Properties

Label 507.2.a.e.1.2
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,2,Mod(1,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} +3.46410 q^{5} +1.73205 q^{7} +1.00000 q^{9} -3.46410 q^{11} +2.00000 q^{12} -3.46410 q^{15} +4.00000 q^{16} +3.46410 q^{19} -6.92820 q^{20} -1.73205 q^{21} +6.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} -3.46410 q^{28} +6.00000 q^{29} +1.73205 q^{31} +3.46410 q^{33} +6.00000 q^{35} -2.00000 q^{36} +6.92820 q^{41} +1.00000 q^{43} +6.92820 q^{44} +3.46410 q^{45} -3.46410 q^{47} -4.00000 q^{48} -4.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} -3.46410 q^{57} +3.46410 q^{59} +6.92820 q^{60} +1.00000 q^{61} +1.73205 q^{63} -8.00000 q^{64} -8.66025 q^{67} -6.00000 q^{69} -10.3923 q^{71} -1.73205 q^{73} -7.00000 q^{75} -6.92820 q^{76} -6.00000 q^{77} -11.0000 q^{79} +13.8564 q^{80} +1.00000 q^{81} -13.8564 q^{83} +3.46410 q^{84} -6.00000 q^{87} +6.92820 q^{89} -12.0000 q^{92} -1.73205 q^{93} +12.0000 q^{95} -5.19615 q^{97} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} + 12 q^{23} + 14 q^{25} - 2 q^{27} + 12 q^{29} + 12 q^{35} - 4 q^{36} + 2 q^{43} - 8 q^{48} - 8 q^{49} + 24 q^{53} - 24 q^{55} + 2 q^{61} - 16 q^{64} - 12 q^{69} - 14 q^{75} - 12 q^{77} - 22 q^{79} + 2 q^{81} - 12 q^{87} - 24 q^{92} + 24 q^{95}+O(q^{100}) \) 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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) −6.92820 −1.54919
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −3.46410 −0.654654
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 1.73205 0.311086 0.155543 0.987829i \(-0.450287\pi\)
0.155543 + 0.987829i \(0.450287\pi\)
\(32\) 0 0
\(33\) 3.46410 0.603023
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) −2.00000 −0.333333
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 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
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 6.92820 1.04447
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) −4.00000 −0.577350
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) −3.46410 −0.458831
\(58\) 0 0
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 6.92820 0.894427
\(61\) 1.00000 0.128037 0.0640184 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 1.73205 0.218218
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.66025 −1.05802 −0.529009 0.848616i \(-0.677436\pi\)
−0.529009 + 0.848616i \(0.677436\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) −1.73205 −0.202721 −0.101361 0.994850i \(-0.532320\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) −6.92820 −0.794719
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 13.8564 1.54919
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −13.8564 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(84\) 3.46410 0.377964
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 6.92820 0.734388 0.367194 0.930144i \(-0.380318\pi\)
0.367194 + 0.930144i \(0.380318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) −1.73205 −0.179605
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) −5.19615 −0.527589 −0.263795 0.964579i \(-0.584974\pi\)
−0.263795 + 0.964579i \(0.584974\pi\)
\(98\) 0 0
\(99\) −3.46410 −0.348155
\(100\) −14.0000 −1.40000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 2.00000 0.192450
\(109\) −15.5885 −1.49310 −0.746552 0.665327i \(-0.768292\pi\)
−0.746552 + 0.665327i \(0.768292\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.92820 0.654654
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 20.7846 1.93817
\(116\) −12.0000 −1.11417
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.92820 −0.624695
\(124\) −3.46410 −0.311086
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) −6.92820 −0.603023
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) −12.0000 −1.01419
\(141\) 3.46410 0.291730
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) 20.7846 1.72607
\(146\) 0 0
\(147\) 4.00000 0.329914
\(148\) 0 0
\(149\) −6.92820 −0.567581 −0.283790 0.958886i \(-0.591592\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(150\) 0 0
\(151\) 3.46410 0.281905 0.140952 0.990016i \(-0.454984\pi\)
0.140952 + 0.990016i \(0.454984\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 11.0000 0.877896 0.438948 0.898513i \(-0.355351\pi\)
0.438948 + 0.898513i \(0.355351\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) 19.0526 1.49231 0.746156 0.665771i \(-0.231897\pi\)
0.746156 + 0.665771i \(0.231897\pi\)
\(164\) −13.8564 −1.08200
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) −6.92820 −0.536120 −0.268060 0.963402i \(-0.586383\pi\)
−0.268060 + 0.963402i \(0.586383\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.46410 0.264906
\(172\) −2.00000 −0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 12.1244 0.916515
\(176\) −13.8564 −1.04447
\(177\) −3.46410 −0.260378
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −6.92820 −0.516398
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −1.00000 −0.0739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.92820 0.505291
\(189\) −1.73205 −0.125988
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 8.00000 0.577350
\(193\) −15.5885 −1.12208 −0.561041 0.827788i \(-0.689599\pi\)
−0.561041 + 0.827788i \(0.689599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 8.00000 0.571429
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) 0 0
\(201\) 8.66025 0.610847
\(202\) 0 0
\(203\) 10.3923 0.729397
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 13.0000 0.894957 0.447478 0.894295i \(-0.352322\pi\)
0.447478 + 0.894295i \(0.352322\pi\)
\(212\) −24.0000 −1.64833
\(213\) 10.3923 0.712069
\(214\) 0 0
\(215\) 3.46410 0.236250
\(216\) 0 0
\(217\) 3.00000 0.203653
\(218\) 0 0
\(219\) 1.73205 0.117041
\(220\) 24.0000 1.61808
\(221\) 0 0
\(222\) 0 0
\(223\) −17.3205 −1.15987 −0.579934 0.814664i \(-0.696921\pi\)
−0.579934 + 0.814664i \(0.696921\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 20.7846 1.37952 0.689761 0.724037i \(-0.257715\pi\)
0.689761 + 0.724037i \(0.257715\pi\)
\(228\) 6.92820 0.458831
\(229\) −27.7128 −1.83131 −0.915657 0.401960i \(-0.868329\pi\)
−0.915657 + 0.401960i \(0.868329\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) −6.92820 −0.450988
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −13.8564 −0.894427
\(241\) −20.7846 −1.33885 −0.669427 0.742878i \(-0.733460\pi\)
−0.669427 + 0.742878i \(0.733460\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −13.8564 −0.885253
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 13.8564 0.878114
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −3.46410 −0.218218
\(253\) −20.7846 −1.30672
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 41.5692 2.55358
\(266\) 0 0
\(267\) −6.92820 −0.423999
\(268\) 17.3205 1.05802
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −5.19615 −0.315644 −0.157822 0.987468i \(-0.550447\pi\)
−0.157822 + 0.987468i \(0.550447\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.2487 −1.46225
\(276\) 12.0000 0.722315
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 1.73205 0.103695
\(280\) 0 0
\(281\) 24.2487 1.44656 0.723278 0.690557i \(-0.242634\pi\)
0.723278 + 0.690557i \(0.242634\pi\)
\(282\) 0 0
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 20.7846 1.23334
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 5.19615 0.304604
\(292\) 3.46410 0.202721
\(293\) 17.3205 1.01187 0.505937 0.862570i \(-0.331147\pi\)
0.505937 + 0.862570i \(0.331147\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 3.46410 0.201008
\(298\) 0 0
\(299\) 0 0
\(300\) 14.0000 0.808290
\(301\) 1.73205 0.0998337
\(302\) 0 0
\(303\) −18.0000 −1.03407
\(304\) 13.8564 0.794719
\(305\) 3.46410 0.198354
\(306\) 0 0
\(307\) 1.73205 0.0988534 0.0494267 0.998778i \(-0.484261\pi\)
0.0494267 + 0.998778i \(0.484261\pi\)
\(308\) 12.0000 0.683763
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) 6.00000 0.338062
\(316\) 22.0000 1.23760
\(317\) 6.92820 0.389127 0.194563 0.980890i \(-0.437671\pi\)
0.194563 + 0.980890i \(0.437671\pi\)
\(318\) 0 0
\(319\) −20.7846 −1.16371
\(320\) −27.7128 −1.54919
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 0 0
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) 15.5885 0.862044
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 5.19615 0.285606 0.142803 0.989751i \(-0.454388\pi\)
0.142803 + 0.989751i \(0.454388\pi\)
\(332\) 27.7128 1.52094
\(333\) 0 0
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) −6.92820 −0.377964
\(337\) 5.00000 0.272367 0.136184 0.990684i \(-0.456516\pi\)
0.136184 + 0.990684i \(0.456516\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) −20.7846 −1.11901
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 12.0000 0.643268
\(349\) −19.0526 −1.01986 −0.509930 0.860216i \(-0.670329\pi\)
−0.509930 + 0.860216i \(0.670329\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.3923 −0.553127 −0.276563 0.960996i \(-0.589196\pi\)
−0.276563 + 0.960996i \(0.589196\pi\)
\(354\) 0 0
\(355\) −36.0000 −1.91068
\(356\) −13.8564 −0.734388
\(357\) 0 0
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 24.0000 1.25109
\(369\) 6.92820 0.360668
\(370\) 0 0
\(371\) 20.7846 1.07908
\(372\) 3.46410 0.179605
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 22.5167 1.15660 0.578302 0.815823i \(-0.303716\pi\)
0.578302 + 0.815823i \(0.303716\pi\)
\(380\) −24.0000 −1.23117
\(381\) 13.0000 0.666010
\(382\) 0 0
\(383\) 27.7128 1.41606 0.708029 0.706183i \(-0.249584\pi\)
0.708029 + 0.706183i \(0.249584\pi\)
\(384\) 0 0
\(385\) −20.7846 −1.05928
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 10.3923 0.527589
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) −38.1051 −1.91728
\(396\) 6.92820 0.348155
\(397\) −15.5885 −0.782362 −0.391181 0.920314i \(-0.627933\pi\)
−0.391181 + 0.920314i \(0.627933\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 28.0000 1.40000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −36.0000 −1.79107
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 8.66025 0.428222 0.214111 0.976809i \(-0.431315\pi\)
0.214111 + 0.976809i \(0.431315\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −48.0000 −2.35623
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 12.0000 0.585540
\(421\) −12.1244 −0.590905 −0.295452 0.955357i \(-0.595470\pi\)
−0.295452 + 0.955357i \(0.595470\pi\)
\(422\) 0 0
\(423\) −3.46410 −0.168430
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.73205 0.0838198
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) −20.7846 −1.00116 −0.500580 0.865690i \(-0.666880\pi\)
−0.500580 + 0.865690i \(0.666880\pi\)
\(432\) −4.00000 −0.192450
\(433\) 23.0000 1.10531 0.552655 0.833410i \(-0.313615\pi\)
0.552655 + 0.833410i \(0.313615\pi\)
\(434\) 0 0
\(435\) −20.7846 −0.996546
\(436\) 31.1769 1.49310
\(437\) 20.7846 0.994263
\(438\) 0 0
\(439\) −35.0000 −1.67046 −0.835229 0.549902i \(-0.814665\pi\)
−0.835229 + 0.549902i \(0.814665\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) 0 0
\(447\) 6.92820 0.327693
\(448\) −13.8564 −0.654654
\(449\) −38.1051 −1.79829 −0.899146 0.437649i \(-0.855811\pi\)
−0.899146 + 0.437649i \(0.855811\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −12.0000 −0.564433
\(453\) −3.46410 −0.162758
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 36.3731 1.70146 0.850730 0.525603i \(-0.176160\pi\)
0.850730 + 0.525603i \(0.176160\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −41.5692 −1.93817
\(461\) −41.5692 −1.93607 −0.968036 0.250812i \(-0.919302\pi\)
−0.968036 + 0.250812i \(0.919302\pi\)
\(462\) 0 0
\(463\) 36.3731 1.69040 0.845200 0.534450i \(-0.179481\pi\)
0.845200 + 0.534450i \(0.179481\pi\)
\(464\) 24.0000 1.11417
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) −11.0000 −0.506853
\(472\) 0 0
\(473\) −3.46410 −0.159280
\(474\) 0 0
\(475\) 24.2487 1.11261
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) 34.6410 1.58279 0.791394 0.611306i \(-0.209356\pi\)
0.791394 + 0.611306i \(0.209356\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −10.3923 −0.472866
\(484\) −2.00000 −0.0909091
\(485\) −18.0000 −0.817338
\(486\) 0 0
\(487\) −24.2487 −1.09881 −0.549407 0.835555i \(-0.685146\pi\)
−0.549407 + 0.835555i \(0.685146\pi\)
\(488\) 0 0
\(489\) −19.0526 −0.861586
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 13.8564 0.624695
\(493\) 0 0
\(494\) 0 0
\(495\) −12.0000 −0.539360
\(496\) 6.92820 0.311086
\(497\) −18.0000 −0.807410
\(498\) 0 0
\(499\) 31.1769 1.39567 0.697835 0.716258i \(-0.254147\pi\)
0.697835 + 0.716258i \(0.254147\pi\)
\(500\) −13.8564 −0.619677
\(501\) 6.92820 0.309529
\(502\) 0 0
\(503\) −30.0000 −1.33763 −0.668817 0.743427i \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 0 0
\(505\) 62.3538 2.77471
\(506\) 0 0
\(507\) 0 0
\(508\) 26.0000 1.15356
\(509\) 17.3205 0.767718 0.383859 0.923392i \(-0.374595\pi\)
0.383859 + 0.923392i \(0.374595\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) −3.46410 −0.152944
\(514\) 0 0
\(515\) 3.46410 0.152647
\(516\) 2.00000 0.0880451
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) −12.1244 −0.529150
\(526\) 0 0
\(527\) 0 0
\(528\) 13.8564 0.603023
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 3.46410 0.150329
\(532\) −12.0000 −0.520266
\(533\) 0 0
\(534\) 0 0
\(535\) −20.7846 −0.898597
\(536\) 0 0
\(537\) 12.0000 0.517838
\(538\) 0 0
\(539\) 13.8564 0.596838
\(540\) 6.92820 0.298142
\(541\) 29.4449 1.26593 0.632967 0.774179i \(-0.281837\pi\)
0.632967 + 0.774179i \(0.281837\pi\)
\(542\) 0 0
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) −54.0000 −2.31311
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) 1.00000 0.0426790
\(550\) 0 0
\(551\) 20.7846 0.885454
\(552\) 0 0
\(553\) −19.0526 −0.810197
\(554\) 0 0
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 24.0000 1.01419
\(561\) 0 0
\(562\) 0 0
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) −6.92820 −0.291730
\(565\) 20.7846 0.874415
\(566\) 0 0
\(567\) 1.73205 0.0727393
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) 18.0000 0.751961
\(574\) 0 0
\(575\) 42.0000 1.75152
\(576\) −8.00000 −0.333333
\(577\) −34.6410 −1.44212 −0.721062 0.692870i \(-0.756346\pi\)
−0.721062 + 0.692870i \(0.756346\pi\)
\(578\) 0 0
\(579\) 15.5885 0.647834
\(580\) −41.5692 −1.72607
\(581\) −24.0000 −0.995688
\(582\) 0 0
\(583\) −41.5692 −1.72162
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.1769 1.28681 0.643404 0.765526i \(-0.277521\pi\)
0.643404 + 0.765526i \(0.277521\pi\)
\(588\) −8.00000 −0.329914
\(589\) 6.00000 0.247226
\(590\) 0 0
\(591\) 13.8564 0.569976
\(592\) 0 0
\(593\) −3.46410 −0.142254 −0.0711268 0.997467i \(-0.522659\pi\)
−0.0711268 + 0.997467i \(0.522659\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.8564 0.567581
\(597\) 7.00000 0.286491
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) −8.66025 −0.352673
\(604\) −6.92820 −0.281905
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) −10.3923 −0.421117
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8.66025 −0.349784 −0.174892 0.984588i \(-0.555958\pi\)
−0.174892 + 0.984588i \(0.555958\pi\)
\(614\) 0 0
\(615\) −24.0000 −0.967773
\(616\) 0 0
\(617\) −10.3923 −0.418378 −0.209189 0.977875i \(-0.567082\pi\)
−0.209189 + 0.977875i \(0.567082\pi\)
\(618\) 0 0
\(619\) −25.9808 −1.04425 −0.522127 0.852867i \(-0.674861\pi\)
−0.522127 + 0.852867i \(0.674861\pi\)
\(620\) −12.0000 −0.481932
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) −22.0000 −0.877896
\(629\) 0 0
\(630\) 0 0
\(631\) −1.73205 −0.0689519 −0.0344759 0.999406i \(-0.510976\pi\)
−0.0344759 + 0.999406i \(0.510976\pi\)
\(632\) 0 0
\(633\) −13.0000 −0.516704
\(634\) 0 0
\(635\) −45.0333 −1.78709
\(636\) 24.0000 0.951662
\(637\) 0 0
\(638\) 0 0
\(639\) −10.3923 −0.411113
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) 19.0526 0.751360 0.375680 0.926750i \(-0.377409\pi\)
0.375680 + 0.926750i \(0.377409\pi\)
\(644\) −20.7846 −0.819028
\(645\) −3.46410 −0.136399
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −3.00000 −0.117579
\(652\) −38.1051 −1.49231
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 20.7846 0.812122
\(656\) 27.7128 1.08200
\(657\) −1.73205 −0.0675737
\(658\) 0 0
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) −24.0000 −0.934199
\(661\) −25.9808 −1.01053 −0.505267 0.862963i \(-0.668606\pi\)
−0.505267 + 0.862963i \(0.668606\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.7846 0.805993
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 13.8564 0.536120
\(669\) 17.3205 0.669650
\(670\) 0 0
\(671\) −3.46410 −0.133730
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) 48.0000 1.84479 0.922395 0.386248i \(-0.126229\pi\)
0.922395 + 0.386248i \(0.126229\pi\)
\(678\) 0 0
\(679\) −9.00000 −0.345388
\(680\) 0 0
\(681\) −20.7846 −0.796468
\(682\) 0 0
\(683\) −24.2487 −0.927851 −0.463926 0.885874i \(-0.653559\pi\)
−0.463926 + 0.885874i \(0.653559\pi\)
\(684\) −6.92820 −0.264906
\(685\) 0 0
\(686\) 0 0
\(687\) 27.7128 1.05731
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) −43.3013 −1.64726 −0.823629 0.567129i \(-0.808054\pi\)
−0.823629 + 0.567129i \(0.808054\pi\)
\(692\) −12.0000 −0.456172
\(693\) −6.00000 −0.227921
\(694\) 0 0
\(695\) 17.3205 0.657004
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) −24.2487 −0.916515
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 27.7128 1.04447
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 31.1769 1.17253
\(708\) 6.92820 0.260378
\(709\) 19.0526 0.715534 0.357767 0.933811i \(-0.383538\pi\)
0.357767 + 0.933811i \(0.383538\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) 10.3923 0.389195
\(714\) 0 0
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 13.8564 0.516398
\(721\) 1.73205 0.0645049
\(722\) 0 0
\(723\) 20.7846 0.772988
\(724\) 28.0000 1.04061
\(725\) 42.0000 1.55984
\(726\) 0 0
\(727\) 35.0000 1.29808 0.649039 0.760755i \(-0.275171\pi\)
0.649039 + 0.760755i \(0.275171\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) 39.8372 1.47142 0.735710 0.677297i \(-0.236849\pi\)
0.735710 + 0.677297i \(0.236849\pi\)
\(734\) 0 0
\(735\) 13.8564 0.511101
\(736\) 0 0
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) 45.0333 1.65658 0.828289 0.560301i \(-0.189315\pi\)
0.828289 + 0.560301i \(0.189315\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.3923 0.381257 0.190628 0.981662i \(-0.438947\pi\)
0.190628 + 0.981662i \(0.438947\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) −13.8564 −0.506979
\(748\) 0 0
\(749\) −10.3923 −0.379727
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −13.8564 −0.505291
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 3.46410 0.125988
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) 20.7846 0.753442 0.376721 0.926327i \(-0.377052\pi\)
0.376721 + 0.926327i \(0.377052\pi\)
\(762\) 0 0
\(763\) −27.0000 −0.977466
\(764\) 36.0000 1.30243
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 −0.577350
\(769\) 6.92820 0.249837 0.124919 0.992167i \(-0.460133\pi\)
0.124919 + 0.992167i \(0.460133\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31.1769 1.12208
\(773\) −51.9615 −1.86893 −0.934463 0.356060i \(-0.884120\pi\)
−0.934463 + 0.356060i \(0.884120\pi\)
\(774\) 0 0
\(775\) 12.1244 0.435520
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −16.0000 −0.571429
\(785\) 38.1051 1.36003
\(786\) 0 0
\(787\) −32.9090 −1.17308 −0.586539 0.809921i \(-0.699510\pi\)
−0.586539 + 0.809921i \(0.699510\pi\)
\(788\) 27.7128 0.987228
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 10.3923 0.369508
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −41.5692 −1.47431
\(796\) 14.0000 0.496217
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 6.92820 0.244796
\(802\) 0 0
\(803\) 6.00000 0.211735
\(804\) −17.3205 −0.610847
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 12.0000 0.421898 0.210949 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) 0 0
\(811\) −25.9808 −0.912308 −0.456154 0.889901i \(-0.650773\pi\)
−0.456154 + 0.889901i \(0.650773\pi\)
\(812\) −20.7846 −0.729397
\(813\) 5.19615 0.182237
\(814\) 0 0
\(815\) 66.0000 2.31188
\(816\) 0 0
\(817\) 3.46410 0.121194
\(818\) 0 0
\(819\) 0 0
\(820\) −48.0000 −1.67623
\(821\) −24.2487 −0.846286 −0.423143 0.906063i \(-0.639073\pi\)
−0.423143 + 0.906063i \(0.639073\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) 24.2487 0.844232
\(826\) 0 0
\(827\) −48.4974 −1.68642 −0.843210 0.537584i \(-0.819337\pi\)
−0.843210 + 0.537584i \(0.819337\pi\)
\(828\) −12.0000 −0.417029
\(829\) −31.0000 −1.07667 −0.538337 0.842729i \(-0.680947\pi\)
−0.538337 + 0.842729i \(0.680947\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) 24.0000 0.830057
\(837\) −1.73205 −0.0598684
\(838\) 0 0
\(839\) −31.1769 −1.07635 −0.538173 0.842834i \(-0.680885\pi\)
−0.538173 + 0.842834i \(0.680885\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −24.2487 −0.835170
\(844\) −26.0000 −0.894957
\(845\) 0 0
\(846\) 0 0
\(847\) 1.73205 0.0595140
\(848\) 48.0000 1.64833
\(849\) −11.0000 −0.377519
\(850\) 0 0
\(851\) 0 0
\(852\) −20.7846 −0.712069
\(853\) 25.9808 0.889564 0.444782 0.895639i \(-0.353281\pi\)
0.444782 + 0.895639i \(0.353281\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) 13.0000 0.443554 0.221777 0.975097i \(-0.428814\pi\)
0.221777 + 0.975097i \(0.428814\pi\)
\(860\) −6.92820 −0.236250
\(861\) −12.0000 −0.408959
\(862\) 0 0
\(863\) −17.3205 −0.589597 −0.294798 0.955559i \(-0.595253\pi\)
−0.294798 + 0.955559i \(0.595253\pi\)
\(864\) 0 0
\(865\) 20.7846 0.706698
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) −6.00000 −0.203653
\(869\) 38.1051 1.29263
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.19615 −0.175863
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) −3.46410 −0.117041
\(877\) 41.5692 1.40369 0.701846 0.712328i \(-0.252359\pi\)
0.701846 + 0.712328i \(0.252359\pi\)
\(878\) 0 0
\(879\) −17.3205 −0.584206
\(880\) −48.0000 −1.61808
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) 0 0
\(883\) 5.00000 0.168263 0.0841317 0.996455i \(-0.473188\pi\)
0.0841317 + 0.996455i \(0.473188\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) 0 0
\(889\) −22.5167 −0.755185
\(890\) 0 0
\(891\) −3.46410 −0.116052
\(892\) 34.6410 1.15987
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −41.5692 −1.38951
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.3923 0.346603
\(900\) −14.0000 −0.466667
\(901\) 0 0
\(902\) 0 0
\(903\) −1.73205 −0.0576390
\(904\) 0 0
\(905\) −48.4974 −1.61211
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −41.5692 −1.37952
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −13.8564 −0.458831
\(913\) 48.0000 1.58857
\(914\) 0 0
\(915\) −3.46410 −0.114520
\(916\) 55.4256 1.83131
\(917\) 10.3923 0.343184
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −1.73205 −0.0570730
\(922\) 0 0
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 0 0
\(927\) 1.00000 0.0328443
\(928\) 0 0
\(929\) −20.7846 −0.681921 −0.340960 0.940078i \(-0.610752\pi\)
−0.340960 + 0.940078i \(0.610752\pi\)
\(930\) 0 0
\(931\) −13.8564 −0.454125
\(932\) −36.0000 −1.17922
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) −13.0000 −0.424239
\(940\) 24.0000 0.782794
\(941\) −10.3923 −0.338779 −0.169390 0.985549i \(-0.554180\pi\)
−0.169390 + 0.985549i \(0.554180\pi\)
\(942\) 0 0
\(943\) 41.5692 1.35368
\(944\) 13.8564 0.450988
\(945\) −6.00000 −0.195180
\(946\) 0 0
\(947\) −48.4974 −1.57595 −0.787977 0.615704i \(-0.788872\pi\)
−0.787977 + 0.615704i \(0.788872\pi\)
\(948\) −22.0000 −0.714527
\(949\) 0 0
\(950\) 0 0
\(951\) −6.92820 −0.224662
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −62.3538 −2.01772
\(956\) 0 0
\(957\) 20.7846 0.671871
\(958\) 0 0
\(959\) 0 0
\(960\) 27.7128 0.894427
\(961\) −28.0000 −0.903226
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 41.5692 1.33885
\(965\) −54.0000 −1.73832
\(966\) 0 0
\(967\) −24.2487 −0.779786 −0.389893 0.920860i \(-0.627488\pi\)
−0.389893 + 0.920860i \(0.627488\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 36.0000 1.15529 0.577647 0.816286i \(-0.303971\pi\)
0.577647 + 0.816286i \(0.303971\pi\)
\(972\) 2.00000 0.0641500
\(973\) 8.66025 0.277635
\(974\) 0 0
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 27.7128 0.886611 0.443306 0.896370i \(-0.353806\pi\)
0.443306 + 0.896370i \(0.353806\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 27.7128 0.885253
\(981\) −15.5885 −0.497701
\(982\) 0 0
\(983\) 10.3923 0.331463 0.165732 0.986171i \(-0.447001\pi\)
0.165732 + 0.986171i \(0.447001\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 6.00000 0.190789
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −5.19615 −0.164895
\(994\) 0 0
\(995\) −24.2487 −0.768736
\(996\) −27.7128 −0.878114
\(997\) 35.0000 1.10846 0.554231 0.832363i \(-0.313013\pi\)
0.554231 + 0.832363i \(0.313013\pi\)
\(998\) 0 0
\(999\) 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 507.2.a.e.1.2 2
3.2 odd 2 1521.2.a.h.1.1 2
4.3 odd 2 8112.2.a.bu.1.2 2
13.2 odd 12 507.2.j.b.316.1 2
13.3 even 3 507.2.e.f.22.2 4
13.4 even 6 507.2.e.f.484.1 4
13.5 odd 4 507.2.b.c.337.1 2
13.6 odd 12 39.2.j.a.10.1 yes 2
13.7 odd 12 507.2.j.b.361.1 2
13.8 odd 4 507.2.b.c.337.2 2
13.9 even 3 507.2.e.f.484.2 4
13.10 even 6 507.2.e.f.22.1 4
13.11 odd 12 39.2.j.a.4.1 2
13.12 even 2 inner 507.2.a.e.1.1 2
39.5 even 4 1521.2.b.f.1351.2 2
39.8 even 4 1521.2.b.f.1351.1 2
39.11 even 12 117.2.q.a.82.1 2
39.32 even 12 117.2.q.a.10.1 2
39.38 odd 2 1521.2.a.h.1.2 2
52.11 even 12 624.2.bv.b.433.1 2
52.19 even 12 624.2.bv.b.49.1 2
52.51 odd 2 8112.2.a.bu.1.1 2
65.19 odd 12 975.2.bc.c.751.1 2
65.24 odd 12 975.2.bc.c.901.1 2
65.32 even 12 975.2.w.d.49.2 4
65.37 even 12 975.2.w.d.199.1 4
65.58 even 12 975.2.w.d.49.1 4
65.63 even 12 975.2.w.d.199.2 4
156.11 odd 12 1872.2.by.f.433.1 2
156.71 odd 12 1872.2.by.f.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.j.a.4.1 2 13.11 odd 12
39.2.j.a.10.1 yes 2 13.6 odd 12
117.2.q.a.10.1 2 39.32 even 12
117.2.q.a.82.1 2 39.11 even 12
507.2.a.e.1.1 2 13.12 even 2 inner
507.2.a.e.1.2 2 1.1 even 1 trivial
507.2.b.c.337.1 2 13.5 odd 4
507.2.b.c.337.2 2 13.8 odd 4
507.2.e.f.22.1 4 13.10 even 6
507.2.e.f.22.2 4 13.3 even 3
507.2.e.f.484.1 4 13.4 even 6
507.2.e.f.484.2 4 13.9 even 3
507.2.j.b.316.1 2 13.2 odd 12
507.2.j.b.361.1 2 13.7 odd 12
624.2.bv.b.49.1 2 52.19 even 12
624.2.bv.b.433.1 2 52.11 even 12
975.2.w.d.49.1 4 65.58 even 12
975.2.w.d.49.2 4 65.32 even 12
975.2.w.d.199.1 4 65.37 even 12
975.2.w.d.199.2 4 65.63 even 12
975.2.bc.c.751.1 2 65.19 odd 12
975.2.bc.c.901.1 2 65.24 odd 12
1521.2.a.h.1.1 2 3.2 odd 2
1521.2.a.h.1.2 2 39.38 odd 2
1521.2.b.f.1351.1 2 39.8 even 4
1521.2.b.f.1351.2 2 39.5 even 4
1872.2.by.f.433.1 2 156.11 odd 12
1872.2.by.f.1297.1 2 156.71 odd 12
8112.2.a.bu.1.1 2 52.51 odd 2
8112.2.a.bu.1.2 2 4.3 odd 2