Properties

Label 6525.2.a.bi.1.4
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.43828\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43828 q^{2} +0.0686587 q^{4} +2.74301 q^{7} -2.77782 q^{8} +O(q^{10})\) \(q+1.43828 q^{2} +0.0686587 q^{4} +2.74301 q^{7} -2.77782 q^{8} -2.74301 q^{11} +5.14744 q^{13} +3.94523 q^{14} -4.13260 q^{16} -3.72913 q^{17} -0.404431 q^{19} -3.94523 q^{22} -5.45825 q^{23} +7.40348 q^{26} +0.188331 q^{28} +1.00000 q^{29} +1.45825 q^{31} -0.388222 q^{32} -5.36354 q^{34} -6.76702 q^{37} -0.581686 q^{38} -9.78090 q^{41} -4.43220 q^{43} -0.188331 q^{44} -7.85051 q^{46} +2.60569 q^{47} +0.524103 q^{49} +0.353416 q^{52} -6.43220 q^{53} -7.61958 q^{56} +1.43828 q^{58} +9.91822 q^{59} -13.0816 q^{61} +2.09738 q^{62} +7.70683 q^{64} -12.4961 q^{67} -0.256037 q^{68} +11.3487 q^{71} +10.7670 q^{73} -9.73289 q^{74} -0.0277677 q^{76} -7.52410 q^{77} +14.1576 q^{79} -14.0677 q^{82} +1.62334 q^{83} -6.37476 q^{86} +7.61958 q^{88} +8.87281 q^{89} +14.1195 q^{91} -0.374756 q^{92} +3.74772 q^{94} -7.82084 q^{97} +0.753809 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} + 5 q^{4} - 2 q^{7} - 12 q^{8} + 2 q^{11} + 8 q^{13} + 3 q^{14} + 11 q^{16} - 10 q^{17} - 2 q^{19} - 3 q^{22} - 12 q^{23} + 7 q^{26} + 9 q^{28} + 4 q^{29} - 4 q^{31} - 17 q^{32} - q^{34} + 16 q^{37} - 10 q^{38} + 12 q^{41} - 2 q^{43} - 9 q^{44} - 8 q^{46} - 12 q^{47} + 6 q^{49} + 3 q^{52} - 10 q^{53} - 3 q^{58} - 2 q^{59} - 26 q^{61} + 20 q^{62} + 34 q^{64} - 2 q^{67} + 9 q^{68} + 10 q^{71} - 48 q^{74} + 16 q^{76} - 34 q^{77} + 22 q^{79} - 38 q^{82} - 10 q^{83} + 4 q^{86} + 4 q^{89} - 8 q^{91} + 28 q^{92} + 39 q^{94} + 22 q^{97} + 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.43828 1.01702 0.508510 0.861056i \(-0.330197\pi\)
0.508510 + 0.861056i \(0.330197\pi\)
\(3\) 0 0
\(4\) 0.0686587 0.0343293
\(5\) 0 0
\(6\) 0 0
\(7\) 2.74301 1.03676 0.518380 0.855150i \(-0.326535\pi\)
0.518380 + 0.855150i \(0.326535\pi\)
\(8\) −2.77782 −0.982106
\(9\) 0 0
\(10\) 0 0
\(11\) −2.74301 −0.827049 −0.413524 0.910493i \(-0.635702\pi\)
−0.413524 + 0.910493i \(0.635702\pi\)
\(12\) 0 0
\(13\) 5.14744 1.42764 0.713822 0.700328i \(-0.246963\pi\)
0.713822 + 0.700328i \(0.246963\pi\)
\(14\) 3.94523 1.05441
\(15\) 0 0
\(16\) −4.13260 −1.03315
\(17\) −3.72913 −0.904446 −0.452223 0.891905i \(-0.649369\pi\)
−0.452223 + 0.891905i \(0.649369\pi\)
\(18\) 0 0
\(19\) −0.404431 −0.0927827 −0.0463914 0.998923i \(-0.514772\pi\)
−0.0463914 + 0.998923i \(0.514772\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.94523 −0.841125
\(23\) −5.45825 −1.13812 −0.569062 0.822295i \(-0.692694\pi\)
−0.569062 + 0.822295i \(0.692694\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.40348 1.45194
\(27\) 0 0
\(28\) 0.188331 0.0355913
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.45825 0.261910 0.130955 0.991388i \(-0.458196\pi\)
0.130955 + 0.991388i \(0.458196\pi\)
\(32\) −0.388222 −0.0686287
\(33\) 0 0
\(34\) −5.36354 −0.919839
\(35\) 0 0
\(36\) 0 0
\(37\) −6.76702 −1.11249 −0.556245 0.831018i \(-0.687759\pi\)
−0.556245 + 0.831018i \(0.687759\pi\)
\(38\) −0.581686 −0.0943619
\(39\) 0 0
\(40\) 0 0
\(41\) −9.78090 −1.52752 −0.763760 0.645500i \(-0.776649\pi\)
−0.763760 + 0.645500i \(0.776649\pi\)
\(42\) 0 0
\(43\) −4.43220 −0.675904 −0.337952 0.941163i \(-0.609734\pi\)
−0.337952 + 0.941163i \(0.609734\pi\)
\(44\) −0.188331 −0.0283920
\(45\) 0 0
\(46\) −7.85051 −1.15749
\(47\) 2.60569 0.380079 0.190040 0.981776i \(-0.439138\pi\)
0.190040 + 0.981776i \(0.439138\pi\)
\(48\) 0 0
\(49\) 0.524103 0.0748719
\(50\) 0 0
\(51\) 0 0
\(52\) 0.353416 0.0490100
\(53\) −6.43220 −0.883530 −0.441765 0.897131i \(-0.645648\pi\)
−0.441765 + 0.897131i \(0.645648\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.61958 −1.01821
\(57\) 0 0
\(58\) 1.43828 0.188856
\(59\) 9.91822 1.29124 0.645621 0.763658i \(-0.276599\pi\)
0.645621 + 0.763658i \(0.276599\pi\)
\(60\) 0 0
\(61\) −13.0816 −1.67493 −0.837463 0.546494i \(-0.815962\pi\)
−0.837463 + 0.546494i \(0.815962\pi\)
\(62\) 2.09738 0.266367
\(63\) 0 0
\(64\) 7.70683 0.963354
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4961 −1.52665 −0.763323 0.646017i \(-0.776434\pi\)
−0.763323 + 0.646017i \(0.776434\pi\)
\(68\) −0.256037 −0.0310490
\(69\) 0 0
\(70\) 0 0
\(71\) 11.3487 1.34684 0.673422 0.739259i \(-0.264824\pi\)
0.673422 + 0.739259i \(0.264824\pi\)
\(72\) 0 0
\(73\) 10.7670 1.26018 0.630092 0.776520i \(-0.283017\pi\)
0.630092 + 0.776520i \(0.283017\pi\)
\(74\) −9.73289 −1.13143
\(75\) 0 0
\(76\) −0.0277677 −0.00318517
\(77\) −7.52410 −0.857451
\(78\) 0 0
\(79\) 14.1576 1.59285 0.796425 0.604737i \(-0.206722\pi\)
0.796425 + 0.604737i \(0.206722\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −14.0677 −1.55352
\(83\) 1.62334 0.178184 0.0890922 0.996023i \(-0.471603\pi\)
0.0890922 + 0.996023i \(0.471603\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.37476 −0.687408
\(87\) 0 0
\(88\) 7.61958 0.812250
\(89\) 8.87281 0.940516 0.470258 0.882529i \(-0.344161\pi\)
0.470258 + 0.882529i \(0.344161\pi\)
\(90\) 0 0
\(91\) 14.1195 1.48012
\(92\) −0.374756 −0.0390711
\(93\) 0 0
\(94\) 3.74772 0.386548
\(95\) 0 0
\(96\) 0 0
\(97\) −7.82084 −0.794086 −0.397043 0.917800i \(-0.629964\pi\)
−0.397043 + 0.917800i \(0.629964\pi\)
\(98\) 0.753809 0.0761462
\(99\) 0 0
\(100\) 0 0
\(101\) 4.88033 0.485611 0.242805 0.970075i \(-0.421932\pi\)
0.242805 + 0.970075i \(0.421932\pi\)
\(102\) 0 0
\(103\) 0.294881 0.0290555 0.0145277 0.999894i \(-0.495376\pi\)
0.0145277 + 0.999894i \(0.495376\pi\)
\(104\) −14.2986 −1.40210
\(105\) 0 0
\(106\) −9.25132 −0.898568
\(107\) −13.7809 −1.33225 −0.666125 0.745840i \(-0.732048\pi\)
−0.666125 + 0.745840i \(0.732048\pi\)
\(108\) 0 0
\(109\) −6.20126 −0.593973 −0.296987 0.954882i \(-0.595982\pi\)
−0.296987 + 0.954882i \(0.595982\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −11.3358 −1.07113
\(113\) −10.5658 −0.993943 −0.496971 0.867767i \(-0.665555\pi\)
−0.496971 + 0.867767i \(0.665555\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.0686587 0.00637480
\(117\) 0 0
\(118\) 14.2652 1.31322
\(119\) −10.2290 −0.937694
\(120\) 0 0
\(121\) −3.47590 −0.315991
\(122\) −18.8150 −1.70343
\(123\) 0 0
\(124\) 0.100122 0.00899119
\(125\) 0 0
\(126\) 0 0
\(127\) −8.54175 −0.757958 −0.378979 0.925405i \(-0.623725\pi\)
−0.378979 + 0.925405i \(0.623725\pi\)
\(128\) 11.8611 1.04838
\(129\) 0 0
\(130\) 0 0
\(131\) −13.3050 −1.16246 −0.581232 0.813738i \(-0.697429\pi\)
−0.581232 + 0.813738i \(0.697429\pi\)
\(132\) 0 0
\(133\) −1.10936 −0.0961934
\(134\) −17.9730 −1.55263
\(135\) 0 0
\(136\) 10.3588 0.888262
\(137\) −18.2253 −1.55709 −0.778545 0.627589i \(-0.784042\pi\)
−0.778545 + 0.627589i \(0.784042\pi\)
\(138\) 0 0
\(139\) −5.66142 −0.480195 −0.240098 0.970749i \(-0.577179\pi\)
−0.240098 + 0.970749i \(0.577179\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.3226 1.36977
\(143\) −14.1195 −1.18073
\(144\) 0 0
\(145\) 0 0
\(146\) 15.4860 1.28163
\(147\) 0 0
\(148\) −0.464614 −0.0381911
\(149\) 11.9182 0.976378 0.488189 0.872738i \(-0.337658\pi\)
0.488189 + 0.872738i \(0.337658\pi\)
\(150\) 0 0
\(151\) 7.19114 0.585207 0.292603 0.956234i \(-0.405478\pi\)
0.292603 + 0.956234i \(0.405478\pi\)
\(152\) 1.12343 0.0911225
\(153\) 0 0
\(154\) −10.8218 −0.872045
\(155\) 0 0
\(156\) 0 0
\(157\) 4.31457 0.344340 0.172170 0.985067i \(-0.444922\pi\)
0.172170 + 0.985067i \(0.444922\pi\)
\(158\) 20.3626 1.61996
\(159\) 0 0
\(160\) 0 0
\(161\) −14.9720 −1.17996
\(162\) 0 0
\(163\) −21.6436 −1.69526 −0.847628 0.530591i \(-0.821970\pi\)
−0.847628 + 0.530591i \(0.821970\pi\)
\(164\) −0.671544 −0.0524388
\(165\) 0 0
\(166\) 2.33482 0.181217
\(167\) −16.4303 −1.27141 −0.635707 0.771930i \(-0.719291\pi\)
−0.635707 + 0.771930i \(0.719291\pi\)
\(168\) 0 0
\(169\) 13.4961 1.03816
\(170\) 0 0
\(171\) 0 0
\(172\) −0.304309 −0.0232033
\(173\) −4.64939 −0.353487 −0.176743 0.984257i \(-0.556556\pi\)
−0.176743 + 0.984257i \(0.556556\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.3358 0.854466
\(177\) 0 0
\(178\) 12.7616 0.956523
\(179\) −7.62334 −0.569795 −0.284897 0.958558i \(-0.591960\pi\)
−0.284897 + 0.958558i \(0.591960\pi\)
\(180\) 0 0
\(181\) −21.3050 −1.58359 −0.791794 0.610788i \(-0.790853\pi\)
−0.791794 + 0.610788i \(0.790853\pi\)
\(182\) 20.3078 1.50532
\(183\) 0 0
\(184\) 15.1620 1.11776
\(185\) 0 0
\(186\) 0 0
\(187\) 10.2290 0.748021
\(188\) 0.178903 0.0130479
\(189\) 0 0
\(190\) 0 0
\(191\) −3.07597 −0.222570 −0.111285 0.993789i \(-0.535497\pi\)
−0.111285 + 0.993789i \(0.535497\pi\)
\(192\) 0 0
\(193\) −6.77454 −0.487642 −0.243821 0.969820i \(-0.578401\pi\)
−0.243821 + 0.969820i \(0.578401\pi\)
\(194\) −11.2486 −0.807601
\(195\) 0 0
\(196\) 0.0359842 0.00257030
\(197\) −13.6455 −0.972201 −0.486100 0.873903i \(-0.661581\pi\)
−0.486100 + 0.873903i \(0.661581\pi\)
\(198\) 0 0
\(199\) −6.33858 −0.449330 −0.224665 0.974436i \(-0.572129\pi\)
−0.224665 + 0.974436i \(0.572129\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.01929 0.493876
\(203\) 2.74301 0.192522
\(204\) 0 0
\(205\) 0 0
\(206\) 0.424122 0.0295500
\(207\) 0 0
\(208\) −21.2723 −1.47497
\(209\) 1.10936 0.0767358
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −0.441626 −0.0303310
\(213\) 0 0
\(214\) −19.8208 −1.35492
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −8.91917 −0.604082
\(219\) 0 0
\(220\) 0 0
\(221\) −19.1955 −1.29123
\(222\) 0 0
\(223\) 2.20126 0.147407 0.0737037 0.997280i \(-0.476518\pi\)
0.0737037 + 0.997280i \(0.476518\pi\)
\(224\) −1.06490 −0.0711515
\(225\) 0 0
\(226\) −15.1965 −1.01086
\(227\) 12.1853 0.808769 0.404384 0.914589i \(-0.367486\pi\)
0.404384 + 0.914589i \(0.367486\pi\)
\(228\) 0 0
\(229\) −3.16337 −0.209041 −0.104521 0.994523i \(-0.533331\pi\)
−0.104521 + 0.994523i \(0.533331\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.77782 −0.182373
\(233\) 23.8087 1.55976 0.779879 0.625930i \(-0.215281\pi\)
0.779879 + 0.625930i \(0.215281\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.680972 0.0443275
\(237\) 0 0
\(238\) −14.7122 −0.953653
\(239\) 6.02025 0.389417 0.194709 0.980861i \(-0.437624\pi\)
0.194709 + 0.980861i \(0.437624\pi\)
\(240\) 0 0
\(241\) 24.5517 1.58151 0.790756 0.612131i \(-0.209688\pi\)
0.790756 + 0.612131i \(0.209688\pi\)
\(242\) −4.99932 −0.321369
\(243\) 0 0
\(244\) −0.898165 −0.0574991
\(245\) 0 0
\(246\) 0 0
\(247\) −2.08178 −0.132461
\(248\) −4.05076 −0.257223
\(249\) 0 0
\(250\) 0 0
\(251\) 27.5162 1.73681 0.868403 0.495858i \(-0.165146\pi\)
0.868403 + 0.495858i \(0.165146\pi\)
\(252\) 0 0
\(253\) 14.9720 0.941284
\(254\) −12.2855 −0.770858
\(255\) 0 0
\(256\) 1.64589 0.102868
\(257\) 15.6436 0.975820 0.487910 0.872894i \(-0.337759\pi\)
0.487910 + 0.872894i \(0.337759\pi\)
\(258\) 0 0
\(259\) −18.5620 −1.15339
\(260\) 0 0
\(261\) 0 0
\(262\) −19.1364 −1.18225
\(263\) −18.6175 −1.14801 −0.574003 0.818853i \(-0.694610\pi\)
−0.574003 + 0.818853i \(0.694610\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.59557 −0.0978306
\(267\) 0 0
\(268\) −0.857969 −0.0524088
\(269\) −10.9006 −0.664620 −0.332310 0.943170i \(-0.607828\pi\)
−0.332310 + 0.943170i \(0.607828\pi\)
\(270\) 0 0
\(271\) 17.7809 1.08011 0.540056 0.841629i \(-0.318403\pi\)
0.540056 + 0.841629i \(0.318403\pi\)
\(272\) 15.4110 0.934429
\(273\) 0 0
\(274\) −26.2131 −1.58359
\(275\) 0 0
\(276\) 0 0
\(277\) 20.6612 1.24141 0.620706 0.784043i \(-0.286846\pi\)
0.620706 + 0.784043i \(0.286846\pi\)
\(278\) −8.14273 −0.488368
\(279\) 0 0
\(280\) 0 0
\(281\) −28.2652 −1.68616 −0.843080 0.537787i \(-0.819260\pi\)
−0.843080 + 0.537787i \(0.819260\pi\)
\(282\) 0 0
\(283\) 3.21139 0.190897 0.0954485 0.995434i \(-0.469571\pi\)
0.0954485 + 0.995434i \(0.469571\pi\)
\(284\) 0.779187 0.0462362
\(285\) 0 0
\(286\) −20.3078 −1.20083
\(287\) −26.8291 −1.58367
\(288\) 0 0
\(289\) −3.09362 −0.181978
\(290\) 0 0
\(291\) 0 0
\(292\) 0.739249 0.0432613
\(293\) −3.99624 −0.233463 −0.116731 0.993164i \(-0.537242\pi\)
−0.116731 + 0.993164i \(0.537242\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 18.7975 1.09258
\(297\) 0 0
\(298\) 17.1418 0.992996
\(299\) −28.0960 −1.62484
\(300\) 0 0
\(301\) −12.1576 −0.700750
\(302\) 10.3429 0.595167
\(303\) 0 0
\(304\) 1.67135 0.0958586
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0555 0.688046 0.344023 0.938961i \(-0.388210\pi\)
0.344023 + 0.938961i \(0.388210\pi\)
\(308\) −0.516595 −0.0294357
\(309\) 0 0
\(310\) 0 0
\(311\) −15.6873 −0.889544 −0.444772 0.895644i \(-0.646715\pi\)
−0.444772 + 0.895644i \(0.646715\pi\)
\(312\) 0 0
\(313\) 33.8726 1.91459 0.957297 0.289107i \(-0.0933585\pi\)
0.957297 + 0.289107i \(0.0933585\pi\)
\(314\) 6.20558 0.350201
\(315\) 0 0
\(316\) 0.972040 0.0546815
\(317\) 1.75689 0.0986770 0.0493385 0.998782i \(-0.484289\pi\)
0.0493385 + 0.998782i \(0.484289\pi\)
\(318\) 0 0
\(319\) −2.74301 −0.153579
\(320\) 0 0
\(321\) 0 0
\(322\) −21.5340 −1.20004
\(323\) 1.50817 0.0839170
\(324\) 0 0
\(325\) 0 0
\(326\) −31.1296 −1.72411
\(327\) 0 0
\(328\) 27.1695 1.50019
\(329\) 7.14744 0.394051
\(330\) 0 0
\(331\) −22.4505 −1.23399 −0.616997 0.786966i \(-0.711651\pi\)
−0.616997 + 0.786966i \(0.711651\pi\)
\(332\) 0.111456 0.00611695
\(333\) 0 0
\(334\) −23.6314 −1.29305
\(335\) 0 0
\(336\) 0 0
\(337\) −15.3886 −0.838273 −0.419136 0.907923i \(-0.637667\pi\)
−0.419136 + 0.907923i \(0.637667\pi\)
\(338\) 19.4113 1.05583
\(339\) 0 0
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −17.7634 −0.959136
\(344\) 12.3118 0.663809
\(345\) 0 0
\(346\) −6.68714 −0.359503
\(347\) −12.3504 −0.663005 −0.331503 0.943454i \(-0.607556\pi\)
−0.331503 + 0.943454i \(0.607556\pi\)
\(348\) 0 0
\(349\) −1.94446 −0.104085 −0.0520424 0.998645i \(-0.516573\pi\)
−0.0520424 + 0.998645i \(0.516573\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.06490 0.0567592
\(353\) 6.72517 0.357945 0.178972 0.983854i \(-0.442723\pi\)
0.178972 + 0.983854i \(0.442723\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.609195 0.0322873
\(357\) 0 0
\(358\) −10.9645 −0.579493
\(359\) −15.2114 −0.802826 −0.401413 0.915897i \(-0.631481\pi\)
−0.401413 + 0.915897i \(0.631481\pi\)
\(360\) 0 0
\(361\) −18.8364 −0.991391
\(362\) −30.6426 −1.61054
\(363\) 0 0
\(364\) 0.969425 0.0508117
\(365\) 0 0
\(366\) 0 0
\(367\) −13.2189 −0.690021 −0.345011 0.938599i \(-0.612125\pi\)
−0.345011 + 0.938599i \(0.612125\pi\)
\(368\) 22.5568 1.17585
\(369\) 0 0
\(370\) 0 0
\(371\) −17.6436 −0.916009
\(372\) 0 0
\(373\) −26.8050 −1.38791 −0.693956 0.720017i \(-0.744134\pi\)
−0.693956 + 0.720017i \(0.744134\pi\)
\(374\) 14.7122 0.760752
\(375\) 0 0
\(376\) −7.23813 −0.373278
\(377\) 5.14744 0.265107
\(378\) 0 0
\(379\) −24.4228 −1.25451 −0.627257 0.778813i \(-0.715822\pi\)
−0.627257 + 0.778813i \(0.715822\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −4.42412 −0.226358
\(383\) 27.2372 1.39176 0.695879 0.718159i \(-0.255015\pi\)
0.695879 + 0.718159i \(0.255015\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.74370 −0.495942
\(387\) 0 0
\(388\) −0.536968 −0.0272604
\(389\) 12.1994 0.618532 0.309266 0.950976i \(-0.399917\pi\)
0.309266 + 0.950976i \(0.399917\pi\)
\(390\) 0 0
\(391\) 20.3545 1.02937
\(392\) −1.45586 −0.0735322
\(393\) 0 0
\(394\) −19.6261 −0.988748
\(395\) 0 0
\(396\) 0 0
\(397\) 17.7254 0.889611 0.444805 0.895627i \(-0.353273\pi\)
0.444805 + 0.895627i \(0.353273\pi\)
\(398\) −9.11667 −0.456977
\(399\) 0 0
\(400\) 0 0
\(401\) 2.48773 0.124231 0.0621157 0.998069i \(-0.480215\pi\)
0.0621157 + 0.998069i \(0.480215\pi\)
\(402\) 0 0
\(403\) 7.50627 0.373914
\(404\) 0.335077 0.0166707
\(405\) 0 0
\(406\) 3.94523 0.195798
\(407\) 18.5620 0.920084
\(408\) 0 0
\(409\) 37.7194 1.86510 0.932551 0.361038i \(-0.117577\pi\)
0.932551 + 0.361038i \(0.117577\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.0202461 0.000997455 0
\(413\) 27.2058 1.33871
\(414\) 0 0
\(415\) 0 0
\(416\) −1.99835 −0.0979773
\(417\) 0 0
\(418\) 1.59557 0.0780419
\(419\) 2.29488 0.112112 0.0560561 0.998428i \(-0.482147\pi\)
0.0560561 + 0.998428i \(0.482147\pi\)
\(420\) 0 0
\(421\) 35.9662 1.75289 0.876443 0.481505i \(-0.159910\pi\)
0.876443 + 0.481505i \(0.159910\pi\)
\(422\) 2.87657 0.140029
\(423\) 0 0
\(424\) 17.8675 0.867721
\(425\) 0 0
\(426\) 0 0
\(427\) −35.8829 −1.73650
\(428\) −0.946178 −0.0457353
\(429\) 0 0
\(430\) 0 0
\(431\) −22.6974 −1.09330 −0.546648 0.837363i \(-0.684096\pi\)
−0.546648 + 0.837363i \(0.684096\pi\)
\(432\) 0 0
\(433\) 11.4108 0.548368 0.274184 0.961677i \(-0.411592\pi\)
0.274184 + 0.961677i \(0.411592\pi\)
\(434\) 5.75313 0.276159
\(435\) 0 0
\(436\) −0.425770 −0.0203907
\(437\) 2.20748 0.105598
\(438\) 0 0
\(439\) −7.36654 −0.351586 −0.175793 0.984427i \(-0.556249\pi\)
−0.175793 + 0.984427i \(0.556249\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −27.6085 −1.31320
\(443\) 36.5423 1.73617 0.868087 0.496411i \(-0.165349\pi\)
0.868087 + 0.496411i \(0.165349\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.16604 0.149916
\(447\) 0 0
\(448\) 21.1399 0.998767
\(449\) 39.6182 1.86970 0.934850 0.355043i \(-0.115534\pi\)
0.934850 + 0.355043i \(0.115534\pi\)
\(450\) 0 0
\(451\) 26.8291 1.26333
\(452\) −0.725431 −0.0341214
\(453\) 0 0
\(454\) 17.5260 0.822534
\(455\) 0 0
\(456\) 0 0
\(457\) −28.5220 −1.33420 −0.667102 0.744967i \(-0.732465\pi\)
−0.667102 + 0.744967i \(0.732465\pi\)
\(458\) −4.54982 −0.212599
\(459\) 0 0
\(460\) 0 0
\(461\) 22.6696 1.05583 0.527915 0.849297i \(-0.322974\pi\)
0.527915 + 0.849297i \(0.322974\pi\)
\(462\) 0 0
\(463\) −28.5517 −1.32691 −0.663455 0.748217i \(-0.730910\pi\)
−0.663455 + 0.748217i \(0.730910\pi\)
\(464\) −4.13260 −0.191851
\(465\) 0 0
\(466\) 34.2436 1.58630
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −34.2770 −1.58277
\(470\) 0 0
\(471\) 0 0
\(472\) −27.5510 −1.26814
\(473\) 12.1576 0.559005
\(474\) 0 0
\(475\) 0 0
\(476\) −0.702312 −0.0321904
\(477\) 0 0
\(478\) 8.65882 0.396045
\(479\) 4.43410 0.202599 0.101300 0.994856i \(-0.467700\pi\)
0.101300 + 0.994856i \(0.467700\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) 35.3123 1.60843
\(483\) 0 0
\(484\) −0.238650 −0.0108477
\(485\) 0 0
\(486\) 0 0
\(487\) −14.1035 −0.639093 −0.319546 0.947571i \(-0.603531\pi\)
−0.319546 + 0.947571i \(0.603531\pi\)
\(488\) 36.3382 1.64496
\(489\) 0 0
\(490\) 0 0
\(491\) −39.6376 −1.78882 −0.894410 0.447249i \(-0.852404\pi\)
−0.894410 + 0.447249i \(0.852404\pi\)
\(492\) 0 0
\(493\) −3.72913 −0.167951
\(494\) −2.99419 −0.134715
\(495\) 0 0
\(496\) −6.02638 −0.270592
\(497\) 31.1296 1.39635
\(498\) 0 0
\(499\) 6.33858 0.283754 0.141877 0.989884i \(-0.454686\pi\)
0.141877 + 0.989884i \(0.454686\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 39.5761 1.76637
\(503\) −19.0638 −0.850011 −0.425005 0.905191i \(-0.639728\pi\)
−0.425005 + 0.905191i \(0.639728\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 21.5340 0.957305
\(507\) 0 0
\(508\) −0.586465 −0.0260202
\(509\) 12.8145 0.567992 0.283996 0.958826i \(-0.408340\pi\)
0.283996 + 0.958826i \(0.408340\pi\)
\(510\) 0 0
\(511\) 29.5340 1.30651
\(512\) −21.3549 −0.943760
\(513\) 0 0
\(514\) 22.4999 0.992428
\(515\) 0 0
\(516\) 0 0
\(517\) −7.14744 −0.314344
\(518\) −26.6974 −1.17302
\(519\) 0 0
\(520\) 0 0
\(521\) 35.4800 1.55441 0.777204 0.629249i \(-0.216637\pi\)
0.777204 + 0.629249i \(0.216637\pi\)
\(522\) 0 0
\(523\) 14.9227 0.652525 0.326263 0.945279i \(-0.394211\pi\)
0.326263 + 0.945279i \(0.394211\pi\)
\(524\) −0.913504 −0.0399066
\(525\) 0 0
\(526\) −26.7773 −1.16754
\(527\) −5.43801 −0.236883
\(528\) 0 0
\(529\) 6.79252 0.295327
\(530\) 0 0
\(531\) 0 0
\(532\) −0.0761670 −0.00330226
\(533\) −50.3466 −2.18075
\(534\) 0 0
\(535\) 0 0
\(536\) 34.7120 1.49933
\(537\) 0 0
\(538\) −15.6781 −0.675931
\(539\) −1.43762 −0.0619227
\(540\) 0 0
\(541\) −22.8644 −0.983017 −0.491509 0.870873i \(-0.663554\pi\)
−0.491509 + 0.870873i \(0.663554\pi\)
\(542\) 25.5740 1.09850
\(543\) 0 0
\(544\) 1.44773 0.0620709
\(545\) 0 0
\(546\) 0 0
\(547\) 35.3290 1.51056 0.755279 0.655404i \(-0.227502\pi\)
0.755279 + 0.655404i \(0.227502\pi\)
\(548\) −1.25132 −0.0534539
\(549\) 0 0
\(550\) 0 0
\(551\) −0.404431 −0.0172293
\(552\) 0 0
\(553\) 38.8343 1.65140
\(554\) 29.7167 1.26254
\(555\) 0 0
\(556\) −0.388706 −0.0164848
\(557\) 32.7994 1.38976 0.694878 0.719127i \(-0.255458\pi\)
0.694878 + 0.719127i \(0.255458\pi\)
\(558\) 0 0
\(559\) −22.8145 −0.964950
\(560\) 0 0
\(561\) 0 0
\(562\) −40.6534 −1.71486
\(563\) 16.8169 0.708747 0.354374 0.935104i \(-0.384694\pi\)
0.354374 + 0.935104i \(0.384694\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.61888 0.194146
\(567\) 0 0
\(568\) −31.5246 −1.32274
\(569\) −8.88033 −0.372283 −0.186141 0.982523i \(-0.559598\pi\)
−0.186141 + 0.982523i \(0.559598\pi\)
\(570\) 0 0
\(571\) 25.1240 1.05141 0.525703 0.850668i \(-0.323802\pi\)
0.525703 + 0.850668i \(0.323802\pi\)
\(572\) −0.969425 −0.0405337
\(573\) 0 0
\(574\) −38.5879 −1.61063
\(575\) 0 0
\(576\) 0 0
\(577\) 13.9026 0.578774 0.289387 0.957212i \(-0.406549\pi\)
0.289387 + 0.957212i \(0.406549\pi\)
\(578\) −4.44950 −0.185075
\(579\) 0 0
\(580\) 0 0
\(581\) 4.45283 0.184735
\(582\) 0 0
\(583\) 17.6436 0.730723
\(584\) −29.9088 −1.23763
\(585\) 0 0
\(586\) −5.74772 −0.237436
\(587\) −20.9942 −0.866523 −0.433262 0.901268i \(-0.642637\pi\)
−0.433262 + 0.901268i \(0.642637\pi\)
\(588\) 0 0
\(589\) −0.589762 −0.0243007
\(590\) 0 0
\(591\) 0 0
\(592\) 27.9654 1.14937
\(593\) 3.54003 0.145372 0.0726859 0.997355i \(-0.476843\pi\)
0.0726859 + 0.997355i \(0.476843\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.818289 0.0335184
\(597\) 0 0
\(598\) −40.4100 −1.65249
\(599\) 32.4886 1.32745 0.663725 0.747977i \(-0.268975\pi\)
0.663725 + 0.747977i \(0.268975\pi\)
\(600\) 0 0
\(601\) −12.5620 −0.512414 −0.256207 0.966622i \(-0.582473\pi\)
−0.256207 + 0.966622i \(0.582473\pi\)
\(602\) −17.4860 −0.712677
\(603\) 0 0
\(604\) 0.493734 0.0200898
\(605\) 0 0
\(606\) 0 0
\(607\) 0.369141 0.0149830 0.00749149 0.999972i \(-0.497615\pi\)
0.00749149 + 0.999972i \(0.497615\pi\)
\(608\) 0.157009 0.00636756
\(609\) 0 0
\(610\) 0 0
\(611\) 13.4126 0.542618
\(612\) 0 0
\(613\) −44.7017 −1.80549 −0.902743 0.430181i \(-0.858450\pi\)
−0.902743 + 0.430181i \(0.858450\pi\)
\(614\) 17.3393 0.699756
\(615\) 0 0
\(616\) 20.9006 0.842108
\(617\) −5.91014 −0.237933 −0.118967 0.992898i \(-0.537958\pi\)
−0.118967 + 0.992898i \(0.537958\pi\)
\(618\) 0 0
\(619\) −44.2187 −1.77730 −0.888650 0.458586i \(-0.848356\pi\)
−0.888650 + 0.458586i \(0.848356\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −22.5628 −0.904684
\(623\) 24.3382 0.975089
\(624\) 0 0
\(625\) 0 0
\(626\) 48.7184 1.94718
\(627\) 0 0
\(628\) 0.296233 0.0118210
\(629\) 25.2351 1.00619
\(630\) 0 0
\(631\) −29.5702 −1.17717 −0.588586 0.808435i \(-0.700315\pi\)
−0.588586 + 0.808435i \(0.700315\pi\)
\(632\) −39.3271 −1.56435
\(633\) 0 0
\(634\) 2.52691 0.100356
\(635\) 0 0
\(636\) 0 0
\(637\) 2.69779 0.106890
\(638\) −3.94523 −0.156193
\(639\) 0 0
\(640\) 0 0
\(641\) −26.6335 −1.05196 −0.525979 0.850497i \(-0.676301\pi\)
−0.525979 + 0.850497i \(0.676301\pi\)
\(642\) 0 0
\(643\) −8.16597 −0.322035 −0.161017 0.986952i \(-0.551477\pi\)
−0.161017 + 0.986952i \(0.551477\pi\)
\(644\) −1.02796 −0.0405073
\(645\) 0 0
\(646\) 2.16918 0.0853452
\(647\) 20.8381 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(648\) 0 0
\(649\) −27.2058 −1.06792
\(650\) 0 0
\(651\) 0 0
\(652\) −1.48602 −0.0581970
\(653\) 13.4267 0.525428 0.262714 0.964874i \(-0.415382\pi\)
0.262714 + 0.964874i \(0.415382\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 40.4206 1.57816
\(657\) 0 0
\(658\) 10.2800 0.400758
\(659\) 42.9744 1.67405 0.837023 0.547167i \(-0.184294\pi\)
0.837023 + 0.547167i \(0.184294\pi\)
\(660\) 0 0
\(661\) −10.0178 −0.389649 −0.194824 0.980838i \(-0.562414\pi\)
−0.194824 + 0.980838i \(0.562414\pi\)
\(662\) −32.2902 −1.25500
\(663\) 0 0
\(664\) −4.50933 −0.174996
\(665\) 0 0
\(666\) 0 0
\(667\) −5.45825 −0.211344
\(668\) −1.12808 −0.0436468
\(669\) 0 0
\(670\) 0 0
\(671\) 35.8829 1.38525
\(672\) 0 0
\(673\) −23.0878 −0.889970 −0.444985 0.895538i \(-0.646791\pi\)
−0.444985 + 0.895538i \(0.646791\pi\)
\(674\) −22.1332 −0.852540
\(675\) 0 0
\(676\) 0.926627 0.0356395
\(677\) 4.95555 0.190457 0.0952287 0.995455i \(-0.469642\pi\)
0.0952287 + 0.995455i \(0.469642\pi\)
\(678\) 0 0
\(679\) −21.4526 −0.823277
\(680\) 0 0
\(681\) 0 0
\(682\) −5.75313 −0.220299
\(683\) −36.8010 −1.40815 −0.704075 0.710126i \(-0.748638\pi\)
−0.704075 + 0.710126i \(0.748638\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −25.5489 −0.975460
\(687\) 0 0
\(688\) 18.3165 0.698311
\(689\) −33.1094 −1.26137
\(690\) 0 0
\(691\) 15.8246 0.601996 0.300998 0.953625i \(-0.402680\pi\)
0.300998 + 0.953625i \(0.402680\pi\)
\(692\) −0.319221 −0.0121350
\(693\) 0 0
\(694\) −17.7634 −0.674289
\(695\) 0 0
\(696\) 0 0
\(697\) 36.4742 1.38156
\(698\) −2.79669 −0.105856
\(699\) 0 0
\(700\) 0 0
\(701\) 42.6156 1.60957 0.804785 0.593567i \(-0.202281\pi\)
0.804785 + 0.593567i \(0.202281\pi\)
\(702\) 0 0
\(703\) 2.73679 0.103220
\(704\) −21.1399 −0.796741
\(705\) 0 0
\(706\) 9.67270 0.364037
\(707\) 13.3868 0.503462
\(708\) 0 0
\(709\) 35.1795 1.32119 0.660597 0.750740i \(-0.270303\pi\)
0.660597 + 0.750740i \(0.270303\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −24.6470 −0.923686
\(713\) −7.95951 −0.298086
\(714\) 0 0
\(715\) 0 0
\(716\) −0.523408 −0.0195607
\(717\) 0 0
\(718\) −21.8783 −0.816490
\(719\) 29.4563 1.09854 0.549268 0.835646i \(-0.314907\pi\)
0.549268 + 0.835646i \(0.314907\pi\)
\(720\) 0 0
\(721\) 0.808861 0.0301236
\(722\) −27.0921 −1.00826
\(723\) 0 0
\(724\) −1.46277 −0.0543635
\(725\) 0 0
\(726\) 0 0
\(727\) −46.3391 −1.71862 −0.859311 0.511454i \(-0.829107\pi\)
−0.859311 + 0.511454i \(0.829107\pi\)
\(728\) −39.2213 −1.45364
\(729\) 0 0
\(730\) 0 0
\(731\) 16.5282 0.611319
\(732\) 0 0
\(733\) 3.73344 0.137898 0.0689489 0.997620i \(-0.478035\pi\)
0.0689489 + 0.997620i \(0.478035\pi\)
\(734\) −19.0125 −0.701765
\(735\) 0 0
\(736\) 2.11902 0.0781080
\(737\) 34.2770 1.26261
\(738\) 0 0
\(739\) −6.48021 −0.238378 −0.119189 0.992872i \(-0.538030\pi\)
−0.119189 + 0.992872i \(0.538030\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −25.3765 −0.931600
\(743\) −51.0638 −1.87335 −0.936674 0.350203i \(-0.886112\pi\)
−0.936674 + 0.350203i \(0.886112\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −38.5533 −1.41153
\(747\) 0 0
\(748\) 0.702312 0.0256791
\(749\) −37.8011 −1.38122
\(750\) 0 0
\(751\) 24.6494 0.899469 0.449735 0.893162i \(-0.351519\pi\)
0.449735 + 0.893162i \(0.351519\pi\)
\(752\) −10.7683 −0.392679
\(753\) 0 0
\(754\) 7.40348 0.269619
\(755\) 0 0
\(756\) 0 0
\(757\) 38.6833 1.40597 0.702985 0.711205i \(-0.251850\pi\)
0.702985 + 0.711205i \(0.251850\pi\)
\(758\) −35.1269 −1.27587
\(759\) 0 0
\(760\) 0 0
\(761\) 3.23725 0.117350 0.0586750 0.998277i \(-0.481312\pi\)
0.0586750 + 0.998277i \(0.481312\pi\)
\(762\) 0 0
\(763\) −17.0101 −0.615808
\(764\) −0.211192 −0.00764067
\(765\) 0 0
\(766\) 39.1749 1.41545
\(767\) 51.0534 1.84343
\(768\) 0 0
\(769\) 46.3166 1.67022 0.835111 0.550082i \(-0.185404\pi\)
0.835111 + 0.550082i \(0.185404\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.465131 −0.0167404
\(773\) −27.0341 −0.972350 −0.486175 0.873861i \(-0.661608\pi\)
−0.486175 + 0.873861i \(0.661608\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 21.7248 0.779877
\(777\) 0 0
\(778\) 17.5461 0.629059
\(779\) 3.95569 0.141727
\(780\) 0 0
\(781\) −31.1296 −1.11390
\(782\) 29.2756 1.04689
\(783\) 0 0
\(784\) −2.16591 −0.0773540
\(785\) 0 0
\(786\) 0 0
\(787\) 39.1870 1.39687 0.698434 0.715675i \(-0.253881\pi\)
0.698434 + 0.715675i \(0.253881\pi\)
\(788\) −0.936881 −0.0333750
\(789\) 0 0
\(790\) 0 0
\(791\) −28.9820 −1.03048
\(792\) 0 0
\(793\) −67.3367 −2.39120
\(794\) 25.4941 0.904752
\(795\) 0 0
\(796\) −0.435198 −0.0154252
\(797\) −53.2933 −1.88775 −0.943873 0.330307i \(-0.892848\pi\)
−0.943873 + 0.330307i \(0.892848\pi\)
\(798\) 0 0
\(799\) −9.71696 −0.343761
\(800\) 0 0
\(801\) 0 0
\(802\) 3.57807 0.126346
\(803\) −29.5340 −1.04223
\(804\) 0 0
\(805\) 0 0
\(806\) 10.7961 0.380278
\(807\) 0 0
\(808\) −13.5567 −0.476921
\(809\) −0.613214 −0.0215595 −0.0107797 0.999942i \(-0.503431\pi\)
−0.0107797 + 0.999942i \(0.503431\pi\)
\(810\) 0 0
\(811\) −0.230936 −0.00810927 −0.00405463 0.999992i \(-0.501291\pi\)
−0.00405463 + 0.999992i \(0.501291\pi\)
\(812\) 0.188331 0.00660914
\(813\) 0 0
\(814\) 26.6974 0.935744
\(815\) 0 0
\(816\) 0 0
\(817\) 1.79252 0.0627122
\(818\) 54.2511 1.89685
\(819\) 0 0
\(820\) 0 0
\(821\) −25.7254 −0.897821 −0.448911 0.893577i \(-0.648188\pi\)
−0.448911 + 0.893577i \(0.648188\pi\)
\(822\) 0 0
\(823\) 38.2258 1.33247 0.666234 0.745743i \(-0.267905\pi\)
0.666234 + 0.745743i \(0.267905\pi\)
\(824\) −0.819125 −0.0285356
\(825\) 0 0
\(826\) 39.1296 1.36149
\(827\) 24.9199 0.866551 0.433275 0.901262i \(-0.357358\pi\)
0.433275 + 0.901262i \(0.357358\pi\)
\(828\) 0 0
\(829\) −1.65301 −0.0574115 −0.0287057 0.999588i \(-0.509139\pi\)
−0.0287057 + 0.999588i \(0.509139\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 39.6705 1.37533
\(833\) −1.95445 −0.0677176
\(834\) 0 0
\(835\) 0 0
\(836\) 0.0761670 0.00263429
\(837\) 0 0
\(838\) 3.30069 0.114020
\(839\) 31.4757 1.08666 0.543331 0.839519i \(-0.317163\pi\)
0.543331 + 0.839519i \(0.317163\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 51.7296 1.78272
\(843\) 0 0
\(844\) 0.137317 0.00472666
\(845\) 0 0
\(846\) 0 0
\(847\) −9.53442 −0.327607
\(848\) 26.5817 0.912820
\(849\) 0 0
\(850\) 0 0
\(851\) 36.9361 1.26615
\(852\) 0 0
\(853\) 30.3753 1.04003 0.520015 0.854157i \(-0.325926\pi\)
0.520015 + 0.854157i \(0.325926\pi\)
\(854\) −51.6098 −1.76605
\(855\) 0 0
\(856\) 38.2808 1.30841
\(857\) −6.53423 −0.223205 −0.111602 0.993753i \(-0.535598\pi\)
−0.111602 + 0.993753i \(0.535598\pi\)
\(858\) 0 0
\(859\) 27.0220 0.921977 0.460989 0.887406i \(-0.347495\pi\)
0.460989 + 0.887406i \(0.347495\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −32.6453 −1.11190
\(863\) 31.9587 1.08789 0.543944 0.839122i \(-0.316931\pi\)
0.543944 + 0.839122i \(0.316931\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 16.4120 0.557701
\(867\) 0 0
\(868\) 0.274635 0.00932171
\(869\) −38.8343 −1.31736
\(870\) 0 0
\(871\) −64.3232 −2.17951
\(872\) 17.2260 0.583345
\(873\) 0 0
\(874\) 3.17499 0.107396
\(875\) 0 0
\(876\) 0 0
\(877\) 50.0679 1.69067 0.845336 0.534235i \(-0.179400\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(878\) −10.5952 −0.357570
\(879\) 0 0
\(880\) 0 0
\(881\) 13.6817 0.460947 0.230474 0.973079i \(-0.425972\pi\)
0.230474 + 0.973079i \(0.425972\pi\)
\(882\) 0 0
\(883\) −29.5693 −0.995087 −0.497543 0.867439i \(-0.665764\pi\)
−0.497543 + 0.867439i \(0.665764\pi\)
\(884\) −1.31793 −0.0443269
\(885\) 0 0
\(886\) 52.5581 1.76572
\(887\) 4.35130 0.146102 0.0730512 0.997328i \(-0.476726\pi\)
0.0730512 + 0.997328i \(0.476726\pi\)
\(888\) 0 0
\(889\) −23.4301 −0.785820
\(890\) 0 0
\(891\) 0 0
\(892\) 0.151136 0.00506040
\(893\) −1.05382 −0.0352648
\(894\) 0 0
\(895\) 0 0
\(896\) 32.5350 1.08692
\(897\) 0 0
\(898\) 56.9822 1.90152
\(899\) 1.45825 0.0486354
\(900\) 0 0
\(901\) 23.9865 0.799105
\(902\) 38.5879 1.28484
\(903\) 0 0
\(904\) 29.3497 0.976157
\(905\) 0 0
\(906\) 0 0
\(907\) 52.3965 1.73980 0.869899 0.493230i \(-0.164184\pi\)
0.869899 + 0.493230i \(0.164184\pi\)
\(908\) 0.836629 0.0277645
\(909\) 0 0
\(910\) 0 0
\(911\) −7.85085 −0.260110 −0.130055 0.991507i \(-0.541515\pi\)
−0.130055 + 0.991507i \(0.541515\pi\)
\(912\) 0 0
\(913\) −4.45283 −0.147367
\(914\) −41.0227 −1.35691
\(915\) 0 0
\(916\) −0.217193 −0.00717625
\(917\) −36.4958 −1.20520
\(918\) 0 0
\(919\) 13.4979 0.445253 0.222627 0.974904i \(-0.428537\pi\)
0.222627 + 0.974904i \(0.428537\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 32.6054 1.07380
\(923\) 58.4168 1.92281
\(924\) 0 0
\(925\) 0 0
\(926\) −41.0654 −1.34949
\(927\) 0 0
\(928\) −0.388222 −0.0127440
\(929\) −11.6177 −0.381165 −0.190583 0.981671i \(-0.561038\pi\)
−0.190583 + 0.981671i \(0.561038\pi\)
\(930\) 0 0
\(931\) −0.211963 −0.00694682
\(932\) 1.63467 0.0535455
\(933\) 0 0
\(934\) −11.5063 −0.376497
\(935\) 0 0
\(936\) 0 0
\(937\) 42.6740 1.39410 0.697049 0.717024i \(-0.254496\pi\)
0.697049 + 0.717024i \(0.254496\pi\)
\(938\) −49.3001 −1.60971
\(939\) 0 0
\(940\) 0 0
\(941\) 5.91479 0.192817 0.0964083 0.995342i \(-0.469265\pi\)
0.0964083 + 0.995342i \(0.469265\pi\)
\(942\) 0 0
\(943\) 53.3866 1.73851
\(944\) −40.9881 −1.33405
\(945\) 0 0
\(946\) 17.4860 0.568520
\(947\) −45.0915 −1.46528 −0.732639 0.680618i \(-0.761711\pi\)
−0.732639 + 0.680618i \(0.761711\pi\)
\(948\) 0 0
\(949\) 55.4226 1.79909
\(950\) 0 0
\(951\) 0 0
\(952\) 28.4144 0.920915
\(953\) 30.3166 0.982053 0.491026 0.871145i \(-0.336622\pi\)
0.491026 + 0.871145i \(0.336622\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.413342 0.0133684
\(957\) 0 0
\(958\) 6.37750 0.206048
\(959\) −49.9921 −1.61433
\(960\) 0 0
\(961\) −28.8735 −0.931403
\(962\) −50.0995 −1.61527
\(963\) 0 0
\(964\) 1.68569 0.0542923
\(965\) 0 0
\(966\) 0 0
\(967\) −5.99809 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(968\) 9.65540 0.310336
\(969\) 0 0
\(970\) 0 0
\(971\) −28.5140 −0.915057 −0.457529 0.889195i \(-0.651265\pi\)
−0.457529 + 0.889195i \(0.651265\pi\)
\(972\) 0 0
\(973\) −15.5293 −0.497848
\(974\) −20.2849 −0.649970
\(975\) 0 0
\(976\) 54.0610 1.73045
\(977\) −5.53813 −0.177180 −0.0885902 0.996068i \(-0.528236\pi\)
−0.0885902 + 0.996068i \(0.528236\pi\)
\(978\) 0 0
\(979\) −24.3382 −0.777852
\(980\) 0 0
\(981\) 0 0
\(982\) −57.0101 −1.81926
\(983\) 7.37086 0.235094 0.117547 0.993067i \(-0.462497\pi\)
0.117547 + 0.993067i \(0.462497\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −5.36354 −0.170810
\(987\) 0 0
\(988\) −0.142932 −0.00454729
\(989\) 24.1921 0.769263
\(990\) 0 0
\(991\) 3.68167 0.116952 0.0584760 0.998289i \(-0.481376\pi\)
0.0584760 + 0.998289i \(0.481376\pi\)
\(992\) −0.566126 −0.0179745
\(993\) 0 0
\(994\) 44.7732 1.42012
\(995\) 0 0
\(996\) 0 0
\(997\) −36.8747 −1.16783 −0.583916 0.811814i \(-0.698480\pi\)
−0.583916 + 0.811814i \(0.698480\pi\)
\(998\) 9.11667 0.288583
\(999\) 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 6525.2.a.bi.1.4 4
3.2 odd 2 2175.2.a.v.1.1 4
5.4 even 2 1305.2.a.r.1.1 4
15.2 even 4 2175.2.c.n.349.3 8
15.8 even 4 2175.2.c.n.349.6 8
15.14 odd 2 435.2.a.j.1.4 4
60.59 even 2 6960.2.a.co.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.4 4 15.14 odd 2
1305.2.a.r.1.1 4 5.4 even 2
2175.2.a.v.1.1 4 3.2 odd 2
2175.2.c.n.349.3 8 15.2 even 4
2175.2.c.n.349.6 8 15.8 even 4
6525.2.a.bi.1.4 4 1.1 even 1 trivial
6960.2.a.co.1.4 4 60.59 even 2