Properties

Label 912.2.f.e.113.2
Level $912$
Weight $2$
Character 912.113
Analytic conductor $7.282$
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] = [912,2,Mod(113,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.f (of order \(2\), degree \(1\), 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: \(7.28235666434\)
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 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 113.2
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 912.113
Dual form 912.2.f.e.113.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} +O(q^{10})\) \(q+(1.50000 + 0.866025i) q^{3} -3.46410i q^{5} -1.00000 q^{7} +(1.50000 + 2.59808i) q^{9} -3.46410i q^{11} +1.73205i q^{13} +(3.00000 - 5.19615i) q^{15} -1.73205i q^{17} +(4.00000 + 1.73205i) q^{19} +(-1.50000 - 0.866025i) q^{21} -5.19615i q^{23} -7.00000 q^{25} +5.19615i q^{27} +9.00000 q^{29} -10.3923i q^{31} +(3.00000 - 5.19615i) q^{33} +3.46410i q^{35} -6.92820i q^{37} +(-1.50000 + 2.59808i) q^{39} -2.00000 q^{43} +(9.00000 - 5.19615i) q^{45} +3.46410i q^{47} -6.00000 q^{49} +(1.50000 - 2.59808i) q^{51} -9.00000 q^{53} -12.0000 q^{55} +(4.50000 + 6.06218i) q^{57} +3.00000 q^{59} -8.00000 q^{61} +(-1.50000 - 2.59808i) q^{63} +6.00000 q^{65} +8.66025i q^{67} +(4.50000 - 7.79423i) q^{69} +12.0000 q^{71} +11.0000 q^{73} +(-10.5000 - 6.06218i) q^{75} +3.46410i q^{77} +6.92820i q^{79} +(-4.50000 + 7.79423i) q^{81} +10.3923i q^{83} -6.00000 q^{85} +(13.5000 + 7.79423i) q^{87} +6.00000 q^{89} -1.73205i q^{91} +(9.00000 - 15.5885i) q^{93} +(6.00000 - 13.8564i) q^{95} +13.8564i q^{97} +(9.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{15} + 8 q^{19} - 3 q^{21} - 14 q^{25} + 18 q^{29} + 6 q^{33} - 3 q^{39} - 4 q^{43} + 18 q^{45} - 12 q^{49} + 3 q^{51} - 18 q^{53} - 24 q^{55} + 9 q^{57} + 6 q^{59} - 16 q^{61} - 3 q^{63} + 12 q^{65} + 9 q^{69} + 24 q^{71} + 22 q^{73} - 21 q^{75} - 9 q^{81} - 12 q^{85} + 27 q^{87} + 12 q^{89} + 18 q^{93} + 12 q^{95} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.50000 + 0.866025i 0.866025 + 0.500000i
\(4\) 0 0
\(5\) 3.46410i 1.54919i −0.632456 0.774597i \(-0.717953\pi\)
0.632456 0.774597i \(-0.282047\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 3.46410i 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 3.00000 5.19615i 0.774597 1.34164i
\(16\) 0 0
\(17\) 1.73205i 0.420084i −0.977692 0.210042i \(-0.932640\pi\)
0.977692 0.210042i \(-0.0673601\pi\)
\(18\) 0 0
\(19\) 4.00000 + 1.73205i 0.917663 + 0.397360i
\(20\) 0 0
\(21\) −1.50000 0.866025i −0.327327 0.188982i
\(22\) 0 0
\(23\) 5.19615i 1.08347i −0.840548 0.541736i \(-0.817767\pi\)
0.840548 0.541736i \(-0.182233\pi\)
\(24\) 0 0
\(25\) −7.00000 −1.40000
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 10.3923i 1.86651i −0.359211 0.933257i \(-0.616954\pi\)
0.359211 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 3.00000 5.19615i 0.522233 0.904534i
\(34\) 0 0
\(35\) 3.46410i 0.585540i
\(36\) 0 0
\(37\) 6.92820i 1.13899i −0.821995 0.569495i \(-0.807139\pi\)
0.821995 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) −1.50000 + 2.59808i −0.240192 + 0.416025i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 9.00000 5.19615i 1.34164 0.774597i
\(46\) 0 0
\(47\) 3.46410i 0.505291i 0.967559 + 0.252646i \(0.0813007\pi\)
−0.967559 + 0.252646i \(0.918699\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.50000 2.59808i 0.210042 0.363803i
\(52\) 0 0
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 4.50000 + 6.06218i 0.596040 + 0.802955i
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −1.50000 2.59808i −0.188982 0.327327i
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.66025i 1.05802i 0.848616 + 0.529009i \(0.177436\pi\)
−0.848616 + 0.529009i \(0.822564\pi\)
\(68\) 0 0
\(69\) 4.50000 7.79423i 0.541736 0.938315i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −10.5000 6.06218i −1.21244 0.700000i
\(76\) 0 0
\(77\) 3.46410i 0.394771i
\(78\) 0 0
\(79\) 6.92820i 0.779484i 0.920924 + 0.389742i \(0.127436\pi\)
−0.920924 + 0.389742i \(0.872564\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 10.3923i 1.14070i 0.821401 + 0.570352i \(0.193193\pi\)
−0.821401 + 0.570352i \(0.806807\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 13.5000 + 7.79423i 1.44735 + 0.835629i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 1.73205i 0.181568i
\(92\) 0 0
\(93\) 9.00000 15.5885i 0.933257 1.61645i
\(94\) 0 0
\(95\) 6.00000 13.8564i 0.615587 1.42164i
\(96\) 0 0
\(97\) 13.8564i 1.40690i 0.710742 + 0.703452i \(0.248359\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 9.00000 5.19615i 0.904534 0.522233i
\(100\) 0 0
\(101\) 10.3923i 1.03407i 0.855963 + 0.517036i \(0.172965\pi\)
−0.855963 + 0.517036i \(0.827035\pi\)
\(102\) 0 0
\(103\) 3.46410i 0.341328i 0.985329 + 0.170664i \(0.0545913\pi\)
−0.985329 + 0.170664i \(0.945409\pi\)
\(104\) 0 0
\(105\) −3.00000 + 5.19615i −0.292770 + 0.507093i
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) 15.5885i 1.49310i 0.665327 + 0.746552i \(0.268292\pi\)
−0.665327 + 0.746552i \(0.731708\pi\)
\(110\) 0 0
\(111\) 6.00000 10.3923i 0.569495 0.986394i
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −18.0000 −1.67851
\(116\) 0 0
\(117\) −4.50000 + 2.59808i −0.416025 + 0.240192i
\(118\) 0 0
\(119\) 1.73205i 0.158777i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −3.00000 1.73205i −0.264135 0.152499i
\(130\) 0 0
\(131\) 3.46410i 0.302660i 0.988483 + 0.151330i \(0.0483556\pi\)
−0.988483 + 0.151330i \(0.951644\pi\)
\(132\) 0 0
\(133\) −4.00000 1.73205i −0.346844 0.150188i
\(134\) 0 0
\(135\) 18.0000 1.54919
\(136\) 0 0
\(137\) 8.66025i 0.739895i −0.929053 0.369948i \(-0.879376\pi\)
0.929053 0.369948i \(-0.120624\pi\)
\(138\) 0 0
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) −3.00000 + 5.19615i −0.252646 + 0.437595i
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 31.1769i 2.58910i
\(146\) 0 0
\(147\) −9.00000 5.19615i −0.742307 0.428571i
\(148\) 0 0
\(149\) 6.92820i 0.567581i −0.958886 0.283790i \(-0.908408\pi\)
0.958886 0.283790i \(-0.0915919\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i −0.990016 0.140952i \(-0.954984\pi\)
0.990016 0.140952i \(-0.0450164\pi\)
\(152\) 0 0
\(153\) 4.50000 2.59808i 0.363803 0.210042i
\(154\) 0 0
\(155\) −36.0000 −2.89159
\(156\) 0 0
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) 0 0
\(159\) −13.5000 7.79423i −1.07062 0.618123i
\(160\) 0 0
\(161\) 5.19615i 0.409514i
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) −18.0000 10.3923i −1.40130 0.809040i
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) 1.50000 + 12.9904i 0.114708 + 0.993399i
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 7.00000 0.529150
\(176\) 0 0
\(177\) 4.50000 + 2.59808i 0.338241 + 0.195283i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 13.8564i 1.02994i 0.857209 + 0.514969i \(0.172197\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(182\) 0 0
\(183\) −12.0000 6.92820i −0.887066 0.512148i
\(184\) 0 0
\(185\) −24.0000 −1.76452
\(186\) 0 0
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) 5.19615i 0.377964i
\(190\) 0 0
\(191\) 19.0526i 1.37859i 0.724479 + 0.689297i \(0.242081\pi\)
−0.724479 + 0.689297i \(0.757919\pi\)
\(192\) 0 0
\(193\) 3.46410i 0.249351i 0.992198 + 0.124676i \(0.0397891\pi\)
−0.992198 + 0.124676i \(0.960211\pi\)
\(194\) 0 0
\(195\) 9.00000 + 5.19615i 0.644503 + 0.372104i
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) −7.50000 + 12.9904i −0.529009 + 0.916271i
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 13.5000 7.79423i 0.938315 0.541736i
\(208\) 0 0
\(209\) 6.00000 13.8564i 0.415029 0.958468i
\(210\) 0 0
\(211\) 12.1244i 0.834675i 0.908752 + 0.417338i \(0.137037\pi\)
−0.908752 + 0.417338i \(0.862963\pi\)
\(212\) 0 0
\(213\) 18.0000 + 10.3923i 1.23334 + 0.712069i
\(214\) 0 0
\(215\) 6.92820i 0.472500i
\(216\) 0 0
\(217\) 10.3923i 0.705476i
\(218\) 0 0
\(219\) 16.5000 + 9.52628i 1.11497 + 0.643726i
\(220\) 0 0
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) 10.3923i 0.695920i 0.937509 + 0.347960i \(0.113126\pi\)
−0.937509 + 0.347960i \(0.886874\pi\)
\(224\) 0 0
\(225\) −10.5000 18.1865i −0.700000 1.21244i
\(226\) 0 0
\(227\) 3.00000 0.199117 0.0995585 0.995032i \(-0.468257\pi\)
0.0995585 + 0.995032i \(0.468257\pi\)
\(228\) 0 0
\(229\) −8.00000 −0.528655 −0.264327 0.964433i \(-0.585150\pi\)
−0.264327 + 0.964433i \(0.585150\pi\)
\(230\) 0 0
\(231\) −3.00000 + 5.19615i −0.197386 + 0.341882i
\(232\) 0 0
\(233\) 6.92820i 0.453882i −0.973909 0.226941i \(-0.927128\pi\)
0.973909 0.226941i \(-0.0728724\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) −6.00000 + 10.3923i −0.389742 + 0.675053i
\(238\) 0 0
\(239\) 12.1244i 0.784259i −0.919910 0.392130i \(-0.871738\pi\)
0.919910 0.392130i \(-0.128262\pi\)
\(240\) 0 0
\(241\) 13.8564i 0.892570i −0.894891 0.446285i \(-0.852747\pi\)
0.894891 0.446285i \(-0.147253\pi\)
\(242\) 0 0
\(243\) −13.5000 + 7.79423i −0.866025 + 0.500000i
\(244\) 0 0
\(245\) 20.7846i 1.32788i
\(246\) 0 0
\(247\) −3.00000 + 6.92820i −0.190885 + 0.440831i
\(248\) 0 0
\(249\) −9.00000 + 15.5885i −0.570352 + 0.987878i
\(250\) 0 0
\(251\) 13.8564i 0.874609i −0.899314 0.437304i \(-0.855933\pi\)
0.899314 0.437304i \(-0.144067\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) −9.00000 5.19615i −0.563602 0.325396i
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 6.92820i 0.430498i
\(260\) 0 0
\(261\) 13.5000 + 23.3827i 0.835629 + 1.44735i
\(262\) 0 0
\(263\) 3.46410i 0.213606i −0.994280 0.106803i \(-0.965939\pi\)
0.994280 0.106803i \(-0.0340614\pi\)
\(264\) 0 0
\(265\) 31.1769i 1.91518i
\(266\) 0 0
\(267\) 9.00000 + 5.19615i 0.550791 + 0.317999i
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 1.00000 0.0607457 0.0303728 0.999539i \(-0.490331\pi\)
0.0303728 + 0.999539i \(0.490331\pi\)
\(272\) 0 0
\(273\) 1.50000 2.59808i 0.0907841 0.157243i
\(274\) 0 0
\(275\) 24.2487i 1.46225i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) 27.0000 15.5885i 1.61645 0.933257i
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 10.0000 0.594438 0.297219 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 0 0
\(285\) 21.0000 15.5885i 1.24393 0.923381i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 14.0000 0.823529
\(290\) 0 0
\(291\) −12.0000 + 20.7846i −0.703452 + 1.21842i
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 10.3923i 0.605063i
\(296\) 0 0
\(297\) 18.0000 1.04447
\(298\) 0 0
\(299\) 9.00000 0.520483
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) −9.00000 + 15.5885i −0.517036 + 0.895533i
\(304\) 0 0
\(305\) 27.7128i 1.58683i
\(306\) 0 0
\(307\) 10.3923i 0.593120i 0.955014 + 0.296560i \(0.0958395\pi\)
−0.955014 + 0.296560i \(0.904160\pi\)
\(308\) 0 0
\(309\) −3.00000 + 5.19615i −0.170664 + 0.295599i
\(310\) 0 0
\(311\) 22.5167i 1.27680i −0.769704 0.638401i \(-0.779596\pi\)
0.769704 0.638401i \(-0.220404\pi\)
\(312\) 0 0
\(313\) 13.0000 0.734803 0.367402 0.930062i \(-0.380247\pi\)
0.367402 + 0.930062i \(0.380247\pi\)
\(314\) 0 0
\(315\) −9.00000 + 5.19615i −0.507093 + 0.292770i
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) 31.1769i 1.74557i
\(320\) 0 0
\(321\) −4.50000 2.59808i −0.251166 0.145010i
\(322\) 0 0
\(323\) 3.00000 6.92820i 0.166924 0.385496i
\(324\) 0 0
\(325\) 12.1244i 0.672538i
\(326\) 0 0
\(327\) −13.5000 + 23.3827i −0.746552 + 1.29307i
\(328\) 0 0
\(329\) 3.46410i 0.190982i
\(330\) 0 0
\(331\) 19.0526i 1.04722i 0.851957 + 0.523612i \(0.175416\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 0 0
\(333\) 18.0000 10.3923i 0.986394 0.569495i
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 0 0
\(337\) 24.2487i 1.32091i −0.750865 0.660456i \(-0.770363\pi\)
0.750865 0.660456i \(-0.229637\pi\)
\(338\) 0 0
\(339\) −18.0000 10.3923i −0.977626 0.564433i
\(340\) 0 0
\(341\) −36.0000 −1.94951
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) −27.0000 15.5885i −1.45363 0.839254i
\(346\) 0 0
\(347\) 27.7128i 1.48770i −0.668346 0.743851i \(-0.732997\pi\)
0.668346 0.743851i \(-0.267003\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) 25.9808i 1.38282i −0.722464 0.691408i \(-0.756991\pi\)
0.722464 0.691408i \(-0.243009\pi\)
\(354\) 0 0
\(355\) 41.5692i 2.20627i
\(356\) 0 0
\(357\) −1.50000 + 2.59808i −0.0793884 + 0.137505i
\(358\) 0 0
\(359\) 19.0526i 1.00556i 0.864416 + 0.502778i \(0.167689\pi\)
−0.864416 + 0.502778i \(0.832311\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) −1.50000 0.866025i −0.0787296 0.0454545i
\(364\) 0 0
\(365\) 38.1051i 1.99451i
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 0 0
\(373\) 29.4449i 1.52460i 0.647225 + 0.762299i \(0.275929\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) −6.00000 + 10.3923i −0.309839 + 0.536656i
\(376\) 0 0
\(377\) 15.5885i 0.802846i
\(378\) 0 0
\(379\) 12.1244i 0.622786i 0.950281 + 0.311393i \(0.100796\pi\)
−0.950281 + 0.311393i \(0.899204\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) −3.00000 5.19615i −0.152499 0.264135i
\(388\) 0 0
\(389\) 3.46410i 0.175637i 0.996136 + 0.0878185i \(0.0279895\pi\)
−0.996136 + 0.0878185i \(0.972010\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) −3.00000 + 5.19615i −0.151330 + 0.262111i
\(394\) 0 0
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) −4.50000 6.06218i −0.225282 0.303488i
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 27.0000 + 15.5885i 1.34164 + 0.774597i
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) 38.1051i 1.88418i −0.335365 0.942088i \(-0.608860\pi\)
0.335365 0.942088i \(-0.391140\pi\)
\(410\) 0 0
\(411\) 7.50000 12.9904i 0.369948 0.640768i
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) 21.0000 + 12.1244i 1.02837 + 0.593732i
\(418\) 0 0
\(419\) 10.3923i 0.507697i 0.967244 + 0.253849i \(0.0816965\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 12.1244i 0.590905i 0.955357 + 0.295452i \(0.0954704\pi\)
−0.955357 + 0.295452i \(0.904530\pi\)
\(422\) 0 0
\(423\) −9.00000 + 5.19615i −0.437595 + 0.252646i
\(424\) 0 0
\(425\) 12.1244i 0.588118i
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) 9.00000 + 5.19615i 0.434524 + 0.250873i
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 10.3923i 0.499422i −0.968320 0.249711i \(-0.919664\pi\)
0.968320 0.249711i \(-0.0803357\pi\)
\(434\) 0 0
\(435\) 27.0000 46.7654i 1.29455 2.24223i
\(436\) 0 0
\(437\) 9.00000 20.7846i 0.430528 0.994263i
\(438\) 0 0
\(439\) 20.7846i 0.991995i −0.868324 0.495998i \(-0.834802\pi\)
0.868324 0.495998i \(-0.165198\pi\)
\(440\) 0 0
\(441\) −9.00000 15.5885i −0.428571 0.742307i
\(442\) 0 0
\(443\) 17.3205i 0.822922i −0.911427 0.411461i \(-0.865019\pi\)
0.911427 0.411461i \(-0.134981\pi\)
\(444\) 0 0
\(445\) 20.7846i 0.985285i
\(446\) 0 0
\(447\) 6.00000 10.3923i 0.283790 0.491539i
\(448\) 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
\(453\) 3.00000 5.19615i 0.140952 0.244137i
\(454\) 0 0
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −25.0000 −1.16945 −0.584725 0.811231i \(-0.698798\pi\)
−0.584725 + 0.811231i \(0.698798\pi\)
\(458\) 0 0
\(459\) 9.00000 0.420084
\(460\) 0 0
\(461\) 13.8564i 0.645357i 0.946509 + 0.322679i \(0.104583\pi\)
−0.946509 + 0.322679i \(0.895417\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) −54.0000 31.1769i −2.50419 1.44579i
\(466\) 0 0
\(467\) 27.7128i 1.28240i 0.767375 + 0.641198i \(0.221562\pi\)
−0.767375 + 0.641198i \(0.778438\pi\)
\(468\) 0 0
\(469\) 8.66025i 0.399893i
\(470\) 0 0
\(471\) 6.00000 + 3.46410i 0.276465 + 0.159617i
\(472\) 0 0
\(473\) 6.92820i 0.318559i
\(474\) 0 0
\(475\) −28.0000 12.1244i −1.28473 0.556304i
\(476\) 0 0
\(477\) −13.5000 23.3827i −0.618123 1.07062i
\(478\) 0 0
\(479\) 10.3923i 0.474837i −0.971408 0.237418i \(-0.923699\pi\)
0.971408 0.237418i \(-0.0763012\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) 0 0
\(483\) −4.50000 + 7.79423i −0.204757 + 0.354650i
\(484\) 0 0
\(485\) 48.0000 2.17957
\(486\) 0 0
\(487\) 3.46410i 0.156973i −0.996915 0.0784867i \(-0.974991\pi\)
0.996915 0.0784867i \(-0.0250088\pi\)
\(488\) 0 0
\(489\) −15.0000 8.66025i −0.678323 0.391630i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 15.5885i 0.702069i
\(494\) 0 0
\(495\) −18.0000 31.1769i −0.809040 1.40130i
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 25.9808i 1.15842i −0.815177 0.579212i \(-0.803360\pi\)
0.815177 0.579212i \(-0.196640\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 15.0000 + 8.66025i 0.666173 + 0.384615i
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −11.0000 −0.486611
\(512\) 0 0
\(513\) −9.00000 + 20.7846i −0.397360 + 0.917663i
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) 9.00000 + 5.19615i 0.395056 + 0.228086i
\(520\) 0 0
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) 32.9090i 1.43901i −0.694488 0.719504i \(-0.744369\pi\)
0.694488 0.719504i \(-0.255631\pi\)
\(524\) 0 0
\(525\) 10.5000 + 6.06218i 0.458258 + 0.264575i
\(526\) 0 0
\(527\) −18.0000 −0.784092
\(528\) 0 0
\(529\) −4.00000 −0.173913
\(530\) 0 0
\(531\) 4.50000 + 7.79423i 0.195283 + 0.338241i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 10.3923i 0.449299i
\(536\) 0 0
\(537\) −18.0000 10.3923i −0.776757 0.448461i
\(538\) 0 0
\(539\) 20.7846i 0.895257i
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) −12.0000 + 20.7846i −0.514969 + 0.891953i
\(544\) 0 0
\(545\) 54.0000 2.31311
\(546\) 0 0
\(547\) 24.2487i 1.03680i 0.855138 + 0.518400i \(0.173472\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) −12.0000 20.7846i −0.512148 0.887066i
\(550\) 0 0
\(551\) 36.0000 + 15.5885i 1.53365 + 0.664091i
\(552\) 0 0
\(553\) 6.92820i 0.294617i
\(554\) 0 0
\(555\) −36.0000 20.7846i −1.52811 0.882258i
\(556\) 0 0
\(557\) 45.0333i 1.90812i −0.299611 0.954062i \(-0.596857\pi\)
0.299611 0.954062i \(-0.403143\pi\)
\(558\) 0 0
\(559\) 3.46410i 0.146516i
\(560\) 0 0
\(561\) −9.00000 5.19615i −0.379980 0.219382i
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 41.5692i 1.74883i
\(566\) 0 0
\(567\) 4.50000 7.79423i 0.188982 0.327327i
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) −16.5000 + 28.5788i −0.689297 + 1.19390i
\(574\) 0 0
\(575\) 36.3731i 1.51686i
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) 0 0
\(579\) −3.00000 + 5.19615i −0.124676 + 0.215945i
\(580\) 0 0
\(581\) 10.3923i 0.431145i
\(582\) 0 0
\(583\) 31.1769i 1.29122i
\(584\) 0 0
\(585\) 9.00000 + 15.5885i 0.372104 + 0.644503i
\(586\) 0 0
\(587\) 13.8564i 0.571915i −0.958242 0.285958i \(-0.907688\pi\)
0.958242 0.285958i \(-0.0923116\pi\)
\(588\) 0 0
\(589\) 18.0000 41.5692i 0.741677 1.71283i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 20.7846i 0.853522i 0.904365 + 0.426761i \(0.140345\pi\)
−0.904365 + 0.426761i \(0.859655\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 0 0
\(597\) −16.5000 9.52628i −0.675300 0.389885i
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 0 0
\(601\) 10.3923i 0.423911i −0.977279 0.211955i \(-0.932017\pi\)
0.977279 0.211955i \(-0.0679832\pi\)
\(602\) 0 0
\(603\) −22.5000 + 12.9904i −0.916271 + 0.529009i
\(604\) 0 0
\(605\) 3.46410i 0.140836i
\(606\) 0 0
\(607\) 27.7128i 1.12483i 0.826856 + 0.562414i \(0.190127\pi\)
−0.826856 + 0.562414i \(0.809873\pi\)
\(608\) 0 0
\(609\) −13.5000 7.79423i −0.547048 0.315838i
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.4974i 1.95243i 0.216799 + 0.976216i \(0.430439\pi\)
−0.216799 + 0.976216i \(0.569561\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 27.0000 1.08347
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 21.0000 15.5885i 0.838659 0.622543i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −10.5000 + 18.1865i −0.417338 + 0.722850i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 10.3923i 0.411758i
\(638\) 0 0
\(639\) 18.0000 + 31.1769i 0.712069 + 1.23334i
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 28.0000 1.10421 0.552106 0.833774i \(-0.313824\pi\)
0.552106 + 0.833774i \(0.313824\pi\)
\(644\) 0 0
\(645\) −6.00000 + 10.3923i −0.236250 + 0.409197i
\(646\) 0 0
\(647\) 1.73205i 0.0680939i 0.999420 + 0.0340470i \(0.0108396\pi\)
−0.999420 + 0.0340470i \(0.989160\pi\)
\(648\) 0 0
\(649\) 10.3923i 0.407934i
\(650\) 0 0
\(651\) −9.00000 + 15.5885i −0.352738 + 0.610960i
\(652\) 0 0
\(653\) 17.3205i 0.677804i 0.940822 + 0.338902i \(0.110055\pi\)
−0.940822 + 0.338902i \(0.889945\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) 16.5000 + 28.5788i 0.643726 + 1.11497i
\(658\) 0 0
\(659\) 33.0000 1.28550 0.642749 0.766077i \(-0.277794\pi\)
0.642749 + 0.766077i \(0.277794\pi\)
\(660\) 0 0
\(661\) 22.5167i 0.875797i 0.899025 + 0.437898i \(0.144277\pi\)
−0.899025 + 0.437898i \(0.855723\pi\)
\(662\) 0 0
\(663\) 4.50000 + 2.59808i 0.174766 + 0.100901i
\(664\) 0 0
\(665\) −6.00000 + 13.8564i −0.232670 + 0.537328i
\(666\) 0 0
\(667\) 46.7654i 1.81076i
\(668\) 0 0
\(669\) −9.00000 + 15.5885i −0.347960 + 0.602685i
\(670\) 0 0
\(671\) 27.7128i 1.06984i
\(672\) 0 0
\(673\) 48.4974i 1.86944i 0.355387 + 0.934719i \(0.384349\pi\)
−0.355387 + 0.934719i \(0.615651\pi\)
\(674\) 0 0
\(675\) 36.3731i 1.40000i
\(676\) 0 0
\(677\) −15.0000 −0.576497 −0.288248 0.957556i \(-0.593073\pi\)
−0.288248 + 0.957556i \(0.593073\pi\)
\(678\) 0 0
\(679\) 13.8564i 0.531760i
\(680\) 0 0
\(681\) 4.50000 + 2.59808i 0.172440 + 0.0995585i
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −30.0000 −1.14624
\(686\) 0 0
\(687\) −12.0000 6.92820i −0.457829 0.264327i
\(688\) 0 0
\(689\) 15.5885i 0.593873i
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 0 0
\(693\) −9.00000 + 5.19615i −0.341882 + 0.197386i
\(694\) 0 0
\(695\) 48.4974i 1.83961i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000 10.3923i 0.226941 0.393073i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 12.0000 27.7128i 0.452589 1.04521i
\(704\) 0 0
\(705\) 18.0000 + 10.3923i 0.677919 + 0.391397i
\(706\) 0 0
\(707\) 10.3923i 0.390843i
\(708\) 0 0
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −18.0000 + 10.3923i −0.675053 + 0.389742i
\(712\) 0 0
\(713\) −54.0000 −2.02232
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) 10.5000 18.1865i 0.392130 0.679189i
\(718\) 0 0
\(719\) 8.66025i 0.322973i −0.986875 0.161486i \(-0.948371\pi\)
0.986875 0.161486i \(-0.0516288\pi\)
\(720\) 0 0
\(721\) 3.46410i 0.129010i
\(722\) 0 0
\(723\) 12.0000 20.7846i 0.446285 0.772988i
\(724\) 0 0
\(725\) −63.0000 −2.33976
\(726\) 0 0
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 3.46410i 0.128124i
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) −18.0000 + 31.1769i −0.663940 + 1.14998i
\(736\) 0 0
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) −10.5000 + 7.79423i −0.385727 + 0.286328i
\(742\) 0 0
\(743\) −30.0000 −1.10059 −0.550297 0.834969i \(-0.685485\pi\)
−0.550297 + 0.834969i \(0.685485\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) −27.0000 + 15.5885i −0.987878 + 0.570352i
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) 6.92820i 0.252814i 0.991978 + 0.126407i \(0.0403445\pi\)
−0.991978 + 0.126407i \(0.959656\pi\)
\(752\) 0 0
\(753\) 12.0000 20.7846i 0.437304 0.757433i
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 16.0000 0.581530 0.290765 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 0 0
\(759\) −27.0000 15.5885i −0.980038 0.565825i
\(760\) 0 0
\(761\) 8.66025i 0.313934i 0.987604 + 0.156967i \(0.0501716\pi\)
−0.987604 + 0.156967i \(0.949828\pi\)
\(762\) 0 0
\(763\) 15.5885i 0.564340i
\(764\) 0 0
\(765\) −9.00000 15.5885i −0.325396 0.563602i
\(766\) 0 0
\(767\) 5.19615i 0.187622i
\(768\) 0 0
\(769\) 49.0000 1.76699 0.883493 0.468445i \(-0.155186\pi\)
0.883493 + 0.468445i \(0.155186\pi\)
\(770\) 0 0
\(771\) 9.00000 + 5.19615i 0.324127 + 0.187135i
\(772\) 0 0
\(773\) −3.00000 −0.107903 −0.0539513 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) 0 0
\(775\) 72.7461i 2.61312i
\(776\) 0 0
\(777\) −6.00000 + 10.3923i −0.215249 + 0.372822i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 41.5692i 1.48746i
\(782\) 0 0
\(783\) 46.7654i 1.67126i
\(784\) 0 0
\(785\) 13.8564i 0.494556i
\(786\) 0 0
\(787\) 22.5167i 0.802632i −0.915940 0.401316i \(-0.868553\pi\)
0.915940 0.401316i \(-0.131447\pi\)
\(788\) 0 0
\(789\) 3.00000 5.19615i 0.106803 0.184988i
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 13.8564i 0.492055i
\(794\) 0 0
\(795\) −27.0000 + 46.7654i −0.957591 + 1.65860i
\(796\) 0 0
\(797\) −21.0000 −0.743858 −0.371929 0.928261i \(-0.621304\pi\)
−0.371929 + 0.928261i \(0.621304\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 9.00000 + 15.5885i 0.317999 + 0.550791i
\(802\) 0 0
\(803\) 38.1051i 1.34470i
\(804\) 0 0
\(805\) 18.0000 0.634417
\(806\) 0 0
\(807\) 27.0000 + 15.5885i 0.950445 + 0.548740i
\(808\) 0 0
\(809\) 32.9090i 1.15702i −0.815676 0.578509i \(-0.803635\pi\)
0.815676 0.578509i \(-0.196365\pi\)
\(810\) 0 0
\(811\) 46.7654i 1.64215i 0.570817 + 0.821077i \(0.306626\pi\)
−0.570817 + 0.821077i \(0.693374\pi\)
\(812\) 0 0
\(813\) 1.50000 + 0.866025i 0.0526073 + 0.0303728i
\(814\) 0 0
\(815\) 34.6410i 1.21342i
\(816\) 0 0
\(817\) −8.00000 3.46410i −0.279885 0.121194i
\(818\) 0 0
\(819\) 4.50000 2.59808i 0.157243 0.0907841i
\(820\) 0 0
\(821\) 17.3205i 0.604490i 0.953230 + 0.302245i \(0.0977361\pi\)
−0.953230 + 0.302245i \(0.902264\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 0 0
\(825\) −21.0000 + 36.3731i −0.731126 + 1.26635i
\(826\) 0 0
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) 0 0
\(829\) 1.73205i 0.0601566i −0.999548 0.0300783i \(-0.990424\pi\)
0.999548 0.0300783i \(-0.00957567\pi\)
\(830\) 0 0
\(831\) −3.00000 1.73205i −0.104069 0.0600842i
\(832\) 0 0
\(833\) 10.3923i 0.360072i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) 0 0
\(839\) −42.0000 −1.45000 −0.725001 0.688748i \(-0.758161\pi\)
−0.725001 + 0.688748i \(0.758161\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) −18.0000 10.3923i −0.619953 0.357930i
\(844\) 0 0
\(845\) 34.6410i 1.19169i
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) 15.0000 + 8.66025i 0.514799 + 0.297219i
\(850\) 0 0
\(851\) −36.0000 −1.23406
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 45.0000 5.19615i 1.53897 0.177705i
\(856\) 0 0
\(857\) −24.0000 −0.819824 −0.409912 0.912125i \(-0.634441\pi\)
−0.409912 + 0.912125i \(0.634441\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 20.7846i 0.706698i
\(866\) 0 0
\(867\) 21.0000 + 12.1244i 0.713197 + 0.411765i
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) −15.0000 −0.508256
\(872\) 0 0
\(873\) −36.0000 + 20.7846i −1.21842 + 0.703452i
\(874\) 0 0
\(875\) 6.92820i 0.234216i
\(876\) 0 0
\(877\) 53.6936i 1.81310i 0.422095 + 0.906552i \(0.361295\pi\)
−0.422095 + 0.906552i \(0.638705\pi\)
\(878\) 0 0
\(879\) −31.5000 18.1865i −1.06247 0.613417i
\(880\) 0 0
\(881\) 34.6410i 1.16709i −0.812082 0.583543i \(-0.801666\pi\)
0.812082 0.583543i \(-0.198334\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 9.00000 15.5885i 0.302532 0.524000i
\(886\) 0 0
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 27.0000 + 15.5885i 0.904534 + 0.522233i
\(892\) 0 0
\(893\) −6.00000 + 13.8564i −0.200782 + 0.463687i
\(894\) 0 0
\(895\) 41.5692i 1.38951i
\(896\) 0 0
\(897\) 13.5000 + 7.79423i 0.450752 + 0.260242i
\(898\) 0 0
\(899\) 93.5307i 3.11942i
\(900\) 0 0
\(901\) 15.5885i 0.519327i
\(902\) 0 0
\(903\) 3.00000 + 1.73205i 0.0998337 + 0.0576390i
\(904\) 0 0
\(905\) 48.0000 1.59557
\(906\) 0 0
\(907\) 8.66025i 0.287559i 0.989610 + 0.143780i \(0.0459256\pi\)
−0.989610 + 0.143780i \(0.954074\pi\)
\(908\) 0 0
\(909\) −27.0000 + 15.5885i −0.895533 + 0.517036i
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) −24.0000 + 41.5692i −0.793416 + 1.37424i
\(916\) 0 0
\(917\) 3.46410i 0.114395i
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) −9.00000 + 15.5885i −0.296560 + 0.513657i
\(922\) 0 0
\(923\) 20.7846i 0.684134i
\(924\) 0 0
\(925\) 48.4974i 1.59459i
\(926\) 0 0
\(927\) −9.00000 + 5.19615i −0.295599 + 0.170664i
\(928\) 0 0
\(929\) 15.5885i 0.511441i −0.966751 0.255720i \(-0.917687\pi\)
0.966751 0.255720i \(-0.0823126\pi\)
\(930\) 0 0
\(931\) −24.0000 10.3923i −0.786568 0.340594i
\(932\) 0 0
\(933\) 19.5000 33.7750i 0.638401 1.10574i
\(934\) 0 0
\(935\) 20.7846i 0.679729i
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 19.5000 + 11.2583i 0.636358 + 0.367402i
\(940\) 0 0
\(941\) 39.0000 1.27136 0.635682 0.771951i \(-0.280719\pi\)
0.635682 + 0.771951i \(0.280719\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −18.0000 −0.585540
\(946\) 0 0
\(947\) 31.1769i 1.01311i 0.862207 + 0.506557i \(0.169082\pi\)
−0.862207 + 0.506557i \(0.830918\pi\)
\(948\) 0 0
\(949\) 19.0526i 0.618472i
\(950\) 0 0
\(951\) 4.50000 + 2.59808i 0.145922 + 0.0842484i
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 66.0000 2.13571
\(956\) 0 0
\(957\) 27.0000 46.7654i 0.872786 1.51171i
\(958\) 0 0
\(959\) 8.66025i 0.279654i
\(960\) 0 0
\(961\) −77.0000 −2.48387
\(962\) 0 0
\(963\) −4.50000 7.79423i −0.145010 0.251166i
\(964\) 0 0
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 10.5000 7.79423i 0.337309 0.250387i
\(970\) 0 0
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) −14.0000 −0.448819
\(974\) 0 0
\(975\) 10.5000 18.1865i 0.336269 0.582435i
\(976\) 0 0
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 20.7846i 0.664279i
\(980\) 0 0
\(981\) −40.5000 + 23.3827i −1.29307 + 0.746552i
\(982\) 0 0
\(983\) −60.0000 −1.91370 −0.956851 0.290578i \(-0.906153\pi\)
−0.956851 + 0.290578i \(0.906153\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.00000 5.19615i 0.0954911 0.165395i
\(988\) 0 0
\(989\) 10.3923i 0.330456i
\(990\) 0 0
\(991\) 3.46410i 0.110041i −0.998485 0.0550204i \(-0.982478\pi\)
0.998485 0.0550204i \(-0.0175224\pi\)
\(992\) 0 0
\(993\) −16.5000 + 28.5788i −0.523612 + 0.906922i
\(994\) 0 0
\(995\) 38.1051i 1.20801i
\(996\) 0 0
\(997\) −38.0000 −1.20347 −0.601736 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 0 0
\(999\) 36.0000 1.13899
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 912.2.f.e.113.2 2
3.2 odd 2 912.2.f.a.113.2 2
4.3 odd 2 114.2.b.c.113.1 yes 2
12.11 even 2 114.2.b.b.113.1 2
19.18 odd 2 912.2.f.a.113.1 2
57.56 even 2 inner 912.2.f.e.113.1 2
76.75 even 2 114.2.b.b.113.2 yes 2
228.227 odd 2 114.2.b.c.113.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.b.b.113.1 2 12.11 even 2
114.2.b.b.113.2 yes 2 76.75 even 2
114.2.b.c.113.1 yes 2 4.3 odd 2
114.2.b.c.113.2 yes 2 228.227 odd 2
912.2.f.a.113.1 2 19.18 odd 2
912.2.f.a.113.2 2 3.2 odd 2
912.2.f.e.113.1 2 57.56 even 2 inner
912.2.f.e.113.2 2 1.1 even 1 trivial