Properties

Label 625.2.a.e.1.6
Level $625$
Weight $2$
Character 625.1
Self dual yes
Analytic conductor $4.991$
Analytic rank $1$
Dimension $8$
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] = [625,2,Mod(1,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.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: \(4.99065012633\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6152203125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.32675\) of defining polynomial
Character \(\chi\) \(=\) 625.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.326751 q^{2} +1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} -3.42409 q^{7} -1.27212 q^{8} -0.0574791 q^{9} +O(q^{10})\) \(q+0.326751 q^{2} +1.71538 q^{3} -1.89323 q^{4} +0.560502 q^{6} -3.42409 q^{7} -1.27212 q^{8} -0.0574791 q^{9} -5.34111 q^{11} -3.24761 q^{12} +3.52114 q^{13} -1.11883 q^{14} +3.37080 q^{16} -2.55787 q^{17} -0.0187814 q^{18} -2.02579 q^{19} -5.87362 q^{21} -1.74522 q^{22} -7.57082 q^{23} -2.18216 q^{24} +1.15054 q^{26} -5.24473 q^{27} +6.48261 q^{28} +4.74270 q^{29} +1.62421 q^{31} +3.64565 q^{32} -9.16203 q^{33} -0.835788 q^{34} +0.108821 q^{36} +0.0134290 q^{37} -0.661929 q^{38} +6.04008 q^{39} +9.67740 q^{41} -1.91921 q^{42} -2.32645 q^{43} +10.1120 q^{44} -2.47377 q^{46} -6.94647 q^{47} +5.78220 q^{48} +4.72443 q^{49} -4.38772 q^{51} -6.66634 q^{52} -1.72246 q^{53} -1.71372 q^{54} +4.35585 q^{56} -3.47500 q^{57} +1.54968 q^{58} -0.0221830 q^{59} +3.91768 q^{61} +0.530712 q^{62} +0.196814 q^{63} -5.55038 q^{64} -2.99370 q^{66} -4.11832 q^{67} +4.84265 q^{68} -12.9868 q^{69} +2.33894 q^{71} +0.0731202 q^{72} -1.51373 q^{73} +0.00438793 q^{74} +3.83530 q^{76} +18.2885 q^{77} +1.97360 q^{78} +0.426831 q^{79} -8.82426 q^{81} +3.16210 q^{82} +6.04187 q^{83} +11.1201 q^{84} -0.760170 q^{86} +8.13552 q^{87} +6.79453 q^{88} -6.09362 q^{89} -12.0567 q^{91} +14.3333 q^{92} +2.78613 q^{93} -2.26977 q^{94} +6.25367 q^{96} +16.0018 q^{97} +1.54371 q^{98} +0.307002 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} - 5 q^{3} + 11 q^{4} - 4 q^{6} - 10 q^{7} - 15 q^{8} + 9 q^{9} + q^{11} - 10 q^{12} - 10 q^{13} - 8 q^{14} + 13 q^{16} - 15 q^{17} + 5 q^{18} - 10 q^{19} - 14 q^{21} + 5 q^{22} - 30 q^{23} + 5 q^{24} + 11 q^{26} - 20 q^{27} + 5 q^{28} + 10 q^{29} - 9 q^{31} - 30 q^{32} - 5 q^{33} + 7 q^{34} + 3 q^{36} + 10 q^{37} - 20 q^{38} + 8 q^{39} - 4 q^{41} + 35 q^{42} - 18 q^{44} - 9 q^{46} - 30 q^{47} - 5 q^{48} - 4 q^{49} - 14 q^{51} - 5 q^{52} - 10 q^{53} - 20 q^{54} + 10 q^{57} + 30 q^{58} - 5 q^{59} + 6 q^{61} - 10 q^{62} - 9 q^{64} - 18 q^{66} - 10 q^{67} - 40 q^{68} + 3 q^{69} - 9 q^{71} + 15 q^{72} - 18 q^{74} - 10 q^{76} - 5 q^{77} + 30 q^{78} - 20 q^{79} + 8 q^{81} + 45 q^{82} - 40 q^{83} - 28 q^{84} - 24 q^{86} - 40 q^{87} + 40 q^{88} - 5 q^{89} + 6 q^{91} - 15 q^{92} + 40 q^{93} + 47 q^{94} + 71 q^{96} + 30 q^{98} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.326751 0.231048 0.115524 0.993305i \(-0.463145\pi\)
0.115524 + 0.993305i \(0.463145\pi\)
\(3\) 1.71538 0.990374 0.495187 0.868786i \(-0.335100\pi\)
0.495187 + 0.868786i \(0.335100\pi\)
\(4\) −1.89323 −0.946617
\(5\) 0 0
\(6\) 0.560502 0.228824
\(7\) −3.42409 −1.29419 −0.647093 0.762411i \(-0.724016\pi\)
−0.647093 + 0.762411i \(0.724016\pi\)
\(8\) −1.27212 −0.449762
\(9\) −0.0574791 −0.0191597
\(10\) 0 0
\(11\) −5.34111 −1.61041 −0.805203 0.592999i \(-0.797944\pi\)
−0.805203 + 0.592999i \(0.797944\pi\)
\(12\) −3.24761 −0.937505
\(13\) 3.52114 0.976588 0.488294 0.872679i \(-0.337619\pi\)
0.488294 + 0.872679i \(0.337619\pi\)
\(14\) −1.11883 −0.299019
\(15\) 0 0
\(16\) 3.37080 0.842700
\(17\) −2.55787 −0.620375 −0.310188 0.950675i \(-0.600392\pi\)
−0.310188 + 0.950675i \(0.600392\pi\)
\(18\) −0.0187814 −0.00442681
\(19\) −2.02579 −0.464748 −0.232374 0.972626i \(-0.574649\pi\)
−0.232374 + 0.972626i \(0.574649\pi\)
\(20\) 0 0
\(21\) −5.87362 −1.28173
\(22\) −1.74522 −0.372081
\(23\) −7.57082 −1.57863 −0.789313 0.613991i \(-0.789563\pi\)
−0.789313 + 0.613991i \(0.789563\pi\)
\(24\) −2.18216 −0.445432
\(25\) 0 0
\(26\) 1.15054 0.225639
\(27\) −5.24473 −1.00935
\(28\) 6.48261 1.22510
\(29\) 4.74270 0.880697 0.440348 0.897827i \(-0.354855\pi\)
0.440348 + 0.897827i \(0.354855\pi\)
\(30\) 0 0
\(31\) 1.62421 0.291716 0.145858 0.989306i \(-0.453406\pi\)
0.145858 + 0.989306i \(0.453406\pi\)
\(32\) 3.64565 0.644466
\(33\) −9.16203 −1.59490
\(34\) −0.835788 −0.143336
\(35\) 0 0
\(36\) 0.108821 0.0181369
\(37\) 0.0134290 0.00220771 0.00110385 0.999999i \(-0.499649\pi\)
0.00110385 + 0.999999i \(0.499649\pi\)
\(38\) −0.661929 −0.107379
\(39\) 6.04008 0.967187
\(40\) 0 0
\(41\) 9.67740 1.51136 0.755678 0.654943i \(-0.227307\pi\)
0.755678 + 0.654943i \(0.227307\pi\)
\(42\) −1.91921 −0.296141
\(43\) −2.32645 −0.354780 −0.177390 0.984141i \(-0.556765\pi\)
−0.177390 + 0.984141i \(0.556765\pi\)
\(44\) 10.1120 1.52444
\(45\) 0 0
\(46\) −2.47377 −0.364738
\(47\) −6.94647 −1.01325 −0.506624 0.862167i \(-0.669107\pi\)
−0.506624 + 0.862167i \(0.669107\pi\)
\(48\) 5.78220 0.834588
\(49\) 4.72443 0.674918
\(50\) 0 0
\(51\) −4.38772 −0.614403
\(52\) −6.66634 −0.924454
\(53\) −1.72246 −0.236598 −0.118299 0.992978i \(-0.537744\pi\)
−0.118299 + 0.992978i \(0.537744\pi\)
\(54\) −1.71372 −0.233208
\(55\) 0 0
\(56\) 4.35585 0.582075
\(57\) −3.47500 −0.460275
\(58\) 1.54968 0.203483
\(59\) −0.0221830 −0.00288798 −0.00144399 0.999999i \(-0.500460\pi\)
−0.00144399 + 0.999999i \(0.500460\pi\)
\(60\) 0 0
\(61\) 3.91768 0.501607 0.250804 0.968038i \(-0.419305\pi\)
0.250804 + 0.968038i \(0.419305\pi\)
\(62\) 0.530712 0.0674004
\(63\) 0.196814 0.0247962
\(64\) −5.55038 −0.693798
\(65\) 0 0
\(66\) −2.99370 −0.368499
\(67\) −4.11832 −0.503133 −0.251566 0.967840i \(-0.580946\pi\)
−0.251566 + 0.967840i \(0.580946\pi\)
\(68\) 4.84265 0.587258
\(69\) −12.9868 −1.56343
\(70\) 0 0
\(71\) 2.33894 0.277581 0.138790 0.990322i \(-0.455679\pi\)
0.138790 + 0.990322i \(0.455679\pi\)
\(72\) 0.0731202 0.00861730
\(73\) −1.51373 −0.177169 −0.0885843 0.996069i \(-0.528234\pi\)
−0.0885843 + 0.996069i \(0.528234\pi\)
\(74\) 0.00438793 0.000510086 0
\(75\) 0 0
\(76\) 3.83530 0.439939
\(77\) 18.2885 2.08417
\(78\) 1.97360 0.223467
\(79\) 0.426831 0.0480222 0.0240111 0.999712i \(-0.492356\pi\)
0.0240111 + 0.999712i \(0.492356\pi\)
\(80\) 0 0
\(81\) −8.82426 −0.980473
\(82\) 3.16210 0.349196
\(83\) 6.04187 0.663181 0.331591 0.943423i \(-0.392415\pi\)
0.331591 + 0.943423i \(0.392415\pi\)
\(84\) 11.1201 1.21331
\(85\) 0 0
\(86\) −0.760170 −0.0819712
\(87\) 8.13552 0.872219
\(88\) 6.79453 0.724299
\(89\) −6.09362 −0.645922 −0.322961 0.946412i \(-0.604678\pi\)
−0.322961 + 0.946412i \(0.604678\pi\)
\(90\) 0 0
\(91\) −12.0567 −1.26389
\(92\) 14.3333 1.49435
\(93\) 2.78613 0.288908
\(94\) −2.26977 −0.234109
\(95\) 0 0
\(96\) 6.25367 0.638262
\(97\) 16.0018 1.62474 0.812370 0.583143i \(-0.198177\pi\)
0.812370 + 0.583143i \(0.198177\pi\)
\(98\) 1.54371 0.155938
\(99\) 0.307002 0.0308549
\(100\) 0 0
\(101\) 1.44418 0.143701 0.0718505 0.997415i \(-0.477110\pi\)
0.0718505 + 0.997415i \(0.477110\pi\)
\(102\) −1.43369 −0.141957
\(103\) −14.6657 −1.44506 −0.722529 0.691341i \(-0.757020\pi\)
−0.722529 + 0.691341i \(0.757020\pi\)
\(104\) −4.47930 −0.439232
\(105\) 0 0
\(106\) −0.562816 −0.0546655
\(107\) −12.2169 −1.18106 −0.590528 0.807017i \(-0.701081\pi\)
−0.590528 + 0.807017i \(0.701081\pi\)
\(108\) 9.92950 0.955467
\(109\) −15.3516 −1.47041 −0.735207 0.677843i \(-0.762915\pi\)
−0.735207 + 0.677843i \(0.762915\pi\)
\(110\) 0 0
\(111\) 0.0230357 0.00218646
\(112\) −11.5419 −1.09061
\(113\) −18.5544 −1.74545 −0.872727 0.488209i \(-0.837650\pi\)
−0.872727 + 0.488209i \(0.837650\pi\)
\(114\) −1.13546 −0.106345
\(115\) 0 0
\(116\) −8.97904 −0.833683
\(117\) −0.202392 −0.0187111
\(118\) −0.00724831 −0.000667261 0
\(119\) 8.75840 0.802881
\(120\) 0 0
\(121\) 17.5275 1.59341
\(122\) 1.28011 0.115895
\(123\) 16.6004 1.49681
\(124\) −3.07500 −0.276144
\(125\) 0 0
\(126\) 0.0643092 0.00572911
\(127\) −0.677902 −0.0601541 −0.0300771 0.999548i \(-0.509575\pi\)
−0.0300771 + 0.999548i \(0.509575\pi\)
\(128\) −9.10489 −0.804766
\(129\) −3.99074 −0.351365
\(130\) 0 0
\(131\) 7.05058 0.616012 0.308006 0.951384i \(-0.400338\pi\)
0.308006 + 0.951384i \(0.400338\pi\)
\(132\) 17.3459 1.50976
\(133\) 6.93650 0.601471
\(134\) −1.34567 −0.116248
\(135\) 0 0
\(136\) 3.25392 0.279021
\(137\) 10.9186 0.932837 0.466418 0.884564i \(-0.345544\pi\)
0.466418 + 0.884564i \(0.345544\pi\)
\(138\) −4.24346 −0.361227
\(139\) 19.5102 1.65483 0.827416 0.561589i \(-0.189810\pi\)
0.827416 + 0.561589i \(0.189810\pi\)
\(140\) 0 0
\(141\) −11.9158 −1.00349
\(142\) 0.764251 0.0641345
\(143\) −18.8068 −1.57270
\(144\) −0.193751 −0.0161459
\(145\) 0 0
\(146\) −0.494613 −0.0409345
\(147\) 8.10418 0.668421
\(148\) −0.0254242 −0.00208985
\(149\) −12.7945 −1.04817 −0.524085 0.851666i \(-0.675593\pi\)
−0.524085 + 0.851666i \(0.675593\pi\)
\(150\) 0 0
\(151\) 2.15617 0.175466 0.0877331 0.996144i \(-0.472038\pi\)
0.0877331 + 0.996144i \(0.472038\pi\)
\(152\) 2.57705 0.209026
\(153\) 0.147024 0.0118862
\(154\) 5.97578 0.481542
\(155\) 0 0
\(156\) −11.4353 −0.915556
\(157\) 15.7474 1.25678 0.628389 0.777899i \(-0.283715\pi\)
0.628389 + 0.777899i \(0.283715\pi\)
\(158\) 0.139467 0.0110954
\(159\) −2.95467 −0.234321
\(160\) 0 0
\(161\) 25.9232 2.04304
\(162\) −2.88334 −0.226536
\(163\) −7.39219 −0.579001 −0.289500 0.957178i \(-0.593489\pi\)
−0.289500 + 0.957178i \(0.593489\pi\)
\(164\) −18.3216 −1.43068
\(165\) 0 0
\(166\) 1.97419 0.153227
\(167\) 10.8165 0.837007 0.418504 0.908215i \(-0.362555\pi\)
0.418504 + 0.908215i \(0.362555\pi\)
\(168\) 7.47194 0.576472
\(169\) −0.601591 −0.0462762
\(170\) 0 0
\(171\) 0.116441 0.00890444
\(172\) 4.40451 0.335841
\(173\) −14.9983 −1.14030 −0.570149 0.821541i \(-0.693115\pi\)
−0.570149 + 0.821541i \(0.693115\pi\)
\(174\) 2.65829 0.201524
\(175\) 0 0
\(176\) −18.0038 −1.35709
\(177\) −0.0380522 −0.00286018
\(178\) −1.99110 −0.149239
\(179\) 7.39841 0.552983 0.276492 0.961016i \(-0.410828\pi\)
0.276492 + 0.961016i \(0.410828\pi\)
\(180\) 0 0
\(181\) −10.9177 −0.811503 −0.405752 0.913983i \(-0.632990\pi\)
−0.405752 + 0.913983i \(0.632990\pi\)
\(182\) −3.93954 −0.292018
\(183\) 6.72030 0.496779
\(184\) 9.63098 0.710006
\(185\) 0 0
\(186\) 0.910371 0.0667516
\(187\) 13.6619 0.999056
\(188\) 13.1513 0.959157
\(189\) 17.9585 1.30629
\(190\) 0 0
\(191\) −1.75142 −0.126728 −0.0633642 0.997990i \(-0.520183\pi\)
−0.0633642 + 0.997990i \(0.520183\pi\)
\(192\) −9.52100 −0.687119
\(193\) 9.53146 0.686089 0.343045 0.939319i \(-0.388542\pi\)
0.343045 + 0.939319i \(0.388542\pi\)
\(194\) 5.22862 0.375393
\(195\) 0 0
\(196\) −8.94444 −0.638889
\(197\) −23.9169 −1.70401 −0.852004 0.523535i \(-0.824613\pi\)
−0.852004 + 0.523535i \(0.824613\pi\)
\(198\) 0.100313 0.00712896
\(199\) 23.8281 1.68913 0.844566 0.535451i \(-0.179858\pi\)
0.844566 + 0.535451i \(0.179858\pi\)
\(200\) 0 0
\(201\) −7.06448 −0.498290
\(202\) 0.471886 0.0332018
\(203\) −16.2394 −1.13979
\(204\) 8.30697 0.581605
\(205\) 0 0
\(206\) −4.79204 −0.333878
\(207\) 0.435164 0.0302460
\(208\) 11.8691 0.822971
\(209\) 10.8200 0.748434
\(210\) 0 0
\(211\) −15.3923 −1.05965 −0.529826 0.848107i \(-0.677743\pi\)
−0.529826 + 0.848107i \(0.677743\pi\)
\(212\) 3.26102 0.223968
\(213\) 4.01216 0.274909
\(214\) −3.99190 −0.272881
\(215\) 0 0
\(216\) 6.67192 0.453967
\(217\) −5.56144 −0.377535
\(218\) −5.01614 −0.339736
\(219\) −2.59662 −0.175463
\(220\) 0 0
\(221\) −9.00662 −0.605851
\(222\) 0.00752696 0.000505176 0
\(223\) 19.5753 1.31086 0.655429 0.755257i \(-0.272488\pi\)
0.655429 + 0.755257i \(0.272488\pi\)
\(224\) −12.4831 −0.834059
\(225\) 0 0
\(226\) −6.06268 −0.403283
\(227\) −1.34207 −0.0890760 −0.0445380 0.999008i \(-0.514182\pi\)
−0.0445380 + 0.999008i \(0.514182\pi\)
\(228\) 6.57898 0.435704
\(229\) −22.3702 −1.47827 −0.739133 0.673560i \(-0.764764\pi\)
−0.739133 + 0.673560i \(0.764764\pi\)
\(230\) 0 0
\(231\) 31.3717 2.06410
\(232\) −6.03327 −0.396104
\(233\) −18.3651 −1.20314 −0.601568 0.798822i \(-0.705457\pi\)
−0.601568 + 0.798822i \(0.705457\pi\)
\(234\) −0.0661317 −0.00432317
\(235\) 0 0
\(236\) 0.0419976 0.00273381
\(237\) 0.732176 0.0475600
\(238\) 2.86182 0.185504
\(239\) −12.0037 −0.776458 −0.388229 0.921563i \(-0.626913\pi\)
−0.388229 + 0.921563i \(0.626913\pi\)
\(240\) 0 0
\(241\) 10.1170 0.651692 0.325846 0.945423i \(-0.394351\pi\)
0.325846 + 0.945423i \(0.394351\pi\)
\(242\) 5.72713 0.368154
\(243\) 0.597258 0.0383141
\(244\) −7.41708 −0.474830
\(245\) 0 0
\(246\) 5.42420 0.345834
\(247\) −7.13309 −0.453867
\(248\) −2.06618 −0.131203
\(249\) 10.3641 0.656797
\(250\) 0 0
\(251\) 16.7258 1.05573 0.527863 0.849330i \(-0.322994\pi\)
0.527863 + 0.849330i \(0.322994\pi\)
\(252\) −0.372615 −0.0234725
\(253\) 40.4366 2.54223
\(254\) −0.221505 −0.0138985
\(255\) 0 0
\(256\) 8.12573 0.507858
\(257\) −15.0170 −0.936732 −0.468366 0.883535i \(-0.655157\pi\)
−0.468366 + 0.883535i \(0.655157\pi\)
\(258\) −1.30398 −0.0811822
\(259\) −0.0459821 −0.00285719
\(260\) 0 0
\(261\) −0.272606 −0.0168739
\(262\) 2.30379 0.142328
\(263\) −6.17182 −0.380571 −0.190285 0.981729i \(-0.560941\pi\)
−0.190285 + 0.981729i \(0.560941\pi\)
\(264\) 11.6552 0.717327
\(265\) 0 0
\(266\) 2.26651 0.138969
\(267\) −10.4529 −0.639704
\(268\) 7.79694 0.476274
\(269\) 11.1052 0.677098 0.338549 0.940949i \(-0.390064\pi\)
0.338549 + 0.940949i \(0.390064\pi\)
\(270\) 0 0
\(271\) −1.16149 −0.0705554 −0.0352777 0.999378i \(-0.511232\pi\)
−0.0352777 + 0.999378i \(0.511232\pi\)
\(272\) −8.62208 −0.522790
\(273\) −20.6818 −1.25172
\(274\) 3.56766 0.215530
\(275\) 0 0
\(276\) 24.5871 1.47997
\(277\) 2.17486 0.130675 0.0653374 0.997863i \(-0.479188\pi\)
0.0653374 + 0.997863i \(0.479188\pi\)
\(278\) 6.37497 0.382346
\(279\) −0.0933580 −0.00558920
\(280\) 0 0
\(281\) −24.1177 −1.43874 −0.719370 0.694627i \(-0.755569\pi\)
−0.719370 + 0.694627i \(0.755569\pi\)
\(282\) −3.89351 −0.231855
\(283\) −16.2144 −0.963845 −0.481923 0.876214i \(-0.660061\pi\)
−0.481923 + 0.876214i \(0.660061\pi\)
\(284\) −4.42816 −0.262763
\(285\) 0 0
\(286\) −6.14514 −0.363370
\(287\) −33.1363 −1.95598
\(288\) −0.209549 −0.0123478
\(289\) −10.4573 −0.615135
\(290\) 0 0
\(291\) 27.4492 1.60910
\(292\) 2.86584 0.167711
\(293\) −3.48929 −0.203846 −0.101923 0.994792i \(-0.532500\pi\)
−0.101923 + 0.994792i \(0.532500\pi\)
\(294\) 2.64805 0.154437
\(295\) 0 0
\(296\) −0.0170832 −0.000992943 0
\(297\) 28.0127 1.62546
\(298\) −4.18063 −0.242178
\(299\) −26.6579 −1.54167
\(300\) 0 0
\(301\) 7.96599 0.459152
\(302\) 0.704529 0.0405411
\(303\) 2.47731 0.142318
\(304\) −6.82854 −0.391644
\(305\) 0 0
\(306\) 0.0480403 0.00274628
\(307\) −8.92690 −0.509485 −0.254742 0.967009i \(-0.581991\pi\)
−0.254742 + 0.967009i \(0.581991\pi\)
\(308\) −34.6244 −1.97291
\(309\) −25.1573 −1.43115
\(310\) 0 0
\(311\) −27.1101 −1.53727 −0.768635 0.639687i \(-0.779064\pi\)
−0.768635 + 0.639687i \(0.779064\pi\)
\(312\) −7.68370 −0.435004
\(313\) 20.1073 1.13653 0.568267 0.822844i \(-0.307614\pi\)
0.568267 + 0.822844i \(0.307614\pi\)
\(314\) 5.14548 0.290376
\(315\) 0 0
\(316\) −0.808091 −0.0454587
\(317\) −9.67993 −0.543679 −0.271840 0.962343i \(-0.587632\pi\)
−0.271840 + 0.962343i \(0.587632\pi\)
\(318\) −0.965442 −0.0541393
\(319\) −25.3313 −1.41828
\(320\) 0 0
\(321\) −20.9567 −1.16969
\(322\) 8.47044 0.472039
\(323\) 5.18171 0.288318
\(324\) 16.7064 0.928132
\(325\) 0 0
\(326\) −2.41540 −0.133777
\(327\) −26.3337 −1.45626
\(328\) −12.3108 −0.679750
\(329\) 23.7854 1.31133
\(330\) 0 0
\(331\) 19.6759 1.08148 0.540742 0.841189i \(-0.318144\pi\)
0.540742 + 0.841189i \(0.318144\pi\)
\(332\) −11.4387 −0.627778
\(333\) −0.000771885 0 −4.22990e−5 0
\(334\) 3.53431 0.193389
\(335\) 0 0
\(336\) −19.7988 −1.08011
\(337\) −18.8552 −1.02711 −0.513554 0.858057i \(-0.671671\pi\)
−0.513554 + 0.858057i \(0.671671\pi\)
\(338\) −0.196571 −0.0106920
\(339\) −31.8279 −1.72865
\(340\) 0 0
\(341\) −8.67508 −0.469782
\(342\) 0.0380471 0.00205735
\(343\) 7.79178 0.420717
\(344\) 2.95952 0.159567
\(345\) 0 0
\(346\) −4.90071 −0.263464
\(347\) −25.9753 −1.39442 −0.697212 0.716865i \(-0.745577\pi\)
−0.697212 + 0.716865i \(0.745577\pi\)
\(348\) −15.4024 −0.825657
\(349\) −26.1490 −1.39972 −0.699861 0.714279i \(-0.746755\pi\)
−0.699861 + 0.714279i \(0.746755\pi\)
\(350\) 0 0
\(351\) −18.4674 −0.985718
\(352\) −19.4718 −1.03785
\(353\) 5.32892 0.283630 0.141815 0.989893i \(-0.454706\pi\)
0.141815 + 0.989893i \(0.454706\pi\)
\(354\) −0.0124336 −0.000660838 0
\(355\) 0 0
\(356\) 11.5366 0.611441
\(357\) 15.0240 0.795152
\(358\) 2.41744 0.127766
\(359\) 9.89929 0.522465 0.261232 0.965276i \(-0.415871\pi\)
0.261232 + 0.965276i \(0.415871\pi\)
\(360\) 0 0
\(361\) −14.8962 −0.784009
\(362\) −3.56736 −0.187496
\(363\) 30.0663 1.57807
\(364\) 22.8262 1.19642
\(365\) 0 0
\(366\) 2.19587 0.114780
\(367\) −16.4392 −0.858118 −0.429059 0.903276i \(-0.641155\pi\)
−0.429059 + 0.903276i \(0.641155\pi\)
\(368\) −25.5197 −1.33031
\(369\) −0.556248 −0.0289571
\(370\) 0 0
\(371\) 5.89787 0.306202
\(372\) −5.27479 −0.273485
\(373\) −22.9933 −1.19055 −0.595273 0.803524i \(-0.702956\pi\)
−0.595273 + 0.803524i \(0.702956\pi\)
\(374\) 4.46404 0.230830
\(375\) 0 0
\(376\) 8.83674 0.455720
\(377\) 16.6997 0.860078
\(378\) 5.86795 0.301815
\(379\) 16.4246 0.843674 0.421837 0.906672i \(-0.361385\pi\)
0.421837 + 0.906672i \(0.361385\pi\)
\(380\) 0 0
\(381\) −1.16286 −0.0595751
\(382\) −0.572279 −0.0292803
\(383\) −4.70503 −0.240416 −0.120208 0.992749i \(-0.538356\pi\)
−0.120208 + 0.992749i \(0.538356\pi\)
\(384\) −15.6183 −0.797020
\(385\) 0 0
\(386\) 3.11441 0.158520
\(387\) 0.133722 0.00679748
\(388\) −30.2952 −1.53801
\(389\) 3.53865 0.179417 0.0897083 0.995968i \(-0.471407\pi\)
0.0897083 + 0.995968i \(0.471407\pi\)
\(390\) 0 0
\(391\) 19.3652 0.979340
\(392\) −6.01003 −0.303552
\(393\) 12.0944 0.610082
\(394\) −7.81487 −0.393707
\(395\) 0 0
\(396\) −0.581227 −0.0292078
\(397\) 8.76374 0.439839 0.219920 0.975518i \(-0.429420\pi\)
0.219920 + 0.975518i \(0.429420\pi\)
\(398\) 7.78587 0.390270
\(399\) 11.8987 0.595681
\(400\) 0 0
\(401\) 22.7677 1.13697 0.568483 0.822695i \(-0.307531\pi\)
0.568483 + 0.822695i \(0.307531\pi\)
\(402\) −2.30833 −0.115129
\(403\) 5.71906 0.284887
\(404\) −2.73416 −0.136030
\(405\) 0 0
\(406\) −5.30626 −0.263345
\(407\) −0.0717256 −0.00355531
\(408\) 5.58170 0.276335
\(409\) 28.0426 1.38662 0.693309 0.720640i \(-0.256152\pi\)
0.693309 + 0.720640i \(0.256152\pi\)
\(410\) 0 0
\(411\) 18.7295 0.923857
\(412\) 27.7657 1.36792
\(413\) 0.0759566 0.00373758
\(414\) 0.142190 0.00698827
\(415\) 0 0
\(416\) 12.8368 0.629378
\(417\) 33.4673 1.63890
\(418\) 3.53544 0.172924
\(419\) 34.0901 1.66541 0.832705 0.553717i \(-0.186791\pi\)
0.832705 + 0.553717i \(0.186791\pi\)
\(420\) 0 0
\(421\) −32.1390 −1.56636 −0.783180 0.621796i \(-0.786403\pi\)
−0.783180 + 0.621796i \(0.786403\pi\)
\(422\) −5.02946 −0.244830
\(423\) 0.399277 0.0194135
\(424\) 2.19117 0.106413
\(425\) 0 0
\(426\) 1.31098 0.0635171
\(427\) −13.4145 −0.649173
\(428\) 23.1295 1.11801
\(429\) −32.2608 −1.55756
\(430\) 0 0
\(431\) −17.7549 −0.855222 −0.427611 0.903963i \(-0.640645\pi\)
−0.427611 + 0.903963i \(0.640645\pi\)
\(432\) −17.6789 −0.850579
\(433\) −3.06764 −0.147421 −0.0737107 0.997280i \(-0.523484\pi\)
−0.0737107 + 0.997280i \(0.523484\pi\)
\(434\) −1.81721 −0.0872287
\(435\) 0 0
\(436\) 29.0641 1.39192
\(437\) 15.3369 0.733664
\(438\) −0.848448 −0.0405404
\(439\) 14.7475 0.703861 0.351931 0.936026i \(-0.385525\pi\)
0.351931 + 0.936026i \(0.385525\pi\)
\(440\) 0 0
\(441\) −0.271556 −0.0129312
\(442\) −2.94292 −0.139981
\(443\) 10.1857 0.483935 0.241968 0.970284i \(-0.422207\pi\)
0.241968 + 0.970284i \(0.422207\pi\)
\(444\) −0.0436121 −0.00206974
\(445\) 0 0
\(446\) 6.39624 0.302871
\(447\) −21.9475 −1.03808
\(448\) 19.0050 0.897904
\(449\) 18.9484 0.894230 0.447115 0.894477i \(-0.352451\pi\)
0.447115 + 0.894477i \(0.352451\pi\)
\(450\) 0 0
\(451\) −51.6881 −2.43390
\(452\) 35.1279 1.65228
\(453\) 3.69864 0.173777
\(454\) −0.438522 −0.0205808
\(455\) 0 0
\(456\) 4.42061 0.207014
\(457\) 30.4392 1.42389 0.711943 0.702237i \(-0.247815\pi\)
0.711943 + 0.702237i \(0.247815\pi\)
\(458\) −7.30949 −0.341550
\(459\) 13.4154 0.626175
\(460\) 0 0
\(461\) 2.36972 0.110369 0.0551844 0.998476i \(-0.482425\pi\)
0.0551844 + 0.998476i \(0.482425\pi\)
\(462\) 10.2507 0.476907
\(463\) −0.320982 −0.0149173 −0.00745865 0.999972i \(-0.502374\pi\)
−0.00745865 + 0.999972i \(0.502374\pi\)
\(464\) 15.9867 0.742164
\(465\) 0 0
\(466\) −6.00081 −0.277982
\(467\) −12.8071 −0.592644 −0.296322 0.955088i \(-0.595760\pi\)
−0.296322 + 0.955088i \(0.595760\pi\)
\(468\) 0.383175 0.0177123
\(469\) 14.1015 0.651148
\(470\) 0 0
\(471\) 27.0127 1.24468
\(472\) 0.0282194 0.00129890
\(473\) 12.4258 0.571341
\(474\) 0.239239 0.0109886
\(475\) 0 0
\(476\) −16.5817 −0.760021
\(477\) 0.0990055 0.00453315
\(478\) −3.92224 −0.179399
\(479\) −20.3446 −0.929569 −0.464784 0.885424i \(-0.653868\pi\)
−0.464784 + 0.885424i \(0.653868\pi\)
\(480\) 0 0
\(481\) 0.0472852 0.00215602
\(482\) 3.30573 0.150572
\(483\) 44.4681 2.02337
\(484\) −33.1837 −1.50835
\(485\) 0 0
\(486\) 0.195155 0.00885239
\(487\) −15.4919 −0.702003 −0.351002 0.936375i \(-0.614159\pi\)
−0.351002 + 0.936375i \(0.614159\pi\)
\(488\) −4.98375 −0.225604
\(489\) −12.6804 −0.573427
\(490\) 0 0
\(491\) −27.5085 −1.24144 −0.620722 0.784031i \(-0.713160\pi\)
−0.620722 + 0.784031i \(0.713160\pi\)
\(492\) −31.4284 −1.41690
\(493\) −12.1312 −0.546362
\(494\) −2.33074 −0.104865
\(495\) 0 0
\(496\) 5.47488 0.245829
\(497\) −8.00875 −0.359241
\(498\) 3.38648 0.151752
\(499\) 4.91044 0.219821 0.109911 0.993941i \(-0.464943\pi\)
0.109911 + 0.993941i \(0.464943\pi\)
\(500\) 0 0
\(501\) 18.5544 0.828950
\(502\) 5.46518 0.243923
\(503\) −41.0454 −1.83012 −0.915062 0.403314i \(-0.867858\pi\)
−0.915062 + 0.403314i \(0.867858\pi\)
\(504\) −0.250371 −0.0111524
\(505\) 0 0
\(506\) 13.2127 0.587377
\(507\) −1.03196 −0.0458308
\(508\) 1.28343 0.0569429
\(509\) 41.4404 1.83681 0.918407 0.395637i \(-0.129476\pi\)
0.918407 + 0.395637i \(0.129476\pi\)
\(510\) 0 0
\(511\) 5.18315 0.229289
\(512\) 20.8649 0.922106
\(513\) 10.6247 0.469093
\(514\) −4.90681 −0.216430
\(515\) 0 0
\(516\) 7.55540 0.332608
\(517\) 37.1019 1.63174
\(518\) −0.0150247 −0.000660147 0
\(519\) −25.7277 −1.12932
\(520\) 0 0
\(521\) −1.62447 −0.0711691 −0.0355846 0.999367i \(-0.511329\pi\)
−0.0355846 + 0.999367i \(0.511329\pi\)
\(522\) −0.0890743 −0.00389868
\(523\) 19.0009 0.830853 0.415427 0.909627i \(-0.363632\pi\)
0.415427 + 0.909627i \(0.363632\pi\)
\(524\) −13.3484 −0.583128
\(525\) 0 0
\(526\) −2.01665 −0.0879301
\(527\) −4.15451 −0.180973
\(528\) −30.8834 −1.34403
\(529\) 34.3174 1.49206
\(530\) 0 0
\(531\) 0.00127506 5.53328e−5 0
\(532\) −13.1324 −0.569362
\(533\) 34.0755 1.47597
\(534\) −3.41548 −0.147802
\(535\) 0 0
\(536\) 5.23899 0.226290
\(537\) 12.6911 0.547660
\(538\) 3.62865 0.156442
\(539\) −25.2337 −1.08689
\(540\) 0 0
\(541\) −33.5195 −1.44112 −0.720558 0.693394i \(-0.756114\pi\)
−0.720558 + 0.693394i \(0.756114\pi\)
\(542\) −0.379518 −0.0163017
\(543\) −18.7279 −0.803691
\(544\) −9.32511 −0.399811
\(545\) 0 0
\(546\) −6.75780 −0.289207
\(547\) 17.4258 0.745072 0.372536 0.928018i \(-0.378488\pi\)
0.372536 + 0.928018i \(0.378488\pi\)
\(548\) −20.6714 −0.883039
\(549\) −0.225185 −0.00961065
\(550\) 0 0
\(551\) −9.60771 −0.409302
\(552\) 16.5208 0.703171
\(553\) −1.46151 −0.0621497
\(554\) 0.710638 0.0301921
\(555\) 0 0
\(556\) −36.9373 −1.56649
\(557\) 16.2929 0.690351 0.345175 0.938538i \(-0.387819\pi\)
0.345175 + 0.938538i \(0.387819\pi\)
\(558\) −0.0305048 −0.00129137
\(559\) −8.19175 −0.346474
\(560\) 0 0
\(561\) 23.4353 0.989439
\(562\) −7.88048 −0.332418
\(563\) −23.0999 −0.973545 −0.486773 0.873529i \(-0.661826\pi\)
−0.486773 + 0.873529i \(0.661826\pi\)
\(564\) 22.5594 0.949924
\(565\) 0 0
\(566\) −5.29807 −0.222694
\(567\) 30.2151 1.26891
\(568\) −2.97541 −0.124845
\(569\) −9.28334 −0.389178 −0.194589 0.980885i \(-0.562337\pi\)
−0.194589 + 0.980885i \(0.562337\pi\)
\(570\) 0 0
\(571\) 22.2143 0.929640 0.464820 0.885405i \(-0.346119\pi\)
0.464820 + 0.885405i \(0.346119\pi\)
\(572\) 35.6057 1.48875
\(573\) −3.00435 −0.125509
\(574\) −10.8273 −0.451924
\(575\) 0 0
\(576\) 0.319031 0.0132930
\(577\) −14.6093 −0.608192 −0.304096 0.952641i \(-0.598354\pi\)
−0.304096 + 0.952641i \(0.598354\pi\)
\(578\) −3.41693 −0.142126
\(579\) 16.3500 0.679485
\(580\) 0 0
\(581\) −20.6879 −0.858280
\(582\) 8.96905 0.371779
\(583\) 9.19986 0.381019
\(584\) 1.92564 0.0796837
\(585\) 0 0
\(586\) −1.14013 −0.0470983
\(587\) −23.3435 −0.963489 −0.481744 0.876312i \(-0.659997\pi\)
−0.481744 + 0.876312i \(0.659997\pi\)
\(588\) −15.3431 −0.632739
\(589\) −3.29030 −0.135575
\(590\) 0 0
\(591\) −41.0265 −1.68760
\(592\) 0.0452664 0.00186044
\(593\) 41.9815 1.72397 0.861986 0.506932i \(-0.169220\pi\)
0.861986 + 0.506932i \(0.169220\pi\)
\(594\) 9.15319 0.375560
\(595\) 0 0
\(596\) 24.2231 0.992215
\(597\) 40.8743 1.67287
\(598\) −8.71050 −0.356199
\(599\) −25.4160 −1.03847 −0.519236 0.854631i \(-0.673783\pi\)
−0.519236 + 0.854631i \(0.673783\pi\)
\(600\) 0 0
\(601\) 37.1379 1.51489 0.757444 0.652900i \(-0.226448\pi\)
0.757444 + 0.652900i \(0.226448\pi\)
\(602\) 2.60289 0.106086
\(603\) 0.236717 0.00963988
\(604\) −4.08213 −0.166099
\(605\) 0 0
\(606\) 0.809463 0.0328822
\(607\) 18.9242 0.768109 0.384054 0.923310i \(-0.374527\pi\)
0.384054 + 0.923310i \(0.374527\pi\)
\(608\) −7.38532 −0.299514
\(609\) −27.8568 −1.12881
\(610\) 0 0
\(611\) −24.4595 −0.989525
\(612\) −0.278351 −0.0112517
\(613\) −9.20317 −0.371713 −0.185856 0.982577i \(-0.559506\pi\)
−0.185856 + 0.982577i \(0.559506\pi\)
\(614\) −2.91687 −0.117715
\(615\) 0 0
\(616\) −23.2651 −0.937378
\(617\) 32.7375 1.31796 0.658981 0.752159i \(-0.270988\pi\)
0.658981 + 0.752159i \(0.270988\pi\)
\(618\) −8.22017 −0.330664
\(619\) 36.3952 1.46285 0.731424 0.681923i \(-0.238856\pi\)
0.731424 + 0.681923i \(0.238856\pi\)
\(620\) 0 0
\(621\) 39.7069 1.59338
\(622\) −8.85824 −0.355183
\(623\) 20.8651 0.835943
\(624\) 20.3599 0.815049
\(625\) 0 0
\(626\) 6.57010 0.262594
\(627\) 18.5604 0.741229
\(628\) −29.8135 −1.18969
\(629\) −0.0343496 −0.00136961
\(630\) 0 0
\(631\) 12.1083 0.482024 0.241012 0.970522i \(-0.422521\pi\)
0.241012 + 0.970522i \(0.422521\pi\)
\(632\) −0.542980 −0.0215986
\(633\) −26.4037 −1.04945
\(634\) −3.16293 −0.125616
\(635\) 0 0
\(636\) 5.59388 0.221812
\(637\) 16.6354 0.659117
\(638\) −8.27703 −0.327691
\(639\) −0.134440 −0.00531837
\(640\) 0 0
\(641\) 9.75177 0.385172 0.192586 0.981280i \(-0.438313\pi\)
0.192586 + 0.981280i \(0.438313\pi\)
\(642\) −6.84762 −0.270254
\(643\) −6.77862 −0.267323 −0.133661 0.991027i \(-0.542673\pi\)
−0.133661 + 0.991027i \(0.542673\pi\)
\(644\) −49.0787 −1.93397
\(645\) 0 0
\(646\) 1.69313 0.0666153
\(647\) 40.1188 1.57723 0.788616 0.614886i \(-0.210798\pi\)
0.788616 + 0.614886i \(0.210798\pi\)
\(648\) 11.2255 0.440979
\(649\) 0.118482 0.00465082
\(650\) 0 0
\(651\) −9.53997 −0.373901
\(652\) 13.9951 0.548092
\(653\) −6.14210 −0.240359 −0.120179 0.992752i \(-0.538347\pi\)
−0.120179 + 0.992752i \(0.538347\pi\)
\(654\) −8.60458 −0.336466
\(655\) 0 0
\(656\) 32.6206 1.27362
\(657\) 0.0870078 0.00339450
\(658\) 7.77190 0.302980
\(659\) −44.1645 −1.72040 −0.860202 0.509954i \(-0.829662\pi\)
−0.860202 + 0.509954i \(0.829662\pi\)
\(660\) 0 0
\(661\) 27.7447 1.07915 0.539573 0.841939i \(-0.318586\pi\)
0.539573 + 0.841939i \(0.318586\pi\)
\(662\) 6.42911 0.249875
\(663\) −15.4498 −0.600019
\(664\) −7.68597 −0.298274
\(665\) 0 0
\(666\) −0.000252214 0 −9.77310e−6 0
\(667\) −35.9061 −1.39029
\(668\) −20.4782 −0.792325
\(669\) 33.5790 1.29824
\(670\) 0 0
\(671\) −20.9248 −0.807792
\(672\) −21.4131 −0.826030
\(673\) 41.8324 1.61252 0.806259 0.591562i \(-0.201489\pi\)
0.806259 + 0.591562i \(0.201489\pi\)
\(674\) −6.16096 −0.237311
\(675\) 0 0
\(676\) 1.13895 0.0438059
\(677\) −22.6274 −0.869643 −0.434821 0.900517i \(-0.643188\pi\)
−0.434821 + 0.900517i \(0.643188\pi\)
\(678\) −10.3998 −0.399401
\(679\) −54.7918 −2.10272
\(680\) 0 0
\(681\) −2.30215 −0.0882186
\(682\) −2.83459 −0.108542
\(683\) 49.0024 1.87502 0.937512 0.347953i \(-0.113123\pi\)
0.937512 + 0.347953i \(0.113123\pi\)
\(684\) −0.220449 −0.00842909
\(685\) 0 0
\(686\) 2.54597 0.0972057
\(687\) −38.3734 −1.46404
\(688\) −7.84200 −0.298973
\(689\) −6.06502 −0.231059
\(690\) 0 0
\(691\) 36.2583 1.37933 0.689665 0.724128i \(-0.257758\pi\)
0.689665 + 0.724128i \(0.257758\pi\)
\(692\) 28.3953 1.07943
\(693\) −1.05121 −0.0399320
\(694\) −8.48744 −0.322179
\(695\) 0 0
\(696\) −10.3493 −0.392291
\(697\) −24.7536 −0.937608
\(698\) −8.54421 −0.323403
\(699\) −31.5030 −1.19155
\(700\) 0 0
\(701\) −8.32362 −0.314379 −0.157189 0.987568i \(-0.550243\pi\)
−0.157189 + 0.987568i \(0.550243\pi\)
\(702\) −6.03425 −0.227748
\(703\) −0.0272043 −0.00102603
\(704\) 29.6452 1.11730
\(705\) 0 0
\(706\) 1.74123 0.0655321
\(707\) −4.94500 −0.185976
\(708\) 0.0720417 0.00270749
\(709\) 37.3097 1.40119 0.700597 0.713557i \(-0.252917\pi\)
0.700597 + 0.713557i \(0.252917\pi\)
\(710\) 0 0
\(711\) −0.0245339 −0.000920092 0
\(712\) 7.75180 0.290511
\(713\) −12.2966 −0.460511
\(714\) 4.90910 0.183718
\(715\) 0 0
\(716\) −14.0069 −0.523463
\(717\) −20.5910 −0.768983
\(718\) 3.23460 0.120714
\(719\) −7.66524 −0.285865 −0.142933 0.989732i \(-0.545653\pi\)
−0.142933 + 0.989732i \(0.545653\pi\)
\(720\) 0 0
\(721\) 50.2169 1.87017
\(722\) −4.86734 −0.181144
\(723\) 17.3544 0.645419
\(724\) 20.6697 0.768183
\(725\) 0 0
\(726\) 9.82419 0.364610
\(727\) 45.9089 1.70267 0.851334 0.524624i \(-0.175794\pi\)
0.851334 + 0.524624i \(0.175794\pi\)
\(728\) 15.3376 0.568448
\(729\) 27.4973 1.01842
\(730\) 0 0
\(731\) 5.95076 0.220097
\(732\) −12.7231 −0.470259
\(733\) −23.7131 −0.875864 −0.437932 0.899008i \(-0.644289\pi\)
−0.437932 + 0.899008i \(0.644289\pi\)
\(734\) −5.37152 −0.198266
\(735\) 0 0
\(736\) −27.6006 −1.01737
\(737\) 21.9964 0.810249
\(738\) −0.181755 −0.00669048
\(739\) −3.53683 −0.130104 −0.0650522 0.997882i \(-0.520721\pi\)
−0.0650522 + 0.997882i \(0.520721\pi\)
\(740\) 0 0
\(741\) −12.2359 −0.449498
\(742\) 1.92714 0.0707474
\(743\) 15.7201 0.576715 0.288358 0.957523i \(-0.406891\pi\)
0.288358 + 0.957523i \(0.406891\pi\)
\(744\) −3.54429 −0.129940
\(745\) 0 0
\(746\) −7.51307 −0.275073
\(747\) −0.347281 −0.0127063
\(748\) −25.8651 −0.945724
\(749\) 41.8320 1.52851
\(750\) 0 0
\(751\) −14.1856 −0.517642 −0.258821 0.965925i \(-0.583334\pi\)
−0.258821 + 0.965925i \(0.583334\pi\)
\(752\) −23.4152 −0.853864
\(753\) 28.6911 1.04556
\(754\) 5.45664 0.198719
\(755\) 0 0
\(756\) −33.9996 −1.23655
\(757\) −12.7388 −0.463000 −0.231500 0.972835i \(-0.574363\pi\)
−0.231500 + 0.972835i \(0.574363\pi\)
\(758\) 5.36675 0.194929
\(759\) 69.3641 2.51776
\(760\) 0 0
\(761\) 32.2299 1.16833 0.584167 0.811633i \(-0.301421\pi\)
0.584167 + 0.811633i \(0.301421\pi\)
\(762\) −0.379965 −0.0137647
\(763\) 52.5652 1.90299
\(764\) 3.31585 0.119963
\(765\) 0 0
\(766\) −1.53737 −0.0555476
\(767\) −0.0781093 −0.00282036
\(768\) 13.9387 0.502969
\(769\) −11.3687 −0.409965 −0.204983 0.978766i \(-0.565714\pi\)
−0.204983 + 0.978766i \(0.565714\pi\)
\(770\) 0 0
\(771\) −25.7597 −0.927715
\(772\) −18.0453 −0.649464
\(773\) −27.1994 −0.978295 −0.489147 0.872201i \(-0.662692\pi\)
−0.489147 + 0.872201i \(0.662692\pi\)
\(774\) 0.0436939 0.00157054
\(775\) 0 0
\(776\) −20.3562 −0.730746
\(777\) −0.0788766 −0.00282968
\(778\) 1.15626 0.0414538
\(779\) −19.6044 −0.702400
\(780\) 0 0
\(781\) −12.4925 −0.447018
\(782\) 6.32760 0.226275
\(783\) −24.8742 −0.888931
\(784\) 15.9251 0.568754
\(785\) 0 0
\(786\) 3.95186 0.140958
\(787\) 25.7266 0.917054 0.458527 0.888680i \(-0.348377\pi\)
0.458527 + 0.888680i \(0.348377\pi\)
\(788\) 45.2803 1.61304
\(789\) −10.5870 −0.376908
\(790\) 0 0
\(791\) 63.5321 2.25894
\(792\) −0.390543 −0.0138774
\(793\) 13.7947 0.489864
\(794\) 2.86356 0.101624
\(795\) 0 0
\(796\) −45.1122 −1.59896
\(797\) −17.6713 −0.625951 −0.312975 0.949761i \(-0.601326\pi\)
−0.312975 + 0.949761i \(0.601326\pi\)
\(798\) 3.88792 0.137631
\(799\) 17.7682 0.628593
\(800\) 0 0
\(801\) 0.350256 0.0123757
\(802\) 7.43938 0.262694
\(803\) 8.08500 0.285314
\(804\) 13.3747 0.471689
\(805\) 0 0
\(806\) 1.86871 0.0658224
\(807\) 19.0497 0.670580
\(808\) −1.83716 −0.0646312
\(809\) −40.8576 −1.43648 −0.718238 0.695798i \(-0.755051\pi\)
−0.718238 + 0.695798i \(0.755051\pi\)
\(810\) 0 0
\(811\) 16.6214 0.583656 0.291828 0.956471i \(-0.405737\pi\)
0.291828 + 0.956471i \(0.405737\pi\)
\(812\) 30.7451 1.07894
\(813\) −1.99239 −0.0698763
\(814\) −0.0234364 −0.000821447 0
\(815\) 0 0
\(816\) −14.7901 −0.517758
\(817\) 4.71290 0.164884
\(818\) 9.16296 0.320375
\(819\) 0.693009 0.0242157
\(820\) 0 0
\(821\) −43.8983 −1.53206 −0.766031 0.642803i \(-0.777771\pi\)
−0.766031 + 0.642803i \(0.777771\pi\)
\(822\) 6.11988 0.213455
\(823\) −26.1962 −0.913143 −0.456571 0.889687i \(-0.650923\pi\)
−0.456571 + 0.889687i \(0.650923\pi\)
\(824\) 18.6565 0.649932
\(825\) 0 0
\(826\) 0.0248189 0.000863561 0
\(827\) −22.3827 −0.778321 −0.389161 0.921170i \(-0.627235\pi\)
−0.389161 + 0.921170i \(0.627235\pi\)
\(828\) −0.823867 −0.0286314
\(829\) 0.841282 0.0292189 0.0146095 0.999893i \(-0.495349\pi\)
0.0146095 + 0.999893i \(0.495349\pi\)
\(830\) 0 0
\(831\) 3.73071 0.129417
\(832\) −19.5437 −0.677554
\(833\) −12.0845 −0.418702
\(834\) 10.9355 0.378665
\(835\) 0 0
\(836\) −20.4848 −0.708480
\(837\) −8.51853 −0.294444
\(838\) 11.1390 0.384790
\(839\) 35.5528 1.22742 0.613710 0.789532i \(-0.289677\pi\)
0.613710 + 0.789532i \(0.289677\pi\)
\(840\) 0 0
\(841\) −6.50681 −0.224373
\(842\) −10.5015 −0.361904
\(843\) −41.3709 −1.42489
\(844\) 29.1413 1.00308
\(845\) 0 0
\(846\) 0.130464 0.00448545
\(847\) −60.0158 −2.06217
\(848\) −5.80608 −0.199381
\(849\) −27.8138 −0.954567
\(850\) 0 0
\(851\) −0.101668 −0.00348515
\(852\) −7.59596 −0.260233
\(853\) 19.5406 0.669058 0.334529 0.942386i \(-0.391423\pi\)
0.334529 + 0.942386i \(0.391423\pi\)
\(854\) −4.38320 −0.149990
\(855\) 0 0
\(856\) 15.5414 0.531194
\(857\) 0.570622 0.0194921 0.00974604 0.999953i \(-0.496898\pi\)
0.00974604 + 0.999953i \(0.496898\pi\)
\(858\) −10.5412 −0.359872
\(859\) −20.0983 −0.685744 −0.342872 0.939382i \(-0.611400\pi\)
−0.342872 + 0.939382i \(0.611400\pi\)
\(860\) 0 0
\(861\) −56.8413 −1.93715
\(862\) −5.80142 −0.197597
\(863\) −13.9645 −0.475358 −0.237679 0.971344i \(-0.576387\pi\)
−0.237679 + 0.971344i \(0.576387\pi\)
\(864\) −19.1205 −0.650491
\(865\) 0 0
\(866\) −1.00236 −0.0340614
\(867\) −17.9382 −0.609213
\(868\) 10.5291 0.357381
\(869\) −2.27975 −0.0773353
\(870\) 0 0
\(871\) −14.5012 −0.491353
\(872\) 19.5290 0.661336
\(873\) −0.919771 −0.0311295
\(874\) 5.01135 0.169511
\(875\) 0 0
\(876\) 4.91601 0.166096
\(877\) −23.4512 −0.791891 −0.395946 0.918274i \(-0.629583\pi\)
−0.395946 + 0.918274i \(0.629583\pi\)
\(878\) 4.81877 0.162626
\(879\) −5.98545 −0.201884
\(880\) 0 0
\(881\) 44.4714 1.49828 0.749139 0.662412i \(-0.230467\pi\)
0.749139 + 0.662412i \(0.230467\pi\)
\(882\) −0.0887311 −0.00298773
\(883\) 0.758169 0.0255144 0.0127572 0.999919i \(-0.495939\pi\)
0.0127572 + 0.999919i \(0.495939\pi\)
\(884\) 17.0516 0.573509
\(885\) 0 0
\(886\) 3.32818 0.111812
\(887\) −55.4695 −1.86248 −0.931242 0.364402i \(-0.881273\pi\)
−0.931242 + 0.364402i \(0.881273\pi\)
\(888\) −0.0293042 −0.000983385 0
\(889\) 2.32120 0.0778506
\(890\) 0 0
\(891\) 47.1314 1.57896
\(892\) −37.0606 −1.24088
\(893\) 14.0721 0.470905
\(894\) −7.17136 −0.239846
\(895\) 0 0
\(896\) 31.1760 1.04152
\(897\) −45.7284 −1.52683
\(898\) 6.19141 0.206610
\(899\) 7.70312 0.256914
\(900\) 0 0
\(901\) 4.40584 0.146780
\(902\) −16.8891 −0.562347
\(903\) 13.6647 0.454732
\(904\) 23.6034 0.785038
\(905\) 0 0
\(906\) 1.20853 0.0401509
\(907\) −15.8193 −0.525271 −0.262636 0.964895i \(-0.584592\pi\)
−0.262636 + 0.964895i \(0.584592\pi\)
\(908\) 2.54084 0.0843209
\(909\) −0.0830100 −0.00275327
\(910\) 0 0
\(911\) −8.02411 −0.265851 −0.132925 0.991126i \(-0.542437\pi\)
−0.132925 + 0.991126i \(0.542437\pi\)
\(912\) −11.7135 −0.387873
\(913\) −32.2703 −1.06799
\(914\) 9.94605 0.328986
\(915\) 0 0
\(916\) 42.3520 1.39935
\(917\) −24.1419 −0.797235
\(918\) 4.38348 0.144676
\(919\) 22.8402 0.753428 0.376714 0.926330i \(-0.377054\pi\)
0.376714 + 0.926330i \(0.377054\pi\)
\(920\) 0 0
\(921\) −15.3130 −0.504581
\(922\) 0.774308 0.0255005
\(923\) 8.23572 0.271082
\(924\) −59.3939 −1.95392
\(925\) 0 0
\(926\) −0.104881 −0.00344661
\(927\) 0.842973 0.0276869
\(928\) 17.2902 0.567579
\(929\) −39.7278 −1.30343 −0.651713 0.758465i \(-0.725950\pi\)
−0.651713 + 0.758465i \(0.725950\pi\)
\(930\) 0 0
\(931\) −9.57070 −0.313667
\(932\) 34.7694 1.13891
\(933\) −46.5040 −1.52247
\(934\) −4.18475 −0.136929
\(935\) 0 0
\(936\) 0.257466 0.00841555
\(937\) −30.4367 −0.994325 −0.497162 0.867658i \(-0.665625\pi\)
−0.497162 + 0.867658i \(0.665625\pi\)
\(938\) 4.60769 0.150446
\(939\) 34.4917 1.12559
\(940\) 0 0
\(941\) 25.6117 0.834916 0.417458 0.908696i \(-0.362921\pi\)
0.417458 + 0.908696i \(0.362921\pi\)
\(942\) 8.82643 0.287581
\(943\) −73.2659 −2.38587
\(944\) −0.0747744 −0.00243370
\(945\) 0 0
\(946\) 4.06016 0.132007
\(947\) 58.0741 1.88716 0.943578 0.331151i \(-0.107437\pi\)
0.943578 + 0.331151i \(0.107437\pi\)
\(948\) −1.38618 −0.0450211
\(949\) −5.33005 −0.173021
\(950\) 0 0
\(951\) −16.6047 −0.538446
\(952\) −11.1417 −0.361105
\(953\) 45.6216 1.47783 0.738915 0.673799i \(-0.235339\pi\)
0.738915 + 0.673799i \(0.235339\pi\)
\(954\) 0.0323502 0.00104738
\(955\) 0 0
\(956\) 22.7259 0.735008
\(957\) −43.4527 −1.40463
\(958\) −6.64762 −0.214775
\(959\) −37.3862 −1.20726
\(960\) 0 0
\(961\) −28.3620 −0.914902
\(962\) 0.0154505 0.000498144 0
\(963\) 0.702219 0.0226287
\(964\) −19.1538 −0.616903
\(965\) 0 0
\(966\) 14.5300 0.467495
\(967\) 42.4449 1.36494 0.682468 0.730916i \(-0.260907\pi\)
0.682468 + 0.730916i \(0.260907\pi\)
\(968\) −22.2971 −0.716655
\(969\) 8.88860 0.285543
\(970\) 0 0
\(971\) −14.4332 −0.463182 −0.231591 0.972813i \(-0.574393\pi\)
−0.231591 + 0.972813i \(0.574393\pi\)
\(972\) −1.13075 −0.0362688
\(973\) −66.8047 −2.14166
\(974\) −5.06198 −0.162196
\(975\) 0 0
\(976\) 13.2057 0.422705
\(977\) −16.5748 −0.530276 −0.265138 0.964211i \(-0.585417\pi\)
−0.265138 + 0.964211i \(0.585417\pi\)
\(978\) −4.14333 −0.132489
\(979\) 32.5467 1.04020
\(980\) 0 0
\(981\) 0.882394 0.0281727
\(982\) −8.98845 −0.286833
\(983\) −33.3174 −1.06266 −0.531330 0.847165i \(-0.678308\pi\)
−0.531330 + 0.847165i \(0.678308\pi\)
\(984\) −21.1177 −0.673207
\(985\) 0 0
\(986\) −3.96389 −0.126236
\(987\) 40.8009 1.29871
\(988\) 13.5046 0.429639
\(989\) 17.6131 0.560065
\(990\) 0 0
\(991\) −24.4962 −0.778146 −0.389073 0.921207i \(-0.627205\pi\)
−0.389073 + 0.921207i \(0.627205\pi\)
\(992\) 5.92129 0.188001
\(993\) 33.7516 1.07107
\(994\) −2.61687 −0.0830020
\(995\) 0 0
\(996\) −19.6216 −0.621735
\(997\) 59.8575 1.89571 0.947854 0.318706i \(-0.103248\pi\)
0.947854 + 0.318706i \(0.103248\pi\)
\(998\) 1.60449 0.0507893
\(999\) −0.0704313 −0.00222835
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 625.2.a.e.1.6 8
3.2 odd 2 5625.2.a.be.1.3 8
4.3 odd 2 10000.2.a.bn.1.2 8
5.2 odd 4 625.2.b.d.624.9 16
5.3 odd 4 625.2.b.d.624.8 16
5.4 even 2 625.2.a.g.1.3 yes 8
15.14 odd 2 5625.2.a.s.1.6 8
20.19 odd 2 10000.2.a.be.1.7 8
25.2 odd 20 625.2.e.j.124.5 32
25.3 odd 20 625.2.e.k.249.4 32
25.4 even 10 625.2.d.n.376.3 16
25.6 even 5 625.2.d.p.251.2 16
25.8 odd 20 625.2.e.k.374.5 32
25.9 even 10 625.2.d.m.126.2 16
25.11 even 5 625.2.d.q.501.3 16
25.12 odd 20 625.2.e.j.499.4 32
25.13 odd 20 625.2.e.j.499.5 32
25.14 even 10 625.2.d.m.501.2 16
25.16 even 5 625.2.d.q.126.3 16
25.17 odd 20 625.2.e.k.374.4 32
25.19 even 10 625.2.d.n.251.3 16
25.21 even 5 625.2.d.p.376.2 16
25.22 odd 20 625.2.e.k.249.5 32
25.23 odd 20 625.2.e.j.124.4 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
625.2.a.e.1.6 8 1.1 even 1 trivial
625.2.a.g.1.3 yes 8 5.4 even 2
625.2.b.d.624.8 16 5.3 odd 4
625.2.b.d.624.9 16 5.2 odd 4
625.2.d.m.126.2 16 25.9 even 10
625.2.d.m.501.2 16 25.14 even 10
625.2.d.n.251.3 16 25.19 even 10
625.2.d.n.376.3 16 25.4 even 10
625.2.d.p.251.2 16 25.6 even 5
625.2.d.p.376.2 16 25.21 even 5
625.2.d.q.126.3 16 25.16 even 5
625.2.d.q.501.3 16 25.11 even 5
625.2.e.j.124.4 32 25.23 odd 20
625.2.e.j.124.5 32 25.2 odd 20
625.2.e.j.499.4 32 25.12 odd 20
625.2.e.j.499.5 32 25.13 odd 20
625.2.e.k.249.4 32 25.3 odd 20
625.2.e.k.249.5 32 25.22 odd 20
625.2.e.k.374.4 32 25.17 odd 20
625.2.e.k.374.5 32 25.8 odd 20
5625.2.a.s.1.6 8 15.14 odd 2
5625.2.a.be.1.3 8 3.2 odd 2
10000.2.a.be.1.7 8 20.19 odd 2
10000.2.a.bn.1.2 8 4.3 odd 2