Properties

Label 3042.2.a.p.1.2
Level $3042$
Weight $2$
Character 3042.1
Self dual yes
Analytic conductor $24.290$
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] = [3042,2,Mod(1,3042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3042.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: \(24.2904922949\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} +0.732051 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.267949 q^{5} +0.732051 q^{7} -1.00000 q^{8} +0.267949 q^{10} -4.73205 q^{11} -0.732051 q^{14} +1.00000 q^{16} +2.26795 q^{17} +1.26795 q^{19} -0.267949 q^{20} +4.73205 q^{22} +6.19615 q^{23} -4.92820 q^{25} +0.732051 q^{28} -2.46410 q^{29} -5.46410 q^{31} -1.00000 q^{32} -2.26795 q^{34} -0.196152 q^{35} +10.4641 q^{37} -1.26795 q^{38} +0.267949 q^{40} -11.3923 q^{41} +7.66025 q^{43} -4.73205 q^{44} -6.19615 q^{46} +8.19615 q^{47} -6.46410 q^{49} +4.92820 q^{50} -0.464102 q^{53} +1.26795 q^{55} -0.732051 q^{56} +2.46410 q^{58} -8.00000 q^{59} +1.19615 q^{61} +5.46410 q^{62} +1.00000 q^{64} -11.1244 q^{67} +2.26795 q^{68} +0.196152 q^{70} -1.26795 q^{71} -9.73205 q^{73} -10.4641 q^{74} +1.26795 q^{76} -3.46410 q^{77} -9.46410 q^{79} -0.267949 q^{80} +11.3923 q^{82} -10.1962 q^{83} -0.607695 q^{85} -7.66025 q^{86} +4.73205 q^{88} +2.53590 q^{89} +6.19615 q^{92} -8.19615 q^{94} -0.339746 q^{95} -6.00000 q^{97} +6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{7} - 2 q^{8} + 4 q^{10} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} + 6 q^{19} - 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{28} + 2 q^{29} - 4 q^{31} - 2 q^{32} - 8 q^{34} + 10 q^{35} + 14 q^{37} - 6 q^{38} + 4 q^{40} - 2 q^{41} - 2 q^{43} - 6 q^{44} - 2 q^{46} + 6 q^{47} - 6 q^{49} - 4 q^{50} + 6 q^{53} + 6 q^{55} + 2 q^{56} - 2 q^{58} - 16 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{64} + 2 q^{67} + 8 q^{68} - 10 q^{70} - 6 q^{71} - 16 q^{73} - 14 q^{74} + 6 q^{76} - 12 q^{79} - 4 q^{80} + 2 q^{82} - 10 q^{83} - 22 q^{85} + 2 q^{86} + 6 q^{88} + 12 q^{89} + 2 q^{92} - 6 q^{94} - 18 q^{95} - 12 q^{97} + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.267949 −0.119831 −0.0599153 0.998203i \(-0.519083\pi\)
−0.0599153 + 0.998203i \(0.519083\pi\)
\(6\) 0 0
\(7\) 0.732051 0.276689 0.138345 0.990384i \(-0.455822\pi\)
0.138345 + 0.990384i \(0.455822\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.267949 0.0847330
\(11\) −4.73205 −1.42677 −0.713384 0.700774i \(-0.752838\pi\)
−0.713384 + 0.700774i \(0.752838\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −0.732051 −0.195649
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.26795 0.550058 0.275029 0.961436i \(-0.411312\pi\)
0.275029 + 0.961436i \(0.411312\pi\)
\(18\) 0 0
\(19\) 1.26795 0.290887 0.145444 0.989367i \(-0.453539\pi\)
0.145444 + 0.989367i \(0.453539\pi\)
\(20\) −0.267949 −0.0599153
\(21\) 0 0
\(22\) 4.73205 1.00888
\(23\) 6.19615 1.29199 0.645994 0.763343i \(-0.276443\pi\)
0.645994 + 0.763343i \(0.276443\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 0 0
\(27\) 0 0
\(28\) 0.732051 0.138345
\(29\) −2.46410 −0.457572 −0.228786 0.973477i \(-0.573476\pi\)
−0.228786 + 0.973477i \(0.573476\pi\)
\(30\) 0 0
\(31\) −5.46410 −0.981382 −0.490691 0.871334i \(-0.663256\pi\)
−0.490691 + 0.871334i \(0.663256\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.26795 −0.388950
\(35\) −0.196152 −0.0331558
\(36\) 0 0
\(37\) 10.4641 1.72029 0.860144 0.510052i \(-0.170374\pi\)
0.860144 + 0.510052i \(0.170374\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) −11.3923 −1.77918 −0.889590 0.456761i \(-0.849010\pi\)
−0.889590 + 0.456761i \(0.849010\pi\)
\(42\) 0 0
\(43\) 7.66025 1.16818 0.584089 0.811690i \(-0.301452\pi\)
0.584089 + 0.811690i \(0.301452\pi\)
\(44\) −4.73205 −0.713384
\(45\) 0 0
\(46\) −6.19615 −0.913573
\(47\) 8.19615 1.19553 0.597766 0.801671i \(-0.296055\pi\)
0.597766 + 0.801671i \(0.296055\pi\)
\(48\) 0 0
\(49\) −6.46410 −0.923443
\(50\) 4.92820 0.696953
\(51\) 0 0
\(52\) 0 0
\(53\) −0.464102 −0.0637493 −0.0318746 0.999492i \(-0.510148\pi\)
−0.0318746 + 0.999492i \(0.510148\pi\)
\(54\) 0 0
\(55\) 1.26795 0.170970
\(56\) −0.732051 −0.0978244
\(57\) 0 0
\(58\) 2.46410 0.323552
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 1.19615 0.153152 0.0765758 0.997064i \(-0.475601\pi\)
0.0765758 + 0.997064i \(0.475601\pi\)
\(62\) 5.46410 0.693942
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −11.1244 −1.35906 −0.679528 0.733649i \(-0.737815\pi\)
−0.679528 + 0.733649i \(0.737815\pi\)
\(68\) 2.26795 0.275029
\(69\) 0 0
\(70\) 0.196152 0.0234447
\(71\) −1.26795 −0.150478 −0.0752389 0.997166i \(-0.523972\pi\)
−0.0752389 + 0.997166i \(0.523972\pi\)
\(72\) 0 0
\(73\) −9.73205 −1.13905 −0.569525 0.821974i \(-0.692873\pi\)
−0.569525 + 0.821974i \(0.692873\pi\)
\(74\) −10.4641 −1.21643
\(75\) 0 0
\(76\) 1.26795 0.145444
\(77\) −3.46410 −0.394771
\(78\) 0 0
\(79\) −9.46410 −1.06479 −0.532397 0.846495i \(-0.678709\pi\)
−0.532397 + 0.846495i \(0.678709\pi\)
\(80\) −0.267949 −0.0299576
\(81\) 0 0
\(82\) 11.3923 1.25807
\(83\) −10.1962 −1.11917 −0.559587 0.828772i \(-0.689040\pi\)
−0.559587 + 0.828772i \(0.689040\pi\)
\(84\) 0 0
\(85\) −0.607695 −0.0659138
\(86\) −7.66025 −0.826026
\(87\) 0 0
\(88\) 4.73205 0.504438
\(89\) 2.53590 0.268805 0.134402 0.990927i \(-0.457089\pi\)
0.134402 + 0.990927i \(0.457089\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.19615 0.645994
\(93\) 0 0
\(94\) −8.19615 −0.845369
\(95\) −0.339746 −0.0348572
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 6.46410 0.652973
\(99\) 0 0
\(100\) −4.92820 −0.492820
\(101\) 11.9282 1.18690 0.593450 0.804871i \(-0.297765\pi\)
0.593450 + 0.804871i \(0.297765\pi\)
\(102\) 0 0
\(103\) −18.7321 −1.84572 −0.922862 0.385131i \(-0.874156\pi\)
−0.922862 + 0.385131i \(0.874156\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0.464102 0.0450775
\(107\) 0.196152 0.0189628 0.00948139 0.999955i \(-0.496982\pi\)
0.00948139 + 0.999955i \(0.496982\pi\)
\(108\) 0 0
\(109\) −5.46410 −0.523366 −0.261683 0.965154i \(-0.584277\pi\)
−0.261683 + 0.965154i \(0.584277\pi\)
\(110\) −1.26795 −0.120894
\(111\) 0 0
\(112\) 0.732051 0.0691723
\(113\) 18.6603 1.75541 0.877705 0.479202i \(-0.159074\pi\)
0.877705 + 0.479202i \(0.159074\pi\)
\(114\) 0 0
\(115\) −1.66025 −0.154819
\(116\) −2.46410 −0.228786
\(117\) 0 0
\(118\) 8.00000 0.736460
\(119\) 1.66025 0.152195
\(120\) 0 0
\(121\) 11.3923 1.03566
\(122\) −1.19615 −0.108295
\(123\) 0 0
\(124\) −5.46410 −0.490691
\(125\) 2.66025 0.237940
\(126\) 0 0
\(127\) 17.8564 1.58450 0.792250 0.610197i \(-0.208910\pi\)
0.792250 + 0.610197i \(0.208910\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −13.4641 −1.17636 −0.588182 0.808729i \(-0.700156\pi\)
−0.588182 + 0.808729i \(0.700156\pi\)
\(132\) 0 0
\(133\) 0.928203 0.0804854
\(134\) 11.1244 0.960998
\(135\) 0 0
\(136\) −2.26795 −0.194475
\(137\) 1.92820 0.164738 0.0823688 0.996602i \(-0.473751\pi\)
0.0823688 + 0.996602i \(0.473751\pi\)
\(138\) 0 0
\(139\) −9.85641 −0.836009 −0.418005 0.908445i \(-0.637270\pi\)
−0.418005 + 0.908445i \(0.637270\pi\)
\(140\) −0.196152 −0.0165779
\(141\) 0 0
\(142\) 1.26795 0.106404
\(143\) 0 0
\(144\) 0 0
\(145\) 0.660254 0.0548311
\(146\) 9.73205 0.805430
\(147\) 0 0
\(148\) 10.4641 0.860144
\(149\) −2.80385 −0.229700 −0.114850 0.993383i \(-0.536639\pi\)
−0.114850 + 0.993383i \(0.536639\pi\)
\(150\) 0 0
\(151\) 3.26795 0.265942 0.132971 0.991120i \(-0.457548\pi\)
0.132971 + 0.991120i \(0.457548\pi\)
\(152\) −1.26795 −0.102844
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) 1.46410 0.117599
\(156\) 0 0
\(157\) −23.5885 −1.88256 −0.941282 0.337622i \(-0.890378\pi\)
−0.941282 + 0.337622i \(0.890378\pi\)
\(158\) 9.46410 0.752923
\(159\) 0 0
\(160\) 0.267949 0.0211832
\(161\) 4.53590 0.357479
\(162\) 0 0
\(163\) −6.53590 −0.511931 −0.255966 0.966686i \(-0.582393\pi\)
−0.255966 + 0.966686i \(0.582393\pi\)
\(164\) −11.3923 −0.889590
\(165\) 0 0
\(166\) 10.1962 0.791375
\(167\) 2.53590 0.196234 0.0981169 0.995175i \(-0.468718\pi\)
0.0981169 + 0.995175i \(0.468718\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0.607695 0.0466081
\(171\) 0 0
\(172\) 7.66025 0.584089
\(173\) 16.3923 1.24628 0.623142 0.782109i \(-0.285856\pi\)
0.623142 + 0.782109i \(0.285856\pi\)
\(174\) 0 0
\(175\) −3.60770 −0.272716
\(176\) −4.73205 −0.356692
\(177\) 0 0
\(178\) −2.53590 −0.190074
\(179\) −22.0526 −1.64829 −0.824143 0.566382i \(-0.808343\pi\)
−0.824143 + 0.566382i \(0.808343\pi\)
\(180\) 0 0
\(181\) 8.80385 0.654385 0.327192 0.944958i \(-0.393897\pi\)
0.327192 + 0.944958i \(0.393897\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.19615 −0.456786
\(185\) −2.80385 −0.206143
\(186\) 0 0
\(187\) −10.7321 −0.784805
\(188\) 8.19615 0.597766
\(189\) 0 0
\(190\) 0.339746 0.0246478
\(191\) −6.92820 −0.501307 −0.250654 0.968077i \(-0.580646\pi\)
−0.250654 + 0.968077i \(0.580646\pi\)
\(192\) 0 0
\(193\) −8.26795 −0.595140 −0.297570 0.954700i \(-0.596176\pi\)
−0.297570 + 0.954700i \(0.596176\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) −9.85641 −0.702240 −0.351120 0.936330i \(-0.614199\pi\)
−0.351120 + 0.936330i \(0.614199\pi\)
\(198\) 0 0
\(199\) −3.80385 −0.269648 −0.134824 0.990870i \(-0.543047\pi\)
−0.134824 + 0.990870i \(0.543047\pi\)
\(200\) 4.92820 0.348477
\(201\) 0 0
\(202\) −11.9282 −0.839265
\(203\) −1.80385 −0.126605
\(204\) 0 0
\(205\) 3.05256 0.213200
\(206\) 18.7321 1.30512
\(207\) 0 0
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) −4.39230 −0.302379 −0.151189 0.988505i \(-0.548310\pi\)
−0.151189 + 0.988505i \(0.548310\pi\)
\(212\) −0.464102 −0.0318746
\(213\) 0 0
\(214\) −0.196152 −0.0134087
\(215\) −2.05256 −0.139983
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 5.46410 0.370076
\(219\) 0 0
\(220\) 1.26795 0.0854851
\(221\) 0 0
\(222\) 0 0
\(223\) −13.0718 −0.875352 −0.437676 0.899133i \(-0.644198\pi\)
−0.437676 + 0.899133i \(0.644198\pi\)
\(224\) −0.732051 −0.0489122
\(225\) 0 0
\(226\) −18.6603 −1.24126
\(227\) 1.80385 0.119726 0.0598628 0.998207i \(-0.480934\pi\)
0.0598628 + 0.998207i \(0.480934\pi\)
\(228\) 0 0
\(229\) 15.8564 1.04782 0.523910 0.851773i \(-0.324473\pi\)
0.523910 + 0.851773i \(0.324473\pi\)
\(230\) 1.66025 0.109474
\(231\) 0 0
\(232\) 2.46410 0.161776
\(233\) −19.8564 −1.30084 −0.650418 0.759576i \(-0.725406\pi\)
−0.650418 + 0.759576i \(0.725406\pi\)
\(234\) 0 0
\(235\) −2.19615 −0.143261
\(236\) −8.00000 −0.520756
\(237\) 0 0
\(238\) −1.66025 −0.107618
\(239\) 9.66025 0.624870 0.312435 0.949939i \(-0.398855\pi\)
0.312435 + 0.949939i \(0.398855\pi\)
\(240\) 0 0
\(241\) −17.5885 −1.13297 −0.566486 0.824071i \(-0.691698\pi\)
−0.566486 + 0.824071i \(0.691698\pi\)
\(242\) −11.3923 −0.732325
\(243\) 0 0
\(244\) 1.19615 0.0765758
\(245\) 1.73205 0.110657
\(246\) 0 0
\(247\) 0 0
\(248\) 5.46410 0.346971
\(249\) 0 0
\(250\) −2.66025 −0.168249
\(251\) −6.53590 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(252\) 0 0
\(253\) −29.3205 −1.84336
\(254\) −17.8564 −1.12041
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −26.6603 −1.66302 −0.831510 0.555509i \(-0.812523\pi\)
−0.831510 + 0.555509i \(0.812523\pi\)
\(258\) 0 0
\(259\) 7.66025 0.475985
\(260\) 0 0
\(261\) 0 0
\(262\) 13.4641 0.831815
\(263\) −28.0526 −1.72979 −0.864897 0.501949i \(-0.832617\pi\)
−0.864897 + 0.501949i \(0.832617\pi\)
\(264\) 0 0
\(265\) 0.124356 0.00763911
\(266\) −0.928203 −0.0569118
\(267\) 0 0
\(268\) −11.1244 −0.679528
\(269\) 1.46410 0.0892679 0.0446339 0.999003i \(-0.485788\pi\)
0.0446339 + 0.999003i \(0.485788\pi\)
\(270\) 0 0
\(271\) 5.85641 0.355751 0.177876 0.984053i \(-0.443078\pi\)
0.177876 + 0.984053i \(0.443078\pi\)
\(272\) 2.26795 0.137515
\(273\) 0 0
\(274\) −1.92820 −0.116487
\(275\) 23.3205 1.40628
\(276\) 0 0
\(277\) −2.26795 −0.136268 −0.0681339 0.997676i \(-0.521705\pi\)
−0.0681339 + 0.997676i \(0.521705\pi\)
\(278\) 9.85641 0.591148
\(279\) 0 0
\(280\) 0.196152 0.0117223
\(281\) 22.3205 1.33153 0.665765 0.746162i \(-0.268105\pi\)
0.665765 + 0.746162i \(0.268105\pi\)
\(282\) 0 0
\(283\) 8.33975 0.495746 0.247873 0.968792i \(-0.420268\pi\)
0.247873 + 0.968792i \(0.420268\pi\)
\(284\) −1.26795 −0.0752389
\(285\) 0 0
\(286\) 0 0
\(287\) −8.33975 −0.492280
\(288\) 0 0
\(289\) −11.8564 −0.697436
\(290\) −0.660254 −0.0387715
\(291\) 0 0
\(292\) −9.73205 −0.569525
\(293\) 14.5167 0.848072 0.424036 0.905645i \(-0.360613\pi\)
0.424036 + 0.905645i \(0.360613\pi\)
\(294\) 0 0
\(295\) 2.14359 0.124805
\(296\) −10.4641 −0.608214
\(297\) 0 0
\(298\) 2.80385 0.162423
\(299\) 0 0
\(300\) 0 0
\(301\) 5.60770 0.323222
\(302\) −3.26795 −0.188049
\(303\) 0 0
\(304\) 1.26795 0.0727219
\(305\) −0.320508 −0.0183522
\(306\) 0 0
\(307\) 8.58846 0.490169 0.245085 0.969502i \(-0.421184\pi\)
0.245085 + 0.969502i \(0.421184\pi\)
\(308\) −3.46410 −0.197386
\(309\) 0 0
\(310\) −1.46410 −0.0831554
\(311\) −15.6603 −0.888012 −0.444006 0.896024i \(-0.646443\pi\)
−0.444006 + 0.896024i \(0.646443\pi\)
\(312\) 0 0
\(313\) 13.4641 0.761036 0.380518 0.924774i \(-0.375746\pi\)
0.380518 + 0.924774i \(0.375746\pi\)
\(314\) 23.5885 1.33117
\(315\) 0 0
\(316\) −9.46410 −0.532397
\(317\) −3.33975 −0.187579 −0.0937894 0.995592i \(-0.529898\pi\)
−0.0937894 + 0.995592i \(0.529898\pi\)
\(318\) 0 0
\(319\) 11.6603 0.652849
\(320\) −0.267949 −0.0149788
\(321\) 0 0
\(322\) −4.53590 −0.252776
\(323\) 2.87564 0.160005
\(324\) 0 0
\(325\) 0 0
\(326\) 6.53590 0.361990
\(327\) 0 0
\(328\) 11.3923 0.629035
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −10.1962 −0.559587
\(333\) 0 0
\(334\) −2.53590 −0.138758
\(335\) 2.98076 0.162856
\(336\) 0 0
\(337\) 6.85641 0.373492 0.186746 0.982408i \(-0.440206\pi\)
0.186746 + 0.982408i \(0.440206\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −0.607695 −0.0329569
\(341\) 25.8564 1.40020
\(342\) 0 0
\(343\) −9.85641 −0.532196
\(344\) −7.66025 −0.413013
\(345\) 0 0
\(346\) −16.3923 −0.881256
\(347\) −8.87564 −0.476470 −0.238235 0.971208i \(-0.576569\pi\)
−0.238235 + 0.971208i \(0.576569\pi\)
\(348\) 0 0
\(349\) −19.3205 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(350\) 3.60770 0.192839
\(351\) 0 0
\(352\) 4.73205 0.252219
\(353\) −19.7846 −1.05303 −0.526514 0.850166i \(-0.676501\pi\)
−0.526514 + 0.850166i \(0.676501\pi\)
\(354\) 0 0
\(355\) 0.339746 0.0180318
\(356\) 2.53590 0.134402
\(357\) 0 0
\(358\) 22.0526 1.16551
\(359\) −23.1244 −1.22046 −0.610228 0.792226i \(-0.708922\pi\)
−0.610228 + 0.792226i \(0.708922\pi\)
\(360\) 0 0
\(361\) −17.3923 −0.915384
\(362\) −8.80385 −0.462720
\(363\) 0 0
\(364\) 0 0
\(365\) 2.60770 0.136493
\(366\) 0 0
\(367\) −14.7321 −0.769007 −0.384503 0.923124i \(-0.625627\pi\)
−0.384503 + 0.923124i \(0.625627\pi\)
\(368\) 6.19615 0.322997
\(369\) 0 0
\(370\) 2.80385 0.145765
\(371\) −0.339746 −0.0176387
\(372\) 0 0
\(373\) −10.2679 −0.531654 −0.265827 0.964021i \(-0.585645\pi\)
−0.265827 + 0.964021i \(0.585645\pi\)
\(374\) 10.7321 0.554941
\(375\) 0 0
\(376\) −8.19615 −0.422684
\(377\) 0 0
\(378\) 0 0
\(379\) 1.46410 0.0752058 0.0376029 0.999293i \(-0.488028\pi\)
0.0376029 + 0.999293i \(0.488028\pi\)
\(380\) −0.339746 −0.0174286
\(381\) 0 0
\(382\) 6.92820 0.354478
\(383\) −5.46410 −0.279203 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(384\) 0 0
\(385\) 0.928203 0.0473056
\(386\) 8.26795 0.420828
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) 29.7846 1.51014 0.755070 0.655644i \(-0.227603\pi\)
0.755070 + 0.655644i \(0.227603\pi\)
\(390\) 0 0
\(391\) 14.0526 0.710668
\(392\) 6.46410 0.326486
\(393\) 0 0
\(394\) 9.85641 0.496559
\(395\) 2.53590 0.127595
\(396\) 0 0
\(397\) −0.392305 −0.0196892 −0.00984461 0.999952i \(-0.503134\pi\)
−0.00984461 + 0.999952i \(0.503134\pi\)
\(398\) 3.80385 0.190670
\(399\) 0 0
\(400\) −4.92820 −0.246410
\(401\) −21.9282 −1.09504 −0.547521 0.836792i \(-0.684428\pi\)
−0.547521 + 0.836792i \(0.684428\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 11.9282 0.593450
\(405\) 0 0
\(406\) 1.80385 0.0895235
\(407\) −49.5167 −2.45445
\(408\) 0 0
\(409\) 14.2679 0.705505 0.352752 0.935717i \(-0.385246\pi\)
0.352752 + 0.935717i \(0.385246\pi\)
\(410\) −3.05256 −0.150755
\(411\) 0 0
\(412\) −18.7321 −0.922862
\(413\) −5.85641 −0.288175
\(414\) 0 0
\(415\) 2.73205 0.134111
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 10.5359 0.514712 0.257356 0.966317i \(-0.417149\pi\)
0.257356 + 0.966317i \(0.417149\pi\)
\(420\) 0 0
\(421\) 32.7128 1.59432 0.797162 0.603765i \(-0.206333\pi\)
0.797162 + 0.603765i \(0.206333\pi\)
\(422\) 4.39230 0.213814
\(423\) 0 0
\(424\) 0.464102 0.0225388
\(425\) −11.1769 −0.542160
\(426\) 0 0
\(427\) 0.875644 0.0423754
\(428\) 0.196152 0.00948139
\(429\) 0 0
\(430\) 2.05256 0.0989832
\(431\) 11.1244 0.535841 0.267921 0.963441i \(-0.413663\pi\)
0.267921 + 0.963441i \(0.413663\pi\)
\(432\) 0 0
\(433\) −14.8564 −0.713953 −0.356977 0.934113i \(-0.616192\pi\)
−0.356977 + 0.934113i \(0.616192\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) −5.46410 −0.261683
\(437\) 7.85641 0.375823
\(438\) 0 0
\(439\) −17.6603 −0.842878 −0.421439 0.906857i \(-0.638475\pi\)
−0.421439 + 0.906857i \(0.638475\pi\)
\(440\) −1.26795 −0.0604471
\(441\) 0 0
\(442\) 0 0
\(443\) −36.3923 −1.72905 −0.864525 0.502589i \(-0.832381\pi\)
−0.864525 + 0.502589i \(0.832381\pi\)
\(444\) 0 0
\(445\) −0.679492 −0.0322110
\(446\) 13.0718 0.618968
\(447\) 0 0
\(448\) 0.732051 0.0345861
\(449\) 23.3205 1.10056 0.550281 0.834979i \(-0.314520\pi\)
0.550281 + 0.834979i \(0.314520\pi\)
\(450\) 0 0
\(451\) 53.9090 2.53847
\(452\) 18.6603 0.877705
\(453\) 0 0
\(454\) −1.80385 −0.0846588
\(455\) 0 0
\(456\) 0 0
\(457\) 18.6603 0.872890 0.436445 0.899731i \(-0.356237\pi\)
0.436445 + 0.899731i \(0.356237\pi\)
\(458\) −15.8564 −0.740921
\(459\) 0 0
\(460\) −1.66025 −0.0774097
\(461\) −25.7321 −1.19846 −0.599231 0.800577i \(-0.704527\pi\)
−0.599231 + 0.800577i \(0.704527\pi\)
\(462\) 0 0
\(463\) −28.0526 −1.30371 −0.651856 0.758342i \(-0.726010\pi\)
−0.651856 + 0.758342i \(0.726010\pi\)
\(464\) −2.46410 −0.114393
\(465\) 0 0
\(466\) 19.8564 0.919830
\(467\) 12.5885 0.582524 0.291262 0.956643i \(-0.405925\pi\)
0.291262 + 0.956643i \(0.405925\pi\)
\(468\) 0 0
\(469\) −8.14359 −0.376036
\(470\) 2.19615 0.101301
\(471\) 0 0
\(472\) 8.00000 0.368230
\(473\) −36.2487 −1.66672
\(474\) 0 0
\(475\) −6.24871 −0.286711
\(476\) 1.66025 0.0760976
\(477\) 0 0
\(478\) −9.66025 −0.441850
\(479\) −26.5359 −1.21246 −0.606228 0.795291i \(-0.707318\pi\)
−0.606228 + 0.795291i \(0.707318\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 17.5885 0.801132
\(483\) 0 0
\(484\) 11.3923 0.517832
\(485\) 1.60770 0.0730017
\(486\) 0 0
\(487\) 21.1244 0.957236 0.478618 0.878023i \(-0.341138\pi\)
0.478618 + 0.878023i \(0.341138\pi\)
\(488\) −1.19615 −0.0541473
\(489\) 0 0
\(490\) −1.73205 −0.0782461
\(491\) −5.26795 −0.237739 −0.118870 0.992910i \(-0.537927\pi\)
−0.118870 + 0.992910i \(0.537927\pi\)
\(492\) 0 0
\(493\) −5.58846 −0.251691
\(494\) 0 0
\(495\) 0 0
\(496\) −5.46410 −0.245345
\(497\) −0.928203 −0.0416356
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 2.66025 0.118970
\(501\) 0 0
\(502\) 6.53590 0.291711
\(503\) 10.9808 0.489608 0.244804 0.969573i \(-0.421276\pi\)
0.244804 + 0.969573i \(0.421276\pi\)
\(504\) 0 0
\(505\) −3.19615 −0.142227
\(506\) 29.3205 1.30346
\(507\) 0 0
\(508\) 17.8564 0.792250
\(509\) −10.2679 −0.455119 −0.227559 0.973764i \(-0.573075\pi\)
−0.227559 + 0.973764i \(0.573075\pi\)
\(510\) 0 0
\(511\) −7.12436 −0.315163
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 26.6603 1.17593
\(515\) 5.01924 0.221174
\(516\) 0 0
\(517\) −38.7846 −1.70575
\(518\) −7.66025 −0.336572
\(519\) 0 0
\(520\) 0 0
\(521\) 17.4449 0.764273 0.382137 0.924106i \(-0.375188\pi\)
0.382137 + 0.924106i \(0.375188\pi\)
\(522\) 0 0
\(523\) 36.4449 1.59362 0.796811 0.604228i \(-0.206518\pi\)
0.796811 + 0.604228i \(0.206518\pi\)
\(524\) −13.4641 −0.588182
\(525\) 0 0
\(526\) 28.0526 1.22315
\(527\) −12.3923 −0.539817
\(528\) 0 0
\(529\) 15.3923 0.669231
\(530\) −0.124356 −0.00540166
\(531\) 0 0
\(532\) 0.928203 0.0402427
\(533\) 0 0
\(534\) 0 0
\(535\) −0.0525589 −0.00227232
\(536\) 11.1244 0.480499
\(537\) 0 0
\(538\) −1.46410 −0.0631219
\(539\) 30.5885 1.31754
\(540\) 0 0
\(541\) 40.3205 1.73351 0.866757 0.498731i \(-0.166200\pi\)
0.866757 + 0.498731i \(0.166200\pi\)
\(542\) −5.85641 −0.251554
\(543\) 0 0
\(544\) −2.26795 −0.0972375
\(545\) 1.46410 0.0627152
\(546\) 0 0
\(547\) 6.19615 0.264928 0.132464 0.991188i \(-0.457711\pi\)
0.132464 + 0.991188i \(0.457711\pi\)
\(548\) 1.92820 0.0823688
\(549\) 0 0
\(550\) −23.3205 −0.994390
\(551\) −3.12436 −0.133102
\(552\) 0 0
\(553\) −6.92820 −0.294617
\(554\) 2.26795 0.0963559
\(555\) 0 0
\(556\) −9.85641 −0.418005
\(557\) 30.3731 1.28695 0.643474 0.765468i \(-0.277492\pi\)
0.643474 + 0.765468i \(0.277492\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.196152 −0.00828895
\(561\) 0 0
\(562\) −22.3205 −0.941534
\(563\) −21.0718 −0.888070 −0.444035 0.896009i \(-0.646453\pi\)
−0.444035 + 0.896009i \(0.646453\pi\)
\(564\) 0 0
\(565\) −5.00000 −0.210352
\(566\) −8.33975 −0.350546
\(567\) 0 0
\(568\) 1.26795 0.0532020
\(569\) 38.6410 1.61992 0.809958 0.586488i \(-0.199490\pi\)
0.809958 + 0.586488i \(0.199490\pi\)
\(570\) 0 0
\(571\) −24.0526 −1.00657 −0.503284 0.864121i \(-0.667875\pi\)
−0.503284 + 0.864121i \(0.667875\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 8.33975 0.348094
\(575\) −30.5359 −1.27343
\(576\) 0 0
\(577\) 0.267949 0.0111549 0.00557744 0.999984i \(-0.498225\pi\)
0.00557744 + 0.999984i \(0.498225\pi\)
\(578\) 11.8564 0.493161
\(579\) 0 0
\(580\) 0.660254 0.0274156
\(581\) −7.46410 −0.309663
\(582\) 0 0
\(583\) 2.19615 0.0909553
\(584\) 9.73205 0.402715
\(585\) 0 0
\(586\) −14.5167 −0.599678
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) −6.92820 −0.285472
\(590\) −2.14359 −0.0882503
\(591\) 0 0
\(592\) 10.4641 0.430072
\(593\) 36.8564 1.51351 0.756756 0.653698i \(-0.226783\pi\)
0.756756 + 0.653698i \(0.226783\pi\)
\(594\) 0 0
\(595\) −0.444864 −0.0182376
\(596\) −2.80385 −0.114850
\(597\) 0 0
\(598\) 0 0
\(599\) 9.46410 0.386693 0.193346 0.981131i \(-0.438066\pi\)
0.193346 + 0.981131i \(0.438066\pi\)
\(600\) 0 0
\(601\) −5.92820 −0.241816 −0.120908 0.992664i \(-0.538581\pi\)
−0.120908 + 0.992664i \(0.538581\pi\)
\(602\) −5.60770 −0.228553
\(603\) 0 0
\(604\) 3.26795 0.132971
\(605\) −3.05256 −0.124104
\(606\) 0 0
\(607\) 0.784610 0.0318463 0.0159232 0.999873i \(-0.494931\pi\)
0.0159232 + 0.999873i \(0.494931\pi\)
\(608\) −1.26795 −0.0514221
\(609\) 0 0
\(610\) 0.320508 0.0129770
\(611\) 0 0
\(612\) 0 0
\(613\) 11.3923 0.460131 0.230065 0.973175i \(-0.426106\pi\)
0.230065 + 0.973175i \(0.426106\pi\)
\(614\) −8.58846 −0.346602
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) −35.2487 −1.41906 −0.709530 0.704675i \(-0.751093\pi\)
−0.709530 + 0.704675i \(0.751093\pi\)
\(618\) 0 0
\(619\) 10.5359 0.423474 0.211737 0.977327i \(-0.432088\pi\)
0.211737 + 0.977327i \(0.432088\pi\)
\(620\) 1.46410 0.0587997
\(621\) 0 0
\(622\) 15.6603 0.627919
\(623\) 1.85641 0.0743754
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) −13.4641 −0.538134
\(627\) 0 0
\(628\) −23.5885 −0.941282
\(629\) 23.7321 0.946259
\(630\) 0 0
\(631\) 47.7128 1.89942 0.949709 0.313135i \(-0.101379\pi\)
0.949709 + 0.313135i \(0.101379\pi\)
\(632\) 9.46410 0.376462
\(633\) 0 0
\(634\) 3.33975 0.132638
\(635\) −4.78461 −0.189871
\(636\) 0 0
\(637\) 0 0
\(638\) −11.6603 −0.461634
\(639\) 0 0
\(640\) 0.267949 0.0105916
\(641\) −25.9808 −1.02618 −0.513089 0.858335i \(-0.671499\pi\)
−0.513089 + 0.858335i \(0.671499\pi\)
\(642\) 0 0
\(643\) 13.8564 0.546443 0.273222 0.961951i \(-0.411911\pi\)
0.273222 + 0.961951i \(0.411911\pi\)
\(644\) 4.53590 0.178739
\(645\) 0 0
\(646\) −2.87564 −0.113141
\(647\) −26.2487 −1.03194 −0.515972 0.856606i \(-0.672569\pi\)
−0.515972 + 0.856606i \(0.672569\pi\)
\(648\) 0 0
\(649\) 37.8564 1.48599
\(650\) 0 0
\(651\) 0 0
\(652\) −6.53590 −0.255966
\(653\) 10.5359 0.412302 0.206151 0.978520i \(-0.433906\pi\)
0.206151 + 0.978520i \(0.433906\pi\)
\(654\) 0 0
\(655\) 3.60770 0.140964
\(656\) −11.3923 −0.444795
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) −38.2487 −1.48996 −0.744979 0.667088i \(-0.767541\pi\)
−0.744979 + 0.667088i \(0.767541\pi\)
\(660\) 0 0
\(661\) −9.39230 −0.365318 −0.182659 0.983176i \(-0.558470\pi\)
−0.182659 + 0.983176i \(0.558470\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 10.1962 0.395687
\(665\) −0.248711 −0.00964461
\(666\) 0 0
\(667\) −15.2679 −0.591177
\(668\) 2.53590 0.0981169
\(669\) 0 0
\(670\) −2.98076 −0.115157
\(671\) −5.66025 −0.218512
\(672\) 0 0
\(673\) 14.0718 0.542428 0.271214 0.962519i \(-0.412575\pi\)
0.271214 + 0.962519i \(0.412575\pi\)
\(674\) −6.85641 −0.264099
\(675\) 0 0
\(676\) 0 0
\(677\) 38.5359 1.48105 0.740527 0.672026i \(-0.234576\pi\)
0.740527 + 0.672026i \(0.234576\pi\)
\(678\) 0 0
\(679\) −4.39230 −0.168561
\(680\) 0.607695 0.0233040
\(681\) 0 0
\(682\) −25.8564 −0.990093
\(683\) 37.8564 1.44854 0.724268 0.689519i \(-0.242178\pi\)
0.724268 + 0.689519i \(0.242178\pi\)
\(684\) 0 0
\(685\) −0.516660 −0.0197406
\(686\) 9.85641 0.376319
\(687\) 0 0
\(688\) 7.66025 0.292044
\(689\) 0 0
\(690\) 0 0
\(691\) 26.3397 1.00201 0.501006 0.865444i \(-0.332964\pi\)
0.501006 + 0.865444i \(0.332964\pi\)
\(692\) 16.3923 0.623142
\(693\) 0 0
\(694\) 8.87564 0.336915
\(695\) 2.64102 0.100179
\(696\) 0 0
\(697\) −25.8372 −0.978653
\(698\) 19.3205 0.731292
\(699\) 0 0
\(700\) −3.60770 −0.136358
\(701\) −31.3205 −1.18296 −0.591480 0.806320i \(-0.701456\pi\)
−0.591480 + 0.806320i \(0.701456\pi\)
\(702\) 0 0
\(703\) 13.2679 0.500410
\(704\) −4.73205 −0.178346
\(705\) 0 0
\(706\) 19.7846 0.744604
\(707\) 8.73205 0.328403
\(708\) 0 0
\(709\) 40.8564 1.53439 0.767197 0.641411i \(-0.221651\pi\)
0.767197 + 0.641411i \(0.221651\pi\)
\(710\) −0.339746 −0.0127504
\(711\) 0 0
\(712\) −2.53590 −0.0950368
\(713\) −33.8564 −1.26793
\(714\) 0 0
\(715\) 0 0
\(716\) −22.0526 −0.824143
\(717\) 0 0
\(718\) 23.1244 0.862993
\(719\) 22.5359 0.840447 0.420224 0.907421i \(-0.361952\pi\)
0.420224 + 0.907421i \(0.361952\pi\)
\(720\) 0 0
\(721\) −13.7128 −0.510692
\(722\) 17.3923 0.647275
\(723\) 0 0
\(724\) 8.80385 0.327192
\(725\) 12.1436 0.451002
\(726\) 0 0
\(727\) 20.9808 0.778133 0.389067 0.921210i \(-0.372798\pi\)
0.389067 + 0.921210i \(0.372798\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.60770 −0.0965151
\(731\) 17.3731 0.642566
\(732\) 0 0
\(733\) 19.0000 0.701781 0.350891 0.936416i \(-0.385879\pi\)
0.350891 + 0.936416i \(0.385879\pi\)
\(734\) 14.7321 0.543770
\(735\) 0 0
\(736\) −6.19615 −0.228393
\(737\) 52.6410 1.93906
\(738\) 0 0
\(739\) −10.9282 −0.402000 −0.201000 0.979591i \(-0.564419\pi\)
−0.201000 + 0.979591i \(0.564419\pi\)
\(740\) −2.80385 −0.103071
\(741\) 0 0
\(742\) 0.339746 0.0124725
\(743\) 27.6077 1.01283 0.506414 0.862290i \(-0.330971\pi\)
0.506414 + 0.862290i \(0.330971\pi\)
\(744\) 0 0
\(745\) 0.751289 0.0275251
\(746\) 10.2679 0.375936
\(747\) 0 0
\(748\) −10.7321 −0.392403
\(749\) 0.143594 0.00524679
\(750\) 0 0
\(751\) 15.9090 0.580526 0.290263 0.956947i \(-0.406257\pi\)
0.290263 + 0.956947i \(0.406257\pi\)
\(752\) 8.19615 0.298883
\(753\) 0 0
\(754\) 0 0
\(755\) −0.875644 −0.0318680
\(756\) 0 0
\(757\) 7.07180 0.257029 0.128514 0.991708i \(-0.458979\pi\)
0.128514 + 0.991708i \(0.458979\pi\)
\(758\) −1.46410 −0.0531786
\(759\) 0 0
\(760\) 0.339746 0.0123239
\(761\) 23.3205 0.845368 0.422684 0.906277i \(-0.361088\pi\)
0.422684 + 0.906277i \(0.361088\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) −6.92820 −0.250654
\(765\) 0 0
\(766\) 5.46410 0.197426
\(767\) 0 0
\(768\) 0 0
\(769\) −16.1436 −0.582153 −0.291076 0.956700i \(-0.594013\pi\)
−0.291076 + 0.956700i \(0.594013\pi\)
\(770\) −0.928203 −0.0334501
\(771\) 0 0
\(772\) −8.26795 −0.297570
\(773\) 35.0718 1.26144 0.630722 0.776009i \(-0.282759\pi\)
0.630722 + 0.776009i \(0.282759\pi\)
\(774\) 0 0
\(775\) 26.9282 0.967290
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −29.7846 −1.06783
\(779\) −14.4449 −0.517541
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) −14.0526 −0.502518
\(783\) 0 0
\(784\) −6.46410 −0.230861
\(785\) 6.32051 0.225589
\(786\) 0 0
\(787\) 39.3205 1.40162 0.700812 0.713346i \(-0.252821\pi\)
0.700812 + 0.713346i \(0.252821\pi\)
\(788\) −9.85641 −0.351120
\(789\) 0 0
\(790\) −2.53590 −0.0902232
\(791\) 13.6603 0.485703
\(792\) 0 0
\(793\) 0 0
\(794\) 0.392305 0.0139224
\(795\) 0 0
\(796\) −3.80385 −0.134824
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) 18.5885 0.657612
\(800\) 4.92820 0.174238
\(801\) 0 0
\(802\) 21.9282 0.774312
\(803\) 46.0526 1.62516
\(804\) 0 0
\(805\) −1.21539 −0.0428369
\(806\) 0 0
\(807\) 0 0
\(808\) −11.9282 −0.419633
\(809\) 22.4115 0.787948 0.393974 0.919122i \(-0.371100\pi\)
0.393974 + 0.919122i \(0.371100\pi\)
\(810\) 0 0
\(811\) −45.1769 −1.58638 −0.793188 0.608977i \(-0.791580\pi\)
−0.793188 + 0.608977i \(0.791580\pi\)
\(812\) −1.80385 −0.0633026
\(813\) 0 0
\(814\) 49.5167 1.73556
\(815\) 1.75129 0.0613450
\(816\) 0 0
\(817\) 9.71281 0.339808
\(818\) −14.2679 −0.498867
\(819\) 0 0
\(820\) 3.05256 0.106600
\(821\) −12.9282 −0.451197 −0.225599 0.974220i \(-0.572434\pi\)
−0.225599 + 0.974220i \(0.572434\pi\)
\(822\) 0 0
\(823\) −41.5692 −1.44901 −0.724506 0.689269i \(-0.757932\pi\)
−0.724506 + 0.689269i \(0.757932\pi\)
\(824\) 18.7321 0.652562
\(825\) 0 0
\(826\) 5.85641 0.203770
\(827\) −33.4641 −1.16366 −0.581830 0.813310i \(-0.697663\pi\)
−0.581830 + 0.813310i \(0.697663\pi\)
\(828\) 0 0
\(829\) 12.1244 0.421096 0.210548 0.977583i \(-0.432475\pi\)
0.210548 + 0.977583i \(0.432475\pi\)
\(830\) −2.73205 −0.0948309
\(831\) 0 0
\(832\) 0 0
\(833\) −14.6603 −0.507948
\(834\) 0 0
\(835\) −0.679492 −0.0235148
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −10.5359 −0.363957
\(839\) −14.1436 −0.488291 −0.244146 0.969739i \(-0.578507\pi\)
−0.244146 + 0.969739i \(0.578507\pi\)
\(840\) 0 0
\(841\) −22.9282 −0.790628
\(842\) −32.7128 −1.12736
\(843\) 0 0
\(844\) −4.39230 −0.151189
\(845\) 0 0
\(846\) 0 0
\(847\) 8.33975 0.286557
\(848\) −0.464102 −0.0159373
\(849\) 0 0
\(850\) 11.1769 0.383365
\(851\) 64.8372 2.22259
\(852\) 0 0
\(853\) 8.17691 0.279972 0.139986 0.990153i \(-0.455294\pi\)
0.139986 + 0.990153i \(0.455294\pi\)
\(854\) −0.875644 −0.0299639
\(855\) 0 0
\(856\) −0.196152 −0.00670435
\(857\) 19.4449 0.664224 0.332112 0.943240i \(-0.392239\pi\)
0.332112 + 0.943240i \(0.392239\pi\)
\(858\) 0 0
\(859\) −22.8756 −0.780507 −0.390253 0.920707i \(-0.627613\pi\)
−0.390253 + 0.920707i \(0.627613\pi\)
\(860\) −2.05256 −0.0699917
\(861\) 0 0
\(862\) −11.1244 −0.378897
\(863\) −7.12436 −0.242516 −0.121258 0.992621i \(-0.538693\pi\)
−0.121258 + 0.992621i \(0.538693\pi\)
\(864\) 0 0
\(865\) −4.39230 −0.149343
\(866\) 14.8564 0.504841
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 44.7846 1.51921
\(870\) 0 0
\(871\) 0 0
\(872\) 5.46410 0.185038
\(873\) 0 0
\(874\) −7.85641 −0.265747
\(875\) 1.94744 0.0658355
\(876\) 0 0
\(877\) 10.0718 0.340100 0.170050 0.985435i \(-0.445607\pi\)
0.170050 + 0.985435i \(0.445607\pi\)
\(878\) 17.6603 0.596005
\(879\) 0 0
\(880\) 1.26795 0.0427426
\(881\) −51.8372 −1.74644 −0.873219 0.487327i \(-0.837972\pi\)
−0.873219 + 0.487327i \(0.837972\pi\)
\(882\) 0 0
\(883\) 29.0718 0.978344 0.489172 0.872187i \(-0.337299\pi\)
0.489172 + 0.872187i \(0.337299\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.3923 1.22262
\(887\) −10.1436 −0.340589 −0.170294 0.985393i \(-0.554472\pi\)
−0.170294 + 0.985393i \(0.554472\pi\)
\(888\) 0 0
\(889\) 13.0718 0.438414
\(890\) 0.679492 0.0227766
\(891\) 0 0
\(892\) −13.0718 −0.437676
\(893\) 10.3923 0.347765
\(894\) 0 0
\(895\) 5.90897 0.197515
\(896\) −0.732051 −0.0244561
\(897\) 0 0
\(898\) −23.3205 −0.778215
\(899\) 13.4641 0.449053
\(900\) 0 0
\(901\) −1.05256 −0.0350658
\(902\) −53.9090 −1.79497
\(903\) 0 0
\(904\) −18.6603 −0.620631
\(905\) −2.35898 −0.0784153
\(906\) 0 0
\(907\) 15.6077 0.518245 0.259123 0.965844i \(-0.416567\pi\)
0.259123 + 0.965844i \(0.416567\pi\)
\(908\) 1.80385 0.0598628
\(909\) 0 0
\(910\) 0 0
\(911\) 9.46410 0.313560 0.156780 0.987634i \(-0.449889\pi\)
0.156780 + 0.987634i \(0.449889\pi\)
\(912\) 0 0
\(913\) 48.2487 1.59680
\(914\) −18.6603 −0.617226
\(915\) 0 0
\(916\) 15.8564 0.523910
\(917\) −9.85641 −0.325487
\(918\) 0 0
\(919\) −57.9615 −1.91197 −0.955987 0.293409i \(-0.905210\pi\)
−0.955987 + 0.293409i \(0.905210\pi\)
\(920\) 1.66025 0.0547370
\(921\) 0 0
\(922\) 25.7321 0.847440
\(923\) 0 0
\(924\) 0 0
\(925\) −51.5692 −1.69559
\(926\) 28.0526 0.921864
\(927\) 0 0
\(928\) 2.46410 0.0808881
\(929\) −9.24871 −0.303440 −0.151720 0.988423i \(-0.548481\pi\)
−0.151720 + 0.988423i \(0.548481\pi\)
\(930\) 0 0
\(931\) −8.19615 −0.268618
\(932\) −19.8564 −0.650418
\(933\) 0 0
\(934\) −12.5885 −0.411907
\(935\) 2.87564 0.0940436
\(936\) 0 0
\(937\) 43.2487 1.41287 0.706437 0.707776i \(-0.250301\pi\)
0.706437 + 0.707776i \(0.250301\pi\)
\(938\) 8.14359 0.265898
\(939\) 0 0
\(940\) −2.19615 −0.0716306
\(941\) 56.6410 1.84644 0.923222 0.384267i \(-0.125546\pi\)
0.923222 + 0.384267i \(0.125546\pi\)
\(942\) 0 0
\(943\) −70.5885 −2.29868
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 36.2487 1.17855
\(947\) −34.9282 −1.13501 −0.567507 0.823369i \(-0.692092\pi\)
−0.567507 + 0.823369i \(0.692092\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 6.24871 0.202735
\(951\) 0 0
\(952\) −1.66025 −0.0538091
\(953\) 41.5692 1.34656 0.673280 0.739388i \(-0.264885\pi\)
0.673280 + 0.739388i \(0.264885\pi\)
\(954\) 0 0
\(955\) 1.85641 0.0600719
\(956\) 9.66025 0.312435
\(957\) 0 0
\(958\) 26.5359 0.857336
\(959\) 1.41154 0.0455811
\(960\) 0 0
\(961\) −1.14359 −0.0368901
\(962\) 0 0
\(963\) 0 0
\(964\) −17.5885 −0.566486
\(965\) 2.21539 0.0713159
\(966\) 0 0
\(967\) −18.8756 −0.607000 −0.303500 0.952831i \(-0.598155\pi\)
−0.303500 + 0.952831i \(0.598155\pi\)
\(968\) −11.3923 −0.366163
\(969\) 0 0
\(970\) −1.60770 −0.0516200
\(971\) 18.2487 0.585629 0.292815 0.956169i \(-0.405408\pi\)
0.292815 + 0.956169i \(0.405408\pi\)
\(972\) 0 0
\(973\) −7.21539 −0.231315
\(974\) −21.1244 −0.676868
\(975\) 0 0
\(976\) 1.19615 0.0382879
\(977\) −32.0718 −1.02607 −0.513034 0.858368i \(-0.671478\pi\)
−0.513034 + 0.858368i \(0.671478\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 1.73205 0.0553283
\(981\) 0 0
\(982\) 5.26795 0.168107
\(983\) 20.7846 0.662926 0.331463 0.943468i \(-0.392458\pi\)
0.331463 + 0.943468i \(0.392458\pi\)
\(984\) 0 0
\(985\) 2.64102 0.0841498
\(986\) 5.58846 0.177973
\(987\) 0 0
\(988\) 0 0
\(989\) 47.4641 1.50927
\(990\) 0 0
\(991\) −8.58846 −0.272821 −0.136411 0.990652i \(-0.543557\pi\)
−0.136411 + 0.990652i \(0.543557\pi\)
\(992\) 5.46410 0.173485
\(993\) 0 0
\(994\) 0.928203 0.0294408
\(995\) 1.01924 0.0323120
\(996\) 0 0
\(997\) 38.6603 1.22438 0.612191 0.790710i \(-0.290288\pi\)
0.612191 + 0.790710i \(0.290288\pi\)
\(998\) −32.0000 −1.01294
\(999\) 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 3042.2.a.p.1.2 2
3.2 odd 2 1014.2.a.k.1.1 2
12.11 even 2 8112.2.a.bp.1.1 2
13.2 odd 12 234.2.l.c.199.1 4
13.5 odd 4 3042.2.b.i.1351.3 4
13.7 odd 12 234.2.l.c.127.1 4
13.8 odd 4 3042.2.b.i.1351.2 4
13.12 even 2 3042.2.a.y.1.1 2
39.2 even 12 78.2.i.a.43.2 4
39.5 even 4 1014.2.b.e.337.2 4
39.8 even 4 1014.2.b.e.337.3 4
39.11 even 12 1014.2.i.a.823.1 4
39.17 odd 6 1014.2.e.i.991.2 4
39.20 even 12 78.2.i.a.49.2 yes 4
39.23 odd 6 1014.2.e.i.529.2 4
39.29 odd 6 1014.2.e.g.529.1 4
39.32 even 12 1014.2.i.a.361.1 4
39.35 odd 6 1014.2.e.g.991.1 4
39.38 odd 2 1014.2.a.i.1.2 2
52.7 even 12 1872.2.by.h.1297.1 4
52.15 even 12 1872.2.by.h.433.2 4
156.59 odd 12 624.2.bv.e.49.2 4
156.119 odd 12 624.2.bv.e.433.1 4
156.155 even 2 8112.2.a.bj.1.2 2
195.2 odd 12 1950.2.y.b.199.2 4
195.59 even 12 1950.2.bc.d.751.1 4
195.98 odd 12 1950.2.y.b.49.2 4
195.119 even 12 1950.2.bc.d.901.1 4
195.137 odd 12 1950.2.y.g.49.1 4
195.158 odd 12 1950.2.y.g.199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 4 39.2 even 12
78.2.i.a.49.2 yes 4 39.20 even 12
234.2.l.c.127.1 4 13.7 odd 12
234.2.l.c.199.1 4 13.2 odd 12
624.2.bv.e.49.2 4 156.59 odd 12
624.2.bv.e.433.1 4 156.119 odd 12
1014.2.a.i.1.2 2 39.38 odd 2
1014.2.a.k.1.1 2 3.2 odd 2
1014.2.b.e.337.2 4 39.5 even 4
1014.2.b.e.337.3 4 39.8 even 4
1014.2.e.g.529.1 4 39.29 odd 6
1014.2.e.g.991.1 4 39.35 odd 6
1014.2.e.i.529.2 4 39.23 odd 6
1014.2.e.i.991.2 4 39.17 odd 6
1014.2.i.a.361.1 4 39.32 even 12
1014.2.i.a.823.1 4 39.11 even 12
1872.2.by.h.433.2 4 52.15 even 12
1872.2.by.h.1297.1 4 52.7 even 12
1950.2.y.b.49.2 4 195.98 odd 12
1950.2.y.b.199.2 4 195.2 odd 12
1950.2.y.g.49.1 4 195.137 odd 12
1950.2.y.g.199.1 4 195.158 odd 12
1950.2.bc.d.751.1 4 195.59 even 12
1950.2.bc.d.901.1 4 195.119 even 12
3042.2.a.p.1.2 2 1.1 even 1 trivial
3042.2.a.y.1.1 2 13.12 even 2
3042.2.b.i.1351.2 4 13.8 odd 4
3042.2.b.i.1351.3 4 13.5 odd 4
8112.2.a.bj.1.2 2 156.155 even 2
8112.2.a.bp.1.1 2 12.11 even 2