Properties

Label 1200.3.c.k.449.6
Level $1200$
Weight $3$
Character 1200.449
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(449,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), 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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.6
Root \(-0.437016 + 0.437016i\) of defining polynomial
Character \(\chi\) \(=\) 1200.449
Dual form 1200.3.c.k.449.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.52896 + 2.58114i) q^{3} +7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +O(q^{10})\) \(q+(1.52896 + 2.58114i) q^{3} +7.48683i q^{7} +(-4.32456 + 7.89292i) q^{9} +8.48528i q^{11} +10.0000i q^{13} -30.3870 q^{17} +26.9737 q^{19} +(-19.3246 + 11.4471i) q^{21} +9.17377 q^{23} +(-26.9848 + 0.905694i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(-21.9017 + 12.9737i) q^{33} +15.9473i q^{37} +(-25.8114 + 15.2896i) q^{39} -47.3575i q^{41} -14.4605i q^{43} -45.8688 q^{47} -7.05267 q^{49} +(-46.4605 - 78.4330i) q^{51} +30.3870 q^{53} +(41.2417 + 69.6228i) q^{57} +24.0789i q^{59} -53.9473 q^{61} +(-59.0930 - 32.3772i) q^{63} +110.460i q^{67} +(14.0263 + 23.6788i) q^{69} +15.5936i q^{71} -87.9473i q^{73} -63.5279 q^{77} -46.9737 q^{79} +(-43.5964 - 68.2668i) q^{81} +26.1443 q^{83} +(69.2592 - 41.0263i) q^{87} +60.7739i q^{89} -74.8683 q^{91} +(-12.2317 - 20.6491i) q^{93} +36.0527i q^{97} +(-66.9737 - 36.6951i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 64 q^{19} - 104 q^{21} - 64 q^{31} - 80 q^{39} - 360 q^{49} - 144 q^{51} - 128 q^{61} + 264 q^{69} - 224 q^{79} + 56 q^{81} + 160 q^{91} - 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.52896 + 2.58114i 0.509654 + 0.860380i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 7.48683i 1.06955i 0.844995 + 0.534774i \(0.179603\pi\)
−0.844995 + 0.534774i \(0.820397\pi\)
\(8\) 0 0
\(9\) −4.32456 + 7.89292i −0.480506 + 0.876991i
\(10\) 0 0
\(11\) 8.48528i 0.771389i 0.922627 + 0.385695i \(0.126038\pi\)
−0.922627 + 0.385695i \(0.873962\pi\)
\(12\) 0 0
\(13\) 10.0000i 0.769231i 0.923077 + 0.384615i \(0.125666\pi\)
−0.923077 + 0.384615i \(0.874334\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −30.3870 −1.78747 −0.893734 0.448596i \(-0.851924\pi\)
−0.893734 + 0.448596i \(0.851924\pi\)
\(18\) 0 0
\(19\) 26.9737 1.41967 0.709833 0.704370i \(-0.248770\pi\)
0.709833 + 0.704370i \(0.248770\pi\)
\(20\) 0 0
\(21\) −19.3246 + 11.4471i −0.920217 + 0.545099i
\(22\) 0 0
\(23\) 9.17377 0.398859 0.199430 0.979912i \(-0.436091\pi\)
0.199430 + 0.979912i \(0.436091\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −26.9848 + 0.905694i −0.999437 + 0.0335442i
\(28\) 0 0
\(29\) 26.8328i 0.925270i −0.886549 0.462635i \(-0.846904\pi\)
0.886549 0.462635i \(-0.153096\pi\)
\(30\) 0 0
\(31\) −8.00000 −0.258065 −0.129032 0.991640i \(-0.541187\pi\)
−0.129032 + 0.991640i \(0.541187\pi\)
\(32\) 0 0
\(33\) −21.9017 + 12.9737i −0.663688 + 0.393141i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 15.9473i 0.431009i 0.976503 + 0.215504i \(0.0691396\pi\)
−0.976503 + 0.215504i \(0.930860\pi\)
\(38\) 0 0
\(39\) −25.8114 + 15.2896i −0.661830 + 0.392041i
\(40\) 0 0
\(41\) 47.3575i 1.15506i −0.816369 0.577531i \(-0.804016\pi\)
0.816369 0.577531i \(-0.195984\pi\)
\(42\) 0 0
\(43\) 14.4605i 0.336291i −0.985762 0.168145i \(-0.946222\pi\)
0.985762 0.168145i \(-0.0537778\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −45.8688 −0.975933 −0.487966 0.872862i \(-0.662261\pi\)
−0.487966 + 0.872862i \(0.662261\pi\)
\(48\) 0 0
\(49\) −7.05267 −0.143932
\(50\) 0 0
\(51\) −46.4605 78.4330i −0.910990 1.53790i
\(52\) 0 0
\(53\) 30.3870 0.573339 0.286670 0.958030i \(-0.407452\pi\)
0.286670 + 0.958030i \(0.407452\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 41.2417 + 69.6228i 0.723538 + 1.22145i
\(58\) 0 0
\(59\) 24.0789i 0.408116i 0.978959 + 0.204058i \(0.0654132\pi\)
−0.978959 + 0.204058i \(0.934587\pi\)
\(60\) 0 0
\(61\) −53.9473 −0.884382 −0.442191 0.896921i \(-0.645799\pi\)
−0.442191 + 0.896921i \(0.645799\pi\)
\(62\) 0 0
\(63\) −59.0930 32.3772i −0.937984 0.513924i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 110.460i 1.64866i 0.566107 + 0.824332i \(0.308449\pi\)
−0.566107 + 0.824332i \(0.691551\pi\)
\(68\) 0 0
\(69\) 14.0263 + 23.6788i 0.203280 + 0.343171i
\(70\) 0 0
\(71\) 15.5936i 0.219628i 0.993952 + 0.109814i \(0.0350255\pi\)
−0.993952 + 0.109814i \(0.964974\pi\)
\(72\) 0 0
\(73\) 87.9473i 1.20476i −0.798210 0.602379i \(-0.794220\pi\)
0.798210 0.602379i \(-0.205780\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −63.5279 −0.825037
\(78\) 0 0
\(79\) −46.9737 −0.594603 −0.297302 0.954784i \(-0.596087\pi\)
−0.297302 + 0.954784i \(0.596087\pi\)
\(80\) 0 0
\(81\) −43.5964 68.2668i −0.538228 0.842799i
\(82\) 0 0
\(83\) 26.1443 0.314992 0.157496 0.987520i \(-0.449658\pi\)
0.157496 + 0.987520i \(0.449658\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 69.2592 41.0263i 0.796083 0.471567i
\(88\) 0 0
\(89\) 60.7739i 0.682853i 0.939908 + 0.341427i \(0.110910\pi\)
−0.939908 + 0.341427i \(0.889090\pi\)
\(90\) 0 0
\(91\) −74.8683 −0.822729
\(92\) 0 0
\(93\) −12.2317 20.6491i −0.131524 0.222033i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 36.0527i 0.371677i 0.982580 + 0.185838i \(0.0595001\pi\)
−0.982580 + 0.185838i \(0.940500\pi\)
\(98\) 0 0
\(99\) −66.9737 36.6951i −0.676502 0.370657i
\(100\) 0 0
\(101\) 48.1577i 0.476809i 0.971166 + 0.238405i \(0.0766245\pi\)
−0.971166 + 0.238405i \(0.923376\pi\)
\(102\) 0 0
\(103\) 140.408i 1.36318i 0.731733 + 0.681591i \(0.238712\pi\)
−0.731733 + 0.681591i \(0.761288\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 43.1149 0.402943 0.201471 0.979494i \(-0.435428\pi\)
0.201471 + 0.979494i \(0.435428\pi\)
\(108\) 0 0
\(109\) −133.842 −1.22791 −0.613954 0.789342i \(-0.710422\pi\)
−0.613954 + 0.789342i \(0.710422\pi\)
\(110\) 0 0
\(111\) −41.1623 + 24.3829i −0.370831 + 0.219665i
\(112\) 0 0
\(113\) −7.90852 −0.0699869 −0.0349935 0.999388i \(-0.511141\pi\)
−0.0349935 + 0.999388i \(0.511141\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −78.9292 43.2456i −0.674609 0.369620i
\(118\) 0 0
\(119\) 227.502i 1.91178i
\(120\) 0 0
\(121\) 49.0000 0.404959
\(122\) 0 0
\(123\) 122.236 72.4078i 0.993792 0.588682i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 134.460i 1.05874i 0.848390 + 0.529372i \(0.177572\pi\)
−0.848390 + 0.529372i \(0.822428\pi\)
\(128\) 0 0
\(129\) 37.3246 22.1095i 0.289338 0.171392i
\(130\) 0 0
\(131\) 220.394i 1.68240i −0.540727 0.841198i \(-0.681851\pi\)
0.540727 0.841198i \(-0.318149\pi\)
\(132\) 0 0
\(133\) 201.947i 1.51840i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 95.5153 0.697192 0.348596 0.937273i \(-0.386659\pi\)
0.348596 + 0.937273i \(0.386659\pi\)
\(138\) 0 0
\(139\) −76.8157 −0.552631 −0.276315 0.961067i \(-0.589113\pi\)
−0.276315 + 0.961067i \(0.589113\pi\)
\(140\) 0 0
\(141\) −70.1317 118.394i −0.497388 0.839673i
\(142\) 0 0
\(143\) −84.8528 −0.593376
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.7833 18.2039i −0.0733555 0.123836i
\(148\) 0 0
\(149\) 276.237i 1.85394i −0.375139 0.926969i \(-0.622405\pi\)
0.375139 0.926969i \(-0.377595\pi\)
\(150\) 0 0
\(151\) −18.0527 −0.119554 −0.0597770 0.998212i \(-0.519039\pi\)
−0.0597770 + 0.998212i \(0.519039\pi\)
\(152\) 0 0
\(153\) 131.410 239.842i 0.858890 1.56759i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 103.842i 0.661414i 0.943733 + 0.330707i \(0.107287\pi\)
−0.943733 + 0.330707i \(0.892713\pi\)
\(158\) 0 0
\(159\) 46.4605 + 78.4330i 0.292204 + 0.493289i
\(160\) 0 0
\(161\) 68.6825i 0.426599i
\(162\) 0 0
\(163\) 11.3815i 0.0698251i −0.999390 0.0349126i \(-0.988885\pi\)
0.999390 0.0349126i \(-0.0111153\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −252.270 −1.51060 −0.755298 0.655382i \(-0.772508\pi\)
−0.755298 + 0.655382i \(0.772508\pi\)
\(168\) 0 0
\(169\) 69.0000 0.408284
\(170\) 0 0
\(171\) −116.649 + 212.901i −0.682159 + 1.24504i
\(172\) 0 0
\(173\) −11.8160 −0.0683005 −0.0341502 0.999417i \(-0.510872\pi\)
−0.0341502 + 0.999417i \(0.510872\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −62.1509 + 36.8157i −0.351135 + 0.207998i
\(178\) 0 0
\(179\) 69.0358i 0.385675i 0.981231 + 0.192837i \(0.0617690\pi\)
−0.981231 + 0.192837i \(0.938231\pi\)
\(180\) 0 0
\(181\) −189.684 −1.04798 −0.523989 0.851725i \(-0.675557\pi\)
−0.523989 + 0.851725i \(0.675557\pi\)
\(182\) 0 0
\(183\) −82.4834 139.246i −0.450729 0.760905i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 257.842i 1.37883i
\(188\) 0 0
\(189\) −6.78078 202.031i −0.0358771 1.06895i
\(190\) 0 0
\(191\) 108.708i 0.569153i 0.958653 + 0.284577i \(0.0918530\pi\)
−0.958653 + 0.284577i \(0.908147\pi\)
\(192\) 0 0
\(193\) 167.947i 0.870193i 0.900384 + 0.435097i \(0.143286\pi\)
−0.900384 + 0.435097i \(0.856714\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 171.659 0.871367 0.435684 0.900100i \(-0.356507\pi\)
0.435684 + 0.900100i \(0.356507\pi\)
\(198\) 0 0
\(199\) 35.0790 0.176276 0.0881382 0.996108i \(-0.471908\pi\)
0.0881382 + 0.996108i \(0.471908\pi\)
\(200\) 0 0
\(201\) −285.114 + 168.890i −1.41848 + 0.840248i
\(202\) 0 0
\(203\) 200.893 0.989620
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −39.6725 + 72.4078i −0.191654 + 0.349796i
\(208\) 0 0
\(209\) 228.879i 1.09512i
\(210\) 0 0
\(211\) 58.1580 0.275630 0.137815 0.990458i \(-0.455992\pi\)
0.137815 + 0.990458i \(0.455992\pi\)
\(212\) 0 0
\(213\) −40.2492 + 23.8420i −0.188963 + 0.111934i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 59.8947i 0.276012i
\(218\) 0 0
\(219\) 227.004 134.468i 1.03655 0.614009i
\(220\) 0 0
\(221\) 303.870i 1.37498i
\(222\) 0 0
\(223\) 99.3815i 0.445657i 0.974858 + 0.222828i \(0.0715290\pi\)
−0.974858 + 0.222828i \(0.928471\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 216.951 0.955733 0.477867 0.878432i \(-0.341410\pi\)
0.477867 + 0.878432i \(0.341410\pi\)
\(228\) 0 0
\(229\) −325.684 −1.42220 −0.711100 0.703090i \(-0.751803\pi\)
−0.711100 + 0.703090i \(0.751803\pi\)
\(230\) 0 0
\(231\) −97.1317 163.974i −0.420483 0.709845i
\(232\) 0 0
\(233\) 51.7119 0.221939 0.110970 0.993824i \(-0.464604\pi\)
0.110970 + 0.993824i \(0.464604\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −71.8209 121.246i −0.303042 0.511585i
\(238\) 0 0
\(239\) 410.047i 1.71568i 0.513917 + 0.857840i \(0.328194\pi\)
−0.513917 + 0.857840i \(0.671806\pi\)
\(240\) 0 0
\(241\) 445.526 1.84866 0.924328 0.381599i \(-0.124627\pi\)
0.924328 + 0.381599i \(0.124627\pi\)
\(242\) 0 0
\(243\) 109.549 216.906i 0.450818 0.892616i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 269.737i 1.09205i
\(248\) 0 0
\(249\) 39.9737 + 67.4821i 0.160537 + 0.271013i
\(250\) 0 0
\(251\) 237.364i 0.945675i −0.881150 0.472838i \(-0.843230\pi\)
0.881150 0.472838i \(-0.156770\pi\)
\(252\) 0 0
\(253\) 77.8420i 0.307676i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −318.887 −1.24080 −0.620402 0.784284i \(-0.713030\pi\)
−0.620402 + 0.784284i \(0.713030\pi\)
\(258\) 0 0
\(259\) −119.395 −0.460985
\(260\) 0 0
\(261\) 211.789 + 116.040i 0.811453 + 0.444598i
\(262\) 0 0
\(263\) −36.2300 −0.137757 −0.0688784 0.997625i \(-0.521942\pi\)
−0.0688784 + 0.997625i \(0.521942\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −156.866 + 92.9210i −0.587513 + 0.348019i
\(268\) 0 0
\(269\) 528.041i 1.96298i 0.191518 + 0.981489i \(0.438659\pi\)
−0.191518 + 0.981489i \(0.561341\pi\)
\(270\) 0 0
\(271\) 475.895 1.75607 0.878034 0.478597i \(-0.158855\pi\)
0.878034 + 0.478597i \(0.158855\pi\)
\(272\) 0 0
\(273\) −114.471 193.246i −0.419307 0.707859i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 188.158i 0.679271i 0.940557 + 0.339635i \(0.110304\pi\)
−0.940557 + 0.339635i \(0.889696\pi\)
\(278\) 0 0
\(279\) 34.5964 63.1434i 0.124002 0.226320i
\(280\) 0 0
\(281\) 24.4322i 0.0869473i 0.999055 + 0.0434736i \(0.0138424\pi\)
−0.999055 + 0.0434736i \(0.986158\pi\)
\(282\) 0 0
\(283\) 198.460i 0.701274i 0.936511 + 0.350637i \(0.114035\pi\)
−0.936511 + 0.350637i \(0.885965\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 354.558 1.23539
\(288\) 0 0
\(289\) 634.368 2.19504
\(290\) 0 0
\(291\) −93.0569 + 55.1231i −0.319783 + 0.189427i
\(292\) 0 0
\(293\) −513.825 −1.75367 −0.876834 0.480794i \(-0.840349\pi\)
−0.876834 + 0.480794i \(0.840349\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.68507 228.974i −0.0258757 0.770955i
\(298\) 0 0
\(299\) 91.7377i 0.306815i
\(300\) 0 0
\(301\) 108.263 0.359679
\(302\) 0 0
\(303\) −124.302 + 73.6313i −0.410237 + 0.243008i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.3815i 0.0370733i 0.999828 + 0.0185366i \(0.00590073\pi\)
−0.999828 + 0.0185366i \(0.994099\pi\)
\(308\) 0 0
\(309\) −362.412 + 214.678i −1.17285 + 0.694751i
\(310\) 0 0
\(311\) 518.756i 1.66802i 0.551746 + 0.834012i \(0.313962\pi\)
−0.551746 + 0.834012i \(0.686038\pi\)
\(312\) 0 0
\(313\) 46.3160i 0.147974i 0.997259 + 0.0739872i \(0.0235724\pi\)
−0.997259 + 0.0739872i \(0.976428\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −39.0957 −0.123330 −0.0616651 0.998097i \(-0.519641\pi\)
−0.0616651 + 0.998097i \(0.519641\pi\)
\(318\) 0 0
\(319\) 227.684 0.713743
\(320\) 0 0
\(321\) 65.9210 + 111.286i 0.205361 + 0.346684i
\(322\) 0 0
\(323\) −819.648 −2.53761
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −204.639 345.465i −0.625808 1.05647i
\(328\) 0 0
\(329\) 343.412i 1.04381i
\(330\) 0 0
\(331\) 445.421 1.34568 0.672841 0.739787i \(-0.265074\pi\)
0.672841 + 0.739787i \(0.265074\pi\)
\(332\) 0 0
\(333\) −125.871 68.9651i −0.377991 0.207102i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 325.684i 0.966421i 0.875504 + 0.483211i \(0.160529\pi\)
−0.875504 + 0.483211i \(0.839471\pi\)
\(338\) 0 0
\(339\) −12.0918 20.4130i −0.0356691 0.0602153i
\(340\) 0 0
\(341\) 67.8823i 0.199068i
\(342\) 0 0
\(343\) 314.053i 0.915605i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −51.8236 −0.149348 −0.0746738 0.997208i \(-0.523792\pi\)
−0.0746738 + 0.997208i \(0.523792\pi\)
\(348\) 0 0
\(349\) 97.5787 0.279595 0.139798 0.990180i \(-0.455355\pi\)
0.139798 + 0.990180i \(0.455355\pi\)
\(350\) 0 0
\(351\) −9.05694 269.848i −0.0258033 0.768798i
\(352\) 0 0
\(353\) 569.797 1.61416 0.807078 0.590445i \(-0.201048\pi\)
0.807078 + 0.590445i \(0.201048\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 587.215 347.842i 1.64486 0.974347i
\(358\) 0 0
\(359\) 274.283i 0.764019i −0.924158 0.382010i \(-0.875232\pi\)
0.924158 0.382010i \(-0.124768\pi\)
\(360\) 0 0
\(361\) 366.579 1.01545
\(362\) 0 0
\(363\) 74.9191 + 126.476i 0.206389 + 0.348418i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 461.828i 1.25839i −0.777248 0.629194i \(-0.783385\pi\)
0.777248 0.629194i \(-0.216615\pi\)
\(368\) 0 0
\(369\) 373.789 + 204.800i 1.01298 + 0.555014i
\(370\) 0 0
\(371\) 227.502i 0.613213i
\(372\) 0 0
\(373\) 491.947i 1.31889i 0.751751 + 0.659447i \(0.229209\pi\)
−0.751751 + 0.659447i \(0.770791\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 268.328 0.711746
\(378\) 0 0
\(379\) −258.763 −0.682752 −0.341376 0.939927i \(-0.610893\pi\)
−0.341376 + 0.939927i \(0.610893\pi\)
\(380\) 0 0
\(381\) −347.061 + 205.585i −0.910922 + 0.539593i
\(382\) 0 0
\(383\) 522.422 1.36402 0.682012 0.731341i \(-0.261105\pi\)
0.682012 + 0.731341i \(0.261105\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 114.136 + 62.5352i 0.294924 + 0.161590i
\(388\) 0 0
\(389\) 610.847i 1.57030i −0.619306 0.785150i \(-0.712586\pi\)
0.619306 0.785150i \(-0.287414\pi\)
\(390\) 0 0
\(391\) −278.763 −0.712949
\(392\) 0 0
\(393\) 568.867 336.974i 1.44750 0.857439i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 214.000i 0.539043i −0.962994 0.269521i \(-0.913135\pi\)
0.962994 0.269521i \(-0.0868655\pi\)
\(398\) 0 0
\(399\) −521.254 + 308.770i −1.30640 + 0.773859i
\(400\) 0 0
\(401\) 454.557i 1.13356i 0.823869 + 0.566780i \(0.191811\pi\)
−0.823869 + 0.566780i \(0.808189\pi\)
\(402\) 0 0
\(403\) 80.0000i 0.198511i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −135.318 −0.332476
\(408\) 0 0
\(409\) 573.842 1.40304 0.701518 0.712651i \(-0.252506\pi\)
0.701518 + 0.712651i \(0.252506\pi\)
\(410\) 0 0
\(411\) 146.039 + 246.538i 0.355326 + 0.599850i
\(412\) 0 0
\(413\) −180.274 −0.436500
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −117.448 198.272i −0.281650 0.475472i
\(418\) 0 0
\(419\) 97.9159i 0.233690i 0.993150 + 0.116845i \(0.0372780\pi\)
−0.993150 + 0.116845i \(0.962722\pi\)
\(420\) 0 0
\(421\) 717.315 1.70384 0.851918 0.523675i \(-0.175439\pi\)
0.851918 + 0.523675i \(0.175439\pi\)
\(422\) 0 0
\(423\) 198.362 362.039i 0.468942 0.855885i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 403.895i 0.945889i
\(428\) 0 0
\(429\) −129.737 219.017i −0.302416 0.510529i
\(430\) 0 0
\(431\) 293.077i 0.679994i −0.940427 0.339997i \(-0.889574\pi\)
0.940427 0.339997i \(-0.110426\pi\)
\(432\) 0 0
\(433\) 487.526i 1.12593i −0.826482 0.562963i \(-0.809661\pi\)
0.826482 0.562963i \(-0.190339\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 247.450 0.566247
\(438\) 0 0
\(439\) 257.237 0.585961 0.292981 0.956118i \(-0.405353\pi\)
0.292981 + 0.956118i \(0.405353\pi\)
\(440\) 0 0
\(441\) 30.4997 55.6662i 0.0691602 0.126227i
\(442\) 0 0
\(443\) 293.096 0.661615 0.330808 0.943698i \(-0.392679\pi\)
0.330808 + 0.943698i \(0.392679\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 713.005 422.355i 1.59509 0.944866i
\(448\) 0 0
\(449\) 585.614i 1.30426i 0.758106 + 0.652132i \(0.226125\pi\)
−0.758106 + 0.652132i \(0.773875\pi\)
\(450\) 0 0
\(451\) 401.842 0.891002
\(452\) 0 0
\(453\) −27.6018 46.5964i −0.0609312 0.102862i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 813.052i 1.77911i −0.456831 0.889554i \(-0.651016\pi\)
0.456831 0.889554i \(-0.348984\pi\)
\(458\) 0 0
\(459\) 819.986 27.5213i 1.78646 0.0599593i
\(460\) 0 0
\(461\) 554.074i 1.20190i 0.799288 + 0.600948i \(0.205210\pi\)
−0.799288 + 0.600948i \(0.794790\pi\)
\(462\) 0 0
\(463\) 449.723i 0.971324i −0.874147 0.485662i \(-0.838579\pi\)
0.874147 0.485662i \(-0.161421\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.7221 −0.0657862 −0.0328931 0.999459i \(-0.510472\pi\)
−0.0328931 + 0.999459i \(0.510472\pi\)
\(468\) 0 0
\(469\) −826.999 −1.76332
\(470\) 0 0
\(471\) −268.031 + 158.770i −0.569067 + 0.337092i
\(472\) 0 0
\(473\) 122.701 0.259411
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −131.410 + 239.842i −0.275493 + 0.502813i
\(478\) 0 0
\(479\) 735.242i 1.53495i −0.641078 0.767476i \(-0.721512\pi\)
0.641078 0.767476i \(-0.278488\pi\)
\(480\) 0 0
\(481\) −159.473 −0.331545
\(482\) 0 0
\(483\) −177.279 + 105.013i −0.367037 + 0.217418i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 92.6185i 0.190182i 0.995469 + 0.0950909i \(0.0303142\pi\)
−0.995469 + 0.0950909i \(0.969686\pi\)
\(488\) 0 0
\(489\) 29.3772 17.4019i 0.0600761 0.0355866i
\(490\) 0 0
\(491\) 898.323i 1.82958i 0.403933 + 0.914789i \(0.367643\pi\)
−0.403933 + 0.914789i \(0.632357\pi\)
\(492\) 0 0
\(493\) 815.368i 1.65389i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −116.747 −0.234903
\(498\) 0 0
\(499\) 136.921 0.274391 0.137195 0.990544i \(-0.456191\pi\)
0.137195 + 0.990544i \(0.456191\pi\)
\(500\) 0 0
\(501\) −385.710 651.143i −0.769881 1.29969i
\(502\) 0 0
\(503\) 443.077 0.880868 0.440434 0.897785i \(-0.354825\pi\)
0.440434 + 0.897785i \(0.354825\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 105.498 + 178.099i 0.208083 + 0.351279i
\(508\) 0 0
\(509\) 213.062i 0.418590i −0.977853 0.209295i \(-0.932883\pi\)
0.977853 0.209295i \(-0.0671168\pi\)
\(510\) 0 0
\(511\) 658.447 1.28855
\(512\) 0 0
\(513\) −727.879 + 24.4299i −1.41887 + 0.0476216i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 389.210i 0.752824i
\(518\) 0 0
\(519\) −18.0662 30.4987i −0.0348096 0.0587643i
\(520\) 0 0
\(521\) 3.20085i 0.00614366i 0.999995 + 0.00307183i \(0.000977795\pi\)
−0.999995 + 0.00307183i \(0.999022\pi\)
\(522\) 0 0
\(523\) 966.644i 1.84827i 0.382069 + 0.924134i \(0.375212\pi\)
−0.382069 + 0.924134i \(0.624788\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 243.096 0.461282
\(528\) 0 0
\(529\) −444.842 −0.840911
\(530\) 0 0
\(531\) −190.053 104.130i −0.357915 0.196102i
\(532\) 0 0
\(533\) 473.575 0.888509
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −178.191 + 105.553i −0.331827 + 0.196561i
\(538\) 0 0
\(539\) 59.8439i 0.111028i
\(540\) 0 0
\(541\) −186.105 −0.344002 −0.172001 0.985097i \(-0.555023\pi\)
−0.172001 + 0.985097i \(0.555023\pi\)
\(542\) 0 0
\(543\) −290.019 489.601i −0.534106 0.901659i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 309.434i 0.565693i 0.959165 + 0.282847i \(0.0912787\pi\)
−0.959165 + 0.282847i \(0.908721\pi\)
\(548\) 0 0
\(549\) 233.298 425.802i 0.424951 0.775596i
\(550\) 0 0
\(551\) 723.779i 1.31357i
\(552\) 0 0
\(553\) 351.684i 0.635957i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.00106 −0.00718323 −0.00359161 0.999994i \(-0.501143\pi\)
−0.00359161 + 0.999994i \(0.501143\pi\)
\(558\) 0 0
\(559\) 144.605 0.258685
\(560\) 0 0
\(561\) 665.526 394.230i 1.18632 0.702728i
\(562\) 0 0
\(563\) 166.970 0.296572 0.148286 0.988945i \(-0.452624\pi\)
0.148286 + 0.988945i \(0.452624\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 511.102 326.399i 0.901414 0.575660i
\(568\) 0 0
\(569\) 156.289i 0.274673i 0.990524 + 0.137337i \(0.0438542\pi\)
−0.990524 + 0.137337i \(0.956146\pi\)
\(570\) 0 0
\(571\) 144.105 0.252374 0.126187 0.992006i \(-0.459726\pi\)
0.126187 + 0.992006i \(0.459726\pi\)
\(572\) 0 0
\(573\) −280.591 + 166.211i −0.489688 + 0.290071i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 532.947i 0.923651i 0.886971 + 0.461826i \(0.152805\pi\)
−0.886971 + 0.461826i \(0.847195\pi\)
\(578\) 0 0
\(579\) −433.495 + 256.785i −0.748697 + 0.443497i
\(580\) 0 0
\(581\) 195.738i 0.336899i
\(582\) 0 0
\(583\) 257.842i 0.442268i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −190.342 −0.324262 −0.162131 0.986769i \(-0.551837\pi\)
−0.162131 + 0.986769i \(0.551837\pi\)
\(588\) 0 0
\(589\) −215.789 −0.366366
\(590\) 0 0
\(591\) 262.460 + 443.077i 0.444096 + 0.749707i
\(592\) 0 0
\(593\) −345.719 −0.583001 −0.291500 0.956571i \(-0.594154\pi\)
−0.291500 + 0.956571i \(0.594154\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 53.6344 + 90.5438i 0.0898399 + 0.151665i
\(598\) 0 0
\(599\) 704.055i 1.17538i 0.809085 + 0.587692i \(0.199963\pi\)
−0.809085 + 0.587692i \(0.800037\pi\)
\(600\) 0 0
\(601\) 338.474 0.563185 0.281592 0.959534i \(-0.409137\pi\)
0.281592 + 0.959534i \(0.409137\pi\)
\(602\) 0 0
\(603\) −871.856 477.693i −1.44586 0.792193i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 816.513i 1.34516i 0.740024 + 0.672581i \(0.234814\pi\)
−0.740024 + 0.672581i \(0.765186\pi\)
\(608\) 0 0
\(609\) 307.157 + 518.532i 0.504363 + 0.851449i
\(610\) 0 0
\(611\) 458.688i 0.750717i
\(612\) 0 0
\(613\) 229.263i 0.374001i 0.982360 + 0.187001i \(0.0598766\pi\)
−0.982360 + 0.187001i \(0.940123\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1072.25 1.73785 0.868924 0.494945i \(-0.164812\pi\)
0.868924 + 0.494945i \(0.164812\pi\)
\(618\) 0 0
\(619\) −80.7103 −0.130388 −0.0651941 0.997873i \(-0.520767\pi\)
−0.0651941 + 0.997873i \(0.520767\pi\)
\(620\) 0 0
\(621\) −247.552 + 8.30863i −0.398635 + 0.0133794i
\(622\) 0 0
\(623\) −455.004 −0.730344
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −590.769 + 349.947i −0.942215 + 0.558130i
\(628\) 0 0
\(629\) 484.591i 0.770415i
\(630\) 0 0
\(631\) −492.894 −0.781131 −0.390566 0.920575i \(-0.627721\pi\)
−0.390566 + 0.920575i \(0.627721\pi\)
\(632\) 0 0
\(633\) 88.9213 + 150.114i 0.140476 + 0.237147i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 70.5267i 0.110717i
\(638\) 0 0
\(639\) −123.079 67.4353i −0.192612 0.105533i
\(640\) 0 0
\(641\) 65.4816i 0.102155i 0.998695 + 0.0510777i \(0.0162656\pi\)
−0.998695 + 0.0510777i \(0.983734\pi\)
\(642\) 0 0
\(643\) 428.619i 0.666592i −0.942822 0.333296i \(-0.891839\pi\)
0.942822 0.333296i \(-0.108161\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −462.801 −0.715303 −0.357652 0.933855i \(-0.616422\pi\)
−0.357652 + 0.933855i \(0.616422\pi\)
\(648\) 0 0
\(649\) −204.316 −0.314817
\(650\) 0 0
\(651\) 154.596 91.5766i 0.237475 0.140671i
\(652\) 0 0
\(653\) −425.064 −0.650941 −0.325470 0.945552i \(-0.605523\pi\)
−0.325470 + 0.945552i \(0.605523\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 694.161 + 380.333i 1.05656 + 0.578894i
\(658\) 0 0
\(659\) 182.769i 0.277343i 0.990338 + 0.138671i \(0.0442831\pi\)
−0.990338 + 0.138671i \(0.955717\pi\)
\(660\) 0 0
\(661\) −482.053 −0.729278 −0.364639 0.931149i \(-0.618808\pi\)
−0.364639 + 0.931149i \(0.618808\pi\)
\(662\) 0 0
\(663\) 784.330 464.605i 1.18300 0.700762i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 246.158i 0.369052i
\(668\) 0 0
\(669\) −256.517 + 151.950i −0.383434 + 0.227131i
\(670\) 0 0
\(671\) 457.758i 0.682203i
\(672\) 0 0
\(673\) 184.579i 0.274264i −0.990553 0.137132i \(-0.956212\pi\)
0.990553 0.137132i \(-0.0437884\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1065.85 −1.57437 −0.787187 0.616715i \(-0.788463\pi\)
−0.787187 + 0.616715i \(0.788463\pi\)
\(678\) 0 0
\(679\) −269.920 −0.397526
\(680\) 0 0
\(681\) 331.710 + 559.982i 0.487093 + 0.822293i
\(682\) 0 0
\(683\) −788.926 −1.15509 −0.577545 0.816359i \(-0.695989\pi\)
−0.577545 + 0.816359i \(0.695989\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −497.958 840.636i −0.724830 1.22363i
\(688\) 0 0
\(689\) 303.870i 0.441030i
\(690\) 0 0
\(691\) −932.000 −1.34877 −0.674385 0.738380i \(-0.735591\pi\)
−0.674385 + 0.738380i \(0.735591\pi\)
\(692\) 0 0
\(693\) 274.730 501.421i 0.396436 0.723551i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1439.05i 2.06464i
\(698\) 0 0
\(699\) 79.0655 + 133.476i 0.113112 + 0.190952i
\(700\) 0 0
\(701\) 1352.75i 1.92974i −0.262721 0.964872i \(-0.584620\pi\)
0.262721 0.964872i \(-0.415380\pi\)
\(702\) 0 0
\(703\) 430.158i 0.611889i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −360.549 −0.509970
\(708\) 0 0
\(709\) 269.473 0.380075 0.190038 0.981777i \(-0.439139\pi\)
0.190038 + 0.981777i \(0.439139\pi\)
\(710\) 0 0
\(711\) 203.140 370.759i 0.285711 0.521462i
\(712\) 0 0
\(713\) −73.3901 −0.102931
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1058.39 + 626.947i −1.47614 + 0.874403i
\(718\) 0 0
\(719\) 537.103i 0.747014i 0.927627 + 0.373507i \(0.121845\pi\)
−0.927627 + 0.373507i \(0.878155\pi\)
\(720\) 0 0
\(721\) −1051.21 −1.45799
\(722\) 0 0
\(723\) 681.192 + 1149.96i 0.942174 + 1.59055i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1117.83i 1.53759i 0.639495 + 0.768795i \(0.279144\pi\)
−0.639495 + 0.768795i \(0.720856\pi\)
\(728\) 0 0
\(729\) 727.359 48.8800i 0.997750 0.0670507i
\(730\) 0 0
\(731\) 439.411i 0.601109i
\(732\) 0 0
\(733\) 7.52599i 0.0102674i −0.999987 0.00513369i \(-0.998366\pi\)
0.999987 0.00513369i \(-0.00163411\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −937.288 −1.27176
\(738\) 0 0
\(739\) −823.079 −1.11377 −0.556887 0.830588i \(-0.688004\pi\)
−0.556887 + 0.830588i \(0.688004\pi\)
\(740\) 0 0
\(741\) −696.228 + 412.417i −0.939579 + 0.556568i
\(742\) 0 0
\(743\) −3.21898 −0.00433241 −0.00216620 0.999998i \(-0.500690\pi\)
−0.00216620 + 0.999998i \(0.500690\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −113.063 + 206.355i −0.151356 + 0.276245i
\(748\) 0 0
\(749\) 322.794i 0.430967i
\(750\) 0 0
\(751\) −1185.63 −1.57874 −0.789368 0.613920i \(-0.789592\pi\)
−0.789368 + 0.613920i \(0.789592\pi\)
\(752\) 0 0
\(753\) 612.671 362.921i 0.813639 0.481967i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 863.315i 1.14044i −0.821491 0.570221i \(-0.806857\pi\)
0.821491 0.570221i \(-0.193143\pi\)
\(758\) 0 0
\(759\) −200.921 + 119.017i −0.264718 + 0.156808i
\(760\) 0 0
\(761\) 570.597i 0.749800i −0.927065 0.374900i \(-0.877677\pi\)
0.927065 0.374900i \(-0.122323\pi\)
\(762\) 0 0
\(763\) 1002.05i 1.31331i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −240.789 −0.313936
\(768\) 0 0
\(769\) 741.684 0.964479 0.482239 0.876040i \(-0.339824\pi\)
0.482239 + 0.876040i \(0.339824\pi\)
\(770\) 0 0
\(771\) −487.565 823.090i −0.632380 1.06756i
\(772\) 0 0
\(773\) 623.203 0.806214 0.403107 0.915153i \(-0.367930\pi\)
0.403107 + 0.915153i \(0.367930\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −182.550 308.175i −0.234943 0.396622i
\(778\) 0 0
\(779\) 1277.41i 1.63980i
\(780\) 0 0
\(781\) −132.316 −0.169419
\(782\) 0 0
\(783\) 24.3023 + 724.078i 0.0310375 + 0.924749i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 335.303i 0.426052i −0.977046 0.213026i \(-0.931668\pi\)
0.977046 0.213026i \(-0.0683320\pi\)
\(788\) 0 0
\(789\) −55.3943 93.5147i −0.0702083 0.118523i
\(790\) 0 0
\(791\) 59.2098i 0.0748543i
\(792\) 0 0
\(793\) 539.473i 0.680294i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −550.520 −0.690740 −0.345370 0.938467i \(-0.612247\pi\)
−0.345370 + 0.938467i \(0.612247\pi\)
\(798\) 0 0
\(799\) 1393.81 1.74445
\(800\) 0 0
\(801\) −479.684 262.820i −0.598856 0.328115i
\(802\) 0 0
\(803\) 746.258 0.929337
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1362.95 + 807.354i −1.68891 + 1.00044i
\(808\) 0 0
\(809\) 560.288i 0.692569i 0.938130 + 0.346284i \(0.112557\pi\)
−0.938130 + 0.346284i \(0.887443\pi\)
\(810\) 0 0
\(811\) 237.842 0.293270 0.146635 0.989191i \(-0.453156\pi\)
0.146635 + 0.989191i \(0.453156\pi\)
\(812\) 0 0
\(813\) 727.624 + 1228.35i 0.894987 + 1.51089i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 390.053i 0.477421i
\(818\) 0 0
\(819\) 323.772 590.930i 0.395326 0.721526i
\(820\) 0 0
\(821\) 65.4816i 0.0797584i 0.999205 + 0.0398792i \(0.0126973\pi\)
−0.999205 + 0.0398792i \(0.987303\pi\)
\(822\) 0 0
\(823\) 521.512i 0.633673i 0.948480 + 0.316836i \(0.102621\pi\)
−0.948480 + 0.316836i \(0.897379\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 987.512 1.19409 0.597045 0.802208i \(-0.296342\pi\)
0.597045 + 0.802208i \(0.296342\pi\)
\(828\) 0 0
\(829\) −333.631 −0.402450 −0.201225 0.979545i \(-0.564492\pi\)
−0.201225 + 0.979545i \(0.564492\pi\)
\(830\) 0 0
\(831\) −485.662 + 287.686i −0.584431 + 0.346193i
\(832\) 0 0
\(833\) 214.309 0.257274
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 215.878 7.24555i 0.257919 0.00865657i
\(838\) 0 0
\(839\) 129.363i 0.154187i −0.997024 0.0770934i \(-0.975436\pi\)
0.997024 0.0770934i \(-0.0245640\pi\)
\(840\) 0 0
\(841\) 121.000 0.143876
\(842\) 0 0
\(843\) −63.0629 + 37.3559i −0.0748077 + 0.0443130i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 366.855i 0.433123i
\(848\) 0 0
\(849\) −512.254 + 303.438i −0.603362 + 0.357407i
\(850\) 0 0
\(851\) 146.297i 0.171912i
\(852\) 0 0
\(853\) 1080.42i 1.26661i −0.773902 0.633306i \(-0.781697\pi\)
0.773902 0.633306i \(-0.218303\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −702.548 −0.819776 −0.409888 0.912136i \(-0.634432\pi\)
−0.409888 + 0.912136i \(0.634432\pi\)
\(858\) 0 0
\(859\) −281.132 −0.327278 −0.163639 0.986520i \(-0.552323\pi\)
−0.163639 + 0.986520i \(0.552323\pi\)
\(860\) 0 0
\(861\) 542.105 + 915.163i 0.629623 + 1.06291i
\(862\) 0 0
\(863\) −419.221 −0.485772 −0.242886 0.970055i \(-0.578094\pi\)
−0.242886 + 0.970055i \(0.578094\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 969.924 + 1637.39i 1.11871 + 1.88857i
\(868\) 0 0
\(869\) 398.585i 0.458671i
\(870\) 0 0
\(871\) −1104.60 −1.26820
\(872\) 0 0
\(873\) −284.561 155.912i −0.325958 0.178593i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1079.42i 1.23081i 0.788211 + 0.615405i \(0.211008\pi\)
−0.788211 + 0.615405i \(0.788992\pi\)
\(878\) 0 0
\(879\) −785.618 1326.25i −0.893763 1.50882i
\(880\) 0 0
\(881\) 748.212i 0.849275i −0.905363 0.424638i \(-0.860401\pi\)
0.905363 0.424638i \(-0.139599\pi\)
\(882\) 0 0
\(883\) 875.749i 0.991788i −0.868383 0.495894i \(-0.834840\pi\)
0.868383 0.495894i \(-0.165160\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1015.05 1.14436 0.572182 0.820127i \(-0.306097\pi\)
0.572182 + 0.820127i \(0.306097\pi\)
\(888\) 0 0
\(889\) −1006.68 −1.13238
\(890\) 0 0
\(891\) 579.263 369.928i 0.650126 0.415183i
\(892\) 0 0
\(893\) −1237.25 −1.38550
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −236.788 + 140.263i −0.263977 + 0.156369i
\(898\) 0 0
\(899\) 214.663i 0.238779i
\(900\) 0 0
\(901\) −923.368 −1.02483
\(902\) 0 0
\(903\) 165.530 + 279.443i 0.183312 + 0.309460i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1504.70i 1.65898i −0.558520 0.829491i \(-0.688631\pi\)
0.558520 0.829491i \(-0.311369\pi\)
\(908\) 0 0
\(909\) −380.105 208.261i −0.418158 0.229110i
\(910\) 0 0
\(911\) 1002.19i 1.10010i 0.835131 + 0.550051i \(0.185392\pi\)
−0.835131 + 0.550051i \(0.814608\pi\)
\(912\) 0 0
\(913\) 221.842i 0.242981i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1650.05 1.79940
\(918\) 0 0
\(919\) −780.289 −0.849063 −0.424532 0.905413i \(-0.639561\pi\)
−0.424532 + 0.905413i \(0.639561\pi\)
\(920\) 0 0
\(921\) −29.3772 + 17.4019i −0.0318971 + 0.0188945i
\(922\) 0 0
\(923\) −155.936 −0.168945
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1108.23 607.201i −1.19550 0.655018i
\(928\) 0 0
\(929\) 1093.84i 1.17744i −0.808339 0.588718i \(-0.799633\pi\)
0.808339 0.588718i \(-0.200367\pi\)
\(930\) 0 0
\(931\) −190.236 −0.204335
\(932\) 0 0
\(933\) −1338.98 + 793.157i −1.43513 + 0.850115i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 407.947i 0.435376i −0.976018 0.217688i \(-0.930148\pi\)
0.976018 0.217688i \(-0.0698515\pi\)
\(938\) 0 0
\(939\) −119.548 + 70.8154i −0.127314 + 0.0754157i
\(940\) 0 0
\(941\) 671.008i 0.713079i 0.934280 + 0.356540i \(0.116044\pi\)
−0.934280 + 0.356540i \(0.883956\pi\)
\(942\) 0 0
\(943\) 434.447i 0.460707i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1608.13 −1.69813 −0.849064 0.528290i \(-0.822833\pi\)
−0.849064 + 0.528290i \(0.822833\pi\)
\(948\) 0 0
\(949\) 879.473 0.926737
\(950\) 0 0
\(951\) −59.7758 100.911i −0.0628557 0.106111i
\(952\) 0 0
\(953\) −695.440 −0.729737 −0.364869 0.931059i \(-0.618886\pi\)
−0.364869 + 0.931059i \(0.618886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 348.120 + 587.684i 0.363762 + 0.614090i
\(958\) 0 0
\(959\) 715.107i 0.745680i
\(960\) 0 0
\(961\) −897.000 −0.933403
\(962\) 0 0
\(963\) −186.453 + 340.302i −0.193617 + 0.353377i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1030.07i 1.06522i 0.846361 + 0.532609i \(0.178788\pi\)
−0.846361 + 0.532609i \(0.821212\pi\)
\(968\) 0 0
\(969\) −1253.21 2115.63i −1.29330 2.18331i
\(970\) 0 0
\(971\) 1165.24i 1.20004i −0.799986 0.600019i \(-0.795160\pi\)
0.799986 0.600019i \(-0.204840\pi\)
\(972\) 0 0
\(973\) 575.106i 0.591065i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −726.440 −0.743541 −0.371771 0.928325i \(-0.621249\pi\)
−0.371771 + 0.928325i \(0.621249\pi\)
\(978\) 0 0
\(979\) −515.684 −0.526746
\(980\) 0 0
\(981\) 578.807 1056.40i 0.590017 1.07686i
\(982\) 0 0
\(983\) −1024.21 −1.04192 −0.520960 0.853581i \(-0.674426\pi\)
−0.520960 + 0.853581i \(0.674426\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 886.395 525.064i 0.898070 0.531980i
\(988\) 0 0
\(989\) 132.657i 0.134133i
\(990\) 0 0
\(991\) 1797.89 1.81422 0.907111 0.420892i \(-0.138283\pi\)
0.907111 + 0.420892i \(0.138283\pi\)
\(992\) 0 0
\(993\) 681.031 + 1149.69i 0.685832 + 1.15780i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 901.368i 0.904080i 0.891998 + 0.452040i \(0.149304\pi\)
−0.891998 + 0.452040i \(0.850696\pi\)
\(998\) 0 0
\(999\) −14.4434 430.336i −0.0144579 0.430766i
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 1200.3.c.k.449.6 8
3.2 odd 2 inner 1200.3.c.k.449.4 8
4.3 odd 2 150.3.b.b.149.1 8
5.2 odd 4 1200.3.l.u.401.3 4
5.3 odd 4 240.3.l.c.161.2 4
5.4 even 2 inner 1200.3.c.k.449.3 8
12.11 even 2 150.3.b.b.149.7 8
15.2 even 4 1200.3.l.u.401.4 4
15.8 even 4 240.3.l.c.161.1 4
15.14 odd 2 inner 1200.3.c.k.449.5 8
20.3 even 4 30.3.d.a.11.4 yes 4
20.7 even 4 150.3.d.c.101.1 4
20.19 odd 2 150.3.b.b.149.8 8
40.3 even 4 960.3.l.e.641.2 4
40.13 odd 4 960.3.l.f.641.3 4
60.23 odd 4 30.3.d.a.11.2 4
60.47 odd 4 150.3.d.c.101.3 4
60.59 even 2 150.3.b.b.149.2 8
120.53 even 4 960.3.l.f.641.4 4
120.83 odd 4 960.3.l.e.641.1 4
180.23 odd 12 810.3.h.a.431.3 8
180.43 even 12 810.3.h.a.701.3 8
180.83 odd 12 810.3.h.a.701.2 8
180.103 even 12 810.3.h.a.431.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.3.d.a.11.2 4 60.23 odd 4
30.3.d.a.11.4 yes 4 20.3 even 4
150.3.b.b.149.1 8 4.3 odd 2
150.3.b.b.149.2 8 60.59 even 2
150.3.b.b.149.7 8 12.11 even 2
150.3.b.b.149.8 8 20.19 odd 2
150.3.d.c.101.1 4 20.7 even 4
150.3.d.c.101.3 4 60.47 odd 4
240.3.l.c.161.1 4 15.8 even 4
240.3.l.c.161.2 4 5.3 odd 4
810.3.h.a.431.2 8 180.103 even 12
810.3.h.a.431.3 8 180.23 odd 12
810.3.h.a.701.2 8 180.83 odd 12
810.3.h.a.701.3 8 180.43 even 12
960.3.l.e.641.1 4 120.83 odd 4
960.3.l.e.641.2 4 40.3 even 4
960.3.l.f.641.3 4 40.13 odd 4
960.3.l.f.641.4 4 120.53 even 4
1200.3.c.k.449.3 8 5.4 even 2 inner
1200.3.c.k.449.4 8 3.2 odd 2 inner
1200.3.c.k.449.5 8 15.14 odd 2 inner
1200.3.c.k.449.6 8 1.1 even 1 trivial
1200.3.l.u.401.3 4 5.2 odd 4
1200.3.l.u.401.4 4 15.2 even 4