Properties

Label 1152.3.m.d.415.8
Level $1152$
Weight $3$
Character 1152.415
Analytic conductor $31.390$
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] = [1152,3,Mod(415,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.415");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.m (of order \(4\), degree \(2\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6x^{14} + 10x^{12} + 88x^{10} - 752x^{8} + 1408x^{6} + 2560x^{4} - 24576x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 415.8
Root \(1.99750 - 0.0999235i\) of defining polynomial
Character \(\chi\) \(=\) 1152.415
Dual form 1152.3.m.d.991.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(6.01265 + 6.01265i) q^{5} -8.23187 q^{7} +O(q^{10})\) \(q+(6.01265 + 6.01265i) q^{5} -8.23187 q^{7} +(6.51529 - 6.51529i) q^{11} +(-8.82865 + 8.82865i) q^{13} -14.1753 q^{17} +(-23.7488 - 23.7488i) q^{19} +9.42125 q^{23} +47.3040i q^{25} +(-23.7973 + 23.7973i) q^{29} +24.4148i q^{31} +(-49.4954 - 49.4954i) q^{35} +(-24.2052 - 24.2052i) q^{37} -6.67771i q^{41} +(0.897918 - 0.897918i) q^{43} -25.2401i q^{47} +18.7636 q^{49} +(-32.6251 - 32.6251i) q^{53} +78.3484 q^{55} +(-8.31871 + 8.31871i) q^{59} +(68.4028 - 68.4028i) q^{61} -106.167 q^{65} +(-7.71922 - 7.71922i) q^{67} -137.259 q^{71} -52.8655i q^{73} +(-53.6330 + 53.6330i) q^{77} +87.2269i q^{79} +(9.53893 + 9.53893i) q^{83} +(-85.2314 - 85.2314i) q^{85} +146.488i q^{89} +(72.6762 - 72.6762i) q^{91} -285.586i q^{95} +101.170 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{19} - 96 q^{37} - 32 q^{43} + 112 q^{49} + 256 q^{55} + 32 q^{61} - 256 q^{67} - 160 q^{85} + 288 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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 0
\(4\) 0 0
\(5\) 6.01265 + 6.01265i 1.20253 + 1.20253i 0.973394 + 0.229136i \(0.0735901\pi\)
0.229136 + 0.973394i \(0.426410\pi\)
\(6\) 0 0
\(7\) −8.23187 −1.17598 −0.587990 0.808868i \(-0.700081\pi\)
−0.587990 + 0.808868i \(0.700081\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.51529 6.51529i 0.592299 0.592299i −0.345953 0.938252i \(-0.612444\pi\)
0.938252 + 0.345953i \(0.112444\pi\)
\(12\) 0 0
\(13\) −8.82865 + 8.82865i −0.679127 + 0.679127i −0.959803 0.280676i \(-0.909441\pi\)
0.280676 + 0.959803i \(0.409441\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −14.1753 −0.833844 −0.416922 0.908942i \(-0.636891\pi\)
−0.416922 + 0.908942i \(0.636891\pi\)
\(18\) 0 0
\(19\) −23.7488 23.7488i −1.24994 1.24994i −0.955748 0.294187i \(-0.904951\pi\)
−0.294187 0.955748i \(-0.595049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.42125 0.409619 0.204810 0.978802i \(-0.434342\pi\)
0.204810 + 0.978802i \(0.434342\pi\)
\(24\) 0 0
\(25\) 47.3040i 1.89216i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −23.7973 + 23.7973i −0.820598 + 0.820598i −0.986194 0.165596i \(-0.947045\pi\)
0.165596 + 0.986194i \(0.447045\pi\)
\(30\) 0 0
\(31\) 24.4148i 0.787575i 0.919202 + 0.393787i \(0.128835\pi\)
−0.919202 + 0.393787i \(0.871165\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −49.4954 49.4954i −1.41415 1.41415i
\(36\) 0 0
\(37\) −24.2052 24.2052i −0.654193 0.654193i 0.299807 0.954000i \(-0.403078\pi\)
−0.954000 + 0.299807i \(0.903078\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.67771i 0.162871i −0.996679 0.0814355i \(-0.974050\pi\)
0.996679 0.0814355i \(-0.0259505\pi\)
\(42\) 0 0
\(43\) 0.897918 0.897918i 0.0208818 0.0208818i −0.696589 0.717471i \(-0.745300\pi\)
0.717471 + 0.696589i \(0.245300\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 25.2401i 0.537023i −0.963276 0.268512i \(-0.913468\pi\)
0.963276 0.268512i \(-0.0865318\pi\)
\(48\) 0 0
\(49\) 18.7636 0.382931
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −32.6251 32.6251i −0.615568 0.615568i 0.328824 0.944391i \(-0.393348\pi\)
−0.944391 + 0.328824i \(0.893348\pi\)
\(54\) 0 0
\(55\) 78.3484 1.42452
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.31871 + 8.31871i −0.140995 + 0.140995i −0.774081 0.633086i \(-0.781788\pi\)
0.633086 + 0.774081i \(0.281788\pi\)
\(60\) 0 0
\(61\) 68.4028 68.4028i 1.12136 1.12136i 0.129820 0.991538i \(-0.458560\pi\)
0.991538 0.129820i \(-0.0414399\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −106.167 −1.63334
\(66\) 0 0
\(67\) −7.71922 7.71922i −0.115212 0.115212i 0.647150 0.762362i \(-0.275961\pi\)
−0.762362 + 0.647150i \(0.775961\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −137.259 −1.93322 −0.966612 0.256245i \(-0.917515\pi\)
−0.966612 + 0.256245i \(0.917515\pi\)
\(72\) 0 0
\(73\) 52.8655i 0.724185i −0.932142 0.362093i \(-0.882062\pi\)
0.932142 0.362093i \(-0.117938\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −53.6330 + 53.6330i −0.696533 + 0.696533i
\(78\) 0 0
\(79\) 87.2269i 1.10414i 0.833798 + 0.552069i \(0.186162\pi\)
−0.833798 + 0.552069i \(0.813838\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.53893 + 9.53893i 0.114927 + 0.114927i 0.762231 0.647305i \(-0.224104\pi\)
−0.647305 + 0.762231i \(0.724104\pi\)
\(84\) 0 0
\(85\) −85.2314 85.2314i −1.00272 1.00272i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 146.488i 1.64593i 0.568089 + 0.822967i \(0.307683\pi\)
−0.568089 + 0.822967i \(0.692317\pi\)
\(90\) 0 0
\(91\) 72.6762 72.6762i 0.798640 0.798640i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 285.586i 3.00617i
\(96\) 0 0
\(97\) 101.170 1.04298 0.521492 0.853256i \(-0.325376\pi\)
0.521492 + 0.853256i \(0.325376\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −47.4912 47.4912i −0.470210 0.470210i 0.431773 0.901982i \(-0.357888\pi\)
−0.901982 + 0.431773i \(0.857888\pi\)
\(102\) 0 0
\(103\) −7.58518 −0.0736425 −0.0368213 0.999322i \(-0.511723\pi\)
−0.0368213 + 0.999322i \(0.511723\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 119.757 119.757i 1.11922 1.11922i 0.127367 0.991856i \(-0.459347\pi\)
0.991856 0.127367i \(-0.0406525\pi\)
\(108\) 0 0
\(109\) −61.0263 + 61.0263i −0.559874 + 0.559874i −0.929272 0.369397i \(-0.879564\pi\)
0.369397 + 0.929272i \(0.379564\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −31.6377 −0.279980 −0.139990 0.990153i \(-0.544707\pi\)
−0.139990 + 0.990153i \(0.544707\pi\)
\(114\) 0 0
\(115\) 56.6467 + 56.6467i 0.492580 + 0.492580i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 116.690 0.980585
\(120\) 0 0
\(121\) 36.1019i 0.298363i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −134.106 + 134.106i −1.07285 + 1.07285i
\(126\) 0 0
\(127\) 87.8736i 0.691918i −0.938250 0.345959i \(-0.887554\pi\)
0.938250 0.345959i \(-0.112446\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −176.351 176.351i −1.34619 1.34619i −0.889761 0.456428i \(-0.849129\pi\)
−0.456428 0.889761i \(-0.650871\pi\)
\(132\) 0 0
\(133\) 195.497 + 195.497i 1.46990 + 1.46990i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 97.0224i 0.708192i 0.935209 + 0.354096i \(0.115211\pi\)
−0.935209 + 0.354096i \(0.884789\pi\)
\(138\) 0 0
\(139\) −130.221 + 130.221i −0.936841 + 0.936841i −0.998121 0.0612800i \(-0.980482\pi\)
0.0612800 + 0.998121i \(0.480482\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 115.042i 0.804493i
\(144\) 0 0
\(145\) −286.170 −1.97359
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −26.3592 26.3592i −0.176907 0.176907i 0.613099 0.790006i \(-0.289923\pi\)
−0.790006 + 0.613099i \(0.789923\pi\)
\(150\) 0 0
\(151\) −196.107 −1.29872 −0.649360 0.760481i \(-0.724963\pi\)
−0.649360 + 0.760481i \(0.724963\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −146.798 + 146.798i −0.947083 + 0.947083i
\(156\) 0 0
\(157\) −98.5323 + 98.5323i −0.627594 + 0.627594i −0.947462 0.319868i \(-0.896362\pi\)
0.319868 + 0.947462i \(0.396362\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −77.5545 −0.481705
\(162\) 0 0
\(163\) −90.6677 90.6677i −0.556244 0.556244i 0.371992 0.928236i \(-0.378675\pi\)
−0.928236 + 0.371992i \(0.878675\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −181.602 −1.08744 −0.543719 0.839267i \(-0.682984\pi\)
−0.543719 + 0.839267i \(0.682984\pi\)
\(168\) 0 0
\(169\) 13.1100i 0.0775741i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.66532 + 2.66532i −0.0154065 + 0.0154065i −0.714768 0.699362i \(-0.753468\pi\)
0.699362 + 0.714768i \(0.253468\pi\)
\(174\) 0 0
\(175\) 389.400i 2.22514i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.7491 + 12.7491i 0.0712241 + 0.0712241i 0.741822 0.670597i \(-0.233962\pi\)
−0.670597 + 0.741822i \(0.733962\pi\)
\(180\) 0 0
\(181\) 61.3009 + 61.3009i 0.338679 + 0.338679i 0.855870 0.517191i \(-0.173022\pi\)
−0.517191 + 0.855870i \(0.673022\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 291.074i 1.57338i
\(186\) 0 0
\(187\) −92.3566 + 92.3566i −0.493885 + 0.493885i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 80.1204i 0.419479i −0.977757 0.209739i \(-0.932738\pi\)
0.977757 0.209739i \(-0.0672615\pi\)
\(192\) 0 0
\(193\) 28.7555 0.148992 0.0744961 0.997221i \(-0.476265\pi\)
0.0744961 + 0.997221i \(0.476265\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 92.8030 + 92.8030i 0.471081 + 0.471081i 0.902264 0.431183i \(-0.141904\pi\)
−0.431183 + 0.902264i \(0.641904\pi\)
\(198\) 0 0
\(199\) −165.555 −0.831934 −0.415967 0.909380i \(-0.636557\pi\)
−0.415967 + 0.909380i \(0.636557\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 195.897 195.897i 0.965008 0.965008i
\(204\) 0 0
\(205\) 40.1507 40.1507i 0.195857 0.195857i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −309.460 −1.48067
\(210\) 0 0
\(211\) 187.769 + 187.769i 0.889902 + 0.889902i 0.994513 0.104611i \(-0.0333598\pi\)
−0.104611 + 0.994513i \(0.533360\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.7977 0.0502220
\(216\) 0 0
\(217\) 200.980i 0.926173i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 125.149 125.149i 0.566286 0.566286i
\(222\) 0 0
\(223\) 307.192i 1.37754i 0.724979 + 0.688771i \(0.241849\pi\)
−0.724979 + 0.688771i \(0.758151\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −112.960 112.960i −0.497621 0.497621i 0.413075 0.910697i \(-0.364455\pi\)
−0.910697 + 0.413075i \(0.864455\pi\)
\(228\) 0 0
\(229\) 7.32210 + 7.32210i 0.0319742 + 0.0319742i 0.722913 0.690939i \(-0.242803\pi\)
−0.690939 + 0.722913i \(0.742803\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 115.408i 0.495315i −0.968848 0.247657i \(-0.920339\pi\)
0.968848 0.247657i \(-0.0796607\pi\)
\(234\) 0 0
\(235\) 151.760 151.760i 0.645787 0.645787i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 323.271i 1.35260i 0.736627 + 0.676300i \(0.236417\pi\)
−0.736627 + 0.676300i \(0.763583\pi\)
\(240\) 0 0
\(241\) 118.056 0.489860 0.244930 0.969541i \(-0.421235\pi\)
0.244930 + 0.969541i \(0.421235\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 112.819 + 112.819i 0.460486 + 0.460486i
\(246\) 0 0
\(247\) 419.339 1.69773
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −166.229 + 166.229i −0.662265 + 0.662265i −0.955913 0.293648i \(-0.905130\pi\)
0.293648 + 0.955913i \(0.405130\pi\)
\(252\) 0 0
\(253\) 61.3822 61.3822i 0.242617 0.242617i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 342.745 1.33364 0.666820 0.745219i \(-0.267655\pi\)
0.666820 + 0.745219i \(0.267655\pi\)
\(258\) 0 0
\(259\) 199.254 + 199.254i 0.769319 + 0.769319i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 219.637 0.835123 0.417562 0.908649i \(-0.362885\pi\)
0.417562 + 0.908649i \(0.362885\pi\)
\(264\) 0 0
\(265\) 392.327i 1.48048i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −77.9061 + 77.9061i −0.289614 + 0.289614i −0.836928 0.547314i \(-0.815650\pi\)
0.547314 + 0.836928i \(0.315650\pi\)
\(270\) 0 0
\(271\) 155.092i 0.572294i −0.958186 0.286147i \(-0.907625\pi\)
0.958186 0.286147i \(-0.0923746\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 308.199 + 308.199i 1.12072 + 1.12072i
\(276\) 0 0
\(277\) 30.0601 + 30.0601i 0.108520 + 0.108520i 0.759282 0.650762i \(-0.225550\pi\)
−0.650762 + 0.759282i \(0.725550\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 265.590i 0.945162i 0.881287 + 0.472581i \(0.156677\pi\)
−0.881287 + 0.472581i \(0.843323\pi\)
\(282\) 0 0
\(283\) 160.221 160.221i 0.566153 0.566153i −0.364895 0.931049i \(-0.618895\pi\)
0.931049 + 0.364895i \(0.118895\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 54.9700i 0.191533i
\(288\) 0 0
\(289\) −88.0595 −0.304704
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −173.762 173.762i −0.593044 0.593044i 0.345408 0.938453i \(-0.387740\pi\)
−0.938453 + 0.345408i \(0.887740\pi\)
\(294\) 0 0
\(295\) −100.035 −0.339102
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −83.1769 + 83.1769i −0.278183 + 0.278183i
\(300\) 0 0
\(301\) −7.39154 + 7.39154i −0.0245566 + 0.0245566i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 822.565 2.69693
\(306\) 0 0
\(307\) 136.842 + 136.842i 0.445741 + 0.445741i 0.893936 0.448195i \(-0.147933\pi\)
−0.448195 + 0.893936i \(0.647933\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 458.117 1.47305 0.736523 0.676412i \(-0.236466\pi\)
0.736523 + 0.676412i \(0.236466\pi\)
\(312\) 0 0
\(313\) 170.865i 0.545893i 0.962029 + 0.272947i \(0.0879982\pi\)
−0.962029 + 0.272947i \(0.912002\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 372.560 372.560i 1.17527 1.17527i 0.194334 0.980935i \(-0.437746\pi\)
0.980935 0.194334i \(-0.0622544\pi\)
\(318\) 0 0
\(319\) 310.093i 0.972079i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 336.647 + 336.647i 1.04225 + 1.04225i
\(324\) 0 0
\(325\) −417.630 417.630i −1.28502 1.28502i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 207.773i 0.631529i
\(330\) 0 0
\(331\) −177.685 + 177.685i −0.536812 + 0.536812i −0.922591 0.385779i \(-0.873933\pi\)
0.385779 + 0.922591i \(0.373933\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 92.8260i 0.277092i
\(336\) 0 0
\(337\) 491.680 1.45899 0.729496 0.683985i \(-0.239755\pi\)
0.729496 + 0.683985i \(0.239755\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 159.070 + 159.070i 0.466480 + 0.466480i
\(342\) 0 0
\(343\) 248.902 0.725661
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 176.788 176.788i 0.509477 0.509477i −0.404889 0.914366i \(-0.632690\pi\)
0.914366 + 0.404889i \(0.132690\pi\)
\(348\) 0 0
\(349\) 272.298 272.298i 0.780223 0.780223i −0.199645 0.979868i \(-0.563979\pi\)
0.979868 + 0.199645i \(0.0639789\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 168.099 0.476200 0.238100 0.971241i \(-0.423475\pi\)
0.238100 + 0.971241i \(0.423475\pi\)
\(354\) 0 0
\(355\) −825.290 825.290i −2.32476 2.32476i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −476.009 −1.32593 −0.662965 0.748651i \(-0.730702\pi\)
−0.662965 + 0.748651i \(0.730702\pi\)
\(360\) 0 0
\(361\) 767.008i 2.12468i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 317.862 317.862i 0.870855 0.870855i
\(366\) 0 0
\(367\) 104.894i 0.285814i 0.989736 + 0.142907i \(0.0456450\pi\)
−0.989736 + 0.142907i \(0.954355\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 268.565 + 268.565i 0.723896 + 0.723896i
\(372\) 0 0
\(373\) 122.301 + 122.301i 0.327884 + 0.327884i 0.851781 0.523898i \(-0.175523\pi\)
−0.523898 + 0.851781i \(0.675523\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 420.197i 1.11458i
\(378\) 0 0
\(379\) 75.5625 75.5625i 0.199373 0.199373i −0.600358 0.799731i \(-0.704975\pi\)
0.799731 + 0.600358i \(0.204975\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 501.509i 1.30942i 0.755878 + 0.654712i \(0.227210\pi\)
−0.755878 + 0.654712i \(0.772790\pi\)
\(384\) 0 0
\(385\) −644.953 −1.67520
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −22.6053 22.6053i −0.0581114 0.0581114i 0.677454 0.735565i \(-0.263083\pi\)
−0.735565 + 0.677454i \(0.763083\pi\)
\(390\) 0 0
\(391\) −133.549 −0.341559
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −524.465 + 524.465i −1.32776 + 1.32776i
\(396\) 0 0
\(397\) 6.09563 6.09563i 0.0153542 0.0153542i −0.699388 0.714742i \(-0.746544\pi\)
0.714742 + 0.699388i \(0.246544\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −598.130 −1.49160 −0.745798 0.666172i \(-0.767932\pi\)
−0.745798 + 0.666172i \(0.767932\pi\)
\(402\) 0 0
\(403\) −215.550 215.550i −0.534863 0.534863i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −315.407 −0.774957
\(408\) 0 0
\(409\) 271.241i 0.663181i 0.943423 + 0.331591i \(0.107585\pi\)
−0.943423 + 0.331591i \(0.892415\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 68.4785 68.4785i 0.165808 0.165808i
\(414\) 0 0
\(415\) 114.709i 0.276406i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 412.163 + 412.163i 0.983681 + 0.983681i 0.999869 0.0161876i \(-0.00515289\pi\)
−0.0161876 + 0.999869i \(0.505153\pi\)
\(420\) 0 0
\(421\) −135.609 135.609i −0.322111 0.322111i 0.527466 0.849576i \(-0.323142\pi\)
−0.849576 + 0.527466i \(0.823142\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 670.550i 1.57777i
\(426\) 0 0
\(427\) −563.083 + 563.083i −1.31869 + 1.31869i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 204.169i 0.473710i 0.971545 + 0.236855i \(0.0761166\pi\)
−0.971545 + 0.236855i \(0.923883\pi\)
\(432\) 0 0
\(433\) 317.530 0.733325 0.366663 0.930354i \(-0.380500\pi\)
0.366663 + 0.930354i \(0.380500\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −223.743 223.743i −0.511998 0.511998i
\(438\) 0 0
\(439\) −226.050 −0.514921 −0.257461 0.966289i \(-0.582886\pi\)
−0.257461 + 0.966289i \(0.582886\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 157.388 157.388i 0.355278 0.355278i −0.506791 0.862069i \(-0.669169\pi\)
0.862069 + 0.506791i \(0.169169\pi\)
\(444\) 0 0
\(445\) −880.782 + 880.782i −1.97929 + 1.97929i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 458.567 1.02131 0.510653 0.859787i \(-0.329404\pi\)
0.510653 + 0.859787i \(0.329404\pi\)
\(450\) 0 0
\(451\) −43.5072 43.5072i −0.0964684 0.0964684i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 873.954 1.92078
\(456\) 0 0
\(457\) 548.682i 1.20062i 0.799768 + 0.600309i \(0.204956\pi\)
−0.799768 + 0.600309i \(0.795044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 359.058 359.058i 0.778868 0.778868i −0.200770 0.979638i \(-0.564344\pi\)
0.979638 + 0.200770i \(0.0643444\pi\)
\(462\) 0 0
\(463\) 126.697i 0.273643i −0.990596 0.136822i \(-0.956311\pi\)
0.990596 0.136822i \(-0.0436887\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −526.834 526.834i −1.12812 1.12812i −0.990482 0.137643i \(-0.956047\pi\)
−0.137643 0.990482i \(-0.543953\pi\)
\(468\) 0 0
\(469\) 63.5436 + 63.5436i 0.135487 + 0.135487i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.7004i 0.0247366i
\(474\) 0 0
\(475\) 1123.41 1123.41i 2.36508 2.36508i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 712.831i 1.48817i 0.668087 + 0.744083i \(0.267113\pi\)
−0.668087 + 0.744083i \(0.732887\pi\)
\(480\) 0 0
\(481\) 427.398 0.888560
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 608.297 + 608.297i 1.25422 + 1.25422i
\(486\) 0 0
\(487\) −579.851 −1.19066 −0.595330 0.803481i \(-0.702979\pi\)
−0.595330 + 0.803481i \(0.702979\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −131.309 + 131.309i −0.267432 + 0.267432i −0.828065 0.560632i \(-0.810558\pi\)
0.560632 + 0.828065i \(0.310558\pi\)
\(492\) 0 0
\(493\) 337.336 337.336i 0.684251 0.684251i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1129.90 2.27343
\(498\) 0 0
\(499\) 329.208 + 329.208i 0.659736 + 0.659736i 0.955318 0.295582i \(-0.0955134\pi\)
−0.295582 + 0.955318i \(0.595513\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −254.299 −0.505564 −0.252782 0.967523i \(-0.581346\pi\)
−0.252782 + 0.967523i \(0.581346\pi\)
\(504\) 0 0
\(505\) 571.096i 1.13088i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −326.512 + 326.512i −0.641477 + 0.641477i −0.950918 0.309442i \(-0.899858\pi\)
0.309442 + 0.950918i \(0.399858\pi\)
\(510\) 0 0
\(511\) 435.182i 0.851628i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −45.6071 45.6071i −0.0885574 0.0885574i
\(516\) 0 0
\(517\) −164.447 164.447i −0.318079 0.318079i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 775.042i 1.48760i 0.668400 + 0.743802i \(0.266980\pi\)
−0.668400 + 0.743802i \(0.733020\pi\)
\(522\) 0 0
\(523\) −461.875 + 461.875i −0.883126 + 0.883126i −0.993851 0.110725i \(-0.964683\pi\)
0.110725 + 0.993851i \(0.464683\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 346.089i 0.656715i
\(528\) 0 0
\(529\) −440.240 −0.832212
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 58.9551 + 58.9551i 0.110610 + 0.110610i
\(534\) 0 0
\(535\) 1440.11 2.69180
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 122.251 122.251i 0.226810 0.226810i
\(540\) 0 0
\(541\) 426.071 426.071i 0.787563 0.787563i −0.193531 0.981094i \(-0.561994\pi\)
0.981094 + 0.193531i \(0.0619941\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −733.860 −1.34653
\(546\) 0 0
\(547\) −111.138 111.138i −0.203178 0.203178i 0.598182 0.801360i \(-0.295890\pi\)
−0.801360 + 0.598182i \(0.795890\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1130.32 2.05139
\(552\) 0 0
\(553\) 718.041i 1.29845i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −233.731 + 233.731i −0.419625 + 0.419625i −0.885075 0.465449i \(-0.845893\pi\)
0.465449 + 0.885075i \(0.345893\pi\)
\(558\) 0 0
\(559\) 15.8548i 0.0283628i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −261.362 261.362i −0.464231 0.464231i 0.435809 0.900039i \(-0.356462\pi\)
−0.900039 + 0.435809i \(0.856462\pi\)
\(564\) 0 0
\(565\) −190.227 190.227i −0.336684 0.336684i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 453.881i 0.797682i −0.917020 0.398841i \(-0.869413\pi\)
0.917020 0.398841i \(-0.130587\pi\)
\(570\) 0 0
\(571\) −289.009 + 289.009i −0.506146 + 0.506146i −0.913341 0.407195i \(-0.866507\pi\)
0.407195 + 0.913341i \(0.366507\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 445.662i 0.775065i
\(576\) 0 0
\(577\) 588.041 1.01913 0.509567 0.860431i \(-0.329806\pi\)
0.509567 + 0.860431i \(0.329806\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −78.5232 78.5232i −0.135152 0.135152i
\(582\) 0 0
\(583\) −425.124 −0.729201
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 392.596 392.596i 0.668818 0.668818i −0.288624 0.957442i \(-0.593198\pi\)
0.957442 + 0.288624i \(0.0931978\pi\)
\(588\) 0 0
\(589\) 579.822 579.822i 0.984417 0.984417i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1008.70 −1.70101 −0.850505 0.525967i \(-0.823703\pi\)
−0.850505 + 0.525967i \(0.823703\pi\)
\(594\) 0 0
\(595\) 701.614 + 701.614i 1.17918 + 1.17918i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −526.819 −0.879498 −0.439749 0.898121i \(-0.644933\pi\)
−0.439749 + 0.898121i \(0.644933\pi\)
\(600\) 0 0
\(601\) 867.891i 1.44408i −0.691853 0.722039i \(-0.743205\pi\)
0.691853 0.722039i \(-0.256795\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −217.068 + 217.068i −0.358790 + 0.358790i
\(606\) 0 0
\(607\) 98.8497i 0.162850i −0.996679 0.0814248i \(-0.974053\pi\)
0.996679 0.0814248i \(-0.0259470\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 222.836 + 222.836i 0.364707 + 0.364707i
\(612\) 0 0
\(613\) 616.976 + 616.976i 1.00649 + 1.00649i 0.999979 + 0.00650651i \(0.00207110\pi\)
0.00650651 + 0.999979i \(0.497929\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1073.62i 1.74007i −0.492993 0.870033i \(-0.664097\pi\)
0.492993 0.870033i \(-0.335903\pi\)
\(618\) 0 0
\(619\) 397.119 397.119i 0.641549 0.641549i −0.309387 0.950936i \(-0.600124\pi\)
0.950936 + 0.309387i \(0.100124\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1205.87i 1.93559i
\(624\) 0 0
\(625\) −430.067 −0.688107
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 343.117 + 343.117i 0.545495 + 0.545495i
\(630\) 0 0
\(631\) 381.081 0.603931 0.301966 0.953319i \(-0.402357\pi\)
0.301966 + 0.953319i \(0.402357\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 528.354 528.354i 0.832053 0.832053i
\(636\) 0 0
\(637\) −165.657 + 165.657i −0.260059 + 0.260059i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −485.443 −0.757321 −0.378661 0.925536i \(-0.623615\pi\)
−0.378661 + 0.925536i \(0.623615\pi\)
\(642\) 0 0
\(643\) 457.641 + 457.641i 0.711728 + 0.711728i 0.966897 0.255168i \(-0.0821308\pi\)
−0.255168 + 0.966897i \(0.582131\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −845.780 −1.30723 −0.653617 0.756826i \(-0.726749\pi\)
−0.653617 + 0.756826i \(0.726749\pi\)
\(648\) 0 0
\(649\) 108.398i 0.167023i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11.1039 11.1039i 0.0170044 0.0170044i −0.698553 0.715558i \(-0.746173\pi\)
0.715558 + 0.698553i \(0.246173\pi\)
\(654\) 0 0
\(655\) 2120.67i 3.23766i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −296.711 296.711i −0.450245 0.450245i 0.445191 0.895436i \(-0.353136\pi\)
−0.895436 + 0.445191i \(0.853136\pi\)
\(660\) 0 0
\(661\) 0.564899 + 0.564899i 0.000854613 + 0.000854613i 0.707534 0.706679i \(-0.249808\pi\)
−0.706679 + 0.707534i \(0.749808\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2350.91i 3.53520i
\(666\) 0 0
\(667\) −224.201 + 224.201i −0.336133 + 0.336133i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 891.329i 1.32836i
\(672\) 0 0
\(673\) −1091.24 −1.62146 −0.810731 0.585419i \(-0.800930\pi\)
−0.810731 + 0.585419i \(0.800930\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −494.409 494.409i −0.730294 0.730294i 0.240384 0.970678i \(-0.422727\pi\)
−0.970678 + 0.240384i \(0.922727\pi\)
\(678\) 0 0
\(679\) −832.814 −1.22653
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −845.268 + 845.268i −1.23758 + 1.23758i −0.276594 + 0.960987i \(0.589206\pi\)
−0.960987 + 0.276594i \(0.910794\pi\)
\(684\) 0 0
\(685\) −583.362 + 583.362i −0.851623 + 0.851623i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 576.071 0.836097
\(690\) 0 0
\(691\) 457.715 + 457.715i 0.662395 + 0.662395i 0.955944 0.293549i \(-0.0948364\pi\)
−0.293549 + 0.955944i \(0.594836\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1565.95 −2.25316
\(696\) 0 0
\(697\) 94.6589i 0.135809i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 199.252 199.252i 0.284239 0.284239i −0.550558 0.834797i \(-0.685585\pi\)
0.834797 + 0.550558i \(0.185585\pi\)
\(702\) 0 0
\(703\) 1149.69i 1.63540i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 390.941 + 390.941i 0.552958 + 0.552958i
\(708\) 0 0
\(709\) 547.374 + 547.374i 0.772036 + 0.772036i 0.978462 0.206426i \(-0.0661832\pi\)
−0.206426 + 0.978462i \(0.566183\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 230.018i 0.322606i
\(714\) 0 0
\(715\) −691.710 + 691.710i −0.967427 + 0.967427i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 90.5082i 0.125881i −0.998017 0.0629403i \(-0.979952\pi\)
0.998017 0.0629403i \(-0.0200478\pi\)
\(720\) 0 0
\(721\) 62.4402 0.0866022
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1125.71 1125.71i −1.55270 1.55270i
\(726\) 0 0
\(727\) 222.068 0.305459 0.152729 0.988268i \(-0.451194\pi\)
0.152729 + 0.988268i \(0.451194\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −12.7283 + 12.7283i −0.0174122 + 0.0174122i
\(732\) 0 0
\(733\) −201.142 + 201.142i −0.274410 + 0.274410i −0.830873 0.556463i \(-0.812158\pi\)
0.556463 + 0.830873i \(0.312158\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −100.586 −0.136480
\(738\) 0 0
\(739\) −771.595 771.595i −1.04411 1.04411i −0.998981 0.0451261i \(-0.985631\pi\)
−0.0451261 0.998981i \(-0.514369\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 754.171 1.01503 0.507517 0.861641i \(-0.330563\pi\)
0.507517 + 0.861641i \(0.330563\pi\)
\(744\) 0 0
\(745\) 316.977i 0.425473i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −985.822 + 985.822i −1.31618 + 1.31618i
\(750\) 0 0
\(751\) 606.477i 0.807559i −0.914856 0.403780i \(-0.867696\pi\)
0.914856 0.403780i \(-0.132304\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1179.12 1179.12i −1.56175 1.56175i
\(756\) 0 0
\(757\) 852.254 + 852.254i 1.12583 + 1.12583i 0.990848 + 0.134983i \(0.0430981\pi\)
0.134983 + 0.990848i \(0.456902\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1373.25i 1.80453i −0.431180 0.902266i \(-0.641903\pi\)
0.431180 0.902266i \(-0.358097\pi\)
\(762\) 0 0
\(763\) 502.360 502.360i 0.658401 0.658401i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 146.886i 0.191507i
\(768\) 0 0
\(769\) 515.727 0.670647 0.335323 0.942103i \(-0.391154\pi\)
0.335323 + 0.942103i \(0.391154\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −448.985 448.985i −0.580834 0.580834i 0.354299 0.935132i \(-0.384720\pi\)
−0.935132 + 0.354299i \(0.884720\pi\)
\(774\) 0 0
\(775\) −1154.92 −1.49022
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −158.587 + 158.587i −0.203578 + 0.203578i
\(780\) 0 0
\(781\) −894.282 + 894.282i −1.14505 + 1.14505i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1184.88 −1.50940
\(786\) 0 0
\(787\) 70.0485 + 70.0485i 0.0890070 + 0.0890070i 0.750208 0.661201i \(-0.229953\pi\)
−0.661201 + 0.750208i \(0.729953\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 260.437 0.329251
\(792\) 0 0
\(793\) 1207.81i 1.52309i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −584.026 + 584.026i −0.732780 + 0.732780i −0.971170 0.238390i \(-0.923380\pi\)
0.238390 + 0.971170i \(0.423380\pi\)
\(798\) 0 0
\(799\) 357.787i 0.447794i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −344.434 344.434i −0.428935 0.428935i
\(804\) 0 0
\(805\) −466.308 466.308i −0.579265 0.579265i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1252.15i 1.54778i 0.633321 + 0.773889i \(0.281691\pi\)
−0.633321 + 0.773889i \(0.718309\pi\)
\(810\) 0 0
\(811\) −255.775 + 255.775i −0.315382 + 0.315382i −0.846990 0.531608i \(-0.821588\pi\)
0.531608 + 0.846990i \(0.321588\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1090.31i 1.33780i
\(816\) 0 0
\(817\) −42.6489 −0.0522018
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −228.380 228.380i −0.278173 0.278173i 0.554206 0.832379i \(-0.313022\pi\)
−0.832379 + 0.554206i \(0.813022\pi\)
\(822\) 0 0
\(823\) −1050.55 −1.27649 −0.638246 0.769832i \(-0.720340\pi\)
−0.638246 + 0.769832i \(0.720340\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 78.0300 78.0300i 0.0943531 0.0943531i −0.658355 0.752708i \(-0.728747\pi\)
0.752708 + 0.658355i \(0.228747\pi\)
\(828\) 0 0
\(829\) 517.848 517.848i 0.624665 0.624665i −0.322056 0.946721i \(-0.604374\pi\)
0.946721 + 0.322056i \(0.104374\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −265.981 −0.319305
\(834\) 0 0
\(835\) −1091.91 1091.91i −1.30768 1.30768i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −649.581 −0.774232 −0.387116 0.922031i \(-0.626529\pi\)
−0.387116 + 0.922031i \(0.626529\pi\)
\(840\) 0 0
\(841\) 291.627i 0.346762i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −78.8260 + 78.8260i −0.0932852 + 0.0932852i
\(846\) 0 0
\(847\) 297.186i 0.350869i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −228.043 228.043i −0.267970 0.267970i
\(852\) 0 0
\(853\) −1124.04 1124.04i −1.31775 1.31775i −0.915551 0.402202i \(-0.868245\pi\)
−0.402202 0.915551i \(-0.631755\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 596.973i 0.696585i 0.937386 + 0.348292i \(0.113238\pi\)
−0.937386 + 0.348292i \(0.886762\pi\)
\(858\) 0 0
\(859\) 701.595 701.595i 0.816758 0.816758i −0.168879 0.985637i \(-0.554015\pi\)
0.985637 + 0.168879i \(0.0540147\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 437.703i 0.507187i −0.967311 0.253594i \(-0.918387\pi\)
0.967311 0.253594i \(-0.0816126\pi\)
\(864\) 0 0
\(865\) −32.0513 −0.0370535
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 568.309 + 568.309i 0.653981 + 0.653981i
\(870\) 0 0
\(871\) 136.301 0.156487
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1103.94 1103.94i 1.26165 1.26165i
\(876\) 0 0
\(877\) −769.110 + 769.110i −0.876978 + 0.876978i −0.993221 0.116243i \(-0.962915\pi\)
0.116243 + 0.993221i \(0.462915\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −351.923 −0.399459 −0.199729 0.979851i \(-0.564006\pi\)
−0.199729 + 0.979851i \(0.564006\pi\)
\(882\) 0 0
\(883\) 661.014 + 661.014i 0.748601 + 0.748601i 0.974216 0.225616i \(-0.0724394\pi\)
−0.225616 + 0.974216i \(0.572439\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 794.853 0.896114 0.448057 0.894005i \(-0.352116\pi\)
0.448057 + 0.894005i \(0.352116\pi\)
\(888\) 0 0
\(889\) 723.364i 0.813683i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −599.421 + 599.421i −0.671244 + 0.671244i
\(894\) 0 0
\(895\) 153.312i 0.171298i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −581.008 581.008i −0.646282 0.646282i
\(900\) 0 0
\(901\) 462.472 + 462.472i 0.513288 + 0.513288i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 737.162i 0.814544i
\(906\) 0 0
\(907\) −943.805 + 943.805i −1.04058 + 1.04058i −0.0414379 + 0.999141i \(0.513194\pi\)
−0.999141 + 0.0414379i \(0.986806\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 190.842i 0.209487i −0.994499 0.104743i \(-0.966598\pi\)
0.994499 0.104743i \(-0.0334021\pi\)
\(912\) 0 0
\(913\) 124.298 0.136142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1451.70 + 1451.70i 1.58309 + 1.58309i
\(918\) 0 0
\(919\) 925.230 1.00678 0.503389 0.864060i \(-0.332086\pi\)
0.503389 + 0.864060i \(0.332086\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1211.81 1211.81i 1.31290 1.31290i
\(924\) 0 0
\(925\) 1145.00 1145.00i 1.23784 1.23784i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −91.8388 −0.0988577 −0.0494289 0.998778i \(-0.515740\pi\)
−0.0494289 + 0.998778i \(0.515740\pi\)
\(930\) 0 0
\(931\) −445.613 445.613i −0.478639 0.478639i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1110.62 −1.18782
\(936\) 0 0
\(937\) 1615.44i 1.72406i −0.506857 0.862030i \(-0.669193\pi\)
0.506857 0.862030i \(-0.330807\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 484.931 484.931i 0.515336 0.515336i −0.400820 0.916157i \(-0.631275\pi\)
0.916157 + 0.400820i \(0.131275\pi\)
\(942\) 0 0
\(943\) 62.9124i 0.0667151i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1053.02 + 1053.02i 1.11195 + 1.11195i 0.992887 + 0.119064i \(0.0379895\pi\)
0.119064 + 0.992887i \(0.462010\pi\)
\(948\) 0 0
\(949\) 466.731 + 466.731i 0.491814 + 0.491814i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1256.72i 1.31870i −0.751838 0.659348i \(-0.770832\pi\)
0.751838 0.659348i \(-0.229168\pi\)
\(954\) 0 0
\(955\) 481.736 481.736i 0.504436 0.504436i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 798.675i 0.832821i
\(960\) 0 0
\(961\) 364.917 0.379726
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 172.897 + 172.897i 0.179168 + 0.179168i
\(966\) 0 0
\(967\) 1842.43 1.90530 0.952651 0.304066i \(-0.0983444\pi\)
0.952651 + 0.304066i \(0.0983444\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −802.530 + 802.530i −0.826498 + 0.826498i −0.987031 0.160532i \(-0.948679\pi\)
0.160532 + 0.987031i \(0.448679\pi\)
\(972\) 0 0
\(973\) 1071.96 1071.96i 1.10171 1.10171i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1505.64 −1.54109 −0.770544 0.637387i \(-0.780015\pi\)
−0.770544 + 0.637387i \(0.780015\pi\)
\(978\) 0 0
\(979\) 954.413 + 954.413i 0.974886 + 0.974886i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 764.708 0.777933 0.388966 0.921252i \(-0.372832\pi\)
0.388966 + 0.921252i \(0.372832\pi\)
\(984\) 0 0
\(985\) 1115.98i 1.13298i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.45951 8.45951i 0.00855360 0.00855360i
\(990\) 0 0
\(991\) 1683.22i 1.69851i 0.527984 + 0.849254i \(0.322948\pi\)
−0.527984 + 0.849254i \(0.677052\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −995.424 995.424i −1.00043 1.00043i
\(996\) 0 0
\(997\) −942.082 942.082i −0.944917 0.944917i 0.0536431 0.998560i \(-0.482917\pi\)
−0.998560 + 0.0536431i \(0.982917\pi\)
\(998\) 0 0
\(999\) 0 0
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 1152.3.m.d.415.8 16
3.2 odd 2 inner 1152.3.m.d.415.1 16
4.3 odd 2 1152.3.m.e.415.8 16
8.3 odd 2 144.3.m.b.91.4 yes 16
8.5 even 2 576.3.m.b.271.1 16
12.11 even 2 1152.3.m.e.415.1 16
16.3 odd 4 inner 1152.3.m.d.991.8 16
16.5 even 4 144.3.m.b.19.4 16
16.11 odd 4 576.3.m.b.559.1 16
16.13 even 4 1152.3.m.e.991.8 16
24.5 odd 2 576.3.m.b.271.8 16
24.11 even 2 144.3.m.b.91.5 yes 16
48.5 odd 4 144.3.m.b.19.5 yes 16
48.11 even 4 576.3.m.b.559.8 16
48.29 odd 4 1152.3.m.e.991.1 16
48.35 even 4 inner 1152.3.m.d.991.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.m.b.19.4 16 16.5 even 4
144.3.m.b.19.5 yes 16 48.5 odd 4
144.3.m.b.91.4 yes 16 8.3 odd 2
144.3.m.b.91.5 yes 16 24.11 even 2
576.3.m.b.271.1 16 8.5 even 2
576.3.m.b.271.8 16 24.5 odd 2
576.3.m.b.559.1 16 16.11 odd 4
576.3.m.b.559.8 16 48.11 even 4
1152.3.m.d.415.1 16 3.2 odd 2 inner
1152.3.m.d.415.8 16 1.1 even 1 trivial
1152.3.m.d.991.1 16 48.35 even 4 inner
1152.3.m.d.991.8 16 16.3 odd 4 inner
1152.3.m.e.415.1 16 12.11 even 2
1152.3.m.e.415.8 16 4.3 odd 2
1152.3.m.e.991.1 16 48.29 odd 4
1152.3.m.e.991.8 16 16.13 even 4