Properties

Label 435.2.a.j.1.4
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43828 q^{2} -1.00000 q^{3} +0.0686587 q^{4} -1.00000 q^{5} -1.43828 q^{6} -2.74301 q^{7} -2.77782 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.43828 q^{2} -1.00000 q^{3} +0.0686587 q^{4} -1.00000 q^{5} -1.43828 q^{6} -2.74301 q^{7} -2.77782 q^{8} +1.00000 q^{9} -1.43828 q^{10} +2.74301 q^{11} -0.0686587 q^{12} -5.14744 q^{13} -3.94523 q^{14} +1.00000 q^{15} -4.13260 q^{16} -3.72913 q^{17} +1.43828 q^{18} -0.404431 q^{19} -0.0686587 q^{20} +2.74301 q^{21} +3.94523 q^{22} -5.45825 q^{23} +2.77782 q^{24} +1.00000 q^{25} -7.40348 q^{26} -1.00000 q^{27} -0.188331 q^{28} -1.00000 q^{29} +1.43828 q^{30} +1.45825 q^{31} -0.388222 q^{32} -2.74301 q^{33} -5.36354 q^{34} +2.74301 q^{35} +0.0686587 q^{36} +6.76702 q^{37} -0.581686 q^{38} +5.14744 q^{39} +2.77782 q^{40} +9.78090 q^{41} +3.94523 q^{42} +4.43220 q^{43} +0.188331 q^{44} -1.00000 q^{45} -7.85051 q^{46} +2.60569 q^{47} +4.13260 q^{48} +0.524103 q^{49} +1.43828 q^{50} +3.72913 q^{51} -0.353416 q^{52} -6.43220 q^{53} -1.43828 q^{54} -2.74301 q^{55} +7.61958 q^{56} +0.404431 q^{57} -1.43828 q^{58} -9.91822 q^{59} +0.0686587 q^{60} -13.0816 q^{61} +2.09738 q^{62} -2.74301 q^{63} +7.70683 q^{64} +5.14744 q^{65} -3.94523 q^{66} +12.4961 q^{67} -0.256037 q^{68} +5.45825 q^{69} +3.94523 q^{70} -11.3487 q^{71} -2.77782 q^{72} -10.7670 q^{73} +9.73289 q^{74} -1.00000 q^{75} -0.0277677 q^{76} -7.52410 q^{77} +7.40348 q^{78} +14.1576 q^{79} +4.13260 q^{80} +1.00000 q^{81} +14.0677 q^{82} +1.62334 q^{83} +0.188331 q^{84} +3.72913 q^{85} +6.37476 q^{86} +1.00000 q^{87} -7.61958 q^{88} -8.87281 q^{89} -1.43828 q^{90} +14.1195 q^{91} -0.374756 q^{92} -1.45825 q^{93} +3.74772 q^{94} +0.404431 q^{95} +0.388222 q^{96} +7.82084 q^{97} +0.753809 q^{98} +2.74301 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 2 q^{11} - 5 q^{12} - 8 q^{13} - 3 q^{14} + 4 q^{15} + 11 q^{16} - 10 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} - 2 q^{21} + 3 q^{22} - 12 q^{23} + 12 q^{24} + 4 q^{25} - 7 q^{26} - 4 q^{27} - 9 q^{28} - 4 q^{29} - 3 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 5 q^{36} - 16 q^{37} - 10 q^{38} + 8 q^{39} + 12 q^{40} - 12 q^{41} + 3 q^{42} + 2 q^{43} + 9 q^{44} - 4 q^{45} - 8 q^{46} - 12 q^{47} - 11 q^{48} + 6 q^{49} - 3 q^{50} + 10 q^{51} - 3 q^{52} - 10 q^{53} + 3 q^{54} + 2 q^{55} + 2 q^{57} + 3 q^{58} + 2 q^{59} + 5 q^{60} - 26 q^{61} + 20 q^{62} + 2 q^{63} + 34 q^{64} + 8 q^{65} - 3 q^{66} + 2 q^{67} + 9 q^{68} + 12 q^{69} + 3 q^{70} - 10 q^{71} - 12 q^{72} + 48 q^{74} - 4 q^{75} + 16 q^{76} - 34 q^{77} + 7 q^{78} + 22 q^{79} - 11 q^{80} + 4 q^{81} + 38 q^{82} - 10 q^{83} + 9 q^{84} + 10 q^{85} - 4 q^{86} + 4 q^{87} - 4 q^{89} + 3 q^{90} - 8 q^{91} + 28 q^{92} + 4 q^{93} + 39 q^{94} + 2 q^{95} + 17 q^{96} - 22 q^{97} + 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.43828 1.01702 0.508510 0.861056i \(-0.330197\pi\)
0.508510 + 0.861056i \(0.330197\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.0686587 0.0343293
\(5\) −1.00000 −0.447214
\(6\) −1.43828 −0.587177
\(7\) −2.74301 −1.03676 −0.518380 0.855150i \(-0.673465\pi\)
−0.518380 + 0.855150i \(0.673465\pi\)
\(8\) −2.77782 −0.982106
\(9\) 1.00000 0.333333
\(10\) −1.43828 −0.454825
\(11\) 2.74301 0.827049 0.413524 0.910493i \(-0.364298\pi\)
0.413524 + 0.910493i \(0.364298\pi\)
\(12\) −0.0686587 −0.0198201
\(13\) −5.14744 −1.42764 −0.713822 0.700328i \(-0.753037\pi\)
−0.713822 + 0.700328i \(0.753037\pi\)
\(14\) −3.94523 −1.05441
\(15\) 1.00000 0.258199
\(16\) −4.13260 −1.03315
\(17\) −3.72913 −0.904446 −0.452223 0.891905i \(-0.649369\pi\)
−0.452223 + 0.891905i \(0.649369\pi\)
\(18\) 1.43828 0.339007
\(19\) −0.404431 −0.0927827 −0.0463914 0.998923i \(-0.514772\pi\)
−0.0463914 + 0.998923i \(0.514772\pi\)
\(20\) −0.0686587 −0.0153525
\(21\) 2.74301 0.598574
\(22\) 3.94523 0.841125
\(23\) −5.45825 −1.13812 −0.569062 0.822295i \(-0.692694\pi\)
−0.569062 + 0.822295i \(0.692694\pi\)
\(24\) 2.77782 0.567019
\(25\) 1.00000 0.200000
\(26\) −7.40348 −1.45194
\(27\) −1.00000 −0.192450
\(28\) −0.188331 −0.0355913
\(29\) −1.00000 −0.185695
\(30\) 1.43828 0.262593
\(31\) 1.45825 0.261910 0.130955 0.991388i \(-0.458196\pi\)
0.130955 + 0.991388i \(0.458196\pi\)
\(32\) −0.388222 −0.0686287
\(33\) −2.74301 −0.477497
\(34\) −5.36354 −0.919839
\(35\) 2.74301 0.463653
\(36\) 0.0686587 0.0114431
\(37\) 6.76702 1.11249 0.556245 0.831018i \(-0.312241\pi\)
0.556245 + 0.831018i \(0.312241\pi\)
\(38\) −0.581686 −0.0943619
\(39\) 5.14744 0.824250
\(40\) 2.77782 0.439211
\(41\) 9.78090 1.52752 0.763760 0.645500i \(-0.223351\pi\)
0.763760 + 0.645500i \(0.223351\pi\)
\(42\) 3.94523 0.608761
\(43\) 4.43220 0.675904 0.337952 0.941163i \(-0.390266\pi\)
0.337952 + 0.941163i \(0.390266\pi\)
\(44\) 0.188331 0.0283920
\(45\) −1.00000 −0.149071
\(46\) −7.85051 −1.15749
\(47\) 2.60569 0.380079 0.190040 0.981776i \(-0.439138\pi\)
0.190040 + 0.981776i \(0.439138\pi\)
\(48\) 4.13260 0.596490
\(49\) 0.524103 0.0748719
\(50\) 1.43828 0.203404
\(51\) 3.72913 0.522182
\(52\) −0.353416 −0.0490100
\(53\) −6.43220 −0.883530 −0.441765 0.897131i \(-0.645648\pi\)
−0.441765 + 0.897131i \(0.645648\pi\)
\(54\) −1.43828 −0.195726
\(55\) −2.74301 −0.369867
\(56\) 7.61958 1.01821
\(57\) 0.404431 0.0535681
\(58\) −1.43828 −0.188856
\(59\) −9.91822 −1.29124 −0.645621 0.763658i \(-0.723401\pi\)
−0.645621 + 0.763658i \(0.723401\pi\)
\(60\) 0.0686587 0.00886380
\(61\) −13.0816 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(62\) 2.09738 0.266367
\(63\) −2.74301 −0.345587
\(64\) 7.70683 0.963354
\(65\) 5.14744 0.638461
\(66\) −3.94523 −0.485624
\(67\) 12.4961 1.52665 0.763323 0.646017i \(-0.223566\pi\)
0.763323 + 0.646017i \(0.223566\pi\)
\(68\) −0.256037 −0.0310490
\(69\) 5.45825 0.657096
\(70\) 3.94523 0.471545
\(71\) −11.3487 −1.34684 −0.673422 0.739259i \(-0.735176\pi\)
−0.673422 + 0.739259i \(0.735176\pi\)
\(72\) −2.77782 −0.327369
\(73\) −10.7670 −1.26018 −0.630092 0.776520i \(-0.716983\pi\)
−0.630092 + 0.776520i \(0.716983\pi\)
\(74\) 9.73289 1.13143
\(75\) −1.00000 −0.115470
\(76\) −0.0277677 −0.00318517
\(77\) −7.52410 −0.857451
\(78\) 7.40348 0.838279
\(79\) 14.1576 1.59285 0.796425 0.604737i \(-0.206722\pi\)
0.796425 + 0.604737i \(0.206722\pi\)
\(80\) 4.13260 0.462039
\(81\) 1.00000 0.111111
\(82\) 14.0677 1.55352
\(83\) 1.62334 0.178184 0.0890922 0.996023i \(-0.471603\pi\)
0.0890922 + 0.996023i \(0.471603\pi\)
\(84\) 0.188331 0.0205486
\(85\) 3.72913 0.404481
\(86\) 6.37476 0.687408
\(87\) 1.00000 0.107211
\(88\) −7.61958 −0.812250
\(89\) −8.87281 −0.940516 −0.470258 0.882529i \(-0.655839\pi\)
−0.470258 + 0.882529i \(0.655839\pi\)
\(90\) −1.43828 −0.151608
\(91\) 14.1195 1.48012
\(92\) −0.374756 −0.0390711
\(93\) −1.45825 −0.151214
\(94\) 3.74772 0.386548
\(95\) 0.404431 0.0414937
\(96\) 0.388222 0.0396228
\(97\) 7.82084 0.794086 0.397043 0.917800i \(-0.370036\pi\)
0.397043 + 0.917800i \(0.370036\pi\)
\(98\) 0.753809 0.0761462
\(99\) 2.74301 0.275683
\(100\) 0.0686587 0.00686587
\(101\) −4.88033 −0.485611 −0.242805 0.970075i \(-0.578068\pi\)
−0.242805 + 0.970075i \(0.578068\pi\)
\(102\) 5.36354 0.531070
\(103\) −0.294881 −0.0290555 −0.0145277 0.999894i \(-0.504624\pi\)
−0.0145277 + 0.999894i \(0.504624\pi\)
\(104\) 14.2986 1.40210
\(105\) −2.74301 −0.267690
\(106\) −9.25132 −0.898568
\(107\) −13.7809 −1.33225 −0.666125 0.745840i \(-0.732048\pi\)
−0.666125 + 0.745840i \(0.732048\pi\)
\(108\) −0.0686587 −0.00660668
\(109\) −6.20126 −0.593973 −0.296987 0.954882i \(-0.595982\pi\)
−0.296987 + 0.954882i \(0.595982\pi\)
\(110\) −3.94523 −0.376162
\(111\) −6.76702 −0.642297
\(112\) 11.3358 1.07113
\(113\) −10.5658 −0.993943 −0.496971 0.867767i \(-0.665555\pi\)
−0.496971 + 0.867767i \(0.665555\pi\)
\(114\) 0.581686 0.0544799
\(115\) 5.45825 0.508985
\(116\) −0.0686587 −0.00637480
\(117\) −5.14744 −0.475881
\(118\) −14.2652 −1.31322
\(119\) 10.2290 0.937694
\(120\) −2.77782 −0.253579
\(121\) −3.47590 −0.315991
\(122\) −18.8150 −1.70343
\(123\) −9.78090 −0.881914
\(124\) 0.100122 0.00899119
\(125\) −1.00000 −0.0894427
\(126\) −3.94523 −0.351469
\(127\) 8.54175 0.757958 0.378979 0.925405i \(-0.376275\pi\)
0.378979 + 0.925405i \(0.376275\pi\)
\(128\) 11.8611 1.04838
\(129\) −4.43220 −0.390233
\(130\) 7.40348 0.649328
\(131\) 13.3050 1.16246 0.581232 0.813738i \(-0.302571\pi\)
0.581232 + 0.813738i \(0.302571\pi\)
\(132\) −0.188331 −0.0163921
\(133\) 1.10936 0.0961934
\(134\) 17.9730 1.55263
\(135\) 1.00000 0.0860663
\(136\) 10.3588 0.888262
\(137\) −18.2253 −1.55709 −0.778545 0.627589i \(-0.784042\pi\)
−0.778545 + 0.627589i \(0.784042\pi\)
\(138\) 7.85051 0.668280
\(139\) −5.66142 −0.480195 −0.240098 0.970749i \(-0.577179\pi\)
−0.240098 + 0.970749i \(0.577179\pi\)
\(140\) 0.188331 0.0159169
\(141\) −2.60569 −0.219439
\(142\) −16.3226 −1.36977
\(143\) −14.1195 −1.18073
\(144\) −4.13260 −0.344384
\(145\) 1.00000 0.0830455
\(146\) −15.4860 −1.28163
\(147\) −0.524103 −0.0432273
\(148\) 0.464614 0.0381911
\(149\) −11.9182 −0.976378 −0.488189 0.872738i \(-0.662342\pi\)
−0.488189 + 0.872738i \(0.662342\pi\)
\(150\) −1.43828 −0.117435
\(151\) 7.19114 0.585207 0.292603 0.956234i \(-0.405478\pi\)
0.292603 + 0.956234i \(0.405478\pi\)
\(152\) 1.12343 0.0911225
\(153\) −3.72913 −0.301482
\(154\) −10.8218 −0.872045
\(155\) −1.45825 −0.117130
\(156\) 0.353416 0.0282960
\(157\) −4.31457 −0.344340 −0.172170 0.985067i \(-0.555078\pi\)
−0.172170 + 0.985067i \(0.555078\pi\)
\(158\) 20.3626 1.61996
\(159\) 6.43220 0.510106
\(160\) 0.388222 0.0306917
\(161\) 14.9720 1.17996
\(162\) 1.43828 0.113002
\(163\) 21.6436 1.69526 0.847628 0.530591i \(-0.178030\pi\)
0.847628 + 0.530591i \(0.178030\pi\)
\(164\) 0.671544 0.0524388
\(165\) 2.74301 0.213543
\(166\) 2.33482 0.181217
\(167\) −16.4303 −1.27141 −0.635707 0.771930i \(-0.719291\pi\)
−0.635707 + 0.771930i \(0.719291\pi\)
\(168\) −7.61958 −0.587863
\(169\) 13.4961 1.03816
\(170\) 5.36354 0.411365
\(171\) −0.404431 −0.0309276
\(172\) 0.304309 0.0232033
\(173\) −4.64939 −0.353487 −0.176743 0.984257i \(-0.556556\pi\)
−0.176743 + 0.984257i \(0.556556\pi\)
\(174\) 1.43828 0.109036
\(175\) −2.74301 −0.207352
\(176\) −11.3358 −0.854466
\(177\) 9.91822 0.745499
\(178\) −12.7616 −0.956523
\(179\) 7.62334 0.569795 0.284897 0.958558i \(-0.408040\pi\)
0.284897 + 0.958558i \(0.408040\pi\)
\(180\) −0.0686587 −0.00511752
\(181\) −21.3050 −1.58359 −0.791794 0.610788i \(-0.790853\pi\)
−0.791794 + 0.610788i \(0.790853\pi\)
\(182\) 20.3078 1.50532
\(183\) 13.0816 0.967019
\(184\) 15.1620 1.11776
\(185\) −6.76702 −0.497521
\(186\) −2.09738 −0.153787
\(187\) −10.2290 −0.748021
\(188\) 0.178903 0.0130479
\(189\) 2.74301 0.199525
\(190\) 0.581686 0.0421999
\(191\) 3.07597 0.222570 0.111285 0.993789i \(-0.464503\pi\)
0.111285 + 0.993789i \(0.464503\pi\)
\(192\) −7.70683 −0.556193
\(193\) 6.77454 0.487642 0.243821 0.969820i \(-0.421599\pi\)
0.243821 + 0.969820i \(0.421599\pi\)
\(194\) 11.2486 0.807601
\(195\) −5.14744 −0.368616
\(196\) 0.0359842 0.00257030
\(197\) −13.6455 −0.972201 −0.486100 0.873903i \(-0.661581\pi\)
−0.486100 + 0.873903i \(0.661581\pi\)
\(198\) 3.94523 0.280375
\(199\) −6.33858 −0.449330 −0.224665 0.974436i \(-0.572129\pi\)
−0.224665 + 0.974436i \(0.572129\pi\)
\(200\) −2.77782 −0.196421
\(201\) −12.4961 −0.881410
\(202\) −7.01929 −0.493876
\(203\) 2.74301 0.192522
\(204\) 0.256037 0.0179262
\(205\) −9.78090 −0.683128
\(206\) −0.424122 −0.0295500
\(207\) −5.45825 −0.379375
\(208\) 21.2723 1.47497
\(209\) −1.10936 −0.0767358
\(210\) −3.94523 −0.272246
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −0.441626 −0.0303310
\(213\) 11.3487 0.777600
\(214\) −19.8208 −1.35492
\(215\) −4.43220 −0.302273
\(216\) 2.77782 0.189006
\(217\) −4.00000 −0.271538
\(218\) −8.91917 −0.604082
\(219\) 10.7670 0.727568
\(220\) −0.188331 −0.0126973
\(221\) 19.1955 1.29123
\(222\) −9.73289 −0.653229
\(223\) −2.20126 −0.147407 −0.0737037 0.997280i \(-0.523482\pi\)
−0.0737037 + 0.997280i \(0.523482\pi\)
\(224\) 1.06490 0.0711515
\(225\) 1.00000 0.0666667
\(226\) −15.1965 −1.01086
\(227\) 12.1853 0.808769 0.404384 0.914589i \(-0.367486\pi\)
0.404384 + 0.914589i \(0.367486\pi\)
\(228\) 0.0277677 0.00183896
\(229\) −3.16337 −0.209041 −0.104521 0.994523i \(-0.533331\pi\)
−0.104521 + 0.994523i \(0.533331\pi\)
\(230\) 7.85051 0.517647
\(231\) 7.52410 0.495050
\(232\) 2.77782 0.182373
\(233\) 23.8087 1.55976 0.779879 0.625930i \(-0.215281\pi\)
0.779879 + 0.625930i \(0.215281\pi\)
\(234\) −7.40348 −0.483980
\(235\) −2.60569 −0.169977
\(236\) −0.680972 −0.0443275
\(237\) −14.1576 −0.919633
\(238\) 14.7122 0.953653
\(239\) −6.02025 −0.389417 −0.194709 0.980861i \(-0.562376\pi\)
−0.194709 + 0.980861i \(0.562376\pi\)
\(240\) −4.13260 −0.266758
\(241\) 24.5517 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(242\) −4.99932 −0.321369
\(243\) −1.00000 −0.0641500
\(244\) −0.898165 −0.0574991
\(245\) −0.524103 −0.0334837
\(246\) −14.0677 −0.896924
\(247\) 2.08178 0.132461
\(248\) −4.05076 −0.257223
\(249\) −1.62334 −0.102875
\(250\) −1.43828 −0.0909650
\(251\) −27.5162 −1.73681 −0.868403 0.495858i \(-0.834854\pi\)
−0.868403 + 0.495858i \(0.834854\pi\)
\(252\) −0.188331 −0.0118638
\(253\) −14.9720 −0.941284
\(254\) 12.2855 0.770858
\(255\) −3.72913 −0.233527
\(256\) 1.64589 0.102868
\(257\) 15.6436 0.975820 0.487910 0.872894i \(-0.337759\pi\)
0.487910 + 0.872894i \(0.337759\pi\)
\(258\) −6.37476 −0.396875
\(259\) −18.5620 −1.15339
\(260\) 0.353416 0.0219180
\(261\) −1.00000 −0.0618984
\(262\) 19.1364 1.18225
\(263\) −18.6175 −1.14801 −0.574003 0.818853i \(-0.694610\pi\)
−0.574003 + 0.818853i \(0.694610\pi\)
\(264\) 7.61958 0.468953
\(265\) 6.43220 0.395127
\(266\) 1.59557 0.0978306
\(267\) 8.87281 0.543007
\(268\) 0.857969 0.0524088
\(269\) 10.9006 0.664620 0.332310 0.943170i \(-0.392172\pi\)
0.332310 + 0.943170i \(0.392172\pi\)
\(270\) 1.43828 0.0875311
\(271\) 17.7809 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(272\) 15.4110 0.934429
\(273\) −14.1195 −0.854550
\(274\) −26.2131 −1.58359
\(275\) 2.74301 0.165410
\(276\) 0.374756 0.0225577
\(277\) −20.6612 −1.24141 −0.620706 0.784043i \(-0.713154\pi\)
−0.620706 + 0.784043i \(0.713154\pi\)
\(278\) −8.14273 −0.488368
\(279\) 1.45825 0.0873033
\(280\) −7.61958 −0.455357
\(281\) 28.2652 1.68616 0.843080 0.537787i \(-0.180740\pi\)
0.843080 + 0.537787i \(0.180740\pi\)
\(282\) −3.74772 −0.223174
\(283\) −3.21139 −0.190897 −0.0954485 0.995434i \(-0.530429\pi\)
−0.0954485 + 0.995434i \(0.530429\pi\)
\(284\) −0.779187 −0.0462362
\(285\) −0.404431 −0.0239564
\(286\) −20.3078 −1.20083
\(287\) −26.8291 −1.58367
\(288\) −0.388222 −0.0228762
\(289\) −3.09362 −0.181978
\(290\) 1.43828 0.0844589
\(291\) −7.82084 −0.458466
\(292\) −0.739249 −0.0432613
\(293\) −3.99624 −0.233463 −0.116731 0.993164i \(-0.537242\pi\)
−0.116731 + 0.993164i \(0.537242\pi\)
\(294\) −0.753809 −0.0439630
\(295\) 9.91822 0.577461
\(296\) −18.7975 −1.09258
\(297\) −2.74301 −0.159166
\(298\) −17.1418 −0.992996
\(299\) 28.0960 1.62484
\(300\) −0.0686587 −0.00396401
\(301\) −12.1576 −0.700750
\(302\) 10.3429 0.595167
\(303\) 4.88033 0.280367
\(304\) 1.67135 0.0958586
\(305\) 13.0816 0.749050
\(306\) −5.36354 −0.306613
\(307\) −12.0555 −0.688046 −0.344023 0.938961i \(-0.611790\pi\)
−0.344023 + 0.938961i \(0.611790\pi\)
\(308\) −0.516595 −0.0294357
\(309\) 0.294881 0.0167752
\(310\) −2.09738 −0.119123
\(311\) 15.6873 0.889544 0.444772 0.895644i \(-0.353285\pi\)
0.444772 + 0.895644i \(0.353285\pi\)
\(312\) −14.2986 −0.809501
\(313\) −33.8726 −1.91459 −0.957297 0.289107i \(-0.906641\pi\)
−0.957297 + 0.289107i \(0.906641\pi\)
\(314\) −6.20558 −0.350201
\(315\) 2.74301 0.154551
\(316\) 0.972040 0.0546815
\(317\) 1.75689 0.0986770 0.0493385 0.998782i \(-0.484289\pi\)
0.0493385 + 0.998782i \(0.484289\pi\)
\(318\) 9.25132 0.518788
\(319\) −2.74301 −0.153579
\(320\) −7.70683 −0.430825
\(321\) 13.7809 0.769175
\(322\) 21.5340 1.20004
\(323\) 1.50817 0.0839170
\(324\) 0.0686587 0.00381437
\(325\) −5.14744 −0.285529
\(326\) 31.1296 1.72411
\(327\) 6.20126 0.342931
\(328\) −27.1695 −1.50019
\(329\) −7.14744 −0.394051
\(330\) 3.94523 0.217177
\(331\) −22.4505 −1.23399 −0.616997 0.786966i \(-0.711651\pi\)
−0.616997 + 0.786966i \(0.711651\pi\)
\(332\) 0.111456 0.00611695
\(333\) 6.76702 0.370830
\(334\) −23.6314 −1.29305
\(335\) −12.4961 −0.682737
\(336\) −11.3358 −0.618417
\(337\) 15.3886 0.838273 0.419136 0.907923i \(-0.362333\pi\)
0.419136 + 0.907923i \(0.362333\pi\)
\(338\) 19.4113 1.05583
\(339\) 10.5658 0.573853
\(340\) 0.256037 0.0138855
\(341\) 4.00000 0.216612
\(342\) −0.581686 −0.0314540
\(343\) 17.7634 0.959136
\(344\) −12.3118 −0.663809
\(345\) −5.45825 −0.293862
\(346\) −6.68714 −0.359503
\(347\) −12.3504 −0.663005 −0.331503 0.943454i \(-0.607556\pi\)
−0.331503 + 0.943454i \(0.607556\pi\)
\(348\) 0.0686587 0.00368049
\(349\) −1.94446 −0.104085 −0.0520424 0.998645i \(-0.516573\pi\)
−0.0520424 + 0.998645i \(0.516573\pi\)
\(350\) −3.94523 −0.210881
\(351\) 5.14744 0.274750
\(352\) −1.06490 −0.0567592
\(353\) 6.72517 0.357945 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(354\) 14.2652 0.758187
\(355\) 11.3487 0.602327
\(356\) −0.609195 −0.0322873
\(357\) −10.2290 −0.541378
\(358\) 10.9645 0.579493
\(359\) 15.2114 0.802826 0.401413 0.915897i \(-0.368519\pi\)
0.401413 + 0.915897i \(0.368519\pi\)
\(360\) 2.77782 0.146404
\(361\) −18.8364 −0.991391
\(362\) −30.6426 −1.61054
\(363\) 3.47590 0.182437
\(364\) 0.969425 0.0508117
\(365\) 10.7670 0.563571
\(366\) 18.8150 0.983477
\(367\) 13.2189 0.690021 0.345011 0.938599i \(-0.387875\pi\)
0.345011 + 0.938599i \(0.387875\pi\)
\(368\) 22.5568 1.17585
\(369\) 9.78090 0.509173
\(370\) −9.73289 −0.505989
\(371\) 17.6436 0.916009
\(372\) −0.100122 −0.00519107
\(373\) 26.8050 1.38791 0.693956 0.720017i \(-0.255866\pi\)
0.693956 + 0.720017i \(0.255866\pi\)
\(374\) −14.7122 −0.760752
\(375\) 1.00000 0.0516398
\(376\) −7.23813 −0.373278
\(377\) 5.14744 0.265107
\(378\) 3.94523 0.202920
\(379\) −24.4228 −1.25451 −0.627257 0.778813i \(-0.715822\pi\)
−0.627257 + 0.778813i \(0.715822\pi\)
\(380\) 0.0277677 0.00142445
\(381\) −8.54175 −0.437607
\(382\) 4.42412 0.226358
\(383\) 27.2372 1.39176 0.695879 0.718159i \(-0.255015\pi\)
0.695879 + 0.718159i \(0.255015\pi\)
\(384\) −11.8611 −0.605282
\(385\) 7.52410 0.383464
\(386\) 9.74370 0.495942
\(387\) 4.43220 0.225301
\(388\) 0.536968 0.0272604
\(389\) −12.1994 −0.618532 −0.309266 0.950976i \(-0.600083\pi\)
−0.309266 + 0.950976i \(0.600083\pi\)
\(390\) −7.40348 −0.374890
\(391\) 20.3545 1.02937
\(392\) −1.45586 −0.0735322
\(393\) −13.3050 −0.671149
\(394\) −19.6261 −0.988748
\(395\) −14.1576 −0.712344
\(396\) 0.188331 0.00946401
\(397\) −17.7254 −0.889611 −0.444805 0.895627i \(-0.646727\pi\)
−0.444805 + 0.895627i \(0.646727\pi\)
\(398\) −9.11667 −0.456977
\(399\) −1.10936 −0.0555373
\(400\) −4.13260 −0.206630
\(401\) −2.48773 −0.124231 −0.0621157 0.998069i \(-0.519785\pi\)
−0.0621157 + 0.998069i \(0.519785\pi\)
\(402\) −17.9730 −0.896411
\(403\) −7.50627 −0.373914
\(404\) −0.335077 −0.0166707
\(405\) −1.00000 −0.0496904
\(406\) 3.94523 0.195798
\(407\) 18.5620 0.920084
\(408\) −10.3588 −0.512838
\(409\) 37.7194 1.86510 0.932551 0.361038i \(-0.117577\pi\)
0.932551 + 0.361038i \(0.117577\pi\)
\(410\) −14.0677 −0.694754
\(411\) 18.2253 0.898986
\(412\) −0.0202461 −0.000997455 0
\(413\) 27.2058 1.33871
\(414\) −7.85051 −0.385832
\(415\) −1.62334 −0.0796865
\(416\) 1.99835 0.0979773
\(417\) 5.66142 0.277241
\(418\) −1.59557 −0.0780419
\(419\) −2.29488 −0.112112 −0.0560561 0.998428i \(-0.517853\pi\)
−0.0560561 + 0.998428i \(0.517853\pi\)
\(420\) −0.188331 −0.00918963
\(421\) 35.9662 1.75289 0.876443 0.481505i \(-0.159910\pi\)
0.876443 + 0.481505i \(0.159910\pi\)
\(422\) 2.87657 0.140029
\(423\) 2.60569 0.126693
\(424\) 17.8675 0.867721
\(425\) −3.72913 −0.180889
\(426\) 16.3226 0.790835
\(427\) 35.8829 1.73650
\(428\) −0.946178 −0.0457353
\(429\) 14.1195 0.681695
\(430\) −6.37476 −0.307418
\(431\) 22.6974 1.09330 0.546648 0.837363i \(-0.315904\pi\)
0.546648 + 0.837363i \(0.315904\pi\)
\(432\) 4.13260 0.198830
\(433\) −11.4108 −0.548368 −0.274184 0.961677i \(-0.588408\pi\)
−0.274184 + 0.961677i \(0.588408\pi\)
\(434\) −5.75313 −0.276159
\(435\) −1.00000 −0.0479463
\(436\) −0.425770 −0.0203907
\(437\) 2.20748 0.105598
\(438\) 15.4860 0.739951
\(439\) −7.36654 −0.351586 −0.175793 0.984427i \(-0.556249\pi\)
−0.175793 + 0.984427i \(0.556249\pi\)
\(440\) 7.61958 0.363249
\(441\) 0.524103 0.0249573
\(442\) 27.6085 1.31320
\(443\) 36.5423 1.73617 0.868087 0.496411i \(-0.165349\pi\)
0.868087 + 0.496411i \(0.165349\pi\)
\(444\) −0.464614 −0.0220496
\(445\) 8.87281 0.420611
\(446\) −3.16604 −0.149916
\(447\) 11.9182 0.563712
\(448\) −21.1399 −0.998767
\(449\) −39.6182 −1.86970 −0.934850 0.355043i \(-0.884466\pi\)
−0.934850 + 0.355043i \(0.884466\pi\)
\(450\) 1.43828 0.0678013
\(451\) 26.8291 1.26333
\(452\) −0.725431 −0.0341214
\(453\) −7.19114 −0.337869
\(454\) 17.5260 0.822534
\(455\) −14.1195 −0.661931
\(456\) −1.12343 −0.0526096
\(457\) 28.5220 1.33420 0.667102 0.744967i \(-0.267535\pi\)
0.667102 + 0.744967i \(0.267535\pi\)
\(458\) −4.54982 −0.212599
\(459\) 3.72913 0.174061
\(460\) 0.374756 0.0174731
\(461\) −22.6696 −1.05583 −0.527915 0.849297i \(-0.677026\pi\)
−0.527915 + 0.849297i \(0.677026\pi\)
\(462\) 10.8218 0.503475
\(463\) 28.5517 1.32691 0.663455 0.748217i \(-0.269090\pi\)
0.663455 + 0.748217i \(0.269090\pi\)
\(464\) 4.13260 0.191851
\(465\) 1.45825 0.0676248
\(466\) 34.2436 1.58630
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −0.353416 −0.0163367
\(469\) −34.2770 −1.58277
\(470\) −3.74772 −0.172870
\(471\) 4.31457 0.198805
\(472\) 27.5510 1.26814
\(473\) 12.1576 0.559005
\(474\) −20.3626 −0.935285
\(475\) −0.404431 −0.0185565
\(476\) 0.702312 0.0321904
\(477\) −6.43220 −0.294510
\(478\) −8.65882 −0.396045
\(479\) −4.43410 −0.202599 −0.101300 0.994856i \(-0.532300\pi\)
−0.101300 + 0.994856i \(0.532300\pi\)
\(480\) −0.388222 −0.0177198
\(481\) −34.8328 −1.58824
\(482\) 35.3123 1.60843
\(483\) −14.9720 −0.681251
\(484\) −0.238650 −0.0108477
\(485\) −7.82084 −0.355126
\(486\) −1.43828 −0.0652419
\(487\) 14.1035 0.639093 0.319546 0.947571i \(-0.396469\pi\)
0.319546 + 0.947571i \(0.396469\pi\)
\(488\) 36.3382 1.64496
\(489\) −21.6436 −0.978757
\(490\) −0.753809 −0.0340536
\(491\) 39.6376 1.78882 0.894410 0.447249i \(-0.147596\pi\)
0.894410 + 0.447249i \(0.147596\pi\)
\(492\) −0.671544 −0.0302755
\(493\) 3.72913 0.167951
\(494\) 2.99419 0.134715
\(495\) −2.74301 −0.123289
\(496\) −6.02638 −0.270592
\(497\) 31.1296 1.39635
\(498\) −2.33482 −0.104626
\(499\) 6.33858 0.283754 0.141877 0.989884i \(-0.454686\pi\)
0.141877 + 0.989884i \(0.454686\pi\)
\(500\) −0.0686587 −0.00307051
\(501\) 16.4303 0.734051
\(502\) −39.5761 −1.76637
\(503\) −19.0638 −0.850011 −0.425005 0.905191i \(-0.639728\pi\)
−0.425005 + 0.905191i \(0.639728\pi\)
\(504\) 7.61958 0.339403
\(505\) 4.88033 0.217172
\(506\) −21.5340 −0.957305
\(507\) −13.4961 −0.599385
\(508\) 0.586465 0.0260202
\(509\) −12.8145 −0.567992 −0.283996 0.958826i \(-0.591660\pi\)
−0.283996 + 0.958826i \(0.591660\pi\)
\(510\) −5.36354 −0.237502
\(511\) 29.5340 1.30651
\(512\) −21.3549 −0.943760
\(513\) 0.404431 0.0178560
\(514\) 22.4999 0.992428
\(515\) 0.294881 0.0129940
\(516\) −0.304309 −0.0133965
\(517\) 7.14744 0.314344
\(518\) −26.6974 −1.17302
\(519\) 4.64939 0.204086
\(520\) −14.2986 −0.627037
\(521\) −35.4800 −1.55441 −0.777204 0.629249i \(-0.783363\pi\)
−0.777204 + 0.629249i \(0.783363\pi\)
\(522\) −1.43828 −0.0629519
\(523\) −14.9227 −0.652525 −0.326263 0.945279i \(-0.605789\pi\)
−0.326263 + 0.945279i \(0.605789\pi\)
\(524\) 0.913504 0.0399066
\(525\) 2.74301 0.119715
\(526\) −26.7773 −1.16754
\(527\) −5.43801 −0.236883
\(528\) 11.3358 0.493326
\(529\) 6.79252 0.295327
\(530\) 9.25132 0.401852
\(531\) −9.91822 −0.430414
\(532\) 0.0761670 0.00330226
\(533\) −50.3466 −2.18075
\(534\) 12.7616 0.552249
\(535\) 13.7809 0.595800
\(536\) −34.7120 −1.49933
\(537\) −7.62334 −0.328971
\(538\) 15.6781 0.675931
\(539\) 1.43762 0.0619227
\(540\) 0.0686587 0.00295460
\(541\) −22.8644 −0.983017 −0.491509 0.870873i \(-0.663554\pi\)
−0.491509 + 0.870873i \(0.663554\pi\)
\(542\) 25.5740 1.09850
\(543\) 21.3050 0.914285
\(544\) 1.44773 0.0620709
\(545\) 6.20126 0.265633
\(546\) −20.3078 −0.869094
\(547\) −35.3290 −1.51056 −0.755279 0.655404i \(-0.772498\pi\)
−0.755279 + 0.655404i \(0.772498\pi\)
\(548\) −1.25132 −0.0534539
\(549\) −13.0816 −0.558309
\(550\) 3.94523 0.168225
\(551\) 0.404431 0.0172293
\(552\) −15.1620 −0.645338
\(553\) −38.8343 −1.65140
\(554\) −29.7167 −1.26254
\(555\) 6.76702 0.287244
\(556\) −0.388706 −0.0164848
\(557\) 32.7994 1.38976 0.694878 0.719127i \(-0.255458\pi\)
0.694878 + 0.719127i \(0.255458\pi\)
\(558\) 2.09738 0.0887892
\(559\) −22.8145 −0.964950
\(560\) −11.3358 −0.479024
\(561\) 10.2290 0.431870
\(562\) 40.6534 1.71486
\(563\) 16.8169 0.708747 0.354374 0.935104i \(-0.384694\pi\)
0.354374 + 0.935104i \(0.384694\pi\)
\(564\) −0.178903 −0.00753319
\(565\) 10.5658 0.444505
\(566\) −4.61888 −0.194146
\(567\) −2.74301 −0.115196
\(568\) 31.5246 1.32274
\(569\) 8.88033 0.372283 0.186141 0.982523i \(-0.440402\pi\)
0.186141 + 0.982523i \(0.440402\pi\)
\(570\) −0.581686 −0.0243641
\(571\) 25.1240 1.05141 0.525703 0.850668i \(-0.323802\pi\)
0.525703 + 0.850668i \(0.323802\pi\)
\(572\) −0.969425 −0.0405337
\(573\) −3.07597 −0.128501
\(574\) −38.5879 −1.61063
\(575\) −5.45825 −0.227625
\(576\) 7.70683 0.321118
\(577\) −13.9026 −0.578774 −0.289387 0.957212i \(-0.593451\pi\)
−0.289387 + 0.957212i \(0.593451\pi\)
\(578\) −4.44950 −0.185075
\(579\) −6.77454 −0.281540
\(580\) 0.0686587 0.00285090
\(581\) −4.45283 −0.184735
\(582\) −11.2486 −0.466269
\(583\) −17.6436 −0.730723
\(584\) 29.9088 1.23763
\(585\) 5.14744 0.212820
\(586\) −5.74772 −0.237436
\(587\) −20.9942 −0.866523 −0.433262 0.901268i \(-0.642637\pi\)
−0.433262 + 0.901268i \(0.642637\pi\)
\(588\) −0.0359842 −0.00148396
\(589\) −0.589762 −0.0243007
\(590\) 14.2652 0.587289
\(591\) 13.6455 0.561300
\(592\) −27.9654 −1.14937
\(593\) 3.54003 0.145372 0.0726859 0.997355i \(-0.476843\pi\)
0.0726859 + 0.997355i \(0.476843\pi\)
\(594\) −3.94523 −0.161875
\(595\) −10.2290 −0.419349
\(596\) −0.818289 −0.0335184
\(597\) 6.33858 0.259421
\(598\) 40.4100 1.65249
\(599\) −32.4886 −1.32745 −0.663725 0.747977i \(-0.731025\pi\)
−0.663725 + 0.747977i \(0.731025\pi\)
\(600\) 2.77782 0.113404
\(601\) −12.5620 −0.512414 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(602\) −17.4860 −0.712677
\(603\) 12.4961 0.508882
\(604\) 0.493734 0.0200898
\(605\) 3.47590 0.141315
\(606\) 7.01929 0.285139
\(607\) −0.369141 −0.0149830 −0.00749149 0.999972i \(-0.502385\pi\)
−0.00749149 + 0.999972i \(0.502385\pi\)
\(608\) 0.157009 0.00636756
\(609\) −2.74301 −0.111152
\(610\) 18.8150 0.761798
\(611\) −13.4126 −0.542618
\(612\) −0.256037 −0.0103497
\(613\) 44.7017 1.80549 0.902743 0.430181i \(-0.141550\pi\)
0.902743 + 0.430181i \(0.141550\pi\)
\(614\) −17.3393 −0.699756
\(615\) 9.78090 0.394404
\(616\) 20.9006 0.842108
\(617\) −5.91014 −0.237933 −0.118967 0.992898i \(-0.537958\pi\)
−0.118967 + 0.992898i \(0.537958\pi\)
\(618\) 0.424122 0.0170607
\(619\) −44.2187 −1.77730 −0.888650 0.458586i \(-0.848356\pi\)
−0.888650 + 0.458586i \(0.848356\pi\)
\(620\) −0.100122 −0.00402098
\(621\) 5.45825 0.219032
\(622\) 22.5628 0.904684
\(623\) 24.3382 0.975089
\(624\) −21.2723 −0.851575
\(625\) 1.00000 0.0400000
\(626\) −48.7184 −1.94718
\(627\) 1.10936 0.0443034
\(628\) −0.296233 −0.0118210
\(629\) −25.2351 −1.00619
\(630\) 3.94523 0.157182
\(631\) −29.5702 −1.17717 −0.588586 0.808435i \(-0.700315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(632\) −39.3271 −1.56435
\(633\) −2.00000 −0.0794929
\(634\) 2.52691 0.100356
\(635\) −8.54175 −0.338969
\(636\) 0.441626 0.0175116
\(637\) −2.69779 −0.106890
\(638\) −3.94523 −0.156193
\(639\) −11.3487 −0.448948
\(640\) −11.8611 −0.468849
\(641\) 26.6335 1.05196 0.525979 0.850497i \(-0.323699\pi\)
0.525979 + 0.850497i \(0.323699\pi\)
\(642\) 19.8208 0.782266
\(643\) 8.16597 0.322035 0.161017 0.986952i \(-0.448523\pi\)
0.161017 + 0.986952i \(0.448523\pi\)
\(644\) 1.02796 0.0405073
\(645\) 4.43220 0.174518
\(646\) 2.16918 0.0853452
\(647\) 20.8381 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(648\) −2.77782 −0.109123
\(649\) −27.2058 −1.06792
\(650\) −7.40348 −0.290388
\(651\) 4.00000 0.156772
\(652\) 1.48602 0.0581970
\(653\) 13.4267 0.525428 0.262714 0.964874i \(-0.415382\pi\)
0.262714 + 0.964874i \(0.415382\pi\)
\(654\) 8.91917 0.348767
\(655\) −13.3050 −0.519870
\(656\) −40.4206 −1.57816
\(657\) −10.7670 −0.420061
\(658\) −10.2800 −0.400758
\(659\) −42.9744 −1.67405 −0.837023 0.547167i \(-0.815706\pi\)
−0.837023 + 0.547167i \(0.815706\pi\)
\(660\) 0.188331 0.00733079
\(661\) −10.0178 −0.389649 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(662\) −32.2902 −1.25500
\(663\) −19.1955 −0.745490
\(664\) −4.50933 −0.174996
\(665\) −1.10936 −0.0430190
\(666\) 9.73289 0.377142
\(667\) 5.45825 0.211344
\(668\) −1.12808 −0.0436468
\(669\) 2.20126 0.0851057
\(670\) −17.9730 −0.694357
\(671\) −35.8829 −1.38525
\(672\) −1.06490 −0.0410793
\(673\) 23.0878 0.889970 0.444985 0.895538i \(-0.353209\pi\)
0.444985 + 0.895538i \(0.353209\pi\)
\(674\) 22.1332 0.852540
\(675\) −1.00000 −0.0384900
\(676\) 0.926627 0.0356395
\(677\) 4.95555 0.190457 0.0952287 0.995455i \(-0.469642\pi\)
0.0952287 + 0.995455i \(0.469642\pi\)
\(678\) 15.1965 0.583620
\(679\) −21.4526 −0.823277
\(680\) −10.3588 −0.397243
\(681\) −12.1853 −0.466943
\(682\) 5.75313 0.220299
\(683\) −36.8010 −1.40815 −0.704075 0.710126i \(-0.748638\pi\)
−0.704075 + 0.710126i \(0.748638\pi\)
\(684\) −0.0277677 −0.00106172
\(685\) 18.2253 0.696352
\(686\) 25.5489 0.975460
\(687\) 3.16337 0.120690
\(688\) −18.3165 −0.698311
\(689\) 33.1094 1.26137
\(690\) −7.85051 −0.298864
\(691\) 15.8246 0.601996 0.300998 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(692\) −0.319221 −0.0121350
\(693\) −7.52410 −0.285817
\(694\) −17.7634 −0.674289
\(695\) 5.66142 0.214750
\(696\) −2.77782 −0.105293
\(697\) −36.4742 −1.38156
\(698\) −2.79669 −0.105856
\(699\) −23.8087 −0.900527
\(700\) −0.188331 −0.00711826
\(701\) −42.6156 −1.60957 −0.804785 0.593567i \(-0.797719\pi\)
−0.804785 + 0.593567i \(0.797719\pi\)
\(702\) 7.40348 0.279426
\(703\) −2.73679 −0.103220
\(704\) 21.1399 0.796741
\(705\) 2.60569 0.0981361
\(706\) 9.67270 0.364037
\(707\) 13.3868 0.503462
\(708\) 0.680972 0.0255925
\(709\) 35.1795 1.32119 0.660597 0.750740i \(-0.270303\pi\)
0.660597 + 0.750740i \(0.270303\pi\)
\(710\) 16.3226 0.612578
\(711\) 14.1576 0.530950
\(712\) 24.6470 0.923686
\(713\) −7.95951 −0.298086
\(714\) −14.7122 −0.550592
\(715\) 14.1195 0.528039
\(716\) 0.523408 0.0195607
\(717\) 6.02025 0.224830
\(718\) 21.8783 0.816490
\(719\) −29.4563 −1.09854 −0.549268 0.835646i \(-0.685093\pi\)
−0.549268 + 0.835646i \(0.685093\pi\)
\(720\) 4.13260 0.154013
\(721\) 0.808861 0.0301236
\(722\) −27.0921 −1.00826
\(723\) −24.5517 −0.913087
\(724\) −1.46277 −0.0543635
\(725\) −1.00000 −0.0371391
\(726\) 4.99932 0.185542
\(727\) 46.3391 1.71862 0.859311 0.511454i \(-0.170893\pi\)
0.859311 + 0.511454i \(0.170893\pi\)
\(728\) −39.2213 −1.45364
\(729\) 1.00000 0.0370370
\(730\) 15.4860 0.573163
\(731\) −16.5282 −0.611319
\(732\) 0.898165 0.0331971
\(733\) −3.73344 −0.137898 −0.0689489 0.997620i \(-0.521965\pi\)
−0.0689489 + 0.997620i \(0.521965\pi\)
\(734\) 19.0125 0.701765
\(735\) 0.524103 0.0193318
\(736\) 2.11902 0.0781080
\(737\) 34.2770 1.26261
\(738\) 14.0677 0.517839
\(739\) −6.48021 −0.238378 −0.119189 0.992872i \(-0.538030\pi\)
−0.119189 + 0.992872i \(0.538030\pi\)
\(740\) −0.464614 −0.0170796
\(741\) −2.08178 −0.0764762
\(742\) 25.3765 0.931600
\(743\) −51.0638 −1.87335 −0.936674 0.350203i \(-0.886112\pi\)
−0.936674 + 0.350203i \(0.886112\pi\)
\(744\) 4.05076 0.148508
\(745\) 11.9182 0.436650
\(746\) 38.5533 1.41153
\(747\) 1.62334 0.0593948
\(748\) −0.702312 −0.0256791
\(749\) 37.8011 1.38122
\(750\) 1.43828 0.0525187
\(751\) 24.6494 0.899469 0.449735 0.893162i \(-0.351519\pi\)
0.449735 + 0.893162i \(0.351519\pi\)
\(752\) −10.7683 −0.392679
\(753\) 27.5162 1.00275
\(754\) 7.40348 0.269619
\(755\) −7.19114 −0.261712
\(756\) 0.188331 0.00684955
\(757\) −38.6833 −1.40597 −0.702985 0.711205i \(-0.748150\pi\)
−0.702985 + 0.711205i \(0.748150\pi\)
\(758\) −35.1269 −1.27587
\(759\) 14.9720 0.543451
\(760\) −1.12343 −0.0407512
\(761\) −3.23725 −0.117350 −0.0586750 0.998277i \(-0.518688\pi\)
−0.0586750 + 0.998277i \(0.518688\pi\)
\(762\) −12.2855 −0.445055
\(763\) 17.0101 0.615808
\(764\) 0.211192 0.00764067
\(765\) 3.72913 0.134827
\(766\) 39.1749 1.41545
\(767\) 51.0534 1.84343
\(768\) −1.64589 −0.0593909
\(769\) 46.3166 1.67022 0.835111 0.550082i \(-0.185404\pi\)
0.835111 + 0.550082i \(0.185404\pi\)
\(770\) 10.8218 0.389990
\(771\) −15.6436 −0.563390
\(772\) 0.465131 0.0167404
\(773\) −27.0341 −0.972350 −0.486175 0.873861i \(-0.661608\pi\)
−0.486175 + 0.873861i \(0.661608\pi\)
\(774\) 6.37476 0.229136
\(775\) 1.45825 0.0523820
\(776\) −21.7248 −0.779877
\(777\) 18.5620 0.665908
\(778\) −17.5461 −0.629059
\(779\) −3.95569 −0.141727
\(780\) −0.353416 −0.0126543
\(781\) −31.1296 −1.11390
\(782\) 29.2756 1.04689
\(783\) 1.00000 0.0357371
\(784\) −2.16591 −0.0773540
\(785\) 4.31457 0.153994
\(786\) −19.1364 −0.682572
\(787\) −39.1870 −1.39687 −0.698434 0.715675i \(-0.746119\pi\)
−0.698434 + 0.715675i \(0.746119\pi\)
\(788\) −0.936881 −0.0333750
\(789\) 18.6175 0.662802
\(790\) −20.3626 −0.724468
\(791\) 28.9820 1.03048
\(792\) −7.61958 −0.270750
\(793\) 67.3367 2.39120
\(794\) −25.4941 −0.904752
\(795\) −6.43220 −0.228127
\(796\) −0.435198 −0.0154252
\(797\) −53.2933 −1.88775 −0.943873 0.330307i \(-0.892848\pi\)
−0.943873 + 0.330307i \(0.892848\pi\)
\(798\) −1.59557 −0.0564825
\(799\) −9.71696 −0.343761
\(800\) −0.388222 −0.0137257
\(801\) −8.87281 −0.313505
\(802\) −3.57807 −0.126346
\(803\) −29.5340 −1.04223
\(804\) −0.857969 −0.0302582
\(805\) −14.9720 −0.527695
\(806\) −10.7961 −0.380278
\(807\) −10.9006 −0.383718
\(808\) 13.5567 0.476921
\(809\) 0.613214 0.0215595 0.0107797 0.999942i \(-0.496569\pi\)
0.0107797 + 0.999942i \(0.496569\pi\)
\(810\) −1.43828 −0.0505361
\(811\) −0.230936 −0.00810927 −0.00405463 0.999992i \(-0.501291\pi\)
−0.00405463 + 0.999992i \(0.501291\pi\)
\(812\) 0.188331 0.00660914
\(813\) −17.7809 −0.623603
\(814\) 26.6974 0.935744
\(815\) −21.6436 −0.758142
\(816\) −15.4110 −0.539493
\(817\) −1.79252 −0.0627122
\(818\) 54.2511 1.89685
\(819\) 14.1195 0.493375
\(820\) −0.671544 −0.0234513
\(821\) 25.7254 0.897821 0.448911 0.893577i \(-0.351812\pi\)
0.448911 + 0.893577i \(0.351812\pi\)
\(822\) 26.2131 0.914287
\(823\) −38.2258 −1.33247 −0.666234 0.745743i \(-0.732095\pi\)
−0.666234 + 0.745743i \(0.732095\pi\)
\(824\) 0.819125 0.0285356
\(825\) −2.74301 −0.0954993
\(826\) 39.1296 1.36149
\(827\) 24.9199 0.866551 0.433275 0.901262i \(-0.357358\pi\)
0.433275 + 0.901262i \(0.357358\pi\)
\(828\) −0.374756 −0.0130237
\(829\) −1.65301 −0.0574115 −0.0287057 0.999588i \(-0.509139\pi\)
−0.0287057 + 0.999588i \(0.509139\pi\)
\(830\) −2.33482 −0.0810427
\(831\) 20.6612 0.716730
\(832\) −39.6705 −1.37533
\(833\) −1.95445 −0.0677176
\(834\) 8.14273 0.281960
\(835\) 16.4303 0.568594
\(836\) −0.0761670 −0.00263429
\(837\) −1.45825 −0.0504046
\(838\) −3.30069 −0.114020
\(839\) −31.4757 −1.08666 −0.543331 0.839519i \(-0.682837\pi\)
−0.543331 + 0.839519i \(0.682837\pi\)
\(840\) 7.61958 0.262900
\(841\) 1.00000 0.0344828
\(842\) 51.7296 1.78272
\(843\) −28.2652 −0.973505
\(844\) 0.137317 0.00472666
\(845\) −13.4961 −0.464281
\(846\) 3.74772 0.128849
\(847\) 9.53442 0.327607
\(848\) 26.5817 0.912820
\(849\) 3.21139 0.110214
\(850\) −5.36354 −0.183968
\(851\) −36.9361 −1.26615
\(852\) 0.779187 0.0266945
\(853\) −30.3753 −1.04003 −0.520015 0.854157i \(-0.674074\pi\)
−0.520015 + 0.854157i \(0.674074\pi\)
\(854\) 51.6098 1.76605
\(855\) 0.404431 0.0138312
\(856\) 38.2808 1.30841
\(857\) −6.53423 −0.223205 −0.111602 0.993753i \(-0.535598\pi\)
−0.111602 + 0.993753i \(0.535598\pi\)
\(858\) 20.3078 0.693297
\(859\) 27.0220 0.921977 0.460989 0.887406i \(-0.347495\pi\)
0.460989 + 0.887406i \(0.347495\pi\)
\(860\) −0.304309 −0.0103768
\(861\) 26.8291 0.914334
\(862\) 32.6453 1.11190
\(863\) 31.9587 1.08789 0.543944 0.839122i \(-0.316931\pi\)
0.543944 + 0.839122i \(0.316931\pi\)
\(864\) 0.388222 0.0132076
\(865\) 4.64939 0.158084
\(866\) −16.4120 −0.557701
\(867\) 3.09362 0.105065
\(868\) −0.274635 −0.00932171
\(869\) 38.8343 1.31736
\(870\) −1.43828 −0.0487624
\(871\) −64.3232 −2.17951
\(872\) 17.2260 0.583345
\(873\) 7.82084 0.264695
\(874\) 3.17499 0.107396
\(875\) 2.74301 0.0927307
\(876\) 0.739249 0.0249769
\(877\) −50.0679 −1.69067 −0.845336 0.534235i \(-0.820600\pi\)
−0.845336 + 0.534235i \(0.820600\pi\)
\(878\) −10.5952 −0.357570
\(879\) 3.99624 0.134790
\(880\) 11.3358 0.382129
\(881\) −13.6817 −0.460947 −0.230474 0.973079i \(-0.574028\pi\)
−0.230474 + 0.973079i \(0.574028\pi\)
\(882\) 0.753809 0.0253821
\(883\) 29.5693 0.995087 0.497543 0.867439i \(-0.334236\pi\)
0.497543 + 0.867439i \(0.334236\pi\)
\(884\) 1.31793 0.0443269
\(885\) −9.91822 −0.333397
\(886\) 52.5581 1.76572
\(887\) 4.35130 0.146102 0.0730512 0.997328i \(-0.476726\pi\)
0.0730512 + 0.997328i \(0.476726\pi\)
\(888\) 18.7975 0.630804
\(889\) −23.4301 −0.785820
\(890\) 12.7616 0.427770
\(891\) 2.74301 0.0918943
\(892\) −0.151136 −0.00506040
\(893\) −1.05382 −0.0352648
\(894\) 17.1418 0.573307
\(895\) −7.62334 −0.254820
\(896\) −32.5350 −1.08692
\(897\) −28.0960 −0.938099
\(898\) −56.9822 −1.90152
\(899\) −1.45825 −0.0486354
\(900\) 0.0686587 0.00228862
\(901\) 23.9865 0.799105
\(902\) 38.5879 1.28484
\(903\) 12.1576 0.404578
\(904\) 29.3497 0.976157
\(905\) 21.3050 0.708202
\(906\) −10.3429 −0.343620
\(907\) −52.3965 −1.73980 −0.869899 0.493230i \(-0.835816\pi\)
−0.869899 + 0.493230i \(0.835816\pi\)
\(908\) 0.836629 0.0277645
\(909\) −4.88033 −0.161870
\(910\) −20.3078 −0.673197
\(911\) 7.85085 0.260110 0.130055 0.991507i \(-0.458485\pi\)
0.130055 + 0.991507i \(0.458485\pi\)
\(912\) −1.67135 −0.0553440
\(913\) 4.45283 0.147367
\(914\) 41.0227 1.35691
\(915\) −13.0816 −0.432464
\(916\) −0.217193 −0.00717625
\(917\) −36.4958 −1.20520
\(918\) 5.36354 0.177023
\(919\) 13.4979 0.445253 0.222627 0.974904i \(-0.428537\pi\)
0.222627 + 0.974904i \(0.428537\pi\)
\(920\) −15.1620 −0.499877
\(921\) 12.0555 0.397243
\(922\) −32.6054 −1.07380
\(923\) 58.4168 1.92281
\(924\) 0.516595 0.0169947
\(925\) 6.76702 0.222498
\(926\) 41.0654 1.34949
\(927\) −0.294881 −0.00968516
\(928\) 0.388222 0.0127440
\(929\) 11.6177 0.381165 0.190583 0.981671i \(-0.438962\pi\)
0.190583 + 0.981671i \(0.438962\pi\)
\(930\) 2.09738 0.0687758
\(931\) −0.211963 −0.00694682
\(932\) 1.63467 0.0535455
\(933\) −15.6873 −0.513579
\(934\) −11.5063 −0.376497
\(935\) 10.2290 0.334525
\(936\) 14.2986 0.467366
\(937\) −42.6740 −1.39410 −0.697049 0.717024i \(-0.745504\pi\)
−0.697049 + 0.717024i \(0.745504\pi\)
\(938\) −49.3001 −1.60971
\(939\) 33.8726 1.10539
\(940\) −0.178903 −0.00583519
\(941\) −5.91479 −0.192817 −0.0964083 0.995342i \(-0.530735\pi\)
−0.0964083 + 0.995342i \(0.530735\pi\)
\(942\) 6.20558 0.202189
\(943\) −53.3866 −1.73851
\(944\) 40.9881 1.33405
\(945\) −2.74301 −0.0892301
\(946\) 17.4860 0.568520
\(947\) −45.0915 −1.46528 −0.732639 0.680618i \(-0.761711\pi\)
−0.732639 + 0.680618i \(0.761711\pi\)
\(948\) −0.972040 −0.0315704
\(949\) 55.4226 1.79909
\(950\) −0.581686 −0.0188724
\(951\) −1.75689 −0.0569712
\(952\) −28.4144 −0.920915
\(953\) 30.3166 0.982053 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(954\) −9.25132 −0.299523
\(955\) −3.07597 −0.0995362
\(956\) −0.413342 −0.0133684
\(957\) 2.74301 0.0886689
\(958\) −6.37750 −0.206048
\(959\) 49.9921 1.61433
\(960\) 7.70683 0.248737
\(961\) −28.8735 −0.931403
\(962\) −50.0995 −1.61527
\(963\) −13.7809 −0.444083
\(964\) 1.68569 0.0542923
\(965\) −6.77454 −0.218080
\(966\) −21.5340 −0.692846
\(967\) 5.99809 0.192886 0.0964428 0.995339i \(-0.469254\pi\)
0.0964428 + 0.995339i \(0.469254\pi\)
\(968\) 9.65540 0.310336
\(969\) −1.50817 −0.0484495
\(970\) −11.2486 −0.361170
\(971\) 28.5140 0.915057 0.457529 0.889195i \(-0.348735\pi\)
0.457529 + 0.889195i \(0.348735\pi\)
\(972\) −0.0686587 −0.00220223
\(973\) 15.5293 0.497848
\(974\) 20.2849 0.649970
\(975\) 5.14744 0.164850
\(976\) 54.0610 1.73045
\(977\) −5.53813 −0.177180 −0.0885902 0.996068i \(-0.528236\pi\)
−0.0885902 + 0.996068i \(0.528236\pi\)
\(978\) −31.1296 −0.995415
\(979\) −24.3382 −0.777852
\(980\) −0.0359842 −0.00114947
\(981\) −6.20126 −0.197991
\(982\) 57.0101 1.81926
\(983\) 7.37086 0.235094 0.117547 0.993067i \(-0.462497\pi\)
0.117547 + 0.993067i \(0.462497\pi\)
\(984\) 27.1695 0.866133
\(985\) 13.6455 0.434781
\(986\) 5.36354 0.170810
\(987\) 7.14744 0.227506
\(988\) 0.142932 0.00454729
\(989\) −24.1921 −0.769263
\(990\) −3.94523 −0.125387
\(991\) 3.68167 0.116952 0.0584760 0.998289i \(-0.481376\pi\)
0.0584760 + 0.998289i \(0.481376\pi\)
\(992\) −0.566126 −0.0179745
\(993\) 22.4505 0.712446
\(994\) 44.7732 1.42012
\(995\) 6.33858 0.200946
\(996\) −0.111456 −0.00353162
\(997\) 36.8747 1.16783 0.583916 0.811814i \(-0.301520\pi\)
0.583916 + 0.811814i \(0.301520\pi\)
\(998\) 9.11667 0.288583
\(999\) −6.76702 −0.214099
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 435.2.a.j.1.4 4
3.2 odd 2 1305.2.a.r.1.1 4
4.3 odd 2 6960.2.a.co.1.4 4
5.2 odd 4 2175.2.c.n.349.6 8
5.3 odd 4 2175.2.c.n.349.3 8
5.4 even 2 2175.2.a.v.1.1 4
15.14 odd 2 6525.2.a.bi.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 1.1 even 1 trivial
1305.2.a.r.1.1 4 3.2 odd 2
2175.2.a.v.1.1 4 5.4 even 2
2175.2.c.n.349.3 8 5.3 odd 4
2175.2.c.n.349.6 8 5.2 odd 4
6525.2.a.bi.1.4 4 15.14 odd 2
6960.2.a.co.1.4 4 4.3 odd 2