Properties

Label 1008.2.s.m.865.1
Level $1008$
Weight $2$
Character 1008.865
Analytic conductor $8.049$
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] = [1008,2,Mod(289,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.s (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: \(8.04892052375\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1008.865
Dual form 1008.2.s.m.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 + 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +O(q^{10})\) \(q+(1.00000 + 1.73205i) q^{5} +(-2.50000 - 0.866025i) q^{7} +(3.00000 - 5.19615i) q^{11} -3.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-2.50000 - 4.33013i) q^{19} +(2.00000 + 3.46410i) q^{23} +(0.500000 - 0.866025i) q^{25} +4.00000 q^{29} +(3.50000 - 6.06218i) q^{31} +(-1.00000 - 5.19615i) q^{35} +(4.50000 + 7.79423i) q^{37} +2.00000 q^{41} +1.00000 q^{43} +(-1.00000 - 1.73205i) q^{47} +(5.50000 + 4.33013i) q^{49} +(4.00000 - 6.92820i) q^{53} +12.0000 q^{55} +(-5.00000 - 8.66025i) q^{61} +(-3.00000 - 5.19615i) q^{65} +(-7.50000 + 12.9904i) q^{67} -6.00000 q^{71} +(5.50000 - 9.52628i) q^{73} +(-12.0000 + 10.3923i) q^{77} +(0.500000 + 0.866025i) q^{79} +6.00000 q^{83} +8.00000 q^{85} +(-4.00000 - 6.92820i) q^{89} +(7.50000 + 2.59808i) q^{91} +(5.00000 - 8.66025i) q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 5 q^{7} + 6 q^{11} - 6 q^{13} + 4 q^{17} - 5 q^{19} + 4 q^{23} + q^{25} + 8 q^{29} + 7 q^{31} - 2 q^{35} + 9 q^{37} + 4 q^{41} + 2 q^{43} - 2 q^{47} + 11 q^{49} + 8 q^{53} + 24 q^{55} - 10 q^{61} - 6 q^{65} - 15 q^{67} - 12 q^{71} + 11 q^{73} - 24 q^{77} + q^{79} + 12 q^{83} + 16 q^{85} - 8 q^{89} + 15 q^{91} + 10 q^{95} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 + 1.73205i 0.447214 + 0.774597i 0.998203 0.0599153i \(-0.0190830\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(6\) 0 0
\(7\) −2.50000 0.866025i −0.944911 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 5.19615i 0.904534 1.56670i 0.0829925 0.996550i \(-0.473552\pi\)
0.821541 0.570149i \(-0.193114\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.00000 3.46410i 0.485071 0.840168i −0.514782 0.857321i \(-0.672127\pi\)
0.999853 + 0.0171533i \(0.00546033\pi\)
\(18\) 0 0
\(19\) −2.50000 4.33013i −0.573539 0.993399i −0.996199 0.0871106i \(-0.972237\pi\)
0.422659 0.906289i \(-0.361097\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 + 3.46410i 0.417029 + 0.722315i 0.995639 0.0932891i \(-0.0297381\pi\)
−0.578610 + 0.815604i \(0.696405\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 3.50000 6.06218i 0.628619 1.08880i −0.359211 0.933257i \(-0.616954\pi\)
0.987829 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.00000 5.19615i −0.169031 0.878310i
\(36\) 0 0
\(37\) 4.50000 + 7.79423i 0.739795 + 1.28136i 0.952587 + 0.304266i \(0.0984111\pi\)
−0.212792 + 0.977098i \(0.568256\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 1.73205i −0.145865 0.252646i 0.783830 0.620975i \(-0.213263\pi\)
−0.929695 + 0.368329i \(0.879930\pi\)
\(48\) 0 0
\(49\) 5.50000 + 4.33013i 0.785714 + 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 6.92820i 0.549442 0.951662i −0.448871 0.893597i \(-0.648174\pi\)
0.998313 0.0580651i \(-0.0184931\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −5.00000 8.66025i −0.640184 1.10883i −0.985391 0.170305i \(-0.945525\pi\)
0.345207 0.938527i \(-0.387809\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.00000 5.19615i −0.372104 0.644503i
\(66\) 0 0
\(67\) −7.50000 + 12.9904i −0.916271 + 1.58703i −0.111241 + 0.993793i \(0.535483\pi\)
−0.805030 + 0.593234i \(0.797851\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 5.50000 9.52628i 0.643726 1.11497i −0.340868 0.940111i \(-0.610721\pi\)
0.984594 0.174855i \(-0.0559458\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −12.0000 + 10.3923i −1.36753 + 1.18431i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) −4.00000 6.92820i −0.423999 0.734388i 0.572327 0.820025i \(-0.306041\pi\)
−0.996326 + 0.0856373i \(0.972707\pi\)
\(90\) 0 0
\(91\) 7.50000 + 2.59808i 0.786214 + 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.00000 8.66025i 0.512989 0.888523i
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 + 5.19615i −0.298511 + 0.517036i −0.975796 0.218685i \(-0.929823\pi\)
0.677284 + 0.735721i \(0.263157\pi\)
\(102\) 0 0
\(103\) −4.50000 7.79423i −0.443398 0.767988i 0.554541 0.832156i \(-0.312894\pi\)
−0.997939 + 0.0641683i \(0.979561\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 + 10.3923i 0.580042 + 1.00466i 0.995474 + 0.0950377i \(0.0302972\pi\)
−0.415432 + 0.909624i \(0.636370\pi\)
\(108\) 0 0
\(109\) 5.50000 9.52628i 0.526804 0.912452i −0.472708 0.881219i \(-0.656723\pi\)
0.999512 0.0312328i \(-0.00994332\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 + 6.92820i −0.373002 + 0.646058i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.00000 + 6.92820i −0.733359 + 0.635107i
\(120\) 0 0
\(121\) −12.5000 21.6506i −1.13636 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 1.00000 0.0887357 0.0443678 0.999015i \(-0.485873\pi\)
0.0443678 + 0.999015i \(0.485873\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.00000 12.1244i −0.611593 1.05931i −0.990972 0.134069i \(-0.957196\pi\)
0.379379 0.925241i \(-0.376138\pi\)
\(132\) 0 0
\(133\) 2.50000 + 12.9904i 0.216777 + 1.12641i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 17.3205i 0.854358 1.47979i −0.0228820 0.999738i \(-0.507284\pi\)
0.877240 0.480053i \(-0.159382\pi\)
\(138\) 0 0
\(139\) 9.00000 0.763370 0.381685 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.00000 + 15.5885i −0.752618 + 1.30357i
\(144\) 0 0
\(145\) 4.00000 + 6.92820i 0.332182 + 0.575356i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.00000 + 3.46410i 0.163846 + 0.283790i 0.936245 0.351348i \(-0.114277\pi\)
−0.772399 + 0.635138i \(0.780943\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 14.0000 1.12451
\(156\) 0 0
\(157\) −9.00000 + 15.5885i −0.718278 + 1.24409i 0.243403 + 0.969925i \(0.421736\pi\)
−0.961681 + 0.274169i \(0.911597\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.00000 10.3923i −0.157622 0.819028i
\(162\) 0 0
\(163\) 2.00000 + 3.46410i 0.156652 + 0.271329i 0.933659 0.358162i \(-0.116597\pi\)
−0.777007 + 0.629492i \(0.783263\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.0000 + 17.3205i 0.760286 + 1.31685i 0.942703 + 0.333633i \(0.108275\pi\)
−0.182417 + 0.983221i \(0.558392\pi\)
\(174\) 0 0
\(175\) −2.00000 + 1.73205i −0.151186 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.0000 + 22.5167i −0.971666 + 1.68297i −0.281139 + 0.959667i \(0.590712\pi\)
−0.690526 + 0.723307i \(0.742621\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.00000 + 15.5885i −0.661693 + 1.14609i
\(186\) 0 0
\(187\) −12.0000 20.7846i −0.877527 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −5.00000 8.66025i −0.361787 0.626634i 0.626468 0.779447i \(-0.284500\pi\)
−0.988255 + 0.152813i \(0.951167\pi\)
\(192\) 0 0
\(193\) −1.50000 + 2.59808i −0.107972 + 0.187014i −0.914949 0.403570i \(-0.867769\pi\)
0.806976 + 0.590584i \(0.201102\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −8.00000 + 13.8564i −0.567105 + 0.982255i 0.429745 + 0.902950i \(0.358603\pi\)
−0.996850 + 0.0793045i \(0.974730\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −10.0000 3.46410i −0.701862 0.243132i
\(204\) 0 0
\(205\) 2.00000 + 3.46410i 0.139686 + 0.241943i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.00000 + 1.73205i 0.0681994 + 0.118125i
\(216\) 0 0
\(217\) −14.0000 + 12.1244i −0.950382 + 0.823055i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −7.00000 + 12.1244i −0.464606 + 0.804722i −0.999184 0.0403978i \(-0.987137\pi\)
0.534577 + 0.845120i \(0.320471\pi\)
\(228\) 0 0
\(229\) 3.50000 + 6.06218i 0.231287 + 0.400600i 0.958187 0.286143i \(-0.0923732\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.0000 + 22.5167i 0.851658 + 1.47512i 0.879711 + 0.475509i \(0.157736\pi\)
−0.0280525 + 0.999606i \(0.508931\pi\)
\(234\) 0 0
\(235\) 2.00000 3.46410i 0.130466 0.225973i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 1.00000 1.73205i 0.0644157 0.111571i −0.832019 0.554747i \(-0.812815\pi\)
0.896435 + 0.443176i \(0.146148\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.00000 + 13.8564i −0.127775 + 0.885253i
\(246\) 0 0
\(247\) 7.50000 + 12.9904i 0.477214 + 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.00000 15.5885i −0.561405 0.972381i −0.997374 0.0724199i \(-0.976928\pi\)
0.435970 0.899961i \(-0.356405\pi\)
\(258\) 0 0
\(259\) −4.50000 23.3827i −0.279616 1.45293i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 10.3923i 0.369976 0.640817i −0.619586 0.784929i \(-0.712699\pi\)
0.989561 + 0.144112i \(0.0460326\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.00000 + 15.5885i −0.548740 + 0.950445i 0.449622 + 0.893219i \(0.351559\pi\)
−0.998361 + 0.0572259i \(0.981774\pi\)
\(270\) 0 0
\(271\) 4.00000 + 6.92820i 0.242983 + 0.420858i 0.961563 0.274586i \(-0.0885408\pi\)
−0.718580 + 0.695444i \(0.755208\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.00000 5.19615i −0.180907 0.313340i
\(276\) 0 0
\(277\) −0.500000 + 0.866025i −0.0300421 + 0.0520344i −0.880656 0.473757i \(-0.842897\pi\)
0.850613 + 0.525792i \(0.176231\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −0.500000 + 0.866025i −0.0297219 + 0.0514799i −0.880504 0.474039i \(-0.842796\pi\)
0.850782 + 0.525519i \(0.176129\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.00000 1.73205i −0.295141 0.102240i
\(288\) 0 0
\(289\) 0.500000 + 0.866025i 0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 10.3923i −0.346989 0.601003i
\(300\) 0 0
\(301\) −2.50000 0.866025i −0.144098 0.0499169i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 10.0000 17.3205i 0.572598 0.991769i
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.00000 15.5885i 0.510343 0.883940i −0.489585 0.871956i \(-0.662852\pi\)
0.999928 0.0119847i \(-0.00381495\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(318\) 0 0
\(319\) 12.0000 20.7846i 0.671871 1.16371i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −20.0000 −1.11283
\(324\) 0 0
\(325\) −1.50000 + 2.59808i −0.0832050 + 0.144115i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.00000 + 5.19615i 0.0551318 + 0.286473i
\(330\) 0 0
\(331\) 2.50000 + 4.33013i 0.137412 + 0.238005i 0.926516 0.376254i \(-0.122788\pi\)
−0.789104 + 0.614260i \(0.789455\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) 29.0000 1.57973 0.789865 0.613280i \(-0.210150\pi\)
0.789865 + 0.613280i \(0.210150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −21.0000 36.3731i −1.13721 1.96971i
\(342\) 0 0
\(343\) −10.0000 15.5885i −0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) −22.0000 −1.17763 −0.588817 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.00000 5.19615i 0.159674 0.276563i −0.775077 0.631867i \(-0.782289\pi\)
0.934751 + 0.355303i \(0.115622\pi\)
\(354\) 0 0
\(355\) −6.00000 10.3923i −0.318447 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 + 10.3923i 0.316668 + 0.548485i 0.979791 0.200026i \(-0.0641026\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 22.0000 1.15153
\(366\) 0 0
\(367\) −3.50000 + 6.06218i −0.182699 + 0.316443i −0.942799 0.333363i \(-0.891817\pi\)
0.760100 + 0.649806i \(0.225150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −16.0000 + 13.8564i −0.830679 + 0.719389i
\(372\) 0 0
\(373\) 6.50000 + 11.2583i 0.336557 + 0.582934i 0.983783 0.179364i \(-0.0574041\pi\)
−0.647225 + 0.762299i \(0.724071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 15.0000 0.770498 0.385249 0.922813i \(-0.374116\pi\)
0.385249 + 0.922813i \(0.374116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) −30.0000 10.3923i −1.52894 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.0000 + 22.5167i −0.659126 + 1.14164i 0.321716 + 0.946836i \(0.395740\pi\)
−0.980842 + 0.194804i \(0.937593\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.00000 + 1.73205i −0.0503155 + 0.0871489i
\(396\) 0 0
\(397\) 2.50000 + 4.33013i 0.125471 + 0.217323i 0.921917 0.387387i \(-0.126622\pi\)
−0.796446 + 0.604710i \(0.793289\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) −10.5000 + 18.1865i −0.523042 + 0.905936i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 54.0000 2.67668
\(408\) 0 0
\(409\) 1.50000 2.59808i 0.0741702 0.128467i −0.826555 0.562856i \(-0.809703\pi\)
0.900725 + 0.434389i \(0.143036\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000 + 10.3923i 0.294528 + 0.510138i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 3.46410i −0.0970143 0.168034i
\(426\) 0 0
\(427\) 5.00000 + 25.9808i 0.241967 + 1.25730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.00000 15.5885i 0.433515 0.750870i −0.563658 0.826008i \(-0.690607\pi\)
0.997173 + 0.0751385i \(0.0239399\pi\)
\(432\) 0 0
\(433\) 31.0000 1.48976 0.744882 0.667196i \(-0.232506\pi\)
0.744882 + 0.667196i \(0.232506\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.0000 17.3205i 0.478365 0.828552i
\(438\) 0 0
\(439\) 12.0000 + 20.7846i 0.572729 + 0.991995i 0.996284 + 0.0861252i \(0.0274485\pi\)
−0.423556 + 0.905870i \(0.639218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.00000 + 13.8564i 0.380091 + 0.658338i 0.991075 0.133306i \(-0.0425592\pi\)
−0.610984 + 0.791643i \(0.709226\pi\)
\(444\) 0 0
\(445\) 8.00000 13.8564i 0.379236 0.656857i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 38.0000 1.79333 0.896665 0.442709i \(-0.145982\pi\)
0.896665 + 0.442709i \(0.145982\pi\)
\(450\) 0 0
\(451\) 6.00000 10.3923i 0.282529 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 + 15.5885i 0.140642 + 0.730798i
\(456\) 0 0
\(457\) −6.50000 11.2583i −0.304057 0.526642i 0.672994 0.739648i \(-0.265008\pi\)
−0.977051 + 0.213006i \(0.931675\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −17.0000 −0.790057 −0.395029 0.918669i \(-0.629265\pi\)
−0.395029 + 0.918669i \(0.629265\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.0000 25.9808i −0.694117 1.20225i −0.970477 0.241192i \(-0.922462\pi\)
0.276360 0.961054i \(-0.410872\pi\)
\(468\) 0 0
\(469\) 30.0000 25.9808i 1.38527 1.19968i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 5.19615i 0.137940 0.238919i
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.00000 + 13.8564i −0.365529 + 0.633115i −0.988861 0.148842i \(-0.952445\pi\)
0.623332 + 0.781958i \(0.285779\pi\)
\(480\) 0 0
\(481\) −13.5000 23.3827i −0.615547 1.06616i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −14.0000 24.2487i −0.635707 1.10108i
\(486\) 0 0
\(487\) 12.5000 21.6506i 0.566429 0.981084i −0.430486 0.902597i \(-0.641658\pi\)
0.996915 0.0784867i \(-0.0250088\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) 0 0
\(493\) 8.00000 13.8564i 0.360302 0.624061i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15.0000 + 5.19615i 0.672842 + 0.233079i
\(498\) 0 0
\(499\) −8.50000 14.7224i −0.380512 0.659067i 0.610623 0.791921i \(-0.290919\pi\)
−0.991136 + 0.132855i \(0.957586\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.0000 −0.624229 −0.312115 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.00000 + 5.19615i 0.132973 + 0.230315i 0.924821 0.380402i \(-0.124214\pi\)
−0.791849 + 0.610718i \(0.790881\pi\)
\(510\) 0 0
\(511\) −22.0000 + 19.0526i −0.973223 + 0.842836i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.00000 15.5885i 0.396587 0.686909i
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 + 10.3923i −0.262865 + 0.455295i −0.967002 0.254769i \(-0.918001\pi\)
0.704137 + 0.710064i \(0.251334\pi\)
\(522\) 0 0
\(523\) 14.5000 + 25.1147i 0.634041 + 1.09819i 0.986718 + 0.162446i \(0.0519382\pi\)
−0.352677 + 0.935745i \(0.614728\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.0000 24.2487i −0.609850 1.05629i
\(528\) 0 0
\(529\) 3.50000 6.06218i 0.152174 0.263573i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) −12.0000 + 20.7846i −0.518805 + 0.898597i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 39.0000 15.5885i 1.67985 0.671442i
\(540\) 0 0
\(541\) −0.500000 0.866025i −0.0214967 0.0372333i 0.855077 0.518501i \(-0.173510\pi\)
−0.876574 + 0.481268i \(0.840176\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 22.0000 0.942376
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.0000 17.3205i −0.426014 0.737878i
\(552\) 0 0
\(553\) −0.500000 2.59808i −0.0212622 0.110481i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.00000 + 1.73205i −0.0423714 + 0.0733893i −0.886433 0.462856i \(-0.846825\pi\)
0.844062 + 0.536246i \(0.180158\pi\)
\(558\) 0 0
\(559\) −3.00000 −0.126886
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.00000 1.73205i 0.0421450 0.0729972i −0.844183 0.536054i \(-0.819914\pi\)
0.886328 + 0.463057i \(0.153248\pi\)
\(564\) 0 0
\(565\) −6.00000 10.3923i −0.252422 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.00000 15.5885i −0.377300 0.653502i 0.613369 0.789797i \(-0.289814\pi\)
−0.990668 + 0.136295i \(0.956481\pi\)
\(570\) 0 0
\(571\) 11.5000 19.9186i 0.481260 0.833567i −0.518509 0.855072i \(-0.673513\pi\)
0.999769 + 0.0215055i \(0.00684595\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −19.5000 + 33.7750i −0.811796 + 1.40607i 0.0998105 + 0.995006i \(0.468176\pi\)
−0.911606 + 0.411065i \(0.865157\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.0000 5.19615i −0.622305 0.215573i
\(582\) 0 0
\(583\) −24.0000 41.5692i −0.993978 1.72162i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −35.0000 −1.44215
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −15.0000 25.9808i −0.615976 1.06690i −0.990212 0.139569i \(-0.955428\pi\)
0.374236 0.927333i \(-0.377905\pi\)
\(594\) 0 0
\(595\) −20.0000 6.92820i −0.819920 0.284029i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.00000 + 3.46410i −0.0817178 + 0.141539i −0.903988 0.427558i \(-0.859374\pi\)
0.822270 + 0.569097i \(0.192707\pi\)
\(600\) 0 0
\(601\) 31.0000 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 25.0000 43.3013i 1.01639 1.76045i
\(606\) 0 0
\(607\) 0.500000 + 0.866025i 0.0202944 + 0.0351509i 0.875994 0.482322i \(-0.160206\pi\)
−0.855700 + 0.517472i \(0.826873\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.00000 + 5.19615i 0.121367 + 0.210214i
\(612\) 0 0
\(613\) 19.0000 32.9090i 0.767403 1.32918i −0.171564 0.985173i \(-0.554882\pi\)
0.938967 0.344008i \(-0.111785\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 4.50000 7.79423i 0.180870 0.313276i −0.761307 0.648392i \(-0.775442\pi\)
0.942177 + 0.335115i \(0.108775\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.00000 + 20.7846i 0.160257 + 0.832718i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.0000 1.43541
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.00000 + 1.73205i 0.0396838 + 0.0687343i
\(636\) 0 0
\(637\) −16.5000 12.9904i −0.653754 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0000 + 17.3205i −0.394976 + 0.684119i −0.993098 0.117286i \(-0.962581\pi\)
0.598122 + 0.801405i \(0.295914\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.00000 15.5885i 0.353827 0.612845i −0.633090 0.774078i \(-0.718214\pi\)
0.986916 + 0.161233i \(0.0515470\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 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
\(655\) 14.0000 24.2487i 0.547025 0.947476i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) −17.5000 + 30.3109i −0.680671 + 1.17896i 0.294105 + 0.955773i \(0.404978\pi\)
−0.974776 + 0.223184i \(0.928355\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −20.0000 + 17.3205i −0.775567 + 0.671660i
\(666\) 0 0
\(667\) 8.00000 + 13.8564i 0.309761 + 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −60.0000 −2.31627
\(672\) 0 0
\(673\) 7.00000 0.269830 0.134915 0.990857i \(-0.456924\pi\)
0.134915 + 0.990857i \(0.456924\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 + 10.3923i 0.230599 + 0.399409i 0.957984 0.286820i \(-0.0925982\pi\)
−0.727386 + 0.686229i \(0.759265\pi\)
\(678\) 0 0
\(679\) 35.0000 + 12.1244i 1.34318 + 0.465290i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 40.0000 1.52832
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 + 20.7846i −0.457164 + 0.791831i
\(690\) 0 0
\(691\) −3.50000 6.06218i −0.133146 0.230616i 0.791742 0.610856i \(-0.209175\pi\)
−0.924888 + 0.380240i \(0.875841\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.00000 + 15.5885i 0.341389 + 0.591304i
\(696\) 0 0
\(697\) 4.00000 6.92820i 0.151511 0.262424i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) 0 0
\(703\) 22.5000 38.9711i 0.848604 1.46982i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 10.3923i 0.451306 0.390843i
\(708\) 0 0
\(709\) 25.0000 + 43.3013i 0.938895 + 1.62621i 0.767537 + 0.641004i \(0.221482\pi\)
0.171358 + 0.985209i \(0.445185\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.0000 + 25.9808i 0.559406 + 0.968919i 0.997546 + 0.0700124i \(0.0223039\pi\)
−0.438141 + 0.898906i \(0.644363\pi\)
\(720\) 0 0
\(721\) 4.50000 + 23.3827i 0.167589 + 0.870817i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 3.46410i 0.0742781 0.128654i
\(726\) 0 0
\(727\) −5.00000 −0.185440 −0.0927199 0.995692i \(-0.529556\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.00000 3.46410i 0.0739727 0.128124i
\(732\) 0 0
\(733\) 5.50000 + 9.52628i 0.203147 + 0.351861i 0.949541 0.313644i \(-0.101550\pi\)
−0.746394 + 0.665505i \(0.768216\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 45.0000 + 77.9423i 1.65760 + 2.87104i
\(738\) 0 0
\(739\) −2.50000 + 4.33013i −0.0919640 + 0.159286i −0.908337 0.418238i \(-0.862648\pi\)
0.816373 + 0.577524i \(0.195981\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) −4.00000 + 6.92820i −0.146549 + 0.253830i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 31.1769i −0.219235 1.13918i
\(750\) 0 0
\(751\) −18.5000 32.0429i −0.675075 1.16926i −0.976447 0.215757i \(-0.930778\pi\)
0.301373 0.953506i \(-0.402555\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 10.3923i −0.217500 0.376721i 0.736543 0.676391i \(-0.236457\pi\)
−0.954043 + 0.299670i \(0.903123\pi\)
\(762\) 0 0
\(763\) −22.0000 + 19.0526i −0.796453 + 0.689749i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 7.00000 0.252426 0.126213 0.992003i \(-0.459718\pi\)
0.126213 + 0.992003i \(0.459718\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −25.0000 + 43.3013i −0.899188 + 1.55744i −0.0706526 + 0.997501i \(0.522508\pi\)
−0.828535 + 0.559937i \(0.810825\pi\)
\(774\) 0 0
\(775\) −3.50000 6.06218i −0.125724 0.217760i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.00000 8.66025i −0.179144 0.310286i
\(780\) 0 0
\(781\) −18.0000 + 31.1769i −0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −36.0000 −1.28490
\(786\) 0 0
\(787\) −16.0000 + 27.7128i −0.570338 + 0.987855i 0.426193 + 0.904632i \(0.359855\pi\)
−0.996531 + 0.0832226i \(0.973479\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 15.0000 + 5.19615i 0.533339 + 0.184754i
\(792\) 0 0
\(793\) 15.0000 + 25.9808i 0.532666 + 0.922604i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −33.0000 57.1577i −1.16454 2.01705i
\(804\) 0 0
\(805\) 16.0000 13.8564i 0.563926 0.488374i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15.0000 + 25.9808i −0.527372 + 0.913435i 0.472119 + 0.881535i \(0.343489\pi\)
−0.999491 + 0.0319002i \(0.989844\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.00000 + 6.92820i −0.140114 + 0.242684i
\(816\) 0 0
\(817\) −2.50000 4.33013i −0.0874639 0.151492i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9.00000 15.5885i −0.314102 0.544041i 0.665144 0.746715i \(-0.268370\pi\)
−0.979246 + 0.202674i \(0.935037\pi\)
\(822\) 0 0
\(823\) 4.00000 6.92820i 0.139431 0.241502i −0.787850 0.615867i \(-0.788806\pi\)
0.927281 + 0.374365i \(0.122139\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) 0 0
\(829\) 21.5000 37.2391i 0.746726 1.29337i −0.202658 0.979250i \(-0.564958\pi\)
0.949384 0.314118i \(-0.101709\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 26.0000 10.3923i 0.900847 0.360072i
\(834\) 0 0
\(835\) 18.0000 + 31.1769i 0.622916 + 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.00000 6.92820i −0.137604 0.238337i
\(846\) 0 0
\(847\) 12.5000 + 64.9519i 0.429505 + 2.23177i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −18.0000 + 31.1769i −0.617032 + 1.06873i
\(852\) 0 0
\(853\) −17.0000 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) 8.00000 + 13.8564i 0.272956 + 0.472774i 0.969618 0.244626i \(-0.0786652\pi\)
−0.696661 + 0.717400i \(0.745332\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −7.00000 12.1244i −0.238283 0.412718i 0.721939 0.691957i \(-0.243251\pi\)
−0.960222 + 0.279239i \(0.909918\pi\)
\(864\) 0 0
\(865\) −20.0000 + 34.6410i −0.680020 + 1.17783i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) 22.5000 38.9711i 0.762383 1.32049i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −30.0000 10.3923i −1.01419 0.351324i
\(876\) 0 0
\(877\) −1.00000 1.73205i −0.0337676 0.0584872i 0.848648 0.528958i \(-0.177417\pi\)
−0.882415 + 0.470471i \(0.844084\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 27.0000 + 46.7654i 0.906571 + 1.57023i 0.818794 + 0.574087i \(0.194643\pi\)
0.0877772 + 0.996140i \(0.472024\pi\)
\(888\) 0 0
\(889\) −2.50000 0.866025i −0.0838473 0.0290456i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −5.00000 + 8.66025i −0.167319 + 0.289804i
\(894\) 0 0
\(895\) −52.0000 −1.73817
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.0000 24.2487i 0.466926 0.808740i
\(900\) 0 0
\(901\) −16.0000 27.7128i −0.533037 0.923248i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.00000 12.1244i −0.232688 0.403027i
\(906\) 0 0
\(907\) 8.50000 14.7224i 0.282238 0.488850i −0.689698 0.724097i \(-0.742257\pi\)
0.971936 + 0.235247i \(0.0755899\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 18.0000 31.1769i 0.595713 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.00000 + 36.3731i 0.231160 + 1.20114i
\(918\) 0 0
\(919\) 0.500000 + 0.866025i 0.0164935 + 0.0285675i 0.874154 0.485648i \(-0.161416\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) 9.00000 0.295918
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 29.0000 + 50.2295i 0.951459 + 1.64798i 0.742271 + 0.670100i \(0.233749\pi\)
0.209189 + 0.977875i \(0.432918\pi\)
\(930\) 0 0
\(931\) 5.00000 34.6410i 0.163868 1.13531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.0000 41.5692i 0.784884 1.35946i
\(936\) 0 0
\(937\) −33.0000 −1.07806 −0.539032 0.842286i \(-0.681210\pi\)
−0.539032 + 0.842286i \(0.681210\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24.0000 41.5692i 0.782378 1.35512i −0.148176 0.988961i \(-0.547340\pi\)
0.930553 0.366157i \(-0.119327\pi\)
\(942\) 0 0
\(943\) 4.00000 + 6.92820i 0.130258 + 0.225613i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19.0000 + 32.9090i 0.617417 + 1.06940i 0.989955 + 0.141381i \(0.0451542\pi\)
−0.372538 + 0.928017i \(0.621512\pi\)
\(948\) 0 0
\(949\) −16.5000 + 28.5788i −0.535613 + 0.927708i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.00000 0.259145 0.129573 0.991570i \(-0.458639\pi\)
0.129573 + 0.991570i \(0.458639\pi\)
\(954\) 0 0
\(955\) 10.0000 17.3205i 0.323592 0.560478i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −40.0000 + 34.6410i −1.29167 + 1.11862i
\(960\) 0 0
\(961\) −9.00000 15.5885i −0.290323 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 27.0000 0.868261 0.434131 0.900850i \(-0.357056\pi\)
0.434131 + 0.900850i \(0.357056\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 28.0000 + 48.4974i 0.898563 + 1.55636i 0.829332 + 0.558756i \(0.188721\pi\)
0.0692304 + 0.997601i \(0.477946\pi\)
\(972\) 0 0
\(973\) −22.5000 7.79423i −0.721317 0.249871i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.00000 + 15.5885i −0.287936 + 0.498719i −0.973317 0.229465i \(-0.926302\pi\)
0.685381 + 0.728184i \(0.259636\pi\)
\(978\) 0 0
\(979\) −48.0000 −1.53409
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(984\) 0 0
\(985\) 12.0000 + 20.7846i 0.382352 + 0.662253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.00000 + 3.46410i 0.0635963 + 0.110152i
\(990\) 0 0
\(991\) −16.5000 + 28.5788i −0.524140 + 0.907837i 0.475465 + 0.879734i \(0.342280\pi\)
−0.999605 + 0.0281022i \(0.991054\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −32.0000 −1.01447
\(996\) 0 0
\(997\) 8.50000 14.7224i 0.269198 0.466264i −0.699457 0.714675i \(-0.746575\pi\)
0.968655 + 0.248410i \(0.0799082\pi\)
\(998\) 0 0
\(999\) 0 0
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 1008.2.s.m.865.1 2
3.2 odd 2 336.2.q.a.193.1 2
4.3 odd 2 504.2.s.g.361.1 2
7.2 even 3 inner 1008.2.s.m.289.1 2
7.3 odd 6 7056.2.a.bn.1.1 1
7.4 even 3 7056.2.a.i.1.1 1
12.11 even 2 168.2.q.b.25.1 2
21.2 odd 6 336.2.q.a.289.1 2
21.5 even 6 2352.2.q.v.961.1 2
21.11 odd 6 2352.2.a.x.1.1 1
21.17 even 6 2352.2.a.e.1.1 1
21.20 even 2 2352.2.q.v.1537.1 2
24.5 odd 2 1344.2.q.t.193.1 2
24.11 even 2 1344.2.q.i.193.1 2
28.3 even 6 3528.2.a.y.1.1 1
28.11 odd 6 3528.2.a.f.1.1 1
28.19 even 6 3528.2.s.d.3313.1 2
28.23 odd 6 504.2.s.g.289.1 2
28.27 even 2 3528.2.s.d.361.1 2
84.11 even 6 1176.2.a.d.1.1 1
84.23 even 6 168.2.q.b.121.1 yes 2
84.47 odd 6 1176.2.q.e.961.1 2
84.59 odd 6 1176.2.a.e.1.1 1
84.83 odd 2 1176.2.q.e.361.1 2
168.11 even 6 9408.2.a.cd.1.1 1
168.53 odd 6 9408.2.a.f.1.1 1
168.59 odd 6 9408.2.a.bk.1.1 1
168.101 even 6 9408.2.a.cs.1.1 1
168.107 even 6 1344.2.q.i.961.1 2
168.149 odd 6 1344.2.q.t.961.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.b.25.1 2 12.11 even 2
168.2.q.b.121.1 yes 2 84.23 even 6
336.2.q.a.193.1 2 3.2 odd 2
336.2.q.a.289.1 2 21.2 odd 6
504.2.s.g.289.1 2 28.23 odd 6
504.2.s.g.361.1 2 4.3 odd 2
1008.2.s.m.289.1 2 7.2 even 3 inner
1008.2.s.m.865.1 2 1.1 even 1 trivial
1176.2.a.d.1.1 1 84.11 even 6
1176.2.a.e.1.1 1 84.59 odd 6
1176.2.q.e.361.1 2 84.83 odd 2
1176.2.q.e.961.1 2 84.47 odd 6
1344.2.q.i.193.1 2 24.11 even 2
1344.2.q.i.961.1 2 168.107 even 6
1344.2.q.t.193.1 2 24.5 odd 2
1344.2.q.t.961.1 2 168.149 odd 6
2352.2.a.e.1.1 1 21.17 even 6
2352.2.a.x.1.1 1 21.11 odd 6
2352.2.q.v.961.1 2 21.5 even 6
2352.2.q.v.1537.1 2 21.20 even 2
3528.2.a.f.1.1 1 28.11 odd 6
3528.2.a.y.1.1 1 28.3 even 6
3528.2.s.d.361.1 2 28.27 even 2
3528.2.s.d.3313.1 2 28.19 even 6
7056.2.a.i.1.1 1 7.4 even 3
7056.2.a.bn.1.1 1 7.3 odd 6
9408.2.a.f.1.1 1 168.53 odd 6
9408.2.a.bk.1.1 1 168.59 odd 6
9408.2.a.cd.1.1 1 168.11 even 6
9408.2.a.cs.1.1 1 168.101 even 6