Properties

Label 1200.4.f.i.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(49,1200)]
 
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, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.i.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +5.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +5.00000i q^{7} -9.00000 q^{9} -14.0000 q^{11} +1.00000i q^{13} -46.0000i q^{17} +19.0000 q^{19} +15.0000 q^{21} +46.0000i q^{23} +27.0000i q^{27} -14.0000 q^{29} -133.000 q^{31} +42.0000i q^{33} -258.000i q^{37} +3.00000 q^{39} +84.0000 q^{41} +167.000i q^{43} +410.000i q^{47} +318.000 q^{49} -138.000 q^{51} +456.000i q^{53} -57.0000i q^{57} -194.000 q^{59} -17.0000 q^{61} -45.0000i q^{63} +653.000i q^{67} +138.000 q^{69} -828.000 q^{71} +570.000i q^{73} -70.0000i q^{77} -552.000 q^{79} +81.0000 q^{81} -142.000i q^{83} +42.0000i q^{87} +1104.00 q^{89} -5.00000 q^{91} +399.000i q^{93} -841.000i q^{97} +126.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 28 q^{11} + 38 q^{19} + 30 q^{21} - 28 q^{29} - 266 q^{31} + 6 q^{39} + 168 q^{41} + 636 q^{49} - 276 q^{51} - 388 q^{59} - 34 q^{61} + 276 q^{69} - 1656 q^{71} - 1104 q^{79} + 162 q^{81} + 2208 q^{89} - 10 q^{91} + 252 q^{99}+O(q^{100}) \) 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
\(3\) − 3.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.00000i 0.269975i 0.990847 + 0.134987i \(0.0430994\pi\)
−0.990847 + 0.134987i \(0.956901\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −14.0000 −0.383742 −0.191871 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.0213346i 0.999943 + 0.0106673i \(0.00339558\pi\)
−0.999943 + 0.0106673i \(0.996604\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 46.0000i − 0.656273i −0.944630 0.328136i \(-0.893579\pi\)
0.944630 0.328136i \(-0.106421\pi\)
\(18\) 0 0
\(19\) 19.0000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 15.0000 0.155870
\(22\) 0 0
\(23\) 46.0000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −14.0000 −0.0896460 −0.0448230 0.998995i \(-0.514272\pi\)
−0.0448230 + 0.998995i \(0.514272\pi\)
\(30\) 0 0
\(31\) −133.000 −0.770565 −0.385282 0.922799i \(-0.625896\pi\)
−0.385282 + 0.922799i \(0.625896\pi\)
\(32\) 0 0
\(33\) 42.0000i 0.221553i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 258.000i − 1.14635i −0.819433 0.573175i \(-0.805712\pi\)
0.819433 0.573175i \(-0.194288\pi\)
\(38\) 0 0
\(39\) 3.00000 0.0123176
\(40\) 0 0
\(41\) 84.0000 0.319966 0.159983 0.987120i \(-0.448856\pi\)
0.159983 + 0.987120i \(0.448856\pi\)
\(42\) 0 0
\(43\) 167.000i 0.592262i 0.955147 + 0.296131i \(0.0956965\pi\)
−0.955147 + 0.296131i \(0.904304\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 410.000i 1.27244i 0.771508 + 0.636220i \(0.219503\pi\)
−0.771508 + 0.636220i \(0.780497\pi\)
\(48\) 0 0
\(49\) 318.000 0.927114
\(50\) 0 0
\(51\) −138.000 −0.378899
\(52\) 0 0
\(53\) 456.000i 1.18182i 0.806738 + 0.590910i \(0.201231\pi\)
−0.806738 + 0.590910i \(0.798769\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 57.0000i − 0.132453i
\(58\) 0 0
\(59\) −194.000 −0.428079 −0.214039 0.976825i \(-0.568662\pi\)
−0.214039 + 0.976825i \(0.568662\pi\)
\(60\) 0 0
\(61\) −17.0000 −0.0356824 −0.0178412 0.999841i \(-0.505679\pi\)
−0.0178412 + 0.999841i \(0.505679\pi\)
\(62\) 0 0
\(63\) − 45.0000i − 0.0899915i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 653.000i 1.19070i 0.803468 + 0.595348i \(0.202986\pi\)
−0.803468 + 0.595348i \(0.797014\pi\)
\(68\) 0 0
\(69\) 138.000 0.240772
\(70\) 0 0
\(71\) −828.000 −1.38402 −0.692011 0.721887i \(-0.743275\pi\)
−0.692011 + 0.721887i \(0.743275\pi\)
\(72\) 0 0
\(73\) 570.000i 0.913883i 0.889497 + 0.456941i \(0.151055\pi\)
−0.889497 + 0.456941i \(0.848945\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 70.0000i − 0.103601i
\(78\) 0 0
\(79\) −552.000 −0.786137 −0.393069 0.919509i \(-0.628587\pi\)
−0.393069 + 0.919509i \(0.628587\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) − 142.000i − 0.187789i −0.995582 0.0938947i \(-0.970068\pi\)
0.995582 0.0938947i \(-0.0299317\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 42.0000i 0.0517572i
\(88\) 0 0
\(89\) 1104.00 1.31487 0.657437 0.753510i \(-0.271641\pi\)
0.657437 + 0.753510i \(0.271641\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.00575981
\(92\) 0 0
\(93\) 399.000i 0.444886i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 841.000i − 0.880316i −0.897920 0.440158i \(-0.854923\pi\)
0.897920 0.440158i \(-0.145077\pi\)
\(98\) 0 0
\(99\) 126.000 0.127914
\(100\) 0 0
\(101\) 552.000 0.543822 0.271911 0.962322i \(-0.412344\pi\)
0.271911 + 0.962322i \(0.412344\pi\)
\(102\) 0 0
\(103\) 308.000i 0.294642i 0.989089 + 0.147321i \(0.0470651\pi\)
−0.989089 + 0.147321i \(0.952935\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 984.000i 0.889036i 0.895770 + 0.444518i \(0.146625\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(108\) 0 0
\(109\) 1843.00 1.61952 0.809759 0.586763i \(-0.199598\pi\)
0.809759 + 0.586763i \(0.199598\pi\)
\(110\) 0 0
\(111\) −774.000 −0.661845
\(112\) 0 0
\(113\) 876.000i 0.729267i 0.931151 + 0.364633i \(0.118806\pi\)
−0.931151 + 0.364633i \(0.881194\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 9.00000i − 0.00711154i
\(118\) 0 0
\(119\) 230.000 0.177177
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) − 252.000i − 0.184732i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2376.00i 1.66013i 0.557670 + 0.830063i \(0.311695\pi\)
−0.557670 + 0.830063i \(0.688305\pi\)
\(128\) 0 0
\(129\) 501.000 0.341943
\(130\) 0 0
\(131\) −1056.00 −0.704299 −0.352149 0.935944i \(-0.614549\pi\)
−0.352149 + 0.935944i \(0.614549\pi\)
\(132\) 0 0
\(133\) 95.0000i 0.0619364i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 778.000i 0.485175i 0.970129 + 0.242588i \(0.0779962\pi\)
−0.970129 + 0.242588i \(0.922004\pi\)
\(138\) 0 0
\(139\) 1692.00 1.03247 0.516236 0.856446i \(-0.327333\pi\)
0.516236 + 0.856446i \(0.327333\pi\)
\(140\) 0 0
\(141\) 1230.00 0.734643
\(142\) 0 0
\(143\) − 14.0000i − 0.00818698i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 954.000i − 0.535269i
\(148\) 0 0
\(149\) 494.000 0.271611 0.135806 0.990736i \(-0.456638\pi\)
0.135806 + 0.990736i \(0.456638\pi\)
\(150\) 0 0
\(151\) 841.000 0.453242 0.226621 0.973983i \(-0.427232\pi\)
0.226621 + 0.973983i \(0.427232\pi\)
\(152\) 0 0
\(153\) 414.000i 0.218758i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 19.0000i − 0.00965838i −0.999988 0.00482919i \(-0.998463\pi\)
0.999988 0.00482919i \(-0.00153718\pi\)
\(158\) 0 0
\(159\) 1368.00 0.682324
\(160\) 0 0
\(161\) −230.000 −0.112587
\(162\) 0 0
\(163\) 2261.00i 1.08647i 0.839580 + 0.543237i \(0.182801\pi\)
−0.839580 + 0.543237i \(0.817199\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2112.00i 0.978632i 0.872107 + 0.489316i \(0.162753\pi\)
−0.872107 + 0.489316i \(0.837247\pi\)
\(168\) 0 0
\(169\) 2196.00 0.999545
\(170\) 0 0
\(171\) −171.000 −0.0764719
\(172\) 0 0
\(173\) 562.000i 0.246983i 0.992346 + 0.123492i \(0.0394092\pi\)
−0.992346 + 0.123492i \(0.960591\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 582.000i 0.247151i
\(178\) 0 0
\(179\) 3718.00 1.55249 0.776247 0.630429i \(-0.217121\pi\)
0.776247 + 0.630429i \(0.217121\pi\)
\(180\) 0 0
\(181\) −1639.00 −0.673071 −0.336536 0.941671i \(-0.609255\pi\)
−0.336536 + 0.941671i \(0.609255\pi\)
\(182\) 0 0
\(183\) 51.0000i 0.0206012i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 644.000i 0.251839i
\(188\) 0 0
\(189\) −135.000 −0.0519566
\(190\) 0 0
\(191\) −2410.00 −0.912992 −0.456496 0.889725i \(-0.650896\pi\)
−0.456496 + 0.889725i \(0.650896\pi\)
\(192\) 0 0
\(193\) − 2621.00i − 0.977532i −0.872415 0.488766i \(-0.837447\pi\)
0.872415 0.488766i \(-0.162553\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4954.00i 1.79166i 0.444392 + 0.895832i \(0.353420\pi\)
−0.444392 + 0.895832i \(0.646580\pi\)
\(198\) 0 0
\(199\) 1739.00 0.619470 0.309735 0.950823i \(-0.399760\pi\)
0.309735 + 0.950823i \(0.399760\pi\)
\(200\) 0 0
\(201\) 1959.00 0.687449
\(202\) 0 0
\(203\) − 70.0000i − 0.0242022i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 414.000i − 0.139010i
\(208\) 0 0
\(209\) −266.000 −0.0880364
\(210\) 0 0
\(211\) 4525.00 1.47637 0.738184 0.674599i \(-0.235683\pi\)
0.738184 + 0.674599i \(0.235683\pi\)
\(212\) 0 0
\(213\) 2484.00i 0.799065i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 665.000i − 0.208033i
\(218\) 0 0
\(219\) 1710.00 0.527631
\(220\) 0 0
\(221\) 46.0000 0.0140013
\(222\) 0 0
\(223\) 3211.00i 0.964235i 0.876106 + 0.482118i \(0.160132\pi\)
−0.876106 + 0.482118i \(0.839868\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2484.00i − 0.726295i −0.931732 0.363147i \(-0.881702\pi\)
0.931732 0.363147i \(-0.118298\pi\)
\(228\) 0 0
\(229\) 1847.00 0.532983 0.266492 0.963837i \(-0.414136\pi\)
0.266492 + 0.963837i \(0.414136\pi\)
\(230\) 0 0
\(231\) −210.000 −0.0598138
\(232\) 0 0
\(233\) − 1020.00i − 0.286792i −0.989665 0.143396i \(-0.954198\pi\)
0.989665 0.143396i \(-0.0458022\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1656.00i 0.453877i
\(238\) 0 0
\(239\) 1176.00 0.318281 0.159140 0.987256i \(-0.449128\pi\)
0.159140 + 0.987256i \(0.449128\pi\)
\(240\) 0 0
\(241\) −6967.00 −1.86217 −0.931087 0.364797i \(-0.881138\pi\)
−0.931087 + 0.364797i \(0.881138\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 19.0000i 0.00489450i
\(248\) 0 0
\(249\) −426.000 −0.108420
\(250\) 0 0
\(251\) −1380.00 −0.347031 −0.173516 0.984831i \(-0.555513\pi\)
−0.173516 + 0.984831i \(0.555513\pi\)
\(252\) 0 0
\(253\) − 644.000i − 0.160031i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6924.00i 1.68057i 0.542143 + 0.840286i \(0.317613\pi\)
−0.542143 + 0.840286i \(0.682387\pi\)
\(258\) 0 0
\(259\) 1290.00 0.309485
\(260\) 0 0
\(261\) 126.000 0.0298820
\(262\) 0 0
\(263\) − 1884.00i − 0.441720i −0.975305 0.220860i \(-0.929114\pi\)
0.975305 0.220860i \(-0.0708864\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 3312.00i − 0.759143i
\(268\) 0 0
\(269\) −3610.00 −0.818236 −0.409118 0.912481i \(-0.634164\pi\)
−0.409118 + 0.912481i \(0.634164\pi\)
\(270\) 0 0
\(271\) 6072.00 1.36106 0.680531 0.732719i \(-0.261749\pi\)
0.680531 + 0.732719i \(0.261749\pi\)
\(272\) 0 0
\(273\) 15.0000i 0.00332543i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2803.00i 0.608000i 0.952672 + 0.304000i \(0.0983223\pi\)
−0.952672 + 0.304000i \(0.901678\pi\)
\(278\) 0 0
\(279\) 1197.00 0.256855
\(280\) 0 0
\(281\) −6694.00 −1.42111 −0.710553 0.703644i \(-0.751555\pi\)
−0.710553 + 0.703644i \(0.751555\pi\)
\(282\) 0 0
\(283\) 6481.00i 1.36133i 0.732596 + 0.680663i \(0.238308\pi\)
−0.732596 + 0.680663i \(0.761692\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 420.000i 0.0863826i
\(288\) 0 0
\(289\) 2797.00 0.569306
\(290\) 0 0
\(291\) −2523.00 −0.508250
\(292\) 0 0
\(293\) 3014.00i 0.600955i 0.953789 + 0.300477i \(0.0971460\pi\)
−0.953789 + 0.300477i \(0.902854\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 378.000i − 0.0738511i
\(298\) 0 0
\(299\) −46.0000 −0.00889715
\(300\) 0 0
\(301\) −835.000 −0.159896
\(302\) 0 0
\(303\) − 1656.00i − 0.313976i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5369.00i − 0.998127i −0.866565 0.499064i \(-0.833677\pi\)
0.866565 0.499064i \(-0.166323\pi\)
\(308\) 0 0
\(309\) 924.000 0.170112
\(310\) 0 0
\(311\) −4846.00 −0.883574 −0.441787 0.897120i \(-0.645655\pi\)
−0.441787 + 0.897120i \(0.645655\pi\)
\(312\) 0 0
\(313\) − 757.000i − 0.136703i −0.997661 0.0683517i \(-0.978226\pi\)
0.997661 0.0683517i \(-0.0217740\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7632.00i 1.35223i 0.736798 + 0.676113i \(0.236337\pi\)
−0.736798 + 0.676113i \(0.763663\pi\)
\(318\) 0 0
\(319\) 196.000 0.0344009
\(320\) 0 0
\(321\) 2952.00 0.513285
\(322\) 0 0
\(323\) − 874.000i − 0.150559i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 5529.00i − 0.935029i
\(328\) 0 0
\(329\) −2050.00 −0.343526
\(330\) 0 0
\(331\) 6780.00 1.12587 0.562934 0.826502i \(-0.309672\pi\)
0.562934 + 0.826502i \(0.309672\pi\)
\(332\) 0 0
\(333\) 2322.00i 0.382117i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 7849.00i − 1.26873i −0.773033 0.634365i \(-0.781261\pi\)
0.773033 0.634365i \(-0.218739\pi\)
\(338\) 0 0
\(339\) 2628.00 0.421042
\(340\) 0 0
\(341\) 1862.00 0.295698
\(342\) 0 0
\(343\) 3305.00i 0.520272i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 634.000i 0.0980833i 0.998797 + 0.0490416i \(0.0156167\pi\)
−0.998797 + 0.0490416i \(0.984383\pi\)
\(348\) 0 0
\(349\) −930.000 −0.142641 −0.0713206 0.997453i \(-0.522721\pi\)
−0.0713206 + 0.997453i \(0.522721\pi\)
\(350\) 0 0
\(351\) −27.0000 −0.00410585
\(352\) 0 0
\(353\) − 4286.00i − 0.646234i −0.946359 0.323117i \(-0.895269\pi\)
0.946359 0.323117i \(-0.104731\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 690.000i − 0.102293i
\(358\) 0 0
\(359\) −4236.00 −0.622751 −0.311375 0.950287i \(-0.600790\pi\)
−0.311375 + 0.950287i \(0.600790\pi\)
\(360\) 0 0
\(361\) −6498.00 −0.947368
\(362\) 0 0
\(363\) 3405.00i 0.492331i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1451.00i 0.206380i 0.994662 + 0.103190i \(0.0329050\pi\)
−0.994662 + 0.103190i \(0.967095\pi\)
\(368\) 0 0
\(369\) −756.000 −0.106655
\(370\) 0 0
\(371\) −2280.00 −0.319061
\(372\) 0 0
\(373\) 3115.00i 0.432409i 0.976348 + 0.216205i \(0.0693678\pi\)
−0.976348 + 0.216205i \(0.930632\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14.0000i − 0.00191256i
\(378\) 0 0
\(379\) −1415.00 −0.191777 −0.0958887 0.995392i \(-0.530569\pi\)
−0.0958887 + 0.995392i \(0.530569\pi\)
\(380\) 0 0
\(381\) 7128.00 0.958474
\(382\) 0 0
\(383\) 180.000i 0.0240145i 0.999928 + 0.0120073i \(0.00382213\pi\)
−0.999928 + 0.0120073i \(0.996178\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1503.00i − 0.197421i
\(388\) 0 0
\(389\) −12372.0 −1.61256 −0.806279 0.591535i \(-0.798522\pi\)
−0.806279 + 0.591535i \(0.798522\pi\)
\(390\) 0 0
\(391\) 2116.00 0.273685
\(392\) 0 0
\(393\) 3168.00i 0.406627i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 5767.00i − 0.729062i −0.931191 0.364531i \(-0.881229\pi\)
0.931191 0.364531i \(-0.118771\pi\)
\(398\) 0 0
\(399\) 285.000 0.0357590
\(400\) 0 0
\(401\) −3120.00 −0.388542 −0.194271 0.980948i \(-0.562234\pi\)
−0.194271 + 0.980948i \(0.562234\pi\)
\(402\) 0 0
\(403\) − 133.000i − 0.0164397i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3612.00i 0.439902i
\(408\) 0 0
\(409\) −1501.00 −0.181466 −0.0907331 0.995875i \(-0.528921\pi\)
−0.0907331 + 0.995875i \(0.528921\pi\)
\(410\) 0 0
\(411\) 2334.00 0.280116
\(412\) 0 0
\(413\) − 970.000i − 0.115570i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5076.00i − 0.596098i
\(418\) 0 0
\(419\) −9072.00 −1.05775 −0.528874 0.848701i \(-0.677386\pi\)
−0.528874 + 0.848701i \(0.677386\pi\)
\(420\) 0 0
\(421\) 7350.00 0.850872 0.425436 0.904989i \(-0.360121\pi\)
0.425436 + 0.904989i \(0.360121\pi\)
\(422\) 0 0
\(423\) − 3690.00i − 0.424146i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 85.0000i − 0.00963334i
\(428\) 0 0
\(429\) −42.0000 −0.00472676
\(430\) 0 0
\(431\) −5962.00 −0.666310 −0.333155 0.942872i \(-0.608113\pi\)
−0.333155 + 0.942872i \(0.608113\pi\)
\(432\) 0 0
\(433\) − 10093.0i − 1.12018i −0.828431 0.560091i \(-0.810766\pi\)
0.828431 0.560091i \(-0.189234\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 874.000i 0.0956730i
\(438\) 0 0
\(439\) −2555.00 −0.277776 −0.138888 0.990308i \(-0.544353\pi\)
−0.138888 + 0.990308i \(0.544353\pi\)
\(440\) 0 0
\(441\) −2862.00 −0.309038
\(442\) 0 0
\(443\) − 6240.00i − 0.669236i −0.942354 0.334618i \(-0.891393\pi\)
0.942354 0.334618i \(-0.108607\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 1482.00i − 0.156815i
\(448\) 0 0
\(449\) −3324.00 −0.349375 −0.174687 0.984624i \(-0.555891\pi\)
−0.174687 + 0.984624i \(0.555891\pi\)
\(450\) 0 0
\(451\) −1176.00 −0.122784
\(452\) 0 0
\(453\) − 2523.00i − 0.261680i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16774.0i − 1.71697i −0.512840 0.858484i \(-0.671407\pi\)
0.512840 0.858484i \(-0.328593\pi\)
\(458\) 0 0
\(459\) 1242.00 0.126300
\(460\) 0 0
\(461\) 14304.0 1.44513 0.722564 0.691304i \(-0.242964\pi\)
0.722564 + 0.691304i \(0.242964\pi\)
\(462\) 0 0
\(463\) − 6936.00i − 0.696206i −0.937456 0.348103i \(-0.886826\pi\)
0.937456 0.348103i \(-0.113174\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15622.0i 1.54797i 0.633207 + 0.773983i \(0.281738\pi\)
−0.633207 + 0.773983i \(0.718262\pi\)
\(468\) 0 0
\(469\) −3265.00 −0.321458
\(470\) 0 0
\(471\) −57.0000 −0.00557627
\(472\) 0 0
\(473\) − 2338.00i − 0.227276i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4104.00i − 0.393940i
\(478\) 0 0
\(479\) −13354.0 −1.27382 −0.636910 0.770938i \(-0.719788\pi\)
−0.636910 + 0.770938i \(0.719788\pi\)
\(480\) 0 0
\(481\) 258.000 0.0244569
\(482\) 0 0
\(483\) 690.000i 0.0650023i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 461.000i − 0.0428951i −0.999770 0.0214475i \(-0.993173\pi\)
0.999770 0.0214475i \(-0.00682749\pi\)
\(488\) 0 0
\(489\) 6783.00 0.627276
\(490\) 0 0
\(491\) −3768.00 −0.346329 −0.173164 0.984893i \(-0.555399\pi\)
−0.173164 + 0.984893i \(0.555399\pi\)
\(492\) 0 0
\(493\) 644.000i 0.0588323i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 4140.00i − 0.373651i
\(498\) 0 0
\(499\) −14317.0 −1.28440 −0.642201 0.766536i \(-0.721979\pi\)
−0.642201 + 0.766536i \(0.721979\pi\)
\(500\) 0 0
\(501\) 6336.00 0.565013
\(502\) 0 0
\(503\) − 9228.00i − 0.818004i −0.912533 0.409002i \(-0.865877\pi\)
0.912533 0.409002i \(-0.134123\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6588.00i − 0.577087i
\(508\) 0 0
\(509\) −4574.00 −0.398308 −0.199154 0.979968i \(-0.563819\pi\)
−0.199154 + 0.979968i \(0.563819\pi\)
\(510\) 0 0
\(511\) −2850.00 −0.246725
\(512\) 0 0
\(513\) 513.000i 0.0441511i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 5740.00i − 0.488288i
\(518\) 0 0
\(519\) 1686.00 0.142596
\(520\) 0 0
\(521\) −8494.00 −0.714259 −0.357129 0.934055i \(-0.616245\pi\)
−0.357129 + 0.934055i \(0.616245\pi\)
\(522\) 0 0
\(523\) − 8263.00i − 0.690852i −0.938446 0.345426i \(-0.887734\pi\)
0.938446 0.345426i \(-0.112266\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6118.00i 0.505701i
\(528\) 0 0
\(529\) 10051.0 0.826087
\(530\) 0 0
\(531\) 1746.00 0.142693
\(532\) 0 0
\(533\) 84.0000i 0.00682635i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 11154.0i − 0.896333i
\(538\) 0 0
\(539\) −4452.00 −0.355772
\(540\) 0 0
\(541\) 21157.0 1.68135 0.840675 0.541540i \(-0.182158\pi\)
0.840675 + 0.541540i \(0.182158\pi\)
\(542\) 0 0
\(543\) 4917.00i 0.388598i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 4048.00i − 0.316417i −0.987406 0.158208i \(-0.949428\pi\)
0.987406 0.158208i \(-0.0505718\pi\)
\(548\) 0 0
\(549\) 153.000 0.0118941
\(550\) 0 0
\(551\) −266.000 −0.0205662
\(552\) 0 0
\(553\) − 2760.00i − 0.212237i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6758.00i 0.514086i 0.966400 + 0.257043i \(0.0827481\pi\)
−0.966400 + 0.257043i \(0.917252\pi\)
\(558\) 0 0
\(559\) −167.000 −0.0126357
\(560\) 0 0
\(561\) 1932.00 0.145399
\(562\) 0 0
\(563\) − 24506.0i − 1.83447i −0.398350 0.917233i \(-0.630417\pi\)
0.398350 0.917233i \(-0.369583\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 405.000i 0.0299972i
\(568\) 0 0
\(569\) 1430.00 0.105358 0.0526790 0.998611i \(-0.483224\pi\)
0.0526790 + 0.998611i \(0.483224\pi\)
\(570\) 0 0
\(571\) −3691.00 −0.270514 −0.135257 0.990811i \(-0.543186\pi\)
−0.135257 + 0.990811i \(0.543186\pi\)
\(572\) 0 0
\(573\) 7230.00i 0.527116i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 4571.00i − 0.329798i −0.986310 0.164899i \(-0.947270\pi\)
0.986310 0.164899i \(-0.0527298\pi\)
\(578\) 0 0
\(579\) −7863.00 −0.564378
\(580\) 0 0
\(581\) 710.000 0.0506984
\(582\) 0 0
\(583\) − 6384.00i − 0.453513i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14808.0i 1.04121i 0.853797 + 0.520606i \(0.174294\pi\)
−0.853797 + 0.520606i \(0.825706\pi\)
\(588\) 0 0
\(589\) −2527.00 −0.176780
\(590\) 0 0
\(591\) 14862.0 1.03442
\(592\) 0 0
\(593\) − 24588.0i − 1.70271i −0.524588 0.851356i \(-0.675781\pi\)
0.524588 0.851356i \(-0.324219\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 5217.00i − 0.357651i
\(598\) 0 0
\(599\) −27564.0 −1.88019 −0.940096 0.340911i \(-0.889265\pi\)
−0.940096 + 0.340911i \(0.889265\pi\)
\(600\) 0 0
\(601\) 10987.0 0.745706 0.372853 0.927891i \(-0.378380\pi\)
0.372853 + 0.927891i \(0.378380\pi\)
\(602\) 0 0
\(603\) − 5877.00i − 0.396899i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 13200.0i − 0.882655i −0.897346 0.441327i \(-0.854508\pi\)
0.897346 0.441327i \(-0.145492\pi\)
\(608\) 0 0
\(609\) −210.000 −0.0139731
\(610\) 0 0
\(611\) −410.000 −0.0271470
\(612\) 0 0
\(613\) 21066.0i 1.38801i 0.719972 + 0.694003i \(0.244155\pi\)
−0.719972 + 0.694003i \(0.755845\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 12336.0i − 0.804909i −0.915440 0.402454i \(-0.868157\pi\)
0.915440 0.402454i \(-0.131843\pi\)
\(618\) 0 0
\(619\) −1441.00 −0.0935681 −0.0467841 0.998905i \(-0.514897\pi\)
−0.0467841 + 0.998905i \(0.514897\pi\)
\(620\) 0 0
\(621\) −1242.00 −0.0802572
\(622\) 0 0
\(623\) 5520.00i 0.354983i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 798.000i 0.0508278i
\(628\) 0 0
\(629\) −11868.0 −0.752318
\(630\) 0 0
\(631\) 9839.00 0.620736 0.310368 0.950616i \(-0.399548\pi\)
0.310368 + 0.950616i \(0.399548\pi\)
\(632\) 0 0
\(633\) − 13575.0i − 0.852382i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 318.000i 0.0197796i
\(638\) 0 0
\(639\) 7452.00 0.461340
\(640\) 0 0
\(641\) −21564.0 −1.32875 −0.664373 0.747401i \(-0.731302\pi\)
−0.664373 + 0.747401i \(0.731302\pi\)
\(642\) 0 0
\(643\) − 8604.00i − 0.527696i −0.964564 0.263848i \(-0.915008\pi\)
0.964564 0.263848i \(-0.0849918\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3444.00i 0.209270i 0.994511 + 0.104635i \(0.0333674\pi\)
−0.994511 + 0.104635i \(0.966633\pi\)
\(648\) 0 0
\(649\) 2716.00 0.164272
\(650\) 0 0
\(651\) −1995.00 −0.120108
\(652\) 0 0
\(653\) − 3518.00i − 0.210827i −0.994428 0.105413i \(-0.966383\pi\)
0.994428 0.105413i \(-0.0336166\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 5130.00i − 0.304628i
\(658\) 0 0
\(659\) −12612.0 −0.745514 −0.372757 0.927929i \(-0.621588\pi\)
−0.372757 + 0.927929i \(0.621588\pi\)
\(660\) 0 0
\(661\) 27090.0 1.59407 0.797034 0.603935i \(-0.206401\pi\)
0.797034 + 0.603935i \(0.206401\pi\)
\(662\) 0 0
\(663\) − 138.000i − 0.00808367i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 644.000i − 0.0373850i
\(668\) 0 0
\(669\) 9633.00 0.556701
\(670\) 0 0
\(671\) 238.000 0.0136928
\(672\) 0 0
\(673\) − 5442.00i − 0.311699i −0.987781 0.155850i \(-0.950188\pi\)
0.987781 0.155850i \(-0.0498115\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15226.0i 0.864376i 0.901783 + 0.432188i \(0.142258\pi\)
−0.901783 + 0.432188i \(0.857742\pi\)
\(678\) 0 0
\(679\) 4205.00 0.237663
\(680\) 0 0
\(681\) −7452.00 −0.419326
\(682\) 0 0
\(683\) 552.000i 0.0309249i 0.999880 + 0.0154624i \(0.00492204\pi\)
−0.999880 + 0.0154624i \(0.995078\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 5541.00i − 0.307718i
\(688\) 0 0
\(689\) −456.000 −0.0252137
\(690\) 0 0
\(691\) −9776.00 −0.538201 −0.269100 0.963112i \(-0.586726\pi\)
−0.269100 + 0.963112i \(0.586726\pi\)
\(692\) 0 0
\(693\) 630.000i 0.0345335i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 3864.00i − 0.209985i
\(698\) 0 0
\(699\) −3060.00 −0.165579
\(700\) 0 0
\(701\) 13066.0 0.703989 0.351994 0.936002i \(-0.385504\pi\)
0.351994 + 0.936002i \(0.385504\pi\)
\(702\) 0 0
\(703\) − 4902.00i − 0.262991i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2760.00i 0.146818i
\(708\) 0 0
\(709\) −28985.0 −1.53534 −0.767669 0.640847i \(-0.778583\pi\)
−0.767669 + 0.640847i \(0.778583\pi\)
\(710\) 0 0
\(711\) 4968.00 0.262046
\(712\) 0 0
\(713\) − 6118.00i − 0.321348i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3528.00i − 0.183760i
\(718\) 0 0
\(719\) 15722.0 0.815482 0.407741 0.913098i \(-0.366317\pi\)
0.407741 + 0.913098i \(0.366317\pi\)
\(720\) 0 0
\(721\) −1540.00 −0.0795459
\(722\) 0 0
\(723\) 20901.0i 1.07513i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 32611.0i − 1.66365i −0.555036 0.831826i \(-0.687296\pi\)
0.555036 0.831826i \(-0.312704\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 7682.00 0.388685
\(732\) 0 0
\(733\) 8358.00i 0.421159i 0.977577 + 0.210580i \(0.0675351\pi\)
−0.977577 + 0.210580i \(0.932465\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 9142.00i − 0.456920i
\(738\) 0 0
\(739\) 20604.0 1.02562 0.512808 0.858503i \(-0.328605\pi\)
0.512808 + 0.858503i \(0.328605\pi\)
\(740\) 0 0
\(741\) 57.0000 0.00282584
\(742\) 0 0
\(743\) 19476.0i 0.961649i 0.876817 + 0.480824i \(0.159663\pi\)
−0.876817 + 0.480824i \(0.840337\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1278.00i 0.0625965i
\(748\) 0 0
\(749\) −4920.00 −0.240017
\(750\) 0 0
\(751\) −3864.00 −0.187749 −0.0938744 0.995584i \(-0.529925\pi\)
−0.0938744 + 0.995584i \(0.529925\pi\)
\(752\) 0 0
\(753\) 4140.00i 0.200359i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 18871.0i 0.906048i 0.891498 + 0.453024i \(0.149655\pi\)
−0.891498 + 0.453024i \(0.850345\pi\)
\(758\) 0 0
\(759\) −1932.00 −0.0923941
\(760\) 0 0
\(761\) −36372.0 −1.73257 −0.866284 0.499552i \(-0.833498\pi\)
−0.866284 + 0.499552i \(0.833498\pi\)
\(762\) 0 0
\(763\) 9215.00i 0.437229i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 194.000i − 0.00913290i
\(768\) 0 0
\(769\) −4603.00 −0.215850 −0.107925 0.994159i \(-0.534421\pi\)
−0.107925 + 0.994159i \(0.534421\pi\)
\(770\) 0 0
\(771\) 20772.0 0.970279
\(772\) 0 0
\(773\) − 36.0000i − 0.00167507i −1.00000 0.000837536i \(-0.999733\pi\)
1.00000 0.000837536i \(-0.000266596\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3870.00i − 0.178681i
\(778\) 0 0
\(779\) 1596.00 0.0734052
\(780\) 0 0
\(781\) 11592.0 0.531107
\(782\) 0 0
\(783\) − 378.000i − 0.0172524i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 20281.0i − 0.918602i −0.888281 0.459301i \(-0.848100\pi\)
0.888281 0.459301i \(-0.151900\pi\)
\(788\) 0 0
\(789\) −5652.00 −0.255027
\(790\) 0 0
\(791\) −4380.00 −0.196884
\(792\) 0 0
\(793\) − 17.0000i 0 0.000761271i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 37524.0i 1.66771i 0.551980 + 0.833857i \(0.313872\pi\)
−0.551980 + 0.833857i \(0.686128\pi\)
\(798\) 0 0
\(799\) 18860.0 0.835067
\(800\) 0 0
\(801\) −9936.00 −0.438291
\(802\) 0 0
\(803\) − 7980.00i − 0.350695i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10830.0i 0.472409i
\(808\) 0 0
\(809\) −31224.0 −1.35696 −0.678478 0.734621i \(-0.737360\pi\)
−0.678478 + 0.734621i \(0.737360\pi\)
\(810\) 0 0
\(811\) −32579.0 −1.41061 −0.705304 0.708905i \(-0.749190\pi\)
−0.705304 + 0.708905i \(0.749190\pi\)
\(812\) 0 0
\(813\) − 18216.0i − 0.785809i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3173.00i 0.135874i
\(818\) 0 0
\(819\) 45.0000 0.00191994
\(820\) 0 0
\(821\) −19810.0 −0.842112 −0.421056 0.907035i \(-0.638340\pi\)
−0.421056 + 0.907035i \(0.638340\pi\)
\(822\) 0 0
\(823\) − 10273.0i − 0.435108i −0.976048 0.217554i \(-0.930192\pi\)
0.976048 0.217554i \(-0.0698079\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16656.0i 0.700346i 0.936685 + 0.350173i \(0.113877\pi\)
−0.936685 + 0.350173i \(0.886123\pi\)
\(828\) 0 0
\(829\) 4790.00 0.200680 0.100340 0.994953i \(-0.468007\pi\)
0.100340 + 0.994953i \(0.468007\pi\)
\(830\) 0 0
\(831\) 8409.00 0.351029
\(832\) 0 0
\(833\) − 14628.0i − 0.608440i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3591.00i − 0.148295i
\(838\) 0 0
\(839\) 7414.00 0.305077 0.152539 0.988298i \(-0.451255\pi\)
0.152539 + 0.988298i \(0.451255\pi\)
\(840\) 0 0
\(841\) −24193.0 −0.991964
\(842\) 0 0
\(843\) 20082.0i 0.820475i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 5675.00i − 0.230219i
\(848\) 0 0
\(849\) 19443.0 0.785962
\(850\) 0 0
\(851\) 11868.0 0.478061
\(852\) 0 0
\(853\) 30155.0i 1.21042i 0.796066 + 0.605210i \(0.206911\pi\)
−0.796066 + 0.605210i \(0.793089\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8244.00i 0.328599i 0.986410 + 0.164300i \(0.0525364\pi\)
−0.986410 + 0.164300i \(0.947464\pi\)
\(858\) 0 0
\(859\) 17552.0 0.697167 0.348584 0.937278i \(-0.386663\pi\)
0.348584 + 0.937278i \(0.386663\pi\)
\(860\) 0 0
\(861\) 1260.00 0.0498730
\(862\) 0 0
\(863\) 34104.0i 1.34521i 0.740003 + 0.672604i \(0.234824\pi\)
−0.740003 + 0.672604i \(0.765176\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 8391.00i − 0.328689i
\(868\) 0 0
\(869\) 7728.00 0.301674
\(870\) 0 0
\(871\) −653.000 −0.0254031
\(872\) 0 0
\(873\) 7569.00i 0.293439i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46229.0i 1.77998i 0.455980 + 0.889990i \(0.349289\pi\)
−0.455980 + 0.889990i \(0.650711\pi\)
\(878\) 0 0
\(879\) 9042.00 0.346961
\(880\) 0 0
\(881\) −22440.0 −0.858142 −0.429071 0.903271i \(-0.641159\pi\)
−0.429071 + 0.903271i \(0.641159\pi\)
\(882\) 0 0
\(883\) 17143.0i 0.653350i 0.945137 + 0.326675i \(0.105928\pi\)
−0.945137 + 0.326675i \(0.894072\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 23626.0i − 0.894344i −0.894448 0.447172i \(-0.852431\pi\)
0.894448 0.447172i \(-0.147569\pi\)
\(888\) 0 0
\(889\) −11880.0 −0.448192
\(890\) 0 0
\(891\) −1134.00 −0.0426380
\(892\) 0 0
\(893\) 7790.00i 0.291918i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 138.000i 0.00513677i
\(898\) 0 0
\(899\) 1862.00 0.0690781
\(900\) 0 0
\(901\) 20976.0 0.775596
\(902\) 0 0
\(903\) 2505.00i 0.0923158i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 11268.0i − 0.412511i −0.978498 0.206256i \(-0.933872\pi\)
0.978498 0.206256i \(-0.0661279\pi\)
\(908\) 0 0
\(909\) −4968.00 −0.181274
\(910\) 0 0
\(911\) −10046.0 −0.365355 −0.182678 0.983173i \(-0.558476\pi\)
−0.182678 + 0.983173i \(0.558476\pi\)
\(912\) 0 0
\(913\) 1988.00i 0.0720626i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 5280.00i − 0.190143i
\(918\) 0 0
\(919\) 15359.0 0.551302 0.275651 0.961258i \(-0.411107\pi\)
0.275651 + 0.961258i \(0.411107\pi\)
\(920\) 0 0
\(921\) −16107.0 −0.576269
\(922\) 0 0
\(923\) − 828.000i − 0.0295276i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 2772.00i − 0.0982141i
\(928\) 0 0
\(929\) 39790.0 1.40524 0.702620 0.711565i \(-0.252014\pi\)
0.702620 + 0.711565i \(0.252014\pi\)
\(930\) 0 0
\(931\) 6042.00 0.212694
\(932\) 0 0
\(933\) 14538.0i 0.510132i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17009.0i 0.593020i 0.955030 + 0.296510i \(0.0958228\pi\)
−0.955030 + 0.296510i \(0.904177\pi\)
\(938\) 0 0
\(939\) −2271.00 −0.0789258
\(940\) 0 0
\(941\) −11674.0 −0.404422 −0.202211 0.979342i \(-0.564813\pi\)
−0.202211 + 0.979342i \(0.564813\pi\)
\(942\) 0 0
\(943\) 3864.00i 0.133435i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6026.00i 0.206778i 0.994641 + 0.103389i \(0.0329686\pi\)
−0.994641 + 0.103389i \(0.967031\pi\)
\(948\) 0 0
\(949\) −570.000 −0.0194973
\(950\) 0 0
\(951\) 22896.0 0.780708
\(952\) 0 0
\(953\) − 14088.0i − 0.478862i −0.970913 0.239431i \(-0.923039\pi\)
0.970913 0.239431i \(-0.0769608\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 588.000i − 0.0198614i
\(958\) 0 0
\(959\) −3890.00 −0.130985
\(960\) 0 0
\(961\) −12102.0 −0.406230
\(962\) 0 0
\(963\) − 8856.00i − 0.296345i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 11208.0i − 0.372725i −0.982481 0.186362i \(-0.940330\pi\)
0.982481 0.186362i \(-0.0596699\pi\)
\(968\) 0 0
\(969\) −2622.00 −0.0869255
\(970\) 0 0
\(971\) 26054.0 0.861084 0.430542 0.902571i \(-0.358322\pi\)
0.430542 + 0.902571i \(0.358322\pi\)
\(972\) 0 0
\(973\) 8460.00i 0.278741i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26870.0i 0.879885i 0.898026 + 0.439942i \(0.145001\pi\)
−0.898026 + 0.439942i \(0.854999\pi\)
\(978\) 0 0
\(979\) −15456.0 −0.504572
\(980\) 0 0
\(981\) −16587.0 −0.539839
\(982\) 0 0
\(983\) 23388.0i 0.758862i 0.925220 + 0.379431i \(0.123880\pi\)
−0.925220 + 0.379431i \(0.876120\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6150.00i 0.198335i
\(988\) 0 0
\(989\) −7682.00 −0.246990
\(990\) 0 0
\(991\) −17345.0 −0.555986 −0.277993 0.960583i \(-0.589669\pi\)
−0.277993 + 0.960583i \(0.589669\pi\)
\(992\) 0 0
\(993\) − 20340.0i − 0.650021i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25998.0i 0.825842i 0.910767 + 0.412921i \(0.135492\pi\)
−0.910767 + 0.412921i \(0.864508\pi\)
\(998\) 0 0
\(999\) 6966.00 0.220615
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 1200.4.f.i.49.1 2
4.3 odd 2 600.4.f.e.49.2 2
5.2 odd 4 1200.4.a.g.1.1 1
5.3 odd 4 1200.4.a.bd.1.1 1
5.4 even 2 inner 1200.4.f.i.49.2 2
12.11 even 2 1800.4.f.l.649.1 2
20.3 even 4 600.4.a.e.1.1 1
20.7 even 4 600.4.a.n.1.1 yes 1
20.19 odd 2 600.4.f.e.49.1 2
60.23 odd 4 1800.4.a.m.1.1 1
60.47 odd 4 1800.4.a.v.1.1 1
60.59 even 2 1800.4.f.l.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.e.1.1 1 20.3 even 4
600.4.a.n.1.1 yes 1 20.7 even 4
600.4.f.e.49.1 2 20.19 odd 2
600.4.f.e.49.2 2 4.3 odd 2
1200.4.a.g.1.1 1 5.2 odd 4
1200.4.a.bd.1.1 1 5.3 odd 4
1200.4.f.i.49.1 2 1.1 even 1 trivial
1200.4.f.i.49.2 2 5.4 even 2 inner
1800.4.a.m.1.1 1 60.23 odd 4
1800.4.a.v.1.1 1 60.47 odd 4
1800.4.f.l.649.1 2 12.11 even 2
1800.4.f.l.649.2 2 60.59 even 2