Properties

Label 432.3.q.c.17.2
Level $432$
Weight $3$
Character 432.17
Analytic conductor $11.771$
Analytic rank $0$
Dimension $4$
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] = [432,3,Mod(17,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.q (of order \(6\), 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: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 17.2
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 432.17
Dual form 432.3.q.c.305.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.39898 + 1.96240i) q^{5} +(6.39898 + 11.0834i) q^{7} +O(q^{10})\) \(q+(3.39898 + 1.96240i) q^{5} +(6.39898 + 11.0834i) q^{7} +(-14.2980 + 8.25493i) q^{11} +(1.39898 - 2.42310i) q^{13} +2.54270i q^{17} -21.5959 q^{19} +(2.60102 + 1.50170i) q^{23} +(-4.79796 - 8.31031i) q^{25} +(13.1969 - 7.61926i) q^{29} +(-13.6010 + 23.5577i) q^{31} +50.2295i q^{35} -10.4041 q^{37} +(34.5000 + 19.9186i) q^{41} +(17.0959 + 29.6110i) q^{43} +(58.1969 - 33.6000i) q^{47} +(-57.3939 + 99.4091i) q^{49} +100.680i q^{53} -64.7980 q^{55} +(5.29796 + 3.05878i) q^{59} +(-20.3990 - 35.3321i) q^{61} +(9.51021 - 5.49072i) q^{65} +(54.4898 - 94.3791i) q^{67} +52.8829i q^{71} +68.7878 q^{73} +(-182.985 - 105.646i) q^{77} +(-12.7929 - 22.1579i) q^{79} +(52.0857 - 30.0717i) q^{83} +(-4.98979 + 8.64258i) q^{85} +7.62809i q^{89} +35.8082 q^{91} +(-73.4041 - 42.3799i) q^{95} +(50.7020 + 87.8185i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{5} + 6 q^{7} - 18 q^{11} - 14 q^{13} - 8 q^{19} + 30 q^{23} + 20 q^{25} - 6 q^{29} - 74 q^{31} - 120 q^{37} + 138 q^{41} - 10 q^{43} + 174 q^{47} - 112 q^{49} - 220 q^{55} - 18 q^{59} - 62 q^{61} + 234 q^{65} + 22 q^{67} + 40 q^{73} - 438 q^{77} + 86 q^{79} - 66 q^{83} + 176 q^{85} + 300 q^{91} - 372 q^{95} + 242 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 3.39898 + 1.96240i 0.679796 + 0.392480i 0.799778 0.600296i \(-0.204950\pi\)
−0.119982 + 0.992776i \(0.538284\pi\)
\(6\) 0 0
\(7\) 6.39898 + 11.0834i 0.914140 + 1.58334i 0.808156 + 0.588969i \(0.200466\pi\)
0.105984 + 0.994368i \(0.466201\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −14.2980 + 8.25493i −1.29981 + 0.750448i −0.980372 0.197157i \(-0.936829\pi\)
−0.319443 + 0.947606i \(0.603496\pi\)
\(12\) 0 0
\(13\) 1.39898 2.42310i 0.107614 0.186393i −0.807189 0.590293i \(-0.799012\pi\)
0.914803 + 0.403900i \(0.132346\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.54270i 0.149570i 0.997200 + 0.0747852i \(0.0238271\pi\)
−0.997200 + 0.0747852i \(0.976173\pi\)
\(18\) 0 0
\(19\) −21.5959 −1.13663 −0.568314 0.822812i \(-0.692404\pi\)
−0.568314 + 0.822812i \(0.692404\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.60102 + 1.50170i 0.113088 + 0.0652913i 0.555477 0.831532i \(-0.312536\pi\)
−0.442389 + 0.896823i \(0.645869\pi\)
\(24\) 0 0
\(25\) −4.79796 8.31031i −0.191918 0.332412i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 13.1969 7.61926i 0.455067 0.262733i −0.254901 0.966967i \(-0.582043\pi\)
0.709968 + 0.704234i \(0.248710\pi\)
\(30\) 0 0
\(31\) −13.6010 + 23.5577i −0.438743 + 0.759924i −0.997593 0.0693442i \(-0.977909\pi\)
0.558850 + 0.829269i \(0.311243\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 50.2295i 1.43513i
\(36\) 0 0
\(37\) −10.4041 −0.281191 −0.140596 0.990067i \(-0.544902\pi\)
−0.140596 + 0.990067i \(0.544902\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 34.5000 + 19.9186i 0.841463 + 0.485819i 0.857761 0.514048i \(-0.171855\pi\)
−0.0162980 + 0.999867i \(0.505188\pi\)
\(42\) 0 0
\(43\) 17.0959 + 29.6110i 0.397579 + 0.688628i 0.993427 0.114470i \(-0.0365170\pi\)
−0.595847 + 0.803098i \(0.703184\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 58.1969 33.6000i 1.23823 0.714894i 0.269500 0.963000i \(-0.413142\pi\)
0.968733 + 0.248106i \(0.0798083\pi\)
\(48\) 0 0
\(49\) −57.3939 + 99.4091i −1.17130 + 2.02876i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 100.680i 1.89963i 0.312810 + 0.949816i \(0.398730\pi\)
−0.312810 + 0.949816i \(0.601270\pi\)
\(54\) 0 0
\(55\) −64.7980 −1.17814
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.29796 + 3.05878i 0.0897959 + 0.0518437i 0.544226 0.838939i \(-0.316824\pi\)
−0.454430 + 0.890783i \(0.650157\pi\)
\(60\) 0 0
\(61\) −20.3990 35.3321i −0.334409 0.579214i 0.648962 0.760821i \(-0.275204\pi\)
−0.983371 + 0.181607i \(0.941870\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.51021 5.49072i 0.146311 0.0844726i
\(66\) 0 0
\(67\) 54.4898 94.3791i 0.813281 1.40864i −0.0972755 0.995257i \(-0.531013\pi\)
0.910556 0.413386i \(-0.135654\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 52.8829i 0.744830i 0.928066 + 0.372415i \(0.121470\pi\)
−0.928066 + 0.372415i \(0.878530\pi\)
\(72\) 0 0
\(73\) 68.7878 0.942298 0.471149 0.882054i \(-0.343839\pi\)
0.471149 + 0.882054i \(0.343839\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −182.985 105.646i −2.37642 1.37203i
\(78\) 0 0
\(79\) −12.7929 22.1579i −0.161935 0.280479i 0.773628 0.633640i \(-0.218440\pi\)
−0.935563 + 0.353161i \(0.885107\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 52.0857 30.0717i 0.627539 0.362310i −0.152260 0.988341i \(-0.548655\pi\)
0.779798 + 0.626031i \(0.215322\pi\)
\(84\) 0 0
\(85\) −4.98979 + 8.64258i −0.0587035 + 0.101677i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.62809i 0.0857089i 0.999081 + 0.0428545i \(0.0136452\pi\)
−0.999081 + 0.0428545i \(0.986355\pi\)
\(90\) 0 0
\(91\) 35.8082 0.393496
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −73.4041 42.3799i −0.772675 0.446104i
\(96\) 0 0
\(97\) 50.7020 + 87.8185i 0.522701 + 0.905345i 0.999651 + 0.0264148i \(0.00840908\pi\)
−0.476950 + 0.878931i \(0.658258\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.19694 0.691053i 0.0118509 0.00684211i −0.494063 0.869426i \(-0.664489\pi\)
0.505914 + 0.862584i \(0.331155\pi\)
\(102\) 0 0
\(103\) −0.792856 + 1.37327i −0.00769763 + 0.0133327i −0.869849 0.493319i \(-0.835784\pi\)
0.862151 + 0.506652i \(0.169117\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 103.223i 0.964702i −0.875978 0.482351i \(-0.839783\pi\)
0.875978 0.482351i \(-0.160217\pi\)
\(108\) 0 0
\(109\) 14.4041 0.132148 0.0660738 0.997815i \(-0.478953\pi\)
0.0660738 + 0.997815i \(0.478953\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 92.0755 + 53.1598i 0.814828 + 0.470441i 0.848630 0.528988i \(-0.177428\pi\)
−0.0338020 + 0.999429i \(0.510762\pi\)
\(114\) 0 0
\(115\) 5.89388 + 10.2085i 0.0512511 + 0.0887695i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −28.1816 + 16.2707i −0.236820 + 0.136728i
\(120\) 0 0
\(121\) 75.7878 131.268i 0.626345 1.08486i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 135.782i 1.08626i
\(126\) 0 0
\(127\) 183.980 1.44866 0.724329 0.689454i \(-0.242150\pi\)
0.724329 + 0.689454i \(0.242150\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −91.6918 52.9383i −0.699938 0.404109i 0.107387 0.994217i \(-0.465752\pi\)
−0.807324 + 0.590108i \(0.799085\pi\)
\(132\) 0 0
\(133\) −138.192 239.355i −1.03904 1.79966i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 165.288 95.4289i 1.20648 0.696562i 0.244491 0.969651i \(-0.421379\pi\)
0.961989 + 0.273090i \(0.0880457\pi\)
\(138\) 0 0
\(139\) −47.4898 + 82.2547i −0.341653 + 0.591761i −0.984740 0.174033i \(-0.944320\pi\)
0.643087 + 0.765793i \(0.277653\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 46.1939i 0.323034i
\(144\) 0 0
\(145\) 59.8082 0.412470
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −79.6112 45.9636i −0.534304 0.308480i 0.208464 0.978030i \(-0.433154\pi\)
−0.742767 + 0.669550i \(0.766487\pi\)
\(150\) 0 0
\(151\) −8.20714 14.2152i −0.0543519 0.0941403i 0.837569 0.546331i \(-0.183976\pi\)
−0.891921 + 0.452191i \(0.850643\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −92.4592 + 53.3813i −0.596511 + 0.344396i
\(156\) 0 0
\(157\) 105.985 183.571i 0.675062 1.16924i −0.301389 0.953501i \(-0.597450\pi\)
0.976451 0.215740i \(-0.0692164\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 38.4374i 0.238742i
\(162\) 0 0
\(163\) −172.788 −1.06005 −0.530024 0.847983i \(-0.677817\pi\)
−0.530024 + 0.847983i \(0.677817\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −163.197 94.2218i −0.977227 0.564202i −0.0757953 0.997123i \(-0.524150\pi\)
−0.901432 + 0.432921i \(0.857483\pi\)
\(168\) 0 0
\(169\) 80.5857 + 139.579i 0.476839 + 0.825909i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −52.1969 + 30.1359i −0.301716 + 0.174196i −0.643214 0.765687i \(-0.722399\pi\)
0.341497 + 0.939883i \(0.389066\pi\)
\(174\) 0 0
\(175\) 61.4041 106.355i 0.350880 0.607743i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 72.7108i 0.406205i 0.979157 + 0.203103i \(0.0651025\pi\)
−0.979157 + 0.203103i \(0.934897\pi\)
\(180\) 0 0
\(181\) −97.5959 −0.539204 −0.269602 0.962972i \(-0.586892\pi\)
−0.269602 + 0.962972i \(0.586892\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −35.3633 20.4170i −0.191153 0.110362i
\(186\) 0 0
\(187\) −20.9898 36.3554i −0.112245 0.194414i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 282.560 163.136i 1.47937 0.854116i 0.479645 0.877462i \(-0.340765\pi\)
0.999727 + 0.0233462i \(0.00743200\pi\)
\(192\) 0 0
\(193\) −10.5102 + 18.2042i −0.0544570 + 0.0943223i −0.891969 0.452097i \(-0.850676\pi\)
0.837512 + 0.546419i \(0.184009\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 32.5413i 0.165184i −0.996583 0.0825922i \(-0.973680\pi\)
0.996583 0.0825922i \(-0.0263199\pi\)
\(198\) 0 0
\(199\) 62.0000 0.311558 0.155779 0.987792i \(-0.450211\pi\)
0.155779 + 0.987792i \(0.450211\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 168.894 + 97.5109i 0.831990 + 0.480349i
\(204\) 0 0
\(205\) 78.1765 + 135.406i 0.381349 + 0.660516i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 308.778 178.273i 1.47740 0.852980i
\(210\) 0 0
\(211\) −84.2980 + 146.008i −0.399516 + 0.691983i −0.993666 0.112372i \(-0.964155\pi\)
0.594150 + 0.804354i \(0.297489\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 134.196i 0.624169i
\(216\) 0 0
\(217\) −348.131 −1.60429
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.16122 + 3.55718i 0.0278788 + 0.0160958i
\(222\) 0 0
\(223\) −78.1867 135.423i −0.350613 0.607280i 0.635744 0.771900i \(-0.280693\pi\)
−0.986357 + 0.164620i \(0.947360\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 86.6510 50.0280i 0.381723 0.220388i −0.296845 0.954926i \(-0.595934\pi\)
0.678567 + 0.734538i \(0.262601\pi\)
\(228\) 0 0
\(229\) 104.995 181.856i 0.458493 0.794133i −0.540389 0.841416i \(-0.681723\pi\)
0.998882 + 0.0472824i \(0.0150561\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 307.127i 1.31814i 0.752081 + 0.659070i \(0.229050\pi\)
−0.752081 + 0.659070i \(0.770950\pi\)
\(234\) 0 0
\(235\) 263.747 1.12233
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −70.3888 40.6390i −0.294514 0.170038i 0.345462 0.938433i \(-0.387722\pi\)
−0.639976 + 0.768395i \(0.721056\pi\)
\(240\) 0 0
\(241\) 180.096 + 311.935i 0.747286 + 1.29434i 0.949119 + 0.314917i \(0.101977\pi\)
−0.201833 + 0.979420i \(0.564690\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −390.161 + 225.260i −1.59249 + 0.919427i
\(246\) 0 0
\(247\) −30.2122 + 52.3291i −0.122317 + 0.211859i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 400.179i 1.59434i −0.603755 0.797170i \(-0.706330\pi\)
0.603755 0.797170i \(-0.293670\pi\)
\(252\) 0 0
\(253\) −49.5857 −0.195991
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −315.035 181.885i −1.22582 0.707725i −0.259664 0.965699i \(-0.583612\pi\)
−0.966152 + 0.257974i \(0.916945\pi\)
\(258\) 0 0
\(259\) −66.5755 115.312i −0.257048 0.445221i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −326.570 + 188.546i −1.24171 + 0.716903i −0.969443 0.245318i \(-0.921108\pi\)
−0.272270 + 0.962221i \(0.587774\pi\)
\(264\) 0 0
\(265\) −197.576 + 342.211i −0.745568 + 1.29136i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 170.849i 0.635125i 0.948237 + 0.317562i \(0.102864\pi\)
−0.948237 + 0.317562i \(0.897136\pi\)
\(270\) 0 0
\(271\) 50.4041 0.185993 0.0929965 0.995666i \(-0.470355\pi\)
0.0929965 + 0.995666i \(0.470355\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 137.202 + 79.2136i 0.498917 + 0.288050i
\(276\) 0 0
\(277\) −12.8031 22.1756i −0.0462204 0.0800561i 0.841990 0.539494i \(-0.181384\pi\)
−0.888210 + 0.459438i \(0.848051\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −290.267 + 167.586i −1.03298 + 0.596391i −0.917837 0.396958i \(-0.870066\pi\)
−0.115143 + 0.993349i \(0.536733\pi\)
\(282\) 0 0
\(283\) −7.48979 + 12.9727i −0.0264657 + 0.0458399i −0.878955 0.476905i \(-0.841759\pi\)
0.852489 + 0.522745i \(0.175092\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 509.834i 1.77643i
\(288\) 0 0
\(289\) 282.535 0.977629
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 261.015 + 150.697i 0.890837 + 0.514325i 0.874216 0.485537i \(-0.161376\pi\)
0.0166210 + 0.999862i \(0.494709\pi\)
\(294\) 0 0
\(295\) 12.0051 + 20.7934i 0.0406953 + 0.0704863i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.27755 4.20169i 0.0243396 0.0140525i
\(300\) 0 0
\(301\) −218.793 + 378.960i −0.726887 + 1.25900i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 160.124i 0.524997i
\(306\) 0 0
\(307\) 44.7469 0.145755 0.0728777 0.997341i \(-0.476782\pi\)
0.0728777 + 0.997341i \(0.476782\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 281.348 + 162.436i 0.904656 + 0.522303i 0.878708 0.477360i \(-0.158406\pi\)
0.0259480 + 0.999663i \(0.491740\pi\)
\(312\) 0 0
\(313\) −156.894 271.748i −0.501258 0.868205i −0.999999 0.00145368i \(-0.999537\pi\)
0.498741 0.866751i \(-0.333796\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 47.9847 27.7040i 0.151371 0.0873942i −0.422401 0.906409i \(-0.638813\pi\)
0.573773 + 0.819015i \(0.305479\pi\)
\(318\) 0 0
\(319\) −125.793 + 217.880i −0.394335 + 0.683008i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 54.9119i 0.170006i
\(324\) 0 0
\(325\) −26.8490 −0.0826123
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 744.802 + 430.012i 2.26384 + 1.30703i
\(330\) 0 0
\(331\) 7.70204 + 13.3403i 0.0232690 + 0.0403031i 0.877425 0.479713i \(-0.159259\pi\)
−0.854156 + 0.520016i \(0.825926\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 370.419 213.862i 1.10573 0.638393i
\(336\) 0 0
\(337\) −37.8837 + 65.6164i −0.112414 + 0.194708i −0.916743 0.399477i \(-0.869192\pi\)
0.804329 + 0.594184i \(0.202525\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 449.102i 1.31701i
\(342\) 0 0
\(343\) −841.949 −2.45466
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 578.651 + 334.084i 1.66758 + 0.962779i 0.968936 + 0.247312i \(0.0795472\pi\)
0.698646 + 0.715467i \(0.253786\pi\)
\(348\) 0 0
\(349\) 123.015 + 213.069i 0.352479 + 0.610512i 0.986683 0.162654i \(-0.0520053\pi\)
−0.634204 + 0.773166i \(0.718672\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −156.510 + 90.3612i −0.443372 + 0.255981i −0.705027 0.709181i \(-0.749065\pi\)
0.261655 + 0.965161i \(0.415732\pi\)
\(354\) 0 0
\(355\) −103.778 + 179.748i −0.292331 + 0.506332i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 256.273i 0.713852i −0.934133 0.356926i \(-0.883825\pi\)
0.934133 0.356926i \(-0.116175\pi\)
\(360\) 0 0
\(361\) 105.384 0.291922
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 233.808 + 134.989i 0.640570 + 0.369833i
\(366\) 0 0
\(367\) 270.358 + 468.274i 0.736671 + 1.27595i 0.953986 + 0.299850i \(0.0969366\pi\)
−0.217316 + 0.976101i \(0.569730\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1115.88 + 644.252i −3.00776 + 1.73653i
\(372\) 0 0
\(373\) 85.9847 148.930i 0.230522 0.399276i −0.727440 0.686171i \(-0.759290\pi\)
0.957962 + 0.286896i \(0.0926233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 42.6367i 0.113095i
\(378\) 0 0
\(379\) 170.849 0.450789 0.225394 0.974268i \(-0.427633\pi\)
0.225394 + 0.974268i \(0.427633\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −505.681 291.955i −1.32031 0.762284i −0.336536 0.941671i \(-0.609255\pi\)
−0.983779 + 0.179386i \(0.942589\pi\)
\(384\) 0 0
\(385\) −414.641 718.179i −1.07699 1.86540i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 384.752 222.137i 0.989080 0.571045i 0.0840807 0.996459i \(-0.473205\pi\)
0.904999 + 0.425413i \(0.139871\pi\)
\(390\) 0 0
\(391\) −3.81837 + 6.61361i −0.00976565 + 0.0169146i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 100.419i 0.254225i
\(396\) 0 0
\(397\) −575.090 −1.44859 −0.724294 0.689491i \(-0.757834\pi\)
−0.724294 + 0.689491i \(0.757834\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −141.318 81.5902i −0.352415 0.203467i 0.313333 0.949643i \(-0.398554\pi\)
−0.665748 + 0.746176i \(0.731888\pi\)
\(402\) 0 0
\(403\) 38.0551 + 65.9134i 0.0944295 + 0.163557i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 148.757 85.8850i 0.365497 0.211020i
\(408\) 0 0
\(409\) 300.641 520.725i 0.735063 1.27317i −0.219633 0.975583i \(-0.570486\pi\)
0.954696 0.297584i \(-0.0961808\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 78.2922i 0.189570i
\(414\) 0 0
\(415\) 236.051 0.568798
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −158.620 91.5795i −0.378569 0.218567i 0.298626 0.954370i \(-0.403472\pi\)
−0.677195 + 0.735803i \(0.736805\pi\)
\(420\) 0 0
\(421\) −83.7724 145.098i −0.198984 0.344651i 0.749215 0.662327i \(-0.230431\pi\)
−0.948199 + 0.317676i \(0.897098\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 21.1306 12.1998i 0.0497191 0.0287053i
\(426\) 0 0
\(427\) 261.065 452.178i 0.611394 1.05897i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 644.766i 1.49598i −0.663712 0.747988i \(-0.731020\pi\)
0.663712 0.747988i \(-0.268980\pi\)
\(432\) 0 0
\(433\) 133.514 0.308347 0.154174 0.988044i \(-0.450729\pi\)
0.154174 + 0.988044i \(0.450729\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −56.1714 32.4306i −0.128539 0.0742119i
\(438\) 0 0
\(439\) −46.2276 80.0685i −0.105302 0.182388i 0.808560 0.588414i \(-0.200248\pi\)
−0.913862 + 0.406026i \(0.866914\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −150.157 + 86.6933i −0.338955 + 0.195696i −0.659810 0.751433i \(-0.729363\pi\)
0.320855 + 0.947128i \(0.396030\pi\)
\(444\) 0 0
\(445\) −14.9694 + 25.9277i −0.0336391 + 0.0582646i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 430.692i 0.959224i −0.877481 0.479612i \(-0.840777\pi\)
0.877481 0.479612i \(-0.159223\pi\)
\(450\) 0 0
\(451\) −657.706 −1.45833
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 121.711 + 70.2700i 0.267497 + 0.154440i
\(456\) 0 0
\(457\) 262.843 + 455.257i 0.575148 + 0.996186i 0.996025 + 0.0890687i \(0.0283891\pi\)
−0.420877 + 0.907118i \(0.638278\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.54999 + 0.894890i −0.00336224 + 0.00194119i −0.501680 0.865053i \(-0.667285\pi\)
0.498318 + 0.866994i \(0.333951\pi\)
\(462\) 0 0
\(463\) 263.166 455.817i 0.568394 0.984487i −0.428331 0.903622i \(-0.640898\pi\)
0.996725 0.0808651i \(-0.0257683\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 409.322i 0.876494i 0.898855 + 0.438247i \(0.144400\pi\)
−0.898855 + 0.438247i \(0.855600\pi\)
\(468\) 0 0
\(469\) 1394.72 2.97381
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −488.873 282.251i −1.03356 0.596726i
\(474\) 0 0
\(475\) 103.616 + 179.469i 0.218140 + 0.377829i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 525.207 303.228i 1.09647 0.633045i 0.161175 0.986926i \(-0.448472\pi\)
0.935290 + 0.353881i \(0.115138\pi\)
\(480\) 0 0
\(481\) −14.5551 + 25.2102i −0.0302601 + 0.0524120i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 397.991i 0.820600i
\(486\) 0 0
\(487\) 275.131 0.564950 0.282475 0.959275i \(-0.408845\pi\)
0.282475 + 0.959275i \(0.408845\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −127.469 73.5945i −0.259612 0.149887i 0.364546 0.931186i \(-0.381224\pi\)
−0.624157 + 0.781299i \(0.714558\pi\)
\(492\) 0 0
\(493\) 19.3735 + 33.5558i 0.0392971 + 0.0680646i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −586.120 + 338.397i −1.17932 + 0.680879i
\(498\) 0 0
\(499\) 226.308 391.977i 0.453523 0.785526i −0.545079 0.838385i \(-0.683500\pi\)
0.998602 + 0.0528594i \(0.0168335\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 571.541i 1.13627i 0.822937 + 0.568133i \(0.192334\pi\)
−0.822937 + 0.568133i \(0.807666\pi\)
\(504\) 0 0
\(505\) 5.42449 0.0107416
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −394.136 227.554i −0.774333 0.447062i 0.0600849 0.998193i \(-0.480863\pi\)
−0.834418 + 0.551132i \(0.814196\pi\)
\(510\) 0 0
\(511\) 440.171 + 762.399i 0.861392 + 1.49198i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.38981 + 3.11181i −0.0104656 + 0.00604234i
\(516\) 0 0
\(517\) −554.732 + 960.823i −1.07298 + 1.85846i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 846.127i 1.62404i −0.583627 0.812022i \(-0.698367\pi\)
0.583627 0.812022i \(-0.301633\pi\)
\(522\) 0 0
\(523\) 487.616 0.932345 0.466172 0.884694i \(-0.345633\pi\)
0.466172 + 0.884694i \(0.345633\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −59.9000 34.5833i −0.113662 0.0656229i
\(528\) 0 0
\(529\) −259.990 450.316i −0.491474 0.851258i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 96.5296 55.7314i 0.181106 0.104562i
\(534\) 0 0
\(535\) 202.565 350.853i 0.378627 0.655801i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1895.13i 3.51601i
\(540\) 0 0
\(541\) 608.302 1.12440 0.562202 0.827000i \(-0.309955\pi\)
0.562202 + 0.827000i \(0.309955\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 48.9592 + 28.2666i 0.0898334 + 0.0518653i
\(546\) 0 0
\(547\) −431.671 747.677i −0.789162 1.36687i −0.926481 0.376341i \(-0.877182\pi\)
0.137319 0.990527i \(-0.456151\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −285.000 + 164.545i −0.517241 + 0.298629i
\(552\) 0 0
\(553\) 163.722 283.576i 0.296062 0.512795i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 331.526i 0.595200i −0.954691 0.297600i \(-0.903814\pi\)
0.954691 0.297600i \(-0.0961861\pi\)
\(558\) 0 0
\(559\) 95.6674 0.171140
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 514.024 + 296.772i 0.913010 + 0.527126i 0.881398 0.472374i \(-0.156603\pi\)
0.0316114 + 0.999500i \(0.489936\pi\)
\(564\) 0 0
\(565\) 208.642 + 361.378i 0.369278 + 0.639608i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −889.155 + 513.354i −1.56266 + 0.902204i −0.565676 + 0.824627i \(0.691385\pi\)
−0.996986 + 0.0775764i \(0.975282\pi\)
\(570\) 0 0
\(571\) 191.843 332.282i 0.335977 0.581929i −0.647695 0.761900i \(-0.724267\pi\)
0.983672 + 0.179971i \(0.0576002\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.8204i 0.0501224i
\(576\) 0 0
\(577\) −777.433 −1.34737 −0.673685 0.739019i \(-0.735290\pi\)
−0.673685 + 0.739019i \(0.735290\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 666.591 + 384.856i 1.14732 + 0.662403i
\(582\) 0 0
\(583\) −831.110 1439.53i −1.42557 2.46917i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 581.520 335.741i 0.990665 0.571961i 0.0851921 0.996365i \(-0.472850\pi\)
0.905473 + 0.424404i \(0.139516\pi\)
\(588\) 0 0
\(589\) 293.727 508.749i 0.498687 0.863751i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 536.457i 0.904650i 0.891853 + 0.452325i \(0.149405\pi\)
−0.891853 + 0.452325i \(0.850595\pi\)
\(594\) 0 0
\(595\) −127.718 −0.214653
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −735.438 424.605i −1.22778 0.708857i −0.261212 0.965282i \(-0.584122\pi\)
−0.966564 + 0.256425i \(0.917455\pi\)
\(600\) 0 0
\(601\) −330.692 572.775i −0.550236 0.953037i −0.998257 0.0590138i \(-0.981204\pi\)
0.448021 0.894023i \(-0.352129\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 515.202 297.452i 0.851574 0.491656i
\(606\) 0 0
\(607\) −10.9541 + 18.9730i −0.0180463 + 0.0312570i −0.874908 0.484290i \(-0.839078\pi\)
0.856861 + 0.515547i \(0.172411\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 188.023i 0.307730i
\(612\) 0 0
\(613\) −1164.75 −1.90008 −0.950038 0.312133i \(-0.898956\pi\)
−0.950038 + 0.312133i \(0.898956\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −558.227 322.292i −0.904743 0.522354i −0.0260071 0.999662i \(-0.508279\pi\)
−0.878736 + 0.477308i \(0.841613\pi\)
\(618\) 0 0
\(619\) 564.469 + 977.690i 0.911905 + 1.57947i 0.811370 + 0.584533i \(0.198722\pi\)
0.100535 + 0.994933i \(0.467944\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −84.5449 + 48.8120i −0.135706 + 0.0783499i
\(624\) 0 0
\(625\) 146.510 253.763i 0.234416 0.406021i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 26.4544i 0.0420579i
\(630\) 0 0
\(631\) −143.212 −0.226961 −0.113480 0.993540i \(-0.536200\pi\)
−0.113480 + 0.993540i \(0.536200\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 625.343 + 361.042i 0.984792 + 0.568570i
\(636\) 0 0
\(637\) 160.586 + 278.143i 0.252097 + 0.436645i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 198.418 114.557i 0.309545 0.178716i −0.337178 0.941441i \(-0.609472\pi\)
0.646723 + 0.762725i \(0.276139\pi\)
\(642\) 0 0
\(643\) −554.682 + 960.737i −0.862646 + 1.49415i 0.00671893 + 0.999977i \(0.497861\pi\)
−0.869365 + 0.494170i \(0.835472\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1052.57i 1.62685i 0.581669 + 0.813426i \(0.302400\pi\)
−0.581669 + 0.813426i \(0.697600\pi\)
\(648\) 0 0
\(649\) −101.000 −0.155624
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 795.499 + 459.282i 1.21822 + 0.703341i 0.964537 0.263946i \(-0.0850241\pi\)
0.253685 + 0.967287i \(0.418357\pi\)
\(654\) 0 0
\(655\) −207.772 359.872i −0.317210 0.549424i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −207.288 + 119.678i −0.314549 + 0.181605i −0.648960 0.760822i \(-0.724796\pi\)
0.334411 + 0.942427i \(0.391463\pi\)
\(660\) 0 0
\(661\) −177.793 + 307.946i −0.268976 + 0.465879i −0.968598 0.248634i \(-0.920018\pi\)
0.699622 + 0.714513i \(0.253352\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1084.75i 1.63121i
\(666\) 0 0
\(667\) 45.7673 0.0686167
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 583.328 + 336.784i 0.869341 + 0.501914i
\(672\) 0 0
\(673\) 291.429 + 504.769i 0.433029 + 0.750028i 0.997132 0.0756758i \(-0.0241114\pi\)
−0.564103 + 0.825704i \(0.690778\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 154.450 89.1718i 0.228139 0.131716i −0.381574 0.924338i \(-0.624618\pi\)
0.609713 + 0.792622i \(0.291285\pi\)
\(678\) 0 0
\(679\) −648.883 + 1123.90i −0.955645 + 1.65522i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 660.510i 0.967072i 0.875325 + 0.483536i \(0.160648\pi\)
−0.875325 + 0.483536i \(0.839352\pi\)
\(684\) 0 0
\(685\) 749.080 1.09355
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 243.959 + 140.850i 0.354077 + 0.204427i
\(690\) 0 0
\(691\) 43.2571 + 74.9236i 0.0626008 + 0.108428i 0.895627 0.444805i \(-0.146727\pi\)
−0.833026 + 0.553233i \(0.813394\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −322.834 + 186.388i −0.464509 + 0.268184i
\(696\) 0 0
\(697\) −50.6469 + 87.7231i −0.0726642 + 0.125858i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1204.11i 1.71770i 0.512227 + 0.858850i \(0.328821\pi\)
−0.512227 + 0.858850i \(0.671179\pi\)
\(702\) 0 0
\(703\) 224.686 0.319610
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.3184 + 8.84406i 0.0216667 + 0.0125093i
\(708\) 0 0
\(709\) 401.944 + 696.187i 0.566917 + 0.981928i 0.996869 + 0.0790766i \(0.0251972\pi\)
−0.429952 + 0.902852i \(0.641470\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −70.7531 + 40.8493i −0.0992329 + 0.0572921i
\(714\) 0 0
\(715\) −90.6510 + 157.012i −0.126785 + 0.219597i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 66.1101i 0.0919474i −0.998943 0.0459737i \(-0.985361\pi\)
0.998943 0.0459737i \(-0.0146390\pi\)
\(720\) 0 0
\(721\) −20.2939 −0.0281469
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −126.637 73.1138i −0.174671 0.100847i
\(726\) 0 0
\(727\) 259.834 + 450.045i 0.357405 + 0.619044i 0.987527 0.157453i \(-0.0503281\pi\)
−0.630121 + 0.776497i \(0.716995\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −75.2918 + 43.4698i −0.102998 + 0.0594662i
\(732\) 0 0
\(733\) 80.1459 138.817i 0.109340 0.189382i −0.806163 0.591693i \(-0.798460\pi\)
0.915503 + 0.402311i \(0.131793\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1799.24i 2.44130i
\(738\) 0 0
\(739\) 122.849 0.166237 0.0831184 0.996540i \(-0.473512\pi\)
0.0831184 + 0.996540i \(0.473512\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −585.944 338.295i −0.788619 0.455309i 0.0508572 0.998706i \(-0.483805\pi\)
−0.839476 + 0.543397i \(0.817138\pi\)
\(744\) 0 0
\(745\) −180.398 312.458i −0.242145 0.419407i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1144.06 660.523i 1.52745 0.881873i
\(750\) 0 0
\(751\) 171.430 296.925i 0.228268 0.395373i −0.729027 0.684485i \(-0.760027\pi\)
0.957295 + 0.289113i \(0.0933603\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 64.4229i 0.0853283i
\(756\) 0 0
\(757\) −452.220 −0.597385 −0.298692 0.954349i \(-0.596550\pi\)
−0.298692 + 0.954349i \(0.596550\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −203.520 117.503i −0.267438 0.154405i 0.360285 0.932842i \(-0.382680\pi\)
−0.627723 + 0.778437i \(0.716013\pi\)
\(762\) 0 0
\(763\) 92.1714 + 159.646i 0.120801 + 0.209234i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.8235 8.55834i 0.0193266 0.0111582i
\(768\) 0 0
\(769\) 318.439 551.552i 0.414095 0.717233i −0.581238 0.813733i \(-0.697432\pi\)
0.995333 + 0.0965005i \(0.0307649\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 592.885i 0.766992i −0.923543 0.383496i \(-0.874720\pi\)
0.923543 0.383496i \(-0.125280\pi\)
\(774\) 0 0
\(775\) 261.029 0.336811
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −745.059 430.160i −0.956430 0.552195i
\(780\) 0 0
\(781\) −436.545 756.118i −0.558956 0.968141i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 720.480 415.969i 0.917808 0.529897i
\(786\) 0 0
\(787\) −666.904 + 1155.11i −0.847400 + 1.46774i 0.0361199 + 0.999347i \(0.488500\pi\)
−0.883520 + 0.468393i \(0.844833\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1360.67i 1.72020i
\(792\) 0 0
\(793\) −114.151 −0.143948
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −638.540 368.661i −0.801179 0.462561i 0.0427041 0.999088i \(-0.486403\pi\)
−0.843883 + 0.536527i \(0.819736\pi\)
\(798\) 0 0
\(799\) 85.4347 + 147.977i 0.106927 + 0.185203i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −983.524 + 567.838i −1.22481 + 0.707146i
\(804\) 0 0
\(805\) −75.4296 + 130.648i −0.0937014 + 0.162296i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 71.6833i 0.0886073i 0.999018 + 0.0443037i \(0.0141069\pi\)
−0.999018 + 0.0443037i \(0.985893\pi\)
\(810\) 0 0
\(811\) −1086.24 −1.33938 −0.669692 0.742639i \(-0.733574\pi\)
−0.669692 + 0.742639i \(0.733574\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −587.302 339.079i −0.720616 0.416048i
\(816\) 0 0
\(817\) −369.202 639.477i −0.451900 0.782713i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1232.92 + 711.829i −1.50173 + 0.867026i −0.501736 + 0.865021i \(0.667305\pi\)
−0.999998 + 0.00200552i \(0.999362\pi\)
\(822\) 0 0
\(823\) −217.379 + 376.511i −0.264129 + 0.457486i −0.967335 0.253501i \(-0.918418\pi\)
0.703206 + 0.710986i \(0.251751\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 677.821i 0.819614i 0.912172 + 0.409807i \(0.134404\pi\)
−0.912172 + 0.409807i \(0.865596\pi\)
\(828\) 0 0
\(829\) −995.775 −1.20118 −0.600588 0.799558i \(-0.705067\pi\)
−0.600588 + 0.799558i \(0.705067\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −252.767 145.935i −0.303442 0.175192i
\(834\) 0 0
\(835\) −369.802 640.516i −0.442877 0.767085i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1149.53 663.681i 1.37012 0.791038i 0.379176 0.925325i \(-0.376207\pi\)
0.990943 + 0.134286i \(0.0428741\pi\)
\(840\) 0 0
\(841\) −304.394 + 527.226i −0.361943 + 0.626903i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 632.566i 0.748599i
\(846\) 0 0
\(847\) 1939.86 2.29027
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27.0612 15.6238i −0.0317993 0.0183594i
\(852\) 0 0
\(853\) 685.317 + 1187.00i 0.803420 + 1.39156i 0.917352 + 0.398076i \(0.130322\pi\)
−0.113932 + 0.993489i \(0.536345\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1038.74 599.717i 1.21207 0.699787i 0.248857 0.968540i \(-0.419945\pi\)
0.963209 + 0.268753i \(0.0866117\pi\)
\(858\) 0 0
\(859\) 31.1163 53.8951i 0.0362239 0.0627416i −0.847345 0.531043i \(-0.821800\pi\)
0.883569 + 0.468301i \(0.155134\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 154.565i 0.179102i −0.995982 0.0895509i \(-0.971457\pi\)
0.995982 0.0895509i \(-0.0285432\pi\)
\(864\) 0 0
\(865\) −236.555 −0.273474
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 365.823 + 211.208i 0.420971 + 0.243048i
\(870\) 0 0
\(871\) −152.460 264.069i −0.175040 0.303179i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1504.92 868.867i 1.71991 0.992991i
\(876\) 0 0
\(877\) 548.187 949.487i 0.625070 1.08265i −0.363457 0.931611i \(-0.618404\pi\)
0.988527 0.151043i \(-0.0482631\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1241.22i 1.40888i 0.709765 + 0.704438i \(0.248801\pi\)
−0.709765 + 0.704438i \(0.751199\pi\)
\(882\) 0 0
\(883\) −553.555 −0.626903 −0.313451 0.949604i \(-0.601485\pi\)
−0.313451 + 0.949604i \(0.601485\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −351.905 203.173i −0.396736 0.229056i 0.288338 0.957529i \(-0.406897\pi\)
−0.685075 + 0.728473i \(0.740231\pi\)
\(888\) 0 0
\(889\) 1177.28 + 2039.11i 1.32428 + 2.29371i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1256.82 + 725.623i −1.40741 + 0.812568i
\(894\) 0 0
\(895\) −142.688 + 247.142i −0.159428 + 0.276137i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 414.519i 0.461089i
\(900\) 0 0
\(901\) −256.000 −0.284129
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −331.727 191.522i −0.366549 0.211627i
\(906\) 0 0
\(907\) −231.712 401.337i −0.255471 0.442489i 0.709552 0.704653i \(-0.248897\pi\)
−0.965023 + 0.262164i \(0.915564\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 997.489 575.900i 1.09494 0.632163i 0.160051 0.987109i \(-0.448834\pi\)
0.934887 + 0.354946i \(0.115501\pi\)
\(912\) 0 0
\(913\) −496.480 + 859.928i −0.543789 + 0.941871i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1355.00i 1.47765i
\(918\) 0 0
\(919\) −944.665 −1.02793 −0.513964 0.857812i \(-0.671823\pi\)
−0.513964 + 0.857812i \(0.671823\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 128.141 + 73.9821i 0.138831 + 0.0801540i
\(924\) 0 0
\(925\) 49.9184 + 86.4611i 0.0539658 + 0.0934715i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 912.702 526.949i 0.982456 0.567221i 0.0794456 0.996839i \(-0.474685\pi\)
0.903011 + 0.429618i \(0.141352\pi\)
\(930\) 0 0
\(931\) 1239.47 2146.83i 1.33134 2.30594i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 164.762i 0.176216i
\(936\) 0 0
\(937\) 561.392 0.599137 0.299569 0.954075i \(-0.403157\pi\)
0.299569 + 0.954075i \(0.403157\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 98.4092 + 56.8166i 0.104579 + 0.0603789i 0.551377 0.834256i \(-0.314102\pi\)
−0.446798 + 0.894635i \(0.647436\pi\)
\(942\) 0 0
\(943\) 59.8235 + 103.617i 0.0634395 + 0.109880i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 62.6510 36.1716i 0.0661574 0.0381960i −0.466556 0.884491i \(-0.654506\pi\)
0.532714 + 0.846295i \(0.321172\pi\)
\(948\) 0 0
\(949\) 96.2327 166.680i 0.101404 0.175637i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 776.447i 0.814739i 0.913263 + 0.407370i \(0.133554\pi\)
−0.913263 + 0.407370i \(0.866446\pi\)
\(954\) 0 0
\(955\) 1280.56 1.34090
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2115.35 + 1221.30i 2.20578 + 1.27351i
\(960\) 0 0
\(961\) 110.524 + 191.434i 0.115010 + 0.199203i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −71.4479 + 41.2505i −0.0740393 + 0.0427466i
\(966\) 0 0
\(967\) 11.3888 19.7259i 0.0117774 0.0203991i −0.860077 0.510165i \(-0.829584\pi\)
0.871854 + 0.489766i \(0.162918\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1016.98i 1.04735i 0.851919 + 0.523674i \(0.175439\pi\)
−0.851919 + 0.523674i \(0.824561\pi\)
\(972\) 0 0
\(973\) −1215.54 −1.24928
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −975.802 563.380i −0.998774 0.576642i −0.0908886 0.995861i \(-0.528971\pi\)
−0.907885 + 0.419219i \(0.862304\pi\)
\(978\) 0 0
\(979\) −62.9694 109.066i −0.0643201 0.111406i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −280.105 + 161.719i −0.284949 + 0.164516i −0.635662 0.771968i \(-0.719273\pi\)
0.350713 + 0.936483i \(0.385939\pi\)
\(984\) 0 0
\(985\) 63.8592 110.607i 0.0648317 0.112292i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 102.692i 0.103834i
\(990\) 0 0
\(991\) −1961.47 −1.97929 −0.989644 0.143547i \(-0.954149\pi\)
−0.989644 + 0.143547i \(0.954149\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 210.737 + 121.669i 0.211796 + 0.122280i
\(996\) 0 0
\(997\) −559.954 969.869i −0.561639 0.972787i −0.997354 0.0727024i \(-0.976838\pi\)
0.435715 0.900085i \(-0.356496\pi\)
\(998\) 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 432.3.q.c.17.2 4
3.2 odd 2 144.3.q.d.113.1 4
4.3 odd 2 216.3.m.a.17.2 4
8.3 odd 2 1728.3.q.e.449.1 4
8.5 even 2 1728.3.q.f.449.1 4
9.2 odd 6 inner 432.3.q.c.305.2 4
9.4 even 3 1296.3.e.c.161.2 4
9.5 odd 6 1296.3.e.c.161.3 4
9.7 even 3 144.3.q.d.65.1 4
12.11 even 2 72.3.m.a.41.1 4
24.5 odd 2 576.3.q.c.257.2 4
24.11 even 2 576.3.q.h.257.2 4
36.7 odd 6 72.3.m.a.65.1 yes 4
36.11 even 6 216.3.m.a.89.2 4
36.23 even 6 648.3.e.b.161.3 4
36.31 odd 6 648.3.e.b.161.2 4
72.11 even 6 1728.3.q.e.1601.1 4
72.29 odd 6 1728.3.q.f.1601.1 4
72.43 odd 6 576.3.q.h.65.2 4
72.61 even 6 576.3.q.c.65.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.m.a.41.1 4 12.11 even 2
72.3.m.a.65.1 yes 4 36.7 odd 6
144.3.q.d.65.1 4 9.7 even 3
144.3.q.d.113.1 4 3.2 odd 2
216.3.m.a.17.2 4 4.3 odd 2
216.3.m.a.89.2 4 36.11 even 6
432.3.q.c.17.2 4 1.1 even 1 trivial
432.3.q.c.305.2 4 9.2 odd 6 inner
576.3.q.c.65.2 4 72.61 even 6
576.3.q.c.257.2 4 24.5 odd 2
576.3.q.h.65.2 4 72.43 odd 6
576.3.q.h.257.2 4 24.11 even 2
648.3.e.b.161.2 4 36.31 odd 6
648.3.e.b.161.3 4 36.23 even 6
1296.3.e.c.161.2 4 9.4 even 3
1296.3.e.c.161.3 4 9.5 odd 6
1728.3.q.e.449.1 4 8.3 odd 2
1728.3.q.e.1601.1 4 72.11 even 6
1728.3.q.f.449.1 4 8.5 even 2
1728.3.q.f.1601.1 4 72.29 odd 6