Properties

Label 966.2.f.a.461.4
Level $966$
Weight $2$
Character 966.461
Analytic conductor $7.714$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [966,2,Mod(461,966)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(966, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("966.461");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.f (of order \(2\), degree \(1\), 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: \(7.71354883526\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 461.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 966.461
Dual form 966.2.f.a.461.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.73205i q^{3} -1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{6} +(2.00000 - 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.73205i q^{3} -1.00000 q^{4} +1.73205 q^{5} -1.73205 q^{6} +(2.00000 - 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} +1.73205i q^{10} -1.73205i q^{12} +1.73205i q^{13} +(1.73205 + 2.00000i) q^{14} +3.00000i q^{15} +1.00000 q^{16} +3.46410 q^{17} -3.00000i q^{18} +6.92820i q^{19} -1.73205 q^{20} +(3.00000 + 3.46410i) q^{21} -1.00000i q^{23} +1.73205 q^{24} -2.00000 q^{25} -1.73205 q^{26} -5.19615i q^{27} +(-2.00000 + 1.73205i) q^{28} +9.00000i q^{29} -3.00000 q^{30} +10.3923i q^{31} +1.00000i q^{32} +3.46410i q^{34} +(3.46410 - 3.00000i) q^{35} +3.00000 q^{36} -5.00000 q^{37} -6.92820 q^{38} -3.00000 q^{39} -1.73205i q^{40} +12.1244 q^{41} +(-3.46410 + 3.00000i) q^{42} +1.00000 q^{43} -5.19615 q^{45} +1.00000 q^{46} +8.66025 q^{47} +1.73205i q^{48} +(1.00000 - 6.92820i) q^{49} -2.00000i q^{50} +6.00000i q^{51} -1.73205i q^{52} -12.0000i q^{53} +5.19615 q^{54} +(-1.73205 - 2.00000i) q^{56} -12.0000 q^{57} -9.00000 q^{58} -3.46410 q^{59} -3.00000i q^{60} -10.3923 q^{62} +(-6.00000 + 5.19615i) q^{63} -1.00000 q^{64} +3.00000i q^{65} -4.00000 q^{67} -3.46410 q^{68} +1.73205 q^{69} +(3.00000 + 3.46410i) q^{70} -12.0000i q^{71} +3.00000i q^{72} +6.92820i q^{73} -5.00000i q^{74} -3.46410i q^{75} -6.92820i q^{76} -3.00000i q^{78} -10.0000 q^{79} +1.73205 q^{80} +9.00000 q^{81} +12.1244i q^{82} +3.46410 q^{83} +(-3.00000 - 3.46410i) q^{84} +6.00000 q^{85} +1.00000i q^{86} -15.5885 q^{87} -3.46410 q^{89} -5.19615i q^{90} +(3.00000 + 3.46410i) q^{91} +1.00000i q^{92} -18.0000 q^{93} +8.66025i q^{94} +12.0000i q^{95} -1.73205 q^{96} -12.1244i q^{97} +(6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9} + 4 q^{16} + 12 q^{21} - 8 q^{25} - 8 q^{28} - 12 q^{30} + 12 q^{36} - 20 q^{37} - 12 q^{39} + 4 q^{43} + 4 q^{46} + 4 q^{49} - 48 q^{57} - 36 q^{58} - 24 q^{63} - 4 q^{64} - 16 q^{67} + 12 q^{70} - 40 q^{79} + 36 q^{81} - 12 q^{84} + 24 q^{85} + 12 q^{91} - 72 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/966\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(829\) \(925\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.73205i 1.00000i
\(4\) −1.00000 −0.500000
\(5\) 1.73205 0.774597 0.387298 0.921954i \(-0.373408\pi\)
0.387298 + 0.921954i \(0.373408\pi\)
\(6\) −1.73205 −0.707107
\(7\) 2.00000 1.73205i 0.755929 0.654654i
\(8\) 1.00000i 0.353553i
\(9\) −3.00000 −1.00000
\(10\) 1.73205i 0.547723i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.73205i 0.500000i
\(13\) 1.73205i 0.480384i 0.970725 + 0.240192i \(0.0772105\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 1.73205 + 2.00000i 0.462910 + 0.534522i
\(15\) 3.00000i 0.774597i
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) −1.73205 −0.387298
\(21\) 3.00000 + 3.46410i 0.654654 + 0.755929i
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 1.73205 0.353553
\(25\) −2.00000 −0.400000
\(26\) −1.73205 −0.339683
\(27\) 5.19615i 1.00000i
\(28\) −2.00000 + 1.73205i −0.377964 + 0.327327i
\(29\) 9.00000i 1.67126i 0.549294 + 0.835629i \(0.314897\pi\)
−0.549294 + 0.835629i \(0.685103\pi\)
\(30\) −3.00000 −0.547723
\(31\) 10.3923i 1.86651i 0.359211 + 0.933257i \(0.383046\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 3.46410i 0.594089i
\(35\) 3.46410 3.00000i 0.585540 0.507093i
\(36\) 3.00000 0.500000
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) −6.92820 −1.12390
\(39\) −3.00000 −0.480384
\(40\) 1.73205i 0.273861i
\(41\) 12.1244 1.89351 0.946753 0.321960i \(-0.104342\pi\)
0.946753 + 0.321960i \(0.104342\pi\)
\(42\) −3.46410 + 3.00000i −0.534522 + 0.462910i
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) −5.19615 −0.774597
\(46\) 1.00000 0.147442
\(47\) 8.66025 1.26323 0.631614 0.775283i \(-0.282393\pi\)
0.631614 + 0.775283i \(0.282393\pi\)
\(48\) 1.73205i 0.250000i
\(49\) 1.00000 6.92820i 0.142857 0.989743i
\(50\) 2.00000i 0.282843i
\(51\) 6.00000i 0.840168i
\(52\) 1.73205i 0.240192i
\(53\) 12.0000i 1.64833i −0.566352 0.824163i \(-0.691646\pi\)
0.566352 0.824163i \(-0.308354\pi\)
\(54\) 5.19615 0.707107
\(55\) 0 0
\(56\) −1.73205 2.00000i −0.231455 0.267261i
\(57\) −12.0000 −1.58944
\(58\) −9.00000 −1.18176
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 3.00000i 0.387298i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −10.3923 −1.31982
\(63\) −6.00000 + 5.19615i −0.755929 + 0.654654i
\(64\) −1.00000 −0.125000
\(65\) 3.00000i 0.372104i
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −3.46410 −0.420084
\(69\) 1.73205 0.208514
\(70\) 3.00000 + 3.46410i 0.358569 + 0.414039i
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 5.00000i 0.581238i
\(75\) 3.46410i 0.400000i
\(76\) 6.92820i 0.794719i
\(77\) 0 0
\(78\) 3.00000i 0.339683i
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.73205 0.193649
\(81\) 9.00000 1.00000
\(82\) 12.1244i 1.33891i
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) −3.00000 3.46410i −0.327327 0.377964i
\(85\) 6.00000 0.650791
\(86\) 1.00000i 0.107833i
\(87\) −15.5885 −1.67126
\(88\) 0 0
\(89\) −3.46410 −0.367194 −0.183597 0.983002i \(-0.558774\pi\)
−0.183597 + 0.983002i \(0.558774\pi\)
\(90\) 5.19615i 0.547723i
\(91\) 3.00000 + 3.46410i 0.314485 + 0.363137i
\(92\) 1.00000i 0.104257i
\(93\) −18.0000 −1.86651
\(94\) 8.66025i 0.893237i
\(95\) 12.0000i 1.23117i
\(96\) −1.73205 −0.176777
\(97\) 12.1244i 1.23104i −0.788121 0.615521i \(-0.788946\pi\)
0.788121 0.615521i \(-0.211054\pi\)
\(98\) 6.92820 + 1.00000i 0.699854 + 0.101015i
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.00000 −0.594089
\(103\) 5.19615i 0.511992i −0.966678 0.255996i \(-0.917597\pi\)
0.966678 0.255996i \(-0.0824034\pi\)
\(104\) 1.73205 0.169842
\(105\) 5.19615 + 6.00000i 0.507093 + 0.585540i
\(106\) 12.0000 1.16554
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 5.19615i 0.500000i
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 8.66025i 0.821995i
\(112\) 2.00000 1.73205i 0.188982 0.163663i
\(113\) 9.00000i 0.846649i −0.905978 0.423324i \(-0.860863\pi\)
0.905978 0.423324i \(-0.139137\pi\)
\(114\) 12.0000i 1.12390i
\(115\) 1.73205i 0.161515i
\(116\) 9.00000i 0.835629i
\(117\) 5.19615i 0.480384i
\(118\) 3.46410i 0.318896i
\(119\) 6.92820 6.00000i 0.635107 0.550019i
\(120\) 3.00000 0.273861
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 21.0000i 1.89351i
\(124\) 10.3923i 0.933257i
\(125\) −12.1244 −1.08444
\(126\) −5.19615 6.00000i −0.462910 0.534522i
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 1.73205i 0.152499i
\(130\) −3.00000 −0.263117
\(131\) −13.8564 −1.21064 −0.605320 0.795982i \(-0.706955\pi\)
−0.605320 + 0.795982i \(0.706955\pi\)
\(132\) 0 0
\(133\) 12.0000 + 13.8564i 1.04053 + 1.20150i
\(134\) 4.00000i 0.345547i
\(135\) 9.00000i 0.774597i
\(136\) 3.46410i 0.297044i
\(137\) 15.0000i 1.28154i −0.767734 0.640768i \(-0.778616\pi\)
0.767734 0.640768i \(-0.221384\pi\)
\(138\) 1.73205i 0.147442i
\(139\) 5.19615i 0.440732i −0.975417 0.220366i \(-0.929275\pi\)
0.975417 0.220366i \(-0.0707252\pi\)
\(140\) −3.46410 + 3.00000i −0.292770 + 0.253546i
\(141\) 15.0000i 1.26323i
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 15.5885i 1.29455i
\(146\) −6.92820 −0.573382
\(147\) 12.0000 + 1.73205i 0.989743 + 0.142857i
\(148\) 5.00000 0.410997
\(149\) 18.0000i 1.47462i −0.675556 0.737309i \(-0.736096\pi\)
0.675556 0.737309i \(-0.263904\pi\)
\(150\) 3.46410 0.282843
\(151\) 19.0000 1.54620 0.773099 0.634285i \(-0.218706\pi\)
0.773099 + 0.634285i \(0.218706\pi\)
\(152\) 6.92820 0.561951
\(153\) −10.3923 −0.840168
\(154\) 0 0
\(155\) 18.0000i 1.44579i
\(156\) 3.00000 0.240192
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 20.7846 1.64833
\(160\) 1.73205i 0.136931i
\(161\) −1.73205 2.00000i −0.136505 0.157622i
\(162\) 9.00000i 0.707107i
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −12.1244 −0.946753
\(165\) 0 0
\(166\) 3.46410i 0.268866i
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 3.46410 3.00000i 0.267261 0.231455i
\(169\) 10.0000 0.769231
\(170\) 6.00000i 0.460179i
\(171\) 20.7846i 1.58944i
\(172\) −1.00000 −0.0762493
\(173\) −3.46410 −0.263371 −0.131685 0.991292i \(-0.542039\pi\)
−0.131685 + 0.991292i \(0.542039\pi\)
\(174\) 15.5885i 1.18176i
\(175\) −4.00000 + 3.46410i −0.302372 + 0.261861i
\(176\) 0 0
\(177\) 6.00000i 0.450988i
\(178\) 3.46410i 0.259645i
\(179\) 21.0000i 1.56961i 0.619740 + 0.784807i \(0.287238\pi\)
−0.619740 + 0.784807i \(0.712762\pi\)
\(180\) 5.19615 0.387298
\(181\) 17.3205i 1.28742i −0.765268 0.643712i \(-0.777394\pi\)
0.765268 0.643712i \(-0.222606\pi\)
\(182\) −3.46410 + 3.00000i −0.256776 + 0.222375i
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) −8.66025 −0.636715
\(186\) 18.0000i 1.31982i
\(187\) 0 0
\(188\) −8.66025 −0.631614
\(189\) −9.00000 10.3923i −0.654654 0.755929i
\(190\) −12.0000 −0.870572
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.73205i 0.125000i
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 12.1244 0.870478
\(195\) −5.19615 −0.372104
\(196\) −1.00000 + 6.92820i −0.0714286 + 0.494872i
\(197\) 15.0000i 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 0 0
\(199\) 12.1244i 0.859473i 0.902954 + 0.429736i \(0.141394\pi\)
−0.902954 + 0.429736i \(0.858606\pi\)
\(200\) 2.00000i 0.141421i
\(201\) 6.92820i 0.488678i
\(202\) 0 0
\(203\) 15.5885 + 18.0000i 1.09410 + 1.26335i
\(204\) 6.00000i 0.420084i
\(205\) 21.0000 1.46670
\(206\) 5.19615 0.362033
\(207\) 3.00000i 0.208514i
\(208\) 1.73205i 0.120096i
\(209\) 0 0
\(210\) −6.00000 + 5.19615i −0.414039 + 0.358569i
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 20.7846 1.42414
\(214\) −18.0000 −1.23045
\(215\) 1.73205 0.118125
\(216\) −5.19615 −0.353553
\(217\) 18.0000 + 20.7846i 1.22192 + 1.41095i
\(218\) 11.0000i 0.745014i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 6.00000i 0.403604i
\(222\) 8.66025 0.581238
\(223\) 6.92820i 0.463947i 0.972722 + 0.231973i \(0.0745182\pi\)
−0.972722 + 0.231973i \(0.925482\pi\)
\(224\) 1.73205 + 2.00000i 0.115728 + 0.133631i
\(225\) 6.00000 0.400000
\(226\) 9.00000 0.598671
\(227\) −15.5885 −1.03464 −0.517321 0.855791i \(-0.673071\pi\)
−0.517321 + 0.855791i \(0.673071\pi\)
\(228\) 12.0000 0.794719
\(229\) 24.2487i 1.60240i −0.598397 0.801200i \(-0.704195\pi\)
0.598397 0.801200i \(-0.295805\pi\)
\(230\) 1.73205 0.114208
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 12.0000i 0.786146i −0.919507 0.393073i \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) 5.19615 0.339683
\(235\) 15.0000 0.978492
\(236\) 3.46410 0.225494
\(237\) 17.3205i 1.12509i
\(238\) 6.00000 + 6.92820i 0.388922 + 0.449089i
\(239\) 6.00000i 0.388108i −0.980991 0.194054i \(-0.937836\pi\)
0.980991 0.194054i \(-0.0621637\pi\)
\(240\) 3.00000i 0.193649i
\(241\) 15.5885i 1.00414i 0.864827 + 0.502070i \(0.167428\pi\)
−0.864827 + 0.502070i \(0.832572\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 15.5885i 1.00000i
\(244\) 0 0
\(245\) 1.73205 12.0000i 0.110657 0.766652i
\(246\) −21.0000 −1.33891
\(247\) −12.0000 −0.763542
\(248\) 10.3923 0.659912
\(249\) 6.00000i 0.380235i
\(250\) 12.1244i 0.766812i
\(251\) 22.5167 1.42124 0.710620 0.703577i \(-0.248415\pi\)
0.710620 + 0.703577i \(0.248415\pi\)
\(252\) 6.00000 5.19615i 0.377964 0.327327i
\(253\) 0 0
\(254\) 17.0000i 1.06667i
\(255\) 10.3923i 0.650791i
\(256\) 1.00000 0.0625000
\(257\) −6.92820 −0.432169 −0.216085 0.976375i \(-0.569329\pi\)
−0.216085 + 0.976375i \(0.569329\pi\)
\(258\) −1.73205 −0.107833
\(259\) −10.0000 + 8.66025i −0.621370 + 0.538122i
\(260\) 3.00000i 0.186052i
\(261\) 27.0000i 1.67126i
\(262\) 13.8564i 0.856052i
\(263\) 27.0000i 1.66489i 0.554107 + 0.832446i \(0.313060\pi\)
−0.554107 + 0.832446i \(0.686940\pi\)
\(264\) 0 0
\(265\) 20.7846i 1.27679i
\(266\) −13.8564 + 12.0000i −0.849591 + 0.735767i
\(267\) 6.00000i 0.367194i
\(268\) 4.00000 0.244339
\(269\) −13.8564 −0.844840 −0.422420 0.906400i \(-0.638819\pi\)
−0.422420 + 0.906400i \(0.638819\pi\)
\(270\) 9.00000 0.547723
\(271\) 20.7846i 1.26258i −0.775549 0.631288i \(-0.782527\pi\)
0.775549 0.631288i \(-0.217473\pi\)
\(272\) 3.46410 0.210042
\(273\) −6.00000 + 5.19615i −0.363137 + 0.314485i
\(274\) 15.0000 0.906183
\(275\) 0 0
\(276\) −1.73205 −0.104257
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 5.19615 0.311645
\(279\) 31.1769i 1.86651i
\(280\) −3.00000 3.46410i −0.179284 0.207020i
\(281\) 9.00000i 0.536895i −0.963294 0.268447i \(-0.913489\pi\)
0.963294 0.268447i \(-0.0865106\pi\)
\(282\) −15.0000 −0.893237
\(283\) 3.46410i 0.205919i 0.994686 + 0.102960i \(0.0328313\pi\)
−0.994686 + 0.102960i \(0.967169\pi\)
\(284\) 12.0000i 0.712069i
\(285\) −20.7846 −1.23117
\(286\) 0 0
\(287\) 24.2487 21.0000i 1.43136 1.23959i
\(288\) 3.00000i 0.176777i
\(289\) −5.00000 −0.294118
\(290\) −15.5885 −0.915386
\(291\) 21.0000 1.23104
\(292\) 6.92820i 0.405442i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −1.73205 + 12.0000i −0.101015 + 0.699854i
\(295\) −6.00000 −0.349334
\(296\) 5.00000i 0.290619i
\(297\) 0 0
\(298\) 18.0000 1.04271
\(299\) 1.73205 0.100167
\(300\) 3.46410i 0.200000i
\(301\) 2.00000 1.73205i 0.115278 0.0998337i
\(302\) 19.0000i 1.09333i
\(303\) 0 0
\(304\) 6.92820i 0.397360i
\(305\) 0 0
\(306\) 10.3923i 0.594089i
\(307\) 1.73205i 0.0988534i −0.998778 0.0494267i \(-0.984261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) 0 0
\(309\) 9.00000 0.511992
\(310\) −18.0000 −1.02233
\(311\) 10.3923 0.589294 0.294647 0.955606i \(-0.404798\pi\)
0.294647 + 0.955606i \(0.404798\pi\)
\(312\) 3.00000i 0.169842i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −10.3923 + 9.00000i −0.585540 + 0.507093i
\(316\) 10.0000 0.562544
\(317\) 3.00000i 0.168497i 0.996445 + 0.0842484i \(0.0268489\pi\)
−0.996445 + 0.0842484i \(0.973151\pi\)
\(318\) 20.7846i 1.16554i
\(319\) 0 0
\(320\) −1.73205 −0.0968246
\(321\) −31.1769 −1.74013
\(322\) 2.00000 1.73205i 0.111456 0.0965234i
\(323\) 24.0000i 1.33540i
\(324\) −9.00000 −0.500000
\(325\) 3.46410i 0.192154i
\(326\) 14.0000i 0.775388i
\(327\) 19.0526i 1.05361i
\(328\) 12.1244i 0.669456i
\(329\) 17.3205 15.0000i 0.954911 0.826977i
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −3.46410 −0.190117
\(333\) 15.0000 0.821995
\(334\) 17.3205i 0.947736i
\(335\) −6.92820 −0.378528
\(336\) 3.00000 + 3.46410i 0.163663 + 0.188982i
\(337\) 4.00000 0.217894 0.108947 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) 10.0000i 0.543928i
\(339\) 15.5885 0.846649
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 20.7846 1.12390
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 1.00000i 0.0539164i
\(345\) 3.00000 0.161515
\(346\) 3.46410i 0.186231i
\(347\) 9.00000i 0.483145i −0.970383 0.241573i \(-0.922337\pi\)
0.970383 0.241573i \(-0.0776632\pi\)
\(348\) 15.5885 0.835629
\(349\) 13.8564i 0.741716i 0.928689 + 0.370858i \(0.120936\pi\)
−0.928689 + 0.370858i \(0.879064\pi\)
\(350\) −3.46410 4.00000i −0.185164 0.213809i
\(351\) 9.00000 0.480384
\(352\) 0 0
\(353\) 5.19615 0.276563 0.138282 0.990393i \(-0.455842\pi\)
0.138282 + 0.990393i \(0.455842\pi\)
\(354\) 6.00000 0.318896
\(355\) 20.7846i 1.10313i
\(356\) 3.46410 0.183597
\(357\) 10.3923 + 12.0000i 0.550019 + 0.635107i
\(358\) −21.0000 −1.10988
\(359\) 15.0000i 0.791670i −0.918322 0.395835i \(-0.870455\pi\)
0.918322 0.395835i \(-0.129545\pi\)
\(360\) 5.19615i 0.273861i
\(361\) −29.0000 −1.52632
\(362\) 17.3205 0.910346
\(363\) 19.0526i 1.00000i
\(364\) −3.00000 3.46410i −0.157243 0.181568i
\(365\) 12.0000i 0.628109i
\(366\) 0 0
\(367\) 1.73205i 0.0904123i −0.998978 0.0452062i \(-0.985606\pi\)
0.998978 0.0452062i \(-0.0143945\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −36.3731 −1.89351
\(370\) 8.66025i 0.450225i
\(371\) −20.7846 24.0000i −1.07908 1.24602i
\(372\) 18.0000 0.933257
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 21.0000i 1.08444i
\(376\) 8.66025i 0.446619i
\(377\) −15.5885 −0.802846
\(378\) 10.3923 9.00000i 0.534522 0.462910i
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 12.0000i 0.615587i
\(381\) 29.4449i 1.50851i
\(382\) 0 0
\(383\) −27.7128 −1.41606 −0.708029 0.706183i \(-0.750416\pi\)
−0.708029 + 0.706183i \(0.750416\pi\)
\(384\) 1.73205 0.0883883
\(385\) 0 0
\(386\) 5.00000i 0.254493i
\(387\) −3.00000 −0.152499
\(388\) 12.1244i 0.615521i
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 5.19615i 0.263117i
\(391\) 3.46410i 0.175187i
\(392\) −6.92820 1.00000i −0.349927 0.0505076i
\(393\) 24.0000i 1.21064i
\(394\) 15.0000 0.755689
\(395\) −17.3205 −0.871489
\(396\) 0 0
\(397\) 13.8564i 0.695433i −0.937600 0.347717i \(-0.886957\pi\)
0.937600 0.347717i \(-0.113043\pi\)
\(398\) −12.1244 −0.607739
\(399\) −24.0000 + 20.7846i −1.20150 + 1.04053i
\(400\) −2.00000 −0.100000
\(401\) 18.0000i 0.898877i 0.893311 + 0.449439i \(0.148376\pi\)
−0.893311 + 0.449439i \(0.851624\pi\)
\(402\) 6.92820 0.345547
\(403\) −18.0000 −0.896644
\(404\) 0 0
\(405\) 15.5885 0.774597
\(406\) −18.0000 + 15.5885i −0.893325 + 0.773642i
\(407\) 0 0
\(408\) 6.00000 0.297044
\(409\) 13.8564i 0.685155i 0.939490 + 0.342578i \(0.111300\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 21.0000i 1.03712i
\(411\) 25.9808 1.28154
\(412\) 5.19615i 0.255996i
\(413\) −6.92820 + 6.00000i −0.340915 + 0.295241i
\(414\) −3.00000 −0.147442
\(415\) 6.00000 0.294528
\(416\) −1.73205 −0.0849208
\(417\) 9.00000 0.440732
\(418\) 0 0
\(419\) 3.46410 0.169232 0.0846162 0.996414i \(-0.473034\pi\)
0.0846162 + 0.996414i \(0.473034\pi\)
\(420\) −5.19615 6.00000i −0.253546 0.292770i
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 26.0000i 1.26566i
\(423\) −25.9808 −1.26323
\(424\) −12.0000 −0.582772
\(425\) −6.92820 −0.336067
\(426\) 20.7846i 1.00702i
\(427\) 0 0
\(428\) 18.0000i 0.870063i
\(429\) 0 0
\(430\) 1.73205i 0.0835269i
\(431\) 27.0000i 1.30054i −0.759701 0.650272i \(-0.774655\pi\)
0.759701 0.650272i \(-0.225345\pi\)
\(432\) 5.19615i 0.250000i
\(433\) 19.0526i 0.915608i −0.889053 0.457804i \(-0.848636\pi\)
0.889053 0.457804i \(-0.151364\pi\)
\(434\) −20.7846 + 18.0000i −0.997693 + 0.864028i
\(435\) −27.0000 −1.29455
\(436\) 11.0000 0.526804
\(437\) 6.92820 0.331421
\(438\) 12.0000i 0.573382i
\(439\) 27.7128i 1.32266i 0.750095 + 0.661330i \(0.230008\pi\)
−0.750095 + 0.661330i \(0.769992\pi\)
\(440\) 0 0
\(441\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(442\) −6.00000 −0.285391
\(443\) 9.00000i 0.427603i 0.976877 + 0.213801i \(0.0685846\pi\)
−0.976877 + 0.213801i \(0.931415\pi\)
\(444\) 8.66025i 0.410997i
\(445\) −6.00000 −0.284427
\(446\) −6.92820 −0.328060
\(447\) 31.1769 1.47462
\(448\) −2.00000 + 1.73205i −0.0944911 + 0.0818317i
\(449\) 24.0000i 1.13263i −0.824189 0.566315i \(-0.808369\pi\)
0.824189 0.566315i \(-0.191631\pi\)
\(450\) 6.00000i 0.282843i
\(451\) 0 0
\(452\) 9.00000i 0.423324i
\(453\) 32.9090i 1.54620i
\(454\) 15.5885i 0.731603i
\(455\) 5.19615 + 6.00000i 0.243599 + 0.281284i
\(456\) 12.0000i 0.561951i
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 24.2487 1.13307
\(459\) 18.0000i 0.840168i
\(460\) 1.73205i 0.0807573i
\(461\) 31.1769 1.45205 0.726027 0.687666i \(-0.241365\pi\)
0.726027 + 0.687666i \(0.241365\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 9.00000i 0.417815i
\(465\) −31.1769 −1.44579
\(466\) 12.0000 0.555889
\(467\) 19.0526 0.881647 0.440824 0.897594i \(-0.354686\pi\)
0.440824 + 0.897594i \(0.354686\pi\)
\(468\) 5.19615i 0.240192i
\(469\) −8.00000 + 6.92820i −0.369406 + 0.319915i
\(470\) 15.0000i 0.691898i
\(471\) 0 0
\(472\) 3.46410i 0.159448i
\(473\) 0 0
\(474\) 17.3205 0.795557
\(475\) 13.8564i 0.635776i
\(476\) −6.92820 + 6.00000i −0.317554 + 0.275010i
\(477\) 36.0000i 1.64833i
\(478\) 6.00000 0.274434
\(479\) 27.7128 1.26623 0.633115 0.774057i \(-0.281776\pi\)
0.633115 + 0.774057i \(0.281776\pi\)
\(480\) −3.00000 −0.136931
\(481\) 8.66025i 0.394874i
\(482\) −15.5885 −0.710035
\(483\) 3.46410 3.00000i 0.157622 0.136505i
\(484\) −11.0000 −0.500000
\(485\) 21.0000i 0.953561i
\(486\) −15.5885 −0.707107
\(487\) −29.0000 −1.31412 −0.657058 0.753840i \(-0.728199\pi\)
−0.657058 + 0.753840i \(0.728199\pi\)
\(488\) 0 0
\(489\) 24.2487i 1.09656i
\(490\) 12.0000 + 1.73205i 0.542105 + 0.0782461i
\(491\) 36.0000i 1.62466i −0.583200 0.812329i \(-0.698200\pi\)
0.583200 0.812329i \(-0.301800\pi\)
\(492\) 21.0000i 0.946753i
\(493\) 31.1769i 1.40414i
\(494\) 12.0000i 0.539906i
\(495\) 0 0
\(496\) 10.3923i 0.466628i
\(497\) −20.7846 24.0000i −0.932317 1.07655i
\(498\) −6.00000 −0.268866
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 12.1244 0.542218
\(501\) 30.0000i 1.34030i
\(502\) 22.5167i 1.00497i
\(503\) 20.7846 0.926740 0.463370 0.886165i \(-0.346640\pi\)
0.463370 + 0.886165i \(0.346640\pi\)
\(504\) 5.19615 + 6.00000i 0.231455 + 0.267261i
\(505\) 0 0
\(506\) 0 0
\(507\) 17.3205i 0.769231i
\(508\) −17.0000 −0.754253
\(509\) −24.2487 −1.07481 −0.537403 0.843326i \(-0.680594\pi\)
−0.537403 + 0.843326i \(0.680594\pi\)
\(510\) −10.3923 −0.460179
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 1.00000i 0.0441942i
\(513\) 36.0000 1.58944
\(514\) 6.92820i 0.305590i
\(515\) 9.00000i 0.396587i
\(516\) 1.73205i 0.0762493i
\(517\) 0 0
\(518\) −8.66025 10.0000i −0.380510 0.439375i
\(519\) 6.00000i 0.263371i
\(520\) 3.00000 0.131559
\(521\) −10.3923 −0.455295 −0.227648 0.973744i \(-0.573103\pi\)
−0.227648 + 0.973744i \(0.573103\pi\)
\(522\) 27.0000 1.18176
\(523\) 3.46410i 0.151475i 0.997128 + 0.0757373i \(0.0241310\pi\)
−0.997128 + 0.0757373i \(0.975869\pi\)
\(524\) 13.8564 0.605320
\(525\) −6.00000 6.92820i −0.261861 0.302372i
\(526\) −27.0000 −1.17726
\(527\) 36.0000i 1.56818i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 20.7846 0.902826
\(531\) 10.3923 0.450988
\(532\) −12.0000 13.8564i −0.520266 0.600751i
\(533\) 21.0000i 0.909611i
\(534\) 6.00000 0.259645
\(535\) 31.1769i 1.34790i
\(536\) 4.00000i 0.172774i
\(537\) −36.3731 −1.56961
\(538\) 13.8564i 0.597392i
\(539\) 0 0
\(540\) 9.00000i 0.387298i
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 20.7846 0.892775
\(543\) 30.0000 1.28742
\(544\) 3.46410i 0.148522i
\(545\) −19.0526 −0.816122
\(546\) −5.19615 6.00000i −0.222375 0.256776i
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) 15.0000i 0.640768i
\(549\) 0 0
\(550\) 0 0
\(551\) −62.3538 −2.65636
\(552\) 1.73205i 0.0737210i
\(553\) −20.0000 + 17.3205i −0.850487 + 0.736543i
\(554\) 10.0000i 0.424859i
\(555\) 15.0000i 0.636715i
\(556\) 5.19615i 0.220366i
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 31.1769 1.31982
\(559\) 1.73205i 0.0732579i
\(560\) 3.46410 3.00000i 0.146385 0.126773i
\(561\) 0 0
\(562\) 9.00000 0.379642
\(563\) −32.9090 −1.38695 −0.693474 0.720482i \(-0.743921\pi\)
−0.693474 + 0.720482i \(0.743921\pi\)
\(564\) 15.0000i 0.631614i
\(565\) 15.5885i 0.655811i
\(566\) −3.46410 −0.145607
\(567\) 18.0000 15.5885i 0.755929 0.654654i
\(568\) −12.0000 −0.503509
\(569\) 9.00000i 0.377300i 0.982044 + 0.188650i \(0.0604111\pi\)
−0.982044 + 0.188650i \(0.939589\pi\)
\(570\) 20.7846i 0.870572i
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 21.0000 + 24.2487i 0.876523 + 1.01212i
\(575\) 2.00000i 0.0834058i
\(576\) 3.00000 0.125000
\(577\) 13.8564i 0.576850i −0.957503 0.288425i \(-0.906868\pi\)
0.957503 0.288425i \(-0.0931316\pi\)
\(578\) 5.00000i 0.207973i
\(579\) 8.66025i 0.359908i
\(580\) 15.5885i 0.647275i
\(581\) 6.92820 6.00000i 0.287430 0.248922i
\(582\) 21.0000i 0.870478i
\(583\) 0 0
\(584\) 6.92820 0.286691
\(585\) 9.00000i 0.372104i
\(586\) 0 0
\(587\) −38.1051 −1.57277 −0.786383 0.617739i \(-0.788049\pi\)
−0.786383 + 0.617739i \(0.788049\pi\)
\(588\) −12.0000 1.73205i −0.494872 0.0714286i
\(589\) −72.0000 −2.96671
\(590\) 6.00000i 0.247016i
\(591\) 25.9808 1.06871
\(592\) −5.00000 −0.205499
\(593\) 5.19615 0.213380 0.106690 0.994292i \(-0.465975\pi\)
0.106690 + 0.994292i \(0.465975\pi\)
\(594\) 0 0
\(595\) 12.0000 10.3923i 0.491952 0.426043i
\(596\) 18.0000i 0.737309i
\(597\) −21.0000 −0.859473
\(598\) 1.73205i 0.0708288i
\(599\) 18.0000i 0.735460i −0.929933 0.367730i \(-0.880135\pi\)
0.929933 0.367730i \(-0.119865\pi\)
\(600\) −3.46410 −0.141421
\(601\) 24.2487i 0.989126i 0.869142 + 0.494563i \(0.164672\pi\)
−0.869142 + 0.494563i \(0.835328\pi\)
\(602\) 1.73205 + 2.00000i 0.0705931 + 0.0815139i
\(603\) 12.0000 0.488678
\(604\) −19.0000 −0.773099
\(605\) 19.0526 0.774597
\(606\) 0 0
\(607\) 17.3205i 0.703018i 0.936185 + 0.351509i \(0.114331\pi\)
−0.936185 + 0.351509i \(0.885669\pi\)
\(608\) −6.92820 −0.280976
\(609\) −31.1769 + 27.0000i −1.26335 + 1.09410i
\(610\) 0 0
\(611\) 15.0000i 0.606835i
\(612\) 10.3923 0.420084
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) 1.73205 0.0698999
\(615\) 36.3731i 1.46670i
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 9.00000i 0.362033i
\(619\) 6.92820i 0.278468i 0.990260 + 0.139234i \(0.0444640\pi\)
−0.990260 + 0.139234i \(0.955536\pi\)
\(620\) 18.0000i 0.722897i
\(621\) −5.19615 −0.208514
\(622\) 10.3923i 0.416693i
\(623\) −6.92820 + 6.00000i −0.277573 + 0.240385i
\(624\) −3.00000 −0.120096
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −17.3205 −0.690614
\(630\) −9.00000 10.3923i −0.358569 0.414039i
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 45.0333i 1.78991i
\(634\) −3.00000 −0.119145
\(635\) 29.4449 1.16848
\(636\) −20.7846 −0.824163
\(637\) 12.0000 + 1.73205i 0.475457 + 0.0686264i
\(638\) 0 0
\(639\) 36.0000i 1.42414i
\(640\) 1.73205i 0.0684653i
\(641\) 27.0000i 1.06644i −0.845978 0.533218i \(-0.820983\pi\)
0.845978 0.533218i \(-0.179017\pi\)
\(642\) 31.1769i 1.23045i
\(643\) 38.1051i 1.50272i −0.659893 0.751360i \(-0.729398\pi\)
0.659893 0.751360i \(-0.270602\pi\)
\(644\) 1.73205 + 2.00000i 0.0682524 + 0.0788110i
\(645\) 3.00000i 0.118125i
\(646\) −24.0000 −0.944267
\(647\) 17.3205 0.680939 0.340470 0.940255i \(-0.389414\pi\)
0.340470 + 0.940255i \(0.389414\pi\)
\(648\) 9.00000i 0.353553i
\(649\) 0 0
\(650\) 3.46410 0.135873
\(651\) −36.0000 + 31.1769i −1.41095 + 1.22192i
\(652\) −14.0000 −0.548282
\(653\) 21.0000i 0.821794i −0.911682 0.410897i \(-0.865216\pi\)
0.911682 0.410897i \(-0.134784\pi\)
\(654\) 19.0526 0.745014
\(655\) −24.0000 −0.937758
\(656\) 12.1244 0.473377
\(657\) 20.7846i 0.810885i
\(658\) 15.0000 + 17.3205i 0.584761 + 0.675224i
\(659\) 6.00000i 0.233727i 0.993148 + 0.116863i \(0.0372840\pi\)
−0.993148 + 0.116863i \(0.962716\pi\)
\(660\) 0 0
\(661\) 6.92820i 0.269476i −0.990881 0.134738i \(-0.956981\pi\)
0.990881 0.134738i \(-0.0430193\pi\)
\(662\) 10.0000i 0.388661i
\(663\) −10.3923 −0.403604
\(664\) 3.46410i 0.134433i
\(665\) 20.7846 + 24.0000i 0.805993 + 0.930680i
\(666\) 15.0000i 0.581238i
\(667\) 9.00000 0.348481
\(668\) −17.3205 −0.670151
\(669\) −12.0000 −0.463947
\(670\) 6.92820i 0.267660i
\(671\) 0 0
\(672\) −3.46410 + 3.00000i −0.133631 + 0.115728i
\(673\) 25.0000 0.963679 0.481840 0.876259i \(-0.339969\pi\)
0.481840 + 0.876259i \(0.339969\pi\)
\(674\) 4.00000i 0.154074i
\(675\) 10.3923i 0.400000i
\(676\) −10.0000 −0.384615
\(677\) 27.7128 1.06509 0.532545 0.846402i \(-0.321236\pi\)
0.532545 + 0.846402i \(0.321236\pi\)
\(678\) 15.5885i 0.598671i
\(679\) −21.0000 24.2487i −0.805906 0.930580i
\(680\) 6.00000i 0.230089i
\(681\) 27.0000i 1.03464i
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 20.7846i 0.794719i
\(685\) 25.9808i 0.992674i
\(686\) 15.5885 10.0000i 0.595170 0.381802i
\(687\) 42.0000 1.60240
\(688\) 1.00000 0.0381246
\(689\) 20.7846 0.791831
\(690\) 3.00000i 0.114208i
\(691\) 43.3013i 1.64726i 0.567129 + 0.823629i \(0.308054\pi\)
−0.567129 + 0.823629i \(0.691946\pi\)
\(692\) 3.46410 0.131685
\(693\) 0 0
\(694\) 9.00000 0.341635
\(695\) 9.00000i 0.341389i
\(696\) 15.5885i 0.590879i
\(697\) 42.0000 1.59086
\(698\) −13.8564 −0.524473
\(699\) 20.7846 0.786146
\(700\) 4.00000 3.46410i 0.151186 0.130931i
\(701\) 6.00000i 0.226617i 0.993560 + 0.113308i \(0.0361448\pi\)
−0.993560 + 0.113308i \(0.963855\pi\)
\(702\) 9.00000i 0.339683i
\(703\) 34.6410i 1.30651i
\(704\) 0 0
\(705\) 25.9808i 0.978492i
\(706\) 5.19615i 0.195560i
\(707\) 0 0
\(708\) 6.00000i 0.225494i
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 20.7846 0.780033
\(711\) 30.0000 1.12509
\(712\) 3.46410i 0.129823i
\(713\) 10.3923 0.389195
\(714\) −12.0000 + 10.3923i −0.449089 + 0.388922i
\(715\) 0 0
\(716\) 21.0000i 0.784807i
\(717\) 10.3923 0.388108
\(718\) 15.0000 0.559795
\(719\) −12.1244 −0.452162 −0.226081 0.974108i \(-0.572591\pi\)
−0.226081 + 0.974108i \(0.572591\pi\)
\(720\) −5.19615 −0.193649
\(721\) −9.00000 10.3923i −0.335178 0.387030i
\(722\) 29.0000i 1.07927i
\(723\) −27.0000 −1.00414
\(724\) 17.3205i 0.643712i
\(725\) 18.0000i 0.668503i
\(726\) −19.0526 −0.707107
\(727\) 31.1769i 1.15629i −0.815935 0.578144i \(-0.803777\pi\)
0.815935 0.578144i \(-0.196223\pi\)
\(728\) 3.46410 3.00000i 0.128388 0.111187i
\(729\) −27.0000 −1.00000
\(730\) −12.0000 −0.444140
\(731\) 3.46410 0.128124
\(732\) 0 0
\(733\) 3.46410i 0.127950i −0.997952 0.0639748i \(-0.979622\pi\)
0.997952 0.0639748i \(-0.0203777\pi\)
\(734\) 1.73205 0.0639312
\(735\) 20.7846 + 3.00000i 0.766652 + 0.110657i
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 36.3731i 1.33891i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 8.66025 0.318357
\(741\) 20.7846i 0.763542i
\(742\) 24.0000 20.7846i 0.881068 0.763027i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 18.0000i 0.659912i
\(745\) 31.1769i 1.14223i
\(746\) 10.0000i 0.366126i
\(747\) −10.3923 −0.380235
\(748\) 0 0
\(749\) 31.1769 + 36.0000i 1.13918 + 1.31541i
\(750\) 21.0000 0.766812
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 8.66025 0.315807
\(753\) 39.0000i 1.42124i
\(754\) 15.5885i 0.567698i
\(755\) 32.9090 1.19768
\(756\) 9.00000 + 10.3923i 0.327327 + 0.377964i
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 11.0000i 0.399538i
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −27.7128 −1.00459 −0.502294 0.864697i \(-0.667511\pi\)
−0.502294 + 0.864697i \(0.667511\pi\)
\(762\) −29.4449 −1.06667
\(763\) −22.0000 + 19.0526i −0.796453 + 0.689749i
\(764\) 0 0
\(765\) −18.0000 −0.650791
\(766\) 27.7128i 1.00130i
\(767\) 6.00000i 0.216647i
\(768\) 1.73205i 0.0625000i
\(769\) 8.66025i 0.312297i 0.987734 + 0.156148i \(0.0499078\pi\)
−0.987734 + 0.156148i \(0.950092\pi\)
\(770\) 0 0
\(771\) 12.0000i 0.432169i
\(772\) 5.00000 0.179954
\(773\) 1.73205 0.0622975 0.0311488 0.999515i \(-0.490083\pi\)
0.0311488 + 0.999515i \(0.490083\pi\)
\(774\) 3.00000i 0.107833i
\(775\) 20.7846i 0.746605i
\(776\) −12.1244 −0.435239
\(777\) −15.0000 17.3205i −0.538122 0.621370i
\(778\) 0 0
\(779\) 84.0000i 3.00961i
\(780\) 5.19615 0.186052
\(781\) 0 0
\(782\) 3.46410 0.123876
\(783\) 46.7654 1.67126
\(784\) 1.00000 6.92820i 0.0357143 0.247436i
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) 17.3205i 0.617409i −0.951158 0.308705i \(-0.900105\pi\)
0.951158 0.308705i \(-0.0998955\pi\)
\(788\) 15.0000i 0.534353i
\(789\) −46.7654 −1.66489
\(790\) 17.3205i 0.616236i
\(791\) −15.5885 18.0000i −0.554262 0.640006i
\(792\) 0 0
\(793\) 0 0
\(794\) 13.8564 0.491745
\(795\) 36.0000 1.27679
\(796\) 12.1244i 0.429736i
\(797\) −15.5885 −0.552171 −0.276086 0.961133i \(-0.589037\pi\)
−0.276086 + 0.961133i \(0.589037\pi\)
\(798\) −20.7846 24.0000i −0.735767 0.849591i
\(799\) 30.0000 1.06132
\(800\) 2.00000i 0.0707107i
\(801\) 10.3923 0.367194
\(802\) −18.0000 −0.635602
\(803\) 0 0
\(804\) 6.92820i 0.244339i
\(805\) −3.00000 3.46410i −0.105736 0.122094i
\(806\) 18.0000i 0.634023i
\(807\) 24.0000i 0.844840i
\(808\) 0 0
\(809\) 48.0000i 1.68759i −0.536666 0.843795i \(-0.680316\pi\)
0.536666 0.843795i \(-0.319684\pi\)
\(810\) 15.5885i 0.547723i
\(811\) 1.73205i 0.0608205i −0.999538 0.0304103i \(-0.990319\pi\)
0.999538 0.0304103i \(-0.00968138\pi\)
\(812\) −15.5885 18.0000i −0.547048 0.631676i
\(813\) 36.0000 1.26258
\(814\) 0 0
\(815\) 24.2487 0.849395
\(816\) 6.00000i 0.210042i
\(817\) 6.92820i 0.242387i
\(818\) −13.8564 −0.484478
\(819\) −9.00000 10.3923i −0.314485 0.363137i
\(820\) −21.0000 −0.733352
\(821\) 6.00000i 0.209401i −0.994504 0.104701i \(-0.966612\pi\)
0.994504 0.104701i \(-0.0333885\pi\)
\(822\) 25.9808i 0.906183i
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) −5.19615 −0.181017
\(825\) 0 0
\(826\) −6.00000 6.92820i −0.208767 0.241063i
\(827\) 30.0000i 1.04320i −0.853189 0.521601i \(-0.825335\pi\)
0.853189 0.521601i \(-0.174665\pi\)
\(828\) 3.00000i 0.104257i
\(829\) 6.92820i 0.240626i −0.992736 0.120313i \(-0.961610\pi\)
0.992736 0.120313i \(-0.0383899\pi\)
\(830\) 6.00000i 0.208263i
\(831\) 17.3205i 0.600842i
\(832\) 1.73205i 0.0600481i
\(833\) 3.46410 24.0000i 0.120024 0.831551i
\(834\) 9.00000i 0.311645i
\(835\) 30.0000 1.03819
\(836\) 0 0
\(837\) 54.0000 1.86651
\(838\) 3.46410i 0.119665i
\(839\) 45.0333 1.55472 0.777361 0.629054i \(-0.216558\pi\)
0.777361 + 0.629054i \(0.216558\pi\)
\(840\) 6.00000 5.19615i 0.207020 0.179284i
\(841\) −52.0000 −1.79310
\(842\) 19.0000i 0.654783i
\(843\) 15.5885 0.536895
\(844\) 26.0000 0.894957
\(845\) 17.3205 0.595844
\(846\) 25.9808i 0.893237i
\(847\) 22.0000 19.0526i 0.755929 0.654654i
\(848\) 12.0000i 0.412082i
\(849\) −6.00000 −0.205919
\(850\) 6.92820i 0.237635i
\(851\) 5.00000i 0.171398i
\(852\) −20.7846 −0.712069
\(853\) 22.5167i 0.770956i 0.922717 + 0.385478i \(0.125963\pi\)
−0.922717 + 0.385478i \(0.874037\pi\)
\(854\) 0 0
\(855\) 36.0000i 1.23117i
\(856\) 18.0000 0.615227
\(857\) 50.2295 1.71581 0.857903 0.513812i \(-0.171767\pi\)
0.857903 + 0.513812i \(0.171767\pi\)
\(858\) 0 0
\(859\) 8.66025i 0.295484i −0.989026 0.147742i \(-0.952799\pi\)
0.989026 0.147742i \(-0.0472005\pi\)
\(860\) −1.73205 −0.0590624
\(861\) 36.3731 + 42.0000i 1.23959 + 1.43136i
\(862\) 27.0000 0.919624
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 5.19615 0.176777
\(865\) −6.00000 −0.204006
\(866\) 19.0526 0.647432
\(867\) 8.66025i 0.294118i
\(868\) −18.0000 20.7846i −0.610960 0.705476i
\(869\) 0 0
\(870\) 27.0000i 0.915386i
\(871\) 6.92820i 0.234753i
\(872\) 11.0000i 0.372507i
\(873\) 36.3731i 1.23104i
\(874\) 6.92820i 0.234350i
\(875\) −24.2487 + 21.0000i −0.819756 + 0.709930i
\(876\) 12.0000 0.405442
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −27.7128 −0.935262
\(879\) 0 0
\(880\) 0 0
\(881\) −3.46410 −0.116709 −0.0583543 0.998296i \(-0.518585\pi\)
−0.0583543 + 0.998296i \(0.518585\pi\)
\(882\) −20.7846 3.00000i −0.699854 0.101015i
\(883\) 10.0000 0.336527 0.168263 0.985742i \(-0.446184\pi\)
0.168263 + 0.985742i \(0.446184\pi\)
\(884\) 6.00000i 0.201802i
\(885\) 10.3923i 0.349334i
\(886\) −9.00000 −0.302361
\(887\) 51.9615 1.74470 0.872349 0.488884i \(-0.162596\pi\)
0.872349 + 0.488884i \(0.162596\pi\)
\(888\) −8.66025 −0.290619
\(889\) 34.0000 29.4449i 1.14032 0.987549i
\(890\) 6.00000i 0.201120i
\(891\) 0 0
\(892\) 6.92820i 0.231973i
\(893\) 60.0000i 2.00782i
\(894\) 31.1769i 1.04271i
\(895\) 36.3731i 1.21582i
\(896\) −1.73205 2.00000i −0.0578638 0.0668153i
\(897\) 3.00000i 0.100167i
\(898\) 24.0000 0.800890
\(899\) −93.5307 −3.11942
\(900\) −6.00000 −0.200000
\(901\) 41.5692i 1.38487i
\(902\) 0 0
\(903\) 3.00000 + 3.46410i 0.0998337 + 0.115278i
\(904\) −9.00000 −0.299336
\(905\) 30.0000i 0.997234i
\(906\) −32.9090 −1.09333
\(907\) 37.0000 1.22856 0.614282 0.789086i \(-0.289446\pi\)
0.614282 + 0.789086i \(0.289446\pi\)
\(908\) 15.5885 0.517321
\(909\) 0 0
\(910\) −6.00000 + 5.19615i −0.198898 + 0.172251i
\(911\) 9.00000i 0.298183i 0.988823 + 0.149092i \(0.0476349\pi\)
−0.988823 + 0.149092i \(0.952365\pi\)
\(912\) −12.0000 −0.397360
\(913\) 0 0
\(914\) 22.0000i 0.727695i
\(915\) 0 0
\(916\) 24.2487i 0.801200i
\(917\) −27.7128 + 24.0000i −0.915158 + 0.792550i
\(918\) 18.0000 0.594089
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) −1.73205 −0.0571040
\(921\) 3.00000 0.0988534
\(922\) 31.1769i 1.02676i
\(923\) 20.7846 0.684134
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 17.0000i 0.558655i
\(927\) 15.5885i 0.511992i
\(928\) −9.00000 −0.295439
\(929\) −50.2295 −1.64798 −0.823988 0.566608i \(-0.808256\pi\)
−0.823988 + 0.566608i \(0.808256\pi\)
\(930\) 31.1769i 1.02233i
\(931\) 48.0000 + 6.92820i 1.57314 + 0.227063i
\(932\) 12.0000i 0.393073i
\(933\) 18.0000i 0.589294i
\(934\) 19.0526i 0.623419i
\(935\) 0 0
\(936\) −5.19615 −0.169842
\(937\) 5.19615i 0.169751i 0.996392 + 0.0848755i \(0.0270492\pi\)
−0.996392 + 0.0848755i \(0.972951\pi\)
\(938\) −6.92820 8.00000i −0.226214 0.261209i
\(939\) 0 0
\(940\) −15.0000 −0.489246
\(941\) 39.8372 1.29865 0.649327 0.760509i \(-0.275051\pi\)
0.649327 + 0.760509i \(0.275051\pi\)
\(942\) 0 0
\(943\) 12.1244i 0.394823i
\(944\) −3.46410 −0.112747
\(945\) −15.5885 18.0000i −0.507093 0.585540i
\(946\) 0 0
\(947\) 27.0000i 0.877382i 0.898638 + 0.438691i \(0.144558\pi\)
−0.898638 + 0.438691i \(0.855442\pi\)
\(948\) 17.3205i 0.562544i
\(949\) −12.0000 −0.389536
\(950\) 13.8564 0.449561
\(951\) −5.19615 −0.168497
\(952\) −6.00000 6.92820i −0.194461 0.224544i
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) −36.0000 −1.16554
\(955\) 0 0
\(956\) 6.00000i 0.194054i
\(957\) 0 0
\(958\) 27.7128i 0.895360i
\(959\) −25.9808 30.0000i −0.838963 0.968751i
\(960\) 3.00000i 0.0968246i
\(961\) −77.0000 −2.48387
\(962\) 8.66025 0.279218
\(963\) 54.0000i 1.74013i
\(964\) 15.5885i 0.502070i
\(965\) −8.66025 −0.278783
\(966\) 3.00000 + 3.46410i 0.0965234 + 0.111456i
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 11.0000i 0.353553i
\(969\) −41.5692 −1.33540
\(970\) 21.0000 0.674269
\(971\) −10.3923 −0.333505 −0.166752 0.985999i \(-0.553328\pi\)
−0.166752 + 0.985999i \(0.553328\pi\)
\(972\) 15.5885i 0.500000i
\(973\) −9.00000 10.3923i −0.288527 0.333162i
\(974\) 29.0000i 0.929220i
\(975\) 6.00000 0.192154
\(976\) 0 0
\(977\) 57.0000i 1.82359i −0.410644 0.911796i \(-0.634696\pi\)
0.410644 0.911796i \(-0.365304\pi\)
\(978\) −24.2487 −0.775388
\(979\) 0 0
\(980\) −1.73205 + 12.0000i −0.0553283 + 0.383326i
\(981\) 33.0000 1.05361
\(982\) 36.0000 1.14881
\(983\) 17.3205 0.552438 0.276219 0.961095i \(-0.410918\pi\)
0.276219 + 0.961095i \(0.410918\pi\)
\(984\) 21.0000 0.669456
\(985\) 25.9808i 0.827816i
\(986\) −31.1769 −0.992875
\(987\) 25.9808 + 30.0000i 0.826977 + 0.954911i
\(988\) 12.0000 0.381771
\(989\) 1.00000i 0.0317982i
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −10.3923 −0.329956
\(993\) 17.3205i 0.549650i
\(994\) 24.0000 20.7846i 0.761234 0.659248i
\(995\) 21.0000i 0.665745i
\(996\) 6.00000i 0.190117i
\(997\) 6.92820i 0.219418i −0.993964 0.109709i \(-0.965008\pi\)
0.993964 0.109709i \(-0.0349920\pi\)
\(998\) 4.00000i 0.126618i
\(999\) 25.9808i 0.821995i
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 966.2.f.a.461.4 yes 4
3.2 odd 2 inner 966.2.f.a.461.2 yes 4
7.6 odd 2 inner 966.2.f.a.461.3 yes 4
21.20 even 2 inner 966.2.f.a.461.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.f.a.461.1 4 21.20 even 2 inner
966.2.f.a.461.2 yes 4 3.2 odd 2 inner
966.2.f.a.461.3 yes 4 7.6 odd 2 inner
966.2.f.a.461.4 yes 4 1.1 even 1 trivial