Properties

Label 7569.2.a.bl.1.4
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $9$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.15439\) of defining polynomial
Character \(\chi\) \(=\) 7569.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} +O(q^{10})\) \(q-1.15439 q^{2} -0.667390 q^{4} +3.44957 q^{5} +4.51754 q^{7} +3.07920 q^{8} -3.98215 q^{10} +0.554210 q^{11} +2.75998 q^{13} -5.21499 q^{14} -2.21981 q^{16} +2.99688 q^{17} -0.278196 q^{19} -2.30221 q^{20} -0.639773 q^{22} -0.927891 q^{23} +6.89956 q^{25} -3.18609 q^{26} -3.01496 q^{28} +9.12071 q^{31} -3.59588 q^{32} -3.45957 q^{34} +15.5836 q^{35} -6.87516 q^{37} +0.321146 q^{38} +10.6219 q^{40} +2.85317 q^{41} -11.2189 q^{43} -0.369874 q^{44} +1.07115 q^{46} -4.19552 q^{47} +13.4081 q^{49} -7.96477 q^{50} -1.84198 q^{52} +2.07886 q^{53} +1.91179 q^{55} +13.9104 q^{56} +14.2670 q^{59} -7.23144 q^{61} -10.5288 q^{62} +8.59066 q^{64} +9.52076 q^{65} -12.2857 q^{67} -2.00009 q^{68} -17.9895 q^{70} +2.38207 q^{71} -0.913120 q^{73} +7.93660 q^{74} +0.185665 q^{76} +2.50366 q^{77} +2.06809 q^{79} -7.65740 q^{80} -3.29367 q^{82} +10.9704 q^{83} +10.3380 q^{85} +12.9510 q^{86} +1.70652 q^{88} +6.80692 q^{89} +12.4683 q^{91} +0.619265 q^{92} +4.84325 q^{94} -0.959659 q^{95} -3.51641 q^{97} -15.4782 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.15439 −0.816275 −0.408138 0.912920i \(-0.633822\pi\)
−0.408138 + 0.912920i \(0.633822\pi\)
\(3\) 0 0
\(4\) −0.667390 −0.333695
\(5\) 3.44957 1.54270 0.771348 0.636413i \(-0.219583\pi\)
0.771348 + 0.636413i \(0.219583\pi\)
\(6\) 0 0
\(7\) 4.51754 1.70747 0.853734 0.520710i \(-0.174333\pi\)
0.853734 + 0.520710i \(0.174333\pi\)
\(8\) 3.07920 1.08866
\(9\) 0 0
\(10\) −3.98215 −1.25926
\(11\) 0.554210 0.167101 0.0835503 0.996504i \(-0.473374\pi\)
0.0835503 + 0.996504i \(0.473374\pi\)
\(12\) 0 0
\(13\) 2.75998 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(14\) −5.21499 −1.39376
\(15\) 0 0
\(16\) −2.21981 −0.554953
\(17\) 2.99688 0.726851 0.363426 0.931623i \(-0.381607\pi\)
0.363426 + 0.931623i \(0.381607\pi\)
\(18\) 0 0
\(19\) −0.278196 −0.0638226 −0.0319113 0.999491i \(-0.510159\pi\)
−0.0319113 + 0.999491i \(0.510159\pi\)
\(20\) −2.30221 −0.514790
\(21\) 0 0
\(22\) −0.639773 −0.136400
\(23\) −0.927891 −0.193479 −0.0967393 0.995310i \(-0.530841\pi\)
−0.0967393 + 0.995310i \(0.530841\pi\)
\(24\) 0 0
\(25\) 6.89956 1.37991
\(26\) −3.18609 −0.624843
\(27\) 0 0
\(28\) −3.01496 −0.569773
\(29\) 0 0
\(30\) 0 0
\(31\) 9.12071 1.63813 0.819064 0.573702i \(-0.194493\pi\)
0.819064 + 0.573702i \(0.194493\pi\)
\(32\) −3.59588 −0.635668
\(33\) 0 0
\(34\) −3.45957 −0.593311
\(35\) 15.5836 2.63410
\(36\) 0 0
\(37\) −6.87516 −1.13027 −0.565135 0.824999i \(-0.691176\pi\)
−0.565135 + 0.824999i \(0.691176\pi\)
\(38\) 0.321146 0.0520968
\(39\) 0 0
\(40\) 10.6219 1.67948
\(41\) 2.85317 0.445591 0.222795 0.974865i \(-0.428482\pi\)
0.222795 + 0.974865i \(0.428482\pi\)
\(42\) 0 0
\(43\) −11.2189 −1.71087 −0.855433 0.517914i \(-0.826709\pi\)
−0.855433 + 0.517914i \(0.826709\pi\)
\(44\) −0.369874 −0.0557606
\(45\) 0 0
\(46\) 1.07115 0.157932
\(47\) −4.19552 −0.611979 −0.305989 0.952035i \(-0.598987\pi\)
−0.305989 + 0.952035i \(0.598987\pi\)
\(48\) 0 0
\(49\) 13.4081 1.91545
\(50\) −7.96477 −1.12639
\(51\) 0 0
\(52\) −1.84198 −0.255437
\(53\) 2.07886 0.285554 0.142777 0.989755i \(-0.454397\pi\)
0.142777 + 0.989755i \(0.454397\pi\)
\(54\) 0 0
\(55\) 1.91179 0.257785
\(56\) 13.9104 1.85886
\(57\) 0 0
\(58\) 0 0
\(59\) 14.2670 1.85741 0.928703 0.370824i \(-0.120925\pi\)
0.928703 + 0.370824i \(0.120925\pi\)
\(60\) 0 0
\(61\) −7.23144 −0.925892 −0.462946 0.886387i \(-0.653208\pi\)
−0.462946 + 0.886387i \(0.653208\pi\)
\(62\) −10.5288 −1.33716
\(63\) 0 0
\(64\) 8.59066 1.07383
\(65\) 9.52076 1.18091
\(66\) 0 0
\(67\) −12.2857 −1.50094 −0.750469 0.660905i \(-0.770172\pi\)
−0.750469 + 0.660905i \(0.770172\pi\)
\(68\) −2.00009 −0.242547
\(69\) 0 0
\(70\) −17.9895 −2.15015
\(71\) 2.38207 0.282699 0.141350 0.989960i \(-0.454856\pi\)
0.141350 + 0.989960i \(0.454856\pi\)
\(72\) 0 0
\(73\) −0.913120 −0.106873 −0.0534363 0.998571i \(-0.517017\pi\)
−0.0534363 + 0.998571i \(0.517017\pi\)
\(74\) 7.93660 0.922611
\(75\) 0 0
\(76\) 0.185665 0.0212973
\(77\) 2.50366 0.285319
\(78\) 0 0
\(79\) 2.06809 0.232678 0.116339 0.993210i \(-0.462884\pi\)
0.116339 + 0.993210i \(0.462884\pi\)
\(80\) −7.65740 −0.856124
\(81\) 0 0
\(82\) −3.29367 −0.363725
\(83\) 10.9704 1.20416 0.602080 0.798436i \(-0.294339\pi\)
0.602080 + 0.798436i \(0.294339\pi\)
\(84\) 0 0
\(85\) 10.3380 1.12131
\(86\) 12.9510 1.39654
\(87\) 0 0
\(88\) 1.70652 0.181916
\(89\) 6.80692 0.721532 0.360766 0.932656i \(-0.382515\pi\)
0.360766 + 0.932656i \(0.382515\pi\)
\(90\) 0 0
\(91\) 12.4683 1.30703
\(92\) 0.619265 0.0645628
\(93\) 0 0
\(94\) 4.84325 0.499543
\(95\) −0.959659 −0.0984590
\(96\) 0 0
\(97\) −3.51641 −0.357037 −0.178518 0.983937i \(-0.557130\pi\)
−0.178518 + 0.983937i \(0.557130\pi\)
\(98\) −15.4782 −1.56353
\(99\) 0 0
\(100\) −4.60470 −0.460470
\(101\) 12.6707 1.26078 0.630390 0.776279i \(-0.282895\pi\)
0.630390 + 0.776279i \(0.282895\pi\)
\(102\) 0 0
\(103\) 2.01754 0.198794 0.0993972 0.995048i \(-0.468309\pi\)
0.0993972 + 0.995048i \(0.468309\pi\)
\(104\) 8.49854 0.833350
\(105\) 0 0
\(106\) −2.39981 −0.233091
\(107\) 0.572045 0.0553017 0.0276508 0.999618i \(-0.491197\pi\)
0.0276508 + 0.999618i \(0.491197\pi\)
\(108\) 0 0
\(109\) 19.4684 1.86474 0.932368 0.361510i \(-0.117739\pi\)
0.932368 + 0.361510i \(0.117739\pi\)
\(110\) −2.20694 −0.210424
\(111\) 0 0
\(112\) −10.0281 −0.947564
\(113\) −2.43686 −0.229241 −0.114620 0.993409i \(-0.536565\pi\)
−0.114620 + 0.993409i \(0.536565\pi\)
\(114\) 0 0
\(115\) −3.20083 −0.298479
\(116\) 0 0
\(117\) 0 0
\(118\) −16.4697 −1.51615
\(119\) 13.5385 1.24108
\(120\) 0 0
\(121\) −10.6929 −0.972077
\(122\) 8.34789 0.755782
\(123\) 0 0
\(124\) −6.08707 −0.546635
\(125\) 6.55269 0.586090
\(126\) 0 0
\(127\) −9.18908 −0.815399 −0.407699 0.913116i \(-0.633669\pi\)
−0.407699 + 0.913116i \(0.633669\pi\)
\(128\) −2.72519 −0.240875
\(129\) 0 0
\(130\) −10.9906 −0.963944
\(131\) 13.0939 1.14402 0.572010 0.820247i \(-0.306164\pi\)
0.572010 + 0.820247i \(0.306164\pi\)
\(132\) 0 0
\(133\) −1.25676 −0.108975
\(134\) 14.1825 1.22518
\(135\) 0 0
\(136\) 9.22801 0.791295
\(137\) 7.60048 0.649353 0.324676 0.945825i \(-0.394745\pi\)
0.324676 + 0.945825i \(0.394745\pi\)
\(138\) 0 0
\(139\) 8.85308 0.750908 0.375454 0.926841i \(-0.377487\pi\)
0.375454 + 0.926841i \(0.377487\pi\)
\(140\) −10.4003 −0.878987
\(141\) 0 0
\(142\) −2.74983 −0.230760
\(143\) 1.52961 0.127912
\(144\) 0 0
\(145\) 0 0
\(146\) 1.05409 0.0872375
\(147\) 0 0
\(148\) 4.58841 0.377165
\(149\) −7.97006 −0.652932 −0.326466 0.945209i \(-0.605858\pi\)
−0.326466 + 0.945209i \(0.605858\pi\)
\(150\) 0 0
\(151\) −0.105522 −0.00858723 −0.00429361 0.999991i \(-0.501367\pi\)
−0.00429361 + 0.999991i \(0.501367\pi\)
\(152\) −0.856623 −0.0694813
\(153\) 0 0
\(154\) −2.89020 −0.232899
\(155\) 31.4626 2.52714
\(156\) 0 0
\(157\) −13.5037 −1.07771 −0.538857 0.842397i \(-0.681144\pi\)
−0.538857 + 0.842397i \(0.681144\pi\)
\(158\) −2.38738 −0.189930
\(159\) 0 0
\(160\) −12.4043 −0.980643
\(161\) −4.19178 −0.330358
\(162\) 0 0
\(163\) −2.61379 −0.204728 −0.102364 0.994747i \(-0.532641\pi\)
−0.102364 + 0.994747i \(0.532641\pi\)
\(164\) −1.90418 −0.148691
\(165\) 0 0
\(166\) −12.6641 −0.982926
\(167\) 10.2188 0.790752 0.395376 0.918519i \(-0.370614\pi\)
0.395376 + 0.918519i \(0.370614\pi\)
\(168\) 0 0
\(169\) −5.38250 −0.414038
\(170\) −11.9340 −0.915298
\(171\) 0 0
\(172\) 7.48738 0.570907
\(173\) −23.0919 −1.75564 −0.877822 0.478987i \(-0.841004\pi\)
−0.877822 + 0.478987i \(0.841004\pi\)
\(174\) 0 0
\(175\) 31.1690 2.35616
\(176\) −1.23024 −0.0927329
\(177\) 0 0
\(178\) −7.85783 −0.588969
\(179\) −14.7887 −1.10536 −0.552678 0.833395i \(-0.686394\pi\)
−0.552678 + 0.833395i \(0.686394\pi\)
\(180\) 0 0
\(181\) 2.88931 0.214761 0.107380 0.994218i \(-0.465754\pi\)
0.107380 + 0.994218i \(0.465754\pi\)
\(182\) −14.3933 −1.06690
\(183\) 0 0
\(184\) −2.85716 −0.210633
\(185\) −23.7164 −1.74366
\(186\) 0 0
\(187\) 1.66090 0.121457
\(188\) 2.80004 0.204214
\(189\) 0 0
\(190\) 1.10782 0.0803696
\(191\) 4.66376 0.337458 0.168729 0.985663i \(-0.446034\pi\)
0.168729 + 0.985663i \(0.446034\pi\)
\(192\) 0 0
\(193\) −14.5839 −1.04977 −0.524887 0.851172i \(-0.675892\pi\)
−0.524887 + 0.851172i \(0.675892\pi\)
\(194\) 4.05929 0.291440
\(195\) 0 0
\(196\) −8.94844 −0.639175
\(197\) −2.89152 −0.206012 −0.103006 0.994681i \(-0.532846\pi\)
−0.103006 + 0.994681i \(0.532846\pi\)
\(198\) 0 0
\(199\) 4.05076 0.287151 0.143575 0.989639i \(-0.454140\pi\)
0.143575 + 0.989639i \(0.454140\pi\)
\(200\) 21.2451 1.50226
\(201\) 0 0
\(202\) −14.6269 −1.02914
\(203\) 0 0
\(204\) 0 0
\(205\) 9.84224 0.687412
\(206\) −2.32903 −0.162271
\(207\) 0 0
\(208\) −6.12664 −0.424806
\(209\) −0.154179 −0.0106648
\(210\) 0 0
\(211\) −13.1686 −0.906565 −0.453283 0.891367i \(-0.649747\pi\)
−0.453283 + 0.891367i \(0.649747\pi\)
\(212\) −1.38741 −0.0952879
\(213\) 0 0
\(214\) −0.660361 −0.0451414
\(215\) −38.7004 −2.63935
\(216\) 0 0
\(217\) 41.2031 2.79705
\(218\) −22.4741 −1.52214
\(219\) 0 0
\(220\) −1.27591 −0.0860217
\(221\) 8.27135 0.556391
\(222\) 0 0
\(223\) −15.2764 −1.02299 −0.511493 0.859287i \(-0.670907\pi\)
−0.511493 + 0.859287i \(0.670907\pi\)
\(224\) −16.2445 −1.08538
\(225\) 0 0
\(226\) 2.81309 0.187124
\(227\) 14.7503 0.979014 0.489507 0.871999i \(-0.337177\pi\)
0.489507 + 0.871999i \(0.337177\pi\)
\(228\) 0 0
\(229\) 10.3737 0.685515 0.342758 0.939424i \(-0.388639\pi\)
0.342758 + 0.939424i \(0.388639\pi\)
\(230\) 3.69500 0.243641
\(231\) 0 0
\(232\) 0 0
\(233\) −0.280226 −0.0183582 −0.00917911 0.999958i \(-0.502922\pi\)
−0.00917911 + 0.999958i \(0.502922\pi\)
\(234\) 0 0
\(235\) −14.4727 −0.944098
\(236\) −9.52166 −0.619807
\(237\) 0 0
\(238\) −15.6287 −1.01306
\(239\) −6.43890 −0.416498 −0.208249 0.978076i \(-0.566776\pi\)
−0.208249 + 0.978076i \(0.566776\pi\)
\(240\) 0 0
\(241\) −15.4251 −0.993617 −0.496809 0.867860i \(-0.665495\pi\)
−0.496809 + 0.867860i \(0.665495\pi\)
\(242\) 12.3437 0.793483
\(243\) 0 0
\(244\) 4.82619 0.308965
\(245\) 46.2523 2.95495
\(246\) 0 0
\(247\) −0.767817 −0.0488550
\(248\) 28.0845 1.78337
\(249\) 0 0
\(250\) −7.56434 −0.478411
\(251\) 3.29145 0.207754 0.103877 0.994590i \(-0.466875\pi\)
0.103877 + 0.994590i \(0.466875\pi\)
\(252\) 0 0
\(253\) −0.514246 −0.0323304
\(254\) 10.6078 0.665590
\(255\) 0 0
\(256\) −14.0354 −0.877212
\(257\) 0.395643 0.0246796 0.0123398 0.999924i \(-0.496072\pi\)
0.0123398 + 0.999924i \(0.496072\pi\)
\(258\) 0 0
\(259\) −31.0588 −1.92990
\(260\) −6.35406 −0.394062
\(261\) 0 0
\(262\) −15.1154 −0.933835
\(263\) 8.14356 0.502153 0.251077 0.967967i \(-0.419215\pi\)
0.251077 + 0.967967i \(0.419215\pi\)
\(264\) 0 0
\(265\) 7.17120 0.440523
\(266\) 1.45079 0.0889537
\(267\) 0 0
\(268\) 8.19936 0.500856
\(269\) −8.97754 −0.547370 −0.273685 0.961819i \(-0.588243\pi\)
−0.273685 + 0.961819i \(0.588243\pi\)
\(270\) 0 0
\(271\) 5.03959 0.306133 0.153066 0.988216i \(-0.451085\pi\)
0.153066 + 0.988216i \(0.451085\pi\)
\(272\) −6.65252 −0.403368
\(273\) 0 0
\(274\) −8.77389 −0.530050
\(275\) 3.82381 0.230584
\(276\) 0 0
\(277\) 2.03513 0.122279 0.0611395 0.998129i \(-0.480527\pi\)
0.0611395 + 0.998129i \(0.480527\pi\)
\(278\) −10.2199 −0.612948
\(279\) 0 0
\(280\) 47.9850 2.86765
\(281\) −23.8243 −1.42124 −0.710618 0.703578i \(-0.751584\pi\)
−0.710618 + 0.703578i \(0.751584\pi\)
\(282\) 0 0
\(283\) −14.9858 −0.890816 −0.445408 0.895328i \(-0.646941\pi\)
−0.445408 + 0.895328i \(0.646941\pi\)
\(284\) −1.58977 −0.0943353
\(285\) 0 0
\(286\) −1.76576 −0.104412
\(287\) 12.8893 0.760832
\(288\) 0 0
\(289\) −8.01868 −0.471687
\(290\) 0 0
\(291\) 0 0
\(292\) 0.609407 0.0356629
\(293\) −30.3700 −1.77424 −0.887118 0.461542i \(-0.847296\pi\)
−0.887118 + 0.461542i \(0.847296\pi\)
\(294\) 0 0
\(295\) 49.2151 2.86541
\(296\) −21.1700 −1.23048
\(297\) 0 0
\(298\) 9.20053 0.532972
\(299\) −2.56096 −0.148104
\(300\) 0 0
\(301\) −50.6818 −2.92125
\(302\) 0.121813 0.00700954
\(303\) 0 0
\(304\) 0.617544 0.0354185
\(305\) −24.9454 −1.42837
\(306\) 0 0
\(307\) 2.10479 0.120127 0.0600635 0.998195i \(-0.480870\pi\)
0.0600635 + 0.998195i \(0.480870\pi\)
\(308\) −1.67092 −0.0952094
\(309\) 0 0
\(310\) −36.3200 −2.06284
\(311\) −16.8761 −0.956958 −0.478479 0.878099i \(-0.658812\pi\)
−0.478479 + 0.878099i \(0.658812\pi\)
\(312\) 0 0
\(313\) 14.8295 0.838213 0.419106 0.907937i \(-0.362343\pi\)
0.419106 + 0.907937i \(0.362343\pi\)
\(314\) 15.5885 0.879712
\(315\) 0 0
\(316\) −1.38022 −0.0776436
\(317\) 5.50623 0.309260 0.154630 0.987972i \(-0.450581\pi\)
0.154630 + 0.987972i \(0.450581\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 29.6341 1.65660
\(321\) 0 0
\(322\) 4.83894 0.269663
\(323\) −0.833723 −0.0463896
\(324\) 0 0
\(325\) 19.0427 1.05630
\(326\) 3.01733 0.167114
\(327\) 0 0
\(328\) 8.78550 0.485098
\(329\) −18.9534 −1.04493
\(330\) 0 0
\(331\) −5.88833 −0.323652 −0.161826 0.986819i \(-0.551738\pi\)
−0.161826 + 0.986819i \(0.551738\pi\)
\(332\) −7.32154 −0.401822
\(333\) 0 0
\(334\) −11.7964 −0.645471
\(335\) −42.3805 −2.31549
\(336\) 0 0
\(337\) 17.4297 0.949457 0.474729 0.880132i \(-0.342546\pi\)
0.474729 + 0.880132i \(0.342546\pi\)
\(338\) 6.21349 0.337969
\(339\) 0 0
\(340\) −6.89946 −0.374176
\(341\) 5.05479 0.273732
\(342\) 0 0
\(343\) 28.9489 1.56309
\(344\) −34.5452 −1.86255
\(345\) 0 0
\(346\) 26.6570 1.43309
\(347\) 2.81562 0.151150 0.0755751 0.997140i \(-0.475921\pi\)
0.0755751 + 0.997140i \(0.475921\pi\)
\(348\) 0 0
\(349\) 23.4949 1.25765 0.628826 0.777546i \(-0.283536\pi\)
0.628826 + 0.777546i \(0.283536\pi\)
\(350\) −35.9811 −1.92327
\(351\) 0 0
\(352\) −1.99287 −0.106220
\(353\) −16.1562 −0.859910 −0.429955 0.902850i \(-0.641471\pi\)
−0.429955 + 0.902850i \(0.641471\pi\)
\(354\) 0 0
\(355\) 8.21712 0.436119
\(356\) −4.54287 −0.240772
\(357\) 0 0
\(358\) 17.0718 0.902274
\(359\) 8.64024 0.456014 0.228007 0.973659i \(-0.426779\pi\)
0.228007 + 0.973659i \(0.426779\pi\)
\(360\) 0 0
\(361\) −18.9226 −0.995927
\(362\) −3.33538 −0.175304
\(363\) 0 0
\(364\) −8.32123 −0.436151
\(365\) −3.14988 −0.164872
\(366\) 0 0
\(367\) 13.4463 0.701892 0.350946 0.936396i \(-0.385860\pi\)
0.350946 + 0.936396i \(0.385860\pi\)
\(368\) 2.05974 0.107371
\(369\) 0 0
\(370\) 27.3779 1.42331
\(371\) 9.39134 0.487574
\(372\) 0 0
\(373\) −3.19324 −0.165340 −0.0826698 0.996577i \(-0.526345\pi\)
−0.0826698 + 0.996577i \(0.526345\pi\)
\(374\) −1.91733 −0.0991425
\(375\) 0 0
\(376\) −12.9188 −0.666238
\(377\) 0 0
\(378\) 0 0
\(379\) −16.5019 −0.847646 −0.423823 0.905745i \(-0.639312\pi\)
−0.423823 + 0.905745i \(0.639312\pi\)
\(380\) 0.640467 0.0328553
\(381\) 0 0
\(382\) −5.38378 −0.275458
\(383\) −1.57818 −0.0806413 −0.0403206 0.999187i \(-0.512838\pi\)
−0.0403206 + 0.999187i \(0.512838\pi\)
\(384\) 0 0
\(385\) 8.63657 0.440160
\(386\) 16.8355 0.856904
\(387\) 0 0
\(388\) 2.34681 0.119141
\(389\) 14.0108 0.710375 0.355188 0.934795i \(-0.384417\pi\)
0.355188 + 0.934795i \(0.384417\pi\)
\(390\) 0 0
\(391\) −2.78078 −0.140630
\(392\) 41.2863 2.08527
\(393\) 0 0
\(394\) 3.33794 0.168163
\(395\) 7.13403 0.358952
\(396\) 0 0
\(397\) 23.9158 1.20030 0.600150 0.799887i \(-0.295107\pi\)
0.600150 + 0.799887i \(0.295107\pi\)
\(398\) −4.67615 −0.234394
\(399\) 0 0
\(400\) −15.3157 −0.765786
\(401\) 35.8795 1.79174 0.895869 0.444318i \(-0.146554\pi\)
0.895869 + 0.444318i \(0.146554\pi\)
\(402\) 0 0
\(403\) 25.1730 1.25396
\(404\) −8.45629 −0.420716
\(405\) 0 0
\(406\) 0 0
\(407\) −3.81028 −0.188869
\(408\) 0 0
\(409\) 15.0182 0.742600 0.371300 0.928513i \(-0.378912\pi\)
0.371300 + 0.928513i \(0.378912\pi\)
\(410\) −11.3618 −0.561117
\(411\) 0 0
\(412\) −1.34649 −0.0663367
\(413\) 64.4517 3.17146
\(414\) 0 0
\(415\) 37.8433 1.85765
\(416\) −9.92456 −0.486592
\(417\) 0 0
\(418\) 0.177983 0.00870541
\(419\) −36.9542 −1.80533 −0.902666 0.430343i \(-0.858393\pi\)
−0.902666 + 0.430343i \(0.858393\pi\)
\(420\) 0 0
\(421\) 1.28657 0.0627036 0.0313518 0.999508i \(-0.490019\pi\)
0.0313518 + 0.999508i \(0.490019\pi\)
\(422\) 15.2017 0.740007
\(423\) 0 0
\(424\) 6.40124 0.310872
\(425\) 20.6772 1.00299
\(426\) 0 0
\(427\) −32.6683 −1.58093
\(428\) −0.381777 −0.0184539
\(429\) 0 0
\(430\) 44.6753 2.15443
\(431\) 25.2826 1.21782 0.608910 0.793239i \(-0.291607\pi\)
0.608910 + 0.793239i \(0.291607\pi\)
\(432\) 0 0
\(433\) −10.3588 −0.497812 −0.248906 0.968528i \(-0.580071\pi\)
−0.248906 + 0.968528i \(0.580071\pi\)
\(434\) −47.5644 −2.28316
\(435\) 0 0
\(436\) −12.9930 −0.622253
\(437\) 0.258136 0.0123483
\(438\) 0 0
\(439\) −13.3949 −0.639306 −0.319653 0.947535i \(-0.603566\pi\)
−0.319653 + 0.947535i \(0.603566\pi\)
\(440\) 5.88678 0.280641
\(441\) 0 0
\(442\) −9.54834 −0.454168
\(443\) −30.7933 −1.46303 −0.731517 0.681823i \(-0.761187\pi\)
−0.731517 + 0.681823i \(0.761187\pi\)
\(444\) 0 0
\(445\) 23.4810 1.11311
\(446\) 17.6349 0.835038
\(447\) 0 0
\(448\) 38.8086 1.83353
\(449\) −6.14934 −0.290205 −0.145103 0.989417i \(-0.546351\pi\)
−0.145103 + 0.989417i \(0.546351\pi\)
\(450\) 0 0
\(451\) 1.58126 0.0744585
\(452\) 1.62634 0.0764965
\(453\) 0 0
\(454\) −17.0276 −0.799145
\(455\) 43.0104 2.01636
\(456\) 0 0
\(457\) −17.0452 −0.797341 −0.398670 0.917094i \(-0.630528\pi\)
−0.398670 + 0.917094i \(0.630528\pi\)
\(458\) −11.9753 −0.559569
\(459\) 0 0
\(460\) 2.13620 0.0996008
\(461\) 16.5645 0.771484 0.385742 0.922607i \(-0.373946\pi\)
0.385742 + 0.922607i \(0.373946\pi\)
\(462\) 0 0
\(463\) 28.9072 1.34343 0.671715 0.740809i \(-0.265558\pi\)
0.671715 + 0.740809i \(0.265558\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0.323489 0.0149854
\(467\) −22.6915 −1.05004 −0.525019 0.851090i \(-0.675942\pi\)
−0.525019 + 0.851090i \(0.675942\pi\)
\(468\) 0 0
\(469\) −55.5011 −2.56280
\(470\) 16.7072 0.770644
\(471\) 0 0
\(472\) 43.9310 2.02209
\(473\) −6.21762 −0.285887
\(474\) 0 0
\(475\) −1.91943 −0.0880697
\(476\) −9.03548 −0.414140
\(477\) 0 0
\(478\) 7.43299 0.339977
\(479\) −6.52355 −0.298068 −0.149034 0.988832i \(-0.547616\pi\)
−0.149034 + 0.988832i \(0.547616\pi\)
\(480\) 0 0
\(481\) −18.9753 −0.865200
\(482\) 17.8065 0.811065
\(483\) 0 0
\(484\) 7.13630 0.324377
\(485\) −12.1301 −0.550800
\(486\) 0 0
\(487\) −23.4649 −1.06329 −0.531647 0.846966i \(-0.678427\pi\)
−0.531647 + 0.846966i \(0.678427\pi\)
\(488\) −22.2671 −1.00798
\(489\) 0 0
\(490\) −53.3931 −2.41205
\(491\) 29.9858 1.35324 0.676619 0.736333i \(-0.263444\pi\)
0.676619 + 0.736333i \(0.263444\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0.886358 0.0398791
\(495\) 0 0
\(496\) −20.2463 −0.909084
\(497\) 10.7611 0.482700
\(498\) 0 0
\(499\) 15.5126 0.694441 0.347220 0.937784i \(-0.387126\pi\)
0.347220 + 0.937784i \(0.387126\pi\)
\(500\) −4.37320 −0.195575
\(501\) 0 0
\(502\) −3.79960 −0.169585
\(503\) 14.6872 0.654871 0.327435 0.944874i \(-0.393816\pi\)
0.327435 + 0.944874i \(0.393816\pi\)
\(504\) 0 0
\(505\) 43.7085 1.94500
\(506\) 0.593639 0.0263905
\(507\) 0 0
\(508\) 6.13270 0.272094
\(509\) −32.6885 −1.44889 −0.724445 0.689332i \(-0.757904\pi\)
−0.724445 + 0.689332i \(0.757904\pi\)
\(510\) 0 0
\(511\) −4.12505 −0.182482
\(512\) 21.6527 0.956922
\(513\) 0 0
\(514\) −0.456726 −0.0201453
\(515\) 6.95967 0.306679
\(516\) 0 0
\(517\) −2.32520 −0.102262
\(518\) 35.8538 1.57533
\(519\) 0 0
\(520\) 29.3163 1.28561
\(521\) 33.5048 1.46787 0.733937 0.679218i \(-0.237681\pi\)
0.733937 + 0.679218i \(0.237681\pi\)
\(522\) 0 0
\(523\) −23.3879 −1.02268 −0.511341 0.859378i \(-0.670851\pi\)
−0.511341 + 0.859378i \(0.670851\pi\)
\(524\) −8.73874 −0.381754
\(525\) 0 0
\(526\) −9.40082 −0.409895
\(527\) 27.3337 1.19068
\(528\) 0 0
\(529\) −22.1390 −0.962566
\(530\) −8.27834 −0.359588
\(531\) 0 0
\(532\) 0.838750 0.0363644
\(533\) 7.87471 0.341092
\(534\) 0 0
\(535\) 1.97331 0.0853137
\(536\) −37.8302 −1.63401
\(537\) 0 0
\(538\) 10.3636 0.446805
\(539\) 7.43091 0.320072
\(540\) 0 0
\(541\) −14.6065 −0.627981 −0.313990 0.949426i \(-0.601666\pi\)
−0.313990 + 0.949426i \(0.601666\pi\)
\(542\) −5.81763 −0.249889
\(543\) 0 0
\(544\) −10.7764 −0.462036
\(545\) 67.1578 2.87672
\(546\) 0 0
\(547\) 12.3265 0.527044 0.263522 0.964653i \(-0.415116\pi\)
0.263522 + 0.964653i \(0.415116\pi\)
\(548\) −5.07248 −0.216686
\(549\) 0 0
\(550\) −4.41415 −0.188220
\(551\) 0 0
\(552\) 0 0
\(553\) 9.34267 0.397291
\(554\) −2.34933 −0.0998134
\(555\) 0 0
\(556\) −5.90845 −0.250574
\(557\) 24.8902 1.05463 0.527316 0.849669i \(-0.323199\pi\)
0.527316 + 0.849669i \(0.323199\pi\)
\(558\) 0 0
\(559\) −30.9640 −1.30964
\(560\) −34.5926 −1.46180
\(561\) 0 0
\(562\) 27.5024 1.16012
\(563\) −16.2217 −0.683662 −0.341831 0.939762i \(-0.611047\pi\)
−0.341831 + 0.939762i \(0.611047\pi\)
\(564\) 0 0
\(565\) −8.40615 −0.353649
\(566\) 17.2995 0.727151
\(567\) 0 0
\(568\) 7.33486 0.307764
\(569\) 44.8298 1.87936 0.939681 0.342052i \(-0.111122\pi\)
0.939681 + 0.342052i \(0.111122\pi\)
\(570\) 0 0
\(571\) −16.9942 −0.711183 −0.355592 0.934641i \(-0.615721\pi\)
−0.355592 + 0.934641i \(0.615721\pi\)
\(572\) −1.02085 −0.0426837
\(573\) 0 0
\(574\) −14.8793 −0.621048
\(575\) −6.40204 −0.266984
\(576\) 0 0
\(577\) −5.54668 −0.230911 −0.115456 0.993313i \(-0.536833\pi\)
−0.115456 + 0.993313i \(0.536833\pi\)
\(578\) 9.25667 0.385027
\(579\) 0 0
\(580\) 0 0
\(581\) 49.5592 2.05606
\(582\) 0 0
\(583\) 1.15213 0.0477162
\(584\) −2.81168 −0.116348
\(585\) 0 0
\(586\) 35.0588 1.44827
\(587\) 19.1436 0.790140 0.395070 0.918651i \(-0.370720\pi\)
0.395070 + 0.918651i \(0.370720\pi\)
\(588\) 0 0
\(589\) −2.53735 −0.104550
\(590\) −56.8133 −2.33897
\(591\) 0 0
\(592\) 15.2616 0.627246
\(593\) −31.7165 −1.30244 −0.651220 0.758889i \(-0.725742\pi\)
−0.651220 + 0.758889i \(0.725742\pi\)
\(594\) 0 0
\(595\) 46.7022 1.91460
\(596\) 5.31913 0.217880
\(597\) 0 0
\(598\) 2.95634 0.120894
\(599\) 47.1580 1.92683 0.963413 0.268023i \(-0.0863703\pi\)
0.963413 + 0.268023i \(0.0863703\pi\)
\(600\) 0 0
\(601\) 14.8050 0.603907 0.301954 0.953323i \(-0.402361\pi\)
0.301954 + 0.953323i \(0.402361\pi\)
\(602\) 58.5064 2.38454
\(603\) 0 0
\(604\) 0.0704241 0.00286551
\(605\) −36.8858 −1.49962
\(606\) 0 0
\(607\) 11.7032 0.475017 0.237508 0.971385i \(-0.423669\pi\)
0.237508 + 0.971385i \(0.423669\pi\)
\(608\) 1.00036 0.0405700
\(609\) 0 0
\(610\) 28.7967 1.16594
\(611\) −11.5795 −0.468458
\(612\) 0 0
\(613\) −45.9483 −1.85583 −0.927917 0.372787i \(-0.878402\pi\)
−0.927917 + 0.372787i \(0.878402\pi\)
\(614\) −2.42975 −0.0980566
\(615\) 0 0
\(616\) 7.70928 0.310616
\(617\) 41.1355 1.65605 0.828026 0.560689i \(-0.189464\pi\)
0.828026 + 0.560689i \(0.189464\pi\)
\(618\) 0 0
\(619\) −2.97032 −0.119387 −0.0596936 0.998217i \(-0.519012\pi\)
−0.0596936 + 0.998217i \(0.519012\pi\)
\(620\) −20.9978 −0.843292
\(621\) 0 0
\(622\) 19.4816 0.781141
\(623\) 30.7505 1.23199
\(624\) 0 0
\(625\) −11.8938 −0.475753
\(626\) −17.1190 −0.684212
\(627\) 0 0
\(628\) 9.01225 0.359628
\(629\) −20.6041 −0.821538
\(630\) 0 0
\(631\) 21.5126 0.856403 0.428202 0.903683i \(-0.359147\pi\)
0.428202 + 0.903683i \(0.359147\pi\)
\(632\) 6.36807 0.253308
\(633\) 0 0
\(634\) −6.35632 −0.252442
\(635\) −31.6984 −1.25791
\(636\) 0 0
\(637\) 37.0062 1.46624
\(638\) 0 0
\(639\) 0 0
\(640\) −9.40075 −0.371597
\(641\) 32.2082 1.27215 0.636075 0.771628i \(-0.280557\pi\)
0.636075 + 0.771628i \(0.280557\pi\)
\(642\) 0 0
\(643\) 39.0467 1.53985 0.769926 0.638133i \(-0.220293\pi\)
0.769926 + 0.638133i \(0.220293\pi\)
\(644\) 2.79755 0.110239
\(645\) 0 0
\(646\) 0.962439 0.0378666
\(647\) 4.12296 0.162090 0.0810452 0.996710i \(-0.474174\pi\)
0.0810452 + 0.996710i \(0.474174\pi\)
\(648\) 0 0
\(649\) 7.90692 0.310374
\(650\) −21.9826 −0.862229
\(651\) 0 0
\(652\) 1.74442 0.0683167
\(653\) 15.2804 0.597966 0.298983 0.954258i \(-0.403353\pi\)
0.298983 + 0.954258i \(0.403353\pi\)
\(654\) 0 0
\(655\) 45.1684 1.76488
\(656\) −6.33351 −0.247282
\(657\) 0 0
\(658\) 21.8796 0.852954
\(659\) 34.5146 1.34450 0.672249 0.740326i \(-0.265329\pi\)
0.672249 + 0.740326i \(0.265329\pi\)
\(660\) 0 0
\(661\) 11.9611 0.465233 0.232616 0.972569i \(-0.425271\pi\)
0.232616 + 0.972569i \(0.425271\pi\)
\(662\) 6.79741 0.264189
\(663\) 0 0
\(664\) 33.7801 1.31092
\(665\) −4.33529 −0.168116
\(666\) 0 0
\(667\) 0 0
\(668\) −6.81990 −0.263870
\(669\) 0 0
\(670\) 48.9235 1.89008
\(671\) −4.00774 −0.154717
\(672\) 0 0
\(673\) −17.6881 −0.681825 −0.340912 0.940095i \(-0.610736\pi\)
−0.340912 + 0.940095i \(0.610736\pi\)
\(674\) −20.1206 −0.775018
\(675\) 0 0
\(676\) 3.59223 0.138163
\(677\) −29.7747 −1.14433 −0.572166 0.820138i \(-0.693897\pi\)
−0.572166 + 0.820138i \(0.693897\pi\)
\(678\) 0 0
\(679\) −15.8855 −0.609629
\(680\) 31.8327 1.22073
\(681\) 0 0
\(682\) −5.83518 −0.223441
\(683\) 16.9703 0.649349 0.324675 0.945826i \(-0.394745\pi\)
0.324675 + 0.945826i \(0.394745\pi\)
\(684\) 0 0
\(685\) 26.2184 1.00175
\(686\) −33.4183 −1.27592
\(687\) 0 0
\(688\) 24.9038 0.949450
\(689\) 5.73763 0.218586
\(690\) 0 0
\(691\) −24.6905 −0.939271 −0.469636 0.882860i \(-0.655615\pi\)
−0.469636 + 0.882860i \(0.655615\pi\)
\(692\) 15.4113 0.585849
\(693\) 0 0
\(694\) −3.25031 −0.123380
\(695\) 30.5394 1.15842
\(696\) 0 0
\(697\) 8.55063 0.323878
\(698\) −27.1222 −1.02659
\(699\) 0 0
\(700\) −20.8019 −0.786238
\(701\) −25.2940 −0.955340 −0.477670 0.878539i \(-0.658519\pi\)
−0.477670 + 0.878539i \(0.658519\pi\)
\(702\) 0 0
\(703\) 1.91264 0.0721367
\(704\) 4.76103 0.179438
\(705\) 0 0
\(706\) 18.6506 0.701923
\(707\) 57.2403 2.15274
\(708\) 0 0
\(709\) 1.99466 0.0749111 0.0374555 0.999298i \(-0.488075\pi\)
0.0374555 + 0.999298i \(0.488075\pi\)
\(710\) −9.48573 −0.355993
\(711\) 0 0
\(712\) 20.9599 0.785505
\(713\) −8.46302 −0.316943
\(714\) 0 0
\(715\) 5.27650 0.197330
\(716\) 9.86979 0.368852
\(717\) 0 0
\(718\) −9.97418 −0.372233
\(719\) −6.32608 −0.235923 −0.117961 0.993018i \(-0.537636\pi\)
−0.117961 + 0.993018i \(0.537636\pi\)
\(720\) 0 0
\(721\) 9.11432 0.339435
\(722\) 21.8440 0.812950
\(723\) 0 0
\(724\) −1.92830 −0.0716646
\(725\) 0 0
\(726\) 0 0
\(727\) 18.1899 0.674626 0.337313 0.941393i \(-0.390482\pi\)
0.337313 + 0.941393i \(0.390482\pi\)
\(728\) 38.3925 1.42292
\(729\) 0 0
\(730\) 3.63618 0.134581
\(731\) −33.6217 −1.24354
\(732\) 0 0
\(733\) −36.2229 −1.33792 −0.668962 0.743296i \(-0.733261\pi\)
−0.668962 + 0.743296i \(0.733261\pi\)
\(734\) −15.5223 −0.572937
\(735\) 0 0
\(736\) 3.33658 0.122988
\(737\) −6.80886 −0.250808
\(738\) 0 0
\(739\) −4.16599 −0.153248 −0.0766242 0.997060i \(-0.524414\pi\)
−0.0766242 + 0.997060i \(0.524414\pi\)
\(740\) 15.8281 0.581851
\(741\) 0 0
\(742\) −10.8412 −0.397995
\(743\) −24.5985 −0.902431 −0.451216 0.892415i \(-0.649009\pi\)
−0.451216 + 0.892415i \(0.649009\pi\)
\(744\) 0 0
\(745\) −27.4933 −1.00728
\(746\) 3.68623 0.134963
\(747\) 0 0
\(748\) −1.10847 −0.0405297
\(749\) 2.58423 0.0944258
\(750\) 0 0
\(751\) 26.1808 0.955350 0.477675 0.878537i \(-0.341480\pi\)
0.477675 + 0.878537i \(0.341480\pi\)
\(752\) 9.31325 0.339619
\(753\) 0 0
\(754\) 0 0
\(755\) −0.364005 −0.0132475
\(756\) 0 0
\(757\) −52.2657 −1.89963 −0.949815 0.312814i \(-0.898729\pi\)
−0.949815 + 0.312814i \(0.898729\pi\)
\(758\) 19.0496 0.691912
\(759\) 0 0
\(760\) −2.95498 −0.107189
\(761\) −4.90272 −0.177724 −0.0888618 0.996044i \(-0.528323\pi\)
−0.0888618 + 0.996044i \(0.528323\pi\)
\(762\) 0 0
\(763\) 87.9493 3.18398
\(764\) −3.11254 −0.112608
\(765\) 0 0
\(766\) 1.82183 0.0658255
\(767\) 39.3767 1.42181
\(768\) 0 0
\(769\) 29.0468 1.04746 0.523728 0.851886i \(-0.324541\pi\)
0.523728 + 0.851886i \(0.324541\pi\)
\(770\) −9.96995 −0.359292
\(771\) 0 0
\(772\) 9.73316 0.350304
\(773\) −53.5664 −1.92665 −0.963325 0.268337i \(-0.913526\pi\)
−0.963325 + 0.268337i \(0.913526\pi\)
\(774\) 0 0
\(775\) 62.9289 2.26047
\(776\) −10.8277 −0.388692
\(777\) 0 0
\(778\) −16.1739 −0.579862
\(779\) −0.793743 −0.0284388
\(780\) 0 0
\(781\) 1.32016 0.0472392
\(782\) 3.21010 0.114793
\(783\) 0 0
\(784\) −29.7635 −1.06298
\(785\) −46.5821 −1.66259
\(786\) 0 0
\(787\) −42.8418 −1.52715 −0.763573 0.645722i \(-0.776557\pi\)
−0.763573 + 0.645722i \(0.776557\pi\)
\(788\) 1.92977 0.0687453
\(789\) 0 0
\(790\) −8.23544 −0.293004
\(791\) −11.0086 −0.391422
\(792\) 0 0
\(793\) −19.9587 −0.708753
\(794\) −27.6081 −0.979775
\(795\) 0 0
\(796\) −2.70344 −0.0958208
\(797\) −24.6827 −0.874307 −0.437154 0.899387i \(-0.644013\pi\)
−0.437154 + 0.899387i \(0.644013\pi\)
\(798\) 0 0
\(799\) −12.5735 −0.444818
\(800\) −24.8100 −0.877166
\(801\) 0 0
\(802\) −41.4189 −1.46255
\(803\) −0.506060 −0.0178585
\(804\) 0 0
\(805\) −14.4599 −0.509643
\(806\) −29.0594 −1.02357
\(807\) 0 0
\(808\) 39.0156 1.37256
\(809\) −30.2713 −1.06428 −0.532141 0.846656i \(-0.678612\pi\)
−0.532141 + 0.846656i \(0.678612\pi\)
\(810\) 0 0
\(811\) 33.6697 1.18230 0.591151 0.806561i \(-0.298674\pi\)
0.591151 + 0.806561i \(0.298674\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 4.39854 0.154169
\(815\) −9.01647 −0.315833
\(816\) 0 0
\(817\) 3.12106 0.109192
\(818\) −17.3368 −0.606166
\(819\) 0 0
\(820\) −6.56861 −0.229386
\(821\) 39.4748 1.37768 0.688840 0.724913i \(-0.258120\pi\)
0.688840 + 0.724913i \(0.258120\pi\)
\(822\) 0 0
\(823\) 30.7129 1.07058 0.535292 0.844667i \(-0.320202\pi\)
0.535292 + 0.844667i \(0.320202\pi\)
\(824\) 6.21242 0.216420
\(825\) 0 0
\(826\) −74.4022 −2.58879
\(827\) −50.6699 −1.76196 −0.880982 0.473150i \(-0.843117\pi\)
−0.880982 + 0.473150i \(0.843117\pi\)
\(828\) 0 0
\(829\) 17.7320 0.615858 0.307929 0.951409i \(-0.400364\pi\)
0.307929 + 0.951409i \(0.400364\pi\)
\(830\) −43.6858 −1.51636
\(831\) 0 0
\(832\) 23.7101 0.821999
\(833\) 40.1826 1.39224
\(834\) 0 0
\(835\) 35.2504 1.21989
\(836\) 0.102898 0.00355879
\(837\) 0 0
\(838\) 42.6595 1.47365
\(839\) −23.4750 −0.810447 −0.405223 0.914218i \(-0.632806\pi\)
−0.405223 + 0.914218i \(0.632806\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −1.48520 −0.0511834
\(843\) 0 0
\(844\) 8.78860 0.302516
\(845\) −18.5673 −0.638736
\(846\) 0 0
\(847\) −48.3053 −1.65979
\(848\) −4.61469 −0.158469
\(849\) 0 0
\(850\) −23.8695 −0.818717
\(851\) 6.37939 0.218683
\(852\) 0 0
\(853\) −24.8185 −0.849771 −0.424885 0.905247i \(-0.639686\pi\)
−0.424885 + 0.905247i \(0.639686\pi\)
\(854\) 37.7119 1.29047
\(855\) 0 0
\(856\) 1.76144 0.0602048
\(857\) 42.3415 1.44636 0.723180 0.690660i \(-0.242680\pi\)
0.723180 + 0.690660i \(0.242680\pi\)
\(858\) 0 0
\(859\) 22.2450 0.758990 0.379495 0.925194i \(-0.376098\pi\)
0.379495 + 0.925194i \(0.376098\pi\)
\(860\) 25.8283 0.880737
\(861\) 0 0
\(862\) −29.1859 −0.994076
\(863\) −36.5854 −1.24538 −0.622691 0.782468i \(-0.713961\pi\)
−0.622691 + 0.782468i \(0.713961\pi\)
\(864\) 0 0
\(865\) −79.6572 −2.70843
\(866\) 11.9581 0.406352
\(867\) 0 0
\(868\) −27.4986 −0.933362
\(869\) 1.14616 0.0388807
\(870\) 0 0
\(871\) −33.9083 −1.14894
\(872\) 59.9472 2.03007
\(873\) 0 0
\(874\) −0.297989 −0.0100796
\(875\) 29.6020 1.00073
\(876\) 0 0
\(877\) 36.0234 1.21642 0.608211 0.793775i \(-0.291887\pi\)
0.608211 + 0.793775i \(0.291887\pi\)
\(878\) 15.4630 0.521849
\(879\) 0 0
\(880\) −4.24381 −0.143059
\(881\) 20.8449 0.702281 0.351141 0.936323i \(-0.385794\pi\)
0.351141 + 0.936323i \(0.385794\pi\)
\(882\) 0 0
\(883\) 27.9120 0.939314 0.469657 0.882849i \(-0.344378\pi\)
0.469657 + 0.882849i \(0.344378\pi\)
\(884\) −5.52021 −0.185665
\(885\) 0 0
\(886\) 35.5474 1.19424
\(887\) 54.6365 1.83451 0.917257 0.398296i \(-0.130398\pi\)
0.917257 + 0.398296i \(0.130398\pi\)
\(888\) 0 0
\(889\) −41.5120 −1.39227
\(890\) −27.1062 −0.908600
\(891\) 0 0
\(892\) 10.1953 0.341365
\(893\) 1.16718 0.0390581
\(894\) 0 0
\(895\) −51.0146 −1.70523
\(896\) −12.3111 −0.411286
\(897\) 0 0
\(898\) 7.09872 0.236887
\(899\) 0 0
\(900\) 0 0
\(901\) 6.23012 0.207555
\(902\) −1.82538 −0.0607786
\(903\) 0 0
\(904\) −7.50360 −0.249566
\(905\) 9.96689 0.331311
\(906\) 0 0
\(907\) 16.1102 0.534931 0.267465 0.963567i \(-0.413814\pi\)
0.267465 + 0.963567i \(0.413814\pi\)
\(908\) −9.84422 −0.326692
\(909\) 0 0
\(910\) −49.6506 −1.64590
\(911\) −42.9222 −1.42207 −0.711037 0.703154i \(-0.751774\pi\)
−0.711037 + 0.703154i \(0.751774\pi\)
\(912\) 0 0
\(913\) 6.07991 0.201216
\(914\) 19.6768 0.650849
\(915\) 0 0
\(916\) −6.92332 −0.228753
\(917\) 59.1522 1.95338
\(918\) 0 0
\(919\) −19.3983 −0.639892 −0.319946 0.947436i \(-0.603665\pi\)
−0.319946 + 0.947436i \(0.603665\pi\)
\(920\) −9.85599 −0.324942
\(921\) 0 0
\(922\) −19.1218 −0.629743
\(923\) 6.57446 0.216401
\(924\) 0 0
\(925\) −47.4356 −1.55967
\(926\) −33.3701 −1.09661
\(927\) 0 0
\(928\) 0 0
\(929\) −15.9251 −0.522484 −0.261242 0.965273i \(-0.584132\pi\)
−0.261242 + 0.965273i \(0.584132\pi\)
\(930\) 0 0
\(931\) −3.73009 −0.122249
\(932\) 0.187020 0.00612604
\(933\) 0 0
\(934\) 26.1948 0.857120
\(935\) 5.72941 0.187372
\(936\) 0 0
\(937\) −40.1366 −1.31121 −0.655603 0.755105i \(-0.727586\pi\)
−0.655603 + 0.755105i \(0.727586\pi\)
\(938\) 64.0698 2.09195
\(939\) 0 0
\(940\) 9.65896 0.315041
\(941\) −2.46839 −0.0804673 −0.0402337 0.999190i \(-0.512810\pi\)
−0.0402337 + 0.999190i \(0.512810\pi\)
\(942\) 0 0
\(943\) −2.64743 −0.0862123
\(944\) −31.6701 −1.03077
\(945\) 0 0
\(946\) 7.17755 0.233362
\(947\) −26.3388 −0.855896 −0.427948 0.903803i \(-0.640763\pi\)
−0.427948 + 0.903803i \(0.640763\pi\)
\(948\) 0 0
\(949\) −2.52020 −0.0818090
\(950\) 2.21577 0.0718891
\(951\) 0 0
\(952\) 41.6879 1.35111
\(953\) −47.1256 −1.52655 −0.763275 0.646074i \(-0.776410\pi\)
−0.763275 + 0.646074i \(0.776410\pi\)
\(954\) 0 0
\(955\) 16.0880 0.520595
\(956\) 4.29726 0.138983
\(957\) 0 0
\(958\) 7.53070 0.243306
\(959\) 34.3354 1.10875
\(960\) 0 0
\(961\) 52.1874 1.68346
\(962\) 21.9049 0.706241
\(963\) 0 0
\(964\) 10.2945 0.331565
\(965\) −50.3083 −1.61948
\(966\) 0 0
\(967\) −28.5319 −0.917524 −0.458762 0.888559i \(-0.651707\pi\)
−0.458762 + 0.888559i \(0.651707\pi\)
\(968\) −32.9254 −1.05826
\(969\) 0 0
\(970\) 14.0028 0.449604
\(971\) −0.490076 −0.0157273 −0.00786364 0.999969i \(-0.502503\pi\)
−0.00786364 + 0.999969i \(0.502503\pi\)
\(972\) 0 0
\(973\) 39.9941 1.28215
\(974\) 27.0876 0.867941
\(975\) 0 0
\(976\) 16.0524 0.513826
\(977\) 18.1436 0.580465 0.290232 0.956956i \(-0.406267\pi\)
0.290232 + 0.956956i \(0.406267\pi\)
\(978\) 0 0
\(979\) 3.77246 0.120568
\(980\) −30.8683 −0.986053
\(981\) 0 0
\(982\) −34.6152 −1.10462
\(983\) 55.6548 1.77511 0.887556 0.460699i \(-0.152401\pi\)
0.887556 + 0.460699i \(0.152401\pi\)
\(984\) 0 0
\(985\) −9.97452 −0.317815
\(986\) 0 0
\(987\) 0 0
\(988\) 0.512433 0.0163027
\(989\) 10.4099 0.331016
\(990\) 0 0
\(991\) 20.2428 0.643034 0.321517 0.946904i \(-0.395807\pi\)
0.321517 + 0.946904i \(0.395807\pi\)
\(992\) −32.7970 −1.04131
\(993\) 0 0
\(994\) −12.4224 −0.394016
\(995\) 13.9734 0.442987
\(996\) 0 0
\(997\) 29.2318 0.925780 0.462890 0.886416i \(-0.346812\pi\)
0.462890 + 0.886416i \(0.346812\pi\)
\(998\) −17.9076 −0.566855
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7569.2.a.bl.1.4 9
3.2 odd 2 2523.2.a.p.1.6 9
29.7 even 7 261.2.k.b.136.2 18
29.25 even 7 261.2.k.b.190.2 18
29.28 even 2 7569.2.a.bk.1.6 9
87.65 odd 14 87.2.g.b.49.2 yes 18
87.83 odd 14 87.2.g.b.16.2 18
87.86 odd 2 2523.2.a.q.1.4 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.b.16.2 18 87.83 odd 14
87.2.g.b.49.2 yes 18 87.65 odd 14
261.2.k.b.136.2 18 29.7 even 7
261.2.k.b.190.2 18 29.25 even 7
2523.2.a.p.1.6 9 3.2 odd 2
2523.2.a.q.1.4 9 87.86 odd 2
7569.2.a.bk.1.6 9 29.28 even 2
7569.2.a.bl.1.4 9 1.1 even 1 trivial