Properties

Label 1080.2.k.d.541.6
Level $1080$
Weight $2$
Character 1080.541
Analytic conductor $8.624$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(541,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.k (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 541.6
Root \(-1.17425 - 0.788128i\) of defining polynomial
Character \(\chi\) \(=\) 1080.541
Dual form 1080.2.k.d.541.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.17425 + 0.788128i) q^{2} +(0.757708 - 1.85091i) q^{4} -1.00000i q^{5} +2.12726 q^{7} +(0.569020 + 2.77060i) q^{8} +O(q^{10})\) \(q+(-1.17425 + 0.788128i) q^{2} +(0.757708 - 1.85091i) q^{4} -1.00000i q^{5} +2.12726 q^{7} +(0.569020 + 2.77060i) q^{8} +(0.788128 + 1.17425i) q^{10} +5.48036i q^{11} +4.91228i q^{13} +(-2.49792 + 1.67655i) q^{14} +(-2.85176 - 2.80490i) q^{16} +0.235541 q^{17} -5.23139i q^{19} +(-1.85091 - 0.757708i) q^{20} +(-4.31922 - 6.43529i) q^{22} -7.42390 q^{23} -1.00000 q^{25} +(-3.87150 - 5.76822i) q^{26} +(1.61184 - 3.93737i) q^{28} +4.24894i q^{29} +5.05609 q^{31} +(5.55929 + 1.04610i) q^{32} +(-0.276583 + 0.185636i) q^{34} -2.12726i q^{35} +2.27608i q^{37} +(4.12300 + 6.14294i) q^{38} +(2.77060 - 0.569020i) q^{40} -3.26132 q^{41} +9.90812i q^{43} +(10.1437 + 4.15251i) q^{44} +(8.71749 - 5.85098i) q^{46} +8.17359 q^{47} -2.47478 q^{49} +(1.17425 - 0.788128i) q^{50} +(9.09220 + 3.72207i) q^{52} +10.6637i q^{53} +5.48036 q^{55} +(1.21045 + 5.89377i) q^{56} +(-3.34871 - 4.98930i) q^{58} -0.702988i q^{59} -0.319109i q^{61} +(-5.93709 + 3.98485i) q^{62} +(-7.35243 + 3.15305i) q^{64} +4.91228 q^{65} +2.70667i q^{67} +(0.178471 - 0.435965i) q^{68} +(1.67655 + 2.49792i) q^{70} -7.18836 q^{71} +9.66370 q^{73} +(-1.79384 - 2.67268i) q^{74} +(-9.68284 - 3.96386i) q^{76} +11.6581i q^{77} -10.8565 q^{79} +(-2.80490 + 2.85176i) q^{80} +(3.82959 - 2.57033i) q^{82} -8.53645i q^{83} -0.235541i q^{85} +(-7.80887 - 11.6346i) q^{86} +(-15.1839 + 3.11843i) q^{88} +9.90812 q^{89} +10.4497i q^{91} +(-5.62515 + 13.7410i) q^{92} +(-9.59781 + 6.44184i) q^{94} -5.23139 q^{95} +14.8895 q^{97} +(2.90600 - 1.95044i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{4} - 2 q^{10} - 18 q^{16} + 16 q^{22} - 20 q^{25} + 16 q^{28} + 20 q^{31} - 6 q^{34} - 4 q^{40} + 54 q^{46} + 36 q^{49} + 56 q^{52} - 72 q^{58} - 28 q^{64} - 40 q^{73} + 58 q^{76} - 4 q^{79} - 92 q^{82} - 116 q^{88} + 72 q^{94} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17425 + 0.788128i −0.830317 + 0.557291i
\(3\) 0 0
\(4\) 0.757708 1.85091i 0.378854 0.925456i
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.12726 0.804027 0.402014 0.915634i \(-0.368310\pi\)
0.402014 + 0.915634i \(0.368310\pi\)
\(8\) 0.569020 + 2.77060i 0.201179 + 0.979554i
\(9\) 0 0
\(10\) 0.788128 + 1.17425i 0.249228 + 0.371329i
\(11\) 5.48036i 1.65239i 0.563384 + 0.826195i \(0.309499\pi\)
−0.563384 + 0.826195i \(0.690501\pi\)
\(12\) 0 0
\(13\) 4.91228i 1.36242i 0.732088 + 0.681210i \(0.238546\pi\)
−0.732088 + 0.681210i \(0.761454\pi\)
\(14\) −2.49792 + 1.67655i −0.667598 + 0.448077i
\(15\) 0 0
\(16\) −2.85176 2.80490i −0.712939 0.701226i
\(17\) 0.235541 0.0571270 0.0285635 0.999592i \(-0.490907\pi\)
0.0285635 + 0.999592i \(0.490907\pi\)
\(18\) 0 0
\(19\) 5.23139i 1.20016i −0.799939 0.600081i \(-0.795135\pi\)
0.799939 0.600081i \(-0.204865\pi\)
\(20\) −1.85091 0.757708i −0.413877 0.169429i
\(21\) 0 0
\(22\) −4.31922 6.43529i −0.920862 1.37201i
\(23\) −7.42390 −1.54799 −0.773995 0.633192i \(-0.781744\pi\)
−0.773995 + 0.633192i \(0.781744\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −3.87150 5.76822i −0.759264 1.13124i
\(27\) 0 0
\(28\) 1.61184 3.93737i 0.304609 0.744092i
\(29\) 4.24894i 0.789008i 0.918894 + 0.394504i \(0.129083\pi\)
−0.918894 + 0.394504i \(0.870917\pi\)
\(30\) 0 0
\(31\) 5.05609 0.908100 0.454050 0.890976i \(-0.349979\pi\)
0.454050 + 0.890976i \(0.349979\pi\)
\(32\) 5.55929 + 1.04610i 0.982752 + 0.184926i
\(33\) 0 0
\(34\) −0.276583 + 0.185636i −0.0474335 + 0.0318363i
\(35\) 2.12726i 0.359572i
\(36\) 0 0
\(37\) 2.27608i 0.374185i 0.982342 + 0.187093i \(0.0599065\pi\)
−0.982342 + 0.187093i \(0.940094\pi\)
\(38\) 4.12300 + 6.14294i 0.668839 + 0.996516i
\(39\) 0 0
\(40\) 2.77060 0.569020i 0.438070 0.0899700i
\(41\) −3.26132 −0.509332 −0.254666 0.967029i \(-0.581966\pi\)
−0.254666 + 0.967029i \(0.581966\pi\)
\(42\) 0 0
\(43\) 9.90812i 1.51097i 0.655163 + 0.755487i \(0.272600\pi\)
−0.655163 + 0.755487i \(0.727400\pi\)
\(44\) 10.1437 + 4.15251i 1.52921 + 0.626015i
\(45\) 0 0
\(46\) 8.71749 5.85098i 1.28532 0.862680i
\(47\) 8.17359 1.19224 0.596121 0.802895i \(-0.296708\pi\)
0.596121 + 0.802895i \(0.296708\pi\)
\(48\) 0 0
\(49\) −2.47478 −0.353540
\(50\) 1.17425 0.788128i 0.166063 0.111458i
\(51\) 0 0
\(52\) 9.09220 + 3.72207i 1.26086 + 0.516159i
\(53\) 10.6637i 1.46477i 0.680890 + 0.732386i \(0.261593\pi\)
−0.680890 + 0.732386i \(0.738407\pi\)
\(54\) 0 0
\(55\) 5.48036 0.738971
\(56\) 1.21045 + 5.89377i 0.161753 + 0.787589i
\(57\) 0 0
\(58\) −3.34871 4.98930i −0.439707 0.655127i
\(59\) 0.702988i 0.0915212i −0.998952 0.0457606i \(-0.985429\pi\)
0.998952 0.0457606i \(-0.0145711\pi\)
\(60\) 0 0
\(61\) 0.319109i 0.0408577i −0.999791 0.0204288i \(-0.993497\pi\)
0.999791 0.0204288i \(-0.00650315\pi\)
\(62\) −5.93709 + 3.98485i −0.754012 + 0.506076i
\(63\) 0 0
\(64\) −7.35243 + 3.15305i −0.919054 + 0.394132i
\(65\) 4.91228 0.609293
\(66\) 0 0
\(67\) 2.70667i 0.330672i 0.986237 + 0.165336i \(0.0528708\pi\)
−0.986237 + 0.165336i \(0.947129\pi\)
\(68\) 0.178471 0.435965i 0.0216428 0.0528685i
\(69\) 0 0
\(70\) 1.67655 + 2.49792i 0.200386 + 0.298559i
\(71\) −7.18836 −0.853101 −0.426551 0.904464i \(-0.640271\pi\)
−0.426551 + 0.904464i \(0.640271\pi\)
\(72\) 0 0
\(73\) 9.66370 1.13105 0.565525 0.824731i \(-0.308673\pi\)
0.565525 + 0.824731i \(0.308673\pi\)
\(74\) −1.79384 2.67268i −0.208530 0.310693i
\(75\) 0 0
\(76\) −9.68284 3.96386i −1.11070 0.454686i
\(77\) 11.6581i 1.32857i
\(78\) 0 0
\(79\) −10.8565 −1.22146 −0.610729 0.791840i \(-0.709123\pi\)
−0.610729 + 0.791840i \(0.709123\pi\)
\(80\) −2.80490 + 2.85176i −0.313598 + 0.318836i
\(81\) 0 0
\(82\) 3.82959 2.57033i 0.422907 0.283846i
\(83\) 8.53645i 0.936997i −0.883464 0.468498i \(-0.844795\pi\)
0.883464 0.468498i \(-0.155205\pi\)
\(84\) 0 0
\(85\) 0.235541i 0.0255480i
\(86\) −7.80887 11.6346i −0.842052 1.25459i
\(87\) 0 0
\(88\) −15.1839 + 3.11843i −1.61861 + 0.332426i
\(89\) 9.90812 1.05026 0.525129 0.851022i \(-0.324017\pi\)
0.525129 + 0.851022i \(0.324017\pi\)
\(90\) 0 0
\(91\) 10.4497i 1.09542i
\(92\) −5.62515 + 13.7410i −0.586462 + 1.43260i
\(93\) 0 0
\(94\) −9.59781 + 6.44184i −0.989939 + 0.664425i
\(95\) −5.23139 −0.536729
\(96\) 0 0
\(97\) 14.8895 1.51180 0.755902 0.654685i \(-0.227199\pi\)
0.755902 + 0.654685i \(0.227199\pi\)
\(98\) 2.90600 1.95044i 0.293550 0.197025i
\(99\) 0 0
\(100\) −0.757708 + 1.85091i −0.0757708 + 0.185091i
\(101\) 6.13283i 0.610240i −0.952314 0.305120i \(-0.901303\pi\)
0.952314 0.305120i \(-0.0986965\pi\)
\(102\) 0 0
\(103\) −8.82788 −0.869837 −0.434918 0.900470i \(-0.643223\pi\)
−0.434918 + 0.900470i \(0.643223\pi\)
\(104\) −13.6099 + 2.79519i −1.33457 + 0.274090i
\(105\) 0 0
\(106\) −8.40436 12.5218i −0.816304 1.21623i
\(107\) 7.48593i 0.723693i 0.932238 + 0.361846i \(0.117853\pi\)
−0.932238 + 0.361846i \(0.882147\pi\)
\(108\) 0 0
\(109\) 17.8462i 1.70935i 0.519160 + 0.854677i \(0.326245\pi\)
−0.519160 + 0.854677i \(0.673755\pi\)
\(110\) −6.43529 + 4.31922i −0.613581 + 0.411822i
\(111\) 0 0
\(112\) −6.06642 5.96675i −0.573223 0.563805i
\(113\) 14.8609 1.39800 0.698998 0.715124i \(-0.253630\pi\)
0.698998 + 0.715124i \(0.253630\pi\)
\(114\) 0 0
\(115\) 7.42390i 0.692282i
\(116\) 7.86441 + 3.21945i 0.730192 + 0.298919i
\(117\) 0 0
\(118\) 0.554045 + 0.825481i 0.0510039 + 0.0759917i
\(119\) 0.501055 0.0459316
\(120\) 0 0
\(121\) −19.0343 −1.73039
\(122\) 0.251498 + 0.374712i 0.0227696 + 0.0339248i
\(123\) 0 0
\(124\) 3.83104 9.35838i 0.344038 0.840407i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 6.82788 0.605876 0.302938 0.953010i \(-0.402032\pi\)
0.302938 + 0.953010i \(0.402032\pi\)
\(128\) 6.14856 9.49712i 0.543461 0.839435i
\(129\) 0 0
\(130\) −5.76822 + 3.87150i −0.505907 + 0.339553i
\(131\) 21.3874i 1.86863i −0.356452 0.934314i \(-0.616014\pi\)
0.356452 0.934314i \(-0.383986\pi\)
\(132\) 0 0
\(133\) 11.1285i 0.964963i
\(134\) −2.13320 3.17829i −0.184280 0.274563i
\(135\) 0 0
\(136\) 0.134027 + 0.652588i 0.0114927 + 0.0559590i
\(137\) 19.6081 1.67523 0.837617 0.546259i \(-0.183948\pi\)
0.837617 + 0.546259i \(0.183948\pi\)
\(138\) 0 0
\(139\) 0.901666i 0.0764784i 0.999269 + 0.0382392i \(0.0121749\pi\)
−0.999269 + 0.0382392i \(0.987825\pi\)
\(140\) −3.93737 1.61184i −0.332768 0.136225i
\(141\) 0 0
\(142\) 8.44090 5.66535i 0.708345 0.475425i
\(143\) −26.9210 −2.25125
\(144\) 0 0
\(145\) 4.24894 0.352855
\(146\) −11.3476 + 7.61624i −0.939131 + 0.630324i
\(147\) 0 0
\(148\) 4.21283 + 1.72461i 0.346292 + 0.141762i
\(149\) 1.75106i 0.143453i −0.997424 0.0717264i \(-0.977149\pi\)
0.997424 0.0717264i \(-0.0228508\pi\)
\(150\) 0 0
\(151\) 2.90141 0.236114 0.118057 0.993007i \(-0.462334\pi\)
0.118057 + 0.993007i \(0.462334\pi\)
\(152\) 14.4941 2.97676i 1.17562 0.241447i
\(153\) 0 0
\(154\) −9.18810 13.6895i −0.740398 1.10313i
\(155\) 5.05609i 0.406115i
\(156\) 0 0
\(157\) 19.7601i 1.57703i 0.615018 + 0.788513i \(0.289149\pi\)
−0.615018 + 0.788513i \(0.710851\pi\)
\(158\) 12.7483 8.55635i 1.01420 0.680707i
\(159\) 0 0
\(160\) 1.04610 5.55929i 0.0827014 0.439500i
\(161\) −15.7925 −1.24463
\(162\) 0 0
\(163\) 21.4946i 1.68359i 0.539799 + 0.841794i \(0.318500\pi\)
−0.539799 + 0.841794i \(0.681500\pi\)
\(164\) −2.47113 + 6.03641i −0.192963 + 0.471364i
\(165\) 0 0
\(166\) 6.72781 + 10.0239i 0.522180 + 0.778005i
\(167\) −19.0666 −1.47541 −0.737707 0.675121i \(-0.764091\pi\)
−0.737707 + 0.675121i \(0.764091\pi\)
\(168\) 0 0
\(169\) −11.1305 −0.856190
\(170\) 0.185636 + 0.276583i 0.0142376 + 0.0212129i
\(171\) 0 0
\(172\) 18.3391 + 7.50747i 1.39834 + 0.572439i
\(173\) 9.31060i 0.707872i 0.935270 + 0.353936i \(0.115157\pi\)
−0.935270 + 0.353936i \(0.884843\pi\)
\(174\) 0 0
\(175\) −2.12726 −0.160805
\(176\) 15.3719 15.6286i 1.15870 1.17805i
\(177\) 0 0
\(178\) −11.6346 + 7.80887i −0.872048 + 0.585300i
\(179\) 2.70620i 0.202271i −0.994873 0.101136i \(-0.967752\pi\)
0.994873 0.101136i \(-0.0322476\pi\)
\(180\) 0 0
\(181\) 7.42390i 0.551814i 0.961184 + 0.275907i \(0.0889782\pi\)
−0.961184 + 0.275907i \(0.911022\pi\)
\(182\) −8.23568 12.2705i −0.610469 0.909549i
\(183\) 0 0
\(184\) −4.22435 20.5686i −0.311423 1.51634i
\(185\) 2.27608 0.167341
\(186\) 0 0
\(187\) 1.29085i 0.0943960i
\(188\) 6.19320 15.1286i 0.451685 1.10337i
\(189\) 0 0
\(190\) 6.14294 4.12300i 0.445655 0.299114i
\(191\) −13.2099 −0.955837 −0.477919 0.878404i \(-0.658609\pi\)
−0.477919 + 0.878404i \(0.658609\pi\)
\(192\) 0 0
\(193\) −3.73487 −0.268842 −0.134421 0.990924i \(-0.542917\pi\)
−0.134421 + 0.990924i \(0.542917\pi\)
\(194\) −17.4840 + 11.7349i −1.25528 + 0.842515i
\(195\) 0 0
\(196\) −1.87516 + 4.58060i −0.133940 + 0.327186i
\(197\) 7.00321i 0.498958i −0.968380 0.249479i \(-0.919741\pi\)
0.968380 0.249479i \(-0.0802594\pi\)
\(198\) 0 0
\(199\) 21.2769 1.50828 0.754140 0.656714i \(-0.228054\pi\)
0.754140 + 0.656714i \(0.228054\pi\)
\(200\) −0.569020 2.77060i −0.0402358 0.195911i
\(201\) 0 0
\(202\) 4.83346 + 7.20146i 0.340081 + 0.506693i
\(203\) 9.03858i 0.634384i
\(204\) 0 0
\(205\) 3.26132i 0.227780i
\(206\) 10.3661 6.95750i 0.722241 0.484752i
\(207\) 0 0
\(208\) 13.7785 14.0086i 0.955365 0.971323i
\(209\) 28.6699 1.98314
\(210\) 0 0
\(211\) 24.4773i 1.68509i −0.538628 0.842544i \(-0.681057\pi\)
0.538628 0.842544i \(-0.318943\pi\)
\(212\) 19.7376 + 8.07998i 1.35558 + 0.554935i
\(213\) 0 0
\(214\) −5.89988 8.79033i −0.403307 0.600895i
\(215\) 9.90812 0.675728
\(216\) 0 0
\(217\) 10.7556 0.730138
\(218\) −14.0651 20.9558i −0.952607 1.41931i
\(219\) 0 0
\(220\) 4.15251 10.1437i 0.279962 0.683886i
\(221\) 1.15704i 0.0778310i
\(222\) 0 0
\(223\) −16.9862 −1.13748 −0.568741 0.822516i \(-0.692569\pi\)
−0.568741 + 0.822516i \(0.692569\pi\)
\(224\) 11.8260 + 2.22532i 0.790160 + 0.148685i
\(225\) 0 0
\(226\) −17.4504 + 11.7123i −1.16078 + 0.779090i
\(227\) 20.5203i 1.36198i −0.732294 0.680988i \(-0.761550\pi\)
0.732294 0.680988i \(-0.238450\pi\)
\(228\) 0 0
\(229\) 3.01268i 0.199083i −0.995033 0.0995416i \(-0.968262\pi\)
0.995033 0.0995416i \(-0.0317376\pi\)
\(230\) −5.85098 8.71749i −0.385802 0.574814i
\(231\) 0 0
\(232\) −11.7721 + 2.41773i −0.772876 + 0.158732i
\(233\) −15.2379 −0.998267 −0.499134 0.866525i \(-0.666348\pi\)
−0.499134 + 0.866525i \(0.666348\pi\)
\(234\) 0 0
\(235\) 8.17359i 0.533186i
\(236\) −1.30117 0.532660i −0.0846989 0.0346732i
\(237\) 0 0
\(238\) −0.588362 + 0.394896i −0.0381379 + 0.0255973i
\(239\) 15.8761 1.02694 0.513470 0.858108i \(-0.328360\pi\)
0.513470 + 0.858108i \(0.328360\pi\)
\(240\) 0 0
\(241\) 20.9838 1.35169 0.675843 0.737046i \(-0.263780\pi\)
0.675843 + 0.737046i \(0.263780\pi\)
\(242\) 22.3510 15.0015i 1.43678 0.964332i
\(243\) 0 0
\(244\) −0.590642 0.241791i −0.0378120 0.0154791i
\(245\) 2.47478i 0.158108i
\(246\) 0 0
\(247\) 25.6980 1.63513
\(248\) 2.87702 + 14.0084i 0.182691 + 0.889534i
\(249\) 0 0
\(250\) −0.788128 1.17425i −0.0498456 0.0742659i
\(251\) 8.49787i 0.536381i −0.963366 0.268190i \(-0.913574\pi\)
0.963366 0.268190i \(-0.0864256\pi\)
\(252\) 0 0
\(253\) 40.6856i 2.55788i
\(254\) −8.01761 + 5.38124i −0.503070 + 0.337649i
\(255\) 0 0
\(256\) 0.265028 + 15.9978i 0.0165642 + 0.999863i
\(257\) −0.222442 −0.0138756 −0.00693778 0.999976i \(-0.502208\pi\)
−0.00693778 + 0.999976i \(0.502208\pi\)
\(258\) 0 0
\(259\) 4.84181i 0.300855i
\(260\) 3.72207 9.09220i 0.230833 0.563874i
\(261\) 0 0
\(262\) 16.8560 + 25.1141i 1.04137 + 1.55155i
\(263\) 7.74301 0.477454 0.238727 0.971087i \(-0.423270\pi\)
0.238727 + 0.971087i \(0.423270\pi\)
\(264\) 0 0
\(265\) 10.6637 0.655066
\(266\) 8.77068 + 13.0676i 0.537765 + 0.801226i
\(267\) 0 0
\(268\) 5.00980 + 2.05086i 0.306022 + 0.125276i
\(269\) 2.50903i 0.152978i −0.997070 0.0764890i \(-0.975629\pi\)
0.997070 0.0764890i \(-0.0243710\pi\)
\(270\) 0 0
\(271\) −7.00558 −0.425559 −0.212779 0.977100i \(-0.568252\pi\)
−0.212779 + 0.977100i \(0.568252\pi\)
\(272\) −0.671704 0.660669i −0.0407281 0.0400589i
\(273\) 0 0
\(274\) −23.0247 + 15.4537i −1.39098 + 0.933592i
\(275\) 5.48036i 0.330478i
\(276\) 0 0
\(277\) 8.13052i 0.488516i −0.969710 0.244258i \(-0.921456\pi\)
0.969710 0.244258i \(-0.0785443\pi\)
\(278\) −0.710629 1.05878i −0.0426207 0.0635013i
\(279\) 0 0
\(280\) 5.89377 1.21045i 0.352220 0.0723383i
\(281\) 20.2038 1.20526 0.602628 0.798023i \(-0.294120\pi\)
0.602628 + 0.798023i \(0.294120\pi\)
\(282\) 0 0
\(283\) 27.4626i 1.63248i −0.577712 0.816241i \(-0.696054\pi\)
0.577712 0.816241i \(-0.303946\pi\)
\(284\) −5.44668 + 13.3050i −0.323201 + 0.789508i
\(285\) 0 0
\(286\) 31.6119 21.2172i 1.86925 1.25460i
\(287\) −6.93765 −0.409517
\(288\) 0 0
\(289\) −16.9445 −0.996737
\(290\) −4.98930 + 3.34871i −0.292982 + 0.196643i
\(291\) 0 0
\(292\) 7.32227 17.8867i 0.428503 1.04674i
\(293\) 5.00321i 0.292291i −0.989263 0.146145i \(-0.953313\pi\)
0.989263 0.146145i \(-0.0466867\pi\)
\(294\) 0 0
\(295\) −0.702988 −0.0409295
\(296\) −6.30611 + 1.29514i −0.366535 + 0.0752783i
\(297\) 0 0
\(298\) 1.38006 + 2.05618i 0.0799449 + 0.119111i
\(299\) 36.4683i 2.10901i
\(300\) 0 0
\(301\) 21.0771i 1.21486i
\(302\) −3.40697 + 2.28668i −0.196049 + 0.131584i
\(303\) 0 0
\(304\) −14.6735 + 14.9186i −0.841585 + 0.855643i
\(305\) −0.319109 −0.0182721
\(306\) 0 0
\(307\) 9.18634i 0.524292i −0.965028 0.262146i \(-0.915570\pi\)
0.965028 0.262146i \(-0.0844302\pi\)
\(308\) 21.5782 + 8.83346i 1.22953 + 0.503333i
\(309\) 0 0
\(310\) 3.98485 + 5.93709i 0.226324 + 0.337204i
\(311\) −1.99798 −0.113295 −0.0566475 0.998394i \(-0.518041\pi\)
−0.0566475 + 0.998394i \(0.518041\pi\)
\(312\) 0 0
\(313\) −10.6052 −0.599440 −0.299720 0.954027i \(-0.596893\pi\)
−0.299720 + 0.954027i \(0.596893\pi\)
\(314\) −15.5735 23.2032i −0.878862 1.30943i
\(315\) 0 0
\(316\) −8.22610 + 20.0945i −0.462754 + 1.13041i
\(317\) 9.76229i 0.548305i −0.961686 0.274152i \(-0.911603\pi\)
0.961686 0.274152i \(-0.0883973\pi\)
\(318\) 0 0
\(319\) −23.2857 −1.30375
\(320\) 3.15305 + 7.35243i 0.176261 + 0.411013i
\(321\) 0 0
\(322\) 18.5443 12.4465i 1.03343 0.693619i
\(323\) 1.23220i 0.0685616i
\(324\) 0 0
\(325\) 4.91228i 0.272484i
\(326\) −16.9405 25.2400i −0.938248 1.39791i
\(327\) 0 0
\(328\) −1.85575 9.03580i −0.102467 0.498918i
\(329\) 17.3873 0.958594
\(330\) 0 0
\(331\) 5.62096i 0.308956i −0.987996 0.154478i \(-0.950630\pi\)
0.987996 0.154478i \(-0.0493696\pi\)
\(332\) −15.8002 6.46814i −0.867150 0.354985i
\(333\) 0 0
\(334\) 22.3888 15.0269i 1.22506 0.822235i
\(335\) 2.70667 0.147881
\(336\) 0 0
\(337\) −0.804011 −0.0437973 −0.0218986 0.999760i \(-0.506971\pi\)
−0.0218986 + 0.999760i \(0.506971\pi\)
\(338\) 13.0699 8.77224i 0.710910 0.477147i
\(339\) 0 0
\(340\) −0.435965 0.178471i −0.0236435 0.00967895i
\(341\) 27.7092i 1.50054i
\(342\) 0 0
\(343\) −20.1553 −1.08828
\(344\) −27.4514 + 5.63792i −1.48008 + 0.303976i
\(345\) 0 0
\(346\) −7.33795 10.9329i −0.394490 0.587758i
\(347\) 10.7019i 0.574507i 0.957855 + 0.287253i \(0.0927421\pi\)
−0.957855 + 0.287253i \(0.907258\pi\)
\(348\) 0 0
\(349\) 24.9915i 1.33776i −0.743369 0.668881i \(-0.766774\pi\)
0.743369 0.668881i \(-0.233226\pi\)
\(350\) 2.49792 1.67655i 0.133520 0.0896154i
\(351\) 0 0
\(352\) −5.73299 + 30.4669i −0.305570 + 1.62389i
\(353\) −15.4860 −0.824237 −0.412119 0.911130i \(-0.635211\pi\)
−0.412119 + 0.911130i \(0.635211\pi\)
\(354\) 0 0
\(355\) 7.18836i 0.381518i
\(356\) 7.50747 18.3391i 0.397895 0.971969i
\(357\) 0 0
\(358\) 2.13283 + 3.17775i 0.112724 + 0.167949i
\(359\) 11.9335 0.629826 0.314913 0.949121i \(-0.398025\pi\)
0.314913 + 0.949121i \(0.398025\pi\)
\(360\) 0 0
\(361\) −8.36740 −0.440390
\(362\) −5.85098 8.71749i −0.307521 0.458181i
\(363\) 0 0
\(364\) 19.3414 + 7.91780i 1.01377 + 0.415006i
\(365\) 9.66370i 0.505821i
\(366\) 0 0
\(367\) 6.95514 0.363055 0.181528 0.983386i \(-0.441896\pi\)
0.181528 + 0.983386i \(0.441896\pi\)
\(368\) 21.1712 + 20.8233i 1.10362 + 1.08549i
\(369\) 0 0
\(370\) −2.67268 + 1.79384i −0.138946 + 0.0932575i
\(371\) 22.6844i 1.17772i
\(372\) 0 0
\(373\) 11.6296i 0.602156i −0.953600 0.301078i \(-0.902654\pi\)
0.953600 0.301078i \(-0.0973464\pi\)
\(374\) −1.01735 1.51577i −0.0526060 0.0783787i
\(375\) 0 0
\(376\) 4.65094 + 22.6457i 0.239854 + 1.16787i
\(377\) −20.8720 −1.07496
\(378\) 0 0
\(379\) 0.235541i 0.0120989i −0.999982 0.00604945i \(-0.998074\pi\)
0.999982 0.00604945i \(-0.00192561\pi\)
\(380\) −3.96386 + 9.68284i −0.203342 + 0.496719i
\(381\) 0 0
\(382\) 15.5117 10.4111i 0.793648 0.532679i
\(383\) −4.23305 −0.216299 −0.108149 0.994135i \(-0.534493\pi\)
−0.108149 + 0.994135i \(0.534493\pi\)
\(384\) 0 0
\(385\) 11.6581 0.594153
\(386\) 4.38566 2.94356i 0.223224 0.149823i
\(387\) 0 0
\(388\) 11.2819 27.5593i 0.572753 1.39911i
\(389\) 27.0991i 1.37398i −0.726667 0.686990i \(-0.758932\pi\)
0.726667 0.686990i \(-0.241068\pi\)
\(390\) 0 0
\(391\) −1.74863 −0.0884320
\(392\) −1.40820 6.85662i −0.0711248 0.346312i
\(393\) 0 0
\(394\) 5.51943 + 8.22350i 0.278065 + 0.414294i
\(395\) 10.8565i 0.546252i
\(396\) 0 0
\(397\) 21.9552i 1.10190i −0.834539 0.550949i \(-0.814266\pi\)
0.834539 0.550949i \(-0.185734\pi\)
\(398\) −24.9843 + 16.7689i −1.25235 + 0.840550i
\(399\) 0 0
\(400\) 2.85176 + 2.80490i 0.142588 + 0.140245i
\(401\) 8.36813 0.417884 0.208942 0.977928i \(-0.432998\pi\)
0.208942 + 0.977928i \(0.432998\pi\)
\(402\) 0 0
\(403\) 24.8369i 1.23721i
\(404\) −11.3513 4.64690i −0.564750 0.231192i
\(405\) 0 0
\(406\) −7.12355 10.6135i −0.353536 0.526740i
\(407\) −12.4737 −0.618300
\(408\) 0 0
\(409\) −28.2474 −1.39675 −0.698373 0.715734i \(-0.746092\pi\)
−0.698373 + 0.715734i \(0.746092\pi\)
\(410\) −2.57033 3.82959i −0.126940 0.189130i
\(411\) 0 0
\(412\) −6.68896 + 16.3396i −0.329541 + 0.804996i
\(413\) 1.49544i 0.0735856i
\(414\) 0 0
\(415\) −8.53645 −0.419038
\(416\) −5.13873 + 27.3088i −0.251947 + 1.33892i
\(417\) 0 0
\(418\) −33.6655 + 22.5955i −1.64663 + 1.10518i
\(419\) 8.68443i 0.424262i 0.977241 + 0.212131i \(0.0680404\pi\)
−0.977241 + 0.212131i \(0.931960\pi\)
\(420\) 0 0
\(421\) 4.73033i 0.230542i 0.993334 + 0.115271i \(0.0367737\pi\)
−0.993334 + 0.115271i \(0.963226\pi\)
\(422\) 19.2913 + 28.7424i 0.939084 + 1.39916i
\(423\) 0 0
\(424\) −29.5448 + 6.06786i −1.43482 + 0.294681i
\(425\) −0.235541 −0.0114254
\(426\) 0 0
\(427\) 0.678826i 0.0328507i
\(428\) 13.8558 + 5.67215i 0.669746 + 0.274174i
\(429\) 0 0
\(430\) −11.6346 + 7.80887i −0.561069 + 0.376577i
\(431\) −39.1052 −1.88363 −0.941817 0.336127i \(-0.890883\pi\)
−0.941817 + 0.336127i \(0.890883\pi\)
\(432\) 0 0
\(433\) 33.8221 1.62538 0.812692 0.582693i \(-0.198001\pi\)
0.812692 + 0.582693i \(0.198001\pi\)
\(434\) −12.6297 + 8.47679i −0.606246 + 0.406899i
\(435\) 0 0
\(436\) 33.0317 + 13.5222i 1.58193 + 0.647596i
\(437\) 38.8373i 1.85784i
\(438\) 0 0
\(439\) 1.59960 0.0763448 0.0381724 0.999271i \(-0.487846\pi\)
0.0381724 + 0.999271i \(0.487846\pi\)
\(440\) 3.11843 + 15.1839i 0.148666 + 0.723863i
\(441\) 0 0
\(442\) −0.911896 1.35865i −0.0433745 0.0646244i
\(443\) 8.64862i 0.410909i 0.978667 + 0.205454i \(0.0658672\pi\)
−0.978667 + 0.205454i \(0.934133\pi\)
\(444\) 0 0
\(445\) 9.90812i 0.469690i
\(446\) 19.9460 13.3873i 0.944472 0.633909i
\(447\) 0 0
\(448\) −15.6405 + 6.70735i −0.738945 + 0.316893i
\(449\) 15.9597 0.753184 0.376592 0.926379i \(-0.377096\pi\)
0.376592 + 0.926379i \(0.377096\pi\)
\(450\) 0 0
\(451\) 17.8732i 0.841615i
\(452\) 11.2602 27.5062i 0.529636 1.29378i
\(453\) 0 0
\(454\) 16.1726 + 24.0958i 0.759017 + 1.13087i
\(455\) 10.4497 0.489888
\(456\) 0 0
\(457\) −22.6462 −1.05934 −0.529672 0.848203i \(-0.677685\pi\)
−0.529672 + 0.848203i \(0.677685\pi\)
\(458\) 2.37438 + 3.53762i 0.110947 + 0.165302i
\(459\) 0 0
\(460\) 13.7410 + 5.62515i 0.640677 + 0.262274i
\(461\) 22.4400i 1.04513i 0.852599 + 0.522566i \(0.175025\pi\)
−0.852599 + 0.522566i \(0.824975\pi\)
\(462\) 0 0
\(463\) 30.8710 1.43470 0.717348 0.696715i \(-0.245356\pi\)
0.717348 + 0.696715i \(0.245356\pi\)
\(464\) 11.9179 12.1169i 0.553273 0.562514i
\(465\) 0 0
\(466\) 17.8930 12.0094i 0.828879 0.556325i
\(467\) 3.05734i 0.141477i −0.997495 0.0707383i \(-0.977464\pi\)
0.997495 0.0707383i \(-0.0225355\pi\)
\(468\) 0 0
\(469\) 5.75777i 0.265869i
\(470\) 6.44184 + 9.59781i 0.297140 + 0.442714i
\(471\) 0 0
\(472\) 1.94770 0.400014i 0.0896500 0.0184122i
\(473\) −54.3001 −2.49672
\(474\) 0 0
\(475\) 5.23139i 0.240032i
\(476\) 0.379654 0.927409i 0.0174014 0.0425077i
\(477\) 0 0
\(478\) −18.6425 + 12.5124i −0.852686 + 0.572304i
\(479\) 22.4262 1.02468 0.512341 0.858782i \(-0.328779\pi\)
0.512341 + 0.858782i \(0.328779\pi\)
\(480\) 0 0
\(481\) −11.1807 −0.509798
\(482\) −24.6402 + 16.5379i −1.12233 + 0.753282i
\(483\) 0 0
\(484\) −14.4225 + 35.2309i −0.655566 + 1.60140i
\(485\) 14.8895i 0.676100i
\(486\) 0 0
\(487\) −11.9961 −0.543594 −0.271797 0.962355i \(-0.587618\pi\)
−0.271797 + 0.962355i \(0.587618\pi\)
\(488\) 0.884122 0.181579i 0.0400223 0.00821971i
\(489\) 0 0
\(490\) −1.95044 2.90600i −0.0881121 0.131280i
\(491\) 0.261916i 0.0118201i 0.999983 + 0.00591005i \(0.00188124\pi\)
−0.999983 + 0.00591005i \(0.998119\pi\)
\(492\) 0 0
\(493\) 1.00080i 0.0450736i
\(494\) −30.1758 + 20.2533i −1.35767 + 0.911240i
\(495\) 0 0
\(496\) −14.4187 14.1818i −0.647420 0.636784i
\(497\) −15.2915 −0.685917
\(498\) 0 0
\(499\) 43.0171i 1.92571i −0.270018 0.962855i \(-0.587030\pi\)
0.270018 0.962855i \(-0.412970\pi\)
\(500\) 1.85091 + 0.757708i 0.0827753 + 0.0338857i
\(501\) 0 0
\(502\) 6.69741 + 9.97859i 0.298920 + 0.445366i
\(503\) −21.3295 −0.951037 −0.475518 0.879706i \(-0.657740\pi\)
−0.475518 + 0.879706i \(0.657740\pi\)
\(504\) 0 0
\(505\) −6.13283 −0.272908
\(506\) 32.0655 + 47.7749i 1.42548 + 2.12385i
\(507\) 0 0
\(508\) 5.17354 12.6378i 0.229539 0.560712i
\(509\) 15.0919i 0.668936i 0.942407 + 0.334468i \(0.108557\pi\)
−0.942407 + 0.334468i \(0.891443\pi\)
\(510\) 0 0
\(511\) 20.5572 0.909396
\(512\) −12.9195 18.5765i −0.570968 0.820972i
\(513\) 0 0
\(514\) 0.261202 0.175313i 0.0115211 0.00773272i
\(515\) 8.82788i 0.389003i
\(516\) 0 0
\(517\) 44.7942i 1.97005i
\(518\) −3.81596 5.68547i −0.167664 0.249805i
\(519\) 0 0
\(520\) 2.79519 + 13.6099i 0.122577 + 0.596836i
\(521\) 33.8443 1.48274 0.741372 0.671094i \(-0.234175\pi\)
0.741372 + 0.671094i \(0.234175\pi\)
\(522\) 0 0
\(523\) 10.5034i 0.459281i 0.973276 + 0.229640i \(0.0737550\pi\)
−0.973276 + 0.229640i \(0.926245\pi\)
\(524\) −39.5862 16.2054i −1.72933 0.707937i
\(525\) 0 0
\(526\) −9.09220 + 6.10248i −0.396439 + 0.266081i
\(527\) 1.19091 0.0518770
\(528\) 0 0
\(529\) 32.1143 1.39627
\(530\) −12.5218 + 8.40436i −0.543913 + 0.365062i
\(531\) 0 0
\(532\) −20.5979 8.43216i −0.893031 0.365580i
\(533\) 16.0205i 0.693924i
\(534\) 0 0
\(535\) 7.48593 0.323645
\(536\) −7.49909 + 1.54015i −0.323911 + 0.0665243i
\(537\) 0 0
\(538\) 1.97743 + 2.94621i 0.0852532 + 0.127020i
\(539\) 13.5627i 0.584186i
\(540\) 0 0
\(541\) 18.2058i 0.782728i 0.920236 + 0.391364i \(0.127997\pi\)
−0.920236 + 0.391364i \(0.872003\pi\)
\(542\) 8.22627 5.52129i 0.353349 0.237160i
\(543\) 0 0
\(544\) 1.30944 + 0.246399i 0.0561417 + 0.0105643i
\(545\) 17.8462 0.764446
\(546\) 0 0
\(547\) 7.81348i 0.334080i −0.985950 0.167040i \(-0.946579\pi\)
0.985950 0.167040i \(-0.0534209\pi\)
\(548\) 14.8572 36.2929i 0.634669 1.55036i
\(549\) 0 0
\(550\) 4.31922 + 6.43529i 0.184172 + 0.274402i
\(551\) 22.2278 0.946937
\(552\) 0 0
\(553\) −23.0947 −0.982085
\(554\) 6.40789 + 9.54723i 0.272245 + 0.405623i
\(555\) 0 0
\(556\) 1.66891 + 0.683200i 0.0707774 + 0.0289741i
\(557\) 14.7679i 0.625735i 0.949797 + 0.312867i \(0.101290\pi\)
−0.949797 + 0.312867i \(0.898710\pi\)
\(558\) 0 0
\(559\) −48.6715 −2.05858
\(560\) −5.96675 + 6.06642i −0.252141 + 0.256353i
\(561\) 0 0
\(562\) −23.7242 + 15.9231i −1.00074 + 0.671678i
\(563\) 29.5041i 1.24345i 0.783236 + 0.621724i \(0.213567\pi\)
−0.783236 + 0.621724i \(0.786433\pi\)
\(564\) 0 0
\(565\) 14.8609i 0.625203i
\(566\) 21.6440 + 32.2478i 0.909767 + 1.35548i
\(567\) 0 0
\(568\) −4.09032 19.9161i −0.171626 0.835659i
\(569\) 29.0862 1.21935 0.609677 0.792650i \(-0.291299\pi\)
0.609677 + 0.792650i \(0.291299\pi\)
\(570\) 0 0
\(571\) 42.1417i 1.76357i −0.471649 0.881787i \(-0.656341\pi\)
0.471649 0.881787i \(-0.343659\pi\)
\(572\) −20.3983 + 49.8285i −0.852895 + 2.08343i
\(573\) 0 0
\(574\) 8.14651 5.46776i 0.340029 0.228220i
\(575\) 7.42390 0.309598
\(576\) 0 0
\(577\) −33.8258 −1.40819 −0.704094 0.710107i \(-0.748646\pi\)
−0.704094 + 0.710107i \(0.748646\pi\)
\(578\) 19.8970 13.3545i 0.827608 0.555472i
\(579\) 0 0
\(580\) 3.21945 7.86441i 0.133681 0.326552i
\(581\) 18.1592i 0.753371i
\(582\) 0 0
\(583\) −58.4409 −2.42037
\(584\) 5.49884 + 26.7742i 0.227544 + 1.10793i
\(585\) 0 0
\(586\) 3.94317 + 5.87501i 0.162891 + 0.242694i
\(587\) 2.36693i 0.0976937i −0.998806 0.0488469i \(-0.984445\pi\)
0.998806 0.0488469i \(-0.0155546\pi\)
\(588\) 0 0
\(589\) 26.4504i 1.08987i
\(590\) 0.825481 0.554045i 0.0339845 0.0228097i
\(591\) 0 0
\(592\) 6.38419 6.49083i 0.262389 0.266771i
\(593\) 14.0133 0.575459 0.287729 0.957712i \(-0.407100\pi\)
0.287729 + 0.957712i \(0.407100\pi\)
\(594\) 0 0
\(595\) 0.501055i 0.0205413i
\(596\) −3.24107 1.32680i −0.132759 0.0543477i
\(597\) 0 0
\(598\) 28.7417 + 42.8227i 1.17533 + 1.75115i
\(599\) 0.194646 0.00795304 0.00397652 0.999992i \(-0.498734\pi\)
0.00397652 + 0.999992i \(0.498734\pi\)
\(600\) 0 0
\(601\) −40.3962 −1.64780 −0.823898 0.566738i \(-0.808205\pi\)
−0.823898 + 0.566738i \(0.808205\pi\)
\(602\) −16.6115 24.7497i −0.677033 1.00872i
\(603\) 0 0
\(604\) 2.19842 5.37026i 0.0894526 0.218513i
\(605\) 19.0343i 0.773855i
\(606\) 0 0
\(607\) 34.9823 1.41989 0.709944 0.704258i \(-0.248720\pi\)
0.709944 + 0.704258i \(0.248720\pi\)
\(608\) 5.47255 29.0828i 0.221941 1.17946i
\(609\) 0 0
\(610\) 0.374712 0.251498i 0.0151716 0.0101829i
\(611\) 40.1510i 1.62433i
\(612\) 0 0
\(613\) 6.46784i 0.261233i 0.991433 + 0.130617i \(0.0416957\pi\)
−0.991433 + 0.130617i \(0.958304\pi\)
\(614\) 7.24001 + 10.7870i 0.292183 + 0.435329i
\(615\) 0 0
\(616\) −32.3000 + 6.63371i −1.30140 + 0.267280i
\(617\) 20.9261 0.842451 0.421226 0.906956i \(-0.361600\pi\)
0.421226 + 0.906956i \(0.361600\pi\)
\(618\) 0 0
\(619\) 11.1568i 0.448430i −0.974540 0.224215i \(-0.928018\pi\)
0.974540 0.224215i \(-0.0719818\pi\)
\(620\) −9.35838 3.83104i −0.375842 0.153858i
\(621\) 0 0
\(622\) 2.34612 1.57466i 0.0940708 0.0631383i
\(623\) 21.0771 0.844437
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 12.4531 8.35824i 0.497726 0.334062i
\(627\) 0 0
\(628\) 36.5742 + 14.9724i 1.45947 + 0.597463i
\(629\) 0.536109i 0.0213761i
\(630\) 0 0
\(631\) 4.79278 0.190798 0.0953989 0.995439i \(-0.469587\pi\)
0.0953989 + 0.995439i \(0.469587\pi\)
\(632\) −6.17760 30.0791i −0.245732 1.19648i
\(633\) 0 0
\(634\) 7.69394 + 11.4633i 0.305565 + 0.455267i
\(635\) 6.82788i 0.270956i
\(636\) 0 0
\(637\) 12.1568i 0.481670i
\(638\) 27.3431 18.3521i 1.08252 0.726567i
\(639\) 0 0
\(640\) −9.49712 6.14856i −0.375407 0.243043i
\(641\) −42.9892 −1.69797 −0.848986 0.528416i \(-0.822786\pi\)
−0.848986 + 0.528416i \(0.822786\pi\)
\(642\) 0 0
\(643\) 28.0578i 1.10649i 0.833017 + 0.553247i \(0.186611\pi\)
−0.833017 + 0.553247i \(0.813389\pi\)
\(644\) −11.9661 + 29.2306i −0.471532 + 1.15185i
\(645\) 0 0
\(646\) 0.971134 + 1.44691i 0.0382088 + 0.0569279i
\(647\) −5.30186 −0.208437 −0.104219 0.994554i \(-0.533234\pi\)
−0.104219 + 0.994554i \(0.533234\pi\)
\(648\) 0 0
\(649\) 3.85263 0.151229
\(650\) 3.87150 + 5.76822i 0.151853 + 0.226248i
\(651\) 0 0
\(652\) 39.7846 + 16.2866i 1.55809 + 0.637834i
\(653\) 10.6725i 0.417647i 0.977953 + 0.208823i \(0.0669634\pi\)
−0.977953 + 0.208823i \(0.933037\pi\)
\(654\) 0 0
\(655\) −21.3874 −0.835676
\(656\) 9.30048 + 9.14768i 0.363123 + 0.357157i
\(657\) 0 0
\(658\) −20.4170 + 13.7034i −0.795938 + 0.534216i
\(659\) 15.6338i 0.609008i 0.952511 + 0.304504i \(0.0984907\pi\)
−0.952511 + 0.304504i \(0.901509\pi\)
\(660\) 0 0
\(661\) 35.2867i 1.37249i 0.727369 + 0.686247i \(0.240743\pi\)
−0.727369 + 0.686247i \(0.759257\pi\)
\(662\) 4.43004 + 6.60040i 0.172178 + 0.256532i
\(663\) 0 0
\(664\) 23.6511 4.85741i 0.917839 0.188504i
\(665\) −11.1285 −0.431545
\(666\) 0 0
\(667\) 31.5437i 1.22138i
\(668\) −14.4469 + 35.2905i −0.558967 + 1.36543i
\(669\) 0 0
\(670\) −3.17829 + 2.13320i −0.122788 + 0.0824127i
\(671\) 1.74883 0.0675128
\(672\) 0 0
\(673\) 43.2233 1.66614 0.833068 0.553170i \(-0.186582\pi\)
0.833068 + 0.553170i \(0.186582\pi\)
\(674\) 0.944107 0.633664i 0.0363657 0.0244078i
\(675\) 0 0
\(676\) −8.43365 + 20.6015i −0.324371 + 0.792367i
\(677\) 39.8297i 1.53078i −0.643566 0.765391i \(-0.722546\pi\)
0.643566 0.765391i \(-0.277454\pi\)
\(678\) 0 0
\(679\) 31.6739 1.21553
\(680\) 0.652588 0.134027i 0.0250256 0.00513971i
\(681\) 0 0
\(682\) −21.8384 32.5374i −0.836235 1.24592i
\(683\) 14.7286i 0.563574i −0.959477 0.281787i \(-0.909073\pi\)
0.959477 0.281787i \(-0.0909271\pi\)
\(684\) 0 0
\(685\) 19.6081i 0.749187i
\(686\) 23.6673 15.8849i 0.903621 0.606490i
\(687\) 0 0
\(688\) 27.7913 28.2556i 1.05953 1.07723i
\(689\) −52.3831 −1.99564
\(690\) 0 0
\(691\) 31.2491i 1.18877i −0.804180 0.594386i \(-0.797395\pi\)
0.804180 0.594386i \(-0.202605\pi\)
\(692\) 17.2331 + 7.05472i 0.655105 + 0.268180i
\(693\) 0 0
\(694\) −8.43445 12.5666i −0.320167 0.477023i
\(695\) 0.901666 0.0342022
\(696\) 0 0
\(697\) −0.768172 −0.0290966
\(698\) 19.6965 + 29.3461i 0.745522 + 1.11077i
\(699\) 0 0
\(700\) −1.61184 + 3.93737i −0.0609218 + 0.148818i
\(701\) 4.52968i 0.171084i −0.996335 0.0855419i \(-0.972738\pi\)
0.996335 0.0855419i \(-0.0272621\pi\)
\(702\) 0 0
\(703\) 11.9071 0.449083
\(704\) −17.2799 40.2940i −0.651259 1.51864i
\(705\) 0 0
\(706\) 18.1844 12.2050i 0.684379 0.459340i
\(707\) 13.0461i 0.490649i
\(708\) 0 0
\(709\) 15.1260i 0.568069i −0.958814 0.284035i \(-0.908327\pi\)
0.958814 0.284035i \(-0.0916731\pi\)
\(710\) −5.66535 8.44090i −0.212617 0.316781i
\(711\) 0 0
\(712\) 5.63792 + 27.4514i 0.211290 + 1.02879i
\(713\) −37.5359 −1.40573
\(714\) 0 0
\(715\) 26.9210i 1.00679i
\(716\) −5.00894 2.05051i −0.187193 0.0766312i
\(717\) 0 0
\(718\) −14.0129 + 9.40513i −0.522956 + 0.350996i
\(719\) 12.8773 0.480243 0.240122 0.970743i \(-0.422813\pi\)
0.240122 + 0.970743i \(0.422813\pi\)
\(720\) 0 0
\(721\) −18.7792 −0.699373
\(722\) 9.82539 6.59458i 0.365663 0.245425i
\(723\) 0 0
\(724\) 13.7410 + 5.62515i 0.510680 + 0.209057i
\(725\) 4.24894i 0.157802i
\(726\) 0 0
\(727\) 7.50213 0.278239 0.139119 0.990276i \(-0.455573\pi\)
0.139119 + 0.990276i \(0.455573\pi\)
\(728\) −28.9519 + 5.94608i −1.07303 + 0.220376i
\(729\) 0 0
\(730\) 7.61624 + 11.3476i 0.281890 + 0.419992i
\(731\) 2.33376i 0.0863174i
\(732\) 0 0
\(733\) 12.4046i 0.458174i −0.973406 0.229087i \(-0.926426\pi\)
0.973406 0.229087i \(-0.0735740\pi\)
\(734\) −8.16704 + 5.48154i −0.301451 + 0.202327i
\(735\) 0 0
\(736\) −41.2716 7.76613i −1.52129 0.286263i
\(737\) −14.8335 −0.546399
\(738\) 0 0
\(739\) 49.6664i 1.82701i 0.406830 + 0.913504i \(0.366634\pi\)
−0.406830 + 0.913504i \(0.633366\pi\)
\(740\) 1.72461 4.21283i 0.0633978 0.154867i
\(741\) 0 0
\(742\) −17.8782 26.6371i −0.656331 0.977879i
\(743\) −26.1312 −0.958662 −0.479331 0.877634i \(-0.659121\pi\)
−0.479331 + 0.877634i \(0.659121\pi\)
\(744\) 0 0
\(745\) −1.75106 −0.0641540
\(746\) 9.16558 + 13.6560i 0.335576 + 0.499980i
\(747\) 0 0
\(748\) 2.38924 + 0.978085i 0.0873594 + 0.0357623i
\(749\) 15.9245i 0.581869i
\(750\) 0 0
\(751\) 42.6818 1.55748 0.778741 0.627346i \(-0.215859\pi\)
0.778741 + 0.627346i \(0.215859\pi\)
\(752\) −23.3091 22.9261i −0.849995 0.836030i
\(753\) 0 0
\(754\) 24.5088 16.4498i 0.892558 0.599065i
\(755\) 2.90141i 0.105593i
\(756\) 0 0
\(757\) 37.3850i 1.35878i −0.733777 0.679391i \(-0.762244\pi\)
0.733777 0.679391i \(-0.237756\pi\)
\(758\) 0.185636 + 0.276583i 0.00674261 + 0.0100459i
\(759\) 0 0
\(760\) −2.97676 14.4941i −0.107979 0.525755i
\(761\) −20.2038 −0.732386 −0.366193 0.930539i \(-0.619339\pi\)
−0.366193 + 0.930539i \(0.619339\pi\)
\(762\) 0 0
\(763\) 37.9634i 1.37437i
\(764\) −10.0093 + 24.4504i −0.362123 + 0.884586i
\(765\) 0 0
\(766\) 4.97065 3.33619i 0.179597 0.120541i
\(767\) 3.45327 0.124690
\(768\) 0 0
\(769\) 28.9265 1.04312 0.521558 0.853216i \(-0.325351\pi\)
0.521558 + 0.853216i \(0.325351\pi\)
\(770\) −13.6895 + 9.18810i −0.493336 + 0.331116i
\(771\) 0 0
\(772\) −2.82994 + 6.91292i −0.101852 + 0.248801i
\(773\) 47.0349i 1.69173i 0.533400 + 0.845863i \(0.320914\pi\)
−0.533400 + 0.845863i \(0.679086\pi\)
\(774\) 0 0
\(775\) −5.05609 −0.181620
\(776\) 8.47245 + 41.2530i 0.304143 + 1.48089i
\(777\) 0 0
\(778\) 21.3576 + 31.8210i 0.765706 + 1.14084i
\(779\) 17.0612i 0.611281i
\(780\) 0 0
\(781\) 39.3948i 1.40966i
\(782\) 2.05332 1.37814i 0.0734266 0.0492823i
\(783\) 0 0
\(784\) 7.05747 + 6.94152i 0.252052 + 0.247911i
\(785\) 19.7601 0.705267
\(786\) 0 0
\(787\) 27.5698i 0.982756i −0.870946 0.491378i \(-0.836493\pi\)
0.870946 0.491378i \(-0.163507\pi\)
\(788\) −12.9623 5.30639i −0.461764 0.189032i
\(789\) 0 0
\(790\) −8.55635 12.7483i −0.304421 0.453563i
\(791\) 31.6129 1.12403
\(792\) 0 0
\(793\) 1.56755 0.0556653
\(794\) 17.3035 + 25.7808i 0.614078 + 0.914925i
\(795\) 0 0
\(796\) 16.1217 39.3817i 0.571418 1.39585i
\(797\) 46.4890i 1.64672i −0.567516 0.823362i \(-0.692096\pi\)
0.567516 0.823362i \(-0.307904\pi\)
\(798\) 0 0
\(799\) 1.92521 0.0681091
\(800\) −5.55929 1.04610i −0.196550 0.0369852i
\(801\) 0 0
\(802\) −9.82624 + 6.59516i −0.346977 + 0.232883i
\(803\) 52.9605i 1.86894i
\(804\) 0 0
\(805\) 15.7925i 0.556614i
\(806\) −19.5747 29.1647i −0.689488 1.02728i
\(807\) 0 0
\(808\) 16.9916 3.48971i 0.597763 0.122767i
\(809\) −47.2189 −1.66013 −0.830064 0.557668i \(-0.811696\pi\)
−0.830064 + 0.557668i \(0.811696\pi\)
\(810\) 0 0
\(811\) 9.83765i 0.345447i −0.984970 0.172723i \(-0.944743\pi\)
0.984970 0.172723i \(-0.0552567\pi\)
\(812\) 16.7296 + 6.84860i 0.587094 + 0.240339i
\(813\) 0 0
\(814\) 14.6472 9.83090i 0.513385 0.344573i
\(815\) 21.4946 0.752923
\(816\) 0 0
\(817\) 51.8332 1.81341
\(818\) 33.1695 22.2626i 1.15974 0.778394i
\(819\) 0 0
\(820\) 6.03641 + 2.47113i 0.210801 + 0.0862955i
\(821\) 35.6781i 1.24517i 0.782551 + 0.622587i \(0.213918\pi\)
−0.782551 + 0.622587i \(0.786082\pi\)
\(822\) 0 0
\(823\) −23.7092 −0.826450 −0.413225 0.910629i \(-0.635598\pi\)
−0.413225 + 0.910629i \(0.635598\pi\)
\(824\) −5.02324 24.4585i −0.174993 0.852053i
\(825\) 0 0
\(826\) 1.17859 + 1.75601i 0.0410086 + 0.0610994i
\(827\) 9.64454i 0.335373i −0.985840 0.167687i \(-0.946370\pi\)
0.985840 0.167687i \(-0.0536297\pi\)
\(828\) 0 0
\(829\) 30.3639i 1.05458i 0.849685 + 0.527291i \(0.176792\pi\)
−0.849685 + 0.527291i \(0.823208\pi\)
\(830\) 10.0239 6.72781i 0.347934 0.233526i
\(831\) 0 0
\(832\) −15.4887 36.1172i −0.536973 1.25214i
\(833\) −0.582911 −0.0201967
\(834\) 0 0
\(835\) 19.0666i 0.659825i
\(836\) 21.7234 53.0654i 0.751319 1.83531i
\(837\) 0 0
\(838\) −6.84444 10.1977i −0.236437 0.352272i
\(839\) 5.58047 0.192659 0.0963296 0.995349i \(-0.469290\pi\)
0.0963296 + 0.995349i \(0.469290\pi\)
\(840\) 0 0
\(841\) 10.9465 0.377467
\(842\) −3.72811 5.55457i −0.128479 0.191423i
\(843\) 0 0
\(844\) −45.3054 18.5467i −1.55947 0.638402i
\(845\) 11.1305i 0.382900i
\(846\) 0 0
\(847\) −40.4909 −1.39128
\(848\) 29.9107 30.4103i 1.02714 1.04429i
\(849\) 0 0
\(850\) 0.276583 0.185636i 0.00948670 0.00636727i
\(851\) 16.8974i 0.579235i
\(852\) 0 0
\(853\) 50.6473i 1.73413i −0.498193 0.867066i \(-0.666003\pi\)
0.498193 0.867066i \(-0.333997\pi\)
\(854\) 0.535002 + 0.797109i 0.0183074 + 0.0272765i
\(855\) 0 0
\(856\) −20.7405 + 4.25965i −0.708896 + 0.145592i
\(857\) −19.1906 −0.655539 −0.327769 0.944758i \(-0.606297\pi\)
−0.327769 + 0.944758i \(0.606297\pi\)
\(858\) 0 0
\(859\) 1.31864i 0.0449915i −0.999747 0.0224957i \(-0.992839\pi\)
0.999747 0.0224957i \(-0.00716122\pi\)
\(860\) 7.50747 18.3391i 0.256003 0.625357i
\(861\) 0 0
\(862\) 45.9192 30.8199i 1.56401 1.04973i
\(863\) 12.3923 0.421840 0.210920 0.977503i \(-0.432354\pi\)
0.210920 + 0.977503i \(0.432354\pi\)
\(864\) 0 0
\(865\) 9.31060 0.316570
\(866\) −39.7154 + 26.6561i −1.34959 + 0.905812i
\(867\) 0 0
\(868\) 8.14961 19.9077i 0.276616 0.675710i
\(869\) 59.4978i 2.01832i
\(870\) 0 0
\(871\) −13.2959 −0.450514
\(872\) −49.4446 + 10.1548i −1.67440 + 0.343886i
\(873\) 0 0
\(874\) −30.6088 45.6045i −1.03536 1.54260i
\(875\) 2.12726i 0.0719144i
\(876\) 0 0
\(877\) 36.8292i 1.24363i 0.783163 + 0.621816i \(0.213605\pi\)
−0.783163 + 0.621816i \(0.786395\pi\)
\(878\) −1.87833 + 1.26069i −0.0633904 + 0.0425462i
\(879\) 0 0
\(880\) −15.6286 15.3719i −0.526841 0.518186i
\(881\) −21.4110 −0.721356 −0.360678 0.932690i \(-0.617455\pi\)
−0.360678 + 0.932690i \(0.617455\pi\)
\(882\) 0 0
\(883\) 1.59479i 0.0536690i 0.999640 + 0.0268345i \(0.00854271\pi\)
−0.999640 + 0.0268345i \(0.991457\pi\)
\(884\) 2.14158 + 0.876699i 0.0720292 + 0.0294866i
\(885\) 0 0
\(886\) −6.81622 10.1556i −0.228996 0.341185i
\(887\) 28.6129 0.960727 0.480363 0.877070i \(-0.340505\pi\)
0.480363 + 0.877070i \(0.340505\pi\)
\(888\) 0 0
\(889\) 14.5247 0.487141
\(890\) 7.80887 + 11.6346i 0.261754 + 0.389992i
\(891\) 0 0
\(892\) −12.8706 + 31.4401i −0.430940 + 1.05269i
\(893\) 42.7592i 1.43088i
\(894\) 0 0
\(895\) −2.70620 −0.0904584
\(896\) 13.0796 20.2028i 0.436957 0.674928i
\(897\) 0 0
\(898\) −18.7406 + 12.5783i −0.625381 + 0.419742i
\(899\) 21.4830i 0.716498i
\(900\) 0 0
\(901\) 2.51173i 0.0836780i
\(902\) 14.0864 + 20.9875i 0.469024 + 0.698808i
\(903\) 0 0
\(904\) 8.45615 + 41.1736i 0.281247 + 1.36941i
\(905\) 7.42390 0.246779
\(906\) 0 0
\(907\) 39.3412i 1.30630i 0.757228 + 0.653151i \(0.226553\pi\)
−0.757228 + 0.653151i \(0.773447\pi\)
\(908\) −37.9812 15.5484i −1.26045 0.515991i
\(909\) 0 0
\(910\) −12.2705 + 8.23568i −0.406763 + 0.273010i
\(911\) 33.6919 1.11626 0.558131 0.829753i \(-0.311519\pi\)
0.558131 + 0.829753i \(0.311519\pi\)
\(912\) 0 0
\(913\) 46.7828 1.54828
\(914\) 26.5922 17.8481i 0.879592 0.590363i
\(915\) 0 0
\(916\) −5.57620 2.28273i −0.184243 0.0754235i
\(917\) 45.4965i 1.50243i
\(918\) 0 0
\(919\) 39.2196 1.29373 0.646867 0.762603i \(-0.276079\pi\)
0.646867 + 0.762603i \(0.276079\pi\)
\(920\) −20.5686 + 4.22435i −0.678128 + 0.139273i
\(921\) 0 0
\(922\) −17.6856 26.3500i −0.582443 0.867792i
\(923\) 35.3112i 1.16228i
\(924\) 0 0
\(925\) 2.27608i 0.0748371i
\(926\) −36.2501 + 24.3303i −1.19125 + 0.799543i
\(927\) 0 0
\(928\) −4.44481 + 23.6211i −0.145908 + 0.775399i
\(929\) 45.1242 1.48048 0.740239 0.672344i \(-0.234712\pi\)
0.740239 + 0.672344i \(0.234712\pi\)
\(930\) 0 0
\(931\) 12.9465i 0.424305i
\(932\) −11.5459 + 28.2040i −0.378198 + 0.923853i
\(933\) 0 0
\(934\) 2.40957 + 3.59007i 0.0788436 + 0.117471i
\(935\) 1.29085 0.0422152
\(936\) 0 0
\(937\) −56.9008 −1.85887 −0.929435 0.368986i \(-0.879705\pi\)
−0.929435 + 0.368986i \(0.879705\pi\)
\(938\) −4.53786 6.76104i −0.148166 0.220756i
\(939\) 0 0
\(940\) −15.1286 6.19320i −0.493441 0.202000i
\(941\) 56.6437i 1.84653i −0.384163 0.923265i \(-0.625510\pi\)
0.384163 0.923265i \(-0.374490\pi\)
\(942\) 0 0
\(943\) 24.2117 0.788441
\(944\) −1.97181 + 2.00475i −0.0641771 + 0.0652491i
\(945\) 0 0
\(946\) 63.7616 42.7954i 2.07307 1.39140i
\(947\) 19.1789i 0.623230i 0.950208 + 0.311615i \(0.100870\pi\)
−0.950208 + 0.311615i \(0.899130\pi\)
\(948\) 0 0
\(949\) 47.4708i 1.54097i
\(950\) −4.12300 6.14294i −0.133768 0.199303i
\(951\) 0 0
\(952\) 0.285111 + 1.38822i 0.00924048 + 0.0449926i
\(953\) 29.3512 0.950777 0.475389 0.879776i \(-0.342307\pi\)
0.475389 + 0.879776i \(0.342307\pi\)
\(954\) 0 0
\(955\) 13.2099i 0.427463i
\(956\) 12.0295 29.3853i 0.389060 0.950388i
\(957\) 0 0
\(958\) −26.3339 + 17.6748i −0.850811 + 0.571045i
\(959\) 41.7115 1.34693
\(960\) 0 0
\(961\) −5.43596 −0.175354
\(962\) 13.1289 8.81186i 0.423294 0.284106i
\(963\) 0 0
\(964\) 15.8996 38.8392i 0.512092 1.25093i
\(965\) 3.73487i 0.120230i
\(966\) 0 0
\(967\) −57.6440 −1.85371 −0.926854 0.375423i \(-0.877497\pi\)
−0.926854 + 0.375423i \(0.877497\pi\)
\(968\) −10.8309 52.7365i −0.348119 1.69501i
\(969\) 0 0
\(970\) 11.7349 + 17.4840i 0.376784 + 0.561377i
\(971\) 6.42670i 0.206243i −0.994669 0.103121i \(-0.967117\pi\)
0.994669 0.103121i \(-0.0328830\pi\)
\(972\) 0 0
\(973\) 1.91808i 0.0614907i
\(974\) 14.0863 9.45444i 0.451356 0.302940i
\(975\) 0 0
\(976\) −0.895069 + 0.910020i −0.0286505 + 0.0291290i
\(977\) −29.6408 −0.948293 −0.474147 0.880446i \(-0.657243\pi\)
−0.474147 + 0.880446i \(0.657243\pi\)
\(978\) 0 0
\(979\) 54.3001i 1.73544i
\(980\) 4.58060 + 1.87516i 0.146322 + 0.0598998i
\(981\) 0 0
\(982\) −0.206423 0.307554i −0.00658723 0.00981443i
\(983\) −20.4287 −0.651573 −0.325787 0.945443i \(-0.605629\pi\)
−0.325787 + 0.945443i \(0.605629\pi\)
\(984\) 0 0
\(985\) −7.00321 −0.223141
\(986\) −0.788756 1.17518i −0.0251191 0.0374254i
\(987\) 0 0
\(988\) 19.4716 47.5648i 0.619474 1.51324i
\(989\) 73.5569i 2.33897i
\(990\) 0 0
\(991\) 43.0135 1.36637 0.683184 0.730246i \(-0.260595\pi\)
0.683184 + 0.730246i \(0.260595\pi\)
\(992\) 28.1083 + 5.28917i 0.892438 + 0.167931i
\(993\) 0 0
\(994\) 17.9560 12.0516i 0.569529 0.382255i
\(995\) 21.2769i 0.674523i
\(996\) 0 0
\(997\) 51.4261i 1.62868i 0.580387 + 0.814341i \(0.302901\pi\)
−0.580387 + 0.814341i \(0.697099\pi\)
\(998\) 33.9030 + 50.5127i 1.07318 + 1.59895i
\(999\) 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 1080.2.k.d.541.6 yes 20
3.2 odd 2 inner 1080.2.k.d.541.15 yes 20
4.3 odd 2 4320.2.k.d.2161.4 20
8.3 odd 2 4320.2.k.d.2161.13 20
8.5 even 2 inner 1080.2.k.d.541.5 20
12.11 even 2 4320.2.k.d.2161.14 20
24.5 odd 2 inner 1080.2.k.d.541.16 yes 20
24.11 even 2 4320.2.k.d.2161.3 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.2.k.d.541.5 20 8.5 even 2 inner
1080.2.k.d.541.6 yes 20 1.1 even 1 trivial
1080.2.k.d.541.15 yes 20 3.2 odd 2 inner
1080.2.k.d.541.16 yes 20 24.5 odd 2 inner
4320.2.k.d.2161.3 20 24.11 even 2
4320.2.k.d.2161.4 20 4.3 odd 2
4320.2.k.d.2161.13 20 8.3 odd 2
4320.2.k.d.2161.14 20 12.11 even 2