Properties

Label 1216.3.d.d.191.8
Level $1216$
Weight $3$
Character 1216.191
Analytic conductor $33.134$
Analytic rank $0$
Dimension $14$
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] = [1216,3,Mod(191,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1216.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.8
Root \(-1.92254 - 0.551226i\) of defining polynomial
Character \(\chi\) \(=\) 1216.191
Dual form 1216.3.d.d.191.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.644704i q^{3} +2.32715 q^{5} -8.62924i q^{7} +8.58436 q^{9} +O(q^{10})\) \(q+0.644704i q^{3} +2.32715 q^{5} -8.62924i q^{7} +8.58436 q^{9} +19.2717i q^{11} -13.8067 q^{13} +1.50032i q^{15} -12.5180 q^{17} +4.35890i q^{19} +5.56330 q^{21} +37.4981i q^{23} -19.5844 q^{25} +11.3367i q^{27} +6.36930 q^{29} +5.44851i q^{31} -12.4245 q^{33} -20.0816i q^{35} -20.9250 q^{37} -8.90120i q^{39} +72.8337 q^{41} +10.1553i q^{43} +19.9771 q^{45} +32.4450i q^{47} -25.4638 q^{49} -8.07040i q^{51} +42.9339 q^{53} +44.8482i q^{55} -2.81020 q^{57} +38.9728i q^{59} -25.5611 q^{61} -74.0765i q^{63} -32.1302 q^{65} -65.3183i q^{67} -24.1751 q^{69} +18.7557i q^{71} +72.8251 q^{73} -12.6261i q^{75} +166.300 q^{77} +139.874i q^{79} +69.9504 q^{81} +94.7116i q^{83} -29.1313 q^{85} +4.10631i q^{87} +33.3546 q^{89} +119.141i q^{91} -3.51267 q^{93} +10.1438i q^{95} -150.817 q^{97} +165.435i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 68 q^{9} - 54 q^{13} + 34 q^{17} + 38 q^{21} - 86 q^{25} - 54 q^{29} + 20 q^{33} - 100 q^{37} + 224 q^{41} + 168 q^{45} - 220 q^{49} - 14 q^{53} + 38 q^{57} - 28 q^{61} - 472 q^{65} - 122 q^{69} + 70 q^{73} - 228 q^{77} + 334 q^{81} - 48 q^{85} + 176 q^{93} + 308 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.644704i 0.214901i 0.994210 + 0.107451i \(0.0342688\pi\)
−0.994210 + 0.107451i \(0.965731\pi\)
\(4\) 0 0
\(5\) 2.32715 0.465431 0.232715 0.972545i \(-0.425239\pi\)
0.232715 + 0.972545i \(0.425239\pi\)
\(6\) 0 0
\(7\) − 8.62924i − 1.23275i −0.787453 0.616374i \(-0.788601\pi\)
0.787453 0.616374i \(-0.211399\pi\)
\(8\) 0 0
\(9\) 8.58436 0.953817
\(10\) 0 0
\(11\) 19.2717i 1.75197i 0.482334 + 0.875987i \(0.339789\pi\)
−0.482334 + 0.875987i \(0.660211\pi\)
\(12\) 0 0
\(13\) −13.8067 −1.06205 −0.531025 0.847356i \(-0.678193\pi\)
−0.531025 + 0.847356i \(0.678193\pi\)
\(14\) 0 0
\(15\) 1.50032i 0.100022i
\(16\) 0 0
\(17\) −12.5180 −0.736353 −0.368177 0.929756i \(-0.620018\pi\)
−0.368177 + 0.929756i \(0.620018\pi\)
\(18\) 0 0
\(19\) 4.35890i 0.229416i
\(20\) 0 0
\(21\) 5.56330 0.264919
\(22\) 0 0
\(23\) 37.4981i 1.63035i 0.579214 + 0.815175i \(0.303360\pi\)
−0.579214 + 0.815175i \(0.696640\pi\)
\(24\) 0 0
\(25\) −19.5844 −0.783374
\(26\) 0 0
\(27\) 11.3367i 0.419878i
\(28\) 0 0
\(29\) 6.36930 0.219631 0.109816 0.993952i \(-0.464974\pi\)
0.109816 + 0.993952i \(0.464974\pi\)
\(30\) 0 0
\(31\) 5.44851i 0.175758i 0.996131 + 0.0878791i \(0.0280089\pi\)
−0.996131 + 0.0878791i \(0.971991\pi\)
\(32\) 0 0
\(33\) −12.4245 −0.376502
\(34\) 0 0
\(35\) − 20.0816i − 0.573759i
\(36\) 0 0
\(37\) −20.9250 −0.565542 −0.282771 0.959187i \(-0.591254\pi\)
−0.282771 + 0.959187i \(0.591254\pi\)
\(38\) 0 0
\(39\) − 8.90120i − 0.228236i
\(40\) 0 0
\(41\) 72.8337 1.77643 0.888216 0.459425i \(-0.151945\pi\)
0.888216 + 0.459425i \(0.151945\pi\)
\(42\) 0 0
\(43\) 10.1553i 0.236171i 0.993003 + 0.118085i \(0.0376757\pi\)
−0.993003 + 0.118085i \(0.962324\pi\)
\(44\) 0 0
\(45\) 19.9771 0.443936
\(46\) 0 0
\(47\) 32.4450i 0.690318i 0.938544 + 0.345159i \(0.112175\pi\)
−0.938544 + 0.345159i \(0.887825\pi\)
\(48\) 0 0
\(49\) −25.4638 −0.519669
\(50\) 0 0
\(51\) − 8.07040i − 0.158243i
\(52\) 0 0
\(53\) 42.9339 0.810074 0.405037 0.914300i \(-0.367259\pi\)
0.405037 + 0.914300i \(0.367259\pi\)
\(54\) 0 0
\(55\) 44.8482i 0.815423i
\(56\) 0 0
\(57\) −2.81020 −0.0493017
\(58\) 0 0
\(59\) 38.9728i 0.660556i 0.943884 + 0.330278i \(0.107142\pi\)
−0.943884 + 0.330278i \(0.892858\pi\)
\(60\) 0 0
\(61\) −25.5611 −0.419034 −0.209517 0.977805i \(-0.567189\pi\)
−0.209517 + 0.977805i \(0.567189\pi\)
\(62\) 0 0
\(63\) − 74.0765i − 1.17582i
\(64\) 0 0
\(65\) −32.1302 −0.494311
\(66\) 0 0
\(67\) − 65.3183i − 0.974900i −0.873151 0.487450i \(-0.837927\pi\)
0.873151 0.487450i \(-0.162073\pi\)
\(68\) 0 0
\(69\) −24.1751 −0.350364
\(70\) 0 0
\(71\) 18.7557i 0.264165i 0.991239 + 0.132083i \(0.0421665\pi\)
−0.991239 + 0.132083i \(0.957834\pi\)
\(72\) 0 0
\(73\) 72.8251 0.997604 0.498802 0.866716i \(-0.333773\pi\)
0.498802 + 0.866716i \(0.333773\pi\)
\(74\) 0 0
\(75\) − 12.6261i − 0.168348i
\(76\) 0 0
\(77\) 166.300 2.15974
\(78\) 0 0
\(79\) 139.874i 1.77056i 0.465060 + 0.885279i \(0.346033\pi\)
−0.465060 + 0.885279i \(0.653967\pi\)
\(80\) 0 0
\(81\) 69.9504 0.863585
\(82\) 0 0
\(83\) 94.7116i 1.14110i 0.821262 + 0.570552i \(0.193271\pi\)
−0.821262 + 0.570552i \(0.806729\pi\)
\(84\) 0 0
\(85\) −29.1313 −0.342721
\(86\) 0 0
\(87\) 4.10631i 0.0471990i
\(88\) 0 0
\(89\) 33.3546 0.374770 0.187385 0.982287i \(-0.439999\pi\)
0.187385 + 0.982287i \(0.439999\pi\)
\(90\) 0 0
\(91\) 119.141i 1.30924i
\(92\) 0 0
\(93\) −3.51267 −0.0377707
\(94\) 0 0
\(95\) 10.1438i 0.106777i
\(96\) 0 0
\(97\) −150.817 −1.55482 −0.777408 0.628996i \(-0.783466\pi\)
−0.777408 + 0.628996i \(0.783466\pi\)
\(98\) 0 0
\(99\) 165.435i 1.67106i
\(100\) 0 0
\(101\) −77.7745 −0.770045 −0.385022 0.922907i \(-0.625806\pi\)
−0.385022 + 0.922907i \(0.625806\pi\)
\(102\) 0 0
\(103\) 14.9481i 0.145127i 0.997364 + 0.0725637i \(0.0231180\pi\)
−0.997364 + 0.0725637i \(0.976882\pi\)
\(104\) 0 0
\(105\) 12.9467 0.123302
\(106\) 0 0
\(107\) − 80.5205i − 0.752528i −0.926512 0.376264i \(-0.877208\pi\)
0.926512 0.376264i \(-0.122792\pi\)
\(108\) 0 0
\(109\) −53.8512 −0.494048 −0.247024 0.969009i \(-0.579453\pi\)
−0.247024 + 0.969009i \(0.579453\pi\)
\(110\) 0 0
\(111\) − 13.4905i − 0.121536i
\(112\) 0 0
\(113\) 105.921 0.937355 0.468677 0.883369i \(-0.344731\pi\)
0.468677 + 0.883369i \(0.344731\pi\)
\(114\) 0 0
\(115\) 87.2638i 0.758815i
\(116\) 0 0
\(117\) −118.521 −1.01300
\(118\) 0 0
\(119\) 108.021i 0.907739i
\(120\) 0 0
\(121\) −250.399 −2.06941
\(122\) 0 0
\(123\) 46.9562i 0.381758i
\(124\) 0 0
\(125\) −103.755 −0.830037
\(126\) 0 0
\(127\) − 17.1409i − 0.134968i −0.997720 0.0674839i \(-0.978503\pi\)
0.997720 0.0674839i \(-0.0214971\pi\)
\(128\) 0 0
\(129\) −6.54719 −0.0507534
\(130\) 0 0
\(131\) 229.336i 1.75066i 0.483526 + 0.875330i \(0.339356\pi\)
−0.483526 + 0.875330i \(0.660644\pi\)
\(132\) 0 0
\(133\) 37.6140 0.282812
\(134\) 0 0
\(135\) 26.3822i 0.195424i
\(136\) 0 0
\(137\) −120.567 −0.880054 −0.440027 0.897985i \(-0.645031\pi\)
−0.440027 + 0.897985i \(0.645031\pi\)
\(138\) 0 0
\(139\) − 145.864i − 1.04938i −0.851292 0.524692i \(-0.824181\pi\)
0.851292 0.524692i \(-0.175819\pi\)
\(140\) 0 0
\(141\) −20.9174 −0.148350
\(142\) 0 0
\(143\) − 266.078i − 1.86069i
\(144\) 0 0
\(145\) 14.8223 0.102223
\(146\) 0 0
\(147\) − 16.4166i − 0.111678i
\(148\) 0 0
\(149\) −19.2547 −0.129226 −0.0646132 0.997910i \(-0.520581\pi\)
−0.0646132 + 0.997910i \(0.520581\pi\)
\(150\) 0 0
\(151\) 165.526i 1.09620i 0.836414 + 0.548099i \(0.184648\pi\)
−0.836414 + 0.548099i \(0.815352\pi\)
\(152\) 0 0
\(153\) −107.459 −0.702347
\(154\) 0 0
\(155\) 12.6795i 0.0818033i
\(156\) 0 0
\(157\) −149.449 −0.951907 −0.475954 0.879470i \(-0.657897\pi\)
−0.475954 + 0.879470i \(0.657897\pi\)
\(158\) 0 0
\(159\) 27.6796i 0.174086i
\(160\) 0 0
\(161\) 323.580 2.00981
\(162\) 0 0
\(163\) − 146.763i − 0.900384i −0.892932 0.450192i \(-0.851356\pi\)
0.892932 0.450192i \(-0.148644\pi\)
\(164\) 0 0
\(165\) −28.9138 −0.175235
\(166\) 0 0
\(167\) 213.493i 1.27840i 0.769040 + 0.639201i \(0.220735\pi\)
−0.769040 + 0.639201i \(0.779265\pi\)
\(168\) 0 0
\(169\) 21.6237 0.127951
\(170\) 0 0
\(171\) 37.4183i 0.218821i
\(172\) 0 0
\(173\) 280.715 1.62263 0.811314 0.584610i \(-0.198753\pi\)
0.811314 + 0.584610i \(0.198753\pi\)
\(174\) 0 0
\(175\) 168.998i 0.965704i
\(176\) 0 0
\(177\) −25.1259 −0.141954
\(178\) 0 0
\(179\) 35.1878i 0.196580i 0.995158 + 0.0982899i \(0.0313372\pi\)
−0.995158 + 0.0982899i \(0.968663\pi\)
\(180\) 0 0
\(181\) 345.539 1.90906 0.954529 0.298120i \(-0.0963594\pi\)
0.954529 + 0.298120i \(0.0963594\pi\)
\(182\) 0 0
\(183\) − 16.4793i − 0.0900509i
\(184\) 0 0
\(185\) −48.6958 −0.263220
\(186\) 0 0
\(187\) − 241.244i − 1.29007i
\(188\) 0 0
\(189\) 97.8271 0.517604
\(190\) 0 0
\(191\) − 100.233i − 0.524782i −0.964962 0.262391i \(-0.915489\pi\)
0.964962 0.262391i \(-0.0845110\pi\)
\(192\) 0 0
\(193\) 168.119 0.871085 0.435543 0.900168i \(-0.356557\pi\)
0.435543 + 0.900168i \(0.356557\pi\)
\(194\) 0 0
\(195\) − 20.7145i − 0.106228i
\(196\) 0 0
\(197\) −216.705 −1.10002 −0.550012 0.835157i \(-0.685377\pi\)
−0.550012 + 0.835157i \(0.685377\pi\)
\(198\) 0 0
\(199\) 82.2545i 0.413339i 0.978411 + 0.206670i \(0.0662625\pi\)
−0.978411 + 0.206670i \(0.933738\pi\)
\(200\) 0 0
\(201\) 42.1109 0.209507
\(202\) 0 0
\(203\) − 54.9623i − 0.270750i
\(204\) 0 0
\(205\) 169.495 0.826806
\(206\) 0 0
\(207\) 321.897i 1.55506i
\(208\) 0 0
\(209\) −84.0035 −0.401931
\(210\) 0 0
\(211\) − 16.4195i − 0.0778173i −0.999243 0.0389087i \(-0.987612\pi\)
0.999243 0.0389087i \(-0.0123881\pi\)
\(212\) 0 0
\(213\) −12.0919 −0.0567695
\(214\) 0 0
\(215\) 23.6331i 0.109921i
\(216\) 0 0
\(217\) 47.0165 0.216666
\(218\) 0 0
\(219\) 46.9506i 0.214386i
\(220\) 0 0
\(221\) 172.832 0.782044
\(222\) 0 0
\(223\) − 195.225i − 0.875450i −0.899109 0.437725i \(-0.855784\pi\)
0.899109 0.437725i \(-0.144216\pi\)
\(224\) 0 0
\(225\) −168.119 −0.747196
\(226\) 0 0
\(227\) 305.403i 1.34539i 0.739920 + 0.672695i \(0.234863\pi\)
−0.739920 + 0.672695i \(0.765137\pi\)
\(228\) 0 0
\(229\) 229.686 1.00300 0.501499 0.865158i \(-0.332782\pi\)
0.501499 + 0.865158i \(0.332782\pi\)
\(230\) 0 0
\(231\) 107.214i 0.464132i
\(232\) 0 0
\(233\) 187.667 0.805436 0.402718 0.915324i \(-0.368065\pi\)
0.402718 + 0.915324i \(0.368065\pi\)
\(234\) 0 0
\(235\) 75.5044i 0.321295i
\(236\) 0 0
\(237\) −90.1774 −0.380495
\(238\) 0 0
\(239\) − 69.8504i − 0.292261i −0.989265 0.146130i \(-0.953318\pi\)
0.989265 0.146130i \(-0.0466819\pi\)
\(240\) 0 0
\(241\) 297.561 1.23469 0.617346 0.786691i \(-0.288208\pi\)
0.617346 + 0.786691i \(0.288208\pi\)
\(242\) 0 0
\(243\) 147.128i 0.605463i
\(244\) 0 0
\(245\) −59.2582 −0.241870
\(246\) 0 0
\(247\) − 60.1818i − 0.243651i
\(248\) 0 0
\(249\) −61.0609 −0.245225
\(250\) 0 0
\(251\) 36.4667i 0.145286i 0.997358 + 0.0726429i \(0.0231433\pi\)
−0.997358 + 0.0726429i \(0.976857\pi\)
\(252\) 0 0
\(253\) −722.652 −2.85633
\(254\) 0 0
\(255\) − 18.7811i − 0.0736513i
\(256\) 0 0
\(257\) −175.943 −0.684604 −0.342302 0.939590i \(-0.611207\pi\)
−0.342302 + 0.939590i \(0.611207\pi\)
\(258\) 0 0
\(259\) 180.567i 0.697171i
\(260\) 0 0
\(261\) 54.6764 0.209488
\(262\) 0 0
\(263\) − 444.887i − 1.69158i −0.533512 0.845792i \(-0.679128\pi\)
0.533512 0.845792i \(-0.320872\pi\)
\(264\) 0 0
\(265\) 99.9138 0.377033
\(266\) 0 0
\(267\) 21.5038i 0.0805386i
\(268\) 0 0
\(269\) −149.744 −0.556670 −0.278335 0.960484i \(-0.589783\pi\)
−0.278335 + 0.960484i \(0.589783\pi\)
\(270\) 0 0
\(271\) − 439.685i − 1.62245i −0.584731 0.811227i \(-0.698800\pi\)
0.584731 0.811227i \(-0.301200\pi\)
\(272\) 0 0
\(273\) −76.8106 −0.281358
\(274\) 0 0
\(275\) − 377.424i − 1.37245i
\(276\) 0 0
\(277\) −83.2452 −0.300524 −0.150262 0.988646i \(-0.548012\pi\)
−0.150262 + 0.988646i \(0.548012\pi\)
\(278\) 0 0
\(279\) 46.7719i 0.167641i
\(280\) 0 0
\(281\) −12.9851 −0.0462105 −0.0231052 0.999733i \(-0.507355\pi\)
−0.0231052 + 0.999733i \(0.507355\pi\)
\(282\) 0 0
\(283\) 111.022i 0.392303i 0.980574 + 0.196152i \(0.0628445\pi\)
−0.980574 + 0.196152i \(0.937156\pi\)
\(284\) 0 0
\(285\) −6.53976 −0.0229465
\(286\) 0 0
\(287\) − 628.500i − 2.18990i
\(288\) 0 0
\(289\) −132.300 −0.457784
\(290\) 0 0
\(291\) − 97.2324i − 0.334132i
\(292\) 0 0
\(293\) 141.572 0.483179 0.241590 0.970378i \(-0.422331\pi\)
0.241590 + 0.970378i \(0.422331\pi\)
\(294\) 0 0
\(295\) 90.6956i 0.307443i
\(296\) 0 0
\(297\) −218.478 −0.735615
\(298\) 0 0
\(299\) − 517.723i − 1.73151i
\(300\) 0 0
\(301\) 87.6329 0.291139
\(302\) 0 0
\(303\) − 50.1415i − 0.165484i
\(304\) 0 0
\(305\) −59.4846 −0.195031
\(306\) 0 0
\(307\) 357.744i 1.16529i 0.812727 + 0.582645i \(0.197982\pi\)
−0.812727 + 0.582645i \(0.802018\pi\)
\(308\) 0 0
\(309\) −9.63710 −0.0311880
\(310\) 0 0
\(311\) 472.383i 1.51892i 0.650557 + 0.759458i \(0.274536\pi\)
−0.650557 + 0.759458i \(0.725464\pi\)
\(312\) 0 0
\(313\) 63.7596 0.203705 0.101852 0.994800i \(-0.467523\pi\)
0.101852 + 0.994800i \(0.467523\pi\)
\(314\) 0 0
\(315\) − 172.387i − 0.547261i
\(316\) 0 0
\(317\) −198.721 −0.626880 −0.313440 0.949608i \(-0.601481\pi\)
−0.313440 + 0.949608i \(0.601481\pi\)
\(318\) 0 0
\(319\) 122.747i 0.384788i
\(320\) 0 0
\(321\) 51.9119 0.161719
\(322\) 0 0
\(323\) − 54.5647i − 0.168931i
\(324\) 0 0
\(325\) 270.394 0.831983
\(326\) 0 0
\(327\) − 34.7181i − 0.106172i
\(328\) 0 0
\(329\) 279.975 0.850989
\(330\) 0 0
\(331\) − 438.736i − 1.32549i −0.748846 0.662744i \(-0.769392\pi\)
0.748846 0.662744i \(-0.230608\pi\)
\(332\) 0 0
\(333\) −179.628 −0.539424
\(334\) 0 0
\(335\) − 152.006i − 0.453748i
\(336\) 0 0
\(337\) −538.699 −1.59851 −0.799257 0.600989i \(-0.794773\pi\)
−0.799257 + 0.600989i \(0.794773\pi\)
\(338\) 0 0
\(339\) 68.2877i 0.201439i
\(340\) 0 0
\(341\) −105.002 −0.307924
\(342\) 0 0
\(343\) − 203.100i − 0.592127i
\(344\) 0 0
\(345\) −56.2593 −0.163070
\(346\) 0 0
\(347\) − 584.405i − 1.68417i −0.539349 0.842083i \(-0.681329\pi\)
0.539349 0.842083i \(-0.318671\pi\)
\(348\) 0 0
\(349\) 300.832 0.861983 0.430991 0.902356i \(-0.358164\pi\)
0.430991 + 0.902356i \(0.358164\pi\)
\(350\) 0 0
\(351\) − 156.522i − 0.445931i
\(352\) 0 0
\(353\) −316.023 −0.895250 −0.447625 0.894221i \(-0.647730\pi\)
−0.447625 + 0.894221i \(0.647730\pi\)
\(354\) 0 0
\(355\) 43.6475i 0.122951i
\(356\) 0 0
\(357\) −69.6415 −0.195074
\(358\) 0 0
\(359\) 116.055i 0.323272i 0.986850 + 0.161636i \(0.0516770\pi\)
−0.986850 + 0.161636i \(0.948323\pi\)
\(360\) 0 0
\(361\) −19.0000 −0.0526316
\(362\) 0 0
\(363\) − 161.433i − 0.444720i
\(364\) 0 0
\(365\) 169.475 0.464316
\(366\) 0 0
\(367\) 67.8773i 0.184952i 0.995715 + 0.0924759i \(0.0294781\pi\)
−0.995715 + 0.0924759i \(0.970522\pi\)
\(368\) 0 0
\(369\) 625.231 1.69439
\(370\) 0 0
\(371\) − 370.487i − 0.998617i
\(372\) 0 0
\(373\) 267.337 0.716721 0.358361 0.933583i \(-0.383336\pi\)
0.358361 + 0.933583i \(0.383336\pi\)
\(374\) 0 0
\(375\) − 66.8910i − 0.178376i
\(376\) 0 0
\(377\) −87.9388 −0.233259
\(378\) 0 0
\(379\) − 372.834i − 0.983731i −0.870671 0.491866i \(-0.836315\pi\)
0.870671 0.491866i \(-0.163685\pi\)
\(380\) 0 0
\(381\) 11.0508 0.0290047
\(382\) 0 0
\(383\) − 19.6246i − 0.0512392i −0.999672 0.0256196i \(-0.991844\pi\)
0.999672 0.0256196i \(-0.00815587\pi\)
\(384\) 0 0
\(385\) 387.006 1.00521
\(386\) 0 0
\(387\) 87.1771i 0.225264i
\(388\) 0 0
\(389\) −198.727 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(390\) 0 0
\(391\) − 469.401i − 1.20051i
\(392\) 0 0
\(393\) −147.854 −0.376219
\(394\) 0 0
\(395\) 325.509i 0.824072i
\(396\) 0 0
\(397\) −295.448 −0.744203 −0.372101 0.928192i \(-0.621363\pi\)
−0.372101 + 0.928192i \(0.621363\pi\)
\(398\) 0 0
\(399\) 24.2499i 0.0607766i
\(400\) 0 0
\(401\) −63.9986 −0.159597 −0.0797987 0.996811i \(-0.525428\pi\)
−0.0797987 + 0.996811i \(0.525428\pi\)
\(402\) 0 0
\(403\) − 75.2256i − 0.186664i
\(404\) 0 0
\(405\) 162.785 0.401939
\(406\) 0 0
\(407\) − 403.262i − 0.990815i
\(408\) 0 0
\(409\) 504.473 1.23343 0.616715 0.787186i \(-0.288463\pi\)
0.616715 + 0.787186i \(0.288463\pi\)
\(410\) 0 0
\(411\) − 77.7302i − 0.189125i
\(412\) 0 0
\(413\) 336.306 0.814299
\(414\) 0 0
\(415\) 220.408i 0.531105i
\(416\) 0 0
\(417\) 94.0393 0.225514
\(418\) 0 0
\(419\) − 470.299i − 1.12243i −0.827670 0.561215i \(-0.810334\pi\)
0.827670 0.561215i \(-0.189666\pi\)
\(420\) 0 0
\(421\) 470.034 1.11647 0.558235 0.829683i \(-0.311479\pi\)
0.558235 + 0.829683i \(0.311479\pi\)
\(422\) 0 0
\(423\) 278.519i 0.658437i
\(424\) 0 0
\(425\) 245.157 0.576840
\(426\) 0 0
\(427\) 220.573i 0.516564i
\(428\) 0 0
\(429\) 171.541 0.399864
\(430\) 0 0
\(431\) 224.005i 0.519733i 0.965645 + 0.259866i \(0.0836785\pi\)
−0.965645 + 0.259866i \(0.916321\pi\)
\(432\) 0 0
\(433\) 180.128 0.415999 0.207999 0.978129i \(-0.433305\pi\)
0.207999 + 0.978129i \(0.433305\pi\)
\(434\) 0 0
\(435\) 9.55602i 0.0219679i
\(436\) 0 0
\(437\) −163.450 −0.374028
\(438\) 0 0
\(439\) 590.004i 1.34397i 0.740563 + 0.671987i \(0.234559\pi\)
−0.740563 + 0.671987i \(0.765441\pi\)
\(440\) 0 0
\(441\) −218.590 −0.495670
\(442\) 0 0
\(443\) − 157.459i − 0.355439i −0.984081 0.177719i \(-0.943128\pi\)
0.984081 0.177719i \(-0.0568720\pi\)
\(444\) 0 0
\(445\) 77.6212 0.174430
\(446\) 0 0
\(447\) − 12.4136i − 0.0277709i
\(448\) 0 0
\(449\) −715.434 −1.59339 −0.796697 0.604379i \(-0.793421\pi\)
−0.796697 + 0.604379i \(0.793421\pi\)
\(450\) 0 0
\(451\) 1403.63i 3.11226i
\(452\) 0 0
\(453\) −106.715 −0.235574
\(454\) 0 0
\(455\) 277.259i 0.609361i
\(456\) 0 0
\(457\) 31.3497 0.0685988 0.0342994 0.999412i \(-0.489080\pi\)
0.0342994 + 0.999412i \(0.489080\pi\)
\(458\) 0 0
\(459\) − 141.913i − 0.309178i
\(460\) 0 0
\(461\) −368.719 −0.799824 −0.399912 0.916554i \(-0.630959\pi\)
−0.399912 + 0.916554i \(0.630959\pi\)
\(462\) 0 0
\(463\) − 817.405i − 1.76545i −0.469887 0.882726i \(-0.655705\pi\)
0.469887 0.882726i \(-0.344295\pi\)
\(464\) 0 0
\(465\) −8.17453 −0.0175796
\(466\) 0 0
\(467\) − 86.9194i − 0.186123i −0.995660 0.0930614i \(-0.970335\pi\)
0.995660 0.0930614i \(-0.0296653\pi\)
\(468\) 0 0
\(469\) −563.647 −1.20181
\(470\) 0 0
\(471\) − 96.3506i − 0.204566i
\(472\) 0 0
\(473\) −195.711 −0.413765
\(474\) 0 0
\(475\) − 85.3662i − 0.179718i
\(476\) 0 0
\(477\) 368.560 0.772662
\(478\) 0 0
\(479\) − 333.107i − 0.695421i −0.937602 0.347710i \(-0.886959\pi\)
0.937602 0.347710i \(-0.113041\pi\)
\(480\) 0 0
\(481\) 288.905 0.600634
\(482\) 0 0
\(483\) 208.613i 0.431911i
\(484\) 0 0
\(485\) −350.975 −0.723659
\(486\) 0 0
\(487\) − 264.690i − 0.543511i −0.962366 0.271755i \(-0.912396\pi\)
0.962366 0.271755i \(-0.0876042\pi\)
\(488\) 0 0
\(489\) 94.6183 0.193494
\(490\) 0 0
\(491\) − 402.874i − 0.820518i −0.911969 0.410259i \(-0.865438\pi\)
0.911969 0.410259i \(-0.134562\pi\)
\(492\) 0 0
\(493\) −79.7310 −0.161726
\(494\) 0 0
\(495\) 384.993i 0.777764i
\(496\) 0 0
\(497\) 161.848 0.325650
\(498\) 0 0
\(499\) − 350.536i − 0.702477i −0.936286 0.351238i \(-0.885761\pi\)
0.936286 0.351238i \(-0.114239\pi\)
\(500\) 0 0
\(501\) −137.640 −0.274730
\(502\) 0 0
\(503\) − 339.615i − 0.675179i −0.941293 0.337589i \(-0.890388\pi\)
0.941293 0.337589i \(-0.109612\pi\)
\(504\) 0 0
\(505\) −180.993 −0.358403
\(506\) 0 0
\(507\) 13.9409i 0.0274969i
\(508\) 0 0
\(509\) 933.003 1.83301 0.916506 0.400020i \(-0.130997\pi\)
0.916506 + 0.400020i \(0.130997\pi\)
\(510\) 0 0
\(511\) − 628.426i − 1.22980i
\(512\) 0 0
\(513\) −49.4155 −0.0963266
\(514\) 0 0
\(515\) 34.7866i 0.0675467i
\(516\) 0 0
\(517\) −625.270 −1.20942
\(518\) 0 0
\(519\) 180.978i 0.348705i
\(520\) 0 0
\(521\) −415.284 −0.797091 −0.398545 0.917149i \(-0.630485\pi\)
−0.398545 + 0.917149i \(0.630485\pi\)
\(522\) 0 0
\(523\) 241.598i 0.461947i 0.972960 + 0.230973i \(0.0741911\pi\)
−0.972960 + 0.230973i \(0.925809\pi\)
\(524\) 0 0
\(525\) −108.954 −0.207531
\(526\) 0 0
\(527\) − 68.2044i − 0.129420i
\(528\) 0 0
\(529\) −877.105 −1.65804
\(530\) 0 0
\(531\) 334.556i 0.630050i
\(532\) 0 0
\(533\) −1005.59 −1.88666
\(534\) 0 0
\(535\) − 187.384i − 0.350250i
\(536\) 0 0
\(537\) −22.6857 −0.0422452
\(538\) 0 0
\(539\) − 490.731i − 0.910447i
\(540\) 0 0
\(541\) 304.787 0.563377 0.281689 0.959506i \(-0.409106\pi\)
0.281689 + 0.959506i \(0.409106\pi\)
\(542\) 0 0
\(543\) 222.770i 0.410259i
\(544\) 0 0
\(545\) −125.320 −0.229945
\(546\) 0 0
\(547\) − 555.915i − 1.01630i −0.861269 0.508149i \(-0.830330\pi\)
0.861269 0.508149i \(-0.169670\pi\)
\(548\) 0 0
\(549\) −219.425 −0.399682
\(550\) 0 0
\(551\) 27.7632i 0.0503869i
\(552\) 0 0
\(553\) 1207.01 2.18265
\(554\) 0 0
\(555\) − 31.3944i − 0.0565664i
\(556\) 0 0
\(557\) 192.055 0.344802 0.172401 0.985027i \(-0.444848\pi\)
0.172401 + 0.985027i \(0.444848\pi\)
\(558\) 0 0
\(559\) − 140.211i − 0.250825i
\(560\) 0 0
\(561\) 155.531 0.277238
\(562\) 0 0
\(563\) 107.771i 0.191423i 0.995409 + 0.0957114i \(0.0305126\pi\)
−0.995409 + 0.0957114i \(0.969487\pi\)
\(564\) 0 0
\(565\) 246.495 0.436274
\(566\) 0 0
\(567\) − 603.619i − 1.06458i
\(568\) 0 0
\(569\) −647.934 −1.13872 −0.569362 0.822087i \(-0.692810\pi\)
−0.569362 + 0.822087i \(0.692810\pi\)
\(570\) 0 0
\(571\) 285.402i 0.499828i 0.968268 + 0.249914i \(0.0804023\pi\)
−0.968268 + 0.249914i \(0.919598\pi\)
\(572\) 0 0
\(573\) 64.6208 0.112776
\(574\) 0 0
\(575\) − 734.376i − 1.27717i
\(576\) 0 0
\(577\) −92.4624 −0.160247 −0.0801234 0.996785i \(-0.525531\pi\)
−0.0801234 + 0.996785i \(0.525531\pi\)
\(578\) 0 0
\(579\) 108.387i 0.187197i
\(580\) 0 0
\(581\) 817.289 1.40669
\(582\) 0 0
\(583\) 827.410i 1.41923i
\(584\) 0 0
\(585\) −275.817 −0.471482
\(586\) 0 0
\(587\) − 654.717i − 1.11536i −0.830056 0.557681i \(-0.811691\pi\)
0.830056 0.557681i \(-0.188309\pi\)
\(588\) 0 0
\(589\) −23.7495 −0.0403217
\(590\) 0 0
\(591\) − 139.710i − 0.236396i
\(592\) 0 0
\(593\) 106.297 0.179253 0.0896264 0.995975i \(-0.471433\pi\)
0.0896264 + 0.995975i \(0.471433\pi\)
\(594\) 0 0
\(595\) 251.381i 0.422489i
\(596\) 0 0
\(597\) −53.0298 −0.0888271
\(598\) 0 0
\(599\) 478.675i 0.799124i 0.916706 + 0.399562i \(0.130838\pi\)
−0.916706 + 0.399562i \(0.869162\pi\)
\(600\) 0 0
\(601\) 808.024 1.34447 0.672233 0.740340i \(-0.265335\pi\)
0.672233 + 0.740340i \(0.265335\pi\)
\(602\) 0 0
\(603\) − 560.715i − 0.929876i
\(604\) 0 0
\(605\) −582.717 −0.963169
\(606\) 0 0
\(607\) − 694.694i − 1.14447i −0.820089 0.572236i \(-0.806076\pi\)
0.820089 0.572236i \(-0.193924\pi\)
\(608\) 0 0
\(609\) 35.4344 0.0581845
\(610\) 0 0
\(611\) − 447.956i − 0.733153i
\(612\) 0 0
\(613\) 496.830 0.810489 0.405245 0.914208i \(-0.367186\pi\)
0.405245 + 0.914208i \(0.367186\pi\)
\(614\) 0 0
\(615\) 109.274i 0.177682i
\(616\) 0 0
\(617\) 387.951 0.628770 0.314385 0.949296i \(-0.398202\pi\)
0.314385 + 0.949296i \(0.398202\pi\)
\(618\) 0 0
\(619\) 882.105i 1.42505i 0.701647 + 0.712525i \(0.252448\pi\)
−0.701647 + 0.712525i \(0.747552\pi\)
\(620\) 0 0
\(621\) −425.104 −0.684548
\(622\) 0 0
\(623\) − 287.824i − 0.461998i
\(624\) 0 0
\(625\) 248.156 0.397050
\(626\) 0 0
\(627\) − 54.1574i − 0.0863754i
\(628\) 0 0
\(629\) 261.940 0.416439
\(630\) 0 0
\(631\) 502.601i 0.796515i 0.917274 + 0.398258i \(0.130385\pi\)
−0.917274 + 0.398258i \(0.869615\pi\)
\(632\) 0 0
\(633\) 10.5857 0.0167230
\(634\) 0 0
\(635\) − 39.8895i − 0.0628181i
\(636\) 0 0
\(637\) 351.570 0.551915
\(638\) 0 0
\(639\) 161.006i 0.251966i
\(640\) 0 0
\(641\) 385.625 0.601600 0.300800 0.953687i \(-0.402746\pi\)
0.300800 + 0.953687i \(0.402746\pi\)
\(642\) 0 0
\(643\) 925.293i 1.43902i 0.694480 + 0.719512i \(0.255635\pi\)
−0.694480 + 0.719512i \(0.744365\pi\)
\(644\) 0 0
\(645\) −15.2363 −0.0236222
\(646\) 0 0
\(647\) 656.956i 1.01539i 0.861538 + 0.507694i \(0.169502\pi\)
−0.861538 + 0.507694i \(0.830498\pi\)
\(648\) 0 0
\(649\) −751.073 −1.15728
\(650\) 0 0
\(651\) 30.3117i 0.0465617i
\(652\) 0 0
\(653\) −392.897 −0.601680 −0.300840 0.953675i \(-0.597267\pi\)
−0.300840 + 0.953675i \(0.597267\pi\)
\(654\) 0 0
\(655\) 533.701i 0.814811i
\(656\) 0 0
\(657\) 625.157 0.951533
\(658\) 0 0
\(659\) 695.576i 1.05550i 0.849399 + 0.527751i \(0.176965\pi\)
−0.849399 + 0.527751i \(0.823035\pi\)
\(660\) 0 0
\(661\) −1179.05 −1.78373 −0.891866 0.452299i \(-0.850604\pi\)
−0.891866 + 0.452299i \(0.850604\pi\)
\(662\) 0 0
\(663\) 111.425i 0.168062i
\(664\) 0 0
\(665\) 87.5335 0.131629
\(666\) 0 0
\(667\) 238.837i 0.358076i
\(668\) 0 0
\(669\) 125.863 0.188135
\(670\) 0 0
\(671\) − 492.606i − 0.734137i
\(672\) 0 0
\(673\) −881.170 −1.30932 −0.654658 0.755925i \(-0.727187\pi\)
−0.654658 + 0.755925i \(0.727187\pi\)
\(674\) 0 0
\(675\) − 222.022i − 0.328921i
\(676\) 0 0
\(677\) 126.536 0.186907 0.0934533 0.995624i \(-0.470209\pi\)
0.0934533 + 0.995624i \(0.470209\pi\)
\(678\) 0 0
\(679\) 1301.44i 1.91670i
\(680\) 0 0
\(681\) −196.895 −0.289126
\(682\) 0 0
\(683\) − 873.042i − 1.27825i −0.769105 0.639123i \(-0.779298\pi\)
0.769105 0.639123i \(-0.220702\pi\)
\(684\) 0 0
\(685\) −280.579 −0.409604
\(686\) 0 0
\(687\) 148.080i 0.215545i
\(688\) 0 0
\(689\) −592.774 −0.860339
\(690\) 0 0
\(691\) − 610.841i − 0.883996i −0.897016 0.441998i \(-0.854270\pi\)
0.897016 0.441998i \(-0.145730\pi\)
\(692\) 0 0
\(693\) 1427.58 2.06000
\(694\) 0 0
\(695\) − 339.449i − 0.488415i
\(696\) 0 0
\(697\) −911.733 −1.30808
\(698\) 0 0
\(699\) 120.989i 0.173089i
\(700\) 0 0
\(701\) 506.393 0.722387 0.361193 0.932491i \(-0.382369\pi\)
0.361193 + 0.932491i \(0.382369\pi\)
\(702\) 0 0
\(703\) − 91.2102i − 0.129744i
\(704\) 0 0
\(705\) −48.6780 −0.0690467
\(706\) 0 0
\(707\) 671.135i 0.949272i
\(708\) 0 0
\(709\) 1036.42 1.46180 0.730902 0.682483i \(-0.239100\pi\)
0.730902 + 0.682483i \(0.239100\pi\)
\(710\) 0 0
\(711\) 1200.73i 1.68879i
\(712\) 0 0
\(713\) −204.308 −0.286548
\(714\) 0 0
\(715\) − 619.204i − 0.866020i
\(716\) 0 0
\(717\) 45.0328 0.0628072
\(718\) 0 0
\(719\) 834.943i 1.16126i 0.814169 + 0.580628i \(0.197193\pi\)
−0.814169 + 0.580628i \(0.802807\pi\)
\(720\) 0 0
\(721\) 128.991 0.178905
\(722\) 0 0
\(723\) 191.839i 0.265337i
\(724\) 0 0
\(725\) −124.739 −0.172053
\(726\) 0 0
\(727\) 225.252i 0.309838i 0.987927 + 0.154919i \(0.0495116\pi\)
−0.987927 + 0.154919i \(0.950488\pi\)
\(728\) 0 0
\(729\) 534.700 0.733470
\(730\) 0 0
\(731\) − 127.125i − 0.173905i
\(732\) 0 0
\(733\) −697.401 −0.951434 −0.475717 0.879598i \(-0.657811\pi\)
−0.475717 + 0.879598i \(0.657811\pi\)
\(734\) 0 0
\(735\) − 38.2040i − 0.0519782i
\(736\) 0 0
\(737\) 1258.80 1.70800
\(738\) 0 0
\(739\) − 1065.87i − 1.44231i −0.692772 0.721157i \(-0.743611\pi\)
0.692772 0.721157i \(-0.256389\pi\)
\(740\) 0 0
\(741\) 38.7994 0.0523609
\(742\) 0 0
\(743\) 1050.43i 1.41376i 0.707332 + 0.706881i \(0.249898\pi\)
−0.707332 + 0.706881i \(0.750102\pi\)
\(744\) 0 0
\(745\) −44.8087 −0.0601459
\(746\) 0 0
\(747\) 813.038i 1.08840i
\(748\) 0 0
\(749\) −694.831 −0.927678
\(750\) 0 0
\(751\) 26.6606i 0.0355002i 0.999842 + 0.0177501i \(0.00565033\pi\)
−0.999842 + 0.0177501i \(0.994350\pi\)
\(752\) 0 0
\(753\) −23.5102 −0.0312221
\(754\) 0 0
\(755\) 385.204i 0.510204i
\(756\) 0 0
\(757\) −445.989 −0.589153 −0.294577 0.955628i \(-0.595179\pi\)
−0.294577 + 0.955628i \(0.595179\pi\)
\(758\) 0 0
\(759\) − 465.897i − 0.613830i
\(760\) 0 0
\(761\) 294.799 0.387384 0.193692 0.981062i \(-0.437954\pi\)
0.193692 + 0.981062i \(0.437954\pi\)
\(762\) 0 0
\(763\) 464.695i 0.609037i
\(764\) 0 0
\(765\) −250.074 −0.326894
\(766\) 0 0
\(767\) − 538.084i − 0.701543i
\(768\) 0 0
\(769\) −279.363 −0.363281 −0.181641 0.983365i \(-0.558141\pi\)
−0.181641 + 0.983365i \(0.558141\pi\)
\(770\) 0 0
\(771\) − 113.431i − 0.147122i
\(772\) 0 0
\(773\) 1092.48 1.41329 0.706647 0.707566i \(-0.250207\pi\)
0.706647 + 0.707566i \(0.250207\pi\)
\(774\) 0 0
\(775\) − 106.705i − 0.137684i
\(776\) 0 0
\(777\) −116.412 −0.149823
\(778\) 0 0
\(779\) 317.475i 0.407542i
\(780\) 0 0
\(781\) −361.456 −0.462811
\(782\) 0 0
\(783\) 72.2069i 0.0922183i
\(784\) 0 0
\(785\) −347.792 −0.443047
\(786\) 0 0
\(787\) 231.147i 0.293706i 0.989158 + 0.146853i \(0.0469145\pi\)
−0.989158 + 0.146853i \(0.953086\pi\)
\(788\) 0 0
\(789\) 286.820 0.363524
\(790\) 0 0
\(791\) − 914.019i − 1.15552i
\(792\) 0 0
\(793\) 352.913 0.445035
\(794\) 0 0
\(795\) 64.4148i 0.0810249i
\(796\) 0 0
\(797\) −453.389 −0.568869 −0.284435 0.958695i \(-0.591806\pi\)
−0.284435 + 0.958695i \(0.591806\pi\)
\(798\) 0 0
\(799\) − 406.146i − 0.508318i
\(800\) 0 0
\(801\) 286.327 0.357462
\(802\) 0 0
\(803\) 1403.47i 1.74778i
\(804\) 0 0
\(805\) 753.020 0.935429
\(806\) 0 0
\(807\) − 96.5406i − 0.119629i
\(808\) 0 0
\(809\) 528.874 0.653738 0.326869 0.945070i \(-0.394006\pi\)
0.326869 + 0.945070i \(0.394006\pi\)
\(810\) 0 0
\(811\) 1185.82i 1.46217i 0.682287 + 0.731084i \(0.260985\pi\)
−0.682287 + 0.731084i \(0.739015\pi\)
\(812\) 0 0
\(813\) 283.467 0.348668
\(814\) 0 0
\(815\) − 341.539i − 0.419066i
\(816\) 0 0
\(817\) −44.2661 −0.0541813
\(818\) 0 0
\(819\) 1022.75i 1.24878i
\(820\) 0 0
\(821\) 570.471 0.694849 0.347424 0.937708i \(-0.387056\pi\)
0.347424 + 0.937708i \(0.387056\pi\)
\(822\) 0 0
\(823\) 1386.72i 1.68496i 0.538728 + 0.842480i \(0.318905\pi\)
−0.538728 + 0.842480i \(0.681095\pi\)
\(824\) 0 0
\(825\) 243.327 0.294942
\(826\) 0 0
\(827\) 1073.59i 1.29818i 0.760712 + 0.649089i \(0.224850\pi\)
−0.760712 + 0.649089i \(0.775150\pi\)
\(828\) 0 0
\(829\) −1109.24 −1.33804 −0.669021 0.743243i \(-0.733287\pi\)
−0.669021 + 0.743243i \(0.733287\pi\)
\(830\) 0 0
\(831\) − 53.6685i − 0.0645830i
\(832\) 0 0
\(833\) 318.756 0.382660
\(834\) 0 0
\(835\) 496.831i 0.595007i
\(836\) 0 0
\(837\) −61.7681 −0.0737970
\(838\) 0 0
\(839\) − 1547.29i − 1.84421i −0.386943 0.922104i \(-0.626469\pi\)
0.386943 0.922104i \(-0.373531\pi\)
\(840\) 0 0
\(841\) −800.432 −0.951762
\(842\) 0 0
\(843\) − 8.37157i − 0.00993069i
\(844\) 0 0
\(845\) 50.3218 0.0595524
\(846\) 0 0
\(847\) 2160.76i 2.55107i
\(848\) 0 0
\(849\) −71.5762 −0.0843065
\(850\) 0 0
\(851\) − 784.649i − 0.922031i
\(852\) 0 0
\(853\) 341.108 0.399892 0.199946 0.979807i \(-0.435923\pi\)
0.199946 + 0.979807i \(0.435923\pi\)
\(854\) 0 0
\(855\) 87.0782i 0.101846i
\(856\) 0 0
\(857\) 1535.74 1.79200 0.896000 0.444054i \(-0.146460\pi\)
0.896000 + 0.444054i \(0.146460\pi\)
\(858\) 0 0
\(859\) − 1223.16i − 1.42393i −0.702213 0.711967i \(-0.747804\pi\)
0.702213 0.711967i \(-0.252196\pi\)
\(860\) 0 0
\(861\) 405.196 0.470611
\(862\) 0 0
\(863\) − 672.603i − 0.779377i −0.920947 0.389689i \(-0.872583\pi\)
0.920947 0.389689i \(-0.127417\pi\)
\(864\) 0 0
\(865\) 653.266 0.755221
\(866\) 0 0
\(867\) − 85.2940i − 0.0983783i
\(868\) 0 0
\(869\) −2695.62 −3.10197
\(870\) 0 0
\(871\) 901.827i 1.03539i
\(872\) 0 0
\(873\) −1294.67 −1.48301
\(874\) 0 0
\(875\) 895.324i 1.02323i
\(876\) 0 0
\(877\) 1328.68 1.51503 0.757514 0.652819i \(-0.226414\pi\)
0.757514 + 0.652819i \(0.226414\pi\)
\(878\) 0 0
\(879\) 91.2717i 0.103836i
\(880\) 0 0
\(881\) 372.924 0.423296 0.211648 0.977346i \(-0.432117\pi\)
0.211648 + 0.977346i \(0.432117\pi\)
\(882\) 0 0
\(883\) 669.314i 0.758000i 0.925397 + 0.379000i \(0.123732\pi\)
−0.925397 + 0.379000i \(0.876268\pi\)
\(884\) 0 0
\(885\) −58.4718 −0.0660699
\(886\) 0 0
\(887\) − 874.225i − 0.985598i −0.870143 0.492799i \(-0.835974\pi\)
0.870143 0.492799i \(-0.164026\pi\)
\(888\) 0 0
\(889\) −147.913 −0.166381
\(890\) 0 0
\(891\) 1348.06i 1.51298i
\(892\) 0 0
\(893\) −141.424 −0.158370
\(894\) 0 0
\(895\) 81.8873i 0.0914942i
\(896\) 0 0
\(897\) 333.778 0.372105
\(898\) 0 0
\(899\) 34.7032i 0.0386020i
\(900\) 0 0
\(901\) −537.447 −0.596500
\(902\) 0 0
\(903\) 56.4973i 0.0625662i
\(904\) 0 0
\(905\) 804.123 0.888534
\(906\) 0 0
\(907\) 33.1952i 0.0365989i 0.999833 + 0.0182994i \(0.00582522\pi\)
−0.999833 + 0.0182994i \(0.994175\pi\)
\(908\) 0 0
\(909\) −667.644 −0.734482
\(910\) 0 0
\(911\) 25.4273i 0.0279114i 0.999903 + 0.0139557i \(0.00444238\pi\)
−0.999903 + 0.0139557i \(0.995558\pi\)
\(912\) 0 0
\(913\) −1825.26 −1.99918
\(914\) 0 0
\(915\) − 38.3499i − 0.0419125i
\(916\) 0 0
\(917\) 1979.00 2.15812
\(918\) 0 0
\(919\) − 445.408i − 0.484666i −0.970193 0.242333i \(-0.922087\pi\)
0.970193 0.242333i \(-0.0779126\pi\)
\(920\) 0 0
\(921\) −230.639 −0.250422
\(922\) 0 0
\(923\) − 258.954i − 0.280557i
\(924\) 0 0
\(925\) 409.804 0.443031
\(926\) 0 0
\(927\) 128.320i 0.138425i
\(928\) 0 0
\(929\) −408.473 −0.439691 −0.219846 0.975535i \(-0.570555\pi\)
−0.219846 + 0.975535i \(0.570555\pi\)
\(930\) 0 0
\(931\) − 110.994i − 0.119220i
\(932\) 0 0
\(933\) −304.547 −0.326417
\(934\) 0 0
\(935\) − 561.411i − 0.600439i
\(936\) 0 0
\(937\) −1101.68 −1.17576 −0.587878 0.808950i \(-0.700037\pi\)
−0.587878 + 0.808950i \(0.700037\pi\)
\(938\) 0 0
\(939\) 41.1060i 0.0437764i
\(940\) 0 0
\(941\) 1342.45 1.42662 0.713309 0.700850i \(-0.247196\pi\)
0.713309 + 0.700850i \(0.247196\pi\)
\(942\) 0 0
\(943\) 2731.12i 2.89621i
\(944\) 0 0
\(945\) 227.659 0.240909
\(946\) 0 0
\(947\) − 786.294i − 0.830300i −0.909753 0.415150i \(-0.863729\pi\)
0.909753 0.415150i \(-0.136271\pi\)
\(948\) 0 0
\(949\) −1005.47 −1.05951
\(950\) 0 0
\(951\) − 128.116i − 0.134717i
\(952\) 0 0
\(953\) 1134.64 1.19060 0.595301 0.803503i \(-0.297033\pi\)
0.595301 + 0.803503i \(0.297033\pi\)
\(954\) 0 0
\(955\) − 233.258i − 0.244250i
\(956\) 0 0
\(957\) −79.1357 −0.0826915
\(958\) 0 0
\(959\) 1040.40i 1.08488i
\(960\) 0 0
\(961\) 931.314 0.969109
\(962\) 0 0
\(963\) − 691.217i − 0.717775i
\(964\) 0 0
\(965\) 391.240 0.405430
\(966\) 0 0
\(967\) − 963.333i − 0.996208i −0.867117 0.498104i \(-0.834030\pi\)
0.867117 0.498104i \(-0.165970\pi\)
\(968\) 0 0
\(969\) 35.1781 0.0363035
\(970\) 0 0
\(971\) − 1320.90i − 1.36035i −0.733050 0.680174i \(-0.761904\pi\)
0.733050 0.680174i \(-0.238096\pi\)
\(972\) 0 0
\(973\) −1258.70 −1.29363
\(974\) 0 0
\(975\) 174.324i 0.178794i
\(976\) 0 0
\(977\) −1159.80 −1.18710 −0.593550 0.804797i \(-0.702274\pi\)
−0.593550 + 0.804797i \(0.702274\pi\)
\(978\) 0 0
\(979\) 642.800i 0.656588i
\(980\) 0 0
\(981\) −462.278 −0.471232
\(982\) 0 0
\(983\) − 8.37645i − 0.00852131i −0.999991 0.00426065i \(-0.998644\pi\)
0.999991 0.00426065i \(-0.00135621\pi\)
\(984\) 0 0
\(985\) −504.305 −0.511985
\(986\) 0 0
\(987\) 180.501i 0.182879i
\(988\) 0 0
\(989\) −380.806 −0.385041
\(990\) 0 0
\(991\) 854.355i 0.862114i 0.902325 + 0.431057i \(0.141859\pi\)
−0.902325 + 0.431057i \(0.858141\pi\)
\(992\) 0 0
\(993\) 282.855 0.284849
\(994\) 0 0
\(995\) 191.419i 0.192381i
\(996\) 0 0
\(997\) 77.4664 0.0776995 0.0388497 0.999245i \(-0.487631\pi\)
0.0388497 + 0.999245i \(0.487631\pi\)
\(998\) 0 0
\(999\) − 237.221i − 0.237458i
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 1216.3.d.d.191.8 14
4.3 odd 2 inner 1216.3.d.d.191.7 14
8.3 odd 2 76.3.b.b.39.1 14
8.5 even 2 76.3.b.b.39.2 yes 14
24.5 odd 2 684.3.g.b.343.13 14
24.11 even 2 684.3.g.b.343.14 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.b.b.39.1 14 8.3 odd 2
76.3.b.b.39.2 yes 14 8.5 even 2
684.3.g.b.343.13 14 24.5 odd 2
684.3.g.b.343.14 14 24.11 even 2
1216.3.d.d.191.7 14 4.3 odd 2 inner
1216.3.d.d.191.8 14 1.1 even 1 trivial