78\) −9.78903 −1.10839
\(79\) −2.56401 −0.288474 −0.144237 0.989543i \(-0.546073\pi\)
−0.144237 + 0.989543i \(0.546073\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.68684 0.738438
\(83\) −16.0805 −1.76506 −0.882532 0.470252i \(-0.844163\pi\)
−0.882532 + 0.470252i \(0.844163\pi\)
\(84\) 2.33826 0.255125
\(85\) 0 0
\(86\) −3.59845 −0.388031
\(87\) −5.53818 −0.593755
\(88\) 0 0
\(89\) −14.5062 −1.53765 −0.768825 0.639459i \(-0.779159\pi\)
−0.768825 + 0.639459i \(0.779159\pi\)
\(90\) 0 0
\(91\) −9.36121 −0.981321
\(92\) −11.1848 −1.16610
\(93\) −0.0213642 −0.00221536
\(94\) −6.97864 −0.719791
\(95\) 0 0
\(96\) −6.50828 −0.664249
\(97\) 1.89972 0.192887 0.0964435 0.995338i \(-0.469253\pi\)
0.0964435 + 0.995338i \(0.469253\pi\)
\(98\) −7.21182 −0.728504
\(99\) 0 0
\(100\) 0 0
\(101\) −6.71011 −0.667681 −0.333841 0.942630i \(-0.608345\pi\)
−0.333841 + 0.942630i \(0.608345\pi\)
\(102\) 13.3939 1.32619
\(103\) 4.24678 0.418448 0.209224 0.977868i \(-0.432906\pi\)
0.209224 + 0.977868i \(0.432906\pi\)
\(104\) 6.47778 0.635199
\(105\) 0 0
\(106\) −3.05284 −0.296518
\(107\) −15.7230 −1.52000 −0.759999 0.649924i \(-0.774801\pi\)
−0.759999 + 0.649924i \(0.774801\pi\)
\(108\) 1.33826 0.128774
\(109\) −4.23895 −0.406018 −0.203009 0.979177i \(-0.565072\pi\)
−0.203009 + 0.979177i \(0.565072\pi\)
\(110\) 0 0
\(111\) 1.12165 0.106462
\(112\) −8.53627 −0.806602
\(113\) 7.19643 0.676983 0.338492 0.940969i \(-0.390083\pi\)
0.338492 + 0.940969i \(0.390083\pi\)
\(114\) −10.9896 −1.02927
\(115\) 0 0
\(116\) −7.41153 −0.688144
\(117\) −5.35772 −0.495321
\(118\) 10.3474 0.952555
\(119\) 12.8085 1.17415
\(120\) 0 0
\(121\) 0 0
\(122\) −16.9550 −1.53503
\(123\) 3.65983 0.329996
\(124\) −0.0285909 −0.00256754
\(125\) 0 0
\(126\) 3.19236 0.284398
\(127\) 12.6870 1.12579 0.562897 0.826527i \(-0.309687\pi\)
0.562897 + 0.826527i \(0.309687\pi\)
\(128\) 9.14301 0.808136
\(129\) −1.96950 −0.173405
\(130\) 0 0
\(131\) 0.970102 0.0847582 0.0423791 0.999102i \(-0.486506\pi\)
0.0423791 + 0.999102i \(0.486506\pi\)
\(132\) 0 0
\(133\) −10.5093 −0.911268
\(134\) −0.640375 −0.0553199
\(135\) 0 0
\(136\) −8.86324 −0.760016
\(137\) −4.22035 −0.360569 −0.180284 0.983615i \(-0.557702\pi\)
−0.180284 + 0.983615i \(0.557702\pi\)
\(138\) −15.2703 −1.29989
\(139\) −8.84846 −0.750516 −0.375258 0.926920i \(-0.622446\pi\)
−0.375258 + 0.926920i \(0.622446\pi\)
\(140\) 0 0
\(141\) −3.81953 −0.321663
\(142\) 8.29167 0.695821
\(143\) 0 0
\(144\) −4.88558 −0.407132
\(145\) 0 0
\(146\) 18.5258 1.53321
\(147\) −3.94716 −0.325556
\(148\) 1.50106 0.123386
\(149\) −9.55357 −0.782659 −0.391329 0.920251i \(-0.627985\pi\)
−0.391329 + 0.920251i \(0.627985\pi\)
\(150\) 0 0
\(151\) 6.75158 0.549436 0.274718 0.961525i \(-0.411416\pi\)
0.274718 + 0.961525i \(0.411416\pi\)
\(152\) 7.27222 0.589855
\(153\) 7.33070 0.592652
\(154\) 0 0
\(155\) 0 0
\(156\) −7.17002 −0.574061
\(157\) 12.6604 1.01041 0.505206 0.862999i \(-0.331416\pi\)
0.505206 + 0.862999i \(0.331416\pi\)
\(158\) −4.68468 −0.372693
\(159\) −1.67088 −0.132509
\(160\) 0 0
\(161\) −14.6029 −1.15087
\(162\) 1.82709 0.143550
\(163\) 23.4058 1.83328 0.916640 0.399713i \(-0.130890\pi\)
0.916640 + 0.399713i \(0.130890\pi\)
\(164\) 4.89781 0.382454
\(165\) 0 0
\(166\) −29.3805 −2.28037
\(167\) 3.05191 0.236164 0.118082 0.993004i \(-0.462325\pi\)
0.118082 + 0.993004i \(0.462325\pi\)
\(168\) −2.11251 −0.162984
\(169\) 15.7051 1.20809
\(170\) 0 0
\(171\) −6.01478 −0.459962
\(172\) −2.63570 −0.200971
\(173\) 16.0074 1.21702 0.608511 0.793545i \(-0.291767\pi\)
0.608511 + 0.793545i \(0.291767\pi\)
\(174\) −10.1188 −0.767101
\(175\) 0 0
\(176\) 0 0
\(177\) 5.66332 0.425681
\(178\) −26.5041 −1.98657
\(179\) −19.2991 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(180\) 0 0
\(181\) −12.1911 −0.906154 −0.453077 0.891471i \(-0.649674\pi\)
−0.453077 + 0.891471i \(0.649674\pi\)
\(182\) −17.1038 −1.26782
\(183\) −9.27977 −0.685981
\(184\) 10.1050 0.744947
\(185\) 0 0
\(186\) −0.0390343 −0.00286214
\(187\) 0 0
\(188\) −5.11153 −0.372797
\(189\) 1.74724 0.127093
\(190\) 0 0
\(191\) −13.4244 −0.971353 −0.485676 0.874139i \(-0.661427\pi\)
−0.485676 + 0.874139i \(0.661427\pi\)
\(192\) −2.12007 −0.153003
\(193\) 7.57782 0.545463 0.272732 0.962090i \(-0.412073\pi\)
0.272732 + 0.962090i \(0.412073\pi\)
\(194\) 3.47096 0.249200
\(195\) 0 0
\(196\) −5.28233 −0.377309
\(197\) −6.89118 −0.490976 −0.245488 0.969400i \(-0.578948\pi\)
−0.245488 + 0.969400i \(0.578948\pi\)
\(198\) 0 0
\(199\) 0.639398 0.0453257 0.0226629 0.999743i \(-0.492786\pi\)
0.0226629 + 0.999743i \(0.492786\pi\)
\(200\) 0 0
\(201\) −0.350489 −0.0247216
\(202\) −12.2600 −0.862610
\(203\) −9.67652 −0.679159
\(204\) 9.81040 0.686865
\(205\) 0 0
\(206\) 7.75926 0.540613
\(207\) −8.35772 −0.580901
\(208\) 26.1755 1.81495
\(209\) 0 0
\(210\) 0 0
\(211\) 3.82149 0.263082 0.131541 0.991311i \(-0.458008\pi\)
0.131541 + 0.991311i \(0.458008\pi\)
\(212\) −2.23607 −0.153574
\(213\) 4.53818 0.310951
\(214\) −28.7273 −1.96376
\(215\) 0 0
\(216\) −1.20906 −0.0822659
\(217\) −0.0373284 −0.00253401
\(218\) −7.74496 −0.524555
\(219\) 10.1395 0.685165
\(220\) 0 0
\(221\) −39.2758 −2.64198
\(222\) 2.04935 0.137543
\(223\) 26.7201 1.78931 0.894657 0.446755i \(-0.147420\pi\)
0.894657 + 0.446755i \(0.147420\pi\)
\(224\) −11.3715 −0.759792
\(225\) 0 0
\(226\) 13.1485 0.874627
\(227\) 12.2373 0.812219 0.406110 0.913824i \(-0.366885\pi\)
0.406110 + 0.913824i \(0.366885\pi\)
\(228\) −8.04935 −0.533081
\(229\) −6.68664 −0.441865 −0.220933 0.975289i \(-0.570910\pi\)
−0.220933 + 0.975289i \(0.570910\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.69598 0.439612
\(233\) −6.51742 −0.426970 −0.213485 0.976946i \(-0.568482\pi\)
−0.213485 + 0.976946i \(0.568482\pi\)
\(234\) −9.78903 −0.639929
\(235\) 0 0
\(236\) 7.57900 0.493351
\(237\) −2.56401 −0.166550
\(238\) 23.4023 1.51694
\(239\) 10.8419 0.701303 0.350651 0.936506i \(-0.385960\pi\)
0.350651 + 0.936506i \(0.385960\pi\)
\(240\) 0 0
\(241\) −27.1803 −1.75084 −0.875420 0.483363i \(-0.839415\pi\)
−0.875420 + 0.483363i \(0.839415\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −12.4188 −0.795030
\(245\) 0 0
\(246\) 6.68684 0.426337
\(247\) 32.2255 2.05046
\(248\) 0.0258305 0.00164024
\(249\) −16.0805 −1.01906
\(250\) 0 0
\(251\) 5.54443 0.349961 0.174981 0.984572i \(-0.444014\pi\)
0.174981 + 0.984572i \(0.444014\pi\)
\(252\) 2.33826 0.147297
\(253\) 0 0
\(254\) 23.1804 1.45447
\(255\) 0 0
\(256\) 20.9452 1.30908
\(257\) 18.9439 1.18169 0.590845 0.806785i \(-0.298794\pi\)
0.590845 + 0.806785i \(0.298794\pi\)
\(258\) −3.59845 −0.224030
\(259\) 1.95979 0.121775
\(260\) 0 0
\(261\) −5.53818 −0.342805
\(262\) 1.77247 0.109503
\(263\) 3.20873 0.197859 0.0989293 0.995094i \(-0.468458\pi\)
0.0989293 + 0.995094i \(0.468458\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −19.2014 −1.17731
\(267\) −14.5062 −0.887763
\(268\) −0.469045 −0.0286515
\(269\) −3.24423 −0.197804 −0.0989020 0.995097i \(-0.531533\pi\)
−0.0989020 + 0.995097i \(0.531533\pi\)
\(270\) 0 0
\(271\) 2.02977 0.123300 0.0616499 0.998098i \(-0.480364\pi\)
0.0616499 + 0.998098i \(0.480364\pi\)
\(272\) −35.8147 −2.17159
\(273\) −9.36121 −0.566566
\(274\) −7.71096 −0.465836
\(275\) 0 0
\(276\) −11.1848 −0.673246
\(277\) 11.3166 0.679947 0.339973 0.940435i \(-0.389582\pi\)
0.339973 + 0.940435i \(0.389582\pi\)
\(278\) −16.1669 −0.969628
\(279\) −0.0213642 −0.00127904
\(280\) 0 0
\(281\) 10.3177 0.615503 0.307751 0.951467i \(-0.400424\pi\)
0.307751 + 0.951467i \(0.400424\pi\)
\(282\) −6.97864 −0.415572
\(283\) −2.10589 −0.125182 −0.0625910 0.998039i \(-0.519936\pi\)
−0.0625910 + 0.998039i \(0.519936\pi\)
\(284\) 6.07327 0.360382
\(285\) 0 0
\(286\) 0 0
\(287\) 6.39459 0.377461
\(288\) −6.50828 −0.383504
\(289\) 36.7392 2.16113
\(290\) 0 0
\(291\) 1.89972 0.111363
\(292\) 13.5693 0.794085
\(293\) −23.3755 −1.36561 −0.682805 0.730601i \(-0.739240\pi\)
−0.682805 + 0.730601i \(0.739240\pi\)
\(294\) −7.21182 −0.420602
\(295\) 0 0
\(296\) −1.35614 −0.0788238
\(297\) 0 0
\(298\) −17.4552 −1.01115
\(299\) 44.7783 2.58959
\(300\) 0 0
\(301\) −3.44118 −0.198347
\(302\) 12.3357 0.709843
\(303\) −6.71011 −0.385486
\(304\) 29.3857 1.68539
\(305\) 0 0
\(306\) 13.3939 0.765676
\(307\) −32.0518 −1.82929 −0.914646 0.404256i \(-0.867531\pi\)
−0.914646 + 0.404256i \(0.867531\pi\)
\(308\) 0 0
\(309\) 4.24678 0.241591
\(310\) 0 0
\(311\) 1.63433 0.0926743 0.0463371 0.998926i \(-0.485245\pi\)
0.0463371 + 0.998926i \(0.485245\pi\)
\(312\) 6.47778 0.366732
\(313\) −30.4130 −1.71904 −0.859522 0.511099i \(-0.829239\pi\)
−0.859522 + 0.511099i \(0.829239\pi\)
\(314\) 23.1318 1.30540
\(315\) 0 0
\(316\) −3.43132 −0.193027
\(317\) 29.4340 1.65318 0.826590 0.562805i \(-0.190278\pi\)
0.826590 + 0.562805i \(0.190278\pi\)
\(318\) −3.05284 −0.171195
\(319\) 0 0
\(320\) 0 0
\(321\) −15.7230 −0.877572
\(322\) −26.6809 −1.48687
\(323\) −44.0926 −2.45338
\(324\) 1.33826 0.0743478
\(325\) 0 0
\(326\) 42.7645 2.36850
\(327\) −4.23895 −0.234415
\(328\) −4.42494 −0.244327
\(329\) −6.67364 −0.367929
\(330\) 0 0
\(331\) −22.7902 −1.25266 −0.626332 0.779557i \(-0.715444\pi\)
−0.626332 + 0.779557i \(0.715444\pi\)
\(332\) −21.5199 −1.18106
\(333\) 1.12165 0.0614659
\(334\) 5.57611 0.305111
\(335\) 0 0
\(336\) −8.53627 −0.465692
\(337\) −14.4901 −0.789325 −0.394662 0.918826i \(-0.629138\pi\)
−0.394662 + 0.918826i \(0.629138\pi\)
\(338\) 28.6947 1.56078
\(339\) 7.19643 0.390856
\(340\) 0 0
\(341\) 0 0
\(342\) −10.9896 −0.594247
\(343\) −19.1273 −1.03278
\(344\) 2.38124 0.128388
\(345\) 0 0
\(346\) 29.2470 1.57233
\(347\) 4.38728 0.235522 0.117761 0.993042i \(-0.462428\pi\)
0.117761 + 0.993042i \(0.462428\pi\)
\(348\) −7.41153 −0.397300
\(349\) −14.4061 −0.771140 −0.385570 0.922679i \(-0.625995\pi\)
−0.385570 + 0.922679i \(0.625995\pi\)
\(350\) 0 0
\(351\) −5.35772 −0.285974
\(352\) 0 0
\(353\) 6.01065 0.319914 0.159957 0.987124i \(-0.448864\pi\)
0.159957 + 0.987124i \(0.448864\pi\)
\(354\) 10.3474 0.549958
\(355\) 0 0
\(356\) −19.4130 −1.02889
\(357\) 12.8085 0.677897
\(358\) −35.2612 −1.86361
\(359\) 2.45715 0.129683 0.0648417 0.997896i \(-0.479346\pi\)
0.0648417 + 0.997896i \(0.479346\pi\)
\(360\) 0 0
\(361\) 17.1776 0.904085
\(362\) −22.2742 −1.17070
\(363\) 0 0
\(364\) −12.5277 −0.656632
\(365\) 0 0
\(366\) −16.9550 −0.886251
\(367\) −1.47990 −0.0772500 −0.0386250 0.999254i \(-0.512298\pi\)
−0.0386250 + 0.999254i \(0.512298\pi\)
\(368\) 40.8323 2.12853
\(369\) 3.65983 0.190523
\(370\) 0 0
\(371\) −2.91942 −0.151569
\(372\) −0.0285909 −0.00148237
\(373\) −24.3331 −1.25992 −0.629961 0.776627i \(-0.716929\pi\)
−0.629961 + 0.776627i \(0.716929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 4.61803 0.238157
\(377\) 29.6720 1.52819
\(378\) 3.19236 0.164197
\(379\) 15.9311 0.818328 0.409164 0.912461i \(-0.365820\pi\)
0.409164 + 0.912461i \(0.365820\pi\)
\(380\) 0 0
\(381\) 12.6870 0.649977
\(382\) −24.5275 −1.25494
\(383\) 3.91994 0.200300 0.100150 0.994972i \(-0.468068\pi\)
0.100150 + 0.994972i \(0.468068\pi\)
\(384\) 9.14301 0.466577
\(385\) 0 0
\(386\) 13.8454 0.704710
\(387\) −1.96950 −0.100115
\(388\) 2.54232 0.129067
\(389\) −35.2910 −1.78933 −0.894663 0.446742i \(-0.852584\pi\)
−0.894663 + 0.446742i \(0.852584\pi\)
\(390\) 0 0
\(391\) −61.2679 −3.09845
\(392\) 4.77234 0.241040
\(393\) 0.970102 0.0489352
\(394\) −12.5908 −0.634316
\(395\) 0 0
\(396\) 0 0
\(397\) −23.2231 −1.16553 −0.582767 0.812639i \(-0.698030\pi\)
−0.582767 + 0.812639i \(0.698030\pi\)
\(398\) 1.16824 0.0585585
\(399\) −10.5093 −0.526121
\(400\) 0 0
\(401\) 5.69460 0.284375 0.142187 0.989840i \(-0.454586\pi\)
0.142187 + 0.989840i \(0.454586\pi\)
\(402\) −0.640375 −0.0319390
\(403\) 0.114463 0.00570182
\(404\) −8.97989 −0.446766
\(405\) 0 0
\(406\) −17.6799 −0.877438
\(407\) 0 0
\(408\) −8.86324 −0.438796
\(409\) −25.4530 −1.25857 −0.629284 0.777176i \(-0.716652\pi\)
−0.629284 + 0.777176i \(0.716652\pi\)
\(410\) 0 0
\(411\) −4.22035 −0.208175
\(412\) 5.68331 0.279996
\(413\) 9.89517 0.486909
\(414\) −15.2703 −0.750495
\(415\) 0 0
\(416\) 34.8695 1.70962
\(417\) −8.84846 −0.433311
\(418\) 0 0
\(419\) −13.4850 −0.658786 −0.329393 0.944193i \(-0.606844\pi\)
−0.329393 + 0.944193i \(0.606844\pi\)
\(420\) 0 0
\(421\) 2.94967 0.143758 0.0718791 0.997413i \(-0.477100\pi\)
0.0718791 + 0.997413i \(0.477100\pi\)
\(422\) 6.98220 0.339888
\(423\) −3.81953 −0.185712
\(424\) 2.02018 0.0981088
\(425\) 0 0
\(426\) 8.29167 0.401733
\(427\) −16.2140 −0.784649
\(428\) −21.0415 −1.01708
\(429\) 0 0
\(430\) 0 0
\(431\) −11.5227 −0.555031 −0.277515 0.960721i \(-0.589511\pi\)
−0.277515 + 0.960721i \(0.589511\pi\)
\(432\) −4.88558 −0.235058
\(433\) −11.1638 −0.536500 −0.268250 0.963349i \(-0.586445\pi\)
−0.268250 + 0.963349i \(0.586445\pi\)
\(434\) −0.0682023 −0.00327382
\(435\) 0 0
\(436\) −5.67283 −0.271679
\(437\) 50.2698 2.40473
\(438\) 18.5258 0.885198
\(439\) −0.993624 −0.0474231 −0.0237115 0.999719i \(-0.507548\pi\)
−0.0237115 + 0.999719i \(0.507548\pi\)
\(440\) 0 0
\(441\) −3.94716 −0.187960
\(442\) −71.7605 −3.41330
\(443\) 10.9825 0.521796 0.260898 0.965366i \(-0.415981\pi\)
0.260898 + 0.965366i \(0.415981\pi\)
\(444\) 1.50106 0.0712370
\(445\) 0 0
\(446\) 48.8201 2.31170
\(447\) −9.55357 −0.451868
\(448\) −3.70426 −0.175010
\(449\) 29.0203 1.36955 0.684776 0.728753i \(-0.259900\pi\)
0.684776 + 0.728753i \(0.259900\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.63070 0.452990
\(453\) 6.75158 0.317217
\(454\) 22.3587 1.04935
\(455\) 0 0
\(456\) 7.27222 0.340553
\(457\) −3.55670 −0.166375 −0.0831877 0.996534i \(-0.526510\pi\)
−0.0831877 + 0.996534i \(0.526510\pi\)
\(458\) −12.2171 −0.570867
\(459\) 7.33070 0.342168
\(460\) 0 0
\(461\) −25.1829 −1.17288 −0.586441 0.809992i \(-0.699472\pi\)
−0.586441 + 0.809992i \(0.699472\pi\)
\(462\) 0 0
\(463\) 17.8743 0.830690 0.415345 0.909664i \(-0.363661\pi\)
0.415345 + 0.909664i \(0.363661\pi\)
\(464\) 27.0572 1.25610
\(465\) 0 0
\(466\) −11.9079 −0.551624
\(467\) 25.2252 1.16728 0.583642 0.812011i \(-0.301627\pi\)
0.583642 + 0.812011i \(0.301627\pi\)
\(468\) −7.17002 −0.331434
\(469\) −0.612387 −0.0282774
\(470\) 0 0
\(471\) 12.6604 0.583362
\(472\) −6.84727 −0.315171
\(473\) 0 0
\(474\) −4.68468 −0.215175
\(475\) 0 0
\(476\) 17.1411 0.785661
\(477\) −1.67088 −0.0765041
\(478\) 19.8091 0.906047
\(479\) 5.23574 0.239227 0.119613 0.992821i \(-0.461834\pi\)
0.119613 + 0.992821i \(0.461834\pi\)
\(480\) 0 0
\(481\) −6.00947 −0.274008
\(482\) −49.6610 −2.26199
\(483\) −14.6029 −0.664456
\(484\) 0 0
\(485\) 0 0
\(486\) 1.82709 0.0828785
\(487\) −9.04508 −0.409872 −0.204936 0.978775i \(-0.565699\pi\)
−0.204936 + 0.978775i \(0.565699\pi\)
\(488\) 11.2198 0.507895
\(489\) 23.4058 1.05845
\(490\) 0 0
\(491\) 34.8112 1.57101 0.785505 0.618856i \(-0.212403\pi\)
0.785505 + 0.618856i \(0.212403\pi\)
\(492\) 4.89781 0.220810
\(493\) −40.5988 −1.82848
\(494\) 58.8789 2.64909
\(495\) 0 0
\(496\) 0.104377 0.00468664
\(497\) 7.92928 0.355677
\(498\) −29.3805 −1.31657
\(499\) 19.5992 0.877380 0.438690 0.898638i \(-0.355443\pi\)
0.438690 + 0.898638i \(0.355443\pi\)
\(500\) 0 0
\(501\) 3.05191 0.136349
\(502\) 10.1302 0.452132
\(503\) 18.4160 0.821129 0.410564 0.911832i \(-0.365332\pi\)
0.410564 + 0.911832i \(0.365332\pi\)
\(504\) −2.11251 −0.0940987
\(505\) 0 0
\(506\) 0 0
\(507\) 15.7051 0.697489
\(508\) 16.9786 0.753303
\(509\) 1.23838 0.0548901 0.0274451 0.999623i \(-0.491263\pi\)
0.0274451 + 0.999623i \(0.491263\pi\)
\(510\) 0 0
\(511\) 17.7162 0.783717
\(512\) 19.9828 0.883126
\(513\) −6.01478 −0.265559
\(514\) 34.6123 1.52668
\(515\) 0 0
\(516\) −2.63570 −0.116030
\(517\) 0 0
\(518\) 3.58071 0.157327
\(519\) 16.0074 0.702648
\(520\) 0 0
\(521\) 33.1850 1.45386 0.726931 0.686711i \(-0.240946\pi\)
0.726931 + 0.686711i \(0.240946\pi\)
\(522\) −10.1188 −0.442886
\(523\) 13.1508 0.575046 0.287523 0.957774i \(-0.407168\pi\)
0.287523 + 0.957774i \(0.407168\pi\)
\(524\) 1.29825 0.0567143
\(525\) 0 0
\(526\) 5.86264 0.255623
\(527\) −0.156615 −0.00682224
\(528\) 0 0
\(529\) 46.8514 2.03702
\(530\) 0 0
\(531\) 5.66332 0.245767
\(532\) −14.0641 −0.609758
\(533\) −19.6083 −0.849331
\(534\) −26.5041 −1.14694
\(535\) 0 0
\(536\) 0.423761 0.0183037
\(537\) −19.2991 −0.832818
\(538\) −5.92750 −0.255553
\(539\) 0 0
\(540\) 0 0
\(541\) −32.6519 −1.40381 −0.701907 0.712269i \(-0.747668\pi\)
−0.701907 + 0.712269i \(0.747668\pi\)
\(542\) 3.70858 0.159297
\(543\) −12.1911 −0.523168
\(544\) −47.7103 −2.04556
\(545\) 0 0
\(546\) −17.1038 −0.731974
\(547\) 27.7048 1.18457 0.592285 0.805728i \(-0.298226\pi\)
0.592285 + 0.805728i \(0.298226\pi\)
\(548\) −5.64793 −0.241268
\(549\) −9.27977 −0.396051
\(550\) 0 0
\(551\) 33.3110 1.41909
\(552\) 10.1050 0.430095
\(553\) −4.47994 −0.190506
\(554\) 20.6764 0.878456
\(555\) 0 0
\(556\) −11.8415 −0.502193
\(557\) −23.4252 −0.992558 −0.496279 0.868163i \(-0.665301\pi\)
−0.496279 + 0.868163i \(0.665301\pi\)
\(558\) −0.0390343 −0.00165246
\(559\) 10.5520 0.446303
\(560\) 0 0
\(561\) 0 0
\(562\) 18.8514 0.795198
\(563\) 22.0856 0.930796 0.465398 0.885101i \(-0.345911\pi\)
0.465398 + 0.885101i \(0.345911\pi\)
\(564\) −5.11153 −0.215234
\(565\) 0 0
\(566\) −3.84765 −0.161729
\(567\) 1.74724 0.0733771
\(568\) −5.48692 −0.230226
\(569\) 13.7351 0.575805 0.287902 0.957660i \(-0.407042\pi\)
0.287902 + 0.957660i \(0.407042\pi\)
\(570\) 0 0
\(571\) 26.5201 1.10983 0.554915 0.831907i \(-0.312751\pi\)
0.554915 + 0.831907i \(0.312751\pi\)
\(572\) 0 0
\(573\) −13.4244 −0.560811
\(574\) 11.6835 0.487660
\(575\) 0 0
\(576\) −2.12007 −0.0883361
\(577\) −5.68765 −0.236780 −0.118390 0.992967i \(-0.537773\pi\)
−0.118390 + 0.992967i \(0.537773\pi\)
\(578\) 67.1259 2.79207
\(579\) 7.57782 0.314923
\(580\) 0 0
\(581\) −28.0965 −1.16564
\(582\) 3.47096 0.143876
\(583\) 0 0
\(584\) −12.2593 −0.507292
\(585\) 0 0
\(586\) −42.7091 −1.76430
\(587\) −13.5460 −0.559104 −0.279552 0.960131i \(-0.590186\pi\)
−0.279552 + 0.960131i \(0.590186\pi\)
\(588\) −5.28233 −0.217840
\(589\) 0.128501 0.00529479
\(590\) 0 0
\(591\) −6.89118 −0.283465
\(592\) −5.47990 −0.225222
\(593\) 21.1786 0.869702 0.434851 0.900502i \(-0.356801\pi\)
0.434851 + 0.900502i \(0.356801\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −12.7852 −0.523701
\(597\) 0.639398 0.0261688
\(598\) 81.8140 3.34562
\(599\) −33.8404 −1.38268 −0.691341 0.722528i \(-0.742980\pi\)
−0.691341 + 0.722528i \(0.742980\pi\)
\(600\) 0 0
\(601\) −29.3177 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(602\) −6.28736 −0.256254
\(603\) −0.350489 −0.0142730
\(604\) 9.03538 0.367644
\(605\) 0 0
\(606\) −12.2600 −0.498028
\(607\) 7.17486 0.291219 0.145609 0.989342i \(-0.453486\pi\)
0.145609 + 0.989342i \(0.453486\pi\)
\(608\) 39.1459 1.58758
\(609\) −9.67652 −0.392112
\(610\) 0 0
\(611\) 20.4640 0.827884
\(612\) 9.81040 0.396562
\(613\) −27.5044 −1.11089 −0.555447 0.831552i \(-0.687453\pi\)
−0.555447 + 0.831552i \(0.687453\pi\)
\(614\) −58.5615 −2.36335
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0712 −0.969068 −0.484534 0.874772i \(-0.661011\pi\)
−0.484534 + 0.874772i \(0.661011\pi\)
\(618\) 7.75926 0.312123
\(619\) 26.9729 1.08413 0.542067 0.840335i \(-0.317642\pi\)
0.542067 + 0.840335i \(0.317642\pi\)
\(620\) 0 0
\(621\) −8.35772 −0.335384
\(622\) 2.98607 0.119730
\(623\) −25.3457 −1.01546
\(624\) 26.1755 1.04786
\(625\) 0 0
\(626\) −55.5673 −2.22092
\(627\) 0 0
\(628\) 16.9430 0.676098
\(629\) 8.22246 0.327851
\(630\) 0 0
\(631\) 9.73627 0.387595 0.193797 0.981042i \(-0.437920\pi\)
0.193797 + 0.981042i \(0.437920\pi\)
\(632\) 3.10004 0.123313
\(633\) 3.82149 0.151890
\(634\) 53.7786 2.13582
\(635\) 0 0
\(636\) −2.23607 −0.0886659
\(637\) 21.1478 0.837904
\(638\) 0 0
\(639\) 4.53818 0.179528
\(640\) 0 0
\(641\) 25.1165 0.992040 0.496020 0.868311i \(-0.334794\pi\)
0.496020 + 0.868311i \(0.334794\pi\)
\(642\) −28.7273 −1.13378
\(643\) 26.9257 1.06185 0.530924 0.847420i \(-0.321845\pi\)
0.530924 + 0.847420i \(0.321845\pi\)
\(644\) −19.5425 −0.770083
\(645\) 0 0
\(646\) −80.5612 −3.16964
\(647\) 26.8011 1.05366 0.526829 0.849971i \(-0.323381\pi\)
0.526829 + 0.849971i \(0.323381\pi\)
\(648\) −1.20906 −0.0474962
\(649\) 0 0
\(650\) 0 0
\(651\) −0.0373284 −0.00146301
\(652\) 31.3230 1.22670
\(653\) −28.7633 −1.12559 −0.562797 0.826595i \(-0.690275\pi\)
−0.562797 + 0.826595i \(0.690275\pi\)
\(654\) −7.74496 −0.302852
\(655\) 0 0
\(656\) −17.8804 −0.698112
\(657\) 10.1395 0.395580
\(658\) −12.1933 −0.475346
\(659\) −36.5327 −1.42311 −0.711556 0.702629i \(-0.752009\pi\)
−0.711556 + 0.702629i \(0.752009\pi\)
\(660\) 0 0
\(661\) −35.7750 −1.39149 −0.695743 0.718291i \(-0.744925\pi\)
−0.695743 + 0.718291i \(0.744925\pi\)
\(662\) −41.6398 −1.61838
\(663\) −39.2758 −1.52535
\(664\) 19.4422 0.754505
\(665\) 0 0
\(666\) 2.04935 0.0794108
\(667\) 46.2865 1.79222
\(668\) 4.08425 0.158024
\(669\) 26.7201 1.03306
\(670\) 0 0
\(671\) 0 0
\(672\) −11.3715 −0.438666
\(673\) −11.2915 −0.435254 −0.217627 0.976032i \(-0.569832\pi\)
−0.217627 + 0.976032i \(0.569832\pi\)
\(674\) −26.4747 −1.01977
\(675\) 0 0
\(676\) 21.0175 0.808367
\(677\) −43.8574 −1.68558 −0.842788 0.538246i \(-0.819087\pi\)
−0.842788 + 0.538246i \(0.819087\pi\)
\(678\) 13.1485 0.504966
\(679\) 3.31926 0.127381
\(680\) 0 0
\(681\) 12.2373 0.468935
\(682\) 0 0
\(683\) 18.5355 0.709241 0.354620 0.935010i \(-0.384610\pi\)
0.354620 + 0.935010i \(0.384610\pi\)
\(684\) −8.04935 −0.307775
\(685\) 0 0
\(686\) −34.9473 −1.33429
\(687\) −6.68664 −0.255111
\(688\) 9.62214 0.366841
\(689\) 8.95208 0.341047
\(690\) 0 0
\(691\) −28.2055 −1.07299 −0.536494 0.843904i \(-0.680251\pi\)
−0.536494 + 0.843904i \(0.680251\pi\)
\(692\) 21.4221 0.814347
\(693\) 0 0
\(694\) 8.01596 0.304282
\(695\) 0 0
\(696\) 6.69598 0.253810
\(697\) 26.8291 1.01623
\(698\) −26.3212 −0.996274
\(699\) −6.51742 −0.246511
\(700\) 0 0
\(701\) −0.315386 −0.0119120 −0.00595599 0.999982i \(-0.501896\pi\)
−0.00595599 + 0.999982i \(0.501896\pi\)
\(702\) −9.78903 −0.369463
\(703\) −6.74647 −0.254448
\(704\) 0 0
\(705\) 0 0
\(706\) 10.9820 0.413313
\(707\) −11.7242 −0.440933
\(708\) 7.57900 0.284836
\(709\) 39.0246 1.46560 0.732800 0.680444i \(-0.238213\pi\)
0.732800 + 0.680444i \(0.238213\pi\)
\(710\) 0 0
\(711\) −2.56401 −0.0961580
\(712\) 17.5388 0.657294
\(713\) 0.178556 0.00668697
\(714\) 23.4023 0.875808
\(715\) 0 0
\(716\) −25.8272 −0.965209
\(717\) 10.8419 0.404897
\(718\) 4.48943 0.167544
\(719\) 28.4691 1.06172 0.530859 0.847460i \(-0.321869\pi\)
0.530859 + 0.847460i \(0.321869\pi\)
\(720\) 0 0
\(721\) 7.42014 0.276341
\(722\) 31.3851 1.16803
\(723\) −27.1803 −1.01085
\(724\) −16.3148 −0.606335
\(725\) 0 0
\(726\) 0 0
\(727\) −45.8400 −1.70011 −0.850057 0.526691i \(-0.823432\pi\)
−0.850057 + 0.526691i \(0.823432\pi\)
\(728\) 11.3182 0.419482
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −14.4378 −0.534002
\(732\) −12.4188 −0.459011
\(733\) 37.9681 1.40239 0.701193 0.712972i \(-0.252651\pi\)
0.701193 + 0.712972i \(0.252651\pi\)
\(734\) −2.70391 −0.0998030
\(735\) 0 0
\(736\) 54.3944 2.00500
\(737\) 0 0
\(738\) 6.68684 0.246146
\(739\) −21.1226 −0.777008 −0.388504 0.921447i \(-0.627008\pi\)
−0.388504 + 0.921447i \(0.627008\pi\)
\(740\) 0 0
\(741\) 32.2255 1.18383
\(742\) −5.33404 −0.195819
\(743\) 7.98035 0.292771 0.146385 0.989228i \(-0.453236\pi\)
0.146385 + 0.989228i \(0.453236\pi\)
\(744\) 0.0258305 0.000946993 0
\(745\) 0 0
\(746\) −44.4588 −1.62775
\(747\) −16.0805 −0.588355
\(748\) 0 0
\(749\) −27.4718 −1.00380
\(750\) 0 0
\(751\) −9.62482 −0.351215 −0.175607 0.984460i \(-0.556189\pi\)
−0.175607 + 0.984460i \(0.556189\pi\)
\(752\) 18.6606 0.680483
\(753\) 5.54443 0.202050
\(754\) 54.2134 1.97434
\(755\) 0 0
\(756\) 2.33826 0.0850417
\(757\) 46.6153 1.69426 0.847130 0.531385i \(-0.178328\pi\)
0.847130 + 0.531385i \(0.178328\pi\)
\(758\) 29.1076 1.05724
\(759\) 0 0
\(760\) 0 0
\(761\) 21.9943 0.797293 0.398646 0.917105i \(-0.369480\pi\)
0.398646 + 0.917105i \(0.369480\pi\)
\(762\) 23.1804 0.839737
\(763\) −7.40646 −0.268132
\(764\) −17.9653 −0.649962
\(765\) 0 0
\(766\) 7.16209 0.258777
\(767\) −30.3424 −1.09560
\(768\) 20.9452 0.755797
\(769\) −48.0330 −1.73211 −0.866057 0.499946i \(-0.833353\pi\)
−0.866057 + 0.499946i \(0.833353\pi\)
\(770\) 0 0
\(771\) 18.9439 0.682249
\(772\) 10.1411 0.364986
\(773\) 17.0851 0.614509 0.307255 0.951627i \(-0.400590\pi\)
0.307255 + 0.951627i \(0.400590\pi\)
\(774\) −3.59845 −0.129344
\(775\) 0 0
\(776\) −2.29687 −0.0824527
\(777\) 1.95979 0.0703069
\(778\) −64.4799 −2.31172
\(779\) −22.0131 −0.788700
\(780\) 0 0
\(781\) 0 0
\(782\) −111.942 −4.00304
\(783\) −5.53818 −0.197918
\(784\) 19.2842 0.688720
\(785\) 0 0
\(786\) 1.77247 0.0632217
\(787\) −22.4918 −0.801746 −0.400873 0.916134i \(-0.631293\pi\)
−0.400873 + 0.916134i \(0.631293\pi\)
\(788\) −9.22220 −0.328527
\(789\) 3.20873 0.114234
\(790\) 0 0
\(791\) 12.5739 0.447076
\(792\) 0 0
\(793\) 49.7184 1.76555
\(794\) −42.4307 −1.50581
\(795\) 0 0
\(796\) 0.855682 0.0303288
\(797\) 3.41778 0.121064 0.0605321 0.998166i \(-0.480720\pi\)
0.0605321 + 0.998166i \(0.480720\pi\)
\(798\) −19.2014 −0.679721
\(799\) −27.9999 −0.990564
\(800\) 0 0
\(801\) −14.5062 −0.512550
\(802\) 10.4046 0.367398
\(803\) 0 0
\(804\) −0.469045 −0.0165420
\(805\) 0 0
\(806\) 0.209135 0.00736646
\(807\) −3.24423 −0.114202
\(808\) 8.11291 0.285411
\(809\) 10.1185 0.355748 0.177874 0.984053i \(-0.443078\pi\)
0.177874 + 0.984053i \(0.443078\pi\)
\(810\) 0 0
\(811\) −25.3749 −0.891034 −0.445517 0.895273i \(-0.646980\pi\)
−0.445517 + 0.895273i \(0.646980\pi\)
\(812\) −12.9497 −0.454446
\(813\) 2.02977 0.0711872
\(814\) 0 0
\(815\) 0 0
\(816\) −35.8147 −1.25377
\(817\) 11.8461 0.414443
\(818\) −46.5049 −1.62600
\(819\) −9.36121 −0.327107
\(820\) 0 0
\(821\) −1.49915 −0.0523206 −0.0261603 0.999658i \(-0.508328\pi\)
−0.0261603 + 0.999658i \(0.508328\pi\)
\(822\) −7.71096 −0.268951
\(823\) 19.1246 0.666641 0.333320 0.942814i \(-0.391831\pi\)
0.333320 + 0.942814i \(0.391831\pi\)
\(824\) −5.13460 −0.178872
\(825\) 0 0
\(826\) 18.0794 0.629062
\(827\) −16.6305 −0.578298 −0.289149 0.957284i \(-0.593372\pi\)
−0.289149 + 0.957284i \(0.593372\pi\)
\(828\) −11.1848 −0.388699
\(829\) 25.6955 0.892442 0.446221 0.894923i \(-0.352769\pi\)
0.446221 + 0.894923i \(0.352769\pi\)
\(830\) 0 0
\(831\) 11.3166 0.392567
\(832\) 11.3587 0.393793
\(833\) −28.9355 −1.00255
\(834\) −16.1669 −0.559815
\(835\) 0 0
\(836\) 0 0
\(837\) −0.0213642 −0.000738455 0
\(838\) −24.6383 −0.851117
\(839\) −28.3971 −0.980376 −0.490188 0.871617i \(-0.663072\pi\)
−0.490188 + 0.871617i \(0.663072\pi\)
\(840\) 0 0
\(841\) 1.67145 0.0576363
\(842\) 5.38932 0.185728
\(843\) 10.3177 0.355361
\(844\) 5.11415 0.176036
\(845\) 0 0
\(846\) −6.97864 −0.239930
\(847\) 0 0
\(848\) 8.16320 0.280325
\(849\) −2.10589 −0.0722739
\(850\) 0 0
\(851\) −9.37441 −0.321351
\(852\) 6.07327 0.208067
\(853\) 15.1072 0.517260 0.258630 0.965976i \(-0.416729\pi\)
0.258630 + 0.965976i \(0.416729\pi\)
\(854\) −29.6244 −1.01373
\(855\) 0 0
\(856\) 19.0100 0.649748
\(857\) 2.37040 0.0809713 0.0404856 0.999180i \(-0.487109\pi\)
0.0404856 + 0.999180i \(0.487109\pi\)
\(858\) 0 0
\(859\) 45.0423 1.53682 0.768411 0.639956i \(-0.221048\pi\)
0.768411 + 0.639956i \(0.221048\pi\)
\(860\) 0 0
\(861\) 6.39459 0.217927
\(862\) −21.0531 −0.717071
\(863\) 25.0210 0.851724 0.425862 0.904788i \(-0.359971\pi\)
0.425862 + 0.904788i \(0.359971\pi\)
\(864\) −6.50828 −0.221416
\(865\) 0 0
\(866\) −20.3974 −0.693130
\(867\) 36.7392 1.24773
\(868\) −0.0499551 −0.00169559
\(869\) 0 0
\(870\) 0 0
\(871\) 1.87782 0.0636274
\(872\) 5.12514 0.173559
\(873\) 1.89972 0.0642957
\(874\) 91.8476 3.10679
\(875\) 0 0
\(876\) 13.5693 0.458465
\(877\) −48.8755 −1.65041 −0.825203 0.564836i \(-0.808940\pi\)
−0.825203 + 0.564836i \(0.808940\pi\)
\(878\) −1.81544 −0.0612682
\(879\) −23.3755 −0.788435
\(880\) 0 0
\(881\) 44.5530 1.50103 0.750515 0.660853i \(-0.229805\pi\)
0.750515 + 0.660853i \(0.229805\pi\)
\(882\) −7.21182 −0.242835
\(883\) 44.0547 1.48256 0.741279 0.671197i \(-0.234220\pi\)
0.741279 + 0.671197i \(0.234220\pi\)
\(884\) −52.5613 −1.76783
\(885\) 0 0
\(886\) 20.0661 0.674133
\(887\) −12.1293 −0.407261 −0.203631 0.979048i \(-0.565274\pi\)
−0.203631 + 0.979048i \(0.565274\pi\)
\(888\) −1.35614 −0.0455089
\(889\) 22.1673 0.743467
\(890\) 0 0
\(891\) 0 0
\(892\) 35.7585 1.19728
\(893\) 22.9737 0.768785
\(894\) −17.4552 −0.583790
\(895\) 0 0
\(896\) 15.9750 0.533688
\(897\) 44.7783 1.49510
\(898\) 53.0227 1.76939
\(899\) 0.118319 0.00394615
\(900\) 0 0
\(901\) −12.2487 −0.408063
\(902\) 0 0
\(903\) −3.44118 −0.114515
\(904\) −8.70089 −0.289387
\(905\) 0 0
\(906\) 12.3357 0.409828
\(907\) −32.8672 −1.09134 −0.545670 0.838000i \(-0.683725\pi\)
−0.545670 + 0.838000i \(0.683725\pi\)
\(908\) 16.3767 0.543481
\(909\) −6.71011 −0.222560
\(910\) 0 0
\(911\) −46.7054 −1.54742 −0.773710 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(912\) 29.3857 0.973058
\(913\) 0 0
\(914\) −6.49842 −0.214949
\(915\) 0 0
\(916\) −8.94847 −0.295666
\(917\) 1.69500 0.0559738
\(918\) 13.3939 0.442063
\(919\) −7.33826 −0.242067 −0.121033 0.992648i \(-0.538621\pi\)
−0.121033 + 0.992648i \(0.538621\pi\)
\(920\) 0 0
\(921\) −32.0518 −1.05614
\(922\) −46.0114 −1.51530
\(923\) −24.3143 −0.800314
\(924\) 0 0
\(925\) 0 0
\(926\) 32.6580 1.07321
\(927\) 4.24678 0.139483
\(928\) 36.0441 1.18320
\(929\) 12.7285 0.417609 0.208804 0.977957i \(-0.433043\pi\)
0.208804 + 0.977957i \(0.433043\pi\)
\(930\) 0 0
\(931\) 23.7413 0.778090
\(932\) −8.72201 −0.285699
\(933\) 1.63433 0.0535055
\(934\) 46.0888 1.50807
\(935\) 0 0
\(936\) 6.47778 0.211733
\(937\) 19.2533 0.628979 0.314490 0.949261i \(-0.398167\pi\)
0.314490 + 0.949261i \(0.398167\pi\)
\(938\) −1.11889 −0.0365330
\(939\) −30.4130 −0.992490
\(940\) 0 0
\(941\) 41.2694 1.34535 0.672673 0.739940i \(-0.265146\pi\)
0.672673 + 0.739940i \(0.265146\pi\)
\(942\) 23.1318 0.753673
\(943\) −30.5878 −0.996076
\(944\) −27.6686 −0.900536
\(945\) 0 0
\(946\) 0 0
\(947\) 50.3012 1.63457 0.817285 0.576233i \(-0.195478\pi\)
0.817285 + 0.576233i \(0.195478\pi\)
\(948\) −3.43132 −0.111444
\(949\) −54.3247 −1.76345
\(950\) 0 0
\(951\) 29.4340 0.954464
\(952\) −15.4862 −0.501910
\(953\) −9.02580 −0.292374 −0.146187 0.989257i \(-0.546700\pi\)
−0.146187 + 0.989257i \(0.546700\pi\)
\(954\) −3.05284 −0.0988394
\(955\) 0 0
\(956\) 14.5093 0.469263
\(957\) 0 0
\(958\) 9.56617 0.309069
\(959\) −7.37396 −0.238118
\(960\) 0 0
\(961\) −30.9995 −0.999985
\(962\) −10.9798 −0.354004
\(963\) −15.7230 −0.506666
\(964\) −36.3744 −1.17154
\(965\) 0 0
\(966\) −26.6809 −0.858443
\(967\) −30.2503 −0.972785 −0.486392 0.873740i \(-0.661687\pi\)
−0.486392 + 0.873740i \(0.661687\pi\)
\(968\) 0 0
\(969\) −44.0926 −1.41646
\(970\) 0 0
\(971\) 12.8285 0.411686 0.205843 0.978585i \(-0.434006\pi\)
0.205843 + 0.978585i \(0.434006\pi\)
\(972\) 1.33826 0.0429247
\(973\) −15.4604 −0.495636
\(974\) −16.5262 −0.529533
\(975\) 0 0
\(976\) 45.3371 1.45120
\(977\) 13.1592 0.420999 0.210500 0.977594i \(-0.432491\pi\)
0.210500 + 0.977594i \(0.432491\pi\)
\(978\) 42.7645 1.36746
\(979\) 0 0
\(980\) 0 0
\(981\) −4.23895 −0.135339
\(982\) 63.6033 2.02966
\(983\) −22.4369 −0.715627 −0.357813 0.933793i \(-0.616478\pi\)
−0.357813 + 0.933793i \(0.616478\pi\)
\(984\) −4.42494 −0.141062
\(985\) 0 0
\(986\) −74.1776 −2.36230
\(987\) −6.67364 −0.212424
\(988\) 43.1261 1.37203
\(989\) 16.4605 0.523414
\(990\) 0 0
\(991\) 22.9146 0.727907 0.363953 0.931417i \(-0.381427\pi\)
0.363953 + 0.931417i \(0.381427\pi\)
\(992\) 0.139044 0.00441466
\(993\) −22.7902 −0.723226
\(994\) 14.4875 0.459516
\(995\) 0 0
\(996\) −21.5199 −0.681884
\(997\) −16.8394 −0.533309 −0.266654 0.963792i \(-0.585918\pi\)
−0.266654 + 0.963792i \(0.585918\pi\)
\(998\) 35.8095 1.13353
\(999\) 1.12165 0.0354874
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 9075.2.a.df.1.4 4
5.4 even 2 1815.2.a.q.1.1 4
11.7 odd 10 825.2.n.j.676.1 8
11.8 odd 10 825.2.n.j.526.1 8
11.10 odd 2 9075.2.a.co.1.1 4
15.14 odd 2 5445.2.a.bq.1.4 4
55.7 even 20 825.2.bx.e.49.4 16
55.8 even 20 825.2.bx.e.724.4 16
55.18 even 20 825.2.bx.e.49.1 16
55.19 odd 10 165.2.m.c.31.2 yes 8
55.29 odd 10 165.2.m.c.16.2 8
55.52 even 20 825.2.bx.e.724.1 16
55.54 odd 2 1815.2.a.u.1.4 4
165.29 even 10 495.2.n.c.181.1 8
165.74 even 10 495.2.n.c.361.1 8
165.164 even 2 5445.2.a.bj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.c.16.2 8 55.29 odd 10
165.2.m.c.31.2 yes 8 55.19 odd 10
495.2.n.c.181.1 8 165.29 even 10
495.2.n.c.361.1 8 165.74 even 10
825.2.n.j.526.1 8 11.8 odd 10
825.2.n.j.676.1 8 11.7 odd 10
825.2.bx.e.49.1 16 55.18 even 20
825.2.bx.e.49.4 16 55.7 even 20
825.2.bx.e.724.1 16 55.52 even 20
825.2.bx.e.724.4 16 55.8 even 20
1815.2.a.q.1.1 4 5.4 even 2
1815.2.a.u.1.4 4 55.54 odd 2
5445.2.a.bj.1.1 4 165.164 even 2
5445.2.a.bq.1.4 4 15.14 odd 2
9075.2.a.co.1.1 4 11.10 odd 2
9075.2.a.df.1.4 4 1.1 even 1 trivial