Properties

Label 720.2.q.c.481.1
Level $720$
Weight $2$
Character 720.481
Analytic conductor $5.749$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(241,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 481.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 720.481
Dual form 720.2.q.c.241.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(1.50000 - 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{11} +(2.00000 - 3.46410i) q^{13} +1.73205i q^{15} +3.00000 q^{17} -5.00000 q^{19} +(-6.00000 - 3.46410i) q^{21} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} -5.19615i q^{27} +(-3.00000 - 5.19615i) q^{29} +(1.00000 - 1.73205i) q^{31} +(4.50000 + 2.59808i) q^{33} +4.00000 q^{35} -4.00000 q^{37} -6.92820i q^{39} +(1.50000 - 2.59808i) q^{41} +(5.50000 + 9.52628i) q^{43} +(1.50000 + 2.59808i) q^{45} +(-4.50000 + 7.79423i) q^{49} +(4.50000 - 2.59808i) q^{51} +6.00000 q^{53} -3.00000 q^{55} +(-7.50000 + 4.33013i) q^{57} +(-1.50000 + 2.59808i) q^{59} +(5.00000 + 8.66025i) q^{61} -12.0000 q^{63} +(2.00000 + 3.46410i) q^{65} +(2.50000 - 4.33013i) q^{67} -10.3923i q^{69} -6.00000 q^{71} -7.00000 q^{73} +(-1.50000 - 0.866025i) q^{75} +(6.00000 - 10.3923i) q^{77} +(7.00000 + 12.1244i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(6.00000 + 10.3923i) q^{83} +(-1.50000 + 2.59808i) q^{85} +(-9.00000 - 5.19615i) q^{87} +6.00000 q^{89} -16.0000 q^{91} -3.46410i q^{93} +(2.50000 - 4.33013i) q^{95} +(-5.50000 - 9.52628i) q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - q^{5} - 4 q^{7} + 3 q^{9} + 3 q^{11} + 4 q^{13} + 6 q^{17} - 10 q^{19} - 12 q^{21} + 6 q^{23} - q^{25} - 6 q^{29} + 2 q^{31} + 9 q^{33} + 8 q^{35} - 8 q^{37} + 3 q^{41} + 11 q^{43} + 3 q^{45} - 9 q^{49} + 9 q^{51} + 12 q^{53} - 6 q^{55} - 15 q^{57} - 3 q^{59} + 10 q^{61} - 24 q^{63} + 4 q^{65} + 5 q^{67} - 12 q^{71} - 14 q^{73} - 3 q^{75} + 12 q^{77} + 14 q^{79} - 9 q^{81} + 12 q^{83} - 3 q^{85} - 18 q^{87} + 12 q^{89} - 32 q^{91} + 5 q^{95} - 11 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −2.00000 3.46410i −0.755929 1.30931i −0.944911 0.327327i \(-0.893852\pi\)
0.188982 0.981981i \(-0.439481\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0 0
\(13\) 2.00000 3.46410i 0.554700 0.960769i −0.443227 0.896410i \(-0.646166\pi\)
0.997927 0.0643593i \(-0.0205004\pi\)
\(14\) 0 0
\(15\) 1.73205i 0.447214i
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) −6.00000 3.46410i −1.30931 0.755929i
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −3.00000 5.19615i −0.557086 0.964901i −0.997738 0.0672232i \(-0.978586\pi\)
0.440652 0.897678i \(-0.354747\pi\)
\(30\) 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
\(33\) 4.50000 + 2.59808i 0.783349 + 0.452267i
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 6.92820i 1.10940i
\(40\) 0 0
\(41\) 1.50000 2.59808i 0.234261 0.405751i −0.724797 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 0 0
\(43\) 5.50000 + 9.52628i 0.838742 + 1.45274i 0.890947 + 0.454108i \(0.150042\pi\)
−0.0522047 + 0.998636i \(0.516625\pi\)
\(44\) 0 0
\(45\) 1.50000 + 2.59808i 0.223607 + 0.387298i
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −4.50000 + 7.79423i −0.642857 + 1.11346i
\(50\) 0 0
\(51\) 4.50000 2.59808i 0.630126 0.363803i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) −7.50000 + 4.33013i −0.993399 + 0.573539i
\(58\) 0 0
\(59\) −1.50000 + 2.59808i −0.195283 + 0.338241i −0.946993 0.321253i \(-0.895896\pi\)
0.751710 + 0.659494i \(0.229229\pi\)
\(60\) 0 0
\(61\) 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i \(0.0544754\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 0 0
\(63\) −12.0000 −1.51186
\(64\) 0 0
\(65\) 2.00000 + 3.46410i 0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.50000 4.33013i 0.305424 0.529009i −0.671932 0.740613i \(-0.734535\pi\)
0.977356 + 0.211604i \(0.0678686\pi\)
\(68\) 0 0
\(69\) 10.3923i 1.25109i
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) −1.50000 0.866025i −0.173205 0.100000i
\(76\) 0 0
\(77\) 6.00000 10.3923i 0.683763 1.18431i
\(78\) 0 0
\(79\) 7.00000 + 12.1244i 0.787562 + 1.36410i 0.927457 + 0.373930i \(0.121990\pi\)
−0.139895 + 0.990166i \(0.544677\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −1.50000 + 2.59808i −0.162698 + 0.281801i
\(86\) 0 0
\(87\) −9.00000 5.19615i −0.964901 0.557086i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) 0 0
\(93\) 3.46410i 0.359211i
\(94\) 0 0
\(95\) 2.50000 4.33013i 0.256495 0.444262i
\(96\) 0 0
\(97\) −5.50000 9.52628i −0.558440 0.967247i −0.997627 0.0688512i \(-0.978067\pi\)
0.439187 0.898396i \(-0.355267\pi\)
\(98\) 0 0
\(99\) 9.00000 0.904534
\(100\) 0 0
\(101\) 6.00000 + 10.3923i 0.597022 + 1.03407i 0.993258 + 0.115924i \(0.0369830\pi\)
−0.396236 + 0.918149i \(0.629684\pi\)
\(102\) 0 0
\(103\) −2.00000 + 3.46410i −0.197066 + 0.341328i −0.947576 0.319531i \(-0.896475\pi\)
0.750510 + 0.660859i \(0.229808\pi\)
\(104\) 0 0
\(105\) 6.00000 3.46410i 0.585540 0.338062i
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −6.00000 + 3.46410i −0.569495 + 0.328798i
\(112\) 0 0
\(113\) −9.00000 + 15.5885i −0.846649 + 1.46644i 0.0375328 + 0.999295i \(0.488050\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) 3.00000 + 5.19615i 0.279751 + 0.484544i
\(116\) 0 0
\(117\) −6.00000 10.3923i −0.554700 0.960769i
\(118\) 0 0
\(119\) −6.00000 10.3923i −0.550019 0.952661i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 5.19615i 0.468521i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 16.5000 + 9.52628i 1.45274 + 0.838742i
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) 10.0000 + 17.3205i 0.867110 + 1.50188i
\(134\) 0 0
\(135\) 4.50000 + 2.59808i 0.387298 + 0.223607i
\(136\) 0 0
\(137\) 4.50000 + 7.79423i 0.384461 + 0.665906i 0.991694 0.128618i \(-0.0410540\pi\)
−0.607233 + 0.794524i \(0.707721\pi\)
\(138\) 0 0
\(139\) −0.500000 + 0.866025i −0.0424094 + 0.0734553i −0.886451 0.462822i \(-0.846837\pi\)
0.844042 + 0.536278i \(0.180170\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 15.5885i 1.28571i
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.406894 0.704761i 0.587646 0.809118i \(-0.300055\pi\)
−0.994540 + 0.104357i \(0.966722\pi\)
\(152\) 0 0
\(153\) 4.50000 7.79423i 0.363803 0.630126i
\(154\) 0 0
\(155\) 1.00000 + 1.73205i 0.0803219 + 0.139122i
\(156\) 0 0
\(157\) −4.00000 + 6.92820i −0.319235 + 0.552931i −0.980329 0.197372i \(-0.936759\pi\)
0.661094 + 0.750303i \(0.270093\pi\)
\(158\) 0 0
\(159\) 9.00000 5.19615i 0.713746 0.412082i
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −4.50000 + 2.59808i −0.350325 + 0.202260i
\(166\) 0 0
\(167\) −9.00000 + 15.5885i −0.696441 + 1.20627i 0.273252 + 0.961943i \(0.411901\pi\)
−0.969693 + 0.244328i \(0.921432\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) −7.50000 + 12.9904i −0.573539 + 0.993399i
\(172\) 0 0
\(173\) 9.00000 + 15.5885i 0.684257 + 1.18517i 0.973670 + 0.227964i \(0.0732068\pi\)
−0.289412 + 0.957205i \(0.593460\pi\)
\(174\) 0 0
\(175\) −2.00000 + 3.46410i −0.151186 + 0.261861i
\(176\) 0 0
\(177\) 5.19615i 0.390567i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) 15.0000 + 8.66025i 1.10883 + 0.640184i
\(184\) 0 0
\(185\) 2.00000 3.46410i 0.147043 0.254686i
\(186\) 0 0
\(187\) 4.50000 + 7.79423i 0.329073 + 0.569970i
\(188\) 0 0
\(189\) −18.0000 + 10.3923i −1.30931 + 0.755929i
\(190\) 0 0
\(191\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(192\) 0 0
\(193\) 6.50000 11.2583i 0.467880 0.810392i −0.531446 0.847092i \(-0.678351\pi\)
0.999326 + 0.0366998i \(0.0116845\pi\)
\(194\) 0 0
\(195\) 6.00000 + 3.46410i 0.429669 + 0.248069i
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 8.66025i 0.610847i
\(202\) 0 0
\(203\) −12.0000 + 20.7846i −0.842235 + 1.45879i
\(204\) 0 0
\(205\) 1.50000 + 2.59808i 0.104765 + 0.181458i
\(206\) 0 0
\(207\) −9.00000 15.5885i −0.625543 1.08347i
\(208\) 0 0
\(209\) −7.50000 12.9904i −0.518786 0.898563i
\(210\) 0 0
\(211\) −2.00000 + 3.46410i −0.137686 + 0.238479i −0.926620 0.375999i \(-0.877300\pi\)
0.788935 + 0.614477i \(0.210633\pi\)
\(212\) 0 0
\(213\) −9.00000 + 5.19615i −0.616670 + 0.356034i
\(214\) 0 0
\(215\) −11.0000 −0.750194
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −10.5000 + 6.06218i −0.709524 + 0.409644i
\(220\) 0 0
\(221\) 6.00000 10.3923i 0.403604 0.699062i
\(222\) 0 0
\(223\) −11.0000 19.0526i −0.736614 1.27585i −0.954011 0.299770i \(-0.903090\pi\)
0.217397 0.976083i \(-0.430243\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −1.50000 2.59808i −0.0995585 0.172440i 0.811943 0.583736i \(-0.198410\pi\)
−0.911502 + 0.411296i \(0.865076\pi\)
\(228\) 0 0
\(229\) −10.0000 + 17.3205i −0.660819 + 1.14457i 0.319582 + 0.947559i \(0.396457\pi\)
−0.980401 + 0.197013i \(0.936876\pi\)
\(230\) 0 0
\(231\) 20.7846i 1.36753i
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 21.0000 + 12.1244i 1.36410 + 0.787562i
\(238\) 0 0
\(239\) 3.00000 5.19615i 0.194054 0.336111i −0.752536 0.658551i \(-0.771170\pi\)
0.946590 + 0.322440i \(0.104503\pi\)
\(240\) 0 0
\(241\) −8.50000 14.7224i −0.547533 0.948355i −0.998443 0.0557856i \(-0.982234\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) −4.50000 7.79423i −0.287494 0.497955i
\(246\) 0 0
\(247\) −10.0000 + 17.3205i −0.636285 + 1.10208i
\(248\) 0 0
\(249\) 18.0000 + 10.3923i 1.14070 + 0.658586i
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 5.19615i 0.325396i
\(256\) 0 0
\(257\) −4.50000 + 7.79423i −0.280702 + 0.486191i −0.971558 0.236802i \(-0.923901\pi\)
0.690856 + 0.722993i \(0.257234\pi\)
\(258\) 0 0
\(259\) 8.00000 + 13.8564i 0.497096 + 0.860995i
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 12.0000 + 20.7846i 0.739952 + 1.28163i 0.952517 + 0.304487i \(0.0984850\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(264\) 0 0
\(265\) −3.00000 + 5.19615i −0.184289 + 0.319197i
\(266\) 0 0
\(267\) 9.00000 5.19615i 0.550791 0.317999i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) −24.0000 + 13.8564i −1.45255 + 0.838628i
\(274\) 0 0
\(275\) 1.50000 2.59808i 0.0904534 0.156670i
\(276\) 0 0
\(277\) 11.0000 + 19.0526i 0.660926 + 1.14476i 0.980373 + 0.197153i \(0.0631696\pi\)
−0.319447 + 0.947604i \(0.603497\pi\)
\(278\) 0 0
\(279\) −3.00000 5.19615i −0.179605 0.311086i
\(280\) 0 0
\(281\) 3.00000 + 5.19615i 0.178965 + 0.309976i 0.941526 0.336939i \(-0.109392\pi\)
−0.762561 + 0.646916i \(0.776058\pi\)
\(282\) 0 0
\(283\) 10.0000 17.3205i 0.594438 1.02960i −0.399188 0.916869i \(-0.630708\pi\)
0.993626 0.112728i \(-0.0359589\pi\)
\(284\) 0 0
\(285\) 8.66025i 0.512989i
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −16.5000 9.52628i −0.967247 0.558440i
\(292\) 0 0
\(293\) 9.00000 15.5885i 0.525786 0.910687i −0.473763 0.880652i \(-0.657105\pi\)
0.999549 0.0300351i \(-0.00956192\pi\)
\(294\) 0 0
\(295\) −1.50000 2.59808i −0.0873334 0.151266i
\(296\) 0 0
\(297\) 13.5000 7.79423i 0.783349 0.452267i
\(298\) 0 0
\(299\) −12.0000 20.7846i −0.693978 1.20201i
\(300\) 0 0
\(301\) 22.0000 38.1051i 1.26806 2.19634i
\(302\) 0 0
\(303\) 18.0000 + 10.3923i 1.03407 + 0.597022i
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 6.92820i 0.394132i
\(310\) 0 0
\(311\) −3.00000 + 5.19615i −0.170114 + 0.294647i −0.938460 0.345389i \(-0.887747\pi\)
0.768345 + 0.640036i \(0.221080\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 6.00000 10.3923i 0.338062 0.585540i
\(316\) 0 0
\(317\) −12.0000 20.7846i −0.673987 1.16738i −0.976764 0.214318i \(-0.931247\pi\)
0.302777 0.953062i \(-0.402086\pi\)
\(318\) 0 0
\(319\) 9.00000 15.5885i 0.503903 0.872786i
\(320\) 0 0
\(321\) 13.5000 7.79423i 0.753497 0.435031i
\(322\) 0 0
\(323\) −15.0000 −0.834622
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −6.00000 + 3.46410i −0.331801 + 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.00000 3.46410i −0.109930 0.190404i 0.805812 0.592172i \(-0.201729\pi\)
−0.915742 + 0.401768i \(0.868396\pi\)
\(332\) 0 0
\(333\) −6.00000 + 10.3923i −0.328798 + 0.569495i
\(334\) 0 0
\(335\) 2.50000 + 4.33013i 0.136590 + 0.236580i
\(336\) 0 0
\(337\) 15.5000 26.8468i 0.844339 1.46244i −0.0418554 0.999124i \(-0.513327\pi\)
0.886194 0.463314i \(-0.153340\pi\)
\(338\) 0 0
\(339\) 31.1769i 1.69330i
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 9.00000 + 5.19615i 0.484544 + 0.279751i
\(346\) 0 0
\(347\) 10.5000 18.1865i 0.563670 0.976304i −0.433503 0.901152i \(-0.642722\pi\)
0.997172 0.0751519i \(-0.0239442\pi\)
\(348\) 0 0
\(349\) 8.00000 + 13.8564i 0.428230 + 0.741716i 0.996716 0.0809766i \(-0.0258039\pi\)
−0.568486 + 0.822693i \(0.692471\pi\)
\(350\) 0 0
\(351\) −18.0000 10.3923i −0.960769 0.554700i
\(352\) 0 0
\(353\) −4.50000 7.79423i −0.239511 0.414845i 0.721063 0.692869i \(-0.243654\pi\)
−0.960574 + 0.278024i \(0.910320\pi\)
\(354\) 0 0
\(355\) 3.00000 5.19615i 0.159223 0.275783i
\(356\) 0 0
\(357\) −18.0000 10.3923i −0.952661 0.550019i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 3.46410i 0.181818i
\(364\) 0 0
\(365\) 3.50000 6.06218i 0.183198 0.317309i
\(366\) 0 0
\(367\) 4.00000 + 6.92820i 0.208798 + 0.361649i 0.951336 0.308155i \(-0.0997115\pi\)
−0.742538 + 0.669804i \(0.766378\pi\)
\(368\) 0 0
\(369\) −4.50000 7.79423i −0.234261 0.405751i
\(370\) 0 0
\(371\) −12.0000 20.7846i −0.623009 1.07908i
\(372\) 0 0
\(373\) 5.00000 8.66025i 0.258890 0.448411i −0.707055 0.707159i \(-0.749977\pi\)
0.965945 + 0.258748i \(0.0833099\pi\)
\(374\) 0 0
\(375\) 1.50000 0.866025i 0.0774597 0.0447214i
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) −3.00000 + 1.73205i −0.153695 + 0.0887357i
\(382\) 0 0
\(383\) 6.00000 10.3923i 0.306586 0.531022i −0.671027 0.741433i \(-0.734147\pi\)
0.977613 + 0.210411i \(0.0674801\pi\)
\(384\) 0 0
\(385\) 6.00000 + 10.3923i 0.305788 + 0.529641i
\(386\) 0 0
\(387\) 33.0000 1.67748
\(388\) 0 0
\(389\) −18.0000 31.1769i −0.912636 1.58073i −0.810326 0.585980i \(-0.800710\pi\)
−0.102311 0.994753i \(-0.532624\pi\)
\(390\) 0 0
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) 20.7846i 1.04844i
\(394\) 0 0
\(395\) −14.0000 −0.704416
\(396\) 0 0
\(397\) 8.00000 0.401508 0.200754 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 0 0
\(399\) 30.0000 + 17.3205i 1.50188 + 0.867110i
\(400\) 0 0
\(401\) −16.5000 + 28.5788i −0.823971 + 1.42716i 0.0787327 + 0.996896i \(0.474913\pi\)
−0.902703 + 0.430263i \(0.858421\pi\)
\(402\) 0 0
\(403\) −4.00000 6.92820i −0.199254 0.345118i
\(404\) 0 0
\(405\) 9.00000 0.447214
\(406\) 0 0
\(407\) −6.00000 10.3923i −0.297409 0.515127i
\(408\) 0 0
\(409\) 15.5000 26.8468i 0.766426 1.32749i −0.173064 0.984911i \(-0.555367\pi\)
0.939490 0.342578i \(-0.111300\pi\)
\(410\) 0 0
\(411\) 13.5000 + 7.79423i 0.665906 + 0.384461i
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 1.73205i 0.0848189i
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −1.00000 1.73205i −0.0487370 0.0844150i 0.840628 0.541613i \(-0.182186\pi\)
−0.889365 + 0.457198i \(0.848853\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.50000 2.59808i −0.0727607 0.126025i
\(426\) 0 0
\(427\) 20.0000 34.6410i 0.967868 1.67640i
\(428\) 0 0
\(429\) 18.0000 10.3923i 0.869048 0.501745i
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −13.0000 −0.624740 −0.312370 0.949960i \(-0.601123\pi\)
−0.312370 + 0.949960i \(0.601123\pi\)
\(434\) 0 0
\(435\) 9.00000 5.19615i 0.431517 0.249136i
\(436\) 0 0
\(437\) −15.0000 + 25.9808i −0.717547 + 1.24283i
\(438\) 0 0
\(439\) −5.00000 8.66025i −0.238637 0.413331i 0.721686 0.692220i \(-0.243367\pi\)
−0.960323 + 0.278889i \(0.910034\pi\)
\(440\) 0 0
\(441\) 13.5000 + 23.3827i 0.642857 + 1.11346i
\(442\) 0 0
\(443\) 1.50000 + 2.59808i 0.0712672 + 0.123438i 0.899457 0.437009i \(-0.143962\pi\)
−0.828190 + 0.560448i \(0.810629\pi\)
\(444\) 0 0
\(445\) −3.00000 + 5.19615i −0.142214 + 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −15.0000 −0.707894 −0.353947 0.935266i \(-0.615161\pi\)
−0.353947 + 0.935266i \(0.615161\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) −15.0000 8.66025i −0.704761 0.406894i
\(454\) 0 0
\(455\) 8.00000 13.8564i 0.375046 0.649598i
\(456\) 0 0
\(457\) 0.500000 + 0.866025i 0.0233890 + 0.0405110i 0.877483 0.479608i \(-0.159221\pi\)
−0.854094 + 0.520119i \(0.825888\pi\)
\(458\) 0 0
\(459\) 15.5885i 0.727607i
\(460\) 0 0
\(461\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(462\) 0 0
\(463\) 10.0000 17.3205i 0.464739 0.804952i −0.534450 0.845200i \(-0.679481\pi\)
0.999190 + 0.0402476i \(0.0128147\pi\)
\(464\) 0 0
\(465\) 3.00000 + 1.73205i 0.139122 + 0.0803219i
\(466\) 0 0
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 13.8564i 0.638470i
\(472\) 0 0
\(473\) −16.5000 + 28.5788i −0.758671 + 1.31406i
\(474\) 0 0
\(475\) 2.50000 + 4.33013i 0.114708 + 0.198680i
\(476\) 0 0
\(477\) 9.00000 15.5885i 0.412082 0.713746i
\(478\) 0 0
\(479\) −3.00000 5.19615i −0.137073 0.237418i 0.789314 0.613990i \(-0.210436\pi\)
−0.926388 + 0.376571i \(0.877103\pi\)
\(480\) 0 0
\(481\) −8.00000 + 13.8564i −0.364769 + 0.631798i
\(482\) 0 0
\(483\) −36.0000 + 20.7846i −1.63806 + 0.945732i
\(484\) 0 0
\(485\) 11.0000 0.499484
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 24.0000 13.8564i 1.08532 0.626608i
\(490\) 0 0
\(491\) −16.5000 + 28.5788i −0.744635 + 1.28974i 0.205731 + 0.978609i \(0.434043\pi\)
−0.950365 + 0.311136i \(0.899290\pi\)
\(492\) 0 0
\(493\) −9.00000 15.5885i −0.405340 0.702069i
\(494\) 0 0
\(495\) −4.50000 + 7.79423i −0.202260 + 0.350325i
\(496\) 0 0
\(497\) 12.0000 + 20.7846i 0.538274 + 0.932317i
\(498\) 0 0
\(499\) −15.5000 + 26.8468i −0.693875 + 1.20183i 0.276683 + 0.960961i \(0.410765\pi\)
−0.970558 + 0.240866i \(0.922569\pi\)
\(500\) 0 0
\(501\) 31.1769i 1.39288i
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) −4.50000 2.59808i −0.199852 0.115385i
\(508\) 0 0
\(509\) 6.00000 10.3923i 0.265945 0.460631i −0.701866 0.712309i \(-0.747649\pi\)
0.967811 + 0.251679i \(0.0809826\pi\)
\(510\) 0 0
\(511\) 14.0000 + 24.2487i 0.619324 + 1.07270i
\(512\) 0 0
\(513\) 25.9808i 1.14708i
\(514\) 0 0
\(515\) −2.00000 3.46410i −0.0881305 0.152647i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 27.0000 + 15.5885i 1.18517 + 0.684257i
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) 6.92820i 0.302372i
\(526\) 0 0
\(527\) 3.00000 5.19615i 0.130682 0.226348i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 4.50000 + 7.79423i 0.195283 + 0.338241i
\(532\) 0 0
\(533\) −6.00000 10.3923i −0.259889 0.450141i
\(534\) 0 0
\(535\) −4.50000 + 7.79423i −0.194552 + 0.336974i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −27.0000 −1.16297
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) −24.0000 + 13.8564i −1.02994 + 0.594635i
\(544\) 0 0
\(545\) 2.00000 3.46410i 0.0856706 0.148386i
\(546\) 0 0
\(547\) −0.500000 0.866025i −0.0213785 0.0370286i 0.855138 0.518400i \(-0.173472\pi\)
−0.876517 + 0.481371i \(0.840139\pi\)
\(548\) 0 0
\(549\) 30.0000 1.28037
\(550\) 0 0
\(551\) 15.0000 + 25.9808i 0.639021 + 1.10682i
\(552\) 0 0
\(553\) 28.0000 48.4974i 1.19068 2.06232i
\(554\) 0 0
\(555\) 6.92820i 0.294086i
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 44.0000 1.86100
\(560\) 0 0
\(561\) 13.5000 + 7.79423i 0.569970 + 0.329073i
\(562\) 0 0
\(563\) −1.50000 + 2.59808i −0.0632175 + 0.109496i −0.895902 0.444252i \(-0.853470\pi\)
0.832684 + 0.553748i \(0.186803\pi\)
\(564\) 0 0
\(565\) −9.00000 15.5885i −0.378633 0.655811i
\(566\) 0 0
\(567\) −18.0000 + 31.1769i −0.755929 + 1.30931i
\(568\) 0 0
\(569\) 19.5000 + 33.7750i 0.817483 + 1.41592i 0.907532 + 0.419984i \(0.137964\pi\)
−0.0900490 + 0.995937i \(0.528702\pi\)
\(570\) 0 0
\(571\) 14.5000 25.1147i 0.606806 1.05102i −0.384957 0.922934i \(-0.625784\pi\)
0.991763 0.128085i \(-0.0408829\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −7.00000 −0.291414 −0.145707 0.989328i \(-0.546546\pi\)
−0.145707 + 0.989328i \(0.546546\pi\)
\(578\) 0 0
\(579\) 22.5167i 0.935760i
\(580\) 0 0
\(581\) 24.0000 41.5692i 0.995688 1.72458i
\(582\) 0 0
\(583\) 9.00000 + 15.5885i 0.372742 + 0.645608i
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) 19.5000 + 33.7750i 0.804851 + 1.39404i 0.916392 + 0.400283i \(0.131088\pi\)
−0.111540 + 0.993760i \(0.535578\pi\)
\(588\) 0 0
\(589\) −5.00000 + 8.66025i −0.206021 + 0.356840i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) 12.0000 0.491952
\(596\) 0 0
\(597\) −30.0000 + 17.3205i −1.22782 + 0.708881i
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 12.5000 + 21.6506i 0.509886 + 0.883148i 0.999934 + 0.0114528i \(0.00364562\pi\)
−0.490049 + 0.871695i \(0.663021\pi\)
\(602\) 0 0
\(603\) −7.50000 12.9904i −0.305424 0.529009i
\(604\) 0 0
\(605\) 1.00000 + 1.73205i 0.0406558 + 0.0704179i
\(606\) 0 0
\(607\) 4.00000 6.92820i 0.162355 0.281207i −0.773358 0.633970i \(-0.781424\pi\)
0.935713 + 0.352763i \(0.114758\pi\)
\(608\) 0 0
\(609\) 41.5692i 1.68447i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) 4.50000 + 2.59808i 0.181458 + 0.104765i
\(616\) 0 0
\(617\) 19.5000 33.7750i 0.785040 1.35973i −0.143934 0.989587i \(-0.545975\pi\)
0.928975 0.370143i \(-0.120691\pi\)
\(618\) 0 0
\(619\) −9.50000 16.4545i −0.381837 0.661361i 0.609488 0.792796i \(-0.291375\pi\)
−0.991325 + 0.131434i \(0.958042\pi\)
\(620\) 0 0
\(621\) −27.0000 15.5885i −1.08347 0.625543i
\(622\) 0 0
\(623\) −12.0000 20.7846i −0.480770 0.832718i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −22.5000 12.9904i −0.898563 0.518786i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) 6.92820i 0.275371i
\(634\) 0 0
\(635\) 1.00000 1.73205i 0.0396838 0.0687343i
\(636\) 0 0
\(637\) 18.0000 + 31.1769i 0.713186 + 1.23527i
\(638\) 0 0
\(639\) −9.00000 + 15.5885i −0.356034 + 0.616670i
\(640\) 0 0
\(641\) −4.50000 7.79423i −0.177739 0.307854i 0.763367 0.645966i \(-0.223545\pi\)
−0.941106 + 0.338112i \(0.890212\pi\)
\(642\) 0 0
\(643\) 11.5000 19.9186i 0.453516 0.785512i −0.545086 0.838380i \(-0.683503\pi\)
0.998602 + 0.0528680i \(0.0168363\pi\)
\(644\) 0 0
\(645\) −16.5000 + 9.52628i −0.649687 + 0.375097i
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) −12.0000 + 6.92820i −0.470317 + 0.271538i
\(652\) 0 0
\(653\) −21.0000 + 36.3731i −0.821794 + 1.42339i 0.0825519 + 0.996587i \(0.473693\pi\)
−0.904345 + 0.426801i \(0.859640\pi\)
\(654\) 0 0
\(655\) 6.00000 + 10.3923i 0.234439 + 0.406061i
\(656\) 0 0
\(657\) −10.5000 + 18.1865i −0.409644 + 0.709524i
\(658\) 0 0
\(659\) −18.0000 31.1769i −0.701180 1.21448i −0.968052 0.250748i \(-0.919323\pi\)
0.266872 0.963732i \(-0.414010\pi\)
\(660\) 0 0
\(661\) −16.0000 + 27.7128i −0.622328 + 1.07790i 0.366723 + 0.930330i \(0.380480\pi\)
−0.989051 + 0.147573i \(0.952854\pi\)
\(662\) 0 0
\(663\) 20.7846i 0.807207i
\(664\) 0 0
\(665\) −20.0000 −0.775567
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) −33.0000 19.0526i −1.27585 0.736614i
\(670\) 0 0
\(671\) −15.0000 + 25.9808i −0.579069 + 1.00298i
\(672\) 0 0
\(673\) −7.00000 12.1244i −0.269830 0.467360i 0.698988 0.715134i \(-0.253634\pi\)
−0.968818 + 0.247774i \(0.920301\pi\)
\(674\) 0 0
\(675\) −4.50000 + 2.59808i −0.173205 + 0.100000i
\(676\) 0 0
\(677\) 18.0000 + 31.1769i 0.691796 + 1.19823i 0.971249 + 0.238067i \(0.0765137\pi\)
−0.279453 + 0.960159i \(0.590153\pi\)
\(678\) 0 0
\(679\) −22.0000 + 38.1051i −0.844283 + 1.46234i
\(680\) 0 0
\(681\) −4.50000 2.59808i −0.172440 0.0995585i
\(682\) 0 0
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) 34.6410i 1.32164i
\(688\) 0 0
\(689\) 12.0000 20.7846i 0.457164 0.791831i
\(690\) 0 0
\(691\) 4.00000 + 6.92820i 0.152167 + 0.263561i 0.932024 0.362397i \(-0.118041\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 0 0
\(693\) −18.0000 31.1769i −0.683763 1.18431i
\(694\) 0 0
\(695\) −0.500000 0.866025i −0.0189661 0.0328502i
\(696\) 0 0
\(697\) 4.50000 7.79423i 0.170450 0.295227i
\(698\) 0 0
\(699\) 31.5000 18.1865i 1.19144 0.687878i
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000 41.5692i 0.902613 1.56337i
\(708\) 0 0
\(709\) 17.0000 + 29.4449i 0.638448 + 1.10583i 0.985773 + 0.168080i \(0.0537568\pi\)
−0.347325 + 0.937745i \(0.612910\pi\)
\(710\) 0 0
\(711\) 42.0000 1.57512
\(712\) 0 0
\(713\) −6.00000 10.3923i −0.224702 0.389195i
\(714\) 0 0
\(715\) −6.00000 + 10.3923i −0.224387 + 0.388650i
\(716\) 0 0
\(717\) 10.3923i 0.388108i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −25.5000 14.7224i −0.948355 0.547533i
\(724\) 0 0
\(725\) −3.00000 + 5.19615i −0.111417 + 0.192980i
\(726\) 0 0
\(727\) −14.0000 24.2487i −0.519231 0.899335i −0.999750 0.0223506i \(-0.992885\pi\)
0.480519 0.876984i \(-0.340448\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 16.5000 + 28.5788i 0.610275 + 1.05703i
\(732\) 0 0
\(733\) −16.0000 + 27.7128i −0.590973 + 1.02360i 0.403128 + 0.915144i \(0.367923\pi\)
−0.994102 + 0.108453i \(0.965410\pi\)
\(734\) 0 0
\(735\) −13.5000 7.79423i −0.497955 0.287494i
\(736\) 0 0
\(737\) 15.0000 0.552532
\(738\) 0 0
\(739\) −29.0000 −1.06678 −0.533391 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 34.6410i 1.27257i
\(742\) 0 0
\(743\) −3.00000 + 5.19615i −0.110059 + 0.190628i −0.915794 0.401648i \(-0.868437\pi\)
0.805735 + 0.592277i \(0.201771\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 36.0000 1.31717
\(748\) 0 0
\(749\) −18.0000 31.1769i −0.657706 1.13918i
\(750\) 0 0
\(751\) −14.0000 + 24.2487i −0.510867 + 0.884848i 0.489053 + 0.872254i \(0.337342\pi\)
−0.999921 + 0.0125942i \(0.995991\pi\)
\(752\) 0 0
\(753\) 31.5000 18.1865i 1.14792 0.662754i
\(754\) 0 0
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 27.0000 15.5885i 0.980038 0.565825i
\(760\) 0 0
\(761\) −9.00000 + 15.5885i −0.326250 + 0.565081i −0.981764 0.190101i \(-0.939118\pi\)
0.655515 + 0.755182i \(0.272452\pi\)
\(762\) 0 0
\(763\) 8.00000 + 13.8564i 0.289619 + 0.501636i
\(764\) 0 0
\(765\) 4.50000 + 7.79423i 0.162698 + 0.281801i
\(766\) 0 0
\(767\) 6.00000 + 10.3923i 0.216647 + 0.375244i
\(768\) 0 0
\(769\) −25.0000 + 43.3013i −0.901523 + 1.56148i −0.0760054 + 0.997107i \(0.524217\pi\)
−0.825518 + 0.564376i \(0.809117\pi\)
\(770\) 0 0
\(771\) 15.5885i 0.561405i
\(772\) 0 0
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 24.0000 + 13.8564i 0.860995 + 0.497096i
\(778\) 0 0
\(779\) −7.50000 + 12.9904i −0.268715 + 0.465429i
\(780\) 0 0
\(781\) −9.00000 15.5885i −0.322045 0.557799i
\(782\) 0 0
\(783\) −27.0000 + 15.5885i −0.964901 + 0.557086i
\(784\) 0 0
\(785\) −4.00000 6.92820i −0.142766 0.247278i
\(786\) 0 0
\(787\) −2.00000 + 3.46410i −0.0712923 + 0.123482i −0.899468 0.436987i \(-0.856046\pi\)
0.828176 + 0.560469i \(0.189379\pi\)
\(788\) 0 0
\(789\) 36.0000 + 20.7846i 1.28163 + 0.739952i
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 10.3923i 0.368577i
\(796\) 0 0
\(797\) 24.0000 41.5692i 0.850124 1.47246i −0.0309726 0.999520i \(-0.509860\pi\)
0.881096 0.472937i \(-0.156806\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 9.00000 15.5885i 0.317999 0.550791i
\(802\) 0 0
\(803\) −10.5000 18.1865i −0.370537 0.641789i
\(804\) 0 0
\(805\) 12.0000 20.7846i 0.422944 0.732561i
\(806\) 0 0
\(807\) 9.00000 5.19615i 0.316815 0.182913i
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) 24.0000 13.8564i 0.841717 0.485965i
\(814\) 0 0
\(815\) −8.00000 + 13.8564i −0.280228 + 0.485369i
\(816\) 0 0
\(817\) −27.5000 47.6314i −0.962103 1.66641i
\(818\) 0 0
\(819\) −24.0000 + 41.5692i −0.838628 + 1.45255i
\(820\) 0 0
\(821\) 24.0000 + 41.5692i 0.837606 + 1.45078i 0.891891 + 0.452250i \(0.149379\pi\)
−0.0542853 + 0.998525i \(0.517288\pi\)
\(822\) 0 0
\(823\) −2.00000 + 3.46410i −0.0697156 + 0.120751i −0.898776 0.438408i \(-0.855543\pi\)
0.829060 + 0.559159i \(0.188876\pi\)
\(824\) 0 0
\(825\) 5.19615i 0.180907i
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 44.0000 1.52818 0.764092 0.645108i \(-0.223188\pi\)
0.764092 + 0.645108i \(0.223188\pi\)
\(830\) 0 0
\(831\) 33.0000 + 19.0526i 1.14476 + 0.660926i
\(832\) 0 0
\(833\) −13.5000 + 23.3827i −0.467747 + 0.810162i
\(834\) 0 0
\(835\) −9.00000 15.5885i −0.311458 0.539461i
\(836\) 0 0
\(837\) −9.00000 5.19615i −0.311086 0.179605i
\(838\) 0 0
\(839\) −15.0000 25.9808i −0.517858 0.896956i −0.999785 0.0207443i \(-0.993396\pi\)
0.481927 0.876211i \(-0.339937\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) 9.00000 + 5.19615i 0.309976 + 0.178965i
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −8.00000 −0.274883
\(848\) 0 0
\(849\) 34.6410i 1.18888i
\(850\) 0 0
\(851\) −12.0000 + 20.7846i −0.411355 + 0.712487i
\(852\) 0 0
\(853\) −22.0000 38.1051i −0.753266 1.30469i −0.946232 0.323489i \(-0.895144\pi\)
0.192966 0.981205i \(-0.438189\pi\)
\(854\) 0 0
\(855\) −7.50000 12.9904i −0.256495 0.444262i
\(856\) 0 0
\(857\) 9.00000 + 15.5885i 0.307434 + 0.532492i 0.977800 0.209539i \(-0.0671963\pi\)
−0.670366 + 0.742030i \(0.733863\pi\)
\(858\) 0 0
\(859\) −9.50000 + 16.4545i −0.324136 + 0.561420i −0.981337 0.192295i \(-0.938407\pi\)
0.657201 + 0.753715i \(0.271740\pi\)
\(860\) 0 0
\(861\) −18.0000 + 10.3923i −0.613438 + 0.354169i
\(862\) 0 0
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −12.0000 + 6.92820i −0.407541 + 0.235294i
\(868\) 0 0
\(869\) −21.0000 + 36.3731i −0.712376 + 1.23387i
\(870\) 0 0
\(871\) −10.0000 17.3205i −0.338837 0.586883i
\(872\) 0 0
\(873\) −33.0000 −1.11688
\(874\) 0 0
\(875\) −2.00000 3.46410i −0.0676123 0.117108i
\(876\) 0 0
\(877\) −1.00000 + 1.73205i −0.0337676 + 0.0584872i −0.882415 0.470471i \(-0.844084\pi\)
0.848648 + 0.528958i \(0.177417\pi\)
\(878\) 0 0
\(879\) 31.1769i 1.05157i
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 0 0
\(885\) −4.50000 2.59808i −0.151266 0.0873334i
\(886\) 0 0
\(887\) −21.0000 + 36.3731i −0.705111 + 1.22129i 0.261540 + 0.965193i \(0.415770\pi\)
−0.966651 + 0.256096i \(0.917564\pi\)
\(888\) 0 0
\(889\) 4.00000 + 6.92820i 0.134156 + 0.232364i
\(890\) 0 0
\(891\) 13.5000 23.3827i 0.452267 0.783349i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −36.0000 20.7846i −1.20201 0.693978i
\(898\) 0 0
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 76.2102i 2.53612i
\(904\) 0 0
\(905\) 8.00000 13.8564i 0.265929 0.460603i
\(906\) 0 0
\(907\) 2.50000 + 4.33013i 0.0830111 + 0.143780i 0.904542 0.426385i \(-0.140213\pi\)
−0.821531 + 0.570164i \(0.806880\pi\)
\(908\) 0 0
\(909\) 36.0000 1.19404
\(910\) 0 0
\(911\) −15.0000 25.9808i −0.496972 0.860781i 0.503022 0.864274i \(-0.332222\pi\)
−0.999994 + 0.00349271i \(0.998888\pi\)
\(912\) 0 0
\(913\) −18.0000 + 31.1769i −0.595713 + 1.03181i
\(914\) 0 0
\(915\) −15.0000 + 8.66025i −0.495885 + 0.286299i
\(916\) 0 0
\(917\) −48.0000 −1.58510
\(918\) 0 0
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) 10.5000 6.06218i 0.345987 0.199756i
\(922\) 0 0
\(923\) −12.0000 + 20.7846i −0.394985 + 0.684134i
\(924\) 0 0
\(925\) 2.00000 + 3.46410i 0.0657596 + 0.113899i
\(926\) 0 0
\(927\) 6.00000 + 10.3923i 0.197066 + 0.341328i
\(928\) 0 0
\(929\) −15.0000 25.9808i −0.492134 0.852401i 0.507825 0.861460i \(-0.330450\pi\)
−0.999959 + 0.00905914i \(0.997116\pi\)
\(930\) 0 0
\(931\) 22.5000 38.9711i 0.737408 1.27723i
\(932\) 0 0
\(933\) 10.3923i 0.340229i
\(934\) 0 0
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 1.50000 + 0.866025i 0.0489506 + 0.0282617i
\(940\) 0 0
\(941\) −12.0000 + 20.7846i −0.391189 + 0.677559i −0.992607 0.121376i \(-0.961269\pi\)
0.601418 + 0.798935i \(0.294603\pi\)
\(942\) 0 0
\(943\) −9.00000 15.5885i −0.293080 0.507630i
\(944\) 0 0
\(945\) 20.7846i 0.676123i
\(946\) 0 0
\(947\) 13.5000 + 23.3827i 0.438691 + 0.759835i 0.997589 0.0694014i \(-0.0221089\pi\)
−0.558898 + 0.829237i \(0.688776\pi\)
\(948\) 0 0
\(949\) −14.0000 + 24.2487i −0.454459 + 0.787146i
\(950\) 0 0
\(951\) −36.0000 20.7846i −1.16738 0.673987i
\(952\) 0 0
\(953\) 51.0000 1.65205 0.826026 0.563632i \(-0.190596\pi\)
0.826026 + 0.563632i \(0.190596\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.1769i 1.00781i
\(958\) 0 0
\(959\) 18.0000 31.1769i 0.581250 1.00676i
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) 0 0
\(963\) 13.5000 23.3827i 0.435031 0.753497i
\(964\) 0 0
\(965\) 6.50000 + 11.2583i 0.209242 + 0.362418i
\(966\) 0 0
\(967\) −11.0000 + 19.0526i −0.353736 + 0.612689i −0.986901 0.161328i \(-0.948422\pi\)
0.633165 + 0.774017i \(0.281756\pi\)
\(968\) 0 0
\(969\) −22.5000 + 12.9904i −0.722804 + 0.417311i
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 4.00000 0.128234
\(974\) 0 0
\(975\) −6.00000 + 3.46410i −0.192154 + 0.110940i
\(976\) 0 0
\(977\) 4.50000 7.79423i 0.143968 0.249359i −0.785020 0.619471i \(-0.787347\pi\)
0.928987 + 0.370111i \(0.120681\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) −6.00000 + 10.3923i −0.191565 + 0.331801i
\(982\) 0 0
\(983\) 18.0000 + 31.1769i 0.574111 + 0.994389i 0.996138 + 0.0878058i \(0.0279855\pi\)
−0.422027 + 0.906583i \(0.638681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 66.0000 2.09868
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) −6.00000 3.46410i −0.190404 0.109930i
\(994\) 0 0
\(995\) 10.0000 17.3205i 0.317021 0.549097i
\(996\) 0 0
\(997\) −13.0000 22.5167i −0.411714 0.713110i 0.583363 0.812211i \(-0.301736\pi\)
−0.995077 + 0.0991016i \(0.968403\pi\)
\(998\) 0 0
\(999\) 20.7846i 0.657596i
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 720.2.q.c.481.1 2
3.2 odd 2 2160.2.q.d.1441.1 2
4.3 odd 2 90.2.e.b.31.1 2
9.2 odd 6 2160.2.q.d.721.1 2
9.4 even 3 6480.2.a.z.1.1 1
9.5 odd 6 6480.2.a.l.1.1 1
9.7 even 3 inner 720.2.q.c.241.1 2
12.11 even 2 270.2.e.a.91.1 2
20.3 even 4 450.2.j.a.49.2 4
20.7 even 4 450.2.j.a.49.1 4
20.19 odd 2 450.2.e.d.301.1 2
36.7 odd 6 90.2.e.b.61.1 yes 2
36.11 even 6 270.2.e.a.181.1 2
36.23 even 6 810.2.a.e.1.1 1
36.31 odd 6 810.2.a.a.1.1 1
60.23 odd 4 1350.2.j.c.199.1 4
60.47 odd 4 1350.2.j.c.199.2 4
60.59 even 2 1350.2.e.g.901.1 2
180.7 even 12 450.2.j.a.349.2 4
180.23 odd 12 4050.2.c.d.649.1 2
180.43 even 12 450.2.j.a.349.1 4
180.47 odd 12 1350.2.j.c.1099.1 4
180.59 even 6 4050.2.a.q.1.1 1
180.67 even 12 4050.2.c.p.649.1 2
180.79 odd 6 450.2.e.d.151.1 2
180.83 odd 12 1350.2.j.c.1099.2 4
180.103 even 12 4050.2.c.p.649.2 2
180.119 even 6 1350.2.e.g.451.1 2
180.139 odd 6 4050.2.a.bi.1.1 1
180.167 odd 12 4050.2.c.d.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.e.b.31.1 2 4.3 odd 2
90.2.e.b.61.1 yes 2 36.7 odd 6
270.2.e.a.91.1 2 12.11 even 2
270.2.e.a.181.1 2 36.11 even 6
450.2.e.d.151.1 2 180.79 odd 6
450.2.e.d.301.1 2 20.19 odd 2
450.2.j.a.49.1 4 20.7 even 4
450.2.j.a.49.2 4 20.3 even 4
450.2.j.a.349.1 4 180.43 even 12
450.2.j.a.349.2 4 180.7 even 12
720.2.q.c.241.1 2 9.7 even 3 inner
720.2.q.c.481.1 2 1.1 even 1 trivial
810.2.a.a.1.1 1 36.31 odd 6
810.2.a.e.1.1 1 36.23 even 6
1350.2.e.g.451.1 2 180.119 even 6
1350.2.e.g.901.1 2 60.59 even 2
1350.2.j.c.199.1 4 60.23 odd 4
1350.2.j.c.199.2 4 60.47 odd 4
1350.2.j.c.1099.1 4 180.47 odd 12
1350.2.j.c.1099.2 4 180.83 odd 12
2160.2.q.d.721.1 2 9.2 odd 6
2160.2.q.d.1441.1 2 3.2 odd 2
4050.2.a.q.1.1 1 180.59 even 6
4050.2.a.bi.1.1 1 180.139 odd 6
4050.2.c.d.649.1 2 180.23 odd 12
4050.2.c.d.649.2 2 180.167 odd 12
4050.2.c.p.649.1 2 180.67 even 12
4050.2.c.p.649.2 2 180.103 even 12
6480.2.a.l.1.1 1 9.5 odd 6
6480.2.a.z.1.1 1 9.4 even 3