Properties

Label 7728.2.a.bh.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
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] = [7728,2,Mod(1,7728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.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: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.37228 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +1.37228 q^{13} +1.37228 q^{15} -4.74456 q^{17} -4.00000 q^{19} +1.00000 q^{21} +1.00000 q^{23} -3.11684 q^{25} +1.00000 q^{27} +9.37228 q^{29} -6.74456 q^{31} -4.00000 q^{33} +1.37228 q^{35} +2.62772 q^{37} +1.37228 q^{39} -8.11684 q^{41} -6.11684 q^{43} +1.37228 q^{45} -4.62772 q^{47} +1.00000 q^{49} -4.74456 q^{51} +4.74456 q^{53} -5.48913 q^{55} -4.00000 q^{57} +2.74456 q^{59} -2.00000 q^{61} +1.00000 q^{63} +1.88316 q^{65} +4.00000 q^{67} +1.00000 q^{69} -14.7446 q^{71} -12.7446 q^{73} -3.11684 q^{75} -4.00000 q^{77} +13.4891 q^{79} +1.00000 q^{81} -4.00000 q^{83} -6.51087 q^{85} +9.37228 q^{87} +7.48913 q^{89} +1.37228 q^{91} -6.74456 q^{93} -5.48913 q^{95} -9.37228 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} - 8 q^{11} - 3 q^{13} - 3 q^{15} + 2 q^{17} - 8 q^{19} + 2 q^{21} + 2 q^{23} + 11 q^{25} + 2 q^{27} + 13 q^{29} - 2 q^{31} - 8 q^{33} - 3 q^{35} + 11 q^{37} - 3 q^{39} + q^{41} + 5 q^{43} - 3 q^{45} - 15 q^{47} + 2 q^{49} + 2 q^{51} - 2 q^{53} + 12 q^{55} - 8 q^{57} - 6 q^{59} - 4 q^{61} + 2 q^{63} + 21 q^{65} + 8 q^{67} + 2 q^{69} - 18 q^{71} - 14 q^{73} + 11 q^{75} - 8 q^{77} + 4 q^{79} + 2 q^{81} - 8 q^{83} - 36 q^{85} + 13 q^{87} - 8 q^{89} - 3 q^{91} - 2 q^{93} + 12 q^{95} - 13 q^{97} - 8 q^{99}+O(q^{100}) \) 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 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.37228 0.613703 0.306851 0.951757i \(-0.400725\pi\)
0.306851 + 0.951757i \(0.400725\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 1.37228 0.380602 0.190301 0.981726i \(-0.439054\pi\)
0.190301 + 0.981726i \(0.439054\pi\)
\(14\) 0 0
\(15\) 1.37228 0.354322
\(16\) 0 0
\(17\) −4.74456 −1.15073 −0.575363 0.817898i \(-0.695139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.11684 −0.623369
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.37228 1.74039 0.870194 0.492708i \(-0.163993\pi\)
0.870194 + 0.492708i \(0.163993\pi\)
\(30\) 0 0
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 1.37228 0.231958
\(36\) 0 0
\(37\) 2.62772 0.431994 0.215997 0.976394i \(-0.430700\pi\)
0.215997 + 0.976394i \(0.430700\pi\)
\(38\) 0 0
\(39\) 1.37228 0.219741
\(40\) 0 0
\(41\) −8.11684 −1.26764 −0.633819 0.773481i \(-0.718514\pi\)
−0.633819 + 0.773481i \(0.718514\pi\)
\(42\) 0 0
\(43\) −6.11684 −0.932810 −0.466405 0.884571i \(-0.654451\pi\)
−0.466405 + 0.884571i \(0.654451\pi\)
\(44\) 0 0
\(45\) 1.37228 0.204568
\(46\) 0 0
\(47\) −4.62772 −0.675022 −0.337511 0.941322i \(-0.609585\pi\)
−0.337511 + 0.941322i \(0.609585\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.74456 −0.664372
\(52\) 0 0
\(53\) 4.74456 0.651716 0.325858 0.945419i \(-0.394347\pi\)
0.325858 + 0.945419i \(0.394347\pi\)
\(54\) 0 0
\(55\) −5.48913 −0.740154
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 1.88316 0.233577
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −14.7446 −1.74986 −0.874929 0.484252i \(-0.839092\pi\)
−0.874929 + 0.484252i \(0.839092\pi\)
\(72\) 0 0
\(73\) −12.7446 −1.49164 −0.745819 0.666149i \(-0.767942\pi\)
−0.745819 + 0.666149i \(0.767942\pi\)
\(74\) 0 0
\(75\) −3.11684 −0.359902
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 13.4891 1.51765 0.758823 0.651297i \(-0.225775\pi\)
0.758823 + 0.651297i \(0.225775\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −6.51087 −0.706204
\(86\) 0 0
\(87\) 9.37228 1.00481
\(88\) 0 0
\(89\) 7.48913 0.793846 0.396923 0.917852i \(-0.370078\pi\)
0.396923 + 0.917852i \(0.370078\pi\)
\(90\) 0 0
\(91\) 1.37228 0.143854
\(92\) 0 0
\(93\) −6.74456 −0.699379
\(94\) 0 0
\(95\) −5.48913 −0.563172
\(96\) 0 0
\(97\) −9.37228 −0.951611 −0.475805 0.879551i \(-0.657843\pi\)
−0.475805 + 0.879551i \(0.657843\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −15.4891 −1.54123 −0.770613 0.637304i \(-0.780050\pi\)
−0.770613 + 0.637304i \(0.780050\pi\)
\(102\) 0 0
\(103\) 10.1168 0.996842 0.498421 0.866935i \(-0.333913\pi\)
0.498421 + 0.866935i \(0.333913\pi\)
\(104\) 0 0
\(105\) 1.37228 0.133921
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 8.11684 0.777453 0.388726 0.921353i \(-0.372915\pi\)
0.388726 + 0.921353i \(0.372915\pi\)
\(110\) 0 0
\(111\) 2.62772 0.249412
\(112\) 0 0
\(113\) −16.1168 −1.51615 −0.758073 0.652170i \(-0.773859\pi\)
−0.758073 + 0.652170i \(0.773859\pi\)
\(114\) 0 0
\(115\) 1.37228 0.127966
\(116\) 0 0
\(117\) 1.37228 0.126867
\(118\) 0 0
\(119\) −4.74456 −0.434933
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −8.11684 −0.731871
\(124\) 0 0
\(125\) −11.1386 −0.996266
\(126\) 0 0
\(127\) 11.3723 1.00913 0.504563 0.863375i \(-0.331653\pi\)
0.504563 + 0.863375i \(0.331653\pi\)
\(128\) 0 0
\(129\) −6.11684 −0.538558
\(130\) 0 0
\(131\) −18.7446 −1.63772 −0.818860 0.573993i \(-0.805394\pi\)
−0.818860 + 0.573993i \(0.805394\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) 1.37228 0.118107
\(136\) 0 0
\(137\) −8.11684 −0.693469 −0.346734 0.937963i \(-0.612709\pi\)
−0.346734 + 0.937963i \(0.612709\pi\)
\(138\) 0 0
\(139\) −8.62772 −0.731794 −0.365897 0.930655i \(-0.619238\pi\)
−0.365897 + 0.930655i \(0.619238\pi\)
\(140\) 0 0
\(141\) −4.62772 −0.389724
\(142\) 0 0
\(143\) −5.48913 −0.459024
\(144\) 0 0
\(145\) 12.8614 1.06808
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 18.2337 1.49376 0.746881 0.664958i \(-0.231550\pi\)
0.746881 + 0.664958i \(0.231550\pi\)
\(150\) 0 0
\(151\) 3.37228 0.274432 0.137216 0.990541i \(-0.456184\pi\)
0.137216 + 0.990541i \(0.456184\pi\)
\(152\) 0 0
\(153\) −4.74456 −0.383575
\(154\) 0 0
\(155\) −9.25544 −0.743415
\(156\) 0 0
\(157\) −11.2554 −0.898282 −0.449141 0.893461i \(-0.648270\pi\)
−0.449141 + 0.893461i \(0.648270\pi\)
\(158\) 0 0
\(159\) 4.74456 0.376268
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −5.48913 −0.427328
\(166\) 0 0
\(167\) 5.48913 0.424761 0.212381 0.977187i \(-0.431878\pi\)
0.212381 + 0.977187i \(0.431878\pi\)
\(168\) 0 0
\(169\) −11.1168 −0.855142
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −7.48913 −0.569388 −0.284694 0.958618i \(-0.591892\pi\)
−0.284694 + 0.958618i \(0.591892\pi\)
\(174\) 0 0
\(175\) −3.11684 −0.235611
\(176\) 0 0
\(177\) 2.74456 0.206294
\(178\) 0 0
\(179\) 20.8614 1.55925 0.779627 0.626244i \(-0.215408\pi\)
0.779627 + 0.626244i \(0.215408\pi\)
\(180\) 0 0
\(181\) −20.9783 −1.55930 −0.779651 0.626215i \(-0.784603\pi\)
−0.779651 + 0.626215i \(0.784603\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 3.60597 0.265116
\(186\) 0 0
\(187\) 18.9783 1.38783
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −2.51087 −0.181681 −0.0908403 0.995865i \(-0.528955\pi\)
−0.0908403 + 0.995865i \(0.528955\pi\)
\(192\) 0 0
\(193\) −0.116844 −0.00841061 −0.00420531 0.999991i \(-0.501339\pi\)
−0.00420531 + 0.999991i \(0.501339\pi\)
\(194\) 0 0
\(195\) 1.88316 0.134856
\(196\) 0 0
\(197\) −21.3723 −1.52271 −0.761356 0.648334i \(-0.775466\pi\)
−0.761356 + 0.648334i \(0.775466\pi\)
\(198\) 0 0
\(199\) 16.8614 1.19527 0.597637 0.801767i \(-0.296107\pi\)
0.597637 + 0.801767i \(0.296107\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 9.37228 0.657805
\(204\) 0 0
\(205\) −11.1386 −0.777953
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) −14.7446 −1.01028
\(214\) 0 0
\(215\) −8.39403 −0.572468
\(216\) 0 0
\(217\) −6.74456 −0.457851
\(218\) 0 0
\(219\) −12.7446 −0.861198
\(220\) 0 0
\(221\) −6.51087 −0.437969
\(222\) 0 0
\(223\) 9.25544 0.619790 0.309895 0.950771i \(-0.399706\pi\)
0.309895 + 0.950771i \(0.399706\pi\)
\(224\) 0 0
\(225\) −3.11684 −0.207790
\(226\) 0 0
\(227\) −6.11684 −0.405989 −0.202995 0.979180i \(-0.565067\pi\)
−0.202995 + 0.979180i \(0.565067\pi\)
\(228\) 0 0
\(229\) −0.744563 −0.0492021 −0.0246010 0.999697i \(-0.507832\pi\)
−0.0246010 + 0.999697i \(0.507832\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 23.4891 1.53882 0.769412 0.638753i \(-0.220549\pi\)
0.769412 + 0.638753i \(0.220549\pi\)
\(234\) 0 0
\(235\) −6.35053 −0.414263
\(236\) 0 0
\(237\) 13.4891 0.876213
\(238\) 0 0
\(239\) −13.4891 −0.872539 −0.436269 0.899816i \(-0.643701\pi\)
−0.436269 + 0.899816i \(0.643701\pi\)
\(240\) 0 0
\(241\) −2.62772 −0.169266 −0.0846331 0.996412i \(-0.526972\pi\)
−0.0846331 + 0.996412i \(0.526972\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.37228 0.0876718
\(246\) 0 0
\(247\) −5.48913 −0.349265
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 1.88316 0.118864 0.0594319 0.998232i \(-0.481071\pi\)
0.0594319 + 0.998232i \(0.481071\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) −6.51087 −0.407727
\(256\) 0 0
\(257\) 28.9783 1.80761 0.903807 0.427941i \(-0.140761\pi\)
0.903807 + 0.427941i \(0.140761\pi\)
\(258\) 0 0
\(259\) 2.62772 0.163278
\(260\) 0 0
\(261\) 9.37228 0.580130
\(262\) 0 0
\(263\) −10.1168 −0.623831 −0.311916 0.950110i \(-0.600971\pi\)
−0.311916 + 0.950110i \(0.600971\pi\)
\(264\) 0 0
\(265\) 6.51087 0.399960
\(266\) 0 0
\(267\) 7.48913 0.458327
\(268\) 0 0
\(269\) −20.9783 −1.27907 −0.639533 0.768763i \(-0.720872\pi\)
−0.639533 + 0.768763i \(0.720872\pi\)
\(270\) 0 0
\(271\) 26.9783 1.63881 0.819406 0.573214i \(-0.194303\pi\)
0.819406 + 0.573214i \(0.194303\pi\)
\(272\) 0 0
\(273\) 1.37228 0.0830542
\(274\) 0 0
\(275\) 12.4674 0.751811
\(276\) 0 0
\(277\) 28.7446 1.72709 0.863547 0.504269i \(-0.168238\pi\)
0.863547 + 0.504269i \(0.168238\pi\)
\(278\) 0 0
\(279\) −6.74456 −0.403786
\(280\) 0 0
\(281\) −17.3723 −1.03634 −0.518172 0.855277i \(-0.673387\pi\)
−0.518172 + 0.855277i \(0.673387\pi\)
\(282\) 0 0
\(283\) 13.2554 0.787954 0.393977 0.919120i \(-0.371099\pi\)
0.393977 + 0.919120i \(0.371099\pi\)
\(284\) 0 0
\(285\) −5.48913 −0.325148
\(286\) 0 0
\(287\) −8.11684 −0.479122
\(288\) 0 0
\(289\) 5.51087 0.324169
\(290\) 0 0
\(291\) −9.37228 −0.549413
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 3.76631 0.219283
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 1.37228 0.0793611
\(300\) 0 0
\(301\) −6.11684 −0.352569
\(302\) 0 0
\(303\) −15.4891 −0.889827
\(304\) 0 0
\(305\) −2.74456 −0.157153
\(306\) 0 0
\(307\) −7.37228 −0.420758 −0.210379 0.977620i \(-0.567470\pi\)
−0.210379 + 0.977620i \(0.567470\pi\)
\(308\) 0 0
\(309\) 10.1168 0.575527
\(310\) 0 0
\(311\) −5.48913 −0.311260 −0.155630 0.987815i \(-0.549741\pi\)
−0.155630 + 0.987815i \(0.549741\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 1.37228 0.0773193
\(316\) 0 0
\(317\) 16.1168 0.905212 0.452606 0.891711i \(-0.350494\pi\)
0.452606 + 0.891711i \(0.350494\pi\)
\(318\) 0 0
\(319\) −37.4891 −2.09899
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 18.9783 1.05598
\(324\) 0 0
\(325\) −4.27719 −0.237256
\(326\) 0 0
\(327\) 8.11684 0.448862
\(328\) 0 0
\(329\) −4.62772 −0.255134
\(330\) 0 0
\(331\) −9.48913 −0.521569 −0.260785 0.965397i \(-0.583981\pi\)
−0.260785 + 0.965397i \(0.583981\pi\)
\(332\) 0 0
\(333\) 2.62772 0.143998
\(334\) 0 0
\(335\) 5.48913 0.299903
\(336\) 0 0
\(337\) 22.2337 1.21115 0.605573 0.795790i \(-0.292944\pi\)
0.605573 + 0.795790i \(0.292944\pi\)
\(338\) 0 0
\(339\) −16.1168 −0.875347
\(340\) 0 0
\(341\) 26.9783 1.46095
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.37228 0.0738811
\(346\) 0 0
\(347\) −16.6277 −0.892623 −0.446311 0.894878i \(-0.647263\pi\)
−0.446311 + 0.894878i \(0.647263\pi\)
\(348\) 0 0
\(349\) 35.4891 1.89969 0.949845 0.312722i \(-0.101241\pi\)
0.949845 + 0.312722i \(0.101241\pi\)
\(350\) 0 0
\(351\) 1.37228 0.0732470
\(352\) 0 0
\(353\) 36.1168 1.92231 0.961153 0.276017i \(-0.0890145\pi\)
0.961153 + 0.276017i \(0.0890145\pi\)
\(354\) 0 0
\(355\) −20.2337 −1.07389
\(356\) 0 0
\(357\) −4.74456 −0.251109
\(358\) 0 0
\(359\) −19.3723 −1.02243 −0.511215 0.859453i \(-0.670804\pi\)
−0.511215 + 0.859453i \(0.670804\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) −17.4891 −0.915423
\(366\) 0 0
\(367\) 11.3723 0.593628 0.296814 0.954935i \(-0.404076\pi\)
0.296814 + 0.954935i \(0.404076\pi\)
\(368\) 0 0
\(369\) −8.11684 −0.422546
\(370\) 0 0
\(371\) 4.74456 0.246325
\(372\) 0 0
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) 0 0
\(375\) −11.1386 −0.575194
\(376\) 0 0
\(377\) 12.8614 0.662396
\(378\) 0 0
\(379\) −15.3723 −0.789621 −0.394811 0.918763i \(-0.629190\pi\)
−0.394811 + 0.918763i \(0.629190\pi\)
\(380\) 0 0
\(381\) 11.3723 0.582620
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) −5.48913 −0.279752
\(386\) 0 0
\(387\) −6.11684 −0.310937
\(388\) 0 0
\(389\) 11.4891 0.582522 0.291261 0.956644i \(-0.405925\pi\)
0.291261 + 0.956644i \(0.405925\pi\)
\(390\) 0 0
\(391\) −4.74456 −0.239943
\(392\) 0 0
\(393\) −18.7446 −0.945538
\(394\) 0 0
\(395\) 18.5109 0.931383
\(396\) 0 0
\(397\) −10.0000 −0.501886 −0.250943 0.968002i \(-0.580741\pi\)
−0.250943 + 0.968002i \(0.580741\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −24.9783 −1.24735 −0.623677 0.781682i \(-0.714362\pi\)
−0.623677 + 0.781682i \(0.714362\pi\)
\(402\) 0 0
\(403\) −9.25544 −0.461046
\(404\) 0 0
\(405\) 1.37228 0.0681892
\(406\) 0 0
\(407\) −10.5109 −0.521005
\(408\) 0 0
\(409\) 0.744563 0.0368163 0.0184081 0.999831i \(-0.494140\pi\)
0.0184081 + 0.999831i \(0.494140\pi\)
\(410\) 0 0
\(411\) −8.11684 −0.400374
\(412\) 0 0
\(413\) 2.74456 0.135051
\(414\) 0 0
\(415\) −5.48913 −0.269451
\(416\) 0 0
\(417\) −8.62772 −0.422501
\(418\) 0 0
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) −26.8614 −1.30914 −0.654572 0.755999i \(-0.727151\pi\)
−0.654572 + 0.755999i \(0.727151\pi\)
\(422\) 0 0
\(423\) −4.62772 −0.225007
\(424\) 0 0
\(425\) 14.7881 0.717326
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −5.48913 −0.265017
\(430\) 0 0
\(431\) 8.86141 0.426839 0.213419 0.976961i \(-0.431540\pi\)
0.213419 + 0.976961i \(0.431540\pi\)
\(432\) 0 0
\(433\) 10.8614 0.521966 0.260983 0.965343i \(-0.415953\pi\)
0.260983 + 0.965343i \(0.415953\pi\)
\(434\) 0 0
\(435\) 12.8614 0.616657
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −30.7446 −1.46736 −0.733679 0.679496i \(-0.762198\pi\)
−0.733679 + 0.679496i \(0.762198\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 8.62772 0.409915 0.204958 0.978771i \(-0.434294\pi\)
0.204958 + 0.978771i \(0.434294\pi\)
\(444\) 0 0
\(445\) 10.2772 0.487185
\(446\) 0 0
\(447\) 18.2337 0.862424
\(448\) 0 0
\(449\) 28.9783 1.36757 0.683784 0.729684i \(-0.260333\pi\)
0.683784 + 0.729684i \(0.260333\pi\)
\(450\) 0 0
\(451\) 32.4674 1.52883
\(452\) 0 0
\(453\) 3.37228 0.158444
\(454\) 0 0
\(455\) 1.88316 0.0882837
\(456\) 0 0
\(457\) −10.2337 −0.478712 −0.239356 0.970932i \(-0.576936\pi\)
−0.239356 + 0.970932i \(0.576936\pi\)
\(458\) 0 0
\(459\) −4.74456 −0.221457
\(460\) 0 0
\(461\) −12.5109 −0.582690 −0.291345 0.956618i \(-0.594103\pi\)
−0.291345 + 0.956618i \(0.594103\pi\)
\(462\) 0 0
\(463\) 2.11684 0.0983781 0.0491890 0.998789i \(-0.484336\pi\)
0.0491890 + 0.998789i \(0.484336\pi\)
\(464\) 0 0
\(465\) −9.25544 −0.429211
\(466\) 0 0
\(467\) −28.8614 −1.33555 −0.667773 0.744365i \(-0.732752\pi\)
−0.667773 + 0.744365i \(0.732752\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −11.2554 −0.518623
\(472\) 0 0
\(473\) 24.4674 1.12501
\(474\) 0 0
\(475\) 12.4674 0.572042
\(476\) 0 0
\(477\) 4.74456 0.217239
\(478\) 0 0
\(479\) 6.74456 0.308167 0.154083 0.988058i \(-0.450758\pi\)
0.154083 + 0.988058i \(0.450758\pi\)
\(480\) 0 0
\(481\) 3.60597 0.164418
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −12.8614 −0.584006
\(486\) 0 0
\(487\) −5.88316 −0.266591 −0.133296 0.991076i \(-0.542556\pi\)
−0.133296 + 0.991076i \(0.542556\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 30.9783 1.39803 0.699014 0.715108i \(-0.253622\pi\)
0.699014 + 0.715108i \(0.253622\pi\)
\(492\) 0 0
\(493\) −44.4674 −2.00271
\(494\) 0 0
\(495\) −5.48913 −0.246718
\(496\) 0 0
\(497\) −14.7446 −0.661384
\(498\) 0 0
\(499\) −21.7228 −0.972447 −0.486223 0.873835i \(-0.661626\pi\)
−0.486223 + 0.873835i \(0.661626\pi\)
\(500\) 0 0
\(501\) 5.48913 0.245236
\(502\) 0 0
\(503\) 1.25544 0.0559772 0.0279886 0.999608i \(-0.491090\pi\)
0.0279886 + 0.999608i \(0.491090\pi\)
\(504\) 0 0
\(505\) −21.2554 −0.945855
\(506\) 0 0
\(507\) −11.1168 −0.493716
\(508\) 0 0
\(509\) 26.2337 1.16279 0.581394 0.813622i \(-0.302508\pi\)
0.581394 + 0.813622i \(0.302508\pi\)
\(510\) 0 0
\(511\) −12.7446 −0.563786
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 13.8832 0.611765
\(516\) 0 0
\(517\) 18.5109 0.814107
\(518\) 0 0
\(519\) −7.48913 −0.328736
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) 40.2337 1.75930 0.879648 0.475625i \(-0.157778\pi\)
0.879648 + 0.475625i \(0.157778\pi\)
\(524\) 0 0
\(525\) −3.11684 −0.136030
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 2.74456 0.119104
\(532\) 0 0
\(533\) −11.1386 −0.482466
\(534\) 0 0
\(535\) −5.48913 −0.237316
\(536\) 0 0
\(537\) 20.8614 0.900236
\(538\) 0 0
\(539\) −4.00000 −0.172292
\(540\) 0 0
\(541\) 18.2337 0.783927 0.391964 0.919981i \(-0.371796\pi\)
0.391964 + 0.919981i \(0.371796\pi\)
\(542\) 0 0
\(543\) −20.9783 −0.900263
\(544\) 0 0
\(545\) 11.1386 0.477125
\(546\) 0 0
\(547\) 21.2554 0.908817 0.454408 0.890793i \(-0.349851\pi\)
0.454408 + 0.890793i \(0.349851\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −37.4891 −1.59709
\(552\) 0 0
\(553\) 13.4891 0.573616
\(554\) 0 0
\(555\) 3.60597 0.153065
\(556\) 0 0
\(557\) −14.2337 −0.603101 −0.301550 0.953450i \(-0.597504\pi\)
−0.301550 + 0.953450i \(0.597504\pi\)
\(558\) 0 0
\(559\) −8.39403 −0.355030
\(560\) 0 0
\(561\) 18.9783 0.801262
\(562\) 0 0
\(563\) −24.6277 −1.03793 −0.518967 0.854794i \(-0.673683\pi\)
−0.518967 + 0.854794i \(0.673683\pi\)
\(564\) 0 0
\(565\) −22.1168 −0.930463
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −19.8832 −0.833545 −0.416773 0.909011i \(-0.636839\pi\)
−0.416773 + 0.909011i \(0.636839\pi\)
\(570\) 0 0
\(571\) 22.9783 0.961610 0.480805 0.876828i \(-0.340345\pi\)
0.480805 + 0.876828i \(0.340345\pi\)
\(572\) 0 0
\(573\) −2.51087 −0.104893
\(574\) 0 0
\(575\) −3.11684 −0.129981
\(576\) 0 0
\(577\) 42.4674 1.76794 0.883970 0.467544i \(-0.154861\pi\)
0.883970 + 0.467544i \(0.154861\pi\)
\(578\) 0 0
\(579\) −0.116844 −0.00485587
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −18.9783 −0.785999
\(584\) 0 0
\(585\) 1.88316 0.0778589
\(586\) 0 0
\(587\) −6.51087 −0.268733 −0.134366 0.990932i \(-0.542900\pi\)
−0.134366 + 0.990932i \(0.542900\pi\)
\(588\) 0 0
\(589\) 26.9783 1.11162
\(590\) 0 0
\(591\) −21.3723 −0.879138
\(592\) 0 0
\(593\) −2.62772 −0.107907 −0.0539537 0.998543i \(-0.517182\pi\)
−0.0539537 + 0.998543i \(0.517182\pi\)
\(594\) 0 0
\(595\) −6.51087 −0.266920
\(596\) 0 0
\(597\) 16.8614 0.690091
\(598\) 0 0
\(599\) 48.4674 1.98032 0.990162 0.139928i \(-0.0446872\pi\)
0.990162 + 0.139928i \(0.0446872\pi\)
\(600\) 0 0
\(601\) 30.2337 1.23326 0.616629 0.787254i \(-0.288498\pi\)
0.616629 + 0.787254i \(0.288498\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 6.86141 0.278956
\(606\) 0 0
\(607\) −22.7446 −0.923173 −0.461587 0.887095i \(-0.652720\pi\)
−0.461587 + 0.887095i \(0.652720\pi\)
\(608\) 0 0
\(609\) 9.37228 0.379784
\(610\) 0 0
\(611\) −6.35053 −0.256915
\(612\) 0 0
\(613\) 27.8832 1.12619 0.563095 0.826392i \(-0.309610\pi\)
0.563095 + 0.826392i \(0.309610\pi\)
\(614\) 0 0
\(615\) −11.1386 −0.449151
\(616\) 0 0
\(617\) −35.4891 −1.42874 −0.714369 0.699769i \(-0.753286\pi\)
−0.714369 + 0.699769i \(0.753286\pi\)
\(618\) 0 0
\(619\) −34.7446 −1.39650 −0.698251 0.715853i \(-0.746038\pi\)
−0.698251 + 0.715853i \(0.746038\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 7.48913 0.300045
\(624\) 0 0
\(625\) 0.298936 0.0119574
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) −12.4674 −0.497107
\(630\) 0 0
\(631\) 34.9783 1.39246 0.696231 0.717818i \(-0.254859\pi\)
0.696231 + 0.717818i \(0.254859\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 15.6060 0.619304
\(636\) 0 0
\(637\) 1.37228 0.0543718
\(638\) 0 0
\(639\) −14.7446 −0.583286
\(640\) 0 0
\(641\) −20.3505 −0.803798 −0.401899 0.915684i \(-0.631650\pi\)
−0.401899 + 0.915684i \(0.631650\pi\)
\(642\) 0 0
\(643\) 7.76631 0.306273 0.153137 0.988205i \(-0.451063\pi\)
0.153137 + 0.988205i \(0.451063\pi\)
\(644\) 0 0
\(645\) −8.39403 −0.330515
\(646\) 0 0
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) −10.9783 −0.430934
\(650\) 0 0
\(651\) −6.74456 −0.264340
\(652\) 0 0
\(653\) −31.0951 −1.21685 −0.608423 0.793613i \(-0.708197\pi\)
−0.608423 + 0.793613i \(0.708197\pi\)
\(654\) 0 0
\(655\) −25.7228 −1.00507
\(656\) 0 0
\(657\) −12.7446 −0.497213
\(658\) 0 0
\(659\) −22.9783 −0.895106 −0.447553 0.894258i \(-0.647704\pi\)
−0.447553 + 0.894258i \(0.647704\pi\)
\(660\) 0 0
\(661\) 3.48913 0.135711 0.0678556 0.997695i \(-0.478384\pi\)
0.0678556 + 0.997695i \(0.478384\pi\)
\(662\) 0 0
\(663\) −6.51087 −0.252861
\(664\) 0 0
\(665\) −5.48913 −0.212859
\(666\) 0 0
\(667\) 9.37228 0.362896
\(668\) 0 0
\(669\) 9.25544 0.357836
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −13.6060 −0.524472 −0.262236 0.965004i \(-0.584460\pi\)
−0.262236 + 0.965004i \(0.584460\pi\)
\(674\) 0 0
\(675\) −3.11684 −0.119967
\(676\) 0 0
\(677\) 32.9783 1.26746 0.633729 0.773555i \(-0.281524\pi\)
0.633729 + 0.773555i \(0.281524\pi\)
\(678\) 0 0
\(679\) −9.37228 −0.359675
\(680\) 0 0
\(681\) −6.11684 −0.234398
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −11.1386 −0.425584
\(686\) 0 0
\(687\) −0.744563 −0.0284068
\(688\) 0 0
\(689\) 6.51087 0.248045
\(690\) 0 0
\(691\) −31.8397 −1.21124 −0.605619 0.795755i \(-0.707074\pi\)
−0.605619 + 0.795755i \(0.707074\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −11.8397 −0.449104
\(696\) 0 0
\(697\) 38.5109 1.45870
\(698\) 0 0
\(699\) 23.4891 0.888440
\(700\) 0 0
\(701\) −5.76631 −0.217791 −0.108895 0.994053i \(-0.534731\pi\)
−0.108895 + 0.994053i \(0.534731\pi\)
\(702\) 0 0
\(703\) −10.5109 −0.396425
\(704\) 0 0
\(705\) −6.35053 −0.239175
\(706\) 0 0
\(707\) −15.4891 −0.582529
\(708\) 0 0
\(709\) −4.51087 −0.169409 −0.0847047 0.996406i \(-0.526995\pi\)
−0.0847047 + 0.996406i \(0.526995\pi\)
\(710\) 0 0
\(711\) 13.4891 0.505882
\(712\) 0 0
\(713\) −6.74456 −0.252586
\(714\) 0 0
\(715\) −7.53262 −0.281704
\(716\) 0 0
\(717\) −13.4891 −0.503761
\(718\) 0 0
\(719\) 11.3723 0.424115 0.212057 0.977257i \(-0.431984\pi\)
0.212057 + 0.977257i \(0.431984\pi\)
\(720\) 0 0
\(721\) 10.1168 0.376771
\(722\) 0 0
\(723\) −2.62772 −0.0977259
\(724\) 0 0
\(725\) −29.2119 −1.08490
\(726\) 0 0
\(727\) −10.5109 −0.389827 −0.194913 0.980820i \(-0.562443\pi\)
−0.194913 + 0.980820i \(0.562443\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 29.0217 1.07341
\(732\) 0 0
\(733\) 18.2337 0.673477 0.336738 0.941598i \(-0.390676\pi\)
0.336738 + 0.941598i \(0.390676\pi\)
\(734\) 0 0
\(735\) 1.37228 0.0506174
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 5.25544 0.193324 0.0966622 0.995317i \(-0.469183\pi\)
0.0966622 + 0.995317i \(0.469183\pi\)
\(740\) 0 0
\(741\) −5.48913 −0.201648
\(742\) 0 0
\(743\) −45.9565 −1.68598 −0.842990 0.537929i \(-0.819207\pi\)
−0.842990 + 0.537929i \(0.819207\pi\)
\(744\) 0 0
\(745\) 25.0217 0.916726
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −13.4891 −0.492225 −0.246113 0.969241i \(-0.579153\pi\)
−0.246113 + 0.969241i \(0.579153\pi\)
\(752\) 0 0
\(753\) 1.88316 0.0686260
\(754\) 0 0
\(755\) 4.62772 0.168420
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 4.97825 0.180461 0.0902307 0.995921i \(-0.471240\pi\)
0.0902307 + 0.995921i \(0.471240\pi\)
\(762\) 0 0
\(763\) 8.11684 0.293849
\(764\) 0 0
\(765\) −6.51087 −0.235401
\(766\) 0 0
\(767\) 3.76631 0.135994
\(768\) 0 0
\(769\) −27.8832 −1.00549 −0.502746 0.864434i \(-0.667677\pi\)
−0.502746 + 0.864434i \(0.667677\pi\)
\(770\) 0 0
\(771\) 28.9783 1.04363
\(772\) 0 0
\(773\) −41.6060 −1.49646 −0.748231 0.663438i \(-0.769097\pi\)
−0.748231 + 0.663438i \(0.769097\pi\)
\(774\) 0 0
\(775\) 21.0217 0.755124
\(776\) 0 0
\(777\) 2.62772 0.0942689
\(778\) 0 0
\(779\) 32.4674 1.16326
\(780\) 0 0
\(781\) 58.9783 2.11041
\(782\) 0 0
\(783\) 9.37228 0.334938
\(784\) 0 0
\(785\) −15.4456 −0.551278
\(786\) 0 0
\(787\) −6.51087 −0.232088 −0.116044 0.993244i \(-0.537021\pi\)
−0.116044 + 0.993244i \(0.537021\pi\)
\(788\) 0 0
\(789\) −10.1168 −0.360169
\(790\) 0 0
\(791\) −16.1168 −0.573049
\(792\) 0 0
\(793\) −2.74456 −0.0974623
\(794\) 0 0
\(795\) 6.51087 0.230917
\(796\) 0 0
\(797\) 6.86141 0.243043 0.121522 0.992589i \(-0.461223\pi\)
0.121522 + 0.992589i \(0.461223\pi\)
\(798\) 0 0
\(799\) 21.9565 0.776765
\(800\) 0 0
\(801\) 7.48913 0.264615
\(802\) 0 0
\(803\) 50.9783 1.79898
\(804\) 0 0
\(805\) 1.37228 0.0483666
\(806\) 0 0
\(807\) −20.9783 −0.738469
\(808\) 0 0
\(809\) −12.7446 −0.448075 −0.224037 0.974581i \(-0.571924\pi\)
−0.224037 + 0.974581i \(0.571924\pi\)
\(810\) 0 0
\(811\) −19.6060 −0.688459 −0.344229 0.938886i \(-0.611860\pi\)
−0.344229 + 0.938886i \(0.611860\pi\)
\(812\) 0 0
\(813\) 26.9783 0.946169
\(814\) 0 0
\(815\) −5.48913 −0.192276
\(816\) 0 0
\(817\) 24.4674 0.856005
\(818\) 0 0
\(819\) 1.37228 0.0479514
\(820\) 0 0
\(821\) −23.4891 −0.819776 −0.409888 0.912136i \(-0.634432\pi\)
−0.409888 + 0.912136i \(0.634432\pi\)
\(822\) 0 0
\(823\) 46.3505 1.61568 0.807839 0.589403i \(-0.200637\pi\)
0.807839 + 0.589403i \(0.200637\pi\)
\(824\) 0 0
\(825\) 12.4674 0.434058
\(826\) 0 0
\(827\) 16.2337 0.564501 0.282250 0.959341i \(-0.408919\pi\)
0.282250 + 0.959341i \(0.408919\pi\)
\(828\) 0 0
\(829\) 24.5109 0.851298 0.425649 0.904888i \(-0.360046\pi\)
0.425649 + 0.904888i \(0.360046\pi\)
\(830\) 0 0
\(831\) 28.7446 0.997138
\(832\) 0 0
\(833\) −4.74456 −0.164389
\(834\) 0 0
\(835\) 7.53262 0.260677
\(836\) 0 0
\(837\) −6.74456 −0.233126
\(838\) 0 0
\(839\) −33.2554 −1.14811 −0.574053 0.818818i \(-0.694630\pi\)
−0.574053 + 0.818818i \(0.694630\pi\)
\(840\) 0 0
\(841\) 58.8397 2.02895
\(842\) 0 0
\(843\) −17.3723 −0.598333
\(844\) 0 0
\(845\) −15.2554 −0.524803
\(846\) 0 0
\(847\) 5.00000 0.171802
\(848\) 0 0
\(849\) 13.2554 0.454925
\(850\) 0 0
\(851\) 2.62772 0.0900770
\(852\) 0 0
\(853\) 40.5842 1.38958 0.694789 0.719214i \(-0.255498\pi\)
0.694789 + 0.719214i \(0.255498\pi\)
\(854\) 0 0
\(855\) −5.48913 −0.187724
\(856\) 0 0
\(857\) −24.1168 −0.823816 −0.411908 0.911226i \(-0.635137\pi\)
−0.411908 + 0.911226i \(0.635137\pi\)
\(858\) 0 0
\(859\) −22.1168 −0.754617 −0.377308 0.926088i \(-0.623150\pi\)
−0.377308 + 0.926088i \(0.623150\pi\)
\(860\) 0 0
\(861\) −8.11684 −0.276621
\(862\) 0 0
\(863\) −13.4891 −0.459175 −0.229588 0.973288i \(-0.573738\pi\)
−0.229588 + 0.973288i \(0.573738\pi\)
\(864\) 0 0
\(865\) −10.2772 −0.349435
\(866\) 0 0
\(867\) 5.51087 0.187159
\(868\) 0 0
\(869\) −53.9565 −1.83035
\(870\) 0 0
\(871\) 5.48913 0.185992
\(872\) 0 0
\(873\) −9.37228 −0.317204
\(874\) 0 0
\(875\) −11.1386 −0.376553
\(876\) 0 0
\(877\) −22.2337 −0.750778 −0.375389 0.926867i \(-0.622491\pi\)
−0.375389 + 0.926867i \(0.622491\pi\)
\(878\) 0 0
\(879\) −10.0000 −0.337292
\(880\) 0 0
\(881\) −40.9783 −1.38059 −0.690296 0.723527i \(-0.742520\pi\)
−0.690296 + 0.723527i \(0.742520\pi\)
\(882\) 0 0
\(883\) 21.2554 0.715302 0.357651 0.933855i \(-0.383578\pi\)
0.357651 + 0.933855i \(0.383578\pi\)
\(884\) 0 0
\(885\) 3.76631 0.126603
\(886\) 0 0
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) 11.3723 0.381414
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 18.5109 0.619443
\(894\) 0 0
\(895\) 28.6277 0.956919
\(896\) 0 0
\(897\) 1.37228 0.0458191
\(898\) 0 0
\(899\) −63.2119 −2.10824
\(900\) 0 0
\(901\) −22.5109 −0.749946
\(902\) 0 0
\(903\) −6.11684 −0.203556
\(904\) 0 0
\(905\) −28.7881 −0.956948
\(906\) 0 0
\(907\) 41.0951 1.36454 0.682270 0.731100i \(-0.260993\pi\)
0.682270 + 0.731100i \(0.260993\pi\)
\(908\) 0 0
\(909\) −15.4891 −0.513742
\(910\) 0 0
\(911\) −49.3288 −1.63434 −0.817168 0.576400i \(-0.804457\pi\)
−0.817168 + 0.576400i \(0.804457\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 0 0
\(915\) −2.74456 −0.0907324
\(916\) 0 0
\(917\) −18.7446 −0.619000
\(918\) 0 0
\(919\) 45.9565 1.51597 0.757983 0.652275i \(-0.226185\pi\)
0.757983 + 0.652275i \(0.226185\pi\)
\(920\) 0 0
\(921\) −7.37228 −0.242925
\(922\) 0 0
\(923\) −20.2337 −0.666000
\(924\) 0 0
\(925\) −8.19019 −0.269292
\(926\) 0 0
\(927\) 10.1168 0.332281
\(928\) 0 0
\(929\) 15.8832 0.521109 0.260555 0.965459i \(-0.416095\pi\)
0.260555 + 0.965459i \(0.416095\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 0 0
\(933\) −5.48913 −0.179706
\(934\) 0 0
\(935\) 26.0435 0.851713
\(936\) 0 0
\(937\) −33.3723 −1.09022 −0.545112 0.838363i \(-0.683513\pi\)
−0.545112 + 0.838363i \(0.683513\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −54.6277 −1.78081 −0.890406 0.455166i \(-0.849580\pi\)
−0.890406 + 0.455166i \(0.849580\pi\)
\(942\) 0 0
\(943\) −8.11684 −0.264321
\(944\) 0 0
\(945\) 1.37228 0.0446403
\(946\) 0 0
\(947\) 5.64947 0.183583 0.0917915 0.995778i \(-0.470741\pi\)
0.0917915 + 0.995778i \(0.470741\pi\)
\(948\) 0 0
\(949\) −17.4891 −0.567721
\(950\) 0 0
\(951\) 16.1168 0.522624
\(952\) 0 0
\(953\) 23.4891 0.760887 0.380444 0.924804i \(-0.375771\pi\)
0.380444 + 0.924804i \(0.375771\pi\)
\(954\) 0 0
\(955\) −3.44563 −0.111498
\(956\) 0 0
\(957\) −37.4891 −1.21185
\(958\) 0 0
\(959\) −8.11684 −0.262107
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −0.160343 −0.00516162
\(966\) 0 0
\(967\) 53.4891 1.72009 0.860047 0.510215i \(-0.170434\pi\)
0.860047 + 0.510215i \(0.170434\pi\)
\(968\) 0 0
\(969\) 18.9783 0.609669
\(970\) 0 0
\(971\) 28.4674 0.913562 0.456781 0.889579i \(-0.349002\pi\)
0.456781 + 0.889579i \(0.349002\pi\)
\(972\) 0 0
\(973\) −8.62772 −0.276592
\(974\) 0 0
\(975\) −4.27719 −0.136980
\(976\) 0 0
\(977\) 8.35053 0.267157 0.133579 0.991038i \(-0.457353\pi\)
0.133579 + 0.991038i \(0.457353\pi\)
\(978\) 0 0
\(979\) −29.9565 −0.957414
\(980\) 0 0
\(981\) 8.11684 0.259151
\(982\) 0 0
\(983\) 3.76631 0.120127 0.0600633 0.998195i \(-0.480870\pi\)
0.0600633 + 0.998195i \(0.480870\pi\)
\(984\) 0 0
\(985\) −29.3288 −0.934493
\(986\) 0 0
\(987\) −4.62772 −0.147302
\(988\) 0 0
\(989\) −6.11684 −0.194504
\(990\) 0 0
\(991\) 13.4891 0.428496 0.214248 0.976779i \(-0.431270\pi\)
0.214248 + 0.976779i \(0.431270\pi\)
\(992\) 0 0
\(993\) −9.48913 −0.301128
\(994\) 0 0
\(995\) 23.1386 0.733543
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 0 0
\(999\) 2.62772 0.0831373
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 7728.2.a.bh.1.2 2
4.3 odd 2 966.2.a.o.1.2 2
12.11 even 2 2898.2.a.x.1.1 2
28.27 even 2 6762.2.a.cd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.o.1.2 2 4.3 odd 2
2898.2.a.x.1.1 2 12.11 even 2
6762.2.a.cd.1.1 2 28.27 even 2
7728.2.a.bh.1.2 2 1.1 even 1 trivial