Properties

Label 2475.2.a.bg.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
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] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(0\)
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 495)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.517638 q^{2} -1.73205 q^{4} -2.44949 q^{7} +1.93185 q^{8} +O(q^{10})\) \(q-0.517638 q^{2} -1.73205 q^{4} -2.44949 q^{7} +1.93185 q^{8} -1.00000 q^{11} +4.24264 q^{13} +1.26795 q^{14} +2.46410 q^{16} +0.378937 q^{17} -2.00000 q^{19} +0.517638 q^{22} -7.72741 q^{23} -2.19615 q^{26} +4.24264 q^{28} +2.53590 q^{29} -0.535898 q^{31} -5.13922 q^{32} -0.196152 q^{34} -4.89898 q^{37} +1.03528 q^{38} +2.53590 q^{41} -10.9348 q^{43} +1.73205 q^{44} +4.00000 q^{46} +2.82843 q^{47} -1.00000 q^{49} -7.34847 q^{52} +10.5558 q^{53} -4.73205 q^{56} -1.31268 q^{58} +12.9282 q^{59} -4.92820 q^{61} +0.277401 q^{62} -2.26795 q^{64} +3.58630 q^{67} -0.656339 q^{68} +6.00000 q^{71} +0.656339 q^{73} +2.53590 q^{74} +3.46410 q^{76} +2.44949 q^{77} +11.8564 q^{79} -1.31268 q^{82} -7.07107 q^{83} +5.66025 q^{86} -1.93185 q^{88} +15.4641 q^{89} -10.3923 q^{91} +13.3843 q^{92} -1.46410 q^{94} +13.3843 q^{97} +0.517638 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{11} + 12 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{26} + 24 q^{29} - 16 q^{31} + 20 q^{34} + 24 q^{41} + 16 q^{46} - 4 q^{49} - 12 q^{56} + 24 q^{59} + 8 q^{61} - 16 q^{64} + 24 q^{71} + 24 q^{74} - 8 q^{79} - 12 q^{86} + 48 q^{89} + 8 q^{94}+O(q^{100}) \) 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 \(-0.558585\pi\)
−0.183013 + 0.983111i \(0.558585\pi\)
\(3\) 0 0
\(4\) −1.73205 −0.866025
\(5\) 0 0
\(6\) 0 0
\(7\) −2.44949 −0.925820 −0.462910 0.886405i \(-0.653195\pi\)
−0.462910 + 0.886405i \(0.653195\pi\)
\(8\) 1.93185 0.683013
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 1.26795 0.338874
\(15\) 0 0
\(16\) 2.46410 0.616025
\(17\) 0.378937 0.0919058 0.0459529 0.998944i \(-0.485368\pi\)
0.0459529 + 0.998944i \(0.485368\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.517638 0.110361
\(23\) −7.72741 −1.61128 −0.805638 0.592408i \(-0.798177\pi\)
−0.805638 + 0.592408i \(0.798177\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.19615 −0.430701
\(27\) 0 0
\(28\) 4.24264 0.801784
\(29\) 2.53590 0.470905 0.235452 0.971886i \(-0.424343\pi\)
0.235452 + 0.971886i \(0.424343\pi\)
\(30\) 0 0
\(31\) −0.535898 −0.0962502 −0.0481251 0.998841i \(-0.515325\pi\)
−0.0481251 + 0.998841i \(0.515325\pi\)
\(32\) −5.13922 −0.908494
\(33\) 0 0
\(34\) −0.196152 −0.0336399
\(35\) 0 0
\(36\) 0 0
\(37\) −4.89898 −0.805387 −0.402694 0.915335i \(-0.631926\pi\)
−0.402694 + 0.915335i \(0.631926\pi\)
\(38\) 1.03528 0.167944
\(39\) 0 0
\(40\) 0 0
\(41\) 2.53590 0.396041 0.198020 0.980198i \(-0.436549\pi\)
0.198020 + 0.980198i \(0.436549\pi\)
\(42\) 0 0
\(43\) −10.9348 −1.66754 −0.833768 0.552114i \(-0.813821\pi\)
−0.833768 + 0.552114i \(0.813821\pi\)
\(44\) 1.73205 0.261116
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −7.34847 −1.01905
\(53\) 10.5558 1.44996 0.724978 0.688772i \(-0.241850\pi\)
0.724978 + 0.688772i \(0.241850\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −4.73205 −0.632347
\(57\) 0 0
\(58\) −1.31268 −0.172363
\(59\) 12.9282 1.68311 0.841554 0.540172i \(-0.181641\pi\)
0.841554 + 0.540172i \(0.181641\pi\)
\(60\) 0 0
\(61\) −4.92820 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(62\) 0.277401 0.0352300
\(63\) 0 0
\(64\) −2.26795 −0.283494
\(65\) 0 0
\(66\) 0 0
\(67\) 3.58630 0.438137 0.219068 0.975710i \(-0.429698\pi\)
0.219068 + 0.975710i \(0.429698\pi\)
\(68\) −0.656339 −0.0795928
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 0.656339 0.0768186 0.0384093 0.999262i \(-0.487771\pi\)
0.0384093 + 0.999262i \(0.487771\pi\)
\(74\) 2.53590 0.294792
\(75\) 0 0
\(76\) 3.46410 0.397360
\(77\) 2.44949 0.279145
\(78\) 0 0
\(79\) 11.8564 1.33395 0.666975 0.745080i \(-0.267589\pi\)
0.666975 + 0.745080i \(0.267589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.31268 −0.144961
\(83\) −7.07107 −0.776151 −0.388075 0.921628i \(-0.626860\pi\)
−0.388075 + 0.921628i \(0.626860\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.66025 0.610361
\(87\) 0 0
\(88\) −1.93185 −0.205936
\(89\) 15.4641 1.63919 0.819596 0.572942i \(-0.194198\pi\)
0.819596 + 0.572942i \(0.194198\pi\)
\(90\) 0 0
\(91\) −10.3923 −1.08941
\(92\) 13.3843 1.39541
\(93\) 0 0
\(94\) −1.46410 −0.151011
\(95\) 0 0
\(96\) 0 0
\(97\) 13.3843 1.35897 0.679483 0.733691i \(-0.262204\pi\)
0.679483 + 0.733691i \(0.262204\pi\)
\(98\) 0.517638 0.0522893
\(99\) 0 0
\(100\) 0 0
\(101\) 16.3923 1.63110 0.815548 0.578690i \(-0.196436\pi\)
0.815548 + 0.578690i \(0.196436\pi\)
\(102\) 0 0
\(103\) −8.48528 −0.836080 −0.418040 0.908429i \(-0.637283\pi\)
−0.418040 + 0.908429i \(0.637283\pi\)
\(104\) 8.19615 0.803699
\(105\) 0 0
\(106\) −5.46410 −0.530720
\(107\) −1.41421 −0.136717 −0.0683586 0.997661i \(-0.521776\pi\)
−0.0683586 + 0.997661i \(0.521776\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −6.03579 −0.570329
\(113\) 5.65685 0.532152 0.266076 0.963952i \(-0.414273\pi\)
0.266076 + 0.963952i \(0.414273\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.39230 −0.407815
\(117\) 0 0
\(118\) −6.69213 −0.616061
\(119\) −0.928203 −0.0850883
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.55103 0.230959
\(123\) 0 0
\(124\) 0.928203 0.0833551
\(125\) 0 0
\(126\) 0 0
\(127\) 17.1464 1.52150 0.760750 0.649045i \(-0.224831\pi\)
0.760750 + 0.649045i \(0.224831\pi\)
\(128\) 11.4524 1.01226
\(129\) 0 0
\(130\) 0 0
\(131\) 13.8564 1.21064 0.605320 0.795982i \(-0.293045\pi\)
0.605320 + 0.795982i \(0.293045\pi\)
\(132\) 0 0
\(133\) 4.89898 0.424795
\(134\) −1.85641 −0.160369
\(135\) 0 0
\(136\) 0.732051 0.0627728
\(137\) −6.96953 −0.595447 −0.297724 0.954652i \(-0.596227\pi\)
−0.297724 + 0.954652i \(0.596227\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −3.10583 −0.260635
\(143\) −4.24264 −0.354787
\(144\) 0 0
\(145\) 0 0
\(146\) −0.339746 −0.0281176
\(147\) 0 0
\(148\) 8.48528 0.697486
\(149\) 23.3205 1.91049 0.955245 0.295815i \(-0.0955912\pi\)
0.955245 + 0.295815i \(0.0955912\pi\)
\(150\) 0 0
\(151\) −18.7846 −1.52867 −0.764335 0.644819i \(-0.776933\pi\)
−0.764335 + 0.644819i \(0.776933\pi\)
\(152\) −3.86370 −0.313388
\(153\) 0 0
\(154\) −1.26795 −0.102174
\(155\) 0 0
\(156\) 0 0
\(157\) −18.2832 −1.45916 −0.729581 0.683895i \(-0.760285\pi\)
−0.729581 + 0.683895i \(0.760285\pi\)
\(158\) −6.13733 −0.488260
\(159\) 0 0
\(160\) 0 0
\(161\) 18.9282 1.49175
\(162\) 0 0
\(163\) 1.31268 0.102817 0.0514084 0.998678i \(-0.483629\pi\)
0.0514084 + 0.998678i \(0.483629\pi\)
\(164\) −4.39230 −0.342981
\(165\) 0 0
\(166\) 3.66025 0.284091
\(167\) −23.2838 −1.80175 −0.900876 0.434077i \(-0.857074\pi\)
−0.900876 + 0.434077i \(0.857074\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 18.9396 1.44413
\(173\) −18.6622 −1.41886 −0.709430 0.704776i \(-0.751047\pi\)
−0.709430 + 0.704776i \(0.751047\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.46410 −0.185739
\(177\) 0 0
\(178\) −8.00481 −0.599986
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 16.7846 1.24759 0.623795 0.781588i \(-0.285590\pi\)
0.623795 + 0.781588i \(0.285590\pi\)
\(182\) 5.37945 0.398752
\(183\) 0 0
\(184\) −14.9282 −1.10052
\(185\) 0 0
\(186\) 0 0
\(187\) −0.378937 −0.0277106
\(188\) −4.89898 −0.357295
\(189\) 0 0
\(190\) 0 0
\(191\) −13.8564 −1.00261 −0.501307 0.865269i \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) 9.14162 0.658028 0.329014 0.944325i \(-0.393284\pi\)
0.329014 + 0.944325i \(0.393284\pi\)
\(194\) −6.92820 −0.497416
\(195\) 0 0
\(196\) 1.73205 0.123718
\(197\) 16.3886 1.16764 0.583818 0.811885i \(-0.301558\pi\)
0.583818 + 0.811885i \(0.301558\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −8.48528 −0.597022
\(203\) −6.21166 −0.435973
\(204\) 0 0
\(205\) 0 0
\(206\) 4.39230 0.306026
\(207\) 0 0
\(208\) 10.4543 0.724875
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 15.8564 1.09160 0.545800 0.837915i \(-0.316226\pi\)
0.545800 + 0.837915i \(0.316226\pi\)
\(212\) −18.2832 −1.25570
\(213\) 0 0
\(214\) 0.732051 0.0500420
\(215\) 0 0
\(216\) 0 0
\(217\) 1.31268 0.0891104
\(218\) −5.17638 −0.350589
\(219\) 0 0
\(220\) 0 0
\(221\) 1.60770 0.108145
\(222\) 0 0
\(223\) 14.6969 0.984180 0.492090 0.870544i \(-0.336233\pi\)
0.492090 + 0.870544i \(0.336233\pi\)
\(224\) 12.5885 0.841102
\(225\) 0 0
\(226\) −2.92820 −0.194781
\(227\) −11.2122 −0.744178 −0.372089 0.928197i \(-0.621358\pi\)
−0.372089 + 0.928197i \(0.621358\pi\)
\(228\) 0 0
\(229\) −27.8564 −1.84080 −0.920402 0.390974i \(-0.872138\pi\)
−0.920402 + 0.390974i \(0.872138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.89898 0.321634
\(233\) 21.4906 1.40790 0.703948 0.710251i \(-0.251419\pi\)
0.703948 + 0.710251i \(0.251419\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −22.3923 −1.45761
\(237\) 0 0
\(238\) 0.480473 0.0311445
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) 8.92820 0.575116 0.287558 0.957763i \(-0.407157\pi\)
0.287558 + 0.957763i \(0.407157\pi\)
\(242\) −0.517638 −0.0332750
\(243\) 0 0
\(244\) 8.53590 0.546455
\(245\) 0 0
\(246\) 0 0
\(247\) −8.48528 −0.539906
\(248\) −1.03528 −0.0657401
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 7.72741 0.485818
\(254\) −8.87564 −0.556907
\(255\) 0 0
\(256\) −1.39230 −0.0870191
\(257\) 27.3233 1.70438 0.852191 0.523231i \(-0.175273\pi\)
0.852191 + 0.523231i \(0.175273\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 0 0
\(261\) 0 0
\(262\) −7.17260 −0.443125
\(263\) 24.5964 1.51668 0.758341 0.651859i \(-0.226010\pi\)
0.758341 + 0.651859i \(0.226010\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.53590 −0.155486
\(267\) 0 0
\(268\) −6.21166 −0.379437
\(269\) −1.85641 −0.113187 −0.0565935 0.998397i \(-0.518024\pi\)
−0.0565935 + 0.998397i \(0.518024\pi\)
\(270\) 0 0
\(271\) −4.92820 −0.299367 −0.149684 0.988734i \(-0.547825\pi\)
−0.149684 + 0.988734i \(0.547825\pi\)
\(272\) 0.933740 0.0566163
\(273\) 0 0
\(274\) 3.60770 0.217949
\(275\) 0 0
\(276\) 0 0
\(277\) 15.3533 0.922489 0.461245 0.887273i \(-0.347403\pi\)
0.461245 + 0.887273i \(0.347403\pi\)
\(278\) 1.03528 0.0620917
\(279\) 0 0
\(280\) 0 0
\(281\) −23.3205 −1.39118 −0.695592 0.718437i \(-0.744858\pi\)
−0.695592 + 0.718437i \(0.744858\pi\)
\(282\) 0 0
\(283\) 1.13681 0.0675765 0.0337882 0.999429i \(-0.489243\pi\)
0.0337882 + 0.999429i \(0.489243\pi\)
\(284\) −10.3923 −0.616670
\(285\) 0 0
\(286\) 2.19615 0.129861
\(287\) −6.21166 −0.366663
\(288\) 0 0
\(289\) −16.8564 −0.991553
\(290\) 0 0
\(291\) 0 0
\(292\) −1.13681 −0.0665269
\(293\) 8.10634 0.473578 0.236789 0.971561i \(-0.423905\pi\)
0.236789 + 0.971561i \(0.423905\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.46410 −0.550090
\(297\) 0 0
\(298\) −12.0716 −0.699288
\(299\) −32.7846 −1.89598
\(300\) 0 0
\(301\) 26.7846 1.54384
\(302\) 9.72363 0.559532
\(303\) 0 0
\(304\) −4.92820 −0.282652
\(305\) 0 0
\(306\) 0 0
\(307\) −6.03579 −0.344481 −0.172240 0.985055i \(-0.555101\pi\)
−0.172240 + 0.985055i \(0.555101\pi\)
\(308\) −4.24264 −0.241747
\(309\) 0 0
\(310\) 0 0
\(311\) 11.0718 0.627824 0.313912 0.949452i \(-0.398360\pi\)
0.313912 + 0.949452i \(0.398360\pi\)
\(312\) 0 0
\(313\) −21.8695 −1.23614 −0.618070 0.786123i \(-0.712085\pi\)
−0.618070 + 0.786123i \(0.712085\pi\)
\(314\) 9.46410 0.534090
\(315\) 0 0
\(316\) −20.5359 −1.15523
\(317\) −12.8295 −0.720574 −0.360287 0.932841i \(-0.617321\pi\)
−0.360287 + 0.932841i \(0.617321\pi\)
\(318\) 0 0
\(319\) −2.53590 −0.141983
\(320\) 0 0
\(321\) 0 0
\(322\) −9.79796 −0.546019
\(323\) −0.757875 −0.0421693
\(324\) 0 0
\(325\) 0 0
\(326\) −0.679492 −0.0376336
\(327\) 0 0
\(328\) 4.89898 0.270501
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) −2.39230 −0.131493 −0.0657465 0.997836i \(-0.520943\pi\)
−0.0657465 + 0.997836i \(0.520943\pi\)
\(332\) 12.2474 0.672166
\(333\) 0 0
\(334\) 12.0526 0.659487
\(335\) 0 0
\(336\) 0 0
\(337\) −7.82894 −0.426470 −0.213235 0.977001i \(-0.568400\pi\)
−0.213235 + 0.977001i \(0.568400\pi\)
\(338\) −2.58819 −0.140779
\(339\) 0 0
\(340\) 0 0
\(341\) 0.535898 0.0290205
\(342\) 0 0
\(343\) 19.5959 1.05808
\(344\) −21.1244 −1.13895
\(345\) 0 0
\(346\) 9.66025 0.519338
\(347\) 24.0416 1.29062 0.645311 0.763920i \(-0.276728\pi\)
0.645311 + 0.763920i \(0.276728\pi\)
\(348\) 0 0
\(349\) −32.9282 −1.76261 −0.881303 0.472551i \(-0.843333\pi\)
−0.881303 + 0.472551i \(0.843333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.13922 0.273921
\(353\) 3.38323 0.180071 0.0900356 0.995939i \(-0.471302\pi\)
0.0900356 + 0.995939i \(0.471302\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −26.7846 −1.41958
\(357\) 0 0
\(358\) −3.58630 −0.189542
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −8.68835 −0.456650
\(363\) 0 0
\(364\) 18.0000 0.943456
\(365\) 0 0
\(366\) 0 0
\(367\) 21.8695 1.14158 0.570790 0.821096i \(-0.306637\pi\)
0.570790 + 0.821096i \(0.306637\pi\)
\(368\) −19.0411 −0.992587
\(369\) 0 0
\(370\) 0 0
\(371\) −25.8564 −1.34240
\(372\) 0 0
\(373\) 11.4152 0.591059 0.295529 0.955334i \(-0.404504\pi\)
0.295529 + 0.955334i \(0.404504\pi\)
\(374\) 0.196152 0.0101428
\(375\) 0 0
\(376\) 5.46410 0.281790
\(377\) 10.7589 0.554112
\(378\) 0 0
\(379\) −6.14359 −0.315575 −0.157788 0.987473i \(-0.550436\pi\)
−0.157788 + 0.987473i \(0.550436\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 7.17260 0.366982
\(383\) 3.38323 0.172875 0.0864375 0.996257i \(-0.472452\pi\)
0.0864375 + 0.996257i \(0.472452\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.73205 −0.240855
\(387\) 0 0
\(388\) −23.1822 −1.17690
\(389\) 6.92820 0.351274 0.175637 0.984455i \(-0.443802\pi\)
0.175637 + 0.984455i \(0.443802\pi\)
\(390\) 0 0
\(391\) −2.92820 −0.148086
\(392\) −1.93185 −0.0975732
\(393\) 0 0
\(394\) −8.48334 −0.427384
\(395\) 0 0
\(396\) 0 0
\(397\) −20.5569 −1.03172 −0.515860 0.856673i \(-0.672527\pi\)
−0.515860 + 0.856673i \(0.672527\pi\)
\(398\) −2.07055 −0.103787
\(399\) 0 0
\(400\) 0 0
\(401\) 17.3205 0.864945 0.432472 0.901647i \(-0.357641\pi\)
0.432472 + 0.901647i \(0.357641\pi\)
\(402\) 0 0
\(403\) −2.27362 −0.113257
\(404\) −28.3923 −1.41257
\(405\) 0 0
\(406\) 3.21539 0.159577
\(407\) 4.89898 0.242833
\(408\) 0 0
\(409\) 15.0718 0.745252 0.372626 0.927982i \(-0.378457\pi\)
0.372626 + 0.927982i \(0.378457\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14.6969 0.724066
\(413\) −31.6675 −1.55826
\(414\) 0 0
\(415\) 0 0
\(416\) −21.8038 −1.06902
\(417\) 0 0
\(418\) −1.03528 −0.0506370
\(419\) −6.92820 −0.338465 −0.169232 0.985576i \(-0.554129\pi\)
−0.169232 + 0.985576i \(0.554129\pi\)
\(420\) 0 0
\(421\) −9.07180 −0.442132 −0.221066 0.975259i \(-0.570954\pi\)
−0.221066 + 0.975259i \(0.570954\pi\)
\(422\) −8.20788 −0.399553
\(423\) 0 0
\(424\) 20.3923 0.990338
\(425\) 0 0
\(426\) 0 0
\(427\) 12.0716 0.584185
\(428\) 2.44949 0.118401
\(429\) 0 0
\(430\) 0 0
\(431\) 27.7128 1.33488 0.667440 0.744664i \(-0.267390\pi\)
0.667440 + 0.744664i \(0.267390\pi\)
\(432\) 0 0
\(433\) 32.9802 1.58493 0.792463 0.609920i \(-0.208798\pi\)
0.792463 + 0.609920i \(0.208798\pi\)
\(434\) −0.679492 −0.0326167
\(435\) 0 0
\(436\) −17.3205 −0.829502
\(437\) 15.4548 0.739304
\(438\) 0 0
\(439\) 35.8564 1.71133 0.855666 0.517528i \(-0.173148\pi\)
0.855666 + 0.517528i \(0.173148\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.832204 −0.0395839
\(443\) −9.04008 −0.429507 −0.214754 0.976668i \(-0.568895\pi\)
−0.214754 + 0.976668i \(0.568895\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7.60770 −0.360235
\(447\) 0 0
\(448\) 5.55532 0.262464
\(449\) −15.4641 −0.729796 −0.364898 0.931047i \(-0.618896\pi\)
−0.364898 + 0.931047i \(0.618896\pi\)
\(450\) 0 0
\(451\) −2.53590 −0.119411
\(452\) −9.79796 −0.460857
\(453\) 0 0
\(454\) 5.80385 0.272388
\(455\) 0 0
\(456\) 0 0
\(457\) −10.4543 −0.489031 −0.244516 0.969645i \(-0.578629\pi\)
−0.244516 + 0.969645i \(0.578629\pi\)
\(458\) 14.4195 0.673781
\(459\) 0 0
\(460\) 0 0
\(461\) 7.60770 0.354326 0.177163 0.984182i \(-0.443308\pi\)
0.177163 + 0.984182i \(0.443308\pi\)
\(462\) 0 0
\(463\) −7.52433 −0.349685 −0.174843 0.984596i \(-0.555942\pi\)
−0.174843 + 0.984596i \(0.555942\pi\)
\(464\) 6.24871 0.290089
\(465\) 0 0
\(466\) −11.1244 −0.515326
\(467\) −31.4644 −1.45600 −0.728000 0.685577i \(-0.759550\pi\)
−0.728000 + 0.685577i \(0.759550\pi\)
\(468\) 0 0
\(469\) −8.78461 −0.405636
\(470\) 0 0
\(471\) 0 0
\(472\) 24.9754 1.14958
\(473\) 10.9348 0.502781
\(474\) 0 0
\(475\) 0 0
\(476\) 1.60770 0.0736886
\(477\) 0 0
\(478\) −7.17260 −0.328067
\(479\) −17.0718 −0.780030 −0.390015 0.920808i \(-0.627530\pi\)
−0.390015 + 0.920808i \(0.627530\pi\)
\(480\) 0 0
\(481\) −20.7846 −0.947697
\(482\) −4.62158 −0.210507
\(483\) 0 0
\(484\) −1.73205 −0.0787296
\(485\) 0 0
\(486\) 0 0
\(487\) 40.1528 1.81950 0.909748 0.415161i \(-0.136275\pi\)
0.909748 + 0.415161i \(0.136275\pi\)
\(488\) −9.52056 −0.430975
\(489\) 0 0
\(490\) 0 0
\(491\) −13.8564 −0.625331 −0.312665 0.949863i \(-0.601222\pi\)
−0.312665 + 0.949863i \(0.601222\pi\)
\(492\) 0 0
\(493\) 0.960947 0.0432789
\(494\) 4.39230 0.197619
\(495\) 0 0
\(496\) −1.32051 −0.0592926
\(497\) −14.6969 −0.659248
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −9.31749 −0.415860
\(503\) 25.9091 1.15523 0.577615 0.816309i \(-0.303983\pi\)
0.577615 + 0.816309i \(0.303983\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −4.00000 −0.177822
\(507\) 0 0
\(508\) −29.6985 −1.31766
\(509\) −1.85641 −0.0822838 −0.0411419 0.999153i \(-0.513100\pi\)
−0.0411419 + 0.999153i \(0.513100\pi\)
\(510\) 0 0
\(511\) −1.60770 −0.0711202
\(512\) −22.1841 −0.980408
\(513\) 0 0
\(514\) −14.1436 −0.623847
\(515\) 0 0
\(516\) 0 0
\(517\) −2.82843 −0.124394
\(518\) −6.21166 −0.272925
\(519\) 0 0
\(520\) 0 0
\(521\) 17.3205 0.758825 0.379413 0.925228i \(-0.376126\pi\)
0.379413 + 0.925228i \(0.376126\pi\)
\(522\) 0 0
\(523\) −27.9053 −1.22022 −0.610108 0.792319i \(-0.708874\pi\)
−0.610108 + 0.792319i \(0.708874\pi\)
\(524\) −24.0000 −1.04844
\(525\) 0 0
\(526\) −12.7321 −0.555144
\(527\) −0.203072 −0.00884595
\(528\) 0 0
\(529\) 36.7128 1.59621
\(530\) 0 0
\(531\) 0 0
\(532\) −8.48528 −0.367884
\(533\) 10.7589 0.466020
\(534\) 0 0
\(535\) 0 0
\(536\) 6.92820 0.299253
\(537\) 0 0
\(538\) 0.960947 0.0414294
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 34.7846 1.49551 0.747754 0.663976i \(-0.231132\pi\)
0.747754 + 0.663976i \(0.231132\pi\)
\(542\) 2.55103 0.109576
\(543\) 0 0
\(544\) −1.94744 −0.0834958
\(545\) 0 0
\(546\) 0 0
\(547\) −20.7327 −0.886468 −0.443234 0.896406i \(-0.646169\pi\)
−0.443234 + 0.896406i \(0.646169\pi\)
\(548\) 12.0716 0.515672
\(549\) 0 0
\(550\) 0 0
\(551\) −5.07180 −0.216066
\(552\) 0 0
\(553\) −29.0421 −1.23500
\(554\) −7.94744 −0.337654
\(555\) 0 0
\(556\) 3.46410 0.146911
\(557\) 1.69161 0.0716760 0.0358380 0.999358i \(-0.488590\pi\)
0.0358380 + 0.999358i \(0.488590\pi\)
\(558\) 0 0
\(559\) −46.3923 −1.96219
\(560\) 0 0
\(561\) 0 0
\(562\) 12.0716 0.509209
\(563\) 1.41421 0.0596020 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.588457 −0.0247347
\(567\) 0 0
\(568\) 11.5911 0.486352
\(569\) 2.53590 0.106310 0.0531552 0.998586i \(-0.483072\pi\)
0.0531552 + 0.998586i \(0.483072\pi\)
\(570\) 0 0
\(571\) −4.92820 −0.206239 −0.103119 0.994669i \(-0.532882\pi\)
−0.103119 + 0.994669i \(0.532882\pi\)
\(572\) 7.34847 0.307255
\(573\) 0 0
\(574\) 3.21539 0.134208
\(575\) 0 0
\(576\) 0 0
\(577\) −35.2538 −1.46764 −0.733818 0.679347i \(-0.762263\pi\)
−0.733818 + 0.679347i \(0.762263\pi\)
\(578\) 8.72552 0.362934
\(579\) 0 0
\(580\) 0 0
\(581\) 17.3205 0.718576
\(582\) 0 0
\(583\) −10.5558 −0.437178
\(584\) 1.26795 0.0524681
\(585\) 0 0
\(586\) −4.19615 −0.173341
\(587\) 10.3528 0.427304 0.213652 0.976910i \(-0.431464\pi\)
0.213652 + 0.976910i \(0.431464\pi\)
\(588\) 0 0
\(589\) 1.07180 0.0441626
\(590\) 0 0
\(591\) 0 0
\(592\) −12.0716 −0.496139
\(593\) −27.1475 −1.11481 −0.557406 0.830240i \(-0.688203\pi\)
−0.557406 + 0.830240i \(0.688203\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −40.3923 −1.65453
\(597\) 0 0
\(598\) 16.9706 0.693978
\(599\) −29.0718 −1.18784 −0.593921 0.804524i \(-0.702421\pi\)
−0.593921 + 0.804524i \(0.702421\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) −13.8647 −0.565084
\(603\) 0 0
\(604\) 32.5359 1.32387
\(605\) 0 0
\(606\) 0 0
\(607\) −14.8728 −0.603668 −0.301834 0.953360i \(-0.597599\pi\)
−0.301834 + 0.953360i \(0.597599\pi\)
\(608\) 10.2784 0.416845
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 22.5259 0.909812 0.454906 0.890540i \(-0.349673\pi\)
0.454906 + 0.890540i \(0.349673\pi\)
\(614\) 3.12436 0.126089
\(615\) 0 0
\(616\) 4.73205 0.190660
\(617\) 11.3137 0.455473 0.227736 0.973723i \(-0.426868\pi\)
0.227736 + 0.973723i \(0.426868\pi\)
\(618\) 0 0
\(619\) −18.3923 −0.739249 −0.369625 0.929181i \(-0.620514\pi\)
−0.369625 + 0.929181i \(0.620514\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −5.73118 −0.229800
\(623\) −37.8792 −1.51760
\(624\) 0 0
\(625\) 0 0
\(626\) 11.3205 0.452459
\(627\) 0 0
\(628\) 31.6675 1.26367
\(629\) −1.85641 −0.0740198
\(630\) 0 0
\(631\) 9.85641 0.392377 0.196189 0.980566i \(-0.437143\pi\)
0.196189 + 0.980566i \(0.437143\pi\)
\(632\) 22.9048 0.911105
\(633\) 0 0
\(634\) 6.64102 0.263748
\(635\) 0 0
\(636\) 0 0
\(637\) −4.24264 −0.168100
\(638\) 1.31268 0.0519694
\(639\) 0 0
\(640\) 0 0
\(641\) −15.4641 −0.610795 −0.305398 0.952225i \(-0.598789\pi\)
−0.305398 + 0.952225i \(0.598789\pi\)
\(642\) 0 0
\(643\) 5.85993 0.231093 0.115546 0.993302i \(-0.463138\pi\)
0.115546 + 0.993302i \(0.463138\pi\)
\(644\) −32.7846 −1.29189
\(645\) 0 0
\(646\) 0.392305 0.0154350
\(647\) 23.3853 0.919371 0.459685 0.888082i \(-0.347962\pi\)
0.459685 + 0.888082i \(0.347962\pi\)
\(648\) 0 0
\(649\) −12.9282 −0.507476
\(650\) 0 0
\(651\) 0 0
\(652\) −2.27362 −0.0890420
\(653\) −5.10205 −0.199659 −0.0998294 0.995005i \(-0.531830\pi\)
−0.0998294 + 0.995005i \(0.531830\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.24871 0.243971
\(657\) 0 0
\(658\) 3.58630 0.139809
\(659\) −30.9282 −1.20479 −0.602396 0.798197i \(-0.705787\pi\)
−0.602396 + 0.798197i \(0.705787\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 1.23835 0.0481298
\(663\) 0 0
\(664\) −13.6603 −0.530121
\(665\) 0 0
\(666\) 0 0
\(667\) −19.5959 −0.758757
\(668\) 40.3286 1.56036
\(669\) 0 0
\(670\) 0 0
\(671\) 4.92820 0.190251
\(672\) 0 0
\(673\) 0.656339 0.0253000 0.0126500 0.999920i \(-0.495973\pi\)
0.0126500 + 0.999920i \(0.495973\pi\)
\(674\) 4.05256 0.156099
\(675\) 0 0
\(676\) −8.66025 −0.333087
\(677\) −42.3992 −1.62953 −0.814767 0.579789i \(-0.803135\pi\)
−0.814767 + 0.579789i \(0.803135\pi\)
\(678\) 0 0
\(679\) −32.7846 −1.25816
\(680\) 0 0
\(681\) 0 0
\(682\) −0.277401 −0.0106222
\(683\) −15.2517 −0.583592 −0.291796 0.956481i \(-0.594253\pi\)
−0.291796 + 0.956481i \(0.594253\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.1436 −0.387284
\(687\) 0 0
\(688\) −26.9444 −1.02725
\(689\) 44.7846 1.70616
\(690\) 0 0
\(691\) −17.8564 −0.679290 −0.339645 0.940554i \(-0.610307\pi\)
−0.339645 + 0.940554i \(0.610307\pi\)
\(692\) 32.3238 1.22877
\(693\) 0 0
\(694\) −12.4449 −0.472401
\(695\) 0 0
\(696\) 0 0
\(697\) 0.960947 0.0363985
\(698\) 17.0449 0.645159
\(699\) 0 0
\(700\) 0 0
\(701\) 33.4641 1.26392 0.631961 0.775000i \(-0.282250\pi\)
0.631961 + 0.775000i \(0.282250\pi\)
\(702\) 0 0
\(703\) 9.79796 0.369537
\(704\) 2.26795 0.0854766
\(705\) 0 0
\(706\) −1.75129 −0.0659106
\(707\) −40.1528 −1.51010
\(708\) 0 0
\(709\) −28.7846 −1.08103 −0.540514 0.841335i \(-0.681770\pi\)
−0.540514 + 0.841335i \(0.681770\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 29.8744 1.11959
\(713\) 4.14110 0.155086
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 9.79796 0.365657
\(719\) −41.5692 −1.55027 −0.775135 0.631795i \(-0.782318\pi\)
−0.775135 + 0.631795i \(0.782318\pi\)
\(720\) 0 0
\(721\) 20.7846 0.774059
\(722\) 7.76457 0.288967
\(723\) 0 0
\(724\) −29.0718 −1.08044
\(725\) 0 0
\(726\) 0 0
\(727\) −7.17260 −0.266017 −0.133009 0.991115i \(-0.542464\pi\)
−0.133009 + 0.991115i \(0.542464\pi\)
\(728\) −20.0764 −0.744081
\(729\) 0 0
\(730\) 0 0
\(731\) −4.14359 −0.153256
\(732\) 0 0
\(733\) −8.18067 −0.302160 −0.151080 0.988522i \(-0.548275\pi\)
−0.151080 + 0.988522i \(0.548275\pi\)
\(734\) −11.3205 −0.417848
\(735\) 0 0
\(736\) 39.7128 1.46383
\(737\) −3.58630 −0.132103
\(738\) 0 0
\(739\) 35.8564 1.31900 0.659500 0.751705i \(-0.270768\pi\)
0.659500 + 0.751705i \(0.270768\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 13.3843 0.491352
\(743\) 24.5964 0.902356 0.451178 0.892434i \(-0.351004\pi\)
0.451178 + 0.892434i \(0.351004\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.90897 −0.216343
\(747\) 0 0
\(748\) 0.656339 0.0239981
\(749\) 3.46410 0.126576
\(750\) 0 0
\(751\) −14.1436 −0.516107 −0.258054 0.966131i \(-0.583081\pi\)
−0.258054 + 0.966131i \(0.583081\pi\)
\(752\) 6.96953 0.254153
\(753\) 0 0
\(754\) −5.56922 −0.202819
\(755\) 0 0
\(756\) 0 0
\(757\) −37.8792 −1.37674 −0.688371 0.725359i \(-0.741674\pi\)
−0.688371 + 0.725359i \(0.741674\pi\)
\(758\) 3.18016 0.115509
\(759\) 0 0
\(760\) 0 0
\(761\) −6.24871 −0.226516 −0.113258 0.993566i \(-0.536129\pi\)
−0.113258 + 0.993566i \(0.536129\pi\)
\(762\) 0 0
\(763\) −24.4949 −0.886775
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −1.75129 −0.0632766
\(767\) 54.8497 1.98051
\(768\) 0 0
\(769\) 37.7128 1.35996 0.679979 0.733231i \(-0.261989\pi\)
0.679979 + 0.733231i \(0.261989\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −15.8338 −0.569869
\(773\) −41.6685 −1.49871 −0.749356 0.662167i \(-0.769637\pi\)
−0.749356 + 0.662167i \(0.769637\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 25.8564 0.928191
\(777\) 0 0
\(778\) −3.58630 −0.128575
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) 1.51575 0.0542031
\(783\) 0 0
\(784\) −2.46410 −0.0880036
\(785\) 0 0
\(786\) 0 0
\(787\) −30.5307 −1.08830 −0.544151 0.838987i \(-0.683148\pi\)
−0.544151 + 0.838987i \(0.683148\pi\)
\(788\) −28.3858 −1.01120
\(789\) 0 0
\(790\) 0 0
\(791\) −13.8564 −0.492677
\(792\) 0 0
\(793\) −20.9086 −0.742486
\(794\) 10.6410 0.377636
\(795\) 0 0
\(796\) −6.92820 −0.245564
\(797\) −52.0213 −1.84269 −0.921344 0.388747i \(-0.872908\pi\)
−0.921344 + 0.388747i \(0.872908\pi\)
\(798\) 0 0
\(799\) 1.07180 0.0379174
\(800\) 0 0
\(801\) 0 0
\(802\) −8.96575 −0.316592
\(803\) −0.656339 −0.0231617
\(804\) 0 0
\(805\) 0 0
\(806\) 1.17691 0.0414550
\(807\) 0 0
\(808\) 31.6675 1.11406
\(809\) 9.46410 0.332740 0.166370 0.986063i \(-0.446795\pi\)
0.166370 + 0.986063i \(0.446795\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 10.7589 0.377564
\(813\) 0 0
\(814\) −2.53590 −0.0888832
\(815\) 0 0
\(816\) 0 0
\(817\) 21.8695 0.765118
\(818\) −7.80174 −0.272781
\(819\) 0 0
\(820\) 0 0
\(821\) −37.1769 −1.29748 −0.648742 0.761009i \(-0.724704\pi\)
−0.648742 + 0.761009i \(0.724704\pi\)
\(822\) 0 0
\(823\) −35.2538 −1.22887 −0.614435 0.788967i \(-0.710616\pi\)
−0.614435 + 0.788967i \(0.710616\pi\)
\(824\) −16.3923 −0.571053
\(825\) 0 0
\(826\) 16.3923 0.570361
\(827\) 19.4944 0.677886 0.338943 0.940807i \(-0.389931\pi\)
0.338943 + 0.940807i \(0.389931\pi\)
\(828\) 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
\(832\) −9.62209 −0.333586
\(833\) −0.378937 −0.0131294
\(834\) 0 0
\(835\) 0 0
\(836\) −3.46410 −0.119808
\(837\) 0 0
\(838\) 3.58630 0.123887
\(839\) 14.7846 0.510421 0.255211 0.966885i \(-0.417855\pi\)
0.255211 + 0.966885i \(0.417855\pi\)
\(840\) 0 0
\(841\) −22.5692 −0.778249
\(842\) 4.69591 0.161832
\(843\) 0 0
\(844\) −27.4641 −0.945353
\(845\) 0 0
\(846\) 0 0
\(847\) −2.44949 −0.0841655
\(848\) 26.0106 0.893209
\(849\) 0 0
\(850\) 0 0
\(851\) 37.8564 1.29770
\(852\) 0 0
\(853\) 50.6071 1.73275 0.866377 0.499391i \(-0.166443\pi\)
0.866377 + 0.499391i \(0.166443\pi\)
\(854\) −6.24871 −0.213826
\(855\) 0 0
\(856\) −2.73205 −0.0933796
\(857\) 15.0759 0.514982 0.257491 0.966281i \(-0.417104\pi\)
0.257491 + 0.966281i \(0.417104\pi\)
\(858\) 0 0
\(859\) 47.1769 1.60966 0.804828 0.593508i \(-0.202258\pi\)
0.804828 + 0.593508i \(0.202258\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −14.3452 −0.488600
\(863\) 24.9010 0.847641 0.423821 0.905746i \(-0.360689\pi\)
0.423821 + 0.905746i \(0.360689\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −17.0718 −0.580123
\(867\) 0 0
\(868\) −2.27362 −0.0771718
\(869\) −11.8564 −0.402201
\(870\) 0 0
\(871\) 15.2154 0.515554
\(872\) 19.3185 0.654208
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 0 0
\(877\) 25.1512 0.849297 0.424648 0.905358i \(-0.360398\pi\)
0.424648 + 0.905358i \(0.360398\pi\)
\(878\) −18.5606 −0.626391
\(879\) 0 0
\(880\) 0 0
\(881\) 5.07180 0.170873 0.0854366 0.996344i \(-0.472771\pi\)
0.0854366 + 0.996344i \(0.472771\pi\)
\(882\) 0 0
\(883\) 53.5370 1.80166 0.900832 0.434167i \(-0.142957\pi\)
0.900832 + 0.434167i \(0.142957\pi\)
\(884\) −2.78461 −0.0936566
\(885\) 0 0
\(886\) 4.67949 0.157211
\(887\) 38.7386 1.30071 0.650357 0.759629i \(-0.274619\pi\)
0.650357 + 0.759629i \(0.274619\pi\)
\(888\) 0 0
\(889\) −42.0000 −1.40863
\(890\) 0 0
\(891\) 0 0
\(892\) −25.4558 −0.852325
\(893\) −5.65685 −0.189299
\(894\) 0 0
\(895\) 0 0
\(896\) −28.0526 −0.937170
\(897\) 0 0
\(898\) 8.00481 0.267124
\(899\) −1.35898 −0.0453246
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 1.31268 0.0437074
\(903\) 0 0
\(904\) 10.9282 0.363467
\(905\) 0 0
\(906\) 0 0
\(907\) 48.6381 1.61500 0.807500 0.589867i \(-0.200820\pi\)
0.807500 + 0.589867i \(0.200820\pi\)
\(908\) 19.4201 0.644477
\(909\) 0 0
\(910\) 0 0
\(911\) −37.8564 −1.25424 −0.627119 0.778923i \(-0.715766\pi\)
−0.627119 + 0.778923i \(0.715766\pi\)
\(912\) 0 0
\(913\) 7.07107 0.234018
\(914\) 5.41154 0.178998
\(915\) 0 0
\(916\) 48.2487 1.59418
\(917\) −33.9411 −1.12083
\(918\) 0 0
\(919\) 54.7846 1.80718 0.903589 0.428401i \(-0.140923\pi\)
0.903589 + 0.428401i \(0.140923\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −3.93803 −0.129692
\(923\) 25.4558 0.837889
\(924\) 0 0
\(925\) 0 0
\(926\) 3.89488 0.127994
\(927\) 0 0
\(928\) −13.0325 −0.427814
\(929\) −22.6410 −0.742828 −0.371414 0.928467i \(-0.621127\pi\)
−0.371414 + 0.928467i \(0.621127\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −37.2228 −1.21927
\(933\) 0 0
\(934\) 16.2872 0.532933
\(935\) 0 0
\(936\) 0 0
\(937\) 47.0208 1.53610 0.768051 0.640389i \(-0.221227\pi\)
0.768051 + 0.640389i \(0.221227\pi\)
\(938\) 4.54725 0.148473
\(939\) 0 0
\(940\) 0 0
\(941\) 19.6077 0.639193 0.319596 0.947554i \(-0.396453\pi\)
0.319596 + 0.947554i \(0.396453\pi\)
\(942\) 0 0
\(943\) −19.5959 −0.638131
\(944\) 31.8564 1.03684
\(945\) 0 0
\(946\) −5.66025 −0.184031
\(947\) −16.7675 −0.544870 −0.272435 0.962174i \(-0.587829\pi\)
−0.272435 + 0.962174i \(0.587829\pi\)
\(948\) 0 0
\(949\) 2.78461 0.0903923
\(950\) 0 0
\(951\) 0 0
\(952\) −1.79315 −0.0581164
\(953\) 6.79367 0.220068 0.110034 0.993928i \(-0.464904\pi\)
0.110034 + 0.993928i \(0.464904\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) 8.83701 0.285511
\(959\) 17.0718 0.551277
\(960\) 0 0
\(961\) −30.7128 −0.990736
\(962\) 10.7589 0.346881
\(963\) 0 0
\(964\) −15.4641 −0.498065
\(965\) 0 0
\(966\) 0 0
\(967\) −22.0454 −0.708933 −0.354466 0.935069i \(-0.615337\pi\)
−0.354466 + 0.935069i \(0.615337\pi\)
\(968\) 1.93185 0.0620921
\(969\) 0 0
\(970\) 0 0
\(971\) −39.7128 −1.27444 −0.637222 0.770680i \(-0.719917\pi\)
−0.637222 + 0.770680i \(0.719917\pi\)
\(972\) 0 0
\(973\) 4.89898 0.157054
\(974\) −20.7846 −0.665982
\(975\) 0 0
\(976\) −12.1436 −0.388707
\(977\) −14.1421 −0.452447 −0.226224 0.974075i \(-0.572638\pi\)
−0.226224 + 0.974075i \(0.572638\pi\)
\(978\) 0 0
\(979\) −15.4641 −0.494235
\(980\) 0 0
\(981\) 0 0
\(982\) 7.17260 0.228887
\(983\) 47.1223 1.50297 0.751484 0.659751i \(-0.229338\pi\)
0.751484 + 0.659751i \(0.229338\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −0.497423 −0.0158412
\(987\) 0 0
\(988\) 14.6969 0.467572
\(989\) 84.4974 2.68686
\(990\) 0 0
\(991\) −7.46410 −0.237105 −0.118553 0.992948i \(-0.537825\pi\)
−0.118553 + 0.992948i \(0.537825\pi\)
\(992\) 2.75410 0.0874427
\(993\) 0 0
\(994\) 7.60770 0.241301
\(995\) 0 0
\(996\) 0 0
\(997\) 39.4964 1.25086 0.625432 0.780278i \(-0.284923\pi\)
0.625432 + 0.780278i \(0.284923\pi\)
\(998\) −8.28221 −0.262169
\(999\) 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 2475.2.a.bg.1.2 4
3.2 odd 2 2475.2.a.bh.1.3 4
5.2 odd 4 495.2.c.b.199.2 4
5.3 odd 4 495.2.c.b.199.3 yes 4
5.4 even 2 inner 2475.2.a.bg.1.3 4
15.2 even 4 495.2.c.c.199.3 yes 4
15.8 even 4 495.2.c.c.199.2 yes 4
15.14 odd 2 2475.2.a.bh.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.c.b.199.2 4 5.2 odd 4
495.2.c.b.199.3 yes 4 5.3 odd 4
495.2.c.c.199.2 yes 4 15.8 even 4
495.2.c.c.199.3 yes 4 15.2 even 4
2475.2.a.bg.1.2 4 1.1 even 1 trivial
2475.2.a.bg.1.3 4 5.4 even 2 inner
2475.2.a.bh.1.2 4 15.14 odd 2
2475.2.a.bh.1.3 4 3.2 odd 2