Properties

Label 676.2.e.c.653.1
Level $676$
Weight $2$
Character 676.653
Analytic conductor $5.398$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [676,2,Mod(529,676)]
 
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, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 676.e (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 653.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 676.653
Dual form 676.2.e.c.529.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+2.00000 q^{5} +(1.00000 - 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(1.00000 + 1.73205i) q^{11} +(-3.00000 + 5.19615i) q^{17} +(3.00000 - 5.19615i) q^{19} +(-4.00000 - 6.92820i) q^{23} -1.00000 q^{25} +(-1.00000 - 1.73205i) q^{29} +10.0000 q^{31} +(2.00000 - 3.46410i) q^{35} +(3.00000 + 5.19615i) q^{37} +(3.00000 + 5.19615i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(3.00000 - 5.19615i) q^{45} -2.00000 q^{47} +(1.50000 + 2.59808i) q^{49} +6.00000 q^{53} +(2.00000 + 3.46410i) q^{55} +(5.00000 - 8.66025i) q^{59} +(1.00000 - 1.73205i) q^{61} +(-3.00000 - 5.19615i) q^{63} +(-5.00000 - 8.66025i) q^{67} +(-5.00000 + 8.66025i) q^{71} +2.00000 q^{73} +4.00000 q^{77} -4.00000 q^{79} +(-4.50000 - 7.79423i) q^{81} -6.00000 q^{83} +(-6.00000 + 10.3923i) q^{85} +(3.00000 + 5.19615i) q^{89} +(6.00000 - 10.3923i) q^{95} +(-1.00000 + 1.73205i) q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{11} - 6 q^{17} + 6 q^{19} - 8 q^{23} - 2 q^{25} - 2 q^{29} + 20 q^{31} + 4 q^{35} + 6 q^{37} + 6 q^{41} - 4 q^{43} + 6 q^{45} - 4 q^{47} + 3 q^{49} + 12 q^{53} + 4 q^{55} + 10 q^{59} + 2 q^{61} - 6 q^{63} - 10 q^{67} - 10 q^{71} + 4 q^{73} + 8 q^{77} - 8 q^{79} - 9 q^{81} - 12 q^{83} - 12 q^{85} + 6 q^{89} + 12 q^{95} - 2 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 1.00000 1.73205i 0.377964 0.654654i −0.612801 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 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 \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) 3.00000 5.19615i 0.688247 1.19208i −0.284157 0.958778i \(-0.591714\pi\)
0.972404 0.233301i \(-0.0749529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 6.92820i −0.834058 1.44463i −0.894795 0.446476i \(-0.852679\pi\)
0.0607377 0.998154i \(-0.480655\pi\)
\(24\) 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
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 3.46410i 0.338062 0.585540i
\(36\) 0 0
\(37\) 3.00000 + 5.19615i 0.493197 + 0.854242i 0.999969 0.00783774i \(-0.00249486\pi\)
−0.506772 + 0.862080i \(0.669162\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 + 5.19615i 0.468521 + 0.811503i 0.999353 0.0359748i \(-0.0114536\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) −2.00000 + 3.46410i −0.304997 + 0.528271i −0.977261 0.212041i \(-0.931989\pi\)
0.672264 + 0.740312i \(0.265322\pi\)
\(44\) 0 0
\(45\) 3.00000 5.19615i 0.447214 0.774597i
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 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
\(59\) 5.00000 8.66025i 0.650945 1.12747i −0.331949 0.943297i \(-0.607706\pi\)
0.982894 0.184172i \(-0.0589603\pi\)
\(60\) 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
\(63\) −3.00000 5.19615i −0.377964 0.654654i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 8.66025i −0.610847 1.05802i −0.991098 0.133135i \(-0.957496\pi\)
0.380251 0.924883i \(-0.375838\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 + 8.66025i −0.593391 + 1.02778i 0.400381 + 0.916349i \(0.368878\pi\)
−0.993772 + 0.111434i \(0.964456\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 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
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) −6.00000 + 10.3923i −0.650791 + 1.12720i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 + 5.19615i 0.317999 + 0.550791i 0.980071 0.198650i \(-0.0636557\pi\)
−0.662071 + 0.749441i \(0.730322\pi\)
\(90\) 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
\(97\) −1.00000 + 1.73205i −0.101535 + 0.175863i −0.912317 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 1.00000 + 1.73205i 0.0995037 + 0.172345i 0.911479 0.411346i \(-0.134941\pi\)
−0.811976 + 0.583691i \(0.801608\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 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
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 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
\(115\) −8.00000 13.8564i −0.746004 1.29212i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 + 10.3923i 0.550019 + 0.952661i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 4.00000 + 6.92820i 0.354943 + 0.614779i 0.987108 0.160055i \(-0.0511671\pi\)
−0.632166 + 0.774833i \(0.717834\pi\)
\(128\) 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
\(137\) −9.00000 + 15.5885i −0.768922 + 1.33181i 0.169226 + 0.985577i \(0.445873\pi\)
−0.938148 + 0.346235i \(0.887460\pi\)
\(138\) 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
\(145\) −2.00000 3.46410i −0.166091 0.287678i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 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
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 5.00000 8.66025i 0.391630 0.678323i −0.601035 0.799223i \(-0.705245\pi\)
0.992665 + 0.120900i \(0.0385779\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.00000 5.19615i −0.232147 0.402090i 0.726293 0.687386i \(-0.241242\pi\)
−0.958440 + 0.285295i \(0.907908\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −9.00000 15.5885i −0.688247 1.19208i
\(172\) 0 0
\(173\) 5.00000 8.66025i 0.380143 0.658427i −0.610939 0.791677i \(-0.709208\pi\)
0.991082 + 0.133250i \(0.0425415\pi\)
\(174\) 0 0
\(175\) −1.00000 + 1.73205i −0.0755929 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 10.3923i −0.448461 0.776757i 0.549825 0.835280i \(-0.314694\pi\)
−0.998286 + 0.0585225i \(0.981361\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.00000 + 10.3923i 0.441129 + 0.764057i
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 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
\(193\) −1.00000 1.73205i −0.0719816 0.124676i 0.827788 0.561041i \(-0.189599\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 + 5.19615i 0.213741 + 0.370211i 0.952882 0.303340i \(-0.0981018\pi\)
−0.739141 + 0.673550i \(0.764768\pi\)
\(198\) 0 0
\(199\) 8.00000 13.8564i 0.567105 0.982255i −0.429745 0.902950i \(-0.641397\pi\)
0.996850 0.0793045i \(-0.0252700\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 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
\(215\) −4.00000 + 6.92820i −0.272798 + 0.472500i
\(216\) 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
\(223\) 3.00000 + 5.19615i 0.200895 + 0.347960i 0.948817 0.315826i \(-0.102282\pi\)
−0.747922 + 0.663786i \(0.768948\pi\)
\(224\) 0 0
\(225\) −1.50000 + 2.59808i −0.100000 + 0.173205i
\(226\) 0 0
\(227\) −9.00000 + 15.5885i −0.597351 + 1.03464i 0.395860 + 0.918311i \(0.370447\pi\)
−0.993210 + 0.116331i \(0.962887\pi\)
\(228\) 0 0
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.0000 1.16432 0.582162 0.813073i \(-0.302207\pi\)
0.582162 + 0.813073i \(0.302207\pi\)
\(240\) 0 0
\(241\) 3.00000 5.19615i 0.193247 0.334714i −0.753077 0.657932i \(-0.771431\pi\)
0.946324 + 0.323218i \(0.104765\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.00000 + 5.19615i 0.191663 + 0.331970i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 + 10.3923i −0.378717 + 0.655956i −0.990876 0.134778i \(-0.956968\pi\)
0.612159 + 0.790735i \(0.290301\pi\)
\(252\) 0 0
\(253\) 8.00000 13.8564i 0.502956 0.871145i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.00000 15.5885i −0.561405 0.972381i −0.997374 0.0724199i \(-0.976928\pi\)
0.435970 0.899961i \(-0.356405\pi\)
\(258\) 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
\(265\) 12.0000 0.737154
\(266\) 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
\(271\) 1.00000 + 1.73205i 0.0607457 + 0.105215i 0.894799 0.446469i \(-0.147319\pi\)
−0.834053 + 0.551684i \(0.813985\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 1.73205i −0.0603023 0.104447i
\(276\) 0 0
\(277\) −15.0000 + 25.9808i −0.901263 + 1.56103i −0.0754058 + 0.997153i \(0.524025\pi\)
−0.825857 + 0.563880i \(0.809308\pi\)
\(278\) 0 0
\(279\) 15.0000 25.9808i 0.898027 1.55543i
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −2.00000 3.46410i −0.118888 0.205919i 0.800439 0.599414i \(-0.204600\pi\)
−0.919327 + 0.393494i \(0.871266\pi\)
\(284\) 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
\(293\) 7.00000 12.1244i 0.408944 0.708312i −0.585827 0.810436i \(-0.699230\pi\)
0.994772 + 0.102123i \(0.0325637\pi\)
\(294\) 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
\(301\) 4.00000 + 6.92820i 0.230556 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.00000 3.46410i 0.114520 0.198354i
\(306\) 0 0
\(307\) −34.0000 −1.94048 −0.970241 0.242140i \(-0.922151\pi\)
−0.970241 + 0.242140i \(0.922151\pi\)
\(308\) 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
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 2.00000 3.46410i 0.111979 0.193952i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 18.0000 + 31.1769i 1.00155 + 1.73473i
\(324\) 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
\(331\) 11.0000 19.0526i 0.604615 1.04722i −0.387498 0.921871i \(-0.626660\pi\)
0.992112 0.125353i \(-0.0400062\pi\)
\(332\) 0 0
\(333\) 18.0000 0.986394
\(334\) 0 0
\(335\) −10.0000 17.3205i −0.546358 0.946320i
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.0000 + 17.3205i 0.541530 + 0.937958i
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 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
\(349\) −5.00000 8.66025i −0.267644 0.463573i 0.700609 0.713545i \(-0.252912\pi\)
−0.968253 + 0.249973i \(0.919578\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7.00000 + 12.1244i 0.372572 + 0.645314i 0.989960 0.141344i \(-0.0451425\pi\)
−0.617388 + 0.786659i \(0.711809\pi\)
\(354\) 0 0
\(355\) −10.0000 + 17.3205i −0.530745 + 0.919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 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
\(369\) 18.0000 0.937043
\(370\) 0 0
\(371\) 6.00000 10.3923i 0.311504 0.539542i
\(372\) 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
\(379\) −5.00000 8.66025i −0.256833 0.444847i 0.708559 0.705652i \(-0.249346\pi\)
−0.965392 + 0.260804i \(0.916012\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.00000 + 15.5885i −0.459879 + 0.796533i −0.998954 0.0457244i \(-0.985440\pi\)
0.539076 + 0.842257i \(0.318774\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) 6.00000 + 10.3923i 0.304997 + 0.528271i
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −1.00000 + 1.73205i −0.0501886 + 0.0869291i −0.890028 0.455905i \(-0.849316\pi\)
0.839840 + 0.542834i \(0.182649\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 + 5.19615i 0.149813 + 0.259483i 0.931158 0.364615i \(-0.118800\pi\)
−0.781345 + 0.624099i \(0.785466\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −9.00000 15.5885i −0.447214 0.774597i
\(406\) 0 0
\(407\) −6.00000 + 10.3923i −0.297409 + 0.515127i
\(408\) 0 0
\(409\) 7.00000 12.1244i 0.346128 0.599511i −0.639430 0.768849i \(-0.720830\pi\)
0.985558 + 0.169338i \(0.0541630\pi\)
\(410\) 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
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −3.00000 + 5.19615i −0.145865 + 0.252646i
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −2.00000 3.46410i −0.0967868 0.167640i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 + 25.9808i 0.722525 + 1.25145i 0.959985 + 0.280052i \(0.0903517\pi\)
−0.237460 + 0.971397i \(0.576315\pi\)
\(432\) 0 0
\(433\) 7.00000 12.1244i 0.336399 0.582659i −0.647354 0.762190i \(-0.724124\pi\)
0.983752 + 0.179530i \(0.0574578\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −48.0000 −2.29615
\(438\) 0 0
\(439\) 8.00000 + 13.8564i 0.381819 + 0.661330i 0.991322 0.131453i \(-0.0419644\pi\)
−0.609503 + 0.792784i \(0.708631\pi\)
\(440\) 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
\(449\) 19.0000 32.9090i 0.896665 1.55307i 0.0649356 0.997889i \(-0.479316\pi\)
0.831730 0.555181i \(-0.187351\pi\)
\(450\) 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
\(457\) −17.0000 29.4449i −0.795226 1.37737i −0.922695 0.385530i \(-0.874019\pi\)
0.127469 0.991843i \(-0.459315\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 5.19615i 0.139724 0.242009i −0.787668 0.616100i \(-0.788712\pi\)
0.927392 + 0.374091i \(0.122045\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −3.00000 + 5.19615i −0.137649 + 0.238416i
\(476\) 0 0
\(477\) 9.00000 15.5885i 0.412082 0.713746i
\(478\) 0 0
\(479\) 9.00000 + 15.5885i 0.411220 + 0.712255i 0.995023 0.0996406i \(-0.0317693\pi\)
−0.583803 + 0.811895i \(0.698436\pi\)
\(480\) 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
\(487\) −1.00000 + 1.73205i −0.0453143 + 0.0784867i −0.887793 0.460243i \(-0.847762\pi\)
0.842479 + 0.538730i \(0.181096\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.0000 + 17.3205i 0.451294 + 0.781664i 0.998467 0.0553560i \(-0.0176294\pi\)
−0.547173 + 0.837020i \(0.684296\pi\)
\(492\) 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
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 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
\(505\) 2.00000 + 3.46410i 0.0889988 + 0.154150i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.00000 + 12.1244i 0.310270 + 0.537403i 0.978421 0.206623i \(-0.0662474\pi\)
−0.668151 + 0.744026i \(0.732914\pi\)
\(510\) 0 0
\(511\) 2.00000 3.46410i 0.0884748 0.153243i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 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
\(527\) −30.0000 + 51.9615i −1.30682 + 2.26348i
\(528\) 0 0
\(529\) −20.5000 + 35.5070i −0.891304 + 1.54378i
\(530\) 0 0
\(531\) −15.0000 25.9808i −0.650945 1.12747i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 16.0000 + 27.7128i 0.691740 + 1.19813i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.00000 + 5.19615i −0.129219 + 0.223814i
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 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
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −4.00000 + 6.92820i −0.170097 + 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −21.0000 36.3731i −0.889799 1.54118i −0.840113 0.542411i \(-0.817511\pi\)
−0.0496855 0.998765i \(-0.515822\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.00000 3.46410i 0.0842900 0.145994i −0.820798 0.571218i \(-0.806471\pi\)
0.905088 + 0.425223i \(0.139804\pi\)
\(564\) 0 0
\(565\) −14.0000 + 24.2487i −0.588984 + 1.02015i
\(566\) 0 0
\(567\) −18.0000 −0.755929
\(568\) 0 0
\(569\) −5.00000 8.66025i −0.209611 0.363057i 0.741981 0.670421i \(-0.233886\pi\)
−0.951592 + 0.307364i \(0.900553\pi\)
\(570\) 0 0
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 + 6.92820i 0.166812 + 0.288926i
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.00000 + 10.3923i −0.248922 + 0.431145i
\(582\) 0 0
\(583\) 6.00000 + 10.3923i 0.248495 + 0.430405i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.0000 22.5167i −0.536567 0.929362i −0.999086 0.0427523i \(-0.986387\pi\)
0.462518 0.886610i \(-0.346946\pi\)
\(588\) 0 0
\(589\) 30.0000 51.9615i 1.23613 2.14104i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 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
\(603\) −30.0000 −1.22169
\(604\) 0 0
\(605\) 7.00000 12.1244i 0.284590 0.492925i
\(606\) 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
\(613\) −13.0000 22.5167i −0.525065 0.909439i −0.999574 0.0291886i \(-0.990708\pi\)
0.474509 0.880251i \(-0.342626\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −17.0000 + 29.4449i −0.684394 + 1.18541i 0.289233 + 0.957259i \(0.406600\pi\)
−0.973627 + 0.228147i \(0.926733\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 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
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −11.0000 + 19.0526i −0.437903 + 0.758470i −0.997528 0.0702759i \(-0.977612\pi\)
0.559625 + 0.828746i \(0.310945\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 + 13.8564i 0.317470 + 0.549875i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 15.0000 + 25.9808i 0.593391 + 1.02778i
\(640\) 0 0
\(641\) 13.0000 22.5167i 0.513469 0.889355i −0.486409 0.873731i \(-0.661693\pi\)
0.999878 0.0156233i \(-0.00497325\pi\)
\(642\) 0 0
\(643\) −1.00000 + 1.73205i −0.0394362 + 0.0683054i −0.885070 0.465458i \(-0.845890\pi\)
0.845634 + 0.533764i \(0.179223\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 + 27.7128i 0.629025 + 1.08950i 0.987748 + 0.156059i \(0.0498790\pi\)
−0.358723 + 0.933444i \(0.616788\pi\)
\(648\) 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
\(655\) −32.0000 −1.25034
\(656\) 0 0
\(657\) 3.00000 5.19615i 0.117041 0.202721i
\(658\) 0 0
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) −1.00000 1.73205i −0.0388955 0.0673690i 0.845922 0.533306i \(-0.179051\pi\)
−0.884818 + 0.465937i \(0.845717\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −12.0000 20.7846i −0.465340 0.805993i
\(666\) 0 0
\(667\) −8.00000 + 13.8564i −0.309761 + 0.536522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 13.0000 + 22.5167i 0.501113 + 0.867953i 0.999999 + 0.00128586i \(0.000409302\pi\)
−0.498886 + 0.866668i \(0.666257\pi\)
\(674\) 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
\(683\) 1.00000 1.73205i 0.0382639 0.0662751i −0.846259 0.532771i \(-0.821151\pi\)
0.884523 + 0.466496i \(0.154484\pi\)
\(684\) 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
\(691\) −17.0000 29.4449i −0.646710 1.12014i −0.983904 0.178700i \(-0.942811\pi\)
0.337193 0.941435i \(-0.390522\pi\)
\(692\) 0 0
\(693\) 6.00000 10.3923i 0.227921 0.394771i
\(694\) 0 0
\(695\) −16.0000 + 27.7128i −0.606915 + 1.05121i
\(696\) 0 0
\(697\) −36.0000 −1.36360
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 36.0000 1.35777
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.00000 0.150435
\(708\) 0 0
\(709\) 11.0000 19.0526i 0.413114 0.715534i −0.582115 0.813107i \(-0.697775\pi\)
0.995228 + 0.0975728i \(0.0311079\pi\)
\(710\) 0 0
\(711\) −6.00000 + 10.3923i −0.225018 + 0.389742i
\(712\) 0 0
\(713\) −40.0000 69.2820i −1.49801 2.59463i
\(714\) 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 \(-0.565365\pi\)
0.949785 0.312903i \(-0.101301\pi\)
\(720\) 0 0
\(721\) −8.00000 + 13.8564i −0.297936 + 0.516040i
\(722\) 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 \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −12.0000 20.7846i −0.443836 0.768747i
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.0000 17.3205i 0.368355 0.638009i
\(738\) 0 0
\(739\) −3.00000 5.19615i −0.110357 0.191144i 0.805557 0.592518i \(-0.201866\pi\)
−0.915914 + 0.401374i \(0.868533\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.00000 + 12.1244i 0.256805 + 0.444799i 0.965384 0.260832i \(-0.0839968\pi\)
−0.708579 + 0.705631i \(0.750663\pi\)
\(744\) 0 0
\(745\) −18.0000 + 31.1769i −0.659469 + 1.14223i
\(746\) 0 0
\(747\) −9.00000 + 15.5885i −0.329293 + 0.570352i
\(748\) 0 0
\(749\) 32.0000 1.16925
\(750\) 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
\(761\) −9.00000 + 15.5885i −0.326250 + 0.565081i −0.981764 0.190101i \(-0.939118\pi\)
0.655515 + 0.755182i \(0.272452\pi\)
\(762\) 0 0
\(763\) −14.0000 + 24.2487i −0.506834 + 0.877862i
\(764\) 0 0
\(765\) 18.0000 + 31.1769i 0.650791 + 1.12720i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 27.0000 + 46.7654i 0.973645 + 1.68640i 0.684336 + 0.729167i \(0.260092\pi\)
0.289309 + 0.957236i \(0.406575\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 23.0000 39.8372i 0.827253 1.43284i −0.0729331 0.997337i \(-0.523236\pi\)
0.900186 0.435507i \(-0.143431\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 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
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −19.0000 + 32.9090i −0.677277 + 1.17308i 0.298521 + 0.954403i \(0.403507\pi\)
−0.975798 + 0.218675i \(0.929827\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0000 + 24.2487i 0.497783 + 0.862185i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.00000 + 15.5885i −0.318796 + 0.552171i −0.980237 0.197826i \(-0.936612\pi\)
0.661441 + 0.749997i \(0.269945\pi\)
\(798\) 0 0
\(799\) 6.00000 10.3923i 0.212265 0.367653i
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 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
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10.0000 17.3205i 0.350285 0.606711i
\(816\) 0 0
\(817\) 12.0000 + 20.7846i 0.419827 + 0.727161i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 1.73205i −0.0349002 0.0604490i 0.848048 0.529920i \(-0.177778\pi\)
−0.882948 + 0.469471i \(0.844445\pi\)
\(822\) 0 0
\(823\) 12.0000 20.7846i 0.418294 0.724506i −0.577474 0.816409i \(-0.695962\pi\)
0.995768 + 0.0919029i \(0.0292950\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 0 0
\(829\) −11.0000 19.0526i −0.382046 0.661723i 0.609309 0.792933i \(-0.291447\pi\)
−0.991355 + 0.131210i \(0.958114\pi\)
\(830\) 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
\(839\) 5.00000 8.66025i 0.172619 0.298985i −0.766716 0.641987i \(-0.778110\pi\)
0.939335 + 0.343002i \(0.111444\pi\)
\(840\) 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
\(847\) −7.00000 12.1244i −0.240523 0.416598i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000 41.5692i 0.822709 1.42497i
\(852\) 0 0
\(853\) −38.0000 −1.30110 −0.650548 0.759465i \(-0.725461\pi\)
−0.650548 + 0.759465i \(0.725461\pi\)
\(854\) 0 0
\(855\) −18.0000 31.1769i −0.615587 1.06623i
\(856\) 0 0
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 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
\(863\) 34.0000 1.15737 0.578687 0.815550i \(-0.303565\pi\)
0.578687 + 0.815550i \(0.303565\pi\)
\(864\) 0 0
\(865\) 10.0000 17.3205i 0.340010 0.588915i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.00000 6.92820i −0.135691 0.235023i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.00000 + 5.19615i 0.101535 + 0.175863i
\(874\) 0 0
\(875\) −12.0000 + 20.7846i −0.405674 + 0.702648i
\(876\) 0 0
\(877\) −9.00000 + 15.5885i −0.303908 + 0.526385i −0.977018 0.213158i \(-0.931625\pi\)
0.673109 + 0.739543i \(0.264958\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.00000 15.5885i −0.303218 0.525188i 0.673645 0.739055i \(-0.264728\pi\)
−0.976863 + 0.213866i \(0.931394\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 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
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) 9.00000 15.5885i 0.301511 0.522233i
\(892\) 0 0
\(893\) −6.00000 + 10.3923i −0.200782 + 0.347765i
\(894\) 0 0
\(895\) −12.0000 20.7846i −0.401116 0.694753i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.0000 17.3205i −0.333519 0.577671i
\(900\) 0 0
\(901\) −18.0000 + 31.1769i −0.599667 + 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) −18.0000 31.1769i −0.597680 1.03521i −0.993163 0.116739i \(-0.962756\pi\)
0.395482 0.918474i \(-0.370577\pi\)
\(908\) 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
\(917\) −16.0000 + 27.7128i −0.528367 + 0.915158i
\(918\) 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
\(925\) −3.00000 5.19615i −0.0986394 0.170848i
\(926\) 0 0
\(927\) −12.0000 + 20.7846i −0.394132 + 0.682656i
\(928\) 0 0
\(929\) 27.0000 46.7654i 0.885841 1.53432i 0.0410949 0.999155i \(-0.486915\pi\)
0.844746 0.535167i \(-0.179751\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 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
\(941\) 10.0000 0.325991 0.162995 0.986627i \(-0.447884\pi\)
0.162995 + 0.986627i \(0.447884\pi\)
\(942\) 0 0
\(943\) 24.0000 41.5692i 0.781548 1.35368i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.0000 + 36.3731i 0.682408 + 1.18197i 0.974244 + 0.225497i \(0.0724007\pi\)
−0.291835 + 0.956469i \(0.594266\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −13.0000 + 22.5167i −0.421111 + 0.729386i −0.996048 0.0888114i \(-0.971693\pi\)
0.574937 + 0.818198i \(0.305026\pi\)
\(954\) 0 0
\(955\) −4.00000 + 6.92820i −0.129437 + 0.224191i
\(956\) 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
\(967\) 50.0000 1.60789 0.803946 0.594703i \(-0.202730\pi\)
0.803946 + 0.594703i \(0.202730\pi\)
\(968\) 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
\(973\) 16.0000 + 27.7128i 0.512936 + 0.888432i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.00000 15.5885i −0.287936 0.498719i 0.685381 0.728184i \(-0.259636\pi\)
−0.973317 + 0.229465i \(0.926302\pi\)
\(978\) 0 0
\(979\) −6.00000 + 10.3923i −0.191761 + 0.332140i
\(980\) 0 0
\(981\) −21.0000 + 36.3731i −0.670478 + 1.16130i
\(982\) 0 0
\(983\) 14.0000 0.446531 0.223265 0.974758i \(-0.428328\pi\)
0.223265 + 0.974758i \(0.428328\pi\)
\(984\) 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
\(995\) 16.0000 27.7128i 0.507234 0.878555i
\(996\) 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.e.c.653.1 2
13.2 odd 12 676.2.d.c.337.1 2
13.3 even 3 52.2.a.a.1.1 1
13.4 even 6 676.2.e.b.529.1 2
13.5 odd 4 676.2.h.c.361.1 4
13.6 odd 12 676.2.h.c.485.1 4
13.7 odd 12 676.2.h.c.485.2 4
13.8 odd 4 676.2.h.c.361.2 4
13.9 even 3 inner 676.2.e.c.529.1 2
13.10 even 6 676.2.a.c.1.1 1
13.11 odd 12 676.2.d.c.337.2 2
13.12 even 2 676.2.e.b.653.1 2
39.2 even 12 6084.2.b.m.4393.2 2
39.11 even 12 6084.2.b.m.4393.1 2
39.23 odd 6 6084.2.a.m.1.1 1
39.29 odd 6 468.2.a.b.1.1 1
52.3 odd 6 208.2.a.c.1.1 1
52.11 even 12 2704.2.f.f.337.2 2
52.15 even 12 2704.2.f.f.337.1 2
52.23 odd 6 2704.2.a.g.1.1 1
65.3 odd 12 1300.2.c.c.1249.2 2
65.29 even 6 1300.2.a.d.1.1 1
65.42 odd 12 1300.2.c.c.1249.1 2
91.3 odd 6 2548.2.j.f.1353.1 2
91.16 even 3 2548.2.j.e.1145.1 2
91.55 odd 6 2548.2.a.e.1.1 1
91.68 odd 6 2548.2.j.f.1145.1 2
91.81 even 3 2548.2.j.e.1353.1 2
104.3 odd 6 832.2.a.f.1.1 1
104.29 even 6 832.2.a.e.1.1 1
117.16 even 3 4212.2.i.d.1405.1 2
117.29 odd 6 4212.2.i.i.1405.1 2
117.68 odd 6 4212.2.i.i.2809.1 2
117.94 even 3 4212.2.i.d.2809.1 2
143.120 odd 6 6292.2.a.g.1.1 1
156.107 even 6 1872.2.a.f.1.1 1
208.3 odd 12 3328.2.b.e.1665.1 2
208.29 even 12 3328.2.b.q.1665.1 2
208.107 odd 12 3328.2.b.e.1665.2 2
208.133 even 12 3328.2.b.q.1665.2 2
260.159 odd 6 5200.2.a.q.1.1 1
312.29 odd 6 7488.2.a.bn.1.1 1
312.107 even 6 7488.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
52.2.a.a.1.1 1 13.3 even 3
208.2.a.c.1.1 1 52.3 odd 6
468.2.a.b.1.1 1 39.29 odd 6
676.2.a.c.1.1 1 13.10 even 6
676.2.d.c.337.1 2 13.2 odd 12
676.2.d.c.337.2 2 13.11 odd 12
676.2.e.b.529.1 2 13.4 even 6
676.2.e.b.653.1 2 13.12 even 2
676.2.e.c.529.1 2 13.9 even 3 inner
676.2.e.c.653.1 2 1.1 even 1 trivial
676.2.h.c.361.1 4 13.5 odd 4
676.2.h.c.361.2 4 13.8 odd 4
676.2.h.c.485.1 4 13.6 odd 12
676.2.h.c.485.2 4 13.7 odd 12
832.2.a.e.1.1 1 104.29 even 6
832.2.a.f.1.1 1 104.3 odd 6
1300.2.a.d.1.1 1 65.29 even 6
1300.2.c.c.1249.1 2 65.42 odd 12
1300.2.c.c.1249.2 2 65.3 odd 12
1872.2.a.f.1.1 1 156.107 even 6
2548.2.a.e.1.1 1 91.55 odd 6
2548.2.j.e.1145.1 2 91.16 even 3
2548.2.j.e.1353.1 2 91.81 even 3
2548.2.j.f.1145.1 2 91.68 odd 6
2548.2.j.f.1353.1 2 91.3 odd 6
2704.2.a.g.1.1 1 52.23 odd 6
2704.2.f.f.337.1 2 52.15 even 12
2704.2.f.f.337.2 2 52.11 even 12
3328.2.b.e.1665.1 2 208.3 odd 12
3328.2.b.e.1665.2 2 208.107 odd 12
3328.2.b.q.1665.1 2 208.29 even 12
3328.2.b.q.1665.2 2 208.133 even 12
4212.2.i.d.1405.1 2 117.16 even 3
4212.2.i.d.2809.1 2 117.94 even 3
4212.2.i.i.1405.1 2 117.29 odd 6
4212.2.i.i.2809.1 2 117.68 odd 6
5200.2.a.q.1.1 1 260.159 odd 6
6084.2.a.m.1.1 1 39.23 odd 6
6084.2.b.m.4393.1 2 39.11 even 12
6084.2.b.m.4393.2 2 39.2 even 12
6292.2.a.g.1.1 1 143.120 odd 6
7488.2.a.bn.1.1 1 312.29 odd 6
7488.2.a.bw.1.1 1 312.107 even 6