Properties

Label 7605.2.a.y.1.2
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, 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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.46410 q^{7} -2.53590 q^{8} +O(q^{10})\) \(q+0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.46410 q^{7} -2.53590 q^{8} -0.732051 q^{10} -3.46410 q^{11} -3.26795 q^{14} +1.07180 q^{16} +6.73205 q^{17} +5.46410 q^{19} +1.46410 q^{20} -2.53590 q^{22} +0.535898 q^{23} +1.00000 q^{25} +6.53590 q^{28} +2.73205 q^{29} +3.19615 q^{31} +5.85641 q^{32} +4.92820 q^{34} +4.46410 q^{35} -4.00000 q^{37} +4.00000 q^{38} +2.53590 q^{40} -5.26795 q^{41} -0.267949 q^{43} +5.07180 q^{44} +0.392305 q^{46} -0.196152 q^{47} +12.9282 q^{49} +0.732051 q^{50} +6.92820 q^{53} +3.46410 q^{55} +11.3205 q^{56} +2.00000 q^{58} -7.26795 q^{59} +4.46410 q^{61} +2.33975 q^{62} +2.14359 q^{64} -12.4641 q^{67} -9.85641 q^{68} +3.26795 q^{70} -12.7321 q^{71} +15.3923 q^{73} -2.92820 q^{74} -8.00000 q^{76} +15.4641 q^{77} +1.92820 q^{79} -1.07180 q^{80} -3.85641 q^{82} -2.53590 q^{83} -6.73205 q^{85} -0.196152 q^{86} +8.78461 q^{88} -1.26795 q^{89} -0.784610 q^{92} -0.143594 q^{94} -5.46410 q^{95} +16.4641 q^{97} +9.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 2 q^{7} - 12 q^{8} + 2 q^{10} - 10 q^{14} + 16 q^{16} + 10 q^{17} + 4 q^{19} - 4 q^{20} - 12 q^{22} + 8 q^{23} + 2 q^{25} + 20 q^{28} + 2 q^{29} - 4 q^{31} - 16 q^{32} - 4 q^{34} + 2 q^{35} - 8 q^{37} + 8 q^{38} + 12 q^{40} - 14 q^{41} - 4 q^{43} + 24 q^{44} - 20 q^{46} + 10 q^{47} + 12 q^{49} - 2 q^{50} - 12 q^{56} + 4 q^{58} - 18 q^{59} + 2 q^{61} + 22 q^{62} + 32 q^{64} - 18 q^{67} + 8 q^{68} + 10 q^{70} - 22 q^{71} + 10 q^{73} + 8 q^{74} - 16 q^{76} + 24 q^{77} - 10 q^{79} - 16 q^{80} + 20 q^{82} - 12 q^{83} - 10 q^{85} + 10 q^{86} - 24 q^{88} - 6 q^{89} + 40 q^{92} - 28 q^{94} - 4 q^{95} + 26 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.732051 0.517638 0.258819 0.965926i \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0 0
\(4\) −1.46410 −0.732051
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.46410 −1.68727 −0.843636 0.536916i \(-0.819589\pi\)
−0.843636 + 0.536916i \(0.819589\pi\)
\(8\) −2.53590 −0.896575
\(9\) 0 0
\(10\) −0.732051 −0.231495
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −3.26795 −0.873396
\(15\) 0 0
\(16\) 1.07180 0.267949
\(17\) 6.73205 1.63276 0.816381 0.577514i \(-0.195977\pi\)
0.816381 + 0.577514i \(0.195977\pi\)
\(18\) 0 0
\(19\) 5.46410 1.25355 0.626775 0.779200i \(-0.284374\pi\)
0.626775 + 0.779200i \(0.284374\pi\)
\(20\) 1.46410 0.327383
\(21\) 0 0
\(22\) −2.53590 −0.540655
\(23\) 0.535898 0.111743 0.0558713 0.998438i \(-0.482206\pi\)
0.0558713 + 0.998438i \(0.482206\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 6.53590 1.23517
\(29\) 2.73205 0.507329 0.253665 0.967292i \(-0.418364\pi\)
0.253665 + 0.967292i \(0.418364\pi\)
\(30\) 0 0
\(31\) 3.19615 0.574046 0.287023 0.957924i \(-0.407334\pi\)
0.287023 + 0.957924i \(0.407334\pi\)
\(32\) 5.85641 1.03528
\(33\) 0 0
\(34\) 4.92820 0.845180
\(35\) 4.46410 0.754571
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 2.53590 0.400961
\(41\) −5.26795 −0.822715 −0.411358 0.911474i \(-0.634945\pi\)
−0.411358 + 0.911474i \(0.634945\pi\)
\(42\) 0 0
\(43\) −0.267949 −0.0408619 −0.0204309 0.999791i \(-0.506504\pi\)
−0.0204309 + 0.999791i \(0.506504\pi\)
\(44\) 5.07180 0.764602
\(45\) 0 0
\(46\) 0.392305 0.0578422
\(47\) −0.196152 −0.0286118 −0.0143059 0.999898i \(-0.504554\pi\)
−0.0143059 + 0.999898i \(0.504554\pi\)
\(48\) 0 0
\(49\) 12.9282 1.84689
\(50\) 0.732051 0.103528
\(51\) 0 0
\(52\) 0 0
\(53\) 6.92820 0.951662 0.475831 0.879537i \(-0.342147\pi\)
0.475831 + 0.879537i \(0.342147\pi\)
\(54\) 0 0
\(55\) 3.46410 0.467099
\(56\) 11.3205 1.51277
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) −7.26795 −0.946206 −0.473103 0.881007i \(-0.656866\pi\)
−0.473103 + 0.881007i \(0.656866\pi\)
\(60\) 0 0
\(61\) 4.46410 0.571570 0.285785 0.958294i \(-0.407746\pi\)
0.285785 + 0.958294i \(0.407746\pi\)
\(62\) 2.33975 0.297148
\(63\) 0 0
\(64\) 2.14359 0.267949
\(65\) 0 0
\(66\) 0 0
\(67\) −12.4641 −1.52273 −0.761366 0.648322i \(-0.775471\pi\)
−0.761366 + 0.648322i \(0.775471\pi\)
\(68\) −9.85641 −1.19526
\(69\) 0 0
\(70\) 3.26795 0.390595
\(71\) −12.7321 −1.51102 −0.755508 0.655139i \(-0.772610\pi\)
−0.755508 + 0.655139i \(0.772610\pi\)
\(72\) 0 0
\(73\) 15.3923 1.80153 0.900767 0.434304i \(-0.143006\pi\)
0.900767 + 0.434304i \(0.143006\pi\)
\(74\) −2.92820 −0.340397
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) 15.4641 1.76230
\(78\) 0 0
\(79\) 1.92820 0.216940 0.108470 0.994100i \(-0.465405\pi\)
0.108470 + 0.994100i \(0.465405\pi\)
\(80\) −1.07180 −0.119831
\(81\) 0 0
\(82\) −3.85641 −0.425869
\(83\) −2.53590 −0.278351 −0.139176 0.990268i \(-0.544445\pi\)
−0.139176 + 0.990268i \(0.544445\pi\)
\(84\) 0 0
\(85\) −6.73205 −0.730193
\(86\) −0.196152 −0.0211517
\(87\) 0 0
\(88\) 8.78461 0.936443
\(89\) −1.26795 −0.134402 −0.0672012 0.997739i \(-0.521407\pi\)
−0.0672012 + 0.997739i \(0.521407\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.784610 −0.0818012
\(93\) 0 0
\(94\) −0.143594 −0.0148105
\(95\) −5.46410 −0.560605
\(96\) 0 0
\(97\) 16.4641 1.67168 0.835838 0.548976i \(-0.184982\pi\)
0.835838 + 0.548976i \(0.184982\pi\)
\(98\) 9.46410 0.956019
\(99\) 0 0
\(100\) −1.46410 −0.146410
\(101\) 4.92820 0.490375 0.245187 0.969476i \(-0.421151\pi\)
0.245187 + 0.969476i \(0.421151\pi\)
\(102\) 0 0
\(103\) −3.19615 −0.314926 −0.157463 0.987525i \(-0.550332\pi\)
−0.157463 + 0.987525i \(0.550332\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.07180 0.492616
\(107\) −17.1244 −1.65547 −0.827737 0.561116i \(-0.810372\pi\)
−0.827737 + 0.561116i \(0.810372\pi\)
\(108\) 0 0
\(109\) 8.26795 0.791926 0.395963 0.918266i \(-0.370411\pi\)
0.395963 + 0.918266i \(0.370411\pi\)
\(110\) 2.53590 0.241788
\(111\) 0 0
\(112\) −4.78461 −0.452103
\(113\) −19.4641 −1.83103 −0.915514 0.402285i \(-0.868216\pi\)
−0.915514 + 0.402285i \(0.868216\pi\)
\(114\) 0 0
\(115\) −0.535898 −0.0499728
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) −5.32051 −0.489792
\(119\) −30.0526 −2.75491
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.26795 0.295866
\(123\) 0 0
\(124\) −4.67949 −0.420231
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.6603 1.30089 0.650444 0.759555i \(-0.274583\pi\)
0.650444 + 0.759555i \(0.274583\pi\)
\(128\) −10.1436 −0.896575
\(129\) 0 0
\(130\) 0 0
\(131\) −1.26795 −0.110781 −0.0553906 0.998465i \(-0.517640\pi\)
−0.0553906 + 0.998465i \(0.517640\pi\)
\(132\) 0 0
\(133\) −24.3923 −2.11508
\(134\) −9.12436 −0.788224
\(135\) 0 0
\(136\) −17.0718 −1.46389
\(137\) 9.12436 0.779546 0.389773 0.920911i \(-0.372553\pi\)
0.389773 + 0.920911i \(0.372553\pi\)
\(138\) 0 0
\(139\) −17.9282 −1.52065 −0.760325 0.649543i \(-0.774960\pi\)
−0.760325 + 0.649543i \(0.774960\pi\)
\(140\) −6.53590 −0.552384
\(141\) 0 0
\(142\) −9.32051 −0.782160
\(143\) 0 0
\(144\) 0 0
\(145\) −2.73205 −0.226884
\(146\) 11.2679 0.932542
\(147\) 0 0
\(148\) 5.85641 0.481394
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) 17.8564 1.45313 0.726567 0.687096i \(-0.241115\pi\)
0.726567 + 0.687096i \(0.241115\pi\)
\(152\) −13.8564 −1.12390
\(153\) 0 0
\(154\) 11.3205 0.912233
\(155\) −3.19615 −0.256721
\(156\) 0 0
\(157\) 9.73205 0.776702 0.388351 0.921511i \(-0.373045\pi\)
0.388351 + 0.921511i \(0.373045\pi\)
\(158\) 1.41154 0.112296
\(159\) 0 0
\(160\) −5.85641 −0.462990
\(161\) −2.39230 −0.188540
\(162\) 0 0
\(163\) 5.53590 0.433605 0.216803 0.976215i \(-0.430437\pi\)
0.216803 + 0.976215i \(0.430437\pi\)
\(164\) 7.71281 0.602270
\(165\) 0 0
\(166\) −1.85641 −0.144085
\(167\) −17.4641 −1.35141 −0.675706 0.737171i \(-0.736161\pi\)
−0.675706 + 0.737171i \(0.736161\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −4.92820 −0.377976
\(171\) 0 0
\(172\) 0.392305 0.0299130
\(173\) −3.26795 −0.248458 −0.124229 0.992254i \(-0.539646\pi\)
−0.124229 + 0.992254i \(0.539646\pi\)
\(174\) 0 0
\(175\) −4.46410 −0.337454
\(176\) −3.71281 −0.279864
\(177\) 0 0
\(178\) −0.928203 −0.0695718
\(179\) −9.12436 −0.681986 −0.340993 0.940066i \(-0.610763\pi\)
−0.340993 + 0.940066i \(0.610763\pi\)
\(180\) 0 0
\(181\) −14.5359 −1.08044 −0.540222 0.841522i \(-0.681660\pi\)
−0.540222 + 0.841522i \(0.681660\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.35898 −0.100186
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −23.3205 −1.70536
\(188\) 0.287187 0.0209453
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 27.5167 1.99104 0.995518 0.0945740i \(-0.0301489\pi\)
0.995518 + 0.0945740i \(0.0301489\pi\)
\(192\) 0 0
\(193\) −6.07180 −0.437057 −0.218529 0.975831i \(-0.570126\pi\)
−0.218529 + 0.975831i \(0.570126\pi\)
\(194\) 12.0526 0.865323
\(195\) 0 0
\(196\) −18.9282 −1.35201
\(197\) 2.92820 0.208626 0.104313 0.994545i \(-0.466736\pi\)
0.104313 + 0.994545i \(0.466736\pi\)
\(198\) 0 0
\(199\) −15.0000 −1.06332 −0.531661 0.846957i \(-0.678432\pi\)
−0.531661 + 0.846957i \(0.678432\pi\)
\(200\) −2.53590 −0.179315
\(201\) 0 0
\(202\) 3.60770 0.253837
\(203\) −12.1962 −0.856002
\(204\) 0 0
\(205\) 5.26795 0.367930
\(206\) −2.33975 −0.163018
\(207\) 0 0
\(208\) 0 0
\(209\) −18.9282 −1.30929
\(210\) 0 0
\(211\) 11.5359 0.794164 0.397082 0.917783i \(-0.370023\pi\)
0.397082 + 0.917783i \(0.370023\pi\)
\(212\) −10.1436 −0.696665
\(213\) 0 0
\(214\) −12.5359 −0.856936
\(215\) 0.267949 0.0182740
\(216\) 0 0
\(217\) −14.2679 −0.968572
\(218\) 6.05256 0.409931
\(219\) 0 0
\(220\) −5.07180 −0.341940
\(221\) 0 0
\(222\) 0 0
\(223\) −11.8564 −0.793964 −0.396982 0.917826i \(-0.629942\pi\)
−0.396982 + 0.917826i \(0.629942\pi\)
\(224\) −26.1436 −1.74679
\(225\) 0 0
\(226\) −14.2487 −0.947810
\(227\) 11.6603 0.773918 0.386959 0.922097i \(-0.373525\pi\)
0.386959 + 0.922097i \(0.373525\pi\)
\(228\) 0 0
\(229\) −18.3923 −1.21540 −0.607699 0.794168i \(-0.707907\pi\)
−0.607699 + 0.794168i \(0.707907\pi\)
\(230\) −0.392305 −0.0258678
\(231\) 0 0
\(232\) −6.92820 −0.454859
\(233\) −23.8564 −1.56289 −0.781443 0.623977i \(-0.785516\pi\)
−0.781443 + 0.623977i \(0.785516\pi\)
\(234\) 0 0
\(235\) 0.196152 0.0127956
\(236\) 10.6410 0.692671
\(237\) 0 0
\(238\) −22.0000 −1.42605
\(239\) −5.46410 −0.353443 −0.176722 0.984261i \(-0.556549\pi\)
−0.176722 + 0.984261i \(0.556549\pi\)
\(240\) 0 0
\(241\) 1.60770 0.103561 0.0517804 0.998658i \(-0.483510\pi\)
0.0517804 + 0.998658i \(0.483510\pi\)
\(242\) 0.732051 0.0470580
\(243\) 0 0
\(244\) −6.53590 −0.418418
\(245\) −12.9282 −0.825953
\(246\) 0 0
\(247\) 0 0
\(248\) −8.10512 −0.514675
\(249\) 0 0
\(250\) −0.732051 −0.0462990
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −1.85641 −0.116711
\(254\) 10.7321 0.673389
\(255\) 0 0
\(256\) −11.7128 −0.732051
\(257\) −10.7321 −0.669447 −0.334723 0.942316i \(-0.608643\pi\)
−0.334723 + 0.942316i \(0.608643\pi\)
\(258\) 0 0
\(259\) 17.8564 1.10954
\(260\) 0 0
\(261\) 0 0
\(262\) −0.928203 −0.0573446
\(263\) 6.33975 0.390925 0.195463 0.980711i \(-0.437379\pi\)
0.195463 + 0.980711i \(0.437379\pi\)
\(264\) 0 0
\(265\) −6.92820 −0.425596
\(266\) −17.8564 −1.09485
\(267\) 0 0
\(268\) 18.2487 1.11472
\(269\) −15.6603 −0.954823 −0.477411 0.878680i \(-0.658425\pi\)
−0.477411 + 0.878680i \(0.658425\pi\)
\(270\) 0 0
\(271\) 3.73205 0.226706 0.113353 0.993555i \(-0.463841\pi\)
0.113353 + 0.993555i \(0.463841\pi\)
\(272\) 7.21539 0.437497
\(273\) 0 0
\(274\) 6.67949 0.403523
\(275\) −3.46410 −0.208893
\(276\) 0 0
\(277\) 20.2487 1.21663 0.608314 0.793697i \(-0.291846\pi\)
0.608314 + 0.793697i \(0.291846\pi\)
\(278\) −13.1244 −0.787147
\(279\) 0 0
\(280\) −11.3205 −0.676530
\(281\) −21.1244 −1.26017 −0.630087 0.776525i \(-0.716981\pi\)
−0.630087 + 0.776525i \(0.716981\pi\)
\(282\) 0 0
\(283\) −0.267949 −0.0159279 −0.00796396 0.999968i \(-0.502535\pi\)
−0.00796396 + 0.999968i \(0.502535\pi\)
\(284\) 18.6410 1.10614
\(285\) 0 0
\(286\) 0 0
\(287\) 23.5167 1.38814
\(288\) 0 0
\(289\) 28.3205 1.66591
\(290\) −2.00000 −0.117444
\(291\) 0 0
\(292\) −22.5359 −1.31881
\(293\) −24.5885 −1.43647 −0.718237 0.695799i \(-0.755050\pi\)
−0.718237 + 0.695799i \(0.755050\pi\)
\(294\) 0 0
\(295\) 7.26795 0.423156
\(296\) 10.1436 0.589584
\(297\) 0 0
\(298\) −2.92820 −0.169626
\(299\) 0 0
\(300\) 0 0
\(301\) 1.19615 0.0689451
\(302\) 13.0718 0.752197
\(303\) 0 0
\(304\) 5.85641 0.335888
\(305\) −4.46410 −0.255614
\(306\) 0 0
\(307\) 12.0718 0.688974 0.344487 0.938791i \(-0.388053\pi\)
0.344487 + 0.938791i \(0.388053\pi\)
\(308\) −22.6410 −1.29009
\(309\) 0 0
\(310\) −2.33975 −0.132889
\(311\) −10.1962 −0.578171 −0.289085 0.957303i \(-0.593351\pi\)
−0.289085 + 0.957303i \(0.593351\pi\)
\(312\) 0 0
\(313\) −10.8038 −0.610670 −0.305335 0.952245i \(-0.598768\pi\)
−0.305335 + 0.952245i \(0.598768\pi\)
\(314\) 7.12436 0.402051
\(315\) 0 0
\(316\) −2.82309 −0.158811
\(317\) 17.4641 0.980882 0.490441 0.871474i \(-0.336836\pi\)
0.490441 + 0.871474i \(0.336836\pi\)
\(318\) 0 0
\(319\) −9.46410 −0.529888
\(320\) −2.14359 −0.119831
\(321\) 0 0
\(322\) −1.75129 −0.0975955
\(323\) 36.7846 2.04675
\(324\) 0 0
\(325\) 0 0
\(326\) 4.05256 0.224450
\(327\) 0 0
\(328\) 13.3590 0.737626
\(329\) 0.875644 0.0482758
\(330\) 0 0
\(331\) −8.12436 −0.446555 −0.223277 0.974755i \(-0.571676\pi\)
−0.223277 + 0.974755i \(0.571676\pi\)
\(332\) 3.71281 0.203767
\(333\) 0 0
\(334\) −12.7846 −0.699543
\(335\) 12.4641 0.680987
\(336\) 0 0
\(337\) 31.0526 1.69154 0.845770 0.533547i \(-0.179141\pi\)
0.845770 + 0.533547i \(0.179141\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 9.85641 0.534539
\(341\) −11.0718 −0.599571
\(342\) 0 0
\(343\) −26.4641 −1.42893
\(344\) 0.679492 0.0366357
\(345\) 0 0
\(346\) −2.39230 −0.128611
\(347\) −6.53590 −0.350865 −0.175433 0.984491i \(-0.556132\pi\)
−0.175433 + 0.984491i \(0.556132\pi\)
\(348\) 0 0
\(349\) −29.9808 −1.60483 −0.802417 0.596764i \(-0.796453\pi\)
−0.802417 + 0.596764i \(0.796453\pi\)
\(350\) −3.26795 −0.174679
\(351\) 0 0
\(352\) −20.2872 −1.08131
\(353\) 36.9282 1.96549 0.982745 0.184966i \(-0.0592174\pi\)
0.982745 + 0.184966i \(0.0592174\pi\)
\(354\) 0 0
\(355\) 12.7321 0.675747
\(356\) 1.85641 0.0983893
\(357\) 0 0
\(358\) −6.67949 −0.353022
\(359\) 33.6603 1.77652 0.888260 0.459341i \(-0.151914\pi\)
0.888260 + 0.459341i \(0.151914\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) −10.6410 −0.559279
\(363\) 0 0
\(364\) 0 0
\(365\) −15.3923 −0.805670
\(366\) 0 0
\(367\) −18.1244 −0.946084 −0.473042 0.881040i \(-0.656844\pi\)
−0.473042 + 0.881040i \(0.656844\pi\)
\(368\) 0.574374 0.0299413
\(369\) 0 0
\(370\) 2.92820 0.152230
\(371\) −30.9282 −1.60571
\(372\) 0 0
\(373\) −1.87564 −0.0971172 −0.0485586 0.998820i \(-0.515463\pi\)
−0.0485586 + 0.998820i \(0.515463\pi\)
\(374\) −17.0718 −0.882762
\(375\) 0 0
\(376\) 0.497423 0.0256526
\(377\) 0 0
\(378\) 0 0
\(379\) −19.7321 −1.01357 −0.506784 0.862073i \(-0.669166\pi\)
−0.506784 + 0.862073i \(0.669166\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) 20.1436 1.03064
\(383\) 8.33975 0.426141 0.213071 0.977037i \(-0.431654\pi\)
0.213071 + 0.977037i \(0.431654\pi\)
\(384\) 0 0
\(385\) −15.4641 −0.788124
\(386\) −4.44486 −0.226238
\(387\) 0 0
\(388\) −24.1051 −1.22375
\(389\) −17.4641 −0.885465 −0.442733 0.896654i \(-0.645991\pi\)
−0.442733 + 0.896654i \(0.645991\pi\)
\(390\) 0 0
\(391\) 3.60770 0.182449
\(392\) −32.7846 −1.65587
\(393\) 0 0
\(394\) 2.14359 0.107993
\(395\) −1.92820 −0.0970184
\(396\) 0 0
\(397\) 24.3205 1.22061 0.610306 0.792166i \(-0.291047\pi\)
0.610306 + 0.792166i \(0.291047\pi\)
\(398\) −10.9808 −0.550416
\(399\) 0 0
\(400\) 1.07180 0.0535898
\(401\) −15.3205 −0.765070 −0.382535 0.923941i \(-0.624949\pi\)
−0.382535 + 0.923941i \(0.624949\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −7.21539 −0.358979
\(405\) 0 0
\(406\) −8.92820 −0.443099
\(407\) 13.8564 0.686837
\(408\) 0 0
\(409\) −2.66025 −0.131541 −0.0657705 0.997835i \(-0.520951\pi\)
−0.0657705 + 0.997835i \(0.520951\pi\)
\(410\) 3.85641 0.190454
\(411\) 0 0
\(412\) 4.67949 0.230542
\(413\) 32.4449 1.59651
\(414\) 0 0
\(415\) 2.53590 0.124482
\(416\) 0 0
\(417\) 0 0
\(418\) −13.8564 −0.677739
\(419\) −20.4449 −0.998797 −0.499398 0.866372i \(-0.666446\pi\)
−0.499398 + 0.866372i \(0.666446\pi\)
\(420\) 0 0
\(421\) 23.5885 1.14963 0.574816 0.818283i \(-0.305074\pi\)
0.574816 + 0.818283i \(0.305074\pi\)
\(422\) 8.44486 0.411090
\(423\) 0 0
\(424\) −17.5692 −0.853237
\(425\) 6.73205 0.326552
\(426\) 0 0
\(427\) −19.9282 −0.964393
\(428\) 25.0718 1.21189
\(429\) 0 0
\(430\) 0.196152 0.00945931
\(431\) 23.3205 1.12331 0.561655 0.827372i \(-0.310165\pi\)
0.561655 + 0.827372i \(0.310165\pi\)
\(432\) 0 0
\(433\) 4.80385 0.230858 0.115429 0.993316i \(-0.463176\pi\)
0.115429 + 0.993316i \(0.463176\pi\)
\(434\) −10.4449 −0.501370
\(435\) 0 0
\(436\) −12.1051 −0.579730
\(437\) 2.92820 0.140075
\(438\) 0 0
\(439\) 15.3923 0.734635 0.367317 0.930096i \(-0.380276\pi\)
0.367317 + 0.930096i \(0.380276\pi\)
\(440\) −8.78461 −0.418790
\(441\) 0 0
\(442\) 0 0
\(443\) −3.12436 −0.148443 −0.0742213 0.997242i \(-0.523647\pi\)
−0.0742213 + 0.997242i \(0.523647\pi\)
\(444\) 0 0
\(445\) 1.26795 0.0601066
\(446\) −8.67949 −0.410986
\(447\) 0 0
\(448\) −9.56922 −0.452103
\(449\) −11.0718 −0.522510 −0.261255 0.965270i \(-0.584136\pi\)
−0.261255 + 0.965270i \(0.584136\pi\)
\(450\) 0 0
\(451\) 18.2487 0.859298
\(452\) 28.4974 1.34041
\(453\) 0 0
\(454\) 8.53590 0.400610
\(455\) 0 0
\(456\) 0 0
\(457\) −5.39230 −0.252241 −0.126121 0.992015i \(-0.540253\pi\)
−0.126121 + 0.992015i \(0.540253\pi\)
\(458\) −13.4641 −0.629136
\(459\) 0 0
\(460\) 0.784610 0.0365826
\(461\) 11.5167 0.536384 0.268192 0.963365i \(-0.413574\pi\)
0.268192 + 0.963365i \(0.413574\pi\)
\(462\) 0 0
\(463\) −1.78461 −0.0829378 −0.0414689 0.999140i \(-0.513204\pi\)
−0.0414689 + 0.999140i \(0.513204\pi\)
\(464\) 2.92820 0.135938
\(465\) 0 0
\(466\) −17.4641 −0.809009
\(467\) 23.6603 1.09487 0.547433 0.836850i \(-0.315605\pi\)
0.547433 + 0.836850i \(0.315605\pi\)
\(468\) 0 0
\(469\) 55.6410 2.56926
\(470\) 0.143594 0.00662348
\(471\) 0 0
\(472\) 18.4308 0.848345
\(473\) 0.928203 0.0426788
\(474\) 0 0
\(475\) 5.46410 0.250710
\(476\) 44.0000 2.01674
\(477\) 0 0
\(478\) −4.00000 −0.182956
\(479\) −35.9090 −1.64072 −0.820361 0.571846i \(-0.806228\pi\)
−0.820361 + 0.571846i \(0.806228\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 1.17691 0.0536070
\(483\) 0 0
\(484\) −1.46410 −0.0665501
\(485\) −16.4641 −0.747596
\(486\) 0 0
\(487\) −20.7846 −0.941841 −0.470920 0.882176i \(-0.656078\pi\)
−0.470920 + 0.882176i \(0.656078\pi\)
\(488\) −11.3205 −0.512455
\(489\) 0 0
\(490\) −9.46410 −0.427545
\(491\) −7.12436 −0.321518 −0.160759 0.986994i \(-0.551394\pi\)
−0.160759 + 0.986994i \(0.551394\pi\)
\(492\) 0 0
\(493\) 18.3923 0.828348
\(494\) 0 0
\(495\) 0 0
\(496\) 3.42563 0.153815
\(497\) 56.8372 2.54950
\(498\) 0 0
\(499\) 0.392305 0.0175620 0.00878099 0.999961i \(-0.497205\pi\)
0.00878099 + 0.999961i \(0.497205\pi\)
\(500\) 1.46410 0.0654766
\(501\) 0 0
\(502\) 0 0
\(503\) 30.7846 1.37262 0.686309 0.727310i \(-0.259230\pi\)
0.686309 + 0.727310i \(0.259230\pi\)
\(504\) 0 0
\(505\) −4.92820 −0.219302
\(506\) −1.35898 −0.0604142
\(507\) 0 0
\(508\) −21.4641 −0.952316
\(509\) −12.9282 −0.573033 −0.286516 0.958075i \(-0.592497\pi\)
−0.286516 + 0.958075i \(0.592497\pi\)
\(510\) 0 0
\(511\) −68.7128 −3.03968
\(512\) 11.7128 0.517638
\(513\) 0 0
\(514\) −7.85641 −0.346531
\(515\) 3.19615 0.140839
\(516\) 0 0
\(517\) 0.679492 0.0298840
\(518\) 13.0718 0.574342
\(519\) 0 0
\(520\) 0 0
\(521\) 12.7321 0.557801 0.278901 0.960320i \(-0.410030\pi\)
0.278901 + 0.960320i \(0.410030\pi\)
\(522\) 0 0
\(523\) −37.4641 −1.63819 −0.819095 0.573657i \(-0.805524\pi\)
−0.819095 + 0.573657i \(0.805524\pi\)
\(524\) 1.85641 0.0810975
\(525\) 0 0
\(526\) 4.64102 0.202358
\(527\) 21.5167 0.937280
\(528\) 0 0
\(529\) −22.7128 −0.987514
\(530\) −5.07180 −0.220305
\(531\) 0 0
\(532\) 35.7128 1.54835
\(533\) 0 0
\(534\) 0 0
\(535\) 17.1244 0.740350
\(536\) 31.6077 1.36524
\(537\) 0 0
\(538\) −11.4641 −0.494253
\(539\) −44.7846 −1.92901
\(540\) 0 0
\(541\) 17.5885 0.756187 0.378093 0.925767i \(-0.376580\pi\)
0.378093 + 0.925767i \(0.376580\pi\)
\(542\) 2.73205 0.117352
\(543\) 0 0
\(544\) 39.4256 1.69036
\(545\) −8.26795 −0.354160
\(546\) 0 0
\(547\) −24.6603 −1.05440 −0.527198 0.849742i \(-0.676757\pi\)
−0.527198 + 0.849742i \(0.676757\pi\)
\(548\) −13.3590 −0.570668
\(549\) 0 0
\(550\) −2.53590 −0.108131
\(551\) 14.9282 0.635963
\(552\) 0 0
\(553\) −8.60770 −0.366036
\(554\) 14.8231 0.629773
\(555\) 0 0
\(556\) 26.2487 1.11319
\(557\) −21.4641 −0.909463 −0.454732 0.890629i \(-0.650265\pi\)
−0.454732 + 0.890629i \(0.650265\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 4.78461 0.202187
\(561\) 0 0
\(562\) −15.4641 −0.652314
\(563\) 3.46410 0.145994 0.0729972 0.997332i \(-0.476744\pi\)
0.0729972 + 0.997332i \(0.476744\pi\)
\(564\) 0 0
\(565\) 19.4641 0.818861
\(566\) −0.196152 −0.00824490
\(567\) 0 0
\(568\) 32.2872 1.35474
\(569\) 44.1962 1.85280 0.926400 0.376542i \(-0.122887\pi\)
0.926400 + 0.376542i \(0.122887\pi\)
\(570\) 0 0
\(571\) −3.85641 −0.161386 −0.0806928 0.996739i \(-0.525713\pi\)
−0.0806928 + 0.996739i \(0.525713\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 17.2154 0.718557
\(575\) 0.535898 0.0223485
\(576\) 0 0
\(577\) −11.7128 −0.487611 −0.243805 0.969824i \(-0.578396\pi\)
−0.243805 + 0.969824i \(0.578396\pi\)
\(578\) 20.7321 0.862340
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 11.3205 0.469654
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) −39.0333 −1.61521
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 25.6603 1.05911 0.529556 0.848275i \(-0.322359\pi\)
0.529556 + 0.848275i \(0.322359\pi\)
\(588\) 0 0
\(589\) 17.4641 0.719596
\(590\) 5.32051 0.219042
\(591\) 0 0
\(592\) −4.28719 −0.176202
\(593\) −21.3205 −0.875528 −0.437764 0.899090i \(-0.644230\pi\)
−0.437764 + 0.899090i \(0.644230\pi\)
\(594\) 0 0
\(595\) 30.0526 1.23203
\(596\) 5.85641 0.239888
\(597\) 0 0
\(598\) 0 0
\(599\) 23.1244 0.944836 0.472418 0.881375i \(-0.343381\pi\)
0.472418 + 0.881375i \(0.343381\pi\)
\(600\) 0 0
\(601\) −22.2487 −0.907544 −0.453772 0.891118i \(-0.649922\pi\)
−0.453772 + 0.891118i \(0.649922\pi\)
\(602\) 0.875644 0.0356886
\(603\) 0 0
\(604\) −26.1436 −1.06377
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 17.6077 0.714674 0.357337 0.933975i \(-0.383685\pi\)
0.357337 + 0.933975i \(0.383685\pi\)
\(608\) 32.0000 1.29777
\(609\) 0 0
\(610\) −3.26795 −0.132315
\(611\) 0 0
\(612\) 0 0
\(613\) −2.21539 −0.0894788 −0.0447394 0.998999i \(-0.514246\pi\)
−0.0447394 + 0.998999i \(0.514246\pi\)
\(614\) 8.83717 0.356639
\(615\) 0 0
\(616\) −39.2154 −1.58003
\(617\) −24.6410 −0.992010 −0.496005 0.868320i \(-0.665200\pi\)
−0.496005 + 0.868320i \(0.665200\pi\)
\(618\) 0 0
\(619\) −11.5885 −0.465779 −0.232890 0.972503i \(-0.574818\pi\)
−0.232890 + 0.972503i \(0.574818\pi\)
\(620\) 4.67949 0.187933
\(621\) 0 0
\(622\) −7.46410 −0.299283
\(623\) 5.66025 0.226773
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −7.90897 −0.316106
\(627\) 0 0
\(628\) −14.2487 −0.568585
\(629\) −26.9282 −1.07370
\(630\) 0 0
\(631\) −10.4115 −0.414477 −0.207238 0.978290i \(-0.566448\pi\)
−0.207238 + 0.978290i \(0.566448\pi\)
\(632\) −4.88973 −0.194503
\(633\) 0 0
\(634\) 12.7846 0.507742
\(635\) −14.6603 −0.581774
\(636\) 0 0
\(637\) 0 0
\(638\) −6.92820 −0.274290
\(639\) 0 0
\(640\) 10.1436 0.400961
\(641\) −38.7846 −1.53190 −0.765950 0.642900i \(-0.777731\pi\)
−0.765950 + 0.642900i \(0.777731\pi\)
\(642\) 0 0
\(643\) 3.78461 0.149250 0.0746252 0.997212i \(-0.476224\pi\)
0.0746252 + 0.997212i \(0.476224\pi\)
\(644\) 3.50258 0.138021
\(645\) 0 0
\(646\) 26.9282 1.05948
\(647\) −2.00000 −0.0786281 −0.0393141 0.999227i \(-0.512517\pi\)
−0.0393141 + 0.999227i \(0.512517\pi\)
\(648\) 0 0
\(649\) 25.1769 0.988280
\(650\) 0 0
\(651\) 0 0
\(652\) −8.10512 −0.317421
\(653\) 0.483340 0.0189145 0.00945727 0.999955i \(-0.496990\pi\)
0.00945727 + 0.999955i \(0.496990\pi\)
\(654\) 0 0
\(655\) 1.26795 0.0495429
\(656\) −5.64617 −0.220446
\(657\) 0 0
\(658\) 0.641016 0.0249894
\(659\) 37.1769 1.44821 0.724103 0.689691i \(-0.242254\pi\)
0.724103 + 0.689691i \(0.242254\pi\)
\(660\) 0 0
\(661\) −21.0526 −0.818850 −0.409425 0.912344i \(-0.634271\pi\)
−0.409425 + 0.912344i \(0.634271\pi\)
\(662\) −5.94744 −0.231154
\(663\) 0 0
\(664\) 6.43078 0.249563
\(665\) 24.3923 0.945893
\(666\) 0 0
\(667\) 1.46410 0.0566902
\(668\) 25.5692 0.989303
\(669\) 0 0
\(670\) 9.12436 0.352505
\(671\) −15.4641 −0.596985
\(672\) 0 0
\(673\) −18.6603 −0.719300 −0.359650 0.933087i \(-0.617104\pi\)
−0.359650 + 0.933087i \(0.617104\pi\)
\(674\) 22.7321 0.875606
\(675\) 0 0
\(676\) 0 0
\(677\) −32.7846 −1.26001 −0.630007 0.776589i \(-0.716948\pi\)
−0.630007 + 0.776589i \(0.716948\pi\)
\(678\) 0 0
\(679\) −73.4974 −2.82057
\(680\) 17.0718 0.654674
\(681\) 0 0
\(682\) −8.10512 −0.310361
\(683\) −41.2679 −1.57907 −0.789537 0.613703i \(-0.789679\pi\)
−0.789537 + 0.613703i \(0.789679\pi\)
\(684\) 0 0
\(685\) −9.12436 −0.348624
\(686\) −19.3731 −0.739667
\(687\) 0 0
\(688\) −0.287187 −0.0109489
\(689\) 0 0
\(690\) 0 0
\(691\) −43.5885 −1.65818 −0.829092 0.559113i \(-0.811142\pi\)
−0.829092 + 0.559113i \(0.811142\pi\)
\(692\) 4.78461 0.181884
\(693\) 0 0
\(694\) −4.78461 −0.181621
\(695\) 17.9282 0.680056
\(696\) 0 0
\(697\) −35.4641 −1.34330
\(698\) −21.9474 −0.830723
\(699\) 0 0
\(700\) 6.53590 0.247034
\(701\) 8.78461 0.331790 0.165895 0.986143i \(-0.446949\pi\)
0.165895 + 0.986143i \(0.446949\pi\)
\(702\) 0 0
\(703\) −21.8564 −0.824330
\(704\) −7.42563 −0.279864
\(705\) 0 0
\(706\) 27.0333 1.01741
\(707\) −22.0000 −0.827395
\(708\) 0 0
\(709\) −30.6603 −1.15147 −0.575735 0.817636i \(-0.695284\pi\)
−0.575735 + 0.817636i \(0.695284\pi\)
\(710\) 9.32051 0.349792
\(711\) 0 0
\(712\) 3.21539 0.120502
\(713\) 1.71281 0.0641453
\(714\) 0 0
\(715\) 0 0
\(716\) 13.3590 0.499249
\(717\) 0 0
\(718\) 24.6410 0.919595
\(719\) −34.7321 −1.29529 −0.647643 0.761944i \(-0.724245\pi\)
−0.647643 + 0.761944i \(0.724245\pi\)
\(720\) 0 0
\(721\) 14.2679 0.531366
\(722\) 7.94744 0.295773
\(723\) 0 0
\(724\) 21.2820 0.790941
\(725\) 2.73205 0.101466
\(726\) 0 0
\(727\) −6.26795 −0.232465 −0.116233 0.993222i \(-0.537082\pi\)
−0.116233 + 0.993222i \(0.537082\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −11.2679 −0.417046
\(731\) −1.80385 −0.0667177
\(732\) 0 0
\(733\) −2.32051 −0.0857099 −0.0428550 0.999081i \(-0.513645\pi\)
−0.0428550 + 0.999081i \(0.513645\pi\)
\(734\) −13.2679 −0.489729
\(735\) 0 0
\(736\) 3.13844 0.115684
\(737\) 43.1769 1.59044
\(738\) 0 0
\(739\) −10.3923 −0.382287 −0.191144 0.981562i \(-0.561220\pi\)
−0.191144 + 0.981562i \(0.561220\pi\)
\(740\) −5.85641 −0.215286
\(741\) 0 0
\(742\) −22.6410 −0.831178
\(743\) −39.9090 −1.46412 −0.732059 0.681241i \(-0.761440\pi\)
−0.732059 + 0.681241i \(0.761440\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) −1.37307 −0.0502716
\(747\) 0 0
\(748\) 34.1436 1.24841
\(749\) 76.4449 2.79323
\(750\) 0 0
\(751\) −21.3205 −0.777996 −0.388998 0.921239i \(-0.627179\pi\)
−0.388998 + 0.921239i \(0.627179\pi\)
\(752\) −0.210236 −0.00766650
\(753\) 0 0
\(754\) 0 0
\(755\) −17.8564 −0.649861
\(756\) 0 0
\(757\) 26.9282 0.978722 0.489361 0.872081i \(-0.337230\pi\)
0.489361 + 0.872081i \(0.337230\pi\)
\(758\) −14.4449 −0.524661
\(759\) 0 0
\(760\) 13.8564 0.502625
\(761\) 5.85641 0.212295 0.106147 0.994350i \(-0.466148\pi\)
0.106147 + 0.994350i \(0.466148\pi\)
\(762\) 0 0
\(763\) −36.9090 −1.33619
\(764\) −40.2872 −1.45754
\(765\) 0 0
\(766\) 6.10512 0.220587
\(767\) 0 0
\(768\) 0 0
\(769\) 50.3923 1.81719 0.908596 0.417675i \(-0.137155\pi\)
0.908596 + 0.417675i \(0.137155\pi\)
\(770\) −11.3205 −0.407963
\(771\) 0 0
\(772\) 8.88973 0.319948
\(773\) 7.85641 0.282575 0.141288 0.989969i \(-0.454876\pi\)
0.141288 + 0.989969i \(0.454876\pi\)
\(774\) 0 0
\(775\) 3.19615 0.114809
\(776\) −41.7513 −1.49878
\(777\) 0 0
\(778\) −12.7846 −0.458350
\(779\) −28.7846 −1.03132
\(780\) 0 0
\(781\) 44.1051 1.57821
\(782\) 2.64102 0.0944425
\(783\) 0 0
\(784\) 13.8564 0.494872
\(785\) −9.73205 −0.347352
\(786\) 0 0
\(787\) −21.5359 −0.767672 −0.383836 0.923401i \(-0.625397\pi\)
−0.383836 + 0.923401i \(0.625397\pi\)
\(788\) −4.28719 −0.152725
\(789\) 0 0
\(790\) −1.41154 −0.0502204
\(791\) 86.8897 3.08944
\(792\) 0 0
\(793\) 0 0
\(794\) 17.8038 0.631835
\(795\) 0 0
\(796\) 21.9615 0.778406
\(797\) 30.4449 1.07841 0.539206 0.842174i \(-0.318724\pi\)
0.539206 + 0.842174i \(0.318724\pi\)
\(798\) 0 0
\(799\) −1.32051 −0.0467162
\(800\) 5.85641 0.207055
\(801\) 0 0
\(802\) −11.2154 −0.396029
\(803\) −53.3205 −1.88164
\(804\) 0 0
\(805\) 2.39230 0.0843177
\(806\) 0 0
\(807\) 0 0
\(808\) −12.4974 −0.439658
\(809\) −36.0000 −1.26569 −0.632846 0.774277i \(-0.718114\pi\)
−0.632846 + 0.774277i \(0.718114\pi\)
\(810\) 0 0
\(811\) −0.947441 −0.0332692 −0.0166346 0.999862i \(-0.505295\pi\)
−0.0166346 + 0.999862i \(0.505295\pi\)
\(812\) 17.8564 0.626637
\(813\) 0 0
\(814\) 10.1436 0.355533
\(815\) −5.53590 −0.193914
\(816\) 0 0
\(817\) −1.46410 −0.0512224
\(818\) −1.94744 −0.0680907
\(819\) 0 0
\(820\) −7.71281 −0.269343
\(821\) 0.928203 0.0323945 0.0161973 0.999869i \(-0.494844\pi\)
0.0161973 + 0.999869i \(0.494844\pi\)
\(822\) 0 0
\(823\) −13.1769 −0.459318 −0.229659 0.973271i \(-0.573761\pi\)
−0.229659 + 0.973271i \(0.573761\pi\)
\(824\) 8.10512 0.282355
\(825\) 0 0
\(826\) 23.7513 0.826413
\(827\) 2.58846 0.0900095 0.0450047 0.998987i \(-0.485670\pi\)
0.0450047 + 0.998987i \(0.485670\pi\)
\(828\) 0 0
\(829\) −38.1769 −1.32594 −0.662970 0.748646i \(-0.730704\pi\)
−0.662970 + 0.748646i \(0.730704\pi\)
\(830\) 1.85641 0.0644368
\(831\) 0 0
\(832\) 0 0
\(833\) 87.0333 3.01553
\(834\) 0 0
\(835\) 17.4641 0.604370
\(836\) 27.7128 0.958468
\(837\) 0 0
\(838\) −14.9667 −0.517015
\(839\) 20.5359 0.708978 0.354489 0.935060i \(-0.384655\pi\)
0.354489 + 0.935060i \(0.384655\pi\)
\(840\) 0 0
\(841\) −21.5359 −0.742617
\(842\) 17.2679 0.595093
\(843\) 0 0
\(844\) −16.8897 −0.581368
\(845\) 0 0
\(846\) 0 0
\(847\) −4.46410 −0.153388
\(848\) 7.42563 0.254997
\(849\) 0 0
\(850\) 4.92820 0.169036
\(851\) −2.14359 −0.0734814
\(852\) 0 0
\(853\) −16.6077 −0.568637 −0.284318 0.958730i \(-0.591767\pi\)
−0.284318 + 0.958730i \(0.591767\pi\)
\(854\) −14.5885 −0.499207
\(855\) 0 0
\(856\) 43.4256 1.48426
\(857\) 17.1244 0.584957 0.292478 0.956272i \(-0.405520\pi\)
0.292478 + 0.956272i \(0.405520\pi\)
\(858\) 0 0
\(859\) 7.78461 0.265607 0.132804 0.991142i \(-0.457602\pi\)
0.132804 + 0.991142i \(0.457602\pi\)
\(860\) −0.392305 −0.0133775
\(861\) 0 0
\(862\) 17.0718 0.581468
\(863\) −34.3923 −1.17073 −0.585364 0.810771i \(-0.699048\pi\)
−0.585364 + 0.810771i \(0.699048\pi\)
\(864\) 0 0
\(865\) 3.26795 0.111114
\(866\) 3.51666 0.119501
\(867\) 0 0
\(868\) 20.8897 0.709044
\(869\) −6.67949 −0.226586
\(870\) 0 0
\(871\) 0 0
\(872\) −20.9667 −0.710021
\(873\) 0 0
\(874\) 2.14359 0.0725081
\(875\) 4.46410 0.150914
\(876\) 0 0
\(877\) 18.6410 0.629462 0.314731 0.949181i \(-0.398086\pi\)
0.314731 + 0.949181i \(0.398086\pi\)
\(878\) 11.2679 0.380275
\(879\) 0 0
\(880\) 3.71281 0.125159
\(881\) 25.1769 0.848232 0.424116 0.905608i \(-0.360585\pi\)
0.424116 + 0.905608i \(0.360585\pi\)
\(882\) 0 0
\(883\) −0.660254 −0.0222193 −0.0111097 0.999938i \(-0.503536\pi\)
−0.0111097 + 0.999938i \(0.503536\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.28719 −0.0768396
\(887\) 2.98076 0.100084 0.0500421 0.998747i \(-0.484064\pi\)
0.0500421 + 0.998747i \(0.484064\pi\)
\(888\) 0 0
\(889\) −65.4449 −2.19495
\(890\) 0.928203 0.0311134
\(891\) 0 0
\(892\) 17.3590 0.581222
\(893\) −1.07180 −0.0358663
\(894\) 0 0
\(895\) 9.12436 0.304994
\(896\) 45.2820 1.51277
\(897\) 0 0
\(898\) −8.10512 −0.270471
\(899\) 8.73205 0.291230
\(900\) 0 0
\(901\) 46.6410 1.55384
\(902\) 13.3590 0.444806
\(903\) 0 0
\(904\) 49.3590 1.64166
\(905\) 14.5359 0.483190
\(906\) 0 0
\(907\) −24.5359 −0.814701 −0.407351 0.913272i \(-0.633547\pi\)
−0.407351 + 0.913272i \(0.633547\pi\)
\(908\) −17.0718 −0.566547
\(909\) 0 0
\(910\) 0 0
\(911\) 7.71281 0.255537 0.127768 0.991804i \(-0.459219\pi\)
0.127768 + 0.991804i \(0.459219\pi\)
\(912\) 0 0
\(913\) 8.78461 0.290728
\(914\) −3.94744 −0.130570
\(915\) 0 0
\(916\) 26.9282 0.889733
\(917\) 5.66025 0.186918
\(918\) 0 0
\(919\) −19.0718 −0.629121 −0.314560 0.949238i \(-0.601857\pi\)
−0.314560 + 0.949238i \(0.601857\pi\)
\(920\) 1.35898 0.0448044
\(921\) 0 0
\(922\) 8.43078 0.277653
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −1.30642 −0.0429318
\(927\) 0 0
\(928\) 16.0000 0.525226
\(929\) −15.3205 −0.502650 −0.251325 0.967903i \(-0.580866\pi\)
−0.251325 + 0.967903i \(0.580866\pi\)
\(930\) 0 0
\(931\) 70.6410 2.31517
\(932\) 34.9282 1.14411
\(933\) 0 0
\(934\) 17.3205 0.566744
\(935\) 23.3205 0.762662
\(936\) 0 0
\(937\) 32.2487 1.05352 0.526760 0.850014i \(-0.323407\pi\)
0.526760 + 0.850014i \(0.323407\pi\)
\(938\) 40.7321 1.32995
\(939\) 0 0
\(940\) −0.287187 −0.00936701
\(941\) 9.41154 0.306808 0.153404 0.988164i \(-0.450976\pi\)
0.153404 + 0.988164i \(0.450976\pi\)
\(942\) 0 0
\(943\) −2.82309 −0.0919323
\(944\) −7.78976 −0.253535
\(945\) 0 0
\(946\) 0.679492 0.0220922
\(947\) −5.41154 −0.175852 −0.0879258 0.996127i \(-0.528024\pi\)
−0.0879258 + 0.996127i \(0.528024\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 76.2102 2.46999
\(953\) −24.2487 −0.785493 −0.392746 0.919647i \(-0.628475\pi\)
−0.392746 + 0.919647i \(0.628475\pi\)
\(954\) 0 0
\(955\) −27.5167 −0.890418
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −26.2872 −0.849300
\(959\) −40.7321 −1.31531
\(960\) 0 0
\(961\) −20.7846 −0.670471
\(962\) 0 0
\(963\) 0 0
\(964\) −2.35383 −0.0758117
\(965\) 6.07180 0.195458
\(966\) 0 0
\(967\) 16.2487 0.522523 0.261262 0.965268i \(-0.415861\pi\)
0.261262 + 0.965268i \(0.415861\pi\)
\(968\) −2.53590 −0.0815069
\(969\) 0 0
\(970\) −12.0526 −0.386984
\(971\) −61.1769 −1.96326 −0.981630 0.190793i \(-0.938894\pi\)
−0.981630 + 0.190793i \(0.938894\pi\)
\(972\) 0 0
\(973\) 80.0333 2.56575
\(974\) −15.2154 −0.487533
\(975\) 0 0
\(976\) 4.78461 0.153152
\(977\) 14.9282 0.477596 0.238798 0.971069i \(-0.423247\pi\)
0.238798 + 0.971069i \(0.423247\pi\)
\(978\) 0 0
\(979\) 4.39230 0.140379
\(980\) 18.9282 0.604639
\(981\) 0 0
\(982\) −5.21539 −0.166430
\(983\) 3.85641 0.123000 0.0615001 0.998107i \(-0.480412\pi\)
0.0615001 + 0.998107i \(0.480412\pi\)
\(984\) 0 0
\(985\) −2.92820 −0.0933003
\(986\) 13.4641 0.428784
\(987\) 0 0
\(988\) 0 0
\(989\) −0.143594 −0.00456601
\(990\) 0 0
\(991\) 15.0718 0.478771 0.239386 0.970925i \(-0.423054\pi\)
0.239386 + 0.970925i \(0.423054\pi\)
\(992\) 18.7180 0.594296
\(993\) 0 0
\(994\) 41.6077 1.31972
\(995\) 15.0000 0.475532
\(996\) 0 0
\(997\) −26.4115 −0.836462 −0.418231 0.908341i \(-0.637350\pi\)
−0.418231 + 0.908341i \(0.637350\pi\)
\(998\) 0.287187 0.00909075
\(999\) 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 7605.2.a.y.1.2 2
3.2 odd 2 2535.2.a.s.1.1 2
13.2 odd 12 585.2.bu.a.316.1 4
13.7 odd 12 585.2.bu.a.361.1 4
13.12 even 2 7605.2.a.bk.1.1 2
39.2 even 12 195.2.bb.a.121.2 4
39.20 even 12 195.2.bb.a.166.2 yes 4
39.38 odd 2 2535.2.a.n.1.2 2
195.2 odd 12 975.2.w.a.199.2 4
195.59 even 12 975.2.bc.h.751.1 4
195.98 odd 12 975.2.w.a.49.2 4
195.119 even 12 975.2.bc.h.901.1 4
195.137 odd 12 975.2.w.f.49.1 4
195.158 odd 12 975.2.w.f.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.bb.a.121.2 4 39.2 even 12
195.2.bb.a.166.2 yes 4 39.20 even 12
585.2.bu.a.316.1 4 13.2 odd 12
585.2.bu.a.361.1 4 13.7 odd 12
975.2.w.a.49.2 4 195.98 odd 12
975.2.w.a.199.2 4 195.2 odd 12
975.2.w.f.49.1 4 195.137 odd 12
975.2.w.f.199.1 4 195.158 odd 12
975.2.bc.h.751.1 4 195.59 even 12
975.2.bc.h.901.1 4 195.119 even 12
2535.2.a.n.1.2 2 39.38 odd 2
2535.2.a.s.1.1 2 3.2 odd 2
7605.2.a.y.1.2 2 1.1 even 1 trivial
7605.2.a.bk.1.1 2 13.12 even 2