Properties

Label 896.2.q.c.831.6
Level $896$
Weight $2$
Character 896.831
Analytic conductor $7.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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 831.6
Root \(-0.0777751 - 0.590761i\) of defining polynomial
Character \(\chi\) \(=\) 896.831
Dual form 896.2.q.c.703.6

$q$-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 \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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 \(-0.585834\pi\)
0.701531 + 0.712639i \(0.252500\pi\)
\(4\) 0 0
\(5\) 0.866025 1.50000i 0.387298 0.670820i −0.604787 0.796387i \(-0.706742\pi\)
0.992085 + 0.125567i \(0.0400750\pi\)
\(6\) 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 0.998370 + 0.0570668i \(0.0181748\pi\)
−0.449764 + 0.893148i \(0.648492\pi\)
\(12\) 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 −0.573450 0.819240i \(-0.694395\pi\)
0.422758 + 0.906243i \(0.361062\pi\)
\(18\) 0 0
\(19\) 5.45857 + 3.15150i 1.25228 + 0.723005i 0.971562 0.236785i \(-0.0760938\pi\)
0.280719 + 0.959790i \(0.409427\pi\)
\(20\) 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 \(-0.209465\pi\)
−0.134050 + 0.990975i \(0.542798\pi\)
\(24\) 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 \(0.140000\pi\)
−0.904827 + 0.425780i \(0.860000\pi\)
\(30\) 0 0
\(31\) −3.76687 6.52442i −0.676551 1.17182i −0.976013 0.217712i \(-0.930141\pi\)
0.299463 0.954108i \(-0.403193\pi\)
\(32\) 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 −0.113844 0.993499i \(-0.536317\pi\)
−0.917317 + 0.398157i \(0.869650\pi\)
\(38\) 0 0
\(39\) −0.764683 + 0.441490i −0.122447 + 0.0706950i
\(40\) 0 0
\(41\) 7.34847i 1.14764i −0.818982 0.573819i \(-0.805461\pi\)
0.818982 0.573819i \(-0.194539\pi\)
\(42\) 0 0
\(43\) 10.2928 1.56963 0.784816 0.619728i \(-0.212757\pi\)
0.784816 + 0.619728i \(0.212757\pi\)
\(44\) 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 \(-0.772351\pi\)
0.945387 + 0.325951i \(0.105684\pi\)
\(48\) 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 −0.213916 0.976852i \(-0.568622\pi\)
0.739021 + 0.673683i \(0.235289\pi\)
\(54\) 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 \(-0.648625\pi\)
0.548257 + 0.836310i \(0.315292\pi\)
\(60\) 0 0
\(61\) 6.27231 10.8640i 0.803087 1.39099i −0.114488 0.993425i \(-0.536523\pi\)
0.917575 0.397563i \(-0.130144\pi\)
\(62\) 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 \(-0.0519968\pi\)
−0.634183 + 0.773183i \(0.718663\pi\)
\(68\) 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 −0.678360 0.734730i \(-0.737309\pi\)
0.297115 + 0.954842i \(0.403976\pi\)
\(74\) 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 \(-0.455451\pi\)
−0.787808 + 0.615921i \(0.788784\pi\)
\(80\) 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 0.0817903 0.996650i \(-0.473936\pi\)
−0.822229 + 0.569157i \(0.807270\pi\)
\(90\) 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 \(0.0164031\pi\)
−0.998673 + 0.0515091i \(0.983597\pi\)
\(98\) 0 0
\(99\) −8.16107 −0.820218
\(100\) 0 0
\(101\) 5.76500 + 9.98528i 0.573639 + 0.993573i 0.996188 + 0.0872323i \(0.0278023\pi\)
−0.422549 + 0.906340i \(0.638864\pi\)
\(102\) 0 0
\(103\) −6.99304 + 12.1123i −0.689044 + 1.19346i 0.283103 + 0.959089i \(0.408636\pi\)
−0.972147 + 0.234370i \(0.924697\pi\)
\(104\) 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 \(-0.937561\pi\)
0.659202 + 0.751966i \(0.270894\pi\)
\(108\) 0 0
\(109\) −2.59808 + 1.50000i −0.248851 + 0.143674i −0.619238 0.785203i \(-0.712558\pi\)
0.370387 + 0.928877i \(0.379225\pi\)
\(110\) 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 \(-0.533699\pi\)
−0.914012 + 0.405687i \(0.867032\pi\)
\(132\) 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 0.996984 + 0.0776021i \(0.0247264\pi\)
−0.431287 + 0.902215i \(0.641940\pi\)
\(138\) 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 −0.980165 0.198181i \(-0.936496\pi\)
−0.661713 0.749757i \(-0.730170\pi\)
\(150\) 0 0
\(151\) 13.9114 8.03176i 1.13209 0.653615i 0.187634 0.982239i \(-0.439918\pi\)
0.944461 + 0.328624i \(0.106585\pi\)
\(152\) 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 0.995295 + 0.0968937i \(0.0308907\pi\)
−0.413735 + 0.910397i \(0.635776\pi\)
\(158\) 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 −0.358638 0.933477i \(-0.616759\pi\)
0.987733 0.156149i \(-0.0499081\pi\)
\(164\) 0 0
\(165\) 4.75039 2.74264i 0.369818 0.213514i
\(166\) 0 0
\(167\) −14.1354 −1.09383 −0.546914 0.837189i \(-0.684198\pi\)
−0.546914 + 0.837189i \(0.684198\pi\)
\(168\) 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 −0.244276 0.969706i \(-0.578550\pi\)
0.961928 0.273304i \(-0.0881165\pi\)
\(174\) 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 −0.924320 0.381618i \(-0.875367\pi\)
0.131669 0.991294i \(-0.457967\pi\)
\(180\) 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 \(-0.281022\pi\)
−0.351581 + 0.936158i \(0.614356\pi\)
\(192\) 0 0
\(193\) −8.62132 14.9326i −0.620576 1.07487i −0.989379 0.145362i \(-0.953565\pi\)
0.368802 0.929508i \(-0.379768\pi\)
\(194\) 0 0
\(195\) 1.52937i 0.109520i
\(196\) 0 0
\(197\) 1.07107i 0.0763104i 0.999272 + 0.0381552i \(0.0121481\pi\)
−0.999272 + 0.0381552i \(0.987852\pi\)
\(198\) 0 0
\(199\) 2.53613 + 4.39271i 0.179782 + 0.311391i 0.941806 0.336158i \(-0.109127\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(200\) 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 \(-0.486314\pi\)
0.0429828 + 0.999076i \(0.486314\pi\)
\(212\) 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 \(-0.389687\pi\)
0.339664 + 0.940547i \(0.389687\pi\)
\(224\) 0 0
\(225\) −4.48528 −0.299019
\(226\) 0 0
\(227\) −14.8684 + 8.58425i −0.986847 + 0.569757i −0.904330 0.426833i \(-0.859629\pi\)
−0.0825170 + 0.996590i \(0.526296\pi\)
\(228\) 0 0
\(229\) 9.94655 17.2279i 0.657286 1.13845i −0.324029 0.946047i \(-0.605038\pi\)
0.981315 0.192406i \(-0.0616291\pi\)
\(230\) 0 0
\(231\) −5.76230 + 6.08293i −0.379131 + 0.400227i
\(232\) 0 0
\(233\) 12.1066 20.9692i 0.793130 1.37374i −0.130890 0.991397i \(-0.541783\pi\)
0.924020 0.382344i \(-0.124883\pi\)
\(234\) 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 0.0339839 0.999422i \(-0.489181\pi\)
0.882517 + 0.470280i \(0.155847\pi\)
\(242\) 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 0.121303 0.992615i \(-0.461293\pi\)
−0.798979 + 0.601359i \(0.794626\pi\)
\(258\) 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 \(-0.954197\pi\)
−0.370646 + 0.928774i \(0.620864\pi\)
\(264\) 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 0.974638 + 0.223789i \(0.0718426\pi\)
−0.293512 + 0.955955i \(0.594824\pi\)
\(270\) 0 0
\(271\) −6.99304 + 12.1123i −0.424797 + 0.735769i −0.996401 0.0847597i \(-0.972988\pi\)
0.571605 + 0.820529i \(0.306321\pi\)
\(272\) 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 −0.439721 0.898135i \(-0.644923\pi\)
0.557947 + 0.829877i \(0.311589\pi\)
\(278\) 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 \(-0.412086\pi\)
0.969550 + 0.244894i \(0.0787531\pi\)
\(284\) 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 −0.896286 0.443477i \(-0.853745\pi\)
0.0640807 0.997945i \(-0.479588\pi\)
\(312\) 0 0
\(313\) −9.25736 5.34474i −0.523257 0.302103i 0.215009 0.976612i \(-0.431022\pi\)
−0.738266 + 0.674510i \(0.764355\pi\)
\(314\) 0 0
\(315\) −9.85128 + 2.92753i −0.555057 + 0.164948i
\(316\) 0 0
\(317\) −26.7597 15.4497i −1.50298 0.867744i −0.999994 0.00344803i \(-0.998902\pi\)
−0.502983 0.864296i \(-0.667764\pi\)
\(318\) 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 \(-0.699857\pi\)
0.994569 + 0.104081i \(0.0331902\pi\)
\(332\) 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 0.976783 + 0.214233i \(0.0687251\pi\)
−0.302860 + 0.953035i \(0.597942\pi\)
\(348\) 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 0.700132 0.714013i \(-0.253124\pi\)
0.968420 0.249325i \(-0.0802089\pi\)
\(354\) 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 −0.998994 0.0448462i \(-0.985720\pi\)
−0.538335 0.842731i \(-0.680946\pi\)
\(360\) 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 0.997683 + 0.0680400i \(0.0216745\pi\)
−0.439917 + 0.898038i \(0.644992\pi\)
\(368\) 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 0.172790 0.984959i \(-0.444722\pi\)
−0.766604 + 0.642120i \(0.778055\pi\)
\(374\) 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 \(-0.241829\pi\)
0.725023 + 0.688725i \(0.241829\pi\)
\(380\) 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 \(-0.794415\pi\)
0.920541 + 0.390646i \(0.127748\pi\)
\(384\) 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 −0.366551 0.930398i \(-0.619461\pi\)
0.622473 + 0.782641i \(0.286128\pi\)
\(390\) 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 −0.500797 0.865565i \(-0.666960\pi\)
1.00000 0.000920276i \(0.000292933\pi\)
\(398\) 0 0
\(399\) −3.37548 + 14.1147i −0.168985 + 0.706618i
\(400\) 0 0
\(401\) 6.62132 11.4685i 0.330653 0.572708i −0.651987 0.758230i \(-0.726064\pi\)
0.982640 + 0.185522i \(0.0593977\pi\)
\(402\) 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 −0.895174 0.445717i \(-0.852949\pi\)
−0.0615846 + 0.998102i \(0.519615\pi\)
\(410\) 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 \(0.0605361\pi\)
−0.981970 + 0.189035i \(0.939464\pi\)
\(422\) 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 \(-0.631931\pi\)
0.591344 + 0.806419i \(0.298598\pi\)
\(432\) 0 0
\(433\) 20.6105i 0.990479i 0.868757 + 0.495239i \(0.164920\pi\)
−0.868757 + 0.495239i \(0.835080\pi\)
\(434\) 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 −0.227136 0.973863i \(-0.572936\pi\)
0.956958 0.290226i \(-0.0937304\pi\)
\(440\) 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 \(-0.893446\pi\)
0.756764 + 0.653688i \(0.226779\pi\)
\(444\) 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 −0.835118 0.550071i \(-0.814601\pi\)
0.893935 + 0.448198i \(0.147934\pi\)
\(458\) 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 \(-0.146652\pi\)
0.0628367 + 0.998024i \(0.479985\pi\)
\(468\) 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 0.726003 + 0.687691i \(0.241376\pi\)
0.232557 + 0.972583i \(0.425291\pi\)
\(480\) 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 \(-0.778431\pi\)
0.171627 + 0.985162i \(0.445098\pi\)
\(488\) 0 0
\(489\) 13.9795i 0.632175i
\(490\) 0 0
\(491\) 11.1758 0.504355 0.252177 0.967681i \(-0.418853\pi\)
0.252177 + 0.967681i \(0.418853\pi\)
\(492\) 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 −0.404306 0.914624i \(-0.632487\pi\)
0.994240 0.107173i \(-0.0341799\pi\)
\(500\) 0 0
\(501\) −10.6534 + 6.15076i −0.475960 + 0.274796i
\(502\) 0 0
\(503\) −30.4336 −1.35697 −0.678484 0.734615i \(-0.737363\pi\)
−0.678484 + 0.734615i \(0.737363\pi\)
\(504\) 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 −0.990664 0.136323i \(-0.956471\pi\)
0.613391 + 0.789779i \(0.289805\pi\)
\(510\) 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 −0.743864 0.668331i \(-0.767009\pi\)
0.206859 + 0.978371i \(0.433676\pi\)
\(522\) 0 0
\(523\) 11.8537 + 6.84372i 0.518325 + 0.299255i 0.736249 0.676711i \(-0.236595\pi\)
−0.217924 + 0.975966i \(0.569929\pi\)
\(524\) 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 0.0552031 0.998475i \(-0.482419\pi\)
−0.837103 + 0.547045i \(0.815753\pi\)
\(542\) 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 \(-0.392929\pi\)
0.330065 + 0.943958i \(0.392929\pi\)
\(548\) 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 0.422333 0.906441i \(-0.361211\pi\)
0.996167 + 0.0874688i \(0.0278778\pi\)
\(558\) 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 \(-0.695895\pi\)
0.418485 + 0.908224i \(0.362561\pi\)
\(564\) 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 −0.917771 0.397111i \(-0.870013\pi\)
0.802793 + 0.596257i \(0.203346\pi\)
\(570\) 0 0
\(571\) 4.83420 + 8.37309i 0.202305 + 0.350403i 0.949271 0.314460i \(-0.101823\pi\)
−0.746966 + 0.664863i \(0.768490\pi\)
\(572\) 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 0.135710 0.990749i \(-0.456668\pi\)
0.925869 + 0.377846i \(0.123335\pi\)
\(578\) 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 0.706194 0.708018i \(-0.250411\pi\)
−0.260065 + 0.965591i \(0.583744\pi\)
\(594\) 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 \(-0.583238\pi\)
0.707319 + 0.706895i \(0.249905\pi\)
\(600\) 0 0
\(601\) 42.2357i 1.72283i −0.507903 0.861414i \(-0.669579\pi\)
0.507903 0.861414i \(-0.330421\pi\)
\(602\) 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 \(-0.766343\pi\)
0.951371 + 0.308049i \(0.0996760\pi\)
\(608\) 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 −0.850926 0.525286i \(-0.823959\pi\)
0.0294476 + 0.999566i \(0.490625\pi\)
\(614\) 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 \(-0.553095\pi\)
0.770990 + 0.636847i \(0.219762\pi\)
\(620\) 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 0.749833 0.661627i \(-0.230134\pi\)
−0.947902 + 0.318561i \(0.896800\pi\)
\(642\) 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 \(-0.127127\pi\)
−0.797403 + 0.603447i \(0.793794\pi\)
\(648\) 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 0.583932 0.811802i \(-0.301513\pi\)
−0.411075 + 0.911601i \(0.634847\pi\)
\(654\) 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 \(-0.572172\pi\)
−0.224796 + 0.974406i \(0.572172\pi\)
\(660\) 0 0
\(661\) −18.8169 32.5919i −0.731894 1.26768i −0.956073 0.293129i \(-0.905303\pi\)
0.224179 0.974548i \(-0.428030\pi\)
\(662\) 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 −0.950265 0.311443i \(-0.899188\pi\)
0.744850 + 0.667232i \(0.232521\pi\)
\(678\) 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 −0.874489 0.485045i \(-0.838803\pi\)
0.0171833 0.999852i \(-0.494530\pi\)
\(684\) 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 \(-0.349147\pi\)
−0.542390 + 0.840127i \(0.682480\pi\)
\(692\) 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 \(0.0860557\pi\)
−0.963677 + 0.267071i \(0.913944\pi\)
\(702\) 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 0.903073 0.429487i \(-0.141306\pi\)
0.0795894 + 0.996828i \(0.474639\pi\)
\(710\) 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 \(-0.795835\pi\)
0.918789 + 0.394750i \(0.129169\pi\)
\(720\) 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 \(-0.231521\pi\)
0.746943 + 0.664889i \(0.231521\pi\)
\(728\) 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 −0.864887 0.501967i \(-0.832610\pi\)
0.867160 + 0.498030i \(0.165943\pi\)
\(734\) 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 −0.992458 0.122584i \(-0.960882\pi\)
0.390068 0.920786i \(-0.372451\pi\)
\(740\) 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 \(-0.417340\pi\)
−0.708602 + 0.705609i \(0.750674\pi\)
\(752\) 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 \(0.0510465\pi\)
−0.987169 + 0.159681i \(0.948954\pi\)
\(758\) 0 0
\(759\) 5.76230 + 9.98059i 0.209158 + 0.362272i
\(760\) 0 0
\(761\) −21.1066 12.1859i −0.765114 0.441739i 0.0660150 0.997819i \(-0.478971\pi\)
−0.831129 + 0.556080i \(0.812305\pi\)
\(762\) 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 \(0.0305756\pi\)
−0.995390 + 0.0959084i \(0.969424\pi\)
\(770\) 0 0
\(771\) −10.9171 −0.393171
\(772\) 0 0
\(773\) −2.59808 4.50000i −0.0934463 0.161854i 0.815513 0.578739i \(-0.196455\pi\)
−0.908959 + 0.416885i \(0.863122\pi\)
\(774\) 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 \(-0.905564\pi\)
−0.224977 + 0.974364i \(0.572231\pi\)
\(788\) 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 0.964416 + 0.264389i \(0.0851704\pi\)
−0.253240 + 0.967403i \(0.581496\pi\)
\(810\) 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 0.757137 0.653256i \(-0.226597\pi\)
−0.187168 + 0.982328i \(0.559931\pi\)
\(822\) 0 0
\(823\) −29.8929 + 17.2587i −1.04200 + 0.601600i −0.920399 0.390979i \(-0.872136\pi\)
−0.121602 + 0.992579i \(0.538803\pi\)
\(824\) 0 0
\(825\) 6.33386i 0.220517i
\(826\) 0 0
\(827\) −36.9077 −1.28341 −0.641703 0.766953i \(-0.721772\pi\)
−0.641703 + 0.766953i \(0.721772\pi\)
\(828\) 0 0
\(829\) −12.0989 20.9558i −0.420211 0.727827i 0.575749 0.817627i \(-0.304711\pi\)
−0.995960 + 0.0898000i \(0.971377\pi\)
\(830\) 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 \(-0.309771\pi\)
0.562678 + 0.826676i \(0.309771\pi\)
\(840\) 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 −0.275819 0.961210i \(-0.588949\pi\)
0.694522 + 0.719471i \(0.255616\pi\)
\(858\) 0 0
\(859\) −36.3369 20.9791i −1.23980 0.715798i −0.270745 0.962651i \(-0.587270\pi\)
−0.969053 + 0.246853i \(0.920603\pi\)
\(860\) 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 \(-0.0186613\pi\)
0.448398 + 0.893834i \(0.351995\pi\)
\(864\) 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 0.930975 0.365083i \(-0.118959\pi\)
0.149316 + 0.988789i \(0.452293\pi\)
\(878\) 0 0
\(879\) −4.14016 + 2.39032i −0.139644 + 0.0806235i
\(880\) 0 0
\(881\) 48.7436i 1.64221i 0.570774 + 0.821107i \(0.306643\pi\)
−0.570774 + 0.821107i \(0.693357\pi\)
\(882\) 0 0
\(883\) −1.24872 −0.0420229 −0.0210114 0.999779i \(-0.506689\pi\)
−0.0210114 + 0.999779i \(0.506689\pi\)
\(884\) 0 0
\(885\) −0.655892 1.13604i −0.0220476 0.0381875i
\(886\) 0 0
\(887\) −23.9066 + 41.4075i −0.802706 + 1.39033i 0.115122 + 0.993351i \(0.463274\pi\)
−0.917829 + 0.396977i \(0.870059\pi\)
\(888\) 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 \(-0.142747\pi\)
−0.826044 + 0.563606i \(0.809413\pi\)
\(908\) 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 \(-0.0847666\pi\)
0.254467 + 0.967081i \(0.418100\pi\)
\(920\) 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 −0.301601 0.953434i \(-0.597521\pi\)
−0.976499 + 0.215523i \(0.930854\pi\)
\(930\) 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 \(0.131027\pi\)
−0.916469 + 0.400107i \(0.868973\pi\)
\(938\) 0 0
\(939\) −9.30267 −0.303581
\(940\) 0 0
\(941\) −13.7949 23.8934i −0.449700 0.778903i 0.548667 0.836041i \(-0.315136\pi\)
−0.998366 + 0.0571387i \(0.981802\pi\)
\(942\) 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 0.114166 + 0.993462i \(0.463581\pi\)
−0.917446 + 0.397861i \(0.869753\pi\)
\(948\) 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 \(-0.382422\pi\)
−0.627092 + 0.778945i \(0.715755\pi\)
\(972\) 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 0.927173 0.374633i \(-0.122231\pi\)
−0.788028 + 0.615639i \(0.788898\pi\)
\(978\) 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 \(-0.0886433\pi\)
−0.718804 + 0.695213i \(0.755310\pi\)
\(984\) 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 \(-0.510087\pi\)
0.849748 + 0.527189i \(0.176754\pi\)
\(992\) 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 −0.921861 0.387521i \(-0.873331\pi\)
0.125327 0.992115i \(-0.460002\pi\)
\(998\) 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.831.6 yes 16
4.3 odd 2 inner 896.2.q.c.831.4 yes 16
7.3 odd 6 inner 896.2.q.c.703.5 yes 16
8.3 odd 2 inner 896.2.q.c.831.5 yes 16
8.5 even 2 inner 896.2.q.c.831.3 yes 16
28.3 even 6 inner 896.2.q.c.703.3 16
56.3 even 6 inner 896.2.q.c.703.6 yes 16
56.45 odd 6 inner 896.2.q.c.703.4 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
896.2.q.c.703.3 16 28.3 even 6 inner
896.2.q.c.703.4 yes 16 56.45 odd 6 inner
896.2.q.c.703.5 yes 16 7.3 odd 6 inner
896.2.q.c.703.6 yes 16 56.3 even 6 inner
896.2.q.c.831.3 yes 16 8.5 even 2 inner
896.2.q.c.831.4 yes 16 4.3 odd 2 inner
896.2.q.c.831.5 yes 16 8.3 odd 2 inner
896.2.q.c.831.6 yes 16 1.1 even 1 trivial