Properties

Label 3150.2.bf.b.1151.2
Level $3150$
Weight $2$
Character 3150.1151
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1151.2
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 3150.1151
Dual form 3150.2.bf.b.1601.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.63896 - 0.189469i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(2.63896 - 0.189469i) q^{7} +1.00000i q^{8} +(-4.67303 - 2.69798i) q^{11} +2.51764i q^{13} +(-2.19067 + 1.48356i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-2.24969 + 3.89658i) q^{17} +(-2.48004 + 1.43185i) q^{19} +5.39595 q^{22} +(-0.232051 + 0.133975i) q^{23} +(-1.25882 - 2.18034i) q^{26} +(1.15539 - 2.38014i) q^{28} -8.89898i q^{29} +(4.18154 + 2.41421i) q^{31} +(0.866025 + 0.500000i) q^{32} -4.49938i q^{34} +(-3.25882 - 5.64444i) q^{37} +(1.43185 - 2.48004i) q^{38} +0.760279 q^{41} +5.86370 q^{43} +(-4.67303 + 2.69798i) q^{44} +(0.133975 - 0.232051i) q^{46} +(-3.99768 - 6.92418i) q^{47} +(6.92820 - 1.00000i) q^{49} +(2.18034 + 1.25882i) q^{52} +(7.27319 + 4.19918i) q^{53} +(0.189469 + 2.63896i) q^{56} +(4.44949 + 7.70674i) q^{58} +(6.33573 - 10.9738i) q^{59} +(-2.27035 + 1.31079i) q^{61} -4.82843 q^{62} -1.00000 q^{64} +(4.91119 - 8.50643i) q^{67} +(2.24969 + 3.89658i) q^{68} -4.76268i q^{71} +(-10.0951 - 5.82843i) q^{73} +(5.64444 + 3.25882i) q^{74} +2.86370i q^{76} +(-12.8431 - 6.23445i) q^{77} +(-4.29618 - 7.44120i) q^{79} +(-0.658421 + 0.380139i) q^{82} -9.45001 q^{83} +(-5.07812 + 2.93185i) q^{86} +(2.69798 - 4.67303i) q^{88} +(-3.98502 - 6.90226i) q^{89} +(0.477014 + 6.64394i) q^{91} +0.267949i q^{92} +(6.92418 + 3.99768i) q^{94} +6.16353i q^{97} +(-5.50000 + 4.33013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 24 q^{11} - 4 q^{16} + 12 q^{23} - 8 q^{26} - 24 q^{37} - 4 q^{38} - 32 q^{41} + 16 q^{43} - 24 q^{44} + 8 q^{46} - 8 q^{47} + 24 q^{53} + 16 q^{58} + 24 q^{59} - 16 q^{62} - 8 q^{64} + 24 q^{67} - 16 q^{77} - 24 q^{79} - 16 q^{83} + 16 q^{89} - 20 q^{91} - 12 q^{94} - 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 + 0.500000i −0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 0.866025i 0.250000 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.63896 0.189469i 0.997433 0.0716124i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) −4.67303 2.69798i −1.40897 0.813471i −0.413683 0.910421i \(-0.635758\pi\)
−0.995289 + 0.0969504i \(0.969091\pi\)
\(12\) 0 0
\(13\) 2.51764i 0.698267i 0.937073 + 0.349134i \(0.113524\pi\)
−0.937073 + 0.349134i \(0.886476\pi\)
\(14\) −2.19067 + 1.48356i −0.585481 + 0.396499i
\(15\) 0 0
\(16\) −0.500000 0.866025i −0.125000 0.216506i
\(17\) −2.24969 + 3.89658i −0.545630 + 0.945058i 0.452937 + 0.891542i \(0.350376\pi\)
−0.998567 + 0.0535160i \(0.982957\pi\)
\(18\) 0 0
\(19\) −2.48004 + 1.43185i −0.568960 + 0.328489i −0.756734 0.653723i \(-0.773206\pi\)
0.187774 + 0.982212i \(0.439873\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 5.39595 1.15042
\(23\) −0.232051 + 0.133975i −0.0483859 + 0.0279356i −0.523998 0.851720i \(-0.675560\pi\)
0.475612 + 0.879655i \(0.342227\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.25882 2.18034i −0.246875 0.427600i
\(27\) 0 0
\(28\) 1.15539 2.38014i 0.218349 0.449804i
\(29\) 8.89898i 1.65250i −0.563304 0.826250i \(-0.690470\pi\)
0.563304 0.826250i \(-0.309530\pi\)
\(30\) 0 0
\(31\) 4.18154 + 2.41421i 0.751027 + 0.433606i 0.826065 0.563575i \(-0.190574\pi\)
−0.0750380 + 0.997181i \(0.523908\pi\)
\(32\) 0.866025 + 0.500000i 0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 4.49938i 0.771637i
\(35\) 0 0
\(36\) 0 0
\(37\) −3.25882 5.64444i −0.535747 0.927940i −0.999127 0.0417807i \(-0.986697\pi\)
0.463380 0.886160i \(-0.346636\pi\)
\(38\) 1.43185 2.48004i 0.232277 0.402316i
\(39\) 0 0
\(40\) 0 0
\(41\) 0.760279 0.118736 0.0593678 0.998236i \(-0.481092\pi\)
0.0593678 + 0.998236i \(0.481092\pi\)
\(42\) 0 0
\(43\) 5.86370 0.894206 0.447103 0.894482i \(-0.352456\pi\)
0.447103 + 0.894482i \(0.352456\pi\)
\(44\) −4.67303 + 2.69798i −0.704486 + 0.406735i
\(45\) 0 0
\(46\) 0.133975 0.232051i 0.0197535 0.0342140i
\(47\) −3.99768 6.92418i −0.583121 1.01000i −0.995107 0.0988053i \(-0.968498\pi\)
0.411986 0.911190i \(-0.364835\pi\)
\(48\) 0 0
\(49\) 6.92820 1.00000i 0.989743 0.142857i
\(50\) 0 0
\(51\) 0 0
\(52\) 2.18034 + 1.25882i 0.302359 + 0.174567i
\(53\) 7.27319 + 4.19918i 0.999050 + 0.576802i 0.907967 0.419042i \(-0.137634\pi\)
0.0910826 + 0.995843i \(0.470967\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.189469 + 2.63896i 0.0253188 + 0.352646i
\(57\) 0 0
\(58\) 4.44949 + 7.70674i 0.584247 + 1.01194i
\(59\) 6.33573 10.9738i 0.824842 1.42867i −0.0771977 0.997016i \(-0.524597\pi\)
0.902040 0.431653i \(-0.142069\pi\)
\(60\) 0 0
\(61\) −2.27035 + 1.31079i −0.290689 + 0.167829i −0.638253 0.769827i \(-0.720342\pi\)
0.347564 + 0.937656i \(0.387009\pi\)
\(62\) −4.82843 −0.613211
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 4.91119 8.50643i 0.599997 1.03923i −0.392824 0.919614i \(-0.628502\pi\)
0.992821 0.119612i \(-0.0381649\pi\)
\(68\) 2.24969 + 3.89658i 0.272815 + 0.472529i
\(69\) 0 0
\(70\) 0 0
\(71\) 4.76268i 0.565226i −0.959234 0.282613i \(-0.908799\pi\)
0.959234 0.282613i \(-0.0912013\pi\)
\(72\) 0 0
\(73\) −10.0951 5.82843i −1.18155 0.682166i −0.225174 0.974319i \(-0.572295\pi\)
−0.956372 + 0.292153i \(0.905628\pi\)
\(74\) 5.64444 + 3.25882i 0.656153 + 0.378830i
\(75\) 0 0
\(76\) 2.86370i 0.328489i
\(77\) −12.8431 6.23445i −1.46361 0.710482i
\(78\) 0 0
\(79\) −4.29618 7.44120i −0.483358 0.837200i 0.516460 0.856312i \(-0.327250\pi\)
−0.999817 + 0.0191114i \(0.993916\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.658421 + 0.380139i −0.0727104 + 0.0419794i
\(83\) −9.45001 −1.03727 −0.518636 0.854995i \(-0.673560\pi\)
−0.518636 + 0.854995i \(0.673560\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.07812 + 2.93185i −0.547587 + 0.316150i
\(87\) 0 0
\(88\) 2.69798 4.67303i 0.287605 0.498147i
\(89\) −3.98502 6.90226i −0.422412 0.731638i 0.573763 0.819021i \(-0.305483\pi\)
−0.996175 + 0.0873828i \(0.972150\pi\)
\(90\) 0 0
\(91\) 0.477014 + 6.64394i 0.0500046 + 0.696474i
\(92\) 0.267949i 0.0279356i
\(93\) 0 0
\(94\) 6.92418 + 3.99768i 0.714175 + 0.412329i
\(95\) 0 0
\(96\) 0 0
\(97\) 6.16353i 0.625812i 0.949784 + 0.312906i \(0.101302\pi\)
−0.949784 + 0.312906i \(0.898698\pi\)
\(98\) −5.50000 + 4.33013i −0.555584 + 0.437409i
\(99\) 0 0
\(100\) 0 0
\(101\) −7.02458 + 12.1669i −0.698972 + 1.21065i 0.269852 + 0.962902i \(0.413025\pi\)
−0.968823 + 0.247753i \(0.920308\pi\)
\(102\) 0 0
\(103\) −12.2776 + 7.08845i −1.20974 + 0.698446i −0.962703 0.270560i \(-0.912791\pi\)
−0.247040 + 0.969005i \(0.579458\pi\)
\(104\) −2.51764 −0.246875
\(105\) 0 0
\(106\) −8.39836 −0.815721
\(107\) 1.42178 0.820863i 0.137448 0.0793559i −0.429699 0.902972i \(-0.641380\pi\)
0.567147 + 0.823616i \(0.308047\pi\)
\(108\) 0 0
\(109\) 9.94887 17.2319i 0.952929 1.65052i 0.213890 0.976858i \(-0.431387\pi\)
0.739039 0.673663i \(-0.235280\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.48356 2.19067i −0.140184 0.206999i
\(113\) 5.95867i 0.560545i −0.959921 0.280272i \(-0.909575\pi\)
0.959921 0.280272i \(-0.0904248\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −7.70674 4.44949i −0.715553 0.413125i
\(117\) 0 0
\(118\) 12.6715i 1.16650i
\(119\) −5.19856 + 10.7091i −0.476551 + 0.981706i
\(120\) 0 0
\(121\) 9.05816 + 15.6892i 0.823469 + 1.42629i
\(122\) 1.31079 2.27035i 0.118673 0.205548i
\(123\) 0 0
\(124\) 4.18154 2.41421i 0.375513 0.216803i
\(125\) 0 0
\(126\) 0 0
\(127\) 14.5103 1.28758 0.643792 0.765200i \(-0.277360\pi\)
0.643792 + 0.765200i \(0.277360\pi\)
\(128\) 0.866025 0.500000i 0.0765466 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) 7.73325 + 13.3944i 0.675657 + 1.17027i 0.976276 + 0.216529i \(0.0694735\pi\)
−0.300619 + 0.953744i \(0.597193\pi\)
\(132\) 0 0
\(133\) −6.27343 + 4.24849i −0.543975 + 0.368391i
\(134\) 9.82237i 0.848524i
\(135\) 0 0
\(136\) −3.89658 2.24969i −0.334129 0.192909i
\(137\) −7.46651 4.31079i −0.637907 0.368296i 0.145901 0.989299i \(-0.453392\pi\)
−0.783808 + 0.621004i \(0.786725\pi\)
\(138\) 0 0
\(139\) 10.2512i 0.869495i −0.900552 0.434748i \(-0.856838\pi\)
0.900552 0.434748i \(-0.143162\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.38134 + 4.12460i 0.199838 + 0.346129i
\(143\) 6.79253 11.7650i 0.568020 0.983839i
\(144\) 0 0
\(145\) 0 0
\(146\) 11.6569 0.964728
\(147\) 0 0
\(148\) −6.51764 −0.535747
\(149\) −7.96640 + 4.59940i −0.652633 + 0.376798i −0.789464 0.613797i \(-0.789641\pi\)
0.136831 + 0.990594i \(0.456308\pi\)
\(150\) 0 0
\(151\) 6.37429 11.0406i 0.518733 0.898471i −0.481030 0.876704i \(-0.659737\pi\)
0.999763 0.0217674i \(-0.00692931\pi\)
\(152\) −1.43185 2.48004i −0.116139 0.201158i
\(153\) 0 0
\(154\) 14.2397 1.02236i 1.14747 0.0823845i
\(155\) 0 0
\(156\) 0 0
\(157\) −11.9899 6.92236i −0.956896 0.552464i −0.0616798 0.998096i \(-0.519646\pi\)
−0.895216 + 0.445632i \(0.852979\pi\)
\(158\) 7.44120 + 4.29618i 0.591990 + 0.341786i
\(159\) 0 0
\(160\) 0 0
\(161\) −0.586988 + 0.397520i −0.0462612 + 0.0313289i
\(162\) 0 0
\(163\) −10.3025 17.8444i −0.806954 1.39768i −0.914964 0.403535i \(-0.867781\pi\)
0.108010 0.994150i \(-0.465552\pi\)
\(164\) 0.380139 0.658421i 0.0296839 0.0514140i
\(165\) 0 0
\(166\) 8.18394 4.72500i 0.635197 0.366731i
\(167\) −6.84961 −0.530038 −0.265019 0.964243i \(-0.585378\pi\)
−0.265019 + 0.964243i \(0.585378\pi\)
\(168\) 0 0
\(169\) 6.66150 0.512423
\(170\) 0 0
\(171\) 0 0
\(172\) 2.93185 5.07812i 0.223552 0.387203i
\(173\) 6.37902 + 11.0488i 0.484988 + 0.840024i 0.999851 0.0172486i \(-0.00549068\pi\)
−0.514863 + 0.857272i \(0.672157\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.39595i 0.406735i
\(177\) 0 0
\(178\) 6.90226 + 3.98502i 0.517347 + 0.298690i
\(179\) −16.3390 9.43331i −1.22123 0.705079i −0.256052 0.966663i \(-0.582422\pi\)
−0.965181 + 0.261584i \(0.915755\pi\)
\(180\) 0 0
\(181\) 25.5498i 1.89910i −0.313615 0.949550i \(-0.601540\pi\)
0.313615 0.949550i \(-0.398460\pi\)
\(182\) −3.73508 5.51532i −0.276862 0.408822i
\(183\) 0 0
\(184\) −0.133975 0.232051i −0.00987674 0.0171070i
\(185\) 0 0
\(186\) 0 0
\(187\) 21.0257 12.1392i 1.53755 0.887707i
\(188\) −7.99536 −0.583121
\(189\) 0 0
\(190\) 0 0
\(191\) −7.00657 + 4.04524i −0.506977 + 0.292704i −0.731590 0.681745i \(-0.761222\pi\)
0.224613 + 0.974448i \(0.427888\pi\)
\(192\) 0 0
\(193\) −7.06350 + 12.2343i −0.508442 + 0.880648i 0.491510 + 0.870872i \(0.336445\pi\)
−0.999952 + 0.00977575i \(0.996888\pi\)
\(194\) −3.08176 5.33777i −0.221258 0.383230i
\(195\) 0 0
\(196\) 2.59808 6.50000i 0.185577 0.464286i
\(197\) 14.2738i 1.01697i 0.861072 + 0.508483i \(0.169793\pi\)
−0.861072 + 0.508483i \(0.830207\pi\)
\(198\) 0 0
\(199\) 3.06742 + 1.77098i 0.217444 + 0.125541i 0.604766 0.796403i \(-0.293267\pi\)
−0.387322 + 0.921944i \(0.626600\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 14.0492i 0.988495i
\(203\) −1.68608 23.4840i −0.118339 1.64826i
\(204\) 0 0
\(205\) 0 0
\(206\) 7.08845 12.2776i 0.493876 0.855418i
\(207\) 0 0
\(208\) 2.18034 1.25882i 0.151179 0.0872834i
\(209\) 15.4524 1.06887
\(210\) 0 0
\(211\) −3.92340 −0.270098 −0.135049 0.990839i \(-0.543119\pi\)
−0.135049 + 0.990839i \(0.543119\pi\)
\(212\) 7.27319 4.19918i 0.499525 0.288401i
\(213\) 0 0
\(214\) −0.820863 + 1.42178i −0.0561131 + 0.0971907i
\(215\) 0 0
\(216\) 0 0
\(217\) 11.4923 + 5.57874i 0.780150 + 0.378709i
\(218\) 19.8977i 1.34764i
\(219\) 0 0
\(220\) 0 0
\(221\) −9.81017 5.66390i −0.659903 0.380995i
\(222\) 0 0
\(223\) 14.6904i 0.983740i 0.870669 + 0.491870i \(0.163686\pi\)
−0.870669 + 0.491870i \(0.836314\pi\)
\(224\) 2.38014 + 1.15539i 0.159030 + 0.0771980i
\(225\) 0 0
\(226\) 2.97934 + 5.16036i 0.198182 + 0.343262i
\(227\) 11.3913 19.7303i 0.756068 1.30955i −0.188774 0.982021i \(-0.560451\pi\)
0.944842 0.327527i \(-0.106215\pi\)
\(228\) 0 0
\(229\) −17.5089 + 10.1087i −1.15702 + 0.668005i −0.950588 0.310456i \(-0.899518\pi\)
−0.206431 + 0.978461i \(0.566185\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.89898 0.584247
\(233\) −2.27840 + 1.31543i −0.149263 + 0.0861769i −0.572771 0.819715i \(-0.694132\pi\)
0.423509 + 0.905892i \(0.360798\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.33573 10.9738i −0.412421 0.714334i
\(237\) 0 0
\(238\) −0.852491 11.8737i −0.0552588 0.769656i
\(239\) 16.8766i 1.09165i −0.837898 0.545827i \(-0.816216\pi\)
0.837898 0.545827i \(-0.183784\pi\)
\(240\) 0 0
\(241\) −12.5793 7.26268i −0.810306 0.467831i 0.0367560 0.999324i \(-0.488298\pi\)
−0.847062 + 0.531494i \(0.821631\pi\)
\(242\) −15.6892 9.05816i −1.00854 0.582280i
\(243\) 0 0
\(244\) 2.62158i 0.167829i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.60488 6.24384i −0.229373 0.397286i
\(248\) −2.41421 + 4.18154i −0.153303 + 0.265528i
\(249\) 0 0
\(250\) 0 0
\(251\) 15.7243 0.992507 0.496254 0.868178i \(-0.334709\pi\)
0.496254 + 0.868178i \(0.334709\pi\)
\(252\) 0 0
\(253\) 1.44584 0.0908993
\(254\) −12.5663 + 7.25517i −0.788481 + 0.455230i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.0312500 + 0.0541266i
\(257\) −9.83083 17.0275i −0.613230 1.06215i −0.990692 0.136121i \(-0.956536\pi\)
0.377462 0.926025i \(-0.376797\pi\)
\(258\) 0 0
\(259\) −9.66933 14.2780i −0.600823 0.887192i
\(260\) 0 0
\(261\) 0 0
\(262\) −13.3944 7.73325i −0.827508 0.477762i
\(263\) 3.74275 + 2.16088i 0.230788 + 0.133245i 0.610935 0.791680i \(-0.290794\pi\)
−0.380148 + 0.924926i \(0.624127\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.30871 6.81601i 0.202870 0.417917i
\(267\) 0 0
\(268\) −4.91119 8.50643i −0.299999 0.519613i
\(269\) 8.79895 15.2402i 0.536482 0.929214i −0.462608 0.886563i \(-0.653086\pi\)
0.999090 0.0426509i \(-0.0135803\pi\)
\(270\) 0 0
\(271\) −9.12436 + 5.26795i −0.554265 + 0.320005i −0.750840 0.660484i \(-0.770351\pi\)
0.196575 + 0.980489i \(0.437018\pi\)
\(272\) 4.49938 0.272815
\(273\) 0 0
\(274\) 8.62158 0.520849
\(275\) 0 0
\(276\) 0 0
\(277\) 1.50971 2.61489i 0.0907097 0.157114i −0.817100 0.576496i \(-0.804420\pi\)
0.907810 + 0.419382i \(0.137753\pi\)
\(278\) 5.12560 + 8.87780i 0.307413 + 0.532455i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.6880i 0.637591i −0.947824 0.318795i \(-0.896722\pi\)
0.947824 0.318795i \(-0.103278\pi\)
\(282\) 0 0
\(283\) 15.2149 + 8.78434i 0.904434 + 0.522175i 0.878636 0.477492i \(-0.158454\pi\)
0.0257976 + 0.999667i \(0.491787\pi\)
\(284\) −4.12460 2.38134i −0.244750 0.141307i
\(285\) 0 0
\(286\) 13.5851i 0.803301i
\(287\) 2.00634 0.144049i 0.118431 0.00850295i
\(288\) 0 0
\(289\) −1.62220 2.80973i −0.0954235 0.165278i
\(290\) 0 0
\(291\) 0 0
\(292\) −10.0951 + 5.82843i −0.590773 + 0.341083i
\(293\) 14.6710 0.857086 0.428543 0.903521i \(-0.359027\pi\)
0.428543 + 0.903521i \(0.359027\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.64444 3.25882i 0.328076 0.189415i
\(297\) 0 0
\(298\) 4.59940 7.96640i 0.266436 0.461481i
\(299\) −0.337300 0.584220i −0.0195065 0.0337863i
\(300\) 0 0
\(301\) 15.4741 1.11099i 0.891911 0.0640363i
\(302\) 12.7486i 0.733599i
\(303\) 0 0
\(304\) 2.48004 + 1.43185i 0.142240 + 0.0821223i
\(305\) 0 0
\(306\) 0 0
\(307\) 21.2772i 1.21435i −0.794567 0.607177i \(-0.792302\pi\)
0.794567 0.607177i \(-0.207698\pi\)
\(308\) −11.8208 + 8.00524i −0.673550 + 0.456141i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.91724 + 10.2490i −0.335536 + 0.581165i −0.983588 0.180431i \(-0.942251\pi\)
0.648052 + 0.761596i \(0.275584\pi\)
\(312\) 0 0
\(313\) 3.90551 2.25485i 0.220753 0.127452i −0.385546 0.922689i \(-0.625987\pi\)
0.606299 + 0.795237i \(0.292654\pi\)
\(314\) 13.8447 0.781302
\(315\) 0 0
\(316\) −8.59235 −0.483358
\(317\) 15.9202 9.19151i 0.894165 0.516247i 0.0188626 0.999822i \(-0.493995\pi\)
0.875303 + 0.483576i \(0.160662\pi\)
\(318\) 0 0
\(319\) −24.0092 + 41.5852i −1.34426 + 2.32833i
\(320\) 0 0
\(321\) 0 0
\(322\) 0.309587 0.637756i 0.0172526 0.0355408i
\(323\) 12.8849i 0.716934i
\(324\) 0 0
\(325\) 0 0
\(326\) 17.8444 + 10.3025i 0.988313 + 0.570603i
\(327\) 0 0
\(328\) 0.760279i 0.0419794i
\(329\) −11.8616 17.5152i −0.653952 0.965644i
\(330\) 0 0
\(331\) −3.98066 6.89471i −0.218797 0.378967i 0.735643 0.677369i \(-0.236880\pi\)
−0.954440 + 0.298401i \(0.903547\pi\)
\(332\) −4.72500 + 8.18394i −0.259318 + 0.449152i
\(333\) 0 0
\(334\) 5.93193 3.42480i 0.324581 0.187397i
\(335\) 0 0
\(336\) 0 0
\(337\) 6.89417 0.375549 0.187775 0.982212i \(-0.439873\pi\)
0.187775 + 0.982212i \(0.439873\pi\)
\(338\) −5.76903 + 3.33075i −0.313794 + 0.181169i
\(339\) 0 0
\(340\) 0 0
\(341\) −13.0270 22.5634i −0.705451 1.22188i
\(342\) 0 0
\(343\) 18.0938 3.95164i 0.976972 0.213368i
\(344\) 5.86370i 0.316150i
\(345\) 0 0
\(346\) −11.0488 6.37902i −0.593986 0.342938i
\(347\) 14.1645 + 8.17789i 0.760392 + 0.439012i 0.829436 0.558601i \(-0.188662\pi\)
−0.0690448 + 0.997614i \(0.521995\pi\)
\(348\) 0 0
\(349\) 24.5851i 1.31601i −0.753014 0.658004i \(-0.771401\pi\)
0.753014 0.658004i \(-0.228599\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.69798 4.67303i −0.143803 0.249073i
\(353\) −6.85906 + 11.8802i −0.365071 + 0.632321i −0.988788 0.149329i \(-0.952289\pi\)
0.623717 + 0.781651i \(0.285622\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.97005 −0.422412
\(357\) 0 0
\(358\) 18.8666 0.997132
\(359\) −10.3059 + 5.95011i −0.543924 + 0.314035i −0.746668 0.665197i \(-0.768348\pi\)
0.202744 + 0.979232i \(0.435014\pi\)
\(360\) 0 0
\(361\) −5.39960 + 9.35238i −0.284190 + 0.492231i
\(362\) 12.7749 + 22.1268i 0.671433 + 1.16296i
\(363\) 0 0
\(364\) 5.99233 + 2.90887i 0.314083 + 0.152466i
\(365\) 0 0
\(366\) 0 0
\(367\) 10.9026 + 6.29461i 0.569110 + 0.328576i 0.756794 0.653654i \(-0.226765\pi\)
−0.187684 + 0.982230i \(0.560098\pi\)
\(368\) 0.232051 + 0.133975i 0.0120965 + 0.00698391i
\(369\) 0 0
\(370\) 0 0
\(371\) 19.9893 + 9.70342i 1.03779 + 0.503776i
\(372\) 0 0
\(373\) −13.5868 23.5331i −0.703499 1.21850i −0.967230 0.253900i \(-0.918286\pi\)
0.263731 0.964596i \(-0.415047\pi\)
\(374\) −12.1392 + 21.0257i −0.627704 + 1.08722i
\(375\) 0 0
\(376\) 6.92418 3.99768i 0.357087 0.206164i
\(377\) 22.4044 1.15389
\(378\) 0 0
\(379\) 15.7335 0.808174 0.404087 0.914721i \(-0.367589\pi\)
0.404087 + 0.914721i \(0.367589\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.04524 7.00657i 0.206973 0.358487i
\(383\) 7.89060 + 13.6669i 0.403191 + 0.698347i 0.994109 0.108384i \(-0.0345677\pi\)
−0.590918 + 0.806732i \(0.701234\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.1270i 0.719046i
\(387\) 0 0
\(388\) 5.33777 + 3.08176i 0.270984 + 0.156453i
\(389\) −13.7556 7.94182i −0.697438 0.402666i 0.108954 0.994047i \(-0.465250\pi\)
−0.806393 + 0.591381i \(0.798583\pi\)
\(390\) 0 0
\(391\) 1.20560i 0.0609700i
\(392\) 1.00000 + 6.92820i 0.0505076 + 0.349927i
\(393\) 0 0
\(394\) −7.13689 12.3615i −0.359552 0.622762i
\(395\) 0 0
\(396\) 0 0
\(397\) −32.3557 + 18.6806i −1.62388 + 0.937550i −0.638017 + 0.770022i \(0.720245\pi\)
−0.985867 + 0.167528i \(0.946422\pi\)
\(398\) −3.54195 −0.177542
\(399\) 0 0
\(400\) 0 0
\(401\) 24.4856 14.1368i 1.22275 0.705957i 0.257249 0.966345i \(-0.417184\pi\)
0.965504 + 0.260389i \(0.0838507\pi\)
\(402\) 0 0
\(403\) −6.07812 + 10.5276i −0.302773 + 0.524417i
\(404\) 7.02458 + 12.1669i 0.349486 + 0.605327i
\(405\) 0 0
\(406\) 13.2022 + 19.4947i 0.655214 + 0.967507i
\(407\) 35.1689i 1.74326i
\(408\) 0 0
\(409\) 13.8647 + 8.00481i 0.685567 + 0.395812i 0.801949 0.597392i \(-0.203796\pi\)
−0.116382 + 0.993204i \(0.537130\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 14.1769i 0.698446i
\(413\) 14.6405 30.1599i 0.720414 1.48407i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.25882 + 2.18034i −0.0617187 + 0.106900i
\(417\) 0 0
\(418\) −13.3822 + 7.72620i −0.654544 + 0.377901i
\(419\) 29.5137 1.44184 0.720919 0.693020i \(-0.243720\pi\)
0.720919 + 0.693020i \(0.243720\pi\)
\(420\) 0 0
\(421\) 0.309114 0.0150653 0.00753265 0.999972i \(-0.497602\pi\)
0.00753265 + 0.999972i \(0.497602\pi\)
\(422\) 3.39776 1.96170i 0.165400 0.0954939i
\(423\) 0 0
\(424\) −4.19918 + 7.27319i −0.203930 + 0.353217i
\(425\) 0 0
\(426\) 0 0
\(427\) −5.74301 + 3.88928i −0.277924 + 0.188215i
\(428\) 1.64173i 0.0793559i
\(429\) 0 0
\(430\) 0 0
\(431\) 7.63843 + 4.41005i 0.367930 + 0.212425i 0.672554 0.740048i \(-0.265197\pi\)
−0.304624 + 0.952473i \(0.598531\pi\)
\(432\) 0 0
\(433\) 9.56388i 0.459611i −0.973237 0.229805i \(-0.926191\pi\)
0.973237 0.229805i \(-0.0738089\pi\)
\(434\) −12.7420 + 0.914836i −0.611636 + 0.0439135i
\(435\) 0 0
\(436\) −9.94887 17.2319i −0.476464 0.825260i
\(437\) 0.383663 0.664525i 0.0183531 0.0317885i
\(438\) 0 0
\(439\) −31.3336 + 18.0905i −1.49547 + 0.863412i −0.999986 0.00520362i \(-0.998344\pi\)
−0.495487 + 0.868615i \(0.665010\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.3278 0.538809
\(443\) 3.53830 2.04284i 0.168110 0.0970582i −0.413584 0.910466i \(-0.635723\pi\)
0.581694 + 0.813408i \(0.302390\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −7.34519 12.7222i −0.347805 0.602415i
\(447\) 0 0
\(448\) −2.63896 + 0.189469i −0.124679 + 0.00895155i
\(449\) 19.9377i 0.940918i 0.882422 + 0.470459i \(0.155912\pi\)
−0.882422 + 0.470459i \(0.844088\pi\)
\(450\) 0 0
\(451\) −3.55281 2.05121i −0.167295 0.0965879i
\(452\) −5.16036 2.97934i −0.242723 0.140136i
\(453\) 0 0
\(454\) 22.7826i 1.06924i
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0623 17.4283i −0.470693 0.815264i 0.528745 0.848780i \(-0.322663\pi\)
−0.999438 + 0.0335168i \(0.989329\pi\)
\(458\) 10.1087 17.5089i 0.472351 0.818136i
\(459\) 0 0
\(460\) 0 0
\(461\) −0.909299 −0.0423503 −0.0211751 0.999776i \(-0.506741\pi\)
−0.0211751 + 0.999776i \(0.506741\pi\)
\(462\) 0 0
\(463\) −21.4280 −0.995843 −0.497922 0.867222i \(-0.665903\pi\)
−0.497922 + 0.867222i \(0.665903\pi\)
\(464\) −7.70674 + 4.44949i −0.357777 + 0.206562i
\(465\) 0 0
\(466\) 1.31543 2.27840i 0.0609363 0.105545i
\(467\) 6.34607 + 10.9917i 0.293661 + 0.508636i 0.974672 0.223637i \(-0.0717930\pi\)
−0.681012 + 0.732273i \(0.738460\pi\)
\(468\) 0 0
\(469\) 11.3487 23.3786i 0.524035 1.07952i
\(470\) 0 0
\(471\) 0 0
\(472\) 10.9738 + 6.33573i 0.505111 + 0.291626i
\(473\) −27.4013 15.8201i −1.25991 0.727411i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.67511 + 9.85666i 0.305953 + 0.451779i
\(477\) 0 0
\(478\) 8.43828 + 14.6155i 0.385958 + 0.668499i
\(479\) 6.43828 11.1514i 0.294172 0.509522i −0.680620 0.732637i \(-0.738289\pi\)
0.974792 + 0.223115i \(0.0716227\pi\)
\(480\) 0 0
\(481\) 14.2107 8.20453i 0.647950 0.374094i
\(482\) 14.5254 0.661612
\(483\) 0 0
\(484\) 18.1163 0.823469
\(485\) 0 0
\(486\) 0 0
\(487\) −10.4097 + 18.0301i −0.471708 + 0.817022i −0.999476 0.0323665i \(-0.989696\pi\)
0.527768 + 0.849388i \(0.323029\pi\)
\(488\) −1.31079 2.27035i −0.0593366 0.102774i
\(489\) 0 0
\(490\) 0 0
\(491\) 27.3271i 1.23325i 0.787256 + 0.616627i \(0.211501\pi\)
−0.787256 + 0.616627i \(0.788499\pi\)
\(492\) 0 0
\(493\) 34.6755 + 20.0199i 1.56171 + 0.901653i
\(494\) 6.24384 + 3.60488i 0.280924 + 0.162191i
\(495\) 0 0
\(496\) 4.82843i 0.216803i
\(497\) −0.902379 12.5685i −0.0404772 0.563775i
\(498\) 0 0
\(499\) −16.6802 28.8909i −0.746708 1.29334i −0.949393 0.314092i \(-0.898300\pi\)
0.202685 0.979244i \(-0.435033\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −13.6176 + 7.86214i −0.607784 + 0.350904i
\(503\) 16.2936 0.726494 0.363247 0.931693i \(-0.381668\pi\)
0.363247 + 0.931693i \(0.381668\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.25214 + 0.722921i −0.0556642 + 0.0321377i
\(507\) 0 0
\(508\) 7.25517 12.5663i 0.321896 0.557540i
\(509\) 12.3400 + 21.3735i 0.546961 + 0.947365i 0.998481 + 0.0551036i \(0.0175489\pi\)
−0.451519 + 0.892261i \(0.649118\pi\)
\(510\) 0 0
\(511\) −27.7449 13.4683i −1.22736 0.595801i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 17.0275 + 9.83083i 0.751051 + 0.433619i
\(515\) 0 0
\(516\) 0 0
\(517\) 43.1426i 1.89741i
\(518\) 15.5129 + 7.53044i 0.681597 + 0.330869i
\(519\) 0 0
\(520\) 0 0
\(521\) 0.141663 0.245367i 0.00620635 0.0107497i −0.862906 0.505365i \(-0.831358\pi\)
0.869112 + 0.494616i \(0.164691\pi\)
\(522\) 0 0
\(523\) 9.77021 5.64083i 0.427222 0.246656i −0.270941 0.962596i \(-0.587335\pi\)
0.698162 + 0.715940i \(0.254001\pi\)
\(524\) 15.4665 0.675657
\(525\) 0 0
\(526\) −4.32175 −0.188437
\(527\) −18.8143 + 10.8625i −0.819565 + 0.473176i
\(528\) 0 0
\(529\) −11.4641 + 19.8564i −0.498439 + 0.863322i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.542582 + 7.55719i 0.0235239 + 0.327646i
\(533\) 1.91411i 0.0829092i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.50643 + 4.91119i 0.367422 + 0.212131i
\(537\) 0 0
\(538\) 17.5979i 0.758700i
\(539\) −35.0737 14.0191i −1.51073 0.603845i
\(540\) 0 0
\(541\) 17.4125 + 30.1593i 0.748621 + 1.29665i 0.948484 + 0.316826i \(0.102617\pi\)
−0.199862 + 0.979824i \(0.564049\pi\)
\(542\) 5.26795 9.12436i 0.226278 0.391925i
\(543\) 0 0
\(544\) −3.89658 + 2.24969i −0.167064 + 0.0964546i
\(545\) 0 0
\(546\) 0 0
\(547\) −35.4261 −1.51471 −0.757356 0.653002i \(-0.773509\pi\)
−0.757356 + 0.653002i \(0.773509\pi\)
\(548\) −7.46651 + 4.31079i −0.318953 + 0.184148i
\(549\) 0 0
\(550\) 0 0
\(551\) 12.7420 + 22.0698i 0.542828 + 0.940206i
\(552\) 0 0
\(553\) −12.7473 18.8230i −0.542071 0.800436i
\(554\) 3.01942i 0.128283i
\(555\) 0 0
\(556\) −8.87780 5.12560i −0.376503 0.217374i
\(557\) −7.05105 4.07093i −0.298763 0.172491i 0.343124 0.939290i \(-0.388515\pi\)
−0.641887 + 0.766799i \(0.721848\pi\)
\(558\) 0 0
\(559\) 14.7627i 0.624395i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.34398 + 9.25605i 0.225422 + 0.390443i
\(563\) −10.2088 + 17.6821i −0.430248 + 0.745212i −0.996894 0.0787491i \(-0.974907\pi\)
0.566646 + 0.823961i \(0.308241\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −17.5687 −0.738467
\(567\) 0 0
\(568\) 4.76268 0.199838
\(569\) 22.5542 13.0217i 0.945520 0.545896i 0.0538334 0.998550i \(-0.482856\pi\)
0.891686 + 0.452654i \(0.149523\pi\)
\(570\) 0 0
\(571\) 6.18811 10.7181i 0.258964 0.448539i −0.707000 0.707213i \(-0.749952\pi\)
0.965965 + 0.258674i \(0.0832855\pi\)
\(572\) −6.79253 11.7650i −0.284010 0.491920i
\(573\) 0 0
\(574\) −1.66552 + 1.12792i −0.0695175 + 0.0470786i
\(575\) 0 0
\(576\) 0 0
\(577\) −23.3399 13.4753i −0.971654 0.560985i −0.0719139 0.997411i \(-0.522911\pi\)
−0.899740 + 0.436426i \(0.856244\pi\)
\(578\) 2.80973 + 1.62220i 0.116869 + 0.0674746i
\(579\) 0 0
\(580\) 0 0
\(581\) −24.9382 + 1.79048i −1.03461 + 0.0742816i
\(582\) 0 0
\(583\) −22.6586 39.2458i −0.938422 1.62539i
\(584\) 5.82843 10.0951i 0.241182 0.417740i
\(585\) 0 0
\(586\) −12.7054 + 7.33548i −0.524856 + 0.303026i
\(587\) 35.3511 1.45910 0.729548 0.683930i \(-0.239731\pi\)
0.729548 + 0.683930i \(0.239731\pi\)
\(588\) 0 0
\(589\) −13.8272 −0.569739
\(590\) 0 0
\(591\) 0 0
\(592\) −3.25882 + 5.64444i −0.133937 + 0.231985i
\(593\) −14.7057 25.4711i −0.603893 1.04597i −0.992225 0.124454i \(-0.960282\pi\)
0.388333 0.921519i \(-0.373051\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 9.19881i 0.376798i
\(597\) 0 0
\(598\) 0.584220 + 0.337300i 0.0238905 + 0.0137932i
\(599\) 26.5494 + 15.3283i 1.08478 + 0.626298i 0.932182 0.361990i \(-0.117903\pi\)
0.152598 + 0.988288i \(0.451236\pi\)
\(600\) 0 0
\(601\) 34.3407i 1.40078i 0.713758 + 0.700392i \(0.246992\pi\)
−0.713758 + 0.700392i \(0.753008\pi\)
\(602\) −12.8454 + 8.69918i −0.523541 + 0.354552i
\(603\) 0 0
\(604\) −6.37429 11.0406i −0.259366 0.449236i
\(605\) 0 0
\(606\) 0 0
\(607\) −28.2475 + 16.3087i −1.14653 + 0.661950i −0.948040 0.318153i \(-0.896938\pi\)
−0.198492 + 0.980103i \(0.563604\pi\)
\(608\) −2.86370 −0.116139
\(609\) 0 0
\(610\) 0 0
\(611\) 17.4326 10.0647i 0.705247 0.407174i
\(612\) 0 0
\(613\) −7.99843 + 13.8537i −0.323054 + 0.559545i −0.981117 0.193418i \(-0.938043\pi\)
0.658063 + 0.752963i \(0.271376\pi\)
\(614\) 10.6386 + 18.4266i 0.429339 + 0.743636i
\(615\) 0 0
\(616\) 6.23445 12.8431i 0.251193 0.517464i
\(617\) 25.1429i 1.01221i −0.862471 0.506107i \(-0.831084\pi\)
0.862471 0.506107i \(-0.168916\pi\)
\(618\) 0 0
\(619\) 32.3379 + 18.6703i 1.29977 + 0.750423i 0.980364 0.197195i \(-0.0631832\pi\)
0.319406 + 0.947618i \(0.396517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 11.8345i 0.474519i
\(623\) −11.8241 17.4597i −0.473722 0.699510i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.25485 + 3.90551i −0.0901219 + 0.156096i
\(627\) 0 0
\(628\) −11.9899 + 6.92236i −0.478448 + 0.276232i
\(629\) 29.3253 1.16928
\(630\) 0 0
\(631\) 49.5015 1.97062 0.985311 0.170767i \(-0.0546245\pi\)
0.985311 + 0.170767i \(0.0546245\pi\)
\(632\) 7.44120 4.29618i 0.295995 0.170893i
\(633\) 0 0
\(634\) −9.19151 + 15.9202i −0.365041 + 0.632270i
\(635\) 0 0
\(636\) 0 0
\(637\) 2.51764 + 17.4427i 0.0997525 + 0.691105i
\(638\) 48.0185i 1.90107i
\(639\) 0 0
\(640\) 0 0
\(641\) −31.4439 18.1542i −1.24196 0.717046i −0.272467 0.962165i \(-0.587840\pi\)
−0.969493 + 0.245119i \(0.921173\pi\)
\(642\) 0 0
\(643\) 10.2653i 0.404824i −0.979300 0.202412i \(-0.935122\pi\)
0.979300 0.202412i \(-0.0648780\pi\)
\(644\) 0.0507680 + 0.707107i 0.00200054 + 0.0278639i
\(645\) 0 0
\(646\) 6.44244 + 11.1586i 0.253474 + 0.439031i
\(647\) 10.9108 18.8980i 0.428946 0.742956i −0.567834 0.823143i \(-0.692218\pi\)
0.996780 + 0.0801869i \(0.0255517\pi\)
\(648\) 0 0
\(649\) −59.2142 + 34.1873i −2.32436 + 1.34197i
\(650\) 0 0
\(651\) 0 0
\(652\) −20.6050 −0.806954
\(653\) −40.9556 + 23.6457i −1.60272 + 0.925329i −0.611775 + 0.791032i \(0.709544\pi\)
−0.990941 + 0.134297i \(0.957122\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.380139 0.658421i −0.0148419 0.0257070i
\(657\) 0 0
\(658\) 19.0301 + 9.23779i 0.741869 + 0.360127i
\(659\) 3.58255i 0.139556i 0.997563 + 0.0697782i \(0.0222291\pi\)
−0.997563 + 0.0697782i \(0.977771\pi\)
\(660\) 0 0
\(661\) −1.41761 0.818459i −0.0551388 0.0318344i 0.472177 0.881504i \(-0.343468\pi\)
−0.527316 + 0.849669i \(0.676802\pi\)
\(662\) 6.89471 + 3.98066i 0.267970 + 0.154713i
\(663\) 0 0
\(664\) 9.45001i 0.366731i
\(665\) 0 0
\(666\) 0 0
\(667\) 1.19224 + 2.06502i 0.0461636 + 0.0799577i
\(668\) −3.42480 + 5.93193i −0.132510 + 0.229513i
\(669\) 0 0
\(670\) 0 0
\(671\) 14.1459 0.546097
\(672\) 0 0
\(673\) 2.02242 0.0779587 0.0389794 0.999240i \(-0.487589\pi\)
0.0389794 + 0.999240i \(0.487589\pi\)
\(674\) −5.97053 + 3.44709i −0.229976 + 0.132777i
\(675\) 0 0
\(676\) 3.33075 5.76903i 0.128106 0.221886i
\(677\) −11.4413 19.8169i −0.439725 0.761626i 0.557943 0.829879i \(-0.311591\pi\)
−0.997668 + 0.0682532i \(0.978257\pi\)
\(678\) 0 0
\(679\) 1.16780 + 16.2653i 0.0448159 + 0.624205i
\(680\) 0 0
\(681\) 0 0
\(682\) 22.5634 + 13.0270i 0.863997 + 0.498829i
\(683\) −27.1977 15.7026i −1.04069 0.600844i −0.120663 0.992694i \(-0.538502\pi\)
−0.920029 + 0.391850i \(0.871835\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −13.6938 + 12.4691i −0.522834 + 0.476073i
\(687\) 0 0
\(688\) −2.93185 5.07812i −0.111776 0.193601i
\(689\) −10.5720 + 18.3113i −0.402762 + 0.697604i
\(690\) 0 0
\(691\) 29.3677 16.9554i 1.11720 0.645015i 0.176515 0.984298i \(-0.443518\pi\)
0.940684 + 0.339283i \(0.110184\pi\)
\(692\) 12.7580 0.484988
\(693\) 0 0
\(694\) −16.3558 −0.620857
\(695\) 0 0
\(696\) 0 0
\(697\) −1.71039 + 2.96248i −0.0647857 + 0.112212i
\(698\) 12.2925 + 21.2913i 0.465279 + 0.805887i
\(699\) 0 0
\(700\) 0 0
\(701\) 10.5296i 0.397699i −0.980030 0.198849i \(-0.936280\pi\)
0.980030 0.198849i \(-0.0637205\pi\)
\(702\) 0 0
\(703\) 16.1640 + 9.33229i 0.609637 + 0.351974i
\(704\) 4.67303 + 2.69798i 0.176122 + 0.101684i
\(705\) 0 0
\(706\) 13.7181i 0.516288i
\(707\) −16.2323 + 33.4390i −0.610479 + 1.25760i
\(708\) 0 0
\(709\) −7.52572 13.0349i −0.282634 0.489537i 0.689398 0.724382i \(-0.257875\pi\)
−0.972033 + 0.234845i \(0.924542\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.90226 3.98502i 0.258673 0.149345i
\(713\) −1.29377 −0.0484522
\(714\) 0 0
\(715\) 0 0
\(716\) −16.3390 + 9.43331i −0.610616 + 0.352539i
\(717\) 0 0
\(718\) 5.95011 10.3059i 0.222056 0.384613i
\(719\) 12.2137 + 21.1547i 0.455494 + 0.788938i 0.998716 0.0506506i \(-0.0161295\pi\)
−0.543223 + 0.839589i \(0.682796\pi\)
\(720\) 0 0
\(721\) −31.0569 + 21.0323i −1.15662 + 0.783285i
\(722\) 10.7992i 0.401905i
\(723\) 0 0
\(724\) −22.1268 12.7749i −0.822335 0.474775i
\(725\) 0 0
\(726\) 0 0
\(727\) 43.7349i 1.62204i −0.585020 0.811019i \(-0.698913\pi\)
0.585020 0.811019i \(-0.301087\pi\)
\(728\) −6.64394 + 0.477014i −0.246241 + 0.0176793i
\(729\) 0 0
\(730\) 0 0
\(731\) −13.1915 + 22.8484i −0.487906 + 0.845077i
\(732\) 0 0
\(733\) −38.7280 + 22.3596i −1.43045 + 0.825872i −0.997155 0.0753789i \(-0.975983\pi\)
−0.433297 + 0.901251i \(0.642650\pi\)
\(734\) −12.5892 −0.464677
\(735\) 0 0
\(736\) −0.267949 −0.00987674
\(737\) −45.9003 + 26.5005i −1.69076 + 0.976160i
\(738\) 0 0
\(739\) 10.7360 18.5954i 0.394932 0.684042i −0.598161 0.801376i \(-0.704102\pi\)
0.993092 + 0.117334i \(0.0374348\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −22.1629 + 1.59123i −0.813626 + 0.0584157i
\(743\) 29.5637i 1.08459i 0.840190 + 0.542293i \(0.182444\pi\)
−0.840190 + 0.542293i \(0.817556\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 23.5331 + 13.5868i 0.861607 + 0.497449i
\(747\) 0 0
\(748\) 24.2784i 0.887707i
\(749\) 3.59648 2.43561i 0.131413 0.0889951i
\(750\) 0 0
\(751\) −0.596750 1.03360i −0.0217757 0.0377166i 0.854932 0.518740i \(-0.173599\pi\)
−0.876708 + 0.481023i \(0.840265\pi\)
\(752\) −3.99768 + 6.92418i −0.145780 + 0.252499i
\(753\) 0 0
\(754\) −19.4028 + 11.2022i −0.706608 + 0.407960i
\(755\) 0 0
\(756\) 0 0
\(757\) 26.8915 0.977386 0.488693 0.872456i \(-0.337474\pi\)
0.488693 + 0.872456i \(0.337474\pi\)
\(758\) −13.6256 + 7.86673i −0.494903 + 0.285732i
\(759\) 0 0
\(760\) 0 0
\(761\) −0.939574 1.62739i −0.0340595 0.0589928i 0.848493 0.529206i \(-0.177510\pi\)
−0.882553 + 0.470213i \(0.844177\pi\)
\(762\) 0 0
\(763\) 22.9897 47.3594i 0.832284 1.71452i
\(764\) 8.09049i 0.292704i
\(765\) 0 0
\(766\) −13.6669 7.89060i −0.493806 0.285099i
\(767\) 27.6281 + 15.9511i 0.997592 + 0.575960i
\(768\) 0 0
\(769\) 50.6544i 1.82664i 0.407239 + 0.913322i \(0.366492\pi\)
−0.407239 + 0.913322i \(0.633508\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.06350 + 12.2343i 0.254221 + 0.440324i
\(773\) 24.3353 42.1499i 0.875279 1.51603i 0.0188128 0.999823i \(-0.494011\pi\)
0.856466 0.516204i \(-0.172655\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −6.16353 −0.221258
\(777\) 0 0
\(778\) 15.8836 0.569456
\(779\) −1.88552 + 1.08861i −0.0675558 + 0.0390034i
\(780\) 0 0
\(781\) −12.8496 + 22.2562i −0.459795 + 0.796388i
\(782\) 0.602802 + 1.04408i 0.0215562 + 0.0373364i
\(783\) 0 0
\(784\) −4.33013 5.50000i −0.154647 0.196429i
\(785\) 0 0
\(786\) 0 0
\(787\) 12.9437 + 7.47307i 0.461395 + 0.266386i 0.712630 0.701540i \(-0.247504\pi\)
−0.251236 + 0.967926i \(0.580837\pi\)
\(788\) 12.3615 + 7.13689i 0.440359 + 0.254241i
\(789\) 0 0
\(790\) 0 0
\(791\) −1.12898 15.7247i −0.0401420 0.559105i
\(792\) 0 0
\(793\) −3.30009 5.71593i −0.117190 0.202979i
\(794\) 18.6806 32.3557i 0.662948 1.14826i
\(795\) 0 0
\(796\) 3.06742 1.77098i 0.108722 0.0627706i
\(797\) 15.0557 0.533301 0.266650 0.963793i \(-0.414083\pi\)
0.266650 + 0.963793i \(0.414083\pi\)
\(798\) 0 0
\(799\) 35.9741 1.27267
\(800\) 0 0
\(801\) 0 0
\(802\) −14.1368 + 24.4856i −0.499187 + 0.864617i
\(803\) 31.4499 + 54.4729i 1.10984 + 1.92231i
\(804\) 0 0
\(805\) 0 0
\(806\) 12.1562i 0.428185i
\(807\) 0 0
\(808\) −12.1669 7.02458i −0.428031 0.247124i
\(809\) 30.7426 + 17.7493i 1.08085 + 0.624031i 0.931127 0.364696i \(-0.118827\pi\)
0.149727 + 0.988727i \(0.452160\pi\)
\(810\) 0 0
\(811\) 24.5935i 0.863594i −0.901971 0.431797i \(-0.857880\pi\)
0.901971 0.431797i \(-0.142120\pi\)
\(812\) −21.1808 10.2818i −0.743301 0.360822i
\(813\) 0 0
\(814\) −17.5844 30.4571i −0.616334 1.06752i
\(815\) 0 0
\(816\) 0 0
\(817\) −14.5422 + 8.39595i −0.508768 + 0.293737i
\(818\) −16.0096 −0.559763
\(819\) 0 0
\(820\) 0 0
\(821\) −12.3035 + 7.10342i −0.429395 + 0.247911i −0.699089 0.715035i \(-0.746411\pi\)
0.269694 + 0.962946i \(0.413077\pi\)
\(822\) 0 0
\(823\) 0.945584 1.63780i 0.0329610 0.0570901i −0.849074 0.528273i \(-0.822840\pi\)
0.882035 + 0.471183i \(0.156173\pi\)
\(824\) −7.08845 12.2776i −0.246938 0.427709i
\(825\) 0 0
\(826\) 2.40085 + 33.4395i 0.0835361 + 1.16351i
\(827\) 31.9280i 1.11024i 0.831769 + 0.555122i \(0.187329\pi\)
−0.831769 + 0.555122i \(0.812671\pi\)
\(828\) 0 0
\(829\) 3.38864 + 1.95643i 0.117692 + 0.0679497i 0.557691 0.830049i \(-0.311688\pi\)
−0.439998 + 0.897999i \(0.645021\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.51764i 0.0872834i
\(833\) −11.6897 + 29.2460i −0.405025 + 1.01331i
\(834\) 0 0
\(835\) 0 0
\(836\) 7.72620 13.3822i 0.267216 0.462832i
\(837\) 0 0
\(838\) −25.5596 + 14.7568i −0.882941 + 0.509766i
\(839\) −15.3513 −0.529984 −0.264992 0.964251i \(-0.585369\pi\)
−0.264992 + 0.964251i \(0.585369\pi\)
\(840\) 0 0
\(841\) −50.1918 −1.73075
\(842\) −0.267701 + 0.154557i −0.00922557 + 0.00532639i
\(843\) 0 0
\(844\) −1.96170 + 3.39776i −0.0675244 + 0.116956i
\(845\) 0 0
\(846\) 0 0
\(847\) 26.8767 + 39.6869i 0.923495 + 1.36366i
\(848\) 8.39836i 0.288401i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.51242 + 0.873198i 0.0518452 + 0.0299328i
\(852\) 0 0
\(853\) 22.2302i 0.761148i 0.924750 + 0.380574i \(0.124274\pi\)
−0.924750 + 0.380574i \(0.875726\pi\)
\(854\) 3.02896 6.23972i 0.103649 0.213519i
\(855\) 0 0
\(856\) 0.820863 + 1.42178i 0.0280565 + 0.0485953i
\(857\) −26.4098 + 45.7432i −0.902143 + 1.56256i −0.0774356 + 0.996997i \(0.524673\pi\)
−0.824707 + 0.565560i \(0.808660\pi\)
\(858\) 0 0
\(859\) −15.2916 + 8.82859i −0.521742 + 0.301228i −0.737647 0.675187i \(-0.764063\pi\)
0.215905 + 0.976414i \(0.430730\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −8.82010 −0.300414
\(863\) 14.2452 8.22446i 0.484912 0.279964i −0.237549 0.971375i \(-0.576344\pi\)
0.722461 + 0.691412i \(0.243011\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 4.78194 + 8.28256i 0.162497 + 0.281453i
\(867\) 0 0
\(868\) 10.5775 7.16328i 0.359024 0.243138i
\(869\) 46.3639i 1.57279i
\(870\) 0 0
\(871\) 21.4161 + 12.3646i 0.725657 + 0.418958i
\(872\) 17.2319 + 9.94887i 0.583547 + 0.336911i
\(873\) 0 0
\(874\) 0.767327i 0.0259552i
\(875\) 0 0
\(876\) 0 0
\(877\) 27.1078 + 46.9521i 0.915366 + 1.58546i 0.806365 + 0.591418i \(0.201432\pi\)
0.109001 + 0.994042i \(0.465235\pi\)
\(878\) 18.0905 31.3336i 0.610524 1.05746i
\(879\) 0 0
\(880\) 0 0
\(881\) −50.1647 −1.69009 −0.845046 0.534694i \(-0.820427\pi\)
−0.845046 + 0.534694i \(0.820427\pi\)
\(882\) 0 0
\(883\) 0.841563 0.0283208 0.0141604 0.999900i \(-0.495492\pi\)
0.0141604 + 0.999900i \(0.495492\pi\)
\(884\) −9.81017 + 5.66390i −0.329952 + 0.190498i
\(885\) 0 0
\(886\) −2.04284 + 3.53830i −0.0686305 + 0.118872i
\(887\) 20.0492 + 34.7262i 0.673185 + 1.16599i 0.976996 + 0.213258i \(0.0684075\pi\)
−0.303811 + 0.952732i \(0.598259\pi\)
\(888\) 0 0
\(889\) 38.2922 2.74926i 1.28428 0.0922071i
\(890\) 0 0
\(891\) 0 0
\(892\) 12.7222 + 7.34519i 0.425972 + 0.245935i
\(893\) 19.8288 + 11.4482i 0.663546 + 0.383098i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.19067 1.48356i 0.0731852 0.0495624i
\(897\) 0 0
\(898\) −9.96885 17.2665i −0.332665 0.576192i
\(899\) 21.4840 37.2114i 0.716533 1.24107i
\(900\) 0 0
\(901\) −32.7248 + 18.8937i −1.09022 + 0.629440i
\(902\) 4.10243 0.136596
\(903\) 0 0
\(904\) 5.95867 0.198182
\(905\) 0 0
\(906\) 0 0
\(907\) −22.5945 + 39.1348i −0.750238 + 1.29945i 0.197469 + 0.980309i \(0.436728\pi\)
−0.947707 + 0.319142i \(0.896605\pi\)
\(908\) −11.3913 19.7303i −0.378034 0.654774i
\(909\) 0 0
\(910\) 0 0
\(911\) 58.2281i 1.92918i 0.263746 + 0.964592i \(0.415042\pi\)
−0.263746 + 0.964592i \(0.584958\pi\)
\(912\) 0 0
\(913\) 44.1602 + 25.4959i 1.46149 + 0.843791i
\(914\) 17.4283 + 10.0623i 0.576478 + 0.332830i
\(915\) 0 0
\(916\) 20.2175i 0.668005i
\(917\) 22.9455 + 33.8820i 0.757729 + 1.11888i
\(918\) 0 0
\(919\) −6.61745 11.4618i −0.218290 0.378089i 0.735996 0.676986i \(-0.236714\pi\)
−0.954285 + 0.298898i \(0.903381\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.787476 0.454649i 0.0259341 0.0149731i
\(923\) 11.9907 0.394679
\(924\) 0 0
\(925\) 0 0
\(926\) 18.5572 10.7140i 0.609827 0.352084i
\(927\) 0 0
\(928\) 4.44949 7.70674i 0.146062 0.252986i
\(929\) 10.4434 + 18.0885i 0.342637 + 0.593464i 0.984921 0.173002i \(-0.0553467\pi\)
−0.642285 + 0.766466i \(0.722013\pi\)
\(930\) 0 0
\(931\) −15.7504 + 12.4002i −0.516197 + 0.406400i
\(932\) 2.63087i 0.0861769i
\(933\) 0 0
\(934\) −10.9917 6.34607i −0.359660 0.207650i
\(935\) 0 0
\(936\) 0 0
\(937\) 23.2465i 0.759430i 0.925103 + 0.379715i \(0.123978\pi\)
−0.925103 + 0.379715i \(0.876022\pi\)
\(938\) 1.86103 + 25.9208i 0.0607649 + 0.846345i
\(939\) 0 0
\(940\) 0 0
\(941\) 0.752551 1.30346i 0.0245325 0.0424915i −0.853499 0.521095i \(-0.825524\pi\)
0.878031 + 0.478604i \(0.158857\pi\)
\(942\) 0 0
\(943\) −0.176423 + 0.101858i −0.00574513 + 0.00331695i
\(944\) −12.6715 −0.412421
\(945\) 0 0
\(946\) 31.6403 1.02871
\(947\) 21.0122 12.1314i 0.682805 0.394218i −0.118106 0.993001i \(-0.537682\pi\)
0.800911 + 0.598783i \(0.204349\pi\)
\(948\) 0 0
\(949\) 14.6739 25.4159i 0.476334 0.825035i
\(950\) 0 0
\(951\) 0 0
\(952\) −10.7091 5.19856i −0.347085 0.168486i
\(953\) 56.7061i 1.83689i −0.395547 0.918446i \(-0.629445\pi\)
0.395547 0.918446i \(-0.370555\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −14.6155 8.43828i −0.472700 0.272913i
\(957\) 0 0
\(958\) 12.8766i 0.416023i
\(959\) −20.5206 9.96132i −0.662643 0.321668i
\(960\) 0 0
\(961\) −3.84315 6.65652i −0.123972 0.214727i
\(962\) −8.20453 + 14.2107i −0.264525 + 0.458170i
\(963\) 0 0
\(964\) −12.5793 + 7.26268i −0.405153 + 0.233915i
\(965\) 0 0
\(966\) 0 0
\(967\) −7.23556 −0.232680 −0.116340 0.993209i \(-0.537116\pi\)
−0.116340 + 0.993209i \(0.537116\pi\)
\(968\) −15.6892 + 9.05816i −0.504270 + 0.291140i
\(969\) 0 0
\(970\) 0 0
\(971\) −19.3560 33.5256i −0.621163 1.07589i −0.989269 0.146103i \(-0.953327\pi\)
0.368106 0.929784i \(-0.380006\pi\)
\(972\) 0 0
\(973\) −1.94228 27.0525i −0.0622667 0.867263i
\(974\) 20.8194i 0.667096i
\(975\) 0 0
\(976\) 2.27035 + 1.31079i 0.0726722 + 0.0419573i
\(977\) −15.7279 9.08052i −0.503181 0.290512i 0.226845 0.973931i \(-0.427159\pi\)
−0.730026 + 0.683419i \(0.760492\pi\)
\(978\) 0 0
\(979\) 43.0060i 1.37448i
\(980\) 0 0
\(981\) 0 0
\(982\) −13.6635 23.6659i −0.436021 0.755211i
\(983\) −8.19988 + 14.2026i −0.261536 + 0.452993i −0.966650 0.256100i \(-0.917562\pi\)
0.705115 + 0.709093i \(0.250896\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −40.0399 −1.27513
\(987\) 0 0
\(988\) −7.20977 −0.229373
\(989\) −1.36068 + 0.785587i −0.0432670 + 0.0249802i
\(990\) 0 0
\(991\) 5.44584 9.43247i 0.172993 0.299632i −0.766472 0.642278i \(-0.777990\pi\)
0.939465 + 0.342645i \(0.111323\pi\)
\(992\) 2.41421 + 4.18154i 0.0766514 + 0.132764i
\(993\) 0 0
\(994\) 7.06574 + 10.4335i 0.224112 + 0.330930i
\(995\) 0 0
\(996\) 0 0
\(997\) −30.7856 17.7740i −0.974988 0.562910i −0.0742349 0.997241i \(-0.523651\pi\)
−0.900753 + 0.434331i \(0.856985\pi\)
\(998\) 28.8909 + 16.6802i 0.914527 + 0.528002i
\(999\) 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 3150.2.bf.b.1151.2 8
3.2 odd 2 3150.2.bf.c.1151.4 8
5.2 odd 4 3150.2.bp.a.899.3 8
5.3 odd 4 3150.2.bp.d.899.2 8
5.4 even 2 630.2.be.a.521.3 yes 8
7.5 odd 6 3150.2.bf.c.1601.4 8
15.2 even 4 3150.2.bp.f.899.3 8
15.8 even 4 3150.2.bp.c.899.2 8
15.14 odd 2 630.2.be.b.521.1 yes 8
21.5 even 6 inner 3150.2.bf.b.1601.2 8
35.4 even 6 4410.2.b.e.881.8 8
35.12 even 12 3150.2.bp.c.1349.2 8
35.19 odd 6 630.2.be.b.341.1 yes 8
35.24 odd 6 4410.2.b.b.881.8 8
35.33 even 12 3150.2.bp.f.1349.3 8
105.47 odd 12 3150.2.bp.d.1349.2 8
105.59 even 6 4410.2.b.e.881.1 8
105.68 odd 12 3150.2.bp.a.1349.3 8
105.74 odd 6 4410.2.b.b.881.1 8
105.89 even 6 630.2.be.a.341.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.be.a.341.3 8 105.89 even 6
630.2.be.a.521.3 yes 8 5.4 even 2
630.2.be.b.341.1 yes 8 35.19 odd 6
630.2.be.b.521.1 yes 8 15.14 odd 2
3150.2.bf.b.1151.2 8 1.1 even 1 trivial
3150.2.bf.b.1601.2 8 21.5 even 6 inner
3150.2.bf.c.1151.4 8 3.2 odd 2
3150.2.bf.c.1601.4 8 7.5 odd 6
3150.2.bp.a.899.3 8 5.2 odd 4
3150.2.bp.a.1349.3 8 105.68 odd 12
3150.2.bp.c.899.2 8 15.8 even 4
3150.2.bp.c.1349.2 8 35.12 even 12
3150.2.bp.d.899.2 8 5.3 odd 4
3150.2.bp.d.1349.2 8 105.47 odd 12
3150.2.bp.f.899.3 8 15.2 even 4
3150.2.bp.f.1349.3 8 35.33 even 12
4410.2.b.b.881.1 8 105.74 odd 6
4410.2.b.b.881.8 8 35.24 odd 6
4410.2.b.e.881.1 8 105.59 even 6
4410.2.b.e.881.8 8 35.4 even 6