Properties

Label 1728.3.o.d.1279.4
Level $1728$
Weight $3$
Character 1728.1279
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(127,1728)]
 
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([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (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: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1279.4
Root \(-0.862555 - 0.141174i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1279
Dual form 1728.3.o.d.127.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.81174 - 3.13802i) q^{5} +(1.59422 - 0.920424i) q^{7} +O(q^{10})\) \(q+(1.81174 - 3.13802i) q^{5} +(1.59422 - 0.920424i) q^{7} +(-10.0623 + 5.80948i) q^{11} +(-6.43521 + 11.1461i) q^{13} -12.6235 q^{17} -25.6526i q^{19} +(25.9012 + 14.9541i) q^{23} +(5.93521 + 10.2801i) q^{25} +(-10.8117 - 18.7265i) q^{29} +(52.4027 + 30.2547i) q^{31} -6.67027i q^{35} +25.7409 q^{37} +(33.3704 - 57.7993i) q^{41} +(14.2447 - 8.22418i) q^{43} +(66.1505 - 38.1920i) q^{47} +(-22.8056 + 39.5005i) q^{49} +14.2591 q^{53} +42.1010i q^{55} +(-50.3115 - 29.0474i) q^{59} +(9.43521 + 16.3423i) q^{61} +(23.3178 + 40.3877i) q^{65} +(20.6216 + 11.9059i) q^{67} -46.4758i q^{71} +49.3521 q^{73} +(-10.6944 + 18.5232i) q^{77} +(52.4027 - 30.2547i) q^{79} +(86.2751 - 49.8109i) q^{83} +(-22.8704 + 39.6127i) q^{85} -154.988 q^{89} +23.6925i q^{91} +(-80.4984 - 46.4758i) q^{95} +(-21.1113 - 36.5658i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 1.81174 3.13802i 0.362348 0.627604i −0.625999 0.779824i \(-0.715309\pi\)
0.988347 + 0.152219i \(0.0486420\pi\)
\(6\) 0 0
\(7\) 1.59422 0.920424i 0.227746 0.131489i −0.381786 0.924251i \(-0.624691\pi\)
0.609532 + 0.792762i \(0.291357\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10.0623 + 5.80948i −0.914755 + 0.528134i −0.881958 0.471328i \(-0.843775\pi\)
−0.0327970 + 0.999462i \(0.510441\pi\)
\(12\) 0 0
\(13\) −6.43521 + 11.1461i −0.495016 + 0.857394i −0.999983 0.00574505i \(-0.998171\pi\)
0.504967 + 0.863139i \(0.331505\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −12.6235 −0.742557 −0.371279 0.928521i \(-0.621081\pi\)
−0.371279 + 0.928521i \(0.621081\pi\)
\(18\) 0 0
\(19\) 25.6526i 1.35014i −0.737755 0.675069i \(-0.764114\pi\)
0.737755 0.675069i \(-0.235886\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 25.9012 + 14.9541i 1.12614 + 0.650178i 0.942961 0.332902i \(-0.108028\pi\)
0.183179 + 0.983080i \(0.441361\pi\)
\(24\) 0 0
\(25\) 5.93521 + 10.2801i 0.237409 + 0.411204i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.8117 18.7265i −0.372819 0.645741i 0.617179 0.786822i \(-0.288275\pi\)
−0.989998 + 0.141082i \(0.954942\pi\)
\(30\) 0 0
\(31\) 52.4027 + 30.2547i 1.69041 + 0.975959i 0.954182 + 0.299226i \(0.0967284\pi\)
0.736228 + 0.676733i \(0.236605\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.67027i 0.190579i
\(36\) 0 0
\(37\) 25.7409 0.695699 0.347849 0.937550i \(-0.386912\pi\)
0.347849 + 0.937550i \(0.386912\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 33.3704 57.7993i 0.813913 1.40974i −0.0961931 0.995363i \(-0.530667\pi\)
0.910106 0.414376i \(-0.136000\pi\)
\(42\) 0 0
\(43\) 14.2447 8.22418i 0.331272 0.191260i −0.325134 0.945668i \(-0.605409\pi\)
0.656406 + 0.754408i \(0.272076\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 66.1505 38.1920i 1.40746 0.812595i 0.412314 0.911042i \(-0.364721\pi\)
0.995142 + 0.0984463i \(0.0313873\pi\)
\(48\) 0 0
\(49\) −22.8056 + 39.5005i −0.465421 + 0.806133i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.2591 0.269041 0.134520 0.990911i \(-0.457051\pi\)
0.134520 + 0.990911i \(0.457051\pi\)
\(54\) 0 0
\(55\) 42.1010i 0.765472i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −50.3115 29.0474i −0.852738 0.492328i 0.00883587 0.999961i \(-0.497187\pi\)
−0.861574 + 0.507633i \(0.830521\pi\)
\(60\) 0 0
\(61\) 9.43521 + 16.3423i 0.154676 + 0.267906i 0.932941 0.360030i \(-0.117233\pi\)
−0.778265 + 0.627936i \(0.783900\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 23.3178 + 40.3877i 0.358736 + 0.621349i
\(66\) 0 0
\(67\) 20.6216 + 11.9059i 0.307785 + 0.177700i 0.645935 0.763393i \(-0.276468\pi\)
−0.338150 + 0.941092i \(0.609801\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 46.4758i 0.654589i −0.944922 0.327294i \(-0.893863\pi\)
0.944922 0.327294i \(-0.106137\pi\)
\(72\) 0 0
\(73\) 49.3521 0.676057 0.338028 0.941136i \(-0.390240\pi\)
0.338028 + 0.941136i \(0.390240\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.6944 + 18.5232i −0.138888 + 0.240561i
\(78\) 0 0
\(79\) 52.4027 30.2547i 0.663326 0.382971i −0.130217 0.991485i \(-0.541567\pi\)
0.793543 + 0.608514i \(0.208234\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 86.2751 49.8109i 1.03946 0.600132i 0.119779 0.992801i \(-0.461781\pi\)
0.919680 + 0.392669i \(0.128448\pi\)
\(84\) 0 0
\(85\) −22.8704 + 39.6127i −0.269064 + 0.466032i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −154.988 −1.74144 −0.870718 0.491783i \(-0.836345\pi\)
−0.870718 + 0.491783i \(0.836345\pi\)
\(90\) 0 0
\(91\) 23.6925i 0.260357i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −80.4984 46.4758i −0.847352 0.489219i
\(96\) 0 0
\(97\) −21.1113 36.5658i −0.217642 0.376967i 0.736445 0.676498i \(-0.236503\pi\)
−0.954087 + 0.299531i \(0.903170\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −63.1761 109.424i −0.625506 1.08341i −0.988443 0.151594i \(-0.951559\pi\)
0.362937 0.931814i \(-0.381774\pi\)
\(102\) 0 0
\(103\) −23.9133 13.8064i −0.232168 0.134042i 0.379404 0.925231i \(-0.376129\pi\)
−0.611572 + 0.791189i \(0.709462\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 121.138i 1.13213i −0.824360 0.566066i \(-0.808465\pi\)
0.824360 0.566066i \(-0.191535\pi\)
\(108\) 0 0
\(109\) −61.7409 −0.566430 −0.283215 0.959056i \(-0.591401\pi\)
−0.283215 + 0.959056i \(0.591401\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 43.0709 74.6010i 0.381158 0.660186i −0.610070 0.792348i \(-0.708859\pi\)
0.991228 + 0.132162i \(0.0421920\pi\)
\(114\) 0 0
\(115\) 93.8525 54.1858i 0.816109 0.471180i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −20.1246 + 11.6190i −0.169114 + 0.0976382i
\(120\) 0 0
\(121\) 7.00000 12.1244i 0.0578512 0.100201i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 133.599 1.06879
\(126\) 0 0
\(127\) 183.369i 1.44385i −0.691970 0.721926i \(-0.743257\pi\)
0.691970 0.721926i \(-0.256743\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 149.631 + 86.3894i 1.14222 + 0.659461i 0.946979 0.321295i \(-0.104118\pi\)
0.195240 + 0.980755i \(0.437451\pi\)
\(132\) 0 0
\(133\) −23.6113 40.8959i −0.177528 0.307488i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 80.1235 + 138.778i 0.584843 + 1.01298i 0.994895 + 0.100915i \(0.0321772\pi\)
−0.410052 + 0.912062i \(0.634489\pi\)
\(138\) 0 0
\(139\) 191.971 + 110.835i 1.38109 + 0.797371i 0.992288 0.123951i \(-0.0395566\pi\)
0.388799 + 0.921322i \(0.372890\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 149.541i 1.04574i
\(144\) 0 0
\(145\) −78.3521 −0.540360
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −94.0343 + 162.872i −0.631103 + 1.09310i 0.356224 + 0.934401i \(0.384064\pi\)
−0.987327 + 0.158701i \(0.949269\pi\)
\(150\) 0 0
\(151\) 96.8344 55.9073i 0.641287 0.370247i −0.143823 0.989603i \(-0.545940\pi\)
0.785110 + 0.619356i \(0.212606\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 189.880 109.627i 1.22503 0.707273i
\(156\) 0 0
\(157\) 51.4352 89.0884i 0.327613 0.567442i −0.654425 0.756127i \(-0.727089\pi\)
0.982038 + 0.188685i \(0.0604225\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 55.0564 0.341965
\(162\) 0 0
\(163\) 205.221i 1.25902i 0.776991 + 0.629512i \(0.216745\pi\)
−0.776991 + 0.629512i \(0.783255\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −46.0258 26.5730i −0.275604 0.159120i 0.355828 0.934552i \(-0.384199\pi\)
−0.631432 + 0.775432i \(0.717532\pi\)
\(168\) 0 0
\(169\) 1.67607 + 2.90303i 0.00991755 + 0.0171777i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 94.1517 + 163.075i 0.544229 + 0.942633i 0.998655 + 0.0518482i \(0.0165112\pi\)
−0.454426 + 0.890785i \(0.650155\pi\)
\(174\) 0 0
\(175\) 18.9241 + 10.9258i 0.108138 + 0.0624333i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 92.9516i 0.519283i −0.965705 0.259641i \(-0.916396\pi\)
0.965705 0.259641i \(-0.0836043\pi\)
\(180\) 0 0
\(181\) 240.445 1.32843 0.664213 0.747543i \(-0.268767\pi\)
0.664213 + 0.747543i \(0.268767\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 46.6357 80.7754i 0.252085 0.436624i
\(186\) 0 0
\(187\) 127.021 73.3358i 0.679258 0.392170i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 49.0077 28.2946i 0.256585 0.148139i −0.366191 0.930540i \(-0.619338\pi\)
0.622776 + 0.782400i \(0.286005\pi\)
\(192\) 0 0
\(193\) 50.3704 87.2441i 0.260987 0.452042i −0.705518 0.708692i \(-0.749285\pi\)
0.966504 + 0.256650i \(0.0826188\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −115.247 −0.585010 −0.292505 0.956264i \(-0.594489\pi\)
−0.292505 + 0.956264i \(0.594489\pi\)
\(198\) 0 0
\(199\) 190.733i 0.958455i 0.877691 + 0.479228i \(0.159083\pi\)
−0.877691 + 0.479228i \(0.840917\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −34.4726 19.9028i −0.169816 0.0980432i
\(204\) 0 0
\(205\) −120.917 209.434i −0.589839 1.02163i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 149.028 + 258.124i 0.713054 + 1.23505i
\(210\) 0 0
\(211\) 68.3449 + 39.4590i 0.323910 + 0.187009i 0.653134 0.757242i \(-0.273454\pi\)
−0.329224 + 0.944252i \(0.606787\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 59.6003i 0.277211i
\(216\) 0 0
\(217\) 111.389 0.513312
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 81.2348 140.703i 0.367578 0.636664i
\(222\) 0 0
\(223\) 192.074 110.894i 0.861321 0.497284i −0.00313373 0.999995i \(-0.500997\pi\)
0.864454 + 0.502711i \(0.167664\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.0623 + 5.80948i −0.0443273 + 0.0255924i −0.522000 0.852946i \(-0.674814\pi\)
0.477673 + 0.878538i \(0.341481\pi\)
\(228\) 0 0
\(229\) −24.4352 + 42.3230i −0.106704 + 0.184817i −0.914433 0.404737i \(-0.867363\pi\)
0.807729 + 0.589554i \(0.200696\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −384.785 −1.65144 −0.825719 0.564082i \(-0.809230\pi\)
−0.825719 + 0.564082i \(0.809230\pi\)
\(234\) 0 0
\(235\) 276.775i 1.17777i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 169.755 + 98.0083i 0.710274 + 0.410077i 0.811162 0.584821i \(-0.198835\pi\)
−0.100889 + 0.994898i \(0.532169\pi\)
\(240\) 0 0
\(241\) −130.593 226.194i −0.541880 0.938563i −0.998796 0.0490534i \(-0.984380\pi\)
0.456917 0.889510i \(-0.348954\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 82.6357 + 143.129i 0.337288 + 0.584201i
\(246\) 0 0
\(247\) 285.927 + 165.080i 1.15760 + 0.668340i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 137.922i 0.549490i −0.961517 0.274745i \(-0.911407\pi\)
0.961517 0.274745i \(-0.0885934\pi\)
\(252\) 0 0
\(253\) −347.502 −1.37352
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.7348 23.7893i 0.0534426 0.0925653i −0.838066 0.545568i \(-0.816314\pi\)
0.891509 + 0.453003i \(0.149647\pi\)
\(258\) 0 0
\(259\) 41.0366 23.6925i 0.158443 0.0914768i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 250.254 144.484i 0.951535 0.549369i 0.0579779 0.998318i \(-0.481535\pi\)
0.893558 + 0.448949i \(0.148201\pi\)
\(264\) 0 0
\(265\) 25.8338 44.7455i 0.0974862 0.168851i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −180.235 −0.670018 −0.335009 0.942215i \(-0.608739\pi\)
−0.335009 + 0.942215i \(0.608739\pi\)
\(270\) 0 0
\(271\) 28.9765i 0.106924i 0.998570 + 0.0534622i \(0.0170257\pi\)
−0.998570 + 0.0534622i \(0.982974\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −119.444 68.9609i −0.434341 0.250767i
\(276\) 0 0
\(277\) 146.528 + 253.794i 0.528983 + 0.916225i 0.999429 + 0.0337961i \(0.0107597\pi\)
−0.470446 + 0.882429i \(0.655907\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.6700 25.4091i −0.0522063 0.0904239i 0.838741 0.544530i \(-0.183292\pi\)
−0.890948 + 0.454106i \(0.849959\pi\)
\(282\) 0 0
\(283\) −179.527 103.650i −0.634372 0.366255i 0.148071 0.988977i \(-0.452693\pi\)
−0.782443 + 0.622722i \(0.786027\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 122.860i 0.428083i
\(288\) 0 0
\(289\) −129.648 −0.448609
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 248.905 431.116i 0.849504 1.47138i −0.0321473 0.999483i \(-0.510235\pi\)
0.881651 0.471901i \(-0.156432\pi\)
\(294\) 0 0
\(295\) −182.303 + 105.252i −0.617975 + 0.356788i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −333.360 + 192.465i −1.11492 + 0.643697i
\(300\) 0 0
\(301\) 15.1395 26.2223i 0.0502973 0.0871174i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 68.3765 0.224185
\(306\) 0 0
\(307\) 32.7775i 0.106767i −0.998574 0.0533835i \(-0.982999\pi\)
0.998574 0.0533835i \(-0.0170006\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 330.752 + 190.960i 1.06351 + 0.614019i 0.926402 0.376537i \(-0.122885\pi\)
0.137110 + 0.990556i \(0.456218\pi\)
\(312\) 0 0
\(313\) 247.593 + 428.844i 0.791032 + 1.37011i 0.925329 + 0.379166i \(0.123789\pi\)
−0.134297 + 0.990941i \(0.542878\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 140.200 + 242.834i 0.442273 + 0.766039i 0.997858 0.0654209i \(-0.0208390\pi\)
−0.555585 + 0.831460i \(0.687506\pi\)
\(318\) 0 0
\(319\) 217.582 + 125.621i 0.682075 + 0.393796i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 323.825i 1.00255i
\(324\) 0 0
\(325\) −152.777 −0.470084
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 70.3056 121.773i 0.213695 0.370131i
\(330\) 0 0
\(331\) 4.57609 2.64201i 0.0138251 0.00798190i −0.493072 0.869989i \(-0.664126\pi\)
0.506897 + 0.862007i \(0.330793\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 74.7218 43.1407i 0.223050 0.128778i
\(336\) 0 0
\(337\) −121.945 + 211.215i −0.361855 + 0.626751i −0.988266 0.152742i \(-0.951190\pi\)
0.626411 + 0.779493i \(0.284523\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −703.056 −2.06175
\(342\) 0 0
\(343\) 174.165i 0.507770i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −168.452 97.2556i −0.485451 0.280275i 0.237234 0.971452i \(-0.423759\pi\)
−0.722685 + 0.691177i \(0.757092\pi\)
\(348\) 0 0
\(349\) 202.139 + 350.116i 0.579196 + 1.00320i 0.995572 + 0.0940045i \(0.0299668\pi\)
−0.416376 + 0.909193i \(0.636700\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −145.463 251.950i −0.412078 0.713739i 0.583039 0.812444i \(-0.301863\pi\)
−0.995117 + 0.0987047i \(0.968530\pi\)
\(354\) 0 0
\(355\) −145.842 84.2020i −0.410823 0.237189i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 554.699i 1.54512i 0.634941 + 0.772561i \(0.281024\pi\)
−0.634941 + 0.772561i \(0.718976\pi\)
\(360\) 0 0
\(361\) −297.056 −0.822871
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 89.4131 154.868i 0.244967 0.424296i
\(366\) 0 0
\(367\) −11.3661 + 6.56223i −0.0309703 + 0.0178807i −0.515405 0.856947i \(-0.672359\pi\)
0.484435 + 0.874827i \(0.339025\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 22.7322 13.1245i 0.0612729 0.0353759i
\(372\) 0 0
\(373\) 209.306 362.528i 0.561141 0.971925i −0.436256 0.899823i \(-0.643696\pi\)
0.997397 0.0721024i \(-0.0229708\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 278.303 0.738205
\(378\) 0 0
\(379\) 546.545i 1.44207i −0.692898 0.721036i \(-0.743666\pi\)
0.692898 0.721036i \(-0.256334\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −511.874 295.530i −1.33649 0.771620i −0.350201 0.936675i \(-0.613887\pi\)
−0.986284 + 0.165055i \(0.947220\pi\)
\(384\) 0 0
\(385\) 38.7508 + 67.1183i 0.100651 + 0.174333i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 172.694 + 299.115i 0.443944 + 0.768934i 0.997978 0.0635601i \(-0.0202454\pi\)
−0.554034 + 0.832494i \(0.686912\pi\)
\(390\) 0 0
\(391\) −326.964 188.773i −0.836224 0.482794i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 219.255i 0.555075i
\(396\) 0 0
\(397\) 385.741 0.971639 0.485820 0.874059i \(-0.338521\pi\)
0.485820 + 0.874059i \(0.338521\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 59.0869 102.341i 0.147349 0.255216i −0.782898 0.622150i \(-0.786259\pi\)
0.930247 + 0.366934i \(0.119593\pi\)
\(402\) 0 0
\(403\) −674.445 + 389.391i −1.67356 + 0.966231i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −259.012 + 149.541i −0.636394 + 0.367422i
\(408\) 0 0
\(409\) 167.852 290.728i 0.410396 0.710827i −0.584537 0.811367i \(-0.698724\pi\)
0.994933 + 0.100540i \(0.0320570\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −106.944 −0.258943
\(414\) 0 0
\(415\) 360.977i 0.869825i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.8831 + 16.6757i 0.0689334 + 0.0397987i 0.534071 0.845440i \(-0.320662\pi\)
−0.465137 + 0.885239i \(0.653995\pi\)
\(420\) 0 0
\(421\) −138.824 240.450i −0.329748 0.571140i 0.652714 0.757605i \(-0.273630\pi\)
−0.982462 + 0.186464i \(0.940297\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −74.9230 129.770i −0.176289 0.305342i
\(426\) 0 0
\(427\) 30.0836 + 17.3688i 0.0704535 + 0.0406763i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 368.795i 0.855674i −0.903856 0.427837i \(-0.859276\pi\)
0.903856 0.427837i \(-0.140724\pi\)
\(432\) 0 0
\(433\) 585.093 1.35125 0.675627 0.737244i \(-0.263873\pi\)
0.675627 + 0.737244i \(0.263873\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 383.611 664.434i 0.877829 1.52044i
\(438\) 0 0
\(439\) −569.640 + 328.882i −1.29759 + 0.749161i −0.979987 0.199064i \(-0.936210\pi\)
−0.317599 + 0.948225i \(0.602877\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 96.1502 55.5124i 0.217043 0.125310i −0.387537 0.921854i \(-0.626674\pi\)
0.604581 + 0.796544i \(0.293341\pi\)
\(444\) 0 0
\(445\) −280.797 + 486.355i −0.631005 + 1.09293i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −136.056 −0.303021 −0.151510 0.988456i \(-0.548414\pi\)
−0.151510 + 0.988456i \(0.548414\pi\)
\(450\) 0 0
\(451\) 775.459i 1.71942i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 74.3476 + 42.9246i 0.163401 + 0.0943398i
\(456\) 0 0
\(457\) 29.0747 + 50.3588i 0.0636208 + 0.110194i 0.896081 0.443890i \(-0.146402\pi\)
−0.832461 + 0.554084i \(0.813069\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −51.4916 89.1861i −0.111695 0.193462i 0.804759 0.593602i \(-0.202295\pi\)
−0.916454 + 0.400140i \(0.868961\pi\)
\(462\) 0 0
\(463\) −672.083 388.027i −1.45158 0.838072i −0.453012 0.891504i \(-0.649651\pi\)
−0.998571 + 0.0534320i \(0.982984\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 692.620i 1.48313i 0.670882 + 0.741564i \(0.265915\pi\)
−0.670882 + 0.741564i \(0.734085\pi\)
\(468\) 0 0
\(469\) 43.8338 0.0934623
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −95.5564 + 165.509i −0.202022 + 0.349912i
\(474\) 0 0
\(475\) 263.711 152.254i 0.555181 0.320534i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −630.014 + 363.739i −1.31527 + 0.759371i −0.982963 0.183801i \(-0.941160\pi\)
−0.332305 + 0.943172i \(0.607826\pi\)
\(480\) 0 0
\(481\) −165.648 + 286.911i −0.344382 + 0.596488i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −152.992 −0.315448
\(486\) 0 0
\(487\) 285.503i 0.586248i −0.956074 0.293124i \(-0.905305\pi\)
0.956074 0.293124i \(-0.0946948\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 188.576 + 108.875i 0.384066 + 0.221740i 0.679586 0.733596i \(-0.262160\pi\)
−0.295520 + 0.955337i \(0.595493\pi\)
\(492\) 0 0
\(493\) 136.482 + 236.393i 0.276839 + 0.479499i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −42.7774 74.0927i −0.0860713 0.149080i
\(498\) 0 0
\(499\) −30.1869 17.4284i −0.0604948 0.0349267i 0.469448 0.882960i \(-0.344453\pi\)
−0.529942 + 0.848034i \(0.677786\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 345.125i 0.686134i 0.939311 + 0.343067i \(0.111466\pi\)
−0.939311 + 0.343067i \(0.888534\pi\)
\(504\) 0 0
\(505\) −457.834 −0.906602
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 189.528 328.272i 0.372354 0.644936i −0.617573 0.786513i \(-0.711884\pi\)
0.989927 + 0.141577i \(0.0452174\pi\)
\(510\) 0 0
\(511\) 78.6782 45.4249i 0.153969 0.0888941i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −86.6493 + 50.0270i −0.168251 + 0.0971398i
\(516\) 0 0
\(517\) −443.751 + 768.599i −0.858319 + 1.48665i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −40.2028 −0.0771646 −0.0385823 0.999255i \(-0.512284\pi\)
−0.0385823 + 0.999255i \(0.512284\pi\)
\(522\) 0 0
\(523\) 623.980i 1.19308i 0.802584 + 0.596539i \(0.203458\pi\)
−0.802584 + 0.596539i \(0.796542\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −661.505 381.920i −1.25523 0.724706i
\(528\) 0 0
\(529\) 182.749 + 316.531i 0.345462 + 0.598357i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 429.492 + 743.901i 0.805800 + 1.39569i
\(534\) 0 0
\(535\) −380.134 219.471i −0.710531 0.410225i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 529.955i 0.983219i
\(540\) 0 0
\(541\) −529.741 −0.979188 −0.489594 0.871950i \(-0.662855\pi\)
−0.489594 + 0.871950i \(0.662855\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −111.858 + 193.744i −0.205244 + 0.355494i
\(546\) 0 0
\(547\) −512.558 + 295.925i −0.937035 + 0.540997i −0.889029 0.457850i \(-0.848620\pi\)
−0.0480051 + 0.998847i \(0.515286\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −480.383 + 277.349i −0.871839 + 0.503356i
\(552\) 0 0
\(553\) 55.6944 96.4655i 0.100713 0.174440i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −682.841 −1.22593 −0.612964 0.790111i \(-0.710023\pi\)
−0.612964 + 0.790111i \(0.710023\pi\)
\(558\) 0 0
\(559\) 211.698i 0.378708i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 889.956 + 513.816i 1.58074 + 0.912640i 0.994752 + 0.102319i \(0.0326263\pi\)
0.585987 + 0.810321i \(0.300707\pi\)
\(564\) 0 0
\(565\) −156.066 270.315i −0.276224 0.478433i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 165.913 + 287.370i 0.291587 + 0.505044i 0.974185 0.225750i \(-0.0724833\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(570\) 0 0
\(571\) −289.987 167.424i −0.507858 0.293212i 0.224095 0.974567i \(-0.428057\pi\)
−0.731953 + 0.681356i \(0.761391\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 355.023i 0.617431i
\(576\) 0 0
\(577\) −87.6845 −0.151966 −0.0759831 0.997109i \(-0.524209\pi\)
−0.0759831 + 0.997109i \(0.524209\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 91.6944 158.819i 0.157822 0.273355i
\(582\) 0 0
\(583\) −143.480 + 82.8382i −0.246106 + 0.142089i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −116.275 + 67.1313i −0.198083 + 0.114363i −0.595761 0.803162i \(-0.703149\pi\)
0.397678 + 0.917525i \(0.369816\pi\)
\(588\) 0 0
\(589\) 776.113 1344.27i 1.31768 2.28229i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −256.915 −0.433246 −0.216623 0.976255i \(-0.569504\pi\)
−0.216623 + 0.976255i \(0.569504\pi\)
\(594\) 0 0
\(595\) 84.2020i 0.141516i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 851.385 + 491.547i 1.42134 + 0.820613i 0.996414 0.0846136i \(-0.0269656\pi\)
0.424929 + 0.905227i \(0.360299\pi\)
\(600\) 0 0
\(601\) −558.909 968.058i −0.929964 1.61075i −0.783376 0.621548i \(-0.786504\pi\)
−0.146588 0.989198i \(-0.546829\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −25.3643 43.9323i −0.0419245 0.0726154i
\(606\) 0 0
\(607\) −259.238 149.671i −0.427081 0.246575i 0.271021 0.962573i \(-0.412639\pi\)
−0.698102 + 0.715998i \(0.745972\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 983.094i 1.60899i
\(612\) 0 0
\(613\) 468.332 0.764001 0.382000 0.924162i \(-0.375235\pi\)
0.382000 + 0.924162i \(0.375235\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 396.884 687.423i 0.643248 1.11414i −0.341455 0.939898i \(-0.610920\pi\)
0.984703 0.174240i \(-0.0557470\pi\)
\(618\) 0 0
\(619\) 782.130 451.563i 1.26354 0.729504i 0.289780 0.957093i \(-0.406418\pi\)
0.973757 + 0.227590i \(0.0730845\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −247.085 + 142.654i −0.396605 + 0.228980i
\(624\) 0 0
\(625\) 93.6662 162.235i 0.149866 0.259575i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −324.939 −0.516596
\(630\) 0 0
\(631\) 484.076i 0.767156i 0.923508 + 0.383578i \(0.125308\pi\)
−0.923508 + 0.383578i \(0.874692\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −575.417 332.217i −0.906168 0.523176i
\(636\) 0 0
\(637\) −293.518 508.389i −0.460782 0.798098i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 173.864 + 301.142i 0.271239 + 0.469800i 0.969179 0.246356i \(-0.0792332\pi\)
−0.697940 + 0.716156i \(0.745900\pi\)
\(642\) 0 0
\(643\) −162.862 94.0285i −0.253285 0.146234i 0.367983 0.929833i \(-0.380049\pi\)
−0.621267 + 0.783599i \(0.713382\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 853.348i 1.31893i 0.751735 + 0.659465i \(0.229217\pi\)
−0.751735 + 0.659465i \(0.770783\pi\)
\(648\) 0 0
\(649\) 675.000 1.04006
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −599.678 + 1038.67i −0.918342 + 1.59062i −0.116410 + 0.993201i \(0.537139\pi\)
−0.801933 + 0.597414i \(0.796195\pi\)
\(654\) 0 0
\(655\) 542.183 313.030i 0.827761 0.477908i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −534.606 + 308.655i −0.811238 + 0.468369i −0.847386 0.530978i \(-0.821825\pi\)
0.0361474 + 0.999346i \(0.488491\pi\)
\(660\) 0 0
\(661\) 109.880 190.318i 0.166233 0.287925i −0.770859 0.637006i \(-0.780173\pi\)
0.937093 + 0.349081i \(0.113506\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −171.110 −0.257308
\(666\) 0 0
\(667\) 646.719i 0.969593i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −189.880 109.627i −0.282981 0.163379i
\(672\) 0 0
\(673\) −289.492 501.414i −0.430151 0.745043i 0.566735 0.823900i \(-0.308206\pi\)
−0.996886 + 0.0788568i \(0.974873\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 351.528 + 608.865i 0.519244 + 0.899357i 0.999750 + 0.0223655i \(0.00711975\pi\)
−0.480506 + 0.876992i \(0.659547\pi\)
\(678\) 0 0
\(679\) −67.3121 38.8627i −0.0991342 0.0572351i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 881.535i 1.29068i 0.763895 + 0.645340i \(0.223284\pi\)
−0.763895 + 0.645340i \(0.776716\pi\)
\(684\) 0 0
\(685\) 580.651 0.847666
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −91.7607 + 158.934i −0.133179 + 0.230674i
\(690\) 0 0
\(691\) −890.846 + 514.330i −1.28921 + 0.744328i −0.978514 0.206181i \(-0.933897\pi\)
−0.310699 + 0.950508i \(0.600563\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 695.603 401.607i 1.00087 0.577851i
\(696\) 0 0
\(697\) −421.251 + 729.628i −0.604377 + 1.04681i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 813.287 1.16018 0.580090 0.814552i \(-0.303017\pi\)
0.580090 + 0.814552i \(0.303017\pi\)
\(702\) 0 0
\(703\) 660.320i 0.939289i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −201.433 116.298i −0.284913 0.164494i
\(708\) 0 0
\(709\) −82.4916 142.880i −0.116349 0.201523i 0.801969 0.597366i \(-0.203786\pi\)
−0.918318 + 0.395843i \(0.870453\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 904.864 + 1567.27i 1.26909 + 2.19813i
\(714\) 0 0
\(715\) −469.262 270.929i −0.656311 0.378921i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 179.881i 0.250182i 0.992145 + 0.125091i \(0.0399223\pi\)
−0.992145 + 0.125091i \(0.960078\pi\)
\(720\) 0 0
\(721\) −50.8308 −0.0705004
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 128.340 222.291i 0.177021 0.306609i
\(726\) 0 0
\(727\) −664.674 + 383.749i −0.914269 + 0.527853i −0.881802 0.471619i \(-0.843670\pi\)
−0.0324668 + 0.999473i \(0.510336\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −179.818 + 103.818i −0.245989 + 0.142022i
\(732\) 0 0
\(733\) −505.139 + 874.927i −0.689140 + 1.19363i 0.282977 + 0.959127i \(0.408678\pi\)
−0.972117 + 0.234498i \(0.924655\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −276.668 −0.375397
\(738\) 0 0
\(739\) 941.306i 1.27376i −0.770964 0.636878i \(-0.780225\pi\)
0.770964 0.636878i \(-0.219775\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −94.8464 54.7596i −0.127653 0.0737007i 0.434814 0.900521i \(-0.356814\pi\)
−0.562467 + 0.826820i \(0.690148\pi\)
\(744\) 0 0
\(745\) 340.731 + 590.163i 0.457357 + 0.792166i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −111.498 193.121i −0.148863 0.257839i
\(750\) 0 0
\(751\) 706.330 + 407.800i 0.940519 + 0.543009i 0.890123 0.455720i \(-0.150618\pi\)
0.0503961 + 0.998729i \(0.483952\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 405.158i 0.536633i
\(756\) 0 0
\(757\) −1253.34 −1.65566 −0.827830 0.560978i \(-0.810425\pi\)
−0.827830 + 0.560978i \(0.810425\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 183.095 317.130i 0.240598 0.416728i −0.720287 0.693677i \(-0.755990\pi\)
0.960885 + 0.276948i \(0.0893230\pi\)
\(762\) 0 0
\(763\) −98.4286 + 56.8278i −0.129002 + 0.0744794i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 647.531 373.852i 0.844238 0.487421i
\(768\) 0 0
\(769\) 539.066 933.690i 0.700996 1.21416i −0.267121 0.963663i \(-0.586072\pi\)
0.968117 0.250498i \(-0.0805945\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 283.085 0.366217 0.183108 0.983093i \(-0.441384\pi\)
0.183108 + 0.983093i \(0.441384\pi\)
\(774\) 0 0
\(775\) 718.273i 0.926804i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1482.70 856.039i −1.90334 1.09889i
\(780\) 0 0
\(781\) 270.000 + 467.654i 0.345711 + 0.598788i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −186.374 322.810i −0.237419 0.411222i
\(786\) 0 0
\(787\) −255.224 147.353i −0.324299 0.187234i 0.329008 0.944327i \(-0.393286\pi\)
−0.653307 + 0.757093i \(0.726619\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 158.574i 0.200473i
\(792\) 0 0
\(793\) −242.870 −0.306268
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 396.653 687.023i 0.497683 0.862012i −0.502314 0.864685i \(-0.667518\pi\)
0.999996 + 0.00267363i \(0.000851044\pi\)
\(798\) 0 0
\(799\) −835.049 + 482.116i −1.04512 + 0.603399i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −496.596 + 286.710i −0.618426 + 0.357049i
\(804\) 0 0
\(805\) 99.7477 172.768i 0.123910 0.214619i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1215.12 −1.50200 −0.751000 0.660303i \(-0.770428\pi\)
−0.751000 + 0.660303i \(0.770428\pi\)
\(810\) 0 0
\(811\) 429.447i 0.529527i −0.964313 0.264764i \(-0.914706\pi\)
0.964313 0.264764i \(-0.0852939\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 643.988 + 371.806i 0.790169 + 0.456204i
\(816\) 0 0
\(817\) −210.972 365.414i −0.258227 0.447263i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.68216 + 6.37769i 0.00448497 + 0.00776820i 0.868259 0.496111i \(-0.165239\pi\)
−0.863774 + 0.503879i \(0.831906\pi\)
\(822\) 0 0
\(823\) 1169.36 + 675.132i 1.42085 + 0.820331i 0.996372 0.0851049i \(-0.0271225\pi\)
0.424483 + 0.905436i \(0.360456\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 571.482i 0.691031i −0.938413 0.345515i \(-0.887704\pi\)
0.938413 0.345515i \(-0.112296\pi\)
\(828\) 0 0
\(829\) 888.445 1.07171 0.535854 0.844311i \(-0.319990\pi\)
0.535854 + 0.844311i \(0.319990\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 287.886 498.634i 0.345602 0.598600i
\(834\) 0 0
\(835\) −166.774 + 96.2867i −0.199729 + 0.115313i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −506.659 + 292.519i −0.603884 + 0.348652i −0.770568 0.637358i \(-0.780027\pi\)
0.166684 + 0.986010i \(0.446694\pi\)
\(840\) 0 0
\(841\) 186.713 323.396i 0.222013 0.384537i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.1464 0.0143744
\(846\) 0 0
\(847\) 25.7719i 0.0304272i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 666.720 + 384.931i 0.783455 + 0.452328i
\(852\) 0 0
\(853\) −437.917 758.494i −0.513384 0.889208i −0.999879 0.0155246i \(-0.995058\pi\)
0.486495 0.873683i \(-0.338275\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 186.836 + 323.610i 0.218012 + 0.377608i 0.954200 0.299169i \(-0.0967095\pi\)
−0.736188 + 0.676777i \(0.763376\pi\)
\(858\) 0 0
\(859\) −253.378 146.288i −0.294968 0.170300i 0.345212 0.938525i \(-0.387807\pi\)
−0.640180 + 0.768225i \(0.721140\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1461.41i 1.69341i −0.532066 0.846703i \(-0.678584\pi\)
0.532066 0.846703i \(-0.321416\pi\)
\(864\) 0 0
\(865\) 682.313 0.788801
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −351.528 + 608.865i −0.404520 + 0.700650i
\(870\) 0 0
\(871\) −265.409 + 153.234i −0.304717 + 0.175929i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 212.986 122.968i 0.243413 0.140535i
\(876\) 0 0
\(877\) −38.5648 + 66.7962i −0.0439735 + 0.0761644i −0.887174 0.461434i \(-0.847335\pi\)
0.843201 + 0.537599i \(0.180668\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −371.927 −0.422164 −0.211082 0.977468i \(-0.567699\pi\)
−0.211082 + 0.977468i \(0.567699\pi\)
\(882\) 0 0
\(883\) 875.990i 0.992061i 0.868305 + 0.496030i \(0.165210\pi\)
−0.868305 + 0.496030i \(0.834790\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 247.646 + 142.979i 0.279195 + 0.161193i 0.633059 0.774104i \(-0.281799\pi\)
−0.353864 + 0.935297i \(0.615132\pi\)
\(888\) 0 0
\(889\) −168.777 292.331i −0.189851 0.328831i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −979.724 1696.93i −1.09712 1.90026i
\(894\) 0 0
\(895\) −291.684 168.404i −0.325904 0.188161i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1308.42i 1.45542i
\(900\) 0 0
\(901\) −180.000 −0.199778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 435.623 754.522i 0.481352 0.833726i
\(906\) 0 0
\(907\) −615.208 + 355.190i −0.678289 + 0.391610i −0.799210 0.601052i \(-0.794748\pi\)
0.120921 + 0.992662i \(0.461415\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 167.148 96.5028i 0.183477 0.105931i −0.405448 0.914118i \(-0.632885\pi\)
0.588925 + 0.808187i \(0.299551\pi\)
\(912\) 0 0
\(913\) −578.751 + 1002.43i −0.633900 + 1.09795i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 318.059 0.346848
\(918\) 0 0
\(919\) 1106.39i 1.20390i 0.798533 + 0.601951i \(0.205610\pi\)
−0.798533 + 0.601951i \(0.794390\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 518.025 + 299.082i 0.561240 + 0.324032i
\(924\) 0 0
\(925\) 152.777 + 264.618i 0.165165 + 0.286074i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −54.4108 94.2423i −0.0585692 0.101445i 0.835254 0.549864i \(-0.185321\pi\)
−0.893823 + 0.448419i \(0.851987\pi\)
\(930\) 0 0
\(931\) 1013.29 + 585.024i 1.08839 + 0.628383i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 531.461i 0.568407i
\(936\) 0 0
\(937\) −1530.74 −1.63366 −0.816832 0.576875i \(-0.804272\pi\)
−0.816832 + 0.576875i \(0.804272\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −730.350 + 1265.00i −0.776142 + 1.34432i 0.158008 + 0.987438i \(0.449493\pi\)
−0.934150 + 0.356880i \(0.883841\pi\)
\(942\) 0 0
\(943\) 1728.67 998.048i 1.83316 1.05838i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 292.181 168.691i 0.308533 0.178132i −0.337737 0.941241i \(-0.609661\pi\)
0.646270 + 0.763109i \(0.276328\pi\)
\(948\) 0 0
\(949\) −317.591 + 550.085i −0.334659 + 0.579647i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.6997 0.0122767 0.00613834 0.999981i \(-0.498046\pi\)
0.00613834 + 0.999981i \(0.498046\pi\)
\(954\) 0 0
\(955\) 205.050i 0.214712i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 255.469 + 147.495i 0.266391 + 0.153801i
\(960\) 0 0
\(961\) 1350.20 + 2338.61i 1.40499 + 2.43352i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −182.516 316.127i −0.189136 0.327593i
\(966\) 0 0
\(967\) −1072.38 619.139i −1.10898 0.640268i −0.170413 0.985373i \(-0.554510\pi\)
−0.938564 + 0.345104i \(0.887844\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 52.4978i 0.0540658i −0.999635 0.0270329i \(-0.991394\pi\)
0.999635 0.0270329i \(-0.00860588\pi\)
\(972\) 0 0
\(973\) 408.059 0.419383
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 52.7789 91.4158i 0.0540214 0.0935679i −0.837750 0.546054i \(-0.816129\pi\)
0.891772 + 0.452486i \(0.149463\pi\)
\(978\) 0 0
\(979\) 1559.53 900.398i 1.59299 0.919712i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.77662 + 3.33513i −0.00587652 + 0.00339281i −0.502935 0.864324i \(-0.667747\pi\)
0.497059 + 0.867717i \(0.334413\pi\)
\(984\) 0 0
\(985\) −208.797 + 361.647i −0.211977 + 0.367155i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 491.941 0.497412
\(990\) 0 0
\(991\) 1394.27i 1.40694i −0.710727 0.703468i \(-0.751634\pi\)
0.710727 0.703468i \(-0.248366\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 598.523 + 345.557i 0.601531 + 0.347294i
\(996\) 0 0
\(997\) 634.771 + 1099.45i 0.636681 + 1.10276i 0.986156 + 0.165818i \(0.0530263\pi\)
−0.349476 + 0.936945i \(0.613640\pi\)
\(998\) 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.o.d.1279.4 8
3.2 odd 2 576.3.o.e.319.1 8
4.3 odd 2 inner 1728.3.o.d.1279.3 8
8.3 odd 2 432.3.o.c.415.1 8
8.5 even 2 432.3.o.c.415.2 8
9.2 odd 6 576.3.o.e.511.4 8
9.7 even 3 inner 1728.3.o.d.127.3 8
12.11 even 2 576.3.o.e.319.4 8
24.5 odd 2 144.3.o.b.31.4 yes 8
24.11 even 2 144.3.o.b.31.1 8
36.7 odd 6 inner 1728.3.o.d.127.4 8
36.11 even 6 576.3.o.e.511.1 8
72.5 odd 6 1296.3.g.h.1135.2 4
72.11 even 6 144.3.o.b.79.4 yes 8
72.13 even 6 1296.3.g.d.1135.4 4
72.29 odd 6 144.3.o.b.79.1 yes 8
72.43 odd 6 432.3.o.c.127.2 8
72.59 even 6 1296.3.g.h.1135.1 4
72.61 even 6 432.3.o.c.127.1 8
72.67 odd 6 1296.3.g.d.1135.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.o.b.31.1 8 24.11 even 2
144.3.o.b.31.4 yes 8 24.5 odd 2
144.3.o.b.79.1 yes 8 72.29 odd 6
144.3.o.b.79.4 yes 8 72.11 even 6
432.3.o.c.127.1 8 72.61 even 6
432.3.o.c.127.2 8 72.43 odd 6
432.3.o.c.415.1 8 8.3 odd 2
432.3.o.c.415.2 8 8.5 even 2
576.3.o.e.319.1 8 3.2 odd 2
576.3.o.e.319.4 8 12.11 even 2
576.3.o.e.511.1 8 36.11 even 6
576.3.o.e.511.4 8 9.2 odd 6
1296.3.g.d.1135.3 4 72.67 odd 6
1296.3.g.d.1135.4 4 72.13 even 6
1296.3.g.h.1135.1 4 72.59 even 6
1296.3.g.h.1135.2 4 72.5 odd 6
1728.3.o.d.127.3 8 9.7 even 3 inner
1728.3.o.d.127.4 8 36.7 odd 6 inner
1728.3.o.d.1279.3 8 4.3 odd 2 inner
1728.3.o.d.1279.4 8 1.1 even 1 trivial