Properties

Label 6525.2.a.bj.1.3
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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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: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) 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)
 
\(f(q)\) \(=\) \(q+0.517638 q^{2} -1.73205 q^{4} +2.44949 q^{7} -1.93185 q^{8} +1.26795 q^{11} +1.79315 q^{13} +1.26795 q^{14} +2.46410 q^{16} +1.41421 q^{17} -3.26795 q^{19} +0.656339 q^{22} -6.31319 q^{23} +0.928203 q^{26} -4.24264 q^{28} -1.00000 q^{29} -8.73205 q^{31} +5.13922 q^{32} +0.732051 q^{34} -9.14162 q^{37} -1.69161 q^{38} +6.92820 q^{41} -9.14162 q^{43} -2.19615 q^{44} -3.26795 q^{46} +1.41421 q^{47} -1.00000 q^{49} -3.10583 q^{52} +5.93426 q^{53} -4.73205 q^{56} -0.517638 q^{58} +10.3923 q^{59} +2.92820 q^{61} -4.52004 q^{62} -2.26795 q^{64} -4.24264 q^{67} -2.44949 q^{68} -3.46410 q^{71} +7.34847 q^{73} -4.73205 q^{74} +5.66025 q^{76} +3.10583 q^{77} -4.19615 q^{79} +3.58630 q^{82} +10.1769 q^{83} -4.73205 q^{86} -2.44949 q^{88} -10.3923 q^{89} +4.39230 q^{91} +10.9348 q^{92} +0.732051 q^{94} +10.9348 q^{97} -0.517638 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{11} + 12 q^{14} - 4 q^{16} - 20 q^{19} - 24 q^{26} - 4 q^{29} - 28 q^{31} - 4 q^{34} + 12 q^{44} - 20 q^{46} - 4 q^{49} - 12 q^{56} - 16 q^{61} - 16 q^{64} - 12 q^{74} - 12 q^{76} + 4 q^{79}+ \cdots - 4 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.517638 0.366025 0.183013 0.983111i \(-0.441415\pi\)
0.183013 + 0.983111i \(0.441415\pi\)
\(3\) 0 0
\(4\) −1.73205 −0.866025
\(5\) 0 0
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) −1.93185 −0.683013
\(9\) 0 0
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 1.79315 0.497331 0.248665 0.968589i \(-0.420008\pi\)
0.248665 + 0.968589i \(0.420008\pi\)
\(14\) 1.26795 0.338874
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) −3.26795 −0.749719 −0.374859 0.927082i \(-0.622309\pi\)
−0.374859 + 0.927082i \(0.622309\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.656339 0.139932
\(23\) −6.31319 −1.31639 −0.658196 0.752847i \(-0.728680\pi\)
−0.658196 + 0.752847i \(0.728680\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.928203 0.182036
\(27\) 0 0
\(28\) −4.24264 −0.801784
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.73205 −1.56832 −0.784161 0.620557i \(-0.786907\pi\)
−0.784161 + 0.620557i \(0.786907\pi\)
\(32\) 5.13922 0.908494
\(33\) 0 0
\(34\) 0.732051 0.125546
\(35\) 0 0
\(36\) 0 0
\(37\) −9.14162 −1.50287 −0.751437 0.659805i \(-0.770639\pi\)
−0.751437 + 0.659805i \(0.770639\pi\)
\(38\) −1.69161 −0.274416
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −9.14162 −1.39408 −0.697042 0.717030i \(-0.745501\pi\)
−0.697042 + 0.717030i \(0.745501\pi\)
\(44\) −2.19615 −0.331082
\(45\) 0 0
\(46\) −3.26795 −0.481833
\(47\) 1.41421 0.206284 0.103142 0.994667i \(-0.467110\pi\)
0.103142 + 0.994667i \(0.467110\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −3.10583 −0.430701
\(53\) 5.93426 0.815133 0.407566 0.913176i \(-0.366378\pi\)
0.407566 + 0.913176i \(0.366378\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) −0.517638 −0.0679692
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 2.92820 0.374918 0.187459 0.982272i \(-0.439975\pi\)
0.187459 + 0.982272i \(0.439975\pi\)
\(62\) −4.52004 −0.574046
\(63\) 0 0
\(64\) −2.26795 −0.283494
\(65\) 0 0
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) −2.44949 −0.297044
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 7.34847 0.860073 0.430037 0.902811i \(-0.358501\pi\)
0.430037 + 0.902811i \(0.358501\pi\)
\(74\) −4.73205 −0.550090
\(75\) 0 0
\(76\) 5.66025 0.649276
\(77\) 3.10583 0.353942
\(78\) 0 0
\(79\) −4.19615 −0.472104 −0.236052 0.971740i \(-0.575854\pi\)
−0.236052 + 0.971740i \(0.575854\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.58630 0.396041
\(83\) 10.1769 1.11706 0.558530 0.829484i \(-0.311366\pi\)
0.558530 + 0.829484i \(0.311366\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.73205 −0.510270
\(87\) 0 0
\(88\) −2.44949 −0.261116
\(89\) −10.3923 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 4.39230 0.460439
\(92\) 10.9348 1.14003
\(93\) 0 0
\(94\) 0.732051 0.0755053
\(95\) 0 0
\(96\) 0 0
\(97\) 10.9348 1.11026 0.555129 0.831764i \(-0.312669\pi\)
0.555129 + 0.831764i \(0.312669\pi\)
\(98\) −0.517638 −0.0522893
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −9.14162 −0.900751 −0.450375 0.892839i \(-0.648710\pi\)
−0.450375 + 0.892839i \(0.648710\pi\)
\(104\) −3.46410 −0.339683
\(105\) 0 0
\(106\) 3.07180 0.298359
\(107\) −18.6622 −1.80414 −0.902070 0.431589i \(-0.857953\pi\)
−0.902070 + 0.431589i \(0.857953\pi\)
\(108\) 0 0
\(109\) 17.8564 1.71033 0.855167 0.518353i \(-0.173455\pi\)
0.855167 + 0.518353i \(0.173455\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.03579 0.570329
\(113\) −13.0053 −1.22344 −0.611719 0.791075i \(-0.709522\pi\)
−0.611719 + 0.791075i \(0.709522\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.73205 0.160817
\(117\) 0 0
\(118\) 5.37945 0.495219
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 1.51575 0.137230
\(123\) 0 0
\(124\) 15.1244 1.35821
\(125\) 0 0
\(126\) 0 0
\(127\) −4.24264 −0.376473 −0.188237 0.982124i \(-0.560277\pi\)
−0.188237 + 0.982124i \(0.560277\pi\)
\(128\) −11.4524 −1.01226
\(129\) 0 0
\(130\) 0 0
\(131\) 4.73205 0.413441 0.206721 0.978400i \(-0.433721\pi\)
0.206721 + 0.978400i \(0.433721\pi\)
\(132\) 0 0
\(133\) −8.00481 −0.694105
\(134\) −2.19615 −0.189719
\(135\) 0 0
\(136\) −2.73205 −0.234271
\(137\) 14.7985 1.26432 0.632159 0.774838i \(-0.282169\pi\)
0.632159 + 0.774838i \(0.282169\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.79315 −0.150478
\(143\) 2.27362 0.190130
\(144\) 0 0
\(145\) 0 0
\(146\) 3.80385 0.314809
\(147\) 0 0
\(148\) 15.8338 1.30153
\(149\) 0.928203 0.0760414 0.0380207 0.999277i \(-0.487895\pi\)
0.0380207 + 0.999277i \(0.487895\pi\)
\(150\) 0 0
\(151\) −18.7846 −1.52867 −0.764335 0.644819i \(-0.776933\pi\)
−0.764335 + 0.644819i \(0.776933\pi\)
\(152\) 6.31319 0.512068
\(153\) 0 0
\(154\) 1.60770 0.129552
\(155\) 0 0
\(156\) 0 0
\(157\) −10.9348 −0.872690 −0.436345 0.899780i \(-0.643727\pi\)
−0.436345 + 0.899780i \(0.643727\pi\)
\(158\) −2.17209 −0.172802
\(159\) 0 0
\(160\) 0 0
\(161\) −15.4641 −1.21874
\(162\) 0 0
\(163\) −14.0406 −1.09974 −0.549872 0.835249i \(-0.685324\pi\)
−0.549872 + 0.835249i \(0.685324\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 5.26795 0.408872
\(167\) 11.2122 0.867624 0.433812 0.901003i \(-0.357168\pi\)
0.433812 + 0.901003i \(0.357168\pi\)
\(168\) 0 0
\(169\) −9.78461 −0.752662
\(170\) 0 0
\(171\) 0 0
\(172\) 15.8338 1.20731
\(173\) 4.62158 0.351372 0.175686 0.984446i \(-0.443786\pi\)
0.175686 + 0.984446i \(0.443786\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.12436 0.235507
\(177\) 0 0
\(178\) −5.37945 −0.403207
\(179\) −2.53590 −0.189542 −0.0947710 0.995499i \(-0.530212\pi\)
−0.0947710 + 0.995499i \(0.530212\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 2.27362 0.168532
\(183\) 0 0
\(184\) 12.1962 0.899112
\(185\) 0 0
\(186\) 0 0
\(187\) 1.79315 0.131128
\(188\) −2.44949 −0.178647
\(189\) 0 0
\(190\) 0 0
\(191\) −15.1244 −1.09436 −0.547180 0.837015i \(-0.684299\pi\)
−0.547180 + 0.837015i \(0.684299\pi\)
\(192\) 0 0
\(193\) 0.656339 0.0472443 0.0236222 0.999721i \(-0.492480\pi\)
0.0236222 + 0.999721i \(0.492480\pi\)
\(194\) 5.66025 0.406383
\(195\) 0 0
\(196\) 1.73205 0.123718
\(197\) −22.4243 −1.59767 −0.798834 0.601551i \(-0.794550\pi\)
−0.798834 + 0.601551i \(0.794550\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.10583 0.218525
\(203\) −2.44949 −0.171920
\(204\) 0 0
\(205\) 0 0
\(206\) −4.73205 −0.329698
\(207\) 0 0
\(208\) 4.41851 0.306368
\(209\) −4.14359 −0.286618
\(210\) 0 0
\(211\) −14.7321 −1.01420 −0.507098 0.861888i \(-0.669282\pi\)
−0.507098 + 0.861888i \(0.669282\pi\)
\(212\) −10.2784 −0.705926
\(213\) 0 0
\(214\) −9.66025 −0.660361
\(215\) 0 0
\(216\) 0 0
\(217\) −21.3891 −1.45198
\(218\) 9.24316 0.626026
\(219\) 0 0
\(220\) 0 0
\(221\) 2.53590 0.170583
\(222\) 0 0
\(223\) 15.3533 1.02813 0.514066 0.857751i \(-0.328139\pi\)
0.514066 + 0.857751i \(0.328139\pi\)
\(224\) 12.5885 0.841102
\(225\) 0 0
\(226\) −6.73205 −0.447809
\(227\) −20.4553 −1.35767 −0.678834 0.734292i \(-0.737514\pi\)
−0.678834 + 0.734292i \(0.737514\pi\)
\(228\) 0 0
\(229\) 18.7846 1.24132 0.620661 0.784079i \(-0.286864\pi\)
0.620661 + 0.784079i \(0.286864\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.93185 0.126832
\(233\) −3.38323 −0.221643 −0.110821 0.993840i \(-0.535348\pi\)
−0.110821 + 0.993840i \(0.535348\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −18.0000 −1.17170
\(237\) 0 0
\(238\) 1.79315 0.116233
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) 11.4641 0.738468 0.369234 0.929337i \(-0.379620\pi\)
0.369234 + 0.929337i \(0.379620\pi\)
\(242\) −4.86181 −0.312529
\(243\) 0 0
\(244\) −5.07180 −0.324689
\(245\) 0 0
\(246\) 0 0
\(247\) −5.85993 −0.372858
\(248\) 16.8690 1.07118
\(249\) 0 0
\(250\) 0 0
\(251\) −15.1244 −0.954641 −0.477320 0.878729i \(-0.658392\pi\)
−0.477320 + 0.878729i \(0.658392\pi\)
\(252\) 0 0
\(253\) −8.00481 −0.503258
\(254\) −2.19615 −0.137799
\(255\) 0 0
\(256\) −1.39230 −0.0870191
\(257\) −9.52056 −0.593876 −0.296938 0.954897i \(-0.595965\pi\)
−0.296938 + 0.954897i \(0.595965\pi\)
\(258\) 0 0
\(259\) −22.3923 −1.39139
\(260\) 0 0
\(261\) 0 0
\(262\) 2.44949 0.151330
\(263\) 4.79744 0.295823 0.147912 0.989001i \(-0.452745\pi\)
0.147912 + 0.989001i \(0.452745\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.14359 −0.254060
\(267\) 0 0
\(268\) 7.34847 0.448879
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −26.7321 −1.62386 −0.811928 0.583757i \(-0.801582\pi\)
−0.811928 + 0.583757i \(0.801582\pi\)
\(272\) 3.48477 0.211295
\(273\) 0 0
\(274\) 7.66025 0.462773
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7637 −1.12740 −0.563701 0.825979i \(-0.690623\pi\)
−0.563701 + 0.825979i \(0.690623\pi\)
\(278\) −4.14110 −0.248367
\(279\) 0 0
\(280\) 0 0
\(281\) −32.7846 −1.95577 −0.977883 0.209153i \(-0.932929\pi\)
−0.977883 + 0.209153i \(0.932929\pi\)
\(282\) 0 0
\(283\) 27.4249 1.63024 0.815119 0.579293i \(-0.196671\pi\)
0.815119 + 0.579293i \(0.196671\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 1.17691 0.0695924
\(287\) 16.9706 1.00174
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) −12.7279 −0.744845
\(293\) 30.7338 1.79549 0.897743 0.440520i \(-0.145206\pi\)
0.897743 + 0.440520i \(0.145206\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.6603 1.02648
\(297\) 0 0
\(298\) 0.480473 0.0278331
\(299\) −11.3205 −0.654682
\(300\) 0 0
\(301\) −22.3923 −1.29067
\(302\) −9.72363 −0.559532
\(303\) 0 0
\(304\) −8.05256 −0.461846
\(305\) 0 0
\(306\) 0 0
\(307\) −10.4543 −0.596658 −0.298329 0.954463i \(-0.596429\pi\)
−0.298329 + 0.954463i \(0.596429\pi\)
\(308\) −5.37945 −0.306523
\(309\) 0 0
\(310\) 0 0
\(311\) 6.58846 0.373597 0.186799 0.982398i \(-0.440189\pi\)
0.186799 + 0.982398i \(0.440189\pi\)
\(312\) 0 0
\(313\) 23.6627 1.33749 0.668747 0.743490i \(-0.266831\pi\)
0.668747 + 0.743490i \(0.266831\pi\)
\(314\) −5.66025 −0.319427
\(315\) 0 0
\(316\) 7.26795 0.408854
\(317\) −5.75839 −0.323423 −0.161712 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(318\) 0 0
\(319\) −1.26795 −0.0709915
\(320\) 0 0
\(321\) 0 0
\(322\) −8.00481 −0.446091
\(323\) −4.62158 −0.257151
\(324\) 0 0
\(325\) 0 0
\(326\) −7.26795 −0.402534
\(327\) 0 0
\(328\) −13.3843 −0.739022
\(329\) 3.46410 0.190982
\(330\) 0 0
\(331\) −18.1962 −1.00015 −0.500075 0.865982i \(-0.666694\pi\)
−0.500075 + 0.865982i \(0.666694\pi\)
\(332\) −17.6269 −0.967402
\(333\) 0 0
\(334\) 5.80385 0.317572
\(335\) 0 0
\(336\) 0 0
\(337\) 27.9053 1.52010 0.760050 0.649864i \(-0.225174\pi\)
0.760050 + 0.649864i \(0.225174\pi\)
\(338\) −5.06489 −0.275494
\(339\) 0 0
\(340\) 0 0
\(341\) −11.0718 −0.599571
\(342\) 0 0
\(343\) −19.5959 −1.05808
\(344\) 17.6603 0.952177
\(345\) 0 0
\(346\) 2.39230 0.128611
\(347\) 21.4906 1.15368 0.576838 0.816859i \(-0.304286\pi\)
0.576838 + 0.816859i \(0.304286\pi\)
\(348\) 0 0
\(349\) −22.7846 −1.21963 −0.609816 0.792543i \(-0.708757\pi\)
−0.609816 + 0.792543i \(0.708757\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 6.51626 0.347318
\(353\) −31.9449 −1.70026 −0.850128 0.526576i \(-0.823475\pi\)
−0.850128 + 0.526576i \(0.823475\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) −1.31268 −0.0693772
\(359\) 7.26795 0.383588 0.191794 0.981435i \(-0.438570\pi\)
0.191794 + 0.981435i \(0.438570\pi\)
\(360\) 0 0
\(361\) −8.32051 −0.437921
\(362\) −8.28221 −0.435303
\(363\) 0 0
\(364\) −7.60770 −0.398752
\(365\) 0 0
\(366\) 0 0
\(367\) 33.6365 1.75581 0.877906 0.478833i \(-0.158940\pi\)
0.877906 + 0.478833i \(0.158940\pi\)
\(368\) −15.5563 −0.810931
\(369\) 0 0
\(370\) 0 0
\(371\) 14.5359 0.754666
\(372\) 0 0
\(373\) 19.5959 1.01464 0.507319 0.861758i \(-0.330637\pi\)
0.507319 + 0.861758i \(0.330637\pi\)
\(374\) 0.928203 0.0479962
\(375\) 0 0
\(376\) −2.73205 −0.140895
\(377\) −1.79315 −0.0923520
\(378\) 0 0
\(379\) −16.1962 −0.831940 −0.415970 0.909378i \(-0.636558\pi\)
−0.415970 + 0.909378i \(0.636558\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −7.82894 −0.400564
\(383\) −0.933740 −0.0477119 −0.0238559 0.999715i \(-0.507594\pi\)
−0.0238559 + 0.999715i \(0.507594\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0.339746 0.0172926
\(387\) 0 0
\(388\) −18.9396 −0.961511
\(389\) 20.7846 1.05382 0.526911 0.849921i \(-0.323350\pi\)
0.526911 + 0.849921i \(0.323350\pi\)
\(390\) 0 0
\(391\) −8.92820 −0.451519
\(392\) 1.93185 0.0975732
\(393\) 0 0
\(394\) −11.6077 −0.584787
\(395\) 0 0
\(396\) 0 0
\(397\) −4.89898 −0.245873 −0.122936 0.992415i \(-0.539231\pi\)
−0.122936 + 0.992415i \(0.539231\pi\)
\(398\) −7.24693 −0.363256
\(399\) 0 0
\(400\) 0 0
\(401\) −37.8564 −1.89046 −0.945229 0.326407i \(-0.894162\pi\)
−0.945229 + 0.326407i \(0.894162\pi\)
\(402\) 0 0
\(403\) −15.6579 −0.779975
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) −1.26795 −0.0629273
\(407\) −11.5911 −0.574550
\(408\) 0 0
\(409\) 9.07180 0.448571 0.224286 0.974523i \(-0.427995\pi\)
0.224286 + 0.974523i \(0.427995\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 15.8338 0.780073
\(413\) 25.4558 1.25260
\(414\) 0 0
\(415\) 0 0
\(416\) 9.21539 0.451822
\(417\) 0 0
\(418\) −2.14488 −0.104910
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 5.46410 0.266304 0.133152 0.991096i \(-0.457490\pi\)
0.133152 + 0.991096i \(0.457490\pi\)
\(422\) −7.62587 −0.371222
\(423\) 0 0
\(424\) −11.4641 −0.556746
\(425\) 0 0
\(426\) 0 0
\(427\) 7.17260 0.347107
\(428\) 32.3238 1.56243
\(429\) 0 0
\(430\) 0 0
\(431\) 37.1769 1.79075 0.895374 0.445314i \(-0.146908\pi\)
0.895374 + 0.445314i \(0.146908\pi\)
\(432\) 0 0
\(433\) 2.92996 0.140805 0.0704025 0.997519i \(-0.477572\pi\)
0.0704025 + 0.997519i \(0.477572\pi\)
\(434\) −11.0718 −0.531463
\(435\) 0 0
\(436\) −30.9282 −1.48119
\(437\) 20.6312 0.986924
\(438\) 0 0
\(439\) −7.07180 −0.337518 −0.168759 0.985657i \(-0.553976\pi\)
−0.168759 + 0.985657i \(0.553976\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.31268 0.0624377
\(443\) 18.1817 0.863839 0.431919 0.901912i \(-0.357836\pi\)
0.431919 + 0.901912i \(0.357836\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.94744 0.376322
\(447\) 0 0
\(448\) −5.55532 −0.262464
\(449\) 25.8564 1.22024 0.610120 0.792309i \(-0.291121\pi\)
0.610120 + 0.792309i \(0.291121\pi\)
\(450\) 0 0
\(451\) 8.78461 0.413651
\(452\) 22.5259 1.05953
\(453\) 0 0
\(454\) −10.5885 −0.496941
\(455\) 0 0
\(456\) 0 0
\(457\) 23.6627 1.10689 0.553447 0.832884i \(-0.313312\pi\)
0.553447 + 0.832884i \(0.313312\pi\)
\(458\) 9.72363 0.454355
\(459\) 0 0
\(460\) 0 0
\(461\) −24.9282 −1.16102 −0.580511 0.814252i \(-0.697147\pi\)
−0.580511 + 0.814252i \(0.697147\pi\)
\(462\) 0 0
\(463\) −6.86800 −0.319183 −0.159591 0.987183i \(-0.551018\pi\)
−0.159591 + 0.987183i \(0.551018\pi\)
\(464\) −2.46410 −0.114393
\(465\) 0 0
\(466\) −1.75129 −0.0811269
\(467\) 21.0101 0.972233 0.486116 0.873894i \(-0.338413\pi\)
0.486116 + 0.873894i \(0.338413\pi\)
\(468\) 0 0
\(469\) −10.3923 −0.479872
\(470\) 0 0
\(471\) 0 0
\(472\) −20.0764 −0.924091
\(473\) −11.5911 −0.532960
\(474\) 0 0
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) −5.37945 −0.246050
\(479\) −5.66025 −0.258624 −0.129312 0.991604i \(-0.541277\pi\)
−0.129312 + 0.991604i \(0.541277\pi\)
\(480\) 0 0
\(481\) −16.3923 −0.747425
\(482\) 5.93426 0.270298
\(483\) 0 0
\(484\) 16.2679 0.739452
\(485\) 0 0
\(486\) 0 0
\(487\) 10.9348 0.495502 0.247751 0.968824i \(-0.420309\pi\)
0.247751 + 0.968824i \(0.420309\pi\)
\(488\) −5.65685 −0.256074
\(489\) 0 0
\(490\) 0 0
\(491\) −7.26795 −0.327998 −0.163999 0.986461i \(-0.552439\pi\)
−0.163999 + 0.986461i \(0.552439\pi\)
\(492\) 0 0
\(493\) −1.41421 −0.0636930
\(494\) −3.03332 −0.136476
\(495\) 0 0
\(496\) −21.5167 −0.966127
\(497\) −8.48528 −0.380617
\(498\) 0 0
\(499\) −1.07180 −0.0479802 −0.0239901 0.999712i \(-0.507637\pi\)
−0.0239901 + 0.999712i \(0.507637\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7.82894 −0.349423
\(503\) −12.1731 −0.542773 −0.271386 0.962471i \(-0.587482\pi\)
−0.271386 + 0.962471i \(0.587482\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.14359 −0.184205
\(507\) 0 0
\(508\) 7.34847 0.326036
\(509\) −33.4641 −1.48327 −0.741635 0.670804i \(-0.765949\pi\)
−0.741635 + 0.670804i \(0.765949\pi\)
\(510\) 0 0
\(511\) 18.0000 0.796273
\(512\) 22.1841 0.980408
\(513\) 0 0
\(514\) −4.92820 −0.217374
\(515\) 0 0
\(516\) 0 0
\(517\) 1.79315 0.0788627
\(518\) −11.5911 −0.509284
\(519\) 0 0
\(520\) 0 0
\(521\) −4.39230 −0.192430 −0.0962152 0.995361i \(-0.530674\pi\)
−0.0962152 + 0.995361i \(0.530674\pi\)
\(522\) 0 0
\(523\) −1.61729 −0.0707190 −0.0353595 0.999375i \(-0.511258\pi\)
−0.0353595 + 0.999375i \(0.511258\pi\)
\(524\) −8.19615 −0.358051
\(525\) 0 0
\(526\) 2.48334 0.108279
\(527\) −12.3490 −0.537930
\(528\) 0 0
\(529\) 16.8564 0.732887
\(530\) 0 0
\(531\) 0 0
\(532\) 13.8647 0.601112
\(533\) 12.4233 0.538113
\(534\) 0 0
\(535\) 0 0
\(536\) 8.19615 0.354020
\(537\) 0 0
\(538\) 6.21166 0.267804
\(539\) −1.26795 −0.0546144
\(540\) 0 0
\(541\) 42.3923 1.82259 0.911294 0.411757i \(-0.135085\pi\)
0.911294 + 0.411757i \(0.135085\pi\)
\(542\) −13.8375 −0.594373
\(543\) 0 0
\(544\) 7.26795 0.311611
\(545\) 0 0
\(546\) 0 0
\(547\) 2.44949 0.104733 0.0523663 0.998628i \(-0.483324\pi\)
0.0523663 + 0.998628i \(0.483324\pi\)
\(548\) −25.6317 −1.09493
\(549\) 0 0
\(550\) 0 0
\(551\) 3.26795 0.139219
\(552\) 0 0
\(553\) −10.2784 −0.437083
\(554\) −9.71281 −0.412658
\(555\) 0 0
\(556\) 13.8564 0.587643
\(557\) −0.554803 −0.0235078 −0.0117539 0.999931i \(-0.503741\pi\)
−0.0117539 + 0.999931i \(0.503741\pi\)
\(558\) 0 0
\(559\) −16.3923 −0.693321
\(560\) 0 0
\(561\) 0 0
\(562\) −16.9706 −0.715860
\(563\) 13.2827 0.559800 0.279900 0.960029i \(-0.409699\pi\)
0.279900 + 0.960029i \(0.409699\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.1962 0.596709
\(567\) 0 0
\(568\) 6.69213 0.280796
\(569\) −21.4641 −0.899822 −0.449911 0.893073i \(-0.648544\pi\)
−0.449911 + 0.893073i \(0.648544\pi\)
\(570\) 0 0
\(571\) −39.3205 −1.64551 −0.822756 0.568395i \(-0.807565\pi\)
−0.822756 + 0.568395i \(0.807565\pi\)
\(572\) −3.93803 −0.164657
\(573\) 0 0
\(574\) 8.78461 0.366663
\(575\) 0 0
\(576\) 0 0
\(577\) −9.14162 −0.380571 −0.190285 0.981729i \(-0.560941\pi\)
−0.190285 + 0.981729i \(0.560941\pi\)
\(578\) −7.76457 −0.322964
\(579\) 0 0
\(580\) 0 0
\(581\) 24.9282 1.03420
\(582\) 0 0
\(583\) 7.52433 0.311626
\(584\) −14.1962 −0.587441
\(585\) 0 0
\(586\) 15.9090 0.657193
\(587\) −15.5563 −0.642079 −0.321040 0.947066i \(-0.604032\pi\)
−0.321040 + 0.947066i \(0.604032\pi\)
\(588\) 0 0
\(589\) 28.5359 1.17580
\(590\) 0 0
\(591\) 0 0
\(592\) −22.5259 −0.925808
\(593\) 25.5302 1.04840 0.524199 0.851596i \(-0.324365\pi\)
0.524199 + 0.851596i \(0.324365\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.60770 −0.0658538
\(597\) 0 0
\(598\) −5.85993 −0.239630
\(599\) −1.26795 −0.0518070 −0.0259035 0.999664i \(-0.508246\pi\)
−0.0259035 + 0.999664i \(0.508246\pi\)
\(600\) 0 0
\(601\) 16.7846 0.684659 0.342329 0.939580i \(-0.388784\pi\)
0.342329 + 0.939580i \(0.388784\pi\)
\(602\) −11.5911 −0.472418
\(603\) 0 0
\(604\) 32.5359 1.32387
\(605\) 0 0
\(606\) 0 0
\(607\) −16.3142 −0.662174 −0.331087 0.943600i \(-0.607415\pi\)
−0.331087 + 0.943600i \(0.607415\pi\)
\(608\) −16.7947 −0.681115
\(609\) 0 0
\(610\) 0 0
\(611\) 2.53590 0.102591
\(612\) 0 0
\(613\) −18.2832 −0.738453 −0.369227 0.929339i \(-0.620377\pi\)
−0.369227 + 0.929339i \(0.620377\pi\)
\(614\) −5.41154 −0.218392
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −32.0464 −1.29014 −0.645071 0.764123i \(-0.723172\pi\)
−0.645071 + 0.764123i \(0.723172\pi\)
\(618\) 0 0
\(619\) −23.8038 −0.956757 −0.478379 0.878154i \(-0.658775\pi\)
−0.478379 + 0.878154i \(0.658775\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.41044 0.136746
\(623\) −25.4558 −1.01987
\(624\) 0 0
\(625\) 0 0
\(626\) 12.2487 0.489557
\(627\) 0 0
\(628\) 18.9396 0.755771
\(629\) −12.9282 −0.515481
\(630\) 0 0
\(631\) 15.6077 0.621333 0.310666 0.950519i \(-0.399448\pi\)
0.310666 + 0.950519i \(0.399448\pi\)
\(632\) 8.10634 0.322453
\(633\) 0 0
\(634\) −2.98076 −0.118381
\(635\) 0 0
\(636\) 0 0
\(637\) −1.79315 −0.0710472
\(638\) −0.656339 −0.0259847
\(639\) 0 0
\(640\) 0 0
\(641\) −34.6410 −1.36824 −0.684119 0.729370i \(-0.739813\pi\)
−0.684119 + 0.729370i \(0.739813\pi\)
\(642\) 0 0
\(643\) 10.4543 0.412277 0.206139 0.978523i \(-0.433910\pi\)
0.206139 + 0.978523i \(0.433910\pi\)
\(644\) 26.7846 1.05546
\(645\) 0 0
\(646\) −2.39230 −0.0941240
\(647\) 14.3180 0.562899 0.281449 0.959576i \(-0.409185\pi\)
0.281449 + 0.959576i \(0.409185\pi\)
\(648\) 0 0
\(649\) 13.1769 0.517239
\(650\) 0 0
\(651\) 0 0
\(652\) 24.3190 0.952407
\(653\) −13.0053 −0.508938 −0.254469 0.967081i \(-0.581901\pi\)
−0.254469 + 0.967081i \(0.581901\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.0718 0.666542
\(657\) 0 0
\(658\) 1.79315 0.0699043
\(659\) −32.4449 −1.26387 −0.631936 0.775020i \(-0.717740\pi\)
−0.631936 + 0.775020i \(0.717740\pi\)
\(660\) 0 0
\(661\) −0.784610 −0.0305178 −0.0152589 0.999884i \(-0.504857\pi\)
−0.0152589 + 0.999884i \(0.504857\pi\)
\(662\) −9.41902 −0.366081
\(663\) 0 0
\(664\) −19.6603 −0.762966
\(665\) 0 0
\(666\) 0 0
\(667\) 6.31319 0.244448
\(668\) −19.4201 −0.751384
\(669\) 0 0
\(670\) 0 0
\(671\) 3.71281 0.143332
\(672\) 0 0
\(673\) −46.3644 −1.78722 −0.893609 0.448846i \(-0.851835\pi\)
−0.893609 + 0.448846i \(0.851835\pi\)
\(674\) 14.4449 0.556395
\(675\) 0 0
\(676\) 16.9474 0.651825
\(677\) 32.2495 1.23945 0.619725 0.784819i \(-0.287244\pi\)
0.619725 + 0.784819i \(0.287244\pi\)
\(678\) 0 0
\(679\) 26.7846 1.02790
\(680\) 0 0
\(681\) 0 0
\(682\) −5.73118 −0.219458
\(683\) −21.9711 −0.840700 −0.420350 0.907362i \(-0.638093\pi\)
−0.420350 + 0.907362i \(0.638093\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.1436 −0.387284
\(687\) 0 0
\(688\) −22.5259 −0.858791
\(689\) 10.6410 0.405390
\(690\) 0 0
\(691\) 41.7128 1.58683 0.793415 0.608681i \(-0.208301\pi\)
0.793415 + 0.608681i \(0.208301\pi\)
\(692\) −8.00481 −0.304297
\(693\) 0 0
\(694\) 11.1244 0.422275
\(695\) 0 0
\(696\) 0 0
\(697\) 9.79796 0.371124
\(698\) −11.7942 −0.446416
\(699\) 0 0
\(700\) 0 0
\(701\) −27.7128 −1.04670 −0.523349 0.852118i \(-0.675318\pi\)
−0.523349 + 0.852118i \(0.675318\pi\)
\(702\) 0 0
\(703\) 29.8744 1.12673
\(704\) −2.87564 −0.108380
\(705\) 0 0
\(706\) −16.5359 −0.622337
\(707\) 14.6969 0.552735
\(708\) 0 0
\(709\) 20.3923 0.765849 0.382925 0.923780i \(-0.374917\pi\)
0.382925 + 0.923780i \(0.374917\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 20.0764 0.752395
\(713\) 55.1271 2.06453
\(714\) 0 0
\(715\) 0 0
\(716\) 4.39230 0.164148
\(717\) 0 0
\(718\) 3.76217 0.140403
\(719\) 19.8564 0.740519 0.370260 0.928928i \(-0.379269\pi\)
0.370260 + 0.928928i \(0.379269\pi\)
\(720\) 0 0
\(721\) −22.3923 −0.833933
\(722\) −4.30701 −0.160290
\(723\) 0 0
\(724\) 27.7128 1.02994
\(725\) 0 0
\(726\) 0 0
\(727\) 46.6690 1.73086 0.865430 0.501031i \(-0.167046\pi\)
0.865430 + 0.501031i \(0.167046\pi\)
\(728\) −8.48528 −0.314485
\(729\) 0 0
\(730\) 0 0
\(731\) −12.9282 −0.478167
\(732\) 0 0
\(733\) −12.7279 −0.470117 −0.235058 0.971981i \(-0.575528\pi\)
−0.235058 + 0.971981i \(0.575528\pi\)
\(734\) 17.4115 0.642672
\(735\) 0 0
\(736\) −32.4449 −1.19593
\(737\) −5.37945 −0.198155
\(738\) 0 0
\(739\) −18.9808 −0.698219 −0.349109 0.937082i \(-0.613516\pi\)
−0.349109 + 0.937082i \(0.613516\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7.52433 0.276227
\(743\) −9.89949 −0.363177 −0.181589 0.983375i \(-0.558124\pi\)
−0.181589 + 0.983375i \(0.558124\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 10.1436 0.371383
\(747\) 0 0
\(748\) −3.10583 −0.113560
\(749\) −45.7128 −1.67031
\(750\) 0 0
\(751\) 49.9090 1.82120 0.910602 0.413284i \(-0.135618\pi\)
0.910602 + 0.413284i \(0.135618\pi\)
\(752\) 3.48477 0.127076
\(753\) 0 0
\(754\) −0.928203 −0.0338032
\(755\) 0 0
\(756\) 0 0
\(757\) −10.9348 −0.397431 −0.198716 0.980057i \(-0.563677\pi\)
−0.198716 + 0.980057i \(0.563677\pi\)
\(758\) −8.38375 −0.304511
\(759\) 0 0
\(760\) 0 0
\(761\) 35.5692 1.28938 0.644692 0.764443i \(-0.276986\pi\)
0.644692 + 0.764443i \(0.276986\pi\)
\(762\) 0 0
\(763\) 43.7391 1.58346
\(764\) 26.1962 0.947744
\(765\) 0 0
\(766\) −0.483340 −0.0174638
\(767\) 18.6350 0.672870
\(768\) 0 0
\(769\) −19.0718 −0.687747 −0.343873 0.939016i \(-0.611739\pi\)
−0.343873 + 0.939016i \(0.611739\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.13681 −0.0409148
\(773\) −11.2122 −0.403274 −0.201637 0.979460i \(-0.564626\pi\)
−0.201637 + 0.979460i \(0.564626\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −21.1244 −0.758320
\(777\) 0 0
\(778\) 10.7589 0.385725
\(779\) −22.6410 −0.811199
\(780\) 0 0
\(781\) −4.39230 −0.157169
\(782\) −4.62158 −0.165267
\(783\) 0 0
\(784\) −2.46410 −0.0880036
\(785\) 0 0
\(786\) 0 0
\(787\) −35.0779 −1.25039 −0.625197 0.780467i \(-0.714981\pi\)
−0.625197 + 0.780467i \(0.714981\pi\)
\(788\) 38.8401 1.38362
\(789\) 0 0
\(790\) 0 0
\(791\) −31.8564 −1.13268
\(792\) 0 0
\(793\) 5.25071 0.186458
\(794\) −2.53590 −0.0899957
\(795\) 0 0
\(796\) 24.2487 0.859473
\(797\) 29.0149 1.02776 0.513881 0.857862i \(-0.328207\pi\)
0.513881 + 0.857862i \(0.328207\pi\)
\(798\) 0 0
\(799\) 2.00000 0.0707549
\(800\) 0 0
\(801\) 0 0
\(802\) −19.5959 −0.691956
\(803\) 9.31749 0.328807
\(804\) 0 0
\(805\) 0 0
\(806\) −8.10512 −0.285491
\(807\) 0 0
\(808\) −11.5911 −0.407774
\(809\) −13.6077 −0.478421 −0.239211 0.970968i \(-0.576889\pi\)
−0.239211 + 0.970968i \(0.576889\pi\)
\(810\) 0 0
\(811\) −36.7846 −1.29168 −0.645841 0.763472i \(-0.723493\pi\)
−0.645841 + 0.763472i \(0.723493\pi\)
\(812\) 4.24264 0.148888
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 0 0
\(817\) 29.8744 1.04517
\(818\) 4.69591 0.164189
\(819\) 0 0
\(820\) 0 0
\(821\) −9.21539 −0.321619 −0.160810 0.986985i \(-0.551411\pi\)
−0.160810 + 0.986985i \(0.551411\pi\)
\(822\) 0 0
\(823\) 40.8091 1.42252 0.711258 0.702931i \(-0.248126\pi\)
0.711258 + 0.702931i \(0.248126\pi\)
\(824\) 17.6603 0.615224
\(825\) 0 0
\(826\) 13.1769 0.458483
\(827\) 3.68784 0.128239 0.0641193 0.997942i \(-0.479576\pi\)
0.0641193 + 0.997942i \(0.479576\pi\)
\(828\) 0 0
\(829\) −16.7846 −0.582954 −0.291477 0.956578i \(-0.594147\pi\)
−0.291477 + 0.956578i \(0.594147\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.06678 −0.140990
\(833\) −1.41421 −0.0489996
\(834\) 0 0
\(835\) 0 0
\(836\) 7.17691 0.248219
\(837\) 0 0
\(838\) 3.10583 0.107289
\(839\) −40.9808 −1.41481 −0.707407 0.706807i \(-0.750135\pi\)
−0.707407 + 0.706807i \(0.750135\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 2.82843 0.0974740
\(843\) 0 0
\(844\) 25.5167 0.878320
\(845\) 0 0
\(846\) 0 0
\(847\) −23.0064 −0.790508
\(848\) 14.6226 0.502142
\(849\) 0 0
\(850\) 0 0
\(851\) 57.7128 1.97837
\(852\) 0 0
\(853\) −32.3238 −1.10675 −0.553374 0.832933i \(-0.686660\pi\)
−0.553374 + 0.832933i \(0.686660\pi\)
\(854\) 3.71281 0.127050
\(855\) 0 0
\(856\) 36.0526 1.23225
\(857\) 33.2576 1.13606 0.568029 0.823009i \(-0.307706\pi\)
0.568029 + 0.823009i \(0.307706\pi\)
\(858\) 0 0
\(859\) −18.7321 −0.639129 −0.319565 0.947564i \(-0.603537\pi\)
−0.319565 + 0.947564i \(0.603537\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 19.2442 0.655460
\(863\) 1.69161 0.0575832 0.0287916 0.999585i \(-0.490834\pi\)
0.0287916 + 0.999585i \(0.490834\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.51666 0.0515382
\(867\) 0 0
\(868\) 37.0470 1.25746
\(869\) −5.32051 −0.180486
\(870\) 0 0
\(871\) −7.60770 −0.257777
\(872\) −34.4959 −1.16818
\(873\) 0 0
\(874\) 10.6795 0.361239
\(875\) 0 0
\(876\) 0 0
\(877\) −24.9754 −0.843358 −0.421679 0.906745i \(-0.638559\pi\)
−0.421679 + 0.906745i \(0.638559\pi\)
\(878\) −3.66063 −0.123540
\(879\) 0 0
\(880\) 0 0
\(881\) 50.5359 1.70260 0.851299 0.524681i \(-0.175815\pi\)
0.851299 + 0.524681i \(0.175815\pi\)
\(882\) 0 0
\(883\) −1.48854 −0.0500935 −0.0250467 0.999686i \(-0.507973\pi\)
−0.0250467 + 0.999686i \(0.507973\pi\)
\(884\) −4.39230 −0.147729
\(885\) 0 0
\(886\) 9.41154 0.316187
\(887\) 21.0101 0.705451 0.352726 0.935727i \(-0.385255\pi\)
0.352726 + 0.935727i \(0.385255\pi\)
\(888\) 0 0
\(889\) −10.3923 −0.348547
\(890\) 0 0
\(891\) 0 0
\(892\) −26.5927 −0.890388
\(893\) −4.62158 −0.154655
\(894\) 0 0
\(895\) 0 0
\(896\) −28.0526 −0.937170
\(897\) 0 0
\(898\) 13.3843 0.446639
\(899\) 8.73205 0.291230
\(900\) 0 0
\(901\) 8.39230 0.279588
\(902\) 4.54725 0.151407
\(903\) 0 0
\(904\) 25.1244 0.835624
\(905\) 0 0
\(906\) 0 0
\(907\) 45.7081 1.51771 0.758856 0.651258i \(-0.225758\pi\)
0.758856 + 0.651258i \(0.225758\pi\)
\(908\) 35.4297 1.17577
\(909\) 0 0
\(910\) 0 0
\(911\) 9.80385 0.324816 0.162408 0.986724i \(-0.448074\pi\)
0.162408 + 0.986724i \(0.448074\pi\)
\(912\) 0 0
\(913\) 12.9038 0.427053
\(914\) 12.2487 0.405151
\(915\) 0 0
\(916\) −32.5359 −1.07502
\(917\) 11.5911 0.382772
\(918\) 0 0
\(919\) 43.9615 1.45016 0.725078 0.688666i \(-0.241803\pi\)
0.725078 + 0.688666i \(0.241803\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −12.9038 −0.424964
\(923\) −6.21166 −0.204459
\(924\) 0 0
\(925\) 0 0
\(926\) −3.55514 −0.116829
\(927\) 0 0
\(928\) −5.13922 −0.168703
\(929\) −55.8564 −1.83259 −0.916295 0.400505i \(-0.868835\pi\)
−0.916295 + 0.400505i \(0.868835\pi\)
\(930\) 0 0
\(931\) 3.26795 0.107103
\(932\) 5.85993 0.191948
\(933\) 0 0
\(934\) 10.8756 0.355862
\(935\) 0 0
\(936\) 0 0
\(937\) −2.75410 −0.0899724 −0.0449862 0.998988i \(-0.514324\pi\)
−0.0449862 + 0.998988i \(0.514324\pi\)
\(938\) −5.37945 −0.175645
\(939\) 0 0
\(940\) 0 0
\(941\) 7.85641 0.256112 0.128056 0.991767i \(-0.459126\pi\)
0.128056 + 0.991767i \(0.459126\pi\)
\(942\) 0 0
\(943\) −43.7391 −1.42434
\(944\) 25.6077 0.833459
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −27.9797 −0.909217 −0.454608 0.890691i \(-0.650221\pi\)
−0.454608 + 0.890691i \(0.650221\pi\)
\(948\) 0 0
\(949\) 13.1769 0.427741
\(950\) 0 0
\(951\) 0 0
\(952\) −6.69213 −0.216893
\(953\) −5.65685 −0.183243 −0.0916217 0.995794i \(-0.529205\pi\)
−0.0916217 + 0.995794i \(0.529205\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) −2.92996 −0.0946628
\(959\) 36.2487 1.17053
\(960\) 0 0
\(961\) 45.2487 1.45964
\(962\) −8.48528 −0.273576
\(963\) 0 0
\(964\) −19.8564 −0.639532
\(965\) 0 0
\(966\) 0 0
\(967\) 1.96902 0.0633193 0.0316596 0.999499i \(-0.489921\pi\)
0.0316596 + 0.999499i \(0.489921\pi\)
\(968\) 18.1445 0.583188
\(969\) 0 0
\(970\) 0 0
\(971\) 49.5167 1.58907 0.794533 0.607221i \(-0.207716\pi\)
0.794533 + 0.607221i \(0.207716\pi\)
\(972\) 0 0
\(973\) −19.5959 −0.628216
\(974\) 5.66025 0.181366
\(975\) 0 0
\(976\) 7.21539 0.230959
\(977\) −22.9048 −0.732790 −0.366395 0.930459i \(-0.619408\pi\)
−0.366395 + 0.930459i \(0.619408\pi\)
\(978\) 0 0
\(979\) −13.1769 −0.421136
\(980\) 0 0
\(981\) 0 0
\(982\) −3.76217 −0.120056
\(983\) 30.2533 0.964930 0.482465 0.875915i \(-0.339742\pi\)
0.482465 + 0.875915i \(0.339742\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.732051 −0.0233132
\(987\) 0 0
\(988\) 10.1497 0.322905
\(989\) 57.7128 1.83516
\(990\) 0 0
\(991\) 34.7846 1.10497 0.552485 0.833523i \(-0.313680\pi\)
0.552485 + 0.833523i \(0.313680\pi\)
\(992\) −44.8759 −1.42481
\(993\) 0 0
\(994\) −4.39230 −0.139315
\(995\) 0 0
\(996\) 0 0
\(997\) −26.5927 −0.842198 −0.421099 0.907015i \(-0.638355\pi\)
−0.421099 + 0.907015i \(0.638355\pi\)
\(998\) −0.554803 −0.0175620
\(999\) 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.bj.1.3 4
3.2 odd 2 725.2.a.f.1.2 4
5.2 odd 4 1305.2.c.f.784.3 4
5.3 odd 4 1305.2.c.f.784.2 4
5.4 even 2 inner 6525.2.a.bj.1.2 4
15.2 even 4 145.2.b.b.59.2 4
15.8 even 4 145.2.b.b.59.3 yes 4
15.14 odd 2 725.2.a.f.1.3 4
60.23 odd 4 2320.2.d.f.929.3 4
60.47 odd 4 2320.2.d.f.929.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.b.b.59.2 4 15.2 even 4
145.2.b.b.59.3 yes 4 15.8 even 4
725.2.a.f.1.2 4 3.2 odd 2
725.2.a.f.1.3 4 15.14 odd 2
1305.2.c.f.784.2 4 5.3 odd 4
1305.2.c.f.784.3 4 5.2 odd 4
2320.2.d.f.929.1 4 60.47 odd 4
2320.2.d.f.929.3 4 60.23 odd 4
6525.2.a.bj.1.2 4 5.4 even 2 inner
6525.2.a.bj.1.3 4 1.1 even 1 trivial