Properties

Label 1380.2.r.c.737.15
Level $1380$
Weight $2$
Character 1380.737
Analytic conductor $11.019$
Analytic rank $0$
Dimension $80$
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] = [1380,2,Mod(737,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 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("1380.737");
 
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.r (of order \(4\), degree \(2\), 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: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 737.15
Character \(\chi\) \(=\) 1380.737
Dual form 1380.2.r.c.1013.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.727046 - 1.57207i) q^{3} +(1.08694 - 1.95412i) q^{5} +(2.39152 + 2.39152i) q^{7} +(-1.94281 + 2.28593i) q^{9} +O(q^{10})\) \(q+(-0.727046 - 1.57207i) q^{3} +(1.08694 - 1.95412i) q^{5} +(2.39152 + 2.39152i) q^{7} +(-1.94281 + 2.28593i) q^{9} -3.03033i q^{11} +(0.0771350 - 0.0771350i) q^{13} +(-3.86226 - 0.288012i) q^{15} +(1.23292 - 1.23292i) q^{17} -3.48755i q^{19} +(2.02090 - 5.49839i) q^{21} +(0.707107 + 0.707107i) q^{23} +(-2.63713 - 4.24800i) q^{25} +(5.00616 + 1.39226i) q^{27} +2.96661 q^{29} +1.61431 q^{31} +(-4.76390 + 2.20319i) q^{33} +(7.27275 - 2.07387i) q^{35} +(-3.20541 - 3.20541i) q^{37} +(-0.177342 - 0.0651809i) q^{39} +2.20816i q^{41} +(6.58947 - 6.58947i) q^{43} +(2.35526 + 6.28114i) q^{45} +(4.02234 - 4.02234i) q^{47} +4.43876i q^{49} +(-2.83462 - 1.04184i) q^{51} +(-2.02954 - 2.02954i) q^{53} +(-5.92162 - 3.29379i) q^{55} +(-5.48267 + 2.53560i) q^{57} -5.09767 q^{59} +2.81766 q^{61} +(-10.1131 + 0.820588i) q^{63} +(-0.0668897 - 0.234571i) q^{65} +(0.824194 + 0.824194i) q^{67} +(0.597523 - 1.62572i) q^{69} -0.280915i q^{71} +(-7.78399 + 7.78399i) q^{73} +(-4.76085 + 7.23425i) q^{75} +(7.24711 - 7.24711i) q^{77} +4.52181i q^{79} +(-1.45098 - 8.88227i) q^{81} +(-12.1188 - 12.1188i) q^{83} +(-1.06916 - 3.74936i) q^{85} +(-2.15686 - 4.66372i) q^{87} -7.16782 q^{89} +0.368940 q^{91} +(-1.17368 - 2.53781i) q^{93} +(-6.81507 - 3.79075i) q^{95} +(-10.7971 - 10.7971i) q^{97} +(6.92714 + 5.88736i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + 8 q^{3} + 32 q^{13} + 24 q^{21} - 32 q^{25} - 28 q^{27} - 32 q^{31} - 44 q^{33} + 24 q^{37} - 32 q^{43} + 88 q^{45} + 16 q^{51} + 8 q^{55} + 16 q^{57} - 32 q^{61} - 12 q^{63} - 16 q^{67} - 32 q^{73} + 4 q^{75} - 64 q^{81} - 32 q^{85} + 64 q^{91} + 8 q^{93} - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(1\) \(1\)

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
\(3\) −0.727046 1.57207i −0.419760 0.907635i
\(4\) 0 0
\(5\) 1.08694 1.95412i 0.486093 0.873907i
\(6\) 0 0
\(7\) 2.39152 + 2.39152i 0.903911 + 0.903911i 0.995772 0.0918611i \(-0.0292816\pi\)
−0.0918611 + 0.995772i \(0.529282\pi\)
\(8\) 0 0
\(9\) −1.94281 + 2.28593i −0.647603 + 0.761978i
\(10\) 0 0
\(11\) 3.03033i 0.913680i −0.889549 0.456840i \(-0.848981\pi\)
0.889549 0.456840i \(-0.151019\pi\)
\(12\) 0 0
\(13\) 0.0771350 0.0771350i 0.0213934 0.0213934i −0.696329 0.717723i \(-0.745185\pi\)
0.717723 + 0.696329i \(0.245185\pi\)
\(14\) 0 0
\(15\) −3.86226 0.288012i −0.997231 0.0743645i
\(16\) 0 0
\(17\) 1.23292 1.23292i 0.299026 0.299026i −0.541606 0.840632i \(-0.682184\pi\)
0.840632 + 0.541606i \(0.182184\pi\)
\(18\) 0 0
\(19\) 3.48755i 0.800098i −0.916494 0.400049i \(-0.868993\pi\)
0.916494 0.400049i \(-0.131007\pi\)
\(20\) 0 0
\(21\) 2.02090 5.49839i 0.440996 1.19985i
\(22\) 0 0
\(23\) 0.707107 + 0.707107i 0.147442 + 0.147442i
\(24\) 0 0
\(25\) −2.63713 4.24800i −0.527426 0.849601i
\(26\) 0 0
\(27\) 5.00616 + 1.39226i 0.963436 + 0.267940i
\(28\) 0 0
\(29\) 2.96661 0.550886 0.275443 0.961317i \(-0.411175\pi\)
0.275443 + 0.961317i \(0.411175\pi\)
\(30\) 0 0
\(31\) 1.61431 0.289939 0.144969 0.989436i \(-0.453692\pi\)
0.144969 + 0.989436i \(0.453692\pi\)
\(32\) 0 0
\(33\) −4.76390 + 2.20319i −0.829288 + 0.383526i
\(34\) 0 0
\(35\) 7.27275 2.07387i 1.22932 0.350549i
\(36\) 0 0
\(37\) −3.20541 3.20541i −0.526967 0.526967i 0.392700 0.919667i \(-0.371541\pi\)
−0.919667 + 0.392700i \(0.871541\pi\)
\(38\) 0 0
\(39\) −0.177342 0.0651809i −0.0283975 0.0104373i
\(40\) 0 0
\(41\) 2.20816i 0.344857i 0.985022 + 0.172428i \(0.0551614\pi\)
−0.985022 + 0.172428i \(0.944839\pi\)
\(42\) 0 0
\(43\) 6.58947 6.58947i 1.00488 1.00488i 0.00489686 0.999988i \(-0.498441\pi\)
0.999988 0.00489686i \(-0.00155873\pi\)
\(44\) 0 0
\(45\) 2.35526 + 6.28114i 0.351102 + 0.936337i
\(46\) 0 0
\(47\) 4.02234 4.02234i 0.586719 0.586719i −0.350022 0.936741i \(-0.613826\pi\)
0.936741 + 0.350022i \(0.113826\pi\)
\(48\) 0 0
\(49\) 4.43876i 0.634109i
\(50\) 0 0
\(51\) −2.83462 1.04184i −0.396926 0.145887i
\(52\) 0 0
\(53\) −2.02954 2.02954i −0.278779 0.278779i 0.553842 0.832621i \(-0.313161\pi\)
−0.832621 + 0.553842i \(0.813161\pi\)
\(54\) 0 0
\(55\) −5.92162 3.29379i −0.798471 0.444134i
\(56\) 0 0
\(57\) −5.48267 + 2.53560i −0.726197 + 0.335849i
\(58\) 0 0
\(59\) −5.09767 −0.663661 −0.331830 0.943339i \(-0.607666\pi\)
−0.331830 + 0.943339i \(0.607666\pi\)
\(60\) 0 0
\(61\) 2.81766 0.360764 0.180382 0.983597i \(-0.442267\pi\)
0.180382 + 0.983597i \(0.442267\pi\)
\(62\) 0 0
\(63\) −10.1131 + 0.820588i −1.27414 + 0.103384i
\(64\) 0 0
\(65\) −0.0668897 0.234571i −0.00829664 0.0290950i
\(66\) 0 0
\(67\) 0.824194 + 0.824194i 0.100691 + 0.100691i 0.755658 0.654967i \(-0.227317\pi\)
−0.654967 + 0.755658i \(0.727317\pi\)
\(68\) 0 0
\(69\) 0.597523 1.62572i 0.0719333 0.195714i
\(70\) 0 0
\(71\) 0.280915i 0.0333384i −0.999861 0.0166692i \(-0.994694\pi\)
0.999861 0.0166692i \(-0.00530622\pi\)
\(72\) 0 0
\(73\) −7.78399 + 7.78399i −0.911047 + 0.911047i −0.996355 0.0853074i \(-0.972813\pi\)
0.0853074 + 0.996355i \(0.472813\pi\)
\(74\) 0 0
\(75\) −4.76085 + 7.23425i −0.549735 + 0.835339i
\(76\) 0 0
\(77\) 7.24711 7.24711i 0.825885 0.825885i
\(78\) 0 0
\(79\) 4.52181i 0.508744i 0.967106 + 0.254372i \(0.0818687\pi\)
−0.967106 + 0.254372i \(0.918131\pi\)
\(80\) 0 0
\(81\) −1.45098 8.88227i −0.161220 0.986918i
\(82\) 0 0
\(83\) −12.1188 12.1188i −1.33021 1.33021i −0.905174 0.425041i \(-0.860260\pi\)
−0.425041 0.905174i \(-0.639740\pi\)
\(84\) 0 0
\(85\) −1.06916 3.74936i −0.115966 0.406675i
\(86\) 0 0
\(87\) −2.15686 4.66372i −0.231240 0.500003i
\(88\) 0 0
\(89\) −7.16782 −0.759788 −0.379894 0.925030i \(-0.624040\pi\)
−0.379894 + 0.925030i \(0.624040\pi\)
\(90\) 0 0
\(91\) 0.368940 0.0386754
\(92\) 0 0
\(93\) −1.17368 2.53781i −0.121705 0.263158i
\(94\) 0 0
\(95\) −6.81507 3.79075i −0.699211 0.388922i
\(96\) 0 0
\(97\) −10.7971 10.7971i −1.09628 1.09628i −0.994842 0.101434i \(-0.967657\pi\)
−0.101434 0.994842i \(-0.532343\pi\)
\(98\) 0 0
\(99\) 6.92714 + 5.88736i 0.696204 + 0.591702i
\(100\) 0 0
\(101\) 7.94622i 0.790678i 0.918535 + 0.395339i \(0.129373\pi\)
−0.918535 + 0.395339i \(0.870627\pi\)
\(102\) 0 0
\(103\) 6.30248 6.30248i 0.621002 0.621002i −0.324786 0.945788i \(-0.605292\pi\)
0.945788 + 0.324786i \(0.105292\pi\)
\(104\) 0 0
\(105\) −8.54789 9.92547i −0.834189 0.968627i
\(106\) 0 0
\(107\) 7.89565 7.89565i 0.763301 0.763301i −0.213616 0.976918i \(-0.568524\pi\)
0.976918 + 0.213616i \(0.0685242\pi\)
\(108\) 0 0
\(109\) 9.66158i 0.925411i 0.886512 + 0.462706i \(0.153121\pi\)
−0.886512 + 0.462706i \(0.846879\pi\)
\(110\) 0 0
\(111\) −2.70865 + 7.36961i −0.257094 + 0.699493i
\(112\) 0 0
\(113\) −7.24425 7.24425i −0.681482 0.681482i 0.278852 0.960334i \(-0.410046\pi\)
−0.960334 + 0.278852i \(0.910046\pi\)
\(114\) 0 0
\(115\) 2.15035 0.613187i 0.200521 0.0571800i
\(116\) 0 0
\(117\) 0.0264668 + 0.326184i 0.00244686 + 0.0301557i
\(118\) 0 0
\(119\) 5.89709 0.540585
\(120\) 0 0
\(121\) 1.81707 0.165188
\(122\) 0 0
\(123\) 3.47138 1.60543i 0.313004 0.144757i
\(124\) 0 0
\(125\) −11.1675 + 0.535940i −0.998850 + 0.0479359i
\(126\) 0 0
\(127\) 10.9189 + 10.9189i 0.968899 + 0.968899i 0.999531 0.0306321i \(-0.00975201\pi\)
−0.0306321 + 0.999531i \(0.509752\pi\)
\(128\) 0 0
\(129\) −15.1500 5.56827i −1.33388 0.490258i
\(130\) 0 0
\(131\) 15.3152i 1.33809i 0.743221 + 0.669046i \(0.233297\pi\)
−0.743221 + 0.669046i \(0.766703\pi\)
\(132\) 0 0
\(133\) 8.34055 8.34055i 0.723217 0.723217i
\(134\) 0 0
\(135\) 8.16201 8.26931i 0.702474 0.711709i
\(136\) 0 0
\(137\) −7.66668 + 7.66668i −0.655009 + 0.655009i −0.954195 0.299186i \(-0.903285\pi\)
0.299186 + 0.954195i \(0.403285\pi\)
\(138\) 0 0
\(139\) 19.7232i 1.67290i −0.548045 0.836449i \(-0.684628\pi\)
0.548045 0.836449i \(-0.315372\pi\)
\(140\) 0 0
\(141\) −9.24783 3.39898i −0.778808 0.286246i
\(142\) 0 0
\(143\) −0.233745 0.233745i −0.0195467 0.0195467i
\(144\) 0 0
\(145\) 3.22452 5.79710i 0.267782 0.481423i
\(146\) 0 0
\(147\) 6.97805 3.22718i 0.575540 0.266174i
\(148\) 0 0
\(149\) −7.62678 −0.624810 −0.312405 0.949949i \(-0.601135\pi\)
−0.312405 + 0.949949i \(0.601135\pi\)
\(150\) 0 0
\(151\) −13.8732 −1.12898 −0.564492 0.825439i \(-0.690928\pi\)
−0.564492 + 0.825439i \(0.690928\pi\)
\(152\) 0 0
\(153\) 0.423043 + 5.21368i 0.0342010 + 0.421501i
\(154\) 0 0
\(155\) 1.75465 3.15455i 0.140937 0.253379i
\(156\) 0 0
\(157\) 12.2735 + 12.2735i 0.979533 + 0.979533i 0.999795 0.0202615i \(-0.00644988\pi\)
−0.0202615 + 0.999795i \(0.506450\pi\)
\(158\) 0 0
\(159\) −1.71501 + 4.66615i −0.136009 + 0.370050i
\(160\) 0 0
\(161\) 3.38212i 0.266549i
\(162\) 0 0
\(163\) 14.0286 14.0286i 1.09881 1.09881i 0.104254 0.994551i \(-0.466754\pi\)
0.994551 0.104254i \(-0.0332456\pi\)
\(164\) 0 0
\(165\) −0.872774 + 11.7039i −0.0679454 + 0.911150i
\(166\) 0 0
\(167\) 17.2691 17.2691i 1.33632 1.33632i 0.436725 0.899595i \(-0.356138\pi\)
0.899595 0.436725i \(-0.143862\pi\)
\(168\) 0 0
\(169\) 12.9881i 0.999085i
\(170\) 0 0
\(171\) 7.97230 + 6.77564i 0.609657 + 0.518146i
\(172\) 0 0
\(173\) 7.58768 + 7.58768i 0.576880 + 0.576880i 0.934042 0.357162i \(-0.116256\pi\)
−0.357162 + 0.934042i \(0.616256\pi\)
\(174\) 0 0
\(175\) 3.85244 16.4660i 0.291217 1.24471i
\(176\) 0 0
\(177\) 3.70624 + 8.01390i 0.278578 + 0.602362i
\(178\) 0 0
\(179\) 24.3028 1.81648 0.908239 0.418452i \(-0.137427\pi\)
0.908239 + 0.418452i \(0.137427\pi\)
\(180\) 0 0
\(181\) 7.08837 0.526874 0.263437 0.964677i \(-0.415144\pi\)
0.263437 + 0.964677i \(0.415144\pi\)
\(182\) 0 0
\(183\) −2.04857 4.42956i −0.151434 0.327442i
\(184\) 0 0
\(185\) −9.74783 + 2.77966i −0.716675 + 0.204365i
\(186\) 0 0
\(187\) −3.73615 3.73615i −0.273214 0.273214i
\(188\) 0 0
\(189\) 8.64273 + 15.3020i 0.628666 + 1.11305i
\(190\) 0 0
\(191\) 14.6573i 1.06057i 0.847820 + 0.530284i \(0.177915\pi\)
−0.847820 + 0.530284i \(0.822085\pi\)
\(192\) 0 0
\(193\) 8.09665 8.09665i 0.582810 0.582810i −0.352865 0.935674i \(-0.614792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(194\) 0 0
\(195\) −0.320131 + 0.275699i −0.0229251 + 0.0197432i
\(196\) 0 0
\(197\) −5.59195 + 5.59195i −0.398410 + 0.398410i −0.877672 0.479262i \(-0.840904\pi\)
0.479262 + 0.877672i \(0.340904\pi\)
\(198\) 0 0
\(199\) 13.3085i 0.943418i 0.881754 + 0.471709i \(0.156363\pi\)
−0.881754 + 0.471709i \(0.843637\pi\)
\(200\) 0 0
\(201\) 0.696464 1.89492i 0.0491248 0.133657i
\(202\) 0 0
\(203\) 7.09472 + 7.09472i 0.497952 + 0.497952i
\(204\) 0 0
\(205\) 4.31500 + 2.40013i 0.301373 + 0.167633i
\(206\) 0 0
\(207\) −2.99017 + 0.242625i −0.207831 + 0.0168636i
\(208\) 0 0
\(209\) −10.5684 −0.731034
\(210\) 0 0
\(211\) 14.0331 0.966079 0.483040 0.875599i \(-0.339533\pi\)
0.483040 + 0.875599i \(0.339533\pi\)
\(212\) 0 0
\(213\) −0.441618 + 0.204238i −0.0302591 + 0.0139941i
\(214\) 0 0
\(215\) −5.71424 20.0389i −0.389708 1.36664i
\(216\) 0 0
\(217\) 3.86066 + 3.86066i 0.262079 + 0.262079i
\(218\) 0 0
\(219\) 17.8963 + 6.57767i 1.20932 + 0.444477i
\(220\) 0 0
\(221\) 0.190202i 0.0127944i
\(222\) 0 0
\(223\) −0.706985 + 0.706985i −0.0473432 + 0.0473432i −0.730382 0.683039i \(-0.760658\pi\)
0.683039 + 0.730382i \(0.260658\pi\)
\(224\) 0 0
\(225\) 14.8341 + 2.22476i 0.988940 + 0.148317i
\(226\) 0 0
\(227\) −13.1219 + 13.1219i −0.870934 + 0.870934i −0.992574 0.121641i \(-0.961184\pi\)
0.121641 + 0.992574i \(0.461184\pi\)
\(228\) 0 0
\(229\) 24.6439i 1.62851i 0.580506 + 0.814256i \(0.302855\pi\)
−0.580506 + 0.814256i \(0.697145\pi\)
\(230\) 0 0
\(231\) −16.6620 6.12399i −1.09628 0.402929i
\(232\) 0 0
\(233\) −4.56833 4.56833i −0.299282 0.299282i 0.541451 0.840732i \(-0.317875\pi\)
−0.840732 + 0.541451i \(0.817875\pi\)
\(234\) 0 0
\(235\) −3.48808 12.2322i −0.227537 0.797938i
\(236\) 0 0
\(237\) 7.10861 3.28757i 0.461754 0.213550i
\(238\) 0 0
\(239\) 4.76665 0.308329 0.154164 0.988045i \(-0.450731\pi\)
0.154164 + 0.988045i \(0.450731\pi\)
\(240\) 0 0
\(241\) −5.97265 −0.384732 −0.192366 0.981323i \(-0.561616\pi\)
−0.192366 + 0.981323i \(0.561616\pi\)
\(242\) 0 0
\(243\) −12.9086 + 8.73886i −0.828088 + 0.560598i
\(244\) 0 0
\(245\) 8.67386 + 4.82466i 0.554152 + 0.308236i
\(246\) 0 0
\(247\) −0.269012 0.269012i −0.0171168 0.0171168i
\(248\) 0 0
\(249\) −10.2407 + 27.8626i −0.648979 + 1.76572i
\(250\) 0 0
\(251\) 14.5096i 0.915841i 0.888993 + 0.457920i \(0.151406\pi\)
−0.888993 + 0.457920i \(0.848594\pi\)
\(252\) 0 0
\(253\) 2.14277 2.14277i 0.134715 0.134715i
\(254\) 0 0
\(255\) −5.11693 + 4.40674i −0.320435 + 0.275961i
\(256\) 0 0
\(257\) −7.30591 + 7.30591i −0.455730 + 0.455730i −0.897251 0.441521i \(-0.854439\pi\)
0.441521 + 0.897251i \(0.354439\pi\)
\(258\) 0 0
\(259\) 15.3316i 0.952661i
\(260\) 0 0
\(261\) −5.76356 + 6.78147i −0.356755 + 0.419763i
\(262\) 0 0
\(263\) 18.5283 + 18.5283i 1.14250 + 1.14250i 0.987991 + 0.154510i \(0.0493798\pi\)
0.154510 + 0.987991i \(0.450620\pi\)
\(264\) 0 0
\(265\) −6.17194 + 1.75997i −0.379140 + 0.108114i
\(266\) 0 0
\(267\) 5.21133 + 11.2683i 0.318928 + 0.689610i
\(268\) 0 0
\(269\) 14.3813 0.876846 0.438423 0.898769i \(-0.355537\pi\)
0.438423 + 0.898769i \(0.355537\pi\)
\(270\) 0 0
\(271\) −17.9841 −1.09246 −0.546228 0.837636i \(-0.683937\pi\)
−0.546228 + 0.837636i \(0.683937\pi\)
\(272\) 0 0
\(273\) −0.268236 0.580000i −0.0162344 0.0351032i
\(274\) 0 0
\(275\) −12.8729 + 7.99139i −0.776264 + 0.481899i
\(276\) 0 0
\(277\) 4.62499 + 4.62499i 0.277888 + 0.277888i 0.832266 0.554377i \(-0.187043\pi\)
−0.554377 + 0.832266i \(0.687043\pi\)
\(278\) 0 0
\(279\) −3.13630 + 3.69020i −0.187765 + 0.220927i
\(280\) 0 0
\(281\) 4.31170i 0.257215i 0.991696 + 0.128607i \(0.0410507\pi\)
−0.991696 + 0.128607i \(0.958949\pi\)
\(282\) 0 0
\(283\) 18.0057 18.0057i 1.07033 1.07033i 0.0729938 0.997332i \(-0.476745\pi\)
0.997332 0.0729938i \(-0.0232553\pi\)
\(284\) 0 0
\(285\) −1.00446 + 13.4698i −0.0594989 + 0.797883i
\(286\) 0 0
\(287\) −5.28087 + 5.28087i −0.311720 + 0.311720i
\(288\) 0 0
\(289\) 13.9598i 0.821167i
\(290\) 0 0
\(291\) −9.12379 + 24.8237i −0.534846 + 1.45519i
\(292\) 0 0
\(293\) −20.9659 20.9659i −1.22484 1.22484i −0.965892 0.258947i \(-0.916625\pi\)
−0.258947 0.965892i \(-0.583375\pi\)
\(294\) 0 0
\(295\) −5.54086 + 9.96144i −0.322601 + 0.579978i
\(296\) 0 0
\(297\) 4.21900 15.1703i 0.244811 0.880272i
\(298\) 0 0
\(299\) 0.109085 0.00630857
\(300\) 0 0
\(301\) 31.5177 1.81665
\(302\) 0 0
\(303\) 12.4920 5.77726i 0.717647 0.331895i
\(304\) 0 0
\(305\) 3.06262 5.50603i 0.175365 0.315274i
\(306\) 0 0
\(307\) 2.96829 + 2.96829i 0.169409 + 0.169409i 0.786720 0.617310i \(-0.211778\pi\)
−0.617310 + 0.786720i \(0.711778\pi\)
\(308\) 0 0
\(309\) −14.4901 5.32575i −0.824315 0.302971i
\(310\) 0 0
\(311\) 12.0420i 0.682837i −0.939911 0.341418i \(-0.889093\pi\)
0.939911 0.341418i \(-0.110907\pi\)
\(312\) 0 0
\(313\) 11.9634 11.9634i 0.676213 0.676213i −0.282928 0.959141i \(-0.591306\pi\)
0.959141 + 0.282928i \(0.0913059\pi\)
\(314\) 0 0
\(315\) −9.38883 + 20.6542i −0.529001 + 1.16373i
\(316\) 0 0
\(317\) −20.4069 + 20.4069i −1.14616 + 1.14616i −0.158862 + 0.987301i \(0.550783\pi\)
−0.987301 + 0.158862i \(0.949217\pi\)
\(318\) 0 0
\(319\) 8.98982i 0.503334i
\(320\) 0 0
\(321\) −18.1530 6.67202i −1.01320 0.372396i
\(322\) 0 0
\(323\) −4.29985 4.29985i −0.239250 0.239250i
\(324\) 0 0
\(325\) −0.531085 0.124255i −0.0294593 0.00689241i
\(326\) 0 0
\(327\) 15.1887 7.02441i 0.839936 0.388451i
\(328\) 0 0
\(329\) 19.2391 1.06068
\(330\) 0 0
\(331\) −18.7917 −1.03288 −0.516442 0.856322i \(-0.672744\pi\)
−0.516442 + 0.856322i \(0.672744\pi\)
\(332\) 0 0
\(333\) 13.5549 1.09985i 0.742802 0.0602716i
\(334\) 0 0
\(335\) 2.50642 0.714722i 0.136940 0.0390494i
\(336\) 0 0
\(337\) −0.0203458 0.0203458i −0.00110830 0.00110830i 0.706552 0.707661i \(-0.250249\pi\)
−0.707661 + 0.706552i \(0.750249\pi\)
\(338\) 0 0
\(339\) −6.12157 + 16.6554i −0.332478 + 0.904596i
\(340\) 0 0
\(341\) 4.89190i 0.264911i
\(342\) 0 0
\(343\) 6.12525 6.12525i 0.330733 0.330733i
\(344\) 0 0
\(345\) −2.52737 2.93469i −0.136069 0.157998i
\(346\) 0 0
\(347\) 2.83783 2.83783i 0.152343 0.152343i −0.626821 0.779164i \(-0.715644\pi\)
0.779164 + 0.626821i \(0.215644\pi\)
\(348\) 0 0
\(349\) 13.9751i 0.748069i −0.927415 0.374034i \(-0.877974\pi\)
0.927415 0.374034i \(-0.122026\pi\)
\(350\) 0 0
\(351\) 0.493541 0.278758i 0.0263433 0.0148790i
\(352\) 0 0
\(353\) 6.04195 + 6.04195i 0.321581 + 0.321581i 0.849373 0.527793i \(-0.176980\pi\)
−0.527793 + 0.849373i \(0.676980\pi\)
\(354\) 0 0
\(355\) −0.548940 0.305337i −0.0291347 0.0162056i
\(356\) 0 0
\(357\) −4.28745 9.27064i −0.226916 0.490654i
\(358\) 0 0
\(359\) 12.0378 0.635331 0.317665 0.948203i \(-0.397101\pi\)
0.317665 + 0.948203i \(0.397101\pi\)
\(360\) 0 0
\(361\) 6.83702 0.359843
\(362\) 0 0
\(363\) −1.32109 2.85656i −0.0693394 0.149931i
\(364\) 0 0
\(365\) 6.75010 + 23.6715i 0.353316 + 1.23902i
\(366\) 0 0
\(367\) 0.185005 + 0.185005i 0.00965719 + 0.00965719i 0.711919 0.702262i \(-0.247826\pi\)
−0.702262 + 0.711919i \(0.747826\pi\)
\(368\) 0 0
\(369\) −5.04771 4.29004i −0.262773 0.223330i
\(370\) 0 0
\(371\) 9.70739i 0.503983i
\(372\) 0 0
\(373\) 16.3645 16.3645i 0.847322 0.847322i −0.142476 0.989798i \(-0.545506\pi\)
0.989798 + 0.142476i \(0.0455065\pi\)
\(374\) 0 0
\(375\) 8.96181 + 17.1664i 0.462786 + 0.886470i
\(376\) 0 0
\(377\) 0.228829 0.228829i 0.0117853 0.0117853i
\(378\) 0 0
\(379\) 19.3612i 0.994518i −0.867602 0.497259i \(-0.834340\pi\)
0.867602 0.497259i \(-0.165660\pi\)
\(380\) 0 0
\(381\) 9.22677 25.1039i 0.472702 1.28611i
\(382\) 0 0
\(383\) 10.9821 + 10.9821i 0.561157 + 0.561157i 0.929636 0.368479i \(-0.120121\pi\)
−0.368479 + 0.929636i \(0.620121\pi\)
\(384\) 0 0
\(385\) −6.28453 22.0389i −0.320289 1.12320i
\(386\) 0 0
\(387\) 2.26100 + 27.8652i 0.114933 + 1.41647i
\(388\) 0 0
\(389\) −10.3592 −0.525230 −0.262615 0.964901i \(-0.584585\pi\)
−0.262615 + 0.964901i \(0.584585\pi\)
\(390\) 0 0
\(391\) 1.74361 0.0881779
\(392\) 0 0
\(393\) 24.0765 11.1348i 1.21450 0.561678i
\(394\) 0 0
\(395\) 8.83615 + 4.91493i 0.444595 + 0.247297i
\(396\) 0 0
\(397\) 5.51490 + 5.51490i 0.276785 + 0.276785i 0.831824 0.555039i \(-0.187297\pi\)
−0.555039 + 0.831824i \(0.687297\pi\)
\(398\) 0 0
\(399\) −19.1759 7.04797i −0.959995 0.352840i
\(400\) 0 0
\(401\) 26.0585i 1.30130i 0.759378 + 0.650650i \(0.225503\pi\)
−0.759378 + 0.650650i \(0.774497\pi\)
\(402\) 0 0
\(403\) 0.124520 0.124520i 0.00620277 0.00620277i
\(404\) 0 0
\(405\) −18.9341 6.81909i −0.940843 0.338843i
\(406\) 0 0
\(407\) −9.71347 + 9.71347i −0.481479 + 0.481479i
\(408\) 0 0
\(409\) 8.72894i 0.431618i 0.976436 + 0.215809i \(0.0692389\pi\)
−0.976436 + 0.215809i \(0.930761\pi\)
\(410\) 0 0
\(411\) 17.6266 + 6.47854i 0.869456 + 0.319563i
\(412\) 0 0
\(413\) −12.1912 12.1912i −0.599890 0.599890i
\(414\) 0 0
\(415\) −36.8540 + 10.5092i −1.80909 + 0.515875i
\(416\) 0 0
\(417\) −31.0062 + 14.3396i −1.51838 + 0.702215i
\(418\) 0 0
\(419\) 20.3008 0.991758 0.495879 0.868392i \(-0.334846\pi\)
0.495879 + 0.868392i \(0.334846\pi\)
\(420\) 0 0
\(421\) 17.0673 0.831810 0.415905 0.909408i \(-0.363465\pi\)
0.415905 + 0.909408i \(0.363465\pi\)
\(422\) 0 0
\(423\) 1.38016 + 17.0095i 0.0671057 + 0.827028i
\(424\) 0 0
\(425\) −8.48879 1.98607i −0.411767 0.0963385i
\(426\) 0 0
\(427\) 6.73850 + 6.73850i 0.326099 + 0.326099i
\(428\) 0 0
\(429\) −0.197520 + 0.537406i −0.00953636 + 0.0259462i
\(430\) 0 0
\(431\) 12.7241i 0.612900i 0.951887 + 0.306450i \(0.0991412\pi\)
−0.951887 + 0.306450i \(0.900859\pi\)
\(432\) 0 0
\(433\) 21.8008 21.8008i 1.04768 1.04768i 0.0488738 0.998805i \(-0.484437\pi\)
0.998805 0.0488738i \(-0.0155632\pi\)
\(434\) 0 0
\(435\) −11.4578 0.854421i −0.549361 0.0409663i
\(436\) 0 0
\(437\) 2.46607 2.46607i 0.117968 0.117968i
\(438\) 0 0
\(439\) 25.1349i 1.19963i −0.800140 0.599813i \(-0.795242\pi\)
0.800140 0.599813i \(-0.204758\pi\)
\(440\) 0 0
\(441\) −10.1467 8.62367i −0.483177 0.410651i
\(442\) 0 0
\(443\) 13.8865 + 13.8865i 0.659766 + 0.659766i 0.955325 0.295559i \(-0.0955058\pi\)
−0.295559 + 0.955325i \(0.595506\pi\)
\(444\) 0 0
\(445\) −7.79098 + 14.0068i −0.369328 + 0.663984i
\(446\) 0 0
\(447\) 5.54502 + 11.9898i 0.262270 + 0.567100i
\(448\) 0 0
\(449\) 11.2061 0.528846 0.264423 0.964407i \(-0.414818\pi\)
0.264423 + 0.964407i \(0.414818\pi\)
\(450\) 0 0
\(451\) 6.69147 0.315089
\(452\) 0 0
\(453\) 10.0864 + 21.8096i 0.473902 + 1.02470i
\(454\) 0 0
\(455\) 0.401015 0.720951i 0.0187999 0.0337987i
\(456\) 0 0
\(457\) −0.697802 0.697802i −0.0326418 0.0326418i 0.690598 0.723239i \(-0.257348\pi\)
−0.723239 + 0.690598i \(0.757348\pi\)
\(458\) 0 0
\(459\) 7.88870 4.45564i 0.368213 0.207971i
\(460\) 0 0
\(461\) 7.47325i 0.348064i −0.984740 0.174032i \(-0.944320\pi\)
0.984740 0.174032i \(-0.0556796\pi\)
\(462\) 0 0
\(463\) −19.8330 + 19.8330i −0.921718 + 0.921718i −0.997151 0.0754333i \(-0.975966\pi\)
0.0754333 + 0.997151i \(0.475966\pi\)
\(464\) 0 0
\(465\) −6.23488 0.464941i −0.289136 0.0215611i
\(466\) 0 0
\(467\) −23.5294 + 23.5294i −1.08881 + 1.08881i −0.0931582 + 0.995651i \(0.529696\pi\)
−0.995651 + 0.0931582i \(0.970304\pi\)
\(468\) 0 0
\(469\) 3.94216i 0.182032i
\(470\) 0 0
\(471\) 10.3714 28.2182i 0.477890 1.30023i
\(472\) 0 0
\(473\) −19.9683 19.9683i −0.918143 0.918143i
\(474\) 0 0
\(475\) −14.8151 + 9.19712i −0.679764 + 0.421993i
\(476\) 0 0
\(477\) 8.58241 0.696384i 0.392962 0.0318852i
\(478\) 0 0
\(479\) 15.0000 0.685369 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −0.494499 −0.0225472
\(482\) 0 0
\(483\) 5.31694 2.45896i 0.241929 0.111886i
\(484\) 0 0
\(485\) −32.8345 + 9.36297i −1.49094 + 0.425150i
\(486\) 0 0
\(487\) 14.3336 + 14.3336i 0.649518 + 0.649518i 0.952876 0.303359i \(-0.0981080\pi\)
−0.303359 + 0.952876i \(0.598108\pi\)
\(488\) 0 0
\(489\) −32.2534 11.8545i −1.45855 0.536080i
\(490\) 0 0
\(491\) 5.10767i 0.230506i 0.993336 + 0.115253i \(0.0367678\pi\)
−0.993336 + 0.115253i \(0.963232\pi\)
\(492\) 0 0
\(493\) 3.65758 3.65758i 0.164729 0.164729i
\(494\) 0 0
\(495\) 19.0340 7.13723i 0.855513 0.320795i
\(496\) 0 0
\(497\) 0.671814 0.671814i 0.0301350 0.0301350i
\(498\) 0 0
\(499\) 28.6445i 1.28231i 0.767413 + 0.641153i \(0.221543\pi\)
−0.767413 + 0.641153i \(0.778457\pi\)
\(500\) 0 0
\(501\) −39.7036 14.5928i −1.77382 0.651957i
\(502\) 0 0
\(503\) 2.07906 + 2.07906i 0.0927009 + 0.0927009i 0.751936 0.659236i \(-0.229120\pi\)
−0.659236 + 0.751936i \(0.729120\pi\)
\(504\) 0 0
\(505\) 15.5278 + 8.63705i 0.690979 + 0.384343i
\(506\) 0 0
\(507\) 20.4182 9.44294i 0.906804 0.419376i
\(508\) 0 0
\(509\) 9.28928 0.411740 0.205870 0.978579i \(-0.433998\pi\)
0.205870 + 0.978579i \(0.433998\pi\)
\(510\) 0 0
\(511\) −37.2312 −1.64701
\(512\) 0 0
\(513\) 4.85556 17.4592i 0.214378 0.770843i
\(514\) 0 0
\(515\) −5.46537 19.1662i −0.240833 0.844563i
\(516\) 0 0
\(517\) −12.1890 12.1890i −0.536074 0.536074i
\(518\) 0 0
\(519\) 6.41177 17.4449i 0.281446 0.765748i
\(520\) 0 0
\(521\) 1.84496i 0.0808293i 0.999183 + 0.0404146i \(0.0128679\pi\)
−0.999183 + 0.0404146i \(0.987132\pi\)
\(522\) 0 0
\(523\) −10.5854 + 10.5854i −0.462867 + 0.462867i −0.899594 0.436727i \(-0.856137\pi\)
0.436727 + 0.899594i \(0.356137\pi\)
\(524\) 0 0
\(525\) −28.6865 + 5.91520i −1.25198 + 0.258160i
\(526\) 0 0
\(527\) 1.99031 1.99031i 0.0866991 0.0866991i
\(528\) 0 0
\(529\) 1.00000i 0.0434783i
\(530\) 0 0
\(531\) 9.90381 11.6529i 0.429789 0.505695i
\(532\) 0 0
\(533\) 0.170326 + 0.170326i 0.00737766 + 0.00737766i
\(534\) 0 0
\(535\) −6.84693 24.0111i −0.296018 1.03809i
\(536\) 0 0
\(537\) −17.6693 38.2057i −0.762485 1.64870i
\(538\) 0 0
\(539\) 13.4509 0.579373
\(540\) 0 0
\(541\) −7.75571 −0.333444 −0.166722 0.986004i \(-0.553318\pi\)
−0.166722 + 0.986004i \(0.553318\pi\)
\(542\) 0 0
\(543\) −5.15357 11.1434i −0.221161 0.478210i
\(544\) 0 0
\(545\) 18.8798 + 10.5015i 0.808723 + 0.449837i
\(546\) 0 0
\(547\) 24.1659 + 24.1659i 1.03326 + 1.03326i 0.999427 + 0.0338337i \(0.0107717\pi\)
0.0338337 + 0.999427i \(0.489228\pi\)
\(548\) 0 0
\(549\) −5.47418 + 6.44098i −0.233632 + 0.274894i
\(550\) 0 0
\(551\) 10.3462i 0.440763i
\(552\) 0 0
\(553\) −10.8140 + 10.8140i −0.459859 + 0.459859i
\(554\) 0 0
\(555\) 11.4569 + 13.3033i 0.486320 + 0.564695i
\(556\) 0 0
\(557\) 24.9800 24.9800i 1.05844 1.05844i 0.0602526 0.998183i \(-0.480809\pi\)
0.998183 0.0602526i \(-0.0191906\pi\)
\(558\) 0 0
\(559\) 1.01656i 0.0429958i
\(560\) 0 0
\(561\) −3.15714 + 8.58983i −0.133294 + 0.362663i
\(562\) 0 0
\(563\) −14.7657 14.7657i −0.622301 0.622301i 0.323818 0.946119i \(-0.395033\pi\)
−0.946119 + 0.323818i \(0.895033\pi\)
\(564\) 0 0
\(565\) −22.0302 + 6.28205i −0.926816 + 0.264288i
\(566\) 0 0
\(567\) 17.7721 24.7122i 0.746357 1.03781i
\(568\) 0 0
\(569\) 7.29060 0.305638 0.152819 0.988254i \(-0.451165\pi\)
0.152819 + 0.988254i \(0.451165\pi\)
\(570\) 0 0
\(571\) −46.9287 −1.96390 −0.981952 0.189131i \(-0.939433\pi\)
−0.981952 + 0.189131i \(0.939433\pi\)
\(572\) 0 0
\(573\) 23.0423 10.6565i 0.962608 0.445184i
\(574\) 0 0
\(575\) 1.13906 4.86853i 0.0475021 0.203032i
\(576\) 0 0
\(577\) 25.2223 + 25.2223i 1.05002 + 1.05002i 0.998681 + 0.0513387i \(0.0163488\pi\)
0.0513387 + 0.998681i \(0.483651\pi\)
\(578\) 0 0
\(579\) −18.6151 6.84187i −0.773619 0.284338i
\(580\) 0 0
\(581\) 57.9649i 2.40479i
\(582\) 0 0
\(583\) −6.15019 + 6.15019i −0.254715 + 0.254715i
\(584\) 0 0
\(585\) 0.666169 + 0.302822i 0.0275427 + 0.0125202i
\(586\) 0 0
\(587\) −27.8174 + 27.8174i −1.14815 + 1.14815i −0.161232 + 0.986917i \(0.551547\pi\)
−0.986917 + 0.161232i \(0.948453\pi\)
\(588\) 0 0
\(589\) 5.62998i 0.231979i
\(590\) 0 0
\(591\) 12.8565 + 4.72534i 0.528848 + 0.194374i
\(592\) 0 0
\(593\) 6.05842 + 6.05842i 0.248789 + 0.248789i 0.820474 0.571684i \(-0.193710\pi\)
−0.571684 + 0.820474i \(0.693710\pi\)
\(594\) 0 0
\(595\) 6.40977 11.5236i 0.262775 0.472421i
\(596\) 0 0
\(597\) 20.9220 9.67592i 0.856279 0.396009i
\(598\) 0 0
\(599\) 13.3845 0.546875 0.273438 0.961890i \(-0.411839\pi\)
0.273438 + 0.961890i \(0.411839\pi\)
\(600\) 0 0
\(601\) −42.1321 −1.71860 −0.859302 0.511469i \(-0.829102\pi\)
−0.859302 + 0.511469i \(0.829102\pi\)
\(602\) 0 0
\(603\) −3.48530 + 0.282801i −0.141933 + 0.0115165i
\(604\) 0 0
\(605\) 1.97504 3.55077i 0.0802970 0.144359i
\(606\) 0 0
\(607\) −8.68751 8.68751i −0.352615 0.352615i 0.508467 0.861082i \(-0.330213\pi\)
−0.861082 + 0.508467i \(0.830213\pi\)
\(608\) 0 0
\(609\) 5.99521 16.3116i 0.242938 0.660979i
\(610\) 0 0
\(611\) 0.620527i 0.0251038i
\(612\) 0 0
\(613\) −24.5196 + 24.5196i −0.990336 + 0.990336i −0.999954 0.00961775i \(-0.996939\pi\)
0.00961775 + 0.999954i \(0.496939\pi\)
\(614\) 0 0
\(615\) 0.635978 8.52849i 0.0256451 0.343902i
\(616\) 0 0
\(617\) 12.8654 12.8654i 0.517941 0.517941i −0.399007 0.916948i \(-0.630645\pi\)
0.916948 + 0.399007i \(0.130645\pi\)
\(618\) 0 0
\(619\) 19.6382i 0.789324i 0.918826 + 0.394662i \(0.129138\pi\)
−0.918826 + 0.394662i \(0.870862\pi\)
\(620\) 0 0
\(621\) 2.55542 + 4.52436i 0.102545 + 0.181556i
\(622\) 0 0
\(623\) −17.1420 17.1420i −0.686780 0.686780i
\(624\) 0 0
\(625\) −11.0911 + 22.4051i −0.443643 + 0.896204i
\(626\) 0 0
\(627\) 7.68373 + 16.6143i 0.306859 + 0.663512i
\(628\) 0 0
\(629\) −7.90400 −0.315153
\(630\) 0 0
\(631\) 23.6366 0.940956 0.470478 0.882412i \(-0.344081\pi\)
0.470478 + 0.882412i \(0.344081\pi\)
\(632\) 0 0
\(633\) −10.2027 22.0610i −0.405521 0.876847i
\(634\) 0 0
\(635\) 33.2051 9.46865i 1.31770 0.375752i
\(636\) 0 0
\(637\) 0.342384 + 0.342384i 0.0135657 + 0.0135657i
\(638\) 0 0
\(639\) 0.642152 + 0.545764i 0.0254032 + 0.0215901i
\(640\) 0 0
\(641\) 27.7717i 1.09692i 0.836178 + 0.548458i \(0.184785\pi\)
−0.836178 + 0.548458i \(0.815215\pi\)
\(642\) 0 0
\(643\) 3.98288 3.98288i 0.157069 0.157069i −0.624197 0.781267i \(-0.714574\pi\)
0.781267 + 0.624197i \(0.214574\pi\)
\(644\) 0 0
\(645\) −27.3481 + 23.5524i −1.07683 + 0.927375i
\(646\) 0 0
\(647\) 10.0330 10.0330i 0.394438 0.394438i −0.481828 0.876266i \(-0.660027\pi\)
0.876266 + 0.481828i \(0.160027\pi\)
\(648\) 0 0
\(649\) 15.4477i 0.606374i
\(650\) 0 0
\(651\) 3.26235 8.87610i 0.127862 0.347882i
\(652\) 0 0
\(653\) −22.2164 22.2164i −0.869396 0.869396i 0.123010 0.992405i \(-0.460745\pi\)
−0.992405 + 0.123010i \(0.960745\pi\)
\(654\) 0 0
\(655\) 29.9276 + 16.6466i 1.16937 + 0.650438i
\(656\) 0 0
\(657\) −2.67087 32.9165i −0.104201 1.28419i
\(658\) 0 0
\(659\) −2.69071 −0.104815 −0.0524077 0.998626i \(-0.516690\pi\)
−0.0524077 + 0.998626i \(0.516690\pi\)
\(660\) 0 0
\(661\) 11.8779 0.461996 0.230998 0.972954i \(-0.425801\pi\)
0.230998 + 0.972954i \(0.425801\pi\)
\(662\) 0 0
\(663\) −0.299010 + 0.138285i −0.0116126 + 0.00537056i
\(664\) 0 0
\(665\) −7.23273 25.3640i −0.280473 0.983576i
\(666\) 0 0
\(667\) 2.09771 + 2.09771i 0.0812237 + 0.0812237i
\(668\) 0 0
\(669\) 1.62544 + 0.597420i 0.0628432 + 0.0230976i
\(670\) 0 0
\(671\) 8.53845i 0.329623i
\(672\) 0 0
\(673\) 14.4110 14.4110i 0.555505 0.555505i −0.372519 0.928024i \(-0.621506\pi\)
0.928024 + 0.372519i \(0.121506\pi\)
\(674\) 0 0
\(675\) −7.28759 24.9377i −0.280499 0.959854i
\(676\) 0 0
\(677\) 12.2268 12.2268i 0.469913 0.469913i −0.431974 0.901886i \(-0.642183\pi\)
0.901886 + 0.431974i \(0.142183\pi\)
\(678\) 0 0
\(679\) 51.6429i 1.98187i
\(680\) 0 0
\(681\) 30.1689 + 11.0884i 1.15607 + 0.424907i
\(682\) 0 0
\(683\) 14.4350 + 14.4350i 0.552338 + 0.552338i 0.927115 0.374777i \(-0.122281\pi\)
−0.374777 + 0.927115i \(0.622281\pi\)
\(684\) 0 0
\(685\) 6.64837 + 23.3148i 0.254021 + 0.890813i
\(686\) 0 0
\(687\) 38.7419 17.9172i 1.47810 0.683584i
\(688\) 0 0
\(689\) −0.313097 −0.0119281
\(690\) 0 0
\(691\) −14.2465 −0.541961 −0.270980 0.962585i \(-0.587348\pi\)
−0.270980 + 0.962585i \(0.587348\pi\)
\(692\) 0 0
\(693\) 2.48666 + 30.6462i 0.0944603 + 1.16415i
\(694\) 0 0
\(695\) −38.5413 21.4379i −1.46196 0.813185i
\(696\) 0 0
\(697\) 2.72248 + 2.72248i 0.103121 + 0.103121i
\(698\) 0 0
\(699\) −3.86036 + 10.5031i −0.146012 + 0.397265i
\(700\) 0 0
\(701\) 19.9558i 0.753721i −0.926270 0.376861i \(-0.877004\pi\)
0.926270 0.376861i \(-0.122996\pi\)
\(702\) 0 0
\(703\) −11.1790 + 11.1790i −0.421625 + 0.421625i
\(704\) 0 0
\(705\) −16.6938 + 14.3768i −0.628726 + 0.541463i
\(706\) 0 0
\(707\) −19.0036 + 19.0036i −0.714702 + 0.714702i
\(708\) 0 0
\(709\) 32.1595i 1.20778i −0.797069 0.603888i \(-0.793618\pi\)
0.797069 0.603888i \(-0.206382\pi\)
\(710\) 0 0
\(711\) −10.3366 8.78502i −0.387652 0.329464i
\(712\) 0 0
\(713\) 1.14149 + 1.14149i 0.0427491 + 0.0427491i
\(714\) 0 0
\(715\) −0.710830 + 0.202698i −0.0265835 + 0.00758048i
\(716\) 0 0
\(717\) −3.46557 7.49351i −0.129424 0.279850i
\(718\) 0 0
\(719\) 29.0312 1.08268 0.541341 0.840803i \(-0.317917\pi\)
0.541341 + 0.840803i \(0.317917\pi\)
\(720\) 0 0
\(721\) 30.1451 1.12266
\(722\) 0 0
\(723\) 4.34239 + 9.38942i 0.161495 + 0.349196i
\(724\) 0 0
\(725\) −7.82334 12.6022i −0.290552 0.468033i
\(726\) 0 0
\(727\) 8.23331 + 8.23331i 0.305357 + 0.305357i 0.843105 0.537749i \(-0.180725\pi\)
−0.537749 + 0.843105i \(0.680725\pi\)
\(728\) 0 0
\(729\) 23.1232 + 13.9397i 0.856417 + 0.516285i
\(730\) 0 0
\(731\) 16.2485i 0.600973i
\(732\) 0 0
\(733\) −9.96783 + 9.96783i −0.368170 + 0.368170i −0.866810 0.498639i \(-0.833833\pi\)
0.498639 + 0.866810i \(0.333833\pi\)
\(734\) 0 0
\(735\) 1.27842 17.1437i 0.0471552 0.632353i
\(736\) 0 0
\(737\) 2.49758 2.49758i 0.0919997 0.0919997i
\(738\) 0 0
\(739\) 48.4383i 1.78183i 0.454168 + 0.890916i \(0.349937\pi\)
−0.454168 + 0.890916i \(0.650063\pi\)
\(740\) 0 0
\(741\) −0.227322 + 0.618489i −0.00835087 + 0.0227208i
\(742\) 0 0
\(743\) −23.6600 23.6600i −0.867999 0.867999i 0.124251 0.992251i \(-0.460347\pi\)
−0.992251 + 0.124251i \(0.960347\pi\)
\(744\) 0 0
\(745\) −8.28984 + 14.9036i −0.303716 + 0.546026i
\(746\) 0 0
\(747\) 51.2474 4.15826i 1.87505 0.152143i
\(748\) 0 0
\(749\) 37.7653 1.37991
\(750\) 0 0
\(751\) −20.5562 −0.750106 −0.375053 0.927003i \(-0.622376\pi\)
−0.375053 + 0.927003i \(0.622376\pi\)
\(752\) 0 0
\(753\) 22.8102 10.5492i 0.831249 0.384433i
\(754\) 0 0
\(755\) −15.0793 + 27.1098i −0.548791 + 0.986626i
\(756\) 0 0
\(757\) 2.68775 + 2.68775i 0.0976880 + 0.0976880i 0.754262 0.656574i \(-0.227995\pi\)
−0.656574 + 0.754262i \(0.727995\pi\)
\(758\) 0 0
\(759\) −4.92648 1.81069i −0.178820 0.0657240i
\(760\) 0 0
\(761\) 25.9845i 0.941936i 0.882150 + 0.470968i \(0.156095\pi\)
−0.882150 + 0.470968i \(0.843905\pi\)
\(762\) 0 0
\(763\) −23.1059 + 23.1059i −0.836489 + 0.836489i
\(764\) 0 0
\(765\) 10.6480 + 4.84028i 0.384978 + 0.175001i
\(766\) 0 0
\(767\) −0.393209 + 0.393209i −0.0141979 + 0.0141979i
\(768\) 0 0
\(769\) 26.7478i 0.964552i 0.876019 + 0.482276i \(0.160190\pi\)
−0.876019 + 0.482276i \(0.839810\pi\)
\(770\) 0 0
\(771\) 16.7971 + 6.17368i 0.604934 + 0.222340i
\(772\) 0 0
\(773\) −26.6146 26.6146i −0.957260 0.957260i 0.0418634 0.999123i \(-0.486671\pi\)
−0.999123 + 0.0418634i \(0.986671\pi\)
\(774\) 0 0
\(775\) −4.25715 6.85759i −0.152921 0.246332i
\(776\) 0 0
\(777\) −24.1024 + 11.1468i −0.864669 + 0.399889i
\(778\) 0 0
\(779\) 7.70106 0.275919
\(780\) 0 0
\(781\) −0.851266 −0.0304607
\(782\) 0 0
\(783\) 14.8513 + 4.13028i 0.530743 + 0.147604i
\(784\) 0 0
\(785\) 37.3244 10.6433i 1.33217 0.379876i
\(786\) 0 0
\(787\) −33.6714 33.6714i −1.20025 1.20025i −0.974089 0.226166i \(-0.927381\pi\)
−0.226166 0.974089i \(-0.572619\pi\)
\(788\) 0 0
\(789\) 15.6568 42.5986i 0.557398 1.51655i
\(790\) 0 0
\(791\) 34.6496i 1.23200i
\(792\) 0 0
\(793\) 0.217340 0.217340i 0.00771797 0.00771797i
\(794\) 0 0
\(795\) 7.25408 + 8.42315i 0.257276 + 0.298738i
\(796\) 0 0
\(797\) 8.75184 8.75184i 0.310006 0.310006i −0.534906 0.844912i \(-0.679653\pi\)
0.844912 + 0.534906i \(0.179653\pi\)
\(798\) 0 0
\(799\) 9.91842i 0.350888i
\(800\) 0 0
\(801\) 13.9257 16.3852i 0.492041 0.578941i
\(802\) 0 0
\(803\) 23.5881 + 23.5881i 0.832406 + 0.832406i
\(804\) 0 0
\(805\) 6.60906 + 3.67616i 0.232939 + 0.129568i
\(806\) 0 0
\(807\) −10.4559 22.6085i −0.368065 0.795856i
\(808\) 0 0
\(809\) −41.2305 −1.44959 −0.724793 0.688967i \(-0.758065\pi\)
−0.724793 + 0.688967i \(0.758065\pi\)
\(810\) 0 0
\(811\) 6.02401 0.211532 0.105766 0.994391i \(-0.466271\pi\)
0.105766 + 0.994391i \(0.466271\pi\)
\(812\) 0 0
\(813\) 13.0753 + 28.2723i 0.458570 + 0.991552i
\(814\) 0 0
\(815\) −12.1653 42.6617i −0.426131 1.49438i
\(816\) 0 0
\(817\) −22.9811 22.9811i −0.804006 0.804006i
\(818\) 0 0
\(819\) −0.716780 + 0.843372i −0.0250463 + 0.0294698i
\(820\) 0 0
\(821\) 30.2374i 1.05529i 0.849464 + 0.527646i \(0.176925\pi\)
−0.849464 + 0.527646i \(0.823075\pi\)
\(822\) 0 0
\(823\) −37.7362 + 37.7362i −1.31540 + 1.31540i −0.398030 + 0.917372i \(0.630306\pi\)
−0.917372 + 0.398030i \(0.869694\pi\)
\(824\) 0 0
\(825\) 21.9222 + 14.4270i 0.763233 + 0.502282i
\(826\) 0 0
\(827\) −10.5789 + 10.5789i −0.367864 + 0.367864i −0.866698 0.498834i \(-0.833762\pi\)
0.498834 + 0.866698i \(0.333762\pi\)
\(828\) 0 0
\(829\) 35.3511i 1.22780i 0.789385 + 0.613898i \(0.210399\pi\)
−0.789385 + 0.613898i \(0.789601\pi\)
\(830\) 0 0
\(831\) 3.90823 10.6334i 0.135575 0.368868i
\(832\) 0 0
\(833\) 5.47262 + 5.47262i 0.189615 + 0.189615i
\(834\) 0 0
\(835\) −14.9753 52.5161i −0.518243 1.81740i
\(836\) 0 0
\(837\) 8.08149 + 2.24753i 0.279337 + 0.0776860i
\(838\) 0 0
\(839\) −31.2840 −1.08005 −0.540023 0.841651i \(-0.681584\pi\)
−0.540023 + 0.841651i \(0.681584\pi\)
\(840\) 0 0
\(841\) −20.1992 −0.696525
\(842\) 0 0
\(843\) 6.77830 3.13480i 0.233457 0.107968i
\(844\) 0 0
\(845\) 25.3802 + 14.1173i 0.873107 + 0.485649i
\(846\) 0 0
\(847\) 4.34557 + 4.34557i 0.149316 + 0.149316i
\(848\) 0 0
\(849\) −41.3971 15.2152i −1.42075 0.522186i
\(850\) 0 0
\(851\) 4.53314i 0.155394i
\(852\) 0 0
\(853\) 33.7408 33.7408i 1.15526 1.15526i 0.169781 0.985482i \(-0.445694\pi\)
0.985482 0.169781i \(-0.0543061\pi\)
\(854\) 0 0
\(855\) 21.9058 8.21409i 0.749162 0.280916i
\(856\) 0 0
\(857\) −5.92962 + 5.92962i −0.202552 + 0.202552i −0.801092 0.598541i \(-0.795748\pi\)
0.598541 + 0.801092i \(0.295748\pi\)
\(858\) 0 0
\(859\) 40.4897i 1.38149i −0.723098 0.690745i \(-0.757283\pi\)
0.723098 0.690745i \(-0.242717\pi\)
\(860\) 0 0
\(861\) 12.1413 + 4.46246i 0.413775 + 0.152080i
\(862\) 0 0
\(863\) −37.1926 37.1926i −1.26605 1.26605i −0.948111 0.317939i \(-0.897009\pi\)
−0.317939 0.948111i \(-0.602991\pi\)
\(864\) 0 0
\(865\) 23.0745 6.57986i 0.784558 0.223722i
\(866\) 0 0
\(867\) 21.9458 10.1494i 0.745320 0.344693i
\(868\) 0 0
\(869\) 13.7026 0.464829
\(870\) 0 0
\(871\) 0.127148 0.00430826
\(872\) 0 0
\(873\) 45.6580 3.70473i 1.54529 0.125386i
\(874\) 0 0
\(875\) −27.9890 25.4256i −0.946201 0.859542i
\(876\) 0 0
\(877\) −6.18800 6.18800i −0.208954 0.208954i 0.594869 0.803823i \(-0.297204\pi\)
−0.803823 + 0.594869i \(0.797204\pi\)
\(878\) 0 0
\(879\) −17.7167 + 48.2029i −0.597568 + 1.62584i
\(880\) 0 0
\(881\) 17.5166i 0.590151i −0.955474 0.295075i \(-0.904655\pi\)
0.955474 0.295075i \(-0.0953447\pi\)
\(882\) 0 0
\(883\) 21.1033 21.1033i 0.710183 0.710183i −0.256390 0.966573i \(-0.582533\pi\)
0.966573 + 0.256390i \(0.0825331\pi\)
\(884\) 0 0
\(885\) 19.6885 + 1.46819i 0.661823 + 0.0493528i
\(886\) 0 0
\(887\) 7.12505 7.12505i 0.239236 0.239236i −0.577298 0.816534i \(-0.695893\pi\)
0.816534 + 0.577298i \(0.195893\pi\)
\(888\) 0 0
\(889\) 52.2258i 1.75160i
\(890\) 0 0
\(891\) −26.9162 + 4.39696i −0.901728 + 0.147304i
\(892\) 0 0
\(893\) −14.0281 14.0281i −0.469433 0.469433i
\(894\) 0 0
\(895\) 26.4157 47.4905i 0.882978 1.58743i
\(896\) 0 0
\(897\) −0.0793100 0.171490i −0.00264808 0.00572588i
\(898\) 0 0
\(899\) 4.78903 0.159723
\(900\) 0 0
\(901\) −5.00451 −0.166724
\(902\) 0 0
\(903\) −22.9148 49.5481i −0.762558 1.64886i
\(904\) 0 0
\(905\) 7.70462 13.8515i 0.256110 0.460439i
\(906\) 0 0
\(907\) −25.4316 25.4316i −0.844444 0.844444i 0.144989 0.989433i \(-0.453685\pi\)
−0.989433 + 0.144989i \(0.953685\pi\)
\(908\) 0 0
\(909\) −18.1645 15.4380i −0.602479 0.512046i
\(910\) 0 0
\(911\) 54.9857i 1.82176i −0.412676 0.910878i \(-0.635406\pi\)
0.412676 0.910878i \(-0.364594\pi\)
\(912\) 0 0
\(913\) −36.7241 + 36.7241i −1.21539 + 1.21539i
\(914\) 0 0
\(915\) −10.8825 0.811521i −0.359765 0.0268280i
\(916\) 0 0
\(917\) −36.6266 + 36.6266i −1.20952 + 1.20952i
\(918\) 0 0
\(919\) 3.51450i 0.115933i −0.998319 0.0579663i \(-0.981538\pi\)
0.998319 0.0579663i \(-0.0184616\pi\)
\(920\) 0 0
\(921\) 2.50828 6.82445i 0.0826507 0.224873i
\(922\) 0 0
\(923\) −0.0216683 0.0216683i −0.000713222 0.000713222i
\(924\) 0 0
\(925\) −5.16351 + 22.0697i −0.169775 + 0.725647i
\(926\) 0 0
\(927\) 2.16253 + 26.6516i 0.0710268 + 0.875352i
\(928\) 0 0
\(929\) 48.7325 1.59886 0.799432 0.600757i \(-0.205134\pi\)
0.799432 + 0.600757i \(0.205134\pi\)
\(930\) 0 0
\(931\) 15.4804 0.507349
\(932\) 0 0
\(933\) −18.9308 + 8.75505i −0.619767 + 0.286627i
\(934\) 0 0
\(935\) −11.3618 + 3.23990i −0.371571 + 0.105956i
\(936\) 0 0
\(937\) −23.2348 23.2348i −0.759049 0.759049i 0.217100 0.976149i \(-0.430340\pi\)
−0.976149 + 0.217100i \(0.930340\pi\)
\(938\) 0 0
\(939\) −27.5053 10.1094i −0.897601 0.329907i
\(940\) 0 0
\(941\) 12.8466i 0.418786i 0.977832 + 0.209393i \(0.0671488\pi\)
−0.977832 + 0.209393i \(0.932851\pi\)
\(942\) 0 0
\(943\) −1.56141 + 1.56141i −0.0508464 + 0.0508464i
\(944\) 0 0
\(945\) 39.2959 0.256617i 1.27830 0.00834775i
\(946\) 0 0
\(947\) −26.0951 + 26.0951i −0.847978 + 0.847978i −0.989881 0.141903i \(-0.954678\pi\)
0.141903 + 0.989881i \(0.454678\pi\)
\(948\) 0 0
\(949\) 1.20084i 0.0389808i
\(950\) 0 0
\(951\) 46.9177 + 17.2443i 1.52141 + 0.559185i
\(952\) 0 0
\(953\) 19.4710 + 19.4710i 0.630729 + 0.630729i 0.948251 0.317522i \(-0.102851\pi\)
−0.317522 + 0.948251i \(0.602851\pi\)
\(954\) 0 0
\(955\) 28.6421 + 15.9316i 0.926837 + 0.515535i
\(956\) 0 0
\(957\) −14.1326 + 6.53601i −0.456843 + 0.211279i
\(958\) 0 0
\(959\) −36.6701 −1.18414
\(960\) 0 0
\(961\) −28.3940 −0.915936
\(962\) 0 0
\(963\) 2.70919 + 33.3887i 0.0873023 + 1.07594i
\(964\) 0 0
\(965\) −7.02123 24.6224i −0.226021 0.792622i
\(966\) 0 0
\(967\) −6.68778 6.68778i −0.215065 0.215065i 0.591350 0.806415i \(-0.298595\pi\)
−0.806415 + 0.591350i \(0.798595\pi\)
\(968\) 0 0
\(969\) −3.63348 + 9.88585i −0.116724 + 0.317579i
\(970\) 0 0
\(971\) 41.9373i 1.34583i 0.739718 + 0.672916i \(0.234959\pi\)
−0.739718 + 0.672916i \(0.765041\pi\)
\(972\) 0 0
\(973\) 47.1684 47.1684i 1.51215 1.51215i
\(974\) 0 0
\(975\) 0.190786 + 0.925241i 0.00611003 + 0.0296314i
\(976\) 0 0
\(977\) −4.90198 + 4.90198i −0.156828 + 0.156828i −0.781160 0.624331i \(-0.785371\pi\)
0.624331 + 0.781160i \(0.285371\pi\)
\(978\) 0 0
\(979\) 21.7209i 0.694203i
\(980\) 0 0
\(981\) −22.0857 18.7706i −0.705143 0.599299i
\(982\) 0 0
\(983\) 8.32411 + 8.32411i 0.265498 + 0.265498i 0.827283 0.561785i \(-0.189885\pi\)
−0.561785 + 0.827283i \(0.689885\pi\)
\(984\) 0 0
\(985\) 4.84921 + 17.0054i 0.154509 + 0.541838i
\(986\) 0 0
\(987\) −13.9877 30.2451i −0.445232 0.962713i
\(988\) 0 0
\(989\) 9.31892 0.296324
\(990\) 0 0
\(991\) −15.1511 −0.481289 −0.240645 0.970613i \(-0.577359\pi\)
−0.240645 + 0.970613i \(0.577359\pi\)
\(992\) 0 0
\(993\) 13.6624 + 29.5418i 0.433563 + 0.937481i
\(994\) 0 0
\(995\) 26.0064 + 14.4656i 0.824459 + 0.458589i
\(996\) 0 0
\(997\) −11.3556 11.3556i −0.359635 0.359635i 0.504043 0.863678i \(-0.331845\pi\)
−0.863678 + 0.504043i \(0.831845\pi\)
\(998\) 0 0
\(999\) −11.5841 20.5096i −0.366503 0.648894i
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.r.c.737.15 80
3.2 odd 2 inner 1380.2.r.c.737.36 yes 80
5.3 odd 4 inner 1380.2.r.c.1013.36 yes 80
15.8 even 4 inner 1380.2.r.c.1013.15 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.r.c.737.15 80 1.1 even 1 trivial
1380.2.r.c.737.36 yes 80 3.2 odd 2 inner
1380.2.r.c.1013.15 yes 80 15.8 even 4 inner
1380.2.r.c.1013.36 yes 80 5.3 odd 4 inner