Properties

Label 3042.2.a.y.1.1
Level $3042$
Weight $2$
Character 3042.1
Self dual yes
Analytic conductor $24.290$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +0.267949 q^{5} -0.732051 q^{7} +1.00000 q^{8} +0.267949 q^{10} +4.73205 q^{11} -0.732051 q^{14} +1.00000 q^{16} +2.26795 q^{17} -1.26795 q^{19} +0.267949 q^{20} +4.73205 q^{22} +6.19615 q^{23} -4.92820 q^{25} -0.732051 q^{28} -2.46410 q^{29} +5.46410 q^{31} +1.00000 q^{32} +2.26795 q^{34} -0.196152 q^{35} -10.4641 q^{37} -1.26795 q^{38} +0.267949 q^{40} +11.3923 q^{41} +7.66025 q^{43} +4.73205 q^{44} +6.19615 q^{46} -8.19615 q^{47} -6.46410 q^{49} -4.92820 q^{50} -0.464102 q^{53} +1.26795 q^{55} -0.732051 q^{56} -2.46410 q^{58} +8.00000 q^{59} +1.19615 q^{61} +5.46410 q^{62} +1.00000 q^{64} +11.1244 q^{67} +2.26795 q^{68} -0.196152 q^{70} +1.26795 q^{71} +9.73205 q^{73} -10.4641 q^{74} -1.26795 q^{76} -3.46410 q^{77} -9.46410 q^{79} +0.267949 q^{80} +11.3923 q^{82} +10.1962 q^{83} +0.607695 q^{85} +7.66025 q^{86} +4.73205 q^{88} -2.53590 q^{89} +6.19615 q^{92} -8.19615 q^{94} -0.339746 q^{95} +6.00000 q^{97} -6.46410 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{19} + 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} + 2 q^{28} + 2 q^{29} + 4 q^{31} + 2 q^{32}+ \cdots - 6 q^{98}+O(q^{100}) \) 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 \(-0.480917\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) −0.732051 −0.276689 −0.138345 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 0.267949 0.0847330
\(11\) 4.73205 1.42677 0.713384 0.700774i \(-0.247162\pi\)
0.713384 + 0.700774i \(0.247162\pi\)
\(12\) 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 \(-0.546461\pi\)
−0.145444 + 0.989367i \(0.546461\pi\)
\(20\) 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 \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 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 \(-0.829626\pi\)
−0.860144 + 0.510052i \(0.829626\pi\)
\(38\) −1.26795 −0.205689
\(39\) 0 0
\(40\) 0.267949 0.0423665
\(41\) 11.3923 1.77918 0.889590 0.456761i \(-0.150990\pi\)
0.889590 + 0.456761i \(0.150990\pi\)
\(42\) 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 \(-0.703945\pi\)
−0.597766 + 0.801671i \(0.703945\pi\)
\(48\) 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 \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 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 \(-0.262185\pi\)
0.679528 + 0.733649i \(0.262185\pi\)
\(68\) 2.26795 0.275029
\(69\) 0 0
\(70\) −0.196152 −0.0234447
\(71\) 1.26795 0.150478 0.0752389 0.997166i \(-0.476028\pi\)
0.0752389 + 0.997166i \(0.476028\pi\)
\(72\) 0 0
\(73\) 9.73205 1.13905 0.569525 0.821974i \(-0.307127\pi\)
0.569525 + 0.821974i \(0.307127\pi\)
\(74\) −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 \(-0.310960\pi\)
0.559587 + 0.828772i \(0.310960\pi\)
\(84\) 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 \(-0.542911\pi\)
−0.134402 + 0.990927i \(0.542911\pi\)
\(90\) 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 \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −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 \(-0.415723\pi\)
0.261683 + 0.965154i \(0.415723\pi\)
\(110\) 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 \(-0.526249\pi\)
−0.0823688 + 0.996602i \(0.526249\pi\)
\(138\) 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 \(-0.463361\pi\)
0.114850 + 0.993383i \(0.463361\pi\)
\(150\) 0 0
\(151\) −3.26795 −0.265942 −0.132971 0.991120i \(-0.542452\pi\)
−0.132971 + 0.991120i \(0.542452\pi\)
\(152\) −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 \(-0.417607\pi\)
0.255966 + 0.966686i \(0.417607\pi\)
\(164\) 11.3923 0.889590
\(165\) 0 0
\(166\) 10.1962 0.791375
\(167\) −2.53590 −0.196234 −0.0981169 0.995175i \(-0.531282\pi\)
−0.0981169 + 0.995175i \(0.531282\pi\)
\(168\) 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 \(-0.403824\pi\)
0.297570 + 0.954700i \(0.403824\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) 9.85641 0.702240 0.351120 0.936330i \(-0.385801\pi\)
0.351120 + 0.936330i \(0.385801\pi\)
\(198\) 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 \(-0.355802\pi\)
0.437676 + 0.899133i \(0.355802\pi\)
\(224\) −0.732051 −0.0489122
\(225\) 0 0
\(226\) 18.6603 1.24126
\(227\) −1.80385 −0.119726 −0.0598628 0.998207i \(-0.519066\pi\)
−0.0598628 + 0.998207i \(0.519066\pi\)
\(228\) 0 0
\(229\) −15.8564 −1.04782 −0.523910 0.851773i \(-0.675527\pi\)
−0.523910 + 0.851773i \(0.675527\pi\)
\(230\) 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 \(-0.601145\pi\)
−0.312435 + 0.949939i \(0.601145\pi\)
\(240\) 0 0
\(241\) 17.5885 1.13297 0.566486 0.824071i \(-0.308302\pi\)
0.566486 + 0.824071i \(0.308302\pi\)
\(242\) 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 \(-0.556922\pi\)
−0.177876 + 0.984053i \(0.556922\pi\)
\(272\) 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 \(-0.731895\pi\)
−0.665765 + 0.746162i \(0.731895\pi\)
\(282\) 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 \(-0.639387\pi\)
−0.424036 + 0.905645i \(0.639387\pi\)
\(294\) 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 \(-0.578816\pi\)
−0.245085 + 0.969502i \(0.578816\pi\)
\(308\) −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 \(-0.470102\pi\)
0.0937894 + 0.995592i \(0.470102\pi\)
\(318\) 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 \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 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 \(-0.327011\pi\)
0.517102 + 0.855924i \(0.327011\pi\)
\(350\) 3.60770 0.192839
\(351\) 0 0
\(352\) 4.73205 0.252219
\(353\) 19.7846 1.05303 0.526514 0.850166i \(-0.323499\pi\)
0.526514 + 0.850166i \(0.323499\pi\)
\(354\) 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 \(-0.291078\pi\)
0.610228 + 0.792226i \(0.291078\pi\)
\(360\) 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 \(-0.511972\pi\)
−0.0376029 + 0.999293i \(0.511972\pi\)
\(380\) −0.339746 −0.0174286
\(381\) 0 0
\(382\) −6.92820 −0.354478
\(383\) 5.46410 0.279203 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(384\) 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 \(-0.496866\pi\)
0.00984461 + 0.999952i \(0.496866\pi\)
\(398\) −3.80385 −0.190670
\(399\) 0 0
\(400\) −4.92820 −0.246410
\(401\) 21.9282 1.09504 0.547521 0.836792i \(-0.315572\pi\)
0.547521 + 0.836792i \(0.315572\pi\)
\(402\) 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 \(-0.614754\pi\)
−0.352752 + 0.935717i \(0.614754\pi\)
\(410\) 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 \(-0.793667\pi\)
−0.797162 + 0.603765i \(0.793667\pi\)
\(422\) −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 \(-0.586337\pi\)
−0.267921 + 0.963441i \(0.586337\pi\)
\(432\) 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 \(-0.685480\pi\)
−0.550281 + 0.834979i \(0.685480\pi\)
\(450\) 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 \(-0.643763\pi\)
−0.436445 + 0.899731i \(0.643763\pi\)
\(458\) −15.8564 −0.740921
\(459\) 0 0
\(460\) 1.66025 0.0774097
\(461\) 25.7321 1.19846 0.599231 0.800577i \(-0.295473\pi\)
0.599231 + 0.800577i \(0.295473\pi\)
\(462\) 0 0
\(463\) 28.0526 1.30371 0.651856 0.758342i \(-0.273990\pi\)
0.651856 + 0.758342i \(0.273990\pi\)
\(464\) −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 \(-0.292682\pi\)
0.606228 + 0.795291i \(0.292682\pi\)
\(480\) 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 \(-0.658862\pi\)
−0.478618 + 0.878023i \(0.658862\pi\)
\(488\) 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 \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) −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 \(-0.426925\pi\)
0.227559 + 0.973764i \(0.426925\pi\)
\(510\) 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 \(-0.833800\pi\)
−0.866757 + 0.498731i \(0.833800\pi\)
\(542\) −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 \(-0.722508\pi\)
−0.643474 + 0.765468i \(0.722508\pi\)
\(558\) 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 \(-0.501775\pi\)
−0.00557744 + 0.999984i \(0.501775\pi\)
\(578\) −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 \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 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 \(-0.773217\pi\)
−0.756756 + 0.653698i \(0.773217\pi\)
\(594\) 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 \(-0.573894\pi\)
−0.230065 + 0.973175i \(0.573894\pi\)
\(614\) −8.58846 −0.346602
\(615\) 0 0
\(616\) −3.46410 −0.139573
\(617\) 35.2487 1.41906 0.709530 0.704675i \(-0.248907\pi\)
0.709530 + 0.704675i \(0.248907\pi\)
\(618\) 0 0
\(619\) −10.5359 −0.423474 −0.211737 0.977327i \(-0.567912\pi\)
−0.211737 + 0.977327i \(0.567912\pi\)
\(620\) 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 \(-0.898621\pi\)
−0.949709 + 0.313135i \(0.898621\pi\)
\(632\) −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 \(-0.588089\pi\)
−0.273222 + 0.961951i \(0.588089\pi\)
\(644\) −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 \(-0.441530\pi\)
0.182659 + 0.983176i \(0.441530\pi\)
\(662\) −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 \(-0.757822\pi\)
−0.724268 + 0.689519i \(0.757822\pi\)
\(684\) 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 \(-0.667036\pi\)
−0.501006 + 0.865444i \(0.667036\pi\)
\(692\) 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 \(-0.778349\pi\)
−0.767197 + 0.641411i \(0.778349\pi\)
\(710\) 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 \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) −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 \(-0.435581\pi\)
0.201000 + 0.979591i \(0.435581\pi\)
\(740\) −2.80385 −0.103071
\(741\) 0 0
\(742\) 0.339746 0.0124725
\(743\) −27.6077 −1.01283 −0.506414 0.862290i \(-0.669029\pi\)
−0.506414 + 0.862290i \(0.669029\pi\)
\(744\) 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 \(-0.638912\pi\)
−0.422684 + 0.906277i \(0.638912\pi\)
\(762\) 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 \(-0.405987\pi\)
0.291076 + 0.956700i \(0.405987\pi\)
\(770\) −0.928203 −0.0334501
\(771\) 0 0
\(772\) 8.26795 0.297570
\(773\) −35.0718 −1.26144 −0.630722 0.776009i \(-0.717241\pi\)
−0.630722 + 0.776009i \(0.717241\pi\)
\(774\) 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 \(-0.747179\pi\)
−0.700812 + 0.713346i \(0.747179\pi\)
\(788\) 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 \(-0.208420\pi\)
0.793188 + 0.608977i \(0.208420\pi\)
\(812\) 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 \(-0.427566\pi\)
0.225599 + 0.974220i \(0.427566\pi\)
\(822\) 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 \(-0.302337\pi\)
0.581830 + 0.813310i \(0.302337\pi\)
\(828\) 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 \(-0.421493\pi\)
0.244146 + 0.969739i \(0.421493\pi\)
\(840\) 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 \(-0.544706\pi\)
−0.139986 + 0.990153i \(0.544706\pi\)
\(854\) −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 \(-0.461307\pi\)
0.121258 + 0.992621i \(0.461307\pi\)
\(864\) 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 \(-0.554393\pi\)
−0.170050 + 0.985435i \(0.554393\pi\)
\(878\) −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 \(-0.451519\pi\)
0.151720 + 0.988423i \(0.451519\pi\)
\(930\) 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 \(-0.874454\pi\)
−0.923222 + 0.384267i \(0.874454\pi\)
\(942\) 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 \(-0.307908\pi\)
0.567507 + 0.823369i \(0.307908\pi\)
\(948\) 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 \(-0.401845\pi\)
0.303500 + 0.952831i \(0.401845\pi\)
\(968\) 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 \(-0.328522\pi\)
0.513034 + 0.858368i \(0.328522\pi\)
\(978\) 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 \(-0.607542\pi\)
−0.331463 + 0.943468i \(0.607542\pi\)
\(984\) 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.y.1.1 2
3.2 odd 2 1014.2.a.i.1.2 2
12.11 even 2 8112.2.a.bj.1.2 2
13.5 odd 4 3042.2.b.i.1351.2 4
13.6 odd 12 234.2.l.c.127.1 4
13.8 odd 4 3042.2.b.i.1351.3 4
13.11 odd 12 234.2.l.c.199.1 4
13.12 even 2 3042.2.a.p.1.2 2
39.2 even 12 1014.2.i.a.823.1 4
39.5 even 4 1014.2.b.e.337.3 4
39.8 even 4 1014.2.b.e.337.2 4
39.11 even 12 78.2.i.a.43.2 4
39.17 odd 6 1014.2.e.g.991.1 4
39.20 even 12 1014.2.i.a.361.1 4
39.23 odd 6 1014.2.e.g.529.1 4
39.29 odd 6 1014.2.e.i.529.2 4
39.32 even 12 78.2.i.a.49.2 yes 4
39.35 odd 6 1014.2.e.i.991.2 4
39.38 odd 2 1014.2.a.k.1.1 2
52.11 even 12 1872.2.by.h.433.2 4
52.19 even 12 1872.2.by.h.1297.1 4
156.11 odd 12 624.2.bv.e.433.1 4
156.71 odd 12 624.2.bv.e.49.2 4
156.155 even 2 8112.2.a.bp.1.1 2
195.32 odd 12 1950.2.y.g.49.1 4
195.89 even 12 1950.2.bc.d.901.1 4
195.128 odd 12 1950.2.y.g.199.1 4
195.149 even 12 1950.2.bc.d.751.1 4
195.167 odd 12 1950.2.y.b.199.2 4
195.188 odd 12 1950.2.y.b.49.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.i.a.43.2 4 39.11 even 12
78.2.i.a.49.2 yes 4 39.32 even 12
234.2.l.c.127.1 4 13.6 odd 12
234.2.l.c.199.1 4 13.11 odd 12
624.2.bv.e.49.2 4 156.71 odd 12
624.2.bv.e.433.1 4 156.11 odd 12
1014.2.a.i.1.2 2 3.2 odd 2
1014.2.a.k.1.1 2 39.38 odd 2
1014.2.b.e.337.2 4 39.8 even 4
1014.2.b.e.337.3 4 39.5 even 4
1014.2.e.g.529.1 4 39.23 odd 6
1014.2.e.g.991.1 4 39.17 odd 6
1014.2.e.i.529.2 4 39.29 odd 6
1014.2.e.i.991.2 4 39.35 odd 6
1014.2.i.a.361.1 4 39.20 even 12
1014.2.i.a.823.1 4 39.2 even 12
1872.2.by.h.433.2 4 52.11 even 12
1872.2.by.h.1297.1 4 52.19 even 12
1950.2.y.b.49.2 4 195.188 odd 12
1950.2.y.b.199.2 4 195.167 odd 12
1950.2.y.g.49.1 4 195.32 odd 12
1950.2.y.g.199.1 4 195.128 odd 12
1950.2.bc.d.751.1 4 195.149 even 12
1950.2.bc.d.901.1 4 195.89 even 12
3042.2.a.p.1.2 2 13.12 even 2
3042.2.a.y.1.1 2 1.1 even 1 trivial
3042.2.b.i.1351.2 4 13.5 odd 4
3042.2.b.i.1351.3 4 13.8 odd 4
8112.2.a.bj.1.2 2 12.11 even 2
8112.2.a.bp.1.1 2 156.155 even 2