Properties

Label 1792.3.d.j.1023.11
Level $1792$
Weight $3$
Character 1792.1023
Analytic conductor $48.828$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,3,Mod(1023,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, 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("1792.1023");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1792.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: \(48.8284633734\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 5x^{14} + 48x^{12} + 180x^{10} + 1056x^{8} + 2880x^{6} + 12288x^{4} + 20480x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{38} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1023.11
Root \(1.69931 + 1.05468i\) of defining polynomial
Character \(\chi\) \(=\) 1792.1023
Dual form 1792.3.d.j.1023.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.44128i q^{3} -4.88287 q^{5} -2.64575i q^{7} -2.84239 q^{9} +O(q^{10})\) \(q+3.44128i q^{3} -4.88287 q^{5} -2.64575i q^{7} -2.84239 q^{9} -21.4776i q^{11} +13.0760 q^{13} -16.8033i q^{15} -0.234889 q^{17} -4.55872i q^{19} +9.10476 q^{21} +10.9523i q^{23} -1.15761 q^{25} +21.1900i q^{27} -34.6435 q^{29} +34.1079i q^{31} +73.9103 q^{33} +12.9189i q^{35} -54.2370 q^{37} +44.9982i q^{39} +37.8300 q^{41} -4.84714i q^{43} +13.8790 q^{45} +72.3368i q^{47} -7.00000 q^{49} -0.808319i q^{51} +21.6707 q^{53} +104.872i q^{55} +15.6878 q^{57} +34.9007i q^{59} -63.6012 q^{61} +7.52026i q^{63} -63.8485 q^{65} -18.4344i q^{67} -37.6899 q^{69} -47.5244i q^{71} -55.9103 q^{73} -3.98365i q^{75} -56.8243 q^{77} +95.0135i q^{79} -98.5023 q^{81} -71.5156i q^{83} +1.14693 q^{85} -119.218i q^{87} +159.756 q^{89} -34.5959i q^{91} -117.375 q^{93} +22.2596i q^{95} -90.4794 q^{97} +61.0477i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 96 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 96 q^{9} - 160 q^{17} + 32 q^{25} + 64 q^{33} - 256 q^{41} - 112 q^{49} - 112 q^{57} - 144 q^{65} + 224 q^{73} + 96 q^{81} + 1024 q^{89} + 128 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 3.44128i 1.14709i 0.819173 + 0.573546i \(0.194433\pi\)
−0.819173 + 0.573546i \(0.805567\pi\)
\(4\) 0 0
\(5\) −4.88287 −0.976573 −0.488287 0.872683i \(-0.662378\pi\)
−0.488287 + 0.872683i \(0.662378\pi\)
\(6\) 0 0
\(7\) − 2.64575i − 0.377964i
\(8\) 0 0
\(9\) −2.84239 −0.315821
\(10\) 0 0
\(11\) − 21.4776i − 1.95251i −0.216632 0.976253i \(-0.569507\pi\)
0.216632 0.976253i \(-0.430493\pi\)
\(12\) 0 0
\(13\) 13.0760 1.00585 0.502924 0.864331i \(-0.332258\pi\)
0.502924 + 0.864331i \(0.332258\pi\)
\(14\) 0 0
\(15\) − 16.8033i − 1.12022i
\(16\) 0 0
\(17\) −0.234889 −0.0138170 −0.00690851 0.999976i \(-0.502199\pi\)
−0.00690851 + 0.999976i \(0.502199\pi\)
\(18\) 0 0
\(19\) − 4.55872i − 0.239933i −0.992778 0.119966i \(-0.961721\pi\)
0.992778 0.119966i \(-0.0382787\pi\)
\(20\) 0 0
\(21\) 9.10476 0.433560
\(22\) 0 0
\(23\) 10.9523i 0.476187i 0.971242 + 0.238094i \(0.0765225\pi\)
−0.971242 + 0.238094i \(0.923478\pi\)
\(24\) 0 0
\(25\) −1.15761 −0.0463043
\(26\) 0 0
\(27\) 21.1900i 0.784816i
\(28\) 0 0
\(29\) −34.6435 −1.19460 −0.597302 0.802016i \(-0.703761\pi\)
−0.597302 + 0.802016i \(0.703761\pi\)
\(30\) 0 0
\(31\) 34.1079i 1.10025i 0.835081 + 0.550127i \(0.185421\pi\)
−0.835081 + 0.550127i \(0.814579\pi\)
\(32\) 0 0
\(33\) 73.9103 2.23971
\(34\) 0 0
\(35\) 12.9189i 0.369110i
\(36\) 0 0
\(37\) −54.2370 −1.46586 −0.732932 0.680302i \(-0.761849\pi\)
−0.732932 + 0.680302i \(0.761849\pi\)
\(38\) 0 0
\(39\) 44.9982i 1.15380i
\(40\) 0 0
\(41\) 37.8300 0.922682 0.461341 0.887223i \(-0.347368\pi\)
0.461341 + 0.887223i \(0.347368\pi\)
\(42\) 0 0
\(43\) − 4.84714i − 0.112724i −0.998410 0.0563621i \(-0.982050\pi\)
0.998410 0.0563621i \(-0.0179501\pi\)
\(44\) 0 0
\(45\) 13.8790 0.308423
\(46\) 0 0
\(47\) 72.3368i 1.53908i 0.638598 + 0.769541i \(0.279515\pi\)
−0.638598 + 0.769541i \(0.720485\pi\)
\(48\) 0 0
\(49\) −7.00000 −0.142857
\(50\) 0 0
\(51\) − 0.808319i − 0.0158494i
\(52\) 0 0
\(53\) 21.6707 0.408881 0.204440 0.978879i \(-0.434463\pi\)
0.204440 + 0.978879i \(0.434463\pi\)
\(54\) 0 0
\(55\) 104.872i 1.90677i
\(56\) 0 0
\(57\) 15.6878 0.275225
\(58\) 0 0
\(59\) 34.9007i 0.591537i 0.955260 + 0.295768i \(0.0955757\pi\)
−0.955260 + 0.295768i \(0.904424\pi\)
\(60\) 0 0
\(61\) −63.6012 −1.04264 −0.521321 0.853360i \(-0.674561\pi\)
−0.521321 + 0.853360i \(0.674561\pi\)
\(62\) 0 0
\(63\) 7.52026i 0.119369i
\(64\) 0 0
\(65\) −63.8485 −0.982284
\(66\) 0 0
\(67\) − 18.4344i − 0.275140i −0.990492 0.137570i \(-0.956071\pi\)
0.990492 0.137570i \(-0.0439292\pi\)
\(68\) 0 0
\(69\) −37.6899 −0.546231
\(70\) 0 0
\(71\) − 47.5244i − 0.669358i −0.942332 0.334679i \(-0.891372\pi\)
0.942332 0.334679i \(-0.108628\pi\)
\(72\) 0 0
\(73\) −55.9103 −0.765894 −0.382947 0.923770i \(-0.625091\pi\)
−0.382947 + 0.923770i \(0.625091\pi\)
\(74\) 0 0
\(75\) − 3.98365i − 0.0531153i
\(76\) 0 0
\(77\) −56.8243 −0.737978
\(78\) 0 0
\(79\) 95.0135i 1.20270i 0.798985 + 0.601351i \(0.205371\pi\)
−0.798985 + 0.601351i \(0.794629\pi\)
\(80\) 0 0
\(81\) −98.5023 −1.21608
\(82\) 0 0
\(83\) − 71.5156i − 0.861634i −0.902439 0.430817i \(-0.858225\pi\)
0.902439 0.430817i \(-0.141775\pi\)
\(84\) 0 0
\(85\) 1.14693 0.0134933
\(86\) 0 0
\(87\) − 119.218i − 1.37032i
\(88\) 0 0
\(89\) 159.756 1.79501 0.897504 0.441006i \(-0.145378\pi\)
0.897504 + 0.441006i \(0.145378\pi\)
\(90\) 0 0
\(91\) − 34.5959i − 0.380175i
\(92\) 0 0
\(93\) −117.375 −1.26209
\(94\) 0 0
\(95\) 22.2596i 0.234312i
\(96\) 0 0
\(97\) −90.4794 −0.932777 −0.466389 0.884580i \(-0.654445\pi\)
−0.466389 + 0.884580i \(0.654445\pi\)
\(98\) 0 0
\(99\) 61.0477i 0.616643i
\(100\) 0 0
\(101\) −181.147 −1.79353 −0.896767 0.442503i \(-0.854091\pi\)
−0.896767 + 0.442503i \(0.854091\pi\)
\(102\) 0 0
\(103\) 39.3003i 0.381556i 0.981633 + 0.190778i \(0.0611010\pi\)
−0.981633 + 0.190778i \(0.938899\pi\)
\(104\) 0 0
\(105\) −44.4574 −0.423403
\(106\) 0 0
\(107\) 38.4498i 0.359344i 0.983727 + 0.179672i \(0.0575037\pi\)
−0.983727 + 0.179672i \(0.942496\pi\)
\(108\) 0 0
\(109\) −27.8786 −0.255767 −0.127883 0.991789i \(-0.540818\pi\)
−0.127883 + 0.991789i \(0.540818\pi\)
\(110\) 0 0
\(111\) − 186.644i − 1.68148i
\(112\) 0 0
\(113\) 82.4419 0.729574 0.364787 0.931091i \(-0.381142\pi\)
0.364787 + 0.931091i \(0.381142\pi\)
\(114\) 0 0
\(115\) − 53.4786i − 0.465032i
\(116\) 0 0
\(117\) −37.1672 −0.317668
\(118\) 0 0
\(119\) 0.621458i 0.00522234i
\(120\) 0 0
\(121\) −340.286 −2.81228
\(122\) 0 0
\(123\) 130.183i 1.05840i
\(124\) 0 0
\(125\) 127.724 1.02179
\(126\) 0 0
\(127\) − 25.1408i − 0.197959i −0.995089 0.0989796i \(-0.968442\pi\)
0.995089 0.0989796i \(-0.0315579\pi\)
\(128\) 0 0
\(129\) 16.6804 0.129305
\(130\) 0 0
\(131\) 126.398i 0.964872i 0.875931 + 0.482436i \(0.160248\pi\)
−0.875931 + 0.482436i \(0.839752\pi\)
\(132\) 0 0
\(133\) −12.0612 −0.0906861
\(134\) 0 0
\(135\) − 103.468i − 0.766431i
\(136\) 0 0
\(137\) −34.9456 −0.255078 −0.127539 0.991834i \(-0.540708\pi\)
−0.127539 + 0.991834i \(0.540708\pi\)
\(138\) 0 0
\(139\) − 119.148i − 0.857177i −0.903500 0.428589i \(-0.859011\pi\)
0.903500 0.428589i \(-0.140989\pi\)
\(140\) 0 0
\(141\) −248.931 −1.76547
\(142\) 0 0
\(143\) − 280.841i − 1.96392i
\(144\) 0 0
\(145\) 169.160 1.16662
\(146\) 0 0
\(147\) − 24.0889i − 0.163870i
\(148\) 0 0
\(149\) 121.932 0.818334 0.409167 0.912460i \(-0.365819\pi\)
0.409167 + 0.912460i \(0.365819\pi\)
\(150\) 0 0
\(151\) 220.404i 1.45963i 0.683645 + 0.729815i \(0.260394\pi\)
−0.683645 + 0.729815i \(0.739606\pi\)
\(152\) 0 0
\(153\) 0.667647 0.00436371
\(154\) 0 0
\(155\) − 166.544i − 1.07448i
\(156\) 0 0
\(157\) −6.77014 −0.0431219 −0.0215610 0.999768i \(-0.506864\pi\)
−0.0215610 + 0.999768i \(0.506864\pi\)
\(158\) 0 0
\(159\) 74.5748i 0.469024i
\(160\) 0 0
\(161\) 28.9771 0.179982
\(162\) 0 0
\(163\) 207.243i 1.27143i 0.771924 + 0.635715i \(0.219294\pi\)
−0.771924 + 0.635715i \(0.780706\pi\)
\(164\) 0 0
\(165\) −360.894 −2.18724
\(166\) 0 0
\(167\) − 165.529i − 0.991193i −0.868553 0.495596i \(-0.834950\pi\)
0.868553 0.495596i \(-0.165050\pi\)
\(168\) 0 0
\(169\) 1.98237 0.0117300
\(170\) 0 0
\(171\) 12.9577i 0.0757759i
\(172\) 0 0
\(173\) −88.8530 −0.513601 −0.256800 0.966464i \(-0.582668\pi\)
−0.256800 + 0.966464i \(0.582668\pi\)
\(174\) 0 0
\(175\) 3.06274i 0.0175014i
\(176\) 0 0
\(177\) −120.103 −0.678548
\(178\) 0 0
\(179\) − 80.3791i − 0.449045i −0.974469 0.224523i \(-0.927918\pi\)
0.974469 0.224523i \(-0.0720823\pi\)
\(180\) 0 0
\(181\) −276.353 −1.52681 −0.763406 0.645919i \(-0.776474\pi\)
−0.763406 + 0.645919i \(0.776474\pi\)
\(182\) 0 0
\(183\) − 218.869i − 1.19601i
\(184\) 0 0
\(185\) 264.832 1.43152
\(186\) 0 0
\(187\) 5.04485i 0.0269778i
\(188\) 0 0
\(189\) 56.0636 0.296633
\(190\) 0 0
\(191\) 203.015i 1.06290i 0.847088 + 0.531452i \(0.178353\pi\)
−0.847088 + 0.531452i \(0.821647\pi\)
\(192\) 0 0
\(193\) 87.3328 0.452502 0.226251 0.974069i \(-0.427353\pi\)
0.226251 + 0.974069i \(0.427353\pi\)
\(194\) 0 0
\(195\) − 219.720i − 1.12677i
\(196\) 0 0
\(197\) −21.6639 −0.109969 −0.0549845 0.998487i \(-0.517511\pi\)
−0.0549845 + 0.998487i \(0.517511\pi\)
\(198\) 0 0
\(199\) 181.933i 0.914235i 0.889406 + 0.457118i \(0.151118\pi\)
−0.889406 + 0.457118i \(0.848882\pi\)
\(200\) 0 0
\(201\) 63.4378 0.315611
\(202\) 0 0
\(203\) 91.6582i 0.451518i
\(204\) 0 0
\(205\) −184.719 −0.901067
\(206\) 0 0
\(207\) − 31.1307i − 0.150390i
\(208\) 0 0
\(209\) −97.9103 −0.468470
\(210\) 0 0
\(211\) 21.4204i 0.101519i 0.998711 + 0.0507594i \(0.0161641\pi\)
−0.998711 + 0.0507594i \(0.983836\pi\)
\(212\) 0 0
\(213\) 163.545 0.767815
\(214\) 0 0
\(215\) 23.6680i 0.110084i
\(216\) 0 0
\(217\) 90.2410 0.415857
\(218\) 0 0
\(219\) − 192.403i − 0.878552i
\(220\) 0 0
\(221\) −3.07142 −0.0138978
\(222\) 0 0
\(223\) − 195.958i − 0.878735i −0.898307 0.439367i \(-0.855203\pi\)
0.898307 0.439367i \(-0.144797\pi\)
\(224\) 0 0
\(225\) 3.29038 0.0146239
\(226\) 0 0
\(227\) 27.2652i 0.120111i 0.998195 + 0.0600554i \(0.0191277\pi\)
−0.998195 + 0.0600554i \(0.980872\pi\)
\(228\) 0 0
\(229\) −176.347 −0.770076 −0.385038 0.922901i \(-0.625812\pi\)
−0.385038 + 0.922901i \(0.625812\pi\)
\(230\) 0 0
\(231\) − 195.548i − 0.846529i
\(232\) 0 0
\(233\) −71.8366 −0.308312 −0.154156 0.988047i \(-0.549266\pi\)
−0.154156 + 0.988047i \(0.549266\pi\)
\(234\) 0 0
\(235\) − 353.211i − 1.50303i
\(236\) 0 0
\(237\) −326.968 −1.37961
\(238\) 0 0
\(239\) 71.0926i 0.297459i 0.988878 + 0.148729i \(0.0475183\pi\)
−0.988878 + 0.148729i \(0.952482\pi\)
\(240\) 0 0
\(241\) 56.1113 0.232827 0.116413 0.993201i \(-0.462860\pi\)
0.116413 + 0.993201i \(0.462860\pi\)
\(242\) 0 0
\(243\) − 148.264i − 0.610138i
\(244\) 0 0
\(245\) 34.1801 0.139510
\(246\) 0 0
\(247\) − 59.6100i − 0.241336i
\(248\) 0 0
\(249\) 246.105 0.988374
\(250\) 0 0
\(251\) 368.953i 1.46993i 0.678104 + 0.734966i \(0.262802\pi\)
−0.678104 + 0.734966i \(0.737198\pi\)
\(252\) 0 0
\(253\) 235.229 0.929759
\(254\) 0 0
\(255\) 3.94691i 0.0154781i
\(256\) 0 0
\(257\) 23.7428 0.0923845 0.0461923 0.998933i \(-0.485291\pi\)
0.0461923 + 0.998933i \(0.485291\pi\)
\(258\) 0 0
\(259\) 143.498i 0.554044i
\(260\) 0 0
\(261\) 98.4705 0.377282
\(262\) 0 0
\(263\) − 73.9707i − 0.281257i −0.990062 0.140629i \(-0.955088\pi\)
0.990062 0.140629i \(-0.0449124\pi\)
\(264\) 0 0
\(265\) −105.815 −0.399302
\(266\) 0 0
\(267\) 549.764i 2.05904i
\(268\) 0 0
\(269\) −335.593 −1.24756 −0.623779 0.781601i \(-0.714403\pi\)
−0.623779 + 0.781601i \(0.714403\pi\)
\(270\) 0 0
\(271\) − 187.276i − 0.691054i −0.938409 0.345527i \(-0.887700\pi\)
0.938409 0.345527i \(-0.112300\pi\)
\(272\) 0 0
\(273\) 119.054 0.436096
\(274\) 0 0
\(275\) 24.8626i 0.0904095i
\(276\) 0 0
\(277\) 132.592 0.478670 0.239335 0.970937i \(-0.423071\pi\)
0.239335 + 0.970937i \(0.423071\pi\)
\(278\) 0 0
\(279\) − 96.9480i − 0.347484i
\(280\) 0 0
\(281\) −331.520 −1.17979 −0.589894 0.807481i \(-0.700830\pi\)
−0.589894 + 0.807481i \(0.700830\pi\)
\(282\) 0 0
\(283\) − 66.7158i − 0.235745i −0.993029 0.117873i \(-0.962393\pi\)
0.993029 0.117873i \(-0.0376074\pi\)
\(284\) 0 0
\(285\) −76.6016 −0.268778
\(286\) 0 0
\(287\) − 100.089i − 0.348741i
\(288\) 0 0
\(289\) −288.945 −0.999809
\(290\) 0 0
\(291\) − 311.365i − 1.06998i
\(292\) 0 0
\(293\) −289.215 −0.987082 −0.493541 0.869723i \(-0.664298\pi\)
−0.493541 + 0.869723i \(0.664298\pi\)
\(294\) 0 0
\(295\) − 170.415i − 0.577679i
\(296\) 0 0
\(297\) 455.111 1.53236
\(298\) 0 0
\(299\) 143.213i 0.478972i
\(300\) 0 0
\(301\) −12.8243 −0.0426058
\(302\) 0 0
\(303\) − 623.377i − 2.05735i
\(304\) 0 0
\(305\) 310.556 1.01822
\(306\) 0 0
\(307\) − 0.693177i − 0.00225790i −0.999999 0.00112895i \(-0.999641\pi\)
0.999999 0.00112895i \(-0.000359357\pi\)
\(308\) 0 0
\(309\) −135.243 −0.437680
\(310\) 0 0
\(311\) − 62.1583i − 0.199866i −0.994994 0.0999330i \(-0.968137\pi\)
0.994994 0.0999330i \(-0.0318628\pi\)
\(312\) 0 0
\(313\) −213.594 −0.682408 −0.341204 0.939989i \(-0.610835\pi\)
−0.341204 + 0.939989i \(0.610835\pi\)
\(314\) 0 0
\(315\) − 36.7204i − 0.116573i
\(316\) 0 0
\(317\) 23.4577 0.0739990 0.0369995 0.999315i \(-0.488220\pi\)
0.0369995 + 0.999315i \(0.488220\pi\)
\(318\) 0 0
\(319\) 744.059i 2.33247i
\(320\) 0 0
\(321\) −132.317 −0.412201
\(322\) 0 0
\(323\) 1.07079i 0.00331515i
\(324\) 0 0
\(325\) −15.1369 −0.0465751
\(326\) 0 0
\(327\) − 95.9380i − 0.293388i
\(328\) 0 0
\(329\) 191.385 0.581718
\(330\) 0 0
\(331\) 507.406i 1.53295i 0.642275 + 0.766474i \(0.277991\pi\)
−0.642275 + 0.766474i \(0.722009\pi\)
\(332\) 0 0
\(333\) 154.163 0.462951
\(334\) 0 0
\(335\) 90.0126i 0.268694i
\(336\) 0 0
\(337\) −342.726 −1.01699 −0.508495 0.861065i \(-0.669798\pi\)
−0.508495 + 0.861065i \(0.669798\pi\)
\(338\) 0 0
\(339\) 283.705i 0.836889i
\(340\) 0 0
\(341\) 732.555 2.14825
\(342\) 0 0
\(343\) 18.5203i 0.0539949i
\(344\) 0 0
\(345\) 184.035 0.533434
\(346\) 0 0
\(347\) 136.745i 0.394079i 0.980396 + 0.197039i \(0.0631327\pi\)
−0.980396 + 0.197039i \(0.936867\pi\)
\(348\) 0 0
\(349\) 82.0565 0.235119 0.117559 0.993066i \(-0.462493\pi\)
0.117559 + 0.993066i \(0.462493\pi\)
\(350\) 0 0
\(351\) 277.081i 0.789406i
\(352\) 0 0
\(353\) 507.367 1.43730 0.718651 0.695371i \(-0.244760\pi\)
0.718651 + 0.695371i \(0.244760\pi\)
\(354\) 0 0
\(355\) 232.055i 0.653677i
\(356\) 0 0
\(357\) −2.13861 −0.00599051
\(358\) 0 0
\(359\) 560.809i 1.56214i 0.624442 + 0.781071i \(0.285326\pi\)
−0.624442 + 0.781071i \(0.714674\pi\)
\(360\) 0 0
\(361\) 340.218 0.942432
\(362\) 0 0
\(363\) − 1171.02i − 3.22595i
\(364\) 0 0
\(365\) 273.003 0.747952
\(366\) 0 0
\(367\) − 26.9431i − 0.0734145i −0.999326 0.0367072i \(-0.988313\pi\)
0.999326 0.0367072i \(-0.0116869\pi\)
\(368\) 0 0
\(369\) −107.528 −0.291403
\(370\) 0 0
\(371\) − 57.3352i − 0.154542i
\(372\) 0 0
\(373\) −538.034 −1.44245 −0.721225 0.692701i \(-0.756421\pi\)
−0.721225 + 0.692701i \(0.756421\pi\)
\(374\) 0 0
\(375\) 439.534i 1.17209i
\(376\) 0 0
\(377\) −453.000 −1.20159
\(378\) 0 0
\(379\) − 182.132i − 0.480560i −0.970704 0.240280i \(-0.922761\pi\)
0.970704 0.240280i \(-0.0772393\pi\)
\(380\) 0 0
\(381\) 86.5166 0.227078
\(382\) 0 0
\(383\) 333.271i 0.870160i 0.900392 + 0.435080i \(0.143280\pi\)
−0.900392 + 0.435080i \(0.856720\pi\)
\(384\) 0 0
\(385\) 277.466 0.720690
\(386\) 0 0
\(387\) 13.7775i 0.0356007i
\(388\) 0 0
\(389\) 109.639 0.281847 0.140924 0.990020i \(-0.454993\pi\)
0.140924 + 0.990020i \(0.454993\pi\)
\(390\) 0 0
\(391\) − 2.57258i − 0.00657948i
\(392\) 0 0
\(393\) −434.971 −1.10680
\(394\) 0 0
\(395\) − 463.938i − 1.17453i
\(396\) 0 0
\(397\) 310.938 0.783219 0.391610 0.920131i \(-0.371918\pi\)
0.391610 + 0.920131i \(0.371918\pi\)
\(398\) 0 0
\(399\) − 41.5061i − 0.104025i
\(400\) 0 0
\(401\) −423.903 −1.05711 −0.528557 0.848898i \(-0.677267\pi\)
−0.528557 + 0.848898i \(0.677267\pi\)
\(402\) 0 0
\(403\) 445.995i 1.10669i
\(404\) 0 0
\(405\) 480.974 1.18759
\(406\) 0 0
\(407\) 1164.88i 2.86211i
\(408\) 0 0
\(409\) −444.543 −1.08690 −0.543451 0.839441i \(-0.682883\pi\)
−0.543451 + 0.839441i \(0.682883\pi\)
\(410\) 0 0
\(411\) − 120.258i − 0.292598i
\(412\) 0 0
\(413\) 92.3385 0.223580
\(414\) 0 0
\(415\) 349.201i 0.841449i
\(416\) 0 0
\(417\) 410.020 0.983261
\(418\) 0 0
\(419\) − 457.129i − 1.09100i −0.838111 0.545500i \(-0.816340\pi\)
0.838111 0.545500i \(-0.183660\pi\)
\(420\) 0 0
\(421\) 25.4812 0.0605255 0.0302628 0.999542i \(-0.490366\pi\)
0.0302628 + 0.999542i \(0.490366\pi\)
\(422\) 0 0
\(423\) − 205.610i − 0.486075i
\(424\) 0 0
\(425\) 0.271910 0.000639787 0
\(426\) 0 0
\(427\) 168.273i 0.394082i
\(428\) 0 0
\(429\) 966.453 2.25280
\(430\) 0 0
\(431\) 124.595i 0.289084i 0.989499 + 0.144542i \(0.0461709\pi\)
−0.989499 + 0.144542i \(0.953829\pi\)
\(432\) 0 0
\(433\) −272.271 −0.628802 −0.314401 0.949290i \(-0.601804\pi\)
−0.314401 + 0.949290i \(0.601804\pi\)
\(434\) 0 0
\(435\) 582.126i 1.33822i
\(436\) 0 0
\(437\) 49.9285 0.114253
\(438\) 0 0
\(439\) 255.069i 0.581023i 0.956871 + 0.290512i \(0.0938255\pi\)
−0.956871 + 0.290512i \(0.906174\pi\)
\(440\) 0 0
\(441\) 19.8967 0.0451173
\(442\) 0 0
\(443\) 131.274i 0.296330i 0.988963 + 0.148165i \(0.0473366\pi\)
−0.988963 + 0.148165i \(0.952663\pi\)
\(444\) 0 0
\(445\) −780.066 −1.75296
\(446\) 0 0
\(447\) 419.601i 0.938704i
\(448\) 0 0
\(449\) −642.824 −1.43168 −0.715839 0.698265i \(-0.753956\pi\)
−0.715839 + 0.698265i \(0.753956\pi\)
\(450\) 0 0
\(451\) − 812.496i − 1.80154i
\(452\) 0 0
\(453\) −758.472 −1.67433
\(454\) 0 0
\(455\) 168.927i 0.371269i
\(456\) 0 0
\(457\) −693.088 −1.51660 −0.758302 0.651903i \(-0.773971\pi\)
−0.758302 + 0.651903i \(0.773971\pi\)
\(458\) 0 0
\(459\) − 4.97731i − 0.0108438i
\(460\) 0 0
\(461\) 258.699 0.561170 0.280585 0.959829i \(-0.409472\pi\)
0.280585 + 0.959829i \(0.409472\pi\)
\(462\) 0 0
\(463\) − 637.226i − 1.37630i −0.725569 0.688150i \(-0.758423\pi\)
0.725569 0.688150i \(-0.241577\pi\)
\(464\) 0 0
\(465\) 573.125 1.23253
\(466\) 0 0
\(467\) 199.483i 0.427159i 0.976926 + 0.213580i \(0.0685123\pi\)
−0.976926 + 0.213580i \(0.931488\pi\)
\(468\) 0 0
\(469\) −48.7727 −0.103993
\(470\) 0 0
\(471\) − 23.2979i − 0.0494648i
\(472\) 0 0
\(473\) −104.105 −0.220095
\(474\) 0 0
\(475\) 5.27721i 0.0111099i
\(476\) 0 0
\(477\) −61.5966 −0.129133
\(478\) 0 0
\(479\) 674.160i 1.40743i 0.710481 + 0.703716i \(0.248477\pi\)
−0.710481 + 0.703716i \(0.751523\pi\)
\(480\) 0 0
\(481\) −709.204 −1.47444
\(482\) 0 0
\(483\) 99.7182i 0.206456i
\(484\) 0 0
\(485\) 441.799 0.910926
\(486\) 0 0
\(487\) − 401.718i − 0.824883i −0.910984 0.412442i \(-0.864676\pi\)
0.910984 0.412442i \(-0.135324\pi\)
\(488\) 0 0
\(489\) −713.181 −1.45845
\(490\) 0 0
\(491\) 428.880i 0.873482i 0.899587 + 0.436741i \(0.143867\pi\)
−0.899587 + 0.436741i \(0.856133\pi\)
\(492\) 0 0
\(493\) 8.13739 0.0165059
\(494\) 0 0
\(495\) − 298.088i − 0.602198i
\(496\) 0 0
\(497\) −125.738 −0.252993
\(498\) 0 0
\(499\) − 182.619i − 0.365970i −0.983116 0.182985i \(-0.941424\pi\)
0.983116 0.182985i \(-0.0585760\pi\)
\(500\) 0 0
\(501\) 569.632 1.13699
\(502\) 0 0
\(503\) − 380.158i − 0.755781i −0.925850 0.377891i \(-0.876650\pi\)
0.925850 0.377891i \(-0.123350\pi\)
\(504\) 0 0
\(505\) 884.516 1.75152
\(506\) 0 0
\(507\) 6.82188i 0.0134554i
\(508\) 0 0
\(509\) 289.538 0.568836 0.284418 0.958700i \(-0.408200\pi\)
0.284418 + 0.958700i \(0.408200\pi\)
\(510\) 0 0
\(511\) 147.925i 0.289481i
\(512\) 0 0
\(513\) 96.5995 0.188303
\(514\) 0 0
\(515\) − 191.898i − 0.372617i
\(516\) 0 0
\(517\) 1553.62 3.00507
\(518\) 0 0
\(519\) − 305.768i − 0.589148i
\(520\) 0 0
\(521\) 738.899 1.41823 0.709116 0.705092i \(-0.249094\pi\)
0.709116 + 0.705092i \(0.249094\pi\)
\(522\) 0 0
\(523\) 647.126i 1.23734i 0.785653 + 0.618668i \(0.212327\pi\)
−0.785653 + 0.618668i \(0.787673\pi\)
\(524\) 0 0
\(525\) −10.5397 −0.0200757
\(526\) 0 0
\(527\) − 8.01157i − 0.0152022i
\(528\) 0 0
\(529\) 409.047 0.773246
\(530\) 0 0
\(531\) − 99.2014i − 0.186820i
\(532\) 0 0
\(533\) 494.666 0.928078
\(534\) 0 0
\(535\) − 187.745i − 0.350926i
\(536\) 0 0
\(537\) 276.607 0.515097
\(538\) 0 0
\(539\) 150.343i 0.278930i
\(540\) 0 0
\(541\) −178.722 −0.330355 −0.165178 0.986264i \(-0.552820\pi\)
−0.165178 + 0.986264i \(0.552820\pi\)
\(542\) 0 0
\(543\) − 951.008i − 1.75140i
\(544\) 0 0
\(545\) 136.127 0.249775
\(546\) 0 0
\(547\) − 452.236i − 0.826758i −0.910559 0.413379i \(-0.864349\pi\)
0.910559 0.413379i \(-0.135651\pi\)
\(548\) 0 0
\(549\) 180.780 0.329289
\(550\) 0 0
\(551\) 157.930i 0.286625i
\(552\) 0 0
\(553\) 251.382 0.454579
\(554\) 0 0
\(555\) 911.360i 1.64209i
\(556\) 0 0
\(557\) 854.108 1.53341 0.766704 0.642001i \(-0.221895\pi\)
0.766704 + 0.642001i \(0.221895\pi\)
\(558\) 0 0
\(559\) − 63.3814i − 0.113383i
\(560\) 0 0
\(561\) −17.3607 −0.0309460
\(562\) 0 0
\(563\) − 249.654i − 0.443436i −0.975111 0.221718i \(-0.928834\pi\)
0.975111 0.221718i \(-0.0711664\pi\)
\(564\) 0 0
\(565\) −402.553 −0.712483
\(566\) 0 0
\(567\) 260.613i 0.459634i
\(568\) 0 0
\(569\) −104.353 −0.183396 −0.0916982 0.995787i \(-0.529229\pi\)
−0.0916982 + 0.995787i \(0.529229\pi\)
\(570\) 0 0
\(571\) − 649.705i − 1.13784i −0.822394 0.568919i \(-0.807362\pi\)
0.822394 0.568919i \(-0.192638\pi\)
\(572\) 0 0
\(573\) −698.630 −1.21925
\(574\) 0 0
\(575\) − 12.6785i − 0.0220495i
\(576\) 0 0
\(577\) −346.022 −0.599692 −0.299846 0.953988i \(-0.596935\pi\)
−0.299846 + 0.953988i \(0.596935\pi\)
\(578\) 0 0
\(579\) 300.537i 0.519061i
\(580\) 0 0
\(581\) −189.213 −0.325667
\(582\) 0 0
\(583\) − 465.434i − 0.798342i
\(584\) 0 0
\(585\) 181.482 0.310226
\(586\) 0 0
\(587\) − 1153.54i − 1.96514i −0.185885 0.982572i \(-0.559515\pi\)
0.185885 0.982572i \(-0.440485\pi\)
\(588\) 0 0
\(589\) 155.488 0.263987
\(590\) 0 0
\(591\) − 74.5515i − 0.126145i
\(592\) 0 0
\(593\) 880.135 1.48421 0.742104 0.670285i \(-0.233828\pi\)
0.742104 + 0.670285i \(0.233828\pi\)
\(594\) 0 0
\(595\) − 3.03450i − 0.00510000i
\(596\) 0 0
\(597\) −626.081 −1.04871
\(598\) 0 0
\(599\) − 554.939i − 0.926442i −0.886243 0.463221i \(-0.846694\pi\)
0.886243 0.463221i \(-0.153306\pi\)
\(600\) 0 0
\(601\) 666.057 1.10825 0.554124 0.832434i \(-0.313054\pi\)
0.554124 + 0.832434i \(0.313054\pi\)
\(602\) 0 0
\(603\) 52.3977i 0.0868950i
\(604\) 0 0
\(605\) 1661.57 2.74640
\(606\) 0 0
\(607\) 192.927i 0.317836i 0.987292 + 0.158918i \(0.0508006\pi\)
−0.987292 + 0.158918i \(0.949199\pi\)
\(608\) 0 0
\(609\) −315.421 −0.517933
\(610\) 0 0
\(611\) 945.878i 1.54808i
\(612\) 0 0
\(613\) 608.234 0.992226 0.496113 0.868258i \(-0.334760\pi\)
0.496113 + 0.868258i \(0.334760\pi\)
\(614\) 0 0
\(615\) − 635.668i − 1.03361i
\(616\) 0 0
\(617\) −0.884056 −0.00143283 −0.000716415 1.00000i \(-0.500228\pi\)
−0.000716415 1.00000i \(0.500228\pi\)
\(618\) 0 0
\(619\) 358.525i 0.579200i 0.957148 + 0.289600i \(0.0935222\pi\)
−0.957148 + 0.289600i \(0.906478\pi\)
\(620\) 0 0
\(621\) −232.080 −0.373719
\(622\) 0 0
\(623\) − 422.674i − 0.678449i
\(624\) 0 0
\(625\) −594.720 −0.951552
\(626\) 0 0
\(627\) − 336.937i − 0.537379i
\(628\) 0 0
\(629\) 12.7397 0.0202539
\(630\) 0 0
\(631\) − 390.515i − 0.618883i −0.950918 0.309442i \(-0.899858\pi\)
0.950918 0.309442i \(-0.100142\pi\)
\(632\) 0 0
\(633\) −73.7137 −0.116451
\(634\) 0 0
\(635\) 122.759i 0.193322i
\(636\) 0 0
\(637\) −91.5322 −0.143693
\(638\) 0 0
\(639\) 135.083i 0.211397i
\(640\) 0 0
\(641\) −431.936 −0.673848 −0.336924 0.941532i \(-0.609386\pi\)
−0.336924 + 0.941532i \(0.609386\pi\)
\(642\) 0 0
\(643\) − 49.9370i − 0.0776625i −0.999246 0.0388313i \(-0.987637\pi\)
0.999246 0.0388313i \(-0.0123635\pi\)
\(644\) 0 0
\(645\) −81.4480 −0.126276
\(646\) 0 0
\(647\) − 224.141i − 0.346431i −0.984884 0.173216i \(-0.944584\pi\)
0.984884 0.173216i \(-0.0554157\pi\)
\(648\) 0 0
\(649\) 749.582 1.15498
\(650\) 0 0
\(651\) 310.544i 0.477027i
\(652\) 0 0
\(653\) −80.7637 −0.123681 −0.0618405 0.998086i \(-0.519697\pi\)
−0.0618405 + 0.998086i \(0.519697\pi\)
\(654\) 0 0
\(655\) − 617.186i − 0.942268i
\(656\) 0 0
\(657\) 158.919 0.241886
\(658\) 0 0
\(659\) 940.466i 1.42711i 0.700599 + 0.713555i \(0.252916\pi\)
−0.700599 + 0.713555i \(0.747084\pi\)
\(660\) 0 0
\(661\) 119.930 0.181437 0.0907184 0.995877i \(-0.471084\pi\)
0.0907184 + 0.995877i \(0.471084\pi\)
\(662\) 0 0
\(663\) − 10.5696i − 0.0159421i
\(664\) 0 0
\(665\) 58.8935 0.0885616
\(666\) 0 0
\(667\) − 379.426i − 0.568855i
\(668\) 0 0
\(669\) 674.346 1.00799
\(670\) 0 0
\(671\) 1366.00i 2.03577i
\(672\) 0 0
\(673\) 1085.06 1.61227 0.806136 0.591731i \(-0.201555\pi\)
0.806136 + 0.591731i \(0.201555\pi\)
\(674\) 0 0
\(675\) − 24.5298i − 0.0363404i
\(676\) 0 0
\(677\) 949.901 1.40310 0.701552 0.712618i \(-0.252491\pi\)
0.701552 + 0.712618i \(0.252491\pi\)
\(678\) 0 0
\(679\) 239.386i 0.352557i
\(680\) 0 0
\(681\) −93.8270 −0.137778
\(682\) 0 0
\(683\) − 893.785i − 1.30862i −0.756228 0.654308i \(-0.772960\pi\)
0.756228 0.654308i \(-0.227040\pi\)
\(684\) 0 0
\(685\) 170.635 0.249102
\(686\) 0 0
\(687\) − 606.861i − 0.883349i
\(688\) 0 0
\(689\) 283.366 0.411272
\(690\) 0 0
\(691\) − 1208.56i − 1.74901i −0.485019 0.874504i \(-0.661187\pi\)
0.485019 0.874504i \(-0.338813\pi\)
\(692\) 0 0
\(693\) 161.517 0.233069
\(694\) 0 0
\(695\) 581.782i 0.837096i
\(696\) 0 0
\(697\) −8.88585 −0.0127487
\(698\) 0 0
\(699\) − 247.210i − 0.353662i
\(700\) 0 0
\(701\) −219.477 −0.313091 −0.156546 0.987671i \(-0.550036\pi\)
−0.156546 + 0.987671i \(0.550036\pi\)
\(702\) 0 0
\(703\) 247.251i 0.351709i
\(704\) 0 0
\(705\) 1215.50 1.72411
\(706\) 0 0
\(707\) 479.270i 0.677892i
\(708\) 0 0
\(709\) 1265.13 1.78439 0.892195 0.451651i \(-0.149165\pi\)
0.892195 + 0.451651i \(0.149165\pi\)
\(710\) 0 0
\(711\) − 270.066i − 0.379839i
\(712\) 0 0
\(713\) −373.560 −0.523927
\(714\) 0 0
\(715\) 1371.31i 1.91792i
\(716\) 0 0
\(717\) −244.650 −0.341213
\(718\) 0 0
\(719\) − 1163.47i − 1.61818i −0.587687 0.809089i \(-0.699961\pi\)
0.587687 0.809089i \(-0.300039\pi\)
\(720\) 0 0
\(721\) 103.979 0.144215
\(722\) 0 0
\(723\) 193.094i 0.267074i
\(724\) 0 0
\(725\) 40.1036 0.0553153
\(726\) 0 0
\(727\) 1303.68i 1.79324i 0.442803 + 0.896619i \(0.353984\pi\)
−0.442803 + 0.896619i \(0.646016\pi\)
\(728\) 0 0
\(729\) −376.305 −0.516193
\(730\) 0 0
\(731\) 1.13854i 0.00155751i
\(732\) 0 0
\(733\) 1256.12 1.71367 0.856836 0.515589i \(-0.172427\pi\)
0.856836 + 0.515589i \(0.172427\pi\)
\(734\) 0 0
\(735\) 117.623i 0.160031i
\(736\) 0 0
\(737\) −395.925 −0.537212
\(738\) 0 0
\(739\) − 687.168i − 0.929862i −0.885347 0.464931i \(-0.846079\pi\)
0.885347 0.464931i \(-0.153921\pi\)
\(740\) 0 0
\(741\) 205.134 0.276835
\(742\) 0 0
\(743\) − 362.628i − 0.488059i −0.969768 0.244030i \(-0.921531\pi\)
0.969768 0.244030i \(-0.0784694\pi\)
\(744\) 0 0
\(745\) −595.376 −0.799163
\(746\) 0 0
\(747\) 203.275i 0.272122i
\(748\) 0 0
\(749\) 101.729 0.135819
\(750\) 0 0
\(751\) − 261.366i − 0.348024i −0.984744 0.174012i \(-0.944327\pi\)
0.984744 0.174012i \(-0.0556732\pi\)
\(752\) 0 0
\(753\) −1269.67 −1.68615
\(754\) 0 0
\(755\) − 1076.20i − 1.42544i
\(756\) 0 0
\(757\) −1395.34 −1.84325 −0.921625 0.388081i \(-0.873138\pi\)
−0.921625 + 0.388081i \(0.873138\pi\)
\(758\) 0 0
\(759\) 809.488i 1.06652i
\(760\) 0 0
\(761\) 319.500 0.419843 0.209921 0.977718i \(-0.432679\pi\)
0.209921 + 0.977718i \(0.432679\pi\)
\(762\) 0 0
\(763\) 73.7598i 0.0966708i
\(764\) 0 0
\(765\) −3.26003 −0.00426148
\(766\) 0 0
\(767\) 456.362i 0.594996i
\(768\) 0 0
\(769\) 634.936 0.825664 0.412832 0.910807i \(-0.364540\pi\)
0.412832 + 0.910807i \(0.364540\pi\)
\(770\) 0 0
\(771\) 81.7056i 0.105974i
\(772\) 0 0
\(773\) 96.1663 0.124407 0.0622033 0.998063i \(-0.480187\pi\)
0.0622033 + 0.998063i \(0.480187\pi\)
\(774\) 0 0
\(775\) − 39.4836i − 0.0509465i
\(776\) 0 0
\(777\) −493.815 −0.635540
\(778\) 0 0
\(779\) − 172.456i − 0.221382i
\(780\) 0 0
\(781\) −1020.71 −1.30693
\(782\) 0 0
\(783\) − 734.098i − 0.937545i
\(784\) 0 0
\(785\) 33.0577 0.0421117
\(786\) 0 0
\(787\) 1319.25i 1.67630i 0.545442 + 0.838148i \(0.316362\pi\)
−0.545442 + 0.838148i \(0.683638\pi\)
\(788\) 0 0
\(789\) 254.554 0.322628
\(790\) 0 0
\(791\) − 218.121i − 0.275753i
\(792\) 0 0
\(793\) −831.651 −1.04874
\(794\) 0 0
\(795\) − 364.139i − 0.458036i
\(796\) 0 0
\(797\) 818.575 1.02707 0.513535 0.858068i \(-0.328336\pi\)
0.513535 + 0.858068i \(0.328336\pi\)
\(798\) 0 0
\(799\) − 16.9911i − 0.0212655i
\(800\) 0 0
\(801\) −454.088 −0.566902
\(802\) 0 0
\(803\) 1200.82i 1.49541i
\(804\) 0 0
\(805\) −141.491 −0.175765
\(806\) 0 0
\(807\) − 1154.87i − 1.43106i
\(808\) 0 0
\(809\) −1232.72 −1.52376 −0.761881 0.647717i \(-0.775724\pi\)
−0.761881 + 0.647717i \(0.775724\pi\)
\(810\) 0 0
\(811\) − 1009.05i − 1.24421i −0.782935 0.622103i \(-0.786278\pi\)
0.782935 0.622103i \(-0.213722\pi\)
\(812\) 0 0
\(813\) 644.468 0.792703
\(814\) 0 0
\(815\) − 1011.94i − 1.24164i
\(816\) 0 0
\(817\) −22.0968 −0.0270462
\(818\) 0 0
\(819\) 98.3351i 0.120067i
\(820\) 0 0
\(821\) −939.093 −1.14384 −0.571920 0.820309i \(-0.693801\pi\)
−0.571920 + 0.820309i \(0.693801\pi\)
\(822\) 0 0
\(823\) − 911.100i − 1.10705i −0.832833 0.553524i \(-0.813283\pi\)
0.832833 0.553524i \(-0.186717\pi\)
\(824\) 0 0
\(825\) −85.5591 −0.103708
\(826\) 0 0
\(827\) − 65.6564i − 0.0793910i −0.999212 0.0396955i \(-0.987361\pi\)
0.999212 0.0396955i \(-0.0126388\pi\)
\(828\) 0 0
\(829\) −1515.94 −1.82864 −0.914318 0.404997i \(-0.867272\pi\)
−0.914318 + 0.404997i \(0.867272\pi\)
\(830\) 0 0
\(831\) 456.285i 0.549079i
\(832\) 0 0
\(833\) 1.64422 0.00197386
\(834\) 0 0
\(835\) 808.257i 0.967972i
\(836\) 0 0
\(837\) −722.747 −0.863497
\(838\) 0 0
\(839\) 869.972i 1.03692i 0.855103 + 0.518458i \(0.173494\pi\)
−0.855103 + 0.518458i \(0.826506\pi\)
\(840\) 0 0
\(841\) 359.174 0.427080
\(842\) 0 0
\(843\) − 1140.85i − 1.35333i
\(844\) 0 0
\(845\) −9.67964 −0.0114552
\(846\) 0 0
\(847\) 900.313i 1.06294i
\(848\) 0 0
\(849\) 229.588 0.270421
\(850\) 0 0
\(851\) − 594.020i − 0.698026i
\(852\) 0 0
\(853\) −1643.91 −1.92721 −0.963607 0.267322i \(-0.913861\pi\)
−0.963607 + 0.267322i \(0.913861\pi\)
\(854\) 0 0
\(855\) − 63.2706i − 0.0740007i
\(856\) 0 0
\(857\) −286.059 −0.333791 −0.166895 0.985975i \(-0.553374\pi\)
−0.166895 + 0.985975i \(0.553374\pi\)
\(858\) 0 0
\(859\) 719.782i 0.837930i 0.908002 + 0.418965i \(0.137607\pi\)
−0.908002 + 0.418965i \(0.862393\pi\)
\(860\) 0 0
\(861\) 344.433 0.400038
\(862\) 0 0
\(863\) − 1120.47i − 1.29835i −0.760641 0.649173i \(-0.775115\pi\)
0.760641 0.649173i \(-0.224885\pi\)
\(864\) 0 0
\(865\) 433.857 0.501569
\(866\) 0 0
\(867\) − 994.339i − 1.14687i
\(868\) 0 0
\(869\) 2040.66 2.34828
\(870\) 0 0
\(871\) − 241.048i − 0.276749i
\(872\) 0 0
\(873\) 257.178 0.294591
\(874\) 0 0
\(875\) − 337.926i − 0.386201i
\(876\) 0 0
\(877\) −145.400 −0.165792 −0.0828960 0.996558i \(-0.526417\pi\)
−0.0828960 + 0.996558i \(0.526417\pi\)
\(878\) 0 0
\(879\) − 995.269i − 1.13227i
\(880\) 0 0
\(881\) 476.080 0.540386 0.270193 0.962806i \(-0.412913\pi\)
0.270193 + 0.962806i \(0.412913\pi\)
\(882\) 0 0
\(883\) 1101.22i 1.24714i 0.781769 + 0.623568i \(0.214318\pi\)
−0.781769 + 0.623568i \(0.785682\pi\)
\(884\) 0 0
\(885\) 586.447 0.662652
\(886\) 0 0
\(887\) − 1491.49i − 1.68150i −0.541427 0.840748i \(-0.682116\pi\)
0.541427 0.840748i \(-0.317884\pi\)
\(888\) 0 0
\(889\) −66.5164 −0.0748216
\(890\) 0 0
\(891\) 2115.59i 2.37440i
\(892\) 0 0
\(893\) 329.764 0.369276
\(894\) 0 0
\(895\) 392.481i 0.438526i
\(896\) 0 0
\(897\) −492.834 −0.549425
\(898\) 0 0
\(899\) − 1181.62i − 1.31437i
\(900\) 0 0
\(901\) −5.09021 −0.00564951
\(902\) 0 0
\(903\) − 44.1321i − 0.0488728i
\(904\) 0 0
\(905\) 1349.40 1.49104
\(906\) 0 0
\(907\) 1155.46i 1.27394i 0.770889 + 0.636969i \(0.219812\pi\)
−0.770889 + 0.636969i \(0.780188\pi\)
\(908\) 0 0
\(909\) 514.891 0.566436
\(910\) 0 0
\(911\) 944.690i 1.03698i 0.855083 + 0.518491i \(0.173506\pi\)
−0.855083 + 0.518491i \(0.826494\pi\)
\(912\) 0 0
\(913\) −1535.98 −1.68235
\(914\) 0 0
\(915\) 1068.71i 1.16799i
\(916\) 0 0
\(917\) 334.418 0.364687
\(918\) 0 0
\(919\) 149.150i 0.162296i 0.996702 + 0.0811478i \(0.0258586\pi\)
−0.996702 + 0.0811478i \(0.974141\pi\)
\(920\) 0 0
\(921\) 2.38541 0.00259003
\(922\) 0 0
\(923\) − 621.430i − 0.673272i
\(924\) 0 0
\(925\) 62.7851 0.0678758
\(926\) 0 0
\(927\) − 111.707i − 0.120503i
\(928\) 0 0
\(929\) −58.8399 −0.0633368 −0.0316684 0.999498i \(-0.510082\pi\)
−0.0316684 + 0.999498i \(0.510082\pi\)
\(930\) 0 0
\(931\) 31.9111i 0.0342761i
\(932\) 0 0
\(933\) 213.904 0.229265
\(934\) 0 0
\(935\) − 24.6333i − 0.0263458i
\(936\) 0 0
\(937\) 1700.18 1.81449 0.907246 0.420601i \(-0.138181\pi\)
0.907246 + 0.420601i \(0.138181\pi\)
\(938\) 0 0
\(939\) − 735.036i − 0.782786i
\(940\) 0 0
\(941\) 56.6116 0.0601611 0.0300805 0.999547i \(-0.490424\pi\)
0.0300805 + 0.999547i \(0.490424\pi\)
\(942\) 0 0
\(943\) 414.325i 0.439369i
\(944\) 0 0
\(945\) −273.751 −0.289684
\(946\) 0 0
\(947\) − 242.533i − 0.256107i −0.991767 0.128053i \(-0.959127\pi\)
0.991767 0.128053i \(-0.0408729\pi\)
\(948\) 0 0
\(949\) −731.084 −0.770373
\(950\) 0 0
\(951\) 80.7244i 0.0848837i
\(952\) 0 0
\(953\) 364.070 0.382025 0.191013 0.981588i \(-0.438823\pi\)
0.191013 + 0.981588i \(0.438823\pi\)
\(954\) 0 0
\(955\) − 991.294i − 1.03800i
\(956\) 0 0
\(957\) −2560.51 −2.67556
\(958\) 0 0
\(959\) 92.4575i 0.0964103i
\(960\) 0 0
\(961\) −202.348 −0.210560
\(962\) 0 0
\(963\) − 109.290i − 0.113489i
\(964\) 0 0
\(965\) −426.435 −0.441901
\(966\) 0 0
\(967\) 1221.99i 1.26369i 0.775093 + 0.631847i \(0.217703\pi\)
−0.775093 + 0.631847i \(0.782297\pi\)
\(968\) 0 0
\(969\) −3.68490 −0.00380279
\(970\) 0 0
\(971\) − 1088.53i − 1.12104i −0.828140 0.560521i \(-0.810601\pi\)
0.828140 0.560521i \(-0.189399\pi\)
\(972\) 0 0
\(973\) −315.235 −0.323982
\(974\) 0 0
\(975\) − 52.0903i − 0.0534259i
\(976\) 0 0
\(977\) −1061.51 −1.08650 −0.543252 0.839570i \(-0.682807\pi\)
−0.543252 + 0.839570i \(0.682807\pi\)
\(978\) 0 0
\(979\) − 3431.17i − 3.50477i
\(980\) 0 0
\(981\) 79.2419 0.0807766
\(982\) 0 0
\(983\) 1322.61i 1.34549i 0.739877 + 0.672743i \(0.234884\pi\)
−0.739877 + 0.672743i \(0.765116\pi\)
\(984\) 0 0
\(985\) 105.782 0.107393
\(986\) 0 0
\(987\) 658.610i 0.667285i
\(988\) 0 0
\(989\) 53.0874 0.0536778
\(990\) 0 0
\(991\) 675.806i 0.681944i 0.940073 + 0.340972i \(0.110756\pi\)
−0.940073 + 0.340972i \(0.889244\pi\)
\(992\) 0 0
\(993\) −1746.12 −1.75843
\(994\) 0 0
\(995\) − 888.354i − 0.892818i
\(996\) 0 0
\(997\) −409.220 −0.410452 −0.205226 0.978715i \(-0.565793\pi\)
−0.205226 + 0.978715i \(0.565793\pi\)
\(998\) 0 0
\(999\) − 1149.28i − 1.15043i
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 1792.3.d.j.1023.11 16
4.3 odd 2 inner 1792.3.d.j.1023.5 16
8.3 odd 2 inner 1792.3.d.j.1023.12 16
8.5 even 2 inner 1792.3.d.j.1023.6 16
16.3 odd 4 224.3.g.b.15.6 8
16.5 even 4 224.3.g.b.15.5 8
16.11 odd 4 56.3.g.b.43.4 yes 8
16.13 even 4 56.3.g.b.43.3 8
48.5 odd 4 2016.3.g.b.1135.6 8
48.11 even 4 504.3.g.b.379.5 8
48.29 odd 4 504.3.g.b.379.6 8
48.35 even 4 2016.3.g.b.1135.3 8
112.11 odd 12 392.3.k.o.275.3 16
112.13 odd 4 392.3.g.m.99.3 8
112.27 even 4 392.3.g.m.99.4 8
112.45 odd 12 392.3.k.n.275.8 16
112.59 even 12 392.3.k.n.275.3 16
112.61 odd 12 392.3.k.n.67.3 16
112.69 odd 4 1568.3.g.m.687.4 8
112.75 even 12 392.3.k.n.67.8 16
112.83 even 4 1568.3.g.m.687.3 8
112.93 even 12 392.3.k.o.67.3 16
112.107 odd 12 392.3.k.o.67.8 16
112.109 even 12 392.3.k.o.275.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.g.b.43.3 8 16.13 even 4
56.3.g.b.43.4 yes 8 16.11 odd 4
224.3.g.b.15.5 8 16.5 even 4
224.3.g.b.15.6 8 16.3 odd 4
392.3.g.m.99.3 8 112.13 odd 4
392.3.g.m.99.4 8 112.27 even 4
392.3.k.n.67.3 16 112.61 odd 12
392.3.k.n.67.8 16 112.75 even 12
392.3.k.n.275.3 16 112.59 even 12
392.3.k.n.275.8 16 112.45 odd 12
392.3.k.o.67.3 16 112.93 even 12
392.3.k.o.67.8 16 112.107 odd 12
392.3.k.o.275.3 16 112.11 odd 12
392.3.k.o.275.8 16 112.109 even 12
504.3.g.b.379.5 8 48.11 even 4
504.3.g.b.379.6 8 48.29 odd 4
1568.3.g.m.687.3 8 112.83 even 4
1568.3.g.m.687.4 8 112.69 odd 4
1792.3.d.j.1023.5 16 4.3 odd 2 inner
1792.3.d.j.1023.6 16 8.5 even 2 inner
1792.3.d.j.1023.11 16 1.1 even 1 trivial
1792.3.d.j.1023.12 16 8.3 odd 2 inner
2016.3.g.b.1135.3 8 48.35 even 4
2016.3.g.b.1135.6 8 48.5 odd 4