Properties

Label 180.2.i.a.61.1
Level $180$
Weight $2$
Character 180.61
Analytic conductor $1.437$
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] = [180,2,Mod(61,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("180.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.i (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: \(1.43730723638\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 61.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 180.61
Dual form 180.2.i.a.121.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{9} +(2.00000 + 3.46410i) q^{13} +(-1.50000 - 0.866025i) q^{15} -6.00000 q^{17} +2.00000 q^{19} -1.73205i q^{21} +(1.50000 + 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} -5.19615i q^{27} +(-1.50000 + 2.59808i) q^{29} +(5.00000 + 8.66025i) q^{31} -1.00000 q^{35} -10.0000 q^{37} +(6.00000 + 3.46410i) q^{39} +(-4.50000 - 7.79423i) q^{41} +(2.00000 - 3.46410i) q^{43} -3.00000 q^{45} +(-4.50000 + 7.79423i) q^{47} +(3.00000 + 5.19615i) q^{49} +(-9.00000 + 5.19615i) q^{51} -6.00000 q^{53} +(3.00000 - 1.73205i) q^{57} +(3.00000 + 5.19615i) q^{59} +(0.500000 - 0.866025i) q^{61} +(-1.50000 - 2.59808i) q^{63} +(2.00000 - 3.46410i) q^{65} +(-5.50000 - 9.52628i) q^{67} +(4.50000 + 2.59808i) q^{69} +12.0000 q^{71} -4.00000 q^{73} +1.73205i q^{75} +(5.00000 - 8.66025i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(4.50000 - 7.79423i) q^{83} +(3.00000 + 5.19615i) q^{85} +5.19615i q^{87} +9.00000 q^{89} +4.00000 q^{91} +(15.0000 + 8.66025i) q^{93} +(-1.00000 - 1.73205i) q^{95} +(5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - q^{5} + q^{7} + 3 q^{9} + 4 q^{13} - 3 q^{15} - 12 q^{17} + 4 q^{19} + 3 q^{23} - q^{25} - 3 q^{29} + 10 q^{31} - 2 q^{35} - 20 q^{37} + 12 q^{39} - 9 q^{41} + 4 q^{43} - 6 q^{45} - 9 q^{47} + 6 q^{49} - 18 q^{51} - 12 q^{53} + 6 q^{57} + 6 q^{59} + q^{61} - 3 q^{63} + 4 q^{65} - 11 q^{67} + 9 q^{69} + 24 q^{71} - 8 q^{73} + 10 q^{79} - 9 q^{81} + 9 q^{83} + 6 q^{85} + 18 q^{89} + 8 q^{91} + 30 q^{93} - 2 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/180\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(91\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(3\) 1.50000 0.866025i 0.866025 0.500000i
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 0.500000 0.866025i 0.188982 0.327327i −0.755929 0.654654i \(-0.772814\pi\)
0.944911 + 0.327327i \(0.106148\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 2.00000 + 3.46410i 0.554700 + 0.960769i 0.997927 + 0.0643593i \(0.0205004\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) −1.50000 0.866025i −0.387298 0.223607i
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 1.73205i 0.377964i
\(22\) 0 0
\(23\) 1.50000 + 2.59808i 0.312772 + 0.541736i 0.978961 0.204046i \(-0.0654092\pi\)
−0.666190 + 0.745782i \(0.732076\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) −1.50000 + 2.59808i −0.278543 + 0.482451i −0.971023 0.238987i \(-0.923185\pi\)
0.692480 + 0.721437i \(0.256518\pi\)
\(30\) 0 0
\(31\) 5.00000 + 8.66025i 0.898027 + 1.55543i 0.830014 + 0.557743i \(0.188333\pi\)
0.0680129 + 0.997684i \(0.478334\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 6.00000 + 3.46410i 0.960769 + 0.554700i
\(40\) 0 0
\(41\) −4.50000 7.79423i −0.702782 1.21725i −0.967486 0.252924i \(-0.918608\pi\)
0.264704 0.964330i \(-0.414726\pi\)
\(42\) 0 0
\(43\) 2.00000 3.46410i 0.304997 0.528271i −0.672264 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) −4.50000 + 7.79423i −0.656392 + 1.13691i 0.325150 + 0.945662i \(0.394585\pi\)
−0.981543 + 0.191243i \(0.938748\pi\)
\(48\) 0 0
\(49\) 3.00000 + 5.19615i 0.428571 + 0.742307i
\(50\) 0 0
\(51\) −9.00000 + 5.19615i −1.26025 + 0.727607i
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000 1.73205i 0.397360 0.229416i
\(58\) 0 0
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) 0.500000 0.866025i 0.0640184 0.110883i −0.832240 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −1.50000 2.59808i −0.188982 0.327327i
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 0 0
\(67\) −5.50000 9.52628i −0.671932 1.16382i −0.977356 0.211604i \(-0.932131\pi\)
0.305424 0.952217i \(-0.401202\pi\)
\(68\) 0 0
\(69\) 4.50000 + 2.59808i 0.541736 + 0.312772i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) 1.73205i 0.200000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.00000 8.66025i 0.562544 0.974355i −0.434730 0.900561i \(-0.643156\pi\)
0.997274 0.0737937i \(-0.0235106\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 4.50000 7.79423i 0.493939 0.855528i −0.506036 0.862512i \(-0.668890\pi\)
0.999976 + 0.00698436i \(0.00222321\pi\)
\(84\) 0 0
\(85\) 3.00000 + 5.19615i 0.325396 + 0.563602i
\(86\) 0 0
\(87\) 5.19615i 0.557086i
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 15.0000 + 8.66025i 1.55543 + 0.898027i
\(94\) 0 0
\(95\) −1.00000 1.73205i −0.102598 0.177705i
\(96\) 0 0
\(97\) 5.00000 8.66025i 0.507673 0.879316i −0.492287 0.870433i \(-0.663839\pi\)
0.999961 0.00888289i \(-0.00282755\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 15.5885i 0.895533 1.55111i 0.0623905 0.998052i \(-0.480128\pi\)
0.833143 0.553058i \(-0.186539\pi\)
\(102\) 0 0
\(103\) −4.00000 6.92820i −0.394132 0.682656i 0.598858 0.800855i \(-0.295621\pi\)
−0.992990 + 0.118199i \(0.962288\pi\)
\(104\) 0 0
\(105\) −1.50000 + 0.866025i −0.146385 + 0.0845154i
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) −15.0000 + 8.66025i −1.42374 + 0.821995i
\(112\) 0 0
\(113\) 6.00000 + 10.3923i 0.564433 + 0.977626i 0.997102 + 0.0760733i \(0.0242383\pi\)
−0.432670 + 0.901553i \(0.642428\pi\)
\(114\) 0 0
\(115\) 1.50000 2.59808i 0.139876 0.242272i
\(116\) 0 0
\(117\) 12.0000 1.10940
\(118\) 0 0
\(119\) −3.00000 + 5.19615i −0.275010 + 0.476331i
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 0 0
\(123\) −13.5000 7.79423i −1.21725 0.702782i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 6.92820i 0.609994i
\(130\) 0 0
\(131\) −3.00000 5.19615i −0.262111 0.453990i 0.704692 0.709514i \(-0.251085\pi\)
−0.966803 + 0.255524i \(0.917752\pi\)
\(132\) 0 0
\(133\) 1.00000 1.73205i 0.0867110 0.150188i
\(134\) 0 0
\(135\) −4.50000 + 2.59808i −0.387298 + 0.223607i
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) −10.0000 17.3205i −0.848189 1.46911i −0.882823 0.469706i \(-0.844360\pi\)
0.0346338 0.999400i \(-0.488974\pi\)
\(140\) 0 0
\(141\) 15.5885i 1.31278i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 9.00000 + 5.19615i 0.742307 + 0.428571i
\(148\) 0 0
\(149\) −4.50000 7.79423i −0.368654 0.638528i 0.620701 0.784047i \(-0.286848\pi\)
−0.989355 + 0.145519i \(0.953515\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) −9.00000 + 15.5885i −0.727607 + 1.26025i
\(154\) 0 0
\(155\) 5.00000 8.66025i 0.401610 0.695608i
\(156\) 0 0
\(157\) −1.00000 1.73205i −0.0798087 0.138233i 0.823359 0.567521i \(-0.192098\pi\)
−0.903167 + 0.429289i \(0.858764\pi\)
\(158\) 0 0
\(159\) −9.00000 + 5.19615i −0.713746 + 0.412082i
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.50000 + 12.9904i 0.580367 + 1.00523i 0.995436 + 0.0954356i \(0.0304244\pi\)
−0.415068 + 0.909790i \(0.636242\pi\)
\(168\) 0 0
\(169\) −1.50000 + 2.59808i −0.115385 + 0.199852i
\(170\) 0 0
\(171\) 3.00000 5.19615i 0.229416 0.397360i
\(172\) 0 0
\(173\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(174\) 0 0
\(175\) 0.500000 + 0.866025i 0.0377964 + 0.0654654i
\(176\) 0 0
\(177\) 9.00000 + 5.19615i 0.676481 + 0.390567i
\(178\) 0 0
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 1.73205i 0.128037i
\(184\) 0 0
\(185\) 5.00000 + 8.66025i 0.367607 + 0.636715i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.50000 2.59808i −0.327327 0.188982i
\(190\) 0 0
\(191\) −3.00000 + 5.19615i −0.217072 + 0.375980i −0.953912 0.300088i \(-0.902984\pi\)
0.736839 + 0.676068i \(0.236317\pi\)
\(192\) 0 0
\(193\) 2.00000 + 3.46410i 0.143963 + 0.249351i 0.928986 0.370116i \(-0.120682\pi\)
−0.785022 + 0.619467i \(0.787349\pi\)
\(194\) 0 0
\(195\) 6.92820i 0.496139i
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) −16.5000 9.52628i −1.16382 0.671932i
\(202\) 0 0
\(203\) 1.50000 + 2.59808i 0.105279 + 0.182349i
\(204\) 0 0
\(205\) −4.50000 + 7.79423i −0.314294 + 0.544373i
\(206\) 0 0
\(207\) 9.00000 0.625543
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 2.00000 + 3.46410i 0.137686 + 0.238479i 0.926620 0.375999i \(-0.122700\pi\)
−0.788935 + 0.614477i \(0.789367\pi\)
\(212\) 0 0
\(213\) 18.0000 10.3923i 1.23334 0.712069i
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) 0 0
\(219\) −6.00000 + 3.46410i −0.405442 + 0.234082i
\(220\) 0 0
\(221\) −12.0000 20.7846i −0.807207 1.39812i
\(222\) 0 0
\(223\) −5.50000 + 9.52628i −0.368307 + 0.637927i −0.989301 0.145889i \(-0.953396\pi\)
0.620994 + 0.783815i \(0.286729\pi\)
\(224\) 0 0
\(225\) 1.50000 + 2.59808i 0.100000 + 0.173205i
\(226\) 0 0
\(227\) −6.00000 + 10.3923i −0.398234 + 0.689761i −0.993508 0.113761i \(-0.963710\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(228\) 0 0
\(229\) 0.500000 + 0.866025i 0.0330409 + 0.0572286i 0.882073 0.471113i \(-0.156147\pi\)
−0.849032 + 0.528341i \(0.822814\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −30.0000 −1.96537 −0.982683 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 17.3205i 1.12509i
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 3.50000 6.06218i 0.225455 0.390499i −0.731001 0.682376i \(-0.760947\pi\)
0.956456 + 0.291877i \(0.0942799\pi\)
\(242\) 0 0
\(243\) −13.5000 7.79423i −0.866025 0.500000i
\(244\) 0 0
\(245\) 3.00000 5.19615i 0.191663 0.331970i
\(246\) 0 0
\(247\) 4.00000 + 6.92820i 0.254514 + 0.440831i
\(248\) 0 0
\(249\) 15.5885i 0.987878i
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 9.00000 + 5.19615i 0.563602 + 0.325396i
\(256\) 0 0
\(257\) −3.00000 5.19615i −0.187135 0.324127i 0.757159 0.653231i \(-0.226587\pi\)
−0.944294 + 0.329104i \(0.893253\pi\)
\(258\) 0 0
\(259\) −5.00000 + 8.66025i −0.310685 + 0.538122i
\(260\) 0 0
\(261\) 4.50000 + 7.79423i 0.278543 + 0.482451i
\(262\) 0 0
\(263\) 12.0000 20.7846i 0.739952 1.28163i −0.212565 0.977147i \(-0.568182\pi\)
0.952517 0.304487i \(-0.0984850\pi\)
\(264\) 0 0
\(265\) 3.00000 + 5.19615i 0.184289 + 0.319197i
\(266\) 0 0
\(267\) 13.5000 7.79423i 0.826187 0.476999i
\(268\) 0 0
\(269\) 9.00000 0.548740 0.274370 0.961624i \(-0.411531\pi\)
0.274370 + 0.961624i \(0.411531\pi\)
\(270\) 0 0
\(271\) 14.0000 0.850439 0.425220 0.905090i \(-0.360197\pi\)
0.425220 + 0.905090i \(0.360197\pi\)
\(272\) 0 0
\(273\) 6.00000 3.46410i 0.363137 0.209657i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.00000 + 1.73205i −0.0600842 + 0.104069i −0.894503 0.447062i \(-0.852470\pi\)
0.834419 + 0.551131i \(0.185804\pi\)
\(278\) 0 0
\(279\) 30.0000 1.79605
\(280\) 0 0
\(281\) −16.5000 + 28.5788i −0.984307 + 1.70487i −0.339333 + 0.940666i \(0.610201\pi\)
−0.644974 + 0.764204i \(0.723132\pi\)
\(282\) 0 0
\(283\) 0.500000 + 0.866025i 0.0297219 + 0.0514799i 0.880504 0.474039i \(-0.157204\pi\)
−0.850782 + 0.525519i \(0.823871\pi\)
\(284\) 0 0
\(285\) −3.00000 1.73205i −0.177705 0.102598i
\(286\) 0 0
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 17.3205i 1.01535i
\(292\) 0 0
\(293\) 3.00000 + 5.19615i 0.175262 + 0.303562i 0.940252 0.340480i \(-0.110589\pi\)
−0.764990 + 0.644042i \(0.777256\pi\)
\(294\) 0 0
\(295\) 3.00000 5.19615i 0.174667 0.302532i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 + 10.3923i −0.346989 + 0.601003i
\(300\) 0 0
\(301\) −2.00000 3.46410i −0.115278 0.199667i
\(302\) 0 0
\(303\) 31.1769i 1.79107i
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 0 0
\(309\) −12.0000 6.92820i −0.682656 0.394132i
\(310\) 0 0
\(311\) −9.00000 15.5885i −0.510343 0.883940i −0.999928 0.0119847i \(-0.996185\pi\)
0.489585 0.871956i \(-0.337148\pi\)
\(312\) 0 0
\(313\) 8.00000 13.8564i 0.452187 0.783210i −0.546335 0.837567i \(-0.683977\pi\)
0.998522 + 0.0543564i \(0.0173107\pi\)
\(314\) 0 0
\(315\) −1.50000 + 2.59808i −0.0845154 + 0.146385i
\(316\) 0 0
\(317\) 6.00000 10.3923i 0.336994 0.583690i −0.646872 0.762598i \(-0.723923\pi\)
0.983866 + 0.178908i \(0.0572566\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 13.5000 7.79423i 0.753497 0.435031i
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −28.5000 + 16.4545i −1.57605 + 0.909935i
\(328\) 0 0
\(329\) 4.50000 + 7.79423i 0.248093 + 0.429710i
\(330\) 0 0
\(331\) −13.0000 + 22.5167i −0.714545 + 1.23763i 0.248590 + 0.968609i \(0.420033\pi\)
−0.963135 + 0.269019i \(0.913301\pi\)
\(332\) 0 0
\(333\) −15.0000 + 25.9808i −0.821995 + 1.42374i
\(334\) 0 0
\(335\) −5.50000 + 9.52628i −0.300497 + 0.520476i
\(336\) 0 0
\(337\) −4.00000 6.92820i −0.217894 0.377403i 0.736270 0.676688i \(-0.236585\pi\)
−0.954164 + 0.299285i \(0.903252\pi\)
\(338\) 0 0
\(339\) 18.0000 + 10.3923i 0.977626 + 0.564433i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 5.19615i 0.279751i
\(346\) 0 0
\(347\) −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i \(-0.943860\pi\)
0.340293 0.940319i \(-0.389474\pi\)
\(348\) 0 0
\(349\) −11.5000 + 19.9186i −0.615581 + 1.06622i 0.374701 + 0.927146i \(0.377745\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 18.0000 10.3923i 0.960769 0.554700i
\(352\) 0 0
\(353\) −6.00000 + 10.3923i −0.319348 + 0.553127i −0.980352 0.197256i \(-0.936797\pi\)
0.661004 + 0.750382i \(0.270130\pi\)
\(354\) 0 0
\(355\) −6.00000 10.3923i −0.318447 0.551566i
\(356\) 0 0
\(357\) 10.3923i 0.550019i
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 16.5000 + 9.52628i 0.866025 + 0.500000i
\(364\) 0 0
\(365\) 2.00000 + 3.46410i 0.104685 + 0.181319i
\(366\) 0 0
\(367\) −16.0000 + 27.7128i −0.835193 + 1.44660i 0.0586798 + 0.998277i \(0.481311\pi\)
−0.893873 + 0.448320i \(0.852022\pi\)
\(368\) 0 0
\(369\) −27.0000 −1.40556
\(370\) 0 0
\(371\) −3.00000 + 5.19615i −0.155752 + 0.269771i
\(372\) 0 0
\(373\) −7.00000 12.1244i −0.362446 0.627775i 0.625917 0.779890i \(-0.284725\pi\)
−0.988363 + 0.152115i \(0.951392\pi\)
\(374\) 0 0
\(375\) 1.50000 0.866025i 0.0774597 0.0447214i
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −10.5000 + 6.06218i −0.537931 + 0.310575i
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.00000 10.3923i −0.304997 0.528271i
\(388\) 0 0
\(389\) 7.50000 12.9904i 0.380265 0.658638i −0.610835 0.791758i \(-0.709166\pi\)
0.991100 + 0.133120i \(0.0424994\pi\)
\(390\) 0 0
\(391\) −9.00000 15.5885i −0.455150 0.788342i
\(392\) 0 0
\(393\) −9.00000 5.19615i −0.453990 0.262111i
\(394\) 0 0
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 3.46410i 0.173422i
\(400\) 0 0
\(401\) 9.00000 + 15.5885i 0.449439 + 0.778450i 0.998350 0.0574304i \(-0.0182907\pi\)
−0.548911 + 0.835881i \(0.684957\pi\)
\(402\) 0 0
\(403\) −20.0000 + 34.6410i −0.996271 + 1.72559i
\(404\) 0 0
\(405\) −4.50000 + 7.79423i −0.223607 + 0.387298i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.00000 + 8.66025i 0.247234 + 0.428222i 0.962757 0.270367i \(-0.0871450\pi\)
−0.715523 + 0.698589i \(0.753812\pi\)
\(410\) 0 0
\(411\) 20.7846i 1.02523i
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 0 0
\(415\) −9.00000 −0.441793
\(416\) 0 0
\(417\) −30.0000 17.3205i −1.46911 0.848189i
\(418\) 0 0
\(419\) −6.00000 10.3923i −0.293119 0.507697i 0.681426 0.731887i \(-0.261360\pi\)
−0.974546 + 0.224189i \(0.928027\pi\)
\(420\) 0 0
\(421\) 11.0000 19.0526i 0.536107 0.928565i −0.463002 0.886357i \(-0.653228\pi\)
0.999109 0.0422075i \(-0.0134391\pi\)
\(422\) 0 0
\(423\) 13.5000 + 23.3827i 0.656392 + 1.13691i
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −0.500000 0.866025i −0.0241967 0.0419099i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 4.50000 2.59808i 0.215758 0.124568i
\(436\) 0 0
\(437\) 3.00000 + 5.19615i 0.143509 + 0.248566i
\(438\) 0 0
\(439\) 14.0000 24.2487i 0.668184 1.15733i −0.310228 0.950662i \(-0.600405\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) 0 0
\(443\) 13.5000 23.3827i 0.641404 1.11094i −0.343715 0.939074i \(-0.611685\pi\)
0.985119 0.171871i \(-0.0549812\pi\)
\(444\) 0 0
\(445\) −4.50000 7.79423i −0.213320 0.369482i
\(446\) 0 0
\(447\) −13.5000 7.79423i −0.638528 0.368654i
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.8564i 0.651031i
\(454\) 0 0
\(455\) −2.00000 3.46410i −0.0937614 0.162400i
\(456\) 0 0
\(457\) 5.00000 8.66025i 0.233890 0.405110i −0.725059 0.688686i \(-0.758188\pi\)
0.958950 + 0.283577i \(0.0915211\pi\)
\(458\) 0 0
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) −16.5000 + 28.5788i −0.768482 + 1.33105i 0.169904 + 0.985461i \(0.445654\pi\)
−0.938386 + 0.345589i \(0.887679\pi\)
\(462\) 0 0
\(463\) 2.00000 + 3.46410i 0.0929479 + 0.160990i 0.908750 0.417340i \(-0.137038\pi\)
−0.815802 + 0.578331i \(0.803704\pi\)
\(464\) 0 0
\(465\) 17.3205i 0.803219i
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) −11.0000 −0.507933
\(470\) 0 0
\(471\) −3.00000 1.73205i −0.138233 0.0798087i
\(472\) 0 0
\(473\) 0 0
\(474\) 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
\(479\) 15.0000 25.9808i 0.685367 1.18709i −0.287954 0.957644i \(-0.592975\pi\)
0.973321 0.229447i \(-0.0736918\pi\)
\(480\) 0 0
\(481\) −20.0000 34.6410i −0.911922 1.57949i
\(482\) 0 0
\(483\) 4.50000 2.59808i 0.204757 0.118217i
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −6.00000 + 3.46410i −0.271329 + 0.156652i
\(490\) 0 0
\(491\) 12.0000 + 20.7846i 0.541552 + 0.937996i 0.998815 + 0.0486647i \(0.0154966\pi\)
−0.457263 + 0.889332i \(0.651170\pi\)
\(492\) 0 0
\(493\) 9.00000 15.5885i 0.405340 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 10.3923i 0.269137 0.466159i
\(498\) 0 0
\(499\) 2.00000 + 3.46410i 0.0895323 + 0.155074i 0.907314 0.420455i \(-0.138129\pi\)
−0.817781 + 0.575529i \(0.804796\pi\)
\(500\) 0 0
\(501\) 22.5000 + 12.9904i 1.00523 + 0.580367i
\(502\) 0 0
\(503\) 3.00000 0.133763 0.0668817 0.997761i \(-0.478695\pi\)
0.0668817 + 0.997761i \(0.478695\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 5.19615i 0.230769i
\(508\) 0 0
\(509\) 4.50000 + 7.79423i 0.199459 + 0.345473i 0.948353 0.317217i \(-0.102748\pi\)
−0.748894 + 0.662690i \(0.769415\pi\)
\(510\) 0 0
\(511\) −2.00000 + 3.46410i −0.0884748 + 0.153243i
\(512\) 0 0
\(513\) 10.3923i 0.458831i
\(514\) 0 0
\(515\) −4.00000 + 6.92820i −0.176261 + 0.305293i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) −7.00000 −0.306089 −0.153044 0.988219i \(-0.548908\pi\)
−0.153044 + 0.988219i \(0.548908\pi\)
\(524\) 0 0
\(525\) 1.50000 + 0.866025i 0.0654654 + 0.0377964i
\(526\) 0 0
\(527\) −30.0000 51.9615i −1.30682 2.26348i
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 18.0000 0.781133
\(532\) 0 0
\(533\) 18.0000 31.1769i 0.779667 1.35042i
\(534\) 0 0
\(535\) −4.50000 7.79423i −0.194552 0.336974i
\(536\) 0 0
\(537\) −36.0000 + 20.7846i −1.55351 + 0.896922i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 0 0
\(543\) 7.50000 4.33013i 0.321856 0.185824i
\(544\) 0 0
\(545\) 9.50000 + 16.4545i 0.406935 + 0.704833i
\(546\) 0 0
\(547\) 3.50000 6.06218i 0.149649 0.259200i −0.781449 0.623970i \(-0.785519\pi\)
0.931098 + 0.364770i \(0.118852\pi\)
\(548\) 0 0
\(549\) −1.50000 2.59808i −0.0640184 0.110883i
\(550\) 0 0
\(551\) −3.00000 + 5.19615i −0.127804 + 0.221364i
\(552\) 0 0
\(553\) −5.00000 8.66025i −0.212622 0.368271i
\(554\) 0 0
\(555\) 15.0000 + 8.66025i 0.636715 + 0.367607i
\(556\) 0 0
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.5000 + 33.7750i 0.821827 + 1.42345i 0.904320 + 0.426855i \(0.140378\pi\)
−0.0824933 + 0.996592i \(0.526288\pi\)
\(564\) 0 0
\(565\) 6.00000 10.3923i 0.252422 0.437208i
\(566\) 0 0
\(567\) −9.00000 −0.377964
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) −16.0000 27.7128i −0.669579 1.15975i −0.978022 0.208502i \(-0.933141\pi\)
0.308443 0.951243i \(-0.400192\pi\)
\(572\) 0 0
\(573\) 10.3923i 0.434145i
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 38.0000 1.58196 0.790980 0.611842i \(-0.209571\pi\)
0.790980 + 0.611842i \(0.209571\pi\)
\(578\) 0 0
\(579\) 6.00000 + 3.46410i 0.249351 + 0.143963i
\(580\) 0 0
\(581\) −4.50000 7.79423i −0.186691 0.323359i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 10.3923i −0.248069 0.429669i
\(586\) 0 0
\(587\) 13.5000 23.3827i 0.557205 0.965107i −0.440524 0.897741i \(-0.645207\pi\)
0.997728 0.0673658i \(-0.0214594\pi\)
\(588\) 0 0
\(589\) 10.0000 + 17.3205i 0.412043 + 0.713679i
\(590\) 0 0
\(591\) 18.0000 10.3923i 0.740421 0.427482i
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) 3.00000 1.73205i 0.122782 0.0708881i
\(598\) 0 0
\(599\) −3.00000 5.19615i −0.122577 0.212309i 0.798206 0.602384i \(-0.205782\pi\)
−0.920783 + 0.390075i \(0.872449\pi\)
\(600\) 0 0
\(601\) 11.0000 19.0526i 0.448699 0.777170i −0.549602 0.835426i \(-0.685221\pi\)
0.998302 + 0.0582563i \(0.0185541\pi\)
\(602\) 0 0
\(603\) −33.0000 −1.34386
\(604\) 0 0
\(605\) 5.50000 9.52628i 0.223607 0.387298i
\(606\) 0 0
\(607\) 3.50000 + 6.06218i 0.142061 + 0.246056i 0.928272 0.371901i \(-0.121294\pi\)
−0.786212 + 0.617957i \(0.787961\pi\)
\(608\) 0 0
\(609\) 4.50000 + 2.59808i 0.182349 + 0.105279i
\(610\) 0 0
\(611\) −36.0000 −1.45640
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) 0 0
\(615\) 15.5885i 0.628587i
\(616\) 0 0
\(617\) 6.00000 + 10.3923i 0.241551 + 0.418378i 0.961156 0.276005i \(-0.0890106\pi\)
−0.719605 + 0.694383i \(0.755677\pi\)
\(618\) 0 0
\(619\) −13.0000 + 22.5167i −0.522514 + 0.905021i 0.477143 + 0.878826i \(0.341672\pi\)
−0.999657 + 0.0261952i \(0.991661\pi\)
\(620\) 0 0
\(621\) 13.5000 7.79423i 0.541736 0.312772i
\(622\) 0 0
\(623\) 4.50000 7.79423i 0.180289 0.312269i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 6.00000 + 3.46410i 0.238479 + 0.137686i
\(634\) 0 0
\(635\) 3.50000 + 6.06218i 0.138893 + 0.240570i
\(636\) 0 0
\(637\) −12.0000 + 20.7846i −0.475457 + 0.823516i
\(638\) 0 0
\(639\) 18.0000 31.1769i 0.712069 1.23334i
\(640\) 0 0
\(641\) −10.5000 + 18.1865i −0.414725 + 0.718325i −0.995400 0.0958109i \(-0.969456\pi\)
0.580674 + 0.814136i \(0.302789\pi\)
\(642\) 0 0
\(643\) 15.5000 + 26.8468i 0.611260 + 1.05873i 0.991028 + 0.133652i \(0.0426705\pi\)
−0.379768 + 0.925082i \(0.623996\pi\)
\(644\) 0 0
\(645\) −6.00000 + 3.46410i −0.236250 + 0.136399i
\(646\) 0 0
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 15.0000 8.66025i 0.587896 0.339422i
\(652\) 0 0
\(653\) −3.00000 5.19615i −0.117399 0.203341i 0.801337 0.598213i \(-0.204122\pi\)
−0.918736 + 0.394872i \(0.870789\pi\)
\(654\) 0 0
\(655\) −3.00000 + 5.19615i −0.117220 + 0.203030i
\(656\) 0 0
\(657\) −6.00000 + 10.3923i −0.234082 + 0.405442i
\(658\) 0 0
\(659\) −18.0000 + 31.1769i −0.701180 + 1.21448i 0.266872 + 0.963732i \(0.414010\pi\)
−0.968052 + 0.250748i \(0.919323\pi\)
\(660\) 0 0
\(661\) 5.00000 + 8.66025i 0.194477 + 0.336845i 0.946729 0.322031i \(-0.104366\pi\)
−0.752252 + 0.658876i \(0.771032\pi\)
\(662\) 0 0
\(663\) −36.0000 20.7846i −1.39812 0.807207i
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 0 0
\(667\) −9.00000 −0.348481
\(668\) 0 0
\(669\) 19.0526i 0.736614i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 24.2487i 0.539660 0.934719i −0.459262 0.888301i \(-0.651886\pi\)
0.998922 0.0464181i \(-0.0147807\pi\)
\(674\) 0 0
\(675\) 4.50000 + 2.59808i 0.173205 + 0.100000i
\(676\) 0 0
\(677\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(678\) 0 0
\(679\) −5.00000 8.66025i −0.191882 0.332350i
\(680\) 0 0
\(681\) 20.7846i 0.796468i
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 1.50000 + 0.866025i 0.0572286 + 0.0330409i
\(688\) 0 0
\(689\) −12.0000 20.7846i −0.457164 0.791831i
\(690\) 0 0
\(691\) 5.00000 8.66025i 0.190209 0.329452i −0.755110 0.655598i \(-0.772417\pi\)
0.945319 + 0.326146i \(0.105750\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.0000 + 17.3205i −0.379322 + 0.657004i
\(696\) 0 0
\(697\) 27.0000 + 46.7654i 1.02270 + 1.77136i
\(698\) 0 0
\(699\) −45.0000 + 25.9808i −1.70206 + 0.982683i
\(700\) 0 0
\(701\) −21.0000 −0.793159 −0.396580 0.918000i \(-0.629803\pi\)
−0.396580 + 0.918000i \(0.629803\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 13.5000 7.79423i 0.508439 0.293548i
\(706\) 0 0
\(707\) −9.00000 15.5885i −0.338480 0.586264i
\(708\) 0 0
\(709\) −17.5000 + 30.3109i −0.657226 + 1.13835i 0.324104 + 0.946021i \(0.394937\pi\)
−0.981331 + 0.192328i \(0.938396\pi\)
\(710\) 0 0
\(711\) −15.0000 25.9808i −0.562544 0.974355i
\(712\) 0 0
\(713\) −15.0000 + 25.9808i −0.561754 + 0.972987i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −42.0000 −1.56634 −0.783168 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 12.1244i 0.450910i
\(724\) 0 0
\(725\) −1.50000 2.59808i −0.0557086 0.0964901i
\(726\) 0 0
\(727\) 9.50000 16.4545i 0.352335 0.610263i −0.634323 0.773068i \(-0.718721\pi\)
0.986658 + 0.162805i \(0.0520543\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −12.0000 + 20.7846i −0.443836 + 0.768747i
\(732\) 0 0
\(733\) −7.00000 12.1244i −0.258551 0.447823i 0.707303 0.706910i \(-0.249912\pi\)
−0.965854 + 0.259087i \(0.916578\pi\)
\(734\) 0 0
\(735\) 10.3923i 0.383326i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 2.00000 0.0735712 0.0367856 0.999323i \(-0.488288\pi\)
0.0367856 + 0.999323i \(0.488288\pi\)
\(740\) 0 0
\(741\) 12.0000 + 6.92820i 0.440831 + 0.254514i
\(742\) 0 0
\(743\) −16.5000 28.5788i −0.605326 1.04846i −0.992000 0.126239i \(-0.959709\pi\)
0.386674 0.922217i \(-0.373624\pi\)
\(744\) 0 0
\(745\) −4.50000 + 7.79423i −0.164867 + 0.285558i
\(746\) 0 0
\(747\) −13.5000 23.3827i −0.493939 0.855528i
\(748\) 0 0
\(749\) 4.50000 7.79423i 0.164426 0.284795i
\(750\) 0 0
\(751\) 8.00000 + 13.8564i 0.291924 + 0.505627i 0.974265 0.225407i \(-0.0723712\pi\)
−0.682341 + 0.731034i \(0.739038\pi\)
\(752\) 0 0
\(753\) 27.0000 15.5885i 0.983935 0.568075i
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.5000 + 38.9711i 0.815624 + 1.41270i 0.908879 + 0.417061i \(0.136940\pi\)
−0.0932544 + 0.995642i \(0.529727\pi\)
\(762\) 0 0
\(763\) −9.50000 + 16.4545i −0.343923 + 0.595692i
\(764\) 0 0
\(765\) 18.0000 0.650791
\(766\) 0 0
\(767\) −12.0000 + 20.7846i −0.433295 + 0.750489i
\(768\) 0 0
\(769\) 15.5000 + 26.8468i 0.558944 + 0.968120i 0.997585 + 0.0694574i \(0.0221268\pi\)
−0.438641 + 0.898663i \(0.644540\pi\)
\(770\) 0 0
\(771\) −9.00000 5.19615i −0.324127 0.187135i
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) 17.3205i 0.621370i
\(778\) 0 0
\(779\) −9.00000 15.5885i −0.322458 0.558514i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 13.5000 + 7.79423i 0.482451 + 0.278543i
\(784\) 0 0
\(785\) −1.00000 + 1.73205i −0.0356915 + 0.0618195i
\(786\) 0 0
\(787\) 20.0000 + 34.6410i 0.712923 + 1.23482i 0.963755 + 0.266788i \(0.0859624\pi\)
−0.250832 + 0.968031i \(0.580704\pi\)
\(788\) 0 0
\(789\) 41.5692i 1.47990i
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 9.00000 + 5.19615i 0.319197 + 0.184289i
\(796\) 0 0
\(797\) 12.0000 + 20.7846i 0.425062 + 0.736229i 0.996426 0.0844678i \(-0.0269190\pi\)
−0.571364 + 0.820696i \(0.693586\pi\)
\(798\) 0 0
\(799\) 27.0000 46.7654i 0.955191 1.65444i
\(800\) 0 0
\(801\) 13.5000 23.3827i 0.476999 0.826187i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.50000 2.59808i −0.0528681 0.0915702i
\(806\) 0 0
\(807\) 13.5000 7.79423i 0.475223 0.274370i
\(808\) 0 0
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) 21.0000 12.1244i 0.736502 0.425220i
\(814\) 0 0
\(815\) 2.00000 + 3.46410i 0.0700569 + 0.121342i
\(816\) 0 0
\(817\) 4.00000 6.92820i 0.139942 0.242387i
\(818\) 0 0
\(819\) 6.00000 10.3923i 0.209657 0.363137i
\(820\) 0 0
\(821\) −1.50000 + 2.59808i −0.0523504 + 0.0906735i −0.891013 0.453978i \(-0.850005\pi\)
0.838663 + 0.544651i \(0.183338\pi\)
\(822\) 0 0
\(823\) 24.5000 + 42.4352i 0.854016 + 1.47920i 0.877555 + 0.479477i \(0.159174\pi\)
−0.0235383 + 0.999723i \(0.507493\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 27.0000 0.938882 0.469441 0.882964i \(-0.344455\pi\)
0.469441 + 0.882964i \(0.344455\pi\)
\(828\) 0 0
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 0 0
\(831\) 3.46410i 0.120168i
\(832\) 0 0
\(833\) −18.0000 31.1769i −0.623663 1.08022i
\(834\) 0 0
\(835\) 7.50000 12.9904i 0.259548 0.449551i
\(836\) 0 0
\(837\) 45.0000 25.9808i 1.55543 0.898027i
\(838\) 0 0
\(839\) 15.0000 25.9808i 0.517858 0.896956i −0.481927 0.876211i \(-0.660063\pi\)
0.999785 0.0207443i \(-0.00660359\pi\)
\(840\) 0 0
\(841\) 10.0000 + 17.3205i 0.344828 + 0.597259i
\(842\) 0 0
\(843\) 57.1577i 1.96861i
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 1.50000 + 0.866025i 0.0514799 + 0.0297219i
\(850\) 0 0
\(851\) −15.0000 25.9808i −0.514193 0.890609i
\(852\) 0 0
\(853\) 2.00000 3.46410i 0.0684787 0.118609i −0.829753 0.558131i \(-0.811519\pi\)
0.898232 + 0.439522i \(0.144852\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) 24.0000 41.5692i 0.819824 1.41998i −0.0859870 0.996296i \(-0.527404\pi\)
0.905811 0.423681i \(-0.139262\pi\)
\(858\) 0 0
\(859\) −10.0000 17.3205i −0.341196 0.590968i 0.643459 0.765480i \(-0.277499\pi\)
−0.984655 + 0.174512i \(0.944165\pi\)
\(860\) 0 0
\(861\) −13.5000 + 7.79423i −0.460079 + 0.265627i
\(862\) 0 0
\(863\) 15.0000 0.510606 0.255303 0.966861i \(-0.417825\pi\)
0.255303 + 0.966861i \(0.417825\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 28.5000 16.4545i 0.967911 0.558824i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.0000 38.1051i 0.745442 1.29114i
\(872\) 0 0
\(873\) −15.0000 25.9808i −0.507673 0.879316i
\(874\) 0 0
\(875\) 0.500000 0.866025i 0.0169031 0.0292770i
\(876\) 0 0
\(877\) 2.00000 + 3.46410i 0.0675352 + 0.116974i 0.897816 0.440371i \(-0.145153\pi\)
−0.830281 + 0.557346i \(0.811820\pi\)
\(878\) 0 0
\(879\) 9.00000 + 5.19615i 0.303562 + 0.175262i
\(880\) 0 0
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 29.0000 0.975928 0.487964 0.872864i \(-0.337740\pi\)
0.487964 + 0.872864i \(0.337740\pi\)
\(884\) 0 0
\(885\) 10.3923i 0.349334i
\(886\) 0 0
\(887\) 24.0000 + 41.5692i 0.805841 + 1.39576i 0.915722 + 0.401813i \(0.131620\pi\)
−0.109881 + 0.993945i \(0.535047\pi\)
\(888\) 0 0
\(889\) −3.50000 + 6.06218i −0.117386 + 0.203319i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.00000 + 15.5885i −0.301174 + 0.521648i
\(894\) 0 0
\(895\) 12.0000 + 20.7846i 0.401116 + 0.694753i
\(896\) 0 0
\(897\) 20.7846i 0.693978i
\(898\) 0 0
\(899\) −30.0000 −1.00056
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −6.00000 3.46410i −0.199667 0.115278i
\(904\) 0 0
\(905\) −2.50000 4.33013i −0.0831028 0.143938i
\(906\) 0 0
\(907\) 0.500000 0.866025i 0.0166022 0.0287559i −0.857605 0.514309i \(-0.828048\pi\)
0.874207 + 0.485553i \(0.161382\pi\)
\(908\) 0 0
\(909\) −27.0000 46.7654i −0.895533 1.55111i
\(910\) 0 0
\(911\) 3.00000 5.19615i 0.0993944 0.172156i −0.812040 0.583602i \(-0.801643\pi\)
0.911434 + 0.411446i \(0.134976\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −1.50000 + 0.866025i −0.0495885 + 0.0286299i
\(916\) 0 0
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) 43.5000 25.1147i 1.43337 0.827559i
\(922\) 0 0
\(923\) 24.0000 + 41.5692i 0.789970 + 1.36827i
\(924\) 0 0
\(925\) 5.00000 8.66025i 0.164399 0.284747i
\(926\) 0 0
\(927\) −24.0000 −0.788263
\(928\) 0 0
\(929\) −3.00000 + 5.19615i −0.0984268 + 0.170480i −0.911034 0.412332i \(-0.864714\pi\)
0.812607 + 0.582812i \(0.198048\pi\)
\(930\) 0 0
\(931\) 6.00000 + 10.3923i 0.196642 + 0.340594i
\(932\) 0 0
\(933\) −27.0000 15.5885i −0.883940 0.510343i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 0 0
\(939\) 27.7128i 0.904373i
\(940\) 0 0
\(941\) −22.5000 38.9711i −0.733479 1.27042i −0.955387 0.295355i \(-0.904562\pi\)
0.221908 0.975068i \(-0.428771\pi\)
\(942\) 0 0
\(943\) 13.5000 23.3827i 0.439620 0.761445i
\(944\) 0 0
\(945\) 5.19615i 0.169031i
\(946\) 0 0
\(947\) −7.50000 + 12.9904i −0.243717 + 0.422131i −0.961770 0.273858i \(-0.911700\pi\)
0.718053 + 0.695988i \(0.245034\pi\)
\(948\) 0 0
\(949\) −8.00000 13.8564i −0.259691 0.449798i
\(950\) 0 0
\(951\) 20.7846i 0.673987i
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) 6.00000 0.194155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 + 10.3923i 0.193750 + 0.335585i
\(960\) 0 0
\(961\) −34.5000 + 59.7558i −1.11290 + 1.92760i
\(962\) 0 0
\(963\) 13.5000 23.3827i 0.435031 0.753497i
\(964\) 0 0
\(965\) 2.00000 3.46410i 0.0643823 0.111513i
\(966\) 0 0
\(967\) 0.500000 + 0.866025i 0.0160789 + 0.0278495i 0.873953 0.486011i \(-0.161548\pi\)
−0.857874 + 0.513860i \(0.828215\pi\)
\(968\) 0 0
\(969\) −18.0000 + 10.3923i −0.578243 + 0.333849i
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 0 0
\(975\) −6.00000 + 3.46410i −0.192154 + 0.110940i
\(976\) 0 0
\(977\) −12.0000 20.7846i −0.383914 0.664959i 0.607704 0.794164i \(-0.292091\pi\)
−0.991618 + 0.129205i \(0.958757\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −28.5000 + 49.3634i −0.909935 + 1.57605i
\(982\) 0 0
\(983\) −4.50000 + 7.79423i −0.143528 + 0.248597i −0.928823 0.370525i \(-0.879178\pi\)
0.785295 + 0.619122i \(0.212511\pi\)
\(984\) 0 0
\(985\) −6.00000 10.3923i −0.191176 0.331126i
\(986\) 0 0
\(987\) 13.5000 + 7.79423i 0.429710 + 0.248093i
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 0 0
\(993\) 45.0333i 1.42909i
\(994\) 0 0
\(995\) −1.00000 1.73205i −0.0317021 0.0549097i
\(996\) 0 0
\(997\) 5.00000 8.66025i 0.158352 0.274273i −0.775923 0.630828i \(-0.782715\pi\)
0.934274 + 0.356555i \(0.116049\pi\)
\(998\) 0 0
\(999\) 51.9615i 1.64399i
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 180.2.i.a.61.1 2
3.2 odd 2 540.2.i.a.181.1 2
4.3 odd 2 720.2.q.a.241.1 2
5.2 odd 4 900.2.s.a.349.1 4
5.3 odd 4 900.2.s.a.349.2 4
5.4 even 2 900.2.i.a.601.1 2
9.2 odd 6 1620.2.a.b.1.1 1
9.4 even 3 inner 180.2.i.a.121.1 yes 2
9.5 odd 6 540.2.i.a.361.1 2
9.7 even 3 1620.2.a.e.1.1 1
12.11 even 2 2160.2.q.e.721.1 2
15.2 even 4 2700.2.s.a.2449.2 4
15.8 even 4 2700.2.s.a.2449.1 4
15.14 odd 2 2700.2.i.a.1801.1 2
36.7 odd 6 6480.2.a.t.1.1 1
36.11 even 6 6480.2.a.h.1.1 1
36.23 even 6 2160.2.q.e.1441.1 2
36.31 odd 6 720.2.q.a.481.1 2
45.2 even 12 8100.2.d.f.649.1 2
45.4 even 6 900.2.i.a.301.1 2
45.7 odd 12 8100.2.d.e.649.1 2
45.13 odd 12 900.2.s.a.49.1 4
45.14 odd 6 2700.2.i.a.901.1 2
45.22 odd 12 900.2.s.a.49.2 4
45.23 even 12 2700.2.s.a.1549.2 4
45.29 odd 6 8100.2.a.h.1.1 1
45.32 even 12 2700.2.s.a.1549.1 4
45.34 even 6 8100.2.a.i.1.1 1
45.38 even 12 8100.2.d.f.649.2 2
45.43 odd 12 8100.2.d.e.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.i.a.61.1 2 1.1 even 1 trivial
180.2.i.a.121.1 yes 2 9.4 even 3 inner
540.2.i.a.181.1 2 3.2 odd 2
540.2.i.a.361.1 2 9.5 odd 6
720.2.q.a.241.1 2 4.3 odd 2
720.2.q.a.481.1 2 36.31 odd 6
900.2.i.a.301.1 2 45.4 even 6
900.2.i.a.601.1 2 5.4 even 2
900.2.s.a.49.1 4 45.13 odd 12
900.2.s.a.49.2 4 45.22 odd 12
900.2.s.a.349.1 4 5.2 odd 4
900.2.s.a.349.2 4 5.3 odd 4
1620.2.a.b.1.1 1 9.2 odd 6
1620.2.a.e.1.1 1 9.7 even 3
2160.2.q.e.721.1 2 12.11 even 2
2160.2.q.e.1441.1 2 36.23 even 6
2700.2.i.a.901.1 2 45.14 odd 6
2700.2.i.a.1801.1 2 15.14 odd 2
2700.2.s.a.1549.1 4 45.32 even 12
2700.2.s.a.1549.2 4 45.23 even 12
2700.2.s.a.2449.1 4 15.8 even 4
2700.2.s.a.2449.2 4 15.2 even 4
6480.2.a.h.1.1 1 36.11 even 6
6480.2.a.t.1.1 1 36.7 odd 6
8100.2.a.h.1.1 1 45.29 odd 6
8100.2.a.i.1.1 1 45.34 even 6
8100.2.d.e.649.1 2 45.7 odd 12
8100.2.d.e.649.2 2 45.43 odd 12
8100.2.d.f.649.1 2 45.2 even 12
8100.2.d.f.649.2 2 45.38 even 12