Properties

Label 1200.4.f.s.49.2
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 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +54.0000 q^{11} +55.0000i q^{13} +18.0000i q^{17} -25.0000 q^{19} -21.0000 q^{21} -18.0000i q^{23} -27.0000i q^{27} +54.0000 q^{29} +271.000 q^{31} +162.000i q^{33} +314.000i q^{37} -165.000 q^{39} -360.000 q^{41} -163.000i q^{43} -522.000i q^{47} +294.000 q^{49} -54.0000 q^{51} +36.0000i q^{53} -75.0000i q^{57} +126.000 q^{59} +47.0000 q^{61} -63.0000i q^{63} +343.000i q^{67} +54.0000 q^{69} +1080.00 q^{71} +1054.00i q^{73} +378.000i q^{77} -568.000 q^{79} +81.0000 q^{81} +1422.00i q^{83} +162.000i q^{87} -1440.00 q^{89} -385.000 q^{91} +813.000i q^{93} -439.000i q^{97} -486.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 108 q^{11} - 50 q^{19} - 42 q^{21} + 108 q^{29} + 542 q^{31} - 330 q^{39} - 720 q^{41} + 588 q^{49} - 108 q^{51} + 252 q^{59} + 94 q^{61} + 108 q^{69} + 2160 q^{71} - 1136 q^{79} + 162 q^{81} - 2880 q^{89} - 770 q^{91} - 972 q^{99}+O(q^{100}) \) 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
\(7\) 7.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 54.0000 1.48015 0.740073 0.672526i \(-0.234791\pi\)
0.740073 + 0.672526i \(0.234791\pi\)
\(12\) 0 0
\(13\) 55.0000i 1.17340i 0.809803 + 0.586702i \(0.199574\pi\)
−0.809803 + 0.586702i \(0.800426\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 18.0000i 0.256802i 0.991722 + 0.128401i \(0.0409845\pi\)
−0.991722 + 0.128401i \(0.959015\pi\)
\(18\) 0 0
\(19\) −25.0000 −0.301863 −0.150931 0.988544i \(-0.548227\pi\)
−0.150931 + 0.988544i \(0.548227\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) − 18.0000i − 0.163185i −0.996666 0.0815926i \(-0.973999\pi\)
0.996666 0.0815926i \(-0.0260006\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) 54.0000 0.345778 0.172889 0.984941i \(-0.444690\pi\)
0.172889 + 0.984941i \(0.444690\pi\)
\(30\) 0 0
\(31\) 271.000 1.57010 0.785049 0.619434i \(-0.212638\pi\)
0.785049 + 0.619434i \(0.212638\pi\)
\(32\) 0 0
\(33\) 162.000i 0.854563i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 314.000i 1.39517i 0.716502 + 0.697585i \(0.245742\pi\)
−0.716502 + 0.697585i \(0.754258\pi\)
\(38\) 0 0
\(39\) −165.000 −0.677465
\(40\) 0 0
\(41\) −360.000 −1.37128 −0.685641 0.727940i \(-0.740478\pi\)
−0.685641 + 0.727940i \(0.740478\pi\)
\(42\) 0 0
\(43\) − 163.000i − 0.578076i −0.957318 0.289038i \(-0.906665\pi\)
0.957318 0.289038i \(-0.0933354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 522.000i − 1.62003i −0.586407 0.810016i \(-0.699458\pi\)
0.586407 0.810016i \(-0.300542\pi\)
\(48\) 0 0
\(49\) 294.000 0.857143
\(50\) 0 0
\(51\) −54.0000 −0.148265
\(52\) 0 0
\(53\) 36.0000i 0.0933015i 0.998911 + 0.0466508i \(0.0148548\pi\)
−0.998911 + 0.0466508i \(0.985145\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 75.0000i − 0.174281i
\(58\) 0 0
\(59\) 126.000 0.278031 0.139015 0.990290i \(-0.455606\pi\)
0.139015 + 0.990290i \(0.455606\pi\)
\(60\) 0 0
\(61\) 47.0000 0.0986514 0.0493257 0.998783i \(-0.484293\pi\)
0.0493257 + 0.998783i \(0.484293\pi\)
\(62\) 0 0
\(63\) − 63.0000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 343.000i 0.625435i 0.949846 + 0.312717i \(0.101239\pi\)
−0.949846 + 0.312717i \(0.898761\pi\)
\(68\) 0 0
\(69\) 54.0000 0.0942150
\(70\) 0 0
\(71\) 1080.00 1.80525 0.902623 0.430433i \(-0.141639\pi\)
0.902623 + 0.430433i \(0.141639\pi\)
\(72\) 0 0
\(73\) 1054.00i 1.68988i 0.534860 + 0.844941i \(0.320364\pi\)
−0.534860 + 0.844941i \(0.679636\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 378.000i 0.559443i
\(78\) 0 0
\(79\) −568.000 −0.808924 −0.404462 0.914555i \(-0.632541\pi\)
−0.404462 + 0.914555i \(0.632541\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1422.00i 1.88054i 0.340430 + 0.940270i \(0.389427\pi\)
−0.340430 + 0.940270i \(0.610573\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 162.000i 0.199635i
\(88\) 0 0
\(89\) −1440.00 −1.71505 −0.857526 0.514440i \(-0.828000\pi\)
−0.857526 + 0.514440i \(0.828000\pi\)
\(90\) 0 0
\(91\) −385.000 −0.443505
\(92\) 0 0
\(93\) 813.000i 0.906496i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 439.000i − 0.459523i −0.973247 0.229761i \(-0.926205\pi\)
0.973247 0.229761i \(-0.0737946\pi\)
\(98\) 0 0
\(99\) −486.000 −0.493382
\(100\) 0 0
\(101\) 828.000 0.815733 0.407867 0.913041i \(-0.366273\pi\)
0.407867 + 0.913041i \(0.366273\pi\)
\(102\) 0 0
\(103\) 548.000i 0.524233i 0.965036 + 0.262117i \(0.0844205\pi\)
−0.965036 + 0.262117i \(0.915579\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1476.00i 1.33355i 0.745257 + 0.666777i \(0.232327\pi\)
−0.745257 + 0.666777i \(0.767673\pi\)
\(108\) 0 0
\(109\) −1277.00 −1.12215 −0.561075 0.827765i \(-0.689612\pi\)
−0.561075 + 0.827765i \(0.689612\pi\)
\(110\) 0 0
\(111\) −942.000 −0.805502
\(112\) 0 0
\(113\) − 1836.00i − 1.52846i −0.644942 0.764232i \(-0.723118\pi\)
0.644942 0.764232i \(-0.276882\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 495.000i − 0.391135i
\(118\) 0 0
\(119\) −126.000 −0.0970622
\(120\) 0 0
\(121\) 1585.00 1.19083
\(122\) 0 0
\(123\) − 1080.00i − 0.791710i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 592.000i 0.413634i 0.978380 + 0.206817i \(0.0663105\pi\)
−0.978380 + 0.206817i \(0.933690\pi\)
\(128\) 0 0
\(129\) 489.000 0.333752
\(130\) 0 0
\(131\) 468.000 0.312132 0.156066 0.987747i \(-0.450119\pi\)
0.156066 + 0.987747i \(0.450119\pi\)
\(132\) 0 0
\(133\) − 175.000i − 0.114093i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2574.00i 1.60519i 0.596521 + 0.802597i \(0.296549\pi\)
−0.596521 + 0.802597i \(0.703451\pi\)
\(138\) 0 0
\(139\) −1756.00 −1.07153 −0.535763 0.844369i \(-0.679976\pi\)
−0.535763 + 0.844369i \(0.679976\pi\)
\(140\) 0 0
\(141\) 1566.00 0.935326
\(142\) 0 0
\(143\) 2970.00i 1.73681i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 882.000i 0.494872i
\(148\) 0 0
\(149\) −2682.00 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) −3395.00 −1.82968 −0.914838 0.403820i \(-0.867682\pi\)
−0.914838 + 0.403820i \(0.867682\pi\)
\(152\) 0 0
\(153\) − 162.000i − 0.0856008i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1549.00i − 0.787412i −0.919236 0.393706i \(-0.871193\pi\)
0.919236 0.393706i \(-0.128807\pi\)
\(158\) 0 0
\(159\) −108.000 −0.0538677
\(160\) 0 0
\(161\) 126.000 0.0616782
\(162\) 0 0
\(163\) − 505.000i − 0.242667i −0.992612 0.121333i \(-0.961283\pi\)
0.992612 0.121333i \(-0.0387170\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1476.00i 0.683930i 0.939713 + 0.341965i \(0.111092\pi\)
−0.939713 + 0.341965i \(0.888908\pi\)
\(168\) 0 0
\(169\) −828.000 −0.376878
\(170\) 0 0
\(171\) 225.000 0.100621
\(172\) 0 0
\(173\) − 2358.00i − 1.03627i −0.855298 0.518137i \(-0.826626\pi\)
0.855298 0.518137i \(-0.173374\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 378.000i 0.160521i
\(178\) 0 0
\(179\) 1566.00 0.653901 0.326951 0.945041i \(-0.393979\pi\)
0.326951 + 0.945041i \(0.393979\pi\)
\(180\) 0 0
\(181\) 305.000 0.125251 0.0626256 0.998037i \(-0.480053\pi\)
0.0626256 + 0.998037i \(0.480053\pi\)
\(182\) 0 0
\(183\) 141.000i 0.0569564i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 972.000i 0.380105i
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 1746.00 0.661446 0.330723 0.943728i \(-0.392707\pi\)
0.330723 + 0.943728i \(0.392707\pi\)
\(192\) 0 0
\(193\) 3877.00i 1.44597i 0.690863 + 0.722986i \(0.257231\pi\)
−0.690863 + 0.722986i \(0.742769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2142.00i 0.774676i 0.921938 + 0.387338i \(0.126605\pi\)
−0.921938 + 0.387338i \(0.873395\pi\)
\(198\) 0 0
\(199\) −4033.00 −1.43664 −0.718321 0.695712i \(-0.755089\pi\)
−0.718321 + 0.695712i \(0.755089\pi\)
\(200\) 0 0
\(201\) −1029.00 −0.361095
\(202\) 0 0
\(203\) 378.000i 0.130692i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 162.000i 0.0543951i
\(208\) 0 0
\(209\) −1350.00 −0.446801
\(210\) 0 0
\(211\) 4105.00 1.33934 0.669668 0.742661i \(-0.266436\pi\)
0.669668 + 0.742661i \(0.266436\pi\)
\(212\) 0 0
\(213\) 3240.00i 1.04226i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1897.00i 0.593441i
\(218\) 0 0
\(219\) −3162.00 −0.975654
\(220\) 0 0
\(221\) −990.000 −0.301333
\(222\) 0 0
\(223\) 1385.00i 0.415903i 0.978139 + 0.207952i \(0.0666797\pi\)
−0.978139 + 0.207952i \(0.933320\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2520.00i 0.736821i 0.929663 + 0.368410i \(0.120098\pi\)
−0.929663 + 0.368410i \(0.879902\pi\)
\(228\) 0 0
\(229\) −5129.00 −1.48006 −0.740030 0.672574i \(-0.765189\pi\)
−0.740030 + 0.672574i \(0.765189\pi\)
\(230\) 0 0
\(231\) −1134.00 −0.322994
\(232\) 0 0
\(233\) 3240.00i 0.910985i 0.890240 + 0.455492i \(0.150537\pi\)
−0.890240 + 0.455492i \(0.849463\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 1704.00i − 0.467032i
\(238\) 0 0
\(239\) −2988.00 −0.808693 −0.404347 0.914606i \(-0.632501\pi\)
−0.404347 + 0.914606i \(0.632501\pi\)
\(240\) 0 0
\(241\) −2647.00 −0.707503 −0.353752 0.935339i \(-0.615094\pi\)
−0.353752 + 0.935339i \(0.615094\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1375.00i − 0.354207i
\(248\) 0 0
\(249\) −4266.00 −1.08573
\(250\) 0 0
\(251\) 4212.00 1.05920 0.529600 0.848248i \(-0.322342\pi\)
0.529600 + 0.848248i \(0.322342\pi\)
\(252\) 0 0
\(253\) − 972.000i − 0.241538i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5724.00i 1.38931i 0.719342 + 0.694656i \(0.244444\pi\)
−0.719342 + 0.694656i \(0.755556\pi\)
\(258\) 0 0
\(259\) −2198.00 −0.527325
\(260\) 0 0
\(261\) −486.000 −0.115259
\(262\) 0 0
\(263\) − 4608.00i − 1.08039i −0.841541 0.540193i \(-0.818351\pi\)
0.841541 0.540193i \(-0.181649\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4320.00i − 0.990186i
\(268\) 0 0
\(269\) 6426.00 1.45651 0.728253 0.685308i \(-0.240333\pi\)
0.728253 + 0.685308i \(0.240333\pi\)
\(270\) 0 0
\(271\) 3376.00 0.756743 0.378372 0.925654i \(-0.376484\pi\)
0.378372 + 0.925654i \(0.376484\pi\)
\(272\) 0 0
\(273\) − 1155.00i − 0.256058i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5381.00i 1.16719i 0.812043 + 0.583597i \(0.198355\pi\)
−0.812043 + 0.583597i \(0.801645\pi\)
\(278\) 0 0
\(279\) −2439.00 −0.523366
\(280\) 0 0
\(281\) −3474.00 −0.737514 −0.368757 0.929526i \(-0.620217\pi\)
−0.368757 + 0.929526i \(0.620217\pi\)
\(282\) 0 0
\(283\) − 2269.00i − 0.476601i −0.971191 0.238300i \(-0.923410\pi\)
0.971191 0.238300i \(-0.0765903\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2520.00i − 0.518296i
\(288\) 0 0
\(289\) 4589.00 0.934053
\(290\) 0 0
\(291\) 1317.00 0.265306
\(292\) 0 0
\(293\) 1674.00i 0.333775i 0.985976 + 0.166888i \(0.0533717\pi\)
−0.985976 + 0.166888i \(0.946628\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 1458.00i − 0.284854i
\(298\) 0 0
\(299\) 990.000 0.191482
\(300\) 0 0
\(301\) 1141.00 0.218492
\(302\) 0 0
\(303\) 2484.00i 0.470964i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 539.000i − 0.100203i −0.998744 0.0501016i \(-0.984046\pi\)
0.998744 0.0501016i \(-0.0159545\pi\)
\(308\) 0 0
\(309\) −1644.00 −0.302666
\(310\) 0 0
\(311\) −1494.00 −0.272402 −0.136201 0.990681i \(-0.543489\pi\)
−0.136201 + 0.990681i \(0.543489\pi\)
\(312\) 0 0
\(313\) 3997.00i 0.721801i 0.932604 + 0.360901i \(0.117531\pi\)
−0.932604 + 0.360901i \(0.882469\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 3672.00i − 0.650600i −0.945611 0.325300i \(-0.894535\pi\)
0.945611 0.325300i \(-0.105465\pi\)
\(318\) 0 0
\(319\) 2916.00 0.511801
\(320\) 0 0
\(321\) −4428.00 −0.769928
\(322\) 0 0
\(323\) − 450.000i − 0.0775191i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 3831.00i − 0.647874i
\(328\) 0 0
\(329\) 3654.00 0.612315
\(330\) 0 0
\(331\) −1052.00 −0.174692 −0.0873461 0.996178i \(-0.527839\pi\)
−0.0873461 + 0.996178i \(0.527839\pi\)
\(332\) 0 0
\(333\) − 2826.00i − 0.465057i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3871.00i − 0.625718i −0.949800 0.312859i \(-0.898713\pi\)
0.949800 0.312859i \(-0.101287\pi\)
\(338\) 0 0
\(339\) 5508.00 0.882459
\(340\) 0 0
\(341\) 14634.0 2.32398
\(342\) 0 0
\(343\) 4459.00i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7686.00i 1.18907i 0.804071 + 0.594533i \(0.202663\pi\)
−0.804071 + 0.594533i \(0.797337\pi\)
\(348\) 0 0
\(349\) 46.0000 0.00705537 0.00352768 0.999994i \(-0.498877\pi\)
0.00352768 + 0.999994i \(0.498877\pi\)
\(350\) 0 0
\(351\) 1485.00 0.225822
\(352\) 0 0
\(353\) 6714.00i 1.01232i 0.862439 + 0.506162i \(0.168936\pi\)
−0.862439 + 0.506162i \(0.831064\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 378.000i − 0.0560389i
\(358\) 0 0
\(359\) 1296.00 0.190530 0.0952650 0.995452i \(-0.469630\pi\)
0.0952650 + 0.995452i \(0.469630\pi\)
\(360\) 0 0
\(361\) −6234.00 −0.908879
\(362\) 0 0
\(363\) 4755.00i 0.687528i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 5903.00i − 0.839602i −0.907616 0.419801i \(-0.862100\pi\)
0.907616 0.419801i \(-0.137900\pi\)
\(368\) 0 0
\(369\) 3240.00 0.457094
\(370\) 0 0
\(371\) −252.000 −0.0352647
\(372\) 0 0
\(373\) − 8867.00i − 1.23087i −0.788186 0.615437i \(-0.788980\pi\)
0.788186 0.615437i \(-0.211020\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2970.00i 0.405737i
\(378\) 0 0
\(379\) 8837.00 1.19769 0.598847 0.800863i \(-0.295626\pi\)
0.598847 + 0.800863i \(0.295626\pi\)
\(380\) 0 0
\(381\) −1776.00 −0.238812
\(382\) 0 0
\(383\) − 10044.0i − 1.34001i −0.742356 0.670006i \(-0.766292\pi\)
0.742356 0.670006i \(-0.233708\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1467.00i 0.192692i
\(388\) 0 0
\(389\) 2736.00 0.356609 0.178304 0.983975i \(-0.442939\pi\)
0.178304 + 0.983975i \(0.442939\pi\)
\(390\) 0 0
\(391\) 324.000 0.0419064
\(392\) 0 0
\(393\) 1404.00i 0.180210i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 12241.0i − 1.54750i −0.633490 0.773751i \(-0.718378\pi\)
0.633490 0.773751i \(-0.281622\pi\)
\(398\) 0 0
\(399\) 525.000 0.0658719
\(400\) 0 0
\(401\) 9036.00 1.12528 0.562639 0.826703i \(-0.309786\pi\)
0.562639 + 0.826703i \(0.309786\pi\)
\(402\) 0 0
\(403\) 14905.0i 1.84236i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16956.0i 2.06506i
\(408\) 0 0
\(409\) −8549.00 −1.03355 −0.516774 0.856122i \(-0.672867\pi\)
−0.516774 + 0.856122i \(0.672867\pi\)
\(410\) 0 0
\(411\) −7722.00 −0.926760
\(412\) 0 0
\(413\) 882.000i 0.105086i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5268.00i − 0.618645i
\(418\) 0 0
\(419\) 1548.00 0.180489 0.0902443 0.995920i \(-0.471235\pi\)
0.0902443 + 0.995920i \(0.471235\pi\)
\(420\) 0 0
\(421\) 6110.00 0.707323 0.353662 0.935373i \(-0.384936\pi\)
0.353662 + 0.935373i \(0.384936\pi\)
\(422\) 0 0
\(423\) 4698.00i 0.540011i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 329.000i 0.0372867i
\(428\) 0 0
\(429\) −8910.00 −1.00275
\(430\) 0 0
\(431\) −5958.00 −0.665863 −0.332931 0.942951i \(-0.608038\pi\)
−0.332931 + 0.942951i \(0.608038\pi\)
\(432\) 0 0
\(433\) − 7163.00i − 0.794993i −0.917604 0.397496i \(-0.869879\pi\)
0.917604 0.397496i \(-0.130121\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 450.000i 0.0492595i
\(438\) 0 0
\(439\) 17.0000 0.00184821 0.000924107 1.00000i \(-0.499706\pi\)
0.000924107 1.00000i \(0.499706\pi\)
\(440\) 0 0
\(441\) −2646.00 −0.285714
\(442\) 0 0
\(443\) − 9432.00i − 1.01158i −0.862658 0.505788i \(-0.831202\pi\)
0.862658 0.505788i \(-0.168798\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 8046.00i − 0.851371i
\(448\) 0 0
\(449\) 15228.0 1.60057 0.800283 0.599623i \(-0.204683\pi\)
0.800283 + 0.599623i \(0.204683\pi\)
\(450\) 0 0
\(451\) −19440.0 −2.02970
\(452\) 0 0
\(453\) − 10185.0i − 1.05636i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 250.000i − 0.0255897i −0.999918 0.0127949i \(-0.995927\pi\)
0.999918 0.0127949i \(-0.00407284\pi\)
\(458\) 0 0
\(459\) 486.000 0.0494217
\(460\) 0 0
\(461\) 16956.0 1.71306 0.856529 0.516099i \(-0.172616\pi\)
0.856529 + 0.516099i \(0.172616\pi\)
\(462\) 0 0
\(463\) − 4384.00i − 0.440047i −0.975495 0.220023i \(-0.929387\pi\)
0.975495 0.220023i \(-0.0706134\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5166.00i 0.511893i 0.966691 + 0.255946i \(0.0823871\pi\)
−0.966691 + 0.255946i \(0.917613\pi\)
\(468\) 0 0
\(469\) −2401.00 −0.236392
\(470\) 0 0
\(471\) 4647.00 0.454612
\(472\) 0 0
\(473\) − 8802.00i − 0.855637i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 324.000i − 0.0311005i
\(478\) 0 0
\(479\) 6966.00 0.664477 0.332239 0.943195i \(-0.392196\pi\)
0.332239 + 0.943195i \(0.392196\pi\)
\(480\) 0 0
\(481\) −17270.0 −1.63710
\(482\) 0 0
\(483\) 378.000i 0.0356099i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 18431.0i − 1.71497i −0.514512 0.857483i \(-0.672027\pi\)
0.514512 0.857483i \(-0.327973\pi\)
\(488\) 0 0
\(489\) 1515.00 0.140104
\(490\) 0 0
\(491\) −1224.00 −0.112502 −0.0562509 0.998417i \(-0.517915\pi\)
−0.0562509 + 0.998417i \(0.517915\pi\)
\(492\) 0 0
\(493\) 972.000i 0.0887965i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7560.00i 0.682319i
\(498\) 0 0
\(499\) −2449.00 −0.219704 −0.109852 0.993948i \(-0.535038\pi\)
−0.109852 + 0.993948i \(0.535038\pi\)
\(500\) 0 0
\(501\) −4428.00 −0.394867
\(502\) 0 0
\(503\) − 3312.00i − 0.293588i −0.989167 0.146794i \(-0.953105\pi\)
0.989167 0.146794i \(-0.0468954\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2484.00i − 0.217590i
\(508\) 0 0
\(509\) 9162.00 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −7378.00 −0.638715
\(512\) 0 0
\(513\) 675.000i 0.0580935i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 28188.0i − 2.39789i
\(518\) 0 0
\(519\) 7074.00 0.598293
\(520\) 0 0
\(521\) −5418.00 −0.455599 −0.227799 0.973708i \(-0.573153\pi\)
−0.227799 + 0.973708i \(0.573153\pi\)
\(522\) 0 0
\(523\) − 6829.00i − 0.570959i −0.958385 0.285479i \(-0.907847\pi\)
0.958385 0.285479i \(-0.0921528\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4878.00i 0.403205i
\(528\) 0 0
\(529\) 11843.0 0.973371
\(530\) 0 0
\(531\) −1134.00 −0.0926769
\(532\) 0 0
\(533\) − 19800.0i − 1.60907i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4698.00i 0.377530i
\(538\) 0 0
\(539\) 15876.0 1.26870
\(540\) 0 0
\(541\) 3053.00 0.242622 0.121311 0.992615i \(-0.461290\pi\)
0.121311 + 0.992615i \(0.461290\pi\)
\(542\) 0 0
\(543\) 915.000i 0.0723138i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 20456.0i − 1.59897i −0.600687 0.799484i \(-0.705106\pi\)
0.600687 0.799484i \(-0.294894\pi\)
\(548\) 0 0
\(549\) −423.000 −0.0328838
\(550\) 0 0
\(551\) −1350.00 −0.104377
\(552\) 0 0
\(553\) − 3976.00i − 0.305745i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 10854.0i − 0.825671i −0.910806 0.412835i \(-0.864538\pi\)
0.910806 0.412835i \(-0.135462\pi\)
\(558\) 0 0
\(559\) 8965.00 0.678317
\(560\) 0 0
\(561\) −2916.00 −0.219454
\(562\) 0 0
\(563\) − 24930.0i − 1.86621i −0.359609 0.933103i \(-0.617090\pi\)
0.359609 0.933103i \(-0.382910\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 567.000i 0.0419961i
\(568\) 0 0
\(569\) 24786.0 1.82616 0.913078 0.407784i \(-0.133698\pi\)
0.913078 + 0.407784i \(0.133698\pi\)
\(570\) 0 0
\(571\) 14785.0 1.08360 0.541798 0.840509i \(-0.317744\pi\)
0.541798 + 0.840509i \(0.317744\pi\)
\(572\) 0 0
\(573\) 5238.00i 0.381886i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 15851.0i 1.14365i 0.820376 + 0.571825i \(0.193764\pi\)
−0.820376 + 0.571825i \(0.806236\pi\)
\(578\) 0 0
\(579\) −11631.0 −0.834832
\(580\) 0 0
\(581\) −9954.00 −0.710777
\(582\) 0 0
\(583\) 1944.00i 0.138100i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 23148.0i − 1.62763i −0.581122 0.813816i \(-0.697386\pi\)
0.581122 0.813816i \(-0.302614\pi\)
\(588\) 0 0
\(589\) −6775.00 −0.473954
\(590\) 0 0
\(591\) −6426.00 −0.447259
\(592\) 0 0
\(593\) 21888.0i 1.51574i 0.652407 + 0.757869i \(0.273759\pi\)
−0.652407 + 0.757869i \(0.726241\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 12099.0i − 0.829446i
\(598\) 0 0
\(599\) 10764.0 0.734232 0.367116 0.930175i \(-0.380345\pi\)
0.367116 + 0.930175i \(0.380345\pi\)
\(600\) 0 0
\(601\) −25597.0 −1.73731 −0.868655 0.495417i \(-0.835015\pi\)
−0.868655 + 0.495417i \(0.835015\pi\)
\(602\) 0 0
\(603\) − 3087.00i − 0.208478i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24976.0i 1.67009i 0.550182 + 0.835045i \(0.314558\pi\)
−0.550182 + 0.835045i \(0.685442\pi\)
\(608\) 0 0
\(609\) −1134.00 −0.0754548
\(610\) 0 0
\(611\) 28710.0 1.90095
\(612\) 0 0
\(613\) 2134.00i 0.140606i 0.997526 + 0.0703030i \(0.0223966\pi\)
−0.997526 + 0.0703030i \(0.977603\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13932.0i 0.909046i 0.890735 + 0.454523i \(0.150190\pi\)
−0.890735 + 0.454523i \(0.849810\pi\)
\(618\) 0 0
\(619\) −10429.0 −0.677184 −0.338592 0.940933i \(-0.609951\pi\)
−0.338592 + 0.940933i \(0.609951\pi\)
\(620\) 0 0
\(621\) −486.000 −0.0314050
\(622\) 0 0
\(623\) − 10080.0i − 0.648229i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4050.00i − 0.257961i
\(628\) 0 0
\(629\) −5652.00 −0.358283
\(630\) 0 0
\(631\) 6283.00 0.396390 0.198195 0.980163i \(-0.436492\pi\)
0.198195 + 0.980163i \(0.436492\pi\)
\(632\) 0 0
\(633\) 12315.0i 0.773266i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 16170.0i 1.00578i
\(638\) 0 0
\(639\) −9720.00 −0.601748
\(640\) 0 0
\(641\) −20916.0 −1.28882 −0.644409 0.764681i \(-0.722897\pi\)
−0.644409 + 0.764681i \(0.722897\pi\)
\(642\) 0 0
\(643\) − 23452.0i − 1.43835i −0.694831 0.719173i \(-0.744521\pi\)
0.694831 0.719173i \(-0.255479\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11916.0i 0.724059i 0.932167 + 0.362030i \(0.117916\pi\)
−0.932167 + 0.362030i \(0.882084\pi\)
\(648\) 0 0
\(649\) 6804.00 0.411526
\(650\) 0 0
\(651\) −5691.00 −0.342623
\(652\) 0 0
\(653\) − 31842.0i − 1.90823i −0.299441 0.954115i \(-0.596800\pi\)
0.299441 0.954115i \(-0.403200\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 9486.00i − 0.563294i
\(658\) 0 0
\(659\) 3672.00 0.217057 0.108529 0.994093i \(-0.465386\pi\)
0.108529 + 0.994093i \(0.465386\pi\)
\(660\) 0 0
\(661\) 5138.00 0.302337 0.151169 0.988508i \(-0.451696\pi\)
0.151169 + 0.988508i \(0.451696\pi\)
\(662\) 0 0
\(663\) − 2970.00i − 0.173975i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 972.000i − 0.0564258i
\(668\) 0 0
\(669\) −4155.00 −0.240122
\(670\) 0 0
\(671\) 2538.00 0.146018
\(672\) 0 0
\(673\) 5050.00i 0.289247i 0.989487 + 0.144623i \(0.0461971\pi\)
−0.989487 + 0.144623i \(0.953803\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28458.0i 1.61555i 0.589488 + 0.807777i \(0.299329\pi\)
−0.589488 + 0.807777i \(0.700671\pi\)
\(678\) 0 0
\(679\) 3073.00 0.173683
\(680\) 0 0
\(681\) −7560.00 −0.425404
\(682\) 0 0
\(683\) 24408.0i 1.36742i 0.729755 + 0.683709i \(0.239634\pi\)
−0.729755 + 0.683709i \(0.760366\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 15387.0i − 0.854513i
\(688\) 0 0
\(689\) −1980.00 −0.109480
\(690\) 0 0
\(691\) −16328.0 −0.898909 −0.449455 0.893303i \(-0.648382\pi\)
−0.449455 + 0.893303i \(0.648382\pi\)
\(692\) 0 0
\(693\) − 3402.00i − 0.186481i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6480.00i − 0.352148i
\(698\) 0 0
\(699\) −9720.00 −0.525957
\(700\) 0 0
\(701\) 6246.00 0.336531 0.168265 0.985742i \(-0.446183\pi\)
0.168265 + 0.985742i \(0.446183\pi\)
\(702\) 0 0
\(703\) − 7850.00i − 0.421150i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 5796.00i 0.308318i
\(708\) 0 0
\(709\) 30679.0 1.62507 0.812535 0.582913i \(-0.198087\pi\)
0.812535 + 0.582913i \(0.198087\pi\)
\(710\) 0 0
\(711\) 5112.00 0.269641
\(712\) 0 0
\(713\) − 4878.00i − 0.256217i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 8964.00i − 0.466899i
\(718\) 0 0
\(719\) 15462.0 0.801996 0.400998 0.916079i \(-0.368663\pi\)
0.400998 + 0.916079i \(0.368663\pi\)
\(720\) 0 0
\(721\) −3836.00 −0.198142
\(722\) 0 0
\(723\) − 7941.00i − 0.408477i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 26801.0i − 1.36725i −0.729831 0.683627i \(-0.760401\pi\)
0.729831 0.683627i \(-0.239599\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 2934.00 0.148451
\(732\) 0 0
\(733\) 22858.0i 1.15181i 0.817515 + 0.575907i \(0.195351\pi\)
−0.817515 + 0.575907i \(0.804649\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 18522.0i 0.925735i
\(738\) 0 0
\(739\) −16900.0 −0.841240 −0.420620 0.907237i \(-0.638187\pi\)
−0.420620 + 0.907237i \(0.638187\pi\)
\(740\) 0 0
\(741\) 4125.00 0.204502
\(742\) 0 0
\(743\) 8028.00i 0.396391i 0.980162 + 0.198196i \(0.0635081\pi\)
−0.980162 + 0.198196i \(0.936492\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 12798.0i − 0.626846i
\(748\) 0 0
\(749\) −10332.0 −0.504036
\(750\) 0 0
\(751\) 12448.0 0.604839 0.302419 0.953175i \(-0.402206\pi\)
0.302419 + 0.953175i \(0.402206\pi\)
\(752\) 0 0
\(753\) 12636.0i 0.611529i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 30103.0i − 1.44533i −0.691200 0.722663i \(-0.742918\pi\)
0.691200 0.722663i \(-0.257082\pi\)
\(758\) 0 0
\(759\) 2916.00 0.139452
\(760\) 0 0
\(761\) −17748.0 −0.845420 −0.422710 0.906265i \(-0.638921\pi\)
−0.422710 + 0.906265i \(0.638921\pi\)
\(762\) 0 0
\(763\) − 8939.00i − 0.424133i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6930.00i 0.326242i
\(768\) 0 0
\(769\) −13283.0 −0.622883 −0.311442 0.950265i \(-0.600812\pi\)
−0.311442 + 0.950265i \(0.600812\pi\)
\(770\) 0 0
\(771\) −17172.0 −0.802120
\(772\) 0 0
\(773\) − 26424.0i − 1.22950i −0.788721 0.614751i \(-0.789256\pi\)
0.788721 0.614751i \(-0.210744\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 6594.00i − 0.304451i
\(778\) 0 0
\(779\) 9000.00 0.413939
\(780\) 0 0
\(781\) 58320.0 2.67203
\(782\) 0 0
\(783\) − 1458.00i − 0.0665449i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9709.00i 0.439757i 0.975527 + 0.219878i \(0.0705660\pi\)
−0.975527 + 0.219878i \(0.929434\pi\)
\(788\) 0 0
\(789\) 13824.0 0.623761
\(790\) 0 0
\(791\) 12852.0 0.577705
\(792\) 0 0
\(793\) 2585.00i 0.115758i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 7884.00i − 0.350396i −0.984533 0.175198i \(-0.943943\pi\)
0.984533 0.175198i \(-0.0560566\pi\)
\(798\) 0 0
\(799\) 9396.00 0.416028
\(800\) 0 0
\(801\) 12960.0 0.571684
\(802\) 0 0
\(803\) 56916.0i 2.50127i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19278.0i 0.840914i
\(808\) 0 0
\(809\) 1476.00 0.0641451 0.0320726 0.999486i \(-0.489789\pi\)
0.0320726 + 0.999486i \(0.489789\pi\)
\(810\) 0 0
\(811\) −21455.0 −0.928960 −0.464480 0.885583i \(-0.653759\pi\)
−0.464480 + 0.885583i \(0.653759\pi\)
\(812\) 0 0
\(813\) 10128.0i 0.436906i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4075.00i 0.174500i
\(818\) 0 0
\(819\) 3465.00 0.147835
\(820\) 0 0
\(821\) 31086.0 1.32145 0.660724 0.750629i \(-0.270249\pi\)
0.660724 + 0.750629i \(0.270249\pi\)
\(822\) 0 0
\(823\) 23381.0i 0.990292i 0.868810 + 0.495146i \(0.164885\pi\)
−0.868810 + 0.495146i \(0.835115\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3636.00i − 0.152885i −0.997074 0.0764426i \(-0.975644\pi\)
0.997074 0.0764426i \(-0.0243562\pi\)
\(828\) 0 0
\(829\) −4058.00 −0.170012 −0.0850061 0.996380i \(-0.527091\pi\)
−0.0850061 + 0.996380i \(0.527091\pi\)
\(830\) 0 0
\(831\) −16143.0 −0.673880
\(832\) 0 0
\(833\) 5292.00i 0.220116i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 7317.00i − 0.302165i
\(838\) 0 0
\(839\) 306.000 0.0125915 0.00629576 0.999980i \(-0.497996\pi\)
0.00629576 + 0.999980i \(0.497996\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) − 10422.0i − 0.425804i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 11095.0i 0.450093i
\(848\) 0 0
\(849\) 6807.00 0.275166
\(850\) 0 0
\(851\) 5652.00 0.227671
\(852\) 0 0
\(853\) − 42299.0i − 1.69788i −0.528491 0.848939i \(-0.677242\pi\)
0.528491 0.848939i \(-0.322758\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 11484.0i − 0.457743i −0.973457 0.228872i \(-0.926496\pi\)
0.973457 0.228872i \(-0.0735036\pi\)
\(858\) 0 0
\(859\) 33560.0 1.33301 0.666503 0.745502i \(-0.267790\pi\)
0.666503 + 0.745502i \(0.267790\pi\)
\(860\) 0 0
\(861\) 7560.00 0.299238
\(862\) 0 0
\(863\) − 14976.0i − 0.590717i −0.955386 0.295359i \(-0.904561\pi\)
0.955386 0.295359i \(-0.0954391\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 13767.0i 0.539275i
\(868\) 0 0
\(869\) −30672.0 −1.19733
\(870\) 0 0
\(871\) −18865.0 −0.733888
\(872\) 0 0
\(873\) 3951.00i 0.153174i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 41893.0i − 1.61303i −0.591215 0.806514i \(-0.701351\pi\)
0.591215 0.806514i \(-0.298649\pi\)
\(878\) 0 0
\(879\) −5022.00 −0.192705
\(880\) 0 0
\(881\) −720.000 −0.0275340 −0.0137670 0.999905i \(-0.504382\pi\)
−0.0137670 + 0.999905i \(0.504382\pi\)
\(882\) 0 0
\(883\) 17309.0i 0.659676i 0.944037 + 0.329838i \(0.106994\pi\)
−0.944037 + 0.329838i \(0.893006\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7002.00i 0.265055i 0.991179 + 0.132528i \(0.0423094\pi\)
−0.991179 + 0.132528i \(0.957691\pi\)
\(888\) 0 0
\(889\) −4144.00 −0.156339
\(890\) 0 0
\(891\) 4374.00 0.164461
\(892\) 0 0
\(893\) 13050.0i 0.489028i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2970.00i 0.110552i
\(898\) 0 0
\(899\) 14634.0 0.542905
\(900\) 0 0
\(901\) −648.000 −0.0239601
\(902\) 0 0
\(903\) 3423.00i 0.126147i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1484.00i − 0.0543279i −0.999631 0.0271640i \(-0.991352\pi\)
0.999631 0.0271640i \(-0.00864762\pi\)
\(908\) 0 0
\(909\) −7452.00 −0.271911
\(910\) 0 0
\(911\) 18882.0 0.686705 0.343353 0.939207i \(-0.388437\pi\)
0.343353 + 0.939207i \(0.388437\pi\)
\(912\) 0 0
\(913\) 76788.0i 2.78347i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3276.00i 0.117975i
\(918\) 0 0
\(919\) −50653.0 −1.81816 −0.909080 0.416622i \(-0.863214\pi\)
−0.909080 + 0.416622i \(0.863214\pi\)
\(920\) 0 0
\(921\) 1617.00 0.0578523
\(922\) 0 0
\(923\) 59400.0i 2.11828i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4932.00i − 0.174744i
\(928\) 0 0
\(929\) −35262.0 −1.24533 −0.622663 0.782490i \(-0.713949\pi\)
−0.622663 + 0.782490i \(0.713949\pi\)
\(930\) 0 0
\(931\) −7350.00 −0.258740
\(932\) 0 0
\(933\) − 4482.00i − 0.157271i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 20279.0i 0.707029i 0.935429 + 0.353514i \(0.115013\pi\)
−0.935429 + 0.353514i \(0.884987\pi\)
\(938\) 0 0
\(939\) −11991.0 −0.416732
\(940\) 0 0
\(941\) 42390.0 1.46852 0.734259 0.678870i \(-0.237530\pi\)
0.734259 + 0.678870i \(0.237530\pi\)
\(942\) 0 0
\(943\) 6480.00i 0.223773i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42426.0i 1.45582i 0.685674 + 0.727909i \(0.259508\pi\)
−0.685674 + 0.727909i \(0.740492\pi\)
\(948\) 0 0
\(949\) −57970.0 −1.98291
\(950\) 0 0
\(951\) 11016.0 0.375624
\(952\) 0 0
\(953\) − 48168.0i − 1.63727i −0.574317 0.818633i \(-0.694732\pi\)
0.574317 0.818633i \(-0.305268\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8748.00i 0.295489i
\(958\) 0 0
\(959\) −18018.0 −0.606707
\(960\) 0 0
\(961\) 43650.0 1.46521
\(962\) 0 0
\(963\) − 13284.0i − 0.444518i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 51400.0i 1.70932i 0.519188 + 0.854660i \(0.326234\pi\)
−0.519188 + 0.854660i \(0.673766\pi\)
\(968\) 0 0
\(969\) 1350.00 0.0447557
\(970\) 0 0
\(971\) −33858.0 −1.11901 −0.559503 0.828828i \(-0.689008\pi\)
−0.559503 + 0.828828i \(0.689008\pi\)
\(972\) 0 0
\(973\) − 12292.0i − 0.404998i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47106.0i 1.54253i 0.636513 + 0.771266i \(0.280376\pi\)
−0.636513 + 0.771266i \(0.719624\pi\)
\(978\) 0 0
\(979\) −77760.0 −2.53853
\(980\) 0 0
\(981\) 11493.0 0.374050
\(982\) 0 0
\(983\) − 20844.0i − 0.676318i −0.941089 0.338159i \(-0.890196\pi\)
0.941089 0.338159i \(-0.109804\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 10962.0i 0.353520i
\(988\) 0 0
\(989\) −2934.00 −0.0943334
\(990\) 0 0
\(991\) −4133.00 −0.132481 −0.0662407 0.997804i \(-0.521101\pi\)
−0.0662407 + 0.997804i \(0.521101\pi\)
\(992\) 0 0
\(993\) − 3156.00i − 0.100859i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33122.0i 1.05214i 0.850441 + 0.526070i \(0.176335\pi\)
−0.850441 + 0.526070i \(0.823665\pi\)
\(998\) 0 0
\(999\) 8478.00 0.268501
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.s.49.2 2
4.3 odd 2 300.4.d.a.49.1 2
5.2 odd 4 1200.4.a.y.1.1 1
5.3 odd 4 1200.4.a.m.1.1 1
5.4 even 2 inner 1200.4.f.s.49.1 2
12.11 even 2 900.4.d.j.649.1 2
20.3 even 4 300.4.a.g.1.1 yes 1
20.7 even 4 300.4.a.c.1.1 1
20.19 odd 2 300.4.d.a.49.2 2
60.23 odd 4 900.4.a.h.1.1 1
60.47 odd 4 900.4.a.k.1.1 1
60.59 even 2 900.4.d.j.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.4.a.c.1.1 1 20.7 even 4
300.4.a.g.1.1 yes 1 20.3 even 4
300.4.d.a.49.1 2 4.3 odd 2
300.4.d.a.49.2 2 20.19 odd 2
900.4.a.h.1.1 1 60.23 odd 4
900.4.a.k.1.1 1 60.47 odd 4
900.4.d.j.649.1 2 12.11 even 2
900.4.d.j.649.2 2 60.59 even 2
1200.4.a.m.1.1 1 5.3 odd 4
1200.4.a.y.1.1 1 5.2 odd 4
1200.4.f.s.49.1 2 5.4 even 2 inner
1200.4.f.s.49.2 2 1.1 even 1 trivial