Properties

Label 2300.2.c.j.1749.3
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2300,2,Mod(1749,2300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2300.1749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 53x^{4} + 29x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.3
Root \(-0.631352i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.j.1749.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.631352i q^{3} +3.16780i q^{7} +2.60139 q^{9} +O(q^{10})\) \(q-0.631352i q^{3} +3.16780i q^{7} +2.60139 q^{9} -1.90510 q^{11} -6.76920i q^{13} -0.737296i q^{17} -3.90510 q^{19} +2.00000 q^{21} -1.00000i q^{23} -3.53645i q^{27} +0.935058 q^{29} +1.02996 q^{31} +1.20279i q^{33} -5.94008i q^{37} -4.27375 q^{39} +11.0019 q^{41} -3.90510i q^{43} -5.47151i q^{47} -3.03498 q^{49} -0.465493 q^{51} +3.07290i q^{53} +2.46549i q^{57} -4.90510 q^{59} +11.0729 q^{61} +8.24071i q^{63} +3.20279i q^{67} -0.631352 q^{69} +8.86410 q^{71} -9.40055i q^{73} -6.03498i q^{77} -1.56949 q^{79} +5.57144 q^{81} -2.64240i q^{83} -0.590351i q^{87} +18.6113 q^{89} +21.4435 q^{91} -0.650266i q^{93} +15.3486i q^{97} -4.95592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{9} + 2 q^{11} - 14 q^{19} + 16 q^{21} + 10 q^{29} + 28 q^{31} - 22 q^{39} + 6 q^{41} - 2 q^{49} + 56 q^{51} - 22 q^{59} + 44 q^{61} + 36 q^{71} - 50 q^{79} + 50 q^{91} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) − 0.631352i − 0.364511i −0.983251 0.182256i \(-0.941660\pi\)
0.983251 0.182256i \(-0.0583399\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.16780i 1.19732i 0.801004 + 0.598659i \(0.204299\pi\)
−0.801004 + 0.598659i \(0.795701\pi\)
\(8\) 0 0
\(9\) 2.60139 0.867132
\(10\) 0 0
\(11\) −1.90510 −0.574409 −0.287205 0.957869i \(-0.592726\pi\)
−0.287205 + 0.957869i \(0.592726\pi\)
\(12\) 0 0
\(13\) − 6.76920i − 1.87744i −0.344683 0.938719i \(-0.612014\pi\)
0.344683 0.938719i \(-0.387986\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.737296i − 0.178820i −0.995995 0.0894102i \(-0.971502\pi\)
0.995995 0.0894102i \(-0.0284982\pi\)
\(18\) 0 0
\(19\) −3.90510 −0.895891 −0.447946 0.894061i \(-0.647844\pi\)
−0.447946 + 0.894061i \(0.647844\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 3.53645i − 0.680591i
\(28\) 0 0
\(29\) 0.935058 0.173636 0.0868179 0.996224i \(-0.472330\pi\)
0.0868179 + 0.996224i \(0.472330\pi\)
\(30\) 0 0
\(31\) 1.02996 0.184986 0.0924929 0.995713i \(-0.470516\pi\)
0.0924929 + 0.995713i \(0.470516\pi\)
\(32\) 0 0
\(33\) 1.20279i 0.209379i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.94008i − 0.976544i −0.872692 0.488272i \(-0.837627\pi\)
0.872692 0.488272i \(-0.162373\pi\)
\(38\) 0 0
\(39\) −4.27375 −0.684347
\(40\) 0 0
\(41\) 11.0019 1.71822 0.859108 0.511795i \(-0.171019\pi\)
0.859108 + 0.511795i \(0.171019\pi\)
\(42\) 0 0
\(43\) − 3.90510i − 0.595522i −0.954640 0.297761i \(-0.903760\pi\)
0.954640 0.297761i \(-0.0962399\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 5.47151i − 0.798102i −0.916929 0.399051i \(-0.869340\pi\)
0.916929 0.399051i \(-0.130660\pi\)
\(48\) 0 0
\(49\) −3.03498 −0.433569
\(50\) 0 0
\(51\) −0.465493 −0.0651821
\(52\) 0 0
\(53\) 3.07290i 0.422096i 0.977476 + 0.211048i \(0.0676876\pi\)
−0.977476 + 0.211048i \(0.932312\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.46549i 0.326563i
\(58\) 0 0
\(59\) −4.90510 −0.638590 −0.319295 0.947655i \(-0.603446\pi\)
−0.319295 + 0.947655i \(0.603446\pi\)
\(60\) 0 0
\(61\) 11.0729 1.41774 0.708870 0.705339i \(-0.249205\pi\)
0.708870 + 0.705339i \(0.249205\pi\)
\(62\) 0 0
\(63\) 8.24071i 1.03823i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.20279i 0.391283i 0.980675 + 0.195641i \(0.0626789\pi\)
−0.980675 + 0.195641i \(0.937321\pi\)
\(68\) 0 0
\(69\) −0.631352 −0.0760059
\(70\) 0 0
\(71\) 8.86410 1.05197 0.525987 0.850492i \(-0.323696\pi\)
0.525987 + 0.850492i \(0.323696\pi\)
\(72\) 0 0
\(73\) − 9.40055i − 1.10025i −0.835082 0.550126i \(-0.814580\pi\)
0.835082 0.550126i \(-0.185420\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 6.03498i − 0.687750i
\(78\) 0 0
\(79\) −1.56949 −0.176582 −0.0882908 0.996095i \(-0.528140\pi\)
−0.0882908 + 0.996095i \(0.528140\pi\)
\(80\) 0 0
\(81\) 5.57144 0.619049
\(82\) 0 0
\(83\) − 2.64240i − 0.290041i −0.989429 0.145020i \(-0.953675\pi\)
0.989429 0.145020i \(-0.0463248\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.590351i − 0.0632922i
\(88\) 0 0
\(89\) 18.6113 1.97279 0.986397 0.164380i \(-0.0525624\pi\)
0.986397 + 0.164380i \(0.0525624\pi\)
\(90\) 0 0
\(91\) 21.4435 2.24789
\(92\) 0 0
\(93\) − 0.650266i − 0.0674294i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.3486i 1.55841i 0.626767 + 0.779207i \(0.284378\pi\)
−0.626767 + 0.779207i \(0.715622\pi\)
\(98\) 0 0
\(99\) −4.95592 −0.498088
\(100\) 0 0
\(101\) −9.18079 −0.913523 −0.456762 0.889589i \(-0.650991\pi\)
−0.456762 + 0.889589i \(0.650991\pi\)
\(102\) 0 0
\(103\) 7.50341i 0.739333i 0.929164 + 0.369667i \(0.120528\pi\)
−0.929164 + 0.369667i \(0.879472\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.5384i − 1.50215i −0.660215 0.751077i \(-0.729535\pi\)
0.660215 0.751077i \(-0.270465\pi\)
\(108\) 0 0
\(109\) −3.13282 −0.300070 −0.150035 0.988681i \(-0.547939\pi\)
−0.150035 + 0.988681i \(0.547939\pi\)
\(110\) 0 0
\(111\) −3.75029 −0.355961
\(112\) 0 0
\(113\) − 17.0130i − 1.60045i −0.599702 0.800224i \(-0.704714\pi\)
0.599702 0.800224i \(-0.295286\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 17.6094i − 1.62799i
\(118\) 0 0
\(119\) 2.33561 0.214105
\(120\) 0 0
\(121\) −7.37059 −0.670054
\(122\) 0 0
\(123\) − 6.94610i − 0.626309i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.97004i 0.263549i 0.991280 + 0.131774i \(0.0420674\pi\)
−0.991280 + 0.131774i \(0.957933\pi\)
\(128\) 0 0
\(129\) −2.46549 −0.217075
\(130\) 0 0
\(131\) −4.22967 −0.369548 −0.184774 0.982781i \(-0.559155\pi\)
−0.184774 + 0.982781i \(0.559155\pi\)
\(132\) 0 0
\(133\) − 12.3706i − 1.07267i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.52541i − 0.386632i −0.981137 0.193316i \(-0.938076\pi\)
0.981137 0.193316i \(-0.0619242\pi\)
\(138\) 0 0
\(139\) 7.00991 0.594573 0.297286 0.954788i \(-0.403918\pi\)
0.297286 + 0.954788i \(0.403918\pi\)
\(140\) 0 0
\(141\) −3.45445 −0.290917
\(142\) 0 0
\(143\) 12.8960i 1.07842i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.91614i 0.158041i
\(148\) 0 0
\(149\) −8.73113 −0.715282 −0.357641 0.933859i \(-0.616419\pi\)
−0.357641 + 0.933859i \(0.616419\pi\)
\(150\) 0 0
\(151\) 19.2697 1.56814 0.784072 0.620670i \(-0.213139\pi\)
0.784072 + 0.620670i \(0.213139\pi\)
\(152\) 0 0
\(153\) − 1.91800i − 0.155061i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 6.65529i − 0.531150i −0.964090 0.265575i \(-0.914438\pi\)
0.964090 0.265575i \(-0.0855618\pi\)
\(158\) 0 0
\(159\) 1.94008 0.153859
\(160\) 0 0
\(161\) 3.16780 0.249658
\(162\) 0 0
\(163\) − 9.11585i − 0.714009i −0.934103 0.357004i \(-0.883798\pi\)
0.934103 0.357004i \(-0.116202\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.6463i − 1.67504i −0.546407 0.837520i \(-0.684005\pi\)
0.546407 0.837520i \(-0.315995\pi\)
\(168\) 0 0
\(169\) −32.8221 −2.52477
\(170\) 0 0
\(171\) −10.1587 −0.776856
\(172\) 0 0
\(173\) − 12.8102i − 0.973941i −0.873418 0.486971i \(-0.838102\pi\)
0.873418 0.486971i \(-0.161898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 3.09685i 0.232773i
\(178\) 0 0
\(179\) −25.9769 −1.94160 −0.970801 0.239886i \(-0.922890\pi\)
−0.970801 + 0.239886i \(0.922890\pi\)
\(180\) 0 0
\(181\) −13.1966 −0.980897 −0.490449 0.871470i \(-0.663167\pi\)
−0.490449 + 0.871470i \(0.663167\pi\)
\(182\) 0 0
\(183\) − 6.99090i − 0.516782i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.40462i 0.102716i
\(188\) 0 0
\(189\) 11.2028 0.814883
\(190\) 0 0
\(191\) 19.3136 1.39748 0.698742 0.715374i \(-0.253744\pi\)
0.698742 + 0.715374i \(0.253744\pi\)
\(192\) 0 0
\(193\) 6.70132i 0.482372i 0.970479 + 0.241186i \(0.0775363\pi\)
−0.970479 + 0.241186i \(0.922464\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 10.9320i − 0.778871i −0.921054 0.389436i \(-0.872670\pi\)
0.921054 0.389436i \(-0.127330\pi\)
\(198\) 0 0
\(199\) −8.77228 −0.621850 −0.310925 0.950434i \(-0.600639\pi\)
−0.310925 + 0.950434i \(0.600639\pi\)
\(200\) 0 0
\(201\) 2.02209 0.142627
\(202\) 0 0
\(203\) 2.96208i 0.207897i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 2.60139i − 0.180809i
\(208\) 0 0
\(209\) 7.43961 0.514608
\(210\) 0 0
\(211\) −6.71530 −0.462300 −0.231150 0.972918i \(-0.574249\pi\)
−0.231150 + 0.972918i \(0.574249\pi\)
\(212\) 0 0
\(213\) − 5.59637i − 0.383457i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.26270i 0.221487i
\(218\) 0 0
\(219\) −5.93506 −0.401054
\(220\) 0 0
\(221\) −4.99090 −0.335724
\(222\) 0 0
\(223\) 21.7661i 1.45757i 0.684744 + 0.728784i \(0.259914\pi\)
−0.684744 + 0.728784i \(0.740086\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 23.5485i 1.56297i 0.623927 + 0.781483i \(0.285536\pi\)
−0.623927 + 0.781483i \(0.714464\pi\)
\(228\) 0 0
\(229\) 11.6204 0.767898 0.383949 0.923354i \(-0.374564\pi\)
0.383949 + 0.923354i \(0.374564\pi\)
\(230\) 0 0
\(231\) −3.81020 −0.250693
\(232\) 0 0
\(233\) 12.1178i 0.793863i 0.917848 + 0.396932i \(0.129925\pi\)
−0.917848 + 0.396932i \(0.870075\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.990902i 0.0643660i
\(238\) 0 0
\(239\) 5.63429 0.364452 0.182226 0.983257i \(-0.441670\pi\)
0.182226 + 0.983257i \(0.441670\pi\)
\(240\) 0 0
\(241\) −5.07907 −0.327172 −0.163586 0.986529i \(-0.552306\pi\)
−0.163586 + 0.986529i \(0.552306\pi\)
\(242\) 0 0
\(243\) − 14.1269i − 0.906241i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 26.4344i 1.68198i
\(248\) 0 0
\(249\) −1.66828 −0.105723
\(250\) 0 0
\(251\) 19.4186 1.22569 0.612845 0.790203i \(-0.290025\pi\)
0.612845 + 0.790203i \(0.290025\pi\)
\(252\) 0 0
\(253\) 1.90510i 0.119773i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 8.75507i − 0.546126i −0.961996 0.273063i \(-0.911963\pi\)
0.961996 0.273063i \(-0.0880368\pi\)
\(258\) 0 0
\(259\) 18.8170 1.16923
\(260\) 0 0
\(261\) 2.43245 0.150565
\(262\) 0 0
\(263\) − 16.3457i − 1.00792i −0.863728 0.503958i \(-0.831876\pi\)
0.863728 0.503958i \(-0.168124\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 11.7503i − 0.719106i
\(268\) 0 0
\(269\) 3.14880 0.191986 0.0959928 0.995382i \(-0.469397\pi\)
0.0959928 + 0.995382i \(0.469397\pi\)
\(270\) 0 0
\(271\) 25.1525 1.52791 0.763954 0.645271i \(-0.223255\pi\)
0.763954 + 0.645271i \(0.223255\pi\)
\(272\) 0 0
\(273\) − 13.5384i − 0.819381i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 15.8630i − 0.953113i −0.879144 0.476557i \(-0.841885\pi\)
0.879144 0.476557i \(-0.158115\pi\)
\(278\) 0 0
\(279\) 2.67933 0.160407
\(280\) 0 0
\(281\) −16.2057 −0.966752 −0.483376 0.875413i \(-0.660590\pi\)
−0.483376 + 0.875413i \(0.660590\pi\)
\(282\) 0 0
\(283\) 30.4814i 1.81193i 0.423350 + 0.905966i \(0.360854\pi\)
−0.423350 + 0.905966i \(0.639146\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 34.8520i 2.05725i
\(288\) 0 0
\(289\) 16.4564 0.968023
\(290\) 0 0
\(291\) 9.69037 0.568060
\(292\) 0 0
\(293\) 15.9339i 0.930870i 0.885082 + 0.465435i \(0.154102\pi\)
−0.885082 + 0.465435i \(0.845898\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 6.73730i 0.390938i
\(298\) 0 0
\(299\) −6.76920 −0.391473
\(300\) 0 0
\(301\) 12.3706 0.713029
\(302\) 0 0
\(303\) 5.79631i 0.332990i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.93003i 0.452591i 0.974059 + 0.226295i \(0.0726615\pi\)
−0.974059 + 0.226295i \(0.927339\pi\)
\(308\) 0 0
\(309\) 4.73730 0.269495
\(310\) 0 0
\(311\) −28.9960 −1.64421 −0.822107 0.569333i \(-0.807201\pi\)
−0.822107 + 0.569333i \(0.807201\pi\)
\(312\) 0 0
\(313\) 21.4564i 1.21279i 0.795165 + 0.606394i \(0.207384\pi\)
−0.795165 + 0.606394i \(0.792616\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.70620i 0.264327i 0.991228 + 0.132163i \(0.0421923\pi\)
−0.991228 + 0.132163i \(0.957808\pi\)
\(318\) 0 0
\(319\) −1.78138 −0.0997381
\(320\) 0 0
\(321\) −9.81020 −0.547552
\(322\) 0 0
\(323\) 2.87921i 0.160204i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.97791i 0.109379i
\(328\) 0 0
\(329\) 17.3327 0.955581
\(330\) 0 0
\(331\) −5.98611 −0.329027 −0.164513 0.986375i \(-0.552605\pi\)
−0.164513 + 0.986375i \(0.552605\pi\)
\(332\) 0 0
\(333\) − 15.4525i − 0.846792i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.82319i − 0.371683i −0.982580 0.185841i \(-0.940499\pi\)
0.982580 0.185841i \(-0.0595011\pi\)
\(338\) 0 0
\(339\) −10.7412 −0.583381
\(340\) 0 0
\(341\) −1.96217 −0.106258
\(342\) 0 0
\(343\) 12.5604i 0.678197i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.50048i − 0.187915i −0.995576 0.0939577i \(-0.970048\pi\)
0.995576 0.0939577i \(-0.0299519\pi\)
\(348\) 0 0
\(349\) 7.44042 0.398276 0.199138 0.979971i \(-0.436186\pi\)
0.199138 + 0.979971i \(0.436186\pi\)
\(350\) 0 0
\(351\) −23.9389 −1.27777
\(352\) 0 0
\(353\) − 4.87822i − 0.259642i −0.991537 0.129821i \(-0.958560\pi\)
0.991537 0.129821i \(-0.0414402\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 1.47459i − 0.0780437i
\(358\) 0 0
\(359\) −0.0570722 −0.00301216 −0.00150608 0.999999i \(-0.500479\pi\)
−0.00150608 + 0.999999i \(0.500479\pi\)
\(360\) 0 0
\(361\) −3.75019 −0.197379
\(362\) 0 0
\(363\) 4.65344i 0.244242i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.11082i − 0.214583i −0.994228 0.107292i \(-0.965782\pi\)
0.994228 0.107292i \(-0.0342179\pi\)
\(368\) 0 0
\(369\) 28.6204 1.48992
\(370\) 0 0
\(371\) −9.73436 −0.505383
\(372\) 0 0
\(373\) 26.2057i 1.35688i 0.734655 + 0.678440i \(0.237344\pi\)
−0.734655 + 0.678440i \(0.762656\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 6.32959i − 0.325991i
\(378\) 0 0
\(379\) −27.9938 −1.43795 −0.718973 0.695038i \(-0.755388\pi\)
−0.718973 + 0.695038i \(0.755388\pi\)
\(380\) 0 0
\(381\) 1.87514 0.0960665
\(382\) 0 0
\(383\) 4.16487i 0.212815i 0.994323 + 0.106407i \(0.0339348\pi\)
−0.994323 + 0.106407i \(0.966065\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 10.1587i − 0.516396i
\(388\) 0 0
\(389\) 26.4593 1.34154 0.670771 0.741665i \(-0.265963\pi\)
0.670771 + 0.741665i \(0.265963\pi\)
\(390\) 0 0
\(391\) −0.737296 −0.0372866
\(392\) 0 0
\(393\) 2.67041i 0.134704i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 4.65036i − 0.233395i −0.993168 0.116697i \(-0.962769\pi\)
0.993168 0.116697i \(-0.0372308\pi\)
\(398\) 0 0
\(399\) −7.81020 −0.390999
\(400\) 0 0
\(401\) −25.1328 −1.25507 −0.627537 0.778587i \(-0.715937\pi\)
−0.627537 + 0.778587i \(0.715937\pi\)
\(402\) 0 0
\(403\) − 6.97199i − 0.347299i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.3165i 0.560936i
\(408\) 0 0
\(409\) −16.8659 −0.833965 −0.416982 0.908915i \(-0.636912\pi\)
−0.416982 + 0.908915i \(0.636912\pi\)
\(410\) 0 0
\(411\) −2.85713 −0.140932
\(412\) 0 0
\(413\) − 15.5384i − 0.764595i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 4.42572i − 0.216728i
\(418\) 0 0
\(419\) −19.4594 −0.950655 −0.475328 0.879809i \(-0.657670\pi\)
−0.475328 + 0.879809i \(0.657670\pi\)
\(420\) 0 0
\(421\) −4.41761 −0.215301 −0.107651 0.994189i \(-0.534333\pi\)
−0.107651 + 0.994189i \(0.534333\pi\)
\(422\) 0 0
\(423\) − 14.2336i − 0.692059i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 35.0768i 1.69749i
\(428\) 0 0
\(429\) 8.14192 0.393096
\(430\) 0 0
\(431\) 1.89220 0.0911442 0.0455721 0.998961i \(-0.485489\pi\)
0.0455721 + 0.998961i \(0.485489\pi\)
\(432\) 0 0
\(433\) 25.4147i 1.22135i 0.791881 + 0.610676i \(0.209102\pi\)
−0.791881 + 0.610676i \(0.790898\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.90510i 0.186806i
\(438\) 0 0
\(439\) 15.8979 0.758768 0.379384 0.925239i \(-0.376136\pi\)
0.379384 + 0.925239i \(0.376136\pi\)
\(440\) 0 0
\(441\) −7.89519 −0.375962
\(442\) 0 0
\(443\) 2.34485i 0.111407i 0.998447 + 0.0557037i \(0.0177402\pi\)
−0.998447 + 0.0557037i \(0.982260\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 5.51242i 0.260728i
\(448\) 0 0
\(449\) 36.3774 1.71676 0.858378 0.513017i \(-0.171472\pi\)
0.858378 + 0.513017i \(0.171472\pi\)
\(450\) 0 0
\(451\) −20.9598 −0.986959
\(452\) 0 0
\(453\) − 12.1660i − 0.571606i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 18.5035i − 0.865557i −0.901500 0.432779i \(-0.857533\pi\)
0.901500 0.432779i \(-0.142467\pi\)
\(458\) 0 0
\(459\) −2.60741 −0.121704
\(460\) 0 0
\(461\) 34.0296 1.58492 0.792459 0.609925i \(-0.208801\pi\)
0.792459 + 0.609925i \(0.208801\pi\)
\(462\) 0 0
\(463\) − 16.7153i − 0.776826i −0.921485 0.388413i \(-0.873023\pi\)
0.921485 0.388413i \(-0.126977\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.0480i 0.511239i 0.966777 + 0.255620i \(0.0822795\pi\)
−0.966777 + 0.255620i \(0.917721\pi\)
\(468\) 0 0
\(469\) −10.1458 −0.468490
\(470\) 0 0
\(471\) −4.20183 −0.193610
\(472\) 0 0
\(473\) 7.43961i 0.342073i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 7.99384i 0.366013i
\(478\) 0 0
\(479\) 30.2508 1.38219 0.691096 0.722763i \(-0.257128\pi\)
0.691096 + 0.722763i \(0.257128\pi\)
\(480\) 0 0
\(481\) −40.2096 −1.83340
\(482\) 0 0
\(483\) − 2.00000i − 0.0910032i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17.9211i 0.812082i 0.913855 + 0.406041i \(0.133091\pi\)
−0.913855 + 0.406041i \(0.866909\pi\)
\(488\) 0 0
\(489\) −5.75531 −0.260264
\(490\) 0 0
\(491\) −34.5861 −1.56085 −0.780425 0.625249i \(-0.784997\pi\)
−0.780425 + 0.625249i \(0.784997\pi\)
\(492\) 0 0
\(493\) − 0.689414i − 0.0310497i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 28.0797i 1.25955i
\(498\) 0 0
\(499\) 0.653440 0.0292520 0.0146260 0.999893i \(-0.495344\pi\)
0.0146260 + 0.999893i \(0.495344\pi\)
\(500\) 0 0
\(501\) −13.6664 −0.610571
\(502\) 0 0
\(503\) 6.50048i 0.289842i 0.989443 + 0.144921i \(0.0462928\pi\)
−0.989443 + 0.144921i \(0.953707\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.7223i 0.920308i
\(508\) 0 0
\(509\) −34.6612 −1.53633 −0.768166 0.640251i \(-0.778830\pi\)
−0.768166 + 0.640251i \(0.778830\pi\)
\(510\) 0 0
\(511\) 29.7791 1.31735
\(512\) 0 0
\(513\) 13.8102i 0.609735i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 10.4238i 0.458437i
\(518\) 0 0
\(519\) −8.08775 −0.355013
\(520\) 0 0
\(521\) −12.8073 −0.561096 −0.280548 0.959840i \(-0.590516\pi\)
−0.280548 + 0.959840i \(0.590516\pi\)
\(522\) 0 0
\(523\) 37.9689i 1.66026i 0.557567 + 0.830132i \(0.311735\pi\)
−0.557567 + 0.830132i \(0.688265\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 0.759383i − 0.0330793i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −12.7601 −0.553741
\(532\) 0 0
\(533\) − 74.4744i − 3.22584i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 16.4006i 0.707736i
\(538\) 0 0
\(539\) 5.78195 0.249046
\(540\) 0 0
\(541\) 12.3088 0.529198 0.264599 0.964359i \(-0.414760\pi\)
0.264599 + 0.964359i \(0.414760\pi\)
\(542\) 0 0
\(543\) 8.33172i 0.357548i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.3009i 0.868002i 0.900912 + 0.434001i \(0.142899\pi\)
−0.900912 + 0.434001i \(0.857101\pi\)
\(548\) 0 0
\(549\) 28.8050 1.22937
\(550\) 0 0
\(551\) −3.65149 −0.155559
\(552\) 0 0
\(553\) − 4.97184i − 0.211424i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 6.59149i − 0.279290i −0.990202 0.139645i \(-0.955404\pi\)
0.990202 0.139645i \(-0.0445962\pi\)
\(558\) 0 0
\(559\) −26.4344 −1.11806
\(560\) 0 0
\(561\) 0.886811 0.0374412
\(562\) 0 0
\(563\) − 5.08191i − 0.214177i −0.994249 0.107088i \(-0.965847\pi\)
0.994249 0.107088i \(-0.0341528\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 17.6492i 0.741198i
\(568\) 0 0
\(569\) 16.8672 0.707109 0.353554 0.935414i \(-0.384973\pi\)
0.353554 + 0.935414i \(0.384973\pi\)
\(570\) 0 0
\(571\) 38.3174 1.60353 0.801767 0.597637i \(-0.203894\pi\)
0.801767 + 0.597637i \(0.203894\pi\)
\(572\) 0 0
\(573\) − 12.1937i − 0.509399i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 19.0240i 0.791981i 0.918255 + 0.395990i \(0.129599\pi\)
−0.918255 + 0.395990i \(0.870401\pi\)
\(578\) 0 0
\(579\) 4.23089 0.175830
\(580\) 0 0
\(581\) 8.37059 0.347271
\(582\) 0 0
\(583\) − 5.85419i − 0.242456i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.66747i 0.233922i 0.993137 + 0.116961i \(0.0373152\pi\)
−0.993137 + 0.116961i \(0.962685\pi\)
\(588\) 0 0
\(589\) −4.02209 −0.165727
\(590\) 0 0
\(591\) −6.90193 −0.283907
\(592\) 0 0
\(593\) 35.7130i 1.46656i 0.679928 + 0.733279i \(0.262011\pi\)
−0.679928 + 0.733279i \(0.737989\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.53840i 0.226672i
\(598\) 0 0
\(599\) −20.1717 −0.824193 −0.412097 0.911140i \(-0.635203\pi\)
−0.412097 + 0.911140i \(0.635203\pi\)
\(600\) 0 0
\(601\) −41.9069 −1.70942 −0.854709 0.519107i \(-0.826264\pi\)
−0.854709 + 0.519107i \(0.826264\pi\)
\(602\) 0 0
\(603\) 8.33172i 0.339294i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 35.8678i − 1.45583i −0.685667 0.727915i \(-0.740489\pi\)
0.685667 0.727915i \(-0.259511\pi\)
\(608\) 0 0
\(609\) 1.87012 0.0757809
\(610\) 0 0
\(611\) −37.0377 −1.49839
\(612\) 0 0
\(613\) 21.7819i 0.879765i 0.898055 + 0.439882i \(0.144980\pi\)
−0.898055 + 0.439882i \(0.855020\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 13.8480i − 0.557501i −0.960364 0.278750i \(-0.910080\pi\)
0.960364 0.278750i \(-0.0899202\pi\)
\(618\) 0 0
\(619\) −38.6113 −1.55192 −0.775960 0.630783i \(-0.782734\pi\)
−0.775960 + 0.630783i \(0.782734\pi\)
\(620\) 0 0
\(621\) −3.53645 −0.141913
\(622\) 0 0
\(623\) 58.9570i 2.36206i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 4.69701i − 0.187581i
\(628\) 0 0
\(629\) −4.37960 −0.174626
\(630\) 0 0
\(631\) −0.598221 −0.0238148 −0.0119074 0.999929i \(-0.503790\pi\)
−0.0119074 + 0.999929i \(0.503790\pi\)
\(632\) 0 0
\(633\) 4.23972i 0.168514i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 20.5444i 0.813999i
\(638\) 0 0
\(639\) 23.0590 0.912201
\(640\) 0 0
\(641\) −22.5254 −0.889700 −0.444850 0.895605i \(-0.646743\pi\)
−0.444850 + 0.895605i \(0.646743\pi\)
\(642\) 0 0
\(643\) 36.1908i 1.42723i 0.700539 + 0.713614i \(0.252943\pi\)
−0.700539 + 0.713614i \(0.747057\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 6.33067i − 0.248884i −0.992227 0.124442i \(-0.960286\pi\)
0.992227 0.124442i \(-0.0397141\pi\)
\(648\) 0 0
\(649\) 9.34471 0.366812
\(650\) 0 0
\(651\) 2.05992 0.0807344
\(652\) 0 0
\(653\) 14.3057i 0.559827i 0.960025 + 0.279914i \(0.0903058\pi\)
−0.960025 + 0.279914i \(0.909694\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 24.4545i − 0.954063i
\(658\) 0 0
\(659\) 38.6563 1.50584 0.752919 0.658114i \(-0.228645\pi\)
0.752919 + 0.658114i \(0.228645\pi\)
\(660\) 0 0
\(661\) 12.5954 0.489903 0.244952 0.969535i \(-0.421228\pi\)
0.244952 + 0.969535i \(0.421228\pi\)
\(662\) 0 0
\(663\) 3.15102i 0.122375i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 0.935058i − 0.0362056i
\(668\) 0 0
\(669\) 13.7421 0.531300
\(670\) 0 0
\(671\) −21.0950 −0.814363
\(672\) 0 0
\(673\) − 0.356707i − 0.0137500i −0.999976 0.00687501i \(-0.997812\pi\)
0.999976 0.00687501i \(-0.00218840\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.8831i − 0.725737i −0.931840 0.362868i \(-0.881797\pi\)
0.931840 0.362868i \(-0.118203\pi\)
\(678\) 0 0
\(679\) −48.6214 −1.86592
\(680\) 0 0
\(681\) 14.8674 0.569719
\(682\) 0 0
\(683\) 18.7520i 0.717525i 0.933429 + 0.358763i \(0.116801\pi\)
−0.933429 + 0.358763i \(0.883199\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 7.33656i − 0.279907i
\(688\) 0 0
\(689\) 20.8011 0.792459
\(690\) 0 0
\(691\) −18.3175 −0.696831 −0.348415 0.937340i \(-0.613280\pi\)
−0.348415 + 0.937340i \(0.613280\pi\)
\(692\) 0 0
\(693\) − 15.6994i − 0.596370i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 8.11169i − 0.307252i
\(698\) 0 0
\(699\) 7.65060 0.289372
\(700\) 0 0
\(701\) 27.0007 1.01980 0.509900 0.860233i \(-0.329682\pi\)
0.509900 + 0.860233i \(0.329682\pi\)
\(702\) 0 0
\(703\) 23.1966i 0.874877i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 29.0830i − 1.09378i
\(708\) 0 0
\(709\) −16.1875 −0.607935 −0.303968 0.952682i \(-0.598311\pi\)
−0.303968 + 0.952682i \(0.598311\pi\)
\(710\) 0 0
\(711\) −4.08287 −0.153119
\(712\) 0 0
\(713\) − 1.02996i − 0.0385722i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 3.55722i − 0.132847i
\(718\) 0 0
\(719\) 49.2926 1.83830 0.919151 0.393904i \(-0.128876\pi\)
0.919151 + 0.393904i \(0.128876\pi\)
\(720\) 0 0
\(721\) −23.7693 −0.885217
\(722\) 0 0
\(723\) 3.20668i 0.119258i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 23.2447i − 0.862098i −0.902329 0.431049i \(-0.858144\pi\)
0.902329 0.431049i \(-0.141856\pi\)
\(728\) 0 0
\(729\) 7.79527 0.288714
\(730\) 0 0
\(731\) −2.87921 −0.106492
\(732\) 0 0
\(733\) − 25.8203i − 0.953693i −0.878987 0.476846i \(-0.841780\pi\)
0.878987 0.476846i \(-0.158220\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 6.10163i − 0.224757i
\(738\) 0 0
\(739\) −21.6853 −0.797705 −0.398852 0.917015i \(-0.630591\pi\)
−0.398852 + 0.917015i \(0.630591\pi\)
\(740\) 0 0
\(741\) 16.6894 0.613101
\(742\) 0 0
\(743\) − 27.5932i − 1.01230i −0.862447 0.506148i \(-0.831069\pi\)
0.862447 0.506148i \(-0.168931\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 6.87391i − 0.251503i
\(748\) 0 0
\(749\) 49.2226 1.79855
\(750\) 0 0
\(751\) −48.2644 −1.76119 −0.880597 0.473866i \(-0.842858\pi\)
−0.880597 + 0.473866i \(0.842858\pi\)
\(752\) 0 0
\(753\) − 12.2600i − 0.446778i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 5.87012i − 0.213353i −0.994294 0.106676i \(-0.965979\pi\)
0.994294 0.106676i \(-0.0340209\pi\)
\(758\) 0 0
\(759\) 1.20279 0.0436585
\(760\) 0 0
\(761\) −5.67624 −0.205764 −0.102882 0.994694i \(-0.532806\pi\)
−0.102882 + 0.994694i \(0.532806\pi\)
\(762\) 0 0
\(763\) − 9.92416i − 0.359279i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 33.2036i 1.19891i
\(768\) 0 0
\(769\) −17.3226 −0.624670 −0.312335 0.949972i \(-0.601111\pi\)
−0.312335 + 0.949972i \(0.601111\pi\)
\(770\) 0 0
\(771\) −5.52753 −0.199069
\(772\) 0 0
\(773\) 28.3773i 1.02066i 0.859978 + 0.510331i \(0.170477\pi\)
−0.859978 + 0.510331i \(0.829523\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 11.8802i − 0.426199i
\(778\) 0 0
\(779\) −42.9637 −1.53933
\(780\) 0 0
\(781\) −16.8870 −0.604264
\(782\) 0 0
\(783\) − 3.30679i − 0.118175i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 31.3995i − 1.11927i −0.828739 0.559636i \(-0.810941\pi\)
0.828739 0.559636i \(-0.189059\pi\)
\(788\) 0 0
\(789\) −10.3199 −0.367397
\(790\) 0 0
\(791\) 53.8938 1.91624
\(792\) 0 0
\(793\) − 74.9547i − 2.66172i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0169i 0.744456i 0.928141 + 0.372228i \(0.121406\pi\)
−0.928141 + 0.372228i \(0.878594\pi\)
\(798\) 0 0
\(799\) −4.03412 −0.142717
\(800\) 0 0
\(801\) 48.4153 1.71067
\(802\) 0 0
\(803\) 17.9090i 0.631995i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.98800i − 0.0699809i
\(808\) 0 0
\(809\) −25.6712 −0.902552 −0.451276 0.892384i \(-0.649031\pi\)
−0.451276 + 0.892384i \(0.649031\pi\)
\(810\) 0 0
\(811\) 40.3647 1.41740 0.708698 0.705512i \(-0.249283\pi\)
0.708698 + 0.705512i \(0.249283\pi\)
\(812\) 0 0
\(813\) − 15.8801i − 0.556940i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 15.2498i 0.533523i
\(818\) 0 0
\(819\) 55.7830 1.94922
\(820\) 0 0
\(821\) 25.6396 0.894827 0.447413 0.894327i \(-0.352345\pi\)
0.447413 + 0.894327i \(0.352345\pi\)
\(822\) 0 0
\(823\) 15.6726i 0.546312i 0.961970 + 0.273156i \(0.0880676\pi\)
−0.961970 + 0.273156i \(0.911932\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 45.8429i − 1.59411i −0.603904 0.797057i \(-0.706389\pi\)
0.603904 0.797057i \(-0.293611\pi\)
\(828\) 0 0
\(829\) 15.6214 0.542552 0.271276 0.962502i \(-0.412554\pi\)
0.271276 + 0.962502i \(0.412554\pi\)
\(830\) 0 0
\(831\) −10.0151 −0.347420
\(832\) 0 0
\(833\) 2.23768i 0.0775311i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 3.64240i − 0.125900i
\(838\) 0 0
\(839\) −35.7853 −1.23544 −0.617722 0.786396i \(-0.711944\pi\)
−0.617722 + 0.786396i \(0.711944\pi\)
\(840\) 0 0
\(841\) −28.1257 −0.969851
\(842\) 0 0
\(843\) 10.2315i 0.352392i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 23.3486i − 0.802267i
\(848\) 0 0
\(849\) 19.2445 0.660470
\(850\) 0 0
\(851\) −5.94008 −0.203623
\(852\) 0 0
\(853\) 28.0350i 0.959900i 0.877296 + 0.479950i \(0.159345\pi\)
−0.877296 + 0.479950i \(0.840655\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.37562i 0.320265i 0.987096 + 0.160133i \(0.0511922\pi\)
−0.987096 + 0.160133i \(0.948808\pi\)
\(858\) 0 0
\(859\) 9.85704 0.336318 0.168159 0.985760i \(-0.446218\pi\)
0.168159 + 0.985760i \(0.446218\pi\)
\(860\) 0 0
\(861\) 22.0039 0.749891
\(862\) 0 0
\(863\) − 53.1607i − 1.80961i −0.425827 0.904805i \(-0.640017\pi\)
0.425827 0.904805i \(-0.359983\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 10.3898i − 0.352855i
\(868\) 0 0
\(869\) 2.99004 0.101430
\(870\) 0 0
\(871\) 21.6803 0.734609
\(872\) 0 0
\(873\) 39.9278i 1.35135i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 55.8780i 1.88687i 0.331564 + 0.943433i \(0.392424\pi\)
−0.331564 + 0.943433i \(0.607576\pi\)
\(878\) 0 0
\(879\) 10.0599 0.339313
\(880\) 0 0
\(881\) 35.8961 1.20937 0.604685 0.796465i \(-0.293299\pi\)
0.604685 + 0.796465i \(0.293299\pi\)
\(882\) 0 0
\(883\) − 31.2225i − 1.05072i −0.850880 0.525361i \(-0.823930\pi\)
0.850880 0.525361i \(-0.176070\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.3094i 1.48776i 0.668311 + 0.743882i \(0.267018\pi\)
−0.668311 + 0.743882i \(0.732982\pi\)
\(888\) 0 0
\(889\) −9.40851 −0.315551
\(890\) 0 0
\(891\) −10.6141 −0.355587
\(892\) 0 0
\(893\) 21.3668i 0.715013i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.27375i 0.142696i
\(898\) 0 0
\(899\) 0.963070 0.0321202
\(900\) 0 0
\(901\) 2.26564 0.0754794
\(902\) 0 0
\(903\) − 7.81020i − 0.259907i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 50.5424i 1.67823i 0.543952 + 0.839116i \(0.316927\pi\)
−0.543952 + 0.839116i \(0.683073\pi\)
\(908\) 0 0
\(909\) −23.8829 −0.792145
\(910\) 0 0
\(911\) 0.446726 0.0148007 0.00740034 0.999973i \(-0.497644\pi\)
0.00740034 + 0.999973i \(0.497644\pi\)
\(912\) 0 0
\(913\) 5.03403i 0.166602i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 13.3988i − 0.442466i
\(918\) 0 0
\(919\) −36.6373 −1.20855 −0.604276 0.796775i \(-0.706538\pi\)
−0.604276 + 0.796775i \(0.706538\pi\)
\(920\) 0 0
\(921\) 5.00664 0.164974
\(922\) 0 0
\(923\) − 60.0028i − 1.97502i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 19.5193i 0.641099i
\(928\) 0 0
\(929\) 17.8243 0.584797 0.292399 0.956297i \(-0.405547\pi\)
0.292399 + 0.956297i \(0.405547\pi\)
\(930\) 0 0
\(931\) 11.8519 0.388431
\(932\) 0 0
\(933\) 18.3067i 0.599334i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 0.503505i − 0.0164488i −0.999966 0.00822440i \(-0.997382\pi\)
0.999966 0.00822440i \(-0.00261794\pi\)
\(938\) 0 0
\(939\) 13.5465 0.442075
\(940\) 0 0
\(941\) −42.3593 −1.38087 −0.690437 0.723392i \(-0.742582\pi\)
−0.690437 + 0.723392i \(0.742582\pi\)
\(942\) 0 0
\(943\) − 11.0019i − 0.358273i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.89002i 0.158904i 0.996839 + 0.0794521i \(0.0253171\pi\)
−0.996839 + 0.0794521i \(0.974683\pi\)
\(948\) 0 0
\(949\) −63.6342 −2.06565
\(950\) 0 0
\(951\) 2.97127 0.0963501
\(952\) 0 0
\(953\) − 8.04399i − 0.260570i −0.991477 0.130285i \(-0.958411\pi\)
0.991477 0.130285i \(-0.0415893\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 1.12468i 0.0363557i
\(958\) 0 0
\(959\) 14.3356 0.462921
\(960\) 0 0
\(961\) −29.9392 −0.965780
\(962\) 0 0
\(963\) − 40.4215i − 1.30256i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 21.1938i − 0.681548i −0.940145 0.340774i \(-0.889311\pi\)
0.940145 0.340774i \(-0.110689\pi\)
\(968\) 0 0
\(969\) 1.81780 0.0583961
\(970\) 0 0
\(971\) 61.5773 1.97611 0.988054 0.154106i \(-0.0492496\pi\)
0.988054 + 0.154106i \(0.0492496\pi\)
\(972\) 0 0
\(973\) 22.2060i 0.711892i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 43.8339i − 1.40237i −0.712979 0.701185i \(-0.752654\pi\)
0.712979 0.701185i \(-0.247346\pi\)
\(978\) 0 0
\(979\) −35.4564 −1.13319
\(980\) 0 0
\(981\) −8.14970 −0.260200
\(982\) 0 0
\(983\) − 37.6752i − 1.20165i −0.799380 0.600826i \(-0.794838\pi\)
0.799380 0.600826i \(-0.205162\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 10.9430i − 0.348320i
\(988\) 0 0
\(989\) −3.90510 −0.124175
\(990\) 0 0
\(991\) 17.1831 0.545838 0.272919 0.962037i \(-0.412011\pi\)
0.272919 + 0.962037i \(0.412011\pi\)
\(992\) 0 0
\(993\) 3.77935i 0.119934i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 13.3297i 0.422157i 0.977469 + 0.211079i \(0.0676976\pi\)
−0.977469 + 0.211079i \(0.932302\pi\)
\(998\) 0 0
\(999\) −21.0068 −0.664627
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 2300.2.c.j.1749.3 8
5.2 odd 4 2300.2.a.m.1.2 yes 4
5.3 odd 4 2300.2.a.l.1.3 4
5.4 even 2 inner 2300.2.c.j.1749.6 8
20.3 even 4 9200.2.a.co.1.2 4
20.7 even 4 9200.2.a.cm.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2300.2.a.l.1.3 4 5.3 odd 4
2300.2.a.m.1.2 yes 4 5.2 odd 4
2300.2.c.j.1749.3 8 1.1 even 1 trivial
2300.2.c.j.1749.6 8 5.4 even 2 inner
9200.2.a.cm.1.3 4 20.7 even 4
9200.2.a.co.1.2 4 20.3 even 4