Properties

Label 896.2.q.c.703.4
Level $896$
Weight $2$
Character 896.703
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [896,2,Mod(703,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.703");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.q (of order \(6\), 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: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} - 24 x^{13} + 104 x^{12} - 196 x^{11} + 312 x^{10} - 236 x^{9} + 31 x^{8} + 236 x^{7} + 312 x^{6} + 196 x^{5} + 104 x^{4} + 24 x^{3} + 4 x^{2} + 4 x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 703.4
Root \(0.219056 - 1.66389i\) of defining polynomial
Character \(\chi\) \(=\) 896.703
Dual form 896.2.q.c.831.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.753671 - 0.435132i) q^{3} +(0.866025 + 1.50000i) q^{5} +(0.615370 + 2.57319i) q^{7} +(-1.12132 - 1.94218i) q^{9} +O(q^{10})\) \(q+(-0.753671 - 0.435132i) q^{3} +(0.866025 + 1.50000i) q^{5} +(0.615370 + 2.57319i) q^{7} +(-1.12132 - 1.94218i) q^{9} +(-1.81952 + 3.15150i) q^{11} -1.01461 q^{13} -1.50734i q^{15} +(-0.621320 - 0.358719i) q^{17} +(-5.45857 + 3.15150i) q^{19} +(0.655892 - 2.20711i) q^{21} +(-3.15150 + 1.81952i) q^{23} +(1.00000 - 1.73205i) q^{25} +4.56248i q^{27} -4.58579i q^{29} +(3.76687 - 6.52442i) q^{31} +(2.74264 - 1.58346i) q^{33} +(-3.32686 + 3.15150i) q^{35} +(-6.27231 + 3.62132i) q^{37} +(0.764683 + 0.441490i) q^{39} +7.34847i q^{41} -10.2928 q^{43} +(1.94218 - 3.36396i) q^{45} +(-1.30540 - 2.26101i) q^{47} +(-6.24264 + 3.16693i) q^{49} +(0.312181 + 0.540713i) q^{51} +(3.82282 + 2.20711i) q^{53} -6.30301 q^{55} +5.48528 q^{57} +(-0.753671 - 0.435132i) q^{59} +(6.27231 + 10.8640i) q^{61} +(4.30759 - 4.08053i) q^{63} +(-0.878680 - 1.52192i) q^{65} +(-2.88537 + 4.99761i) q^{67} +3.16693 q^{69} +14.5562i q^{71} +(-3.25736 - 1.88064i) q^{73} +(-1.50734 + 0.870264i) q^{75} +(-9.22911 - 2.74264i) q^{77} +(5.76230 - 3.32686i) q^{79} +(-1.37868 + 2.38794i) q^{81} +3.48106i q^{83} -1.24264i q^{85} +(-1.99542 + 3.45617i) q^{87} +(-6.98528 + 4.03295i) q^{89} +(-0.624361 - 2.61079i) q^{91} +(-5.67796 + 3.27817i) q^{93} +(-9.45451 - 5.45857i) q^{95} -1.01461i q^{97} +8.16107 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{9} + 24 q^{17} + 16 q^{25} - 24 q^{33} - 32 q^{49} - 48 q^{57} - 48 q^{65} - 120 q^{73} - 56 q^{81} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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
\(3\) −0.753671 0.435132i −0.435132 0.251224i 0.266399 0.963863i \(-0.414166\pi\)
−0.701531 + 0.712639i \(0.747500\pi\)
\(4\) 0 0
\(5\) 0.866025 + 1.50000i 0.387298 + 0.670820i 0.992085 0.125567i \(-0.0400750\pi\)
−0.604787 + 0.796387i \(0.706742\pi\)
\(6\) 0 0
\(7\) 0.615370 + 2.57319i 0.232588 + 0.972575i
\(8\) 0 0
\(9\) −1.12132 1.94218i −0.373773 0.647395i
\(10\) 0 0
\(11\) −1.81952 + 3.15150i −0.548607 + 0.950214i 0.449764 + 0.893148i \(0.351508\pi\)
−0.998370 + 0.0570668i \(0.981825\pi\)
\(12\) 0 0
\(13\) −1.01461 −0.281403 −0.140701 0.990052i \(-0.544936\pi\)
−0.140701 + 0.990052i \(0.544936\pi\)
\(14\) 0 0
\(15\) 1.50734i 0.389194i
\(16\) 0 0
\(17\) −0.621320 0.358719i −0.150692 0.0870023i 0.422758 0.906243i \(-0.361062\pi\)
−0.573450 + 0.819240i \(0.694395\pi\)
\(18\) 0 0
\(19\) −5.45857 + 3.15150i −1.25228 + 0.723005i −0.971562 0.236785i \(-0.923906\pi\)
−0.280719 + 0.959790i \(0.590573\pi\)
\(20\) 0 0
\(21\) 0.655892 2.20711i 0.143127 0.481630i
\(22\) 0 0
\(23\) −3.15150 + 1.81952i −0.657134 + 0.379397i −0.791184 0.611578i \(-0.790535\pi\)
0.134050 + 0.990975i \(0.457202\pi\)
\(24\) 0 0
\(25\) 1.00000 1.73205i 0.200000 0.346410i
\(26\) 0 0
\(27\) 4.56248i 0.878050i
\(28\) 0 0
\(29\) 4.58579i 0.851559i −0.904827 0.425780i \(-0.860000\pi\)
0.904827 0.425780i \(-0.140000\pi\)
\(30\) 0 0
\(31\) 3.76687 6.52442i 0.676551 1.17182i −0.299463 0.954108i \(-0.596807\pi\)
0.976013 0.217712i \(-0.0698593\pi\)
\(32\) 0 0
\(33\) 2.74264 1.58346i 0.477432 0.275646i
\(34\) 0 0
\(35\) −3.32686 + 3.15150i −0.562343 + 0.532701i
\(36\) 0 0
\(37\) −6.27231 + 3.62132i −1.03116 + 0.595341i −0.917317 0.398157i \(-0.869650\pi\)
−0.113844 + 0.993499i \(0.536317\pi\)
\(38\) 0 0
\(39\) 0.764683 + 0.441490i 0.122447 + 0.0706950i
\(40\) 0 0
\(41\) 7.34847i 1.14764i 0.818982 + 0.573819i \(0.194539\pi\)
−0.818982 + 0.573819i \(0.805461\pi\)
\(42\) 0 0
\(43\) −10.2928 −1.56963 −0.784816 0.619728i \(-0.787243\pi\)
−0.784816 + 0.619728i \(0.787243\pi\)
\(44\) 0 0
\(45\) 1.94218 3.36396i 0.289524 0.501470i
\(46\) 0 0
\(47\) −1.30540 2.26101i −0.190412 0.329802i 0.754975 0.655753i \(-0.227649\pi\)
−0.945387 + 0.325951i \(0.894316\pi\)
\(48\) 0 0
\(49\) −6.24264 + 3.16693i −0.891806 + 0.452418i
\(50\) 0 0
\(51\) 0.312181 + 0.540713i 0.0437140 + 0.0757149i
\(52\) 0 0
\(53\) 3.82282 + 2.20711i 0.525105 + 0.303169i 0.739021 0.673683i \(-0.235289\pi\)
−0.213916 + 0.976852i \(0.568622\pi\)
\(54\) 0 0
\(55\) −6.30301 −0.849898
\(56\) 0 0
\(57\) 5.48528 0.726543
\(58\) 0 0
\(59\) −0.753671 0.435132i −0.0981196 0.0566494i 0.450137 0.892959i \(-0.351375\pi\)
−0.548257 + 0.836310i \(0.684708\pi\)
\(60\) 0 0
\(61\) 6.27231 + 10.8640i 0.803087 + 1.39099i 0.917575 + 0.397563i \(0.130144\pi\)
−0.114488 + 0.993425i \(0.536523\pi\)
\(62\) 0 0
\(63\) 4.30759 4.08053i 0.542705 0.514099i
\(64\) 0 0
\(65\) −0.878680 1.52192i −0.108987 0.188771i
\(66\) 0 0
\(67\) −2.88537 + 4.99761i −0.352504 + 0.610556i −0.986688 0.162627i \(-0.948003\pi\)
0.634183 + 0.773183i \(0.281337\pi\)
\(68\) 0 0
\(69\) 3.16693 0.381253
\(70\) 0 0
\(71\) 14.5562i 1.72750i 0.503920 + 0.863750i \(0.331890\pi\)
−0.503920 + 0.863750i \(0.668110\pi\)
\(72\) 0 0
\(73\) −3.25736 1.88064i −0.381245 0.220112i 0.297115 0.954842i \(-0.403976\pi\)
−0.678360 + 0.734730i \(0.737309\pi\)
\(74\) 0 0
\(75\) −1.50734 + 0.870264i −0.174053 + 0.100489i
\(76\) 0 0
\(77\) −9.22911 2.74264i −1.05175 0.312553i
\(78\) 0 0
\(79\) 5.76230 3.32686i 0.648309 0.374301i −0.139499 0.990222i \(-0.544549\pi\)
0.787808 + 0.615921i \(0.211216\pi\)
\(80\) 0 0
\(81\) −1.37868 + 2.38794i −0.153187 + 0.265327i
\(82\) 0 0
\(83\) 3.48106i 0.382096i 0.981581 + 0.191048i \(0.0611885\pi\)
−0.981581 + 0.191048i \(0.938811\pi\)
\(84\) 0 0
\(85\) 1.24264i 0.134783i
\(86\) 0 0
\(87\) −1.99542 + 3.45617i −0.213932 + 0.370541i
\(88\) 0 0
\(89\) −6.98528 + 4.03295i −0.740438 + 0.427492i −0.822229 0.569157i \(-0.807270\pi\)
0.0817903 + 0.996650i \(0.473936\pi\)
\(90\) 0 0
\(91\) −0.624361 2.61079i −0.0654508 0.273685i
\(92\) 0 0
\(93\) −5.67796 + 3.27817i −0.588778 + 0.339931i
\(94\) 0 0
\(95\) −9.45451 5.45857i −0.970013 0.560037i
\(96\) 0 0
\(97\) 1.01461i 0.103018i −0.998673 0.0515091i \(-0.983597\pi\)
0.998673 0.0515091i \(-0.0164031\pi\)
\(98\) 0 0
\(99\) 8.16107 0.820218
\(100\) 0 0
\(101\) 5.76500 9.98528i 0.573639 0.993573i −0.422549 0.906340i \(-0.638864\pi\)
0.996188 0.0872323i \(-0.0278023\pi\)
\(102\) 0 0
\(103\) 6.99304 + 12.1123i 0.689044 + 1.19346i 0.972147 + 0.234370i \(0.0753028\pi\)
−0.283103 + 0.959089i \(0.591364\pi\)
\(104\) 0 0
\(105\) 3.87868 0.927572i 0.378520 0.0905218i
\(106\) 0 0
\(107\) 3.32686 + 5.76230i 0.321620 + 0.557062i 0.980822 0.194903i \(-0.0624393\pi\)
−0.659202 + 0.751966i \(0.729106\pi\)
\(108\) 0 0
\(109\) −2.59808 1.50000i −0.248851 0.143674i 0.370387 0.928877i \(-0.379225\pi\)
−0.619238 + 0.785203i \(0.712558\pi\)
\(110\) 0 0
\(111\) 6.30301 0.598255
\(112\) 0 0
\(113\) 4.24264 0.399114 0.199557 0.979886i \(-0.436050\pi\)
0.199557 + 0.979886i \(0.436050\pi\)
\(114\) 0 0
\(115\) −5.45857 3.15150i −0.509014 0.293879i
\(116\) 0 0
\(117\) 1.13770 + 1.97056i 0.105181 + 0.182179i
\(118\) 0 0
\(119\) 0.540713 1.81952i 0.0495671 0.166795i
\(120\) 0 0
\(121\) −1.12132 1.94218i −0.101938 0.176562i
\(122\) 0 0
\(123\) 3.19755 5.53833i 0.288314 0.499374i
\(124\) 0 0
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) 6.02937i 0.535020i −0.963555 0.267510i \(-0.913799\pi\)
0.963555 0.267510i \(-0.0862008\pi\)
\(128\) 0 0
\(129\) 7.75736 + 4.47871i 0.682997 + 0.394329i
\(130\) 0 0
\(131\) 11.6708 6.73814i 1.01968 0.588714i 0.105671 0.994401i \(-0.466301\pi\)
0.914012 + 0.405687i \(0.132968\pi\)
\(132\) 0 0
\(133\) −11.4685 12.1066i −0.994442 1.04978i
\(134\) 0 0
\(135\) −6.84372 + 3.95122i −0.589014 + 0.340067i
\(136\) 0 0
\(137\) 6.62132 11.4685i 0.565698 0.979817i −0.431287 0.902215i \(-0.641940\pi\)
0.996984 0.0776021i \(-0.0247264\pi\)
\(138\) 0 0
\(139\) 17.8276i 1.51212i −0.654504 0.756059i \(-0.727122\pi\)
0.654504 0.756059i \(-0.272878\pi\)
\(140\) 0 0
\(141\) 2.27208i 0.191343i
\(142\) 0 0
\(143\) 1.84611 3.19755i 0.154379 0.267393i
\(144\) 0 0
\(145\) 6.87868 3.97141i 0.571243 0.329807i
\(146\) 0 0
\(147\) 6.08293 + 0.329551i 0.501711 + 0.0271809i
\(148\) 0 0
\(149\) −20.0417 + 11.5711i −1.64188 + 0.947939i −0.661713 + 0.749757i \(0.730170\pi\)
−0.980165 + 0.198181i \(0.936496\pi\)
\(150\) 0 0
\(151\) −13.9114 8.03176i −1.13209 0.653615i −0.187634 0.982239i \(-0.560082\pi\)
−0.944461 + 0.328624i \(0.893415\pi\)
\(152\) 0 0
\(153\) 1.60896i 0.130077i
\(154\) 0 0
\(155\) 13.0488 1.04811
\(156\) 0 0
\(157\) 7.28692 12.6213i 0.581560 1.00729i −0.413735 0.910397i \(-0.635776\pi\)
0.995295 0.0968937i \(-0.0308907\pi\)
\(158\) 0 0
\(159\) −1.92077 3.32686i −0.152327 0.263837i
\(160\) 0 0
\(161\) −6.62132 6.98975i −0.521833 0.550869i
\(162\) 0 0
\(163\) −8.03176 13.9114i −0.629096 1.08963i −0.987733 0.156149i \(-0.950092\pi\)
0.358638 0.933477i \(-0.383241\pi\)
\(164\) 0 0
\(165\) 4.75039 + 2.74264i 0.369818 + 0.213514i
\(166\) 0 0
\(167\) 14.1354 1.09383 0.546914 0.837189i \(-0.315802\pi\)
0.546914 + 0.837189i \(0.315802\pi\)
\(168\) 0 0
\(169\) −11.9706 −0.920813
\(170\) 0 0
\(171\) 12.2416 + 7.06769i 0.936139 + 0.540480i
\(172\) 0 0
\(173\) 9.43924 + 16.3492i 0.717652 + 1.24301i 0.961928 + 0.273304i \(0.0881165\pi\)
−0.244276 + 0.969706i \(0.578550\pi\)
\(174\) 0 0
\(175\) 5.07227 + 1.50734i 0.383428 + 0.113944i
\(176\) 0 0
\(177\) 0.378680 + 0.655892i 0.0284633 + 0.0492999i
\(178\) 0 0
\(179\) 10.6050 18.3683i 0.792651 1.37291i −0.131669 0.991294i \(-0.542033\pi\)
0.924320 0.381618i \(-0.124633\pi\)
\(180\) 0 0
\(181\) 18.7554 1.39408 0.697038 0.717034i \(-0.254501\pi\)
0.697038 + 0.717034i \(0.254501\pi\)
\(182\) 0 0
\(183\) 10.9171i 0.807018i
\(184\) 0 0
\(185\) −10.8640 6.27231i −0.798734 0.461149i
\(186\) 0 0
\(187\) 2.26101 1.30540i 0.165342 0.0954600i
\(188\) 0 0
\(189\) −11.7401 + 2.80761i −0.853970 + 0.204224i
\(190\) 0 0
\(191\) −3.91619 + 2.26101i −0.283365 + 0.163601i −0.634946 0.772557i \(-0.718978\pi\)
0.351581 + 0.936158i \(0.385644\pi\)
\(192\) 0 0
\(193\) −8.62132 + 14.9326i −0.620576 + 1.07487i 0.368802 + 0.929508i \(0.379768\pi\)
−0.989379 + 0.145362i \(0.953565\pi\)
\(194\) 0 0
\(195\) 1.52937i 0.109520i
\(196\) 0 0
\(197\) 1.07107i 0.0763104i −0.999272 0.0381552i \(-0.987852\pi\)
0.999272 0.0381552i \(-0.0121481\pi\)
\(198\) 0 0
\(199\) −2.53613 + 4.39271i −0.179782 + 0.311391i −0.941806 0.336158i \(-0.890873\pi\)
0.762024 + 0.647549i \(0.224206\pi\)
\(200\) 0 0
\(201\) 4.34924 2.51104i 0.306772 0.177115i
\(202\) 0 0
\(203\) 11.8001 2.82195i 0.828205 0.198062i
\(204\) 0 0
\(205\) −11.0227 + 6.36396i −0.769859 + 0.444478i
\(206\) 0 0
\(207\) 7.06769 + 4.08053i 0.491239 + 0.283617i
\(208\) 0 0
\(209\) 22.9369i 1.58658i
\(210\) 0 0
\(211\) −1.24872 −0.0859656 −0.0429828 0.999076i \(-0.513686\pi\)
−0.0429828 + 0.999076i \(0.513686\pi\)
\(212\) 0 0
\(213\) 6.33386 10.9706i 0.433989 0.751691i
\(214\) 0 0
\(215\) −8.91380 15.4392i −0.607916 1.05294i
\(216\) 0 0
\(217\) 19.1066 + 5.67796i 1.29704 + 0.385445i
\(218\) 0 0
\(219\) 1.63665 + 2.83476i 0.110595 + 0.191555i
\(220\) 0 0
\(221\) 0.630399 + 0.363961i 0.0424052 + 0.0244827i
\(222\) 0 0
\(223\) −10.1445 −0.679329 −0.339664 0.940547i \(-0.610313\pi\)
−0.339664 + 0.940547i \(0.610313\pi\)
\(224\) 0 0
\(225\) −4.48528 −0.299019
\(226\) 0 0
\(227\) 14.8684 + 8.58425i 0.986847 + 0.569757i 0.904330 0.426833i \(-0.140371\pi\)
0.0825170 + 0.996590i \(0.473704\pi\)
\(228\) 0 0
\(229\) 9.94655 + 17.2279i 0.657286 + 1.13845i 0.981315 + 0.192406i \(0.0616291\pi\)
−0.324029 + 0.946047i \(0.605038\pi\)
\(230\) 0 0
\(231\) 5.76230 + 6.08293i 0.379131 + 0.400227i
\(232\) 0 0
\(233\) 12.1066 + 20.9692i 0.793130 + 1.37374i 0.924020 + 0.382344i \(0.124883\pi\)
−0.130890 + 0.991397i \(0.541783\pi\)
\(234\) 0 0
\(235\) 2.26101 3.91619i 0.147492 0.255464i
\(236\) 0 0
\(237\) −5.79050 −0.376133
\(238\) 0 0
\(239\) 15.4392i 0.998676i 0.866407 + 0.499338i \(0.166423\pi\)
−0.866407 + 0.499338i \(0.833577\pi\)
\(240\) 0 0
\(241\) 14.2279 + 8.21449i 0.916501 + 0.529142i 0.882517 0.470280i \(-0.155847\pi\)
0.0339839 + 0.999422i \(0.489181\pi\)
\(242\) 0 0
\(243\) 13.9318 8.04354i 0.893726 0.515993i
\(244\) 0 0
\(245\) −10.1567 6.62132i −0.648886 0.423021i
\(246\) 0 0
\(247\) 5.53833 3.19755i 0.352395 0.203455i
\(248\) 0 0
\(249\) 1.51472 2.62357i 0.0959914 0.166262i
\(250\) 0 0
\(251\) 7.17327i 0.452773i −0.974038 0.226386i \(-0.927309\pi\)
0.974038 0.226386i \(-0.0726912\pi\)
\(252\) 0 0
\(253\) 13.2426i 0.832558i
\(254\) 0 0
\(255\) −0.540713 + 0.936542i −0.0338607 + 0.0586485i
\(256\) 0 0
\(257\) −10.8640 + 6.27231i −0.677675 + 0.391256i −0.798979 0.601359i \(-0.794626\pi\)
0.121303 + 0.992615i \(0.461293\pi\)
\(258\) 0 0
\(259\) −13.1781 13.9114i −0.818850 0.864413i
\(260\) 0 0
\(261\) −8.90644 + 5.14214i −0.551295 + 0.318290i
\(262\) 0 0
\(263\) 22.0605 + 12.7367i 1.36031 + 0.785376i 0.989665 0.143399i \(-0.0458031\pi\)
0.370646 + 0.928774i \(0.379136\pi\)
\(264\) 0 0
\(265\) 7.64564i 0.469668i
\(266\) 0 0
\(267\) 7.01947 0.429585
\(268\) 0 0
\(269\) 11.1713 19.3492i 0.681126 1.17974i −0.293512 0.955955i \(-0.594824\pi\)
0.974638 0.223789i \(-0.0718426\pi\)
\(270\) 0 0
\(271\) 6.99304 + 12.1123i 0.424797 + 0.735769i 0.996401 0.0847597i \(-0.0270123\pi\)
−0.571605 + 0.820529i \(0.693679\pi\)
\(272\) 0 0
\(273\) −0.665476 + 2.23936i −0.0402765 + 0.135532i
\(274\) 0 0
\(275\) 3.63904 + 6.30301i 0.219443 + 0.380086i
\(276\) 0 0
\(277\) 1.96768 + 1.13604i 0.118226 + 0.0682580i 0.557947 0.829877i \(-0.311589\pi\)
−0.439721 + 0.898135i \(0.644923\pi\)
\(278\) 0 0
\(279\) −16.8955 −1.01151
\(280\) 0 0
\(281\) −7.75736 −0.462765 −0.231383 0.972863i \(-0.574325\pi\)
−0.231383 + 0.972863i \(0.574325\pi\)
\(282\) 0 0
\(283\) −20.8977 12.0653i −1.24224 0.717208i −0.272691 0.962102i \(-0.587914\pi\)
−0.969550 + 0.244894i \(0.921247\pi\)
\(284\) 0 0
\(285\) 4.75039 + 8.22792i 0.281389 + 0.487380i
\(286\) 0 0
\(287\) −18.9090 + 4.52202i −1.11616 + 0.266927i
\(288\) 0 0
\(289\) −8.24264 14.2767i −0.484861 0.839804i
\(290\) 0 0
\(291\) −0.441490 + 0.764683i −0.0258806 + 0.0448265i
\(292\) 0 0
\(293\) −5.49333 −0.320923 −0.160462 0.987042i \(-0.551298\pi\)
−0.160462 + 0.987042i \(0.551298\pi\)
\(294\) 0 0
\(295\) 1.50734i 0.0877608i
\(296\) 0 0
\(297\) −14.3787 8.30153i −0.834336 0.481704i
\(298\) 0 0
\(299\) 3.19755 1.84611i 0.184919 0.106763i
\(300\) 0 0
\(301\) −6.33386 26.4853i −0.365077 1.52659i
\(302\) 0 0
\(303\) −8.68983 + 5.01708i −0.499218 + 0.288223i
\(304\) 0 0
\(305\) −10.8640 + 18.8169i −0.622069 + 1.07745i
\(306\) 0 0
\(307\) 12.6060i 0.719463i 0.933056 + 0.359732i \(0.117132\pi\)
−0.933056 + 0.359732i \(0.882868\pi\)
\(308\) 0 0
\(309\) 12.1716i 0.692417i
\(310\) 0 0
\(311\) 14.6761 25.4197i 0.832205 1.44142i −0.0640807 0.997945i \(-0.520412\pi\)
0.896286 0.443477i \(-0.146255\pi\)
\(312\) 0 0
\(313\) −9.25736 + 5.34474i −0.523257 + 0.302103i −0.738266 0.674510i \(-0.764355\pi\)
0.215009 + 0.976612i \(0.431022\pi\)
\(314\) 0 0
\(315\) 9.85128 + 2.92753i 0.555057 + 0.164948i
\(316\) 0 0
\(317\) −26.7597 + 15.4497i −1.50298 + 0.867744i −0.502983 + 0.864296i \(0.667764\pi\)
−0.999994 + 0.00344803i \(0.998902\pi\)
\(318\) 0 0
\(319\) 14.4521 + 8.34394i 0.809164 + 0.467171i
\(320\) 0 0
\(321\) 5.79050i 0.323194i
\(322\) 0 0
\(323\) 4.52202 0.251612
\(324\) 0 0
\(325\) −1.01461 + 1.75736i −0.0562805 + 0.0974808i
\(326\) 0 0
\(327\) 1.30540 + 2.26101i 0.0721886 + 0.125034i
\(328\) 0 0
\(329\) 5.01472 4.75039i 0.276470 0.261898i
\(330\) 0 0
\(331\) −7.40740 12.8300i −0.407147 0.705200i 0.587421 0.809281i \(-0.300143\pi\)
−0.994569 + 0.104081i \(0.966810\pi\)
\(332\) 0 0
\(333\) 14.0665 + 8.12132i 0.770842 + 0.445046i
\(334\) 0 0
\(335\) −9.99523 −0.546098
\(336\) 0 0
\(337\) −6.24264 −0.340058 −0.170029 0.985439i \(-0.554386\pi\)
−0.170029 + 0.985439i \(0.554386\pi\)
\(338\) 0 0
\(339\) −3.19755 1.84611i −0.173667 0.100267i
\(340\) 0 0
\(341\) 13.7078 + 23.7426i 0.742320 + 1.28574i
\(342\) 0 0
\(343\) −11.9906 14.1147i −0.647434 0.762121i
\(344\) 0 0
\(345\) 2.74264 + 4.75039i 0.147659 + 0.255753i
\(346\) 0 0
\(347\) −12.5538 + 21.7438i −0.673922 + 1.16727i 0.302860 + 0.953035i \(0.402058\pi\)
−0.976783 + 0.214233i \(0.931275\pi\)
\(348\) 0 0
\(349\) 20.0162 1.07144 0.535721 0.844395i \(-0.320040\pi\)
0.535721 + 0.844395i \(0.320040\pi\)
\(350\) 0 0
\(351\) 4.62915i 0.247086i
\(352\) 0 0
\(353\) 31.3492 + 18.0995i 1.66855 + 0.963339i 0.968420 + 0.249325i \(0.0802089\pi\)
0.700132 + 0.714013i \(0.253124\pi\)
\(354\) 0 0
\(355\) −21.8343 + 12.6060i −1.15884 + 0.669058i
\(356\) 0 0
\(357\) −1.19925 + 1.13604i −0.0634711 + 0.0601256i
\(358\) 0 0
\(359\) 29.1282 16.8172i 1.53733 0.887577i 0.538335 0.842731i \(-0.319054\pi\)
0.998994 0.0448462i \(-0.0142798\pi\)
\(360\) 0 0
\(361\) 10.3640 17.9509i 0.545472 0.944785i
\(362\) 0 0
\(363\) 1.95169i 0.102437i
\(364\) 0 0
\(365\) 6.51472i 0.340996i
\(366\) 0 0
\(367\) −10.6853 + 18.5074i −0.557766 + 0.966078i 0.439917 + 0.898038i \(0.355008\pi\)
−0.997683 + 0.0680400i \(0.978325\pi\)
\(368\) 0 0
\(369\) 14.2721 8.23999i 0.742975 0.428957i
\(370\) 0 0
\(371\) −3.32686 + 11.1950i −0.172722 + 0.581218i
\(372\) 0 0
\(373\) −11.4685 + 6.62132i −0.593815 + 0.342839i −0.766604 0.642120i \(-0.778055\pi\)
0.172790 + 0.984959i \(0.444722\pi\)
\(374\) 0 0
\(375\) −9.13777 5.27569i −0.471872 0.272436i
\(376\) 0 0
\(377\) 4.65279i 0.239631i
\(378\) 0 0
\(379\) −28.2294 −1.45005 −0.725023 0.688725i \(-0.758171\pi\)
−0.725023 + 0.688725i \(0.758171\pi\)
\(380\) 0 0
\(381\) −2.62357 + 4.54416i −0.134410 + 0.232804i
\(382\) 0 0
\(383\) −2.38682 4.13410i −0.121961 0.211242i 0.798580 0.601889i \(-0.205585\pi\)
−0.920541 + 0.390646i \(0.872252\pi\)
\(384\) 0 0
\(385\) −3.87868 16.2189i −0.197676 0.826589i
\(386\) 0 0
\(387\) 11.5415 + 19.9905i 0.586687 + 1.01617i
\(388\) 0 0
\(389\) 5.04757 + 2.91421i 0.255922 + 0.147756i 0.622473 0.782641i \(-0.286128\pi\)
−0.366551 + 0.930398i \(0.619461\pi\)
\(390\) 0 0
\(391\) 2.61079 0.132033
\(392\) 0 0
\(393\) −11.7279 −0.591595
\(394\) 0 0
\(395\) 9.98059 + 5.76230i 0.502178 + 0.289933i
\(396\) 0 0
\(397\) 9.94655 + 17.2279i 0.499203 + 0.864645i 1.00000 0.000920276i \(-0.000292933\pi\)
−0.500797 + 0.865565i \(0.666960\pi\)
\(398\) 0 0
\(399\) 3.37548 + 14.1147i 0.168985 + 0.706618i
\(400\) 0 0
\(401\) 6.62132 + 11.4685i 0.330653 + 0.572708i 0.982640 0.185522i \(-0.0593977\pi\)
−0.651987 + 0.758230i \(0.726064\pi\)
\(402\) 0 0
\(403\) −3.82192 + 6.61975i −0.190383 + 0.329753i
\(404\) 0 0
\(405\) −4.77589 −0.237316
\(406\) 0 0
\(407\) 26.3563i 1.30643i
\(408\) 0 0
\(409\) −19.3492 11.1713i −0.956758 0.552385i −0.0615846 0.998102i \(-0.519615\pi\)
−0.895174 + 0.445717i \(0.852949\pi\)
\(410\) 0 0
\(411\) −9.98059 + 5.76230i −0.492306 + 0.284233i
\(412\) 0 0
\(413\) 0.655892 2.20711i 0.0322744 0.108605i
\(414\) 0 0
\(415\) −5.22158 + 3.01468i −0.256317 + 0.147985i
\(416\) 0 0
\(417\) −7.75736 + 13.4361i −0.379880 + 0.657971i
\(418\) 0 0
\(419\) 1.74053i 0.0850304i −0.999096 0.0425152i \(-0.986463\pi\)
0.999096 0.0425152i \(-0.0135371\pi\)
\(420\) 0 0
\(421\) 7.75736i 0.378071i −0.981970 0.189035i \(-0.939464\pi\)
0.981970 0.189035i \(-0.0605361\pi\)
\(422\) 0 0
\(423\) −2.92753 + 5.07064i −0.142342 + 0.246543i
\(424\) 0 0
\(425\) −1.24264 + 0.717439i −0.0602769 + 0.0348009i
\(426\) 0 0
\(427\) −24.0953 + 22.8252i −1.16605 + 1.10459i
\(428\) 0 0
\(429\) −2.78272 + 1.60660i −0.134351 + 0.0775675i
\(430\) 0 0
\(431\) −3.91619 2.26101i −0.188636 0.108909i 0.402708 0.915329i \(-0.368069\pi\)
−0.591344 + 0.806419i \(0.701402\pi\)
\(432\) 0 0
\(433\) 20.6105i 0.990479i −0.868757 0.495239i \(-0.835080\pi\)
0.868757 0.495239i \(-0.164920\pi\)
\(434\) 0 0
\(435\) −6.91235 −0.331422
\(436\) 0 0
\(437\) 11.4685 19.8640i 0.548611 0.950222i
\(438\) 0 0
\(439\) −15.2915 26.4856i −0.729822 1.26409i −0.956958 0.290226i \(-0.906270\pi\)
0.227136 0.973863i \(-0.427064\pi\)
\(440\) 0 0
\(441\) 13.1508 + 8.57321i 0.626227 + 0.408248i
\(442\) 0 0
\(443\) 3.95122 + 6.84372i 0.187728 + 0.325155i 0.944492 0.328533i \(-0.106554\pi\)
−0.756764 + 0.653688i \(0.773221\pi\)
\(444\) 0 0
\(445\) −12.0989 6.98528i −0.573541 0.331134i
\(446\) 0 0
\(447\) 20.1398 0.952578
\(448\) 0 0
\(449\) 22.2426 1.04970 0.524848 0.851196i \(-0.324122\pi\)
0.524848 + 0.851196i \(0.324122\pi\)
\(450\) 0 0
\(451\) −23.1587 13.3707i −1.09050 0.629602i
\(452\) 0 0
\(453\) 6.98975 + 12.1066i 0.328407 + 0.568818i
\(454\) 0 0
\(455\) 3.37548 3.19755i 0.158245 0.149904i
\(456\) 0 0
\(457\) 1.25736 + 2.17781i 0.0588168 + 0.101874i 0.893935 0.448198i \(-0.147934\pi\)
−0.835118 + 0.550071i \(0.814601\pi\)
\(458\) 0 0
\(459\) 1.63665 2.83476i 0.0763923 0.132315i
\(460\) 0 0
\(461\) −5.31925 −0.247742 −0.123871 0.992298i \(-0.539531\pi\)
−0.123871 + 0.992298i \(0.539531\pi\)
\(462\) 0 0
\(463\) 7.27809i 0.338241i 0.985595 + 0.169121i \(0.0540928\pi\)
−0.985595 + 0.169121i \(0.945907\pi\)
\(464\) 0 0
\(465\) −9.83452 5.67796i −0.456065 0.263309i
\(466\) 0 0
\(467\) −20.7149 + 11.9597i −0.958569 + 0.553430i −0.895732 0.444594i \(-0.853348\pi\)
−0.0628367 + 0.998024i \(0.520015\pi\)
\(468\) 0 0
\(469\) −14.6354 4.34924i −0.675800 0.200829i
\(470\) 0 0
\(471\) −10.9839 + 6.34155i −0.506110 + 0.292203i
\(472\) 0 0
\(473\) 18.7279 32.4377i 0.861111 1.49149i
\(474\) 0 0
\(475\) 12.6060i 0.578404i
\(476\) 0 0
\(477\) 9.89949i 0.453267i
\(478\) 0 0
\(479\) −20.9791 + 36.3369i −0.958560 + 1.66027i −0.232557 + 0.972583i \(0.574709\pi\)
−0.726003 + 0.687691i \(0.758624\pi\)
\(480\) 0 0
\(481\) 6.36396 3.67423i 0.290172 0.167531i
\(482\) 0 0
\(483\) 1.94883 + 8.14912i 0.0886749 + 0.370798i
\(484\) 0 0
\(485\) 1.52192 0.878680i 0.0691067 0.0398988i
\(486\) 0 0
\(487\) 13.1467 + 7.59027i 0.595735 + 0.343948i 0.767362 0.641214i \(-0.221569\pi\)
−0.171627 + 0.985162i \(0.554902\pi\)
\(488\) 0 0
\(489\) 13.9795i 0.632175i
\(490\) 0 0
\(491\) −11.1758 −0.504355 −0.252177 0.967681i \(-0.581147\pi\)
−0.252177 + 0.967681i \(0.581147\pi\)
\(492\) 0 0
\(493\) −1.64501 + 2.84924i −0.0740876 + 0.128323i
\(494\) 0 0
\(495\) 7.06769 + 12.2416i 0.317669 + 0.550219i
\(496\) 0 0
\(497\) −37.4558 + 8.95743i −1.68012 + 0.401796i
\(498\) 0 0
\(499\) −13.1781 22.8252i −0.589935 1.02180i −0.994240 0.107173i \(-0.965820\pi\)
0.404306 0.914624i \(-0.367513\pi\)
\(500\) 0 0
\(501\) −10.6534 6.15076i −0.475960 0.274796i
\(502\) 0 0
\(503\) 30.4336 1.35697 0.678484 0.734615i \(-0.262637\pi\)
0.678484 + 0.734615i \(0.262637\pi\)
\(504\) 0 0
\(505\) 19.9706 0.888678
\(506\) 0 0
\(507\) 9.02186 + 5.20877i 0.400675 + 0.231330i
\(508\) 0 0
\(509\) −8.51167 14.7426i −0.377273 0.653456i 0.613391 0.789779i \(-0.289805\pi\)
−0.990664 + 0.136323i \(0.956471\pi\)
\(510\) 0 0
\(511\) 2.83476 9.53910i 0.125403 0.421985i
\(512\) 0 0
\(513\) −14.3787 24.9046i −0.634834 1.09957i
\(514\) 0 0
\(515\) −12.1123 + 20.9791i −0.533731 + 0.924450i
\(516\) 0 0
\(517\) 9.50079 0.417844
\(518\) 0 0
\(519\) 16.4293i 0.721164i
\(520\) 0 0
\(521\) −12.2574 7.07679i −0.537005 0.310040i 0.206859 0.978371i \(-0.433676\pi\)
−0.743864 + 0.668331i \(0.767009\pi\)
\(522\) 0 0
\(523\) −11.8537 + 6.84372i −0.518325 + 0.299255i −0.736249 0.676711i \(-0.763405\pi\)
0.217924 + 0.975966i \(0.430071\pi\)
\(524\) 0 0
\(525\) −3.16693 3.34315i −0.138216 0.145907i
\(526\) 0 0
\(527\) −4.68087 + 2.70250i −0.203902 + 0.117723i
\(528\) 0 0
\(529\) −4.87868 + 8.45012i −0.212117 + 0.367397i
\(530\) 0 0
\(531\) 1.95169i 0.0846961i
\(532\) 0 0
\(533\) 7.45584i 0.322948i
\(534\) 0 0
\(535\) −5.76230 + 9.98059i −0.249126 + 0.431499i
\(536\) 0 0
\(537\) −15.9853 + 9.22911i −0.689816 + 0.398265i
\(538\) 0 0
\(539\) 1.37803 25.4360i 0.0593560 1.09561i
\(540\) 0 0
\(541\) −18.1865 + 10.5000i −0.781900 + 0.451430i −0.837103 0.547045i \(-0.815753\pi\)
0.0552031 + 0.998475i \(0.482419\pi\)
\(542\) 0 0
\(543\) −14.1354 8.16107i −0.606607 0.350225i
\(544\) 0 0
\(545\) 5.19615i 0.222579i
\(546\) 0 0
\(547\) −15.4392 −0.660131 −0.330065 0.943958i \(-0.607071\pi\)
−0.330065 + 0.943958i \(0.607071\pi\)
\(548\) 0 0
\(549\) 14.0665 24.3640i 0.600345 1.03983i
\(550\) 0 0
\(551\) 14.4521 + 25.0318i 0.615681 + 1.06639i
\(552\) 0 0
\(553\) 12.1066 + 12.7802i 0.514825 + 0.543471i
\(554\) 0 0
\(555\) 5.45857 + 9.45451i 0.231703 + 0.401322i
\(556\) 0 0
\(557\) 33.4778 + 19.3284i 1.41850 + 0.818972i 0.996167 0.0874688i \(-0.0278778\pi\)
0.422333 + 0.906441i \(0.361211\pi\)
\(558\) 0 0
\(559\) 10.4432 0.441699
\(560\) 0 0
\(561\) −2.27208 −0.0959272
\(562\) 0 0
\(563\) 3.76835 + 2.17566i 0.158817 + 0.0916931i 0.577302 0.816531i \(-0.304105\pi\)
−0.418485 + 0.908224i \(0.637439\pi\)
\(564\) 0 0
\(565\) 3.67423 + 6.36396i 0.154576 + 0.267734i
\(566\) 0 0
\(567\) −6.99304 2.07814i −0.293680 0.0872737i
\(568\) 0 0
\(569\) −2.74264 4.75039i −0.114977 0.199147i 0.802793 0.596257i \(-0.203346\pi\)
−0.917771 + 0.397111i \(0.870013\pi\)
\(570\) 0 0
\(571\) −4.83420 + 8.37309i −0.202305 + 0.350403i −0.949271 0.314460i \(-0.898177\pi\)
0.746966 + 0.664863i \(0.231510\pi\)
\(572\) 0 0
\(573\) 3.93535 0.164402
\(574\) 0 0
\(575\) 7.27809i 0.303517i
\(576\) 0 0
\(577\) 25.5000 + 14.7224i 1.06158 + 0.612903i 0.925869 0.377846i \(-0.123335\pi\)
0.135710 + 0.990749i \(0.456668\pi\)
\(578\) 0 0
\(579\) 12.9953 7.50282i 0.540065 0.311807i
\(580\) 0 0
\(581\) −8.95743 + 2.14214i −0.371617 + 0.0888708i
\(582\) 0 0
\(583\) −13.9114 + 8.03176i −0.576152 + 0.332641i
\(584\) 0 0
\(585\) −1.97056 + 3.41311i −0.0814727 + 0.141115i
\(586\) 0 0
\(587\) 44.7802i 1.84828i 0.382060 + 0.924138i \(0.375215\pi\)
−0.382060 + 0.924138i \(0.624785\pi\)
\(588\) 0 0
\(589\) 47.4853i 1.95660i
\(590\) 0 0
\(591\) −0.466056 + 0.807232i −0.0191710 + 0.0332051i
\(592\) 0 0
\(593\) 10.8640 6.27231i 0.446129 0.257573i −0.260065 0.965591i \(-0.583744\pi\)
0.706194 + 0.708018i \(0.250411\pi\)
\(594\) 0 0
\(595\) 3.19755 0.764683i 0.131087 0.0313490i
\(596\) 0 0
\(597\) 3.82282 2.20711i 0.156458 0.0903309i
\(598\) 0 0
\(599\) −10.9839 6.34155i −0.448789 0.259109i 0.258530 0.966003i \(-0.416762\pi\)
−0.707319 + 0.706895i \(0.750095\pi\)
\(600\) 0 0
\(601\) 42.2357i 1.72283i 0.507903 + 0.861414i \(0.330421\pi\)
−0.507903 + 0.861414i \(0.669579\pi\)
\(602\) 0 0
\(603\) 12.9417 0.527027
\(604\) 0 0
\(605\) 1.94218 3.36396i 0.0789610 0.136764i
\(606\) 0 0
\(607\) −5.14693 8.91474i −0.208907 0.361838i 0.742463 0.669887i \(-0.233657\pi\)
−0.951371 + 0.308049i \(0.900324\pi\)
\(608\) 0 0
\(609\) −10.1213 3.00778i −0.410137 0.121881i
\(610\) 0 0
\(611\) 1.32447 + 2.29405i 0.0535823 + 0.0928073i
\(612\) 0 0
\(613\) −20.3389 11.7426i −0.821478 0.474281i 0.0294476 0.999566i \(-0.490625\pi\)
−0.850926 + 0.525286i \(0.823959\pi\)
\(614\) 0 0
\(615\) 11.0767 0.446654
\(616\) 0 0
\(617\) 3.21320 0.129359 0.0646793 0.997906i \(-0.479398\pi\)
0.0646793 + 0.997906i \(0.479398\pi\)
\(618\) 0 0
\(619\) −15.0512 8.68983i −0.604960 0.349274i 0.166030 0.986121i \(-0.446905\pi\)
−0.770990 + 0.636847i \(0.780238\pi\)
\(620\) 0 0
\(621\) −8.30153 14.3787i −0.333129 0.576997i
\(622\) 0 0
\(623\) −14.6761 15.4927i −0.587985 0.620703i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) −9.98059 + 17.2869i −0.398586 + 0.690372i
\(628\) 0 0
\(629\) 5.19615 0.207184
\(630\) 0 0
\(631\) 10.2928i 0.409749i 0.978788 + 0.204874i \(0.0656786\pi\)
−0.978788 + 0.204874i \(0.934321\pi\)
\(632\) 0 0
\(633\) 0.941125 + 0.543359i 0.0374064 + 0.0215966i
\(634\) 0 0
\(635\) 9.04405 5.22158i 0.358902 0.207212i
\(636\) 0 0
\(637\) 6.33386 3.21320i 0.250957 0.127312i
\(638\) 0 0
\(639\) 28.2708 16.3221i 1.11837 0.645694i
\(640\) 0 0
\(641\) −5.01472 + 8.68575i −0.198069 + 0.343066i −0.947902 0.318561i \(-0.896800\pi\)
0.749833 + 0.661627i \(0.230134\pi\)
\(642\) 0 0
\(643\) 23.0492i 0.908971i −0.890754 0.454486i \(-0.849823\pi\)
0.890754 0.454486i \(-0.150177\pi\)
\(644\) 0 0
\(645\) 15.5147i 0.610891i
\(646\) 0 0
\(647\) −3.15150 + 5.45857i −0.123898 + 0.214598i −0.921302 0.388848i \(-0.872873\pi\)
0.797403 + 0.603447i \(0.206206\pi\)
\(648\) 0 0
\(649\) 2.74264 1.58346i 0.107658 0.0621564i
\(650\) 0 0
\(651\) −11.9294 12.5932i −0.467551 0.493567i
\(652\) 0 0
\(653\) 4.41717 2.55025i 0.172857 0.0997991i −0.411075 0.911601i \(-0.634847\pi\)
0.583932 + 0.811802i \(0.301513\pi\)
\(654\) 0 0
\(655\) 20.2144 + 11.6708i 0.789843 + 0.456016i
\(656\) 0 0
\(657\) 8.43519i 0.329088i
\(658\) 0 0
\(659\) 11.5415 0.449593 0.224796 0.974406i \(-0.427828\pi\)
0.224796 + 0.974406i \(0.427828\pi\)
\(660\) 0 0
\(661\) −18.8169 + 32.5919i −0.731894 + 1.26768i 0.224179 + 0.974548i \(0.428030\pi\)
−0.956073 + 0.293129i \(0.905303\pi\)
\(662\) 0 0
\(663\) −0.316742 0.548614i −0.0123012 0.0213064i
\(664\) 0 0
\(665\) 8.22792 27.6873i 0.319065 1.07367i
\(666\) 0 0
\(667\) 8.34394 + 14.4521i 0.323079 + 0.559589i
\(668\) 0 0
\(669\) 7.64564 + 4.41421i 0.295598 + 0.170663i
\(670\) 0 0
\(671\) −45.6504 −1.76232
\(672\) 0 0
\(673\) 6.24264 0.240636 0.120318 0.992735i \(-0.461609\pi\)
0.120318 + 0.992735i \(0.461609\pi\)
\(674\) 0 0
\(675\) 7.90245 + 4.56248i 0.304165 + 0.175610i
\(676\) 0 0
\(677\) −5.34474 9.25736i −0.205415 0.355789i 0.744850 0.667232i \(-0.232521\pi\)
−0.950265 + 0.311443i \(0.899188\pi\)
\(678\) 0 0
\(679\) 2.61079 0.624361i 0.100193 0.0239608i
\(680\) 0 0
\(681\) −7.47056 12.9394i −0.286273 0.495839i
\(682\) 0 0
\(683\) 22.4051 38.8067i 0.857306 1.48490i −0.0171833 0.999852i \(-0.505470\pi\)
0.874489 0.485045i \(-0.161197\pi\)
\(684\) 0 0
\(685\) 22.9369 0.876375
\(686\) 0 0
\(687\) 17.3122i 0.660503i
\(688\) 0 0
\(689\) −3.87868 2.23936i −0.147766 0.0853127i
\(690\) 0 0
\(691\) 2.26101 1.30540i 0.0860130 0.0496596i −0.456377 0.889787i \(-0.650853\pi\)
0.542390 + 0.840127i \(0.317520\pi\)
\(692\) 0 0
\(693\) 5.02207 + 21.0000i 0.190773 + 0.797724i
\(694\) 0 0
\(695\) 26.7414 15.4392i 1.01436 0.585641i
\(696\) 0 0
\(697\) 2.63604 4.56575i 0.0998471 0.172940i
\(698\) 0 0
\(699\) 21.0719i 0.797012i
\(700\) 0 0
\(701\) 14.1421i 0.534141i −0.963677 0.267071i \(-0.913944\pi\)
0.963677 0.267071i \(-0.0860557\pi\)
\(702\) 0 0
\(703\) 22.8252 39.5344i 0.860869 1.49107i
\(704\) 0 0
\(705\) −3.40812 + 1.96768i −0.128357 + 0.0741070i
\(706\) 0 0
\(707\) 29.2417 + 8.68983i 1.09975 + 0.326815i
\(708\) 0 0
\(709\) 26.1654 15.1066i 0.982662 0.567340i 0.0795894 0.996828i \(-0.474639\pi\)
0.903073 + 0.429487i \(0.141306\pi\)
\(710\) 0 0
\(711\) −12.9228 7.46096i −0.484641 0.279808i
\(712\) 0 0
\(713\) 27.4156i 1.02672i
\(714\) 0 0
\(715\) 6.39511 0.239163
\(716\) 0 0
\(717\) 6.71807 11.6360i 0.250891 0.434556i
\(718\) 0 0
\(719\) −3.15150 5.45857i −0.117531 0.203570i 0.801257 0.598320i \(-0.204165\pi\)
−0.918789 + 0.394750i \(0.870831\pi\)
\(720\) 0 0
\(721\) −26.8640 + 25.4480i −1.00047 + 0.947732i
\(722\) 0 0
\(723\) −7.14878 12.3820i −0.265866 0.460493i
\(724\) 0 0
\(725\) −7.94282 4.58579i −0.294989 0.170312i
\(726\) 0 0
\(727\) −40.2795 −1.49389 −0.746943 0.664889i \(-0.768479\pi\)
−0.746943 + 0.664889i \(0.768479\pi\)
\(728\) 0 0
\(729\) −5.72792 −0.212145
\(730\) 0 0
\(731\) 6.39511 + 3.69222i 0.236532 + 0.136562i
\(732\) 0 0
\(733\) 0.0615465 + 0.106602i 0.00227327 + 0.00393742i 0.867160 0.498030i \(-0.165943\pi\)
−0.864887 + 0.501967i \(0.832610\pi\)
\(734\) 0 0
\(735\) 4.77364 + 9.40979i 0.176078 + 0.347085i
\(736\) 0 0
\(737\) −10.5000 18.1865i −0.386772 0.669910i
\(738\) 0 0
\(739\) 16.3757 28.3635i 0.602390 1.04337i −0.390068 0.920786i \(-0.627549\pi\)
0.992458 0.122584i \(-0.0391180\pi\)
\(740\) 0 0
\(741\) −5.56543 −0.204451
\(742\) 0 0
\(743\) 27.3464i 1.00324i 0.865088 + 0.501621i \(0.167263\pi\)
−0.865088 + 0.501621i \(0.832737\pi\)
\(744\) 0 0
\(745\) −34.7132 20.0417i −1.27179 0.734270i
\(746\) 0 0
\(747\) 6.76085 3.90338i 0.247367 0.142817i
\(748\) 0 0
\(749\) −12.7802 + 12.1066i −0.466980 + 0.442366i
\(750\) 0 0
\(751\) 12.3820 7.14878i 0.451827 0.260863i −0.256774 0.966471i \(-0.582660\pi\)
0.708602 + 0.705609i \(0.249326\pi\)
\(752\) 0 0
\(753\) −3.12132 + 5.40629i −0.113747 + 0.197016i
\(754\) 0 0
\(755\) 27.8228i 1.01258i
\(756\) 0 0
\(757\) 8.78680i 0.319362i −0.987169 0.159681i \(-0.948954\pi\)
0.987169 0.159681i \(-0.0510465\pi\)
\(758\) 0 0
\(759\) −5.76230 + 9.98059i −0.209158 + 0.362272i
\(760\) 0 0
\(761\) −21.1066 + 12.1859i −0.765114 + 0.441739i −0.831129 0.556080i \(-0.812305\pi\)
0.0660150 + 0.997819i \(0.478971\pi\)
\(762\) 0 0
\(763\) 2.26101 7.60840i 0.0818541 0.275443i
\(764\) 0 0
\(765\) −2.41344 + 1.39340i −0.0872580 + 0.0503784i
\(766\) 0 0
\(767\) 0.764683 + 0.441490i 0.0276111 + 0.0159413i
\(768\) 0 0
\(769\) 5.31925i 0.191817i −0.995390 0.0959084i \(-0.969424\pi\)
0.995390 0.0959084i \(-0.0305756\pi\)
\(770\) 0 0
\(771\) 10.9171 0.393171
\(772\) 0 0
\(773\) −2.59808 + 4.50000i −0.0934463 + 0.161854i −0.908959 0.416885i \(-0.863122\pi\)
0.815513 + 0.578739i \(0.196455\pi\)
\(774\) 0 0
\(775\) −7.53375 13.0488i −0.270620 0.468728i
\(776\) 0 0
\(777\) 3.87868 + 16.2189i 0.139147 + 0.581848i
\(778\) 0 0
\(779\) −23.1587 40.1121i −0.829748 1.43717i
\(780\) 0 0
\(781\) −45.8739 26.4853i −1.64150 0.947718i
\(782\) 0 0
\(783\) 20.9226 0.747711
\(784\) 0 0
\(785\) 25.2426 0.900948
\(786\) 0 0
\(787\) 33.1393 + 19.1330i 1.18129 + 0.682018i 0.956312 0.292346i \(-0.0944360\pi\)
0.224977 + 0.974364i \(0.427769\pi\)
\(788\) 0 0
\(789\) −11.0843 19.1985i −0.394610 0.683484i
\(790\) 0 0
\(791\) 2.61079 + 10.9171i 0.0928291 + 0.388169i
\(792\) 0 0
\(793\) −6.36396 11.0227i −0.225991 0.391428i
\(794\) 0 0
\(795\) 3.32686 5.76230i 0.117992 0.204368i
\(796\) 0 0
\(797\) −16.3059 −0.577584 −0.288792 0.957392i \(-0.593254\pi\)
−0.288792 + 0.957392i \(0.593254\pi\)
\(798\) 0 0
\(799\) 1.87308i 0.0662649i
\(800\) 0 0
\(801\) 15.6655 + 9.04447i 0.553512 + 0.319571i
\(802\) 0 0
\(803\) 11.8537 6.84372i 0.418307 0.241510i
\(804\) 0 0
\(805\) 4.75039 15.9853i 0.167429 0.563407i
\(806\) 0 0
\(807\) −16.8389 + 9.72197i −0.592759 + 0.342230i
\(808\) 0 0
\(809\) 20.2279 35.0358i 0.711176 1.23179i −0.253240 0.967403i \(-0.581496\pi\)
0.964416 0.264389i \(-0.0851704\pi\)
\(810\) 0 0
\(811\) 19.9905i 0.701960i −0.936383 0.350980i \(-0.885849\pi\)
0.936383 0.350980i \(-0.114151\pi\)
\(812\) 0 0
\(813\) 12.1716i 0.426876i
\(814\) 0 0
\(815\) 13.9114 24.0953i 0.487296 0.844021i
\(816\) 0 0
\(817\) 56.1838 32.4377i 1.96562 1.13485i
\(818\) 0 0
\(819\) −4.37053 + 4.14016i −0.152719 + 0.144669i
\(820\) 0 0
\(821\) 16.3314 9.42893i 0.569969 0.329072i −0.187168 0.982328i \(-0.559931\pi\)
0.757137 + 0.653256i \(0.226597\pi\)
\(822\) 0 0
\(823\) 29.8929 + 17.2587i 1.04200 + 0.601600i 0.920399 0.390979i \(-0.127864\pi\)
0.121602 + 0.992579i \(0.461197\pi\)
\(824\) 0 0
\(825\) 6.33386i 0.220517i
\(826\) 0 0
\(827\) 36.9077 1.28341 0.641703 0.766953i \(-0.278228\pi\)
0.641703 + 0.766953i \(0.278228\pi\)
\(828\) 0 0
\(829\) −12.0989 + 20.9558i −0.420211 + 0.727827i −0.995960 0.0898000i \(-0.971377\pi\)
0.575749 + 0.817627i \(0.304711\pi\)
\(830\) 0 0
\(831\) −0.988654 1.71240i −0.0342960 0.0594024i
\(832\) 0 0
\(833\) 5.01472 + 0.271680i 0.173750 + 0.00941314i
\(834\) 0 0
\(835\) 12.2416 + 21.2031i 0.423638 + 0.733763i
\(836\) 0 0
\(837\) 29.7675 + 17.1863i 1.02892 + 0.594045i
\(838\) 0 0
\(839\) −32.5965 −1.12536 −0.562678 0.826676i \(-0.690229\pi\)
−0.562678 + 0.826676i \(0.690229\pi\)
\(840\) 0 0
\(841\) 7.97056 0.274847
\(842\) 0 0
\(843\) 5.84649 + 3.37548i 0.201364 + 0.116258i
\(844\) 0 0
\(845\) −10.3668 17.9558i −0.356629 0.617700i
\(846\) 0 0
\(847\) 4.30759 4.08053i 0.148010 0.140209i
\(848\) 0 0
\(849\) 10.5000 + 18.1865i 0.360359 + 0.624160i
\(850\) 0 0
\(851\) 13.1781 22.8252i 0.451741 0.782438i
\(852\) 0 0
\(853\) −1.01461 −0.0347396 −0.0173698 0.999849i \(-0.505529\pi\)
−0.0173698 + 0.999849i \(0.505529\pi\)
\(854\) 0 0
\(855\) 24.4832i 0.837308i
\(856\) 0 0
\(857\) 12.2574 + 7.07679i 0.418703 + 0.241739i 0.694522 0.719471i \(-0.255616\pi\)
−0.275819 + 0.961210i \(0.588949\pi\)
\(858\) 0 0
\(859\) 36.3369 20.9791i 1.23980 0.715798i 0.270745 0.962651i \(-0.412730\pi\)
0.969053 + 0.246853i \(0.0793965\pi\)
\(860\) 0 0
\(861\) 16.2189 + 4.81981i 0.552737 + 0.164259i
\(862\) 0 0
\(863\) −42.4989 + 24.5368i −1.44668 + 0.835241i −0.998282 0.0585927i \(-0.981339\pi\)
−0.448398 + 0.893834i \(0.648005\pi\)
\(864\) 0 0
\(865\) −16.3492 + 28.3177i −0.555891 + 0.962831i
\(866\) 0 0
\(867\) 14.3465i 0.487234i
\(868\) 0 0
\(869\) 24.2132i 0.821377i
\(870\) 0 0
\(871\) 2.92753 5.07064i 0.0991957 0.171812i
\(872\) 0 0
\(873\) −1.97056 + 1.13770i −0.0666934 + 0.0385055i
\(874\) 0 0
\(875\) 7.46096 + 31.1983i 0.252226 + 1.05470i
\(876\) 0 0
\(877\) 31.9920 18.4706i 1.08029 0.623707i 0.149316 0.988789i \(-0.452293\pi\)
0.930975 + 0.365083i \(0.118959\pi\)
\(878\) 0 0
\(879\) 4.14016 + 2.39032i 0.139644 + 0.0806235i
\(880\) 0 0
\(881\) 48.7436i 1.64221i −0.570774 0.821107i \(-0.693357\pi\)
0.570774 0.821107i \(-0.306643\pi\)
\(882\) 0 0
\(883\) 1.24872 0.0420229 0.0210114 0.999779i \(-0.493311\pi\)
0.0210114 + 0.999779i \(0.493311\pi\)
\(884\) 0 0
\(885\) −0.655892 + 1.13604i −0.0220476 + 0.0381875i
\(886\) 0 0
\(887\) 23.9066 + 41.4075i 0.802706 + 1.39033i 0.917829 + 0.396977i \(0.129941\pi\)
−0.115122 + 0.993351i \(0.536726\pi\)
\(888\) 0 0
\(889\) 15.5147 3.71029i 0.520347 0.124439i
\(890\) 0 0
\(891\) −5.01708 8.68983i −0.168078 0.291120i
\(892\) 0 0
\(893\) 14.2512 + 8.22792i 0.476898 + 0.275337i
\(894\) 0 0
\(895\) 36.7366 1.22797
\(896\) 0 0
\(897\) −3.21320 −0.107286
\(898\) 0 0
\(899\) −29.9196 17.2741i −0.997874 0.576123i
\(900\) 0 0
\(901\) −1.58346 2.74264i −0.0527528 0.0913706i
\(902\) 0 0
\(903\) −6.75095 + 22.7172i −0.224658 + 0.755983i
\(904\) 0 0
\(905\) 16.2426 + 28.1331i 0.539924 + 0.935175i
\(906\) 0 0
\(907\) −2.26101 + 3.91619i −0.0750757 + 0.130035i −0.901119 0.433571i \(-0.857253\pi\)
0.826044 + 0.563606i \(0.190587\pi\)
\(908\) 0 0
\(909\) −25.8577 −0.857645
\(910\) 0 0
\(911\) 39.9224i 1.32269i −0.750083 0.661343i \(-0.769987\pi\)
0.750083 0.661343i \(-0.230013\pi\)
\(912\) 0 0
\(913\) −10.9706 6.33386i −0.363073 0.209620i
\(914\) 0 0
\(915\) 16.3757 9.45451i 0.541364 0.312557i
\(916\) 0 0
\(917\) 24.5204 + 25.8848i 0.809735 + 0.854791i
\(918\) 0 0
\(919\) −36.9606 + 21.3392i −1.21922 + 0.703916i −0.964751 0.263166i \(-0.915233\pi\)
−0.254467 + 0.967081i \(0.581900\pi\)
\(920\) 0 0
\(921\) 5.48528 9.50079i 0.180746 0.313062i
\(922\) 0 0
\(923\) 14.7689i 0.486123i
\(924\) 0 0
\(925\) 14.4853i 0.476273i
\(926\) 0 0
\(927\) 15.6829 27.1635i 0.515093 0.892167i
\(928\) 0 0
\(929\) −38.9558 + 22.4912i −1.27810 + 0.737911i −0.976499 0.215523i \(-0.930854\pi\)
−0.301601 + 0.953434i \(0.597521\pi\)
\(930\) 0 0
\(931\) 24.0953 36.9606i 0.789691 1.21133i
\(932\) 0 0
\(933\) −22.1219 + 12.7721i −0.724238 + 0.418139i
\(934\) 0 0
\(935\) 3.91619 + 2.26101i 0.128073 + 0.0739430i
\(936\) 0 0
\(937\) 24.4949i 0.800213i −0.916469 0.400107i \(-0.868973\pi\)
0.916469 0.400107i \(-0.131027\pi\)
\(938\) 0 0
\(939\) 9.30267 0.303581
\(940\) 0 0
\(941\) −13.7949 + 23.8934i −0.449700 + 0.778903i −0.998366 0.0571387i \(-0.981802\pi\)
0.548667 + 0.836041i \(0.315136\pi\)
\(942\) 0 0
\(943\) −13.3707 23.1587i −0.435410 0.754152i
\(944\) 0 0
\(945\) −14.3787 15.1788i −0.467738 0.493765i
\(946\) 0 0
\(947\) 24.7196 + 42.8157i 0.803280 + 1.39132i 0.917446 + 0.397861i \(0.130247\pi\)
−0.114166 + 0.993462i \(0.536419\pi\)
\(948\) 0 0
\(949\) 3.30496 + 1.90812i 0.107283 + 0.0619401i
\(950\) 0 0
\(951\) 26.8907 0.871991
\(952\) 0 0
\(953\) −26.1838 −0.848175 −0.424088 0.905621i \(-0.639405\pi\)
−0.424088 + 0.905621i \(0.639405\pi\)
\(954\) 0 0
\(955\) −6.78304 3.91619i −0.219494 0.126725i
\(956\) 0 0
\(957\) −7.26143 12.5772i −0.234729 0.406562i
\(958\) 0 0
\(959\) 33.5851 + 9.98059i 1.08452 + 0.322290i
\(960\) 0 0
\(961\) −12.8787 22.3065i −0.415441 0.719565i
\(962\) 0 0
\(963\) 7.46096 12.9228i 0.240426 0.416430i
\(964\) 0 0
\(965\) −29.8651 −0.961393
\(966\) 0 0
\(967\) 24.8489i 0.799088i 0.916714 + 0.399544i \(0.130832\pi\)
−0.916714 + 0.399544i \(0.869168\pi\)
\(968\) 0 0
\(969\) −3.40812 1.96768i −0.109484 0.0632109i
\(970\) 0 0
\(971\) 8.29038 4.78645i 0.266051 0.153605i −0.361041 0.932550i \(-0.617578\pi\)
0.627092 + 0.778945i \(0.284245\pi\)
\(972\) 0 0
\(973\) 45.8739 10.9706i 1.47065 0.351700i
\(974\) 0 0
\(975\) 1.52937 0.882980i 0.0489789 0.0282780i
\(976\) 0 0
\(977\) 4.34924 7.53311i 0.139145 0.241006i −0.788028 0.615639i \(-0.788898\pi\)
0.927173 + 0.374633i \(0.122231\pi\)
\(978\) 0 0
\(979\) 29.3522i 0.938100i
\(980\) 0 0
\(981\) 6.72792i 0.214806i
\(982\) 0 0
\(983\) −7.60840 + 13.1781i −0.242670 + 0.420318i −0.961474 0.274896i \(-0.911357\pi\)
0.718804 + 0.695213i \(0.244690\pi\)
\(984\) 0 0
\(985\) 1.60660 0.927572i 0.0511906 0.0295549i
\(986\) 0 0
\(987\) −5.84649 + 1.39817i −0.186096 + 0.0445042i
\(988\) 0 0
\(989\) 32.4377 18.7279i 1.03146 0.595513i
\(990\) 0 0
\(991\) −25.7527 14.8684i −0.818063 0.472309i 0.0316850 0.999498i \(-0.489913\pi\)
−0.849748 + 0.527189i \(0.823246\pi\)
\(992\) 0 0
\(993\) 12.8928i 0.409140i
\(994\) 0 0
\(995\) −8.78543 −0.278517
\(996\) 0 0
\(997\) −25.1508 + 43.5624i −0.796534 + 1.37964i 0.125327 + 0.992115i \(0.460002\pi\)
−0.921861 + 0.387521i \(0.873331\pi\)
\(998\) 0 0
\(999\) −16.5222 28.6173i −0.522739 0.905411i
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 896.2.q.c.703.4 yes 16
4.3 odd 2 inner 896.2.q.c.703.6 yes 16
7.5 odd 6 inner 896.2.q.c.831.3 yes 16
8.3 odd 2 inner 896.2.q.c.703.3 16
8.5 even 2 inner 896.2.q.c.703.5 yes 16
28.19 even 6 inner 896.2.q.c.831.5 yes 16
56.5 odd 6 inner 896.2.q.c.831.6 yes 16
56.19 even 6 inner 896.2.q.c.831.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.q.c.703.3 16 8.3 odd 2 inner
896.2.q.c.703.4 yes 16 1.1 even 1 trivial
896.2.q.c.703.5 yes 16 8.5 even 2 inner
896.2.q.c.703.6 yes 16 4.3 odd 2 inner
896.2.q.c.831.3 yes 16 7.5 odd 6 inner
896.2.q.c.831.4 yes 16 56.19 even 6 inner
896.2.q.c.831.5 yes 16 28.19 even 6 inner
896.2.q.c.831.6 yes 16 56.5 odd 6 inner