Properties

Label 192.9.e.e.65.1
Level $192$
Weight $9$
Character 192.65
Analytic conductor $78.217$
Analytic rank $0$
Dimension $2$
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] = [192,9,Mod(65,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.e (of order \(2\), degree \(1\), not 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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-3.74166i\) of defining polynomial
Character \(\chi\) \(=\) 192.65
Dual form 192.9.e.e.65.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-45.0000 - 67.3498i) q^{3} -224.499i q^{5} -1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +O(q^{10})\) \(q+(-45.0000 - 67.3498i) q^{3} -224.499i q^{5} -1750.00 q^{7} +(-2511.00 + 6061.48i) q^{9} +6959.48i q^{11} -25730.0 q^{13} +(-15120.0 + 10102.5i) q^{15} -74893.0i q^{17} -18938.0 q^{19} +(78750.0 + 117862. i) q^{21} +470461. i q^{23} +340225. q^{25} +(521235. - 103651. i) q^{27} +460897. i q^{29} -351478. q^{31} +(468720. - 313177. i) q^{33} +392874. i q^{35} -1.33517e6 q^{37} +(1.15785e6 + 1.73291e6i) q^{39} -1.87547e6i q^{41} +3.52615e6 q^{43} +(1.36080e6 + 563718. i) q^{45} +4.08104e6i q^{47} -2.70230e6 q^{49} +(-5.04403e6 + 3.37019e6i) q^{51} -6.60177e6i q^{53} +1.56240e6 q^{55} +(852210. + 1.27547e6i) q^{57} +1.37149e7i q^{59} -753602. q^{61} +(4.39425e6 - 1.06076e7i) q^{63} +5.77637e6i q^{65} -2.26889e6 q^{67} +(3.16855e7 - 2.11707e7i) q^{69} -1.70220e7i q^{71} +2.76728e7 q^{73} +(-1.53101e7 - 2.29141e7i) q^{75} -1.21791e7i q^{77} -2.29810e7 q^{79} +(-3.04365e7 - 3.04408e7i) q^{81} -4.63952e7i q^{83} -1.68134e7 q^{85} +(3.10414e7 - 2.07404e7i) q^{87} -7.26152e7i q^{89} +4.50275e7 q^{91} +(1.58165e7 + 2.36720e7i) q^{93} +4.25157e6i q^{95} +1.47271e8 q^{97} +(-4.21848e7 - 1.74753e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 90 q^{3} - 3500 q^{7} - 5022 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 90 q^{3} - 3500 q^{7} - 5022 q^{9} - 51460 q^{13} - 30240 q^{15} - 37876 q^{19} + 157500 q^{21} + 680450 q^{25} + 1042470 q^{27} - 702956 q^{31} + 937440 q^{33} - 2670340 q^{37} + 2315700 q^{39} + 7052300 q^{43} + 2721600 q^{45} - 5404602 q^{49} - 10088064 q^{51} + 3124800 q^{55} + 1704420 q^{57} - 1507204 q^{61} + 8788500 q^{63} - 4537780 q^{67} + 63370944 q^{69} + 55345540 q^{73} - 30620250 q^{75} - 45961964 q^{79} - 60872958 q^{81} - 33626880 q^{85} + 62082720 q^{87} + 90055000 q^{91} + 31633020 q^{93} + 294542020 q^{97} - 84369600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −45.0000 67.3498i −0.555556 0.831479i
\(4\) 0 0
\(5\) 224.499i 0.359199i −0.983740 0.179600i \(-0.942520\pi\)
0.983740 0.179600i \(-0.0574802\pi\)
\(6\) 0 0
\(7\) −1750.00 −0.728863 −0.364431 0.931230i \(-0.618737\pi\)
−0.364431 + 0.931230i \(0.618737\pi\)
\(8\) 0 0
\(9\) −2511.00 + 6061.48i −0.382716 + 0.923866i
\(10\) 0 0
\(11\) 6959.48i 0.475342i 0.971346 + 0.237671i \(0.0763840\pi\)
−0.971346 + 0.237671i \(0.923616\pi\)
\(12\) 0 0
\(13\) −25730.0 −0.900879 −0.450439 0.892807i \(-0.648733\pi\)
−0.450439 + 0.892807i \(0.648733\pi\)
\(14\) 0 0
\(15\) −15120.0 + 10102.5i −0.298667 + 0.199555i
\(16\) 0 0
\(17\) 74893.0i 0.896697i −0.893859 0.448348i \(-0.852012\pi\)
0.893859 0.448348i \(-0.147988\pi\)
\(18\) 0 0
\(19\) −18938.0 −0.145318 −0.0726590 0.997357i \(-0.523148\pi\)
−0.0726590 + 0.997357i \(0.523148\pi\)
\(20\) 0 0
\(21\) 78750.0 + 117862.i 0.404924 + 0.606035i
\(22\) 0 0
\(23\) 470461.i 1.68117i 0.541678 + 0.840586i \(0.317789\pi\)
−0.541678 + 0.840586i \(0.682211\pi\)
\(24\) 0 0
\(25\) 340225. 0.870976
\(26\) 0 0
\(27\) 521235. 103651.i 0.980796 0.195038i
\(28\) 0 0
\(29\) 460897.i 0.651647i 0.945431 + 0.325823i \(0.105641\pi\)
−0.945431 + 0.325823i \(0.894359\pi\)
\(30\) 0 0
\(31\) −351478. −0.380585 −0.190292 0.981727i \(-0.560944\pi\)
−0.190292 + 0.981727i \(0.560944\pi\)
\(32\) 0 0
\(33\) 468720. 313177.i 0.395237 0.264079i
\(34\) 0 0
\(35\) 392874.i 0.261807i
\(36\) 0 0
\(37\) −1.33517e6 −0.712409 −0.356205 0.934408i \(-0.615929\pi\)
−0.356205 + 0.934408i \(0.615929\pi\)
\(38\) 0 0
\(39\) 1.15785e6 + 1.73291e6i 0.500488 + 0.749062i
\(40\) 0 0
\(41\) 1.87547e6i 0.663704i −0.943331 0.331852i \(-0.892327\pi\)
0.943331 0.331852i \(-0.107673\pi\)
\(42\) 0 0
\(43\) 3.52615e6 1.03140 0.515700 0.856769i \(-0.327532\pi\)
0.515700 + 0.856769i \(0.327532\pi\)
\(44\) 0 0
\(45\) 1.36080e6 + 563718.i 0.331852 + 0.137471i
\(46\) 0 0
\(47\) 4.08104e6i 0.836333i 0.908370 + 0.418167i \(0.137327\pi\)
−0.908370 + 0.418167i \(0.862673\pi\)
\(48\) 0 0
\(49\) −2.70230e6 −0.468759
\(50\) 0 0
\(51\) −5.04403e6 + 3.37019e6i −0.745585 + 0.498165i
\(52\) 0 0
\(53\) 6.60177e6i 0.836675i −0.908292 0.418337i \(-0.862613\pi\)
0.908292 0.418337i \(-0.137387\pi\)
\(54\) 0 0
\(55\) 1.56240e6 0.170742
\(56\) 0 0
\(57\) 852210. + 1.27547e6i 0.0807323 + 0.120829i
\(58\) 0 0
\(59\) 1.37149e7i 1.13184i 0.824461 + 0.565919i \(0.191479\pi\)
−0.824461 + 0.565919i \(0.808521\pi\)
\(60\) 0 0
\(61\) −753602. −0.0544280 −0.0272140 0.999630i \(-0.508664\pi\)
−0.0272140 + 0.999630i \(0.508664\pi\)
\(62\) 0 0
\(63\) 4.39425e6 1.06076e7i 0.278948 0.673372i
\(64\) 0 0
\(65\) 5.77637e6i 0.323595i
\(66\) 0 0
\(67\) −2.26889e6 −0.112594 −0.0562969 0.998414i \(-0.517929\pi\)
−0.0562969 + 0.998414i \(0.517929\pi\)
\(68\) 0 0
\(69\) 3.16855e7 2.11707e7i 1.39786 0.933985i
\(70\) 0 0
\(71\) 1.70220e7i 0.669849i −0.942245 0.334925i \(-0.891289\pi\)
0.942245 0.334925i \(-0.108711\pi\)
\(72\) 0 0
\(73\) 2.76728e7 0.974454 0.487227 0.873275i \(-0.338008\pi\)
0.487227 + 0.873275i \(0.338008\pi\)
\(74\) 0 0
\(75\) −1.53101e7 2.29141e7i −0.483876 0.724199i
\(76\) 0 0
\(77\) 1.21791e7i 0.346459i
\(78\) 0 0
\(79\) −2.29810e7 −0.590011 −0.295006 0.955496i \(-0.595322\pi\)
−0.295006 + 0.955496i \(0.595322\pi\)
\(80\) 0 0
\(81\) −3.04365e7 3.04408e7i −0.707057 0.707157i
\(82\) 0 0
\(83\) 4.63952e7i 0.977599i −0.872396 0.488799i \(-0.837435\pi\)
0.872396 0.488799i \(-0.162565\pi\)
\(84\) 0 0
\(85\) −1.68134e7 −0.322093
\(86\) 0 0
\(87\) 3.10414e7 2.07404e7i 0.541831 0.362026i
\(88\) 0 0
\(89\) 7.26152e7i 1.15736i −0.815555 0.578679i \(-0.803568\pi\)
0.815555 0.578679i \(-0.196432\pi\)
\(90\) 0 0
\(91\) 4.50275e7 0.656617
\(92\) 0 0
\(93\) 1.58165e7 + 2.36720e7i 0.211436 + 0.316448i
\(94\) 0 0
\(95\) 4.25157e6i 0.0521981i
\(96\) 0 0
\(97\) 1.47271e8 1.66353 0.831764 0.555129i \(-0.187331\pi\)
0.831764 + 0.555129i \(0.187331\pi\)
\(98\) 0 0
\(99\) −4.21848e7 1.74753e7i −0.439152 0.181921i
\(100\) 0 0
\(101\) 1.03545e8i 0.995045i −0.867451 0.497522i \(-0.834243\pi\)
0.867451 0.497522i \(-0.165757\pi\)
\(102\) 0 0
\(103\) −1.66064e8 −1.47545 −0.737726 0.675100i \(-0.764101\pi\)
−0.737726 + 0.675100i \(0.764101\pi\)
\(104\) 0 0
\(105\) 2.64600e7 1.76793e7i 0.217687 0.145448i
\(106\) 0 0
\(107\) 2.25540e7i 0.172063i −0.996292 0.0860316i \(-0.972581\pi\)
0.996292 0.0860316i \(-0.0274186\pi\)
\(108\) 0 0
\(109\) 1.09975e8 0.779091 0.389546 0.921007i \(-0.372632\pi\)
0.389546 + 0.921007i \(0.372632\pi\)
\(110\) 0 0
\(111\) 6.00826e7 + 8.99235e7i 0.395783 + 0.592354i
\(112\) 0 0
\(113\) 2.87748e8i 1.76481i −0.470490 0.882405i \(-0.655923\pi\)
0.470490 0.882405i \(-0.344077\pi\)
\(114\) 0 0
\(115\) 1.05618e8 0.603876
\(116\) 0 0
\(117\) 6.46080e7 1.55962e8i 0.344781 0.832291i
\(118\) 0 0
\(119\) 1.31063e8i 0.653569i
\(120\) 0 0
\(121\) 1.65924e8 0.774050
\(122\) 0 0
\(123\) −1.26312e8 + 8.43961e7i −0.551856 + 0.368724i
\(124\) 0 0
\(125\) 1.64075e8i 0.672053i
\(126\) 0 0
\(127\) 2.75994e8 1.06092 0.530462 0.847708i \(-0.322018\pi\)
0.530462 + 0.847708i \(0.322018\pi\)
\(128\) 0 0
\(129\) −1.58677e8 2.37486e8i −0.573000 0.857588i
\(130\) 0 0
\(131\) 2.89118e8i 0.981725i 0.871237 + 0.490862i \(0.163318\pi\)
−0.871237 + 0.490862i \(0.836682\pi\)
\(132\) 0 0
\(133\) 3.31415e7 0.105917
\(134\) 0 0
\(135\) −2.32697e7 1.17017e8i −0.0700576 0.352301i
\(136\) 0 0
\(137\) 2.07562e8i 0.589205i 0.955620 + 0.294602i \(0.0951872\pi\)
−0.955620 + 0.294602i \(0.904813\pi\)
\(138\) 0 0
\(139\) 1.42668e8 0.382180 0.191090 0.981573i \(-0.438798\pi\)
0.191090 + 0.981573i \(0.438798\pi\)
\(140\) 0 0
\(141\) 2.74857e8 1.83647e8i 0.695394 0.464630i
\(142\) 0 0
\(143\) 1.79067e8i 0.428226i
\(144\) 0 0
\(145\) 1.03471e8 0.234071
\(146\) 0 0
\(147\) 1.21604e8 + 1.82000e8i 0.260422 + 0.389763i
\(148\) 0 0
\(149\) 8.19236e8i 1.66213i −0.556179 0.831063i \(-0.687733\pi\)
0.556179 0.831063i \(-0.312267\pi\)
\(150\) 0 0
\(151\) 4.23861e8 0.815296 0.407648 0.913139i \(-0.366349\pi\)
0.407648 + 0.913139i \(0.366349\pi\)
\(152\) 0 0
\(153\) 4.53963e8 + 1.88056e8i 0.828428 + 0.343180i
\(154\) 0 0
\(155\) 7.89066e7i 0.136706i
\(156\) 0 0
\(157\) 7.59851e8 1.25063 0.625316 0.780371i \(-0.284970\pi\)
0.625316 + 0.780371i \(0.284970\pi\)
\(158\) 0 0
\(159\) −4.44628e8 + 2.97079e8i −0.695678 + 0.464819i
\(160\) 0 0
\(161\) 8.23307e8i 1.22534i
\(162\) 0 0
\(163\) −6.68160e8 −0.946520 −0.473260 0.880923i \(-0.656923\pi\)
−0.473260 + 0.880923i \(0.656923\pi\)
\(164\) 0 0
\(165\) −7.03080e7 1.05227e8i −0.0948569 0.141969i
\(166\) 0 0
\(167\) 1.96306e8i 0.252387i −0.992006 0.126194i \(-0.959724\pi\)
0.992006 0.126194i \(-0.0402761\pi\)
\(168\) 0 0
\(169\) −1.53698e8 −0.188417
\(170\) 0 0
\(171\) 4.75533e7 1.14792e8i 0.0556156 0.134254i
\(172\) 0 0
\(173\) 1.02319e9i 1.14228i 0.820852 + 0.571141i \(0.193499\pi\)
−0.820852 + 0.571141i \(0.806501\pi\)
\(174\) 0 0
\(175\) −5.95394e8 −0.634822
\(176\) 0 0
\(177\) 9.23696e8 6.17170e8i 0.941100 0.628799i
\(178\) 0 0
\(179\) 1.28895e9i 1.25552i −0.778408 0.627759i \(-0.783972\pi\)
0.778408 0.627759i \(-0.216028\pi\)
\(180\) 0 0
\(181\) −4.71707e8 −0.439499 −0.219749 0.975556i \(-0.570524\pi\)
−0.219749 + 0.975556i \(0.570524\pi\)
\(182\) 0 0
\(183\) 3.39121e7 + 5.07550e7i 0.0302378 + 0.0452558i
\(184\) 0 0
\(185\) 2.99745e8i 0.255897i
\(186\) 0 0
\(187\) 5.21217e8 0.426238
\(188\) 0 0
\(189\) −9.12161e8 + 1.81390e8i −0.714866 + 0.142156i
\(190\) 0 0
\(191\) 1.61787e8i 0.121565i 0.998151 + 0.0607827i \(0.0193597\pi\)
−0.998151 + 0.0607827i \(0.980640\pi\)
\(192\) 0 0
\(193\) −1.58840e9 −1.14480 −0.572401 0.819974i \(-0.693988\pi\)
−0.572401 + 0.819974i \(0.693988\pi\)
\(194\) 0 0
\(195\) 3.89038e8 2.59937e8i 0.269062 0.179775i
\(196\) 0 0
\(197\) 5.37769e8i 0.357052i −0.983935 0.178526i \(-0.942867\pi\)
0.983935 0.178526i \(-0.0571328\pi\)
\(198\) 0 0
\(199\) 6.47586e8 0.412938 0.206469 0.978453i \(-0.433803\pi\)
0.206469 + 0.978453i \(0.433803\pi\)
\(200\) 0 0
\(201\) 1.02100e8 + 1.52809e8i 0.0625521 + 0.0936194i
\(202\) 0 0
\(203\) 8.06570e8i 0.474961i
\(204\) 0 0
\(205\) −4.21042e8 −0.238402
\(206\) 0 0
\(207\) −2.85169e9 1.18133e9i −1.55318 0.643412i
\(208\) 0 0
\(209\) 1.31799e8i 0.0690758i
\(210\) 0 0
\(211\) −5.81104e7 −0.0293173 −0.0146586 0.999893i \(-0.504666\pi\)
−0.0146586 + 0.999893i \(0.504666\pi\)
\(212\) 0 0
\(213\) −1.14643e9 + 7.65990e8i −0.556966 + 0.372139i
\(214\) 0 0
\(215\) 7.91619e8i 0.370478i
\(216\) 0 0
\(217\) 6.15086e8 0.277394
\(218\) 0 0
\(219\) −1.24527e9 1.86376e9i −0.541363 0.810238i
\(220\) 0 0
\(221\) 1.92700e9i 0.807815i
\(222\) 0 0
\(223\) 4.40200e9 1.78004 0.890021 0.455920i \(-0.150690\pi\)
0.890021 + 0.455920i \(0.150690\pi\)
\(224\) 0 0
\(225\) −8.54305e8 + 2.06227e9i −0.333336 + 0.804665i
\(226\) 0 0
\(227\) 3.53592e9i 1.33168i 0.746095 + 0.665839i \(0.231926\pi\)
−0.746095 + 0.665839i \(0.768074\pi\)
\(228\) 0 0
\(229\) 1.86569e9 0.678420 0.339210 0.940711i \(-0.389840\pi\)
0.339210 + 0.940711i \(0.389840\pi\)
\(230\) 0 0
\(231\) −8.20260e8 + 5.48059e8i −0.288074 + 0.192477i
\(232\) 0 0
\(233\) 2.72132e9i 0.923328i 0.887055 + 0.461664i \(0.152747\pi\)
−0.887055 + 0.461664i \(0.847253\pi\)
\(234\) 0 0
\(235\) 9.16191e8 0.300410
\(236\) 0 0
\(237\) 1.03414e9 + 1.54777e9i 0.327784 + 0.490582i
\(238\) 0 0
\(239\) 2.27461e9i 0.697132i 0.937284 + 0.348566i \(0.113331\pi\)
−0.937284 + 0.348566i \(0.886669\pi\)
\(240\) 0 0
\(241\) −1.74667e9 −0.517778 −0.258889 0.965907i \(-0.583356\pi\)
−0.258889 + 0.965907i \(0.583356\pi\)
\(242\) 0 0
\(243\) −6.80540e8 + 3.41973e9i −0.195177 + 0.980768i
\(244\) 0 0
\(245\) 6.06665e8i 0.168378i
\(246\) 0 0
\(247\) 4.87275e8 0.130914
\(248\) 0 0
\(249\) −3.12471e9 + 2.08778e9i −0.812853 + 0.543110i
\(250\) 0 0
\(251\) 1.37549e9i 0.346547i 0.984874 + 0.173274i \(0.0554345\pi\)
−0.984874 + 0.173274i \(0.944566\pi\)
\(252\) 0 0
\(253\) −3.27417e9 −0.799132
\(254\) 0 0
\(255\) 7.56605e8 + 1.13238e9i 0.178940 + 0.267813i
\(256\) 0 0
\(257\) 7.93672e9i 1.81932i 0.415356 + 0.909659i \(0.363657\pi\)
−0.415356 + 0.909659i \(0.636343\pi\)
\(258\) 0 0
\(259\) 2.33655e9 0.519249
\(260\) 0 0
\(261\) −2.79372e9 1.15731e9i −0.602034 0.249396i
\(262\) 0 0
\(263\) 3.22555e8i 0.0674187i −0.999432 0.0337093i \(-0.989268\pi\)
0.999432 0.0337093i \(-0.0107320\pi\)
\(264\) 0 0
\(265\) −1.48209e9 −0.300533
\(266\) 0 0
\(267\) −4.89062e9 + 3.26769e9i −0.962319 + 0.642977i
\(268\) 0 0
\(269\) 3.47314e9i 0.663304i 0.943402 + 0.331652i \(0.107606\pi\)
−0.943402 + 0.331652i \(0.892394\pi\)
\(270\) 0 0
\(271\) −1.44216e9 −0.267385 −0.133693 0.991023i \(-0.542683\pi\)
−0.133693 + 0.991023i \(0.542683\pi\)
\(272\) 0 0
\(273\) −2.02624e9 3.03259e9i −0.364787 0.545964i
\(274\) 0 0
\(275\) 2.36779e9i 0.414012i
\(276\) 0 0
\(277\) −3.38046e9 −0.574192 −0.287096 0.957902i \(-0.592690\pi\)
−0.287096 + 0.957902i \(0.592690\pi\)
\(278\) 0 0
\(279\) 8.82561e8 2.13048e9i 0.145656 0.351609i
\(280\) 0 0
\(281\) 4.02262e9i 0.645184i −0.946538 0.322592i \(-0.895446\pi\)
0.946538 0.322592i \(-0.104554\pi\)
\(282\) 0 0
\(283\) 1.04253e10 1.62533 0.812666 0.582730i \(-0.198016\pi\)
0.812666 + 0.582730i \(0.198016\pi\)
\(284\) 0 0
\(285\) 2.86343e8 1.91321e8i 0.0434017 0.0289990i
\(286\) 0 0
\(287\) 3.28207e9i 0.483749i
\(288\) 0 0
\(289\) 1.36679e9 0.195935
\(290\) 0 0
\(291\) −6.62720e9 9.91868e9i −0.924183 1.38319i
\(292\) 0 0
\(293\) 1.03927e10i 1.41012i 0.709146 + 0.705061i \(0.249081\pi\)
−0.709146 + 0.705061i \(0.750919\pi\)
\(294\) 0 0
\(295\) 3.07899e9 0.406555
\(296\) 0 0
\(297\) 7.21360e8 + 3.62753e9i 0.0927099 + 0.466213i
\(298\) 0 0
\(299\) 1.21050e10i 1.51453i
\(300\) 0 0
\(301\) −6.17076e9 −0.751749
\(302\) 0 0
\(303\) −6.97372e9 + 4.65951e9i −0.827359 + 0.552803i
\(304\) 0 0
\(305\) 1.69183e8i 0.0195505i
\(306\) 0 0
\(307\) 2.99309e9 0.336951 0.168476 0.985706i \(-0.446116\pi\)
0.168476 + 0.985706i \(0.446116\pi\)
\(308\) 0 0
\(309\) 7.47286e9 + 1.11843e10i 0.819696 + 1.22681i
\(310\) 0 0
\(311\) 6.44832e9i 0.689295i −0.938732 0.344647i \(-0.887998\pi\)
0.938732 0.344647i \(-0.112002\pi\)
\(312\) 0 0
\(313\) 3.27737e7 0.00341467 0.00170733 0.999999i \(-0.499457\pi\)
0.00170733 + 0.999999i \(0.499457\pi\)
\(314\) 0 0
\(315\) −2.38140e9 9.86507e8i −0.241875 0.100198i
\(316\) 0 0
\(317\) 1.17797e10i 1.16653i −0.812282 0.583264i \(-0.801775\pi\)
0.812282 0.583264i \(-0.198225\pi\)
\(318\) 0 0
\(319\) −3.20761e9 −0.309755
\(320\) 0 0
\(321\) −1.51901e9 + 1.01493e9i −0.143067 + 0.0955907i
\(322\) 0 0
\(323\) 1.41832e9i 0.130306i
\(324\) 0 0
\(325\) −8.75399e9 −0.784644
\(326\) 0 0
\(327\) −4.94888e9 7.40680e9i −0.432829 0.647798i
\(328\) 0 0
\(329\) 7.14182e9i 0.609573i
\(330\) 0 0
\(331\) 1.20100e10 1.00053 0.500265 0.865872i \(-0.333236\pi\)
0.500265 + 0.865872i \(0.333236\pi\)
\(332\) 0 0
\(333\) 3.35261e9 8.09311e9i 0.272651 0.658171i
\(334\) 0 0
\(335\) 5.09365e8i 0.0404436i
\(336\) 0 0
\(337\) 1.59214e10 1.23441 0.617207 0.786801i \(-0.288264\pi\)
0.617207 + 0.786801i \(0.288264\pi\)
\(338\) 0 0
\(339\) −1.93798e10 + 1.29486e10i −1.46740 + 0.980451i
\(340\) 0 0
\(341\) 2.44611e9i 0.180908i
\(342\) 0 0
\(343\) 1.48174e10 1.07052
\(344\) 0 0
\(345\) −4.75282e9 7.11337e9i −0.335487 0.502110i
\(346\) 0 0
\(347\) 4.94792e9i 0.341275i −0.985334 0.170638i \(-0.945417\pi\)
0.985334 0.170638i \(-0.0545828\pi\)
\(348\) 0 0
\(349\) 7.29567e9 0.491772 0.245886 0.969299i \(-0.420921\pi\)
0.245886 + 0.969299i \(0.420921\pi\)
\(350\) 0 0
\(351\) −1.34114e10 + 2.66695e9i −0.883578 + 0.175706i
\(352\) 0 0
\(353\) 6.93875e9i 0.446871i 0.974719 + 0.223436i \(0.0717272\pi\)
−0.974719 + 0.223436i \(0.928273\pi\)
\(354\) 0 0
\(355\) −3.82143e9 −0.240609
\(356\) 0 0
\(357\) 8.82706e9 5.89782e9i 0.543429 0.363094i
\(358\) 0 0
\(359\) 1.60096e10i 0.963838i −0.876216 0.481919i \(-0.839940\pi\)
0.876216 0.481919i \(-0.160060\pi\)
\(360\) 0 0
\(361\) −1.66249e10 −0.978883
\(362\) 0 0
\(363\) −7.46660e9 1.11750e10i −0.430028 0.643607i
\(364\) 0 0
\(365\) 6.21252e9i 0.350023i
\(366\) 0 0
\(367\) 1.36364e10 0.751686 0.375843 0.926683i \(-0.377353\pi\)
0.375843 + 0.926683i \(0.377353\pi\)
\(368\) 0 0
\(369\) 1.13681e10 + 4.70930e9i 0.613173 + 0.254010i
\(370\) 0 0
\(371\) 1.15531e10i 0.609821i
\(372\) 0 0
\(373\) 2.44062e10 1.26085 0.630427 0.776248i \(-0.282880\pi\)
0.630427 + 0.776248i \(0.282880\pi\)
\(374\) 0 0
\(375\) −1.10505e10 + 7.38339e9i −0.558798 + 0.373363i
\(376\) 0 0
\(377\) 1.18589e10i 0.587055i
\(378\) 0 0
\(379\) 1.98392e10 0.961542 0.480771 0.876846i \(-0.340357\pi\)
0.480771 + 0.876846i \(0.340357\pi\)
\(380\) 0 0
\(381\) −1.24197e10 1.85881e10i −0.589403 0.882137i
\(382\) 0 0
\(383\) 1.51133e10i 0.702366i 0.936307 + 0.351183i \(0.114220\pi\)
−0.936307 + 0.351183i \(0.885780\pi\)
\(384\) 0 0
\(385\) −2.73420e9 −0.124448
\(386\) 0 0
\(387\) −8.85416e9 + 2.13737e10i −0.394733 + 0.952875i
\(388\) 0 0
\(389\) 1.79991e10i 0.786056i −0.919527 0.393028i \(-0.871428\pi\)
0.919527 0.393028i \(-0.128572\pi\)
\(390\) 0 0
\(391\) 3.52342e10 1.50750
\(392\) 0 0
\(393\) 1.94720e10 1.30103e10i 0.816284 0.545403i
\(394\) 0 0
\(395\) 5.15922e9i 0.211931i
\(396\) 0 0
\(397\) 2.35673e10 0.948739 0.474370 0.880326i \(-0.342676\pi\)
0.474370 + 0.880326i \(0.342676\pi\)
\(398\) 0 0
\(399\) −1.49137e9 2.23207e9i −0.0588428 0.0880678i
\(400\) 0 0
\(401\) 1.37692e10i 0.532515i −0.963902 0.266257i \(-0.914213\pi\)
0.963902 0.266257i \(-0.0857871\pi\)
\(402\) 0 0
\(403\) 9.04353e9 0.342861
\(404\) 0 0
\(405\) −6.83394e9 + 6.83297e9i −0.254010 + 0.253974i
\(406\) 0 0
\(407\) 9.29209e9i 0.338638i
\(408\) 0 0
\(409\) 3.58480e10 1.28107 0.640533 0.767931i \(-0.278714\pi\)
0.640533 + 0.767931i \(0.278714\pi\)
\(410\) 0 0
\(411\) 1.39793e10 9.34031e9i 0.489912 0.327336i
\(412\) 0 0
\(413\) 2.40011e10i 0.824955i
\(414\) 0 0
\(415\) −1.04157e10 −0.351153
\(416\) 0 0
\(417\) −6.42007e9 9.60868e9i −0.212322 0.317775i
\(418\) 0 0
\(419\) 2.23996e10i 0.726750i 0.931643 + 0.363375i \(0.118376\pi\)
−0.931643 + 0.363375i \(0.881624\pi\)
\(420\) 0 0
\(421\) 1.49535e10 0.476008 0.238004 0.971264i \(-0.423507\pi\)
0.238004 + 0.971264i \(0.423507\pi\)
\(422\) 0 0
\(423\) −2.47372e10 1.02475e10i −0.772660 0.320078i
\(424\) 0 0
\(425\) 2.54805e10i 0.781001i
\(426\) 0 0
\(427\) 1.31880e9 0.0396706
\(428\) 0 0
\(429\) −1.20602e10 + 8.05804e9i −0.356061 + 0.237903i
\(430\) 0 0
\(431\) 6.40436e10i 1.85595i 0.372640 + 0.927976i \(0.378453\pi\)
−0.372640 + 0.927976i \(0.621547\pi\)
\(432\) 0 0
\(433\) −5.22954e9 −0.148769 −0.0743843 0.997230i \(-0.523699\pi\)
−0.0743843 + 0.997230i \(0.523699\pi\)
\(434\) 0 0
\(435\) −4.65620e9 6.96877e9i −0.130039 0.194625i
\(436\) 0 0
\(437\) 8.90959e9i 0.244305i
\(438\) 0 0
\(439\) 4.34801e10 1.17066 0.585332 0.810793i \(-0.300964\pi\)
0.585332 + 0.810793i \(0.300964\pi\)
\(440\) 0 0
\(441\) 6.78548e9 1.63800e10i 0.179402 0.433070i
\(442\) 0 0
\(443\) 3.78737e10i 0.983383i −0.870770 0.491691i \(-0.836379\pi\)
0.870770 0.491691i \(-0.163621\pi\)
\(444\) 0 0
\(445\) −1.63021e10 −0.415722
\(446\) 0 0
\(447\) −5.51754e10 + 3.68656e10i −1.38202 + 0.923403i
\(448\) 0 0
\(449\) 2.95505e10i 0.727076i −0.931579 0.363538i \(-0.881569\pi\)
0.931579 0.363538i \(-0.118431\pi\)
\(450\) 0 0
\(451\) 1.30523e10 0.315486
\(452\) 0 0
\(453\) −1.90737e10 2.85469e10i −0.452942 0.677902i
\(454\) 0 0
\(455\) 1.01086e10i 0.235856i
\(456\) 0 0
\(457\) −2.02181e10 −0.463529 −0.231764 0.972772i \(-0.574450\pi\)
−0.231764 + 0.972772i \(0.574450\pi\)
\(458\) 0 0
\(459\) −7.76277e9 3.90369e10i −0.174890 0.879476i
\(460\) 0 0
\(461\) 7.01826e10i 1.55391i 0.629556 + 0.776955i \(0.283237\pi\)
−0.629556 + 0.776955i \(0.716763\pi\)
\(462\) 0 0
\(463\) 4.16009e9 0.0905271 0.0452635 0.998975i \(-0.485587\pi\)
0.0452635 + 0.998975i \(0.485587\pi\)
\(464\) 0 0
\(465\) 5.31435e9 3.55080e9i 0.113668 0.0759476i
\(466\) 0 0
\(467\) 2.88138e10i 0.605806i 0.953021 + 0.302903i \(0.0979558\pi\)
−0.953021 + 0.302903i \(0.902044\pi\)
\(468\) 0 0
\(469\) 3.97056e9 0.0820654
\(470\) 0 0
\(471\) −3.41933e10 5.11758e10i −0.694796 1.03988i
\(472\) 0 0
\(473\) 2.45402e10i 0.490268i
\(474\) 0 0
\(475\) −6.44318e9 −0.126569
\(476\) 0 0
\(477\) 4.00165e10 + 1.65770e10i 0.772975 + 0.320209i
\(478\) 0 0
\(479\) 4.47149e10i 0.849395i −0.905335 0.424698i \(-0.860380\pi\)
0.905335 0.424698i \(-0.139620\pi\)
\(480\) 0 0
\(481\) 3.43539e10 0.641795
\(482\) 0 0
\(483\) −5.54496e10 + 3.70488e10i −1.01885 + 0.680747i
\(484\) 0 0
\(485\) 3.30623e10i 0.597538i
\(486\) 0 0
\(487\) 5.72836e10 1.01839 0.509195 0.860651i \(-0.329943\pi\)
0.509195 + 0.860651i \(0.329943\pi\)
\(488\) 0 0
\(489\) 3.00672e10 + 4.50005e10i 0.525845 + 0.787012i
\(490\) 0 0
\(491\) 7.25262e10i 1.24787i −0.781477 0.623934i \(-0.785533\pi\)
0.781477 0.623934i \(-0.214467\pi\)
\(492\) 0 0
\(493\) 3.45180e10 0.584330
\(494\) 0 0
\(495\) −3.92319e9 + 9.47046e9i −0.0653459 + 0.157743i
\(496\) 0 0
\(497\) 2.97885e10i 0.488228i
\(498\) 0 0
\(499\) 2.64368e10 0.426389 0.213195 0.977010i \(-0.431613\pi\)
0.213195 + 0.977010i \(0.431613\pi\)
\(500\) 0 0
\(501\) −1.32212e10 + 8.83376e9i −0.209855 + 0.140215i
\(502\) 0 0
\(503\) 7.52828e10i 1.17604i −0.808845 0.588022i \(-0.799907\pi\)
0.808845 0.588022i \(-0.200093\pi\)
\(504\) 0 0
\(505\) −2.32457e10 −0.357419
\(506\) 0 0
\(507\) 6.91640e9 + 1.03515e10i 0.104676 + 0.156665i
\(508\) 0 0
\(509\) 6.45184e10i 0.961197i −0.876941 0.480599i \(-0.840419\pi\)
0.876941 0.480599i \(-0.159581\pi\)
\(510\) 0 0
\(511\) −4.84273e10 −0.710243
\(512\) 0 0
\(513\) −9.87115e9 + 1.96295e9i −0.142527 + 0.0283426i
\(514\) 0 0
\(515\) 3.72812e10i 0.529981i
\(516\) 0 0
\(517\) −2.84019e10 −0.397544
\(518\) 0 0
\(519\) 6.89119e10 4.60437e10i 0.949783 0.634601i
\(520\) 0 0
\(521\) 7.65146e10i 1.03847i 0.854632 + 0.519235i \(0.173783\pi\)
−0.854632 + 0.519235i \(0.826217\pi\)
\(522\) 0 0
\(523\) 8.46771e10 1.13177 0.565886 0.824483i \(-0.308534\pi\)
0.565886 + 0.824483i \(0.308534\pi\)
\(524\) 0 0
\(525\) 2.67927e10 + 4.00997e10i 0.352679 + 0.527842i
\(526\) 0 0
\(527\) 2.63232e10i 0.341269i
\(528\) 0 0
\(529\) −1.43023e11 −1.82634
\(530\) 0 0
\(531\) −8.31326e10 3.44381e10i −1.04567 0.433173i
\(532\) 0 0
\(533\) 4.82558e10i 0.597917i
\(534\) 0 0
\(535\) −5.06336e9 −0.0618050
\(536\) 0 0
\(537\) −8.68104e10 + 5.80026e10i −1.04394 + 0.697510i
\(538\) 0 0
\(539\) 1.88066e10i 0.222821i
\(540\) 0 0
\(541\) −1.43470e11 −1.67483 −0.837415 0.546568i \(-0.815934\pi\)
−0.837415 + 0.546568i \(0.815934\pi\)
\(542\) 0 0
\(543\) 2.12268e10 + 3.17694e10i 0.244166 + 0.365434i
\(544\) 0 0
\(545\) 2.46893e10i 0.279849i
\(546\) 0 0
\(547\) −1.64171e11 −1.83378 −0.916892 0.399134i \(-0.869311\pi\)
−0.916892 + 0.399134i \(0.869311\pi\)
\(548\) 0 0
\(549\) 1.89229e9 4.56795e9i 0.0208305 0.0502842i
\(550\) 0 0
\(551\) 8.72847e9i 0.0946961i
\(552\) 0 0
\(553\) 4.02167e10 0.430037
\(554\) 0 0
\(555\) 2.01878e10 1.34885e10i 0.212773 0.142165i
\(556\) 0 0
\(557\) 1.54420e11i 1.60429i 0.597130 + 0.802145i \(0.296308\pi\)
−0.597130 + 0.802145i \(0.703692\pi\)
\(558\) 0 0
\(559\) −9.07278e10 −0.929166
\(560\) 0 0
\(561\) −2.34547e10 3.51039e10i −0.236799 0.354408i
\(562\) 0 0
\(563\) 1.54622e11i 1.53900i 0.638646 + 0.769500i \(0.279495\pi\)
−0.638646 + 0.769500i \(0.720505\pi\)
\(564\) 0 0
\(565\) −6.45992e10 −0.633919
\(566\) 0 0
\(567\) 5.32638e10 + 5.32714e10i 0.515348 + 0.515420i
\(568\) 0 0
\(569\) 1.15380e11i 1.10073i 0.834925 + 0.550364i \(0.185511\pi\)
−0.834925 + 0.550364i \(0.814489\pi\)
\(570\) 0 0
\(571\) 1.63410e11 1.53722 0.768608 0.639720i \(-0.220950\pi\)
0.768608 + 0.639720i \(0.220950\pi\)
\(572\) 0 0
\(573\) 1.08963e10 7.28041e9i 0.101079 0.0675363i
\(574\) 0 0
\(575\) 1.60063e11i 1.46426i
\(576\) 0 0
\(577\) 7.42282e10 0.669678 0.334839 0.942275i \(-0.391318\pi\)
0.334839 + 0.942275i \(0.391318\pi\)
\(578\) 0 0
\(579\) 7.14779e10 + 1.06978e11i 0.636001 + 0.951879i
\(580\) 0 0
\(581\) 8.11916e10i 0.712535i
\(582\) 0 0
\(583\) 4.59449e10 0.397707
\(584\) 0 0
\(585\) −3.50134e10 1.45045e10i −0.298958 0.123845i
\(586\) 0 0
\(587\) 6.72877e10i 0.566739i −0.959011 0.283369i \(-0.908548\pi\)
0.959011 0.283369i \(-0.0914523\pi\)
\(588\) 0 0
\(589\) 6.65629e9 0.0553059
\(590\) 0 0
\(591\) −3.62187e10 + 2.41996e10i −0.296881 + 0.198362i
\(592\) 0 0
\(593\) 2.36444e10i 0.191210i 0.995419 + 0.0956048i \(0.0304785\pi\)
−0.995419 + 0.0956048i \(0.969521\pi\)
\(594\) 0 0
\(595\) 2.94235e10 0.234761
\(596\) 0 0
\(597\) −2.91414e10 4.36148e10i −0.229410 0.343350i
\(598\) 0 0
\(599\) 3.03370e10i 0.235649i −0.993034 0.117825i \(-0.962408\pi\)
0.993034 0.117825i \(-0.0375921\pi\)
\(600\) 0 0
\(601\) 3.37911e10 0.259003 0.129501 0.991579i \(-0.458662\pi\)
0.129501 + 0.991579i \(0.458662\pi\)
\(602\) 0 0
\(603\) 5.69718e9 1.37528e10i 0.0430914 0.104022i
\(604\) 0 0
\(605\) 3.72500e10i 0.278038i
\(606\) 0 0
\(607\) −3.82366e10 −0.281660 −0.140830 0.990034i \(-0.544977\pi\)
−0.140830 + 0.990034i \(0.544977\pi\)
\(608\) 0 0
\(609\) −5.43224e10 + 3.62957e10i −0.394920 + 0.263867i
\(610\) 0 0
\(611\) 1.05005e11i 0.753435i
\(612\) 0 0
\(613\) −1.08066e11 −0.765330 −0.382665 0.923887i \(-0.624994\pi\)
−0.382665 + 0.923887i \(0.624994\pi\)
\(614\) 0 0
\(615\) 1.89469e10 + 2.83571e10i 0.132445 + 0.198226i
\(616\) 0 0
\(617\) 4.72538e9i 0.0326059i 0.999867 + 0.0163029i \(0.00518962\pi\)
−0.999867 + 0.0163029i \(0.994810\pi\)
\(618\) 0 0
\(619\) −2.29845e10 −0.156557 −0.0782786 0.996932i \(-0.524942\pi\)
−0.0782786 + 0.996932i \(0.524942\pi\)
\(620\) 0 0
\(621\) 4.87639e10 + 2.45221e11i 0.327893 + 1.64889i
\(622\) 0 0
\(623\) 1.27077e11i 0.843555i
\(624\) 0 0
\(625\) 9.60656e10 0.629575
\(626\) 0 0
\(627\) −8.87662e9 + 5.93094e9i −0.0574351 + 0.0383754i
\(628\) 0 0
\(629\) 9.99949e10i 0.638815i
\(630\) 0 0
\(631\) −1.01892e11 −0.642722 −0.321361 0.946957i \(-0.604140\pi\)
−0.321361 + 0.946957i \(0.604140\pi\)
\(632\) 0 0
\(633\) 2.61497e9 + 3.91372e9i 0.0162874 + 0.0243767i
\(634\) 0 0
\(635\) 6.19605e10i 0.381083i
\(636\) 0 0
\(637\) 6.95302e10 0.422295
\(638\) 0 0
\(639\) 1.03179e11 + 4.27422e10i 0.618851 + 0.256362i
\(640\) 0 0
\(641\) 1.17803e11i 0.697791i −0.937162 0.348896i \(-0.886557\pi\)
0.937162 0.348896i \(-0.113443\pi\)
\(642\) 0 0
\(643\) 2.62680e10 0.153668 0.0768339 0.997044i \(-0.475519\pi\)
0.0768339 + 0.997044i \(0.475519\pi\)
\(644\) 0 0
\(645\) −5.33154e10 + 3.56228e10i −0.308045 + 0.205821i
\(646\) 0 0
\(647\) 3.10527e11i 1.77208i −0.463612 0.886038i \(-0.653447\pi\)
0.463612 0.886038i \(-0.346553\pi\)
\(648\) 0 0
\(649\) −9.54486e10 −0.538010
\(650\) 0 0
\(651\) −2.76789e10 4.14260e10i −0.154108 0.230648i
\(652\) 0 0
\(653\) 3.48345e10i 0.191583i 0.995401 + 0.0957914i \(0.0305382\pi\)
−0.995401 + 0.0957914i \(0.969462\pi\)
\(654\) 0 0
\(655\) 6.49068e10 0.352635
\(656\) 0 0
\(657\) −6.94863e10 + 1.67738e11i −0.372939 + 0.900265i
\(658\) 0 0
\(659\) 3.77120e10i 0.199957i 0.994990 + 0.0999787i \(0.0318775\pi\)
−0.994990 + 0.0999787i \(0.968123\pi\)
\(660\) 0 0
\(661\) 1.39619e11 0.731372 0.365686 0.930738i \(-0.380834\pi\)
0.365686 + 0.930738i \(0.380834\pi\)
\(662\) 0 0
\(663\) 1.29783e11 8.67149e10i 0.671682 0.448786i
\(664\) 0 0
\(665\) 7.44025e9i 0.0380453i
\(666\) 0 0
\(667\) −2.16834e11 −1.09553
\(668\) 0 0
\(669\) −1.98090e11 2.96474e11i −0.988912 1.48007i
\(670\) 0 0
\(671\) 5.24468e9i 0.0258719i
\(672\) 0 0
\(673\) −1.29783e11 −0.632642 −0.316321 0.948652i \(-0.602448\pi\)
−0.316321 + 0.948652i \(0.602448\pi\)
\(674\) 0 0
\(675\) 1.77337e11 3.52648e10i 0.854249 0.169874i
\(676\) 0 0
\(677\) 3.40648e11i 1.62163i −0.585306 0.810813i \(-0.699025\pi\)
0.585306 0.810813i \(-0.300975\pi\)
\(678\) 0 0
\(679\) −2.57724e11 −1.21248
\(680\) 0 0
\(681\) 2.38144e11 1.59117e11i 1.10726 0.739821i
\(682\) 0 0
\(683\) 1.02876e11i 0.472750i 0.971662 + 0.236375i \(0.0759593\pi\)
−0.971662 + 0.236375i \(0.924041\pi\)
\(684\) 0 0
\(685\) 4.65976e10 0.211642
\(686\) 0 0
\(687\) −8.39562e10 1.25654e11i −0.376900 0.564092i
\(688\) 0 0
\(689\) 1.69863e11i 0.753742i
\(690\) 0 0
\(691\) −3.58259e11 −1.57140 −0.785698 0.618611i \(-0.787696\pi\)
−0.785698 + 0.618611i \(0.787696\pi\)
\(692\) 0 0
\(693\) 7.38234e10 + 3.05817e10i 0.320082 + 0.132595i
\(694\) 0 0
\(695\) 3.20289e10i 0.137279i
\(696\) 0 0
\(697\) −1.40459e11 −0.595141
\(698\) 0 0
\(699\) 1.83280e11 1.22459e11i 0.767728 0.512960i
\(700\) 0 0
\(701\) 2.49323e11i 1.03250i −0.856438 0.516250i \(-0.827327\pi\)
0.856438 0.516250i \(-0.172673\pi\)
\(702\) 0 0
\(703\) 2.52854e10 0.103526
\(704\) 0 0
\(705\) −4.12286e10 6.17053e10i −0.166895 0.249785i
\(706\) 0 0
\(707\) 1.81203e11i 0.725251i
\(708\) 0 0
\(709\) 3.88874e11 1.53895 0.769474 0.638678i \(-0.220518\pi\)
0.769474 + 0.638678i \(0.220518\pi\)
\(710\) 0 0
\(711\) 5.77052e10 1.39299e11i 0.225807 0.545091i
\(712\) 0 0
\(713\) 1.65357e11i 0.639829i
\(714\) 0 0
\(715\) −4.02006e10 −0.153818
\(716\) 0 0
\(717\) 1.53195e11 1.02357e11i 0.579651 0.387296i
\(718\) 0 0
\(719\) 3.35735e10i 0.125626i −0.998025 0.0628132i \(-0.979993\pi\)
0.998025 0.0628132i \(-0.0200072\pi\)
\(720\) 0 0
\(721\) 2.90611e11 1.07540
\(722\) 0 0
\(723\) 7.86002e10 + 1.17638e11i 0.287654 + 0.430521i
\(724\) 0 0
\(725\) 1.56809e11i 0.567569i
\(726\) 0 0
\(727\) 2.53113e11 0.906102 0.453051 0.891485i \(-0.350336\pi\)
0.453051 + 0.891485i \(0.350336\pi\)
\(728\) 0 0
\(729\) 2.60942e11 1.08053e11i 0.923920 0.382586i
\(730\) 0 0
\(731\) 2.64084e11i 0.924853i
\(732\) 0 0
\(733\) −2.07602e11 −0.719144 −0.359572 0.933117i \(-0.617077\pi\)
−0.359572 + 0.933117i \(0.617077\pi\)
\(734\) 0 0
\(735\) 4.08588e10 2.72999e10i 0.140003 0.0935432i
\(736\) 0 0
\(737\) 1.57903e10i 0.0535205i
\(738\) 0 0
\(739\) −4.15014e11 −1.39151 −0.695753 0.718281i \(-0.744929\pi\)
−0.695753 + 0.718281i \(0.744929\pi\)
\(740\) 0 0
\(741\) −2.19274e10 3.28179e10i −0.0727300 0.108852i
\(742\) 0 0
\(743\) 3.30690e11i 1.08509i 0.840027 + 0.542545i \(0.182539\pi\)
−0.840027 + 0.542545i \(0.817461\pi\)
\(744\) 0 0
\(745\) −1.83918e11 −0.597034
\(746\) 0 0
\(747\) 2.81224e11 + 1.16498e11i 0.903170 + 0.374143i
\(748\) 0 0
\(749\) 3.94695e10i 0.125411i
\(750\) 0 0
\(751\) −4.77978e11 −1.50262 −0.751308 0.659952i \(-0.770577\pi\)
−0.751308 + 0.659952i \(0.770577\pi\)
\(752\) 0 0
\(753\) 9.26390e10 6.18970e10i 0.288147 0.192526i
\(754\) 0 0
\(755\) 9.51565e10i 0.292854i
\(756\) 0 0
\(757\) 8.95066e10 0.272566 0.136283 0.990670i \(-0.456484\pi\)
0.136283 + 0.990670i \(0.456484\pi\)
\(758\) 0 0
\(759\) 1.47337e11 + 2.20514e11i 0.443962 + 0.664462i
\(760\) 0 0
\(761\) 5.71473e11i 1.70395i −0.523581 0.851976i \(-0.675404\pi\)
0.523581 0.851976i \(-0.324596\pi\)
\(762\) 0 0
\(763\) −1.92456e11 −0.567851
\(764\) 0 0
\(765\) 4.22185e10 1.01914e11i 0.123270 0.297570i
\(766\) 0 0
\(767\) 3.52884e11i 1.01965i
\(768\) 0 0
\(769\) 2.56194e11 0.732596 0.366298 0.930498i \(-0.380625\pi\)
0.366298 + 0.930498i \(0.380625\pi\)
\(770\) 0 0
\(771\) 5.34537e11 3.57152e11i 1.51273 1.01073i
\(772\) 0 0
\(773\) 1.00523e11i 0.281543i −0.990042 0.140772i \(-0.955042\pi\)
0.990042 0.140772i \(-0.0449584\pi\)
\(774\) 0 0
\(775\) −1.19582e11 −0.331480
\(776\) 0 0
\(777\) −1.05145e11 1.57366e11i −0.288472 0.431745i
\(778\) 0 0
\(779\) 3.55176e10i 0.0964482i
\(780\) 0 0
\(781\) 1.18464e11 0.318408
\(782\) 0 0
\(783\) 4.77727e10 + 2.40236e11i 0.127096 + 0.639132i
\(784\) 0 0
\(785\) 1.70586e11i 0.449226i
\(786\) 0 0
\(787\) 2.15788e11 0.562508 0.281254 0.959633i \(-0.409250\pi\)
0.281254 + 0.959633i \(0.409250\pi\)
\(788\) 0 0
\(789\) −2.17240e10 + 1.45150e10i −0.0560572 + 0.0374548i
\(790\) 0 0
\(791\) 5.03559e11i 1.28631i
\(792\) 0 0
\(793\) 1.93902e10 0.0490331
\(794\) 0 0
\(795\) 6.66942e10 + 9.98187e10i 0.166963 + 0.249887i
\(796\) 0 0
\(797\) 6.69766e11i 1.65993i −0.557815 0.829966i \(-0.688360\pi\)
0.557815 0.829966i \(-0.311640\pi\)
\(798\) 0 0
\(799\) 3.05641e11 0.749938
\(800\) 0 0
\(801\) 4.40156e11 + 1.82337e11i 1.06924 + 0.442940i
\(802\) 0 0
\(803\) 1.92588e11i 0.463199i
\(804\) 0 0
\(805\) −1.84832e11 −0.440143
\(806\) 0 0
\(807\) 2.33915e11 1.56291e11i 0.551524 0.368502i
\(808\) 0 0
\(809\) 9.93417e10i 0.231920i −0.993254 0.115960i \(-0.963006\pi\)
0.993254 0.115960i \(-0.0369944\pi\)
\(810\) 0 0
\(811\) 2.71549e11 0.627718 0.313859 0.949470i \(-0.398378\pi\)
0.313859 + 0.949470i \(0.398378\pi\)
\(812\) 0 0
\(813\) 6.48974e10 + 9.71295e10i 0.148547 + 0.222325i
\(814\) 0 0
\(815\) 1.50002e11i 0.339989i
\(816\) 0 0
\(817\) −6.67782e10 −0.149881
\(818\) 0 0
\(819\) −1.13064e11 + 2.72934e11i −0.251298 + 0.606626i
\(820\) 0 0
\(821\) 5.63227e11i 1.23968i 0.784727 + 0.619842i \(0.212803\pi\)
−0.784727 + 0.619842i \(0.787197\pi\)
\(822\) 0 0
\(823\) −4.42272e11 −0.964028 −0.482014 0.876164i \(-0.660095\pi\)
−0.482014 + 0.876164i \(0.660095\pi\)
\(824\) 0 0
\(825\) 1.59470e11 1.06551e11i 0.344242 0.230006i
\(826\) 0 0
\(827\) 4.54543e11i 0.971745i 0.874030 + 0.485873i \(0.161498\pi\)
−0.874030 + 0.485873i \(0.838502\pi\)
\(828\) 0 0
\(829\) 1.32535e11 0.280615 0.140308 0.990108i \(-0.455191\pi\)
0.140308 + 0.990108i \(0.455191\pi\)
\(830\) 0 0
\(831\) 1.52121e11 + 2.27674e11i 0.318996 + 0.477429i
\(832\) 0 0
\(833\) 2.02383e11i 0.420334i
\(834\) 0 0
\(835\) −4.40705e10 −0.0906572
\(836\) 0 0
\(837\) −1.83203e11 + 3.64312e10i −0.373276 + 0.0742286i
\(838\) 0 0
\(839\) 4.87205e10i 0.0983250i −0.998791 0.0491625i \(-0.984345\pi\)
0.998791 0.0491625i \(-0.0156552\pi\)
\(840\) 0 0
\(841\) 2.87820e11 0.575357
\(842\) 0 0
\(843\) −2.70923e11 + 1.81018e11i −0.536457 + 0.358435i
\(844\) 0 0
\(845\) 3.45051e10i 0.0676793i
\(846\) 0 0
\(847\) −2.90368e11 −0.564176
\(848\) 0 0
\(849\) −4.69138e11 7.02141e11i −0.902962 1.35143i
\(850\) 0 0
\(851\) 6.28145e11i 1.19768i
\(852\) 0 0
\(853\) 7.24501e11 1.36849 0.684247 0.729250i \(-0.260131\pi\)
0.684247 + 0.729250i \(0.260131\pi\)
\(854\) 0 0
\(855\) −2.57708e10 1.06757e10i −0.0482241 0.0199771i
\(856\) 0 0
\(857\) 4.16662e11i 0.772433i 0.922408 + 0.386216i \(0.126218\pi\)
−0.922408 + 0.386216i \(0.873782\pi\)
\(858\) 0 0
\(859\) −7.54443e11 −1.38565 −0.692826 0.721105i \(-0.743634\pi\)
−0.692826 + 0.721105i \(0.743634\pi\)
\(860\) 0 0
\(861\) 2.21047e11 1.47693e11i 0.402227 0.268750i
\(862\) 0 0
\(863\) 2.06295e10i 0.0371916i 0.999827 + 0.0185958i \(0.00591957\pi\)
−0.999827 + 0.0185958i \(0.994080\pi\)
\(864\) 0 0
\(865\) 2.29706e11 0.410306
\(866\) 0 0
\(867\) −6.15057e10 9.20533e10i −0.108853 0.162916i
\(868\) 0 0
\(869\) 1.59936e11i 0.280457i
\(870\) 0 0
\(871\) 5.83785e10 0.101433
\(872\) 0 0
\(873\) −3.69798e11 + 8.92681e11i −0.636659 + 1.53688i
\(874\) 0 0
\(875\) 2.87132e11i 0.489834i
\(876\) 0 0
\(877\) −5.98158e11 −1.01115 −0.505577 0.862782i \(-0.668720\pi\)
−0.505577 + 0.862782i \(0.668720\pi\)
\(878\) 0 0
\(879\) 6.99945e11 4.67670e11i 1.17249 0.783401i
\(880\) 0 0
\(881\) 1.62306e11i 0.269420i −0.990885 0.134710i \(-0.956990\pi\)
0.990885 0.134710i \(-0.0430103\pi\)
\(882\) 0 0
\(883\) 6.06459e11 0.997605 0.498803 0.866716i \(-0.333773\pi\)
0.498803 + 0.866716i \(0.333773\pi\)
\(884\) 0 0
\(885\) −1.38554e11 2.07369e11i −0.225864 0.338042i
\(886\) 0 0
\(887\) 4.65904e11i 0.752666i −0.926484 0.376333i \(-0.877185\pi\)
0.926484 0.376333i \(-0.122815\pi\)
\(888\) 0 0
\(889\) −4.82989e11 −0.773269
\(890\) 0 0
\(891\) 2.11852e11 2.11822e11i 0.336141 0.336094i
\(892\) 0 0
\(893\) 7.72867e10i 0.121534i
\(894\) 0 0
\(895\) −2.89368e11 −0.450981
\(896\) 0 0
\(897\) −8.15267e11 + 5.44723e11i −1.25930 + 0.841407i
\(898\) 0 0
\(899\) 1.61995e11i 0.248007i
\(900\) 0 0
\(901\) −4.94426e11 −0.750243
\(902\) 0 0
\(903\) 2.77684e11 + 4.15600e11i 0.417638 + 0.625064i
\(904\) 0 0
\(905\) 1.05898e11i 0.157868i
\(906\) 0 0
\(907\) −7.47111e11 −1.10397 −0.551983 0.833855i \(-0.686129\pi\)
−0.551983 + 0.833855i \(0.686129\pi\)
\(908\) 0 0
\(909\) 6.27635e11 + 2.60001e11i 0.919288 + 0.380820i
\(910\) 0 0
\(911\) 9.01051e11i 1.30821i −0.756406 0.654103i \(-0.773046\pi\)
0.756406 0.654103i \(-0.226954\pi\)
\(912\) 0 0
\(913\) 3.22887e11 0.464694
\(914\) 0 0
\(915\) 1.13945e10 7.61325e9i 0.0162558 0.0108614i
\(916\) 0 0
\(917\) 5.05956e11i 0.715543i
\(918\) 0 0
\(919\) −1.14699e12 −1.60804 −0.804021 0.594601i \(-0.797310\pi\)
−0.804021 + 0.594601i \(0.797310\pi\)
\(920\) 0 0
\(921\) −1.34689e11 2.01584e11i −0.187195 0.280168i
\(922\) 0 0
\(923\) 4.37976e11i 0.603453i
\(924\) 0 0
\(925\) −4.54258e11 −0.620492
\(926\) 0 0
\(927\) 4.16985e11 1.00659e12i 0.564679 1.36312i
\(928\) 0 0
\(929\) 4.36054e11i 0.585433i 0.956199 + 0.292716i \(0.0945592\pi\)
−0.956199 + 0.292716i \(0.905441\pi\)
\(930\) 0 0
\(931\) 5.11762e10 0.0681191
\(932\) 0 0
\(933\) −4.34293e11 + 2.90174e11i −0.573135 + 0.382942i
\(934\) 0 0
\(935\) 1.17013e11i 0.153104i
\(936\) 0 0
\(937\) 1.98881e11 0.258009 0.129004 0.991644i \(-0.458822\pi\)
0.129004 + 0.991644i \(0.458822\pi\)
\(938\) 0 0
\(939\) −1.47482e9 2.20731e9i −0.00189704 0.00283923i
\(940\) 0 0
\(941\) 1.28069e12i 1.63338i 0.577079 + 0.816689i \(0.304193\pi\)
−0.577079 + 0.816689i \(0.695807\pi\)
\(942\) 0 0
\(943\) 8.82335e11 1.11580
\(944\) 0 0
\(945\) 4.07219e10 + 2.04780e11i 0.0510624 + 0.256779i
\(946\) 0 0
\(947\) 1.35658e12i 1.68673i −0.537337 0.843367i \(-0.680570\pi\)
0.537337 0.843367i \(-0.319430\pi\)
\(948\) 0 0
\(949\) −7.12020e11 −0.877865
\(950\) 0 0
\(951\) −7.93358e11 + 5.30084e11i −0.969945 + 0.648072i
\(952\) 0 0
\(953\) 5.01575e11i 0.608085i 0.952658 + 0.304042i \(0.0983365\pi\)
−0.952658 + 0.304042i \(0.901664\pi\)
\(954\) 0 0
\(955\) 3.63211e10 0.0436662
\(956\) 0 0
\(957\) 1.44342e11 + 2.16032e11i 0.172086 + 0.257555i
\(958\) 0 0
\(959\) 3.63234e11i 0.429450i
\(960\) 0 0
\(961\) −7.29354e11 −0.855155
\(962\) 0 0
\(963\) 1.36711e11 + 5.66330e10i 0.158963 + 0.0658514i
\(964\) 0 0
\(965\) 3.56594e11i 0.411212i
\(966\) 0 0
\(967\) 1.10988e11 0.126932 0.0634661 0.997984i \(-0.479785\pi\)
0.0634661 + 0.997984i \(0.479785\pi\)
\(968\) 0 0
\(969\) 9.55239e10 6.38246e10i 0.108347 0.0723924i
\(970\) 0 0
\(971\) 2.95380e11i 0.332280i −0.986102 0.166140i \(-0.946870\pi\)
0.986102 0.166140i \(-0.0531304\pi\)
\(972\) 0 0
\(973\) −2.49669e11 −0.278557
\(974\) 0 0
\(975\) 3.93930e11 + 5.89580e11i 0.435913 + 0.652415i
\(976\) 0 0
\(977\) 6.09091e11i 0.668503i 0.942484 + 0.334252i \(0.108484\pi\)
−0.942484 + 0.334252i \(0.891516\pi\)
\(978\) 0 0
\(979\) 5.05365e11 0.550141
\(980\) 0 0
\(981\) −2.76147e11 + 6.66612e11i −0.298171 + 0.719776i
\(982\) 0 0
\(983\) 1.84012e12i 1.97076i 0.170384 + 0.985378i \(0.445499\pi\)
−0.170384 + 0.985378i \(0.554501\pi\)
\(984\) 0 0
\(985\) −1.20729e11 −0.128253
\(986\) 0 0
\(987\) −4.81000e11 + 3.21382e11i −0.506847 + 0.338651i
\(988\) 0 0
\(989\) 1.65892e12i 1.73396i
\(990\) 0 0
\(991\) −5.72336e11 −0.593412 −0.296706 0.954969i \(-0.595888\pi\)
−0.296706 + 0.954969i \(0.595888\pi\)
\(992\) 0 0
\(993\) −5.40449e11 8.08870e11i −0.555850 0.831920i
\(994\) 0 0
\(995\) 1.45383e11i 0.148327i
\(996\) 0 0
\(997\) 1.44614e12 1.46363 0.731813 0.681505i \(-0.238674\pi\)
0.731813 + 0.681505i \(0.238674\pi\)
\(998\) 0 0
\(999\) −6.95937e11 + 1.38392e11i −0.698728 + 0.138947i
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 192.9.e.e.65.1 2
3.2 odd 2 inner 192.9.e.e.65.2 2
4.3 odd 2 192.9.e.f.65.2 2
8.3 odd 2 48.9.e.b.17.1 2
8.5 even 2 3.9.b.a.2.1 2
12.11 even 2 192.9.e.f.65.1 2
24.5 odd 2 3.9.b.a.2.2 yes 2
24.11 even 2 48.9.e.b.17.2 2
40.13 odd 4 75.9.d.b.74.2 4
40.29 even 2 75.9.c.c.26.2 2
40.37 odd 4 75.9.d.b.74.3 4
72.5 odd 6 81.9.d.d.26.1 4
72.13 even 6 81.9.d.d.26.2 4
72.29 odd 6 81.9.d.d.53.2 4
72.61 even 6 81.9.d.d.53.1 4
120.29 odd 2 75.9.c.c.26.1 2
120.53 even 4 75.9.d.b.74.4 4
120.77 even 4 75.9.d.b.74.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.9.b.a.2.1 2 8.5 even 2
3.9.b.a.2.2 yes 2 24.5 odd 2
48.9.e.b.17.1 2 8.3 odd 2
48.9.e.b.17.2 2 24.11 even 2
75.9.c.c.26.1 2 120.29 odd 2
75.9.c.c.26.2 2 40.29 even 2
75.9.d.b.74.1 4 120.77 even 4
75.9.d.b.74.2 4 40.13 odd 4
75.9.d.b.74.3 4 40.37 odd 4
75.9.d.b.74.4 4 120.53 even 4
81.9.d.d.26.1 4 72.5 odd 6
81.9.d.d.26.2 4 72.13 even 6
81.9.d.d.53.1 4 72.61 even 6
81.9.d.d.53.2 4 72.29 odd 6
192.9.e.e.65.1 2 1.1 even 1 trivial
192.9.e.e.65.2 2 3.2 odd 2 inner
192.9.e.f.65.1 2 12.11 even 2
192.9.e.f.65.2 2 4.3 odd 2