Properties

Label 360.2.q.b.121.1
Level $360$
Weight $2$
Character 360.121
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 121.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 360.121
Dual form 360.2.q.b.241.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 1.50000i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.86603 - 3.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 1.50000i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-1.86603 - 3.23205i) q^{7} +(-1.50000 - 2.59808i) q^{9} +(-2.73205 - 4.73205i) q^{11} +(-0.732051 + 1.26795i) q^{13} +(0.866025 + 1.50000i) q^{15} +7.46410 q^{17} -2.00000 q^{19} +6.46410 q^{21} +(-0.133975 + 0.232051i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.19615 q^{27} +(-4.23205 - 7.33013i) q^{29} +(1.00000 - 1.73205i) q^{31} +9.46410 q^{33} -3.73205 q^{35} -10.3923 q^{37} +(-1.26795 - 2.19615i) q^{39} +(1.96410 - 3.40192i) q^{41} +(5.73205 + 9.92820i) q^{43} -3.00000 q^{45} +(-1.86603 - 3.23205i) q^{47} +(-3.46410 + 6.00000i) q^{49} +(-6.46410 + 11.1962i) q^{51} +6.00000 q^{53} -5.46410 q^{55} +(1.73205 - 3.00000i) q^{57} +(3.19615 - 5.53590i) q^{59} +(0.767949 + 1.33013i) q^{61} +(-5.59808 + 9.69615i) q^{63} +(0.732051 + 1.26795i) q^{65} +(-4.86603 + 8.42820i) q^{67} +(-0.232051 - 0.401924i) q^{69} +2.53590 q^{71} -6.92820 q^{73} +1.73205 q^{75} +(-10.1962 + 17.6603i) q^{77} +(-4.26795 - 7.39230i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(1.40192 + 2.42820i) q^{83} +(3.73205 - 6.46410i) q^{85} +14.6603 q^{87} +3.92820 q^{89} +5.46410 q^{91} +(1.73205 + 3.00000i) q^{93} +(-1.00000 + 1.73205i) q^{95} +(-2.46410 - 4.26795i) q^{97} +(-8.19615 + 14.1962i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

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

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
\(7\) −1.86603 3.23205i −0.705291 1.22160i −0.966586 0.256341i \(-0.917483\pi\)
0.261295 0.965259i \(-0.415850\pi\)
\(8\) 0 0
\(9\) −1.50000 2.59808i −0.500000 0.866025i
\(10\) 0 0
\(11\) −2.73205 4.73205i −0.823744 1.42677i −0.902875 0.429903i \(-0.858548\pi\)
0.0791309 0.996864i \(-0.474785\pi\)
\(12\) 0 0
\(13\) −0.732051 + 1.26795i −0.203034 + 0.351666i −0.949505 0.313753i \(-0.898414\pi\)
0.746470 + 0.665419i \(0.231747\pi\)
\(14\) 0 0
\(15\) 0.866025 + 1.50000i 0.223607 + 0.387298i
\(16\) 0 0
\(17\) 7.46410 1.81031 0.905155 0.425081i \(-0.139754\pi\)
0.905155 + 0.425081i \(0.139754\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 6.46410 1.41058
\(22\) 0 0
\(23\) −0.133975 + 0.232051i −0.0279356 + 0.0483859i −0.879655 0.475612i \(-0.842227\pi\)
0.851720 + 0.523998i \(0.175560\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) −4.23205 7.33013i −0.785872 1.36117i −0.928477 0.371391i \(-0.878881\pi\)
0.142605 0.989780i \(-0.454452\pi\)
\(30\) 0 0
\(31\) 1.00000 1.73205i 0.179605 0.311086i −0.762140 0.647412i \(-0.775851\pi\)
0.941745 + 0.336327i \(0.109185\pi\)
\(32\) 0 0
\(33\) 9.46410 1.64749
\(34\) 0 0
\(35\) −3.73205 −0.630832
\(36\) 0 0
\(37\) −10.3923 −1.70848 −0.854242 0.519875i \(-0.825978\pi\)
−0.854242 + 0.519875i \(0.825978\pi\)
\(38\) 0 0
\(39\) −1.26795 2.19615i −0.203034 0.351666i
\(40\) 0 0
\(41\) 1.96410 3.40192i 0.306741 0.531291i −0.670906 0.741542i \(-0.734095\pi\)
0.977647 + 0.210251i \(0.0674281\pi\)
\(42\) 0 0
\(43\) 5.73205 + 9.92820i 0.874130 + 1.51404i 0.857687 + 0.514172i \(0.171901\pi\)
0.0164424 + 0.999865i \(0.494766\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −1.86603 3.23205i −0.272188 0.471443i 0.697234 0.716844i \(-0.254414\pi\)
−0.969422 + 0.245401i \(0.921081\pi\)
\(48\) 0 0
\(49\) −3.46410 + 6.00000i −0.494872 + 0.857143i
\(50\) 0 0
\(51\) −6.46410 + 11.1962i −0.905155 + 1.56777i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −5.46410 −0.736779
\(56\) 0 0
\(57\) 1.73205 3.00000i 0.229416 0.397360i
\(58\) 0 0
\(59\) 3.19615 5.53590i 0.416104 0.720713i −0.579440 0.815015i \(-0.696729\pi\)
0.995544 + 0.0943023i \(0.0300620\pi\)
\(60\) 0 0
\(61\) 0.767949 + 1.33013i 0.0983258 + 0.170305i 0.910992 0.412424i \(-0.135318\pi\)
−0.812666 + 0.582730i \(0.801985\pi\)
\(62\) 0 0
\(63\) −5.59808 + 9.69615i −0.705291 + 1.22160i
\(64\) 0 0
\(65\) 0.732051 + 1.26795i 0.0907997 + 0.157270i
\(66\) 0 0
\(67\) −4.86603 + 8.42820i −0.594480 + 1.02967i 0.399140 + 0.916890i \(0.369309\pi\)
−0.993620 + 0.112779i \(0.964025\pi\)
\(68\) 0 0
\(69\) −0.232051 0.401924i −0.0279356 0.0483859i
\(70\) 0 0
\(71\) 2.53590 0.300956 0.150478 0.988613i \(-0.451919\pi\)
0.150478 + 0.988613i \(0.451919\pi\)
\(72\) 0 0
\(73\) −6.92820 −0.810885 −0.405442 0.914121i \(-0.632883\pi\)
−0.405442 + 0.914121i \(0.632883\pi\)
\(74\) 0 0
\(75\) 1.73205 0.200000
\(76\) 0 0
\(77\) −10.1962 + 17.6603i −1.16196 + 2.01257i
\(78\) 0 0
\(79\) −4.26795 7.39230i −0.480182 0.831699i 0.519560 0.854434i \(-0.326096\pi\)
−0.999742 + 0.0227349i \(0.992763\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 1.40192 + 2.42820i 0.153881 + 0.266530i 0.932651 0.360780i \(-0.117489\pi\)
−0.778770 + 0.627310i \(0.784156\pi\)
\(84\) 0 0
\(85\) 3.73205 6.46410i 0.404798 0.701130i
\(86\) 0 0
\(87\) 14.6603 1.57174
\(88\) 0 0
\(89\) 3.92820 0.416389 0.208194 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) 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
\(97\) −2.46410 4.26795i −0.250192 0.433345i 0.713387 0.700770i \(-0.247160\pi\)
−0.963578 + 0.267426i \(0.913827\pi\)
\(98\) 0 0
\(99\) −8.19615 + 14.1962i −0.823744 + 1.42677i
\(100\) 0 0
\(101\) 6.46410 + 11.1962i 0.643202 + 1.11406i 0.984714 + 0.174181i \(0.0557278\pi\)
−0.341511 + 0.939878i \(0.610939\pi\)
\(102\) 0 0
\(103\) 3.19615 5.53590i 0.314926 0.545468i −0.664496 0.747292i \(-0.731354\pi\)
0.979422 + 0.201824i \(0.0646869\pi\)
\(104\) 0 0
\(105\) 3.23205 5.59808i 0.315416 0.546316i
\(106\) 0 0
\(107\) −5.19615 −0.502331 −0.251166 0.967944i \(-0.580814\pi\)
−0.251166 + 0.967944i \(0.580814\pi\)
\(108\) 0 0
\(109\) 3.39230 0.324924 0.162462 0.986715i \(-0.448057\pi\)
0.162462 + 0.986715i \(0.448057\pi\)
\(110\) 0 0
\(111\) 9.00000 15.5885i 0.854242 1.47959i
\(112\) 0 0
\(113\) 6.92820 12.0000i 0.651751 1.12887i −0.330947 0.943649i \(-0.607368\pi\)
0.982698 0.185216i \(-0.0592984\pi\)
\(114\) 0 0
\(115\) 0.133975 + 0.232051i 0.0124932 + 0.0216388i
\(116\) 0 0
\(117\) 4.39230 0.406069
\(118\) 0 0
\(119\) −13.9282 24.1244i −1.27680 2.21148i
\(120\) 0 0
\(121\) −9.42820 + 16.3301i −0.857109 + 1.48456i
\(122\) 0 0
\(123\) 3.40192 + 5.89230i 0.306741 + 0.531291i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.7321 1.75094 0.875468 0.483276i \(-0.160553\pi\)
0.875468 + 0.483276i \(0.160553\pi\)
\(128\) 0 0
\(129\) −19.8564 −1.74826
\(130\) 0 0
\(131\) −1.73205 + 3.00000i −0.151330 + 0.262111i −0.931717 0.363186i \(-0.881689\pi\)
0.780387 + 0.625297i \(0.215022\pi\)
\(132\) 0 0
\(133\) 3.73205 + 6.46410i 0.323610 + 0.560509i
\(134\) 0 0
\(135\) 2.59808 4.50000i 0.223607 0.387298i
\(136\) 0 0
\(137\) 2.92820 + 5.07180i 0.250173 + 0.433313i 0.963573 0.267444i \(-0.0861791\pi\)
−0.713400 + 0.700757i \(0.752846\pi\)
\(138\) 0 0
\(139\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(140\) 0 0
\(141\) 6.46410 0.544376
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −8.46410 −0.702905
\(146\) 0 0
\(147\) −6.00000 10.3923i −0.494872 0.857143i
\(148\) 0 0
\(149\) −2.23205 + 3.86603i −0.182857 + 0.316717i −0.942852 0.333211i \(-0.891868\pi\)
0.759995 + 0.649928i \(0.225201\pi\)
\(150\) 0 0
\(151\) 0.732051 + 1.26795i 0.0595734 + 0.103184i 0.894274 0.447520i \(-0.147693\pi\)
−0.834701 + 0.550704i \(0.814359\pi\)
\(152\) 0 0
\(153\) −11.1962 19.3923i −0.905155 1.56777i
\(154\) 0 0
\(155\) −1.00000 1.73205i −0.0803219 0.139122i
\(156\) 0 0
\(157\) 4.46410 7.73205i 0.356274 0.617085i −0.631061 0.775733i \(-0.717380\pi\)
0.987335 + 0.158648i \(0.0507136\pi\)
\(158\) 0 0
\(159\) −5.19615 + 9.00000i −0.412082 + 0.713746i
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 21.3205 1.66995 0.834976 0.550287i \(-0.185482\pi\)
0.834976 + 0.550287i \(0.185482\pi\)
\(164\) 0 0
\(165\) 4.73205 8.19615i 0.368390 0.638070i
\(166\) 0 0
\(167\) −4.79423 + 8.30385i −0.370989 + 0.642571i −0.989718 0.143033i \(-0.954314\pi\)
0.618729 + 0.785604i \(0.287648\pi\)
\(168\) 0 0
\(169\) 5.42820 + 9.40192i 0.417554 + 0.723225i
\(170\) 0 0
\(171\) 3.00000 + 5.19615i 0.229416 + 0.397360i
\(172\) 0 0
\(173\) −11.4641 19.8564i −0.871600 1.50965i −0.860342 0.509718i \(-0.829750\pi\)
−0.0112580 0.999937i \(-0.503584\pi\)
\(174\) 0 0
\(175\) −1.86603 + 3.23205i −0.141058 + 0.244320i
\(176\) 0 0
\(177\) 5.53590 + 9.58846i 0.416104 + 0.720713i
\(178\) 0 0
\(179\) 6.53590 0.488516 0.244258 0.969710i \(-0.421456\pi\)
0.244258 + 0.969710i \(0.421456\pi\)
\(180\) 0 0
\(181\) −16.4641 −1.22377 −0.611884 0.790948i \(-0.709588\pi\)
−0.611884 + 0.790948i \(0.709588\pi\)
\(182\) 0 0
\(183\) −2.66025 −0.196652
\(184\) 0 0
\(185\) −5.19615 + 9.00000i −0.382029 + 0.661693i
\(186\) 0 0
\(187\) −20.3923 35.3205i −1.49123 2.58289i
\(188\) 0 0
\(189\) −9.69615 16.7942i −0.705291 1.22160i
\(190\) 0 0
\(191\) −1.53590 2.66025i −0.111134 0.192489i 0.805094 0.593147i \(-0.202115\pi\)
−0.916228 + 0.400658i \(0.868782\pi\)
\(192\) 0 0
\(193\) 5.26795 9.12436i 0.379195 0.656785i −0.611750 0.791051i \(-0.709534\pi\)
0.990945 + 0.134266i \(0.0428675\pi\)
\(194\) 0 0
\(195\) −2.53590 −0.181599
\(196\) 0 0
\(197\) 1.07180 0.0763624 0.0381812 0.999271i \(-0.487844\pi\)
0.0381812 + 0.999271i \(0.487844\pi\)
\(198\) 0 0
\(199\) 20.9282 1.48356 0.741780 0.670643i \(-0.233982\pi\)
0.741780 + 0.670643i \(0.233982\pi\)
\(200\) 0 0
\(201\) −8.42820 14.5981i −0.594480 1.02967i
\(202\) 0 0
\(203\) −15.7942 + 27.3564i −1.10854 + 1.92004i
\(204\) 0 0
\(205\) −1.96410 3.40192i −0.137179 0.237601i
\(206\) 0 0
\(207\) 0.803848 0.0558713
\(208\) 0 0
\(209\) 5.46410 + 9.46410i 0.377960 + 0.654646i
\(210\) 0 0
\(211\) 10.1962 17.6603i 0.701932 1.21578i −0.265855 0.964013i \(-0.585654\pi\)
0.967787 0.251769i \(-0.0810123\pi\)
\(212\) 0 0
\(213\) −2.19615 + 3.80385i −0.150478 + 0.260635i
\(214\) 0 0
\(215\) 11.4641 0.781845
\(216\) 0 0
\(217\) −7.46410 −0.506696
\(218\) 0 0
\(219\) 6.00000 10.3923i 0.405442 0.702247i
\(220\) 0 0
\(221\) −5.46410 + 9.46410i −0.367555 + 0.636624i
\(222\) 0 0
\(223\) −2.66987 4.62436i −0.178788 0.309670i 0.762678 0.646779i \(-0.223884\pi\)
−0.941466 + 0.337109i \(0.890551\pi\)
\(224\) 0 0
\(225\) −1.50000 + 2.59808i −0.100000 + 0.173205i
\(226\) 0 0
\(227\) 12.1244 + 21.0000i 0.804722 + 1.39382i 0.916479 + 0.400083i \(0.131019\pi\)
−0.111757 + 0.993736i \(0.535648\pi\)
\(228\) 0 0
\(229\) 11.6244 20.1340i 0.768159 1.33049i −0.170401 0.985375i \(-0.554506\pi\)
0.938560 0.345116i \(-0.112160\pi\)
\(230\) 0 0
\(231\) −17.6603 30.5885i −1.16196 2.01257i
\(232\) 0 0
\(233\) 10.3923 0.680823 0.340411 0.940277i \(-0.389434\pi\)
0.340411 + 0.940277i \(0.389434\pi\)
\(234\) 0 0
\(235\) −3.73205 −0.243452
\(236\) 0 0
\(237\) 14.7846 0.960364
\(238\) 0 0
\(239\) 10.3923 18.0000i 0.672222 1.16432i −0.305050 0.952336i \(-0.598673\pi\)
0.977273 0.211987i \(-0.0679934\pi\)
\(240\) 0 0
\(241\) 9.96410 + 17.2583i 0.641844 + 1.11171i 0.985021 + 0.172436i \(0.0551638\pi\)
−0.343177 + 0.939271i \(0.611503\pi\)
\(242\) 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
\(247\) 1.46410 2.53590i 0.0931586 0.161355i
\(248\) 0 0
\(249\) −4.85641 −0.307762
\(250\) 0 0
\(251\) 19.8564 1.25333 0.626663 0.779291i \(-0.284420\pi\)
0.626663 + 0.779291i \(0.284420\pi\)
\(252\) 0 0
\(253\) 1.46410 0.0920473
\(254\) 0 0
\(255\) 6.46410 + 11.1962i 0.404798 + 0.701130i
\(256\) 0 0
\(257\) −6.46410 + 11.1962i −0.403220 + 0.698397i −0.994112 0.108353i \(-0.965442\pi\)
0.590893 + 0.806750i \(0.298776\pi\)
\(258\) 0 0
\(259\) 19.3923 + 33.5885i 1.20498 + 2.08709i
\(260\) 0 0
\(261\) −12.6962 + 21.9904i −0.785872 + 1.36117i
\(262\) 0 0
\(263\) −0.803848 1.39230i −0.0495674 0.0858532i 0.840177 0.542312i \(-0.182451\pi\)
−0.889745 + 0.456459i \(0.849118\pi\)
\(264\) 0 0
\(265\) 3.00000 5.19615i 0.184289 0.319197i
\(266\) 0 0
\(267\) −3.40192 + 5.89230i −0.208194 + 0.360603i
\(268\) 0 0
\(269\) −9.39230 −0.572659 −0.286329 0.958131i \(-0.592435\pi\)
−0.286329 + 0.958131i \(0.592435\pi\)
\(270\) 0 0
\(271\) 7.85641 0.477243 0.238621 0.971113i \(-0.423305\pi\)
0.238621 + 0.971113i \(0.423305\pi\)
\(272\) 0 0
\(273\) −4.73205 + 8.19615i −0.286397 + 0.496054i
\(274\) 0 0
\(275\) −2.73205 + 4.73205i −0.164749 + 0.285353i
\(276\) 0 0
\(277\) −3.19615 5.53590i −0.192038 0.332620i 0.753887 0.657004i \(-0.228176\pi\)
−0.945926 + 0.324384i \(0.894843\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −6.42820 11.1340i −0.383474 0.664197i 0.608082 0.793874i \(-0.291939\pi\)
−0.991556 + 0.129677i \(0.958606\pi\)
\(282\) 0 0
\(283\) −9.13397 + 15.8205i −0.542958 + 0.940432i 0.455774 + 0.890096i \(0.349363\pi\)
−0.998732 + 0.0503360i \(0.983971\pi\)
\(284\) 0 0
\(285\) −1.73205 3.00000i −0.102598 0.177705i
\(286\) 0 0
\(287\) −14.6603 −0.865367
\(288\) 0 0
\(289\) 38.7128 2.27722
\(290\) 0 0
\(291\) 8.53590 0.500383
\(292\) 0 0
\(293\) 0.267949 0.464102i 0.0156538 0.0271131i −0.858092 0.513495i \(-0.828350\pi\)
0.873746 + 0.486382i \(0.161684\pi\)
\(294\) 0 0
\(295\) −3.19615 5.53590i −0.186087 0.322312i
\(296\) 0 0
\(297\) −14.1962 24.5885i −0.823744 1.42677i
\(298\) 0 0
\(299\) −0.196152 0.339746i −0.0113438 0.0196480i
\(300\) 0 0
\(301\) 21.3923 37.0526i 1.23303 2.13567i
\(302\) 0 0
\(303\) −22.3923 −1.28640
\(304\) 0 0
\(305\) 1.53590 0.0879453
\(306\) 0 0
\(307\) −20.1244 −1.14856 −0.574279 0.818660i \(-0.694717\pi\)
−0.574279 + 0.818660i \(0.694717\pi\)
\(308\) 0 0
\(309\) 5.53590 + 9.58846i 0.314926 + 0.545468i
\(310\) 0 0
\(311\) −8.46410 + 14.6603i −0.479955 + 0.831307i −0.999736 0.0229931i \(-0.992680\pi\)
0.519780 + 0.854300i \(0.326014\pi\)
\(312\) 0 0
\(313\) 4.19615 + 7.26795i 0.237181 + 0.410809i 0.959904 0.280328i \(-0.0904434\pi\)
−0.722724 + 0.691137i \(0.757110\pi\)
\(314\) 0 0
\(315\) 5.59808 + 9.69615i 0.315416 + 0.546316i
\(316\) 0 0
\(317\) −4.73205 8.19615i −0.265778 0.460342i 0.701989 0.712188i \(-0.252296\pi\)
−0.967767 + 0.251846i \(0.918962\pi\)
\(318\) 0 0
\(319\) −23.1244 + 40.0526i −1.29472 + 2.24251i
\(320\) 0 0
\(321\) 4.50000 7.79423i 0.251166 0.435031i
\(322\) 0 0
\(323\) −14.9282 −0.830627
\(324\) 0 0
\(325\) 1.46410 0.0812137
\(326\) 0 0
\(327\) −2.93782 + 5.08846i −0.162462 + 0.281392i
\(328\) 0 0
\(329\) −6.96410 + 12.0622i −0.383943 + 0.665009i
\(330\) 0 0
\(331\) −1.73205 3.00000i −0.0952021 0.164895i 0.814491 0.580176i \(-0.197016\pi\)
−0.909693 + 0.415282i \(0.863683\pi\)
\(332\) 0 0
\(333\) 15.5885 + 27.0000i 0.854242 + 1.47959i
\(334\) 0 0
\(335\) 4.86603 + 8.42820i 0.265859 + 0.460482i
\(336\) 0 0
\(337\) −11.4641 + 19.8564i −0.624489 + 1.08165i 0.364150 + 0.931340i \(0.381360\pi\)
−0.988639 + 0.150307i \(0.951974\pi\)
\(338\) 0 0
\(339\) 12.0000 + 20.7846i 0.651751 + 1.12887i
\(340\) 0 0
\(341\) −10.9282 −0.591795
\(342\) 0 0
\(343\) −0.267949 −0.0144679
\(344\) 0 0
\(345\) −0.464102 −0.0249864
\(346\) 0 0
\(347\) −1.19615 + 2.07180i −0.0642128 + 0.111220i −0.896345 0.443358i \(-0.853787\pi\)
0.832132 + 0.554578i \(0.187120\pi\)
\(348\) 0 0
\(349\) 3.23205 + 5.59808i 0.173008 + 0.299658i 0.939470 0.342631i \(-0.111318\pi\)
−0.766462 + 0.642289i \(0.777985\pi\)
\(350\) 0 0
\(351\) −3.80385 + 6.58846i −0.203034 + 0.351666i
\(352\) 0 0
\(353\) −2.92820 5.07180i −0.155853 0.269945i 0.777517 0.628862i \(-0.216479\pi\)
−0.933369 + 0.358918i \(0.883146\pi\)
\(354\) 0 0
\(355\) 1.26795 2.19615i 0.0672958 0.116560i
\(356\) 0 0
\(357\) 48.2487 2.55359
\(358\) 0 0
\(359\) 0.928203 0.0489887 0.0244943 0.999700i \(-0.492202\pi\)
0.0244943 + 0.999700i \(0.492202\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −16.3301 28.2846i −0.857109 1.48456i
\(364\) 0 0
\(365\) −3.46410 + 6.00000i −0.181319 + 0.314054i
\(366\) 0 0
\(367\) −3.19615 5.53590i −0.166838 0.288972i 0.770469 0.637478i \(-0.220022\pi\)
−0.937306 + 0.348506i \(0.886689\pi\)
\(368\) 0 0
\(369\) −11.7846 −0.613482
\(370\) 0 0
\(371\) −11.1962 19.3923i −0.581275 1.00680i
\(372\) 0 0
\(373\) −2.46410 + 4.26795i −0.127586 + 0.220986i −0.922741 0.385421i \(-0.874056\pi\)
0.795155 + 0.606407i \(0.207390\pi\)
\(374\) 0 0
\(375\) 0.866025 1.50000i 0.0447214 0.0774597i
\(376\) 0 0
\(377\) 12.3923 0.638236
\(378\) 0 0
\(379\) −27.1769 −1.39598 −0.697992 0.716105i \(-0.745923\pi\)
−0.697992 + 0.716105i \(0.745923\pi\)
\(380\) 0 0
\(381\) −17.0885 + 29.5981i −0.875468 + 1.51636i
\(382\) 0 0
\(383\) 1.73205 3.00000i 0.0885037 0.153293i −0.818375 0.574684i \(-0.805125\pi\)
0.906879 + 0.421392i \(0.138458\pi\)
\(384\) 0 0
\(385\) 10.1962 + 17.6603i 0.519644 + 0.900050i
\(386\) 0 0
\(387\) 17.1962 29.7846i 0.874130 1.51404i
\(388\) 0 0
\(389\) 15.2321 + 26.3827i 0.772296 + 1.33766i 0.936302 + 0.351196i \(0.114225\pi\)
−0.164006 + 0.986459i \(0.552442\pi\)
\(390\) 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
\(395\) −8.53590 −0.429488
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) −12.9282 −0.647220
\(400\) 0 0
\(401\) 13.3923 23.1962i 0.668780 1.15836i −0.309466 0.950911i \(-0.600150\pi\)
0.978246 0.207450i \(-0.0665164\pi\)
\(402\) 0 0
\(403\) 1.46410 + 2.53590i 0.0729321 + 0.126322i
\(404\) 0 0
\(405\) 4.50000 + 7.79423i 0.223607 + 0.387298i
\(406\) 0 0
\(407\) 28.3923 + 49.1769i 1.40735 + 2.43761i
\(408\) 0 0
\(409\) 4.46410 7.73205i 0.220736 0.382325i −0.734296 0.678829i \(-0.762488\pi\)
0.955032 + 0.296504i \(0.0958209\pi\)
\(410\) 0 0
\(411\) −10.1436 −0.500347
\(412\) 0 0
\(413\) −23.8564 −1.17390
\(414\) 0 0
\(415\) 2.80385 0.137635
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.12436 15.8038i 0.445754 0.772068i −0.552350 0.833612i \(-0.686269\pi\)
0.998104 + 0.0615435i \(0.0196023\pi\)
\(420\) 0 0
\(421\) 11.5359 + 19.9808i 0.562225 + 0.973803i 0.997302 + 0.0734093i \(0.0233880\pi\)
−0.435077 + 0.900393i \(0.643279\pi\)
\(422\) 0 0
\(423\) −5.59808 + 9.69615i −0.272188 + 0.471443i
\(424\) 0 0
\(425\) −3.73205 6.46410i −0.181031 0.313555i
\(426\) 0 0
\(427\) 2.86603 4.96410i 0.138697 0.240230i
\(428\) 0 0
\(429\) −6.92820 + 12.0000i −0.334497 + 0.579365i
\(430\) 0 0
\(431\) −11.4641 −0.552206 −0.276103 0.961128i \(-0.589043\pi\)
−0.276103 + 0.961128i \(0.589043\pi\)
\(432\) 0 0
\(433\) −19.4641 −0.935385 −0.467693 0.883891i \(-0.654915\pi\)
−0.467693 + 0.883891i \(0.654915\pi\)
\(434\) 0 0
\(435\) 7.33013 12.6962i 0.351453 0.608734i
\(436\) 0 0
\(437\) 0.267949 0.464102i 0.0128177 0.0222010i
\(438\) 0 0
\(439\) −17.4641 30.2487i −0.833516 1.44369i −0.895233 0.445599i \(-0.852991\pi\)
0.0617168 0.998094i \(-0.480342\pi\)
\(440\) 0 0
\(441\) 20.7846 0.989743
\(442\) 0 0
\(443\) 5.25833 + 9.10770i 0.249831 + 0.432720i 0.963479 0.267785i \(-0.0862917\pi\)
−0.713648 + 0.700505i \(0.752958\pi\)
\(444\) 0 0
\(445\) 1.96410 3.40192i 0.0931073 0.161267i
\(446\) 0 0
\(447\) −3.86603 6.69615i −0.182857 0.316717i
\(448\) 0 0
\(449\) 31.8564 1.50340 0.751698 0.659507i \(-0.229235\pi\)
0.751698 + 0.659507i \(0.229235\pi\)
\(450\) 0 0
\(451\) −21.4641 −1.01071
\(452\) 0 0
\(453\) −2.53590 −0.119147
\(454\) 0 0
\(455\) 2.73205 4.73205i 0.128081 0.221842i
\(456\) 0 0
\(457\) −13.1962 22.8564i −0.617290 1.06918i −0.989978 0.141221i \(-0.954897\pi\)
0.372688 0.927957i \(-0.378436\pi\)
\(458\) 0 0
\(459\) 38.7846 1.81031
\(460\) 0 0
\(461\) −12.6962 21.9904i −0.591319 1.02419i −0.994055 0.108878i \(-0.965274\pi\)
0.402736 0.915316i \(-0.368059\pi\)
\(462\) 0 0
\(463\) −1.73205 + 3.00000i −0.0804952 + 0.139422i −0.903463 0.428667i \(-0.858984\pi\)
0.822968 + 0.568088i \(0.192317\pi\)
\(464\) 0 0
\(465\) 3.46410 0.160644
\(466\) 0 0
\(467\) −9.60770 −0.444591 −0.222296 0.974979i \(-0.571355\pi\)
−0.222296 + 0.974979i \(0.571355\pi\)
\(468\) 0 0
\(469\) 36.3205 1.67713
\(470\) 0 0
\(471\) 7.73205 + 13.3923i 0.356274 + 0.617085i
\(472\) 0 0
\(473\) 31.3205 54.2487i 1.44012 2.49436i
\(474\) 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
\(479\) −3.19615 5.53590i −0.146036 0.252942i 0.783723 0.621110i \(-0.213318\pi\)
−0.929759 + 0.368169i \(0.879985\pi\)
\(480\) 0 0
\(481\) 7.60770 13.1769i 0.346881 0.600816i
\(482\) 0 0
\(483\) −0.866025 + 1.50000i −0.0394055 + 0.0682524i
\(484\) 0 0
\(485\) −4.92820 −0.223778
\(486\) 0 0
\(487\) −8.24871 −0.373785 −0.186892 0.982380i \(-0.559842\pi\)
−0.186892 + 0.982380i \(0.559842\pi\)
\(488\) 0 0
\(489\) −18.4641 + 31.9808i −0.834976 + 1.44622i
\(490\) 0 0
\(491\) −0.928203 + 1.60770i −0.0418892 + 0.0725543i −0.886210 0.463284i \(-0.846671\pi\)
0.844321 + 0.535838i \(0.180004\pi\)
\(492\) 0 0
\(493\) −31.5885 54.7128i −1.42267 2.46414i
\(494\) 0 0
\(495\) 8.19615 + 14.1962i 0.368390 + 0.638070i
\(496\) 0 0
\(497\) −4.73205 8.19615i −0.212261 0.367648i
\(498\) 0 0
\(499\) −4.92820 + 8.53590i −0.220617 + 0.382119i −0.954995 0.296621i \(-0.904140\pi\)
0.734379 + 0.678740i \(0.237474\pi\)
\(500\) 0 0
\(501\) −8.30385 14.3827i −0.370989 0.642571i
\(502\) 0 0
\(503\) 6.12436 0.273072 0.136536 0.990635i \(-0.456403\pi\)
0.136536 + 0.990635i \(0.456403\pi\)
\(504\) 0 0
\(505\) 12.9282 0.575297
\(506\) 0 0
\(507\) −18.8038 −0.835108
\(508\) 0 0
\(509\) 20.6962 35.8468i 0.917341 1.58888i 0.113903 0.993492i \(-0.463665\pi\)
0.803438 0.595389i \(-0.203002\pi\)
\(510\) 0 0
\(511\) 12.9282 + 22.3923i 0.571910 + 0.990577i
\(512\) 0 0
\(513\) −10.3923 −0.458831
\(514\) 0 0
\(515\) −3.19615 5.53590i −0.140839 0.243941i
\(516\) 0 0
\(517\) −10.1962 + 17.6603i −0.448426 + 0.776697i
\(518\) 0 0
\(519\) 39.7128 1.74320
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 0 0
\(523\) 21.1962 0.926843 0.463422 0.886138i \(-0.346622\pi\)
0.463422 + 0.886138i \(0.346622\pi\)
\(524\) 0 0
\(525\) −3.23205 5.59808i −0.141058 0.244320i
\(526\) 0 0
\(527\) 7.46410 12.9282i 0.325141 0.563161i
\(528\) 0 0
\(529\) 11.4641 + 19.8564i 0.498439 + 0.863322i
\(530\) 0 0
\(531\) −19.1769 −0.832207
\(532\) 0 0
\(533\) 2.87564 + 4.98076i 0.124558 + 0.215741i
\(534\) 0 0
\(535\) −2.59808 + 4.50000i −0.112325 + 0.194552i
\(536\) 0 0
\(537\) −5.66025 + 9.80385i −0.244258 + 0.423067i
\(538\) 0 0
\(539\) 37.8564 1.63059
\(540\) 0 0
\(541\) −39.3923 −1.69361 −0.846804 0.531905i \(-0.821476\pi\)
−0.846804 + 0.531905i \(0.821476\pi\)
\(542\) 0 0
\(543\) 14.2583 24.6962i 0.611884 1.05981i
\(544\) 0 0
\(545\) 1.69615 2.93782i 0.0726552 0.125842i
\(546\) 0 0
\(547\) 2.99038 + 5.17949i 0.127859 + 0.221459i 0.922847 0.385167i \(-0.125856\pi\)
−0.794988 + 0.606626i \(0.792523\pi\)
\(548\) 0 0
\(549\) 2.30385 3.99038i 0.0983258 0.170305i
\(550\) 0 0
\(551\) 8.46410 + 14.6603i 0.360583 + 0.624548i
\(552\) 0 0
\(553\) −15.9282 + 27.5885i −0.677336 + 1.17318i
\(554\) 0 0
\(555\) −9.00000 15.5885i −0.382029 0.661693i
\(556\) 0 0
\(557\) −36.6410 −1.55253 −0.776265 0.630407i \(-0.782888\pi\)
−0.776265 + 0.630407i \(0.782888\pi\)
\(558\) 0 0
\(559\) −16.7846 −0.709913
\(560\) 0 0
\(561\) 70.6410 2.98247
\(562\) 0 0
\(563\) 21.7942 37.7487i 0.918517 1.59092i 0.116849 0.993150i \(-0.462721\pi\)
0.801669 0.597769i \(-0.203946\pi\)
\(564\) 0 0
\(565\) −6.92820 12.0000i −0.291472 0.504844i
\(566\) 0 0
\(567\) 33.5885 1.41058
\(568\) 0 0
\(569\) −5.00000 8.66025i −0.209611 0.363057i 0.741981 0.670421i \(-0.233886\pi\)
−0.951592 + 0.307364i \(0.900553\pi\)
\(570\) 0 0
\(571\) −0.392305 + 0.679492i −0.0164174 + 0.0284359i −0.874117 0.485715i \(-0.838559\pi\)
0.857700 + 0.514151i \(0.171893\pi\)
\(572\) 0 0
\(573\) 5.32051 0.222267
\(574\) 0 0
\(575\) 0.267949 0.0111743
\(576\) 0 0
\(577\) 19.0718 0.793969 0.396985 0.917825i \(-0.370057\pi\)
0.396985 + 0.917825i \(0.370057\pi\)
\(578\) 0 0
\(579\) 9.12436 + 15.8038i 0.379195 + 0.656785i
\(580\) 0 0
\(581\) 5.23205 9.06218i 0.217062 0.375962i
\(582\) 0 0
\(583\) −16.3923 28.3923i −0.678900 1.17589i
\(584\) 0 0
\(585\) 2.19615 3.80385i 0.0907997 0.157270i
\(586\) 0 0
\(587\) −19.7942 34.2846i −0.816995 1.41508i −0.907886 0.419216i \(-0.862305\pi\)
0.0908911 0.995861i \(-0.471028\pi\)
\(588\) 0 0
\(589\) −2.00000 + 3.46410i −0.0824086 + 0.142736i
\(590\) 0 0
\(591\) −0.928203 + 1.60770i −0.0381812 + 0.0661317i
\(592\) 0 0
\(593\) −32.5359 −1.33609 −0.668045 0.744121i \(-0.732868\pi\)
−0.668045 + 0.744121i \(0.732868\pi\)
\(594\) 0 0
\(595\) −27.8564 −1.14200
\(596\) 0 0
\(597\) −18.1244 + 31.3923i −0.741780 + 1.28480i
\(598\) 0 0
\(599\) −11.1962 + 19.3923i −0.457462 + 0.792348i −0.998826 0.0484404i \(-0.984575\pi\)
0.541364 + 0.840789i \(0.317908\pi\)
\(600\) 0 0
\(601\) 5.39230 + 9.33975i 0.219957 + 0.380976i 0.954794 0.297267i \(-0.0960751\pi\)
−0.734838 + 0.678243i \(0.762742\pi\)
\(602\) 0 0
\(603\) 29.1962 1.18896
\(604\) 0 0
\(605\) 9.42820 + 16.3301i 0.383311 + 0.663914i
\(606\) 0 0
\(607\) −21.8660 + 37.8731i −0.887515 + 1.53722i −0.0447105 + 0.999000i \(0.514237\pi\)
−0.842804 + 0.538220i \(0.819097\pi\)
\(608\) 0 0
\(609\) −27.3564 47.3827i −1.10854 1.92004i
\(610\) 0 0
\(611\) 5.46410 0.221054
\(612\) 0 0
\(613\) 1.60770 0.0649342 0.0324671 0.999473i \(-0.489664\pi\)
0.0324671 + 0.999473i \(0.489664\pi\)
\(614\) 0 0
\(615\) 6.80385 0.274358
\(616\) 0 0
\(617\) −8.53590 + 14.7846i −0.343642 + 0.595206i −0.985106 0.171947i \(-0.944994\pi\)
0.641464 + 0.767153i \(0.278327\pi\)
\(618\) 0 0
\(619\) −8.85641 15.3397i −0.355969 0.616556i 0.631314 0.775527i \(-0.282516\pi\)
−0.987283 + 0.158971i \(0.949182\pi\)
\(620\) 0 0
\(621\) −0.696152 + 1.20577i −0.0279356 + 0.0483859i
\(622\) 0 0
\(623\) −7.33013 12.6962i −0.293675 0.508661i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −18.9282 −0.755920
\(628\) 0 0
\(629\) −77.5692 −3.09289
\(630\) 0 0
\(631\) −23.7128 −0.943992 −0.471996 0.881601i \(-0.656466\pi\)
−0.471996 + 0.881601i \(0.656466\pi\)
\(632\) 0 0
\(633\) 17.6603 + 30.5885i 0.701932 + 1.21578i
\(634\) 0 0
\(635\) 9.86603 17.0885i 0.391521 0.678135i
\(636\) 0 0
\(637\) −5.07180 8.78461i −0.200952 0.348059i
\(638\) 0 0
\(639\) −3.80385 6.58846i −0.150478 0.260635i
\(640\) 0 0
\(641\) 3.57180 + 6.18653i 0.141077 + 0.244353i 0.927903 0.372823i \(-0.121610\pi\)
−0.786825 + 0.617176i \(0.788277\pi\)
\(642\) 0 0
\(643\) 18.8660 32.6769i 0.744003 1.28865i −0.206656 0.978414i \(-0.566258\pi\)
0.950659 0.310238i \(-0.100409\pi\)
\(644\) 0 0
\(645\) −9.92820 + 17.1962i −0.390923 + 0.677098i
\(646\) 0 0
\(647\) 28.5167 1.12111 0.560553 0.828119i \(-0.310589\pi\)
0.560553 + 0.828119i \(0.310589\pi\)
\(648\) 0 0
\(649\) −34.9282 −1.37105
\(650\) 0 0
\(651\) 6.46410 11.1962i 0.253348 0.438812i
\(652\) 0 0
\(653\) −12.2679 + 21.2487i −0.480082 + 0.831526i −0.999739 0.0228487i \(-0.992726\pi\)
0.519657 + 0.854375i \(0.326060\pi\)
\(654\) 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 \(-0.299335\pi\)
−0.994301 + 0.106606i \(0.966001\pi\)
\(660\) 0 0
\(661\) 16.8564 29.1962i 0.655638 1.13560i −0.326095 0.945337i \(-0.605733\pi\)
0.981733 0.190262i \(-0.0609337\pi\)
\(662\) 0 0
\(663\) −9.46410 16.3923i −0.367555 0.636624i
\(664\) 0 0
\(665\) 7.46410 0.289445
\(666\) 0 0
\(667\) 2.26795 0.0878153
\(668\) 0 0
\(669\) 9.24871 0.357576
\(670\) 0 0
\(671\) 4.19615 7.26795i 0.161991 0.280576i
\(672\) 0 0
\(673\) 23.6603 + 40.9808i 0.912036 + 1.57969i 0.811184 + 0.584791i \(0.198823\pi\)
0.100852 + 0.994901i \(0.467843\pi\)
\(674\) 0 0
\(675\) −2.59808 4.50000i −0.100000 0.173205i
\(676\) 0 0
\(677\) −22.5885 39.1244i −0.868145 1.50367i −0.863890 0.503680i \(-0.831979\pi\)
−0.00425474 0.999991i \(-0.501354\pi\)
\(678\) 0 0
\(679\) −9.19615 + 15.9282i −0.352916 + 0.611268i
\(680\) 0 0
\(681\) −42.0000 −1.60944
\(682\) 0 0
\(683\) 1.60770 0.0615167 0.0307584 0.999527i \(-0.490208\pi\)
0.0307584 + 0.999527i \(0.490208\pi\)
\(684\) 0 0
\(685\) 5.85641 0.223762
\(686\) 0 0
\(687\) 20.1340 + 34.8731i 0.768159 + 1.33049i
\(688\) 0 0
\(689\) −4.39230 + 7.60770i −0.167333 + 0.289830i
\(690\) 0 0
\(691\) 14.8038 + 25.6410i 0.563165 + 0.975430i 0.997218 + 0.0745428i \(0.0237498\pi\)
−0.434053 + 0.900887i \(0.642917\pi\)
\(692\) 0 0
\(693\) 61.1769 2.32392
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 14.6603 25.3923i 0.555297 0.961802i
\(698\) 0 0
\(699\) −9.00000 + 15.5885i −0.340411 + 0.589610i
\(700\) 0 0
\(701\) 38.1769 1.44192 0.720961 0.692976i \(-0.243701\pi\)
0.720961 + 0.692976i \(0.243701\pi\)
\(702\) 0 0
\(703\) 20.7846 0.783906
\(704\) 0 0
\(705\) 3.23205 5.59808i 0.121726 0.210836i
\(706\) 0 0
\(707\) 24.1244 41.7846i 0.907290 1.57147i
\(708\) 0 0
\(709\) 2.69615 + 4.66987i 0.101256 + 0.175381i 0.912202 0.409740i \(-0.134381\pi\)
−0.810946 + 0.585121i \(0.801047\pi\)
\(710\) 0 0
\(711\) −12.8038 + 22.1769i −0.480182 + 0.831699i
\(712\) 0 0
\(713\) 0.267949 + 0.464102i 0.0100348 + 0.0173807i
\(714\) 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
\(719\) 1.32051 0.0492466 0.0246233 0.999697i \(-0.492161\pi\)
0.0246233 + 0.999697i \(0.492161\pi\)
\(720\) 0 0
\(721\) −23.8564 −0.888459
\(722\) 0 0
\(723\) −34.5167 −1.28369
\(724\) 0 0
\(725\) −4.23205 + 7.33013i −0.157174 + 0.272234i
\(726\) 0 0
\(727\) 17.9904 + 31.1603i 0.667226 + 1.15567i 0.978677 + 0.205407i \(0.0658519\pi\)
−0.311450 + 0.950262i \(0.600815\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 42.7846 + 74.1051i 1.58245 + 2.74088i
\(732\) 0 0
\(733\) −21.9282 + 37.9808i −0.809937 + 1.40285i 0.102971 + 0.994684i \(0.467165\pi\)
−0.912907 + 0.408167i \(0.866168\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) 53.1769 1.95880
\(738\) 0 0
\(739\) 48.2487 1.77486 0.887429 0.460945i \(-0.152489\pi\)
0.887429 + 0.460945i \(0.152489\pi\)
\(740\) 0 0
\(741\) 2.53590 + 4.39230i 0.0931586 + 0.161355i
\(742\) 0 0
\(743\) −22.9186 + 39.6962i −0.840801 + 1.45631i 0.0484169 + 0.998827i \(0.484582\pi\)
−0.889218 + 0.457483i \(0.848751\pi\)
\(744\) 0 0
\(745\) 2.23205 + 3.86603i 0.0817760 + 0.141640i
\(746\) 0 0
\(747\) 4.20577 7.28461i 0.153881 0.266530i
\(748\) 0 0
\(749\) 9.69615 + 16.7942i 0.354290 + 0.613648i
\(750\) 0 0
\(751\) 2.19615 3.80385i 0.0801387 0.138804i −0.823171 0.567794i \(-0.807797\pi\)
0.903309 + 0.428990i \(0.141130\pi\)
\(752\) 0 0
\(753\) −17.1962 + 29.7846i −0.626663 + 1.08541i
\(754\) 0 0
\(755\) 1.46410 0.0532841
\(756\) 0 0
\(757\) 0.392305 0.0142586 0.00712928 0.999975i \(-0.497731\pi\)
0.00712928 + 0.999975i \(0.497731\pi\)
\(758\) 0 0
\(759\) −1.26795 + 2.19615i −0.0460236 + 0.0797153i
\(760\) 0 0
\(761\) 10.5000 18.1865i 0.380625 0.659261i −0.610527 0.791995i \(-0.709042\pi\)
0.991152 + 0.132734i \(0.0423756\pi\)
\(762\) 0 0
\(763\) −6.33013 10.9641i −0.229166 0.396927i
\(764\) 0 0
\(765\) −22.3923 −0.809595
\(766\) 0 0
\(767\) 4.67949 + 8.10512i 0.168967 + 0.292659i
\(768\) 0 0
\(769\) 3.89230 6.74167i 0.140360 0.243111i −0.787272 0.616606i \(-0.788507\pi\)
0.927632 + 0.373495i \(0.121841\pi\)
\(770\) 0 0
\(771\) −11.1962 19.3923i −0.403220 0.698397i
\(772\) 0 0
\(773\) 32.7846 1.17918 0.589590 0.807703i \(-0.299289\pi\)
0.589590 + 0.807703i \(0.299289\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) −67.1769 −2.40996
\(778\) 0 0
\(779\) −3.92820 + 6.80385i −0.140742 + 0.243773i
\(780\) 0 0
\(781\) −6.92820 12.0000i −0.247911 0.429394i
\(782\) 0 0
\(783\) −21.9904 38.0885i −0.785872 1.36117i
\(784\) 0 0
\(785\) −4.46410 7.73205i −0.159331 0.275969i
\(786\) 0 0
\(787\) 16.6603 28.8564i 0.593874 1.02862i −0.399831 0.916589i \(-0.630931\pi\)
0.993705 0.112031i \(-0.0357355\pi\)
\(788\) 0 0
\(789\) 2.78461 0.0991347
\(790\) 0 0
\(791\) −51.7128 −1.83870
\(792\) 0 0
\(793\) −2.24871 −0.0798541
\(794\) 0 0
\(795\) 5.19615 + 9.00000i 0.184289 + 0.319197i
\(796\) 0 0
\(797\) 14.1962 24.5885i 0.502854 0.870968i −0.497141 0.867670i \(-0.665617\pi\)
0.999995 0.00329810i \(-0.00104982\pi\)
\(798\) 0 0
\(799\) −13.9282 24.1244i −0.492744 0.853458i
\(800\) 0 0
\(801\) −5.89230 10.2058i −0.208194 0.360603i
\(802\) 0 0
\(803\) 18.9282 + 32.7846i 0.667962 + 1.15694i
\(804\) 0 0
\(805\) 0.500000 0.866025i 0.0176227 0.0305234i
\(806\) 0 0
\(807\) 8.13397 14.0885i 0.286329 0.495937i
\(808\) 0 0
\(809\) −35.0718 −1.23306 −0.616529 0.787332i \(-0.711462\pi\)
−0.616529 + 0.787332i \(0.711462\pi\)
\(810\) 0 0
\(811\) −29.6077 −1.03967 −0.519833 0.854268i \(-0.674006\pi\)
−0.519833 + 0.854268i \(0.674006\pi\)
\(812\) 0 0
\(813\) −6.80385 + 11.7846i −0.238621 + 0.413304i
\(814\) 0 0
\(815\) 10.6603 18.4641i 0.373412 0.646769i
\(816\) 0 0
\(817\) −11.4641 19.8564i −0.401078 0.694688i
\(818\) 0 0
\(819\) −8.19615 14.1962i −0.286397 0.496054i
\(820\) 0 0
\(821\) 12.6244 + 21.8660i 0.440593 + 0.763130i 0.997734 0.0672887i \(-0.0214349\pi\)
−0.557140 + 0.830418i \(0.688102\pi\)
\(822\) 0 0
\(823\) −10.1340 + 17.5526i −0.353248 + 0.611844i −0.986817 0.161843i \(-0.948256\pi\)
0.633568 + 0.773687i \(0.281590\pi\)
\(824\) 0 0
\(825\) −4.73205 8.19615i −0.164749 0.285353i
\(826\) 0 0
\(827\) −55.0526 −1.91437 −0.957183 0.289485i \(-0.906516\pi\)
−0.957183 + 0.289485i \(0.906516\pi\)
\(828\) 0 0
\(829\) −9.39230 −0.326208 −0.163104 0.986609i \(-0.552151\pi\)
−0.163104 + 0.986609i \(0.552151\pi\)
\(830\) 0 0
\(831\) 11.0718 0.384076
\(832\) 0 0
\(833\) −25.8564 + 44.7846i −0.895871 + 1.55169i
\(834\) 0 0
\(835\) 4.79423 + 8.30385i 0.165911 + 0.287366i
\(836\) 0 0
\(837\) 5.19615 9.00000i 0.179605 0.311086i
\(838\) 0 0
\(839\) −4.07180 7.05256i −0.140574 0.243481i 0.787139 0.616776i \(-0.211561\pi\)
−0.927713 + 0.373294i \(0.878228\pi\)
\(840\) 0 0
\(841\) −21.3205 + 36.9282i −0.735190 + 1.27339i
\(842\) 0 0
\(843\) 22.2679 0.766949
\(844\) 0 0
\(845\) 10.8564 0.373472
\(846\) 0 0
\(847\) 70.3731 2.41805
\(848\) 0 0
\(849\) −15.8205 27.4019i −0.542958 0.940432i
\(850\) 0 0
\(851\) 1.39230 2.41154i 0.0477276 0.0826666i
\(852\) 0 0
\(853\) 21.2679 + 36.8372i 0.728201 + 1.26128i 0.957643 + 0.287958i \(0.0929765\pi\)
−0.229442 + 0.973322i \(0.573690\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 15.1244 + 26.1962i 0.516638 + 0.894844i 0.999813 + 0.0193198i \(0.00615008\pi\)
−0.483175 + 0.875524i \(0.660517\pi\)
\(858\) 0 0
\(859\) −0.196152 + 0.339746i −0.00669263 + 0.0115920i −0.869352 0.494193i \(-0.835464\pi\)
0.862660 + 0.505785i \(0.168797\pi\)
\(860\) 0 0
\(861\) 12.6962 21.9904i 0.432684 0.749430i
\(862\) 0 0
\(863\) 9.33975 0.317929 0.158964 0.987284i \(-0.449185\pi\)
0.158964 + 0.987284i \(0.449185\pi\)
\(864\) 0 0
\(865\) −22.9282 −0.779582
\(866\) 0 0
\(867\) −33.5263 + 58.0692i −1.13861 + 1.97213i
\(868\) 0 0
\(869\) −23.3205 + 40.3923i −0.791094 + 1.37022i
\(870\) 0 0
\(871\) −7.12436 12.3397i −0.241400 0.418116i
\(872\) 0 0
\(873\) −7.39230 + 12.8038i −0.250192 + 0.433345i
\(874\) 0 0
\(875\) 1.86603 + 3.23205i 0.0630832 + 0.109263i
\(876\) 0 0
\(877\) −16.1962 + 28.0526i −0.546905 + 0.947268i 0.451579 + 0.892231i \(0.350861\pi\)
−0.998484 + 0.0550365i \(0.982472\pi\)
\(878\) 0 0
\(879\) 0.464102 + 0.803848i 0.0156538 + 0.0271131i
\(880\) 0 0
\(881\) −31.6410 −1.06601 −0.533006 0.846111i \(-0.678938\pi\)
−0.533006 + 0.846111i \(0.678938\pi\)
\(882\) 0 0
\(883\) 1.19615 0.0402537 0.0201269 0.999797i \(-0.493593\pi\)
0.0201269 + 0.999797i \(0.493593\pi\)
\(884\) 0 0
\(885\) 11.0718 0.372174
\(886\) 0 0
\(887\) −12.6603 + 21.9282i −0.425090 + 0.736277i −0.996429 0.0844371i \(-0.973091\pi\)
0.571339 + 0.820714i \(0.306424\pi\)
\(888\) 0 0
\(889\) −36.8205 63.7750i −1.23492 2.13894i
\(890\) 0 0
\(891\) 49.1769 1.64749
\(892\) 0 0
\(893\) 3.73205 + 6.46410i 0.124888 + 0.216313i
\(894\) 0 0
\(895\) 3.26795 5.66025i 0.109235 0.189201i
\(896\) 0 0
\(897\) 0.679492 0.0226876
\(898\) 0 0
\(899\) −16.9282 −0.564587
\(900\) 0 0
\(901\) 44.7846 1.49199
\(902\) 0 0
\(903\) 37.0526 + 64.1769i 1.23303 + 2.13567i
\(904\) 0 0
\(905\) −8.23205 + 14.2583i −0.273643 + 0.473963i
\(906\) 0 0
\(907\) 5.13397 + 8.89230i 0.170471 + 0.295264i 0.938585 0.345049i \(-0.112138\pi\)
−0.768114 + 0.640313i \(0.778804\pi\)
\(908\) 0 0
\(909\) 19.3923 33.5885i 0.643202 1.11406i
\(910\) 0 0
\(911\) 9.73205 + 16.8564i 0.322437 + 0.558478i 0.980990 0.194057i \(-0.0621645\pi\)
−0.658553 + 0.752534i \(0.728831\pi\)
\(912\) 0 0
\(913\) 7.66025 13.2679i 0.253517 0.439105i
\(914\) 0 0
\(915\) −1.33013 + 2.30385i −0.0439726 + 0.0761629i
\(916\) 0 0
\(917\) 12.9282 0.426927
\(918\) 0 0
\(919\) −38.3923 −1.26645 −0.633223 0.773970i \(-0.718268\pi\)
−0.633223 + 0.773970i \(0.718268\pi\)
\(920\) 0 0
\(921\) 17.4282 30.1865i 0.574279 0.994680i
\(922\) 0 0
\(923\) −1.85641 + 3.21539i −0.0611044 + 0.105836i
\(924\) 0 0
\(925\) 5.19615 + 9.00000i 0.170848 + 0.295918i
\(926\) 0 0
\(927\) −19.1769 −0.629853
\(928\) 0 0
\(929\) 22.3205 + 38.6603i 0.732312 + 1.26840i 0.955893 + 0.293716i \(0.0948921\pi\)
−0.223581 + 0.974685i \(0.571775\pi\)
\(930\) 0 0
\(931\) 6.92820 12.0000i 0.227063 0.393284i
\(932\) 0 0
\(933\) −14.6603 25.3923i −0.479955 0.831307i
\(934\) 0 0
\(935\) −40.7846 −1.33380
\(936\) 0 0
\(937\) 43.7128 1.42804 0.714018 0.700128i \(-0.246874\pi\)
0.714018 + 0.700128i \(0.246874\pi\)
\(938\) 0 0
\(939\) −14.5359 −0.474361
\(940\) 0 0
\(941\) 1.30385 2.25833i 0.0425042 0.0736195i −0.843991 0.536358i \(-0.819800\pi\)
0.886495 + 0.462738i \(0.153133\pi\)
\(942\) 0 0
\(943\) 0.526279 + 0.911543i 0.0171380 + 0.0296839i
\(944\) 0 0
\(945\) −19.3923 −0.630832
\(946\) 0 0
\(947\) −15.2583 26.4282i −0.495829 0.858801i 0.504159 0.863611i \(-0.331802\pi\)
−0.999988 + 0.00480945i \(0.998469\pi\)
\(948\) 0 0
\(949\) 5.07180 8.78461i 0.164637 0.285160i
\(950\) 0 0
\(951\) 16.3923 0.531557
\(952\) 0 0
\(953\) 19.6077 0.635156 0.317578 0.948232i \(-0.397131\pi\)
0.317578 + 0.948232i \(0.397131\pi\)
\(954\) 0 0
\(955\) −3.07180 −0.0994010
\(956\) 0 0
\(957\) −40.0526 69.3731i −1.29472 2.24251i
\(958\) 0 0
\(959\) 10.9282 18.9282i 0.352890 0.611224i
\(960\) 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
\(965\) −5.26795 9.12436i −0.169581 0.293723i
\(966\) 0 0
\(967\) −19.3301 + 33.4808i −0.621615 + 1.07667i 0.367570 + 0.929996i \(0.380190\pi\)
−0.989185 + 0.146673i \(0.953144\pi\)
\(968\) 0 0
\(969\) 12.9282 22.3923i 0.415314 0.719344i
\(970\) 0 0
\(971\) −29.7128 −0.953530 −0.476765 0.879031i \(-0.658191\pi\)
−0.476765 + 0.879031i \(0.658191\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.26795 + 2.19615i −0.0406069 + 0.0703332i
\(976\) 0 0
\(977\) 22.5885 39.1244i 0.722669 1.25170i −0.237258 0.971447i \(-0.576249\pi\)
0.959926 0.280252i \(-0.0904181\pi\)
\(978\) 0 0
\(979\) −10.7321 18.5885i −0.342998 0.594090i
\(980\) 0 0
\(981\) −5.08846 8.81347i −0.162462 0.281392i
\(982\) 0 0
\(983\) 2.25833 + 3.91154i 0.0720295 + 0.124759i 0.899791 0.436322i \(-0.143719\pi\)
−0.827761 + 0.561081i \(0.810386\pi\)
\(984\) 0 0
\(985\) 0.535898 0.928203i 0.0170751 0.0295750i
\(986\) 0 0
\(987\) −12.0622 20.8923i −0.383943 0.665009i
\(988\) 0 0
\(989\) −3.07180 −0.0976775
\(990\) 0 0
\(991\) 53.0333 1.68466 0.842329 0.538963i \(-0.181184\pi\)
0.842329 + 0.538963i \(0.181184\pi\)
\(992\) 0 0
\(993\) 6.00000 0.190404
\(994\) 0 0
\(995\) 10.4641 18.1244i 0.331734 0.574581i
\(996\) 0 0
\(997\) 17.7846 + 30.8038i 0.563244 + 0.975568i 0.997211 + 0.0746386i \(0.0237803\pi\)
−0.433966 + 0.900929i \(0.642886\pi\)
\(998\) 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.121.1 4
3.2 odd 2 1080.2.q.b.361.1 4
4.3 odd 2 720.2.q.h.481.2 4
9.2 odd 6 1080.2.q.b.721.1 4
9.4 even 3 3240.2.a.k.1.2 2
9.5 odd 6 3240.2.a.p.1.2 2
9.7 even 3 inner 360.2.q.b.241.1 yes 4
12.11 even 2 2160.2.q.h.1441.2 4
36.7 odd 6 720.2.q.h.241.2 4
36.11 even 6 2160.2.q.h.721.2 4
36.23 even 6 6480.2.a.bk.1.1 2
36.31 odd 6 6480.2.a.ba.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.b.121.1 4 1.1 even 1 trivial
360.2.q.b.241.1 yes 4 9.7 even 3 inner
720.2.q.h.241.2 4 36.7 odd 6
720.2.q.h.481.2 4 4.3 odd 2
1080.2.q.b.361.1 4 3.2 odd 2
1080.2.q.b.721.1 4 9.2 odd 6
2160.2.q.h.721.2 4 36.11 even 6
2160.2.q.h.1441.2 4 12.11 even 2
3240.2.a.k.1.2 2 9.4 even 3
3240.2.a.p.1.2 2 9.5 odd 6
6480.2.a.ba.1.1 2 36.31 odd 6
6480.2.a.bk.1.1 2 36.23 even 6