Properties

Label 676.2.h.c.361.2
Level $676$
Weight $2$
Character 676.361
Analytic conductor $5.398$
Analytic rank $0$
Dimension $4$
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] = [676,2,Mod(361,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.h (of order \(6\), 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: \(5.39788717664\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 361.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 676.361
Dual form 676.2.h.c.485.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{5} +(-1.73205 - 1.00000i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+2.00000i q^{5} +(-1.73205 - 1.00000i) q^{7} +(1.50000 - 2.59808i) q^{9} +(1.73205 - 1.00000i) q^{11} +(3.00000 - 5.19615i) q^{17} +(5.19615 + 3.00000i) q^{19} +(4.00000 + 6.92820i) q^{23} +1.00000 q^{25} +(-1.00000 - 1.73205i) q^{29} +10.0000i q^{31} +(2.00000 - 3.46410i) q^{35} +(5.19615 - 3.00000i) q^{37} +(-5.19615 + 3.00000i) q^{41} +(2.00000 - 3.46410i) q^{43} +(5.19615 + 3.00000i) q^{45} +2.00000i q^{47} +(-1.50000 - 2.59808i) q^{49} +6.00000 q^{53} +(2.00000 + 3.46410i) q^{55} +(-8.66025 - 5.00000i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-5.19615 + 3.00000i) q^{63} +(8.66025 - 5.00000i) q^{67} +(-8.66025 - 5.00000i) q^{71} -2.00000i q^{73} -4.00000 q^{77} -4.00000 q^{79} +(-4.50000 - 7.79423i) q^{81} -6.00000i q^{83} +(10.3923 + 6.00000i) q^{85} +(5.19615 - 3.00000i) q^{89} +(-6.00000 + 10.3923i) q^{95} +(-1.73205 - 1.00000i) q^{97} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} + 12 q^{17} + 16 q^{23} + 4 q^{25} - 4 q^{29} + 8 q^{35} + 8 q^{43} - 6 q^{49} + 24 q^{53} + 8 q^{55} + 4 q^{61} - 16 q^{77} - 16 q^{79} - 18 q^{81} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) 0 0
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) −1.73205 1.00000i −0.654654 0.377964i 0.135583 0.990766i \(-0.456709\pi\)
−0.790237 + 0.612801i \(0.790043\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 1.73205 1.00000i 0.522233 0.301511i −0.215615 0.976478i \(-0.569176\pi\)
0.737848 + 0.674967i \(0.235842\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 5.19615i 0.727607 1.26025i −0.230285 0.973123i \(-0.573966\pi\)
0.957892 0.287129i \(-0.0927008\pi\)
\(18\) 0 0
\(19\) 5.19615 + 3.00000i 1.19208 + 0.688247i 0.958778 0.284157i \(-0.0917138\pi\)
0.233301 + 0.972404i \(0.425047\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 + 6.92820i 0.834058 + 1.44463i 0.894795 + 0.446476i \(0.147321\pi\)
−0.0607377 + 0.998154i \(0.519345\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 1.73205i −0.185695 0.321634i 0.758115 0.652121i \(-0.226120\pi\)
−0.943811 + 0.330487i \(0.892787\pi\)
\(30\) 0 0
\(31\) 10.0000i 1.79605i 0.439941 + 0.898027i \(0.354999\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 3.46410i 0.338062 0.585540i
\(36\) 0 0
\(37\) 5.19615 3.00000i 0.854242 0.493197i −0.00783774 0.999969i \(-0.502495\pi\)
0.862080 + 0.506772i \(0.169162\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.19615 + 3.00000i −0.811503 + 0.468521i −0.847477 0.530831i \(-0.821880\pi\)
0.0359748 + 0.999353i \(0.488546\pi\)
\(42\) 0 0
\(43\) 2.00000 3.46410i 0.304997 0.528271i −0.672264 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) 0 0
\(45\) 5.19615 + 3.00000i 0.774597 + 0.447214i
\(46\) 0 0
\(47\) 2.00000i 0.291730i 0.989305 + 0.145865i \(0.0465965\pi\)
−0.989305 + 0.145865i \(0.953403\pi\)
\(48\) 0 0
\(49\) −1.50000 2.59808i −0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 2.00000 + 3.46410i 0.269680 + 0.467099i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.66025 5.00000i −1.12747 0.650945i −0.184172 0.982894i \(-0.558960\pi\)
−0.943297 + 0.331949i \(0.892294\pi\)
\(60\) 0 0
\(61\) 1.00000 1.73205i 0.128037 0.221766i −0.794879 0.606768i \(-0.792466\pi\)
0.922916 + 0.385002i \(0.125799\pi\)
\(62\) 0 0
\(63\) −5.19615 + 3.00000i −0.654654 + 0.377964i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 8.66025 5.00000i 1.05802 0.610847i 0.133135 0.991098i \(-0.457496\pi\)
0.924883 + 0.380251i \(0.124162\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.66025 5.00000i −1.02778 0.593391i −0.111434 0.993772i \(-0.535544\pi\)
−0.916349 + 0.400381i \(0.868878\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) 6.00000i 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 10.3923 + 6.00000i 1.12720 + 0.650791i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.19615 3.00000i 0.550791 0.317999i −0.198650 0.980071i \(-0.563656\pi\)
0.749441 + 0.662071i \(0.230322\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.00000 + 10.3923i −0.615587 + 1.06623i
\(96\) 0 0
\(97\) −1.73205 1.00000i −0.175863 0.101535i 0.409484 0.912317i \(-0.365709\pi\)
−0.585348 + 0.810782i \(0.699042\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) −1.00000 1.73205i −0.0995037 0.172345i 0.811976 0.583691i \(-0.198392\pi\)
−0.911479 + 0.411346i \(0.865059\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 + 13.8564i 0.773389 + 1.33955i 0.935695 + 0.352809i \(0.114773\pi\)
−0.162306 + 0.986740i \(0.551893\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i −0.741929 0.670478i \(-0.766089\pi\)
0.741929 0.670478i \(-0.233911\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.00000 + 12.1244i −0.658505 + 1.14056i 0.322498 + 0.946570i \(0.395477\pi\)
−0.981003 + 0.193993i \(0.937856\pi\)
\(114\) 0 0
\(115\) −13.8564 + 8.00000i −1.29212 + 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.3923 + 6.00000i −0.952661 + 0.550019i
\(120\) 0 0
\(121\) −3.50000 + 6.06218i −0.318182 + 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) −4.00000 6.92820i −0.354943 0.614779i 0.632166 0.774833i \(-0.282166\pi\)
−0.987108 + 0.160055i \(0.948833\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −6.00000 10.3923i −0.520266 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.5885 + 9.00000i 1.33181 + 0.768922i 0.985577 0.169226i \(-0.0541268\pi\)
0.346235 + 0.938148i \(0.387460\pi\)
\(138\) 0 0
\(139\) −8.00000 + 13.8564i −0.678551 + 1.17529i 0.296866 + 0.954919i \(0.404058\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.46410 2.00000i 0.287678 0.166091i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.5885 9.00000i −1.27706 0.737309i −0.300750 0.953703i \(-0.597237\pi\)
−0.976306 + 0.216394i \(0.930570\pi\)
\(150\) 0 0
\(151\) 6.00000i 0.488273i −0.969741 0.244137i \(-0.921495\pi\)
0.969741 0.244137i \(-0.0785045\pi\)
\(152\) 0 0
\(153\) −9.00000 15.5885i −0.727607 1.26025i
\(154\) 0 0
\(155\) −20.0000 −1.60644
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 16.0000i 1.26098i
\(162\) 0 0
\(163\) −8.66025 5.00000i −0.678323 0.391630i 0.120900 0.992665i \(-0.461422\pi\)
−0.799223 + 0.601035i \(0.794755\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.19615 + 3.00000i −0.402090 + 0.232147i −0.687386 0.726293i \(-0.741242\pi\)
0.285295 + 0.958440i \(0.407908\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 15.5885 9.00000i 1.19208 0.688247i
\(172\) 0 0
\(173\) −5.00000 + 8.66025i −0.380143 + 0.658427i −0.991082 0.133250i \(-0.957459\pi\)
0.610939 + 0.791677i \(0.290792\pi\)
\(174\) 0 0
\(175\) −1.73205 1.00000i −0.130931 0.0755929i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 + 10.3923i 0.441129 + 0.764057i
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.00000 + 3.46410i −0.144715 + 0.250654i −0.929267 0.369410i \(-0.879560\pi\)
0.784552 + 0.620063i \(0.212893\pi\)
\(192\) 0 0
\(193\) −1.73205 + 1.00000i −0.124676 + 0.0719816i −0.561041 0.827788i \(-0.689599\pi\)
0.436365 + 0.899770i \(0.356266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.19615 + 3.00000i −0.370211 + 0.213741i −0.673550 0.739141i \(-0.735232\pi\)
0.303340 + 0.952882i \(0.401898\pi\)
\(198\) 0 0
\(199\) −8.00000 + 13.8564i −0.567105 + 0.982255i 0.429745 + 0.902950i \(0.358603\pi\)
−0.996850 + 0.0793045i \(0.974730\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.00000i 0.280745i
\(204\) 0 0
\(205\) −6.00000 10.3923i −0.419058 0.725830i
\(206\) 0 0
\(207\) 24.0000 1.66812
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −4.00000 6.92820i −0.275371 0.476957i 0.694857 0.719148i \(-0.255467\pi\)
−0.970229 + 0.242190i \(0.922134\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.92820 + 4.00000i 0.472500 + 0.272798i
\(216\) 0 0
\(217\) 10.0000 17.3205i 0.678844 1.17579i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −5.19615 + 3.00000i −0.347960 + 0.200895i −0.663786 0.747922i \(-0.731052\pi\)
0.315826 + 0.948817i \(0.397718\pi\)
\(224\) 0 0
\(225\) 1.50000 2.59808i 0.100000 0.173205i
\(226\) 0 0
\(227\) −15.5885 9.00000i −1.03464 0.597351i −0.116331 0.993210i \(-0.537113\pi\)
−0.918311 + 0.395860i \(0.870447\pi\)
\(228\) 0 0
\(229\) 18.0000i 1.18947i −0.803921 0.594737i \(-0.797256\pi\)
0.803921 0.594737i \(-0.202744\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.0000i 1.16432i 0.813073 + 0.582162i \(0.197793\pi\)
−0.813073 + 0.582162i \(0.802207\pi\)
\(240\) 0 0
\(241\) −5.19615 3.00000i −0.334714 0.193247i 0.323218 0.946324i \(-0.395235\pi\)
−0.657932 + 0.753077i \(0.728569\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.19615 3.00000i 0.331970 0.191663i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 10.3923i 0.378717 0.655956i −0.612159 0.790735i \(-0.709699\pi\)
0.990876 + 0.134778i \(0.0430322\pi\)
\(252\) 0 0
\(253\) 13.8564 + 8.00000i 0.871145 + 0.502956i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.00000 + 15.5885i 0.561405 + 0.972381i 0.997374 + 0.0724199i \(0.0230722\pi\)
−0.435970 + 0.899961i \(0.643595\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 6.00000 + 10.3923i 0.369976 + 0.640817i 0.989561 0.144112i \(-0.0460326\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(264\) 0 0
\(265\) 12.0000i 0.737154i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.00000 + 15.5885i −0.548740 + 0.950445i 0.449622 + 0.893219i \(0.351559\pi\)
−0.998361 + 0.0572259i \(0.981774\pi\)
\(270\) 0 0
\(271\) 1.73205 1.00000i 0.105215 0.0607457i −0.446469 0.894799i \(-0.647319\pi\)
0.551684 + 0.834053i \(0.313985\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.73205 1.00000i 0.104447 0.0603023i
\(276\) 0 0
\(277\) 15.0000 25.9808i 0.901263 1.56103i 0.0754058 0.997153i \(-0.475975\pi\)
0.825857 0.563880i \(-0.190692\pi\)
\(278\) 0 0
\(279\) 25.9808 + 15.0000i 1.55543 + 0.898027i
\(280\) 0 0
\(281\) 14.0000i 0.835170i 0.908638 + 0.417585i \(0.137123\pi\)
−0.908638 + 0.417585i \(0.862877\pi\)
\(282\) 0 0
\(283\) 2.00000 + 3.46410i 0.118888 + 0.205919i 0.919327 0.393494i \(-0.128734\pi\)
−0.800439 + 0.599414i \(0.795400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −12.1244 7.00000i −0.708312 0.408944i 0.102123 0.994772i \(-0.467436\pi\)
−0.810436 + 0.585827i \(0.800770\pi\)
\(294\) 0 0
\(295\) 10.0000 17.3205i 0.582223 1.00844i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.92820 + 4.00000i −0.399335 + 0.230556i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.46410 + 2.00000i 0.198354 + 0.114520i
\(306\) 0 0
\(307\) 34.0000i 1.94048i 0.242140 + 0.970241i \(0.422151\pi\)
−0.242140 + 0.970241i \(0.577849\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −6.00000 10.3923i −0.338062 0.585540i
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) −3.46410 2.00000i −0.193952 0.111979i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 31.1769 18.0000i 1.73473 1.00155i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.00000 3.46410i 0.110264 0.190982i
\(330\) 0 0
\(331\) 19.0526 + 11.0000i 1.04722 + 0.604615i 0.921871 0.387498i \(-0.126660\pi\)
0.125353 + 0.992112i \(0.459994\pi\)
\(332\) 0 0
\(333\) 18.0000i 0.986394i
\(334\) 0 0
\(335\) 10.0000 + 17.3205i 0.546358 + 0.946320i
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.0000 + 17.3205i 0.541530 + 0.937958i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 20.7846i 0.644194 1.11578i −0.340293 0.940319i \(-0.610526\pi\)
0.984487 0.175457i \(-0.0561403\pi\)
\(348\) 0 0
\(349\) −8.66025 + 5.00000i −0.463573 + 0.267644i −0.713545 0.700609i \(-0.752912\pi\)
0.249973 + 0.968253i \(0.419578\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.1244 + 7.00000i −0.645314 + 0.372572i −0.786659 0.617388i \(-0.788191\pi\)
0.141344 + 0.989960i \(0.454858\pi\)
\(354\) 0 0
\(355\) 10.0000 17.3205i 0.530745 0.919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000i 1.58334i −0.610949 0.791670i \(-0.709212\pi\)
0.610949 0.791670i \(-0.290788\pi\)
\(360\) 0 0
\(361\) 8.50000 + 14.7224i 0.447368 + 0.774865i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 2.00000 + 3.46410i 0.104399 + 0.180825i 0.913493 0.406855i \(-0.133375\pi\)
−0.809093 + 0.587680i \(0.800041\pi\)
\(368\) 0 0
\(369\) 18.0000i 0.937043i
\(370\) 0 0
\(371\) −10.3923 6.00000i −0.539542 0.311504i
\(372\) 0 0
\(373\) 11.0000 19.0526i 0.569558 0.986504i −0.427051 0.904227i \(-0.640448\pi\)
0.996610 0.0822766i \(-0.0262191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.66025 5.00000i 0.444847 0.256833i −0.260804 0.965392i \(-0.583988\pi\)
0.705652 + 0.708559i \(0.250654\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −15.5885 9.00000i −0.796533 0.459879i 0.0457244 0.998954i \(-0.485440\pi\)
−0.842257 + 0.539076i \(0.818774\pi\)
\(384\) 0 0
\(385\) 8.00000i 0.407718i
\(386\) 0 0
\(387\) −6.00000 10.3923i −0.304997 0.528271i
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) 1.73205 + 1.00000i 0.0869291 + 0.0501886i 0.542834 0.839840i \(-0.317351\pi\)
−0.455905 + 0.890028i \(0.650684\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.19615 3.00000i 0.259483 0.149813i −0.364615 0.931158i \(-0.618800\pi\)
0.624099 + 0.781345i \(0.285466\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 15.5885 9.00000i 0.774597 0.447214i
\(406\) 0 0
\(407\) 6.00000 10.3923i 0.297409 0.515127i
\(408\) 0 0
\(409\) 12.1244 + 7.00000i 0.599511 + 0.346128i 0.768849 0.639430i \(-0.220830\pi\)
−0.169338 + 0.985558i \(0.554163\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0000 + 17.3205i 0.492068 + 0.852286i
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 20.7846i −0.586238 1.01539i −0.994720 0.102628i \(-0.967275\pi\)
0.408481 0.912767i \(-0.366058\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) 0 0
\(423\) 5.19615 + 3.00000i 0.252646 + 0.145865i
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −3.46410 + 2.00000i −0.167640 + 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.9808 + 15.0000i −1.25145 + 0.722525i −0.971397 0.237460i \(-0.923685\pi\)
−0.280052 + 0.959985i \(0.590352\pi\)
\(432\) 0 0
\(433\) −7.00000 + 12.1244i −0.336399 + 0.582659i −0.983752 0.179530i \(-0.942542\pi\)
0.647354 + 0.762190i \(0.275876\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 48.0000i 2.29615i
\(438\) 0 0
\(439\) −8.00000 13.8564i −0.381819 0.661330i 0.609503 0.792784i \(-0.291369\pi\)
−0.991322 + 0.131453i \(0.958036\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 6.00000 + 10.3923i 0.284427 + 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −32.9090 19.0000i −1.55307 0.896665i −0.997889 0.0649356i \(-0.979316\pi\)
−0.555181 0.831730i \(-0.687351\pi\)
\(450\) 0 0
\(451\) −6.00000 + 10.3923i −0.282529 + 0.489355i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 29.4449 17.0000i 1.37737 0.795226i 0.385530 0.922695i \(-0.374019\pi\)
0.991843 + 0.127469i \(0.0406853\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.19615 + 3.00000i 0.242009 + 0.139724i 0.616100 0.787668i \(-0.288712\pi\)
−0.374091 + 0.927392i \(0.622045\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −20.0000 −0.925490 −0.462745 0.886492i \(-0.653135\pi\)
−0.462745 + 0.886492i \(0.653135\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000i 0.367840i
\(474\) 0 0
\(475\) 5.19615 + 3.00000i 0.238416 + 0.137649i
\(476\) 0 0
\(477\) 9.00000 15.5885i 0.412082 0.713746i
\(478\) 0 0
\(479\) 15.5885 9.00000i 0.712255 0.411220i −0.0996406 0.995023i \(-0.531769\pi\)
0.811895 + 0.583803i \(0.198436\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 3.46410i 0.0908153 0.157297i
\(486\) 0 0
\(487\) −1.73205 1.00000i −0.0784867 0.0453143i 0.460243 0.887793i \(-0.347762\pi\)
−0.538730 + 0.842479i \(0.681096\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −10.0000 17.3205i −0.451294 0.781664i 0.547173 0.837020i \(-0.315704\pi\)
−0.998467 + 0.0553560i \(0.982371\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 10.0000 + 17.3205i 0.448561 + 0.776931i
\(498\) 0 0
\(499\) 14.0000i 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −14.0000 + 24.2487i −0.624229 + 1.08120i 0.364460 + 0.931219i \(0.381254\pi\)
−0.988689 + 0.149978i \(0.952080\pi\)
\(504\) 0 0
\(505\) 3.46410 2.00000i 0.154150 0.0889988i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.1244 + 7.00000i −0.537403 + 0.310270i −0.744026 0.668151i \(-0.767086\pi\)
0.206623 + 0.978421i \(0.433753\pi\)
\(510\) 0 0
\(511\) −2.00000 + 3.46410i −0.0884748 + 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.0000i 0.705044i
\(516\) 0 0
\(517\) 2.00000 + 3.46410i 0.0879599 + 0.152351i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −12.0000 20.7846i −0.524723 0.908848i −0.999586 0.0287874i \(-0.990835\pi\)
0.474862 0.880060i \(-0.342498\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 51.9615 + 30.0000i 2.26348 + 1.30682i
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) −25.9808 + 15.0000i −1.12747 + 0.650945i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −27.7128 + 16.0000i −1.19813 + 0.691740i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −5.19615 3.00000i −0.223814 0.129219i
\(540\) 0 0
\(541\) 26.0000i 1.11783i −0.829226 0.558914i \(-0.811218\pi\)
0.829226 0.558914i \(-0.188782\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 0 0
\(549\) −3.00000 5.19615i −0.128037 0.221766i
\(550\) 0 0
\(551\) 12.0000i 0.511217i
\(552\) 0 0
\(553\) 6.92820 + 4.00000i 0.294617 + 0.170097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −36.3731 + 21.0000i −1.54118 + 0.889799i −0.542411 + 0.840113i \(0.682489\pi\)
−0.998765 + 0.0496855i \(0.984178\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 + 3.46410i −0.0842900 + 0.145994i −0.905088 0.425223i \(-0.860196\pi\)
0.820798 + 0.571218i \(0.193529\pi\)
\(564\) 0 0
\(565\) −24.2487 14.0000i −1.02015 0.588984i
\(566\) 0 0
\(567\) 18.0000i 0.755929i
\(568\) 0 0
\(569\) 5.00000 + 8.66025i 0.209611 + 0.363057i 0.951592 0.307364i \(-0.0994469\pi\)
−0.741981 + 0.670421i \(0.766114\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 + 6.92820i 0.166812 + 0.288926i
\(576\) 0 0
\(577\) 6.00000i 0.249783i −0.992170 0.124892i \(-0.960142\pi\)
0.992170 0.124892i \(-0.0398583\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 + 10.3923i −0.248922 + 0.431145i
\(582\) 0 0
\(583\) 10.3923 6.00000i 0.430405 0.248495i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.5167 13.0000i 0.929362 0.536567i 0.0427523 0.999086i \(-0.486387\pi\)
0.886610 + 0.462518i \(0.153054\pi\)
\(588\) 0 0
\(589\) −30.0000 + 51.9615i −1.23613 + 2.14104i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.0000i 1.06769i −0.845582 0.533846i \(-0.820746\pi\)
0.845582 0.533846i \(-0.179254\pi\)
\(594\) 0 0
\(595\) −12.0000 20.7846i −0.491952 0.852086i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 13.0000 + 22.5167i 0.530281 + 0.918474i 0.999376 + 0.0353259i \(0.0112469\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) 30.0000i 1.22169i
\(604\) 0 0
\(605\) −12.1244 7.00000i −0.492925 0.284590i
\(606\) 0 0
\(607\) 6.00000 10.3923i 0.243532 0.421811i −0.718186 0.695852i \(-0.755027\pi\)
0.961718 + 0.274041i \(0.0883604\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 22.5167 13.0000i 0.909439 0.525065i 0.0291886 0.999574i \(-0.490708\pi\)
0.880251 + 0.474509i \(0.157374\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −29.4449 17.0000i −1.18541 0.684394i −0.228147 0.973627i \(-0.573267\pi\)
−0.957259 + 0.289233i \(0.906600\pi\)
\(618\) 0 0
\(619\) 2.00000i 0.0803868i 0.999192 + 0.0401934i \(0.0127974\pi\)
−0.999192 + 0.0401934i \(0.987203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 36.0000i 1.43541i
\(630\) 0 0
\(631\) 19.0526 + 11.0000i 0.758470 + 0.437903i 0.828746 0.559625i \(-0.189055\pi\)
−0.0702759 + 0.997528i \(0.522388\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.8564 8.00000i 0.549875 0.317470i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −25.9808 + 15.0000i −1.02778 + 0.593391i
\(640\) 0 0
\(641\) −13.0000 + 22.5167i −0.513469 + 0.889355i 0.486409 + 0.873731i \(0.338307\pi\)
−0.999878 + 0.0156233i \(0.995027\pi\)
\(642\) 0 0
\(643\) −1.73205 1.00000i −0.0683054 0.0394362i 0.465458 0.885070i \(-0.345890\pi\)
−0.533764 + 0.845634i \(0.679223\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.0000 27.7128i −0.629025 1.08950i −0.987748 0.156059i \(-0.950121\pi\)
0.358723 0.933444i \(-0.383212\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 15.5885i −0.352197 0.610023i 0.634437 0.772975i \(-0.281232\pi\)
−0.986634 + 0.162951i \(0.947899\pi\)
\(654\) 0 0
\(655\) 32.0000i 1.25034i
\(656\) 0 0
\(657\) −5.19615 3.00000i −0.202721 0.117041i
\(658\) 0 0
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) −1.73205 + 1.00000i −0.0673690 + 0.0388955i −0.533306 0.845922i \(-0.679051\pi\)
0.465937 + 0.884818i \(0.345717\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 20.7846 12.0000i 0.805993 0.465340i
\(666\) 0 0
\(667\) 8.00000 13.8564i 0.309761 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.00000i 0.154418i
\(672\) 0 0
\(673\) −13.0000 22.5167i −0.501113 0.867953i −0.999999 0.00128586i \(-0.999591\pi\)
0.498886 0.866668i \(-0.333743\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 2.00000 + 3.46410i 0.0767530 + 0.132940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.73205 1.00000i −0.0662751 0.0382639i 0.466496 0.884523i \(-0.345516\pi\)
−0.532771 + 0.846259i \(0.678849\pi\)
\(684\) 0 0
\(685\) −18.0000 + 31.1769i −0.687745 + 1.19121i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 29.4449 17.0000i 1.12014 0.646710i 0.178700 0.983904i \(-0.442811\pi\)
0.941435 + 0.337193i \(0.109478\pi\)
\(692\) 0 0
\(693\) −6.00000 + 10.3923i −0.227921 + 0.394771i
\(694\) 0 0
\(695\) −27.7128 16.0000i −1.05121 0.606915i
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.00000i 0.150435i
\(708\) 0 0
\(709\) −19.0526 11.0000i −0.715534 0.413114i 0.0975728 0.995228i \(-0.468892\pi\)
−0.813107 + 0.582115i \(0.802225\pi\)
\(710\) 0 0
\(711\) −6.00000 + 10.3923i −0.225018 + 0.389742i
\(712\) 0 0
\(713\) −69.2820 + 40.0000i −2.59463 + 1.49801i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −20.0000 + 34.6410i −0.745874 + 1.29189i 0.203911 + 0.978989i \(0.434635\pi\)
−0.949785 + 0.312903i \(0.898699\pi\)
\(720\) 0 0
\(721\) −13.8564 8.00000i −0.516040 0.297936i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.00000 1.73205i −0.0371391 0.0643268i
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 0 0
\(733\) 26.0000i 0.960332i 0.877178 + 0.480166i \(0.159424\pi\)
−0.877178 + 0.480166i \(0.840576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 17.3205i 0.368355 0.638009i
\(738\) 0 0
\(739\) −5.19615 + 3.00000i −0.191144 + 0.110357i −0.592518 0.805557i \(-0.701866\pi\)
0.401374 + 0.915914i \(0.368533\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.1244 + 7.00000i −0.444799 + 0.256805i −0.705631 0.708579i \(-0.749337\pi\)
0.260832 + 0.965384i \(0.416003\pi\)
\(744\) 0 0
\(745\) 18.0000 31.1769i 0.659469 1.14223i
\(746\) 0 0
\(747\) −15.5885 9.00000i −0.570352 0.329293i
\(748\) 0 0
\(749\) 32.0000i 1.16925i
\(750\) 0 0
\(751\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 21.0000 + 36.3731i 0.763258 + 1.32200i 0.941163 + 0.337954i \(0.109735\pi\)
−0.177905 + 0.984048i \(0.556932\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.5885 + 9.00000i 0.565081 + 0.326250i 0.755182 0.655515i \(-0.227548\pi\)
−0.190101 + 0.981764i \(0.560882\pi\)
\(762\) 0 0
\(763\) −14.0000 + 24.2487i −0.506834 + 0.877862i
\(764\) 0 0
\(765\) 31.1769 18.0000i 1.12720 0.650791i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −46.7654 + 27.0000i −1.68640 + 0.973645i −0.729167 + 0.684336i \(0.760092\pi\)
−0.957236 + 0.289309i \(0.906575\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.8372 + 23.0000i 1.43284 + 0.827253i 0.997337 0.0729331i \(-0.0232359\pi\)
0.435507 + 0.900186i \(0.356569\pi\)
\(774\) 0 0
\(775\) 10.0000i 0.359211i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −20.0000 −0.715656
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.00000i 0.142766i
\(786\) 0 0
\(787\) 32.9090 + 19.0000i 1.17308 + 0.677277i 0.954403 0.298521i \(-0.0964933\pi\)
0.218675 + 0.975798i \(0.429827\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 24.2487 14.0000i 0.862185 0.497783i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.00000 15.5885i 0.318796 0.552171i −0.661441 0.749997i \(-0.730055\pi\)
0.980237 + 0.197826i \(0.0633881\pi\)
\(798\) 0 0
\(799\) 10.3923 + 6.00000i 0.367653 + 0.212265i
\(800\) 0 0
\(801\) 18.0000i 0.635999i
\(802\) 0 0
\(803\) −2.00000 3.46410i −0.0705785 0.122245i
\(804\) 0 0
\(805\) 32.0000 1.12785
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.00000 5.19615i −0.105474 0.182687i 0.808458 0.588555i \(-0.200303\pi\)
−0.913932 + 0.405868i \(0.866969\pi\)
\(810\) 0 0
\(811\) 22.0000i 0.772524i −0.922389 0.386262i \(-0.873766\pi\)
0.922389 0.386262i \(-0.126234\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.0000 17.3205i 0.350285 0.606711i
\(816\) 0 0
\(817\) 20.7846 12.0000i 0.727161 0.419827i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.73205 1.00000i 0.0604490 0.0349002i −0.469471 0.882948i \(-0.655555\pi\)
0.529920 + 0.848048i \(0.322222\pi\)
\(822\) 0 0
\(823\) −12.0000 + 20.7846i −0.418294 + 0.724506i −0.995768 0.0919029i \(-0.970705\pi\)
0.577474 + 0.816409i \(0.304038\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000i 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 0 0
\(829\) 11.0000 + 19.0526i 0.382046 + 0.661723i 0.991355 0.131210i \(-0.0418863\pi\)
−0.609309 + 0.792933i \(0.708553\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −6.00000 10.3923i −0.207639 0.359641i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.66025 5.00000i −0.298985 0.172619i 0.343002 0.939335i \(-0.388556\pi\)
−0.641987 + 0.766716i \(0.721890\pi\)
\(840\) 0 0
\(841\) 12.5000 21.6506i 0.431034 0.746574i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 12.1244 7.00000i 0.416598 0.240523i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 41.5692 + 24.0000i 1.42497 + 0.822709i
\(852\) 0 0
\(853\) 38.0000i 1.30110i 0.759465 + 0.650548i \(0.225461\pi\)
−0.759465 + 0.650548i \(0.774539\pi\)
\(854\) 0 0
\(855\) 18.0000 + 31.1769i 0.615587 + 1.06623i
\(856\) 0 0
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.0000i 1.15737i 0.815550 + 0.578687i \(0.196435\pi\)
−0.815550 + 0.578687i \(0.803565\pi\)
\(864\) 0 0
\(865\) −17.3205 10.0000i −0.588915 0.340010i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.92820 + 4.00000i −0.235023 + 0.135691i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.19615 + 3.00000i −0.175863 + 0.101535i
\(874\) 0 0
\(875\) 12.0000 20.7846i 0.405674 0.702648i
\(876\) 0 0
\(877\) −15.5885 9.00000i −0.526385 0.303908i 0.213158 0.977018i \(-0.431625\pi\)
−0.739543 + 0.673109i \(0.764958\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.00000 + 15.5885i 0.303218 + 0.525188i 0.976863 0.213866i \(-0.0686057\pi\)
−0.673645 + 0.739055i \(0.735272\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.00000 3.46410i −0.0671534 0.116313i 0.830494 0.557028i \(-0.188058\pi\)
−0.897647 + 0.440715i \(0.854725\pi\)
\(888\) 0 0
\(889\) 16.0000i 0.536623i
\(890\) 0 0
\(891\) −15.5885 9.00000i −0.522233 0.301511i
\(892\) 0 0
\(893\) −6.00000 + 10.3923i −0.200782 + 0.347765i
\(894\) 0 0
\(895\) −20.7846 + 12.0000i −0.694753 + 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 17.3205 10.0000i 0.577671 0.333519i
\(900\) 0 0
\(901\) 18.0000 31.1769i 0.599667 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 12.0000i 0.398893i
\(906\) 0 0
\(907\) 18.0000 + 31.1769i 0.597680 + 1.03521i 0.993163 + 0.116739i \(0.0372441\pi\)
−0.395482 + 0.918474i \(0.629423\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) −6.00000 10.3923i −0.198571 0.343935i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 27.7128 + 16.0000i 0.915158 + 0.528367i
\(918\) 0 0
\(919\) 6.00000 10.3923i 0.197922 0.342811i −0.749933 0.661514i \(-0.769914\pi\)
0.947854 + 0.318704i \(0.103247\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.19615 3.00000i 0.170848 0.0986394i
\(926\) 0 0
\(927\) 12.0000 20.7846i 0.394132 0.682656i
\(928\) 0 0
\(929\) 46.7654 + 27.0000i 1.53432 + 0.885841i 0.999155 + 0.0410949i \(0.0130846\pi\)
0.535167 + 0.844746i \(0.320249\pi\)
\(930\) 0 0
\(931\) 18.0000i 0.589926i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 10.0000i 0.325991i 0.986627 + 0.162995i \(0.0521156\pi\)
−0.986627 + 0.162995i \(0.947884\pi\)
\(942\) 0 0
\(943\) −41.5692 24.0000i −1.35368 0.781548i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.3731 21.0000i 1.18197 0.682408i 0.225497 0.974244i \(-0.427599\pi\)
0.956469 + 0.291835i \(0.0942660\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 13.0000 22.5167i 0.421111 0.729386i −0.574937 0.818198i \(-0.694974\pi\)
0.996048 + 0.0888114i \(0.0283068\pi\)
\(954\) 0 0
\(955\) −6.92820 4.00000i −0.224191 0.129437i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 31.1769i −0.581250 1.00676i
\(960\) 0 0
\(961\) −69.0000 −2.22581
\(962\) 0 0
\(963\) 48.0000 1.54678
\(964\) 0 0
\(965\) −2.00000 3.46410i −0.0643823 0.111513i
\(966\) 0 0
\(967\) 50.0000i 1.60789i 0.594703 + 0.803946i \(0.297270\pi\)
−0.594703 + 0.803946i \(0.702730\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 24.0000 41.5692i 0.770197 1.33402i −0.167258 0.985913i \(-0.553491\pi\)
0.937455 0.348107i \(-0.113175\pi\)
\(972\) 0 0
\(973\) 27.7128 16.0000i 0.888432 0.512936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.5885 9.00000i 0.498719 0.287936i −0.229465 0.973317i \(-0.573698\pi\)
0.728184 + 0.685381i \(0.240364\pi\)
\(978\) 0 0
\(979\) 6.00000 10.3923i 0.191761 0.332140i
\(980\) 0 0
\(981\) −36.3731 21.0000i −1.16130 0.670478i
\(982\) 0 0
\(983\) 14.0000i 0.446531i −0.974758 0.223265i \(-0.928328\pi\)
0.974758 0.223265i \(-0.0716716\pi\)
\(984\) 0 0
\(985\) −6.00000 10.3923i −0.191176 0.331126i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) −6.00000 10.3923i −0.190596 0.330122i 0.754852 0.655895i \(-0.227709\pi\)
−0.945448 + 0.325773i \(0.894375\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −27.7128 16.0000i −0.878555 0.507234i
\(996\) 0 0
\(997\) 29.0000 50.2295i 0.918439 1.59078i 0.116653 0.993173i \(-0.462784\pi\)
0.801786 0.597611i \(-0.203883\pi\)
\(998\) 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 676.2.h.c.361.2 4
13.2 odd 12 52.2.a.a.1.1 1
13.3 even 3 676.2.d.c.337.2 2
13.4 even 6 inner 676.2.h.c.485.1 4
13.5 odd 4 676.2.e.c.653.1 2
13.6 odd 12 676.2.e.c.529.1 2
13.7 odd 12 676.2.e.b.529.1 2
13.8 odd 4 676.2.e.b.653.1 2
13.9 even 3 inner 676.2.h.c.485.2 4
13.10 even 6 676.2.d.c.337.1 2
13.11 odd 12 676.2.a.c.1.1 1
13.12 even 2 inner 676.2.h.c.361.1 4
39.2 even 12 468.2.a.b.1.1 1
39.11 even 12 6084.2.a.m.1.1 1
39.23 odd 6 6084.2.b.m.4393.2 2
39.29 odd 6 6084.2.b.m.4393.1 2
52.3 odd 6 2704.2.f.f.337.2 2
52.11 even 12 2704.2.a.g.1.1 1
52.15 even 12 208.2.a.c.1.1 1
52.23 odd 6 2704.2.f.f.337.1 2
65.2 even 12 1300.2.c.c.1249.1 2
65.28 even 12 1300.2.c.c.1249.2 2
65.54 odd 12 1300.2.a.d.1.1 1
91.2 odd 12 2548.2.j.e.1145.1 2
91.41 even 12 2548.2.a.e.1.1 1
91.54 even 12 2548.2.j.f.1145.1 2
91.67 odd 12 2548.2.j.e.1353.1 2
91.80 even 12 2548.2.j.f.1353.1 2
104.67 even 12 832.2.a.f.1.1 1
104.93 odd 12 832.2.a.e.1.1 1
117.2 even 12 4212.2.i.i.1405.1 2
117.41 even 12 4212.2.i.i.2809.1 2
117.67 odd 12 4212.2.i.d.2809.1 2
117.106 odd 12 4212.2.i.d.1405.1 2
143.54 even 12 6292.2.a.g.1.1 1
156.119 odd 12 1872.2.a.f.1.1 1
208.67 even 12 3328.2.b.e.1665.1 2
208.93 odd 12 3328.2.b.q.1665.1 2
208.171 even 12 3328.2.b.e.1665.2 2
208.197 odd 12 3328.2.b.q.1665.2 2
260.119 even 12 5200.2.a.q.1.1 1
312.197 even 12 7488.2.a.bn.1.1 1
312.275 odd 12 7488.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
52.2.a.a.1.1 1 13.2 odd 12
208.2.a.c.1.1 1 52.15 even 12
468.2.a.b.1.1 1 39.2 even 12
676.2.a.c.1.1 1 13.11 odd 12
676.2.d.c.337.1 2 13.10 even 6
676.2.d.c.337.2 2 13.3 even 3
676.2.e.b.529.1 2 13.7 odd 12
676.2.e.b.653.1 2 13.8 odd 4
676.2.e.c.529.1 2 13.6 odd 12
676.2.e.c.653.1 2 13.5 odd 4
676.2.h.c.361.1 4 13.12 even 2 inner
676.2.h.c.361.2 4 1.1 even 1 trivial
676.2.h.c.485.1 4 13.4 even 6 inner
676.2.h.c.485.2 4 13.9 even 3 inner
832.2.a.e.1.1 1 104.93 odd 12
832.2.a.f.1.1 1 104.67 even 12
1300.2.a.d.1.1 1 65.54 odd 12
1300.2.c.c.1249.1 2 65.2 even 12
1300.2.c.c.1249.2 2 65.28 even 12
1872.2.a.f.1.1 1 156.119 odd 12
2548.2.a.e.1.1 1 91.41 even 12
2548.2.j.e.1145.1 2 91.2 odd 12
2548.2.j.e.1353.1 2 91.67 odd 12
2548.2.j.f.1145.1 2 91.54 even 12
2548.2.j.f.1353.1 2 91.80 even 12
2704.2.a.g.1.1 1 52.11 even 12
2704.2.f.f.337.1 2 52.23 odd 6
2704.2.f.f.337.2 2 52.3 odd 6
3328.2.b.e.1665.1 2 208.67 even 12
3328.2.b.e.1665.2 2 208.171 even 12
3328.2.b.q.1665.1 2 208.93 odd 12
3328.2.b.q.1665.2 2 208.197 odd 12
4212.2.i.d.1405.1 2 117.106 odd 12
4212.2.i.d.2809.1 2 117.67 odd 12
4212.2.i.i.1405.1 2 117.2 even 12
4212.2.i.i.2809.1 2 117.41 even 12
5200.2.a.q.1.1 1 260.119 even 12
6084.2.a.m.1.1 1 39.11 even 12
6084.2.b.m.4393.1 2 39.29 odd 6
6084.2.b.m.4393.2 2 39.23 odd 6
6292.2.a.g.1.1 1 143.54 even 12
7488.2.a.bn.1.1 1 312.197 even 12
7488.2.a.bw.1.1 1 312.275 odd 12