Properties

Label 39.2.k.a.32.1
Level $39$
Weight $2$
Character 39.32
Analytic conductor $0.311$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,2,Mod(2,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.k (of order \(12\), degree \(4\), 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: \(0.311416567883\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 32.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 39.32
Dual form 39.2.k.a.11.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-0.767949 + 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(0.866025 - 1.50000i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(-0.767949 + 2.86603i) q^{7} +(-1.50000 - 2.59808i) q^{9} +3.46410i q^{12} +(-2.59808 - 2.50000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(7.83013 + 2.09808i) q^{19} +(3.63397 + 3.63397i) q^{21} -5.00000i q^{25} -5.19615 q^{27} +(-1.53590 - 5.73205i) q^{28} +(-7.83013 + 7.83013i) q^{31} +(5.19615 + 3.00000i) q^{36} +(2.09808 - 0.562178i) q^{37} +(-6.00000 + 1.73205i) q^{39} +(-1.50000 + 0.866025i) q^{43} +(-3.46410 - 6.00000i) q^{48} +(-1.56218 - 0.901924i) q^{49} +(7.00000 + 1.73205i) q^{52} +(9.92820 - 9.92820i) q^{57} +(4.33013 + 7.50000i) q^{61} +(8.59808 - 2.30385i) q^{63} +8.00000i q^{64} +(0.205771 + 0.767949i) q^{67} +(-9.36603 - 9.36603i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-15.6603 + 4.19615i) q^{76} +12.1244 q^{79} +(-4.50000 + 7.79423i) q^{81} +(-9.92820 - 2.66025i) q^{84} +(9.16025 - 5.52628i) q^{91} +(4.96410 + 18.5263i) q^{93} +(-16.4282 - 4.40192i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{7} - 6 q^{9} + 8 q^{16} + 14 q^{19} + 18 q^{21} - 20 q^{28} - 14 q^{31} - 2 q^{37} - 24 q^{39} - 6 q^{43} + 18 q^{49} + 28 q^{52} + 12 q^{57} + 24 q^{63} + 32 q^{67} - 34 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} - 12 q^{84} + 2 q^{91} + 6 q^{93} - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{12}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0.866025 1.50000i 0.500000 0.866025i
\(4\) −1.73205 + 1.00000i −0.866025 + 0.500000i
\(5\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) −0.767949 + 2.86603i −0.290258 + 1.08326i 0.654654 + 0.755929i \(0.272814\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −1.50000 2.59808i −0.500000 0.866025i
\(10\) 0 0
\(11\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(12\) 3.46410i 1.00000i
\(13\) −2.59808 2.50000i −0.720577 0.693375i
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 3.46410i 0.500000 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 7.83013 + 2.09808i 1.79635 + 0.481332i 0.993399 0.114708i \(-0.0365932\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 3.63397 + 3.63397i 0.792998 + 0.792998i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 5.00000i 1.00000i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) −1.53590 5.73205i −0.290258 1.08326i
\(29\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −7.83013 + 7.83013i −1.40633 + 1.40633i −0.628619 + 0.777714i \(0.716379\pi\)
−0.777714 + 0.628619i \(0.783621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.19615 + 3.00000i 0.866025 + 0.500000i
\(37\) 2.09808 0.562178i 0.344922 0.0924215i −0.0821995 0.996616i \(-0.526194\pi\)
0.427121 + 0.904194i \(0.359528\pi\)
\(38\) 0 0
\(39\) −6.00000 + 1.73205i −0.960769 + 0.277350i
\(40\) 0 0
\(41\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(42\) 0 0
\(43\) −1.50000 + 0.866025i −0.228748 + 0.132068i −0.609994 0.792406i \(-0.708828\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(48\) −3.46410 6.00000i −0.500000 0.866025i
\(49\) −1.56218 0.901924i −0.223168 0.128846i
\(50\) 0 0
\(51\) 0 0
\(52\) 7.00000 + 1.73205i 0.970725 + 0.240192i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 9.92820 9.92820i 1.31502 1.31502i
\(58\) 0 0
\(59\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(60\) 0 0
\(61\) 4.33013 + 7.50000i 0.554416 + 0.960277i 0.997949 + 0.0640184i \(0.0203916\pi\)
−0.443533 + 0.896258i \(0.646275\pi\)
\(62\) 0 0
\(63\) 8.59808 2.30385i 1.08326 0.290258i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.205771 + 0.767949i 0.0251390 + 0.0938199i 0.977356 0.211604i \(-0.0678686\pi\)
−0.952217 + 0.305424i \(0.901202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(72\) 0 0
\(73\) −9.36603 9.36603i −1.09621 1.09621i −0.994850 0.101361i \(-0.967680\pi\)
−0.101361 0.994850i \(-0.532320\pi\)
\(74\) 0 0
\(75\) −7.50000 4.33013i −0.866025 0.500000i
\(76\) −15.6603 + 4.19615i −1.79635 + 0.481332i
\(77\) 0 0
\(78\) 0 0
\(79\) 12.1244 1.36410 0.682048 0.731307i \(-0.261089\pi\)
0.682048 + 0.731307i \(0.261089\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) −9.92820 2.66025i −1.08326 0.290258i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(90\) 0 0
\(91\) 9.16025 5.52628i 0.960256 0.579311i
\(92\) 0 0
\(93\) 4.96410 + 18.5263i 0.514753 + 1.92109i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −16.4282 4.40192i −1.66803 0.446948i −0.703452 0.710742i \(-0.748359\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 5.00000 + 8.66025i 0.500000 + 0.866025i
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 15.5885i 1.53598i −0.640464 0.767988i \(-0.721258\pi\)
0.640464 0.767988i \(-0.278742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 9.00000 5.19615i 0.866025 0.500000i
\(109\) 5.16987 5.16987i 0.495184 0.495184i −0.414751 0.909935i \(-0.636131\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 0.973721 3.63397i 0.0924215 0.344922i
\(112\) 8.39230 + 8.39230i 0.792998 + 0.792998i
\(113\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.59808 + 10.5000i −0.240192 + 0.970725i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.52628 5.50000i 0.866025 0.500000i
\(122\) 0 0
\(123\) 0 0
\(124\) 5.73205 21.3923i 0.514753 1.92109i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.866025 0.500000i −0.0768473 0.0443678i 0.461084 0.887357i \(-0.347461\pi\)
−0.537931 + 0.842989i \(0.680794\pi\)
\(128\) 0 0
\(129\) 3.00000i 0.264135i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −12.0263 + 20.8301i −1.04281 + 1.80620i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(138\) 0 0
\(139\) 3.50000 + 6.06218i 0.296866 + 0.514187i 0.975417 0.220366i \(-0.0707252\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −12.0000 −1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −2.70577 + 1.56218i −0.223168 + 0.128846i
\(148\) −3.07180 + 3.07180i −0.252500 + 0.252500i
\(149\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(150\) 0 0
\(151\) −14.1244 14.1244i −1.14942 1.14942i −0.986666 0.162758i \(-0.947961\pi\)
−0.162758 0.986666i \(-0.552039\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 8.66025 9.00000i 0.693375 0.720577i
\(157\) −11.0000 −0.877896 −0.438948 0.898513i \(-0.644649\pi\)
−0.438948 + 0.898513i \(0.644649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.52628 + 20.6244i −0.432852 + 1.61542i 0.313304 + 0.949653i \(0.398564\pi\)
−0.746156 + 0.665771i \(0.768103\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(168\) 0 0
\(169\) 0.500000 + 12.9904i 0.0384615 + 0.999260i
\(170\) 0 0
\(171\) −6.29423 23.4904i −0.481332 1.79635i
\(172\) 1.73205 3.00000i 0.132068 0.228748i
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) 14.3301 + 3.83975i 1.08326 + 0.290258i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(180\) 0 0
\(181\) 6.92820i 0.514969i 0.966282 + 0.257485i \(0.0828937\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 0 0
\(183\) 15.0000 1.10883
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 3.99038 14.8923i 0.290258 1.08326i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 12.0000 + 6.92820i 0.866025 + 0.500000i
\(193\) 26.3564 7.06218i 1.89718 0.508347i 0.899770 0.436365i \(-0.143734\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.60770 0.257693
\(197\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(198\) 0 0
\(199\) −14.7224 + 8.50000i −1.04365 + 0.602549i −0.920864 0.389885i \(-0.872515\pi\)
−0.122782 + 0.992434i \(0.539182\pi\)
\(200\) 0 0
\(201\) 1.33013 + 0.356406i 0.0938199 + 0.0251390i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −13.8564 + 4.00000i −0.960769 + 0.277350i
\(209\) 0 0
\(210\) 0 0
\(211\) 12.9904 22.5000i 0.894295 1.54896i 0.0596196 0.998221i \(-0.481011\pi\)
0.834675 0.550743i \(-0.185655\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.4282 28.4545i −1.11522 1.93162i
\(218\) 0 0
\(219\) −22.1603 + 5.93782i −1.49745 + 0.401241i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.22243 + 12.0263i 0.215790 + 0.805339i 0.985887 + 0.167412i \(0.0535411\pi\)
−0.770097 + 0.637927i \(0.779792\pi\)
\(224\) 0 0
\(225\) −12.9904 + 7.50000i −0.866025 + 0.500000i
\(226\) 0 0
\(227\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(228\) −7.26795 + 27.1244i −0.481332 + 1.79635i
\(229\) 21.3923 + 21.3923i 1.41364 + 1.41364i 0.726900 + 0.686743i \(0.240960\pi\)
0.686743 + 0.726900i \(0.259040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.5000 18.1865i 0.682048 1.18134i
\(238\) 0 0
\(239\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(240\) 0 0
\(241\) −2.50962 + 9.36603i −0.161659 + 0.603319i 0.836784 + 0.547533i \(0.184433\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 7.79423 + 13.5000i 0.500000 + 0.866025i
\(244\) −15.0000 8.66025i −0.960277 0.554416i
\(245\) 0 0
\(246\) 0 0
\(247\) −15.0981 25.0263i −0.960668 1.59238i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(252\) −12.5885 + 12.5885i −0.792998 + 0.792998i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) 6.44486i 0.400464i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.12436 1.12436i −0.0686810 0.0686810i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −9.16025 + 2.45448i −0.556446 + 0.149099i −0.526073 0.850439i \(-0.676336\pi\)
−0.0303728 + 0.999539i \(0.509669\pi\)
\(272\) 0 0
\(273\) −0.356406 18.5263i −0.0215707 1.12126i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 10.3923i 1.08152 0.624413i 0.150210 0.988654i \(-0.452005\pi\)
0.931305 + 0.364241i \(0.118672\pi\)
\(278\) 0 0
\(279\) 32.0885 + 8.59808i 1.92109 + 0.514753i
\(280\) 0 0
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) 21.6506 + 12.5000i 1.28700 + 0.743048i 0.978117 0.208053i \(-0.0667128\pi\)
0.308879 + 0.951101i \(0.400046\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) −20.8301 + 20.8301i −1.22108 + 1.22108i
\(292\) 25.5885 + 6.85641i 1.49745 + 0.401241i
\(293\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 17.3205 1.00000
\(301\) −1.33013 4.96410i −0.0766672 0.286126i
\(302\) 0 0
\(303\) 0 0
\(304\) 22.9282 22.9282i 1.31502 1.31502i
\(305\) 0 0
\(306\) 0 0
\(307\) 16.6340 + 16.6340i 0.949351 + 0.949351i 0.998778 0.0494267i \(-0.0157394\pi\)
−0.0494267 + 0.998778i \(0.515739\pi\)
\(308\) 0 0
\(309\) −23.3827 13.5000i −1.33019 0.767988i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −32.9090 −1.86012 −0.930062 0.367402i \(-0.880247\pi\)
−0.930062 + 0.367402i \(0.880247\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −21.0000 + 12.1244i −1.18134 + 0.682048i
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) −12.5000 + 12.9904i −0.693375 + 0.720577i
\(326\) 0 0
\(327\) −3.27757 12.2321i −0.181250 0.676434i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −34.1865 9.16025i −1.87906 0.503493i −0.999622 0.0274825i \(-0.991251\pi\)
−0.879440 0.476011i \(-0.842082\pi\)
\(332\) 0 0
\(333\) −4.60770 4.60770i −0.252500 0.252500i
\(334\) 0 0
\(335\) 0 0
\(336\) 19.8564 5.32051i 1.08326 0.290258i
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −10.9019 + 10.9019i −0.588649 + 0.588649i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) −4.40192 + 1.17949i −0.235630 + 0.0631368i −0.374701 0.927146i \(-0.622255\pi\)
0.139072 + 0.990282i \(0.455588\pi\)
\(350\) 0 0
\(351\) 13.5000 + 12.9904i 0.720577 + 0.693375i
\(352\) 0 0
\(353\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(360\) 0 0
\(361\) 40.4545 + 23.3564i 2.12918 + 1.22928i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) −10.3397 + 18.7321i −0.541950 + 0.981826i
\(365\) 0 0
\(366\) 0 0
\(367\) −15.5000 + 26.8468i −0.809093 + 1.40139i 0.104399 + 0.994535i \(0.466708\pi\)
−0.913493 + 0.406855i \(0.866625\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −27.1244 27.1244i −1.40633 1.40633i
\(373\) −18.1865 31.5000i −0.941663 1.63101i −0.762299 0.647225i \(-0.775929\pi\)
−0.179364 0.983783i \(-0.557404\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.55256 16.9904i −0.233849 0.872737i −0.978664 0.205466i \(-0.934129\pi\)
0.744815 0.667271i \(-0.232538\pi\)
\(380\) 0 0
\(381\) −1.50000 + 0.866025i −0.0768473 + 0.0443678i
\(382\) 0 0
\(383\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.50000 + 2.59808i 0.228748 + 0.132068i
\(388\) 32.8564 8.80385i 1.66803 0.446948i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.47372 27.8923i 0.375095 1.39987i −0.478110 0.878300i \(-0.658678\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) 20.8301 + 36.0788i 1.04281 + 1.80620i
\(400\) −17.3205 10.0000i −0.866025 0.500000i
\(401\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(402\) 0 0
\(403\) 39.9186 0.767949i 1.98849 0.0382543i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −3.42820 0.918584i −0.169514 0.0454211i 0.173064 0.984911i \(-0.444633\pi\)
−0.342578 + 0.939490i \(0.611300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 15.5885 + 27.0000i 0.767988 + 1.33019i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 12.1244 0.593732
\(418\) 0 0
\(419\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) 27.6865 27.6865i 1.34936 1.34936i 0.463002 0.886357i \(-0.346772\pi\)
0.886357 0.463002i \(-0.153228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −24.8205 + 6.65064i −1.20115 + 0.321847i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(432\) −10.3923 + 18.0000i −0.500000 + 0.866025i
\(433\) 30.3109 17.5000i 1.45665 0.840996i 0.457804 0.889053i \(-0.348636\pi\)
0.998845 + 0.0480569i \(0.0153029\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.78461 + 14.1244i −0.181250 + 0.676434i
\(437\) 0 0
\(438\) 0 0
\(439\) −34.5000 19.9186i −1.64660 0.950662i −0.978412 0.206666i \(-0.933739\pi\)
−0.668184 0.743996i \(-0.732928\pi\)
\(440\) 0 0
\(441\) 5.41154i 0.257693i
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.94744 + 7.26795i 0.0924215 + 0.344922i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −22.9282 6.14359i −1.08326 0.290258i
\(449\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −33.4186 + 8.95448i −1.57014 + 0.420718i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.72243 + 36.2846i 0.454796 + 1.69732i 0.688686 + 0.725059i \(0.258188\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(462\) 0 0
\(463\) −22.3660 22.3660i −1.03944 1.03944i −0.999190 0.0402476i \(-0.987185\pi\)
−0.0402476 0.999190i \(-0.512815\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −6.00000 20.7846i −0.277350 0.960769i
\(469\) −2.35898 −0.108928
\(470\) 0 0
\(471\) −9.52628 + 16.5000i −0.438948 + 0.760280i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.4904 39.1506i 0.481332 1.79635i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(480\) 0 0
\(481\) −6.85641 3.78461i −0.312625 0.172563i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 + 19.0526i −0.500000 + 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −27.6865 7.41858i −1.25460 0.336168i −0.430486 0.902597i \(-0.641658\pi\)
−0.824110 + 0.566429i \(0.808325\pi\)
\(488\) 0 0
\(489\) 26.1506 + 26.1506i 1.18257 + 1.18257i
\(490\) 0 0
\(491\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 11.4641 + 42.7846i 0.514753 + 1.92109i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.411543 0.411543i 0.0184232 0.0184232i −0.697835 0.716258i \(-0.745853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.9186 + 10.5000i 0.884615 + 0.466321i
\(508\) 2.00000 0.0887357
\(509\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(510\) 0 0
\(511\) 34.0359 19.6506i 1.50566 0.869293i
\(512\) 0 0
\(513\) −40.6865 10.9019i −1.79635 0.481332i
\(514\) 0 0
\(515\) 0 0
\(516\) −3.00000 5.19615i −0.132068 0.228748i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 4.00000 6.92820i 0.174908 0.302949i −0.765222 0.643767i \(-0.777371\pi\)
0.940129 + 0.340818i \(0.110704\pi\)
\(524\) 0 0
\(525\) 18.1699 18.1699i 0.792998 0.792998i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 48.1051i 2.08562i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 13.1506 + 13.1506i 0.565390 + 0.565390i 0.930834 0.365444i \(-0.119083\pi\)
−0.365444 + 0.930834i \(0.619083\pi\)
\(542\) 0 0
\(543\) 10.3923 + 6.00000i 0.445976 + 0.257485i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 41.0000 1.75303 0.876517 0.481371i \(-0.159861\pi\)
0.876517 + 0.481371i \(0.159861\pi\)
\(548\) 0 0
\(549\) 12.9904 22.5000i 0.554416 0.960277i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −9.31089 + 34.7487i −0.395939 + 1.47767i
\(554\) 0 0
\(555\) 0 0
\(556\) −12.1244 7.00000i −0.514187 0.296866i
\(557\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(558\) 0 0
\(559\) 6.06218 + 1.50000i 0.256403 + 0.0634432i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −18.8827 18.8827i −0.792998 0.792998i
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 16.0000i 0.669579i 0.942293 + 0.334790i \(0.108665\pi\)
−0.942293 + 0.334790i \(0.891335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 20.7846 12.0000i 0.866025 0.500000i
\(577\) −16.0718 + 16.0718i −0.669078 + 0.669078i −0.957503 0.288425i \(-0.906868\pi\)
0.288425 + 0.957503i \(0.406868\pi\)
\(578\) 0 0
\(579\) 12.2321 45.6506i 0.508347 1.89718i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(588\) 3.12436 5.41154i 0.128846 0.223168i
\(589\) −77.7391 + 44.8827i −3.20318 + 1.84936i
\(590\) 0 0
\(591\) 0 0
\(592\) 2.24871 8.39230i 0.0924215 0.344922i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 29.4449i 1.20510i
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −20.7846 + 36.0000i −0.847822 + 1.46847i 0.0353259 + 0.999376i \(0.488753\pi\)
−0.883148 + 0.469095i \(0.844580\pi\)
\(602\) 0 0
\(603\) 1.68653 1.68653i 0.0686810 0.0686810i
\(604\) 38.5885 + 10.3397i 1.57014 + 0.420718i
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0000 + 17.3205i 0.405887 + 0.703018i 0.994424 0.105453i \(-0.0336291\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −12.7942 47.7487i −0.516754 1.92855i −0.314806 0.949156i \(-0.601939\pi\)
−0.201948 0.979396i \(-0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(618\) 0 0
\(619\) −31.8827 31.8827i −1.28147 1.28147i −0.939829 0.341644i \(-0.889016\pi\)
−0.341644 0.939829i \(-0.610984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 + 24.2487i −0.240192 + 0.970725i
\(625\) −25.0000 −1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 19.0526 11.0000i 0.760280 0.438948i
\(629\) 0 0
\(630\) 0 0
\(631\) −9.00962 + 33.6244i −0.358667 + 1.33856i 0.517139 + 0.855901i \(0.326997\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) −22.5000 38.9711i −0.894295 1.54896i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.80385 + 6.24871i 0.0714710 + 0.247583i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) 19.0885 + 5.11474i 0.752775 + 0.201706i 0.614749 0.788723i \(-0.289257\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −56.9090 −2.23044
\(652\) −11.0526 41.2487i −0.432852 1.61542i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.2846 + 38.3827i −0.401241 + 1.49745i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 44.1147 11.8205i 1.71586 0.459764i 0.739014 0.673690i \(-0.235292\pi\)
0.976850 + 0.213925i \(0.0686249\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 20.8301 + 5.58142i 0.805339 + 0.215790i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 43.5000 + 25.1147i 1.67680 + 0.968102i 0.963679 + 0.267063i \(0.0860531\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 0 0
\(675\) 25.9808i 1.00000i
\(676\) −13.8564 22.0000i −0.532939 0.846154i
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 25.2321 43.7032i 0.968317 1.67717i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(684\) 34.3923 + 34.3923i 1.31502 + 1.31502i
\(685\) 0 0
\(686\) 0 0
\(687\) 50.6147 13.5622i 1.93107 0.517429i
\(688\) 6.92820i 0.264135i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.48076 + 5.52628i 0.0563308 + 0.210230i 0.988355 0.152167i \(-0.0486252\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −28.6603 + 7.67949i −1.08326 + 0.290258i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 17.6077 0.664087
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.74871 32.6506i 0.328565 1.22622i −0.582115 0.813107i \(-0.697775\pi\)
0.910679 0.413114i \(-0.135559\pi\)
\(710\) 0 0
\(711\) −18.1865 31.5000i −0.682048 1.18134i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) 0 0
\(721\) 44.6769 + 11.9711i 1.66386 + 0.445829i
\(722\) 0 0
\(723\) 11.8756 + 11.8756i 0.441660 + 0.441660i
\(724\) −6.92820 12.0000i −0.257485 0.445976i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i −0.417548 0.908655i \(-0.637111\pi\)
0.417548 0.908655i \(-0.362889\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −25.9808 + 15.0000i −0.960277 + 0.554416i
\(733\) −30.3468 + 30.3468i −1.12088 + 1.12088i −0.129275 + 0.991609i \(0.541265\pi\)
−0.991609 + 0.129275i \(0.958735\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −46.4186 + 12.4378i −1.70754 + 0.457533i −0.974818 0.223001i \(-0.928415\pi\)
−0.732717 + 0.680534i \(0.761748\pi\)
\(740\) 0 0
\(741\) −50.6147 + 0.973721i −1.85938 + 0.0357705i
\(742\) 0 0
\(743\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −15.0000 8.66025i −0.547358 0.316017i 0.200698 0.979653i \(-0.435679\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 7.98076 + 29.7846i 0.290258 + 1.08326i
\(757\) 24.2487 42.0000i 0.881334 1.52652i 0.0314762 0.999505i \(-0.489979\pi\)
0.849858 0.527011i \(-0.176688\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(762\) 0 0
\(763\) 10.8468 + 18.7872i 0.392680 + 0.680142i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −27.7128 −1.00000
\(769\) −9.77757 36.4904i −0.352588 1.31588i −0.883493 0.468445i \(-0.844814\pi\)
0.530904 0.847432i \(-0.321852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −38.5885 + 38.5885i −1.38883 + 1.38883i
\(773\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(774\) 0 0
\(775\) 39.1506 + 39.1506i 1.40633 + 1.40633i
\(776\) 0 0
\(777\) 9.66730 + 5.58142i 0.346812 + 0.200232i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −6.24871 + 3.60770i −0.223168 + 0.128846i
\(785\) 0 0
\(786\) 0 0
\(787\) −13.7679 + 51.3827i −0.490774 + 1.83159i 0.0617409 + 0.998092i \(0.480335\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.50000 30.3109i 0.266333 1.07637i
\(794\) 0 0
\(795\) 0 0
\(796\) 17.0000 29.4449i 0.602549 1.04365i
\(797\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −2.66025 + 0.712813i −0.0938199 + 0.0251390i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(810\) 0 0
\(811\) −17.3468 + 17.3468i −0.609128 + 0.609128i −0.942718 0.333590i \(-0.891740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) −4.25129 + 15.8660i −0.149099 + 0.556446i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −13.5622 + 3.63397i −0.474481 + 0.127137i
\(818\) 0 0
\(819\) −28.0981 15.5096i −0.981826 0.541950i
\(820\) 0 0
\(821\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(822\) 0 0
\(823\) −21.0000 + 12.1244i −0.732014 + 0.422628i −0.819159 0.573567i \(-0.805559\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(828\) 0 0
\(829\) −45.8993 26.5000i −1.59415 0.920383i −0.992584 0.121560i \(-0.961210\pi\)
−0.601566 0.798823i \(-0.705456\pi\)
\(830\) 0 0
\(831\) 36.0000i 1.24883i
\(832\) 20.0000 20.7846i 0.693375 0.720577i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 40.6865 40.6865i 1.40633 1.40633i
\(838\) 0 0
\(839\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(840\) 0 0
\(841\) −14.5000 25.1147i −0.500000 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 51.9615i 1.78859i
\(845\) 0 0
\(846\) 0 0
\(847\) 8.44744 + 31.5263i 0.290258 + 1.08326i
\(848\) 0 0
\(849\) 37.5000 21.6506i 1.28700 0.743048i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −5.88269 5.88269i −0.201419 0.201419i 0.599189 0.800608i \(-0.295490\pi\)
−0.800608 + 0.599189i \(0.795490\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 57.1577 1.95019 0.975097 0.221777i \(-0.0711857\pi\)
0.975097 + 0.221777i \(0.0711857\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −14.7224 25.5000i −0.500000 0.866025i
\(868\) 56.9090 + 32.8564i 1.93162 + 1.11522i
\(869\) 0 0
\(870\) 0 0
\(871\) 1.38526 2.50962i 0.0469379 0.0850352i
\(872\) 0 0
\(873\) 13.2058 + 49.2846i 0.446948 + 1.66803i
\(874\) 0 0
\(875\) 0 0
\(876\) 32.4449 32.4449i 1.09621 1.09621i
\(877\) 56.3468 + 15.0981i 1.90270 + 0.509826i 0.996144 + 0.0877308i \(0.0279615\pi\)
0.906552 + 0.422095i \(0.138705\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(882\) 0 0
\(883\) 55.0000i 1.85090i 0.378873 + 0.925449i \(0.376312\pi\)
−0.378873 + 0.925449i \(0.623688\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 2.09808 2.09808i 0.0703672 0.0703672i
\(890\) 0 0
\(891\) 0 0
\(892\) −17.6077 17.6077i −0.589549 0.589549i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 15.0000 25.9808i 0.500000 0.866025i
\(901\) 0 0
\(902\) 0 0
\(903\) −8.59808 2.30385i −0.286126 0.0766672i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −34.6410 20.0000i −1.15024 0.664089i −0.201291 0.979531i \(-0.564514\pi\)
−0.948945 + 0.315442i \(0.897847\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −14.5359 54.2487i −0.481332 1.79635i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −58.4449 15.6603i −1.93107 0.517429i
\(917\) 0 0
\(918\) 0 0
\(919\) 15.5885 + 27.0000i 0.514216 + 0.890648i 0.999864 + 0.0164935i \(0.00525028\pi\)
−0.485648 + 0.874154i \(0.661416\pi\)
\(920\) 0 0
\(921\) 39.3564 10.5455i 1.29684 0.347487i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −2.81089 10.4904i −0.0924215 0.344922i
\(926\) 0 0
\(927\) −40.5000 + 23.3827i −1.33019 + 0.767988i
\(928\) 0 0
\(929\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(930\) 0 0
\(931\) −10.3397 10.3397i −0.338871 0.338871i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −55.4256 −1.81068 −0.905338 0.424691i \(-0.860383\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 0 0
\(939\) −28.5000 + 49.3634i −0.930062 + 1.61092i
\(940\) 0 0
\(941\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(948\) 42.0000i 1.36410i
\(949\) 0.918584 + 47.7487i 0.0298185 + 1.54999i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 91.6218i 2.95554i
\(962\) 0 0
\(963\) 0 0
\(964\) −5.01924 18.7321i −0.161659 0.603319i
\(965\) 0 0
\(966\) 0 0
\(967\) 19.4449 19.4449i 0.625305 0.625305i −0.321578 0.946883i \(-0.604213\pi\)
0.946883 + 0.321578i \(0.104213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(972\) −27.0000 15.5885i −0.866025 0.500000i
\(973\) −20.0622 + 5.37564i −0.643164 + 0.172335i
\(974\) 0 0
\(975\) 8.66025 + 30.0000i 0.277350 + 0.960769i
\(976\) 34.6410 1.10883
\(977\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −21.1865 5.67691i −0.676434 0.181250i
\(982\) 0 0
\(983\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 51.1769 + 28.2487i 1.62815 + 0.898711i
\(989\) 0 0
\(990\) 0 0
\(991\) −22.0000 + 38.1051i −0.698853 + 1.21045i 0.270011 + 0.962857i \(0.412973\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) −43.3468 + 43.3468i −1.37557 + 1.37557i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 29.5000 + 51.0955i 0.934274 + 1.61821i 0.775923 + 0.630828i \(0.217285\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 0 0
\(999\) −10.9019 + 2.92116i −0.344922 + 0.0924215i
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 39.2.k.a.32.1 yes 4
3.2 odd 2 CM 39.2.k.a.32.1 yes 4
4.3 odd 2 624.2.cn.b.305.1 4
5.2 odd 4 975.2.bp.a.149.1 4
5.3 odd 4 975.2.bp.d.149.1 4
5.4 even 2 975.2.bo.c.851.1 4
12.11 even 2 624.2.cn.b.305.1 4
13.2 odd 12 507.2.k.c.89.1 4
13.3 even 3 507.2.k.b.80.1 4
13.4 even 6 507.2.f.b.239.1 4
13.5 odd 4 507.2.k.a.488.1 4
13.6 odd 12 507.2.f.b.437.1 4
13.7 odd 12 507.2.f.c.437.1 4
13.8 odd 4 507.2.k.b.488.1 4
13.9 even 3 507.2.f.c.239.1 4
13.10 even 6 507.2.k.a.80.1 4
13.11 odd 12 inner 39.2.k.a.11.1 4
13.12 even 2 507.2.k.c.188.1 4
15.2 even 4 975.2.bp.a.149.1 4
15.8 even 4 975.2.bp.d.149.1 4
15.14 odd 2 975.2.bo.c.851.1 4
39.2 even 12 507.2.k.c.89.1 4
39.5 even 4 507.2.k.a.488.1 4
39.8 even 4 507.2.k.b.488.1 4
39.11 even 12 inner 39.2.k.a.11.1 4
39.17 odd 6 507.2.f.b.239.1 4
39.20 even 12 507.2.f.c.437.1 4
39.23 odd 6 507.2.k.a.80.1 4
39.29 odd 6 507.2.k.b.80.1 4
39.32 even 12 507.2.f.b.437.1 4
39.35 odd 6 507.2.f.c.239.1 4
39.38 odd 2 507.2.k.c.188.1 4
52.11 even 12 624.2.cn.b.401.1 4
65.24 odd 12 975.2.bo.c.401.1 4
65.37 even 12 975.2.bp.d.674.1 4
65.63 even 12 975.2.bp.a.674.1 4
156.11 odd 12 624.2.cn.b.401.1 4
195.89 even 12 975.2.bo.c.401.1 4
195.128 odd 12 975.2.bp.a.674.1 4
195.167 odd 12 975.2.bp.d.674.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.k.a.11.1 4 13.11 odd 12 inner
39.2.k.a.11.1 4 39.11 even 12 inner
39.2.k.a.32.1 yes 4 1.1 even 1 trivial
39.2.k.a.32.1 yes 4 3.2 odd 2 CM
507.2.f.b.239.1 4 13.4 even 6
507.2.f.b.239.1 4 39.17 odd 6
507.2.f.b.437.1 4 13.6 odd 12
507.2.f.b.437.1 4 39.32 even 12
507.2.f.c.239.1 4 13.9 even 3
507.2.f.c.239.1 4 39.35 odd 6
507.2.f.c.437.1 4 13.7 odd 12
507.2.f.c.437.1 4 39.20 even 12
507.2.k.a.80.1 4 13.10 even 6
507.2.k.a.80.1 4 39.23 odd 6
507.2.k.a.488.1 4 13.5 odd 4
507.2.k.a.488.1 4 39.5 even 4
507.2.k.b.80.1 4 13.3 even 3
507.2.k.b.80.1 4 39.29 odd 6
507.2.k.b.488.1 4 13.8 odd 4
507.2.k.b.488.1 4 39.8 even 4
507.2.k.c.89.1 4 13.2 odd 12
507.2.k.c.89.1 4 39.2 even 12
507.2.k.c.188.1 4 13.12 even 2
507.2.k.c.188.1 4 39.38 odd 2
624.2.cn.b.305.1 4 4.3 odd 2
624.2.cn.b.305.1 4 12.11 even 2
624.2.cn.b.401.1 4 52.11 even 12
624.2.cn.b.401.1 4 156.11 odd 12
975.2.bo.c.401.1 4 65.24 odd 12
975.2.bo.c.401.1 4 195.89 even 12
975.2.bo.c.851.1 4 5.4 even 2
975.2.bo.c.851.1 4 15.14 odd 2
975.2.bp.a.149.1 4 5.2 odd 4
975.2.bp.a.149.1 4 15.2 even 4
975.2.bp.a.674.1 4 65.63 even 12
975.2.bp.a.674.1 4 195.128 odd 12
975.2.bp.d.149.1 4 5.3 odd 4
975.2.bp.d.149.1 4 15.8 even 4
975.2.bp.d.674.1 4 65.37 even 12
975.2.bp.d.674.1 4 195.167 odd 12