Properties

Label 9075.2.a.dy.1.7
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 18 x^{8} - 220 x^{7} + 61 x^{6} + 330 x^{5} - 145 x^{4} - 148 x^{3} + 71 x^{2} + 10 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.298064\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.298064 q^{2} +1.00000 q^{3} -1.91116 q^{4} -0.298064 q^{6} +2.32107 q^{7} +1.16578 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.298064 q^{2} +1.00000 q^{3} -1.91116 q^{4} -0.298064 q^{6} +2.32107 q^{7} +1.16578 q^{8} +1.00000 q^{9} -1.91116 q^{12} -4.59081 q^{13} -0.691827 q^{14} +3.47484 q^{16} +5.14046 q^{17} -0.298064 q^{18} +5.69187 q^{19} +2.32107 q^{21} -6.02600 q^{23} +1.16578 q^{24} +1.36836 q^{26} +1.00000 q^{27} -4.43592 q^{28} -6.09737 q^{29} -8.43171 q^{31} -3.36728 q^{32} -1.53219 q^{34} -1.91116 q^{36} +3.17279 q^{37} -1.69654 q^{38} -4.59081 q^{39} +0.468592 q^{41} -0.691827 q^{42} -8.08805 q^{43} +1.79614 q^{46} -9.18736 q^{47} +3.47484 q^{48} -1.61265 q^{49} +5.14046 q^{51} +8.77376 q^{52} +4.25892 q^{53} -0.298064 q^{54} +2.70584 q^{56} +5.69187 q^{57} +1.81741 q^{58} +6.65971 q^{59} -1.82041 q^{61} +2.51319 q^{62} +2.32107 q^{63} -5.94601 q^{64} +8.86541 q^{67} -9.82424 q^{68} -6.02600 q^{69} -2.44868 q^{71} +1.16578 q^{72} -9.08299 q^{73} -0.945695 q^{74} -10.8781 q^{76} +1.36836 q^{78} -12.9262 q^{79} +1.00000 q^{81} -0.139670 q^{82} +8.34751 q^{83} -4.43592 q^{84} +2.41076 q^{86} -6.09737 q^{87} -3.73882 q^{89} -10.6556 q^{91} +11.5166 q^{92} -8.43171 q^{93} +2.73842 q^{94} -3.36728 q^{96} -5.58998 q^{97} +0.480673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{2} + 12 q^{3} + 12 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 12 q^{9} + 12 q^{12} - 18 q^{13} + 6 q^{14} + 24 q^{16} - 18 q^{17} - 4 q^{18} - 16 q^{19} - 8 q^{21} - 12 q^{24} + 16 q^{26} + 12 q^{27} - 30 q^{28} - 28 q^{32} + 6 q^{34} + 12 q^{36} - 28 q^{38} - 18 q^{39} + 6 q^{42} - 32 q^{43} - 28 q^{46} - 4 q^{47} + 24 q^{48} + 12 q^{49} - 18 q^{51} - 48 q^{52} - 12 q^{53} - 4 q^{54} + 6 q^{56} - 16 q^{57} - 10 q^{58} + 20 q^{59} - 20 q^{61} - 20 q^{62} - 8 q^{63} - 6 q^{64} + 10 q^{67} - 26 q^{68} + 32 q^{71} - 12 q^{72} - 26 q^{73} + 68 q^{74} - 34 q^{76} + 16 q^{78} - 32 q^{79} + 12 q^{81} - 62 q^{82} - 26 q^{83} - 30 q^{84} - 36 q^{86} - 10 q^{89} + 8 q^{92} + 2 q^{94} - 28 q^{96} - 22 q^{97} - 60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.298064 −0.210763 −0.105382 0.994432i \(-0.533606\pi\)
−0.105382 + 0.994432i \(0.533606\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.91116 −0.955579
\(5\) 0 0
\(6\) −0.298064 −0.121684
\(7\) 2.32107 0.877281 0.438640 0.898663i \(-0.355460\pi\)
0.438640 + 0.898663i \(0.355460\pi\)
\(8\) 1.16578 0.412164
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −1.91116 −0.551704
\(13\) −4.59081 −1.27326 −0.636631 0.771169i \(-0.719672\pi\)
−0.636631 + 0.771169i \(0.719672\pi\)
\(14\) −0.691827 −0.184899
\(15\) 0 0
\(16\) 3.47484 0.868710
\(17\) 5.14046 1.24675 0.623373 0.781925i \(-0.285762\pi\)
0.623373 + 0.781925i \(0.285762\pi\)
\(18\) −0.298064 −0.0702544
\(19\) 5.69187 1.30580 0.652902 0.757442i \(-0.273551\pi\)
0.652902 + 0.757442i \(0.273551\pi\)
\(20\) 0 0
\(21\) 2.32107 0.506498
\(22\) 0 0
\(23\) −6.02600 −1.25651 −0.628254 0.778008i \(-0.716230\pi\)
−0.628254 + 0.778008i \(0.716230\pi\)
\(24\) 1.16578 0.237963
\(25\) 0 0
\(26\) 1.36836 0.268357
\(27\) 1.00000 0.192450
\(28\) −4.43592 −0.838311
\(29\) −6.09737 −1.13225 −0.566127 0.824318i \(-0.691559\pi\)
−0.566127 + 0.824318i \(0.691559\pi\)
\(30\) 0 0
\(31\) −8.43171 −1.51438 −0.757190 0.653195i \(-0.773428\pi\)
−0.757190 + 0.653195i \(0.773428\pi\)
\(32\) −3.36728 −0.595256
\(33\) 0 0
\(34\) −1.53219 −0.262768
\(35\) 0 0
\(36\) −1.91116 −0.318526
\(37\) 3.17279 0.521604 0.260802 0.965392i \(-0.416013\pi\)
0.260802 + 0.965392i \(0.416013\pi\)
\(38\) −1.69654 −0.275215
\(39\) −4.59081 −0.735118
\(40\) 0 0
\(41\) 0.468592 0.0731817 0.0365909 0.999330i \(-0.488350\pi\)
0.0365909 + 0.999330i \(0.488350\pi\)
\(42\) −0.691827 −0.106751
\(43\) −8.08805 −1.23342 −0.616708 0.787192i \(-0.711534\pi\)
−0.616708 + 0.787192i \(0.711534\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.79614 0.264826
\(47\) −9.18736 −1.34011 −0.670057 0.742309i \(-0.733730\pi\)
−0.670057 + 0.742309i \(0.733730\pi\)
\(48\) 3.47484 0.501550
\(49\) −1.61265 −0.230378
\(50\) 0 0
\(51\) 5.14046 0.719809
\(52\) 8.77376 1.21670
\(53\) 4.25892 0.585007 0.292504 0.956264i \(-0.405512\pi\)
0.292504 + 0.956264i \(0.405512\pi\)
\(54\) −0.298064 −0.0405614
\(55\) 0 0
\(56\) 2.70584 0.361584
\(57\) 5.69187 0.753906
\(58\) 1.81741 0.238637
\(59\) 6.65971 0.867021 0.433510 0.901149i \(-0.357275\pi\)
0.433510 + 0.901149i \(0.357275\pi\)
\(60\) 0 0
\(61\) −1.82041 −0.233079 −0.116540 0.993186i \(-0.537180\pi\)
−0.116540 + 0.993186i \(0.537180\pi\)
\(62\) 2.51319 0.319175
\(63\) 2.32107 0.292427
\(64\) −5.94601 −0.743252
\(65\) 0 0
\(66\) 0 0
\(67\) 8.86541 1.08308 0.541541 0.840674i \(-0.317841\pi\)
0.541541 + 0.840674i \(0.317841\pi\)
\(68\) −9.82424 −1.19136
\(69\) −6.02600 −0.725445
\(70\) 0 0
\(71\) −2.44868 −0.290604 −0.145302 0.989387i \(-0.546415\pi\)
−0.145302 + 0.989387i \(0.546415\pi\)
\(72\) 1.16578 0.137388
\(73\) −9.08299 −1.06308 −0.531542 0.847032i \(-0.678387\pi\)
−0.531542 + 0.847032i \(0.678387\pi\)
\(74\) −0.945695 −0.109935
\(75\) 0 0
\(76\) −10.8781 −1.24780
\(77\) 0 0
\(78\) 1.36836 0.154936
\(79\) −12.9262 −1.45431 −0.727154 0.686474i \(-0.759157\pi\)
−0.727154 + 0.686474i \(0.759157\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.139670 −0.0154240
\(83\) 8.34751 0.916259 0.458129 0.888886i \(-0.348520\pi\)
0.458129 + 0.888886i \(0.348520\pi\)
\(84\) −4.43592 −0.483999
\(85\) 0 0
\(86\) 2.41076 0.259959
\(87\) −6.09737 −0.653707
\(88\) 0 0
\(89\) −3.73882 −0.396314 −0.198157 0.980170i \(-0.563496\pi\)
−0.198157 + 0.980170i \(0.563496\pi\)
\(90\) 0 0
\(91\) −10.6556 −1.11701
\(92\) 11.5166 1.20069
\(93\) −8.43171 −0.874327
\(94\) 2.73842 0.282447
\(95\) 0 0
\(96\) −3.36728 −0.343671
\(97\) −5.58998 −0.567576 −0.283788 0.958887i \(-0.591591\pi\)
−0.283788 + 0.958887i \(0.591591\pi\)
\(98\) 0.480673 0.0485553
\(99\) 0 0
\(100\) 0 0
\(101\) −1.61415 −0.160614 −0.0803069 0.996770i \(-0.525590\pi\)
−0.0803069 + 0.996770i \(0.525590\pi\)
\(102\) −1.53219 −0.151709
\(103\) 8.51841 0.839344 0.419672 0.907676i \(-0.362145\pi\)
0.419672 + 0.907676i \(0.362145\pi\)
\(104\) −5.35186 −0.524793
\(105\) 0 0
\(106\) −1.26943 −0.123298
\(107\) −6.11405 −0.591067 −0.295534 0.955332i \(-0.595497\pi\)
−0.295534 + 0.955332i \(0.595497\pi\)
\(108\) −1.91116 −0.183901
\(109\) 5.73259 0.549082 0.274541 0.961575i \(-0.411474\pi\)
0.274541 + 0.961575i \(0.411474\pi\)
\(110\) 0 0
\(111\) 3.17279 0.301148
\(112\) 8.06533 0.762102
\(113\) −8.24031 −0.775183 −0.387592 0.921831i \(-0.626693\pi\)
−0.387592 + 0.921831i \(0.626693\pi\)
\(114\) −1.69654 −0.158896
\(115\) 0 0
\(116\) 11.6530 1.08196
\(117\) −4.59081 −0.424420
\(118\) −1.98502 −0.182736
\(119\) 11.9314 1.09375
\(120\) 0 0
\(121\) 0 0
\(122\) 0.542598 0.0491245
\(123\) 0.468592 0.0422515
\(124\) 16.1143 1.44711
\(125\) 0 0
\(126\) −0.691827 −0.0616328
\(127\) 12.0783 1.07177 0.535887 0.844290i \(-0.319977\pi\)
0.535887 + 0.844290i \(0.319977\pi\)
\(128\) 8.50685 0.751906
\(129\) −8.08805 −0.712113
\(130\) 0 0
\(131\) 3.28131 0.286689 0.143345 0.989673i \(-0.454214\pi\)
0.143345 + 0.989673i \(0.454214\pi\)
\(132\) 0 0
\(133\) 13.2112 1.14556
\(134\) −2.64246 −0.228274
\(135\) 0 0
\(136\) 5.99263 0.513864
\(137\) 4.01740 0.343230 0.171615 0.985164i \(-0.445102\pi\)
0.171615 + 0.985164i \(0.445102\pi\)
\(138\) 1.79614 0.152897
\(139\) −1.77948 −0.150934 −0.0754670 0.997148i \(-0.524045\pi\)
−0.0754670 + 0.997148i \(0.524045\pi\)
\(140\) 0 0
\(141\) −9.18736 −0.773715
\(142\) 0.729862 0.0612487
\(143\) 0 0
\(144\) 3.47484 0.289570
\(145\) 0 0
\(146\) 2.70732 0.224059
\(147\) −1.61265 −0.133009
\(148\) −6.06370 −0.498433
\(149\) −2.20781 −0.180871 −0.0904354 0.995902i \(-0.528826\pi\)
−0.0904354 + 0.995902i \(0.528826\pi\)
\(150\) 0 0
\(151\) −12.8536 −1.04601 −0.523007 0.852329i \(-0.675190\pi\)
−0.523007 + 0.852329i \(0.675190\pi\)
\(152\) 6.63544 0.538205
\(153\) 5.14046 0.415582
\(154\) 0 0
\(155\) 0 0
\(156\) 8.77376 0.702463
\(157\) −0.839648 −0.0670112 −0.0335056 0.999439i \(-0.510667\pi\)
−0.0335056 + 0.999439i \(0.510667\pi\)
\(158\) 3.85283 0.306515
\(159\) 4.25892 0.337754
\(160\) 0 0
\(161\) −13.9868 −1.10231
\(162\) −0.298064 −0.0234181
\(163\) 11.0879 0.868469 0.434234 0.900800i \(-0.357019\pi\)
0.434234 + 0.900800i \(0.357019\pi\)
\(164\) −0.895553 −0.0699309
\(165\) 0 0
\(166\) −2.48810 −0.193114
\(167\) −4.46477 −0.345494 −0.172747 0.984966i \(-0.555264\pi\)
−0.172747 + 0.984966i \(0.555264\pi\)
\(168\) 2.70584 0.208760
\(169\) 8.07553 0.621195
\(170\) 0 0
\(171\) 5.69187 0.435268
\(172\) 15.4575 1.17863
\(173\) −18.0670 −1.37361 −0.686803 0.726843i \(-0.740987\pi\)
−0.686803 + 0.726843i \(0.740987\pi\)
\(174\) 1.81741 0.137777
\(175\) 0 0
\(176\) 0 0
\(177\) 6.65971 0.500575
\(178\) 1.11441 0.0835285
\(179\) −11.9744 −0.895010 −0.447505 0.894281i \(-0.647687\pi\)
−0.447505 + 0.894281i \(0.647687\pi\)
\(180\) 0 0
\(181\) 15.8185 1.17578 0.587890 0.808941i \(-0.299959\pi\)
0.587890 + 0.808941i \(0.299959\pi\)
\(182\) 3.17605 0.235424
\(183\) −1.82041 −0.134568
\(184\) −7.02497 −0.517888
\(185\) 0 0
\(186\) 2.51319 0.184276
\(187\) 0 0
\(188\) 17.5585 1.28059
\(189\) 2.32107 0.168833
\(190\) 0 0
\(191\) 9.00747 0.651757 0.325879 0.945412i \(-0.394340\pi\)
0.325879 + 0.945412i \(0.394340\pi\)
\(192\) −5.94601 −0.429117
\(193\) 3.83930 0.276359 0.138179 0.990407i \(-0.455875\pi\)
0.138179 + 0.990407i \(0.455875\pi\)
\(194\) 1.66617 0.119624
\(195\) 0 0
\(196\) 3.08203 0.220145
\(197\) 8.33739 0.594015 0.297007 0.954875i \(-0.404011\pi\)
0.297007 + 0.954875i \(0.404011\pi\)
\(198\) 0 0
\(199\) 2.22397 0.157653 0.0788266 0.996888i \(-0.474883\pi\)
0.0788266 + 0.996888i \(0.474883\pi\)
\(200\) 0 0
\(201\) 8.86541 0.625318
\(202\) 0.481120 0.0338515
\(203\) −14.1524 −0.993305
\(204\) −9.82424 −0.687834
\(205\) 0 0
\(206\) −2.53903 −0.176903
\(207\) −6.02600 −0.418836
\(208\) −15.9523 −1.10609
\(209\) 0 0
\(210\) 0 0
\(211\) 5.02265 0.345774 0.172887 0.984942i \(-0.444690\pi\)
0.172887 + 0.984942i \(0.444690\pi\)
\(212\) −8.13946 −0.559020
\(213\) −2.44868 −0.167780
\(214\) 1.82238 0.124575
\(215\) 0 0
\(216\) 1.16578 0.0793210
\(217\) −19.5706 −1.32854
\(218\) −1.70868 −0.115726
\(219\) −9.08299 −0.613772
\(220\) 0 0
\(221\) −23.5989 −1.58743
\(222\) −0.945695 −0.0634709
\(223\) −28.3725 −1.89996 −0.949980 0.312310i \(-0.898897\pi\)
−0.949980 + 0.312310i \(0.898897\pi\)
\(224\) −7.81568 −0.522207
\(225\) 0 0
\(226\) 2.45614 0.163380
\(227\) −6.58235 −0.436886 −0.218443 0.975850i \(-0.570098\pi\)
−0.218443 + 0.975850i \(0.570098\pi\)
\(228\) −10.8781 −0.720417
\(229\) −16.5282 −1.09221 −0.546107 0.837715i \(-0.683891\pi\)
−0.546107 + 0.837715i \(0.683891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −7.10817 −0.466674
\(233\) −23.3654 −1.53072 −0.765360 0.643602i \(-0.777439\pi\)
−0.765360 + 0.643602i \(0.777439\pi\)
\(234\) 1.36836 0.0894522
\(235\) 0 0
\(236\) −12.7278 −0.828507
\(237\) −12.9262 −0.839645
\(238\) −3.55631 −0.230521
\(239\) 17.7858 1.15047 0.575233 0.817990i \(-0.304911\pi\)
0.575233 + 0.817990i \(0.304911\pi\)
\(240\) 0 0
\(241\) −6.95713 −0.448148 −0.224074 0.974572i \(-0.571936\pi\)
−0.224074 + 0.974572i \(0.571936\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 3.47909 0.222726
\(245\) 0 0
\(246\) −0.139670 −0.00890506
\(247\) −26.1303 −1.66263
\(248\) −9.82948 −0.624173
\(249\) 8.34751 0.529002
\(250\) 0 0
\(251\) −0.636701 −0.0401882 −0.0200941 0.999798i \(-0.506397\pi\)
−0.0200941 + 0.999798i \(0.506397\pi\)
\(252\) −4.43592 −0.279437
\(253\) 0 0
\(254\) −3.60010 −0.225891
\(255\) 0 0
\(256\) 9.35644 0.584777
\(257\) 0.482055 0.0300698 0.0150349 0.999887i \(-0.495214\pi\)
0.0150349 + 0.999887i \(0.495214\pi\)
\(258\) 2.41076 0.150087
\(259\) 7.36426 0.457593
\(260\) 0 0
\(261\) −6.09737 −0.377418
\(262\) −0.978041 −0.0604236
\(263\) −18.2072 −1.12270 −0.561351 0.827578i \(-0.689718\pi\)
−0.561351 + 0.827578i \(0.689718\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.93779 −0.241441
\(267\) −3.73882 −0.228812
\(268\) −16.9432 −1.03497
\(269\) 26.0546 1.58858 0.794289 0.607540i \(-0.207844\pi\)
0.794289 + 0.607540i \(0.207844\pi\)
\(270\) 0 0
\(271\) 0.294394 0.0178832 0.00894158 0.999960i \(-0.497154\pi\)
0.00894158 + 0.999960i \(0.497154\pi\)
\(272\) 17.8623 1.08306
\(273\) −10.6556 −0.644905
\(274\) −1.19744 −0.0723402
\(275\) 0 0
\(276\) 11.5166 0.693220
\(277\) −30.2893 −1.81991 −0.909953 0.414712i \(-0.863882\pi\)
−0.909953 + 0.414712i \(0.863882\pi\)
\(278\) 0.530401 0.0318113
\(279\) −8.43171 −0.504793
\(280\) 0 0
\(281\) −29.7185 −1.77286 −0.886428 0.462866i \(-0.846821\pi\)
−0.886428 + 0.462866i \(0.846821\pi\)
\(282\) 2.73842 0.163071
\(283\) −1.58988 −0.0945084 −0.0472542 0.998883i \(-0.515047\pi\)
−0.0472542 + 0.998883i \(0.515047\pi\)
\(284\) 4.67980 0.277695
\(285\) 0 0
\(286\) 0 0
\(287\) 1.08763 0.0642009
\(288\) −3.36728 −0.198419
\(289\) 9.42438 0.554375
\(290\) 0 0
\(291\) −5.58998 −0.327690
\(292\) 17.3590 1.01586
\(293\) 0.421054 0.0245982 0.0122991 0.999924i \(-0.496085\pi\)
0.0122991 + 0.999924i \(0.496085\pi\)
\(294\) 0.480673 0.0280334
\(295\) 0 0
\(296\) 3.69876 0.214986
\(297\) 0 0
\(298\) 0.658069 0.0381209
\(299\) 27.6642 1.59986
\(300\) 0 0
\(301\) −18.7729 −1.08205
\(302\) 3.83121 0.220461
\(303\) −1.61415 −0.0927304
\(304\) 19.7783 1.13436
\(305\) 0 0
\(306\) −1.53219 −0.0875894
\(307\) −9.29536 −0.530514 −0.265257 0.964178i \(-0.585457\pi\)
−0.265257 + 0.964178i \(0.585457\pi\)
\(308\) 0 0
\(309\) 8.51841 0.484595
\(310\) 0 0
\(311\) 12.0947 0.685825 0.342912 0.939367i \(-0.388587\pi\)
0.342912 + 0.939367i \(0.388587\pi\)
\(312\) −5.35186 −0.302989
\(313\) −6.97604 −0.394309 −0.197155 0.980372i \(-0.563170\pi\)
−0.197155 + 0.980372i \(0.563170\pi\)
\(314\) 0.250269 0.0141235
\(315\) 0 0
\(316\) 24.7040 1.38971
\(317\) 7.23947 0.406609 0.203304 0.979116i \(-0.434832\pi\)
0.203304 + 0.979116i \(0.434832\pi\)
\(318\) −1.26943 −0.0711861
\(319\) 0 0
\(320\) 0 0
\(321\) −6.11405 −0.341253
\(322\) 4.16895 0.232327
\(323\) 29.2588 1.62801
\(324\) −1.91116 −0.106175
\(325\) 0 0
\(326\) −3.30490 −0.183041
\(327\) 5.73259 0.317013
\(328\) 0.546273 0.0301629
\(329\) −21.3245 −1.17566
\(330\) 0 0
\(331\) 27.6629 1.52049 0.760244 0.649637i \(-0.225079\pi\)
0.760244 + 0.649637i \(0.225079\pi\)
\(332\) −15.9534 −0.875557
\(333\) 3.17279 0.173868
\(334\) 1.33079 0.0728175
\(335\) 0 0
\(336\) 8.06533 0.440000
\(337\) −22.0178 −1.19939 −0.599694 0.800229i \(-0.704711\pi\)
−0.599694 + 0.800229i \(0.704711\pi\)
\(338\) −2.40703 −0.130925
\(339\) −8.24031 −0.447552
\(340\) 0 0
\(341\) 0 0
\(342\) −1.69654 −0.0917385
\(343\) −19.9905 −1.07939
\(344\) −9.42886 −0.508370
\(345\) 0 0
\(346\) 5.38512 0.289506
\(347\) −26.7672 −1.43694 −0.718469 0.695559i \(-0.755157\pi\)
−0.718469 + 0.695559i \(0.755157\pi\)
\(348\) 11.6530 0.624669
\(349\) −21.6412 −1.15843 −0.579215 0.815175i \(-0.696641\pi\)
−0.579215 + 0.815175i \(0.696641\pi\)
\(350\) 0 0
\(351\) −4.59081 −0.245039
\(352\) 0 0
\(353\) 31.5144 1.67734 0.838672 0.544636i \(-0.183332\pi\)
0.838672 + 0.544636i \(0.183332\pi\)
\(354\) −1.98502 −0.105503
\(355\) 0 0
\(356\) 7.14548 0.378710
\(357\) 11.9314 0.631475
\(358\) 3.56915 0.188635
\(359\) −4.46095 −0.235440 −0.117720 0.993047i \(-0.537559\pi\)
−0.117720 + 0.993047i \(0.537559\pi\)
\(360\) 0 0
\(361\) 13.3973 0.705124
\(362\) −4.71493 −0.247811
\(363\) 0 0
\(364\) 20.3645 1.06739
\(365\) 0 0
\(366\) 0.542598 0.0283621
\(367\) −1.03300 −0.0539222 −0.0269611 0.999636i \(-0.508583\pi\)
−0.0269611 + 0.999636i \(0.508583\pi\)
\(368\) −20.9394 −1.09154
\(369\) 0.468592 0.0243939
\(370\) 0 0
\(371\) 9.88523 0.513215
\(372\) 16.1143 0.835489
\(373\) −32.6921 −1.69273 −0.846367 0.532600i \(-0.821215\pi\)
−0.846367 + 0.532600i \(0.821215\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −10.7104 −0.552347
\(377\) 27.9919 1.44166
\(378\) −0.691827 −0.0355837
\(379\) −4.28279 −0.219992 −0.109996 0.993932i \(-0.535084\pi\)
−0.109996 + 0.993932i \(0.535084\pi\)
\(380\) 0 0
\(381\) 12.0783 0.618789
\(382\) −2.68480 −0.137366
\(383\) −26.9383 −1.37648 −0.688241 0.725482i \(-0.741617\pi\)
−0.688241 + 0.725482i \(0.741617\pi\)
\(384\) 8.50685 0.434113
\(385\) 0 0
\(386\) −1.14436 −0.0582462
\(387\) −8.08805 −0.411139
\(388\) 10.6833 0.542364
\(389\) 36.7033 1.86093 0.930465 0.366382i \(-0.119403\pi\)
0.930465 + 0.366382i \(0.119403\pi\)
\(390\) 0 0
\(391\) −30.9765 −1.56655
\(392\) −1.87999 −0.0949537
\(393\) 3.28131 0.165520
\(394\) −2.48508 −0.125196
\(395\) 0 0
\(396\) 0 0
\(397\) −7.52289 −0.377563 −0.188782 0.982019i \(-0.560454\pi\)
−0.188782 + 0.982019i \(0.560454\pi\)
\(398\) −0.662887 −0.0332275
\(399\) 13.2112 0.661387
\(400\) 0 0
\(401\) 15.3995 0.769013 0.384506 0.923122i \(-0.374372\pi\)
0.384506 + 0.923122i \(0.374372\pi\)
\(402\) −2.64246 −0.131794
\(403\) 38.7084 1.92820
\(404\) 3.08489 0.153479
\(405\) 0 0
\(406\) 4.21833 0.209352
\(407\) 0 0
\(408\) 5.99263 0.296679
\(409\) 16.2884 0.805411 0.402706 0.915330i \(-0.368070\pi\)
0.402706 + 0.915330i \(0.368070\pi\)
\(410\) 0 0
\(411\) 4.01740 0.198164
\(412\) −16.2800 −0.802059
\(413\) 15.4576 0.760620
\(414\) 1.79614 0.0882752
\(415\) 0 0
\(416\) 15.4585 0.757917
\(417\) −1.77948 −0.0871417
\(418\) 0 0
\(419\) −11.3868 −0.556281 −0.278140 0.960540i \(-0.589718\pi\)
−0.278140 + 0.960540i \(0.589718\pi\)
\(420\) 0 0
\(421\) −11.0773 −0.539876 −0.269938 0.962878i \(-0.587003\pi\)
−0.269938 + 0.962878i \(0.587003\pi\)
\(422\) −1.49707 −0.0728764
\(423\) −9.18736 −0.446705
\(424\) 4.96494 0.241119
\(425\) 0 0
\(426\) 0.729862 0.0353620
\(427\) −4.22529 −0.204476
\(428\) 11.6849 0.564811
\(429\) 0 0
\(430\) 0 0
\(431\) −30.4478 −1.46662 −0.733311 0.679894i \(-0.762026\pi\)
−0.733311 + 0.679894i \(0.762026\pi\)
\(432\) 3.47484 0.167183
\(433\) 15.1047 0.725885 0.362943 0.931812i \(-0.381772\pi\)
0.362943 + 0.931812i \(0.381772\pi\)
\(434\) 5.83328 0.280006
\(435\) 0 0
\(436\) −10.9559 −0.524691
\(437\) −34.2992 −1.64075
\(438\) 2.70732 0.129361
\(439\) 23.6799 1.13018 0.565090 0.825030i \(-0.308842\pi\)
0.565090 + 0.825030i \(0.308842\pi\)
\(440\) 0 0
\(441\) −1.61265 −0.0767928
\(442\) 7.03399 0.334573
\(443\) 30.7817 1.46248 0.731242 0.682119i \(-0.238941\pi\)
0.731242 + 0.682119i \(0.238941\pi\)
\(444\) −6.06370 −0.287771
\(445\) 0 0
\(446\) 8.45681 0.400442
\(447\) −2.20781 −0.104426
\(448\) −13.8011 −0.652040
\(449\) −4.60851 −0.217489 −0.108744 0.994070i \(-0.534683\pi\)
−0.108744 + 0.994070i \(0.534683\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 15.7485 0.740749
\(453\) −12.8536 −0.603916
\(454\) 1.96196 0.0920795
\(455\) 0 0
\(456\) 6.63544 0.310733
\(457\) −29.2281 −1.36723 −0.683616 0.729842i \(-0.739594\pi\)
−0.683616 + 0.729842i \(0.739594\pi\)
\(458\) 4.92647 0.230199
\(459\) 5.14046 0.239936
\(460\) 0 0
\(461\) 5.76367 0.268441 0.134220 0.990951i \(-0.457147\pi\)
0.134220 + 0.990951i \(0.457147\pi\)
\(462\) 0 0
\(463\) 21.9028 1.01791 0.508954 0.860794i \(-0.330032\pi\)
0.508954 + 0.860794i \(0.330032\pi\)
\(464\) −21.1874 −0.983600
\(465\) 0 0
\(466\) 6.96440 0.322620
\(467\) −14.6356 −0.677253 −0.338627 0.940921i \(-0.609962\pi\)
−0.338627 + 0.940921i \(0.609962\pi\)
\(468\) 8.77376 0.405567
\(469\) 20.5772 0.950167
\(470\) 0 0
\(471\) −0.839648 −0.0386889
\(472\) 7.76373 0.357355
\(473\) 0 0
\(474\) 3.85283 0.176966
\(475\) 0 0
\(476\) −22.8027 −1.04516
\(477\) 4.25892 0.195002
\(478\) −5.30130 −0.242476
\(479\) 28.8633 1.31880 0.659398 0.751794i \(-0.270811\pi\)
0.659398 + 0.751794i \(0.270811\pi\)
\(480\) 0 0
\(481\) −14.5657 −0.664138
\(482\) 2.07367 0.0944531
\(483\) −13.9868 −0.636419
\(484\) 0 0
\(485\) 0 0
\(486\) −0.298064 −0.0135205
\(487\) 22.1587 1.00411 0.502054 0.864836i \(-0.332578\pi\)
0.502054 + 0.864836i \(0.332578\pi\)
\(488\) −2.12219 −0.0960669
\(489\) 11.0879 0.501411
\(490\) 0 0
\(491\) −16.1229 −0.727616 −0.363808 0.931474i \(-0.618524\pi\)
−0.363808 + 0.931474i \(0.618524\pi\)
\(492\) −0.895553 −0.0403746
\(493\) −31.3433 −1.41163
\(494\) 7.78850 0.350421
\(495\) 0 0
\(496\) −29.2988 −1.31556
\(497\) −5.68354 −0.254942
\(498\) −2.48810 −0.111494
\(499\) −30.8626 −1.38160 −0.690799 0.723047i \(-0.742741\pi\)
−0.690799 + 0.723047i \(0.742741\pi\)
\(500\) 0 0
\(501\) −4.46477 −0.199471
\(502\) 0.189778 0.00847020
\(503\) −8.43173 −0.375952 −0.187976 0.982174i \(-0.560193\pi\)
−0.187976 + 0.982174i \(0.560193\pi\)
\(504\) 2.70584 0.120528
\(505\) 0 0
\(506\) 0 0
\(507\) 8.07553 0.358647
\(508\) −23.0835 −1.02416
\(509\) −29.4916 −1.30719 −0.653596 0.756843i \(-0.726741\pi\)
−0.653596 + 0.756843i \(0.726741\pi\)
\(510\) 0 0
\(511\) −21.0822 −0.932623
\(512\) −19.8025 −0.875156
\(513\) 5.69187 0.251302
\(514\) −0.143683 −0.00633761
\(515\) 0 0
\(516\) 15.4575 0.680480
\(517\) 0 0
\(518\) −2.19502 −0.0964437
\(519\) −18.0670 −0.793052
\(520\) 0 0
\(521\) −9.61116 −0.421072 −0.210536 0.977586i \(-0.567521\pi\)
−0.210536 + 0.977586i \(0.567521\pi\)
\(522\) 1.81741 0.0795458
\(523\) −4.14721 −0.181345 −0.0906725 0.995881i \(-0.528902\pi\)
−0.0906725 + 0.995881i \(0.528902\pi\)
\(524\) −6.27110 −0.273954
\(525\) 0 0
\(526\) 5.42690 0.236624
\(527\) −43.3429 −1.88805
\(528\) 0 0
\(529\) 13.3127 0.578813
\(530\) 0 0
\(531\) 6.65971 0.289007
\(532\) −25.2487 −1.09467
\(533\) −2.15122 −0.0931795
\(534\) 1.11441 0.0482252
\(535\) 0 0
\(536\) 10.3351 0.446408
\(537\) −11.9744 −0.516734
\(538\) −7.76595 −0.334814
\(539\) 0 0
\(540\) 0 0
\(541\) 31.2524 1.34364 0.671822 0.740713i \(-0.265512\pi\)
0.671822 + 0.740713i \(0.265512\pi\)
\(542\) −0.0877483 −0.00376911
\(543\) 15.8185 0.678836
\(544\) −17.3094 −0.742133
\(545\) 0 0
\(546\) 3.17605 0.135922
\(547\) −9.02639 −0.385940 −0.192970 0.981205i \(-0.561812\pi\)
−0.192970 + 0.981205i \(0.561812\pi\)
\(548\) −7.67788 −0.327983
\(549\) −1.82041 −0.0776931
\(550\) 0 0
\(551\) −34.7054 −1.47850
\(552\) −7.02497 −0.299003
\(553\) −30.0025 −1.27584
\(554\) 9.02815 0.383569
\(555\) 0 0
\(556\) 3.40088 0.144229
\(557\) −7.20535 −0.305301 −0.152650 0.988280i \(-0.548781\pi\)
−0.152650 + 0.988280i \(0.548781\pi\)
\(558\) 2.51319 0.106392
\(559\) 37.1307 1.57046
\(560\) 0 0
\(561\) 0 0
\(562\) 8.85802 0.373653
\(563\) 26.4111 1.11310 0.556548 0.830815i \(-0.312125\pi\)
0.556548 + 0.830815i \(0.312125\pi\)
\(564\) 17.5585 0.739346
\(565\) 0 0
\(566\) 0.473885 0.0199189
\(567\) 2.32107 0.0974756
\(568\) −2.85461 −0.119777
\(569\) 43.3048 1.81543 0.907715 0.419587i \(-0.137825\pi\)
0.907715 + 0.419587i \(0.137825\pi\)
\(570\) 0 0
\(571\) −1.82495 −0.0763718 −0.0381859 0.999271i \(-0.512158\pi\)
−0.0381859 + 0.999271i \(0.512158\pi\)
\(572\) 0 0
\(573\) 9.00747 0.376292
\(574\) −0.324184 −0.0135312
\(575\) 0 0
\(576\) −5.94601 −0.247751
\(577\) 26.0123 1.08291 0.541454 0.840731i \(-0.317874\pi\)
0.541454 + 0.840731i \(0.317874\pi\)
\(578\) −2.80907 −0.116842
\(579\) 3.83930 0.159556
\(580\) 0 0
\(581\) 19.3751 0.803816
\(582\) 1.66617 0.0690651
\(583\) 0 0
\(584\) −10.5887 −0.438165
\(585\) 0 0
\(586\) −0.125501 −0.00518440
\(587\) 4.00349 0.165242 0.0826209 0.996581i \(-0.473671\pi\)
0.0826209 + 0.996581i \(0.473671\pi\)
\(588\) 3.08203 0.127101
\(589\) −47.9921 −1.97748
\(590\) 0 0
\(591\) 8.33739 0.342954
\(592\) 11.0249 0.453122
\(593\) −32.2116 −1.32277 −0.661386 0.750045i \(-0.730032\pi\)
−0.661386 + 0.750045i \(0.730032\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.21947 0.172836
\(597\) 2.22397 0.0910212
\(598\) −8.24572 −0.337192
\(599\) −13.5168 −0.552281 −0.276140 0.961117i \(-0.589055\pi\)
−0.276140 + 0.961117i \(0.589055\pi\)
\(600\) 0 0
\(601\) 27.4926 1.12145 0.560724 0.828003i \(-0.310523\pi\)
0.560724 + 0.828003i \(0.310523\pi\)
\(602\) 5.59553 0.228057
\(603\) 8.86541 0.361027
\(604\) 24.5653 0.999548
\(605\) 0 0
\(606\) 0.481120 0.0195442
\(607\) −37.6114 −1.52660 −0.763300 0.646044i \(-0.776422\pi\)
−0.763300 + 0.646044i \(0.776422\pi\)
\(608\) −19.1661 −0.777288
\(609\) −14.1524 −0.573485
\(610\) 0 0
\(611\) 42.1774 1.70632
\(612\) −9.82424 −0.397121
\(613\) −29.0831 −1.17466 −0.587328 0.809349i \(-0.699820\pi\)
−0.587328 + 0.809349i \(0.699820\pi\)
\(614\) 2.77062 0.111813
\(615\) 0 0
\(616\) 0 0
\(617\) 9.61911 0.387251 0.193625 0.981076i \(-0.437975\pi\)
0.193625 + 0.981076i \(0.437975\pi\)
\(618\) −2.53903 −0.102135
\(619\) −23.4074 −0.940824 −0.470412 0.882447i \(-0.655895\pi\)
−0.470412 + 0.882447i \(0.655895\pi\)
\(620\) 0 0
\(621\) −6.02600 −0.241815
\(622\) −3.60498 −0.144547
\(623\) −8.67806 −0.347679
\(624\) −15.9523 −0.638604
\(625\) 0 0
\(626\) 2.07931 0.0831059
\(627\) 0 0
\(628\) 1.60470 0.0640345
\(629\) 16.3096 0.650307
\(630\) 0 0
\(631\) −22.6565 −0.901942 −0.450971 0.892539i \(-0.648922\pi\)
−0.450971 + 0.892539i \(0.648922\pi\)
\(632\) −15.0690 −0.599414
\(633\) 5.02265 0.199633
\(634\) −2.15783 −0.0856982
\(635\) 0 0
\(636\) −8.13946 −0.322751
\(637\) 7.40336 0.293332
\(638\) 0 0
\(639\) −2.44868 −0.0968681
\(640\) 0 0
\(641\) 22.9966 0.908313 0.454156 0.890922i \(-0.349941\pi\)
0.454156 + 0.890922i \(0.349941\pi\)
\(642\) 1.82238 0.0719235
\(643\) −11.6129 −0.457970 −0.228985 0.973430i \(-0.573541\pi\)
−0.228985 + 0.973430i \(0.573541\pi\)
\(644\) 26.7309 1.05334
\(645\) 0 0
\(646\) −8.72101 −0.343124
\(647\) −36.1713 −1.42204 −0.711020 0.703172i \(-0.751766\pi\)
−0.711020 + 0.703172i \(0.751766\pi\)
\(648\) 1.16578 0.0457960
\(649\) 0 0
\(650\) 0 0
\(651\) −19.5706 −0.767030
\(652\) −21.1907 −0.829890
\(653\) −8.65293 −0.338615 −0.169308 0.985563i \(-0.554153\pi\)
−0.169308 + 0.985563i \(0.554153\pi\)
\(654\) −1.70868 −0.0668146
\(655\) 0 0
\(656\) 1.62828 0.0635737
\(657\) −9.08299 −0.354361
\(658\) 6.35607 0.247785
\(659\) −17.4599 −0.680143 −0.340071 0.940400i \(-0.610451\pi\)
−0.340071 + 0.940400i \(0.610451\pi\)
\(660\) 0 0
\(661\) 16.2393 0.631636 0.315818 0.948820i \(-0.397721\pi\)
0.315818 + 0.948820i \(0.397721\pi\)
\(662\) −8.24531 −0.320463
\(663\) −23.5989 −0.916505
\(664\) 9.73133 0.377649
\(665\) 0 0
\(666\) −0.945695 −0.0366449
\(667\) 36.7428 1.42269
\(668\) 8.53288 0.330147
\(669\) −28.3725 −1.09694
\(670\) 0 0
\(671\) 0 0
\(672\) −7.81568 −0.301496
\(673\) −26.4394 −1.01916 −0.509582 0.860422i \(-0.670200\pi\)
−0.509582 + 0.860422i \(0.670200\pi\)
\(674\) 6.56273 0.252787
\(675\) 0 0
\(676\) −15.4336 −0.593600
\(677\) 31.3550 1.20507 0.602534 0.798093i \(-0.294158\pi\)
0.602534 + 0.798093i \(0.294158\pi\)
\(678\) 2.45614 0.0943276
\(679\) −12.9747 −0.497924
\(680\) 0 0
\(681\) −6.58235 −0.252236
\(682\) 0 0
\(683\) 3.85934 0.147674 0.0738368 0.997270i \(-0.476476\pi\)
0.0738368 + 0.997270i \(0.476476\pi\)
\(684\) −10.8781 −0.415933
\(685\) 0 0
\(686\) 5.95846 0.227495
\(687\) −16.5282 −0.630590
\(688\) −28.1047 −1.07148
\(689\) −19.5519 −0.744867
\(690\) 0 0
\(691\) −22.0718 −0.839651 −0.419825 0.907605i \(-0.637909\pi\)
−0.419825 + 0.907605i \(0.637909\pi\)
\(692\) 34.5288 1.31259
\(693\) 0 0
\(694\) 7.97835 0.302854
\(695\) 0 0
\(696\) −7.10817 −0.269435
\(697\) 2.40878 0.0912390
\(698\) 6.45048 0.244154
\(699\) −23.3654 −0.883762
\(700\) 0 0
\(701\) 34.4034 1.29940 0.649699 0.760192i \(-0.274895\pi\)
0.649699 + 0.760192i \(0.274895\pi\)
\(702\) 1.36836 0.0516453
\(703\) 18.0591 0.681112
\(704\) 0 0
\(705\) 0 0
\(706\) −9.39333 −0.353523
\(707\) −3.74655 −0.140903
\(708\) −12.7278 −0.478338
\(709\) −25.1938 −0.946172 −0.473086 0.881016i \(-0.656860\pi\)
−0.473086 + 0.881016i \(0.656860\pi\)
\(710\) 0 0
\(711\) −12.9262 −0.484769
\(712\) −4.35863 −0.163347
\(713\) 50.8095 1.90283
\(714\) −3.55631 −0.133092
\(715\) 0 0
\(716\) 22.8850 0.855253
\(717\) 17.7858 0.664222
\(718\) 1.32965 0.0496221
\(719\) −17.4770 −0.651784 −0.325892 0.945407i \(-0.605665\pi\)
−0.325892 + 0.945407i \(0.605665\pi\)
\(720\) 0 0
\(721\) 19.7718 0.736340
\(722\) −3.99327 −0.148614
\(723\) −6.95713 −0.258738
\(724\) −30.2316 −1.12355
\(725\) 0 0
\(726\) 0 0
\(727\) −3.60045 −0.133533 −0.0667666 0.997769i \(-0.521268\pi\)
−0.0667666 + 0.997769i \(0.521268\pi\)
\(728\) −12.4220 −0.460391
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −41.5763 −1.53776
\(732\) 3.47909 0.128591
\(733\) −21.8840 −0.808305 −0.404153 0.914692i \(-0.632434\pi\)
−0.404153 + 0.914692i \(0.632434\pi\)
\(734\) 0.307901 0.0113648
\(735\) 0 0
\(736\) 20.2912 0.747944
\(737\) 0 0
\(738\) −0.139670 −0.00514134
\(739\) −5.74815 −0.211449 −0.105724 0.994395i \(-0.533716\pi\)
−0.105724 + 0.994395i \(0.533716\pi\)
\(740\) 0 0
\(741\) −26.1303 −0.959920
\(742\) −2.94643 −0.108167
\(743\) −18.0014 −0.660406 −0.330203 0.943910i \(-0.607117\pi\)
−0.330203 + 0.943910i \(0.607117\pi\)
\(744\) −9.82948 −0.360366
\(745\) 0 0
\(746\) 9.74435 0.356766
\(747\) 8.34751 0.305420
\(748\) 0 0
\(749\) −14.1911 −0.518532
\(750\) 0 0
\(751\) −19.0167 −0.693928 −0.346964 0.937878i \(-0.612787\pi\)
−0.346964 + 0.937878i \(0.612787\pi\)
\(752\) −31.9246 −1.16417
\(753\) −0.636701 −0.0232027
\(754\) −8.34338 −0.303848
\(755\) 0 0
\(756\) −4.43592 −0.161333
\(757\) −2.39560 −0.0870694 −0.0435347 0.999052i \(-0.513862\pi\)
−0.0435347 + 0.999052i \(0.513862\pi\)
\(758\) 1.27655 0.0463663
\(759\) 0 0
\(760\) 0 0
\(761\) 45.4089 1.64607 0.823035 0.567991i \(-0.192279\pi\)
0.823035 + 0.567991i \(0.192279\pi\)
\(762\) −3.60010 −0.130418
\(763\) 13.3057 0.481699
\(764\) −17.2147 −0.622806
\(765\) 0 0
\(766\) 8.02934 0.290112
\(767\) −30.5735 −1.10394
\(768\) 9.35644 0.337621
\(769\) −49.1931 −1.77395 −0.886975 0.461818i \(-0.847197\pi\)
−0.886975 + 0.461818i \(0.847197\pi\)
\(770\) 0 0
\(771\) 0.482055 0.0173608
\(772\) −7.33750 −0.264082
\(773\) −21.2288 −0.763549 −0.381774 0.924256i \(-0.624687\pi\)
−0.381774 + 0.924256i \(0.624687\pi\)
\(774\) 2.41076 0.0866529
\(775\) 0 0
\(776\) −6.51666 −0.233935
\(777\) 7.36426 0.264191
\(778\) −10.9399 −0.392216
\(779\) 2.66716 0.0955610
\(780\) 0 0
\(781\) 0 0
\(782\) 9.23297 0.330170
\(783\) −6.09737 −0.217902
\(784\) −5.60370 −0.200132
\(785\) 0 0
\(786\) −0.978041 −0.0348856
\(787\) −8.25347 −0.294204 −0.147102 0.989121i \(-0.546995\pi\)
−0.147102 + 0.989121i \(0.546995\pi\)
\(788\) −15.9341 −0.567628
\(789\) −18.2072 −0.648192
\(790\) 0 0
\(791\) −19.1263 −0.680053
\(792\) 0 0
\(793\) 8.35714 0.296771
\(794\) 2.24231 0.0795764
\(795\) 0 0
\(796\) −4.25036 −0.150650
\(797\) −6.29828 −0.223096 −0.111548 0.993759i \(-0.535581\pi\)
−0.111548 + 0.993759i \(0.535581\pi\)
\(798\) −3.93779 −0.139396
\(799\) −47.2273 −1.67078
\(800\) 0 0
\(801\) −3.73882 −0.132105
\(802\) −4.59003 −0.162080
\(803\) 0 0
\(804\) −16.9432 −0.597540
\(805\) 0 0
\(806\) −11.5376 −0.406394
\(807\) 26.0546 0.917166
\(808\) −1.88174 −0.0661992
\(809\) 54.5843 1.91908 0.959540 0.281571i \(-0.0908556\pi\)
0.959540 + 0.281571i \(0.0908556\pi\)
\(810\) 0 0
\(811\) −11.0167 −0.386850 −0.193425 0.981115i \(-0.561960\pi\)
−0.193425 + 0.981115i \(0.561960\pi\)
\(812\) 27.0475 0.949181
\(813\) 0.294394 0.0103248
\(814\) 0 0
\(815\) 0 0
\(816\) 17.8623 0.625305
\(817\) −46.0361 −1.61060
\(818\) −4.85500 −0.169751
\(819\) −10.6556 −0.372336
\(820\) 0 0
\(821\) 41.7127 1.45578 0.727892 0.685692i \(-0.240500\pi\)
0.727892 + 0.685692i \(0.240500\pi\)
\(822\) −1.19744 −0.0417656
\(823\) −3.00669 −0.104806 −0.0524032 0.998626i \(-0.516688\pi\)
−0.0524032 + 0.998626i \(0.516688\pi\)
\(824\) 9.93056 0.345947
\(825\) 0 0
\(826\) −4.60737 −0.160311
\(827\) 44.8126 1.55829 0.779143 0.626846i \(-0.215655\pi\)
0.779143 + 0.626846i \(0.215655\pi\)
\(828\) 11.5166 0.400231
\(829\) −38.6918 −1.34382 −0.671911 0.740631i \(-0.734526\pi\)
−0.671911 + 0.740631i \(0.734526\pi\)
\(830\) 0 0
\(831\) −30.2893 −1.05072
\(832\) 27.2970 0.946354
\(833\) −8.28977 −0.287223
\(834\) 0.530401 0.0183663
\(835\) 0 0
\(836\) 0 0
\(837\) −8.43171 −0.291442
\(838\) 3.39399 0.117244
\(839\) −21.2861 −0.734879 −0.367440 0.930047i \(-0.619766\pi\)
−0.367440 + 0.930047i \(0.619766\pi\)
\(840\) 0 0
\(841\) 8.17796 0.281999
\(842\) 3.30176 0.113786
\(843\) −29.7185 −1.02356
\(844\) −9.59908 −0.330414
\(845\) 0 0
\(846\) 2.73842 0.0941490
\(847\) 0 0
\(848\) 14.7990 0.508201
\(849\) −1.58988 −0.0545644
\(850\) 0 0
\(851\) −19.1192 −0.655399
\(852\) 4.67980 0.160327
\(853\) 23.9145 0.818815 0.409408 0.912352i \(-0.365735\pi\)
0.409408 + 0.912352i \(0.365735\pi\)
\(854\) 1.25941 0.0430960
\(855\) 0 0
\(856\) −7.12761 −0.243617
\(857\) −23.7057 −0.809773 −0.404886 0.914367i \(-0.632689\pi\)
−0.404886 + 0.914367i \(0.632689\pi\)
\(858\) 0 0
\(859\) −29.2131 −0.996739 −0.498369 0.866965i \(-0.666068\pi\)
−0.498369 + 0.866965i \(0.666068\pi\)
\(860\) 0 0
\(861\) 1.08763 0.0370664
\(862\) 9.07541 0.309110
\(863\) −37.2828 −1.26912 −0.634560 0.772874i \(-0.718819\pi\)
−0.634560 + 0.772874i \(0.718819\pi\)
\(864\) −3.36728 −0.114557
\(865\) 0 0
\(866\) −4.50217 −0.152990
\(867\) 9.42438 0.320069
\(868\) 37.4024 1.26952
\(869\) 0 0
\(870\) 0 0
\(871\) −40.6994 −1.37905
\(872\) 6.68291 0.226312
\(873\) −5.58998 −0.189192
\(874\) 10.2234 0.345810
\(875\) 0 0
\(876\) 17.3590 0.586507
\(877\) 26.7080 0.901865 0.450932 0.892558i \(-0.351092\pi\)
0.450932 + 0.892558i \(0.351092\pi\)
\(878\) −7.05813 −0.238200
\(879\) 0.421054 0.0142018
\(880\) 0 0
\(881\) 51.3146 1.72883 0.864417 0.502775i \(-0.167688\pi\)
0.864417 + 0.502775i \(0.167688\pi\)
\(882\) 0.480673 0.0161851
\(883\) 54.9200 1.84820 0.924102 0.382145i \(-0.124815\pi\)
0.924102 + 0.382145i \(0.124815\pi\)
\(884\) 45.1012 1.51692
\(885\) 0 0
\(886\) −9.17493 −0.308238
\(887\) 5.07251 0.170318 0.0851592 0.996367i \(-0.472860\pi\)
0.0851592 + 0.996367i \(0.472860\pi\)
\(888\) 3.69876 0.124122
\(889\) 28.0345 0.940247
\(890\) 0 0
\(891\) 0 0
\(892\) 54.2242 1.81556
\(893\) −52.2932 −1.74993
\(894\) 0.658069 0.0220091
\(895\) 0 0
\(896\) 19.7450 0.659633
\(897\) 27.6642 0.923682
\(898\) 1.37363 0.0458387
\(899\) 51.4113 1.71466
\(900\) 0 0
\(901\) 21.8928 0.729355
\(902\) 0 0
\(903\) −18.7729 −0.624723
\(904\) −9.60636 −0.319503
\(905\) 0 0
\(906\) 3.83121 0.127283
\(907\) 23.8325 0.791346 0.395673 0.918392i \(-0.370511\pi\)
0.395673 + 0.918392i \(0.370511\pi\)
\(908\) 12.5799 0.417479
\(909\) −1.61415 −0.0535379
\(910\) 0 0
\(911\) 15.9409 0.528147 0.264074 0.964503i \(-0.414934\pi\)
0.264074 + 0.964503i \(0.414934\pi\)
\(912\) 19.7783 0.654926
\(913\) 0 0
\(914\) 8.71185 0.288162
\(915\) 0 0
\(916\) 31.5880 1.04370
\(917\) 7.61614 0.251507
\(918\) −1.53219 −0.0505698
\(919\) −34.9656 −1.15341 −0.576705 0.816952i \(-0.695662\pi\)
−0.576705 + 0.816952i \(0.695662\pi\)
\(920\) 0 0
\(921\) −9.29536 −0.306293
\(922\) −1.71794 −0.0565775
\(923\) 11.2414 0.370015
\(924\) 0 0
\(925\) 0 0
\(926\) −6.52843 −0.214538
\(927\) 8.51841 0.279781
\(928\) 20.5315 0.673981
\(929\) −34.3129 −1.12577 −0.562885 0.826535i \(-0.690308\pi\)
−0.562885 + 0.826535i \(0.690308\pi\)
\(930\) 0 0
\(931\) −9.17898 −0.300829
\(932\) 44.6550 1.46272
\(933\) 12.0947 0.395961
\(934\) 4.36234 0.142740
\(935\) 0 0
\(936\) −5.35186 −0.174931
\(937\) 28.9980 0.947322 0.473661 0.880707i \(-0.342932\pi\)
0.473661 + 0.880707i \(0.342932\pi\)
\(938\) −6.13333 −0.200260
\(939\) −6.97604 −0.227655
\(940\) 0 0
\(941\) 16.7926 0.547424 0.273712 0.961812i \(-0.411749\pi\)
0.273712 + 0.961812i \(0.411749\pi\)
\(942\) 0.250269 0.00815421
\(943\) −2.82373 −0.0919534
\(944\) 23.1414 0.753189
\(945\) 0 0
\(946\) 0 0
\(947\) −17.4385 −0.566677 −0.283338 0.959020i \(-0.591442\pi\)
−0.283338 + 0.959020i \(0.591442\pi\)
\(948\) 24.7040 0.802347
\(949\) 41.6983 1.35358
\(950\) 0 0
\(951\) 7.23947 0.234756
\(952\) 13.9093 0.450803
\(953\) −1.47790 −0.0478739 −0.0239369 0.999713i \(-0.507620\pi\)
−0.0239369 + 0.999713i \(0.507620\pi\)
\(954\) −1.26943 −0.0410993
\(955\) 0 0
\(956\) −33.9914 −1.09936
\(957\) 0 0
\(958\) −8.60310 −0.277954
\(959\) 9.32465 0.301109
\(960\) 0 0
\(961\) 40.0937 1.29334
\(962\) 4.34151 0.139976
\(963\) −6.11405 −0.197022
\(964\) 13.2962 0.428241
\(965\) 0 0
\(966\) 4.16895 0.134134
\(967\) −26.6515 −0.857054 −0.428527 0.903529i \(-0.640967\pi\)
−0.428527 + 0.903529i \(0.640967\pi\)
\(968\) 0 0
\(969\) 29.2588 0.939929
\(970\) 0 0
\(971\) −38.7025 −1.24202 −0.621011 0.783802i \(-0.713278\pi\)
−0.621011 + 0.783802i \(0.713278\pi\)
\(972\) −1.91116 −0.0613004
\(973\) −4.13030 −0.132411
\(974\) −6.60473 −0.211629
\(975\) 0 0
\(976\) −6.32562 −0.202478
\(977\) −49.7415 −1.59137 −0.795686 0.605709i \(-0.792890\pi\)
−0.795686 + 0.605709i \(0.792890\pi\)
\(978\) −3.30490 −0.105679
\(979\) 0 0
\(980\) 0 0
\(981\) 5.73259 0.183027
\(982\) 4.80566 0.153355
\(983\) −24.5431 −0.782802 −0.391401 0.920220i \(-0.628009\pi\)
−0.391401 + 0.920220i \(0.628009\pi\)
\(984\) 0.546273 0.0174145
\(985\) 0 0
\(986\) 9.34233 0.297520
\(987\) −21.3245 −0.678766
\(988\) 49.9391 1.58877
\(989\) 48.7386 1.54980
\(990\) 0 0
\(991\) 32.4673 1.03136 0.515679 0.856782i \(-0.327540\pi\)
0.515679 + 0.856782i \(0.327540\pi\)
\(992\) 28.3919 0.901444
\(993\) 27.6629 0.877855
\(994\) 1.69406 0.0537323
\(995\) 0 0
\(996\) −15.9534 −0.505503
\(997\) −39.6946 −1.25714 −0.628571 0.777752i \(-0.716360\pi\)
−0.628571 + 0.777752i \(0.716360\pi\)
\(998\) 9.19902 0.291190
\(999\) 3.17279 0.100383
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9075.2.a.dy.1.7 12
5.2 odd 4 1815.2.c.j.364.11 24
5.3 odd 4 1815.2.c.j.364.14 24
5.4 even 2 9075.2.a.dz.1.6 12
11.3 even 5 825.2.n.p.526.4 24
11.4 even 5 825.2.n.p.676.4 24
11.10 odd 2 9075.2.a.ea.1.6 12
55.3 odd 20 165.2.s.a.64.7 yes 48
55.4 even 10 825.2.n.o.676.3 24
55.14 even 10 825.2.n.o.526.3 24
55.32 even 4 1815.2.c.k.364.14 24
55.37 odd 20 165.2.s.a.49.7 yes 48
55.43 even 4 1815.2.c.k.364.11 24
55.47 odd 20 165.2.s.a.64.6 yes 48
55.48 odd 20 165.2.s.a.49.6 48
55.54 odd 2 9075.2.a.dx.1.7 12
165.47 even 20 495.2.ba.c.64.7 48
165.92 even 20 495.2.ba.c.379.6 48
165.113 even 20 495.2.ba.c.64.6 48
165.158 even 20 495.2.ba.c.379.7 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.s.a.49.6 48 55.48 odd 20
165.2.s.a.49.7 yes 48 55.37 odd 20
165.2.s.a.64.6 yes 48 55.47 odd 20
165.2.s.a.64.7 yes 48 55.3 odd 20
495.2.ba.c.64.6 48 165.113 even 20
495.2.ba.c.64.7 48 165.47 even 20
495.2.ba.c.379.6 48 165.92 even 20
495.2.ba.c.379.7 48 165.158 even 20
825.2.n.o.526.3 24 55.14 even 10
825.2.n.o.676.3 24 55.4 even 10
825.2.n.p.526.4 24 11.3 even 5
825.2.n.p.676.4 24 11.4 even 5
1815.2.c.j.364.11 24 5.2 odd 4
1815.2.c.j.364.14 24 5.3 odd 4
1815.2.c.k.364.11 24 55.43 even 4
1815.2.c.k.364.14 24 55.32 even 4
9075.2.a.dx.1.7 12 55.54 odd 2
9075.2.a.dy.1.7 12 1.1 even 1 trivial
9075.2.a.dz.1.6 12 5.4 even 2
9075.2.a.ea.1.6 12 11.10 odd 2