Properties

Label 1386.2.g.a.881.10
Level $1386$
Weight $2$
Character 1386.881
Analytic conductor $11.067$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(881,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1386.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 8x^{12} + 80x^{10} + 1189x^{8} - 2028x^{6} + 1800x^{4} + 1080x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.10
Root \(2.43348 + 1.00798i\) of defining polynomial
Character \(\chi\) \(=\) 1386.881
Dual form 1386.2.g.a.881.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -2.01596 q^{5} +(1.78450 - 1.95335i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -2.01596 q^{5} +(1.78450 - 1.95335i) q^{7} -1.00000i q^{8} -2.01596i q^{10} -1.00000i q^{11} +(1.95335 + 1.78450i) q^{14} +1.00000 q^{16} -5.92266 q^{17} +0.657927i q^{19} +2.01596 q^{20} +1.00000 q^{22} +7.87575i q^{23} -0.935886 q^{25} +(-1.78450 + 1.95335i) q^{28} +1.32636i q^{29} +10.6942i q^{31} +1.00000i q^{32} -5.92266i q^{34} +(-3.59749 + 3.93788i) q^{35} +5.87575 q^{37} -0.657927 q^{38} +2.01596i q^{40} -4.56462 q^{41} -1.56900 q^{43} +1.00000i q^{44} -7.87575 q^{46} -0.532690 q^{47} +(-0.631122 - 6.97149i) q^{49} -0.935886i q^{50} +7.44475i q^{53} +2.01596i q^{55} +(-1.95335 - 1.78450i) q^{56} -1.32636 q^{58} -4.03193 q^{59} +0.250475i q^{61} -10.6942 q^{62} -1.00000 q^{64} -7.81164 q^{67} +5.92266 q^{68} +(-3.93788 - 3.59749i) q^{70} -0.609527i q^{71} +0.532690i q^{73} +5.87575i q^{74} -0.657927i q^{76} +(-1.95335 - 1.78450i) q^{77} -12.7711 q^{79} -2.01596 q^{80} -4.56462i q^{82} -13.3681 q^{83} +11.9399 q^{85} -1.56900i q^{86} -1.00000 q^{88} -5.22255 q^{89} -7.87575i q^{92} -0.532690i q^{94} -1.32636i q^{95} -2.71607i q^{97} +(6.97149 - 0.631122i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 8 q^{7} + 16 q^{16} + 16 q^{22} + 16 q^{25} + 8 q^{28} - 16 q^{37} + 48 q^{43} - 16 q^{46} + 8 q^{49} - 16 q^{58} - 16 q^{64} + 16 q^{67} - 8 q^{70} - 16 q^{79} + 112 q^{85} - 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −2.01596 −0.901567 −0.450783 0.892633i \(-0.648855\pi\)
−0.450783 + 0.892633i \(0.648855\pi\)
\(6\) 0 0
\(7\) 1.78450 1.95335i 0.674477 0.738295i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 2.01596i 0.637504i
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.95335 + 1.78450i 0.522054 + 0.476928i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.92266 −1.43646 −0.718228 0.695808i \(-0.755046\pi\)
−0.718228 + 0.695808i \(0.755046\pi\)
\(18\) 0 0
\(19\) 0.657927i 0.150939i 0.997148 + 0.0754695i \(0.0240455\pi\)
−0.997148 + 0.0754695i \(0.975954\pi\)
\(20\) 2.01596 0.450783
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 7.87575i 1.64221i 0.570778 + 0.821104i \(0.306642\pi\)
−0.570778 + 0.821104i \(0.693358\pi\)
\(24\) 0 0
\(25\) −0.935886 −0.187177
\(26\) 0 0
\(27\) 0 0
\(28\) −1.78450 + 1.95335i −0.337239 + 0.369148i
\(29\) 1.32636i 0.246299i 0.992388 + 0.123149i \(0.0392994\pi\)
−0.992388 + 0.123149i \(0.960701\pi\)
\(30\) 0 0
\(31\) 10.6942i 1.92074i 0.278732 + 0.960369i \(0.410086\pi\)
−0.278732 + 0.960369i \(0.589914\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 5.92266i 1.01573i
\(35\) −3.59749 + 3.93788i −0.608087 + 0.665623i
\(36\) 0 0
\(37\) 5.87575 0.965968 0.482984 0.875629i \(-0.339553\pi\)
0.482984 + 0.875629i \(0.339553\pi\)
\(38\) −0.657927 −0.106730
\(39\) 0 0
\(40\) 2.01596i 0.318752i
\(41\) −4.56462 −0.712874 −0.356437 0.934319i \(-0.616009\pi\)
−0.356437 + 0.934319i \(0.616009\pi\)
\(42\) 0 0
\(43\) −1.56900 −0.239270 −0.119635 0.992818i \(-0.538172\pi\)
−0.119635 + 0.992818i \(0.538172\pi\)
\(44\) 1.00000i 0.150756i
\(45\) 0 0
\(46\) −7.87575 −1.16122
\(47\) −0.532690 −0.0777008 −0.0388504 0.999245i \(-0.512370\pi\)
−0.0388504 + 0.999245i \(0.512370\pi\)
\(48\) 0 0
\(49\) −0.631122 6.97149i −0.0901603 0.995927i
\(50\) 0.935886i 0.132354i
\(51\) 0 0
\(52\) 0 0
\(53\) 7.44475i 1.02262i 0.859398 + 0.511308i \(0.170839\pi\)
−0.859398 + 0.511308i \(0.829161\pi\)
\(54\) 0 0
\(55\) 2.01596i 0.271833i
\(56\) −1.95335 1.78450i −0.261027 0.238464i
\(57\) 0 0
\(58\) −1.32636 −0.174159
\(59\) −4.03193 −0.524913 −0.262456 0.964944i \(-0.584533\pi\)
−0.262456 + 0.964944i \(0.584533\pi\)
\(60\) 0 0
\(61\) 0.250475i 0.0320700i 0.999871 + 0.0160350i \(0.00510432\pi\)
−0.999871 + 0.0160350i \(0.994896\pi\)
\(62\) −10.6942 −1.35817
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −7.81164 −0.954344 −0.477172 0.878810i \(-0.658338\pi\)
−0.477172 + 0.878810i \(0.658338\pi\)
\(68\) 5.92266 0.718228
\(69\) 0 0
\(70\) −3.93788 3.59749i −0.470666 0.429982i
\(71\) 0.609527i 0.0723376i −0.999346 0.0361688i \(-0.988485\pi\)
0.999346 0.0361688i \(-0.0115154\pi\)
\(72\) 0 0
\(73\) 0.532690i 0.0623466i 0.999514 + 0.0311733i \(0.00992438\pi\)
−0.999514 + 0.0311733i \(0.990076\pi\)
\(74\) 5.87575i 0.683043i
\(75\) 0 0
\(76\) 0.657927i 0.0754695i
\(77\) −1.95335 1.78450i −0.222604 0.203363i
\(78\) 0 0
\(79\) −12.7711 −1.43686 −0.718431 0.695598i \(-0.755139\pi\)
−0.718431 + 0.695598i \(0.755139\pi\)
\(80\) −2.01596 −0.225392
\(81\) 0 0
\(82\) 4.56462i 0.504078i
\(83\) −13.3681 −1.46734 −0.733670 0.679506i \(-0.762194\pi\)
−0.733670 + 0.679506i \(0.762194\pi\)
\(84\) 0 0
\(85\) 11.9399 1.29506
\(86\) 1.56900i 0.169190i
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −5.22255 −0.553589 −0.276794 0.960929i \(-0.589272\pi\)
−0.276794 + 0.960929i \(0.589272\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.87575i 0.821104i
\(93\) 0 0
\(94\) 0.532690i 0.0549428i
\(95\) 1.32636i 0.136082i
\(96\) 0 0
\(97\) 2.71607i 0.275776i −0.990448 0.137888i \(-0.955969\pi\)
0.990448 0.137888i \(-0.0440313\pi\)
\(98\) 6.97149 0.631122i 0.704227 0.0637530i
\(99\) 0 0
\(100\) 0.935886 0.0935886
\(101\) −3.41352 −0.339658 −0.169829 0.985474i \(-0.554322\pi\)
−0.169829 + 0.985474i \(0.554322\pi\)
\(102\) 0 0
\(103\) 13.3681i 1.31720i −0.752494 0.658599i \(-0.771149\pi\)
0.752494 0.658599i \(-0.228851\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −7.44475 −0.723098
\(107\) 14.4002i 1.39212i 0.717982 + 0.696062i \(0.245066\pi\)
−0.717982 + 0.696062i \(0.754934\pi\)
\(108\) 0 0
\(109\) −3.41140 −0.326753 −0.163376 0.986564i \(-0.552238\pi\)
−0.163376 + 0.986564i \(0.552238\pi\)
\(110\) −2.01596 −0.192215
\(111\) 0 0
\(112\) 1.78450 1.95335i 0.168619 0.184574i
\(113\) 4.95947i 0.466548i 0.972411 + 0.233274i \(0.0749439\pi\)
−0.972411 + 0.233274i \(0.925056\pi\)
\(114\) 0 0
\(115\) 15.8772i 1.48056i
\(116\) 1.32636i 0.123149i
\(117\) 0 0
\(118\) 4.03193i 0.371169i
\(119\) −10.5690 + 11.5690i −0.968857 + 1.06053i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) −0.250475 −0.0226769
\(123\) 0 0
\(124\) 10.6942i 0.960369i
\(125\) 11.9665 1.07032
\(126\) 0 0
\(127\) 9.50488 0.843422 0.421711 0.906730i \(-0.361430\pi\)
0.421711 + 0.906730i \(0.361430\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 19.0838 1.66736 0.833681 0.552247i \(-0.186229\pi\)
0.833681 + 0.552247i \(0.186229\pi\)
\(132\) 0 0
\(133\) 1.28516 + 1.17407i 0.111438 + 0.101805i
\(134\) 7.81164i 0.674823i
\(135\) 0 0
\(136\) 5.92266i 0.507864i
\(137\) 8.05428i 0.688124i 0.938947 + 0.344062i \(0.111803\pi\)
−0.938947 + 0.344062i \(0.888197\pi\)
\(138\) 0 0
\(139\) 19.3725i 1.64315i 0.570099 + 0.821576i \(0.306905\pi\)
−0.570099 + 0.821576i \(0.693095\pi\)
\(140\) 3.59749 3.93788i 0.304043 0.332811i
\(141\) 0 0
\(142\) 0.609527 0.0511504
\(143\) 0 0
\(144\) 0 0
\(145\) 2.67389i 0.222055i
\(146\) −0.532690 −0.0440857
\(147\) 0 0
\(148\) −5.87575 −0.482984
\(149\) 3.26224i 0.267253i −0.991032 0.133627i \(-0.957338\pi\)
0.991032 0.133627i \(-0.0426623\pi\)
\(150\) 0 0
\(151\) −14.1785 −1.15383 −0.576916 0.816803i \(-0.695744\pi\)
−0.576916 + 0.816803i \(0.695744\pi\)
\(152\) 0.657927 0.0533650
\(153\) 0 0
\(154\) 1.78450 1.95335i 0.143799 0.157405i
\(155\) 21.5592i 1.73167i
\(156\) 0 0
\(157\) 11.7596i 0.938518i 0.883061 + 0.469259i \(0.155479\pi\)
−0.883061 + 0.469259i \(0.844521\pi\)
\(158\) 12.7711i 1.01602i
\(159\) 0 0
\(160\) 2.01596i 0.159376i
\(161\) 15.3841 + 14.0543i 1.21243 + 1.10763i
\(162\) 0 0
\(163\) 12.5285 0.981306 0.490653 0.871355i \(-0.336758\pi\)
0.490653 + 0.871355i \(0.336758\pi\)
\(164\) 4.56462 0.356437
\(165\) 0 0
\(166\) 13.3681i 1.03757i
\(167\) 23.0722 1.78538 0.892691 0.450669i \(-0.148815\pi\)
0.892691 + 0.450669i \(0.148815\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 11.9399i 0.915746i
\(171\) 0 0
\(172\) 1.56900 0.119635
\(173\) −17.7244 −1.34756 −0.673782 0.738930i \(-0.735331\pi\)
−0.673782 + 0.738930i \(0.735331\pi\)
\(174\) 0 0
\(175\) −1.67009 + 1.82811i −0.126247 + 0.138192i
\(176\) 1.00000i 0.0753778i
\(177\) 0 0
\(178\) 5.22255i 0.391446i
\(179\) 3.81164i 0.284895i 0.989802 + 0.142448i \(0.0454973\pi\)
−0.989802 + 0.142448i \(0.954503\pi\)
\(180\) 0 0
\(181\) 19.1260i 1.42163i 0.703382 + 0.710813i \(0.251673\pi\)
−0.703382 + 0.710813i \(0.748327\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 7.87575 0.580608
\(185\) −11.8453 −0.870885
\(186\) 0 0
\(187\) 5.92266i 0.433108i
\(188\) 0.532690 0.0388504
\(189\) 0 0
\(190\) 1.32636 0.0962242
\(191\) 21.6665i 1.56773i −0.620931 0.783865i \(-0.713245\pi\)
0.620931 0.783865i \(-0.286755\pi\)
\(192\) 0 0
\(193\) −21.6233 −1.55648 −0.778239 0.627968i \(-0.783887\pi\)
−0.778239 + 0.627968i \(0.783887\pi\)
\(194\) 2.71607 0.195003
\(195\) 0 0
\(196\) 0.631122 + 6.97149i 0.0450802 + 0.497964i
\(197\) 21.6914i 1.54545i −0.634743 0.772723i \(-0.718894\pi\)
0.634743 0.772723i \(-0.281106\pi\)
\(198\) 0 0
\(199\) 3.94621i 0.279740i −0.990170 0.139870i \(-0.955332\pi\)
0.990170 0.139870i \(-0.0446684\pi\)
\(200\) 0.935886i 0.0661771i
\(201\) 0 0
\(202\) 3.41352i 0.240175i
\(203\) 2.59084 + 2.36689i 0.181841 + 0.166123i
\(204\) 0 0
\(205\) 9.20211 0.642703
\(206\) 13.3681 0.931400
\(207\) 0 0
\(208\) 0 0
\(209\) 0.657927 0.0455098
\(210\) 0 0
\(211\) 2.30277 0.158529 0.0792647 0.996854i \(-0.474743\pi\)
0.0792647 + 0.996854i \(0.474743\pi\)
\(212\) 7.44475i 0.511308i
\(213\) 0 0
\(214\) −14.4002 −0.984380
\(215\) 3.16305 0.215718
\(216\) 0 0
\(217\) 20.8895 + 19.0838i 1.41807 + 1.29549i
\(218\) 3.41140i 0.231049i
\(219\) 0 0
\(220\) 2.01596i 0.135916i
\(221\) 0 0
\(222\) 0 0
\(223\) 18.2571i 1.22259i −0.791404 0.611294i \(-0.790649\pi\)
0.791404 0.611294i \(-0.209351\pi\)
\(224\) 1.95335 + 1.78450i 0.130513 + 0.119232i
\(225\) 0 0
\(226\) −4.95947 −0.329899
\(227\) 20.0739 1.33235 0.666177 0.745794i \(-0.267930\pi\)
0.666177 + 0.745794i \(0.267930\pi\)
\(228\) 0 0
\(229\) 14.6840i 0.970344i 0.874419 + 0.485172i \(0.161243\pi\)
−0.874419 + 0.485172i \(0.838757\pi\)
\(230\) 15.8772 1.04691
\(231\) 0 0
\(232\) 1.32636 0.0870797
\(233\) 2.48528i 0.162816i −0.996681 0.0814081i \(-0.974058\pi\)
0.996681 0.0814081i \(-0.0259417\pi\)
\(234\) 0 0
\(235\) 1.07388 0.0700525
\(236\) 4.03193 0.262456
\(237\) 0 0
\(238\) −11.5690 10.5690i −0.749907 0.685085i
\(239\) 22.3610i 1.44642i −0.690631 0.723208i \(-0.742667\pi\)
0.690631 0.723208i \(-0.257333\pi\)
\(240\) 0 0
\(241\) 3.45706i 0.222689i −0.993782 0.111344i \(-0.964484\pi\)
0.993782 0.111344i \(-0.0355156\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) 0 0
\(244\) 0.250475i 0.0160350i
\(245\) 1.27232 + 14.0543i 0.0812855 + 0.897895i
\(246\) 0 0
\(247\) 0 0
\(248\) 10.6942 0.679083
\(249\) 0 0
\(250\) 11.9665i 0.756830i
\(251\) −3.78145 −0.238683 −0.119342 0.992853i \(-0.538078\pi\)
−0.119342 + 0.992853i \(0.538078\pi\)
\(252\) 0 0
\(253\) 7.87575 0.495144
\(254\) 9.50488i 0.596390i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.22255 0.325774 0.162887 0.986645i \(-0.447919\pi\)
0.162887 + 0.986645i \(0.447919\pi\)
\(258\) 0 0
\(259\) 10.4853 11.4774i 0.651524 0.713170i
\(260\) 0 0
\(261\) 0 0
\(262\) 19.0838i 1.17900i
\(263\) 13.7306i 0.846664i −0.905975 0.423332i \(-0.860860\pi\)
0.905975 0.423332i \(-0.139140\pi\)
\(264\) 0 0
\(265\) 15.0084i 0.921956i
\(266\) −1.17407 + 1.28516i −0.0719869 + 0.0787982i
\(267\) 0 0
\(268\) 7.81164 0.477172
\(269\) −29.6093 −1.80531 −0.902655 0.430366i \(-0.858385\pi\)
−0.902655 + 0.430366i \(0.858385\pi\)
\(270\) 0 0
\(271\) 15.5015i 0.941651i 0.882226 + 0.470825i \(0.156044\pi\)
−0.882226 + 0.470825i \(0.843956\pi\)
\(272\) −5.92266 −0.359114
\(273\) 0 0
\(274\) −8.05428 −0.486577
\(275\) 0.935886i 0.0564360i
\(276\) 0 0
\(277\) −19.0347 −1.14368 −0.571841 0.820364i \(-0.693771\pi\)
−0.571841 + 0.820364i \(0.693771\pi\)
\(278\) −19.3725 −1.16188
\(279\) 0 0
\(280\) 3.93788 + 3.59749i 0.235333 + 0.214991i
\(281\) 10.9706i 0.654449i −0.944947 0.327224i \(-0.893887\pi\)
0.944947 0.327224i \(-0.106113\pi\)
\(282\) 0 0
\(283\) 19.2091i 1.14186i 0.820999 + 0.570930i \(0.193417\pi\)
−0.820999 + 0.570930i \(0.806583\pi\)
\(284\) 0.609527i 0.0361688i
\(285\) 0 0
\(286\) 0 0
\(287\) −8.14556 + 8.91628i −0.480817 + 0.526311i
\(288\) 0 0
\(289\) 18.0779 1.06340
\(290\) 2.67389 0.157016
\(291\) 0 0
\(292\) 0.532690i 0.0311733i
\(293\) −9.91105 −0.579010 −0.289505 0.957177i \(-0.593491\pi\)
−0.289505 + 0.957177i \(0.593491\pi\)
\(294\) 0 0
\(295\) 8.12823 0.473244
\(296\) 5.87575i 0.341521i
\(297\) 0 0
\(298\) 3.26224 0.188977
\(299\) 0 0
\(300\) 0 0
\(301\) −2.79988 + 3.06480i −0.161382 + 0.176652i
\(302\) 14.1785i 0.815883i
\(303\) 0 0
\(304\) 0.657927i 0.0377347i
\(305\) 0.504949i 0.0289133i
\(306\) 0 0
\(307\) 1.64427i 0.0938433i 0.998899 + 0.0469216i \(0.0149411\pi\)
−0.998899 + 0.0469216i \(0.985059\pi\)
\(308\) 1.95335 + 1.78450i 0.111302 + 0.101681i
\(309\) 0 0
\(310\) 21.5592 1.22448
\(311\) −2.14120 −0.121416 −0.0607082 0.998156i \(-0.519336\pi\)
−0.0607082 + 0.998156i \(0.519336\pi\)
\(312\) 0 0
\(313\) 12.7932i 0.723116i 0.932350 + 0.361558i \(0.117755\pi\)
−0.932350 + 0.361558i \(0.882245\pi\)
\(314\) −11.7596 −0.663632
\(315\) 0 0
\(316\) 12.7711 0.718431
\(317\) 6.51604i 0.365977i −0.983115 0.182989i \(-0.941423\pi\)
0.983115 0.182989i \(-0.0585771\pi\)
\(318\) 0 0
\(319\) 1.32636 0.0742618
\(320\) 2.01596 0.112696
\(321\) 0 0
\(322\) −14.0543 + 15.3841i −0.783214 + 0.857321i
\(323\) 3.89668i 0.216817i
\(324\) 0 0
\(325\) 0 0
\(326\) 12.5285i 0.693888i
\(327\) 0 0
\(328\) 4.56462i 0.252039i
\(329\) −0.950585 + 1.04053i −0.0524074 + 0.0573662i
\(330\) 0 0
\(331\) −30.2159 −1.66081 −0.830407 0.557157i \(-0.811892\pi\)
−0.830407 + 0.557157i \(0.811892\pi\)
\(332\) 13.3681 0.733670
\(333\) 0 0
\(334\) 23.0722i 1.26246i
\(335\) 15.7480 0.860405
\(336\) 0 0
\(337\) 24.8895 1.35582 0.677909 0.735146i \(-0.262886\pi\)
0.677909 + 0.735146i \(0.262886\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 0 0
\(340\) −11.9399 −0.647530
\(341\) 10.6942 0.579124
\(342\) 0 0
\(343\) −14.7440 11.2078i −0.796100 0.605166i
\(344\) 1.56900i 0.0845948i
\(345\) 0 0
\(346\) 17.7244i 0.952872i
\(347\) 9.09481i 0.488235i 0.969746 + 0.244117i \(0.0784982\pi\)
−0.969746 + 0.244117i \(0.921502\pi\)
\(348\) 0 0
\(349\) 31.8335i 1.70401i 0.523534 + 0.852005i \(0.324613\pi\)
−0.523534 + 0.852005i \(0.675387\pi\)
\(350\) −1.82811 1.67009i −0.0977165 0.0892699i
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 10.9052 0.580424 0.290212 0.956962i \(-0.406274\pi\)
0.290212 + 0.956962i \(0.406274\pi\)
\(354\) 0 0
\(355\) 1.22879i 0.0652171i
\(356\) 5.22255 0.276794
\(357\) 0 0
\(358\) −3.81164 −0.201451
\(359\) 23.6874i 1.25017i −0.780556 0.625086i \(-0.785064\pi\)
0.780556 0.625086i \(-0.214936\pi\)
\(360\) 0 0
\(361\) 18.5671 0.977217
\(362\) −19.1260 −1.00524
\(363\) 0 0
\(364\) 0 0
\(365\) 1.07388i 0.0562097i
\(366\) 0 0
\(367\) 13.1176i 0.684735i −0.939566 0.342367i \(-0.888771\pi\)
0.939566 0.342367i \(-0.111229\pi\)
\(368\) 7.87575i 0.410552i
\(369\) 0 0
\(370\) 11.8453i 0.615809i
\(371\) 14.5422 + 13.2852i 0.754992 + 0.689731i
\(372\) 0 0
\(373\) 3.38649 0.175346 0.0876729 0.996149i \(-0.472057\pi\)
0.0876729 + 0.996149i \(0.472057\pi\)
\(374\) −5.92266 −0.306253
\(375\) 0 0
\(376\) 0.532690i 0.0274714i
\(377\) 0 0
\(378\) 0 0
\(379\) 8.52847 0.438078 0.219039 0.975716i \(-0.429708\pi\)
0.219039 + 0.975716i \(0.429708\pi\)
\(380\) 1.32636i 0.0680408i
\(381\) 0 0
\(382\) 21.6665 1.10855
\(383\) 10.1180 0.517005 0.258503 0.966011i \(-0.416771\pi\)
0.258503 + 0.966011i \(0.416771\pi\)
\(384\) 0 0
\(385\) 3.93788 + 3.59749i 0.200693 + 0.183345i
\(386\) 21.6233i 1.10060i
\(387\) 0 0
\(388\) 2.71607i 0.137888i
\(389\) 16.7920i 0.851390i 0.904867 + 0.425695i \(0.139970\pi\)
−0.904867 + 0.425695i \(0.860030\pi\)
\(390\) 0 0
\(391\) 46.6454i 2.35896i
\(392\) −6.97149 + 0.631122i −0.352113 + 0.0318765i
\(393\) 0 0
\(394\) 21.6914 1.09280
\(395\) 25.7461 1.29543
\(396\) 0 0
\(397\) 13.4856i 0.676821i 0.940999 + 0.338411i \(0.109889\pi\)
−0.940999 + 0.338411i \(0.890111\pi\)
\(398\) 3.94621 0.197806
\(399\) 0 0
\(400\) −0.935886 −0.0467943
\(401\) 13.5298i 0.675646i −0.941210 0.337823i \(-0.890310\pi\)
0.941210 0.337823i \(-0.109690\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 3.41352 0.169829
\(405\) 0 0
\(406\) −2.36689 + 2.59084i −0.117467 + 0.128581i
\(407\) 5.87575i 0.291250i
\(408\) 0 0
\(409\) 27.0606i 1.33806i −0.743235 0.669031i \(-0.766709\pi\)
0.743235 0.669031i \(-0.233291\pi\)
\(410\) 9.20211i 0.454460i
\(411\) 0 0
\(412\) 13.3681i 0.658599i
\(413\) −7.19498 + 7.87575i −0.354042 + 0.387541i
\(414\) 0 0
\(415\) 26.9496 1.32291
\(416\) 0 0
\(417\) 0 0
\(418\) 0.657927i 0.0321803i
\(419\) 10.7799 0.526634 0.263317 0.964709i \(-0.415183\pi\)
0.263317 + 0.964709i \(0.415183\pi\)
\(420\) 0 0
\(421\) 23.0948 1.12557 0.562786 0.826603i \(-0.309729\pi\)
0.562786 + 0.826603i \(0.309729\pi\)
\(422\) 2.30277i 0.112097i
\(423\) 0 0
\(424\) 7.44475 0.361549
\(425\) 5.54293 0.268872
\(426\) 0 0
\(427\) 0.489264 + 0.446973i 0.0236772 + 0.0216305i
\(428\) 14.4002i 0.696062i
\(429\) 0 0
\(430\) 3.16305i 0.152536i
\(431\) 13.8508i 0.667172i −0.942720 0.333586i \(-0.891741\pi\)
0.942720 0.333586i \(-0.108259\pi\)
\(432\) 0 0
\(433\) 36.3968i 1.74912i 0.484919 + 0.874559i \(0.338849\pi\)
−0.484919 + 0.874559i \(0.661151\pi\)
\(434\) −19.0838 + 20.8895i −0.916053 + 1.00273i
\(435\) 0 0
\(436\) 3.41140 0.163376
\(437\) −5.18167 −0.247873
\(438\) 0 0
\(439\) 24.9603i 1.19129i −0.803248 0.595645i \(-0.796897\pi\)
0.803248 0.595645i \(-0.203103\pi\)
\(440\) 2.01596 0.0961074
\(441\) 0 0
\(442\) 0 0
\(443\) 7.20610i 0.342372i 0.985239 + 0.171186i \(0.0547599\pi\)
−0.985239 + 0.171186i \(0.945240\pi\)
\(444\) 0 0
\(445\) 10.5285 0.499097
\(446\) 18.2571 0.864500
\(447\) 0 0
\(448\) −1.78450 + 1.95335i −0.0843097 + 0.0922869i
\(449\) 31.8490i 1.50305i 0.659707 + 0.751523i \(0.270680\pi\)
−0.659707 + 0.751523i \(0.729320\pi\)
\(450\) 0 0
\(451\) 4.56462i 0.214940i
\(452\) 4.95947i 0.233274i
\(453\) 0 0
\(454\) 20.0739i 0.942116i
\(455\) 0 0
\(456\) 0 0
\(457\) −10.4853 −0.490481 −0.245240 0.969462i \(-0.578867\pi\)
−0.245240 + 0.969462i \(0.578867\pi\)
\(458\) −14.6840 −0.686137
\(459\) 0 0
\(460\) 15.8772i 0.740280i
\(461\) 24.3881 1.13587 0.567933 0.823075i \(-0.307743\pi\)
0.567933 + 0.823075i \(0.307743\pi\)
\(462\) 0 0
\(463\) −22.6410 −1.05222 −0.526109 0.850417i \(-0.676349\pi\)
−0.526109 + 0.850417i \(0.676349\pi\)
\(464\) 1.32636i 0.0615747i
\(465\) 0 0
\(466\) 2.48528 0.115128
\(467\) −27.6355 −1.27882 −0.639409 0.768867i \(-0.720821\pi\)
−0.639409 + 0.768867i \(0.720821\pi\)
\(468\) 0 0
\(469\) −13.9399 + 15.2588i −0.643683 + 0.704588i
\(470\) 1.07388i 0.0495346i
\(471\) 0 0
\(472\) 4.03193i 0.185585i
\(473\) 1.56900i 0.0721427i
\(474\) 0 0
\(475\) 0.615745i 0.0282523i
\(476\) 10.5690 11.5690i 0.484428 0.530264i
\(477\) 0 0
\(478\) 22.3610 1.02277
\(479\) 34.4166 1.57253 0.786267 0.617887i \(-0.212011\pi\)
0.786267 + 0.617887i \(0.212011\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.45706 0.157465
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 5.47551i 0.248630i
\(486\) 0 0
\(487\) −15.2662 −0.691779 −0.345889 0.938275i \(-0.612423\pi\)
−0.345889 + 0.938275i \(0.612423\pi\)
\(488\) 0.250475 0.0113385
\(489\) 0 0
\(490\) −14.0543 + 1.27232i −0.634908 + 0.0574776i
\(491\) 23.6273i 1.06628i −0.846026 0.533142i \(-0.821011\pi\)
0.846026 0.533142i \(-0.178989\pi\)
\(492\) 0 0
\(493\) 7.85557i 0.353797i
\(494\) 0 0
\(495\) 0 0
\(496\) 10.6942i 0.480185i
\(497\) −1.19062 1.08770i −0.0534065 0.0487901i
\(498\) 0 0
\(499\) −36.0484 −1.61375 −0.806875 0.590723i \(-0.798843\pi\)
−0.806875 + 0.590723i \(0.798843\pi\)
\(500\) −11.9665 −0.535160
\(501\) 0 0
\(502\) 3.78145i 0.168775i
\(503\) −31.1361 −1.38829 −0.694145 0.719836i \(-0.744217\pi\)
−0.694145 + 0.719836i \(0.744217\pi\)
\(504\) 0 0
\(505\) 6.88154 0.306225
\(506\) 7.87575i 0.350120i
\(507\) 0 0
\(508\) −9.50488 −0.421711
\(509\) −21.5454 −0.954984 −0.477492 0.878636i \(-0.658454\pi\)
−0.477492 + 0.878636i \(0.658454\pi\)
\(510\) 0 0
\(511\) 1.04053 + 0.950585i 0.0460302 + 0.0420514i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 5.22255i 0.230357i
\(515\) 26.9496i 1.18754i
\(516\) 0 0
\(517\) 0.532690i 0.0234277i
\(518\) 11.4774 + 10.4853i 0.504287 + 0.460697i
\(519\) 0 0
\(520\) 0 0
\(521\) 24.0663 1.05437 0.527183 0.849752i \(-0.323248\pi\)
0.527183 + 0.849752i \(0.323248\pi\)
\(522\) 0 0
\(523\) 7.11327i 0.311042i −0.987833 0.155521i \(-0.950294\pi\)
0.987833 0.155521i \(-0.0497056\pi\)
\(524\) −19.0838 −0.833681
\(525\) 0 0
\(526\) 13.7306 0.598682
\(527\) 63.3382i 2.75905i
\(528\) 0 0
\(529\) −39.0275 −1.69685
\(530\) 15.0084 0.651922
\(531\) 0 0
\(532\) −1.28516 1.17407i −0.0557188 0.0509024i
\(533\) 0 0
\(534\) 0 0
\(535\) 29.0304i 1.25509i
\(536\) 7.81164i 0.337411i
\(537\) 0 0
\(538\) 29.6093i 1.27655i
\(539\) −6.97149 + 0.631122i −0.300283 + 0.0271844i
\(540\) 0 0
\(541\) 4.24849 0.182657 0.0913285 0.995821i \(-0.470889\pi\)
0.0913285 + 0.995821i \(0.470889\pi\)
\(542\) −15.5015 −0.665848
\(543\) 0 0
\(544\) 5.92266i 0.253932i
\(545\) 6.87726 0.294589
\(546\) 0 0
\(547\) 9.69723 0.414624 0.207312 0.978275i \(-0.433529\pi\)
0.207312 + 0.978275i \(0.433529\pi\)
\(548\) 8.05428i 0.344062i
\(549\) 0 0
\(550\) −0.935886 −0.0399063
\(551\) −0.872648 −0.0371760
\(552\) 0 0
\(553\) −22.7900 + 24.9464i −0.969131 + 1.06083i
\(554\) 19.0347i 0.808706i
\(555\) 0 0
\(556\) 19.3725i 0.821576i
\(557\) 21.6914i 0.919093i −0.888154 0.459547i \(-0.848012\pi\)
0.888154 0.459547i \(-0.151988\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −3.59749 + 3.93788i −0.152022 + 0.166406i
\(561\) 0 0
\(562\) 10.9706 0.462765
\(563\) 1.68243 0.0709062 0.0354531 0.999371i \(-0.488713\pi\)
0.0354531 + 0.999371i \(0.488713\pi\)
\(564\) 0 0
\(565\) 9.99812i 0.420624i
\(566\) −19.2091 −0.807417
\(567\) 0 0
\(568\) −0.609527 −0.0255752
\(569\) 14.4853i 0.607255i −0.952791 0.303627i \(-0.901802\pi\)
0.952791 0.303627i \(-0.0981978\pi\)
\(570\) 0 0
\(571\) −26.7542 −1.11963 −0.559814 0.828619i \(-0.689127\pi\)
−0.559814 + 0.828619i \(0.689127\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.91628 8.14556i −0.372158 0.339989i
\(575\) 7.37081i 0.307384i
\(576\) 0 0
\(577\) 42.3630i 1.76359i 0.471629 + 0.881797i \(0.343666\pi\)
−0.471629 + 0.881797i \(0.656334\pi\)
\(578\) 18.0779i 0.751940i
\(579\) 0 0
\(580\) 2.67389i 0.111027i
\(581\) −23.8554 + 26.1125i −0.989688 + 1.08333i
\(582\) 0 0
\(583\) 7.44475 0.308330
\(584\) 0.532690 0.0220429
\(585\) 0 0
\(586\) 9.91105i 0.409422i
\(587\) 2.05172 0.0846835 0.0423418 0.999103i \(-0.486518\pi\)
0.0423418 + 0.999103i \(0.486518\pi\)
\(588\) 0 0
\(589\) −7.03602 −0.289914
\(590\) 8.12823i 0.334634i
\(591\) 0 0
\(592\) 5.87575 0.241492
\(593\) 40.2640 1.65344 0.826721 0.562611i \(-0.190203\pi\)
0.826721 + 0.562611i \(0.190203\pi\)
\(594\) 0 0
\(595\) 21.3067 23.3227i 0.873489 0.956137i
\(596\) 3.26224i 0.133627i
\(597\) 0 0
\(598\) 0 0
\(599\) 35.6273i 1.45569i 0.685741 + 0.727845i \(0.259478\pi\)
−0.685741 + 0.727845i \(0.740522\pi\)
\(600\) 0 0
\(601\) 39.6574i 1.61766i −0.588044 0.808829i \(-0.700102\pi\)
0.588044 0.808829i \(-0.299898\pi\)
\(602\) −3.06480 2.79988i −0.124912 0.114115i
\(603\) 0 0
\(604\) 14.1785 0.576916
\(605\) 2.01596 0.0819606
\(606\) 0 0
\(607\) 23.6444i 0.959698i −0.877351 0.479849i \(-0.840691\pi\)
0.877351 0.479849i \(-0.159309\pi\)
\(608\) −0.657927 −0.0266825
\(609\) 0 0
\(610\) 0.504949 0.0204448
\(611\) 0 0
\(612\) 0 0
\(613\) −36.0654 −1.45667 −0.728333 0.685223i \(-0.759705\pi\)
−0.728333 + 0.685223i \(0.759705\pi\)
\(614\) −1.64427 −0.0663572
\(615\) 0 0
\(616\) −1.78450 + 1.95335i −0.0718995 + 0.0787026i
\(617\) 2.45192i 0.0987108i −0.998781 0.0493554i \(-0.984283\pi\)
0.998781 0.0493554i \(-0.0157167\pi\)
\(618\) 0 0
\(619\) 12.9107i 0.518925i −0.965753 0.259462i \(-0.916455\pi\)
0.965753 0.259462i \(-0.0835453\pi\)
\(620\) 21.5592i 0.865837i
\(621\) 0 0
\(622\) 2.14120i 0.0858544i
\(623\) −9.31963 + 10.2014i −0.373383 + 0.408712i
\(624\) 0 0
\(625\) −19.4447 −0.777788
\(626\) −12.7932 −0.511321
\(627\) 0 0
\(628\) 11.7596i 0.469259i
\(629\) −34.8001 −1.38757
\(630\) 0 0
\(631\) −6.53245 −0.260053 −0.130026 0.991511i \(-0.541506\pi\)
−0.130026 + 0.991511i \(0.541506\pi\)
\(632\) 12.7711i 0.508008i
\(633\) 0 0
\(634\) 6.51604 0.258785
\(635\) −19.1615 −0.760401
\(636\) 0 0
\(637\) 0 0
\(638\) 1.32636i 0.0525110i
\(639\) 0 0
\(640\) 2.01596i 0.0796880i
\(641\) 6.03336i 0.238303i −0.992876 0.119152i \(-0.961983\pi\)
0.992876 0.119152i \(-0.0380175\pi\)
\(642\) 0 0
\(643\) 9.21360i 0.363349i −0.983359 0.181675i \(-0.941848\pi\)
0.983359 0.181675i \(-0.0581517\pi\)
\(644\) −15.3841 14.0543i −0.606217 0.553816i
\(645\) 0 0
\(646\) 3.89668 0.153313
\(647\) −1.72732 −0.0679080 −0.0339540 0.999423i \(-0.510810\pi\)
−0.0339540 + 0.999423i \(0.510810\pi\)
\(648\) 0 0
\(649\) 4.03193i 0.158267i
\(650\) 0 0
\(651\) 0 0
\(652\) −12.5285 −0.490653
\(653\) 41.9575i 1.64193i 0.570982 + 0.820963i \(0.306563\pi\)
−0.570982 + 0.820963i \(0.693437\pi\)
\(654\) 0 0
\(655\) −38.4723 −1.50324
\(656\) −4.56462 −0.178218
\(657\) 0 0
\(658\) −1.04053 0.950585i −0.0405640 0.0370577i
\(659\) 10.3963i 0.404981i −0.979284 0.202490i \(-0.935097\pi\)
0.979284 0.202490i \(-0.0649035\pi\)
\(660\) 0 0
\(661\) 34.0922i 1.32603i −0.748605 0.663016i \(-0.769276\pi\)
0.748605 0.663016i \(-0.230724\pi\)
\(662\) 30.2159i 1.17437i
\(663\) 0 0
\(664\) 13.3681i 0.518783i
\(665\) −2.59084 2.36689i −0.100468 0.0917839i
\(666\) 0 0
\(667\) −10.4461 −0.404474
\(668\) −23.0722 −0.892691
\(669\) 0 0
\(670\) 15.7480i 0.608398i
\(671\) 0.250475 0.00966948
\(672\) 0 0
\(673\) 9.66513 0.372563 0.186282 0.982496i \(-0.440356\pi\)
0.186282 + 0.982496i \(0.440356\pi\)
\(674\) 24.8895i 0.958708i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 45.8609 1.76258 0.881288 0.472579i \(-0.156677\pi\)
0.881288 + 0.472579i \(0.156677\pi\)
\(678\) 0 0
\(679\) −5.30543 4.84683i −0.203604 0.186004i
\(680\) 11.9399i 0.457873i
\(681\) 0 0
\(682\) 10.6942i 0.409503i
\(683\) 3.93190i 0.150450i −0.997167 0.0752250i \(-0.976032\pi\)
0.997167 0.0752250i \(-0.0239675\pi\)
\(684\) 0 0
\(685\) 16.2371i 0.620389i
\(686\) 11.2078 14.7440i 0.427917 0.562927i
\(687\) 0 0
\(688\) −1.56900 −0.0598175
\(689\) 0 0
\(690\) 0 0
\(691\) 17.6940i 0.673113i 0.941663 + 0.336557i \(0.109262\pi\)
−0.941663 + 0.336557i \(0.890738\pi\)
\(692\) 17.7244 0.673782
\(693\) 0 0
\(694\) −9.09481 −0.345234
\(695\) 39.0542i 1.48141i
\(696\) 0 0
\(697\) 27.0347 1.02401
\(698\) −31.8335 −1.20492
\(699\) 0 0
\(700\) 1.67009 1.82811i 0.0631234 0.0690960i
\(701\) 27.9189i 1.05448i 0.849715 + 0.527242i \(0.176774\pi\)
−0.849715 + 0.527242i \(0.823226\pi\)
\(702\) 0 0
\(703\) 3.86582i 0.145802i
\(704\) 1.00000i 0.0376889i
\(705\) 0 0
\(706\) 10.9052i 0.410422i
\(707\) −6.09143 + 6.66779i −0.229092 + 0.250768i
\(708\) 0 0
\(709\) −25.4951 −0.957487 −0.478743 0.877955i \(-0.658908\pi\)
−0.478743 + 0.877955i \(0.658908\pi\)
\(710\) −1.22879 −0.0461155
\(711\) 0 0
\(712\) 5.22255i 0.195723i
\(713\) −84.2250 −3.15425
\(714\) 0 0
\(715\) 0 0
\(716\) 3.81164i 0.142448i
\(717\) 0 0
\(718\) 23.6874 0.884006
\(719\) −43.0169 −1.60426 −0.802130 0.597150i \(-0.796300\pi\)
−0.802130 + 0.597150i \(0.796300\pi\)
\(720\) 0 0
\(721\) −26.1125 23.8554i −0.972482 0.888421i
\(722\) 18.5671i 0.690997i
\(723\) 0 0
\(724\) 19.1260i 0.710813i
\(725\) 1.24132i 0.0461015i
\(726\) 0 0
\(727\) 30.2686i 1.12260i −0.827613 0.561299i \(-0.810302\pi\)
0.827613 0.561299i \(-0.189698\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.07388 0.0397462
\(731\) 9.29264 0.343701
\(732\) 0 0
\(733\) 7.56291i 0.279342i −0.990198 0.139671i \(-0.955395\pi\)
0.990198 0.139671i \(-0.0446046\pi\)
\(734\) 13.1176 0.484181
\(735\) 0 0
\(736\) −7.87575 −0.290304
\(737\) 7.81164i 0.287745i
\(738\) 0 0
\(739\) −17.1923 −0.632428 −0.316214 0.948688i \(-0.602412\pi\)
−0.316214 + 0.948688i \(0.602412\pi\)
\(740\) 11.8453 0.435442
\(741\) 0 0
\(742\) −13.2852 + 14.5422i −0.487714 + 0.533860i
\(743\) 23.9843i 0.879899i −0.898022 0.439950i \(-0.854996\pi\)
0.898022 0.439950i \(-0.145004\pi\)
\(744\) 0 0
\(745\) 6.57657i 0.240947i
\(746\) 3.38649i 0.123988i
\(747\) 0 0
\(748\) 5.92266i 0.216554i
\(749\) 28.1287 + 25.6972i 1.02780 + 0.938956i
\(750\) 0 0
\(751\) −32.4736 −1.18498 −0.592489 0.805579i \(-0.701855\pi\)
−0.592489 + 0.805579i \(0.701855\pi\)
\(752\) −0.532690 −0.0194252
\(753\) 0 0
\(754\) 0 0
\(755\) 28.5834 1.04026
\(756\) 0 0
\(757\) 40.0315 1.45497 0.727485 0.686124i \(-0.240689\pi\)
0.727485 + 0.686124i \(0.240689\pi\)
\(758\) 8.52847i 0.309768i
\(759\) 0 0
\(760\) −1.32636 −0.0481121
\(761\) −20.6923 −0.750097 −0.375048 0.927005i \(-0.622374\pi\)
−0.375048 + 0.927005i \(0.622374\pi\)
\(762\) 0 0
\(763\) −6.08764 + 6.66364i −0.220387 + 0.241240i
\(764\) 21.6665i 0.783865i
\(765\) 0 0
\(766\) 10.1180i 0.365578i
\(767\) 0 0
\(768\) 0 0
\(769\) 33.6030i 1.21176i −0.795557 0.605878i \(-0.792822\pi\)
0.795557 0.605878i \(-0.207178\pi\)
\(770\) −3.59749 + 3.93788i −0.129644 + 0.141911i
\(771\) 0 0
\(772\) 21.6233 0.778239
\(773\) −13.9825 −0.502916 −0.251458 0.967868i \(-0.580910\pi\)
−0.251458 + 0.967868i \(0.580910\pi\)
\(774\) 0 0
\(775\) 10.0086i 0.359518i
\(776\) −2.71607 −0.0975014
\(777\) 0 0
\(778\) −16.7920 −0.602024
\(779\) 3.00319i 0.107600i
\(780\) 0 0
\(781\) −0.609527 −0.0218106
\(782\) 46.6454 1.66804
\(783\) 0 0
\(784\) −0.631122 6.97149i −0.0225401 0.248982i
\(785\) 23.7069i 0.846137i
\(786\) 0 0
\(787\) 20.4800i 0.730034i 0.931001 + 0.365017i \(0.118937\pi\)
−0.931001 + 0.365017i \(0.881063\pi\)
\(788\) 21.6914i 0.772723i
\(789\) 0 0
\(790\) 25.7461i 0.916006i
\(791\) 9.68757 + 8.85018i 0.344450 + 0.314676i
\(792\) 0 0
\(793\) 0 0
\(794\) −13.4856 −0.478585
\(795\) 0 0
\(796\) 3.94621i 0.139870i
\(797\) 38.1741 1.35220 0.676098 0.736812i \(-0.263670\pi\)
0.676098 + 0.736812i \(0.263670\pi\)
\(798\) 0 0
\(799\) 3.15494 0.111614
\(800\) 0.935886i 0.0330886i
\(801\) 0 0
\(802\) 13.5298 0.477754
\(803\) 0.532690 0.0187982
\(804\) 0 0
\(805\) −31.0138 28.3329i −1.09309 0.998605i
\(806\) 0 0
\(807\) 0 0
\(808\) 3.41352i 0.120087i
\(809\) 22.8503i 0.803374i 0.915777 + 0.401687i \(0.131576\pi\)
−0.915777 + 0.401687i \(0.868424\pi\)
\(810\) 0 0
\(811\) 27.1859i 0.954624i 0.878734 + 0.477312i \(0.158389\pi\)
−0.878734 + 0.477312i \(0.841611\pi\)
\(812\) −2.59084 2.36689i −0.0909206 0.0830614i
\(813\) 0 0
\(814\) 5.87575 0.205945
\(815\) −25.2570 −0.884713
\(816\) 0 0
\(817\) 1.03229i 0.0361152i
\(818\) 27.0606 0.946152
\(819\) 0 0
\(820\) −9.20211 −0.321352
\(821\) 22.6135i 0.789217i −0.918849 0.394608i \(-0.870880\pi\)
0.918849 0.394608i \(-0.129120\pi\)
\(822\) 0 0
\(823\) 36.9706 1.28871 0.644356 0.764725i \(-0.277125\pi\)
0.644356 + 0.764725i \(0.277125\pi\)
\(824\) −13.3681 −0.465700
\(825\) 0 0
\(826\) −7.87575 7.19498i −0.274033 0.250345i
\(827\) 36.8463i 1.28127i 0.767845 + 0.640636i \(0.221329\pi\)
−0.767845 + 0.640636i \(0.778671\pi\)
\(828\) 0 0
\(829\) 34.8858i 1.21163i −0.795604 0.605817i \(-0.792846\pi\)
0.795604 0.605817i \(-0.207154\pi\)
\(830\) 26.9496i 0.935436i
\(831\) 0 0
\(832\) 0 0
\(833\) 3.73792 + 41.2897i 0.129511 + 1.43061i
\(834\) 0 0
\(835\) −46.5128 −1.60964
\(836\) −0.657927 −0.0227549
\(837\) 0 0
\(838\) 10.7799i 0.372387i
\(839\) 41.3715 1.42830 0.714152 0.699991i \(-0.246813\pi\)
0.714152 + 0.699991i \(0.246813\pi\)
\(840\) 0 0
\(841\) 27.2408 0.939337
\(842\) 23.0948i 0.795900i
\(843\) 0 0
\(844\) −2.30277 −0.0792647
\(845\) −26.2075 −0.901567
\(846\) 0 0
\(847\) −1.78450 + 1.95335i −0.0613161 + 0.0671178i
\(848\) 7.44475i 0.255654i
\(849\) 0 0
\(850\) 5.54293i 0.190121i
\(851\) 46.2760i 1.58632i
\(852\) 0 0
\(853\) 6.66093i 0.228066i 0.993477 + 0.114033i \(0.0363770\pi\)
−0.993477 + 0.114033i \(0.963623\pi\)
\(854\) −0.446973 + 0.489264i −0.0152951 + 0.0167423i
\(855\) 0 0
\(856\) 14.4002 0.492190
\(857\) −19.6691 −0.671885 −0.335943 0.941882i \(-0.609055\pi\)
−0.335943 + 0.941882i \(0.609055\pi\)
\(858\) 0 0
\(859\) 33.8062i 1.15345i −0.816937 0.576726i \(-0.804330\pi\)
0.816937 0.576726i \(-0.195670\pi\)
\(860\) −3.16305 −0.107859
\(861\) 0 0
\(862\) 13.8508 0.471762
\(863\) 0.846317i 0.0288090i 0.999896 + 0.0144045i \(0.00458525\pi\)
−0.999896 + 0.0144045i \(0.995415\pi\)
\(864\) 0 0
\(865\) 35.7318 1.21492
\(866\) −36.3968 −1.23681
\(867\) 0 0
\(868\) −20.8895 19.0838i −0.709036 0.647747i
\(869\) 12.7711i 0.433230i
\(870\) 0 0
\(871\) 0 0
\(872\) 3.41140i 0.115524i
\(873\) 0 0
\(874\) 5.18167i 0.175273i
\(875\) 21.3543 23.3748i 0.721906 0.790212i
\(876\) 0 0
\(877\) 49.8039 1.68176 0.840879 0.541222i \(-0.182038\pi\)
0.840879 + 0.541222i \(0.182038\pi\)
\(878\) 24.9603 0.842369
\(879\) 0 0
\(880\) 2.01596i 0.0679582i
\(881\) −39.1207 −1.31801 −0.659004 0.752139i \(-0.729022\pi\)
−0.659004 + 0.752139i \(0.729022\pi\)
\(882\) 0 0
\(883\) 24.4501 0.822810 0.411405 0.911453i \(-0.365038\pi\)
0.411405 + 0.911453i \(0.365038\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −7.20610 −0.242094
\(887\) 28.3025 0.950307 0.475153 0.879903i \(-0.342393\pi\)
0.475153 + 0.879903i \(0.342393\pi\)
\(888\) 0 0
\(889\) 16.9615 18.5663i 0.568869 0.622695i
\(890\) 10.5285i 0.352915i
\(891\) 0 0
\(892\) 18.2571i 0.611294i
\(893\) 0.350471i 0.0117281i
\(894\) 0 0
\(895\) 7.68413i 0.256852i
\(896\) −1.95335 1.78450i −0.0652567 0.0596159i
\(897\) 0 0
\(898\) −31.8490 −1.06281
\(899\) −14.1844 −0.473075
\(900\) 0 0
\(901\) 44.0927i 1.46894i
\(902\) −4.56462 −0.151985
\(903\) 0 0
\(904\) 4.95947 0.164950
\(905\) 38.5574i 1.28169i
\(906\) 0 0
\(907\) 42.3244 1.40536 0.702680 0.711506i \(-0.251987\pi\)
0.702680 + 0.711506i \(0.251987\pi\)
\(908\) −20.0739 −0.666177
\(909\) 0 0
\(910\) 0 0
\(911\) 15.2622i 0.505661i −0.967511 0.252830i \(-0.918639\pi\)
0.967511 0.252830i \(-0.0813615\pi\)
\(912\) 0 0
\(913\) 13.3681i 0.442420i
\(914\) 10.4853i 0.346822i
\(915\) 0 0
\(916\) 14.6840i 0.485172i
\(917\) 34.0551 37.2773i 1.12460 1.23101i
\(918\) 0 0
\(919\) −50.2951 −1.65908 −0.829540 0.558447i \(-0.811398\pi\)
−0.829540 + 0.558447i \(0.811398\pi\)
\(920\) −15.8772 −0.523457
\(921\) 0 0
\(922\) 24.3881i 0.803179i
\(923\) 0 0
\(924\) 0 0
\(925\) −5.49903 −0.180807
\(926\) 22.6410i 0.744030i
\(927\) 0 0
\(928\) −1.32636 −0.0435399
\(929\) 8.43957 0.276893 0.138447 0.990370i \(-0.455789\pi\)
0.138447 + 0.990370i \(0.455789\pi\)
\(930\) 0 0
\(931\) 4.58674 0.415233i 0.150324 0.0136087i
\(932\) 2.48528i 0.0814081i
\(933\) 0 0
\(934\) 27.6355i 0.904261i
\(935\) 11.9399i 0.390475i
\(936\) 0 0
\(937\) 13.7782i 0.450115i −0.974345 0.225057i \(-0.927743\pi\)
0.974345 0.225057i \(-0.0722570\pi\)
\(938\) −15.2588 13.9399i −0.498219 0.455153i
\(939\) 0 0
\(940\) −1.07388 −0.0350262
\(941\) −28.6705 −0.934631 −0.467315 0.884091i \(-0.654779\pi\)
−0.467315 + 0.884091i \(0.654779\pi\)
\(942\) 0 0
\(943\) 35.9498i 1.17069i
\(944\) −4.03193 −0.131228
\(945\) 0 0
\(946\) −1.56900 −0.0510126
\(947\) 19.0609i 0.619397i −0.950835 0.309698i \(-0.899772\pi\)
0.950835 0.309698i \(-0.100228\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.615745 0.0199774
\(951\) 0 0
\(952\) 11.5690 + 10.5690i 0.374953 + 0.342543i
\(953\) 7.72400i 0.250205i 0.992144 + 0.125102i \(0.0399260\pi\)
−0.992144 + 0.125102i \(0.960074\pi\)
\(954\) 0 0
\(955\) 43.6788i 1.41341i
\(956\) 22.3610i 0.723208i
\(957\) 0 0
\(958\) 34.4166i 1.11195i
\(959\) 15.7328 + 14.3729i 0.508039 + 0.464124i
\(960\) 0 0
\(961\) −83.3663 −2.68923
\(962\) 0 0
\(963\) 0 0
\(964\) 3.45706i 0.111344i
\(965\) 43.5918 1.40327
\(966\) 0 0
\(967\) −41.7666 −1.34312 −0.671561 0.740949i \(-0.734376\pi\)
−0.671561 + 0.740949i \(0.734376\pi\)
\(968\) 1.00000i 0.0321412i
\(969\) 0 0
\(970\) −5.47551 −0.175808
\(971\) −52.8054 −1.69461 −0.847303 0.531110i \(-0.821775\pi\)
−0.847303 + 0.531110i \(0.821775\pi\)
\(972\) 0 0
\(973\) 37.8411 + 34.5702i 1.21313 + 1.10827i
\(974\) 15.2662i 0.489161i
\(975\) 0 0
\(976\) 0.250475i 0.00801751i
\(977\) 33.5481i 1.07330i 0.843806 + 0.536649i \(0.180310\pi\)
−0.843806 + 0.536649i \(0.819690\pi\)
\(978\) 0 0
\(979\) 5.22255i 0.166913i
\(980\) −1.27232 14.0543i −0.0406428 0.448948i
\(981\) 0 0
\(982\) 23.6273 0.753976
\(983\) −14.1077 −0.449967 −0.224984 0.974363i \(-0.572233\pi\)
−0.224984 + 0.974363i \(0.572233\pi\)
\(984\) 0 0
\(985\) 43.7291i 1.39332i
\(986\) 7.85557 0.250172
\(987\) 0 0
\(988\) 0 0
\(989\) 12.3571i 0.392931i
\(990\) 0 0
\(991\) 2.39626 0.0761197 0.0380599 0.999275i \(-0.487882\pi\)
0.0380599 + 0.999275i \(0.487882\pi\)
\(992\) −10.6942 −0.339542
\(993\) 0 0
\(994\) 1.08770 1.19062i 0.0344998 0.0377641i
\(995\) 7.95543i 0.252204i
\(996\) 0 0
\(997\) 50.6773i 1.60497i 0.596674 + 0.802483i \(0.296488\pi\)
−0.596674 + 0.802483i \(0.703512\pi\)
\(998\) 36.0484i 1.14109i
\(999\) 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 1386.2.g.a.881.10 yes 16
3.2 odd 2 inner 1386.2.g.a.881.7 yes 16
7.6 odd 2 inner 1386.2.g.a.881.15 yes 16
21.20 even 2 inner 1386.2.g.a.881.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.g.a.881.2 16 21.20 even 2 inner
1386.2.g.a.881.7 yes 16 3.2 odd 2 inner
1386.2.g.a.881.10 yes 16 1.1 even 1 trivial
1386.2.g.a.881.15 yes 16 7.6 odd 2 inner