Properties

Label 360.2.q.b.241.2
Level $360$
Weight $2$
Character 360.241
Analytic conductor $2.875$
Analytic rank $0$
Dimension $4$
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] = [360,2,Mod(121,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.q (of order \(3\), 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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 241.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 360.241
Dual form 360.2.q.b.121.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 1.50000i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.133975 + 0.232051i) q^{7} +(-1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(0.866025 + 1.50000i) q^{3} +(0.500000 + 0.866025i) q^{5} +(-0.133975 + 0.232051i) q^{7} +(-1.50000 + 2.59808i) q^{9} +(0.732051 - 1.26795i) q^{11} +(2.73205 + 4.73205i) q^{13} +(-0.866025 + 1.50000i) q^{15} +0.535898 q^{17} -2.00000 q^{19} -0.464102 q^{21} +(-1.86603 - 3.23205i) q^{23} +(-0.500000 + 0.866025i) q^{25} -5.19615 q^{27} +(-0.767949 + 1.33013i) q^{29} +(1.00000 + 1.73205i) q^{31} +2.53590 q^{33} -0.267949 q^{35} +10.3923 q^{37} +(-4.73205 + 8.19615i) q^{39} +(-4.96410 - 8.59808i) q^{41} +(2.26795 - 3.92820i) q^{43} -3.00000 q^{45} +(-0.133975 + 0.232051i) q^{47} +(3.46410 + 6.00000i) q^{49} +(0.464102 + 0.803848i) q^{51} +6.00000 q^{53} +1.46410 q^{55} +(-1.73205 - 3.00000i) q^{57} +(-7.19615 - 12.4641i) q^{59} +(4.23205 - 7.33013i) q^{61} +(-0.401924 - 0.696152i) q^{63} +(-2.73205 + 4.73205i) q^{65} +(-3.13397 - 5.42820i) q^{67} +(3.23205 - 5.59808i) q^{69} +9.46410 q^{71} +6.92820 q^{73} -1.73205 q^{75} +(0.196152 + 0.339746i) q^{77} +(-7.73205 + 13.3923i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(6.59808 - 11.4282i) q^{83} +(0.267949 + 0.464102i) q^{85} -2.66025 q^{87} -9.92820 q^{89} -1.46410 q^{91} +(-1.73205 + 3.00000i) q^{93} +(-1.00000 - 1.73205i) q^{95} +(4.46410 - 7.73205i) q^{97} +(2.19615 + 3.80385i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 4 q^{7} - 6 q^{9} - 4 q^{11} + 4 q^{13} + 16 q^{17} - 8 q^{19} + 12 q^{21} - 4 q^{23} - 2 q^{25} - 10 q^{29} + 4 q^{31} + 24 q^{33} - 8 q^{35} - 12 q^{39} - 6 q^{41} + 16 q^{43} - 12 q^{45} - 4 q^{47} - 12 q^{51} + 24 q^{53} - 8 q^{55} - 8 q^{59} + 10 q^{61} - 12 q^{63} - 4 q^{65} - 16 q^{67} + 6 q^{69} + 24 q^{71} - 20 q^{77} - 24 q^{79} - 18 q^{81} + 16 q^{83} + 8 q^{85} + 24 q^{87} - 12 q^{89} + 8 q^{91} - 4 q^{95} + 4 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.866025 + 1.50000i 0.500000 + 0.866025i
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −0.133975 + 0.232051i −0.0506376 + 0.0877070i −0.890233 0.455505i \(-0.849459\pi\)
0.839596 + 0.543212i \(0.182792\pi\)
\(8\) 0 0
\(9\) −1.50000 + 2.59808i −0.500000 + 0.866025i
\(10\) 0 0
\(11\) 0.732051 1.26795i 0.220722 0.382301i −0.734306 0.678819i \(-0.762492\pi\)
0.955027 + 0.296518i \(0.0958254\pi\)
\(12\) 0 0
\(13\) 2.73205 + 4.73205i 0.757735 + 1.31243i 0.944003 + 0.329936i \(0.107027\pi\)
−0.186269 + 0.982499i \(0.559640\pi\)
\(14\) 0 0
\(15\) −0.866025 + 1.50000i −0.223607 + 0.387298i
\(16\) 0 0
\(17\) 0.535898 0.129974 0.0649872 0.997886i \(-0.479299\pi\)
0.0649872 + 0.997886i \(0.479299\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −0.464102 −0.101275
\(22\) 0 0
\(23\) −1.86603 3.23205i −0.389093 0.673929i 0.603235 0.797564i \(-0.293878\pi\)
−0.992328 + 0.123635i \(0.960545\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) −0.767949 + 1.33013i −0.142605 + 0.246998i −0.928477 0.371391i \(-0.878881\pi\)
0.785872 + 0.618389i \(0.212214\pi\)
\(30\) 0 0
\(31\) 1.00000 + 1.73205i 0.179605 + 0.311086i 0.941745 0.336327i \(-0.109185\pi\)
−0.762140 + 0.647412i \(0.775851\pi\)
\(32\) 0 0
\(33\) 2.53590 0.441443
\(34\) 0 0
\(35\) −0.267949 −0.0452917
\(36\) 0 0
\(37\) 10.3923 1.70848 0.854242 0.519875i \(-0.174022\pi\)
0.854242 + 0.519875i \(0.174022\pi\)
\(38\) 0 0
\(39\) −4.73205 + 8.19615i −0.757735 + 1.31243i
\(40\) 0 0
\(41\) −4.96410 8.59808i −0.775262 1.34279i −0.934647 0.355577i \(-0.884284\pi\)
0.159384 0.987217i \(-0.449049\pi\)
\(42\) 0 0
\(43\) 2.26795 3.92820i 0.345859 0.599045i −0.639650 0.768666i \(-0.720921\pi\)
0.985509 + 0.169621i \(0.0542542\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −0.133975 + 0.232051i −0.0195422 + 0.0338481i −0.875631 0.482980i \(-0.839554\pi\)
0.856089 + 0.516829i \(0.172888\pi\)
\(48\) 0 0
\(49\) 3.46410 + 6.00000i 0.494872 + 0.857143i
\(50\) 0 0
\(51\) 0.464102 + 0.803848i 0.0649872 + 0.112561i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 1.46410 0.197419
\(56\) 0 0
\(57\) −1.73205 3.00000i −0.229416 0.397360i
\(58\) 0 0
\(59\) −7.19615 12.4641i −0.936859 1.62269i −0.771284 0.636491i \(-0.780385\pi\)
−0.165575 0.986197i \(-0.552948\pi\)
\(60\) 0 0
\(61\) 4.23205 7.33013i 0.541859 0.938527i −0.456939 0.889498i \(-0.651054\pi\)
0.998797 0.0490285i \(-0.0156125\pi\)
\(62\) 0 0
\(63\) −0.401924 0.696152i −0.0506376 0.0877070i
\(64\) 0 0
\(65\) −2.73205 + 4.73205i −0.338869 + 0.586939i
\(66\) 0 0
\(67\) −3.13397 5.42820i −0.382876 0.663161i 0.608596 0.793480i \(-0.291733\pi\)
−0.991472 + 0.130320i \(0.958400\pi\)
\(68\) 0 0
\(69\) 3.23205 5.59808i 0.389093 0.673929i
\(70\) 0 0
\(71\) 9.46410 1.12318 0.561591 0.827415i \(-0.310189\pi\)
0.561591 + 0.827415i \(0.310189\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) −1.73205 −0.200000
\(76\) 0 0
\(77\) 0.196152 + 0.339746i 0.0223536 + 0.0387176i
\(78\) 0 0
\(79\) −7.73205 + 13.3923i −0.869924 + 1.50675i −0.00784992 + 0.999969i \(0.502499\pi\)
−0.862074 + 0.506783i \(0.830835\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 6.59808 11.4282i 0.724233 1.25441i −0.235056 0.971982i \(-0.575527\pi\)
0.959289 0.282426i \(-0.0911393\pi\)
\(84\) 0 0
\(85\) 0.267949 + 0.464102i 0.0290632 + 0.0503389i
\(86\) 0 0
\(87\) −2.66025 −0.285209
\(88\) 0 0
\(89\) −9.92820 −1.05239 −0.526194 0.850365i \(-0.676381\pi\)
−0.526194 + 0.850365i \(0.676381\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 0 0
\(93\) −1.73205 + 3.00000i −0.179605 + 0.311086i
\(94\) 0 0
\(95\) −1.00000 1.73205i −0.102598 0.177705i
\(96\) 0 0
\(97\) 4.46410 7.73205i 0.453261 0.785071i −0.545326 0.838224i \(-0.683594\pi\)
0.998586 + 0.0531536i \(0.0169273\pi\)
\(98\) 0 0
\(99\) 2.19615 + 3.80385i 0.220722 + 0.382301i
\(100\) 0 0
\(101\) −0.464102 + 0.803848i −0.0461798 + 0.0799858i −0.888191 0.459474i \(-0.848038\pi\)
0.842012 + 0.539459i \(0.181371\pi\)
\(102\) 0 0
\(103\) −7.19615 12.4641i −0.709058 1.22812i −0.965207 0.261487i \(-0.915787\pi\)
0.256149 0.966637i \(-0.417546\pi\)
\(104\) 0 0
\(105\) −0.232051 0.401924i −0.0226458 0.0392237i
\(106\) 0 0
\(107\) 5.19615 0.502331 0.251166 0.967944i \(-0.419186\pi\)
0.251166 + 0.967944i \(0.419186\pi\)
\(108\) 0 0
\(109\) −17.3923 −1.66588 −0.832940 0.553363i \(-0.813344\pi\)
−0.832940 + 0.553363i \(0.813344\pi\)
\(110\) 0 0
\(111\) 9.00000 + 15.5885i 0.854242 + 1.47959i
\(112\) 0 0
\(113\) −6.92820 12.0000i −0.651751 1.12887i −0.982698 0.185216i \(-0.940702\pi\)
0.330947 0.943649i \(-0.392632\pi\)
\(114\) 0 0
\(115\) 1.86603 3.23205i 0.174008 0.301390i
\(116\) 0 0
\(117\) −16.3923 −1.51547
\(118\) 0 0
\(119\) −0.0717968 + 0.124356i −0.00658160 + 0.0113997i
\(120\) 0 0
\(121\) 4.42820 + 7.66987i 0.402564 + 0.697261i
\(122\) 0 0
\(123\) 8.59808 14.8923i 0.775262 1.34279i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.2679 1.44355 0.721774 0.692129i \(-0.243327\pi\)
0.721774 + 0.692129i \(0.243327\pi\)
\(128\) 0 0
\(129\) 7.85641 0.691718
\(130\) 0 0
\(131\) 1.73205 + 3.00000i 0.151330 + 0.262111i 0.931717 0.363186i \(-0.118311\pi\)
−0.780387 + 0.625297i \(0.784978\pi\)
\(132\) 0 0
\(133\) 0.267949 0.464102i 0.0232341 0.0402427i
\(134\) 0 0
\(135\) −2.59808 4.50000i −0.223607 0.387298i
\(136\) 0 0
\(137\) −10.9282 + 18.9282i −0.933659 + 1.61715i −0.156652 + 0.987654i \(0.550070\pi\)
−0.777007 + 0.629492i \(0.783263\pi\)
\(138\) 0 0
\(139\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(140\) 0 0
\(141\) −0.464102 −0.0390844
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −1.53590 −0.127549
\(146\) 0 0
\(147\) −6.00000 + 10.3923i −0.494872 + 0.857143i
\(148\) 0 0
\(149\) 1.23205 + 2.13397i 0.100934 + 0.174822i 0.912070 0.410036i \(-0.134484\pi\)
−0.811136 + 0.584858i \(0.801150\pi\)
\(150\) 0 0
\(151\) −2.73205 + 4.73205i −0.222331 + 0.385089i −0.955515 0.294941i \(-0.904700\pi\)
0.733184 + 0.680030i \(0.238033\pi\)
\(152\) 0 0
\(153\) −0.803848 + 1.39230i −0.0649872 + 0.112561i
\(154\) 0 0
\(155\) −1.00000 + 1.73205i −0.0803219 + 0.139122i
\(156\) 0 0
\(157\) −2.46410 4.26795i −0.196657 0.340619i 0.750786 0.660546i \(-0.229675\pi\)
−0.947442 + 0.319926i \(0.896342\pi\)
\(158\) 0 0
\(159\) 5.19615 + 9.00000i 0.412082 + 0.713746i
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −13.3205 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(164\) 0 0
\(165\) 1.26795 + 2.19615i 0.0987097 + 0.170970i
\(166\) 0 0
\(167\) 10.7942 + 18.6962i 0.835282 + 1.44675i 0.893800 + 0.448465i \(0.148029\pi\)
−0.0585177 + 0.998286i \(0.518637\pi\)
\(168\) 0 0
\(169\) −8.42820 + 14.5981i −0.648323 + 1.12293i
\(170\) 0 0
\(171\) 3.00000 5.19615i 0.229416 0.397360i
\(172\) 0 0
\(173\) −4.53590 + 7.85641i −0.344858 + 0.597312i −0.985328 0.170672i \(-0.945406\pi\)
0.640470 + 0.767983i \(0.278740\pi\)
\(174\) 0 0
\(175\) −0.133975 0.232051i −0.0101275 0.0175414i
\(176\) 0 0
\(177\) 12.4641 21.5885i 0.936859 1.62269i
\(178\) 0 0
\(179\) 13.4641 1.00635 0.503177 0.864183i \(-0.332164\pi\)
0.503177 + 0.864183i \(0.332164\pi\)
\(180\) 0 0
\(181\) −9.53590 −0.708798 −0.354399 0.935094i \(-0.615314\pi\)
−0.354399 + 0.935094i \(0.615314\pi\)
\(182\) 0 0
\(183\) 14.6603 1.08372
\(184\) 0 0
\(185\) 5.19615 + 9.00000i 0.382029 + 0.661693i
\(186\) 0 0
\(187\) 0.392305 0.679492i 0.0286882 0.0496894i
\(188\) 0 0
\(189\) 0.696152 1.20577i 0.0506376 0.0877070i
\(190\) 0 0
\(191\) −8.46410 + 14.6603i −0.612441 + 1.06078i 0.378387 + 0.925648i \(0.376479\pi\)
−0.990828 + 0.135131i \(0.956854\pi\)
\(192\) 0 0
\(193\) 8.73205 + 15.1244i 0.628547 + 1.08867i 0.987844 + 0.155452i \(0.0496833\pi\)
−0.359297 + 0.933223i \(0.616983\pi\)
\(194\) 0 0
\(195\) −9.46410 −0.677738
\(196\) 0 0
\(197\) 14.9282 1.06359 0.531795 0.846873i \(-0.321518\pi\)
0.531795 + 0.846873i \(0.321518\pi\)
\(198\) 0 0
\(199\) 7.07180 0.501306 0.250653 0.968077i \(-0.419355\pi\)
0.250653 + 0.968077i \(0.419355\pi\)
\(200\) 0 0
\(201\) 5.42820 9.40192i 0.382876 0.663161i
\(202\) 0 0
\(203\) −0.205771 0.356406i −0.0144423 0.0250148i
\(204\) 0 0
\(205\) 4.96410 8.59808i 0.346708 0.600516i
\(206\) 0 0
\(207\) 11.1962 0.778186
\(208\) 0 0
\(209\) −1.46410 + 2.53590i −0.101274 + 0.175412i
\(210\) 0 0
\(211\) −0.196152 0.339746i −0.0135037 0.0233891i 0.859195 0.511649i \(-0.170965\pi\)
−0.872698 + 0.488260i \(0.837632\pi\)
\(212\) 0 0
\(213\) 8.19615 + 14.1962i 0.561591 + 0.972704i
\(214\) 0 0
\(215\) 4.53590 0.309346
\(216\) 0 0
\(217\) −0.535898 −0.0363792
\(218\) 0 0
\(219\) 6.00000 + 10.3923i 0.405442 + 0.702247i
\(220\) 0 0
\(221\) 1.46410 + 2.53590i 0.0984861 + 0.170583i
\(222\) 0 0
\(223\) −11.3301 + 19.6244i −0.758721 + 1.31414i 0.184781 + 0.982780i \(0.440842\pi\)
−0.943503 + 0.331364i \(0.892491\pi\)
\(224\) 0 0
\(225\) −1.50000 2.59808i −0.100000 0.173205i
\(226\) 0 0
\(227\) −12.1244 + 21.0000i −0.804722 + 1.39382i 0.111757 + 0.993736i \(0.464352\pi\)
−0.916479 + 0.400083i \(0.868981\pi\)
\(228\) 0 0
\(229\) −12.6244 21.8660i −0.834241 1.44495i −0.894647 0.446774i \(-0.852573\pi\)
0.0604061 0.998174i \(-0.480760\pi\)
\(230\) 0 0
\(231\) −0.339746 + 0.588457i −0.0223536 + 0.0387176i
\(232\) 0 0
\(233\) −10.3923 −0.680823 −0.340411 0.940277i \(-0.610566\pi\)
−0.340411 + 0.940277i \(0.610566\pi\)
\(234\) 0 0
\(235\) −0.267949 −0.0174791
\(236\) 0 0
\(237\) −26.7846 −1.73985
\(238\) 0 0
\(239\) −10.3923 18.0000i −0.672222 1.16432i −0.977273 0.211987i \(-0.932007\pi\)
0.305050 0.952336i \(-0.401327\pi\)
\(240\) 0 0
\(241\) 3.03590 5.25833i 0.195559 0.338719i −0.751524 0.659705i \(-0.770681\pi\)
0.947084 + 0.320987i \(0.104014\pi\)
\(242\) 0 0
\(243\) 7.79423 13.5000i 0.500000 0.866025i
\(244\) 0 0
\(245\) −3.46410 + 6.00000i −0.221313 + 0.383326i
\(246\) 0 0
\(247\) −5.46410 9.46410i −0.347672 0.602186i
\(248\) 0 0
\(249\) 22.8564 1.44847
\(250\) 0 0
\(251\) −7.85641 −0.495892 −0.247946 0.968774i \(-0.579756\pi\)
−0.247946 + 0.968774i \(0.579756\pi\)
\(252\) 0 0
\(253\) −5.46410 −0.343525
\(254\) 0 0
\(255\) −0.464102 + 0.803848i −0.0290632 + 0.0503389i
\(256\) 0 0
\(257\) 0.464102 + 0.803848i 0.0289499 + 0.0501426i 0.880137 0.474719i \(-0.157450\pi\)
−0.851187 + 0.524862i \(0.824117\pi\)
\(258\) 0 0
\(259\) −1.39230 + 2.41154i −0.0865136 + 0.149846i
\(260\) 0 0
\(261\) −2.30385 3.99038i −0.142605 0.246998i
\(262\) 0 0
\(263\) −11.1962 + 19.3923i −0.690384 + 1.19578i 0.281328 + 0.959612i \(0.409225\pi\)
−0.971712 + 0.236169i \(0.924108\pi\)
\(264\) 0 0
\(265\) 3.00000 + 5.19615i 0.184289 + 0.319197i
\(266\) 0 0
\(267\) −8.59808 14.8923i −0.526194 0.911394i
\(268\) 0 0
\(269\) 11.3923 0.694601 0.347301 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(270\) 0 0
\(271\) −19.8564 −1.20619 −0.603095 0.797669i \(-0.706066\pi\)
−0.603095 + 0.797669i \(0.706066\pi\)
\(272\) 0 0
\(273\) −1.26795 2.19615i −0.0767398 0.132917i
\(274\) 0 0
\(275\) 0.732051 + 1.26795i 0.0441443 + 0.0764602i
\(276\) 0 0
\(277\) 7.19615 12.4641i 0.432375 0.748895i −0.564702 0.825295i \(-0.691009\pi\)
0.997077 + 0.0763993i \(0.0243424\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 7.42820 12.8660i 0.443129 0.767523i −0.554790 0.831990i \(-0.687202\pi\)
0.997920 + 0.0644675i \(0.0205349\pi\)
\(282\) 0 0
\(283\) −10.8660 18.8205i −0.645918 1.11876i −0.984089 0.177678i \(-0.943141\pi\)
0.338170 0.941085i \(-0.390192\pi\)
\(284\) 0 0
\(285\) 1.73205 3.00000i 0.102598 0.177705i
\(286\) 0 0
\(287\) 2.66025 0.157030
\(288\) 0 0
\(289\) −16.7128 −0.983107
\(290\) 0 0
\(291\) 15.4641 0.906522
\(292\) 0 0
\(293\) 3.73205 + 6.46410i 0.218029 + 0.377637i 0.954205 0.299153i \(-0.0967040\pi\)
−0.736176 + 0.676790i \(0.763371\pi\)
\(294\) 0 0
\(295\) 7.19615 12.4641i 0.418976 0.725688i
\(296\) 0 0
\(297\) −3.80385 + 6.58846i −0.220722 + 0.382301i
\(298\) 0 0
\(299\) 10.1962 17.6603i 0.589659 1.02132i
\(300\) 0 0
\(301\) 0.607695 + 1.05256i 0.0350270 + 0.0606685i
\(302\) 0 0
\(303\) −1.60770 −0.0923597
\(304\) 0 0
\(305\) 8.46410 0.484653
\(306\) 0 0
\(307\) 4.12436 0.235389 0.117695 0.993050i \(-0.462450\pi\)
0.117695 + 0.993050i \(0.462450\pi\)
\(308\) 0 0
\(309\) 12.4641 21.5885i 0.709058 1.22812i
\(310\) 0 0
\(311\) −1.53590 2.66025i −0.0870928 0.150849i 0.819188 0.573525i \(-0.194424\pi\)
−0.906281 + 0.422676i \(0.861091\pi\)
\(312\) 0 0
\(313\) −6.19615 + 10.7321i −0.350227 + 0.606611i −0.986289 0.165027i \(-0.947229\pi\)
0.636062 + 0.771638i \(0.280562\pi\)
\(314\) 0 0
\(315\) 0.401924 0.696152i 0.0226458 0.0392237i
\(316\) 0 0
\(317\) −1.26795 + 2.19615i −0.0712151 + 0.123348i −0.899434 0.437056i \(-0.856021\pi\)
0.828219 + 0.560405i \(0.189354\pi\)
\(318\) 0 0
\(319\) 1.12436 + 1.94744i 0.0629518 + 0.109036i
\(320\) 0 0
\(321\) 4.50000 + 7.79423i 0.251166 + 0.435031i
\(322\) 0 0
\(323\) −1.07180 −0.0596364
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) 0 0
\(327\) −15.0622 26.0885i −0.832940 1.44269i
\(328\) 0 0
\(329\) −0.0358984 0.0621778i −0.00197914 0.00342797i
\(330\) 0 0
\(331\) 1.73205 3.00000i 0.0952021 0.164895i −0.814491 0.580176i \(-0.802984\pi\)
0.909693 + 0.415282i \(0.136317\pi\)
\(332\) 0 0
\(333\) −15.5885 + 27.0000i −0.854242 + 1.47959i
\(334\) 0 0
\(335\) 3.13397 5.42820i 0.171227 0.296574i
\(336\) 0 0
\(337\) −4.53590 7.85641i −0.247086 0.427966i 0.715630 0.698480i \(-0.246140\pi\)
−0.962716 + 0.270514i \(0.912806\pi\)
\(338\) 0 0
\(339\) 12.0000 20.7846i 0.651751 1.12887i
\(340\) 0 0
\(341\) 2.92820 0.158571
\(342\) 0 0
\(343\) −3.73205 −0.201512
\(344\) 0 0
\(345\) 6.46410 0.348016
\(346\) 0 0
\(347\) 9.19615 + 15.9282i 0.493675 + 0.855071i 0.999973 0.00728777i \(-0.00231979\pi\)
−0.506298 + 0.862359i \(0.668986\pi\)
\(348\) 0 0
\(349\) −0.232051 + 0.401924i −0.0124214 + 0.0215145i −0.872169 0.489204i \(-0.837287\pi\)
0.859748 + 0.510719i \(0.170621\pi\)
\(350\) 0 0
\(351\) −14.1962 24.5885i −0.757735 1.31243i
\(352\) 0 0
\(353\) 10.9282 18.9282i 0.581650 1.00745i −0.413634 0.910443i \(-0.635741\pi\)
0.995284 0.0970036i \(-0.0309258\pi\)
\(354\) 0 0
\(355\) 4.73205 + 8.19615i 0.251151 + 0.435007i
\(356\) 0 0
\(357\) −0.248711 −0.0131632
\(358\) 0 0
\(359\) −12.9282 −0.682324 −0.341162 0.940004i \(-0.610821\pi\)
−0.341162 + 0.940004i \(0.610821\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −7.66987 + 13.2846i −0.402564 + 0.697261i
\(364\) 0 0
\(365\) 3.46410 + 6.00000i 0.181319 + 0.314054i
\(366\) 0 0
\(367\) 7.19615 12.4641i 0.375636 0.650621i −0.614786 0.788694i \(-0.710758\pi\)
0.990422 + 0.138073i \(0.0440909\pi\)
\(368\) 0 0
\(369\) 29.7846 1.55052
\(370\) 0 0
\(371\) −0.803848 + 1.39230i −0.0417337 + 0.0722849i
\(372\) 0 0
\(373\) 4.46410 + 7.73205i 0.231142 + 0.400350i 0.958145 0.286285i \(-0.0924203\pi\)
−0.727002 + 0.686635i \(0.759087\pi\)
\(374\) 0 0
\(375\) −0.866025 1.50000i −0.0447214 0.0774597i
\(376\) 0 0
\(377\) −8.39230 −0.432226
\(378\) 0 0
\(379\) 35.1769 1.80692 0.903458 0.428676i \(-0.141020\pi\)
0.903458 + 0.428676i \(0.141020\pi\)
\(380\) 0 0
\(381\) 14.0885 + 24.4019i 0.721774 + 1.25015i
\(382\) 0 0
\(383\) −1.73205 3.00000i −0.0885037 0.153293i 0.818375 0.574684i \(-0.194875\pi\)
−0.906879 + 0.421392i \(0.861542\pi\)
\(384\) 0 0
\(385\) −0.196152 + 0.339746i −0.00999685 + 0.0173151i
\(386\) 0 0
\(387\) 6.80385 + 11.7846i 0.345859 + 0.599045i
\(388\) 0 0
\(389\) 11.7679 20.3827i 0.596659 1.03344i −0.396652 0.917969i \(-0.629828\pi\)
0.993310 0.115474i \(-0.0368387\pi\)
\(390\) 0 0
\(391\) −1.00000 1.73205i −0.0505722 0.0875936i
\(392\) 0 0
\(393\) −3.00000 + 5.19615i −0.151330 + 0.262111i
\(394\) 0 0
\(395\) −15.4641 −0.778083
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0.928203 0.0464683
\(400\) 0 0
\(401\) −7.39230 12.8038i −0.369154 0.639394i 0.620279 0.784381i \(-0.287019\pi\)
−0.989434 + 0.144987i \(0.953686\pi\)
\(402\) 0 0
\(403\) −5.46410 + 9.46410i −0.272186 + 0.471440i
\(404\) 0 0
\(405\) 4.50000 7.79423i 0.223607 0.387298i
\(406\) 0 0
\(407\) 7.60770 13.1769i 0.377099 0.653155i
\(408\) 0 0
\(409\) −2.46410 4.26795i −0.121842 0.211037i 0.798652 0.601793i \(-0.205547\pi\)
−0.920494 + 0.390757i \(0.872213\pi\)
\(410\) 0 0
\(411\) −37.8564 −1.86732
\(412\) 0 0
\(413\) 3.85641 0.189761
\(414\) 0 0
\(415\) 13.1962 0.647774
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −15.1244 26.1962i −0.738873 1.27977i −0.953003 0.302961i \(-0.902025\pi\)
0.214130 0.976805i \(-0.431308\pi\)
\(420\) 0 0
\(421\) 18.4641 31.9808i 0.899885 1.55865i 0.0722458 0.997387i \(-0.476983\pi\)
0.827639 0.561260i \(-0.189683\pi\)
\(422\) 0 0
\(423\) −0.401924 0.696152i −0.0195422 0.0338481i
\(424\) 0 0
\(425\) −0.267949 + 0.464102i −0.0129974 + 0.0225122i
\(426\) 0 0
\(427\) 1.13397 + 1.96410i 0.0548769 + 0.0950495i
\(428\) 0 0
\(429\) 6.92820 + 12.0000i 0.334497 + 0.579365i
\(430\) 0 0
\(431\) −4.53590 −0.218487 −0.109243 0.994015i \(-0.534843\pi\)
−0.109243 + 0.994015i \(0.534843\pi\)
\(432\) 0 0
\(433\) −12.5359 −0.602437 −0.301218 0.953555i \(-0.597393\pi\)
−0.301218 + 0.953555i \(0.597393\pi\)
\(434\) 0 0
\(435\) −1.33013 2.30385i −0.0637747 0.110461i
\(436\) 0 0
\(437\) 3.73205 + 6.46410i 0.178528 + 0.309220i
\(438\) 0 0
\(439\) −10.5359 + 18.2487i −0.502851 + 0.870963i 0.497144 + 0.867668i \(0.334382\pi\)
−0.999995 + 0.00329518i \(0.998951\pi\)
\(440\) 0 0
\(441\) −20.7846 −0.989743
\(442\) 0 0
\(443\) −17.2583 + 29.8923i −0.819968 + 1.42023i 0.0857370 + 0.996318i \(0.472676\pi\)
−0.905705 + 0.423908i \(0.860658\pi\)
\(444\) 0 0
\(445\) −4.96410 8.59808i −0.235321 0.407588i
\(446\) 0 0
\(447\) −2.13397 + 3.69615i −0.100934 + 0.174822i
\(448\) 0 0
\(449\) 4.14359 0.195548 0.0977741 0.995209i \(-0.468828\pi\)
0.0977741 + 0.995209i \(0.468828\pi\)
\(450\) 0 0
\(451\) −14.5359 −0.684469
\(452\) 0 0
\(453\) −9.46410 −0.444662
\(454\) 0 0
\(455\) −0.732051 1.26795i −0.0343191 0.0594424i
\(456\) 0 0
\(457\) −2.80385 + 4.85641i −0.131158 + 0.227173i −0.924123 0.382094i \(-0.875203\pi\)
0.792965 + 0.609267i \(0.208536\pi\)
\(458\) 0 0
\(459\) −2.78461 −0.129974
\(460\) 0 0
\(461\) −2.30385 + 3.99038i −0.107301 + 0.185851i −0.914676 0.404188i \(-0.867554\pi\)
0.807375 + 0.590039i \(0.200887\pi\)
\(462\) 0 0
\(463\) 1.73205 + 3.00000i 0.0804952 + 0.139422i 0.903463 0.428667i \(-0.141016\pi\)
−0.822968 + 0.568088i \(0.807683\pi\)
\(464\) 0 0
\(465\) −3.46410 −0.160644
\(466\) 0 0
\(467\) −30.3923 −1.40639 −0.703194 0.710998i \(-0.748243\pi\)
−0.703194 + 0.710998i \(0.748243\pi\)
\(468\) 0 0
\(469\) 1.67949 0.0775517
\(470\) 0 0
\(471\) 4.26795 7.39230i 0.196657 0.340619i
\(472\) 0 0
\(473\) −3.32051 5.75129i −0.152677 0.264445i
\(474\) 0 0
\(475\) 1.00000 1.73205i 0.0458831 0.0794719i
\(476\) 0 0
\(477\) −9.00000 + 15.5885i −0.412082 + 0.713746i
\(478\) 0 0
\(479\) 7.19615 12.4641i 0.328801 0.569499i −0.653474 0.756949i \(-0.726689\pi\)
0.982274 + 0.187450i \(0.0600223\pi\)
\(480\) 0 0
\(481\) 28.3923 + 49.1769i 1.29458 + 2.24227i
\(482\) 0 0
\(483\) 0.866025 + 1.50000i 0.0394055 + 0.0682524i
\(484\) 0 0
\(485\) 8.92820 0.405409
\(486\) 0 0
\(487\) 40.2487 1.82384 0.911922 0.410364i \(-0.134599\pi\)
0.911922 + 0.410364i \(0.134599\pi\)
\(488\) 0 0
\(489\) −11.5359 19.9808i −0.521671 0.903561i
\(490\) 0 0
\(491\) 12.9282 + 22.3923i 0.583442 + 1.01055i 0.995068 + 0.0991978i \(0.0316276\pi\)
−0.411626 + 0.911353i \(0.635039\pi\)
\(492\) 0 0
\(493\) −0.411543 + 0.712813i −0.0185350 + 0.0321035i
\(494\) 0 0
\(495\) −2.19615 + 3.80385i −0.0987097 + 0.170970i
\(496\) 0 0
\(497\) −1.26795 + 2.19615i −0.0568753 + 0.0985109i
\(498\) 0 0
\(499\) 8.92820 + 15.4641i 0.399681 + 0.692268i 0.993686 0.112193i \(-0.0357875\pi\)
−0.594005 + 0.804461i \(0.702454\pi\)
\(500\) 0 0
\(501\) −18.6962 + 32.3827i −0.835282 + 1.44675i
\(502\) 0 0
\(503\) −18.1244 −0.808125 −0.404063 0.914731i \(-0.632402\pi\)
−0.404063 + 0.914731i \(0.632402\pi\)
\(504\) 0 0
\(505\) −0.928203 −0.0413045
\(506\) 0 0
\(507\) −29.1962 −1.29665
\(508\) 0 0
\(509\) 10.3038 + 17.8468i 0.456710 + 0.791045i 0.998785 0.0492853i \(-0.0156944\pi\)
−0.542075 + 0.840330i \(0.682361\pi\)
\(510\) 0 0
\(511\) −0.928203 + 1.60770i −0.0410613 + 0.0711202i
\(512\) 0 0
\(513\) 10.3923 0.458831
\(514\) 0 0
\(515\) 7.19615 12.4641i 0.317100 0.549234i
\(516\) 0 0
\(517\) 0.196152 + 0.339746i 0.00862677 + 0.0149420i
\(518\) 0 0
\(519\) −15.7128 −0.689716
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 0 0
\(523\) 10.8038 0.472419 0.236210 0.971702i \(-0.424095\pi\)
0.236210 + 0.971702i \(0.424095\pi\)
\(524\) 0 0
\(525\) 0.232051 0.401924i 0.0101275 0.0175414i
\(526\) 0 0
\(527\) 0.535898 + 0.928203i 0.0233441 + 0.0404332i
\(528\) 0 0
\(529\) 4.53590 7.85641i 0.197213 0.341583i
\(530\) 0 0
\(531\) 43.1769 1.87372
\(532\) 0 0
\(533\) 27.1244 46.9808i 1.17489 2.03496i
\(534\) 0 0
\(535\) 2.59808 + 4.50000i 0.112325 + 0.194552i
\(536\) 0 0
\(537\) 11.6603 + 20.1962i 0.503177 + 0.871528i
\(538\) 0 0
\(539\) 10.1436 0.436916
\(540\) 0 0
\(541\) −18.6077 −0.800007 −0.400004 0.916514i \(-0.630991\pi\)
−0.400004 + 0.916514i \(0.630991\pi\)
\(542\) 0 0
\(543\) −8.25833 14.3038i −0.354399 0.613837i
\(544\) 0 0
\(545\) −8.69615 15.0622i −0.372502 0.645193i
\(546\) 0 0
\(547\) −22.9904 + 39.8205i −0.982998 + 1.70260i −0.332481 + 0.943110i \(0.607886\pi\)
−0.650517 + 0.759492i \(0.725448\pi\)
\(548\) 0 0
\(549\) 12.6962 + 21.9904i 0.541859 + 0.938527i
\(550\) 0 0
\(551\) 1.53590 2.66025i 0.0654315 0.113331i
\(552\) 0 0
\(553\) −2.07180 3.58846i −0.0881018 0.152597i
\(554\) 0 0
\(555\) −9.00000 + 15.5885i −0.382029 + 0.661693i
\(556\) 0 0
\(557\) 32.6410 1.38304 0.691522 0.722355i \(-0.256940\pi\)
0.691522 + 0.722355i \(0.256940\pi\)
\(558\) 0 0
\(559\) 24.7846 1.04828
\(560\) 0 0
\(561\) 1.35898 0.0573763
\(562\) 0 0
\(563\) 6.20577 + 10.7487i 0.261542 + 0.453004i 0.966652 0.256094i \(-0.0824356\pi\)
−0.705110 + 0.709098i \(0.749102\pi\)
\(564\) 0 0
\(565\) 6.92820 12.0000i 0.291472 0.504844i
\(566\) 0 0
\(567\) 2.41154 0.101275
\(568\) 0 0
\(569\) −5.00000 + 8.66025i −0.209611 + 0.363057i −0.951592 0.307364i \(-0.900553\pi\)
0.741981 + 0.670421i \(0.233886\pi\)
\(570\) 0 0
\(571\) 20.3923 + 35.3205i 0.853391 + 1.47812i 0.878129 + 0.478423i \(0.158792\pi\)
−0.0247380 + 0.999694i \(0.507875\pi\)
\(572\) 0 0
\(573\) −29.3205 −1.22488
\(574\) 0 0
\(575\) 3.73205 0.155637
\(576\) 0 0
\(577\) 32.9282 1.37082 0.685410 0.728158i \(-0.259623\pi\)
0.685410 + 0.728158i \(0.259623\pi\)
\(578\) 0 0
\(579\) −15.1244 + 26.1962i −0.628547 + 1.08867i
\(580\) 0 0
\(581\) 1.76795 + 3.06218i 0.0733469 + 0.127041i
\(582\) 0 0
\(583\) 4.39230 7.60770i 0.181911 0.315079i
\(584\) 0 0
\(585\) −8.19615 14.1962i −0.338869 0.586939i
\(586\) 0 0
\(587\) −4.20577 + 7.28461i −0.173591 + 0.300668i −0.939673 0.342075i \(-0.888870\pi\)
0.766082 + 0.642743i \(0.222204\pi\)
\(588\) 0 0
\(589\) −2.00000 3.46410i −0.0824086 0.142736i
\(590\) 0 0
\(591\) 12.9282 + 22.3923i 0.531795 + 0.921096i
\(592\) 0 0
\(593\) −39.4641 −1.62060 −0.810298 0.586018i \(-0.800695\pi\)
−0.810298 + 0.586018i \(0.800695\pi\)
\(594\) 0 0
\(595\) −0.143594 −0.00588676
\(596\) 0 0
\(597\) 6.12436 + 10.6077i 0.250653 + 0.434144i
\(598\) 0 0
\(599\) −0.803848 1.39230i −0.0328443 0.0568880i 0.849136 0.528174i \(-0.177123\pi\)
−0.881980 + 0.471286i \(0.843790\pi\)
\(600\) 0 0
\(601\) −15.3923 + 26.6603i −0.627865 + 1.08749i 0.360114 + 0.932908i \(0.382738\pi\)
−0.987979 + 0.154586i \(0.950596\pi\)
\(602\) 0 0
\(603\) 18.8038 0.765752
\(604\) 0 0
\(605\) −4.42820 + 7.66987i −0.180032 + 0.311825i
\(606\) 0 0
\(607\) −20.1340 34.8731i −0.817213 1.41545i −0.907728 0.419559i \(-0.862185\pi\)
0.0905152 0.995895i \(-0.471149\pi\)
\(608\) 0 0
\(609\) 0.356406 0.617314i 0.0144423 0.0250148i
\(610\) 0 0
\(611\) −1.46410 −0.0592312
\(612\) 0 0
\(613\) 22.3923 0.904417 0.452208 0.891912i \(-0.350636\pi\)
0.452208 + 0.891912i \(0.350636\pi\)
\(614\) 0 0
\(615\) 17.1962 0.693416
\(616\) 0 0
\(617\) −15.4641 26.7846i −0.622561 1.07831i −0.989007 0.147868i \(-0.952759\pi\)
0.366446 0.930439i \(-0.380575\pi\)
\(618\) 0 0
\(619\) 18.8564 32.6603i 0.757903 1.31273i −0.186015 0.982547i \(-0.559557\pi\)
0.943918 0.330180i \(-0.107109\pi\)
\(620\) 0 0
\(621\) 9.69615 + 16.7942i 0.389093 + 0.673929i
\(622\) 0 0
\(623\) 1.33013 2.30385i 0.0532904 0.0923017i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) −5.07180 −0.202548
\(628\) 0 0
\(629\) 5.56922 0.222059
\(630\) 0 0
\(631\) 31.7128 1.26247 0.631234 0.775593i \(-0.282549\pi\)
0.631234 + 0.775593i \(0.282549\pi\)
\(632\) 0 0
\(633\) 0.339746 0.588457i 0.0135037 0.0233891i
\(634\) 0 0
\(635\) 8.13397 + 14.0885i 0.322787 + 0.559083i
\(636\) 0 0
\(637\) −18.9282 + 32.7846i −0.749963 + 1.29897i
\(638\) 0 0
\(639\) −14.1962 + 24.5885i −0.561591 + 0.972704i
\(640\) 0 0
\(641\) 17.4282 30.1865i 0.688373 1.19230i −0.283991 0.958827i \(-0.591659\pi\)
0.972364 0.233470i \(-0.0750079\pi\)
\(642\) 0 0
\(643\) 17.1340 + 29.6769i 0.675698 + 1.17034i 0.976264 + 0.216582i \(0.0694910\pi\)
−0.300566 + 0.953761i \(0.597176\pi\)
\(644\) 0 0
\(645\) 3.92820 + 6.80385i 0.154673 + 0.267901i
\(646\) 0 0
\(647\) −16.5167 −0.649337 −0.324668 0.945828i \(-0.605253\pi\)
−0.324668 + 0.945828i \(0.605253\pi\)
\(648\) 0 0
\(649\) −21.0718 −0.827140
\(650\) 0 0
\(651\) −0.464102 0.803848i −0.0181896 0.0315053i
\(652\) 0 0
\(653\) −15.7321 27.2487i −0.615643 1.06632i −0.990271 0.139150i \(-0.955563\pi\)
0.374629 0.927175i \(-0.377770\pi\)
\(654\) 0 0
\(655\) −1.73205 + 3.00000i −0.0676768 + 0.117220i
\(656\) 0 0
\(657\) −10.3923 + 18.0000i −0.405442 + 0.702247i
\(658\) 0 0
\(659\) 10.3923 18.0000i 0.404827 0.701180i −0.589475 0.807787i \(-0.700665\pi\)
0.994301 + 0.106606i \(0.0339985\pi\)
\(660\) 0 0
\(661\) −10.8564 18.8038i −0.422265 0.731385i 0.573895 0.818929i \(-0.305432\pi\)
−0.996161 + 0.0875437i \(0.972098\pi\)
\(662\) 0 0
\(663\) −2.53590 + 4.39230i −0.0984861 + 0.170583i
\(664\) 0 0
\(665\) 0.535898 0.0207812
\(666\) 0 0
\(667\) 5.73205 0.221946
\(668\) 0 0
\(669\) −39.2487 −1.51744
\(670\) 0 0
\(671\) −6.19615 10.7321i −0.239200 0.414306i
\(672\) 0 0
\(673\) 6.33975 10.9808i 0.244379 0.423277i −0.717578 0.696479i \(-0.754749\pi\)
0.961957 + 0.273201i \(0.0880825\pi\)
\(674\) 0 0
\(675\) 2.59808 4.50000i 0.100000 0.173205i
\(676\) 0 0
\(677\) 8.58846 14.8756i 0.330081 0.571717i −0.652446 0.757835i \(-0.726257\pi\)
0.982527 + 0.186118i \(0.0595905\pi\)
\(678\) 0 0
\(679\) 1.19615 + 2.07180i 0.0459041 + 0.0795083i
\(680\) 0 0
\(681\) −42.0000 −1.60944
\(682\) 0 0
\(683\) 22.3923 0.856818 0.428409 0.903585i \(-0.359074\pi\)
0.428409 + 0.903585i \(0.359074\pi\)
\(684\) 0 0
\(685\) −21.8564 −0.835090
\(686\) 0 0
\(687\) 21.8660 37.8731i 0.834241 1.44495i
\(688\) 0 0
\(689\) 16.3923 + 28.3923i 0.624497 + 1.08166i
\(690\) 0 0
\(691\) 25.1962 43.6410i 0.958507 1.66018i 0.232376 0.972626i \(-0.425350\pi\)
0.726131 0.687556i \(-0.241317\pi\)
\(692\) 0 0
\(693\) −1.17691 −0.0447073
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.66025 4.60770i −0.100764 0.174529i
\(698\) 0 0
\(699\) −9.00000 15.5885i −0.340411 0.589610i
\(700\) 0 0
\(701\) −24.1769 −0.913149 −0.456575 0.889685i \(-0.650924\pi\)
−0.456575 + 0.889685i \(0.650924\pi\)
\(702\) 0 0
\(703\) −20.7846 −0.783906
\(704\) 0 0
\(705\) −0.232051 0.401924i −0.00873954 0.0151373i
\(706\) 0 0
\(707\) −0.124356 0.215390i −0.00467688 0.00810059i
\(708\) 0 0
\(709\) −7.69615 + 13.3301i −0.289035 + 0.500623i −0.973580 0.228348i \(-0.926668\pi\)
0.684545 + 0.728971i \(0.260001\pi\)
\(710\) 0 0
\(711\) −23.1962 40.1769i −0.869924 1.50675i
\(712\) 0 0
\(713\) 3.73205 6.46410i 0.139766 0.242083i
\(714\) 0 0
\(715\) 4.00000 + 6.92820i 0.149592 + 0.259100i
\(716\) 0 0
\(717\) 18.0000 31.1769i 0.672222 1.16432i
\(718\) 0 0
\(719\) −33.3205 −1.24265 −0.621323 0.783555i \(-0.713404\pi\)
−0.621323 + 0.783555i \(0.713404\pi\)
\(720\) 0 0
\(721\) 3.85641 0.143620
\(722\) 0 0
\(723\) 10.5167 0.391119
\(724\) 0 0
\(725\) −0.767949 1.33013i −0.0285209 0.0493997i
\(726\) 0 0
\(727\) −7.99038 + 13.8397i −0.296347 + 0.513288i −0.975297 0.220896i \(-0.929102\pi\)
0.678950 + 0.734184i \(0.262435\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 1.21539 2.10512i 0.0449528 0.0778606i
\(732\) 0 0
\(733\) −8.07180 13.9808i −0.298139 0.516391i 0.677571 0.735457i \(-0.263032\pi\)
−0.975710 + 0.219066i \(0.929699\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) −9.17691 −0.338036
\(738\) 0 0
\(739\) −0.248711 −0.00914899 −0.00457450 0.999990i \(-0.501456\pi\)
−0.00457450 + 0.999990i \(0.501456\pi\)
\(740\) 0 0
\(741\) 9.46410 16.3923i 0.347672 0.602186i
\(742\) 0 0
\(743\) 16.9186 + 29.3038i 0.620683 + 1.07505i 0.989359 + 0.145496i \(0.0464778\pi\)
−0.368676 + 0.929558i \(0.620189\pi\)
\(744\) 0 0
\(745\) −1.23205 + 2.13397i −0.0451388 + 0.0781828i
\(746\) 0 0
\(747\) 19.7942 + 34.2846i 0.724233 + 1.25441i
\(748\) 0 0
\(749\) −0.696152 + 1.20577i −0.0254369 + 0.0440579i
\(750\) 0 0
\(751\) −8.19615 14.1962i −0.299082 0.518025i 0.676844 0.736126i \(-0.263347\pi\)
−0.975926 + 0.218101i \(0.930014\pi\)
\(752\) 0 0
\(753\) −6.80385 11.7846i −0.247946 0.429455i
\(754\) 0 0
\(755\) −5.46410 −0.198859
\(756\) 0 0
\(757\) −20.3923 −0.741171 −0.370585 0.928798i \(-0.620843\pi\)
−0.370585 + 0.928798i \(0.620843\pi\)
\(758\) 0 0
\(759\) −4.73205 8.19615i −0.171763 0.297501i
\(760\) 0 0
\(761\) 10.5000 + 18.1865i 0.380625 + 0.659261i 0.991152 0.132734i \(-0.0423756\pi\)
−0.610527 + 0.791995i \(0.709042\pi\)
\(762\) 0 0
\(763\) 2.33013 4.03590i 0.0843563 0.146109i
\(764\) 0 0
\(765\) −1.60770 −0.0581263
\(766\) 0 0
\(767\) 39.3205 68.1051i 1.41978 2.45913i
\(768\) 0 0
\(769\) −16.8923 29.2583i −0.609152 1.05508i −0.991380 0.131014i \(-0.958177\pi\)
0.382228 0.924068i \(-0.375157\pi\)
\(770\) 0 0
\(771\) −0.803848 + 1.39230i −0.0289499 + 0.0501426i
\(772\) 0 0
\(773\) −8.78461 −0.315960 −0.157980 0.987442i \(-0.550498\pi\)
−0.157980 + 0.987442i \(0.550498\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) −4.82309 −0.173027
\(778\) 0 0
\(779\) 9.92820 + 17.1962i 0.355715 + 0.616116i
\(780\) 0 0
\(781\) 6.92820 12.0000i 0.247911 0.429394i
\(782\) 0 0
\(783\) 3.99038 6.91154i 0.142605 0.246998i
\(784\) 0 0
\(785\) 2.46410 4.26795i 0.0879476 0.152330i
\(786\) 0 0
\(787\) −0.660254 1.14359i −0.0235355 0.0407647i 0.854018 0.520244i \(-0.174159\pi\)
−0.877553 + 0.479479i \(0.840826\pi\)
\(788\) 0 0
\(789\) −38.7846 −1.38077
\(790\) 0 0
\(791\) 3.71281 0.132012
\(792\) 0 0
\(793\) 46.2487 1.64234
\(794\) 0 0
\(795\) −5.19615 + 9.00000i −0.184289 + 0.319197i
\(796\) 0 0
\(797\) 3.80385 + 6.58846i 0.134739 + 0.233375i 0.925498 0.378753i \(-0.123647\pi\)
−0.790759 + 0.612128i \(0.790314\pi\)
\(798\) 0 0
\(799\) −0.0717968 + 0.124356i −0.00253999 + 0.00439939i
\(800\) 0 0
\(801\) 14.8923 25.7942i 0.526194 0.911394i
\(802\) 0 0
\(803\) 5.07180 8.78461i 0.178980 0.310002i
\(804\) 0 0
\(805\) 0.500000 + 0.866025i 0.0176227 + 0.0305234i
\(806\) 0 0
\(807\) 9.86603 + 17.0885i 0.347301 + 0.601542i
\(808\) 0 0
\(809\) −48.9282 −1.72022 −0.860112 0.510105i \(-0.829606\pi\)
−0.860112 + 0.510105i \(0.829606\pi\)
\(810\) 0 0
\(811\) −50.3923 −1.76951 −0.884757 0.466053i \(-0.845675\pi\)
−0.884757 + 0.466053i \(0.845675\pi\)
\(812\) 0 0
\(813\) −17.1962 29.7846i −0.603095 1.04459i
\(814\) 0 0
\(815\) −6.66025 11.5359i −0.233299 0.404085i
\(816\) 0 0
\(817\) −4.53590 + 7.85641i −0.158691 + 0.274861i
\(818\) 0 0
\(819\) 2.19615 3.80385i 0.0767398 0.132917i
\(820\) 0 0
\(821\) −11.6244 + 20.1340i −0.405693 + 0.702681i −0.994402 0.105664i \(-0.966303\pi\)
0.588709 + 0.808345i \(0.299636\pi\)
\(822\) 0 0
\(823\) −11.8660 20.5526i −0.413624 0.716417i 0.581659 0.813433i \(-0.302404\pi\)
−0.995283 + 0.0970154i \(0.969070\pi\)
\(824\) 0 0
\(825\) −1.26795 + 2.19615i −0.0441443 + 0.0764602i
\(826\) 0 0
\(827\) −16.9474 −0.589320 −0.294660 0.955602i \(-0.595206\pi\)
−0.294660 + 0.955602i \(0.595206\pi\)
\(828\) 0 0
\(829\) 11.3923 0.395671 0.197836 0.980235i \(-0.436609\pi\)
0.197836 + 0.980235i \(0.436609\pi\)
\(830\) 0 0
\(831\) 24.9282 0.864750
\(832\) 0 0
\(833\) 1.85641 + 3.21539i 0.0643207 + 0.111407i
\(834\) 0 0
\(835\) −10.7942 + 18.6962i −0.373550 + 0.647007i
\(836\) 0 0
\(837\) −5.19615 9.00000i −0.179605 0.311086i
\(838\) 0 0
\(839\) −17.9282 + 31.0526i −0.618950 + 1.07205i 0.370727 + 0.928742i \(0.379108\pi\)
−0.989678 + 0.143312i \(0.954225\pi\)
\(840\) 0 0
\(841\) 13.3205 + 23.0718i 0.459328 + 0.795579i
\(842\) 0 0
\(843\) 25.7321 0.886259
\(844\) 0 0
\(845\) −16.8564 −0.579878
\(846\) 0 0
\(847\) −2.37307 −0.0815395
\(848\) 0 0
\(849\) 18.8205 32.5981i 0.645918 1.11876i
\(850\) 0 0
\(851\) −19.3923 33.5885i −0.664760 1.15140i
\(852\) 0 0
\(853\) 24.7321 42.8372i 0.846809 1.46672i −0.0372313 0.999307i \(-0.511854\pi\)
0.884041 0.467410i \(-0.154813\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) −9.12436 + 15.8038i −0.311682 + 0.539849i −0.978727 0.205169i \(-0.934226\pi\)
0.667045 + 0.745018i \(0.267559\pi\)
\(858\) 0 0
\(859\) 10.1962 + 17.6603i 0.347888 + 0.602560i 0.985874 0.167488i \(-0.0535655\pi\)
−0.637986 + 0.770048i \(0.720232\pi\)
\(860\) 0 0
\(861\) 2.30385 + 3.99038i 0.0785149 + 0.135992i
\(862\) 0 0
\(863\) 26.6603 0.907526 0.453763 0.891123i \(-0.350081\pi\)
0.453763 + 0.891123i \(0.350081\pi\)
\(864\) 0 0
\(865\) −9.07180 −0.308450
\(866\) 0 0
\(867\) −14.4737 25.0692i −0.491553 0.851395i
\(868\) 0 0
\(869\) 11.3205 + 19.6077i 0.384022 + 0.665146i
\(870\) 0 0
\(871\) 17.1244 29.6603i 0.580237 1.00500i
\(872\) 0 0
\(873\) 13.3923 + 23.1962i 0.453261 + 0.785071i
\(874\) 0 0
\(875\) 0.133975 0.232051i 0.00452917 0.00784475i
\(876\) 0 0
\(877\) −5.80385 10.0526i −0.195982 0.339451i 0.751240 0.660029i \(-0.229456\pi\)
−0.947222 + 0.320578i \(0.896123\pi\)
\(878\) 0 0
\(879\) −6.46410 + 11.1962i −0.218029 + 0.377637i
\(880\) 0 0
\(881\) 37.6410 1.26816 0.634079 0.773268i \(-0.281379\pi\)
0.634079 + 0.773268i \(0.281379\pi\)
\(882\) 0 0
\(883\) −9.19615 −0.309475 −0.154738 0.987956i \(-0.549453\pi\)
−0.154738 + 0.987956i \(0.549453\pi\)
\(884\) 0 0
\(885\) 24.9282 0.837952
\(886\) 0 0
\(887\) 4.66025 + 8.07180i 0.156476 + 0.271024i 0.933596 0.358329i \(-0.116653\pi\)
−0.777119 + 0.629353i \(0.783320\pi\)
\(888\) 0 0
\(889\) −2.17949 + 3.77499i −0.0730978 + 0.126609i
\(890\) 0 0
\(891\) −13.1769 −0.441443
\(892\) 0 0
\(893\) 0.267949 0.464102i 0.00896658 0.0155306i
\(894\) 0 0
\(895\) 6.73205 + 11.6603i 0.225028 + 0.389759i
\(896\) 0 0
\(897\) 35.3205 1.17932
\(898\) 0 0
\(899\) −3.07180 −0.102450
\(900\) 0 0
\(901\) 3.21539 0.107120
\(902\) 0 0
\(903\) −1.05256 + 1.82309i −0.0350270 + 0.0606685i
\(904\) 0 0
\(905\) −4.76795 8.25833i −0.158492 0.274516i
\(906\) 0 0
\(907\) 6.86603 11.8923i 0.227983 0.394878i −0.729227 0.684271i \(-0.760120\pi\)
0.957210 + 0.289394i \(0.0934537\pi\)
\(908\) 0 0
\(909\) −1.39230 2.41154i −0.0461798 0.0799858i
\(910\) 0 0
\(911\) 6.26795 10.8564i 0.207666 0.359689i −0.743313 0.668944i \(-0.766746\pi\)
0.950979 + 0.309255i \(0.100080\pi\)
\(912\) 0 0
\(913\) −9.66025 16.7321i −0.319708 0.553750i
\(914\) 0 0
\(915\) 7.33013 + 12.6962i 0.242327 + 0.419722i
\(916\) 0 0
\(917\) −0.928203 −0.0306520
\(918\) 0 0
\(919\) −17.6077 −0.580824 −0.290412 0.956902i \(-0.593792\pi\)
−0.290412 + 0.956902i \(0.593792\pi\)
\(920\) 0 0
\(921\) 3.57180 + 6.18653i 0.117695 + 0.203853i
\(922\) 0 0
\(923\) 25.8564 + 44.7846i 0.851074 + 1.47410i
\(924\) 0 0
\(925\) −5.19615 + 9.00000i −0.170848 + 0.295918i
\(926\) 0 0
\(927\) 43.1769 1.41812
\(928\) 0 0
\(929\) −12.3205 + 21.3397i −0.404223 + 0.700134i −0.994231 0.107263i \(-0.965791\pi\)
0.590008 + 0.807397i \(0.299125\pi\)
\(930\) 0 0
\(931\) −6.92820 12.0000i −0.227063 0.393284i
\(932\) 0 0
\(933\) 2.66025 4.60770i 0.0870928 0.150849i
\(934\) 0 0
\(935\) 0.784610 0.0256595
\(936\) 0 0
\(937\) −11.7128 −0.382641 −0.191320 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(938\) 0 0
\(939\) −21.4641 −0.700454
\(940\) 0 0
\(941\) 11.6962 + 20.2583i 0.381284 + 0.660403i 0.991246 0.132028i \(-0.0421488\pi\)
−0.609962 + 0.792430i \(0.708815\pi\)
\(942\) 0 0
\(943\) −18.5263 + 32.0885i −0.603299 + 1.04494i
\(944\) 0 0
\(945\) 1.39230 0.0452917
\(946\) 0 0
\(947\) 7.25833 12.5718i 0.235864 0.408529i −0.723659 0.690157i \(-0.757541\pi\)
0.959523 + 0.281629i \(0.0908747\pi\)
\(948\) 0 0
\(949\) 18.9282 + 32.7846i 0.614435 + 1.06423i
\(950\) 0 0
\(951\) −4.39230 −0.142430
\(952\) 0 0
\(953\) 40.3923 1.30844 0.654218 0.756306i \(-0.272998\pi\)
0.654218 + 0.756306i \(0.272998\pi\)
\(954\) 0 0
\(955\) −16.9282 −0.547784
\(956\) 0 0
\(957\) −1.94744 + 3.37307i −0.0629518 + 0.109036i
\(958\) 0 0
\(959\) −2.92820 5.07180i −0.0945566 0.163777i
\(960\) 0 0
\(961\) 13.5000 23.3827i 0.435484 0.754280i
\(962\) 0 0
\(963\) −7.79423 + 13.5000i −0.251166 + 0.435031i
\(964\) 0 0
\(965\) −8.73205 + 15.1244i −0.281095 + 0.486870i
\(966\) 0 0
\(967\) −10.6699 18.4808i −0.343120 0.594301i 0.641890 0.766796i \(-0.278150\pi\)
−0.985010 + 0.172495i \(0.944817\pi\)
\(968\) 0 0
\(969\) −0.928203 1.60770i −0.0298182 0.0516466i
\(970\) 0 0
\(971\) 25.7128 0.825163 0.412582 0.910921i \(-0.364627\pi\)
0.412582 + 0.910921i \(0.364627\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.73205 8.19615i −0.151547 0.262487i
\(976\) 0 0
\(977\) −8.58846 14.8756i −0.274769 0.475914i 0.695308 0.718712i \(-0.255268\pi\)
−0.970077 + 0.242798i \(0.921935\pi\)
\(978\) 0 0
\(979\) −7.26795 + 12.5885i −0.232285 + 0.402329i
\(980\) 0 0
\(981\) 26.0885 45.1865i 0.832940 1.44269i
\(982\) 0 0
\(983\) −20.2583 + 35.0885i −0.646140 + 1.11915i 0.337896 + 0.941183i \(0.390285\pi\)
−0.984037 + 0.177965i \(0.943049\pi\)
\(984\) 0 0
\(985\) 7.46410 + 12.9282i 0.237826 + 0.411927i
\(986\) 0 0
\(987\) 0.0621778 0.107695i 0.00197914 0.00342797i
\(988\) 0 0
\(989\) −16.9282 −0.538286
\(990\) 0 0
\(991\) −37.0333 −1.17640 −0.588201 0.808715i \(-0.700164\pi\)
−0.588201 + 0.808715i \(0.700164\pi\)
\(992\) 0 0
\(993\) 6.00000 0.190404
\(994\) 0 0
\(995\) 3.53590 + 6.12436i 0.112096 + 0.194155i
\(996\) 0 0
\(997\) −23.7846 + 41.1962i −0.753266 + 1.30470i 0.192966 + 0.981206i \(0.438189\pi\)
−0.946232 + 0.323490i \(0.895144\pi\)
\(998\) 0 0
\(999\) −54.0000 −1.70848
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 360.2.q.b.241.2 yes 4
3.2 odd 2 1080.2.q.b.721.2 4
4.3 odd 2 720.2.q.h.241.1 4
9.2 odd 6 3240.2.a.p.1.1 2
9.4 even 3 inner 360.2.q.b.121.2 4
9.5 odd 6 1080.2.q.b.361.2 4
9.7 even 3 3240.2.a.k.1.1 2
12.11 even 2 2160.2.q.h.721.1 4
36.7 odd 6 6480.2.a.ba.1.2 2
36.11 even 6 6480.2.a.bk.1.2 2
36.23 even 6 2160.2.q.h.1441.1 4
36.31 odd 6 720.2.q.h.481.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.b.121.2 4 9.4 even 3 inner
360.2.q.b.241.2 yes 4 1.1 even 1 trivial
720.2.q.h.241.1 4 4.3 odd 2
720.2.q.h.481.1 4 36.31 odd 6
1080.2.q.b.361.2 4 9.5 odd 6
1080.2.q.b.721.2 4 3.2 odd 2
2160.2.q.h.721.1 4 12.11 even 2
2160.2.q.h.1441.1 4 36.23 even 6
3240.2.a.k.1.1 2 9.7 even 3
3240.2.a.p.1.1 2 9.2 odd 6
6480.2.a.ba.1.2 2 36.7 odd 6
6480.2.a.bk.1.2 2 36.11 even 6