Properties

Label 4410.2.d.a.4409.13
Level $4410$
Weight $2$
Character 4410.4409
Analytic conductor $35.214$
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] = [4410,2,Mod(4409,4410)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4410.4409");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.d (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: \(35.2140272914\)
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: \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} - 6 x^{8} - 420 x^{7} + 4650 x^{6} - 21000 x^{5} + 70625 x^{4} - 168750 x^{3} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4409.13
Root \(0.104634 - 2.23362i\) of defining polynomial
Character \(\chi\) \(=\) 4410.4409
Dual form 4410.2.d.a.4409.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(1.88205 - 1.20743i) q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(1.88205 - 1.20743i) q^{5} -1.00000 q^{8} +(-1.88205 + 1.20743i) q^{10} -1.59230i q^{11} -0.925091 q^{13} +1.00000 q^{16} -3.95765i q^{17} -0.625477i q^{19} +(1.88205 - 1.20743i) q^{20} +1.59230i q^{22} +7.78564 q^{23} +(2.08425 - 4.54488i) q^{25} +0.925091 q^{26} +9.34805i q^{29} +10.3302i q^{31} -1.00000 q^{32} +3.95765i q^{34} -0.426383i q^{37} +0.625477i q^{38} +(-1.88205 + 1.20743i) q^{40} +8.35463 q^{41} -6.27133i q^{43} -1.59230i q^{44} -7.78564 q^{46} +2.78130i q^{47} +(-2.08425 + 4.54488i) q^{50} -0.925091 q^{52} -3.35114 q^{53} +(-1.92259 - 2.99680i) q^{55} -9.34805i q^{58} +6.21360 q^{59} -11.0005i q^{61} -10.3302i q^{62} +1.00000 q^{64} +(-1.74107 + 1.11698i) q^{65} +0.356178i q^{67} -3.95765i q^{68} -9.07975i q^{71} -6.83023 q^{73} +0.426383i q^{74} -0.625477i q^{76} +9.05164 q^{79} +(1.88205 - 1.20743i) q^{80} -8.35463 q^{82} -0.809898i q^{83} +(-4.77857 - 7.44851i) q^{85} +6.27133i q^{86} +1.59230i q^{88} +4.01442 q^{89} +7.78564 q^{92} -2.78130i q^{94} +(-0.755217 - 1.17718i) q^{95} -7.87721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 16 q^{16} - 16 q^{23} + 12 q^{25} - 16 q^{32} + 16 q^{46} - 12 q^{50} + 32 q^{53} + 16 q^{64} - 40 q^{65} - 8 q^{79} + 64 q^{85} - 16 q^{92} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(2647\) \(3431\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.88205 1.20743i 0.841680 0.539977i
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.88205 + 1.20743i −0.595157 + 0.381821i
\(11\) 1.59230i 0.480097i −0.970761 0.240049i \(-0.922837\pi\)
0.970761 0.240049i \(-0.0771634\pi\)
\(12\) 0 0
\(13\) −0.925091 −0.256574 −0.128287 0.991737i \(-0.540948\pi\)
−0.128287 + 0.991737i \(0.540948\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.95765i 0.959871i −0.877304 0.479936i \(-0.840660\pi\)
0.877304 0.479936i \(-0.159340\pi\)
\(18\) 0 0
\(19\) 0.625477i 0.143494i −0.997423 0.0717472i \(-0.977143\pi\)
0.997423 0.0717472i \(-0.0228575\pi\)
\(20\) 1.88205 1.20743i 0.420840 0.269988i
\(21\) 0 0
\(22\) 1.59230i 0.339480i
\(23\) 7.78564 1.62342 0.811709 0.584062i \(-0.198537\pi\)
0.811709 + 0.584062i \(0.198537\pi\)
\(24\) 0 0
\(25\) 2.08425 4.54488i 0.416850 0.908975i
\(26\) 0.925091 0.181425
\(27\) 0 0
\(28\) 0 0
\(29\) 9.34805i 1.73589i 0.496661 + 0.867945i \(0.334559\pi\)
−0.496661 + 0.867945i \(0.665441\pi\)
\(30\) 0 0
\(31\) 10.3302i 1.85535i 0.373388 + 0.927675i \(0.378196\pi\)
−0.373388 + 0.927675i \(0.621804\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.95765i 0.678732i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.426383i 0.0700970i −0.999386 0.0350485i \(-0.988841\pi\)
0.999386 0.0350485i \(-0.0111586\pi\)
\(38\) 0.625477i 0.101466i
\(39\) 0 0
\(40\) −1.88205 + 1.20743i −0.297579 + 0.190911i
\(41\) 8.35463 1.30477 0.652387 0.757886i \(-0.273768\pi\)
0.652387 + 0.757886i \(0.273768\pi\)
\(42\) 0 0
\(43\) 6.27133i 0.956369i −0.878259 0.478184i \(-0.841295\pi\)
0.878259 0.478184i \(-0.158705\pi\)
\(44\) 1.59230i 0.240049i
\(45\) 0 0
\(46\) −7.78564 −1.14793
\(47\) 2.78130i 0.405695i 0.979210 + 0.202847i \(0.0650195\pi\)
−0.979210 + 0.202847i \(0.934980\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.08425 + 4.54488i −0.294757 + 0.642743i
\(51\) 0 0
\(52\) −0.925091 −0.128287
\(53\) −3.35114 −0.460315 −0.230157 0.973153i \(-0.573924\pi\)
−0.230157 + 0.973153i \(0.573924\pi\)
\(54\) 0 0
\(55\) −1.92259 2.99680i −0.259241 0.404088i
\(56\) 0 0
\(57\) 0 0
\(58\) 9.34805i 1.22746i
\(59\) 6.21360 0.808941 0.404471 0.914551i \(-0.367456\pi\)
0.404471 + 0.914551i \(0.367456\pi\)
\(60\) 0 0
\(61\) 11.0005i 1.40847i −0.709967 0.704235i \(-0.751290\pi\)
0.709967 0.704235i \(-0.248710\pi\)
\(62\) 10.3302i 1.31193i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.74107 + 1.11698i −0.215953 + 0.138544i
\(66\) 0 0
\(67\) 0.356178i 0.0435141i 0.999763 + 0.0217570i \(0.00692603\pi\)
−0.999763 + 0.0217570i \(0.993074\pi\)
\(68\) 3.95765i 0.479936i
\(69\) 0 0
\(70\) 0 0
\(71\) 9.07975i 1.07757i −0.842444 0.538784i \(-0.818884\pi\)
0.842444 0.538784i \(-0.181116\pi\)
\(72\) 0 0
\(73\) −6.83023 −0.799418 −0.399709 0.916642i \(-0.630889\pi\)
−0.399709 + 0.916642i \(0.630889\pi\)
\(74\) 0.426383i 0.0495661i
\(75\) 0 0
\(76\) 0.625477i 0.0717472i
\(77\) 0 0
\(78\) 0 0
\(79\) 9.05164 1.01839 0.509195 0.860651i \(-0.329943\pi\)
0.509195 + 0.860651i \(0.329943\pi\)
\(80\) 1.88205 1.20743i 0.210420 0.134994i
\(81\) 0 0
\(82\) −8.35463 −0.922614
\(83\) 0.809898i 0.0888978i −0.999012 0.0444489i \(-0.985847\pi\)
0.999012 0.0444489i \(-0.0141532\pi\)
\(84\) 0 0
\(85\) −4.77857 7.44851i −0.518308 0.807904i
\(86\) 6.27133i 0.676255i
\(87\) 0 0
\(88\) 1.59230i 0.169740i
\(89\) 4.01442 0.425528 0.212764 0.977104i \(-0.431753\pi\)
0.212764 + 0.977104i \(0.431753\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 7.78564 0.811709
\(93\) 0 0
\(94\) 2.78130i 0.286869i
\(95\) −0.755217 1.17718i −0.0774836 0.120776i
\(96\) 0 0
\(97\) −7.87721 −0.799809 −0.399905 0.916557i \(-0.630957\pi\)
−0.399905 + 0.916557i \(0.630957\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.08425 4.54488i 0.208425 0.454488i
\(101\) 7.52821 0.749085 0.374543 0.927210i \(-0.377800\pi\)
0.374543 + 0.927210i \(0.377800\pi\)
\(102\) 0 0
\(103\) −1.66361 −0.163920 −0.0819601 0.996636i \(-0.526118\pi\)
−0.0819601 + 0.996636i \(0.526118\pi\)
\(104\) 0.925091 0.0907127
\(105\) 0 0
\(106\) 3.35114 0.325492
\(107\) 19.4028 1.87574 0.937869 0.346990i \(-0.112796\pi\)
0.937869 + 0.346990i \(0.112796\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.92259 + 2.99680i 0.183311 + 0.285734i
\(111\) 0 0
\(112\) 0 0
\(113\) 16.0750 1.51221 0.756104 0.654451i \(-0.227100\pi\)
0.756104 + 0.654451i \(0.227100\pi\)
\(114\) 0 0
\(115\) 14.6530 9.40058i 1.36640 0.876608i
\(116\) 9.34805i 0.867945i
\(117\) 0 0
\(118\) −6.21360 −0.572008
\(119\) 0 0
\(120\) 0 0
\(121\) 8.46457 0.769507
\(122\) 11.0005i 0.995938i
\(123\) 0 0
\(124\) 10.3302i 0.927675i
\(125\) −1.56493 11.0703i −0.139972 0.990156i
\(126\) 0 0
\(127\) 13.2173i 1.17285i −0.810004 0.586424i \(-0.800535\pi\)
0.810004 0.586424i \(-0.199465\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.74107 1.11698i 0.152702 0.0979655i
\(131\) −9.40160 −0.821422 −0.410711 0.911766i \(-0.634719\pi\)
−0.410711 + 0.911766i \(0.634719\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.356178i 0.0307691i
\(135\) 0 0
\(136\) 3.95765i 0.339366i
\(137\) −9.37271 −0.800765 −0.400382 0.916348i \(-0.631123\pi\)
−0.400382 + 0.916348i \(0.631123\pi\)
\(138\) 0 0
\(139\) 14.1106i 1.19685i −0.801180 0.598423i \(-0.795794\pi\)
0.801180 0.598423i \(-0.204206\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 9.07975i 0.761956i
\(143\) 1.47303i 0.123181i
\(144\) 0 0
\(145\) 11.2871 + 17.5935i 0.937340 + 1.46106i
\(146\) 6.83023 0.565274
\(147\) 0 0
\(148\) 0.426383i 0.0350485i
\(149\) 2.47225i 0.202535i 0.994859 + 0.101267i \(0.0322897\pi\)
−0.994859 + 0.101267i \(0.967710\pi\)
\(150\) 0 0
\(151\) −20.8849 −1.69959 −0.849796 0.527112i \(-0.823275\pi\)
−0.849796 + 0.527112i \(0.823275\pi\)
\(152\) 0.625477i 0.0507329i
\(153\) 0 0
\(154\) 0 0
\(155\) 12.4729 + 19.4419i 1.00185 + 1.56161i
\(156\) 0 0
\(157\) −24.4475 −1.95113 −0.975563 0.219721i \(-0.929485\pi\)
−0.975563 + 0.219721i \(0.929485\pi\)
\(158\) −9.05164 −0.720110
\(159\) 0 0
\(160\) −1.88205 + 1.20743i −0.148789 + 0.0954553i
\(161\) 0 0
\(162\) 0 0
\(163\) 5.91515i 0.463310i 0.972798 + 0.231655i \(0.0744141\pi\)
−0.972798 + 0.231655i \(0.925586\pi\)
\(164\) 8.35463 0.652387
\(165\) 0 0
\(166\) 0.809898i 0.0628602i
\(167\) 12.1440i 0.939733i −0.882737 0.469867i \(-0.844302\pi\)
0.882737 0.469867i \(-0.155698\pi\)
\(168\) 0 0
\(169\) −12.1442 −0.934170
\(170\) 4.77857 + 7.44851i 0.366499 + 0.571275i
\(171\) 0 0
\(172\) 6.27133i 0.478184i
\(173\) 16.3984i 1.24674i −0.781925 0.623372i \(-0.785762\pi\)
0.781925 0.623372i \(-0.214238\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.59230i 0.120024i
\(177\) 0 0
\(178\) −4.01442 −0.300894
\(179\) 3.15693i 0.235960i −0.993016 0.117980i \(-0.962358\pi\)
0.993016 0.117980i \(-0.0376419\pi\)
\(180\) 0 0
\(181\) 8.17916i 0.607952i −0.952680 0.303976i \(-0.901686\pi\)
0.952680 0.303976i \(-0.0983143\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.78564 −0.573965
\(185\) −0.514826 0.802476i −0.0378508 0.0589992i
\(186\) 0 0
\(187\) −6.30178 −0.460832
\(188\) 2.78130i 0.202847i
\(189\) 0 0
\(190\) 0.755217 + 1.17718i 0.0547892 + 0.0854017i
\(191\) 2.82843i 0.204658i 0.994751 + 0.102329i \(0.0326294\pi\)
−0.994751 + 0.102329i \(0.967371\pi\)
\(192\) 0 0
\(193\) 17.7154i 1.27519i 0.770374 + 0.637593i \(0.220070\pi\)
−0.770374 + 0.637593i \(0.779930\pi\)
\(194\) 7.87721 0.565550
\(195\) 0 0
\(196\) 0 0
\(197\) 4.30350 0.306612 0.153306 0.988179i \(-0.451008\pi\)
0.153306 + 0.988179i \(0.451008\pi\)
\(198\) 0 0
\(199\) 3.46410i 0.245564i −0.992434 0.122782i \(-0.960818\pi\)
0.992434 0.122782i \(-0.0391815\pi\)
\(200\) −2.08425 + 4.54488i −0.147379 + 0.321371i
\(201\) 0 0
\(202\) −7.52821 −0.529683
\(203\) 0 0
\(204\) 0 0
\(205\) 15.7239 10.0876i 1.09820 0.704548i
\(206\) 1.66361 0.115909
\(207\) 0 0
\(208\) −0.925091 −0.0641435
\(209\) −0.995949 −0.0688912
\(210\) 0 0
\(211\) 10.3886 0.715183 0.357592 0.933878i \(-0.383598\pi\)
0.357592 + 0.933878i \(0.383598\pi\)
\(212\) −3.35114 −0.230157
\(213\) 0 0
\(214\) −19.4028 −1.32635
\(215\) −7.57216 11.8030i −0.516417 0.804956i
\(216\) 0 0
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 0 0
\(220\) −1.92259 2.99680i −0.129621 0.202044i
\(221\) 3.66119i 0.246278i
\(222\) 0 0
\(223\) −18.3555 −1.22918 −0.614589 0.788848i \(-0.710678\pi\)
−0.614589 + 0.788848i \(0.710678\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −16.0750 −1.06929
\(227\) 3.02304i 0.200647i −0.994955 0.100323i \(-0.968012\pi\)
0.994955 0.100323i \(-0.0319877\pi\)
\(228\) 0 0
\(229\) 10.6465i 0.703541i −0.936086 0.351770i \(-0.885580\pi\)
0.936086 0.351770i \(-0.114420\pi\)
\(230\) −14.6530 + 9.40058i −0.966190 + 0.619856i
\(231\) 0 0
\(232\) 9.34805i 0.613730i
\(233\) −3.98585 −0.261122 −0.130561 0.991440i \(-0.541678\pi\)
−0.130561 + 0.991440i \(0.541678\pi\)
\(234\) 0 0
\(235\) 3.35821 + 5.23456i 0.219066 + 0.341465i
\(236\) 6.21360 0.404471
\(237\) 0 0
\(238\) 0 0
\(239\) 8.24699i 0.533453i 0.963772 + 0.266727i \(0.0859421\pi\)
−0.963772 + 0.266727i \(0.914058\pi\)
\(240\) 0 0
\(241\) 10.9556i 0.705714i −0.935677 0.352857i \(-0.885210\pi\)
0.935677 0.352857i \(-0.114790\pi\)
\(242\) −8.46457 −0.544123
\(243\) 0 0
\(244\) 11.0005i 0.704235i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.578623i 0.0368169i
\(248\) 10.3302i 0.655965i
\(249\) 0 0
\(250\) 1.56493 + 11.0703i 0.0989750 + 0.700146i
\(251\) 21.7369 1.37202 0.686012 0.727590i \(-0.259360\pi\)
0.686012 + 0.727590i \(0.259360\pi\)
\(252\) 0 0
\(253\) 12.3971i 0.779399i
\(254\) 13.2173i 0.829329i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 17.4352i 1.08757i 0.839223 + 0.543787i \(0.183010\pi\)
−0.839223 + 0.543787i \(0.816990\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.74107 + 1.11698i −0.107977 + 0.0692721i
\(261\) 0 0
\(262\) 9.40160 0.580833
\(263\) 18.3511 1.13158 0.565790 0.824549i \(-0.308571\pi\)
0.565790 + 0.824549i \(0.308571\pi\)
\(264\) 0 0
\(265\) −6.30703 + 4.04625i −0.387438 + 0.248559i
\(266\) 0 0
\(267\) 0 0
\(268\) 0.356178i 0.0217570i
\(269\) −24.8371 −1.51434 −0.757171 0.653217i \(-0.773419\pi\)
−0.757171 + 0.653217i \(0.773419\pi\)
\(270\) 0 0
\(271\) 24.4408i 1.48467i −0.670029 0.742335i \(-0.733718\pi\)
0.670029 0.742335i \(-0.266282\pi\)
\(272\) 3.95765i 0.239968i
\(273\) 0 0
\(274\) 9.37271 0.566226
\(275\) −7.23682 3.31876i −0.436397 0.200128i
\(276\) 0 0
\(277\) 23.1970i 1.39378i 0.717180 + 0.696888i \(0.245432\pi\)
−0.717180 + 0.696888i \(0.754568\pi\)
\(278\) 14.1106i 0.846298i
\(279\) 0 0
\(280\) 0 0
\(281\) 14.0801i 0.839949i 0.907536 + 0.419974i \(0.137961\pi\)
−0.907536 + 0.419974i \(0.862039\pi\)
\(282\) 0 0
\(283\) 2.79820 0.166335 0.0831677 0.996536i \(-0.473496\pi\)
0.0831677 + 0.996536i \(0.473496\pi\)
\(284\) 9.07975i 0.538784i
\(285\) 0 0
\(286\) 1.47303i 0.0871018i
\(287\) 0 0
\(288\) 0 0
\(289\) 1.33700 0.0786468
\(290\) −11.2871 17.5935i −0.662800 1.03313i
\(291\) 0 0
\(292\) −6.83023 −0.399709
\(293\) 25.5598i 1.49322i −0.665263 0.746609i \(-0.731681\pi\)
0.665263 0.746609i \(-0.268319\pi\)
\(294\) 0 0
\(295\) 11.6943 7.50245i 0.680870 0.436810i
\(296\) 0.426383i 0.0247830i
\(297\) 0 0
\(298\) 2.47225i 0.143214i
\(299\) −7.20243 −0.416527
\(300\) 0 0
\(301\) 0 0
\(302\) 20.8849 1.20179
\(303\) 0 0
\(304\) 0.625477i 0.0358736i
\(305\) −13.2823 20.7035i −0.760541 1.18548i
\(306\) 0 0
\(307\) 34.2860 1.95681 0.978403 0.206704i \(-0.0662738\pi\)
0.978403 + 0.206704i \(0.0662738\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −12.4729 19.4419i −0.708412 1.10423i
\(311\) −8.68041 −0.492221 −0.246110 0.969242i \(-0.579153\pi\)
−0.246110 + 0.969242i \(0.579153\pi\)
\(312\) 0 0
\(313\) 30.0213 1.69690 0.848452 0.529273i \(-0.177535\pi\)
0.848452 + 0.529273i \(0.177535\pi\)
\(314\) 24.4475 1.37965
\(315\) 0 0
\(316\) 9.05164 0.509195
\(317\) 27.7256 1.55723 0.778613 0.627504i \(-0.215923\pi\)
0.778613 + 0.627504i \(0.215923\pi\)
\(318\) 0 0
\(319\) 14.8849 0.833396
\(320\) 1.88205 1.20743i 0.105210 0.0674971i
\(321\) 0 0
\(322\) 0 0
\(323\) −2.47542 −0.137736
\(324\) 0 0
\(325\) −1.92812 + 4.20443i −0.106953 + 0.233220i
\(326\) 5.91515i 0.327610i
\(327\) 0 0
\(328\) −8.35463 −0.461307
\(329\) 0 0
\(330\) 0 0
\(331\) 6.30350 0.346472 0.173236 0.984880i \(-0.444578\pi\)
0.173236 + 0.984880i \(0.444578\pi\)
\(332\) 0.809898i 0.0444489i
\(333\) 0 0
\(334\) 12.1440i 0.664492i
\(335\) 0.430058 + 0.670346i 0.0234966 + 0.0366249i
\(336\) 0 0
\(337\) 27.4097i 1.49310i 0.665329 + 0.746550i \(0.268291\pi\)
−0.665329 + 0.746550i \(0.731709\pi\)
\(338\) 12.1442 0.660558
\(339\) 0 0
\(340\) −4.77857 7.44851i −0.259154 0.403952i
\(341\) 16.4487 0.890749
\(342\) 0 0
\(343\) 0 0
\(344\) 6.27133i 0.338127i
\(345\) 0 0
\(346\) 16.3984i 0.881581i
\(347\) 8.16672 0.438412 0.219206 0.975679i \(-0.429653\pi\)
0.219206 + 0.975679i \(0.429653\pi\)
\(348\) 0 0
\(349\) 27.5885i 1.47678i 0.674374 + 0.738390i \(0.264413\pi\)
−0.674374 + 0.738390i \(0.735587\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.59230i 0.0848700i
\(353\) 7.04285i 0.374853i 0.982279 + 0.187426i \(0.0600146\pi\)
−0.982279 + 0.187426i \(0.939985\pi\)
\(354\) 0 0
\(355\) −10.9631 17.0886i −0.581862 0.906967i
\(356\) 4.01442 0.212764
\(357\) 0 0
\(358\) 3.15693i 0.166849i
\(359\) 16.3443i 0.862617i −0.902205 0.431308i \(-0.858052\pi\)
0.902205 0.431308i \(-0.141948\pi\)
\(360\) 0 0
\(361\) 18.6088 0.979409
\(362\) 8.17916i 0.429887i
\(363\) 0 0
\(364\) 0 0
\(365\) −12.8549 + 8.24699i −0.672854 + 0.431667i
\(366\) 0 0
\(367\) −3.00798 −0.157015 −0.0785076 0.996914i \(-0.525015\pi\)
−0.0785076 + 0.996914i \(0.525015\pi\)
\(368\) 7.78564 0.405855
\(369\) 0 0
\(370\) 0.514826 + 0.802476i 0.0267645 + 0.0417188i
\(371\) 0 0
\(372\) 0 0
\(373\) 23.1970i 1.20110i −0.799588 0.600549i \(-0.794949\pi\)
0.799588 0.600549i \(-0.205051\pi\)
\(374\) 6.30178 0.325857
\(375\) 0 0
\(376\) 2.78130i 0.143435i
\(377\) 8.64780i 0.445384i
\(378\) 0 0
\(379\) 27.2718 1.40086 0.700429 0.713722i \(-0.252992\pi\)
0.700429 + 0.713722i \(0.252992\pi\)
\(380\) −0.755217 1.17718i −0.0387418 0.0603881i
\(381\) 0 0
\(382\) 2.82843i 0.144715i
\(383\) 4.26195i 0.217776i 0.994054 + 0.108888i \(0.0347289\pi\)
−0.994054 + 0.108888i \(0.965271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.7154i 0.901692i
\(387\) 0 0
\(388\) −7.87721 −0.399905
\(389\) 2.08172i 0.105547i −0.998607 0.0527736i \(-0.983194\pi\)
0.998607 0.0527736i \(-0.0168062\pi\)
\(390\) 0 0
\(391\) 30.8129i 1.55827i
\(392\) 0 0
\(393\) 0 0
\(394\) −4.30350 −0.216807
\(395\) 17.0357 10.9292i 0.857158 0.549907i
\(396\) 0 0
\(397\) −22.3771 −1.12307 −0.561537 0.827452i \(-0.689790\pi\)
−0.561537 + 0.827452i \(0.689790\pi\)
\(398\) 3.46410i 0.173640i
\(399\) 0 0
\(400\) 2.08425 4.54488i 0.104212 0.227244i
\(401\) 19.6639i 0.981970i −0.871168 0.490985i \(-0.836637\pi\)
0.871168 0.490985i \(-0.163363\pi\)
\(402\) 0 0
\(403\) 9.55634i 0.476035i
\(404\) 7.52821 0.374543
\(405\) 0 0
\(406\) 0 0
\(407\) −0.678932 −0.0336534
\(408\) 0 0
\(409\) 2.53956i 0.125573i 0.998027 + 0.0627866i \(0.0199987\pi\)
−0.998027 + 0.0627866i \(0.980001\pi\)
\(410\) −15.7239 + 10.0876i −0.776546 + 0.498190i
\(411\) 0 0
\(412\) −1.66361 −0.0819601
\(413\) 0 0
\(414\) 0 0
\(415\) −0.977891 1.52427i −0.0480028 0.0748235i
\(416\) 0.925091 0.0453563
\(417\) 0 0
\(418\) 0.995949 0.0487135
\(419\) −32.0568 −1.56608 −0.783039 0.621973i \(-0.786331\pi\)
−0.783039 + 0.621973i \(0.786331\pi\)
\(420\) 0 0
\(421\) −25.2201 −1.22915 −0.614577 0.788857i \(-0.710673\pi\)
−0.614577 + 0.788857i \(0.710673\pi\)
\(422\) −10.3886 −0.505711
\(423\) 0 0
\(424\) 3.35114 0.162746
\(425\) −17.9870 8.24873i −0.872500 0.400122i
\(426\) 0 0
\(427\) 0 0
\(428\) 19.4028 0.937869
\(429\) 0 0
\(430\) 7.57216 + 11.8030i 0.365162 + 0.569190i
\(431\) 33.6710i 1.62187i −0.585133 0.810937i \(-0.698958\pi\)
0.585133 0.810937i \(-0.301042\pi\)
\(432\) 0 0
\(433\) −22.7610 −1.09382 −0.546912 0.837190i \(-0.684197\pi\)
−0.546912 + 0.837190i \(0.684197\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 4.86974i 0.232951i
\(438\) 0 0
\(439\) 2.15100i 0.102661i −0.998682 0.0513307i \(-0.983654\pi\)
0.998682 0.0513307i \(-0.0163463\pi\)
\(440\) 1.92259 + 2.99680i 0.0916557 + 0.142867i
\(441\) 0 0
\(442\) 3.66119i 0.174145i
\(443\) −7.66479 −0.364165 −0.182082 0.983283i \(-0.558284\pi\)
−0.182082 + 0.983283i \(0.558284\pi\)
\(444\) 0 0
\(445\) 7.55535 4.84711i 0.358158 0.229775i
\(446\) 18.3555 0.869160
\(447\) 0 0
\(448\) 0 0
\(449\) 18.8337i 0.888817i 0.895824 + 0.444408i \(0.146586\pi\)
−0.895824 + 0.444408i \(0.853414\pi\)
\(450\) 0 0
\(451\) 13.3031i 0.626418i
\(452\) 16.0750 0.756104
\(453\) 0 0
\(454\) 3.02304i 0.141879i
\(455\) 0 0
\(456\) 0 0
\(457\) 29.2649i 1.36896i 0.729034 + 0.684478i \(0.239970\pi\)
−0.729034 + 0.684478i \(0.760030\pi\)
\(458\) 10.6465i 0.497478i
\(459\) 0 0
\(460\) 14.6530 9.40058i 0.683199 0.438304i
\(461\) 14.1963 0.661188 0.330594 0.943773i \(-0.392751\pi\)
0.330594 + 0.943773i \(0.392751\pi\)
\(462\) 0 0
\(463\) 7.65787i 0.355891i 0.984040 + 0.177946i \(0.0569451\pi\)
−0.984040 + 0.177946i \(0.943055\pi\)
\(464\) 9.34805i 0.433972i
\(465\) 0 0
\(466\) 3.98585 0.184641
\(467\) 17.7143i 0.819718i −0.912149 0.409859i \(-0.865578\pi\)
0.912149 0.409859i \(-0.134422\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −3.35821 5.23456i −0.154903 0.241452i
\(471\) 0 0
\(472\) −6.21360 −0.286004
\(473\) −9.98585 −0.459150
\(474\) 0 0
\(475\) −2.84272 1.30365i −0.130433 0.0598156i
\(476\) 0 0
\(477\) 0 0
\(478\) 8.24699i 0.377209i
\(479\) 12.8399 0.586671 0.293336 0.956010i \(-0.405235\pi\)
0.293336 + 0.956010i \(0.405235\pi\)
\(480\) 0 0
\(481\) 0.394444i 0.0179851i
\(482\) 10.9556i 0.499015i
\(483\) 0 0
\(484\) 8.46457 0.384753
\(485\) −14.8253 + 9.51114i −0.673183 + 0.431878i
\(486\) 0 0
\(487\) 3.77858i 0.171224i 0.996329 + 0.0856119i \(0.0272845\pi\)
−0.996329 + 0.0856119i \(0.972715\pi\)
\(488\) 11.0005i 0.497969i
\(489\) 0 0
\(490\) 0 0
\(491\) 32.1664i 1.45165i −0.687880 0.725824i \(-0.741459\pi\)
0.687880 0.725824i \(-0.258541\pi\)
\(492\) 0 0
\(493\) 36.9963 1.66623
\(494\) 0.578623i 0.0260335i
\(495\) 0 0
\(496\) 10.3302i 0.463838i
\(497\) 0 0
\(498\) 0 0
\(499\) 31.9366 1.42968 0.714839 0.699290i \(-0.246500\pi\)
0.714839 + 0.699290i \(0.246500\pi\)
\(500\) −1.56493 11.0703i −0.0699859 0.495078i
\(501\) 0 0
\(502\) −21.7369 −0.970167
\(503\) 5.29834i 0.236241i −0.992999 0.118121i \(-0.962313\pi\)
0.992999 0.118121i \(-0.0376870\pi\)
\(504\) 0 0
\(505\) 14.1685 9.08975i 0.630490 0.404489i
\(506\) 12.3971i 0.551118i
\(507\) 0 0
\(508\) 13.2173i 0.586424i
\(509\) −37.2643 −1.65171 −0.825854 0.563883i \(-0.809307\pi\)
−0.825854 + 0.563883i \(0.809307\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 17.4352i 0.769032i
\(515\) −3.13100 + 2.00868i −0.137968 + 0.0885132i
\(516\) 0 0
\(517\) 4.42867 0.194773
\(518\) 0 0
\(519\) 0 0
\(520\) 1.74107 1.11698i 0.0763510 0.0489828i
\(521\) −12.0135 −0.526322 −0.263161 0.964752i \(-0.584765\pi\)
−0.263161 + 0.964752i \(0.584765\pi\)
\(522\) 0 0
\(523\) 27.0357 1.18219 0.591093 0.806603i \(-0.298696\pi\)
0.591093 + 0.806603i \(0.298696\pi\)
\(524\) −9.40160 −0.410711
\(525\) 0 0
\(526\) −18.3511 −0.800148
\(527\) 40.8831 1.78090
\(528\) 0 0
\(529\) 37.6162 1.63549
\(530\) 6.30703 4.04625i 0.273960 0.175758i
\(531\) 0 0
\(532\) 0 0
\(533\) −7.72879 −0.334771
\(534\) 0 0
\(535\) 36.5171 23.4274i 1.57877 1.01286i
\(536\) 0.356178i 0.0153846i
\(537\) 0 0
\(538\) 24.8371 1.07080
\(539\) 0 0
\(540\) 0 0
\(541\) −11.7090 −0.503409 −0.251705 0.967804i \(-0.580991\pi\)
−0.251705 + 0.967804i \(0.580991\pi\)
\(542\) 24.4408i 1.04982i
\(543\) 0 0
\(544\) 3.95765i 0.169683i
\(545\) 3.76411 2.41485i 0.161237 0.103441i
\(546\) 0 0
\(547\) 17.6050i 0.752737i 0.926470 + 0.376369i \(0.122827\pi\)
−0.926470 + 0.376369i \(0.877173\pi\)
\(548\) −9.37271 −0.400382
\(549\) 0 0
\(550\) 7.23682 + 3.31876i 0.308579 + 0.141512i
\(551\) 5.84699 0.249090
\(552\) 0 0
\(553\) 0 0
\(554\) 23.1970i 0.985548i
\(555\) 0 0
\(556\) 14.1106i 0.598423i
\(557\) 20.0817 0.850890 0.425445 0.904984i \(-0.360118\pi\)
0.425445 + 0.904984i \(0.360118\pi\)
\(558\) 0 0
\(559\) 5.80155i 0.245380i
\(560\) 0 0
\(561\) 0 0
\(562\) 14.0801i 0.593933i
\(563\) 23.8077i 1.00337i −0.865050 0.501687i \(-0.832713\pi\)
0.865050 0.501687i \(-0.167287\pi\)
\(564\) 0 0
\(565\) 30.2540 19.4094i 1.27280 0.816558i
\(566\) −2.79820 −0.117617
\(567\) 0 0
\(568\) 9.07975i 0.380978i
\(569\) 41.4267i 1.73670i 0.495952 + 0.868350i \(0.334819\pi\)
−0.495952 + 0.868350i \(0.665181\pi\)
\(570\) 0 0
\(571\) 41.1567 1.72235 0.861177 0.508305i \(-0.169728\pi\)
0.861177 + 0.508305i \(0.169728\pi\)
\(572\) 1.47303i 0.0615903i
\(573\) 0 0
\(574\) 0 0
\(575\) 16.2272 35.3848i 0.676722 1.47565i
\(576\) 0 0
\(577\) −12.4167 −0.516914 −0.258457 0.966023i \(-0.583214\pi\)
−0.258457 + 0.966023i \(0.583214\pi\)
\(578\) −1.33700 −0.0556117
\(579\) 0 0
\(580\) 11.2871 + 17.5935i 0.468670 + 0.730532i
\(581\) 0 0
\(582\) 0 0
\(583\) 5.33603i 0.220996i
\(584\) 6.83023 0.282637
\(585\) 0 0
\(586\) 25.5598i 1.05586i
\(587\) 4.01980i 0.165915i −0.996553 0.0829575i \(-0.973563\pi\)
0.996553 0.0829575i \(-0.0264366\pi\)
\(588\) 0 0
\(589\) 6.46127 0.266232
\(590\) −11.6943 + 7.50245i −0.481447 + 0.308871i
\(591\) 0 0
\(592\) 0.426383i 0.0175243i
\(593\) 43.5334i 1.78770i −0.448366 0.893850i \(-0.647994\pi\)
0.448366 0.893850i \(-0.352006\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.47225i 0.101267i
\(597\) 0 0
\(598\) 7.20243 0.294529
\(599\) 32.1259i 1.31263i 0.754488 + 0.656314i \(0.227885\pi\)
−0.754488 + 0.656314i \(0.772115\pi\)
\(600\) 0 0
\(601\) 8.21681i 0.335171i −0.985858 0.167585i \(-0.946403\pi\)
0.985858 0.167585i \(-0.0535970\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −20.8849 −0.849796
\(605\) 15.9308 10.2203i 0.647678 0.415516i
\(606\) 0 0
\(607\) −28.2361 −1.14607 −0.573034 0.819532i \(-0.694234\pi\)
−0.573034 + 0.819532i \(0.694234\pi\)
\(608\) 0.625477i 0.0253664i
\(609\) 0 0
\(610\) 13.2823 + 20.7035i 0.537784 + 0.838261i
\(611\) 2.57296i 0.104091i
\(612\) 0 0
\(613\) 30.7924i 1.24369i 0.783139 + 0.621846i \(0.213617\pi\)
−0.783139 + 0.621846i \(0.786383\pi\)
\(614\) −34.2860 −1.38367
\(615\) 0 0
\(616\) 0 0
\(617\) −4.42613 −0.178189 −0.0890947 0.996023i \(-0.528397\pi\)
−0.0890947 + 0.996023i \(0.528397\pi\)
\(618\) 0 0
\(619\) 10.9356i 0.439537i 0.975552 + 0.219768i \(0.0705302\pi\)
−0.975552 + 0.219768i \(0.929470\pi\)
\(620\) 12.4729 + 19.4419i 0.500923 + 0.780805i
\(621\) 0 0
\(622\) 8.68041 0.348053
\(623\) 0 0
\(624\) 0 0
\(625\) −16.3118 18.9453i −0.652473 0.757812i
\(626\) −30.0213 −1.19989
\(627\) 0 0
\(628\) −24.4475 −0.975563
\(629\) −1.68748 −0.0672841
\(630\) 0 0
\(631\) −38.2293 −1.52189 −0.760943 0.648819i \(-0.775263\pi\)
−0.760943 + 0.648819i \(0.775263\pi\)
\(632\) −9.05164 −0.360055
\(633\) 0 0
\(634\) −27.7256 −1.10113
\(635\) −15.9589 24.8757i −0.633311 0.987163i
\(636\) 0 0
\(637\) 0 0
\(638\) −14.8849 −0.589300
\(639\) 0 0
\(640\) −1.88205 + 1.20743i −0.0743947 + 0.0477277i
\(641\) 33.5254i 1.32418i −0.749426 0.662088i \(-0.769671\pi\)
0.749426 0.662088i \(-0.230329\pi\)
\(642\) 0 0
\(643\) 17.4072 0.686474 0.343237 0.939249i \(-0.388477\pi\)
0.343237 + 0.939249i \(0.388477\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 2.47542 0.0973941
\(647\) 38.2602i 1.50416i 0.659070 + 0.752082i \(0.270950\pi\)
−0.659070 + 0.752082i \(0.729050\pi\)
\(648\) 0 0
\(649\) 9.89393i 0.388371i
\(650\) 1.92812 4.20443i 0.0756271 0.164911i
\(651\) 0 0
\(652\) 5.91515i 0.231655i
\(653\) −32.3343 −1.26534 −0.632669 0.774422i \(-0.718041\pi\)
−0.632669 + 0.774422i \(0.718041\pi\)
\(654\) 0 0
\(655\) −17.6943 + 11.3517i −0.691374 + 0.443549i
\(656\) 8.35463 0.326193
\(657\) 0 0
\(658\) 0 0
\(659\) 16.4516i 0.640865i −0.947271 0.320433i \(-0.896172\pi\)
0.947271 0.320433i \(-0.103828\pi\)
\(660\) 0 0
\(661\) 2.47741i 0.0963600i −0.998839 0.0481800i \(-0.984658\pi\)
0.998839 0.0481800i \(-0.0153421\pi\)
\(662\) −6.30350 −0.244992
\(663\) 0 0
\(664\) 0.809898i 0.0314301i
\(665\) 0 0
\(666\) 0 0
\(667\) 72.7806i 2.81807i
\(668\) 12.1440i 0.469867i
\(669\) 0 0
\(670\) −0.430058 0.670346i −0.0166146 0.0258977i
\(671\) −17.5161 −0.676202
\(672\) 0 0
\(673\) 17.0784i 0.658326i −0.944273 0.329163i \(-0.893233\pi\)
0.944273 0.329163i \(-0.106767\pi\)
\(674\) 27.4097i 1.05578i
\(675\) 0 0
\(676\) −12.1442 −0.467085
\(677\) 46.6082i 1.79130i 0.444759 + 0.895650i \(0.353289\pi\)
−0.444759 + 0.895650i \(0.646711\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 4.77857 + 7.44851i 0.183250 + 0.285637i
\(681\) 0 0
\(682\) −16.4487 −0.629854
\(683\) 2.36707 0.0905734 0.0452867 0.998974i \(-0.485580\pi\)
0.0452867 + 0.998974i \(0.485580\pi\)
\(684\) 0 0
\(685\) −17.6399 + 11.3168i −0.673988 + 0.432395i
\(686\) 0 0
\(687\) 0 0
\(688\) 6.27133i 0.239092i
\(689\) 3.10011 0.118105
\(690\) 0 0
\(691\) 22.0010i 0.836957i 0.908227 + 0.418479i \(0.137437\pi\)
−0.908227 + 0.418479i \(0.862563\pi\)
\(692\) 16.3984i 0.623372i
\(693\) 0 0
\(694\) −8.16672 −0.310004
\(695\) −17.0375 26.5569i −0.646269 1.00736i
\(696\) 0 0
\(697\) 33.0647i 1.25241i
\(698\) 27.5885i 1.04424i
\(699\) 0 0
\(700\) 0 0
\(701\) 28.7909i 1.08742i −0.839274 0.543708i \(-0.817020\pi\)
0.839274 0.543708i \(-0.182980\pi\)
\(702\) 0 0
\(703\) −0.266693 −0.0100585
\(704\) 1.59230i 0.0600122i
\(705\) 0 0
\(706\) 7.04285i 0.265061i
\(707\) 0 0
\(708\) 0 0
\(709\) −15.2810 −0.573890 −0.286945 0.957947i \(-0.592640\pi\)
−0.286945 + 0.957947i \(0.592640\pi\)
\(710\) 10.9631 + 17.0886i 0.411439 + 0.641323i
\(711\) 0 0
\(712\) −4.01442 −0.150447
\(713\) 80.4269i 3.01201i
\(714\) 0 0
\(715\) 1.77857 + 2.77231i 0.0665147 + 0.103679i
\(716\) 3.15693i 0.117980i
\(717\) 0 0
\(718\) 16.3443i 0.609962i
\(719\) 16.6746 0.621858 0.310929 0.950433i \(-0.399360\pi\)
0.310929 + 0.950433i \(0.399360\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.6088 −0.692547
\(723\) 0 0
\(724\) 8.17916i 0.303976i
\(725\) 42.4857 + 19.4837i 1.57788 + 0.723605i
\(726\) 0 0
\(727\) 32.4228 1.20250 0.601248 0.799062i \(-0.294670\pi\)
0.601248 + 0.799062i \(0.294670\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.8549 8.24699i 0.475779 0.305235i
\(731\) −24.8197 −0.917991
\(732\) 0 0
\(733\) −46.3274 −1.71114 −0.855570 0.517687i \(-0.826793\pi\)
−0.855570 + 0.517687i \(0.826793\pi\)
\(734\) 3.00798 0.111026
\(735\) 0 0
\(736\) −7.78564 −0.286983
\(737\) 0.567143 0.0208910
\(738\) 0 0
\(739\) −16.4738 −0.605998 −0.302999 0.952991i \(-0.597988\pi\)
−0.302999 + 0.952991i \(0.597988\pi\)
\(740\) −0.514826 0.802476i −0.0189254 0.0294996i
\(741\) 0 0
\(742\) 0 0
\(743\) −38.4778 −1.41161 −0.705806 0.708405i \(-0.749415\pi\)
−0.705806 + 0.708405i \(0.749415\pi\)
\(744\) 0 0
\(745\) 2.98506 + 4.65290i 0.109364 + 0.170469i
\(746\) 23.1970i 0.849304i
\(747\) 0 0
\(748\) −6.30178 −0.230416
\(749\) 0 0
\(750\) 0 0
\(751\) 37.8289 1.38040 0.690198 0.723620i \(-0.257523\pi\)
0.690198 + 0.723620i \(0.257523\pi\)
\(752\) 2.78130i 0.101424i
\(753\) 0 0
\(754\) 8.64780i 0.314934i
\(755\) −39.3065 + 25.2170i −1.43051 + 0.917740i
\(756\) 0 0
\(757\) 11.1485i 0.405197i −0.979262 0.202599i \(-0.935061\pi\)
0.979262 0.202599i \(-0.0649387\pi\)
\(758\) −27.2718 −0.990556
\(759\) 0 0
\(760\) 0.755217 + 1.17718i 0.0273946 + 0.0427009i
\(761\) 29.2478 1.06023 0.530117 0.847925i \(-0.322148\pi\)
0.530117 + 0.847925i \(0.322148\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 4.26195i 0.153991i
\(767\) −5.74814 −0.207553
\(768\) 0 0
\(769\) 0.892823i 0.0321960i −0.999870 0.0160980i \(-0.994876\pi\)
0.999870 0.0160980i \(-0.00512438\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17.7154i 0.637593i
\(773\) 19.7802i 0.711445i −0.934592 0.355723i \(-0.884235\pi\)
0.934592 0.355723i \(-0.115765\pi\)
\(774\) 0 0
\(775\) 46.9493 + 21.5306i 1.68647 + 0.773402i
\(776\) 7.87721 0.282775
\(777\) 0 0
\(778\) 2.08172i 0.0746332i
\(779\) 5.22563i 0.187228i
\(780\) 0 0
\(781\) −14.4577 −0.517338
\(782\) 30.8129i 1.10187i
\(783\) 0 0
\(784\) 0 0
\(785\) −46.0116 + 29.5186i −1.64222 + 1.05356i
\(786\) 0 0
\(787\) −26.5957 −0.948034 −0.474017 0.880516i \(-0.657196\pi\)
−0.474017 + 0.880516i \(0.657196\pi\)
\(788\) 4.30350 0.153306
\(789\) 0 0
\(790\) −17.0357 + 10.9292i −0.606102 + 0.388843i
\(791\) 0 0
\(792\) 0 0
\(793\) 10.1765i 0.361377i
\(794\) 22.3771 0.794133
\(795\) 0 0
\(796\) 3.46410i 0.122782i
\(797\) 21.4698i 0.760499i 0.924884 + 0.380250i \(0.124162\pi\)
−0.924884 + 0.380250i \(0.875838\pi\)
\(798\) 0 0
\(799\) 11.0074 0.389415
\(800\) −2.08425 + 4.54488i −0.0736893 + 0.160686i
\(801\) 0 0
\(802\) 19.6639i 0.694357i
\(803\) 10.8758i 0.383798i
\(804\) 0 0
\(805\) 0 0
\(806\) 9.55634i 0.336608i
\(807\) 0 0
\(808\) −7.52821 −0.264842
\(809\) 21.1230i 0.742645i 0.928504 + 0.371322i \(0.121096\pi\)
−0.928504 + 0.371322i \(0.878904\pi\)
\(810\) 0 0
\(811\) 13.3784i 0.469779i −0.972022 0.234889i \(-0.924527\pi\)
0.972022 0.234889i \(-0.0754728\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.678932 0.0237965
\(815\) 7.14210 + 11.1326i 0.250177 + 0.389959i
\(816\) 0 0
\(817\) −3.92257 −0.137233
\(818\) 2.53956i 0.0887936i
\(819\) 0 0
\(820\) 15.7239 10.0876i 0.549101 0.352274i
\(821\) 6.55398i 0.228735i −0.993438 0.114368i \(-0.963516\pi\)
0.993438 0.114368i \(-0.0364842\pi\)
\(822\) 0 0
\(823\) 43.8688i 1.52917i −0.644523 0.764585i \(-0.722944\pi\)
0.644523 0.764585i \(-0.277056\pi\)
\(824\) 1.66361 0.0579546
\(825\) 0 0
\(826\) 0 0
\(827\) 19.1611 0.666296 0.333148 0.942875i \(-0.391889\pi\)
0.333148 + 0.942875i \(0.391889\pi\)
\(828\) 0 0
\(829\) 16.9319i 0.588071i 0.955795 + 0.294035i \(0.0949983\pi\)
−0.955795 + 0.294035i \(0.905002\pi\)
\(830\) 0.977891 + 1.52427i 0.0339431 + 0.0529082i
\(831\) 0 0
\(832\) −0.925091 −0.0320718
\(833\) 0 0
\(834\) 0 0
\(835\) −14.6630 22.8557i −0.507434 0.790954i
\(836\) −0.995949 −0.0344456
\(837\) 0 0
\(838\) 32.0568 1.10738
\(839\) −10.5028 −0.362596 −0.181298 0.983428i \(-0.558030\pi\)
−0.181298 + 0.983428i \(0.558030\pi\)
\(840\) 0 0
\(841\) −58.3861 −2.01331
\(842\) 25.2201 0.869143
\(843\) 0 0
\(844\) 10.3886 0.357592
\(845\) −22.8560 + 14.6632i −0.786272 + 0.504430i
\(846\) 0 0
\(847\) 0 0
\(848\) −3.35114 −0.115079
\(849\) 0 0
\(850\) 17.9870 + 8.24873i 0.616950 + 0.282929i
\(851\) 3.31967i 0.113797i
\(852\) 0 0
\(853\) 30.2419 1.03546 0.517731 0.855544i \(-0.326777\pi\)
0.517731 + 0.855544i \(0.326777\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −19.4028 −0.663173
\(857\) 22.2644i 0.760539i 0.924876 + 0.380270i \(0.124169\pi\)
−0.924876 + 0.380270i \(0.875831\pi\)
\(858\) 0 0
\(859\) 30.8374i 1.05216i −0.850436 0.526078i \(-0.823662\pi\)
0.850436 0.526078i \(-0.176338\pi\)
\(860\) −7.57216 11.8030i −0.258209 0.402478i
\(861\) 0 0
\(862\) 33.6710i 1.14684i
\(863\) 23.1159 0.786875 0.393437 0.919351i \(-0.371286\pi\)
0.393437 + 0.919351i \(0.371286\pi\)
\(864\) 0 0
\(865\) −19.7998 30.8626i −0.673213 1.04936i
\(866\) 22.7610 0.773450
\(867\) 0 0
\(868\) 0 0
\(869\) 14.4130i 0.488926i
\(870\) 0 0
\(871\) 0.329497i 0.0111646i
\(872\) −2.00000 −0.0677285
\(873\) 0 0
\(874\) 4.86974i 0.164721i
\(875\) 0 0
\(876\) 0 0
\(877\) 25.2147i 0.851441i 0.904855 + 0.425720i \(0.139979\pi\)
−0.904855 + 0.425720i \(0.860021\pi\)
\(878\) 2.15100i 0.0725926i
\(879\) 0 0
\(880\) −1.92259 2.99680i −0.0648104 0.101022i
\(881\) −20.5142 −0.691140 −0.345570 0.938393i \(-0.612314\pi\)
−0.345570 + 0.938393i \(0.612314\pi\)
\(882\) 0 0
\(883\) 43.0491i 1.44872i 0.689424 + 0.724358i \(0.257864\pi\)
−0.689424 + 0.724358i \(0.742136\pi\)
\(884\) 3.66119i 0.123139i
\(885\) 0 0
\(886\) 7.66479 0.257504
\(887\) 33.3905i 1.12114i 0.828106 + 0.560571i \(0.189418\pi\)
−0.828106 + 0.560571i \(0.810582\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −7.55535 + 4.84711i −0.253256 + 0.162476i
\(891\) 0 0
\(892\) −18.3555 −0.614589
\(893\) 1.73964 0.0582149
\(894\) 0 0
\(895\) −3.81176 5.94151i −0.127413 0.198603i
\(896\) 0 0
\(897\) 0 0
\(898\) 18.8337i 0.628488i
\(899\) −96.5668 −3.22068
\(900\) 0 0
\(901\) 13.2626i 0.441843i
\(902\) 13.3031i 0.442945i
\(903\) 0 0
\(904\) −16.0750 −0.534646
\(905\) −9.87572 15.3936i −0.328280 0.511701i
\(906\) 0 0
\(907\) 27.3797i 0.909127i 0.890714 + 0.454563i \(0.150205\pi\)
−0.890714 + 0.454563i \(0.849795\pi\)
\(908\) 3.02304i 0.100323i
\(909\) 0 0
\(910\) 0 0
\(911\) 33.0422i 1.09474i 0.836892 + 0.547368i \(0.184370\pi\)
−0.836892 + 0.547368i \(0.815630\pi\)
\(912\) 0 0
\(913\) −1.28960 −0.0426796
\(914\) 29.2649i 0.967998i
\(915\) 0 0
\(916\) 10.6465i 0.351770i
\(917\) 0 0
\(918\) 0 0
\(919\) −26.4889 −0.873787 −0.436893 0.899513i \(-0.643921\pi\)
−0.436893 + 0.899513i \(0.643921\pi\)
\(920\) −14.6530 + 9.40058i −0.483095 + 0.309928i
\(921\) 0 0
\(922\) −14.1963 −0.467531
\(923\) 8.39960i 0.276476i
\(924\) 0 0
\(925\) −1.93786 0.888689i −0.0637165 0.0292199i
\(926\) 7.65787i 0.251653i
\(927\) 0 0
\(928\) 9.34805i 0.306865i
\(929\) 31.1904 1.02333 0.511663 0.859186i \(-0.329030\pi\)
0.511663 + 0.859186i \(0.329030\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −3.98585 −0.130561
\(933\) 0 0
\(934\) 17.7143i 0.579628i
\(935\) −11.8603 + 7.60893i −0.387873 + 0.248838i
\(936\) 0 0
\(937\) −6.40017 −0.209084 −0.104542 0.994520i \(-0.533338\pi\)
−0.104542 + 0.994520i \(0.533338\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 3.35821 + 5.23456i 0.109533 + 0.170732i
\(941\) 5.12485 0.167065 0.0835327 0.996505i \(-0.473380\pi\)
0.0835327 + 0.996505i \(0.473380\pi\)
\(942\) 0 0
\(943\) 65.0461 2.11819
\(944\) 6.21360 0.202235
\(945\) 0 0
\(946\) 9.98585 0.324668
\(947\) −42.2250 −1.37213 −0.686064 0.727541i \(-0.740663\pi\)
−0.686064 + 0.727541i \(0.740663\pi\)
\(948\) 0 0
\(949\) 6.31858 0.205110
\(950\) 2.84272 + 1.30365i 0.0922299 + 0.0422960i
\(951\) 0 0
\(952\) 0 0
\(953\) 8.12428 0.263171 0.131586 0.991305i \(-0.457993\pi\)
0.131586 + 0.991305i \(0.457993\pi\)
\(954\) 0 0
\(955\) 3.41511 + 5.32325i 0.110510 + 0.172256i
\(956\) 8.24699i 0.266727i
\(957\) 0 0
\(958\) −12.8399 −0.414839
\(959\) 0 0
\(960\) 0 0
\(961\) −75.7121 −2.44232
\(962\) 0.394444i 0.0127174i
\(963\) 0 0
\(964\) 10.9556i 0.352857i
\(965\) 21.3901 + 33.3414i 0.688571 + 1.07330i
\(966\) 0 0
\(967\) 52.3097i 1.68217i 0.540905 + 0.841084i \(0.318082\pi\)
−0.540905 + 0.841084i \(0.681918\pi\)
\(968\) −8.46457 −0.272062
\(969\) 0 0
\(970\) 14.8253 9.51114i 0.476012 0.305384i
\(971\) −17.1056 −0.548945 −0.274473 0.961595i \(-0.588503\pi\)
−0.274473 + 0.961595i \(0.588503\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 3.77858i 0.121073i
\(975\) 0 0
\(976\) 11.0005i 0.352117i
\(977\) 36.3978 1.16447 0.582235 0.813020i \(-0.302178\pi\)
0.582235 + 0.813020i \(0.302178\pi\)
\(978\) 0 0
\(979\) 6.39217i 0.204295i
\(980\) 0 0
\(981\) 0 0
\(982\) 32.1664i 1.02647i
\(983\) 25.7283i 0.820604i 0.911950 + 0.410302i \(0.134577\pi\)
−0.911950 + 0.410302i \(0.865423\pi\)
\(984\) 0 0
\(985\) 8.09941 5.19615i 0.258069 0.165563i
\(986\) −36.9963 −1.17820
\(987\) 0 0
\(988\) 0.578623i 0.0184085i
\(989\) 48.8263i 1.55259i
\(990\) 0 0
\(991\) −12.1475 −0.385878 −0.192939 0.981211i \(-0.561802\pi\)
−0.192939 + 0.981211i \(0.561802\pi\)
\(992\) 10.3302i 0.327983i
\(993\) 0 0
\(994\) 0 0
\(995\) −4.18264 6.51962i −0.132599 0.206686i
\(996\) 0 0
\(997\) −17.1533 −0.543249 −0.271625 0.962403i \(-0.587561\pi\)
−0.271625 + 0.962403i \(0.587561\pi\)
\(998\) −31.9366 −1.01093
\(999\) 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 4410.2.d.a.4409.13 16
3.2 odd 2 4410.2.d.b.4409.4 16
5.4 even 2 4410.2.d.b.4409.14 16
7.4 even 3 630.2.bo.b.89.4 yes 16
7.5 odd 6 630.2.bo.b.269.7 yes 16
7.6 odd 2 inner 4410.2.d.a.4409.4 16
15.14 odd 2 inner 4410.2.d.a.4409.3 16
21.5 even 6 630.2.bo.a.269.2 yes 16
21.11 odd 6 630.2.bo.a.89.5 yes 16
21.20 even 2 4410.2.d.b.4409.13 16
35.4 even 6 630.2.bo.a.89.2 16
35.12 even 12 3150.2.bf.f.1151.5 32
35.18 odd 12 3150.2.bf.f.1601.6 32
35.19 odd 6 630.2.bo.a.269.5 yes 16
35.32 odd 12 3150.2.bf.f.1601.13 32
35.33 even 12 3150.2.bf.f.1151.16 32
35.34 odd 2 4410.2.d.b.4409.3 16
105.32 even 12 3150.2.bf.f.1601.5 32
105.47 odd 12 3150.2.bf.f.1151.15 32
105.53 even 12 3150.2.bf.f.1601.14 32
105.68 odd 12 3150.2.bf.f.1151.6 32
105.74 odd 6 630.2.bo.b.89.7 yes 16
105.89 even 6 630.2.bo.b.269.4 yes 16
105.104 even 2 inner 4410.2.d.a.4409.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.2 16 35.4 even 6
630.2.bo.a.89.5 yes 16 21.11 odd 6
630.2.bo.a.269.2 yes 16 21.5 even 6
630.2.bo.a.269.5 yes 16 35.19 odd 6
630.2.bo.b.89.4 yes 16 7.4 even 3
630.2.bo.b.89.7 yes 16 105.74 odd 6
630.2.bo.b.269.4 yes 16 105.89 even 6
630.2.bo.b.269.7 yes 16 7.5 odd 6
3150.2.bf.f.1151.5 32 35.12 even 12
3150.2.bf.f.1151.6 32 105.68 odd 12
3150.2.bf.f.1151.15 32 105.47 odd 12
3150.2.bf.f.1151.16 32 35.33 even 12
3150.2.bf.f.1601.5 32 105.32 even 12
3150.2.bf.f.1601.6 32 35.18 odd 12
3150.2.bf.f.1601.13 32 35.32 odd 12
3150.2.bf.f.1601.14 32 105.53 even 12
4410.2.d.a.4409.3 16 15.14 odd 2 inner
4410.2.d.a.4409.4 16 7.6 odd 2 inner
4410.2.d.a.4409.13 16 1.1 even 1 trivial
4410.2.d.a.4409.14 16 105.104 even 2 inner
4410.2.d.b.4409.3 16 35.34 odd 2
4410.2.d.b.4409.4 16 3.2 odd 2
4410.2.d.b.4409.13 16 21.20 even 2
4410.2.d.b.4409.14 16 5.4 even 2