Properties

Label 1380.2.i.a.1241.8
Level $1380$
Weight $2$
Character 1380.1241
Analytic conductor $11.019$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1380,2,Mod(1241,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.i (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: \(11.0193554789\)
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: \( x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.8
Root \(-0.0831550 + 1.73005i\) of defining polynomial
Character \(\chi\) \(=\) 1380.1241
Dual form 1380.2.i.a.1241.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.0831550 + 1.73005i) q^{3} -1.00000 q^{5} +4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} +O(q^{10})\) \(q+(0.0831550 + 1.73005i) q^{3} -1.00000 q^{5} +4.02092i q^{7} +(-2.98617 + 0.287725i) q^{9} +0.641443 q^{11} -2.89996 q^{13} +(-0.0831550 - 1.73005i) q^{15} -3.10358 q^{17} +1.33629i q^{19} +(-6.95640 + 0.334360i) q^{21} +(-3.05339 - 3.69822i) q^{23} +1.00000 q^{25} +(-0.746095 - 5.14231i) q^{27} -3.14000i q^{29} +6.11022 q^{31} +(0.0533392 + 1.10973i) q^{33} -4.02092i q^{35} +7.06427i q^{37} +(-0.241146 - 5.01709i) q^{39} -3.72186i q^{41} +1.20330i q^{43} +(2.98617 - 0.287725i) q^{45} +3.57780i q^{47} -9.16778 q^{49} +(-0.258078 - 5.36936i) q^{51} -5.12728 q^{53} -0.641443 q^{55} +(-2.31186 + 0.111119i) q^{57} -11.3428i q^{59} -0.921789i q^{61} +(-1.15692 - 12.0071i) q^{63} +2.89996 q^{65} -1.27284i q^{67} +(6.14421 - 5.59005i) q^{69} +1.44420i q^{71} -10.9062 q^{73} +(0.0831550 + 1.73005i) q^{75} +2.57919i q^{77} +3.08045i q^{79} +(8.83443 - 1.71839i) q^{81} -15.6650 q^{83} +3.10358 q^{85} +(5.43237 - 0.261107i) q^{87} +4.31384 q^{89} -11.6605i q^{91} +(0.508096 + 10.5710i) q^{93} -1.33629i q^{95} +13.1699i q^{97} +(-1.91546 + 0.184559i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.0831550 + 1.73005i 0.0480096 + 0.998847i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.02092i 1.51976i 0.650061 + 0.759882i \(0.274743\pi\)
−0.650061 + 0.759882i \(0.725257\pi\)
\(8\) 0 0
\(9\) −2.98617 + 0.287725i −0.995390 + 0.0959084i
\(10\) 0 0
\(11\) 0.641443 0.193402 0.0967012 0.995313i \(-0.469171\pi\)
0.0967012 + 0.995313i \(0.469171\pi\)
\(12\) 0 0
\(13\) −2.89996 −0.804305 −0.402152 0.915573i \(-0.631738\pi\)
−0.402152 + 0.915573i \(0.631738\pi\)
\(14\) 0 0
\(15\) −0.0831550 1.73005i −0.0214705 0.446698i
\(16\) 0 0
\(17\) −3.10358 −0.752729 −0.376364 0.926472i \(-0.622826\pi\)
−0.376364 + 0.926472i \(0.622826\pi\)
\(18\) 0 0
\(19\) 1.33629i 0.306567i 0.988182 + 0.153283i \(0.0489847\pi\)
−0.988182 + 0.153283i \(0.951015\pi\)
\(20\) 0 0
\(21\) −6.95640 + 0.334360i −1.51801 + 0.0729633i
\(22\) 0 0
\(23\) −3.05339 3.69822i −0.636676 0.771132i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −0.746095 5.14231i −0.143586 0.989638i
\(28\) 0 0
\(29\) 3.14000i 0.583084i −0.956558 0.291542i \(-0.905832\pi\)
0.956558 0.291542i \(-0.0941683\pi\)
\(30\) 0 0
\(31\) 6.11022 1.09743 0.548714 0.836010i \(-0.315118\pi\)
0.548714 + 0.836010i \(0.315118\pi\)
\(32\) 0 0
\(33\) 0.0533392 + 1.10973i 0.00928517 + 0.193179i
\(34\) 0 0
\(35\) 4.02092i 0.679659i
\(36\) 0 0
\(37\) 7.06427i 1.16136i 0.814132 + 0.580679i \(0.197213\pi\)
−0.814132 + 0.580679i \(0.802787\pi\)
\(38\) 0 0
\(39\) −0.241146 5.01709i −0.0386143 0.803377i
\(40\) 0 0
\(41\) 3.72186i 0.581256i −0.956836 0.290628i \(-0.906136\pi\)
0.956836 0.290628i \(-0.0938643\pi\)
\(42\) 0 0
\(43\) 1.20330i 0.183501i 0.995782 + 0.0917506i \(0.0292462\pi\)
−0.995782 + 0.0917506i \(0.970754\pi\)
\(44\) 0 0
\(45\) 2.98617 0.287725i 0.445152 0.0428916i
\(46\) 0 0
\(47\) 3.57780i 0.521876i 0.965356 + 0.260938i \(0.0840318\pi\)
−0.965356 + 0.260938i \(0.915968\pi\)
\(48\) 0 0
\(49\) −9.16778 −1.30968
\(50\) 0 0
\(51\) −0.258078 5.36936i −0.0361382 0.751861i
\(52\) 0 0
\(53\) −5.12728 −0.704286 −0.352143 0.935946i \(-0.614547\pi\)
−0.352143 + 0.935946i \(0.614547\pi\)
\(54\) 0 0
\(55\) −0.641443 −0.0864922
\(56\) 0 0
\(57\) −2.31186 + 0.111119i −0.306213 + 0.0147181i
\(58\) 0 0
\(59\) 11.3428i 1.47670i −0.674415 0.738352i \(-0.735604\pi\)
0.674415 0.738352i \(-0.264396\pi\)
\(60\) 0 0
\(61\) 0.921789i 0.118023i −0.998257 0.0590115i \(-0.981205\pi\)
0.998257 0.0590115i \(-0.0187949\pi\)
\(62\) 0 0
\(63\) −1.15692 12.0071i −0.145758 1.51276i
\(64\) 0 0
\(65\) 2.89996 0.359696
\(66\) 0 0
\(67\) 1.27284i 0.155502i −0.996973 0.0777509i \(-0.975226\pi\)
0.996973 0.0777509i \(-0.0247739\pi\)
\(68\) 0 0
\(69\) 6.14421 5.59005i 0.739676 0.672963i
\(70\) 0 0
\(71\) 1.44420i 0.171395i 0.996321 + 0.0856976i \(0.0273119\pi\)
−0.996321 + 0.0856976i \(0.972688\pi\)
\(72\) 0 0
\(73\) −10.9062 −1.27647 −0.638235 0.769841i \(-0.720335\pi\)
−0.638235 + 0.769841i \(0.720335\pi\)
\(74\) 0 0
\(75\) 0.0831550 + 1.73005i 0.00960192 + 0.199769i
\(76\) 0 0
\(77\) 2.57919i 0.293926i
\(78\) 0 0
\(79\) 3.08045i 0.346577i 0.984871 + 0.173289i \(0.0554393\pi\)
−0.984871 + 0.173289i \(0.944561\pi\)
\(80\) 0 0
\(81\) 8.83443 1.71839i 0.981603 0.190933i
\(82\) 0 0
\(83\) −15.6650 −1.71946 −0.859728 0.510752i \(-0.829367\pi\)
−0.859728 + 0.510752i \(0.829367\pi\)
\(84\) 0 0
\(85\) 3.10358 0.336631
\(86\) 0 0
\(87\) 5.43237 0.261107i 0.582411 0.0279936i
\(88\) 0 0
\(89\) 4.31384 0.457266 0.228633 0.973513i \(-0.426574\pi\)
0.228633 + 0.973513i \(0.426574\pi\)
\(90\) 0 0
\(91\) 11.6605i 1.22235i
\(92\) 0 0
\(93\) 0.508096 + 10.5710i 0.0526871 + 1.09616i
\(94\) 0 0
\(95\) 1.33629i 0.137101i
\(96\) 0 0
\(97\) 13.1699i 1.33720i 0.743620 + 0.668602i \(0.233107\pi\)
−0.743620 + 0.668602i \(0.766893\pi\)
\(98\) 0 0
\(99\) −1.91546 + 0.184559i −0.192511 + 0.0185489i
\(100\) 0 0
\(101\) 7.15864i 0.712312i 0.934427 + 0.356156i \(0.115913\pi\)
−0.934427 + 0.356156i \(0.884087\pi\)
\(102\) 0 0
\(103\) 0.510025i 0.0502542i −0.999684 0.0251271i \(-0.992001\pi\)
0.999684 0.0251271i \(-0.00799905\pi\)
\(104\) 0 0
\(105\) 6.95640 0.334360i 0.678876 0.0326302i
\(106\) 0 0
\(107\) −14.0806 −1.36122 −0.680611 0.732645i \(-0.738286\pi\)
−0.680611 + 0.732645i \(0.738286\pi\)
\(108\) 0 0
\(109\) 12.5760i 1.20456i −0.798283 0.602282i \(-0.794258\pi\)
0.798283 0.602282i \(-0.205742\pi\)
\(110\) 0 0
\(111\) −12.2216 + 0.587430i −1.16002 + 0.0557563i
\(112\) 0 0
\(113\) −0.170942 −0.0160809 −0.00804045 0.999968i \(-0.502559\pi\)
−0.00804045 + 0.999968i \(0.502559\pi\)
\(114\) 0 0
\(115\) 3.05339 + 3.69822i 0.284730 + 0.344861i
\(116\) 0 0
\(117\) 8.65978 0.834393i 0.800597 0.0771396i
\(118\) 0 0
\(119\) 12.4792i 1.14397i
\(120\) 0 0
\(121\) −10.5886 −0.962596
\(122\) 0 0
\(123\) 6.43901 0.309491i 0.580586 0.0279059i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 15.8686 1.40811 0.704056 0.710145i \(-0.251371\pi\)
0.704056 + 0.710145i \(0.251371\pi\)
\(128\) 0 0
\(129\) −2.08177 + 0.100060i −0.183290 + 0.00880981i
\(130\) 0 0
\(131\) 8.60285i 0.751634i 0.926694 + 0.375817i \(0.122638\pi\)
−0.926694 + 0.375817i \(0.877362\pi\)
\(132\) 0 0
\(133\) −5.37312 −0.465909
\(134\) 0 0
\(135\) 0.746095 + 5.14231i 0.0642137 + 0.442579i
\(136\) 0 0
\(137\) 7.74794 0.661951 0.330975 0.943639i \(-0.392622\pi\)
0.330975 + 0.943639i \(0.392622\pi\)
\(138\) 0 0
\(139\) −10.2656 −0.870715 −0.435358 0.900258i \(-0.643378\pi\)
−0.435358 + 0.900258i \(0.643378\pi\)
\(140\) 0 0
\(141\) −6.18979 + 0.297512i −0.521274 + 0.0250551i
\(142\) 0 0
\(143\) −1.86016 −0.155554
\(144\) 0 0
\(145\) 3.14000i 0.260763i
\(146\) 0 0
\(147\) −0.762347 15.8608i −0.0628774 1.30817i
\(148\) 0 0
\(149\) 9.47232 0.776003 0.388001 0.921659i \(-0.373166\pi\)
0.388001 + 0.921659i \(0.373166\pi\)
\(150\) 0 0
\(151\) −13.8802 −1.12956 −0.564779 0.825242i \(-0.691038\pi\)
−0.564779 + 0.825242i \(0.691038\pi\)
\(152\) 0 0
\(153\) 9.26782 0.892979i 0.749259 0.0721931i
\(154\) 0 0
\(155\) −6.11022 −0.490785
\(156\) 0 0
\(157\) 12.9188i 1.03103i −0.856880 0.515516i \(-0.827600\pi\)
0.856880 0.515516i \(-0.172400\pi\)
\(158\) 0 0
\(159\) −0.426359 8.87047i −0.0338125 0.703474i
\(160\) 0 0
\(161\) 14.8702 12.2774i 1.17194 0.967597i
\(162\) 0 0
\(163\) 21.6758 1.69778 0.848892 0.528567i \(-0.177270\pi\)
0.848892 + 0.528567i \(0.177270\pi\)
\(164\) 0 0
\(165\) −0.0533392 1.10973i −0.00415245 0.0863924i
\(166\) 0 0
\(167\) 3.64677i 0.282196i −0.989996 0.141098i \(-0.954937\pi\)
0.989996 0.141098i \(-0.0450632\pi\)
\(168\) 0 0
\(169\) −4.59022 −0.353094
\(170\) 0 0
\(171\) −0.384485 3.99040i −0.0294023 0.305153i
\(172\) 0 0
\(173\) 9.85550i 0.749300i 0.927166 + 0.374650i \(0.122237\pi\)
−0.927166 + 0.374650i \(0.877763\pi\)
\(174\) 0 0
\(175\) 4.02092i 0.303953i
\(176\) 0 0
\(177\) 19.6236 0.943210i 1.47500 0.0708960i
\(178\) 0 0
\(179\) 16.9605i 1.26769i 0.773460 + 0.633845i \(0.218524\pi\)
−0.773460 + 0.633845i \(0.781476\pi\)
\(180\) 0 0
\(181\) 25.5650i 1.90023i 0.311893 + 0.950117i \(0.399037\pi\)
−0.311893 + 0.950117i \(0.600963\pi\)
\(182\) 0 0
\(183\) 1.59474 0.0766514i 0.117887 0.00566624i
\(184\) 0 0
\(185\) 7.06427i 0.519375i
\(186\) 0 0
\(187\) −1.99077 −0.145580
\(188\) 0 0
\(189\) 20.6768 2.99999i 1.50402 0.218217i
\(190\) 0 0
\(191\) −12.4794 −0.902981 −0.451491 0.892276i \(-0.649108\pi\)
−0.451491 + 0.892276i \(0.649108\pi\)
\(192\) 0 0
\(193\) −21.0097 −1.51231 −0.756155 0.654392i \(-0.772924\pi\)
−0.756155 + 0.654392i \(0.772924\pi\)
\(194\) 0 0
\(195\) 0.241146 + 5.01709i 0.0172689 + 0.359281i
\(196\) 0 0
\(197\) 9.84641i 0.701528i 0.936464 + 0.350764i \(0.114078\pi\)
−0.936464 + 0.350764i \(0.885922\pi\)
\(198\) 0 0
\(199\) 8.69855i 0.616624i 0.951285 + 0.308312i \(0.0997640\pi\)
−0.951285 + 0.308312i \(0.900236\pi\)
\(200\) 0 0
\(201\) 2.20208 0.105843i 0.155323 0.00746558i
\(202\) 0 0
\(203\) 12.6257 0.886150
\(204\) 0 0
\(205\) 3.72186i 0.259946i
\(206\) 0 0
\(207\) 10.1820 + 10.1650i 0.707699 + 0.706514i
\(208\) 0 0
\(209\) 0.857156i 0.0592907i
\(210\) 0 0
\(211\) −19.8422 −1.36600 −0.682998 0.730420i \(-0.739324\pi\)
−0.682998 + 0.730420i \(0.739324\pi\)
\(212\) 0 0
\(213\) −2.49855 + 0.120093i −0.171197 + 0.00822861i
\(214\) 0 0
\(215\) 1.20330i 0.0820642i
\(216\) 0 0
\(217\) 24.5687i 1.66783i
\(218\) 0 0
\(219\) −0.906903 18.8683i −0.0612828 1.27500i
\(220\) 0 0
\(221\) 9.00027 0.605423
\(222\) 0 0
\(223\) −1.99786 −0.133786 −0.0668932 0.997760i \(-0.521309\pi\)
−0.0668932 + 0.997760i \(0.521309\pi\)
\(224\) 0 0
\(225\) −2.98617 + 0.287725i −0.199078 + 0.0191817i
\(226\) 0 0
\(227\) 1.61196 0.106990 0.0534949 0.998568i \(-0.482964\pi\)
0.0534949 + 0.998568i \(0.482964\pi\)
\(228\) 0 0
\(229\) 11.7830i 0.778641i −0.921102 0.389321i \(-0.872710\pi\)
0.921102 0.389321i \(-0.127290\pi\)
\(230\) 0 0
\(231\) −4.46214 + 0.214473i −0.293587 + 0.0141113i
\(232\) 0 0
\(233\) 9.04517i 0.592569i −0.955100 0.296285i \(-0.904252\pi\)
0.955100 0.296285i \(-0.0957477\pi\)
\(234\) 0 0
\(235\) 3.57780i 0.233390i
\(236\) 0 0
\(237\) −5.32934 + 0.256155i −0.346178 + 0.0166390i
\(238\) 0 0
\(239\) 13.0937i 0.846962i 0.905905 + 0.423481i \(0.139192\pi\)
−0.905905 + 0.423481i \(0.860808\pi\)
\(240\) 0 0
\(241\) 21.5418i 1.38763i 0.720155 + 0.693813i \(0.244071\pi\)
−0.720155 + 0.693813i \(0.755929\pi\)
\(242\) 0 0
\(243\) 3.70754 + 15.1411i 0.237839 + 0.971305i
\(244\) 0 0
\(245\) 9.16778 0.585708
\(246\) 0 0
\(247\) 3.87520i 0.246573i
\(248\) 0 0
\(249\) −1.30262 27.1013i −0.0825504 1.71747i
\(250\) 0 0
\(251\) 2.74004 0.172950 0.0864750 0.996254i \(-0.472440\pi\)
0.0864750 + 0.996254i \(0.472440\pi\)
\(252\) 0 0
\(253\) −1.95858 2.37220i −0.123135 0.149139i
\(254\) 0 0
\(255\) 0.258078 + 5.36936i 0.0161615 + 0.336242i
\(256\) 0 0
\(257\) 26.5539i 1.65639i 0.560442 + 0.828194i \(0.310631\pi\)
−0.560442 + 0.828194i \(0.689369\pi\)
\(258\) 0 0
\(259\) −28.4048 −1.76499
\(260\) 0 0
\(261\) 0.903458 + 9.37658i 0.0559227 + 0.580396i
\(262\) 0 0
\(263\) −4.09330 −0.252404 −0.126202 0.992005i \(-0.540279\pi\)
−0.126202 + 0.992005i \(0.540279\pi\)
\(264\) 0 0
\(265\) 5.12728 0.314966
\(266\) 0 0
\(267\) 0.358718 + 7.46317i 0.0219532 + 0.456739i
\(268\) 0 0
\(269\) 5.94400i 0.362412i 0.983445 + 0.181206i \(0.0580001\pi\)
−0.983445 + 0.181206i \(0.942000\pi\)
\(270\) 0 0
\(271\) 2.18049 0.132456 0.0662278 0.997805i \(-0.478904\pi\)
0.0662278 + 0.997805i \(0.478904\pi\)
\(272\) 0 0
\(273\) 20.1733 0.969630i 1.22094 0.0586847i
\(274\) 0 0
\(275\) 0.641443 0.0386805
\(276\) 0 0
\(277\) 25.4414 1.52862 0.764312 0.644846i \(-0.223079\pi\)
0.764312 + 0.644846i \(0.223079\pi\)
\(278\) 0 0
\(279\) −18.2462 + 1.75807i −1.09237 + 0.105253i
\(280\) 0 0
\(281\) 26.1951 1.56267 0.781335 0.624112i \(-0.214539\pi\)
0.781335 + 0.624112i \(0.214539\pi\)
\(282\) 0 0
\(283\) 10.9054i 0.648261i 0.946012 + 0.324130i \(0.105072\pi\)
−0.946012 + 0.324130i \(0.894928\pi\)
\(284\) 0 0
\(285\) 2.31186 0.111119i 0.136943 0.00658215i
\(286\) 0 0
\(287\) 14.9653 0.883373
\(288\) 0 0
\(289\) −7.36779 −0.433399
\(290\) 0 0
\(291\) −22.7847 + 1.09515i −1.33566 + 0.0641986i
\(292\) 0 0
\(293\) −27.0813 −1.58211 −0.791054 0.611746i \(-0.790468\pi\)
−0.791054 + 0.611746i \(0.790468\pi\)
\(294\) 0 0
\(295\) 11.3428i 0.660402i
\(296\) 0 0
\(297\) −0.478578 3.29850i −0.0277699 0.191398i
\(298\) 0 0
\(299\) 8.85471 + 10.7247i 0.512081 + 0.620225i
\(300\) 0 0
\(301\) −4.83836 −0.278878
\(302\) 0 0
\(303\) −12.3848 + 0.595277i −0.711490 + 0.0341978i
\(304\) 0 0
\(305\) 0.921789i 0.0527815i
\(306\) 0 0
\(307\) 14.8147 0.845517 0.422759 0.906242i \(-0.361062\pi\)
0.422759 + 0.906242i \(0.361062\pi\)
\(308\) 0 0
\(309\) 0.882370 0.0424111i 0.0501963 0.00241269i
\(310\) 0 0
\(311\) 0.729791i 0.0413826i 0.999786 + 0.0206913i \(0.00658672\pi\)
−0.999786 + 0.0206913i \(0.993413\pi\)
\(312\) 0 0
\(313\) 17.2658i 0.975921i 0.872866 + 0.487960i \(0.162259\pi\)
−0.872866 + 0.487960i \(0.837741\pi\)
\(314\) 0 0
\(315\) 1.15692 + 12.0071i 0.0651851 + 0.676526i
\(316\) 0 0
\(317\) 26.1142i 1.46672i 0.679841 + 0.733360i \(0.262049\pi\)
−0.679841 + 0.733360i \(0.737951\pi\)
\(318\) 0 0
\(319\) 2.01413i 0.112770i
\(320\) 0 0
\(321\) −1.17087 24.3602i −0.0653517 1.35965i
\(322\) 0 0
\(323\) 4.14729i 0.230762i
\(324\) 0 0
\(325\) −2.89996 −0.160861
\(326\) 0 0
\(327\) 21.7572 1.04576i 1.20318 0.0578306i
\(328\) 0 0
\(329\) −14.3860 −0.793129
\(330\) 0 0
\(331\) 3.80035 0.208886 0.104443 0.994531i \(-0.466694\pi\)
0.104443 + 0.994531i \(0.466694\pi\)
\(332\) 0 0
\(333\) −2.03257 21.0951i −0.111384 1.15601i
\(334\) 0 0
\(335\) 1.27284i 0.0695426i
\(336\) 0 0
\(337\) 31.5995i 1.72133i 0.509169 + 0.860667i \(0.329953\pi\)
−0.509169 + 0.860667i \(0.670047\pi\)
\(338\) 0 0
\(339\) −0.0142147 0.295740i −0.000772038 0.0160624i
\(340\) 0 0
\(341\) 3.91936 0.212245
\(342\) 0 0
\(343\) 8.71648i 0.470646i
\(344\) 0 0
\(345\) −6.14421 + 5.59005i −0.330793 + 0.300958i
\(346\) 0 0
\(347\) 35.4098i 1.90090i −0.310879 0.950449i \(-0.600624\pi\)
0.310879 0.950449i \(-0.399376\pi\)
\(348\) 0 0
\(349\) −6.70111 −0.358702 −0.179351 0.983785i \(-0.557400\pi\)
−0.179351 + 0.983785i \(0.557400\pi\)
\(350\) 0 0
\(351\) 2.16365 + 14.9125i 0.115487 + 0.795970i
\(352\) 0 0
\(353\) 16.1005i 0.856942i 0.903556 + 0.428471i \(0.140948\pi\)
−0.903556 + 0.428471i \(0.859052\pi\)
\(354\) 0 0
\(355\) 1.44420i 0.0766502i
\(356\) 0 0
\(357\) 21.5898 1.03771i 1.14265 0.0549215i
\(358\) 0 0
\(359\) −4.18835 −0.221053 −0.110526 0.993873i \(-0.535254\pi\)
−0.110526 + 0.993873i \(0.535254\pi\)
\(360\) 0 0
\(361\) 17.2143 0.906017
\(362\) 0 0
\(363\) −0.880491 18.3188i −0.0462138 0.961486i
\(364\) 0 0
\(365\) 10.9062 0.570855
\(366\) 0 0
\(367\) 21.2504i 1.10926i 0.832097 + 0.554630i \(0.187140\pi\)
−0.832097 + 0.554630i \(0.812860\pi\)
\(368\) 0 0
\(369\) 1.07087 + 11.1141i 0.0557474 + 0.578577i
\(370\) 0 0
\(371\) 20.6164i 1.07035i
\(372\) 0 0
\(373\) 30.7001i 1.58959i −0.606877 0.794796i \(-0.707578\pi\)
0.606877 0.794796i \(-0.292422\pi\)
\(374\) 0 0
\(375\) −0.0831550 1.73005i −0.00429411 0.0893396i
\(376\) 0 0
\(377\) 9.10589i 0.468977i
\(378\) 0 0
\(379\) 30.0529i 1.54371i −0.635797 0.771857i \(-0.719328\pi\)
0.635797 0.771857i \(-0.280672\pi\)
\(380\) 0 0
\(381\) 1.31955 + 27.4535i 0.0676028 + 1.40649i
\(382\) 0 0
\(383\) 26.4414 1.35109 0.675546 0.737318i \(-0.263908\pi\)
0.675546 + 0.737318i \(0.263908\pi\)
\(384\) 0 0
\(385\) 2.57919i 0.131448i
\(386\) 0 0
\(387\) −0.346219 3.59325i −0.0175993 0.182655i
\(388\) 0 0
\(389\) −15.8089 −0.801542 −0.400771 0.916178i \(-0.631258\pi\)
−0.400771 + 0.916178i \(0.631258\pi\)
\(390\) 0 0
\(391\) 9.47644 + 11.4777i 0.479244 + 0.580453i
\(392\) 0 0
\(393\) −14.8834 + 0.715370i −0.750767 + 0.0360856i
\(394\) 0 0
\(395\) 3.08045i 0.154994i
\(396\) 0 0
\(397\) 12.5795 0.631348 0.315674 0.948868i \(-0.397769\pi\)
0.315674 + 0.948868i \(0.397769\pi\)
\(398\) 0 0
\(399\) −0.446802 9.29579i −0.0223681 0.465372i
\(400\) 0 0
\(401\) 34.7830 1.73698 0.868489 0.495708i \(-0.165091\pi\)
0.868489 + 0.495708i \(0.165091\pi\)
\(402\) 0 0
\(403\) −17.7194 −0.882667
\(404\) 0 0
\(405\) −8.83443 + 1.71839i −0.438986 + 0.0853877i
\(406\) 0 0
\(407\) 4.53133i 0.224610i
\(408\) 0 0
\(409\) −5.98209 −0.295795 −0.147898 0.989003i \(-0.547251\pi\)
−0.147898 + 0.989003i \(0.547251\pi\)
\(410\) 0 0
\(411\) 0.644280 + 13.4043i 0.0317800 + 0.661188i
\(412\) 0 0
\(413\) 45.6084 2.24424
\(414\) 0 0
\(415\) 15.6650 0.768964
\(416\) 0 0
\(417\) −0.853635 17.7600i −0.0418027 0.869711i
\(418\) 0 0
\(419\) −8.04481 −0.393015 −0.196507 0.980502i \(-0.562960\pi\)
−0.196507 + 0.980502i \(0.562960\pi\)
\(420\) 0 0
\(421\) 13.4364i 0.654849i −0.944877 0.327425i \(-0.893819\pi\)
0.944877 0.327425i \(-0.106181\pi\)
\(422\) 0 0
\(423\) −1.02942 10.6839i −0.0500523 0.519470i
\(424\) 0 0
\(425\) −3.10358 −0.150546
\(426\) 0 0
\(427\) 3.70644 0.179367
\(428\) 0 0
\(429\) −0.154682 3.21818i −0.00746811 0.155375i
\(430\) 0 0
\(431\) −2.73846 −0.131907 −0.0659534 0.997823i \(-0.521009\pi\)
−0.0659534 + 0.997823i \(0.521009\pi\)
\(432\) 0 0
\(433\) 19.0187i 0.913979i −0.889472 0.456989i \(-0.848928\pi\)
0.889472 0.456989i \(-0.151072\pi\)
\(434\) 0 0
\(435\) −5.43237 + 0.261107i −0.260462 + 0.0125191i
\(436\) 0 0
\(437\) 4.94190 4.08022i 0.236403 0.195183i
\(438\) 0 0
\(439\) 24.0581 1.14823 0.574114 0.818775i \(-0.305346\pi\)
0.574114 + 0.818775i \(0.305346\pi\)
\(440\) 0 0
\(441\) 27.3766 2.63780i 1.30365 0.125610i
\(442\) 0 0
\(443\) 16.5450i 0.786074i −0.919523 0.393037i \(-0.871424\pi\)
0.919523 0.393037i \(-0.128576\pi\)
\(444\) 0 0
\(445\) −4.31384 −0.204496
\(446\) 0 0
\(447\) 0.787671 + 16.3876i 0.0372556 + 0.775108i
\(448\) 0 0
\(449\) 18.9142i 0.892617i 0.894879 + 0.446309i \(0.147262\pi\)
−0.894879 + 0.446309i \(0.852738\pi\)
\(450\) 0 0
\(451\) 2.38736i 0.112416i
\(452\) 0 0
\(453\) −1.15421 24.0135i −0.0542296 1.12825i
\(454\) 0 0
\(455\) 11.6605i 0.546653i
\(456\) 0 0
\(457\) 3.91674i 0.183217i 0.995795 + 0.0916087i \(0.0292009\pi\)
−0.995795 + 0.0916087i \(0.970799\pi\)
\(458\) 0 0
\(459\) 2.31557 + 15.9596i 0.108081 + 0.744929i
\(460\) 0 0
\(461\) 21.7358i 1.01234i −0.862435 0.506168i \(-0.831062\pi\)
0.862435 0.506168i \(-0.168938\pi\)
\(462\) 0 0
\(463\) −7.15417 −0.332483 −0.166241 0.986085i \(-0.553163\pi\)
−0.166241 + 0.986085i \(0.553163\pi\)
\(464\) 0 0
\(465\) −0.508096 10.5710i −0.0235624 0.490219i
\(466\) 0 0
\(467\) −35.0147 −1.62029 −0.810143 0.586232i \(-0.800611\pi\)
−0.810143 + 0.586232i \(0.800611\pi\)
\(468\) 0 0
\(469\) 5.11798 0.236326
\(470\) 0 0
\(471\) 22.3502 1.07426i 1.02984 0.0494994i
\(472\) 0 0
\(473\) 0.771847i 0.0354896i
\(474\) 0 0
\(475\) 1.33629i 0.0613133i
\(476\) 0 0
\(477\) 15.3109 1.47525i 0.701039 0.0675470i
\(478\) 0 0
\(479\) 12.5386 0.572903 0.286451 0.958095i \(-0.407524\pi\)
0.286451 + 0.958095i \(0.407524\pi\)
\(480\) 0 0
\(481\) 20.4861i 0.934086i
\(482\) 0 0
\(483\) 22.4771 + 24.7054i 1.02275 + 1.12413i
\(484\) 0 0
\(485\) 13.1699i 0.598016i
\(486\) 0 0
\(487\) 16.3582 0.741262 0.370631 0.928780i \(-0.379141\pi\)
0.370631 + 0.928780i \(0.379141\pi\)
\(488\) 0 0
\(489\) 1.80246 + 37.5004i 0.0815099 + 1.69583i
\(490\) 0 0
\(491\) 12.4930i 0.563803i −0.959443 0.281902i \(-0.909035\pi\)
0.959443 0.281902i \(-0.0909651\pi\)
\(492\) 0 0
\(493\) 9.74525i 0.438904i
\(494\) 0 0
\(495\) 1.91546 0.184559i 0.0860935 0.00829533i
\(496\) 0 0
\(497\) −5.80701 −0.260480
\(498\) 0 0
\(499\) −25.5989 −1.14596 −0.572981 0.819568i \(-0.694213\pi\)
−0.572981 + 0.819568i \(0.694213\pi\)
\(500\) 0 0
\(501\) 6.30911 0.303247i 0.281870 0.0135481i
\(502\) 0 0
\(503\) 38.4303 1.71352 0.856762 0.515712i \(-0.172472\pi\)
0.856762 + 0.515712i \(0.172472\pi\)
\(504\) 0 0
\(505\) 7.15864i 0.318556i
\(506\) 0 0
\(507\) −0.381700 7.94132i −0.0169519 0.352687i
\(508\) 0 0
\(509\) 21.0597i 0.933455i 0.884401 + 0.466728i \(0.154567\pi\)
−0.884401 + 0.466728i \(0.845433\pi\)
\(510\) 0 0
\(511\) 43.8528i 1.93993i
\(512\) 0 0
\(513\) 6.87163 0.997002i 0.303390 0.0440187i
\(514\) 0 0
\(515\) 0.510025i 0.0224744i
\(516\) 0 0
\(517\) 2.29496i 0.100932i
\(518\) 0 0
\(519\) −17.0505 + 0.819534i −0.748436 + 0.0359736i
\(520\) 0 0
\(521\) −29.4599 −1.29066 −0.645331 0.763903i \(-0.723280\pi\)
−0.645331 + 0.763903i \(0.723280\pi\)
\(522\) 0 0
\(523\) 24.3365i 1.06416i −0.846694 0.532080i \(-0.821411\pi\)
0.846694 0.532080i \(-0.178589\pi\)
\(524\) 0 0
\(525\) −6.95640 + 0.334360i −0.303602 + 0.0145927i
\(526\) 0 0
\(527\) −18.9636 −0.826066
\(528\) 0 0
\(529\) −4.35364 + 22.5842i −0.189289 + 0.981921i
\(530\) 0 0
\(531\) 3.26361 + 33.8715i 0.141628 + 1.46990i
\(532\) 0 0
\(533\) 10.7932i 0.467507i
\(534\) 0 0
\(535\) 14.0806 0.608757
\(536\) 0 0
\(537\) −29.3427 + 1.41035i −1.26623 + 0.0608613i
\(538\) 0 0
\(539\) −5.88061 −0.253296
\(540\) 0 0
\(541\) 13.0056 0.559157 0.279578 0.960123i \(-0.409805\pi\)
0.279578 + 0.960123i \(0.409805\pi\)
\(542\) 0 0
\(543\) −44.2289 + 2.12586i −1.89804 + 0.0912295i
\(544\) 0 0
\(545\) 12.5760i 0.538698i
\(546\) 0 0
\(547\) −16.2330 −0.694073 −0.347037 0.937852i \(-0.612812\pi\)
−0.347037 + 0.937852i \(0.612812\pi\)
\(548\) 0 0
\(549\) 0.265222 + 2.75262i 0.0113194 + 0.117479i
\(550\) 0 0
\(551\) 4.19596 0.178754
\(552\) 0 0
\(553\) −12.3862 −0.526716
\(554\) 0 0
\(555\) 12.2216 0.587430i 0.518776 0.0249350i
\(556\) 0 0
\(557\) −2.76745 −0.117261 −0.0586303 0.998280i \(-0.518673\pi\)
−0.0586303 + 0.998280i \(0.518673\pi\)
\(558\) 0 0
\(559\) 3.48952i 0.147591i
\(560\) 0 0
\(561\) −0.165543 3.44414i −0.00698921 0.145412i
\(562\) 0 0
\(563\) −32.3086 −1.36165 −0.680823 0.732448i \(-0.738378\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(564\) 0 0
\(565\) 0.170942 0.00719160
\(566\) 0 0
\(567\) 6.90952 + 35.5225i 0.290173 + 1.49181i
\(568\) 0 0
\(569\) −43.6500 −1.82990 −0.914952 0.403563i \(-0.867772\pi\)
−0.914952 + 0.403563i \(0.867772\pi\)
\(570\) 0 0
\(571\) 19.2922i 0.807352i −0.914902 0.403676i \(-0.867732\pi\)
0.914902 0.403676i \(-0.132268\pi\)
\(572\) 0 0
\(573\) −1.03773 21.5901i −0.0433517 0.901940i
\(574\) 0 0
\(575\) −3.05339 3.69822i −0.127335 0.154226i
\(576\) 0 0
\(577\) −32.1157 −1.33700 −0.668498 0.743714i \(-0.733062\pi\)
−0.668498 + 0.743714i \(0.733062\pi\)
\(578\) 0 0
\(579\) −1.74706 36.3479i −0.0726054 1.51057i
\(580\) 0 0
\(581\) 62.9877i 2.61317i
\(582\) 0 0
\(583\) −3.28886 −0.136211
\(584\) 0 0
\(585\) −8.65978 + 0.834393i −0.358038 + 0.0344979i
\(586\) 0 0
\(587\) 7.67739i 0.316880i 0.987369 + 0.158440i \(0.0506464\pi\)
−0.987369 + 0.158440i \(0.949354\pi\)
\(588\) 0 0
\(589\) 8.16505i 0.336435i
\(590\) 0 0
\(591\) −17.0348 + 0.818779i −0.700719 + 0.0336800i
\(592\) 0 0
\(593\) 4.82648i 0.198200i −0.995077 0.0990998i \(-0.968404\pi\)
0.995077 0.0990998i \(-0.0315963\pi\)
\(594\) 0 0
\(595\) 12.4792i 0.511599i
\(596\) 0 0
\(597\) −15.0490 + 0.723328i −0.615913 + 0.0296039i
\(598\) 0 0
\(599\) 32.9192i 1.34504i 0.740078 + 0.672521i \(0.234789\pi\)
−0.740078 + 0.672521i \(0.765211\pi\)
\(600\) 0 0
\(601\) 3.84513 0.156846 0.0784231 0.996920i \(-0.475011\pi\)
0.0784231 + 0.996920i \(0.475011\pi\)
\(602\) 0 0
\(603\) 0.366228 + 3.80091i 0.0149139 + 0.154785i
\(604\) 0 0
\(605\) 10.5886 0.430486
\(606\) 0 0
\(607\) 5.90559 0.239700 0.119850 0.992792i \(-0.461759\pi\)
0.119850 + 0.992792i \(0.461759\pi\)
\(608\) 0 0
\(609\) 1.04989 + 21.8431i 0.0425437 + 0.885128i
\(610\) 0 0
\(611\) 10.3755i 0.419747i
\(612\) 0 0
\(613\) 5.79086i 0.233890i −0.993138 0.116945i \(-0.962690\pi\)
0.993138 0.116945i \(-0.0373102\pi\)
\(614\) 0 0
\(615\) −6.43901 + 0.309491i −0.259646 + 0.0124799i
\(616\) 0 0
\(617\) −22.1573 −0.892018 −0.446009 0.895028i \(-0.647155\pi\)
−0.446009 + 0.895028i \(0.647155\pi\)
\(618\) 0 0
\(619\) 44.6942i 1.79641i −0.439574 0.898206i \(-0.644871\pi\)
0.439574 0.898206i \(-0.355129\pi\)
\(620\) 0 0
\(621\) −16.7393 + 18.4607i −0.671723 + 0.740802i
\(622\) 0 0
\(623\) 17.3456i 0.694937i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.48293 + 0.0712768i −0.0592223 + 0.00284652i
\(628\) 0 0
\(629\) 21.9245i 0.874188i
\(630\) 0 0
\(631\) 8.96253i 0.356793i −0.983959 0.178396i \(-0.942909\pi\)
0.983959 0.178396i \(-0.0570909\pi\)
\(632\) 0 0
\(633\) −1.64998 34.3282i −0.0655809 1.36442i
\(634\) 0 0
\(635\) −15.8686 −0.629727
\(636\) 0 0
\(637\) 26.5862 1.05338
\(638\) 0 0
\(639\) −0.415533 4.31263i −0.0164382 0.170605i
\(640\) 0 0
\(641\) −35.2004 −1.39033 −0.695166 0.718849i \(-0.744669\pi\)
−0.695166 + 0.718849i \(0.744669\pi\)
\(642\) 0 0
\(643\) 38.1653i 1.50509i 0.658538 + 0.752547i \(0.271175\pi\)
−0.658538 + 0.752547i \(0.728825\pi\)
\(644\) 0 0
\(645\) 2.08177 0.100060i 0.0819696 0.00393987i
\(646\) 0 0
\(647\) 16.4962i 0.648533i −0.945966 0.324266i \(-0.894883\pi\)
0.945966 0.324266i \(-0.105117\pi\)
\(648\) 0 0
\(649\) 7.27575i 0.285598i
\(650\) 0 0
\(651\) −42.5052 + 2.04301i −1.66591 + 0.0800719i
\(652\) 0 0
\(653\) 22.1280i 0.865934i 0.901410 + 0.432967i \(0.142533\pi\)
−0.901410 + 0.432967i \(0.857467\pi\)
\(654\) 0 0
\(655\) 8.60285i 0.336141i
\(656\) 0 0
\(657\) 32.5677 3.13798i 1.27059 0.122424i
\(658\) 0 0
\(659\) −42.8854 −1.67058 −0.835288 0.549813i \(-0.814699\pi\)
−0.835288 + 0.549813i \(0.814699\pi\)
\(660\) 0 0
\(661\) 20.0140i 0.778455i 0.921142 + 0.389228i \(0.127258\pi\)
−0.921142 + 0.389228i \(0.872742\pi\)
\(662\) 0 0
\(663\) 0.748418 + 15.5709i 0.0290661 + 0.604725i
\(664\) 0 0
\(665\) 5.37312 0.208361
\(666\) 0 0
\(667\) −11.6124 + 9.58764i −0.449634 + 0.371235i
\(668\) 0 0
\(669\) −0.166132 3.45640i −0.00642303 0.133632i
\(670\) 0 0
\(671\) 0.591275i 0.0228259i
\(672\) 0 0
\(673\) 11.8502 0.456793 0.228396 0.973568i \(-0.426652\pi\)
0.228396 + 0.973568i \(0.426652\pi\)
\(674\) 0 0
\(675\) −0.746095 5.14231i −0.0287172 0.197928i
\(676\) 0 0
\(677\) 28.1072 1.08025 0.540123 0.841586i \(-0.318378\pi\)
0.540123 + 0.841586i \(0.318378\pi\)
\(678\) 0 0
\(679\) −52.9553 −2.03224
\(680\) 0 0
\(681\) 0.134043 + 2.78878i 0.00513653 + 0.106866i
\(682\) 0 0
\(683\) 41.5454i 1.58969i −0.606813 0.794844i \(-0.707552\pi\)
0.606813 0.794844i \(-0.292448\pi\)
\(684\) 0 0
\(685\) −7.74794 −0.296033
\(686\) 0 0
\(687\) 20.3852 0.979814i 0.777744 0.0373823i
\(688\) 0 0
\(689\) 14.8689 0.566461
\(690\) 0 0
\(691\) −13.0681 −0.497133 −0.248566 0.968615i \(-0.579959\pi\)
−0.248566 + 0.968615i \(0.579959\pi\)
\(692\) 0 0
\(693\) −0.742098 7.70190i −0.0281900 0.292571i
\(694\) 0 0
\(695\) 10.2656 0.389396
\(696\) 0 0
\(697\) 11.5511i 0.437529i
\(698\) 0 0
\(699\) 15.6486 0.752152i 0.591886 0.0284490i
\(700\) 0 0
\(701\) 24.3220 0.918629 0.459314 0.888274i \(-0.348095\pi\)
0.459314 + 0.888274i \(0.348095\pi\)
\(702\) 0 0
\(703\) −9.43993 −0.356034
\(704\) 0 0
\(705\) 6.18979 0.297512i 0.233121 0.0112050i
\(706\) 0 0
\(707\) −28.7843 −1.08255
\(708\) 0 0
\(709\) 19.2745i 0.723870i −0.932203 0.361935i \(-0.882116\pi\)
0.932203 0.361935i \(-0.117884\pi\)
\(710\) 0 0
\(711\) −0.886322 9.19874i −0.0332397 0.344980i
\(712\) 0 0
\(713\) −18.6569 22.5969i −0.698706 0.846262i
\(714\) 0 0
\(715\) 1.86016 0.0695661
\(716\) 0 0
\(717\) −22.6528 + 1.08881i −0.845985 + 0.0406623i
\(718\) 0 0
\(719\) 11.2588i 0.419883i 0.977714 + 0.209942i \(0.0673274\pi\)
−0.977714 + 0.209942i \(0.932673\pi\)
\(720\) 0 0
\(721\) 2.05077 0.0763746
\(722\) 0 0
\(723\) −37.2684 + 1.79131i −1.38603 + 0.0666194i
\(724\) 0 0
\(725\) 3.14000i 0.116617i
\(726\) 0 0
\(727\) 14.9001i 0.552614i 0.961069 + 0.276307i \(0.0891107\pi\)
−0.961069 + 0.276307i \(0.910889\pi\)
\(728\) 0 0
\(729\) −25.8867 + 7.67331i −0.958766 + 0.284197i
\(730\) 0 0
\(731\) 3.73453i 0.138127i
\(732\) 0 0
\(733\) 22.6998i 0.838435i 0.907886 + 0.419217i \(0.137696\pi\)
−0.907886 + 0.419217i \(0.862304\pi\)
\(734\) 0 0
\(735\) 0.762347 + 15.8608i 0.0281196 + 0.585033i
\(736\) 0 0
\(737\) 0.816453i 0.0300744i
\(738\) 0 0
\(739\) 47.0904 1.73225 0.866124 0.499829i \(-0.166604\pi\)
0.866124 + 0.499829i \(0.166604\pi\)
\(740\) 0 0
\(741\) 6.70430 0.322242i 0.246289 0.0118379i
\(742\) 0 0
\(743\) 46.1428 1.69282 0.846408 0.532536i \(-0.178761\pi\)
0.846408 + 0.532536i \(0.178761\pi\)
\(744\) 0 0
\(745\) −9.47232 −0.347039
\(746\) 0 0
\(747\) 46.7783 4.50722i 1.71153 0.164910i
\(748\) 0 0
\(749\) 56.6169i 2.06874i
\(750\) 0 0
\(751\) 29.8659i 1.08982i 0.838494 + 0.544911i \(0.183437\pi\)
−0.838494 + 0.544911i \(0.816563\pi\)
\(752\) 0 0
\(753\) 0.227849 + 4.74042i 0.00830326 + 0.172751i
\(754\) 0 0
\(755\) 13.8802 0.505153
\(756\) 0 0
\(757\) 39.1104i 1.42149i −0.703449 0.710746i \(-0.748358\pi\)
0.703449 0.710746i \(-0.251642\pi\)
\(758\) 0 0
\(759\) 3.94116 3.58570i 0.143055 0.130153i
\(760\) 0 0
\(761\) 0.0699634i 0.00253617i 0.999999 + 0.00126809i \(0.000403644\pi\)
−0.999999 + 0.00126809i \(0.999596\pi\)
\(762\) 0 0
\(763\) 50.5672 1.83065
\(764\) 0 0
\(765\) −9.26782 + 0.892979i −0.335079 + 0.0322857i
\(766\) 0 0
\(767\) 32.8936i 1.18772i
\(768\) 0 0
\(769\) 22.1180i 0.797596i 0.917039 + 0.398798i \(0.130573\pi\)
−0.917039 + 0.398798i \(0.869427\pi\)
\(770\) 0 0
\(771\) −45.9397 + 2.20809i −1.65448 + 0.0795225i
\(772\) 0 0
\(773\) −32.8192 −1.18043 −0.590213 0.807248i \(-0.700956\pi\)
−0.590213 + 0.807248i \(0.700956\pi\)
\(774\) 0 0
\(775\) 6.11022 0.219486
\(776\) 0 0
\(777\) −2.36201 49.1419i −0.0847365 1.76296i
\(778\) 0 0
\(779\) 4.97349 0.178194
\(780\) 0 0
\(781\) 0.926373i 0.0331482i
\(782\) 0 0
\(783\) −16.1469 + 2.34274i −0.577042 + 0.0837227i
\(784\) 0 0
\(785\) 12.9188i 0.461091i
\(786\) 0 0
\(787\) 34.8223i 1.24128i −0.784096 0.620640i \(-0.786873\pi\)
0.784096 0.620640i \(-0.213127\pi\)
\(788\) 0 0
\(789\) −0.340379 7.08163i −0.0121178 0.252113i
\(790\) 0 0
\(791\) 0.687345i 0.0244392i
\(792\) 0 0
\(793\) 2.67315i 0.0949265i
\(794\) 0 0
\(795\) 0.426359 + 8.87047i 0.0151214 + 0.314603i
\(796\) 0 0
\(797\) −24.6425 −0.872882 −0.436441 0.899733i \(-0.643761\pi\)
−0.436441 + 0.899733i \(0.643761\pi\)
\(798\) 0 0
\(799\) 11.1040i 0.392831i
\(800\) 0 0
\(801\) −12.8819 + 1.24120i −0.455158 + 0.0438557i
\(802\) 0 0
\(803\) −6.99569 −0.246872
\(804\) 0 0
\(805\) −14.8702 + 12.2774i −0.524107 + 0.432722i
\(806\) 0 0
\(807\) −10.2834 + 0.494274i −0.361994 + 0.0173993i
\(808\) 0 0
\(809\) 17.7305i 0.623372i −0.950185 0.311686i \(-0.899106\pi\)
0.950185 0.311686i \(-0.100894\pi\)
\(810\) 0 0
\(811\) −11.1173 −0.390380 −0.195190 0.980765i \(-0.562532\pi\)
−0.195190 + 0.980765i \(0.562532\pi\)
\(812\) 0 0
\(813\) 0.181319 + 3.77237i 0.00635913 + 0.132303i
\(814\) 0 0
\(815\) −21.6758 −0.759272
\(816\) 0 0
\(817\) −1.60796 −0.0562553
\(818\) 0 0
\(819\) 3.35502 + 34.8203i 0.117234 + 1.21672i
\(820\) 0 0
\(821\) 35.5549i 1.24088i −0.784256 0.620438i \(-0.786955\pi\)
0.784256 0.620438i \(-0.213045\pi\)
\(822\) 0 0
\(823\) 49.0792 1.71079 0.855397 0.517973i \(-0.173313\pi\)
0.855397 + 0.517973i \(0.173313\pi\)
\(824\) 0 0
\(825\) 0.0533392 + 1.10973i 0.00185703 + 0.0386359i
\(826\) 0 0
\(827\) −12.2238 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(828\) 0 0
\(829\) −8.73797 −0.303482 −0.151741 0.988420i \(-0.548488\pi\)
−0.151741 + 0.988420i \(0.548488\pi\)
\(830\) 0 0
\(831\) 2.11558 + 44.0150i 0.0733886 + 1.52686i
\(832\) 0 0
\(833\) 28.4530 0.985837
\(834\) 0 0
\(835\) 3.64677i 0.126202i
\(836\) 0 0
\(837\) −4.55881 31.4206i −0.157575 1.08606i
\(838\) 0 0
\(839\) 1.22017 0.0421250 0.0210625 0.999778i \(-0.493295\pi\)
0.0210625 + 0.999778i \(0.493295\pi\)
\(840\) 0 0
\(841\) 19.1404 0.660013
\(842\) 0 0
\(843\) 2.17826 + 45.3190i 0.0750231 + 1.56087i
\(844\) 0 0
\(845\) 4.59022 0.157908
\(846\) 0 0
\(847\) 42.5757i 1.46292i
\(848\) 0 0
\(849\) −18.8670 + 0.906842i −0.647513 + 0.0311227i
\(850\) 0 0
\(851\) 26.1252 21.5700i 0.895561 0.739409i
\(852\) 0 0
\(853\) −21.6114 −0.739959 −0.369980 0.929040i \(-0.620635\pi\)
−0.369980 + 0.929040i \(0.620635\pi\)
\(854\) 0 0
\(855\) 0.384485 + 3.99040i 0.0131491 + 0.136469i
\(856\) 0 0
\(857\) 32.5300i 1.11120i 0.831448 + 0.555602i \(0.187512\pi\)
−0.831448 + 0.555602i \(0.812488\pi\)
\(858\) 0 0
\(859\) 52.7309 1.79916 0.899578 0.436760i \(-0.143874\pi\)
0.899578 + 0.436760i \(0.143874\pi\)
\(860\) 0 0
\(861\) 1.24444 + 25.8907i 0.0424104 + 0.882354i
\(862\) 0 0
\(863\) 44.1954i 1.50443i 0.658918 + 0.752214i \(0.271014\pi\)
−0.658918 + 0.752214i \(0.728986\pi\)
\(864\) 0 0
\(865\) 9.85550i 0.335097i
\(866\) 0 0
\(867\) −0.612669 12.7467i −0.0208073 0.432899i
\(868\) 0 0
\(869\) 1.97593i 0.0670289i
\(870\) 0 0
\(871\) 3.69118i 0.125071i
\(872\) 0 0
\(873\) −3.78933 39.3277i −0.128249 1.33104i
\(874\) 0 0
\(875\) 4.02092i 0.135932i
\(876\) 0 0
\(877\) 57.0543 1.92659 0.963293 0.268454i \(-0.0865126\pi\)
0.963293 + 0.268454i \(0.0865126\pi\)
\(878\) 0 0
\(879\) −2.25195 46.8522i −0.0759564 1.58028i
\(880\) 0 0
\(881\) −36.7818 −1.23921 −0.619605 0.784914i \(-0.712707\pi\)
−0.619605 + 0.784914i \(0.712707\pi\)
\(882\) 0 0
\(883\) 46.4704 1.56385 0.781926 0.623371i \(-0.214237\pi\)
0.781926 + 0.623371i \(0.214237\pi\)
\(884\) 0 0
\(885\) −19.6236 + 0.943210i −0.659641 + 0.0317056i
\(886\) 0 0
\(887\) 21.1776i 0.711074i −0.934662 0.355537i \(-0.884298\pi\)
0.934662 0.355537i \(-0.115702\pi\)
\(888\) 0 0
\(889\) 63.8064i 2.14000i
\(890\) 0 0
\(891\) 5.66678 1.10225i 0.189844 0.0369268i
\(892\) 0 0
\(893\) −4.78099 −0.159990
\(894\) 0 0
\(895\) 16.9605i 0.566929i
\(896\) 0 0
\(897\) −17.8180 + 16.2109i −0.594925 + 0.541267i
\(898\) 0 0
\(899\) 19.1861i 0.639892i
\(900\) 0 0
\(901\) 15.9129 0.530136
\(902\) 0 0
\(903\) −0.402334 8.37062i −0.0133888 0.278557i
\(904\) 0 0
\(905\) 25.5650i 0.849811i
\(906\) 0 0
\(907\) 45.8979i 1.52402i −0.647567 0.762008i \(-0.724214\pi\)
0.647567 0.762008i \(-0.275786\pi\)
\(908\) 0 0
\(909\) −2.05972 21.3769i −0.0683167 0.709028i
\(910\) 0 0
\(911\) 38.9640 1.29093 0.645467 0.763788i \(-0.276663\pi\)
0.645467 + 0.763788i \(0.276663\pi\)
\(912\) 0 0
\(913\) −10.0482 −0.332547
\(914\) 0 0
\(915\) −1.59474 + 0.0766514i −0.0527206 + 0.00253402i
\(916\) 0 0
\(917\) −34.5913 −1.14231
\(918\) 0 0
\(919\) 41.4674i 1.36788i −0.729537 0.683941i \(-0.760265\pi\)
0.729537 0.683941i \(-0.239735\pi\)
\(920\) 0 0
\(921\) 1.23191 + 25.6302i 0.0405929 + 0.844542i
\(922\) 0 0
\(923\) 4.18813i 0.137854i
\(924\) 0 0
\(925\) 7.06427i 0.232272i
\(926\) 0 0
\(927\) 0.146747 + 1.52302i 0.00481981 + 0.0500226i
\(928\) 0 0
\(929\) 24.0813i 0.790082i 0.918663 + 0.395041i \(0.129270\pi\)
−0.918663 + 0.395041i \(0.870730\pi\)
\(930\) 0 0
\(931\) 12.2508i 0.401505i
\(932\) 0 0
\(933\) −1.26258 + 0.0606858i −0.0413349 + 0.00198676i
\(934\) 0 0
\(935\) 1.99077 0.0651052
\(936\) 0 0
\(937\) 3.45592i 0.112900i −0.998405 0.0564499i \(-0.982022\pi\)
0.998405 0.0564499i \(-0.0179781\pi\)
\(938\) 0 0
\(939\) −29.8708 + 1.43574i −0.974795 + 0.0468535i
\(940\) 0 0
\(941\) −9.70843 −0.316486 −0.158243 0.987400i \(-0.550583\pi\)
−0.158243 + 0.987400i \(0.550583\pi\)
\(942\) 0 0
\(943\) −13.7642 + 11.3643i −0.448225 + 0.370072i
\(944\) 0 0
\(945\) −20.6768 + 2.99999i −0.672616 + 0.0975896i
\(946\) 0 0
\(947\) 25.2341i 0.819998i 0.912086 + 0.409999i \(0.134471\pi\)
−0.912086 + 0.409999i \(0.865529\pi\)
\(948\) 0 0
\(949\) 31.6275 1.02667
\(950\) 0 0
\(951\) −45.1790 + 2.17153i −1.46503 + 0.0704166i
\(952\) 0 0
\(953\) −56.6762 −1.83592 −0.917960 0.396672i \(-0.870165\pi\)
−0.917960 + 0.396672i \(0.870165\pi\)
\(954\) 0 0
\(955\) 12.4794 0.403825
\(956\) 0 0
\(957\) 3.48456 0.167485i 0.112640 0.00541403i
\(958\) 0 0
\(959\) 31.1538i 1.00601i
\(960\) 0 0
\(961\) 6.33481 0.204349
\(962\) 0 0
\(963\) 42.0470 4.05134i 1.35495 0.130553i
\(964\) 0 0
\(965\) 21.0097 0.676326
\(966\) 0 0
\(967\) 2.76298 0.0888513 0.0444257 0.999013i \(-0.485854\pi\)
0.0444257 + 0.999013i \(0.485854\pi\)
\(968\) 0 0
\(969\) 7.17504 0.344868i 0.230495 0.0110788i
\(970\) 0 0
\(971\) −11.8285 −0.379594 −0.189797 0.981823i \(-0.560783\pi\)
−0.189797 + 0.981823i \(0.560783\pi\)
\(972\) 0 0
\(973\) 41.2771i 1.32328i
\(974\) 0 0
\(975\) −0.241146 5.01709i −0.00772287 0.160675i
\(976\) 0 0
\(977\) −43.2315 −1.38310 −0.691549 0.722330i \(-0.743071\pi\)
−0.691549 + 0.722330i \(0.743071\pi\)
\(978\) 0 0
\(979\) 2.76708 0.0884364
\(980\) 0 0
\(981\) 3.61844 + 37.5541i 0.115528 + 1.19901i
\(982\) 0 0
\(983\) −26.4684 −0.844211 −0.422105 0.906547i \(-0.638709\pi\)
−0.422105 + 0.906547i \(0.638709\pi\)
\(984\) 0 0
\(985\) 9.84641i 0.313733i
\(986\) 0 0
\(987\) −1.19627 24.8886i −0.0380778 0.792214i
\(988\) 0 0
\(989\) 4.45006 3.67413i 0.141504 0.116831i
\(990\) 0 0
\(991\) 10.1890 0.323664 0.161832 0.986818i \(-0.448260\pi\)
0.161832 + 0.986818i \(0.448260\pi\)
\(992\) 0 0
\(993\) 0.316018 + 6.57480i 0.0100285 + 0.208645i
\(994\) 0 0
\(995\) 8.69855i 0.275763i
\(996\) 0 0
\(997\) −60.4205 −1.91354 −0.956769 0.290849i \(-0.906062\pi\)
−0.956769 + 0.290849i \(0.906062\pi\)
\(998\) 0 0
\(999\) 36.3267 5.27062i 1.14932 0.166755i
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 1380.2.i.a.1241.8 yes 16
3.2 odd 2 1380.2.i.b.1241.7 yes 16
23.22 odd 2 1380.2.i.b.1241.8 yes 16
69.68 even 2 inner 1380.2.i.a.1241.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.i.a.1241.7 16 69.68 even 2 inner
1380.2.i.a.1241.8 yes 16 1.1 even 1 trivial
1380.2.i.b.1241.7 yes 16 3.2 odd 2
1380.2.i.b.1241.8 yes 16 23.22 odd 2