Properties

Label 4275.2.a.bp.1.3
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
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] = [4275,2,Mod(1,4275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, 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("4275.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.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: \(34.1360468641\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13068.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 171)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.82405\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.27582 q^{2} -0.372281 q^{4} -3.37228 q^{7} -3.02661 q^{8} +O(q^{10})\) \(q+1.27582 q^{2} -0.372281 q^{4} -3.37228 q^{7} -3.02661 q^{8} +4.70285 q^{11} -2.00000 q^{13} -4.30243 q^{14} -3.11684 q^{16} +2.15121 q^{17} +1.00000 q^{19} +6.00000 q^{22} -6.85407 q^{23} -2.55164 q^{26} +1.25544 q^{28} +6.85407 q^{29} -6.74456 q^{31} +2.07668 q^{32} +2.74456 q^{34} +0.744563 q^{37} +1.27582 q^{38} +2.55164 q^{41} -6.11684 q^{43} -1.75079 q^{44} -8.74456 q^{46} +9.00528 q^{47} +4.37228 q^{49} +0.744563 q^{52} +11.9574 q^{53} +10.2066 q^{56} +8.74456 q^{58} -5.10328 q^{59} +12.1168 q^{61} -8.60485 q^{62} +8.88316 q^{64} +4.00000 q^{67} -0.800857 q^{68} +13.7081 q^{71} -12.1168 q^{73} +0.949929 q^{74} -0.372281 q^{76} -15.8593 q^{77} -4.00000 q^{79} +3.25544 q^{82} +1.75079 q^{83} -7.80400 q^{86} -14.2337 q^{88} -11.9574 q^{89} +6.74456 q^{91} +2.55164 q^{92} +11.4891 q^{94} +15.4891 q^{97} +5.57825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{4} - 2 q^{7} - 8 q^{13} + 22 q^{16} + 4 q^{19} + 24 q^{22} + 28 q^{28} - 4 q^{31} - 12 q^{34} - 20 q^{37} + 10 q^{43} - 12 q^{46} + 6 q^{49} - 20 q^{52} + 12 q^{58} + 14 q^{61} + 70 q^{64} + 16 q^{67} - 14 q^{73} + 10 q^{76} - 16 q^{79} + 36 q^{82} + 12 q^{88} + 4 q^{91} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27582 0.902142 0.451071 0.892488i \(-0.351042\pi\)
0.451071 + 0.892488i \(0.351042\pi\)
\(3\) 0 0
\(4\) −0.372281 −0.186141
\(5\) 0 0
\(6\) 0 0
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) −3.02661 −1.07007
\(9\) 0 0
\(10\) 0 0
\(11\) 4.70285 1.41796 0.708982 0.705227i \(-0.249155\pi\)
0.708982 + 0.705227i \(0.249155\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.30243 −1.14987
\(15\) 0 0
\(16\) −3.11684 −0.779211
\(17\) 2.15121 0.521746 0.260873 0.965373i \(-0.415990\pi\)
0.260873 + 0.965373i \(0.415990\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −6.85407 −1.42917 −0.714586 0.699548i \(-0.753385\pi\)
−0.714586 + 0.699548i \(0.753385\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.55164 −0.500418
\(27\) 0 0
\(28\) 1.25544 0.237255
\(29\) 6.85407 1.27277 0.636384 0.771372i \(-0.280429\pi\)
0.636384 + 0.771372i \(0.280429\pi\)
\(30\) 0 0
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) 2.07668 0.367108
\(33\) 0 0
\(34\) 2.74456 0.470689
\(35\) 0 0
\(36\) 0 0
\(37\) 0.744563 0.122405 0.0612027 0.998125i \(-0.480506\pi\)
0.0612027 + 0.998125i \(0.480506\pi\)
\(38\) 1.27582 0.206965
\(39\) 0 0
\(40\) 0 0
\(41\) 2.55164 0.398499 0.199250 0.979949i \(-0.436150\pi\)
0.199250 + 0.979949i \(0.436150\pi\)
\(42\) 0 0
\(43\) −6.11684 −0.932810 −0.466405 0.884571i \(-0.654451\pi\)
−0.466405 + 0.884571i \(0.654451\pi\)
\(44\) −1.75079 −0.263941
\(45\) 0 0
\(46\) −8.74456 −1.28932
\(47\) 9.00528 1.31356 0.656778 0.754084i \(-0.271919\pi\)
0.656778 + 0.754084i \(0.271919\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) 0.744563 0.103252
\(53\) 11.9574 1.64247 0.821234 0.570591i \(-0.193286\pi\)
0.821234 + 0.570591i \(0.193286\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 10.2066 1.36391
\(57\) 0 0
\(58\) 8.74456 1.14822
\(59\) −5.10328 −0.664391 −0.332195 0.943211i \(-0.607789\pi\)
−0.332195 + 0.943211i \(0.607789\pi\)
\(60\) 0 0
\(61\) 12.1168 1.55140 0.775701 0.631100i \(-0.217396\pi\)
0.775701 + 0.631100i \(0.217396\pi\)
\(62\) −8.60485 −1.09282
\(63\) 0 0
\(64\) 8.88316 1.11039
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −0.800857 −0.0971181
\(69\) 0 0
\(70\) 0 0
\(71\) 13.7081 1.62686 0.813428 0.581665i \(-0.197599\pi\)
0.813428 + 0.581665i \(0.197599\pi\)
\(72\) 0 0
\(73\) −12.1168 −1.41817 −0.709085 0.705123i \(-0.750892\pi\)
−0.709085 + 0.705123i \(0.750892\pi\)
\(74\) 0.949929 0.110427
\(75\) 0 0
\(76\) −0.372281 −0.0427036
\(77\) −15.8593 −1.80734
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.25544 0.359503
\(83\) 1.75079 0.192174 0.0960868 0.995373i \(-0.469367\pi\)
0.0960868 + 0.995373i \(0.469367\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −7.80400 −0.841527
\(87\) 0 0
\(88\) −14.2337 −1.51732
\(89\) −11.9574 −1.26748 −0.633738 0.773547i \(-0.718480\pi\)
−0.633738 + 0.773547i \(0.718480\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 2.55164 0.266027
\(93\) 0 0
\(94\) 11.4891 1.18501
\(95\) 0 0
\(96\) 0 0
\(97\) 15.4891 1.57268 0.786341 0.617792i \(-0.211973\pi\)
0.786341 + 0.617792i \(0.211973\pi\)
\(98\) 5.57825 0.563488
\(99\) 0 0
\(100\) 0 0
\(101\) 8.60485 0.856215 0.428107 0.903728i \(-0.359180\pi\)
0.428107 + 0.903728i \(0.359180\pi\)
\(102\) 0 0
\(103\) 18.7446 1.84696 0.923478 0.383651i \(-0.125333\pi\)
0.923478 + 0.383651i \(0.125333\pi\)
\(104\) 6.05321 0.593566
\(105\) 0 0
\(106\) 15.2554 1.48174
\(107\) 9.40571 0.909284 0.454642 0.890674i \(-0.349767\pi\)
0.454642 + 0.890674i \(0.349767\pi\)
\(108\) 0 0
\(109\) 16.7446 1.60384 0.801919 0.597433i \(-0.203812\pi\)
0.801919 + 0.597433i \(0.203812\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.5109 0.993184
\(113\) 1.75079 0.164700 0.0823500 0.996603i \(-0.473757\pi\)
0.0823500 + 0.996603i \(0.473757\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.55164 −0.236914
\(117\) 0 0
\(118\) −6.51087 −0.599375
\(119\) −7.25450 −0.665019
\(120\) 0 0
\(121\) 11.1168 1.01062
\(122\) 15.4589 1.39958
\(123\) 0 0
\(124\) 2.51087 0.225483
\(125\) 0 0
\(126\) 0 0
\(127\) 1.25544 0.111402 0.0557010 0.998447i \(-0.482261\pi\)
0.0557010 + 0.998447i \(0.482261\pi\)
\(128\) 7.17996 0.634625
\(129\) 0 0
\(130\) 0 0
\(131\) −3.90200 −0.340919 −0.170460 0.985365i \(-0.554525\pi\)
−0.170460 + 0.985365i \(0.554525\pi\)
\(132\) 0 0
\(133\) −3.37228 −0.292414
\(134\) 5.10328 0.440857
\(135\) 0 0
\(136\) −6.51087 −0.558303
\(137\) 16.6602 1.42338 0.711689 0.702495i \(-0.247931\pi\)
0.711689 + 0.702495i \(0.247931\pi\)
\(138\) 0 0
\(139\) −14.1168 −1.19738 −0.598688 0.800983i \(-0.704311\pi\)
−0.598688 + 0.800983i \(0.704311\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 17.4891 1.46765
\(143\) −9.40571 −0.786545
\(144\) 0 0
\(145\) 0 0
\(146\) −15.4589 −1.27939
\(147\) 0 0
\(148\) −0.277187 −0.0227846
\(149\) 1.35036 0.110626 0.0553128 0.998469i \(-0.482384\pi\)
0.0553128 + 0.998469i \(0.482384\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) −3.02661 −0.245490
\(153\) 0 0
\(154\) −20.2337 −1.63048
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −5.10328 −0.405995
\(159\) 0 0
\(160\) 0 0
\(161\) 23.1138 1.82163
\(162\) 0 0
\(163\) −1.48913 −0.116637 −0.0583186 0.998298i \(-0.518574\pi\)
−0.0583186 + 0.998298i \(0.518574\pi\)
\(164\) −0.949929 −0.0741770
\(165\) 0 0
\(166\) 2.23369 0.173368
\(167\) 4.30243 0.332932 0.166466 0.986047i \(-0.446764\pi\)
0.166466 + 0.986047i \(0.446764\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 2.27719 0.173634
\(173\) 6.85407 0.521105 0.260553 0.965460i \(-0.416095\pi\)
0.260553 + 0.965460i \(0.416095\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −14.6581 −1.10489
\(177\) 0 0
\(178\) −15.2554 −1.14344
\(179\) 9.40571 0.703016 0.351508 0.936185i \(-0.385669\pi\)
0.351508 + 0.936185i \(0.385669\pi\)
\(180\) 0 0
\(181\) −3.48913 −0.259345 −0.129672 0.991557i \(-0.541393\pi\)
−0.129672 + 0.991557i \(0.541393\pi\)
\(182\) 8.60485 0.637834
\(183\) 0 0
\(184\) 20.7446 1.52931
\(185\) 0 0
\(186\) 0 0
\(187\) 10.1168 0.739817
\(188\) −3.35250 −0.244506
\(189\) 0 0
\(190\) 0 0
\(191\) −8.20442 −0.593651 −0.296826 0.954932i \(-0.595928\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(192\) 0 0
\(193\) 18.2337 1.31249 0.656245 0.754548i \(-0.272144\pi\)
0.656245 + 0.754548i \(0.272144\pi\)
\(194\) 19.7613 1.41878
\(195\) 0 0
\(196\) −1.62772 −0.116266
\(197\) 3.50157 0.249477 0.124738 0.992190i \(-0.460191\pi\)
0.124738 + 0.992190i \(0.460191\pi\)
\(198\) 0 0
\(199\) 18.1168 1.28427 0.642135 0.766592i \(-0.278049\pi\)
0.642135 + 0.766592i \(0.278049\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.9783 0.772427
\(203\) −23.1138 −1.62227
\(204\) 0 0
\(205\) 0 0
\(206\) 23.9147 1.66622
\(207\) 0 0
\(208\) 6.23369 0.432228
\(209\) 4.70285 0.325303
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −4.45150 −0.305730
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 22.7446 1.54400
\(218\) 21.3631 1.44689
\(219\) 0 0
\(220\) 0 0
\(221\) −4.30243 −0.289413
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −7.00314 −0.467917
\(225\) 0 0
\(226\) 2.23369 0.148583
\(227\) −12.9073 −0.856686 −0.428343 0.903616i \(-0.640903\pi\)
−0.428343 + 0.903616i \(0.640903\pi\)
\(228\) 0 0
\(229\) 21.3723 1.41232 0.706160 0.708052i \(-0.250426\pi\)
0.706160 + 0.708052i \(0.250426\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −20.7446 −1.36195
\(233\) −7.25450 −0.475258 −0.237629 0.971356i \(-0.576370\pi\)
−0.237629 + 0.971356i \(0.576370\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.89986 0.123670
\(237\) 0 0
\(238\) −9.25544 −0.599941
\(239\) −27.0158 −1.74751 −0.873755 0.486367i \(-0.838322\pi\)
−0.873755 + 0.486367i \(0.838322\pi\)
\(240\) 0 0
\(241\) 10.2337 0.659210 0.329605 0.944119i \(-0.393084\pi\)
0.329605 + 0.944119i \(0.393084\pi\)
\(242\) 14.1831 0.911724
\(243\) 0 0
\(244\) −4.51087 −0.288779
\(245\) 0 0
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 20.4131 1.29624
\(249\) 0 0
\(250\) 0 0
\(251\) −14.9094 −0.941074 −0.470537 0.882380i \(-0.655940\pi\)
−0.470537 + 0.882380i \(0.655940\pi\)
\(252\) 0 0
\(253\) −32.2337 −2.02651
\(254\) 1.60171 0.100500
\(255\) 0 0
\(256\) −8.60597 −0.537873
\(257\) 2.55164 0.159167 0.0795835 0.996828i \(-0.474641\pi\)
0.0795835 + 0.996828i \(0.474641\pi\)
\(258\) 0 0
\(259\) −2.51087 −0.156018
\(260\) 0 0
\(261\) 0 0
\(262\) −4.97825 −0.307557
\(263\) 9.80614 0.604672 0.302336 0.953201i \(-0.402233\pi\)
0.302336 + 0.953201i \(0.402233\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.30243 −0.263799
\(267\) 0 0
\(268\) −1.48913 −0.0909628
\(269\) −25.6655 −1.56485 −0.782426 0.622743i \(-0.786018\pi\)
−0.782426 + 0.622743i \(0.786018\pi\)
\(270\) 0 0
\(271\) 2.51087 0.152525 0.0762624 0.997088i \(-0.475701\pi\)
0.0762624 + 0.997088i \(0.475701\pi\)
\(272\) −6.70500 −0.406550
\(273\) 0 0
\(274\) 21.2554 1.28409
\(275\) 0 0
\(276\) 0 0
\(277\) 8.11684 0.487694 0.243847 0.969814i \(-0.421591\pi\)
0.243847 + 0.969814i \(0.421591\pi\)
\(278\) −18.0106 −1.08020
\(279\) 0 0
\(280\) 0 0
\(281\) 10.3556 0.617766 0.308883 0.951100i \(-0.400045\pi\)
0.308883 + 0.951100i \(0.400045\pi\)
\(282\) 0 0
\(283\) −0.627719 −0.0373140 −0.0186570 0.999826i \(-0.505939\pi\)
−0.0186570 + 0.999826i \(0.505939\pi\)
\(284\) −5.10328 −0.302824
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −8.60485 −0.507928
\(288\) 0 0
\(289\) −12.3723 −0.727781
\(290\) 0 0
\(291\) 0 0
\(292\) 4.51087 0.263979
\(293\) −26.4663 −1.54618 −0.773090 0.634296i \(-0.781290\pi\)
−0.773090 + 0.634296i \(0.781290\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.25350 −0.130982
\(297\) 0 0
\(298\) 1.72281 0.0997999
\(299\) 13.7081 0.792762
\(300\) 0 0
\(301\) 20.6277 1.18896
\(302\) −5.10328 −0.293661
\(303\) 0 0
\(304\) −3.11684 −0.178763
\(305\) 0 0
\(306\) 0 0
\(307\) −2.51087 −0.143303 −0.0716516 0.997430i \(-0.522827\pi\)
−0.0716516 + 0.997430i \(0.522827\pi\)
\(308\) 5.90414 0.336420
\(309\) 0 0
\(310\) 0 0
\(311\) 27.0158 1.53193 0.765964 0.642883i \(-0.222262\pi\)
0.765964 + 0.642883i \(0.222262\pi\)
\(312\) 0 0
\(313\) 15.4891 0.875497 0.437749 0.899097i \(-0.355776\pi\)
0.437749 + 0.899097i \(0.355776\pi\)
\(314\) −2.55164 −0.143997
\(315\) 0 0
\(316\) 1.48913 0.0837698
\(317\) −35.0712 −1.96979 −0.984897 0.173139i \(-0.944609\pi\)
−0.984897 + 0.173139i \(0.944609\pi\)
\(318\) 0 0
\(319\) 32.2337 1.80474
\(320\) 0 0
\(321\) 0 0
\(322\) 29.4891 1.64336
\(323\) 2.15121 0.119697
\(324\) 0 0
\(325\) 0 0
\(326\) −1.89986 −0.105223
\(327\) 0 0
\(328\) −7.72281 −0.426421
\(329\) −30.3683 −1.67426
\(330\) 0 0
\(331\) −26.9783 −1.48286 −0.741429 0.671031i \(-0.765852\pi\)
−0.741429 + 0.671031i \(0.765852\pi\)
\(332\) −0.651785 −0.0357713
\(333\) 0 0
\(334\) 5.48913 0.300352
\(335\) 0 0
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) −11.4824 −0.624560
\(339\) 0 0
\(340\) 0 0
\(341\) −31.7187 −1.71766
\(342\) 0 0
\(343\) 8.86141 0.478471
\(344\) 18.5133 0.998169
\(345\) 0 0
\(346\) 8.74456 0.470111
\(347\) −3.10114 −0.166478 −0.0832390 0.996530i \(-0.526526\pi\)
−0.0832390 + 0.996530i \(0.526526\pi\)
\(348\) 0 0
\(349\) −10.8614 −0.581398 −0.290699 0.956815i \(-0.593888\pi\)
−0.290699 + 0.956815i \(0.593888\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 9.76631 0.520546
\(353\) −22.3130 −1.18760 −0.593800 0.804612i \(-0.702373\pi\)
−0.593800 + 0.804612i \(0.702373\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 4.45150 0.235929
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 27.8167 1.46811 0.734055 0.679090i \(-0.237626\pi\)
0.734055 + 0.679090i \(0.237626\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −4.45150 −0.233966
\(363\) 0 0
\(364\) −2.51087 −0.131606
\(365\) 0 0
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 21.3631 1.11363
\(369\) 0 0
\(370\) 0 0
\(371\) −40.3236 −2.09349
\(372\) 0 0
\(373\) −10.2337 −0.529880 −0.264940 0.964265i \(-0.585352\pi\)
−0.264940 + 0.964265i \(0.585352\pi\)
\(374\) 12.9073 0.667420
\(375\) 0 0
\(376\) −27.2554 −1.40559
\(377\) −13.7081 −0.706005
\(378\) 0 0
\(379\) 17.2554 0.886352 0.443176 0.896435i \(-0.353852\pi\)
0.443176 + 0.896435i \(0.353852\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −10.4674 −0.535558
\(383\) −0.800857 −0.0409219 −0.0204609 0.999791i \(-0.506513\pi\)
−0.0204609 + 0.999791i \(0.506513\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.2629 1.18405
\(387\) 0 0
\(388\) −5.76631 −0.292740
\(389\) −16.6602 −0.844706 −0.422353 0.906431i \(-0.638796\pi\)
−0.422353 + 0.906431i \(0.638796\pi\)
\(390\) 0 0
\(391\) −14.7446 −0.745665
\(392\) −13.2332 −0.668376
\(393\) 0 0
\(394\) 4.46738 0.225063
\(395\) 0 0
\(396\) 0 0
\(397\) −0.116844 −0.00586423 −0.00293212 0.999996i \(-0.500933\pi\)
−0.00293212 + 0.999996i \(0.500933\pi\)
\(398\) 23.1138 1.15859
\(399\) 0 0
\(400\) 0 0
\(401\) −16.2598 −0.811975 −0.405987 0.913879i \(-0.633072\pi\)
−0.405987 + 0.913879i \(0.633072\pi\)
\(402\) 0 0
\(403\) 13.4891 0.671941
\(404\) −3.20343 −0.159376
\(405\) 0 0
\(406\) −29.4891 −1.46352
\(407\) 3.50157 0.173566
\(408\) 0 0
\(409\) −15.4891 −0.765888 −0.382944 0.923772i \(-0.625090\pi\)
−0.382944 + 0.923772i \(0.625090\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −6.97825 −0.343794
\(413\) 17.2097 0.846834
\(414\) 0 0
\(415\) 0 0
\(416\) −4.15335 −0.203635
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 1.75079 0.0855314 0.0427657 0.999085i \(-0.486383\pi\)
0.0427657 + 0.999085i \(0.486383\pi\)
\(420\) 0 0
\(421\) 36.9783 1.80221 0.901105 0.433601i \(-0.142757\pi\)
0.901105 + 0.433601i \(0.142757\pi\)
\(422\) −5.10328 −0.248424
\(423\) 0 0
\(424\) −36.1902 −1.75755
\(425\) 0 0
\(426\) 0 0
\(427\) −40.8614 −1.97742
\(428\) −3.50157 −0.169255
\(429\) 0 0
\(430\) 0 0
\(431\) −7.80400 −0.375905 −0.187953 0.982178i \(-0.560185\pi\)
−0.187953 + 0.982178i \(0.560185\pi\)
\(432\) 0 0
\(433\) 22.0000 1.05725 0.528626 0.848855i \(-0.322707\pi\)
0.528626 + 0.848855i \(0.322707\pi\)
\(434\) 29.0180 1.39291
\(435\) 0 0
\(436\) −6.23369 −0.298540
\(437\) −6.85407 −0.327875
\(438\) 0 0
\(439\) 16.2337 0.774792 0.387396 0.921913i \(-0.373375\pi\)
0.387396 + 0.921913i \(0.373375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.48913 −0.261091
\(443\) 12.5069 0.594218 0.297109 0.954843i \(-0.403977\pi\)
0.297109 + 0.954843i \(0.403977\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.10328 0.241647
\(447\) 0 0
\(448\) −29.9565 −1.41531
\(449\) 33.4695 1.57952 0.789761 0.613414i \(-0.210204\pi\)
0.789761 + 0.613414i \(0.210204\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −0.651785 −0.0306574
\(453\) 0 0
\(454\) −16.4674 −0.772852
\(455\) 0 0
\(456\) 0 0
\(457\) 2.62772 0.122919 0.0614597 0.998110i \(-0.480424\pi\)
0.0614597 + 0.998110i \(0.480424\pi\)
\(458\) 27.2672 1.27411
\(459\) 0 0
\(460\) 0 0
\(461\) −5.65278 −0.263276 −0.131638 0.991298i \(-0.542024\pi\)
−0.131638 + 0.991298i \(0.542024\pi\)
\(462\) 0 0
\(463\) −3.37228 −0.156723 −0.0783616 0.996925i \(-0.524969\pi\)
−0.0783616 + 0.996925i \(0.524969\pi\)
\(464\) −21.3631 −0.991755
\(465\) 0 0
\(466\) −9.25544 −0.428750
\(467\) 30.5174 1.41218 0.706089 0.708123i \(-0.250458\pi\)
0.706089 + 0.708123i \(0.250458\pi\)
\(468\) 0 0
\(469\) −13.4891 −0.622870
\(470\) 0 0
\(471\) 0 0
\(472\) 15.4456 0.710943
\(473\) −28.7666 −1.32269
\(474\) 0 0
\(475\) 0 0
\(476\) 2.70071 0.123787
\(477\) 0 0
\(478\) −34.4674 −1.57650
\(479\) −8.45578 −0.386355 −0.193177 0.981164i \(-0.561879\pi\)
−0.193177 + 0.981164i \(0.561879\pi\)
\(480\) 0 0
\(481\) −1.48913 −0.0678983
\(482\) 13.0564 0.594701
\(483\) 0 0
\(484\) −4.13859 −0.188118
\(485\) 0 0
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −36.6729 −1.66010
\(489\) 0 0
\(490\) 0 0
\(491\) −6.85407 −0.309320 −0.154660 0.987968i \(-0.549428\pi\)
−0.154660 + 0.987968i \(0.549428\pi\)
\(492\) 0 0
\(493\) 14.7446 0.664062
\(494\) −2.55164 −0.114804
\(495\) 0 0
\(496\) 21.0217 0.943904
\(497\) −46.2277 −2.07360
\(498\) 0 0
\(499\) 6.11684 0.273828 0.136914 0.990583i \(-0.456282\pi\)
0.136914 + 0.990583i \(0.456282\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −19.0217 −0.848982
\(503\) −22.1639 −0.988240 −0.494120 0.869394i \(-0.664510\pi\)
−0.494120 + 0.869394i \(0.664510\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −41.1244 −1.82820
\(507\) 0 0
\(508\) −0.467376 −0.0207365
\(509\) 10.3556 0.459006 0.229503 0.973308i \(-0.426290\pi\)
0.229503 + 0.973308i \(0.426290\pi\)
\(510\) 0 0
\(511\) 40.8614 1.80760
\(512\) −25.3396 −1.11986
\(513\) 0 0
\(514\) 3.25544 0.143591
\(515\) 0 0
\(516\) 0 0
\(517\) 42.3505 1.86257
\(518\) −3.20343 −0.140750
\(519\) 0 0
\(520\) 0 0
\(521\) −7.65492 −0.335368 −0.167684 0.985841i \(-0.553629\pi\)
−0.167684 + 0.985841i \(0.553629\pi\)
\(522\) 0 0
\(523\) −16.2337 −0.709850 −0.354925 0.934895i \(-0.615494\pi\)
−0.354925 + 0.934895i \(0.615494\pi\)
\(524\) 1.45264 0.0634589
\(525\) 0 0
\(526\) 12.5109 0.545500
\(527\) −14.5090 −0.632022
\(528\) 0 0
\(529\) 23.9783 1.04253
\(530\) 0 0
\(531\) 0 0
\(532\) 1.25544 0.0544301
\(533\) −5.10328 −0.221048
\(534\) 0 0
\(535\) 0 0
\(536\) −12.1064 −0.522918
\(537\) 0 0
\(538\) −32.7446 −1.41172
\(539\) 20.5622 0.885677
\(540\) 0 0
\(541\) 3.88316 0.166950 0.0834750 0.996510i \(-0.473398\pi\)
0.0834750 + 0.996510i \(0.473398\pi\)
\(542\) 3.20343 0.137599
\(543\) 0 0
\(544\) 4.46738 0.191537
\(545\) 0 0
\(546\) 0 0
\(547\) 12.2337 0.523075 0.261537 0.965193i \(-0.415771\pi\)
0.261537 + 0.965193i \(0.415771\pi\)
\(548\) −6.20228 −0.264948
\(549\) 0 0
\(550\) 0 0
\(551\) 6.85407 0.291993
\(552\) 0 0
\(553\) 13.4891 0.573616
\(554\) 10.3556 0.439969
\(555\) 0 0
\(556\) 5.25544 0.222880
\(557\) 1.35036 0.0572165 0.0286082 0.999591i \(-0.490892\pi\)
0.0286082 + 0.999591i \(0.490892\pi\)
\(558\) 0 0
\(559\) 12.2337 0.517430
\(560\) 0 0
\(561\) 0 0
\(562\) 13.2119 0.557312
\(563\) 25.8146 1.08795 0.543977 0.839100i \(-0.316918\pi\)
0.543977 + 0.839100i \(0.316918\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.800857 −0.0336625
\(567\) 0 0
\(568\) −41.4891 −1.74085
\(569\) 20.5622 0.862012 0.431006 0.902349i \(-0.358159\pi\)
0.431006 + 0.902349i \(0.358159\pi\)
\(570\) 0 0
\(571\) 25.4891 1.06669 0.533343 0.845899i \(-0.320935\pi\)
0.533343 + 0.845899i \(0.320935\pi\)
\(572\) 3.50157 0.146408
\(573\) 0 0
\(574\) −10.9783 −0.458223
\(575\) 0 0
\(576\) 0 0
\(577\) 23.8832 0.994269 0.497134 0.867674i \(-0.334386\pi\)
0.497134 + 0.867674i \(0.334386\pi\)
\(578\) −15.7848 −0.656562
\(579\) 0 0
\(580\) 0 0
\(581\) −5.90414 −0.244945
\(582\) 0 0
\(583\) 56.2337 2.32896
\(584\) 36.6729 1.51754
\(585\) 0 0
\(586\) −33.7663 −1.39487
\(587\) 32.1191 1.32570 0.662849 0.748753i \(-0.269347\pi\)
0.662849 + 0.748753i \(0.269347\pi\)
\(588\) 0 0
\(589\) −6.74456 −0.277905
\(590\) 0 0
\(591\) 0 0
\(592\) −2.32069 −0.0953796
\(593\) −27.4163 −1.12585 −0.562926 0.826508i \(-0.690324\pi\)
−0.562926 + 0.826508i \(0.690324\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.502713 −0.0205919
\(597\) 0 0
\(598\) 17.4891 0.715184
\(599\) 8.60485 0.351585 0.175792 0.984427i \(-0.443751\pi\)
0.175792 + 0.984427i \(0.443751\pi\)
\(600\) 0 0
\(601\) −36.7446 −1.49884 −0.749421 0.662094i \(-0.769668\pi\)
−0.749421 + 0.662094i \(0.769668\pi\)
\(602\) 26.3173 1.07261
\(603\) 0 0
\(604\) 1.48913 0.0605916
\(605\) 0 0
\(606\) 0 0
\(607\) −17.2554 −0.700377 −0.350188 0.936679i \(-0.613882\pi\)
−0.350188 + 0.936679i \(0.613882\pi\)
\(608\) 2.07668 0.0842204
\(609\) 0 0
\(610\) 0 0
\(611\) −18.0106 −0.728629
\(612\) 0 0
\(613\) 37.6060 1.51889 0.759445 0.650571i \(-0.225470\pi\)
0.759445 + 0.650571i \(0.225470\pi\)
\(614\) −3.20343 −0.129280
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) −2.15121 −0.0866046 −0.0433023 0.999062i \(-0.513788\pi\)
−0.0433023 + 0.999062i \(0.513788\pi\)
\(618\) 0 0
\(619\) −38.9783 −1.56667 −0.783334 0.621601i \(-0.786483\pi\)
−0.783334 + 0.621601i \(0.786483\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 34.4674 1.38202
\(623\) 40.3236 1.61553
\(624\) 0 0
\(625\) 0 0
\(626\) 19.7613 0.789822
\(627\) 0 0
\(628\) 0.744563 0.0297113
\(629\) 1.60171 0.0638645
\(630\) 0 0
\(631\) −2.11684 −0.0842702 −0.0421351 0.999112i \(-0.513416\pi\)
−0.0421351 + 0.999112i \(0.513416\pi\)
\(632\) 12.1064 0.481568
\(633\) 0 0
\(634\) −44.7446 −1.77703
\(635\) 0 0
\(636\) 0 0
\(637\) −8.74456 −0.346472
\(638\) 41.1244 1.62813
\(639\) 0 0
\(640\) 0 0
\(641\) −10.3556 −0.409023 −0.204512 0.978864i \(-0.565561\pi\)
−0.204512 + 0.978864i \(0.565561\pi\)
\(642\) 0 0
\(643\) 17.8832 0.705243 0.352621 0.935766i \(-0.385290\pi\)
0.352621 + 0.935766i \(0.385290\pi\)
\(644\) −8.60485 −0.339079
\(645\) 0 0
\(646\) 2.74456 0.107983
\(647\) 17.6101 0.692326 0.346163 0.938174i \(-0.387484\pi\)
0.346163 + 0.938174i \(0.387484\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0.554374 0.0217109
\(653\) 2.95207 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −7.95307 −0.310515
\(657\) 0 0
\(658\) −38.7446 −1.51042
\(659\) −41.9253 −1.63318 −0.816588 0.577221i \(-0.804137\pi\)
−0.816588 + 0.577221i \(0.804137\pi\)
\(660\) 0 0
\(661\) 4.74456 0.184542 0.0922710 0.995734i \(-0.470587\pi\)
0.0922710 + 0.995734i \(0.470587\pi\)
\(662\) −34.4194 −1.33775
\(663\) 0 0
\(664\) −5.29894 −0.205639
\(665\) 0 0
\(666\) 0 0
\(667\) −46.9783 −1.81901
\(668\) −1.60171 −0.0619721
\(669\) 0 0
\(670\) 0 0
\(671\) 56.9838 2.19983
\(672\) 0 0
\(673\) 36.7446 1.41640 0.708199 0.706012i \(-0.249508\pi\)
0.708199 + 0.706012i \(0.249508\pi\)
\(674\) −17.8615 −0.687999
\(675\) 0 0
\(676\) 3.35053 0.128867
\(677\) 13.5591 0.521117 0.260559 0.965458i \(-0.416093\pi\)
0.260559 + 0.965458i \(0.416093\pi\)
\(678\) 0 0
\(679\) −52.2337 −2.00454
\(680\) 0 0
\(681\) 0 0
\(682\) −40.4674 −1.54958
\(683\) −35.2203 −1.34767 −0.673833 0.738884i \(-0.735353\pi\)
−0.673833 + 0.738884i \(0.735353\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 11.3056 0.431649
\(687\) 0 0
\(688\) 19.0652 0.726856
\(689\) −23.9147 −0.911078
\(690\) 0 0
\(691\) 9.88316 0.375973 0.187986 0.982172i \(-0.439804\pi\)
0.187986 + 0.982172i \(0.439804\pi\)
\(692\) −2.55164 −0.0969989
\(693\) 0 0
\(694\) −3.95650 −0.150187
\(695\) 0 0
\(696\) 0 0
\(697\) 5.48913 0.207915
\(698\) −13.8572 −0.524503
\(699\) 0 0
\(700\) 0 0
\(701\) 29.0180 1.09599 0.547997 0.836480i \(-0.315390\pi\)
0.547997 + 0.836480i \(0.315390\pi\)
\(702\) 0 0
\(703\) 0.744563 0.0280817
\(704\) 41.7762 1.57450
\(705\) 0 0
\(706\) −28.4674 −1.07138
\(707\) −29.0180 −1.09133
\(708\) 0 0
\(709\) 38.0000 1.42712 0.713560 0.700594i \(-0.247082\pi\)
0.713560 + 0.700594i \(0.247082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 36.1902 1.35628
\(713\) 46.2277 1.73124
\(714\) 0 0
\(715\) 0 0
\(716\) −3.50157 −0.130860
\(717\) 0 0
\(718\) 35.4891 1.32444
\(719\) −7.40357 −0.276107 −0.138053 0.990425i \(-0.544085\pi\)
−0.138053 + 0.990425i \(0.544085\pi\)
\(720\) 0 0
\(721\) −63.2119 −2.35414
\(722\) 1.27582 0.0474811
\(723\) 0 0
\(724\) 1.29894 0.0482746
\(725\) 0 0
\(726\) 0 0
\(727\) −14.3505 −0.532232 −0.266116 0.963941i \(-0.585740\pi\)
−0.266116 + 0.963941i \(0.585740\pi\)
\(728\) −20.4131 −0.756561
\(729\) 0 0
\(730\) 0 0
\(731\) −13.1586 −0.486690
\(732\) 0 0
\(733\) 50.4674 1.86406 0.932028 0.362387i \(-0.118038\pi\)
0.932028 + 0.362387i \(0.118038\pi\)
\(734\) −10.2066 −0.376731
\(735\) 0 0
\(736\) −14.2337 −0.524661
\(737\) 18.8114 0.692928
\(738\) 0 0
\(739\) 9.88316 0.363558 0.181779 0.983339i \(-0.441814\pi\)
0.181779 + 0.983339i \(0.441814\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −51.4456 −1.88863
\(743\) 42.7261 1.56747 0.783735 0.621096i \(-0.213312\pi\)
0.783735 + 0.621096i \(0.213312\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −13.0564 −0.478027
\(747\) 0 0
\(748\) −3.76631 −0.137710
\(749\) −31.7187 −1.15898
\(750\) 0 0
\(751\) −44.4674 −1.62264 −0.811319 0.584604i \(-0.801250\pi\)
−0.811319 + 0.584604i \(0.801250\pi\)
\(752\) −28.0681 −1.02354
\(753\) 0 0
\(754\) −17.4891 −0.636916
\(755\) 0 0
\(756\) 0 0
\(757\) −12.1168 −0.440394 −0.220197 0.975455i \(-0.570670\pi\)
−0.220197 + 0.975455i \(0.570670\pi\)
\(758\) 22.0148 0.799615
\(759\) 0 0
\(760\) 0 0
\(761\) 25.2651 0.915858 0.457929 0.888989i \(-0.348591\pi\)
0.457929 + 0.888989i \(0.348591\pi\)
\(762\) 0 0
\(763\) −56.4674 −2.04426
\(764\) 3.05435 0.110503
\(765\) 0 0
\(766\) −1.02175 −0.0369173
\(767\) 10.2066 0.368538
\(768\) 0 0
\(769\) 24.1168 0.869676 0.434838 0.900509i \(-0.356806\pi\)
0.434838 + 0.900509i \(0.356806\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.78806 −0.244308
\(773\) 2.55164 0.0917762 0.0458881 0.998947i \(-0.485388\pi\)
0.0458881 + 0.998947i \(0.485388\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −46.8795 −1.68288
\(777\) 0 0
\(778\) −21.2554 −0.762044
\(779\) 2.55164 0.0914220
\(780\) 0 0
\(781\) 64.4674 2.30682
\(782\) −18.8114 −0.672695
\(783\) 0 0
\(784\) −13.6277 −0.486704
\(785\) 0 0
\(786\) 0 0
\(787\) −41.9565 −1.49559 −0.747794 0.663931i \(-0.768887\pi\)
−0.747794 + 0.663931i \(0.768887\pi\)
\(788\) −1.30357 −0.0464377
\(789\) 0 0
\(790\) 0 0
\(791\) −5.90414 −0.209927
\(792\) 0 0
\(793\) −24.2337 −0.860563
\(794\) −0.149072 −0.00529037
\(795\) 0 0
\(796\) −6.74456 −0.239055
\(797\) 11.1565 0.395183 0.197592 0.980284i \(-0.436688\pi\)
0.197592 + 0.980284i \(0.436688\pi\)
\(798\) 0 0
\(799\) 19.3723 0.685342
\(800\) 0 0
\(801\) 0 0
\(802\) −20.7446 −0.732516
\(803\) −56.9838 −2.01091
\(804\) 0 0
\(805\) 0 0
\(806\) 17.2097 0.606186
\(807\) 0 0
\(808\) −26.0435 −0.916207
\(809\) 34.6708 1.21896 0.609480 0.792802i \(-0.291378\pi\)
0.609480 + 0.792802i \(0.291378\pi\)
\(810\) 0 0
\(811\) −25.2554 −0.886838 −0.443419 0.896314i \(-0.646235\pi\)
−0.443419 + 0.896314i \(0.646235\pi\)
\(812\) 8.60485 0.301971
\(813\) 0 0
\(814\) 4.46738 0.156581
\(815\) 0 0
\(816\) 0 0
\(817\) −6.11684 −0.214001
\(818\) −19.7613 −0.690939
\(819\) 0 0
\(820\) 0 0
\(821\) 17.4611 0.609395 0.304698 0.952449i \(-0.401445\pi\)
0.304698 + 0.952449i \(0.401445\pi\)
\(822\) 0 0
\(823\) 31.6060 1.10171 0.550857 0.834599i \(-0.314301\pi\)
0.550857 + 0.834599i \(0.314301\pi\)
\(824\) −56.7324 −1.97637
\(825\) 0 0
\(826\) 21.9565 0.763964
\(827\) −42.7261 −1.48573 −0.742866 0.669440i \(-0.766534\pi\)
−0.742866 + 0.669440i \(0.766534\pi\)
\(828\) 0 0
\(829\) −12.7446 −0.442637 −0.221318 0.975202i \(-0.571036\pi\)
−0.221318 + 0.975202i \(0.571036\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −17.7663 −0.615936
\(833\) 9.40571 0.325889
\(834\) 0 0
\(835\) 0 0
\(836\) −1.75079 −0.0605522
\(837\) 0 0
\(838\) 2.23369 0.0771615
\(839\) 7.80400 0.269424 0.134712 0.990885i \(-0.456989\pi\)
0.134712 + 0.990885i \(0.456989\pi\)
\(840\) 0 0
\(841\) 17.9783 0.619940
\(842\) 47.1776 1.62585
\(843\) 0 0
\(844\) 1.48913 0.0512578
\(845\) 0 0
\(846\) 0 0
\(847\) −37.4891 −1.28814
\(848\) −37.2692 −1.27983
\(849\) 0 0
\(850\) 0 0
\(851\) −5.10328 −0.174938
\(852\) 0 0
\(853\) −30.4674 −1.04318 −0.521592 0.853195i \(-0.674661\pi\)
−0.521592 + 0.853195i \(0.674661\pi\)
\(854\) −52.1318 −1.78391
\(855\) 0 0
\(856\) −28.4674 −0.972995
\(857\) 42.0743 1.43723 0.718616 0.695407i \(-0.244776\pi\)
0.718616 + 0.695407i \(0.244776\pi\)
\(858\) 0 0
\(859\) −14.1168 −0.481661 −0.240830 0.970567i \(-0.577420\pi\)
−0.240830 + 0.970567i \(0.577420\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9.95650 −0.339120
\(863\) 22.3130 0.759543 0.379771 0.925080i \(-0.376003\pi\)
0.379771 + 0.925080i \(0.376003\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 28.0681 0.953791
\(867\) 0 0
\(868\) −8.46738 −0.287401
\(869\) −18.8114 −0.638134
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −50.6792 −1.71621
\(873\) 0 0
\(874\) −8.74456 −0.295789
\(875\) 0 0
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 20.7113 0.698972
\(879\) 0 0
\(880\) 0 0
\(881\) −25.2651 −0.851201 −0.425601 0.904911i \(-0.639937\pi\)
−0.425601 + 0.904911i \(0.639937\pi\)
\(882\) 0 0
\(883\) 14.1168 0.475070 0.237535 0.971379i \(-0.423661\pi\)
0.237535 + 0.971379i \(0.423661\pi\)
\(884\) 1.60171 0.0538714
\(885\) 0 0
\(886\) 15.9565 0.536069
\(887\) 10.2066 0.342703 0.171351 0.985210i \(-0.445187\pi\)
0.171351 + 0.985210i \(0.445187\pi\)
\(888\) 0 0
\(889\) −4.23369 −0.141993
\(890\) 0 0
\(891\) 0 0
\(892\) −1.48913 −0.0498596
\(893\) 9.00528 0.301350
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2128 −0.808894
\(897\) 0 0
\(898\) 42.7011 1.42495
\(899\) −46.2277 −1.54178
\(900\) 0 0
\(901\) 25.7228 0.856951
\(902\) 15.3098 0.509762
\(903\) 0 0
\(904\) −5.29894 −0.176240
\(905\) 0 0
\(906\) 0 0
\(907\) −34.7446 −1.15367 −0.576837 0.816859i \(-0.695713\pi\)
−0.576837 + 0.816859i \(0.695713\pi\)
\(908\) 4.80514 0.159464
\(909\) 0 0
\(910\) 0 0
\(911\) 24.7156 0.818863 0.409432 0.912341i \(-0.365727\pi\)
0.409432 + 0.912341i \(0.365727\pi\)
\(912\) 0 0
\(913\) 8.23369 0.272495
\(914\) 3.35250 0.110891
\(915\) 0 0
\(916\) −7.95650 −0.262890
\(917\) 13.1586 0.434536
\(918\) 0 0
\(919\) −26.9783 −0.889930 −0.444965 0.895548i \(-0.646784\pi\)
−0.444965 + 0.895548i \(0.646784\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −7.21194 −0.237513
\(923\) −27.4163 −0.902418
\(924\) 0 0
\(925\) 0 0
\(926\) −4.30243 −0.141387
\(927\) 0 0
\(928\) 14.2337 0.467244
\(929\) −46.2277 −1.51668 −0.758341 0.651858i \(-0.773990\pi\)
−0.758341 + 0.651858i \(0.773990\pi\)
\(930\) 0 0
\(931\) 4.37228 0.143296
\(932\) 2.70071 0.0884648
\(933\) 0 0
\(934\) 38.9348 1.27398
\(935\) 0 0
\(936\) 0 0
\(937\) −12.1168 −0.395840 −0.197920 0.980218i \(-0.563419\pi\)
−0.197920 + 0.980218i \(0.563419\pi\)
\(938\) −17.2097 −0.561917
\(939\) 0 0
\(940\) 0 0
\(941\) 10.3556 0.337584 0.168792 0.985652i \(-0.446013\pi\)
0.168792 + 0.985652i \(0.446013\pi\)
\(942\) 0 0
\(943\) −17.4891 −0.569524
\(944\) 15.9061 0.517701
\(945\) 0 0
\(946\) −36.7011 −1.19325
\(947\) −11.9574 −0.388562 −0.194281 0.980946i \(-0.562237\pi\)
−0.194281 + 0.980946i \(0.562237\pi\)
\(948\) 0 0
\(949\) 24.2337 0.786659
\(950\) 0 0
\(951\) 0 0
\(952\) 21.9565 0.711614
\(953\) −24.8646 −0.805444 −0.402722 0.915322i \(-0.631936\pi\)
−0.402722 + 0.915322i \(0.631936\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.0575 0.325283
\(957\) 0 0
\(958\) −10.7881 −0.348546
\(959\) −56.1829 −1.81424
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) −1.89986 −0.0612538
\(963\) 0 0
\(964\) −3.80981 −0.122706
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −33.6463 −1.08143
\(969\) 0 0
\(970\) 0 0
\(971\) −34.4194 −1.10457 −0.552286 0.833655i \(-0.686244\pi\)
−0.552286 + 0.833655i \(0.686244\pi\)
\(972\) 0 0
\(973\) 47.6060 1.52618
\(974\) −10.2066 −0.327039
\(975\) 0 0
\(976\) −37.7663 −1.20887
\(977\) −17.0606 −0.545818 −0.272909 0.962040i \(-0.587986\pi\)
−0.272909 + 0.962040i \(0.587986\pi\)
\(978\) 0 0
\(979\) −56.2337 −1.79724
\(980\) 0 0
\(981\) 0 0
\(982\) −8.74456 −0.279050
\(983\) 39.2246 1.25107 0.625534 0.780197i \(-0.284881\pi\)
0.625534 + 0.780197i \(0.284881\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.8114 0.599078
\(987\) 0 0
\(988\) 0.744563 0.0236877
\(989\) 41.9253 1.33315
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −14.0063 −0.444700
\(993\) 0 0
\(994\) −58.9783 −1.87068
\(995\) 0 0
\(996\) 0 0
\(997\) −32.3505 −1.02455 −0.512276 0.858821i \(-0.671197\pi\)
−0.512276 + 0.858821i \(0.671197\pi\)
\(998\) 7.80400 0.247031
\(999\) 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 4275.2.a.bp.1.3 4
3.2 odd 2 inner 4275.2.a.bp.1.2 4
5.4 even 2 171.2.a.e.1.2 4
15.14 odd 2 171.2.a.e.1.3 yes 4
20.19 odd 2 2736.2.a.bf.1.2 4
35.34 odd 2 8379.2.a.bw.1.2 4
60.59 even 2 2736.2.a.bf.1.3 4
95.94 odd 2 3249.2.a.bf.1.3 4
105.104 even 2 8379.2.a.bw.1.3 4
285.284 even 2 3249.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
171.2.a.e.1.2 4 5.4 even 2
171.2.a.e.1.3 yes 4 15.14 odd 2
2736.2.a.bf.1.2 4 20.19 odd 2
2736.2.a.bf.1.3 4 60.59 even 2
3249.2.a.bf.1.2 4 285.284 even 2
3249.2.a.bf.1.3 4 95.94 odd 2
4275.2.a.bp.1.2 4 3.2 odd 2 inner
4275.2.a.bp.1.3 4 1.1 even 1 trivial
8379.2.a.bw.1.2 4 35.34 odd 2
8379.2.a.bw.1.3 4 105.104 even 2