Properties

Label 3800.2.d.k.3649.4
Level $3800$
Weight $2$
Character 3800.3649
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3800,2,Mod(3649,3800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3800.3649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.d (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: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3649.4
Root \(-1.75233 - 1.75233i\) of defining polynomial
Character \(\chi\) \(=\) 3800.3649
Dual form 3800.2.d.k.3649.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.363328i q^{3} -1.14134i q^{7} +2.86799 q^{9} -2.72666 q^{11} -4.64600i q^{13} +0.858664i q^{17} +1.00000 q^{19} +0.414680 q^{21} -4.41468i q^{23} +2.13201i q^{27} -9.42401 q^{29} -10.2827 q^{31} -0.990671i q^{33} -6.77801i q^{37} +1.68802 q^{39} -7.55602 q^{41} +9.29200i q^{43} +7.00933i q^{47} +5.69735 q^{49} -0.311977 q^{51} -8.64600i q^{53} +0.363328i q^{57} +5.14134 q^{59} +9.45331 q^{61} -3.27334i q^{63} +13.6553i q^{67} +1.60398 q^{69} -5.45331 q^{71} +6.87732i q^{73} +3.11203i q^{77} -17.2920 q^{79} +7.82936 q^{81} -14.2827i q^{83} -3.42401i q^{87} -13.0093 q^{89} -5.30265 q^{91} -3.73599i q^{93} +9.68463i q^{97} -7.82003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{9} - 8 q^{11} + 6 q^{19} - 6 q^{21} - 6 q^{29} - 28 q^{31} + 10 q^{39} - 20 q^{41} - 8 q^{49} - 2 q^{51} + 14 q^{59} + 40 q^{61} - 66 q^{69} - 16 q^{71} - 28 q^{79} + 30 q^{81} - 36 q^{89} - 74 q^{91}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(951\) \(1901\) \(1977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 0.363328i 0.209768i 0.994484 + 0.104884i \(0.0334471\pi\)
−0.994484 + 0.104884i \(0.966553\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.14134i − 0.431385i −0.976461 0.215692i \(-0.930799\pi\)
0.976461 0.215692i \(-0.0692008\pi\)
\(8\) 0 0
\(9\) 2.86799 0.955998
\(10\) 0 0
\(11\) −2.72666 −0.822118 −0.411059 0.911609i \(-0.634841\pi\)
−0.411059 + 0.911609i \(0.634841\pi\)
\(12\) 0 0
\(13\) − 4.64600i − 1.28857i −0.764786 0.644284i \(-0.777155\pi\)
0.764786 0.644284i \(-0.222845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.858664i 0.208257i 0.994564 + 0.104128i \(0.0332053\pi\)
−0.994564 + 0.104128i \(0.966795\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0.414680 0.0904905
\(22\) 0 0
\(23\) − 4.41468i − 0.920524i −0.887783 0.460262i \(-0.847755\pi\)
0.887783 0.460262i \(-0.152245\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.13201i 0.410305i
\(28\) 0 0
\(29\) −9.42401 −1.74999 −0.874997 0.484128i \(-0.839137\pi\)
−0.874997 + 0.484128i \(0.839137\pi\)
\(30\) 0 0
\(31\) −10.2827 −1.84682 −0.923411 0.383812i \(-0.874611\pi\)
−0.923411 + 0.383812i \(0.874611\pi\)
\(32\) 0 0
\(33\) − 0.990671i − 0.172454i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.77801i − 1.11430i −0.830413 0.557149i \(-0.811895\pi\)
0.830413 0.557149i \(-0.188105\pi\)
\(38\) 0 0
\(39\) 1.68802 0.270300
\(40\) 0 0
\(41\) −7.55602 −1.18005 −0.590026 0.807384i \(-0.700882\pi\)
−0.590026 + 0.807384i \(0.700882\pi\)
\(42\) 0 0
\(43\) 9.29200i 1.41702i 0.705702 + 0.708508i \(0.250632\pi\)
−0.705702 + 0.708508i \(0.749368\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.00933i 1.02242i 0.859457 + 0.511208i \(0.170802\pi\)
−0.859457 + 0.511208i \(0.829198\pi\)
\(48\) 0 0
\(49\) 5.69735 0.813907
\(50\) 0 0
\(51\) −0.311977 −0.0436855
\(52\) 0 0
\(53\) − 8.64600i − 1.18762i −0.804605 0.593810i \(-0.797623\pi\)
0.804605 0.593810i \(-0.202377\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.363328i 0.0481240i
\(58\) 0 0
\(59\) 5.14134 0.669345 0.334672 0.942335i \(-0.391374\pi\)
0.334672 + 0.942335i \(0.391374\pi\)
\(60\) 0 0
\(61\) 9.45331 1.21037 0.605186 0.796084i \(-0.293099\pi\)
0.605186 + 0.796084i \(0.293099\pi\)
\(62\) 0 0
\(63\) − 3.27334i − 0.412403i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 13.6553i 1.66826i 0.551565 + 0.834132i \(0.314031\pi\)
−0.551565 + 0.834132i \(0.685969\pi\)
\(68\) 0 0
\(69\) 1.60398 0.193096
\(70\) 0 0
\(71\) −5.45331 −0.647189 −0.323595 0.946196i \(-0.604891\pi\)
−0.323595 + 0.946196i \(0.604891\pi\)
\(72\) 0 0
\(73\) 6.87732i 0.804930i 0.915435 + 0.402465i \(0.131846\pi\)
−0.915435 + 0.402465i \(0.868154\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.11203i 0.354649i
\(78\) 0 0
\(79\) −17.2920 −1.94550 −0.972751 0.231852i \(-0.925521\pi\)
−0.972751 + 0.231852i \(0.925521\pi\)
\(80\) 0 0
\(81\) 7.82936 0.869929
\(82\) 0 0
\(83\) − 14.2827i − 1.56773i −0.620933 0.783863i \(-0.713246\pi\)
0.620933 0.783863i \(-0.286754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 3.42401i − 0.367092i
\(88\) 0 0
\(89\) −13.0093 −1.37899 −0.689493 0.724292i \(-0.742167\pi\)
−0.689493 + 0.724292i \(0.742167\pi\)
\(90\) 0 0
\(91\) −5.30265 −0.555869
\(92\) 0 0
\(93\) − 3.73599i − 0.387404i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.68463i 0.983326i 0.870786 + 0.491663i \(0.163611\pi\)
−0.870786 + 0.491663i \(0.836389\pi\)
\(98\) 0 0
\(99\) −7.82003 −0.785943
\(100\) 0 0
\(101\) 1.45331 0.144610 0.0723050 0.997383i \(-0.476964\pi\)
0.0723050 + 0.997383i \(0.476964\pi\)
\(102\) 0 0
\(103\) 2.23132i 0.219859i 0.993939 + 0.109929i \(0.0350624\pi\)
−0.993939 + 0.109929i \(0.964938\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 13.6553i − 1.32011i −0.751217 0.660055i \(-0.770533\pi\)
0.751217 0.660055i \(-0.229467\pi\)
\(108\) 0 0
\(109\) −20.1693 −1.93187 −0.965935 0.258784i \(-0.916678\pi\)
−0.965935 + 0.258784i \(0.916678\pi\)
\(110\) 0 0
\(111\) 2.46264 0.233744
\(112\) 0 0
\(113\) − 10.0514i − 0.945552i −0.881183 0.472776i \(-0.843252\pi\)
0.881183 0.472776i \(-0.156748\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 13.3247i − 1.23187i
\(118\) 0 0
\(119\) 0.980024 0.0898387
\(120\) 0 0
\(121\) −3.56534 −0.324122
\(122\) 0 0
\(123\) − 2.74531i − 0.247537i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 14.0700i 1.24851i 0.781220 + 0.624256i \(0.214598\pi\)
−0.781220 + 0.624256i \(0.785402\pi\)
\(128\) 0 0
\(129\) −3.37605 −0.297244
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) − 1.14134i − 0.0989664i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.85866i 0.415104i 0.978224 + 0.207552i \(0.0665496\pi\)
−0.978224 + 0.207552i \(0.933450\pi\)
\(138\) 0 0
\(139\) 4.17997 0.354540 0.177270 0.984162i \(-0.443273\pi\)
0.177270 + 0.984162i \(0.443273\pi\)
\(140\) 0 0
\(141\) −2.54669 −0.214470
\(142\) 0 0
\(143\) 12.6680i 1.05936i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.07001i 0.170731i
\(148\) 0 0
\(149\) −1.45331 −0.119060 −0.0595300 0.998227i \(-0.518960\pi\)
−0.0595300 + 0.998227i \(0.518960\pi\)
\(150\) 0 0
\(151\) 7.27334 0.591896 0.295948 0.955204i \(-0.404364\pi\)
0.295948 + 0.955204i \(0.404364\pi\)
\(152\) 0 0
\(153\) 2.46264i 0.199093i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 17.0093i − 1.35749i −0.734373 0.678746i \(-0.762524\pi\)
0.734373 0.678746i \(-0.237476\pi\)
\(158\) 0 0
\(159\) 3.14134 0.249124
\(160\) 0 0
\(161\) −5.03863 −0.397100
\(162\) 0 0
\(163\) − 10.1800i − 0.797357i −0.917091 0.398678i \(-0.869469\pi\)
0.917091 0.398678i \(-0.130531\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 6.07001i − 0.469711i −0.972030 0.234856i \(-0.924538\pi\)
0.972030 0.234856i \(-0.0754618\pi\)
\(168\) 0 0
\(169\) −8.58532 −0.660409
\(170\) 0 0
\(171\) 2.86799 0.219321
\(172\) 0 0
\(173\) − 5.22199i − 0.397021i −0.980099 0.198510i \(-0.936390\pi\)
0.980099 0.198510i \(-0.0636103\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.86799i 0.140407i
\(178\) 0 0
\(179\) 1.55602 0.116302 0.0581510 0.998308i \(-0.481480\pi\)
0.0581510 + 0.998308i \(0.481480\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 3.43466i 0.253897i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.34128i − 0.171211i
\(188\) 0 0
\(189\) 2.43334 0.176999
\(190\) 0 0
\(191\) −5.03863 −0.364583 −0.182291 0.983245i \(-0.558351\pi\)
−0.182291 + 0.983245i \(0.558351\pi\)
\(192\) 0 0
\(193\) − 9.68463i − 0.697115i −0.937287 0.348558i \(-0.886672\pi\)
0.937287 0.348558i \(-0.113328\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) −19.8867 −1.40973 −0.704864 0.709343i \(-0.748992\pi\)
−0.704864 + 0.709343i \(0.748992\pi\)
\(200\) 0 0
\(201\) −4.96137 −0.349948
\(202\) 0 0
\(203\) 10.7560i 0.754920i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 12.6613i − 0.880019i
\(208\) 0 0
\(209\) −2.72666 −0.188607
\(210\) 0 0
\(211\) 4.31198 0.296849 0.148424 0.988924i \(-0.452580\pi\)
0.148424 + 0.988924i \(0.452580\pi\)
\(212\) 0 0
\(213\) − 1.98134i − 0.135759i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 11.7360i 0.796691i
\(218\) 0 0
\(219\) −2.49873 −0.168848
\(220\) 0 0
\(221\) 3.98935 0.268353
\(222\) 0 0
\(223\) − 18.5327i − 1.24104i −0.784191 0.620519i \(-0.786922\pi\)
0.784191 0.620519i \(-0.213078\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.82597i − 0.320311i −0.987092 0.160155i \(-0.948800\pi\)
0.987092 0.160155i \(-0.0511995\pi\)
\(228\) 0 0
\(229\) −23.7360 −1.56852 −0.784259 0.620434i \(-0.786957\pi\)
−0.784259 + 0.620434i \(0.786957\pi\)
\(230\) 0 0
\(231\) −1.13069 −0.0743939
\(232\) 0 0
\(233\) − 14.5653i − 0.954207i −0.878847 0.477104i \(-0.841687\pi\)
0.878847 0.477104i \(-0.158313\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 6.28267i − 0.408103i
\(238\) 0 0
\(239\) −2.43334 −0.157399 −0.0786997 0.996898i \(-0.525077\pi\)
−0.0786997 + 0.996898i \(0.525077\pi\)
\(240\) 0 0
\(241\) 18.1986 1.17228 0.586138 0.810211i \(-0.300648\pi\)
0.586138 + 0.810211i \(0.300648\pi\)
\(242\) 0 0
\(243\) 9.24065i 0.592788i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4.64600i − 0.295618i
\(248\) 0 0
\(249\) 5.18930 0.328858
\(250\) 0 0
\(251\) −15.1120 −0.953863 −0.476931 0.878940i \(-0.658251\pi\)
−0.476931 + 0.878940i \(0.658251\pi\)
\(252\) 0 0
\(253\) 12.0373i 0.756779i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 12.2313i − 0.762969i −0.924375 0.381484i \(-0.875413\pi\)
0.924375 0.381484i \(-0.124587\pi\)
\(258\) 0 0
\(259\) −7.73599 −0.480691
\(260\) 0 0
\(261\) −27.0280 −1.67299
\(262\) 0 0
\(263\) 22.5840i 1.39259i 0.717756 + 0.696295i \(0.245169\pi\)
−0.717756 + 0.696295i \(0.754831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.72666i − 0.289267i
\(268\) 0 0
\(269\) 13.1120 0.799455 0.399727 0.916634i \(-0.369105\pi\)
0.399727 + 0.916634i \(0.369105\pi\)
\(270\) 0 0
\(271\) −20.8960 −1.26934 −0.634670 0.772783i \(-0.718864\pi\)
−0.634670 + 0.772783i \(0.718864\pi\)
\(272\) 0 0
\(273\) − 1.92660i − 0.116603i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 20.1800i 1.21250i 0.795275 + 0.606248i \(0.207326\pi\)
−0.795275 + 0.606248i \(0.792674\pi\)
\(278\) 0 0
\(279\) −29.4906 −1.76556
\(280\) 0 0
\(281\) −22.4626 −1.34001 −0.670004 0.742357i \(-0.733708\pi\)
−0.670004 + 0.742357i \(0.733708\pi\)
\(282\) 0 0
\(283\) − 0.829359i − 0.0493003i −0.999696 0.0246501i \(-0.992153\pi\)
0.999696 0.0246501i \(-0.00784718\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.62395i 0.509056i
\(288\) 0 0
\(289\) 16.2627 0.956629
\(290\) 0 0
\(291\) −3.51870 −0.206270
\(292\) 0 0
\(293\) − 31.2886i − 1.82790i −0.405827 0.913950i \(-0.633016\pi\)
0.405827 0.913950i \(-0.366984\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.81325i − 0.337319i
\(298\) 0 0
\(299\) −20.5106 −1.18616
\(300\) 0 0
\(301\) 10.6053 0.611279
\(302\) 0 0
\(303\) 0.528030i 0.0303345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.11929i 0.178027i 0.996030 + 0.0890136i \(0.0283715\pi\)
−0.996030 + 0.0890136i \(0.971629\pi\)
\(308\) 0 0
\(309\) −0.810702 −0.0461192
\(310\) 0 0
\(311\) −21.4240 −1.21484 −0.607422 0.794379i \(-0.707796\pi\)
−0.607422 + 0.794379i \(0.707796\pi\)
\(312\) 0 0
\(313\) − 12.1320i − 0.685742i −0.939383 0.342871i \(-0.888601\pi\)
0.939383 0.342871i \(-0.111399\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 16.7487i − 0.940701i −0.882480 0.470350i \(-0.844128\pi\)
0.882480 0.470350i \(-0.155872\pi\)
\(318\) 0 0
\(319\) 25.6960 1.43870
\(320\) 0 0
\(321\) 4.96137 0.276916
\(322\) 0 0
\(323\) 0.858664i 0.0477773i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 7.32808i − 0.405244i
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −5.97070 −0.328179 −0.164090 0.986445i \(-0.552469\pi\)
−0.164090 + 0.986445i \(0.552469\pi\)
\(332\) 0 0
\(333\) − 19.4393i − 1.06527i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.3340i 1.10766i 0.832628 + 0.553832i \(0.186835\pi\)
−0.832628 + 0.553832i \(0.813165\pi\)
\(338\) 0 0
\(339\) 3.65194 0.198346
\(340\) 0 0
\(341\) 28.0373 1.51831
\(342\) 0 0
\(343\) − 14.4919i − 0.782492i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 20.8294i − 1.11818i −0.829107 0.559089i \(-0.811151\pi\)
0.829107 0.559089i \(-0.188849\pi\)
\(348\) 0 0
\(349\) 20.0187 1.07157 0.535787 0.844353i \(-0.320015\pi\)
0.535787 + 0.844353i \(0.320015\pi\)
\(350\) 0 0
\(351\) 9.90531 0.528706
\(352\) 0 0
\(353\) − 27.1413i − 1.44459i −0.691586 0.722294i \(-0.743088\pi\)
0.691586 0.722294i \(-0.256912\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.356070i 0.0188452i
\(358\) 0 0
\(359\) 4.59465 0.242496 0.121248 0.992622i \(-0.461310\pi\)
0.121248 + 0.992622i \(0.461310\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) − 1.29539i − 0.0679904i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 34.1214i 1.78112i 0.454865 + 0.890560i \(0.349687\pi\)
−0.454865 + 0.890560i \(0.650313\pi\)
\(368\) 0 0
\(369\) −21.6706 −1.12813
\(370\) 0 0
\(371\) −9.86799 −0.512321
\(372\) 0 0
\(373\) − 7.81664i − 0.404730i −0.979310 0.202365i \(-0.935137\pi\)
0.979310 0.202365i \(-0.0648628\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 43.7839i 2.25499i
\(378\) 0 0
\(379\) 23.0573 1.18437 0.592187 0.805801i \(-0.298265\pi\)
0.592187 + 0.805801i \(0.298265\pi\)
\(380\) 0 0
\(381\) −5.11203 −0.261897
\(382\) 0 0
\(383\) 16.8807i 0.862564i 0.902217 + 0.431282i \(0.141939\pi\)
−0.902217 + 0.431282i \(0.858061\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 26.6494i 1.35466i
\(388\) 0 0
\(389\) 18.3013 0.927914 0.463957 0.885858i \(-0.346429\pi\)
0.463957 + 0.885858i \(0.346429\pi\)
\(390\) 0 0
\(391\) 3.79073 0.191705
\(392\) 0 0
\(393\) − 1.45331i − 0.0733099i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 12.1800i − 0.611295i −0.952145 0.305648i \(-0.901127\pi\)
0.952145 0.305648i \(-0.0988729\pi\)
\(398\) 0 0
\(399\) 0.414680 0.0207599
\(400\) 0 0
\(401\) 16.5840 0.828166 0.414083 0.910239i \(-0.364102\pi\)
0.414083 + 0.910239i \(0.364102\pi\)
\(402\) 0 0
\(403\) 47.7733i 2.37976i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.4813i 0.916084i
\(408\) 0 0
\(409\) −18.5653 −0.917997 −0.458999 0.888437i \(-0.651792\pi\)
−0.458999 + 0.888437i \(0.651792\pi\)
\(410\) 0 0
\(411\) −1.76529 −0.0870753
\(412\) 0 0
\(413\) − 5.86799i − 0.288745i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.51870i 0.0743711i
\(418\) 0 0
\(419\) −2.84802 −0.139135 −0.0695674 0.997577i \(-0.522162\pi\)
−0.0695674 + 0.997577i \(0.522162\pi\)
\(420\) 0 0
\(421\) 24.5360 1.19581 0.597907 0.801566i \(-0.295999\pi\)
0.597907 + 0.801566i \(0.295999\pi\)
\(422\) 0 0
\(423\) 20.1027i 0.977427i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 10.7894i − 0.522136i
\(428\) 0 0
\(429\) −4.60266 −0.222218
\(430\) 0 0
\(431\) −7.53736 −0.363062 −0.181531 0.983385i \(-0.558105\pi\)
−0.181531 + 0.983385i \(0.558105\pi\)
\(432\) 0 0
\(433\) − 0.392633i − 0.0188687i −0.999955 0.00943437i \(-0.996997\pi\)
0.999955 0.00943437i \(-0.00300310\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 4.41468i − 0.211183i
\(438\) 0 0
\(439\) −5.29200 −0.252573 −0.126287 0.991994i \(-0.540306\pi\)
−0.126287 + 0.991994i \(0.540306\pi\)
\(440\) 0 0
\(441\) 16.3400 0.778093
\(442\) 0 0
\(443\) − 22.0187i − 1.04614i −0.852290 0.523069i \(-0.824787\pi\)
0.852290 0.523069i \(-0.175213\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 0.528030i − 0.0249749i
\(448\) 0 0
\(449\) 4.90663 0.231558 0.115779 0.993275i \(-0.463064\pi\)
0.115779 + 0.993275i \(0.463064\pi\)
\(450\) 0 0
\(451\) 20.6027 0.970141
\(452\) 0 0
\(453\) 2.64261i 0.124161i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 26.9800i − 1.26207i −0.775753 0.631036i \(-0.782630\pi\)
0.775753 0.631036i \(-0.217370\pi\)
\(458\) 0 0
\(459\) −1.83068 −0.0854487
\(460\) 0 0
\(461\) −27.1307 −1.26360 −0.631801 0.775131i \(-0.717684\pi\)
−0.631801 + 0.775131i \(0.717684\pi\)
\(462\) 0 0
\(463\) − 16.9253i − 0.786585i −0.919413 0.393292i \(-0.871336\pi\)
0.919413 0.393292i \(-0.128664\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 42.6213i 1.97228i 0.165917 + 0.986140i \(0.446942\pi\)
−0.165917 + 0.986140i \(0.553058\pi\)
\(468\) 0 0
\(469\) 15.5853 0.719663
\(470\) 0 0
\(471\) 6.17997 0.284758
\(472\) 0 0
\(473\) − 25.3361i − 1.16495i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 24.7967i − 1.13536i
\(478\) 0 0
\(479\) 25.3107 1.15647 0.578237 0.815869i \(-0.303741\pi\)
0.578237 + 0.815869i \(0.303741\pi\)
\(480\) 0 0
\(481\) −31.4906 −1.43585
\(482\) 0 0
\(483\) − 1.83068i − 0.0832987i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 25.4020i − 1.15107i −0.817776 0.575536i \(-0.804793\pi\)
0.817776 0.575536i \(-0.195207\pi\)
\(488\) 0 0
\(489\) 3.69867 0.167260
\(490\) 0 0
\(491\) 20.6240 0.930746 0.465373 0.885115i \(-0.345920\pi\)
0.465373 + 0.885115i \(0.345920\pi\)
\(492\) 0 0
\(493\) − 8.09206i − 0.364448i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.22406i 0.279187i
\(498\) 0 0
\(499\) −17.3693 −0.777555 −0.388778 0.921332i \(-0.627103\pi\)
−0.388778 + 0.921332i \(0.627103\pi\)
\(500\) 0 0
\(501\) 2.20541 0.0985303
\(502\) 0 0
\(503\) − 14.9987i − 0.668758i −0.942439 0.334379i \(-0.891473\pi\)
0.942439 0.334379i \(-0.108527\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 3.11929i − 0.138533i
\(508\) 0 0
\(509\) 19.9160 0.882759 0.441380 0.897320i \(-0.354489\pi\)
0.441380 + 0.897320i \(0.354489\pi\)
\(510\) 0 0
\(511\) 7.84934 0.347234
\(512\) 0 0
\(513\) 2.13201i 0.0941304i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 19.1120i − 0.840546i
\(518\) 0 0
\(519\) 1.89730 0.0832821
\(520\) 0 0
\(521\) −14.8294 −0.649686 −0.324843 0.945768i \(-0.605311\pi\)
−0.324843 + 0.945768i \(0.605311\pi\)
\(522\) 0 0
\(523\) 11.9966i 0.524575i 0.964990 + 0.262288i \(0.0844769\pi\)
−0.964990 + 0.262288i \(0.915523\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 8.82936i − 0.384613i
\(528\) 0 0
\(529\) 3.51060 0.152635
\(530\) 0 0
\(531\) 14.7453 0.639892
\(532\) 0 0
\(533\) 35.1053i 1.52058i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.565344i 0.0243964i
\(538\) 0 0
\(539\) −15.5347 −0.669128
\(540\) 0 0
\(541\) −2.01866 −0.0867889 −0.0433944 0.999058i \(-0.513817\pi\)
−0.0433944 + 0.999058i \(0.513817\pi\)
\(542\) 0 0
\(543\) 7.99322i 0.343022i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 18.6940i 0.799296i 0.916669 + 0.399648i \(0.130868\pi\)
−0.916669 + 0.399648i \(0.869132\pi\)
\(548\) 0 0
\(549\) 27.1120 1.15711
\(550\) 0 0
\(551\) −9.42401 −0.401476
\(552\) 0 0
\(553\) 19.7360i 0.839259i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 11.0866i − 0.469754i −0.972025 0.234877i \(-0.924531\pi\)
0.972025 0.234877i \(-0.0754688\pi\)
\(558\) 0 0
\(559\) 43.1706 1.82592
\(560\) 0 0
\(561\) 0.850654 0.0359146
\(562\) 0 0
\(563\) 2.69396i 0.113537i 0.998387 + 0.0567685i \(0.0180797\pi\)
−0.998387 + 0.0567685i \(0.981920\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 8.93593i − 0.375274i
\(568\) 0 0
\(569\) −29.4134 −1.23307 −0.616536 0.787327i \(-0.711465\pi\)
−0.616536 + 0.787327i \(0.711465\pi\)
\(570\) 0 0
\(571\) −33.0466 −1.38296 −0.691479 0.722396i \(-0.743041\pi\)
−0.691479 + 0.722396i \(0.743041\pi\)
\(572\) 0 0
\(573\) − 1.83068i − 0.0764777i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.396022i 0.0164866i 0.999966 + 0.00824331i \(0.00262396\pi\)
−0.999966 + 0.00824331i \(0.997376\pi\)
\(578\) 0 0
\(579\) 3.51870 0.146232
\(580\) 0 0
\(581\) −16.3013 −0.676293
\(582\) 0 0
\(583\) 23.5747i 0.976363i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.0773i 0.581031i 0.956870 + 0.290515i \(0.0938268\pi\)
−0.956870 + 0.290515i \(0.906173\pi\)
\(588\) 0 0
\(589\) −10.2827 −0.423690
\(590\) 0 0
\(591\) −0.726656 −0.0298907
\(592\) 0 0
\(593\) 5.43466i 0.223175i 0.993755 + 0.111587i \(0.0355935\pi\)
−0.993755 + 0.111587i \(0.964407\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 7.22538i − 0.295715i
\(598\) 0 0
\(599\) −11.6333 −0.475323 −0.237662 0.971348i \(-0.576381\pi\)
−0.237662 + 0.971348i \(0.576381\pi\)
\(600\) 0 0
\(601\) 8.12136 0.331277 0.165639 0.986187i \(-0.447031\pi\)
0.165639 + 0.986187i \(0.447031\pi\)
\(602\) 0 0
\(603\) 39.1634i 1.59486i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 10.5327i − 0.427507i −0.976888 0.213754i \(-0.931431\pi\)
0.976888 0.213754i \(-0.0685689\pi\)
\(608\) 0 0
\(609\) −3.90794 −0.158358
\(610\) 0 0
\(611\) 32.5653 1.31745
\(612\) 0 0
\(613\) 47.3293i 1.91161i 0.293996 + 0.955807i \(0.405015\pi\)
−0.293996 + 0.955807i \(0.594985\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.3693i 1.34340i 0.740825 + 0.671698i \(0.234435\pi\)
−0.740825 + 0.671698i \(0.765565\pi\)
\(618\) 0 0
\(619\) 33.8760 1.36159 0.680796 0.732473i \(-0.261634\pi\)
0.680796 + 0.732473i \(0.261634\pi\)
\(620\) 0 0
\(621\) 9.41213 0.377696
\(622\) 0 0
\(623\) 14.8480i 0.594873i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 0.990671i − 0.0395636i
\(628\) 0 0
\(629\) 5.82003 0.232060
\(630\) 0 0
\(631\) 27.8573 1.10898 0.554492 0.832189i \(-0.312913\pi\)
0.554492 + 0.832189i \(0.312913\pi\)
\(632\) 0 0
\(633\) 1.56666i 0.0622693i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 26.4699i − 1.04878i
\(638\) 0 0
\(639\) −15.6401 −0.618711
\(640\) 0 0
\(641\) −4.74531 −0.187429 −0.0937143 0.995599i \(-0.529874\pi\)
−0.0937143 + 0.995599i \(0.529874\pi\)
\(642\) 0 0
\(643\) − 6.28267i − 0.247764i −0.992297 0.123882i \(-0.960466\pi\)
0.992297 0.123882i \(-0.0395345\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 16.1507i − 0.634948i −0.948267 0.317474i \(-0.897165\pi\)
0.948267 0.317474i \(-0.102835\pi\)
\(648\) 0 0
\(649\) −14.0187 −0.550280
\(650\) 0 0
\(651\) −4.26401 −0.167120
\(652\) 0 0
\(653\) 29.8760i 1.16914i 0.811344 + 0.584569i \(0.198736\pi\)
−0.811344 + 0.584569i \(0.801264\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 19.7241i 0.769511i
\(658\) 0 0
\(659\) 17.4427 0.679470 0.339735 0.940521i \(-0.389663\pi\)
0.339735 + 0.940521i \(0.389663\pi\)
\(660\) 0 0
\(661\) −2.87732 −0.111915 −0.0559574 0.998433i \(-0.517821\pi\)
−0.0559574 + 0.998433i \(0.517821\pi\)
\(662\) 0 0
\(663\) 1.44944i 0.0562918i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 41.6040i 1.61091i
\(668\) 0 0
\(669\) 6.73344 0.260330
\(670\) 0 0
\(671\) −25.7759 −0.995069
\(672\) 0 0
\(673\) 10.9393i 0.421680i 0.977521 + 0.210840i \(0.0676199\pi\)
−0.977521 + 0.210840i \(0.932380\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 19.0900i − 0.733688i −0.930283 0.366844i \(-0.880438\pi\)
0.930283 0.366844i \(-0.119562\pi\)
\(678\) 0 0
\(679\) 11.0534 0.424191
\(680\) 0 0
\(681\) 1.75341 0.0671909
\(682\) 0 0
\(683\) 25.7687i 0.986011i 0.870026 + 0.493006i \(0.164102\pi\)
−0.870026 + 0.493006i \(0.835898\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 8.62395i − 0.329024i
\(688\) 0 0
\(689\) −40.1693 −1.53033
\(690\) 0 0
\(691\) −27.2334 −1.03601 −0.518004 0.855378i \(-0.673325\pi\)
−0.518004 + 0.855378i \(0.673325\pi\)
\(692\) 0 0
\(693\) 8.92528i 0.339043i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.48808i − 0.245753i
\(698\) 0 0
\(699\) 5.29200 0.200162
\(700\) 0 0
\(701\) 49.6960 1.87699 0.938497 0.345288i \(-0.112219\pi\)
0.938497 + 0.345288i \(0.112219\pi\)
\(702\) 0 0
\(703\) − 6.77801i − 0.255637i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.65872i − 0.0623825i
\(708\) 0 0
\(709\) −5.64006 −0.211817 −0.105908 0.994376i \(-0.533775\pi\)
−0.105908 + 0.994376i \(0.533775\pi\)
\(710\) 0 0
\(711\) −49.5933 −1.85990
\(712\) 0 0
\(713\) 45.3947i 1.70005i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 0.884100i − 0.0330173i
\(718\) 0 0
\(719\) 33.4240 1.24651 0.623253 0.782021i \(-0.285811\pi\)
0.623253 + 0.782021i \(0.285811\pi\)
\(720\) 0 0
\(721\) 2.54669 0.0948436
\(722\) 0 0
\(723\) 6.61208i 0.245906i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.3306i 0.531494i 0.964043 + 0.265747i \(0.0856185\pi\)
−0.964043 + 0.265747i \(0.914381\pi\)
\(728\) 0 0
\(729\) 20.1307 0.745581
\(730\) 0 0
\(731\) −7.97871 −0.295103
\(732\) 0 0
\(733\) − 40.8853i − 1.51013i −0.655648 0.755067i \(-0.727604\pi\)
0.655648 0.755067i \(-0.272396\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 37.2334i − 1.37151i
\(738\) 0 0
\(739\) 28.8294 1.06051 0.530253 0.847840i \(-0.322097\pi\)
0.530253 + 0.847840i \(0.322097\pi\)
\(740\) 0 0
\(741\) 1.68802 0.0620111
\(742\) 0 0
\(743\) − 8.05135i − 0.295375i −0.989034 0.147688i \(-0.952817\pi\)
0.989034 0.147688i \(-0.0471831\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 40.9626i − 1.49874i
\(748\) 0 0
\(749\) −15.5853 −0.569475
\(750\) 0 0
\(751\) 47.3107 1.72639 0.863195 0.504870i \(-0.168460\pi\)
0.863195 + 0.504870i \(0.168460\pi\)
\(752\) 0 0
\(753\) − 5.49063i − 0.200090i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 6.20541i − 0.225539i −0.993621 0.112770i \(-0.964028\pi\)
0.993621 0.112770i \(-0.0359722\pi\)
\(758\) 0 0
\(759\) −4.37350 −0.158748
\(760\) 0 0
\(761\) 43.8280 1.58877 0.794383 0.607418i \(-0.207795\pi\)
0.794383 + 0.607418i \(0.207795\pi\)
\(762\) 0 0
\(763\) 23.0200i 0.833379i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 23.8867i − 0.862497i
\(768\) 0 0
\(769\) 3.22538 0.116310 0.0581551 0.998308i \(-0.481478\pi\)
0.0581551 + 0.998308i \(0.481478\pi\)
\(770\) 0 0
\(771\) 4.44398 0.160046
\(772\) 0 0
\(773\) 6.16470i 0.221729i 0.993836 + 0.110864i \(0.0353619\pi\)
−0.993836 + 0.110864i \(0.964638\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 2.81070i − 0.100833i
\(778\) 0 0
\(779\) −7.55602 −0.270722
\(780\) 0 0
\(781\) 14.8693 0.532066
\(782\) 0 0
\(783\) − 20.0921i − 0.718031i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 35.1527i − 1.25306i −0.779397 0.626530i \(-0.784475\pi\)
0.779397 0.626530i \(-0.215525\pi\)
\(788\) 0 0
\(789\) −8.20541 −0.292120
\(790\) 0 0
\(791\) −11.4720 −0.407896
\(792\) 0 0
\(793\) − 43.9201i − 1.55965i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 30.3420i 1.07477i 0.843337 + 0.537385i \(0.180588\pi\)
−0.843337 + 0.537385i \(0.819412\pi\)
\(798\) 0 0
\(799\) −6.01866 −0.212925
\(800\) 0 0
\(801\) −37.3107 −1.31831
\(802\) 0 0
\(803\) − 18.7521i − 0.661747i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 4.76397i 0.167700i
\(808\) 0 0
\(809\) 4.96137 0.174432 0.0872162 0.996189i \(-0.472203\pi\)
0.0872162 + 0.996189i \(0.472203\pi\)
\(810\) 0 0
\(811\) 29.8094 1.04675 0.523375 0.852103i \(-0.324673\pi\)
0.523375 + 0.852103i \(0.324673\pi\)
\(812\) 0 0
\(813\) − 7.59210i − 0.266267i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.29200i 0.325086i
\(818\) 0 0
\(819\) −15.2080 −0.531409
\(820\) 0 0
\(821\) −11.1520 −0.389207 −0.194603 0.980882i \(-0.562342\pi\)
−0.194603 + 0.980882i \(0.562342\pi\)
\(822\) 0 0
\(823\) − 17.1413i − 0.597509i −0.954330 0.298755i \(-0.903429\pi\)
0.954330 0.298755i \(-0.0965712\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 24.8633i − 0.864581i −0.901734 0.432291i \(-0.857705\pi\)
0.901734 0.432291i \(-0.142295\pi\)
\(828\) 0 0
\(829\) 18.7746 0.652069 0.326035 0.945358i \(-0.394287\pi\)
0.326035 + 0.945358i \(0.394287\pi\)
\(830\) 0 0
\(831\) −7.33195 −0.254343
\(832\) 0 0
\(833\) 4.89211i 0.169502i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 21.9227i − 0.757761i
\(838\) 0 0
\(839\) −29.0280 −1.00216 −0.501079 0.865402i \(-0.667063\pi\)
−0.501079 + 0.865402i \(0.667063\pi\)
\(840\) 0 0
\(841\) 59.8119 2.06248
\(842\) 0 0
\(843\) − 8.16131i − 0.281091i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.06926i 0.139821i
\(848\) 0 0
\(849\) 0.301330 0.0103416
\(850\) 0 0
\(851\) −29.9227 −1.02574
\(852\) 0 0
\(853\) 40.7894i 1.39660i 0.715804 + 0.698301i \(0.246060\pi\)
−0.715804 + 0.698301i \(0.753940\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.6354i 1.11480i 0.830243 + 0.557401i \(0.188201\pi\)
−0.830243 + 0.557401i \(0.811799\pi\)
\(858\) 0 0
\(859\) −33.4906 −1.14269 −0.571343 0.820712i \(-0.693577\pi\)
−0.571343 + 0.820712i \(0.693577\pi\)
\(860\) 0 0
\(861\) −3.13333 −0.106783
\(862\) 0 0
\(863\) 5.13795i 0.174898i 0.996169 + 0.0874489i \(0.0278714\pi\)
−0.996169 + 0.0874489i \(0.972129\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 5.90870i 0.200670i
\(868\) 0 0
\(869\) 47.1493 1.59943
\(870\) 0 0
\(871\) 63.4427 2.14967
\(872\) 0 0
\(873\) 27.7755i 0.940057i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 16.0220i 0.541026i 0.962716 + 0.270513i \(0.0871933\pi\)
−0.962716 + 0.270513i \(0.912807\pi\)
\(878\) 0 0
\(879\) 11.3680 0.383434
\(880\) 0 0
\(881\) 11.5306 0.388475 0.194238 0.980955i \(-0.437777\pi\)
0.194238 + 0.980955i \(0.437777\pi\)
\(882\) 0 0
\(883\) − 39.5161i − 1.32982i −0.746923 0.664911i \(-0.768470\pi\)
0.746923 0.664911i \(-0.231530\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.8153i 0.698910i 0.936953 + 0.349455i \(0.113633\pi\)
−0.936953 + 0.349455i \(0.886367\pi\)
\(888\) 0 0
\(889\) 16.0586 0.538588
\(890\) 0 0
\(891\) −21.3480 −0.715184
\(892\) 0 0
\(893\) 7.00933i 0.234558i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7.45208i − 0.248818i
\(898\) 0 0
\(899\) 96.9040 3.23193
\(900\) 0 0
\(901\) 7.42401 0.247330
\(902\) 0 0
\(903\) 3.85320i 0.128227i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 32.0339i − 1.06367i −0.846848 0.531835i \(-0.821503\pi\)
0.846848 0.531835i \(-0.178497\pi\)
\(908\) 0 0
\(909\) 4.16809 0.138247
\(910\) 0 0
\(911\) −2.12136 −0.0702838 −0.0351419 0.999382i \(-0.511188\pi\)
−0.0351419 + 0.999382i \(0.511188\pi\)
\(912\) 0 0
\(913\) 38.9439i 1.28886i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.56534i 0.150761i
\(918\) 0 0
\(919\) 26.4333 0.871955 0.435978 0.899957i \(-0.356403\pi\)
0.435978 + 0.899957i \(0.356403\pi\)
\(920\) 0 0
\(921\) −1.13333 −0.0373444
\(922\) 0 0
\(923\) 25.3361i 0.833948i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6.39941i 0.210184i
\(928\) 0 0
\(929\) −17.7907 −0.583695 −0.291847 0.956465i \(-0.594270\pi\)
−0.291847 + 0.956465i \(0.594270\pi\)
\(930\) 0 0
\(931\) 5.69735 0.186723
\(932\) 0 0
\(933\) − 7.78395i − 0.254835i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 7.46396i 0.243837i 0.992540 + 0.121918i \(0.0389046\pi\)
−0.992540 + 0.121918i \(0.961095\pi\)
\(938\) 0 0
\(939\) 4.40790 0.143846
\(940\) 0 0
\(941\) −2.67869 −0.0873229 −0.0436615 0.999046i \(-0.513902\pi\)
−0.0436615 + 0.999046i \(0.513902\pi\)
\(942\) 0 0
\(943\) 33.3574i 1.08627i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 42.5840i − 1.38379i −0.721996 0.691897i \(-0.756775\pi\)
0.721996 0.691897i \(-0.243225\pi\)
\(948\) 0 0
\(949\) 31.9520 1.03721
\(950\) 0 0
\(951\) 6.08528 0.197329
\(952\) 0 0
\(953\) 60.0046i 1.94374i 0.235517 + 0.971870i \(0.424322\pi\)
−0.235517 + 0.971870i \(0.575678\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 9.33609i 0.301793i
\(958\) 0 0
\(959\) 5.54537 0.179069
\(960\) 0 0
\(961\) 74.7333 2.41075
\(962\) 0 0
\(963\) − 39.1634i − 1.26202i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 61.1307i 1.96583i 0.184059 + 0.982915i \(0.441076\pi\)
−0.184059 + 0.982915i \(0.558924\pi\)
\(968\) 0 0
\(969\) −0.311977 −0.0100221
\(970\) 0 0
\(971\) 54.6213 1.75288 0.876441 0.481510i \(-0.159911\pi\)
0.876441 + 0.481510i \(0.159911\pi\)
\(972\) 0 0
\(973\) − 4.77075i − 0.152943i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 19.3060i 0.617655i 0.951118 + 0.308827i \(0.0999366\pi\)
−0.951118 + 0.308827i \(0.900063\pi\)
\(978\) 0 0
\(979\) 35.4720 1.13369
\(980\) 0 0
\(981\) −57.8455 −1.84686
\(982\) 0 0
\(983\) 25.1820i 0.803182i 0.915819 + 0.401591i \(0.131543\pi\)
−0.915819 + 0.401591i \(0.868457\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 2.90663i 0.0925189i
\(988\) 0 0
\(989\) 41.0212 1.30440
\(990\) 0 0
\(991\) −5.82003 −0.184879 −0.0924397 0.995718i \(-0.529467\pi\)
−0.0924397 + 0.995718i \(0.529467\pi\)
\(992\) 0 0
\(993\) − 2.16932i − 0.0688414i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 8.97201i − 0.284147i −0.989856 0.142073i \(-0.954623\pi\)
0.989856 0.142073i \(-0.0453769\pi\)
\(998\) 0 0
\(999\) 14.4508 0.457202
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 3800.2.d.k.3649.4 6
5.2 odd 4 760.2.a.h.1.2 3
5.3 odd 4 3800.2.a.y.1.2 3
5.4 even 2 inner 3800.2.d.k.3649.3 6
15.2 even 4 6840.2.a.bj.1.3 3
20.3 even 4 7600.2.a.bo.1.2 3
20.7 even 4 1520.2.a.r.1.2 3
40.27 even 4 6080.2.a.bs.1.2 3
40.37 odd 4 6080.2.a.bw.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.h.1.2 3 5.2 odd 4
1520.2.a.r.1.2 3 20.7 even 4
3800.2.a.y.1.2 3 5.3 odd 4
3800.2.d.k.3649.3 6 5.4 even 2 inner
3800.2.d.k.3649.4 6 1.1 even 1 trivial
6080.2.a.bs.1.2 3 40.27 even 4
6080.2.a.bw.1.2 3 40.37 odd 4
6840.2.a.bj.1.3 3 15.2 even 4
7600.2.a.bo.1.2 3 20.3 even 4