Properties

Label 1232.2.q.a.529.1
Level $1232$
Weight $2$
Character 1232.529
Analytic conductor $9.838$
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] = [1232,2,Mod(177,1232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1232, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1232.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1232.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.83756952902\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 529.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1232.529
Dual form 1232.2.q.a.177.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{3} +(2.00000 + 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-3.00000 - 5.19615i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{3} +(2.00000 + 3.46410i) q^{5} +(2.50000 + 0.866025i) q^{7} +(-3.00000 - 5.19615i) q^{9} +(-0.500000 + 0.866025i) q^{11} -1.00000 q^{13} -12.0000 q^{15} +(-1.00000 + 1.73205i) q^{17} +(3.00000 + 5.19615i) q^{19} +(-6.00000 + 5.19615i) q^{21} +(-1.00000 - 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +9.00000 q^{27} +1.00000 q^{29} +(2.00000 - 3.46410i) q^{31} +(-1.50000 - 2.59808i) q^{33} +(2.00000 + 10.3923i) q^{35} +(1.00000 + 1.73205i) q^{37} +(1.50000 - 2.59808i) q^{39} -2.00000 q^{41} -4.00000 q^{43} +(12.0000 - 20.7846i) q^{45} +(1.00000 + 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(-3.00000 - 5.19615i) q^{51} +(6.00000 - 10.3923i) q^{53} -4.00000 q^{55} -18.0000 q^{57} +(4.50000 - 7.79423i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-3.00000 - 15.5885i) q^{63} +(-2.00000 - 3.46410i) q^{65} +(-4.50000 + 7.79423i) q^{67} +6.00000 q^{69} -4.00000 q^{71} +(1.00000 - 1.73205i) q^{73} +(-16.5000 - 28.5788i) q^{75} +(-2.00000 + 1.73205i) q^{77} +(-7.50000 - 12.9904i) q^{79} +(-4.50000 + 7.79423i) q^{81} +6.00000 q^{83} -8.00000 q^{85} +(-1.50000 + 2.59808i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(-2.50000 - 0.866025i) q^{91} +(6.00000 + 10.3923i) q^{93} +(-12.0000 + 20.7846i) q^{95} -5.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 4 q^{5} + 5 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 4 q^{5} + 5 q^{7} - 6 q^{9} - q^{11} - 2 q^{13} - 24 q^{15} - 2 q^{17} + 6 q^{19} - 12 q^{21} - 2 q^{23} - 11 q^{25} + 18 q^{27} + 2 q^{29} + 4 q^{31} - 3 q^{33} + 4 q^{35} + 2 q^{37} + 3 q^{39} - 4 q^{41} - 8 q^{43} + 24 q^{45} + 2 q^{47} + 11 q^{49} - 6 q^{51} + 12 q^{53} - 8 q^{55} - 36 q^{57} + 9 q^{59} + 5 q^{61} - 6 q^{63} - 4 q^{65} - 9 q^{67} + 12 q^{69} - 8 q^{71} + 2 q^{73} - 33 q^{75} - 4 q^{77} - 15 q^{79} - 9 q^{81} + 12 q^{83} - 16 q^{85} - 3 q^{87} - 6 q^{89} - 5 q^{91} + 12 q^{93} - 24 q^{95} - 10 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).

\(n\) \(309\) \(353\) \(463\) \(673\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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 + 2.59808i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 2.00000 + 3.46410i 0.894427 + 1.54919i 0.834512 + 0.550990i \(0.185750\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) 2.50000 + 0.866025i 0.944911 + 0.327327i
\(8\) 0 0
\(9\) −3.00000 5.19615i −1.00000 1.73205i
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.150756 + 0.261116i
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −12.0000 −3.09839
\(16\) 0 0
\(17\) −1.00000 + 1.73205i −0.242536 + 0.420084i −0.961436 0.275029i \(-0.911312\pi\)
0.718900 + 0.695113i \(0.244646\pi\)
\(18\) 0 0
\(19\) 3.00000 + 5.19615i 0.688247 + 1.19208i 0.972404 + 0.233301i \(0.0749529\pi\)
−0.284157 + 0.958778i \(0.591714\pi\)
\(20\) 0 0
\(21\) −6.00000 + 5.19615i −1.30931 + 1.13389i
\(22\) 0 0
\(23\) −1.00000 1.73205i −0.208514 0.361158i 0.742732 0.669588i \(-0.233529\pi\)
−0.951247 + 0.308431i \(0.900196\pi\)
\(24\) 0 0
\(25\) −5.50000 + 9.52628i −1.10000 + 1.90526i
\(26\) 0 0
\(27\) 9.00000 1.73205
\(28\) 0 0
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 2.00000 3.46410i 0.359211 0.622171i −0.628619 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) −1.50000 2.59808i −0.261116 0.452267i
\(34\) 0 0
\(35\) 2.00000 + 10.3923i 0.338062 + 1.75662i
\(36\) 0 0
\(37\) 1.00000 + 1.73205i 0.164399 + 0.284747i 0.936442 0.350823i \(-0.114098\pi\)
−0.772043 + 0.635571i \(0.780765\pi\)
\(38\) 0 0
\(39\) 1.50000 2.59808i 0.240192 0.416025i
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 12.0000 20.7846i 1.78885 3.09839i
\(46\) 0 0
\(47\) 1.00000 + 1.73205i 0.145865 + 0.252646i 0.929695 0.368329i \(-0.120070\pi\)
−0.783830 + 0.620975i \(0.786737\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) −3.00000 5.19615i −0.420084 0.727607i
\(52\) 0 0
\(53\) 6.00000 10.3923i 0.824163 1.42749i −0.0783936 0.996922i \(-0.524979\pi\)
0.902557 0.430570i \(-0.141688\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −18.0000 −2.38416
\(58\) 0 0
\(59\) 4.50000 7.79423i 0.585850 1.01472i −0.408919 0.912571i \(-0.634094\pi\)
0.994769 0.102151i \(-0.0325726\pi\)
\(60\) 0 0
\(61\) 2.50000 + 4.33013i 0.320092 + 0.554416i 0.980507 0.196485i \(-0.0629528\pi\)
−0.660415 + 0.750901i \(0.729619\pi\)
\(62\) 0 0
\(63\) −3.00000 15.5885i −0.377964 1.96396i
\(64\) 0 0
\(65\) −2.00000 3.46410i −0.248069 0.429669i
\(66\) 0 0
\(67\) −4.50000 + 7.79423i −0.549762 + 0.952217i 0.448528 + 0.893769i \(0.351948\pi\)
−0.998290 + 0.0584478i \(0.981385\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 1.00000 1.73205i 0.117041 0.202721i −0.801553 0.597924i \(-0.795992\pi\)
0.918594 + 0.395203i \(0.129326\pi\)
\(74\) 0 0
\(75\) −16.5000 28.5788i −1.90526 3.30000i
\(76\) 0 0
\(77\) −2.00000 + 1.73205i −0.227921 + 0.197386i
\(78\) 0 0
\(79\) −7.50000 12.9904i −0.843816 1.46153i −0.886646 0.462450i \(-0.846971\pi\)
0.0428296 0.999082i \(-0.486363\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) −1.50000 + 2.59808i −0.160817 + 0.278543i
\(88\) 0 0
\(89\) −3.00000 5.19615i −0.317999 0.550791i 0.662071 0.749441i \(-0.269678\pi\)
−0.980071 + 0.198650i \(0.936344\pi\)
\(90\) 0 0
\(91\) −2.50000 0.866025i −0.262071 0.0907841i
\(92\) 0 0
\(93\) 6.00000 + 10.3923i 0.622171 + 1.07763i
\(94\) 0 0
\(95\) −12.0000 + 20.7846i −1.23117 + 2.13246i
\(96\) 0 0
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 7.50000 12.9904i 0.746278 1.29259i −0.203317 0.979113i \(-0.565172\pi\)
0.949595 0.313478i \(-0.101494\pi\)
\(102\) 0 0
\(103\) 6.00000 + 10.3923i 0.591198 + 1.02398i 0.994071 + 0.108729i \(0.0346780\pi\)
−0.402874 + 0.915255i \(0.631989\pi\)
\(104\) 0 0
\(105\) −30.0000 10.3923i −2.92770 1.01419i
\(106\) 0 0
\(107\) 4.00000 + 6.92820i 0.386695 + 0.669775i 0.992003 0.126217i \(-0.0402834\pi\)
−0.605308 + 0.795991i \(0.706950\pi\)
\(108\) 0 0
\(109\) −5.00000 + 8.66025i −0.478913 + 0.829502i −0.999708 0.0241802i \(-0.992302\pi\)
0.520794 + 0.853682i \(0.325636\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 17.0000 1.59923 0.799613 0.600516i \(-0.205038\pi\)
0.799613 + 0.600516i \(0.205038\pi\)
\(114\) 0 0
\(115\) 4.00000 6.92820i 0.373002 0.646058i
\(116\) 0 0
\(117\) 3.00000 + 5.19615i 0.277350 + 0.480384i
\(118\) 0 0
\(119\) −4.00000 + 3.46410i −0.366679 + 0.317554i
\(120\) 0 0
\(121\) −0.500000 0.866025i −0.0454545 0.0787296i
\(122\) 0 0
\(123\) 3.00000 5.19615i 0.270501 0.468521i
\(124\) 0 0
\(125\) −24.0000 −2.14663
\(126\) 0 0
\(127\) −5.00000 −0.443678 −0.221839 0.975083i \(-0.571206\pi\)
−0.221839 + 0.975083i \(0.571206\pi\)
\(128\) 0 0
\(129\) 6.00000 10.3923i 0.528271 0.914991i
\(130\) 0 0
\(131\) −9.00000 15.5885i −0.786334 1.36197i −0.928199 0.372084i \(-0.878643\pi\)
0.141865 0.989886i \(-0.454690\pi\)
\(132\) 0 0
\(133\) 3.00000 + 15.5885i 0.260133 + 1.35169i
\(134\) 0 0
\(135\) 18.0000 + 31.1769i 1.54919 + 2.68328i
\(136\) 0 0
\(137\) 4.50000 7.79423i 0.384461 0.665906i −0.607233 0.794524i \(-0.707721\pi\)
0.991694 + 0.128618i \(0.0410540\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 0.500000 0.866025i 0.0418121 0.0724207i
\(144\) 0 0
\(145\) 2.00000 + 3.46410i 0.166091 + 0.287678i
\(146\) 0 0
\(147\) −19.5000 + 7.79423i −1.60833 + 0.642857i
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) 4.50000 7.79423i 0.366205 0.634285i −0.622764 0.782410i \(-0.713990\pi\)
0.988969 + 0.148124i \(0.0473236\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 0 0
\(157\) −2.00000 + 3.46410i −0.159617 + 0.276465i −0.934731 0.355357i \(-0.884359\pi\)
0.775113 + 0.631822i \(0.217693\pi\)
\(158\) 0 0
\(159\) 18.0000 + 31.1769i 1.42749 + 2.47249i
\(160\) 0 0
\(161\) −1.00000 5.19615i −0.0788110 0.409514i
\(162\) 0 0
\(163\) 6.50000 + 11.2583i 0.509119 + 0.881820i 0.999944 + 0.0105623i \(0.00336213\pi\)
−0.490825 + 0.871258i \(0.663305\pi\)
\(164\) 0 0
\(165\) 6.00000 10.3923i 0.467099 0.809040i
\(166\) 0 0
\(167\) 17.0000 1.31550 0.657750 0.753237i \(-0.271508\pi\)
0.657750 + 0.753237i \(0.271508\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 18.0000 31.1769i 1.37649 2.38416i
\(172\) 0 0
\(173\) 2.50000 + 4.33013i 0.190071 + 0.329213i 0.945274 0.326278i \(-0.105795\pi\)
−0.755202 + 0.655492i \(0.772461\pi\)
\(174\) 0 0
\(175\) −22.0000 + 19.0526i −1.66304 + 1.44024i
\(176\) 0 0
\(177\) 13.5000 + 23.3827i 1.01472 + 1.75755i
\(178\) 0 0
\(179\) 6.50000 11.2583i 0.485833 0.841487i −0.514035 0.857769i \(-0.671850\pi\)
0.999867 + 0.0162823i \(0.00518305\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) −15.0000 −1.10883
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) −1.00000 1.73205i −0.0731272 0.126660i
\(188\) 0 0
\(189\) 22.5000 + 7.79423i 1.63663 + 0.566947i
\(190\) 0 0
\(191\) 7.00000 + 12.1244i 0.506502 + 0.877288i 0.999972 + 0.00752447i \(0.00239513\pi\)
−0.493469 + 0.869763i \(0.664272\pi\)
\(192\) 0 0
\(193\) 11.0000 19.0526i 0.791797 1.37143i −0.133056 0.991109i \(-0.542479\pi\)
0.924853 0.380325i \(-0.124188\pi\)
\(194\) 0 0
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) 0 0
\(199\) 5.00000 8.66025i 0.354441 0.613909i −0.632581 0.774494i \(-0.718005\pi\)
0.987022 + 0.160585i \(0.0513380\pi\)
\(200\) 0 0
\(201\) −13.5000 23.3827i −0.952217 1.64929i
\(202\) 0 0
\(203\) 2.50000 + 0.866025i 0.175466 + 0.0607831i
\(204\) 0 0
\(205\) −4.00000 6.92820i −0.279372 0.483887i
\(206\) 0 0
\(207\) −6.00000 + 10.3923i −0.417029 + 0.722315i
\(208\) 0 0
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 0 0
\(213\) 6.00000 10.3923i 0.411113 0.712069i
\(214\) 0 0
\(215\) −8.00000 13.8564i −0.545595 0.944999i
\(216\) 0 0
\(217\) 8.00000 6.92820i 0.543075 0.470317i
\(218\) 0 0
\(219\) 3.00000 + 5.19615i 0.202721 + 0.351123i
\(220\) 0 0
\(221\) 1.00000 1.73205i 0.0672673 0.116510i
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 66.0000 4.40000
\(226\) 0 0
\(227\) 5.00000 8.66025i 0.331862 0.574801i −0.651015 0.759065i \(-0.725657\pi\)
0.982877 + 0.184263i \(0.0589899\pi\)
\(228\) 0 0
\(229\) 8.00000 + 13.8564i 0.528655 + 0.915657i 0.999442 + 0.0334101i \(0.0106368\pi\)
−0.470787 + 0.882247i \(0.656030\pi\)
\(230\) 0 0
\(231\) −1.50000 7.79423i −0.0986928 0.512823i
\(232\) 0 0
\(233\) −3.00000 5.19615i −0.196537 0.340411i 0.750867 0.660454i \(-0.229636\pi\)
−0.947403 + 0.320043i \(0.896303\pi\)
\(234\) 0 0
\(235\) −4.00000 + 6.92820i −0.260931 + 0.451946i
\(236\) 0 0
\(237\) 45.0000 2.92306
\(238\) 0 0
\(239\) −19.0000 −1.22901 −0.614504 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) −15.0000 + 25.9808i −0.966235 + 1.67357i −0.259975 + 0.965615i \(0.583714\pi\)
−0.706260 + 0.707953i \(0.749619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.00000 + 27.7128i −0.255551 + 1.77051i
\(246\) 0 0
\(247\) −3.00000 5.19615i −0.190885 0.330623i
\(248\) 0 0
\(249\) −9.00000 + 15.5885i −0.570352 + 0.987878i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) 12.0000 20.7846i 0.751469 1.30158i
\(256\) 0 0
\(257\) 10.5000 + 18.1865i 0.654972 + 1.13444i 0.981901 + 0.189396i \(0.0606529\pi\)
−0.326929 + 0.945049i \(0.606014\pi\)
\(258\) 0 0
\(259\) 1.00000 + 5.19615i 0.0621370 + 0.322873i
\(260\) 0 0
\(261\) −3.00000 5.19615i −0.185695 0.321634i
\(262\) 0 0
\(263\) 1.50000 2.59808i 0.0924940 0.160204i −0.816066 0.577959i \(-0.803849\pi\)
0.908560 + 0.417755i \(0.137183\pi\)
\(264\) 0 0
\(265\) 48.0000 2.94862
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −6.00000 + 10.3923i −0.365826 + 0.633630i −0.988908 0.148527i \(-0.952547\pi\)
0.623082 + 0.782157i \(0.285880\pi\)
\(270\) 0 0
\(271\) 12.5000 + 21.6506i 0.759321 + 1.31518i 0.943197 + 0.332233i \(0.107802\pi\)
−0.183876 + 0.982949i \(0.558865\pi\)
\(272\) 0 0
\(273\) 6.00000 5.19615i 0.363137 0.314485i
\(274\) 0 0
\(275\) −5.50000 9.52628i −0.331662 0.574456i
\(276\) 0 0
\(277\) 1.50000 2.59808i 0.0901263 0.156103i −0.817438 0.576017i \(-0.804606\pi\)
0.907564 + 0.419914i \(0.137940\pi\)
\(278\) 0 0
\(279\) −24.0000 −1.43684
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −3.00000 + 5.19615i −0.178331 + 0.308879i −0.941309 0.337546i \(-0.890403\pi\)
0.762978 + 0.646425i \(0.223737\pi\)
\(284\) 0 0
\(285\) −36.0000 62.3538i −2.13246 3.69352i
\(286\) 0 0
\(287\) −5.00000 1.73205i −0.295141 0.102240i
\(288\) 0 0
\(289\) 6.50000 + 11.2583i 0.382353 + 0.662255i
\(290\) 0 0
\(291\) 7.50000 12.9904i 0.439658 0.761510i
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 36.0000 2.09600
\(296\) 0 0
\(297\) −4.50000 + 7.79423i −0.261116 + 0.452267i
\(298\) 0 0
\(299\) 1.00000 + 1.73205i 0.0578315 + 0.100167i
\(300\) 0 0
\(301\) −10.0000 3.46410i −0.576390 0.199667i
\(302\) 0 0
\(303\) 22.5000 + 38.9711i 1.29259 + 2.23883i
\(304\) 0 0
\(305\) −10.0000 + 17.3205i −0.572598 + 0.991769i
\(306\) 0 0
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) 0 0
\(309\) −36.0000 −2.04797
\(310\) 0 0
\(311\) −14.0000 + 24.2487i −0.793867 + 1.37502i 0.129689 + 0.991555i \(0.458602\pi\)
−0.923556 + 0.383464i \(0.874731\pi\)
\(312\) 0 0
\(313\) −0.500000 0.866025i −0.0282617 0.0489506i 0.851549 0.524276i \(-0.175664\pi\)
−0.879810 + 0.475325i \(0.842331\pi\)
\(314\) 0 0
\(315\) 48.0000 41.5692i 2.70449 2.34216i
\(316\) 0 0
\(317\) 6.00000 + 10.3923i 0.336994 + 0.583690i 0.983866 0.178908i \(-0.0572566\pi\)
−0.646872 + 0.762598i \(0.723923\pi\)
\(318\) 0 0
\(319\) −0.500000 + 0.866025i −0.0279946 + 0.0484881i
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 5.50000 9.52628i 0.305085 0.528423i
\(326\) 0 0
\(327\) −15.0000 25.9808i −0.829502 1.43674i
\(328\) 0 0
\(329\) 1.00000 + 5.19615i 0.0551318 + 0.286473i
\(330\) 0 0
\(331\) 3.50000 + 6.06218i 0.192377 + 0.333207i 0.946038 0.324057i \(-0.105047\pi\)
−0.753660 + 0.657264i \(0.771714\pi\)
\(332\) 0 0
\(333\) 6.00000 10.3923i 0.328798 0.569495i
\(334\) 0 0
\(335\) −36.0000 −1.96689
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 0 0
\(339\) −25.5000 + 44.1673i −1.38497 + 2.39884i
\(340\) 0 0
\(341\) 2.00000 + 3.46410i 0.108306 + 0.187592i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 12.0000 + 20.7846i 0.646058 + 1.11901i
\(346\) 0 0
\(347\) −14.0000 + 24.2487i −0.751559 + 1.30174i 0.195507 + 0.980702i \(0.437365\pi\)
−0.947067 + 0.321037i \(0.895969\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) 9.00000 15.5885i 0.479022 0.829690i −0.520689 0.853746i \(-0.674325\pi\)
0.999711 + 0.0240566i \(0.00765819\pi\)
\(354\) 0 0
\(355\) −8.00000 13.8564i −0.424596 0.735422i
\(356\) 0 0
\(357\) −3.00000 15.5885i −0.158777 0.825029i
\(358\) 0 0
\(359\) −15.5000 26.8468i −0.818059 1.41692i −0.907111 0.420892i \(-0.861717\pi\)
0.0890519 0.996027i \(-0.471616\pi\)
\(360\) 0 0
\(361\) −8.50000 + 14.7224i −0.447368 + 0.774865i
\(362\) 0 0
\(363\) 3.00000 0.157459
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 0 0
\(367\) −7.00000 + 12.1244i −0.365397 + 0.632886i −0.988840 0.148983i \(-0.952400\pi\)
0.623443 + 0.781869i \(0.285733\pi\)
\(368\) 0 0
\(369\) 6.00000 + 10.3923i 0.312348 + 0.541002i
\(370\) 0 0
\(371\) 24.0000 20.7846i 1.24602 1.07908i
\(372\) 0 0
\(373\) 3.50000 + 6.06218i 0.181223 + 0.313888i 0.942297 0.334777i \(-0.108661\pi\)
−0.761074 + 0.648665i \(0.775328\pi\)
\(374\) 0 0
\(375\) 36.0000 62.3538i 1.85903 3.21994i
\(376\) 0 0
\(377\) −1.00000 −0.0515026
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) 7.50000 12.9904i 0.384237 0.665517i
\(382\) 0 0
\(383\) 4.00000 + 6.92820i 0.204390 + 0.354015i 0.949938 0.312437i \(-0.101145\pi\)
−0.745548 + 0.666452i \(0.767812\pi\)
\(384\) 0 0
\(385\) −10.0000 3.46410i −0.509647 0.176547i
\(386\) 0 0
\(387\) 12.0000 + 20.7846i 0.609994 + 1.05654i
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) 54.0000 2.72394
\(394\) 0 0
\(395\) 30.0000 51.9615i 1.50946 2.61447i
\(396\) 0 0
\(397\) 15.0000 + 25.9808i 0.752828 + 1.30394i 0.946447 + 0.322860i \(0.104644\pi\)
−0.193618 + 0.981077i \(0.562022\pi\)
\(398\) 0 0
\(399\) −45.0000 15.5885i −2.25282 0.780399i
\(400\) 0 0
\(401\) −7.50000 12.9904i −0.374532 0.648709i 0.615725 0.787961i \(-0.288863\pi\)
−0.990257 + 0.139253i \(0.955530\pi\)
\(402\) 0 0
\(403\) −2.00000 + 3.46410i −0.0996271 + 0.172559i
\(404\) 0 0
\(405\) −36.0000 −1.78885
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 16.0000 27.7128i 0.791149 1.37031i −0.134107 0.990967i \(-0.542817\pi\)
0.925256 0.379344i \(-0.123850\pi\)
\(410\) 0 0
\(411\) 13.5000 + 23.3827i 0.665906 + 1.15338i
\(412\) 0 0
\(413\) 18.0000 15.5885i 0.885722 0.767058i
\(414\) 0 0
\(415\) 12.0000 + 20.7846i 0.589057 + 1.02028i
\(416\) 0 0
\(417\) 12.0000 20.7846i 0.587643 1.01783i
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 6.00000 10.3923i 0.291730 0.505291i
\(424\) 0 0
\(425\) −11.0000 19.0526i −0.533578 0.924185i
\(426\) 0 0
\(427\) 2.50000 + 12.9904i 0.120983 + 0.628649i
\(428\) 0 0
\(429\) 1.50000 + 2.59808i 0.0724207 + 0.125436i
\(430\) 0 0
\(431\) 0.500000 0.866025i 0.0240842 0.0417150i −0.853732 0.520712i \(-0.825666\pi\)
0.877816 + 0.478997i \(0.159000\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) 0 0
\(437\) 6.00000 10.3923i 0.287019 0.497131i
\(438\) 0 0
\(439\) −2.50000 4.33013i −0.119318 0.206666i 0.800179 0.599761i \(-0.204738\pi\)
−0.919498 + 0.393095i \(0.871404\pi\)
\(440\) 0 0
\(441\) 6.00000 41.5692i 0.285714 1.97949i
\(442\) 0 0
\(443\) −6.00000 10.3923i −0.285069 0.493753i 0.687557 0.726130i \(-0.258683\pi\)
−0.972626 + 0.232377i \(0.925350\pi\)
\(444\) 0 0
\(445\) 12.0000 20.7846i 0.568855 0.985285i
\(446\) 0 0
\(447\) −30.0000 −1.41895
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 1.00000 1.73205i 0.0470882 0.0815591i
\(452\) 0 0
\(453\) 13.5000 + 23.3827i 0.634285 + 1.09861i
\(454\) 0 0
\(455\) −2.00000 10.3923i −0.0937614 0.487199i
\(456\) 0 0
\(457\) −1.00000 1.73205i −0.0467780 0.0810219i 0.841688 0.539964i \(-0.181562\pi\)
−0.888466 + 0.458942i \(0.848229\pi\)
\(458\) 0 0
\(459\) −9.00000 + 15.5885i −0.420084 + 0.727607i
\(460\) 0 0
\(461\) −3.00000 −0.139724 −0.0698620 0.997557i \(-0.522256\pi\)
−0.0698620 + 0.997557i \(0.522256\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) −24.0000 + 41.5692i −1.11297 + 1.92773i
\(466\) 0 0
\(467\) −6.00000 10.3923i −0.277647 0.480899i 0.693153 0.720791i \(-0.256221\pi\)
−0.970799 + 0.239892i \(0.922888\pi\)
\(468\) 0 0
\(469\) −18.0000 + 15.5885i −0.831163 + 0.719808i
\(470\) 0 0
\(471\) −6.00000 10.3923i −0.276465 0.478852i
\(472\) 0 0
\(473\) 2.00000 3.46410i 0.0919601 0.159280i
\(474\) 0 0
\(475\) −66.0000 −3.02829
\(476\) 0 0
\(477\) −72.0000 −3.29665
\(478\) 0 0
\(479\) −18.5000 + 32.0429i −0.845287 + 1.46408i 0.0400855 + 0.999196i \(0.487237\pi\)
−0.885372 + 0.464883i \(0.846096\pi\)
\(480\) 0 0
\(481\) −1.00000 1.73205i −0.0455961 0.0789747i
\(482\) 0 0
\(483\) 15.0000 + 5.19615i 0.682524 + 0.236433i
\(484\) 0 0
\(485\) −10.0000 17.3205i −0.454077 0.786484i
\(486\) 0 0
\(487\) −2.00000 + 3.46410i −0.0906287 + 0.156973i −0.907776 0.419456i \(-0.862221\pi\)
0.817147 + 0.576429i \(0.195554\pi\)
\(488\) 0 0
\(489\) −39.0000 −1.76364
\(490\) 0 0
\(491\) −18.0000 −0.812329 −0.406164 0.913800i \(-0.633134\pi\)
−0.406164 + 0.913800i \(0.633134\pi\)
\(492\) 0 0
\(493\) −1.00000 + 1.73205i −0.0450377 + 0.0780076i
\(494\) 0 0
\(495\) 12.0000 + 20.7846i 0.539360 + 0.934199i
\(496\) 0 0
\(497\) −10.0000 3.46410i −0.448561 0.155386i
\(498\) 0 0
\(499\) −8.00000 13.8564i −0.358129 0.620298i 0.629519 0.776985i \(-0.283252\pi\)
−0.987648 + 0.156687i \(0.949919\pi\)
\(500\) 0 0
\(501\) −25.5000 + 44.1673i −1.13926 + 1.97325i
\(502\) 0 0
\(503\) 21.0000 0.936344 0.468172 0.883637i \(-0.344913\pi\)
0.468172 + 0.883637i \(0.344913\pi\)
\(504\) 0 0
\(505\) 60.0000 2.66996
\(506\) 0 0
\(507\) 18.0000 31.1769i 0.799408 1.38462i
\(508\) 0 0
\(509\) −1.00000 1.73205i −0.0443242 0.0767718i 0.843012 0.537895i \(-0.180780\pi\)
−0.887336 + 0.461123i \(0.847447\pi\)
\(510\) 0 0
\(511\) 4.00000 3.46410i 0.176950 0.153243i
\(512\) 0 0
\(513\) 27.0000 + 46.7654i 1.19208 + 2.06474i
\(514\) 0 0
\(515\) −24.0000 + 41.5692i −1.05757 + 1.83176i
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) 0 0
\(519\) −15.0000 −0.658427
\(520\) 0 0
\(521\) 13.0000 22.5167i 0.569540 0.986473i −0.427071 0.904218i \(-0.640455\pi\)
0.996611 0.0822547i \(-0.0262121\pi\)
\(522\) 0 0
\(523\) 10.0000 + 17.3205i 0.437269 + 0.757373i 0.997478 0.0709788i \(-0.0226123\pi\)
−0.560208 + 0.828352i \(0.689279\pi\)
\(524\) 0 0
\(525\) −16.5000 85.7365i −0.720119 3.74185i
\(526\) 0 0
\(527\) 4.00000 + 6.92820i 0.174243 + 0.301797i
\(528\) 0 0
\(529\) 9.50000 16.4545i 0.413043 0.715412i
\(530\) 0 0
\(531\) −54.0000 −2.34340
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −16.0000 + 27.7128i −0.691740 + 1.19813i
\(536\) 0 0
\(537\) 19.5000 + 33.7750i 0.841487 + 1.45750i
\(538\) 0 0
\(539\) −6.50000 + 2.59808i −0.279975 + 0.111907i
\(540\) 0 0
\(541\) 12.5000 + 21.6506i 0.537417 + 0.930834i 0.999042 + 0.0437584i \(0.0139332\pi\)
−0.461625 + 0.887075i \(0.652733\pi\)
\(542\) 0 0
\(543\) 33.0000 57.1577i 1.41617 2.45287i
\(544\) 0 0
\(545\) −40.0000 −1.71341
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 15.0000 25.9808i 0.640184 1.10883i
\(550\) 0 0
\(551\) 3.00000 + 5.19615i 0.127804 + 0.221364i
\(552\) 0 0
\(553\) −7.50000 38.9711i −0.318932 1.65722i
\(554\) 0 0
\(555\) −12.0000 20.7846i −0.509372 0.882258i
\(556\) 0 0
\(557\) −19.0000 + 32.9090i −0.805056 + 1.39440i 0.111198 + 0.993798i \(0.464531\pi\)
−0.916253 + 0.400599i \(0.868802\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 0 0
\(563\) 13.0000 22.5167i 0.547885 0.948964i −0.450535 0.892759i \(-0.648767\pi\)
0.998419 0.0562051i \(-0.0179001\pi\)
\(564\) 0 0
\(565\) 34.0000 + 58.8897i 1.43039 + 2.47751i
\(566\) 0 0
\(567\) −18.0000 + 15.5885i −0.755929 + 0.654654i
\(568\) 0 0
\(569\) −18.0000 31.1769i −0.754599 1.30700i −0.945573 0.325409i \(-0.894498\pi\)
0.190974 0.981595i \(-0.438835\pi\)
\(570\) 0 0
\(571\) −10.0000 + 17.3205i −0.418487 + 0.724841i −0.995788 0.0916910i \(-0.970773\pi\)
0.577301 + 0.816532i \(0.304106\pi\)
\(572\) 0 0
\(573\) −42.0000 −1.75458
\(574\) 0 0
\(575\) 22.0000 0.917463
\(576\) 0 0
\(577\) 3.50000 6.06218i 0.145707 0.252372i −0.783930 0.620850i \(-0.786788\pi\)
0.929636 + 0.368478i \(0.120121\pi\)
\(578\) 0 0
\(579\) 33.0000 + 57.1577i 1.37143 + 2.37539i
\(580\) 0 0
\(581\) 15.0000 + 5.19615i 0.622305 + 0.215573i
\(582\) 0 0
\(583\) 6.00000 + 10.3923i 0.248495 + 0.430405i
\(584\) 0 0
\(585\) −12.0000 + 20.7846i −0.496139 + 0.859338i
\(586\) 0 0
\(587\) 3.00000 0.123823 0.0619116 0.998082i \(-0.480280\pi\)
0.0619116 + 0.998082i \(0.480280\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 4.50000 7.79423i 0.185105 0.320612i
\(592\) 0 0
\(593\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(594\) 0 0
\(595\) −20.0000 6.92820i −0.819920 0.284029i
\(596\) 0 0
\(597\) 15.0000 + 25.9808i 0.613909 + 1.06332i
\(598\) 0 0
\(599\) −12.0000 + 20.7846i −0.490307 + 0.849236i −0.999938 0.0111569i \(-0.996449\pi\)
0.509631 + 0.860393i \(0.329782\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 54.0000 2.19905
\(604\) 0 0
\(605\) 2.00000 3.46410i 0.0813116 0.140836i
\(606\) 0 0
\(607\) −8.00000 13.8564i −0.324710 0.562414i 0.656744 0.754114i \(-0.271933\pi\)
−0.981454 + 0.191700i \(0.938600\pi\)
\(608\) 0 0
\(609\) −6.00000 + 5.19615i −0.243132 + 0.210559i
\(610\) 0 0
\(611\) −1.00000 1.73205i −0.0404557 0.0700713i
\(612\) 0 0
\(613\) −11.0000 + 19.0526i −0.444286 + 0.769526i −0.998002 0.0631797i \(-0.979876\pi\)
0.553716 + 0.832705i \(0.313209\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 0 0
\(619\) 10.0000 17.3205i 0.401934 0.696170i −0.592025 0.805919i \(-0.701671\pi\)
0.993959 + 0.109749i \(0.0350048\pi\)
\(620\) 0 0
\(621\) −9.00000 15.5885i −0.361158 0.625543i
\(622\) 0 0
\(623\) −3.00000 15.5885i −0.120192 0.624538i
\(624\) 0 0
\(625\) −20.5000 35.5070i −0.820000 1.42028i
\(626\) 0 0
\(627\) 9.00000 15.5885i 0.359425 0.622543i
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −21.0000 + 36.3731i −0.834675 + 1.44570i
\(634\) 0 0
\(635\) −10.0000 17.3205i −0.396838 0.687343i
\(636\) 0 0
\(637\) −5.50000 4.33013i −0.217918 0.171566i
\(638\) 0 0
\(639\) 12.0000 + 20.7846i 0.474713 + 0.822226i
\(640\) 0 0
\(641\) −4.50000 + 7.79423i −0.177739 + 0.307854i −0.941106 0.338112i \(-0.890212\pi\)
0.763367 + 0.645966i \(0.223545\pi\)
\(642\) 0 0
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 0 0
\(645\) 48.0000 1.89000
\(646\) 0 0
\(647\) 12.0000 20.7846i 0.471769 0.817127i −0.527710 0.849425i \(-0.676949\pi\)
0.999478 + 0.0322975i \(0.0102824\pi\)
\(648\) 0 0
\(649\) 4.50000 + 7.79423i 0.176640 + 0.305950i
\(650\) 0 0
\(651\) 6.00000 + 31.1769i 0.235159 + 1.22192i
\(652\) 0 0
\(653\) −11.0000 19.0526i −0.430463 0.745584i 0.566450 0.824096i \(-0.308316\pi\)
−0.996913 + 0.0785119i \(0.974983\pi\)
\(654\) 0 0
\(655\) 36.0000 62.3538i 1.40664 2.43637i
\(656\) 0 0
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) 32.0000 1.24654 0.623272 0.782006i \(-0.285803\pi\)
0.623272 + 0.782006i \(0.285803\pi\)
\(660\) 0 0
\(661\) 5.00000 8.66025i 0.194477 0.336845i −0.752252 0.658876i \(-0.771032\pi\)
0.946729 + 0.322031i \(0.104366\pi\)
\(662\) 0 0
\(663\) 3.00000 + 5.19615i 0.116510 + 0.201802i
\(664\) 0 0
\(665\) −48.0000 + 41.5692i −1.86136 + 1.61199i
\(666\) 0 0
\(667\) −1.00000 1.73205i −0.0387202 0.0670653i
\(668\) 0 0
\(669\) −39.0000 + 67.5500i −1.50783 + 2.61163i
\(670\) 0 0
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 16.0000 0.616755 0.308377 0.951264i \(-0.400214\pi\)
0.308377 + 0.951264i \(0.400214\pi\)
\(674\) 0 0
\(675\) −49.5000 + 85.7365i −1.90526 + 3.30000i
\(676\) 0 0
\(677\) −19.0000 32.9090i −0.730229 1.26479i −0.956785 0.290796i \(-0.906080\pi\)
0.226556 0.973998i \(-0.427253\pi\)
\(678\) 0 0
\(679\) −12.5000 4.33013i −0.479706 0.166175i
\(680\) 0 0
\(681\) 15.0000 + 25.9808i 0.574801 + 0.995585i
\(682\) 0 0
\(683\) −16.5000 + 28.5788i −0.631355 + 1.09354i 0.355920 + 0.934516i \(0.384168\pi\)
−0.987275 + 0.159022i \(0.949166\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) −48.0000 −1.83131
\(688\) 0 0
\(689\) −6.00000 + 10.3923i −0.228582 + 0.395915i
\(690\) 0 0
\(691\) 7.50000 + 12.9904i 0.285313 + 0.494177i 0.972685 0.232128i \(-0.0745690\pi\)
−0.687372 + 0.726306i \(0.741236\pi\)
\(692\) 0 0
\(693\) 15.0000 + 5.19615i 0.569803 + 0.197386i
\(694\) 0 0
\(695\) −16.0000 27.7128i −0.606915 1.05121i
\(696\) 0 0
\(697\) 2.00000 3.46410i 0.0757554 0.131212i
\(698\) 0 0
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) −6.00000 + 10.3923i −0.226294 + 0.391953i
\(704\) 0 0
\(705\) −12.0000 20.7846i −0.451946 0.782794i
\(706\) 0 0
\(707\) 30.0000 25.9808i 1.12827 0.977107i
\(708\) 0 0
\(709\) −3.00000 5.19615i −0.112667 0.195146i 0.804178 0.594389i \(-0.202606\pi\)
−0.916845 + 0.399244i \(0.869273\pi\)
\(710\) 0 0
\(711\) −45.0000 + 77.9423i −1.68763 + 2.92306i
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 0 0
\(717\) 28.5000 49.3634i 1.06435 1.84351i
\(718\) 0 0
\(719\) −13.0000 22.5167i −0.484818 0.839730i 0.515030 0.857172i \(-0.327781\pi\)
−0.999848 + 0.0174426i \(0.994448\pi\)
\(720\) 0 0
\(721\) 6.00000 + 31.1769i 0.223452 + 1.16109i
\(722\) 0 0
\(723\) −45.0000 77.9423i −1.67357 2.89870i
\(724\) 0 0
\(725\) −5.50000 + 9.52628i −0.204265 + 0.353797i
\(726\) 0 0
\(727\) 34.0000 1.26099 0.630495 0.776193i \(-0.282852\pi\)
0.630495 + 0.776193i \(0.282852\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 4.00000 6.92820i 0.147945 0.256249i
\(732\) 0 0
\(733\) 0.500000 + 0.866025i 0.0184679 + 0.0319874i 0.875112 0.483921i \(-0.160788\pi\)
−0.856644 + 0.515908i \(0.827454\pi\)
\(734\) 0 0
\(735\) −66.0000 51.9615i −2.43445 1.91663i
\(736\) 0 0
\(737\) −4.50000 7.79423i −0.165760 0.287104i
\(738\) 0 0
\(739\) 9.00000 15.5885i 0.331070 0.573431i −0.651652 0.758518i \(-0.725924\pi\)
0.982722 + 0.185088i \(0.0592569\pi\)
\(740\) 0 0
\(741\) 18.0000 0.661247
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) −20.0000 + 34.6410i −0.732743 + 1.26915i
\(746\) 0 0
\(747\) −18.0000 31.1769i −0.658586 1.14070i
\(748\) 0 0
\(749\) 4.00000 + 20.7846i 0.146157 + 0.759453i
\(750\) 0 0
\(751\) −22.0000 38.1051i −0.802791 1.39048i −0.917772 0.397108i \(-0.870014\pi\)
0.114981 0.993368i \(-0.463319\pi\)
\(752\) 0 0
\(753\) 18.0000 31.1769i 0.655956 1.13615i
\(754\) 0 0
\(755\) 36.0000 1.31017
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) −3.00000 + 5.19615i −0.108893 + 0.188608i
\(760\) 0 0
\(761\) −27.0000 46.7654i −0.978749 1.69524i −0.666962 0.745091i \(-0.732406\pi\)
−0.311787 0.950152i \(-0.600927\pi\)
\(762\) 0 0
\(763\) −20.0000 + 17.3205i −0.724049 + 0.627044i
\(764\) 0 0
\(765\) 24.0000 + 41.5692i 0.867722 + 1.50294i
\(766\) 0 0
\(767\) −4.50000 + 7.79423i −0.162486 + 0.281433i
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) −63.0000 −2.26889
\(772\) 0 0
\(773\) 12.0000 20.7846i 0.431610 0.747570i −0.565402 0.824815i \(-0.691279\pi\)
0.997012 + 0.0772449i \(0.0246123\pi\)
\(774\) 0 0
\(775\) 22.0000 + 38.1051i 0.790263 + 1.36878i
\(776\) 0 0
\(777\) −15.0000 5.19615i −0.538122 0.186411i
\(778\) 0 0
\(779\) −6.00000 10.3923i −0.214972 0.372343i
\(780\) 0 0
\(781\) 2.00000 3.46410i 0.0715656 0.123955i
\(782\) 0 0
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) −11.0000 + 19.0526i −0.392108 + 0.679150i −0.992727 0.120384i \(-0.961587\pi\)
0.600620 + 0.799535i \(0.294921\pi\)
\(788\) 0 0
\(789\) 4.50000 + 7.79423i 0.160204 + 0.277482i
\(790\) 0 0
\(791\) 42.5000 + 14.7224i 1.51113 + 0.523469i
\(792\) 0 0
\(793\) −2.50000 4.33013i −0.0887776 0.153767i
\(794\) 0 0
\(795\) −72.0000 + 124.708i −2.55358 + 4.42292i
\(796\) 0 0
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) −4.00000 −0.141510
\(800\) 0 0
\(801\) −18.0000 + 31.1769i −0.635999 + 1.10158i
\(802\) 0 0
\(803\) 1.00000 + 1.73205i 0.0352892 + 0.0611227i
\(804\) 0 0
\(805\) 16.0000 13.8564i 0.563926 0.488374i
\(806\) 0 0
\(807\) −18.0000 31.1769i −0.633630 1.09748i
\(808\) 0 0
\(809\) 24.0000 41.5692i 0.843795 1.46150i −0.0428684 0.999081i \(-0.513650\pi\)
0.886664 0.462415i \(-0.153017\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 0 0
\(813\) −75.0000 −2.63036
\(814\) 0 0
\(815\) −26.0000 + 45.0333i −0.910740 + 1.57745i
\(816\) 0 0
\(817\) −12.0000 20.7846i −0.419827 0.727161i
\(818\) 0 0
\(819\) 3.00000 + 15.5885i 0.104828 + 0.544705i
\(820\) 0 0
\(821\) 7.50000 + 12.9904i 0.261752 + 0.453367i 0.966708 0.255884i \(-0.0823665\pi\)
−0.704956 + 0.709251i \(0.749033\pi\)
\(822\) 0 0
\(823\) 19.0000 32.9090i 0.662298 1.14713i −0.317712 0.948187i \(-0.602914\pi\)
0.980010 0.198947i \(-0.0637522\pi\)
\(824\) 0 0
\(825\) 33.0000 1.14891
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) −8.00000 + 13.8564i −0.277851 + 0.481253i −0.970851 0.239686i \(-0.922956\pi\)
0.692999 + 0.720938i \(0.256289\pi\)
\(830\) 0 0
\(831\) 4.50000 + 7.79423i 0.156103 + 0.270379i
\(832\) 0 0
\(833\) −13.0000 + 5.19615i −0.450423 + 0.180036i
\(834\) 0 0
\(835\) 34.0000 + 58.8897i 1.17662 + 2.03796i
\(836\) 0 0
\(837\) 18.0000 31.1769i 0.622171 1.07763i
\(838\) 0 0
\(839\) −54.0000 −1.86429 −0.932144 0.362089i \(-0.882064\pi\)
−0.932144 + 0.362089i \(0.882064\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 15.0000 25.9808i 0.516627 0.894825i
\(844\) 0 0
\(845\) −24.0000 41.5692i −0.825625 1.43002i
\(846\) 0 0
\(847\) −0.500000 2.59808i −0.0171802 0.0892710i
\(848\) 0 0
\(849\) −9.00000 15.5885i −0.308879 0.534994i
\(850\) 0 0
\(851\) 2.00000 3.46410i 0.0685591 0.118748i
\(852\) 0 0
\(853\) 50.0000 1.71197 0.855984 0.517003i \(-0.172952\pi\)
0.855984 + 0.517003i \(0.172952\pi\)
\(854\) 0 0
\(855\) 144.000 4.92470
\(856\) 0 0
\(857\) 2.00000 3.46410i 0.0683187 0.118331i −0.829843 0.557998i \(-0.811570\pi\)
0.898161 + 0.439666i \(0.144903\pi\)
\(858\) 0 0
\(859\) −8.50000 14.7224i −0.290016 0.502323i 0.683797 0.729672i \(-0.260327\pi\)
−0.973813 + 0.227349i \(0.926994\pi\)
\(860\) 0 0
\(861\) 12.0000 10.3923i 0.408959 0.354169i
\(862\) 0 0
\(863\) 5.00000 + 8.66025i 0.170202 + 0.294798i 0.938490 0.345305i \(-0.112225\pi\)
−0.768288 + 0.640104i \(0.778891\pi\)
\(864\) 0 0
\(865\) −10.0000 + 17.3205i −0.340010 + 0.588915i
\(866\) 0 0
\(867\) −39.0000 −1.32451
\(868\) 0 0
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 4.50000 7.79423i 0.152477 0.264097i
\(872\) 0 0
\(873\) 15.0000 + 25.9808i 0.507673 + 0.879316i
\(874\) 0 0
\(875\) −60.0000 20.7846i −2.02837 0.702648i
\(876\) 0 0
\(877\) −12.5000 21.6506i −0.422095 0.731090i 0.574049 0.818821i \(-0.305372\pi\)
−0.996144 + 0.0877308i \(0.972038\pi\)
\(878\) 0 0
\(879\) 9.00000 15.5885i 0.303562 0.525786i
\(880\) 0 0
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) −54.0000 + 93.5307i −1.81519 + 3.14400i
\(886\) 0 0
\(887\) −8.50000 14.7224i −0.285402 0.494331i 0.687305 0.726369i \(-0.258794\pi\)
−0.972707 + 0.232038i \(0.925460\pi\)
\(888\) 0 0
\(889\) −12.5000 4.33013i −0.419237 0.145228i
\(890\) 0 0
\(891\) −4.50000 7.79423i −0.150756 0.261116i
\(892\) 0 0
\(893\) −6.00000 + 10.3923i −0.200782 + 0.347765i
\(894\) 0 0
\(895\) 52.0000 1.73817
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) 2.00000 3.46410i 0.0667037 0.115534i
\(900\) 0 0
\(901\) 12.0000 + 20.7846i 0.399778 + 0.692436i
\(902\) 0 0
\(903\) 24.0000 20.7846i 0.798670 0.691669i
\(904\) 0 0
\(905\) −44.0000 76.2102i −1.46261 2.53331i
\(906\) 0 0
\(907\) 22.0000 38.1051i 0.730498 1.26526i −0.226173 0.974087i \(-0.572621\pi\)
0.956671 0.291172i \(-0.0940453\pi\)
\(908\) 0 0
\(909\) −90.0000 −2.98511
\(910\) 0 0
\(911\) 54.0000 1.78910 0.894550 0.446968i \(-0.147496\pi\)
0.894550 + 0.446968i \(0.147496\pi\)
\(912\) 0 0
\(913\) −3.00000 + 5.19615i −0.0992855 + 0.171968i
\(914\) 0 0
\(915\) −30.0000 51.9615i −0.991769 1.71780i
\(916\) 0 0
\(917\) −9.00000 46.7654i −0.297206 1.54433i
\(918\) 0 0
\(919\) −12.0000 20.7846i −0.395843 0.685621i 0.597365 0.801970i \(-0.296214\pi\)
−0.993208 + 0.116348i \(0.962881\pi\)
\(920\) 0 0
\(921\) 48.0000 83.1384i 1.58165 2.73950i
\(922\) 0 0
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) −22.0000 −0.723356
\(926\) 0 0
\(927\) 36.0000 62.3538i 1.18240 2.04797i
\(928\) 0 0
\(929\) 16.5000 + 28.5788i 0.541347 + 0.937641i 0.998827 + 0.0484211i \(0.0154190\pi\)
−0.457480 + 0.889220i \(0.651248\pi\)
\(930\) 0 0
\(931\) −6.00000 + 41.5692i −0.196642 + 1.36238i
\(932\) 0 0
\(933\) −42.0000 72.7461i −1.37502 2.38160i
\(934\) 0 0
\(935\) 4.00000 6.92820i 0.130814 0.226576i
\(936\) 0 0
\(937\) −12.0000 −0.392023 −0.196011 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(938\) 0 0
\(939\) 3.00000 0.0979013
\(940\) 0 0
\(941\) −27.5000 + 47.6314i −0.896474 + 1.55274i −0.0645052 + 0.997917i \(0.520547\pi\)
−0.831969 + 0.554822i \(0.812786\pi\)
\(942\) 0 0
\(943\) 2.00000 + 3.46410i 0.0651290 + 0.112807i
\(944\) 0 0
\(945\) 18.0000 + 93.5307i 0.585540 + 3.04256i
\(946\) 0 0
\(947\) −8.00000 13.8564i −0.259965 0.450273i 0.706267 0.707945i \(-0.250378\pi\)
−0.966232 + 0.257673i \(0.917044\pi\)
\(948\) 0 0
\(949\) −1.00000 + 1.73205i −0.0324614 + 0.0562247i
\(950\) 0 0
\(951\) −36.0000 −1.16738
\(952\) 0 0
\(953\) 56.0000 1.81402 0.907009 0.421111i \(-0.138360\pi\)
0.907009 + 0.421111i \(0.138360\pi\)
\(954\) 0 0
\(955\) −28.0000 + 48.4974i −0.906059 + 1.56934i
\(956\) 0 0
\(957\) −1.50000 2.59808i −0.0484881 0.0839839i
\(958\) 0 0
\(959\) 18.0000 15.5885i 0.581250 0.503378i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 24.0000 41.5692i 0.773389 1.33955i
\(964\) 0 0
\(965\) 88.0000 2.83282
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 0 0
\(969\) 18.0000 31.1769i 0.578243 1.00155i
\(970\) 0 0
\(971\) −6.50000 11.2583i −0.208595 0.361297i 0.742677 0.669650i \(-0.233556\pi\)
−0.951272 + 0.308353i \(0.900222\pi\)
\(972\) 0 0
\(973\) −20.0000 6.92820i −0.641171 0.222108i
\(974\) 0 0
\(975\) 16.5000 + 28.5788i 0.528423 + 0.915255i
\(976\) 0 0
\(977\) −7.00000 + 12.1244i −0.223950 + 0.387893i −0.956004 0.293354i \(-0.905229\pi\)
0.732054 + 0.681247i \(0.238562\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 60.0000 1.91565
\(982\) 0 0
\(983\) 9.00000 15.5885i 0.287055 0.497195i −0.686050 0.727554i \(-0.740657\pi\)
0.973106 + 0.230360i \(0.0739903\pi\)
\(984\) 0 0
\(985\) −6.00000 10.3923i −0.191176 0.331126i
\(986\) 0 0
\(987\) −15.0000 5.19615i −0.477455 0.165395i
\(988\) 0 0
\(989\) 4.00000 + 6.92820i 0.127193 + 0.220304i
\(990\) 0 0
\(991\) −10.0000 + 17.3205i −0.317660 + 0.550204i −0.979999 0.199000i \(-0.936231\pi\)
0.662339 + 0.749204i \(0.269564\pi\)
\(992\) 0 0
\(993\) −21.0000 −0.666415
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) 21.0000 36.3731i 0.665077 1.15195i −0.314188 0.949361i \(-0.601732\pi\)
0.979265 0.202586i \(-0.0649345\pi\)
\(998\) 0 0
\(999\) 9.00000 + 15.5885i 0.284747 + 0.493197i
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 1232.2.q.a.529.1 2
4.3 odd 2 154.2.e.d.67.1 yes 2
7.2 even 3 inner 1232.2.q.a.177.1 2
7.3 odd 6 8624.2.a.d.1.1 1
7.4 even 3 8624.2.a.bd.1.1 1
12.11 even 2 1386.2.k.a.991.1 2
28.3 even 6 1078.2.a.f.1.1 1
28.11 odd 6 1078.2.a.a.1.1 1
28.19 even 6 1078.2.e.g.177.1 2
28.23 odd 6 154.2.e.d.23.1 2
28.27 even 2 1078.2.e.g.67.1 2
84.11 even 6 9702.2.a.cg.1.1 1
84.23 even 6 1386.2.k.a.793.1 2
84.59 odd 6 9702.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.d.23.1 2 28.23 odd 6
154.2.e.d.67.1 yes 2 4.3 odd 2
1078.2.a.a.1.1 1 28.11 odd 6
1078.2.a.f.1.1 1 28.3 even 6
1078.2.e.g.67.1 2 28.27 even 2
1078.2.e.g.177.1 2 28.19 even 6
1232.2.q.a.177.1 2 7.2 even 3 inner
1232.2.q.a.529.1 2 1.1 even 1 trivial
1386.2.k.a.793.1 2 84.23 even 6
1386.2.k.a.991.1 2 12.11 even 2
8624.2.a.d.1.1 1 7.3 odd 6
8624.2.a.bd.1.1 1 7.4 even 3
9702.2.a.bb.1.1 1 84.59 odd 6
9702.2.a.cg.1.1 1 84.11 even 6