Properties

Label 1728.3.q.i.449.1
Level $1728$
Weight $3$
Character 1728.449
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
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] = [1728,3,Mod(449,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, 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("1728.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 449.1
Root \(-1.41950 - 2.45865i\) of defining polynomial
Character \(\chi\) \(=\) 1728.449
Dual form 1728.3.q.i.1601.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-8.20800 - 4.73889i) q^{5} +(-1.05671 - 1.83027i) q^{7} +O(q^{10})\) \(q+(-8.20800 - 4.73889i) q^{5} +(-1.05671 - 1.83027i) q^{7} +(-13.7064 + 7.91342i) q^{11} +(-4.70337 + 8.14648i) q^{13} +11.6027i q^{17} +12.9707 q^{19} +(-5.27427 - 3.04510i) q^{23} +(32.4142 + 56.1431i) q^{25} +(-24.7667 + 14.2991i) q^{29} +(-8.75365 + 15.1618i) q^{31} +20.0305i q^{35} +15.6207 q^{37} +(-14.8062 - 8.54836i) q^{41} +(-21.7157 - 37.6127i) q^{43} +(20.6696 - 11.9336i) q^{47} +(22.2667 - 38.5671i) q^{49} -14.1051i q^{53} +150.003 q^{55} +(-38.5788 - 22.2735i) q^{59} +(1.86057 + 3.22260i) q^{61} +(77.2105 - 44.5775i) q^{65} +(21.0090 - 36.3887i) q^{67} +120.440i q^{71} +5.48692 q^{73} +(28.9674 + 16.7243i) q^{77} +(-60.5480 - 104.872i) q^{79} +(-46.5861 + 26.8965i) q^{83} +(54.9840 - 95.2351i) q^{85} -102.195i q^{89} +19.8803 q^{91} +(-106.463 - 61.4667i) q^{95} +(-58.9377 - 102.083i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} - 6 q^{7} - 36 q^{11} - 14 q^{13} + 4 q^{19} - 102 q^{23} + 10 q^{25} - 114 q^{29} + 50 q^{31} - 120 q^{37} - 264 q^{41} - 28 q^{43} + 150 q^{47} + 94 q^{49} + 244 q^{55} + 108 q^{59} - 14 q^{61} + 198 q^{65} - 20 q^{67} - 76 q^{73} + 66 q^{77} - 26 q^{79} - 246 q^{83} + 224 q^{85} + 108 q^{91} - 456 q^{95} - 236 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) −8.20800 4.73889i −1.64160 0.947779i −0.980265 0.197688i \(-0.936657\pi\)
−0.661336 0.750090i \(-0.730010\pi\)
\(6\) 0 0
\(7\) −1.05671 1.83027i −0.150958 0.261467i 0.780622 0.625004i \(-0.214903\pi\)
−0.931580 + 0.363537i \(0.881569\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −13.7064 + 7.91342i −1.24604 + 0.719401i −0.970317 0.241836i \(-0.922250\pi\)
−0.275723 + 0.961237i \(0.588917\pi\)
\(12\) 0 0
\(13\) −4.70337 + 8.14648i −0.361798 + 0.626652i −0.988257 0.152802i \(-0.951170\pi\)
0.626459 + 0.779454i \(0.284504\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11.6027i 0.682513i 0.939970 + 0.341256i \(0.110852\pi\)
−0.939970 + 0.341256i \(0.889148\pi\)
\(18\) 0 0
\(19\) 12.9707 0.682667 0.341334 0.939942i \(-0.389121\pi\)
0.341334 + 0.939942i \(0.389121\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.27427 3.04510i −0.229316 0.132396i 0.380940 0.924600i \(-0.375600\pi\)
−0.610256 + 0.792204i \(0.708934\pi\)
\(24\) 0 0
\(25\) 32.4142 + 56.1431i 1.29657 + 2.24572i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −24.7667 + 14.2991i −0.854026 + 0.493072i −0.862007 0.506896i \(-0.830793\pi\)
0.00798151 + 0.999968i \(0.497459\pi\)
\(30\) 0 0
\(31\) −8.75365 + 15.1618i −0.282376 + 0.489089i −0.971969 0.235107i \(-0.924456\pi\)
0.689594 + 0.724196i \(0.257789\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 20.0305i 0.572299i
\(36\) 0 0
\(37\) 15.6207 0.422181 0.211091 0.977467i \(-0.432298\pi\)
0.211091 + 0.977467i \(0.432298\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −14.8062 8.54836i −0.361127 0.208497i 0.308448 0.951241i \(-0.400190\pi\)
−0.669575 + 0.742745i \(0.733524\pi\)
\(42\) 0 0
\(43\) −21.7157 37.6127i −0.505016 0.874714i −0.999983 0.00580217i \(-0.998153\pi\)
0.494967 0.868912i \(-0.335180\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 20.6696 11.9336i 0.439778 0.253906i −0.263726 0.964598i \(-0.584951\pi\)
0.703503 + 0.710692i \(0.251618\pi\)
\(48\) 0 0
\(49\) 22.2667 38.5671i 0.454423 0.787084i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.1051i 0.266134i −0.991107 0.133067i \(-0.957517\pi\)
0.991107 0.133067i \(-0.0424825\pi\)
\(54\) 0 0
\(55\) 150.003 2.72733
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −38.5788 22.2735i −0.653877 0.377516i 0.136063 0.990700i \(-0.456555\pi\)
−0.789940 + 0.613184i \(0.789888\pi\)
\(60\) 0 0
\(61\) 1.86057 + 3.22260i 0.0305012 + 0.0528296i 0.880873 0.473353i \(-0.156956\pi\)
−0.850372 + 0.526182i \(0.823623\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 77.2105 44.5775i 1.18785 0.685808i
\(66\) 0 0
\(67\) 21.0090 36.3887i 0.313568 0.543115i −0.665564 0.746340i \(-0.731809\pi\)
0.979132 + 0.203225i \(0.0651424\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.440i 1.69634i 0.529724 + 0.848170i \(0.322295\pi\)
−0.529724 + 0.848170i \(0.677705\pi\)
\(72\) 0 0
\(73\) 5.48692 0.0751633 0.0375817 0.999294i \(-0.488035\pi\)
0.0375817 + 0.999294i \(0.488035\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 28.9674 + 16.7243i 0.376200 + 0.217199i
\(78\) 0 0
\(79\) −60.5480 104.872i −0.766430 1.32750i −0.939487 0.342585i \(-0.888698\pi\)
0.173056 0.984912i \(-0.444636\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −46.5861 + 26.8965i −0.561279 + 0.324054i −0.753659 0.657266i \(-0.771713\pi\)
0.192380 + 0.981321i \(0.438379\pi\)
\(84\) 0 0
\(85\) 54.9840 95.2351i 0.646871 1.12041i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 102.195i 1.14825i −0.818766 0.574127i \(-0.805342\pi\)
0.818766 0.574127i \(-0.194658\pi\)
\(90\) 0 0
\(91\) 19.8803 0.218465
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −106.463 61.4667i −1.12067 0.647018i
\(96\) 0 0
\(97\) −58.9377 102.083i −0.607605 1.05240i −0.991634 0.129081i \(-0.958797\pi\)
0.384029 0.923321i \(-0.374536\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 118.181 68.2317i 1.17011 0.675561i 0.216401 0.976304i \(-0.430568\pi\)
0.953705 + 0.300743i \(0.0972347\pi\)
\(102\) 0 0
\(103\) −60.8511 + 105.397i −0.590787 + 1.02327i 0.403340 + 0.915050i \(0.367849\pi\)
−0.994127 + 0.108223i \(0.965484\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 82.1437i 0.767698i −0.923396 0.383849i \(-0.874598\pi\)
0.923396 0.383849i \(-0.125402\pi\)
\(108\) 0 0
\(109\) 165.603 1.51929 0.759646 0.650337i \(-0.225372\pi\)
0.759646 + 0.650337i \(0.225372\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −68.7460 39.6905i −0.608372 0.351244i 0.163956 0.986468i \(-0.447574\pi\)
−0.772328 + 0.635224i \(0.780908\pi\)
\(114\) 0 0
\(115\) 28.8608 + 49.9884i 0.250964 + 0.434682i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 21.2361 12.2607i 0.178455 0.103031i
\(120\) 0 0
\(121\) 64.7443 112.140i 0.535077 0.926781i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 377.485i 3.01988i
\(126\) 0 0
\(127\) −147.235 −1.15933 −0.579666 0.814854i \(-0.696817\pi\)
−0.579666 + 0.814854i \(0.696817\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 145.857 + 84.2103i 1.11341 + 0.642827i 0.939710 0.341972i \(-0.111095\pi\)
0.173699 + 0.984799i \(0.444428\pi\)
\(132\) 0 0
\(133\) −13.7062 23.7398i −0.103054 0.178495i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 174.984 101.027i 1.27726 0.737426i 0.300915 0.953651i \(-0.402708\pi\)
0.976343 + 0.216225i \(0.0693746\pi\)
\(138\) 0 0
\(139\) −129.193 + 223.768i −0.929443 + 1.60984i −0.145189 + 0.989404i \(0.546379\pi\)
−0.784255 + 0.620439i \(0.786954\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 148.879i 1.04111i
\(144\) 0 0
\(145\) 271.047 1.86929
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 68.6316 + 39.6245i 0.460615 + 0.265936i 0.712303 0.701872i \(-0.247652\pi\)
−0.251688 + 0.967808i \(0.580986\pi\)
\(150\) 0 0
\(151\) −4.73094 8.19422i −0.0313307 0.0542664i 0.849935 0.526888i \(-0.176641\pi\)
−0.881266 + 0.472621i \(0.843308\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 143.700 82.9652i 0.927096 0.535259i
\(156\) 0 0
\(157\) −34.3561 + 59.5066i −0.218829 + 0.379023i −0.954450 0.298370i \(-0.903557\pi\)
0.735621 + 0.677393i \(0.236890\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.8711i 0.0799448i
\(162\) 0 0
\(163\) 209.391 1.28461 0.642304 0.766450i \(-0.277979\pi\)
0.642304 + 0.766450i \(0.277979\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.3682 + 12.3369i 0.127953 + 0.0738739i 0.562610 0.826722i \(-0.309797\pi\)
−0.434657 + 0.900596i \(0.643130\pi\)
\(168\) 0 0
\(169\) 40.2566 + 69.7265i 0.238205 + 0.412583i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 129.186 74.5855i 0.746739 0.431130i −0.0777754 0.996971i \(-0.524782\pi\)
0.824514 + 0.565841i \(0.191448\pi\)
\(174\) 0 0
\(175\) 68.5046 118.654i 0.391455 0.678020i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 65.1600i 0.364022i 0.983296 + 0.182011i \(0.0582607\pi\)
−0.983296 + 0.182011i \(0.941739\pi\)
\(180\) 0 0
\(181\) 95.5019 0.527635 0.263817 0.964573i \(-0.415018\pi\)
0.263817 + 0.964573i \(0.415018\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −128.215 74.0248i −0.693053 0.400134i
\(186\) 0 0
\(187\) −91.8171 159.032i −0.491001 0.850438i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 160.947 92.9225i 0.842652 0.486505i −0.0155129 0.999880i \(-0.504938\pi\)
0.858165 + 0.513374i \(0.171605\pi\)
\(192\) 0 0
\(193\) −48.1579 + 83.4119i −0.249523 + 0.432186i −0.963393 0.268091i \(-0.913607\pi\)
0.713871 + 0.700277i \(0.246940\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 126.121i 0.640209i 0.947382 + 0.320105i \(0.103718\pi\)
−0.947382 + 0.320105i \(0.896282\pi\)
\(198\) 0 0
\(199\) −131.718 −0.661899 −0.330950 0.943648i \(-0.607369\pi\)
−0.330950 + 0.943648i \(0.607369\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 52.3424 + 30.2199i 0.257844 + 0.148866i
\(204\) 0 0
\(205\) 81.0195 + 140.330i 0.395217 + 0.684536i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −177.782 + 102.642i −0.850631 + 0.491112i
\(210\) 0 0
\(211\) −5.15331 + 8.92579i −0.0244233 + 0.0423023i −0.877979 0.478700i \(-0.841108\pi\)
0.853555 + 0.521002i \(0.174442\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 411.634i 1.91457i
\(216\) 0 0
\(217\) 37.0001 0.170508
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −94.5212 54.5718i −0.427698 0.246931i
\(222\) 0 0
\(223\) 86.4202 + 149.684i 0.387535 + 0.671230i 0.992117 0.125313i \(-0.0399934\pi\)
−0.604583 + 0.796542i \(0.706660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 173.974 100.444i 0.766403 0.442483i −0.0651869 0.997873i \(-0.520764\pi\)
0.831590 + 0.555390i \(0.187431\pi\)
\(228\) 0 0
\(229\) 130.630 226.259i 0.570439 0.988029i −0.426082 0.904684i \(-0.640107\pi\)
0.996521 0.0833443i \(-0.0265601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 130.530i 0.560214i −0.959969 0.280107i \(-0.909630\pi\)
0.959969 0.280107i \(-0.0903700\pi\)
\(234\) 0 0
\(235\) −226.208 −0.962586
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −231.586 133.706i −0.968981 0.559441i −0.0700553 0.997543i \(-0.522318\pi\)
−0.898925 + 0.438102i \(0.855651\pi\)
\(240\) 0 0
\(241\) −50.8188 88.0207i −0.210866 0.365231i 0.741120 0.671373i \(-0.234295\pi\)
−0.951986 + 0.306142i \(0.900962\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −365.531 + 211.039i −1.49196 + 0.861385i
\(246\) 0 0
\(247\) −61.0059 + 105.665i −0.246987 + 0.427795i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 137.033i 0.545946i 0.962022 + 0.272973i \(0.0880070\pi\)
−0.962022 + 0.272973i \(0.911993\pi\)
\(252\) 0 0
\(253\) 96.3886 0.380983
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 217.737 + 125.711i 0.847226 + 0.489146i 0.859714 0.510776i \(-0.170642\pi\)
−0.0124876 + 0.999922i \(0.503975\pi\)
\(258\) 0 0
\(259\) −16.5065 28.5901i −0.0637317 0.110387i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 416.538 240.488i 1.58379 0.914404i 0.589496 0.807771i \(-0.299326\pi\)
0.994298 0.106633i \(-0.0340069\pi\)
\(264\) 0 0
\(265\) −66.8425 + 115.775i −0.252236 + 0.436885i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 182.939i 0.680071i 0.940413 + 0.340036i \(0.110439\pi\)
−0.940413 + 0.340036i \(0.889561\pi\)
\(270\) 0 0
\(271\) 31.0146 0.114445 0.0572225 0.998361i \(-0.481776\pi\)
0.0572225 + 0.998361i \(0.481776\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −888.567 513.014i −3.23115 1.86551i
\(276\) 0 0
\(277\) 206.382 + 357.464i 0.745060 + 1.29048i 0.950167 + 0.311743i \(0.100913\pi\)
−0.205106 + 0.978740i \(0.565754\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 107.255 61.9236i 0.381690 0.220369i −0.296863 0.954920i \(-0.595941\pi\)
0.678553 + 0.734551i \(0.262607\pi\)
\(282\) 0 0
\(283\) −4.23689 + 7.33850i −0.0149713 + 0.0259311i −0.873414 0.486978i \(-0.838099\pi\)
0.858443 + 0.512910i \(0.171432\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.1324i 0.125897i
\(288\) 0 0
\(289\) 154.377 0.534177
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −93.2120 53.8160i −0.318130 0.183672i 0.332429 0.943128i \(-0.392132\pi\)
−0.650559 + 0.759456i \(0.725465\pi\)
\(294\) 0 0
\(295\) 211.103 + 365.641i 0.715604 + 1.23946i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 49.6137 28.6445i 0.165932 0.0958009i
\(300\) 0 0
\(301\) −45.8943 + 79.4912i −0.152473 + 0.264090i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 35.2682i 0.115633i
\(306\) 0 0
\(307\) −530.715 −1.72871 −0.864357 0.502878i \(-0.832274\pi\)
−0.864357 + 0.502878i \(0.832274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 142.535 + 82.2926i 0.458312 + 0.264606i 0.711334 0.702854i \(-0.248091\pi\)
−0.253022 + 0.967460i \(0.581425\pi\)
\(312\) 0 0
\(313\) 273.833 + 474.293i 0.874866 + 1.51531i 0.856905 + 0.515474i \(0.172384\pi\)
0.0179611 + 0.999839i \(0.494283\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 67.7106 39.0928i 0.213598 0.123321i −0.389384 0.921075i \(-0.627312\pi\)
0.602983 + 0.797754i \(0.293979\pi\)
\(318\) 0 0
\(319\) 226.309 391.979i 0.709433 1.22877i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 150.495i 0.465929i
\(324\) 0 0
\(325\) −609.824 −1.87638
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −43.6833 25.2206i −0.132776 0.0766583i
\(330\) 0 0
\(331\) 274.898 + 476.137i 0.830507 + 1.43848i 0.897637 + 0.440736i \(0.145283\pi\)
−0.0671297 + 0.997744i \(0.521384\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −344.884 + 199.119i −1.02951 + 0.594385i
\(336\) 0 0
\(337\) 36.8057 63.7494i 0.109216 0.189167i −0.806237 0.591593i \(-0.798499\pi\)
0.915453 + 0.402425i \(0.131833\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 277.085i 0.812566i
\(342\) 0 0
\(343\) −197.675 −0.576312
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 367.796 + 212.347i 1.05993 + 0.611951i 0.925413 0.378960i \(-0.123718\pi\)
0.134517 + 0.990911i \(0.457052\pi\)
\(348\) 0 0
\(349\) −267.361 463.082i −0.766077 1.32688i −0.939676 0.342067i \(-0.888873\pi\)
0.173599 0.984816i \(-0.444460\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 155.165 89.5845i 0.439561 0.253781i −0.263851 0.964564i \(-0.584993\pi\)
0.703411 + 0.710783i \(0.251659\pi\)
\(354\) 0 0
\(355\) 570.753 988.573i 1.60775 2.78471i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 351.534i 0.979204i −0.871946 0.489602i \(-0.837142\pi\)
0.871946 0.489602i \(-0.162858\pi\)
\(360\) 0 0
\(361\) −192.761 −0.533965
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −45.0367 26.0019i −0.123388 0.0712382i
\(366\) 0 0
\(367\) −41.9855 72.7210i −0.114402 0.198150i 0.803139 0.595792i \(-0.203162\pi\)
−0.917541 + 0.397642i \(0.869829\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −25.8161 + 14.9050i −0.0695853 + 0.0401751i
\(372\) 0 0
\(373\) 218.337 378.171i 0.585354 1.01386i −0.409477 0.912320i \(-0.634289\pi\)
0.994831 0.101543i \(-0.0323779\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 269.016i 0.713569i
\(378\) 0 0
\(379\) 273.455 0.721516 0.360758 0.932659i \(-0.382518\pi\)
0.360758 + 0.932659i \(0.382518\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −192.544 111.166i −0.502727 0.290249i 0.227112 0.973869i \(-0.427072\pi\)
−0.729839 + 0.683619i \(0.760405\pi\)
\(384\) 0 0
\(385\) −158.510 274.547i −0.411713 0.713108i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 522.471 301.649i 1.34311 0.775447i 0.355851 0.934543i \(-0.384191\pi\)
0.987263 + 0.159096i \(0.0508578\pi\)
\(390\) 0 0
\(391\) 35.3314 61.1958i 0.0903617 0.156511i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1147.72i 2.90563i
\(396\) 0 0
\(397\) 138.804 0.349633 0.174816 0.984601i \(-0.444067\pi\)
0.174816 + 0.984601i \(0.444067\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 281.903 + 162.757i 0.702999 + 0.405877i 0.808464 0.588546i \(-0.200300\pi\)
−0.105464 + 0.994423i \(0.533633\pi\)
\(402\) 0 0
\(403\) −82.3433 142.623i −0.204326 0.353903i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −214.104 + 123.613i −0.526055 + 0.303718i
\(408\) 0 0
\(409\) 257.442 445.903i 0.629443 1.09023i −0.358220 0.933637i \(-0.616616\pi\)
0.987664 0.156591i \(-0.0500504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 94.1461i 0.227957i
\(414\) 0 0
\(415\) 509.839 1.22853
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −107.340 61.9726i −0.256181 0.147906i 0.366410 0.930453i \(-0.380587\pi\)
−0.622591 + 0.782547i \(0.713920\pi\)
\(420\) 0 0
\(421\) −255.924 443.273i −0.607895 1.05291i −0.991587 0.129444i \(-0.958681\pi\)
0.383691 0.923461i \(-0.374653\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −651.412 + 376.093i −1.53273 + 0.884924i
\(426\) 0 0
\(427\) 3.93216 6.81070i 0.00920880 0.0159501i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 650.840i 1.51007i 0.655684 + 0.755035i \(0.272380\pi\)
−0.655684 + 0.755035i \(0.727620\pi\)
\(432\) 0 0
\(433\) −432.455 −0.998742 −0.499371 0.866388i \(-0.666435\pi\)
−0.499371 + 0.866388i \(0.666435\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −68.4108 39.4970i −0.156547 0.0903822i
\(438\) 0 0
\(439\) −190.663 330.238i −0.434312 0.752251i 0.562927 0.826507i \(-0.309675\pi\)
−0.997239 + 0.0742559i \(0.976342\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 33.3781 19.2708i 0.0753455 0.0435007i −0.461854 0.886956i \(-0.652816\pi\)
0.537199 + 0.843455i \(0.319482\pi\)
\(444\) 0 0
\(445\) −484.289 + 838.813i −1.08829 + 1.88497i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 373.577i 0.832019i 0.909360 + 0.416010i \(0.136572\pi\)
−0.909360 + 0.416010i \(0.863428\pi\)
\(450\) 0 0
\(451\) 270.587 0.599971
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −163.178 94.2108i −0.358633 0.207057i
\(456\) 0 0
\(457\) −121.482 210.414i −0.265826 0.460423i 0.701954 0.712222i \(-0.252311\pi\)
−0.967780 + 0.251799i \(0.918978\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −116.211 + 67.0942i −0.252084 + 0.145541i −0.620718 0.784034i \(-0.713159\pi\)
0.368634 + 0.929574i \(0.379825\pi\)
\(462\) 0 0
\(463\) 155.129 268.691i 0.335051 0.580326i −0.648443 0.761263i \(-0.724580\pi\)
0.983495 + 0.180937i \(0.0579131\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 765.680i 1.63957i −0.572670 0.819786i \(-0.694092\pi\)
0.572670 0.819786i \(-0.305908\pi\)
\(468\) 0 0
\(469\) −88.8015 −0.189342
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 595.290 + 343.691i 1.25854 + 0.726619i
\(474\) 0 0
\(475\) 420.434 + 728.214i 0.885125 + 1.53308i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 442.124 255.260i 0.923014 0.532902i 0.0384186 0.999262i \(-0.487768\pi\)
0.884595 + 0.466359i \(0.154435\pi\)
\(480\) 0 0
\(481\) −73.4699 + 127.254i −0.152744 + 0.264561i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1117.20i 2.30350i
\(486\) 0 0
\(487\) −669.532 −1.37481 −0.687405 0.726274i \(-0.741250\pi\)
−0.687405 + 0.726274i \(0.741250\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −640.537 369.814i −1.30456 0.753186i −0.323375 0.946271i \(-0.604817\pi\)
−0.981182 + 0.193085i \(0.938151\pi\)
\(492\) 0 0
\(493\) −165.908 287.361i −0.336528 0.582883i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 220.438 127.270i 0.443537 0.256076i
\(498\) 0 0
\(499\) −461.405 + 799.176i −0.924659 + 1.60156i −0.132550 + 0.991176i \(0.542316\pi\)
−0.792109 + 0.610380i \(0.791017\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 223.098i 0.443534i 0.975100 + 0.221767i \(0.0711824\pi\)
−0.975100 + 0.221767i \(0.928818\pi\)
\(504\) 0 0
\(505\) −1293.37 −2.56113
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 125.233 + 72.3030i 0.246036 + 0.142049i 0.617948 0.786219i \(-0.287964\pi\)
−0.371912 + 0.928268i \(0.621298\pi\)
\(510\) 0 0
\(511\) −5.79807 10.0425i −0.0113465 0.0196527i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 998.932 576.733i 1.93967 1.11987i
\(516\) 0 0
\(517\) −188.871 + 327.134i −0.365320 + 0.632753i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 452.382i 0.868296i 0.900842 + 0.434148i \(0.142950\pi\)
−0.900842 + 0.434148i \(0.857050\pi\)
\(522\) 0 0
\(523\) −168.242 −0.321686 −0.160843 0.986980i \(-0.551421\pi\)
−0.160843 + 0.986980i \(0.551421\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −175.918 101.566i −0.333809 0.192725i
\(528\) 0 0
\(529\) −245.955 426.006i −0.464943 0.805305i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 139.278 80.4122i 0.261309 0.150867i
\(534\) 0 0
\(535\) −389.270 + 674.235i −0.727607 + 1.26025i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 704.824i 1.30765i
\(540\) 0 0
\(541\) 809.693 1.49666 0.748330 0.663327i \(-0.230856\pi\)
0.748330 + 0.663327i \(0.230856\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1359.27 784.774i −2.49407 1.43995i
\(546\) 0 0
\(547\) −468.105 810.781i −0.855767 1.48223i −0.875931 0.482436i \(-0.839752\pi\)
0.0201641 0.999797i \(-0.493581\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −321.242 + 185.469i −0.583015 + 0.336604i
\(552\) 0 0
\(553\) −127.963 + 221.638i −0.231398 + 0.400793i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 318.572i 0.571942i 0.958238 + 0.285971i \(0.0923162\pi\)
−0.958238 + 0.285971i \(0.907684\pi\)
\(558\) 0 0
\(559\) 408.548 0.730855
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −700.038 404.167i −1.24341 0.717882i −0.273621 0.961838i \(-0.588221\pi\)
−0.969786 + 0.243956i \(0.921555\pi\)
\(564\) 0 0
\(565\) 376.178 + 651.560i 0.665802 + 1.15320i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −895.501 + 517.017i −1.57381 + 0.908642i −0.578120 + 0.815952i \(0.696213\pi\)
−0.995695 + 0.0926904i \(0.970453\pi\)
\(570\) 0 0
\(571\) 24.0163 41.5974i 0.0420600 0.0728500i −0.844229 0.535983i \(-0.819941\pi\)
0.886289 + 0.463133i \(0.153275\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 394.818i 0.686640i
\(576\) 0 0
\(577\) 396.617 0.687378 0.343689 0.939084i \(-0.388323\pi\)
0.343689 + 0.939084i \(0.388323\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 98.4558 + 56.8435i 0.169459 + 0.0978373i
\(582\) 0 0
\(583\) 111.619 + 193.331i 0.191457 + 0.331613i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 560.569 323.645i 0.954973 0.551354i 0.0603509 0.998177i \(-0.480778\pi\)
0.894622 + 0.446823i \(0.147445\pi\)
\(588\) 0 0
\(589\) −113.541 + 196.658i −0.192769 + 0.333885i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 322.360i 0.543609i 0.962352 + 0.271805i \(0.0876204\pi\)
−0.962352 + 0.271805i \(0.912380\pi\)
\(594\) 0 0
\(595\) −232.408 −0.390602
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −286.437 165.374i −0.478191 0.276084i 0.241471 0.970408i \(-0.422370\pi\)
−0.719662 + 0.694324i \(0.755703\pi\)
\(600\) 0 0
\(601\) −2.29683 3.97823i −0.00382169 0.00661936i 0.864108 0.503306i \(-0.167883\pi\)
−0.867930 + 0.496687i \(0.834550\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1062.84 + 613.633i −1.75677 + 1.01427i
\(606\) 0 0
\(607\) −100.896 + 174.756i −0.166220 + 0.287902i −0.937088 0.349093i \(-0.886490\pi\)
0.770868 + 0.636995i \(0.219823\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 224.512i 0.367450i
\(612\) 0 0
\(613\) −594.531 −0.969871 −0.484936 0.874550i \(-0.661157\pi\)
−0.484936 + 0.874550i \(0.661157\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 562.741 + 324.899i 0.912060 + 0.526578i 0.881093 0.472942i \(-0.156808\pi\)
0.0309665 + 0.999520i \(0.490141\pi\)
\(618\) 0 0
\(619\) 114.275 + 197.931i 0.184613 + 0.319759i 0.943446 0.331526i \(-0.107564\pi\)
−0.758833 + 0.651285i \(0.774230\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −187.044 + 107.990i −0.300231 + 0.173338i
\(624\) 0 0
\(625\) −978.507 + 1694.82i −1.56561 + 2.71172i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 181.243i 0.288144i
\(630\) 0 0
\(631\) 555.448 0.880266 0.440133 0.897933i \(-0.354931\pi\)
0.440133 + 0.897933i \(0.354931\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1208.51 + 697.732i 1.90316 + 1.09879i
\(636\) 0 0
\(637\) 209.457 + 362.791i 0.328819 + 0.569530i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −374.507 + 216.222i −0.584254 + 0.337319i −0.762822 0.646608i \(-0.776187\pi\)
0.178568 + 0.983928i \(0.442853\pi\)
\(642\) 0 0
\(643\) 170.831 295.888i 0.265678 0.460168i −0.702063 0.712115i \(-0.747738\pi\)
0.967741 + 0.251947i \(0.0810709\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1066.85i 1.64891i 0.565926 + 0.824456i \(0.308519\pi\)
−0.565926 + 0.824456i \(0.691481\pi\)
\(648\) 0 0
\(649\) 705.037 1.08634
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 576.545 + 332.868i 0.882917 + 0.509752i 0.871619 0.490184i \(-0.163070\pi\)
0.0112977 + 0.999936i \(0.496404\pi\)
\(654\) 0 0
\(655\) −798.128 1382.40i −1.21852 2.11053i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −795.462 + 459.260i −1.20707 + 0.696905i −0.962119 0.272630i \(-0.912107\pi\)
−0.244955 + 0.969534i \(0.578773\pi\)
\(660\) 0 0
\(661\) −385.777 + 668.185i −0.583626 + 1.01087i 0.411420 + 0.911446i \(0.365033\pi\)
−0.995045 + 0.0994232i \(0.968300\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 259.809i 0.390690i
\(666\) 0 0
\(667\) 174.169 0.261122
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −51.0036 29.4470i −0.0760114 0.0438852i
\(672\) 0 0
\(673\) 559.767 + 969.546i 0.831750 + 1.44063i 0.896650 + 0.442741i \(0.145994\pi\)
−0.0649002 + 0.997892i \(0.520673\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −339.051 + 195.751i −0.500815 + 0.289145i −0.729050 0.684461i \(-0.760038\pi\)
0.228235 + 0.973606i \(0.426704\pi\)
\(678\) 0 0
\(679\) −124.560 + 215.744i −0.183446 + 0.317737i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 661.278i 0.968196i −0.875014 0.484098i \(-0.839148\pi\)
0.875014 0.484098i \(-0.160852\pi\)
\(684\) 0 0
\(685\) −1915.03 −2.79566
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 114.907 + 66.3415i 0.166773 + 0.0962866i
\(690\) 0 0
\(691\) −412.836 715.053i −0.597447 1.03481i −0.993197 0.116450i \(-0.962849\pi\)
0.395750 0.918358i \(-0.370485\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2120.83 1224.46i 3.05155 1.76181i
\(696\) 0 0
\(697\) 99.1841 171.792i 0.142301 0.246473i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 236.167i 0.336900i 0.985710 + 0.168450i \(0.0538761\pi\)
−0.985710 + 0.168450i \(0.946124\pi\)
\(702\) 0 0
\(703\) 202.611 0.288209
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −249.765 144.202i −0.353274 0.203963i
\(708\) 0 0
\(709\) 247.969 + 429.496i 0.349745 + 0.605777i 0.986204 0.165534i \(-0.0529348\pi\)
−0.636459 + 0.771311i \(0.719601\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 92.3381 53.3115i 0.129507 0.0747706i
\(714\) 0 0
\(715\) −705.521 + 1222.00i −0.986743 + 1.70909i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1065.13i 1.48141i −0.671830 0.740705i \(-0.734492\pi\)
0.671830 0.740705i \(-0.265508\pi\)
\(720\) 0 0
\(721\) 257.207 0.356736
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1605.59 926.987i −2.21460 1.27860i
\(726\) 0 0
\(727\) 549.525 + 951.806i 0.755881 + 1.30922i 0.944935 + 0.327257i \(0.106124\pi\)
−0.189054 + 0.981967i \(0.560542\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 436.409 251.961i 0.597003 0.344680i
\(732\) 0 0
\(733\) 720.569 1248.06i 0.983041 1.70268i 0.332706 0.943031i \(-0.392038\pi\)
0.650336 0.759647i \(-0.274628\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 665.013i 0.902324i
\(738\) 0 0
\(739\) 1095.72 1.48271 0.741356 0.671112i \(-0.234183\pi\)
0.741356 + 0.671112i \(0.234183\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 857.848 + 495.279i 1.15457 + 0.666593i 0.949997 0.312258i \(-0.101085\pi\)
0.204575 + 0.978851i \(0.434419\pi\)
\(744\) 0 0
\(745\) −375.552 650.476i −0.504097 0.873122i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −150.345 + 86.8018i −0.200728 + 0.115890i
\(750\) 0 0
\(751\) −177.884 + 308.103i −0.236862 + 0.410258i −0.959812 0.280643i \(-0.909452\pi\)
0.722950 + 0.690900i \(0.242786\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 89.6776i 0.118778i
\(756\) 0 0
\(757\) 231.917 0.306364 0.153182 0.988198i \(-0.451048\pi\)
0.153182 + 0.988198i \(0.451048\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 368.325 + 212.653i 0.484002 + 0.279438i 0.722083 0.691807i \(-0.243185\pi\)
−0.238081 + 0.971245i \(0.576518\pi\)
\(762\) 0 0
\(763\) −174.994 303.098i −0.229350 0.397245i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 362.900 209.521i 0.473143 0.273169i
\(768\) 0 0
\(769\) 422.147 731.179i 0.548955 0.950819i −0.449391 0.893335i \(-0.648359\pi\)
0.998346 0.0574834i \(-0.0183076\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 816.503i 1.05628i −0.849158 0.528139i \(-0.822890\pi\)
0.849158 0.528139i \(-0.177110\pi\)
\(774\) 0 0
\(775\) −1134.97 −1.46448
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −192.046 110.878i −0.246529 0.142334i
\(780\) 0 0
\(781\) −953.093 1650.81i −1.22035 2.11371i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 563.991 325.620i 0.718459 0.414803i
\(786\) 0 0
\(787\) −178.111 + 308.497i −0.226316 + 0.391991i −0.956713 0.291031i \(-0.906002\pi\)
0.730397 + 0.683022i \(0.239335\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 167.765i 0.212092i
\(792\) 0 0
\(793\) −35.0038 −0.0441410
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 561.666 + 324.278i 0.704725 + 0.406873i 0.809105 0.587664i \(-0.199952\pi\)
−0.104380 + 0.994537i \(0.533286\pi\)
\(798\) 0 0
\(799\) 138.462 + 239.823i 0.173294 + 0.300154i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −75.2062 + 43.4203i −0.0936565 + 0.0540726i
\(804\) 0 0
\(805\) 60.9948 105.646i 0.0757700 0.131237i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 445.244i 0.550363i −0.961392 0.275182i \(-0.911262\pi\)
0.961392 0.275182i \(-0.0887380\pi\)
\(810\) 0 0
\(811\) −373.366 −0.460377 −0.230189 0.973146i \(-0.573934\pi\)
−0.230189 + 0.973146i \(0.573934\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1718.68 992.282i −2.10881 1.21752i
\(816\) 0 0
\(817\) −281.667 487.862i −0.344758 0.597139i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −42.7001 + 24.6529i −0.0520099 + 0.0300279i −0.525779 0.850621i \(-0.676226\pi\)
0.473770 + 0.880649i \(0.342893\pi\)
\(822\) 0 0
\(823\) −289.224 + 500.950i −0.351426 + 0.608688i −0.986500 0.163764i \(-0.947636\pi\)
0.635074 + 0.772452i \(0.280970\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 665.710i 0.804970i 0.915427 + 0.402485i \(0.131853\pi\)
−0.915427 + 0.402485i \(0.868147\pi\)
\(828\) 0 0
\(829\) −133.042 −0.160485 −0.0802423 0.996775i \(-0.525569\pi\)
−0.0802423 + 0.996775i \(0.525569\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 447.483 + 258.355i 0.537195 + 0.310150i
\(834\) 0 0
\(835\) −116.927 202.523i −0.140032 0.242543i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −707.636 + 408.554i −0.843428 + 0.486954i −0.858428 0.512934i \(-0.828559\pi\)
0.0149998 + 0.999887i \(0.495225\pi\)
\(840\) 0 0
\(841\) −11.5724 + 20.0439i −0.0137602 + 0.0238334i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 763.087i 0.903062i
\(846\) 0 0
\(847\) −273.663 −0.323097
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −82.3878 47.5666i −0.0968129 0.0558950i
\(852\) 0 0
\(853\) 14.2616 + 24.7018i 0.0167193 + 0.0289587i 0.874264 0.485451i \(-0.161344\pi\)
−0.857545 + 0.514409i \(0.828011\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1218.49 703.496i 1.42181 0.820883i 0.425357 0.905026i \(-0.360149\pi\)
0.996454 + 0.0841433i \(0.0268154\pi\)
\(858\) 0 0
\(859\) −296.573 + 513.680i −0.345254 + 0.597998i −0.985400 0.170256i \(-0.945541\pi\)
0.640146 + 0.768254i \(0.278874\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 549.058i 0.636220i 0.948054 + 0.318110i \(0.103048\pi\)
−0.948054 + 0.318110i \(0.896952\pi\)
\(864\) 0 0
\(865\) −1413.81 −1.63446
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1659.80 + 958.283i 1.91001 + 1.10274i
\(870\) 0 0
\(871\) 197.626 + 342.299i 0.226896 + 0.392996i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −690.900 + 398.891i −0.789600 + 0.455876i
\(876\) 0 0
\(877\) 481.413 833.831i 0.548931 0.950776i −0.449417 0.893322i \(-0.648368\pi\)
0.998348 0.0574544i \(-0.0182984\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 485.913i 0.551547i −0.961223 0.275774i \(-0.911066\pi\)
0.961223 0.275774i \(-0.0889340\pi\)
\(882\) 0 0
\(883\) 1020.25 1.15544 0.577720 0.816235i \(-0.303943\pi\)
0.577720 + 0.816235i \(0.303943\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 896.965 + 517.863i 1.01123 + 0.583836i 0.911553 0.411183i \(-0.134884\pi\)
0.0996814 + 0.995019i \(0.468218\pi\)
\(888\) 0 0
\(889\) 155.584 + 269.480i 0.175011 + 0.303127i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 268.098 154.787i 0.300222 0.173333i
\(894\) 0 0
\(895\) 308.786 534.834i 0.345013 0.597580i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 500.676i 0.556926i
\(900\) 0 0
\(901\) 163.657 0.181640
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −783.880 452.573i −0.866165 0.500081i
\(906\) 0 0
\(907\) −794.936 1376.87i −0.876445 1.51805i −0.855215 0.518273i \(-0.826575\pi\)
−0.0212301 0.999775i \(-0.506758\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 362.743 209.430i 0.398181 0.229890i −0.287518 0.957775i \(-0.592830\pi\)
0.685699 + 0.727885i \(0.259497\pi\)
\(912\) 0 0
\(913\) 425.687 737.311i 0.466250 0.807569i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 355.943i 0.388160i
\(918\) 0 0
\(919\) 1499.57 1.63174 0.815872 0.578233i \(-0.196257\pi\)
0.815872 + 0.578233i \(0.196257\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −981.162 566.474i −1.06301 0.613732i
\(924\) 0 0
\(925\) 506.333 + 876.994i 0.547387 + 0.948102i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1046.99 604.481i 1.12701 0.650679i 0.183829 0.982958i \(-0.441151\pi\)
0.943181 + 0.332279i \(0.107817\pi\)
\(930\) 0 0
\(931\) 288.815 500.242i 0.310220 0.537317i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1740.45i 1.86144i
\(936\) 0 0
\(937\) 557.393 0.594870 0.297435 0.954742i \(-0.403869\pi\)
0.297435 + 0.954742i \(0.403869\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 465.315 + 268.650i 0.494490 + 0.285494i 0.726435 0.687235i \(-0.241176\pi\)
−0.231945 + 0.972729i \(0.574509\pi\)
\(942\) 0 0
\(943\) 52.0612 + 90.1727i 0.0552081 + 0.0956232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −515.416 + 297.575i −0.544261 + 0.314229i −0.746804 0.665044i \(-0.768413\pi\)
0.202543 + 0.979273i \(0.435079\pi\)
\(948\) 0 0
\(949\) −25.8070 + 44.6991i −0.0271939 + 0.0471012i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 88.9834i 0.0933718i −0.998910 0.0466859i \(-0.985134\pi\)
0.998910 0.0466859i \(-0.0148660\pi\)
\(954\) 0 0
\(955\) −1761.40 −1.84440
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −369.814 213.512i −0.385625 0.222641i
\(960\) 0 0
\(961\) 327.247 + 566.809i 0.340528 + 0.589812i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 790.560 456.430i 0.819233 0.472985i
\(966\) 0 0
\(967\) 658.074 1139.82i 0.680532 1.17872i −0.294287 0.955717i \(-0.595082\pi\)
0.974819 0.222998i \(-0.0715845\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 361.433i 0.372228i −0.982528 0.186114i \(-0.940411\pi\)
0.982528 0.186114i \(-0.0595893\pi\)
\(972\) 0 0
\(973\) 546.075 0.561228
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −883.148 509.885i −0.903938 0.521889i −0.0254624 0.999676i \(-0.508106\pi\)
−0.878476 + 0.477787i \(0.841439\pi\)
\(978\) 0 0
\(979\) 808.708 + 1400.72i 0.826056 + 1.43077i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1144.21 + 660.612i −1.16400 + 0.672037i −0.952260 0.305289i \(-0.901247\pi\)
−0.211742 + 0.977326i \(0.567914\pi\)
\(984\) 0 0
\(985\) 597.675 1035.20i 0.606777 1.05097i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 264.506i 0.267448i
\(990\) 0 0
\(991\) 6.11587 0.00617141 0.00308571 0.999995i \(-0.499018\pi\)
0.00308571 + 0.999995i \(0.499018\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1081.14 + 624.197i 1.08657 + 0.627334i
\(996\) 0 0
\(997\) −315.180 545.908i −0.316129 0.547551i 0.663548 0.748134i \(-0.269050\pi\)
−0.979677 + 0.200583i \(0.935716\pi\)
\(998\) 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 1728.3.q.i.449.1 8
3.2 odd 2 576.3.q.j.257.1 8
4.3 odd 2 1728.3.q.j.449.1 8
8.3 odd 2 216.3.m.b.17.4 8
8.5 even 2 432.3.q.e.17.4 8
9.2 odd 6 inner 1728.3.q.i.1601.1 8
9.7 even 3 576.3.q.j.65.1 8
12.11 even 2 576.3.q.i.257.4 8
24.5 odd 2 144.3.q.e.113.4 8
24.11 even 2 72.3.m.b.41.1 8
36.7 odd 6 576.3.q.i.65.4 8
36.11 even 6 1728.3.q.j.1601.1 8
72.5 odd 6 1296.3.e.i.161.8 8
72.11 even 6 216.3.m.b.89.4 8
72.13 even 6 1296.3.e.i.161.1 8
72.29 odd 6 432.3.q.e.305.4 8
72.43 odd 6 72.3.m.b.65.1 yes 8
72.59 even 6 648.3.e.c.161.8 8
72.61 even 6 144.3.q.e.65.4 8
72.67 odd 6 648.3.e.c.161.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.3.m.b.41.1 8 24.11 even 2
72.3.m.b.65.1 yes 8 72.43 odd 6
144.3.q.e.65.4 8 72.61 even 6
144.3.q.e.113.4 8 24.5 odd 2
216.3.m.b.17.4 8 8.3 odd 2
216.3.m.b.89.4 8 72.11 even 6
432.3.q.e.17.4 8 8.5 even 2
432.3.q.e.305.4 8 72.29 odd 6
576.3.q.i.65.4 8 36.7 odd 6
576.3.q.i.257.4 8 12.11 even 2
576.3.q.j.65.1 8 9.7 even 3
576.3.q.j.257.1 8 3.2 odd 2
648.3.e.c.161.1 8 72.67 odd 6
648.3.e.c.161.8 8 72.59 even 6
1296.3.e.i.161.1 8 72.13 even 6
1296.3.e.i.161.8 8 72.5 odd 6
1728.3.q.i.449.1 8 1.1 even 1 trivial
1728.3.q.i.1601.1 8 9.2 odd 6 inner
1728.3.q.j.449.1 8 4.3 odd 2
1728.3.q.j.1601.1 8 36.11 even 6