Properties

Label 1512.2.bl.d.1025.4
Level $1512$
Weight $2$
Character 1512.1025
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(593,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.bl (of order \(6\), degree \(2\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.81094542259068665856.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + 9x^{12} - 8x^{10} + 44x^{8} - 32x^{6} + 144x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1025.4
Root \(-1.12988 - 0.850516i\) of defining polynomial
Character \(\chi\) \(=\) 1512.1025
Dual form 1512.2.bl.d.593.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.310225 - 0.537326i) q^{5} +(-1.44109 + 2.21884i) q^{7} +(-1.91649 - 1.10649i) q^{11} -0.963711i q^{13} +(0.653486 - 1.13187i) q^{17} +(-1.57941 + 0.911873i) q^{19} +(5.27562 - 3.04588i) q^{23} +(2.30752 - 3.99674i) q^{25} -6.68969i q^{29} +(3.48866 + 2.01418i) q^{31} +(1.63930 + 0.0859913i) q^{35} +(0.165402 + 0.286485i) q^{37} +11.7190 q^{41} -2.18909 q^{43} +(-1.27394 - 2.20652i) q^{47} +(-2.84654 - 6.39509i) q^{49} +(0.985818 + 0.569162i) q^{53} +1.37304i q^{55} +(6.20629 - 10.7496i) q^{59} +(0.603098 - 0.348199i) q^{61} +(-0.517827 + 0.298967i) q^{65} +(-3.61028 + 6.25319i) q^{67} -7.40207i q^{71} +(-2.17396 - 1.25513i) q^{73} +(5.21696 - 2.65786i) q^{77} +(4.52429 + 7.83630i) q^{79} +8.85562 q^{83} -0.810911 q^{85} +(-4.20928 - 7.29069i) q^{89} +(2.13832 + 1.38879i) q^{91} +(0.979946 + 0.565772i) q^{95} -12.4027i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{7} + 12 q^{19} - 12 q^{31} + 16 q^{37} - 16 q^{43} - 28 q^{49} - 60 q^{61} - 4 q^{67} + 12 q^{73} - 32 q^{79} - 32 q^{85} + 24 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.310225 0.537326i −0.138737 0.240299i 0.788282 0.615314i \(-0.210971\pi\)
−0.927019 + 0.375015i \(0.877638\pi\)
\(6\) 0 0
\(7\) −1.44109 + 2.21884i −0.544679 + 0.838644i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.91649 1.10649i −0.577845 0.333619i 0.182432 0.983219i \(-0.441603\pi\)
−0.760276 + 0.649600i \(0.774936\pi\)
\(12\) 0 0
\(13\) 0.963711i 0.267285i −0.991030 0.133643i \(-0.957333\pi\)
0.991030 0.133643i \(-0.0426674\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.653486 1.13187i 0.158494 0.274519i −0.775832 0.630939i \(-0.782670\pi\)
0.934326 + 0.356421i \(0.116003\pi\)
\(18\) 0 0
\(19\) −1.57941 + 0.911873i −0.362342 + 0.209198i −0.670107 0.742264i \(-0.733752\pi\)
0.307766 + 0.951462i \(0.400419\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.27562 3.04588i 1.10004 0.635110i 0.163810 0.986492i \(-0.447622\pi\)
0.936232 + 0.351382i \(0.114288\pi\)
\(24\) 0 0
\(25\) 2.30752 3.99674i 0.461504 0.799349i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.68969i 1.24224i −0.783714 0.621122i \(-0.786677\pi\)
0.783714 0.621122i \(-0.213323\pi\)
\(30\) 0 0
\(31\) 3.48866 + 2.01418i 0.626582 + 0.361757i 0.779427 0.626493i \(-0.215510\pi\)
−0.152845 + 0.988250i \(0.548844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.63930 + 0.0859913i 0.277093 + 0.0145352i
\(36\) 0 0
\(37\) 0.165402 + 0.286485i 0.0271919 + 0.0470978i 0.879301 0.476266i \(-0.158010\pi\)
−0.852109 + 0.523364i \(0.824677\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.7190 1.83020 0.915102 0.403221i \(-0.132110\pi\)
0.915102 + 0.403221i \(0.132110\pi\)
\(42\) 0 0
\(43\) −2.18909 −0.333833 −0.166916 0.985971i \(-0.553381\pi\)
−0.166916 + 0.985971i \(0.553381\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.27394 2.20652i −0.185823 0.321854i 0.758031 0.652219i \(-0.226162\pi\)
−0.943853 + 0.330364i \(0.892828\pi\)
\(48\) 0 0
\(49\) −2.84654 6.39509i −0.406649 0.913585i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.985818 + 0.569162i 0.135413 + 0.0781804i 0.566176 0.824284i \(-0.308422\pi\)
−0.430763 + 0.902465i \(0.641756\pi\)
\(54\) 0 0
\(55\) 1.37304i 0.185141i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.20629 10.7496i 0.807991 1.39948i −0.106263 0.994338i \(-0.533889\pi\)
0.914254 0.405143i \(-0.132778\pi\)
\(60\) 0 0
\(61\) 0.603098 0.348199i 0.0772188 0.0445823i −0.460893 0.887455i \(-0.652471\pi\)
0.538112 + 0.842873i \(0.319138\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.517827 + 0.298967i −0.0642285 + 0.0370823i
\(66\) 0 0
\(67\) −3.61028 + 6.25319i −0.441066 + 0.763949i −0.997769 0.0667628i \(-0.978733\pi\)
0.556703 + 0.830712i \(0.312066\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.40207i 0.878464i −0.898374 0.439232i \(-0.855251\pi\)
0.898374 0.439232i \(-0.144749\pi\)
\(72\) 0 0
\(73\) −2.17396 1.25513i −0.254442 0.146902i 0.367354 0.930081i \(-0.380264\pi\)
−0.621797 + 0.783179i \(0.713597\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.21696 2.65786i 0.594527 0.302891i
\(78\) 0 0
\(79\) 4.52429 + 7.83630i 0.509023 + 0.881653i 0.999945 + 0.0104498i \(0.00332634\pi\)
−0.490923 + 0.871203i \(0.663340\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.85562 0.972031 0.486016 0.873950i \(-0.338450\pi\)
0.486016 + 0.873950i \(0.338450\pi\)
\(84\) 0 0
\(85\) −0.810911 −0.0879556
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.20928 7.29069i −0.446183 0.772812i 0.551951 0.833877i \(-0.313884\pi\)
−0.998134 + 0.0610650i \(0.980550\pi\)
\(90\) 0 0
\(91\) 2.13832 + 1.38879i 0.224157 + 0.145585i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.979946 + 0.565772i 0.100540 + 0.0580470i
\(96\) 0 0
\(97\) 12.4027i 1.25930i −0.776877 0.629652i \(-0.783197\pi\)
0.776877 0.629652i \(-0.216803\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.82480 + 3.16064i −0.181574 + 0.314496i −0.942417 0.334441i \(-0.891453\pi\)
0.760843 + 0.648936i \(0.224786\pi\)
\(102\) 0 0
\(103\) 3.12562 1.80458i 0.307976 0.177810i −0.338044 0.941130i \(-0.609765\pi\)
0.646020 + 0.763320i \(0.276432\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.1599 + 7.02050i −1.17554 + 0.678697i −0.954978 0.296676i \(-0.904122\pi\)
−0.220560 + 0.975373i \(0.570788\pi\)
\(108\) 0 0
\(109\) −0.681140 + 1.17977i −0.0652413 + 0.113001i −0.896801 0.442434i \(-0.854115\pi\)
0.831560 + 0.555435i \(0.187448\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.1266i 1.51707i −0.651635 0.758533i \(-0.725917\pi\)
0.651635 0.758533i \(-0.274083\pi\)
\(114\) 0 0
\(115\) −3.27326 1.88982i −0.305233 0.176226i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.56972 + 3.08111i 0.143896 + 0.282444i
\(120\) 0 0
\(121\) −3.05137 5.28513i −0.277397 0.480466i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.96566 −0.533585
\(126\) 0 0
\(127\) 8.16641 0.724652 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.50391 4.33690i −0.218768 0.378916i 0.735664 0.677347i \(-0.236870\pi\)
−0.954431 + 0.298430i \(0.903537\pi\)
\(132\) 0 0
\(133\) 0.252762 4.81855i 0.0219172 0.417822i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.08103 2.35619i −0.348666 0.201303i 0.315431 0.948948i \(-0.397851\pi\)
−0.664098 + 0.747646i \(0.731184\pi\)
\(138\) 0 0
\(139\) 10.3716i 0.879709i 0.898069 + 0.439854i \(0.144970\pi\)
−0.898069 + 0.439854i \(0.855030\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.06633 + 1.84695i −0.0891714 + 0.154449i
\(144\) 0 0
\(145\) −3.59454 + 2.07531i −0.298511 + 0.172345i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.01781 4.62908i 0.656844 0.379229i −0.134229 0.990950i \(-0.542856\pi\)
0.791074 + 0.611721i \(0.209522\pi\)
\(150\) 0 0
\(151\) −8.62359 + 14.9365i −0.701778 + 1.21552i 0.266064 + 0.963955i \(0.414277\pi\)
−0.967842 + 0.251560i \(0.919056\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.49940i 0.200756i
\(156\) 0 0
\(157\) 3.59454 + 2.07531i 0.286876 + 0.165628i 0.636532 0.771250i \(-0.280368\pi\)
−0.349656 + 0.936878i \(0.613702\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.844287 + 16.0951i −0.0665391 + 1.26848i
\(162\) 0 0
\(163\) −6.38217 11.0542i −0.499890 0.865835i 0.500110 0.865962i \(-0.333293\pi\)
−1.00000 0.000126653i \(0.999960\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.5086 1.58700 0.793501 0.608569i \(-0.208256\pi\)
0.793501 + 0.608569i \(0.208256\pi\)
\(168\) 0 0
\(169\) 12.0713 0.928559
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.81088 + 6.60064i 0.289736 + 0.501837i 0.973747 0.227635i \(-0.0730992\pi\)
−0.684011 + 0.729472i \(0.739766\pi\)
\(174\) 0 0
\(175\) 5.54282 + 10.8797i 0.418997 + 0.822427i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −5.78757 3.34146i −0.432583 0.249752i 0.267863 0.963457i \(-0.413682\pi\)
−0.700447 + 0.713705i \(0.747016\pi\)
\(180\) 0 0
\(181\) 5.28785i 0.393043i −0.980500 0.196521i \(-0.937036\pi\)
0.980500 0.196521i \(-0.0629645\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.102624 0.177749i 0.00754504 0.0130684i
\(186\) 0 0
\(187\) −2.50480 + 1.44615i −0.183169 + 0.105753i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.45434 + 3.72641i −0.467020 + 0.269634i −0.714991 0.699133i \(-0.753569\pi\)
0.247972 + 0.968767i \(0.420236\pi\)
\(192\) 0 0
\(193\) 9.74522 16.8792i 0.701476 1.21499i −0.266472 0.963843i \(-0.585858\pi\)
0.967948 0.251150i \(-0.0808086\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.62061i 0.186711i −0.995633 0.0933555i \(-0.970241\pi\)
0.995633 0.0933555i \(-0.0297593\pi\)
\(198\) 0 0
\(199\) 7.10031 + 4.09937i 0.503328 + 0.290596i 0.730087 0.683355i \(-0.239480\pi\)
−0.226759 + 0.973951i \(0.572813\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 14.8434 + 9.64042i 1.04180 + 0.676625i
\(204\) 0 0
\(205\) −3.63554 6.29694i −0.253917 0.439797i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4.03591 0.279170
\(210\) 0 0
\(211\) −19.6095 −1.34997 −0.674986 0.737831i \(-0.735850\pi\)
−0.674986 + 0.737831i \(0.735850\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.679111 + 1.17625i 0.0463150 + 0.0802199i
\(216\) 0 0
\(217\) −9.49661 + 4.83819i −0.644672 + 0.328438i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.09080 0.629771i −0.0733749 0.0423630i
\(222\) 0 0
\(223\) 23.9924i 1.60665i 0.595539 + 0.803326i \(0.296938\pi\)
−0.595539 + 0.803326i \(0.703062\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.8243 + 22.2123i −0.851177 + 1.47428i 0.0289703 + 0.999580i \(0.490777\pi\)
−0.880147 + 0.474701i \(0.842556\pi\)
\(228\) 0 0
\(229\) 8.65507 4.99701i 0.571943 0.330212i −0.185982 0.982553i \(-0.559547\pi\)
0.757925 + 0.652342i \(0.226213\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −22.4800 + 12.9788i −1.47271 + 0.850271i −0.999529 0.0306937i \(-0.990228\pi\)
−0.473183 + 0.880964i \(0.656895\pi\)
\(234\) 0 0
\(235\) −0.790414 + 1.36904i −0.0515609 + 0.0893062i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.1554i 1.17438i −0.809450 0.587189i \(-0.800235\pi\)
0.809450 0.587189i \(-0.199765\pi\)
\(240\) 0 0
\(241\) 3.44245 + 1.98750i 0.221748 + 0.128026i 0.606759 0.794886i \(-0.292469\pi\)
−0.385011 + 0.922912i \(0.625802\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.55318 + 3.51344i −0.163117 + 0.224465i
\(246\) 0 0
\(247\) 0.878782 + 1.52210i 0.0559156 + 0.0968486i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.403459 −0.0254661 −0.0127331 0.999919i \(-0.504053\pi\)
−0.0127331 + 0.999919i \(0.504053\pi\)
\(252\) 0 0
\(253\) −13.4809 −0.847538
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.22515 + 12.5143i 0.450692 + 0.780622i 0.998429 0.0560288i \(-0.0178439\pi\)
−0.547737 + 0.836651i \(0.684511\pi\)
\(258\) 0 0
\(259\) −0.874023 0.0458477i −0.0543091 0.00284884i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.3400 9.43390i −1.00757 0.581719i −0.0970889 0.995276i \(-0.530953\pi\)
−0.910478 + 0.413556i \(0.864286\pi\)
\(264\) 0 0
\(265\) 0.706274i 0.0433861i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −4.65868 + 8.06908i −0.284045 + 0.491980i −0.972377 0.233415i \(-0.925010\pi\)
0.688332 + 0.725396i \(0.258343\pi\)
\(270\) 0 0
\(271\) 23.0776 13.3239i 1.40187 0.809368i 0.407282 0.913302i \(-0.366477\pi\)
0.994584 + 0.103934i \(0.0331432\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.84470 + 5.10649i −0.533355 + 0.307933i
\(276\) 0 0
\(277\) 9.63457 16.6876i 0.578885 1.00266i −0.416722 0.909034i \(-0.636821\pi\)
0.995608 0.0936250i \(-0.0298455\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.20740i 0.0720276i 0.999351 + 0.0360138i \(0.0114660\pi\)
−0.999351 + 0.0360138i \(0.988534\pi\)
\(282\) 0 0
\(283\) −13.2610 7.65622i −0.788282 0.455115i 0.0510754 0.998695i \(-0.483735\pi\)
−0.839357 + 0.543580i \(0.817068\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −16.8881 + 26.0027i −0.996875 + 1.53489i
\(288\) 0 0
\(289\) 7.64591 + 13.2431i 0.449760 + 0.779006i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.9303 1.10592 0.552960 0.833208i \(-0.313498\pi\)
0.552960 + 0.833208i \(0.313498\pi\)
\(294\) 0 0
\(295\) −7.70139 −0.448392
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.93535 5.08417i −0.169755 0.294025i
\(300\) 0 0
\(301\) 3.15467 4.85725i 0.181832 0.279967i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.374192 0.216040i −0.0214262 0.0123704i
\(306\) 0 0
\(307\) 19.4828i 1.11194i −0.831202 0.555971i \(-0.812346\pi\)
0.831202 0.555971i \(-0.187654\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.06853 + 15.7072i −0.514229 + 0.890671i 0.485634 + 0.874162i \(0.338589\pi\)
−0.999864 + 0.0165093i \(0.994745\pi\)
\(312\) 0 0
\(313\) −16.2088 + 9.35817i −0.916177 + 0.528955i −0.882413 0.470475i \(-0.844083\pi\)
−0.0337634 + 0.999430i \(0.510749\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −29.4904 + 17.0263i −1.65634 + 0.956290i −0.681962 + 0.731388i \(0.738873\pi\)
−0.974381 + 0.224903i \(0.927794\pi\)
\(318\) 0 0
\(319\) −7.40207 + 12.8208i −0.414436 + 0.717824i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.38358i 0.132626i
\(324\) 0 0
\(325\) −3.85170 2.22378i −0.213654 0.123353i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.73178 + 0.353122i 0.371135 + 0.0194683i
\(330\) 0 0
\(331\) −8.76511 15.1816i −0.481774 0.834457i 0.518007 0.855376i \(-0.326674\pi\)
−0.999781 + 0.0209192i \(0.993341\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.48000 0.244769
\(336\) 0 0
\(337\) 9.53700 0.519513 0.259757 0.965674i \(-0.416358\pi\)
0.259757 + 0.965674i \(0.416358\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −4.45733 7.72032i −0.241378 0.418079i
\(342\) 0 0
\(343\) 18.2918 + 2.89985i 0.987666 + 0.156577i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.58768 2.64870i −0.246280 0.142190i 0.371780 0.928321i \(-0.378748\pi\)
−0.618060 + 0.786131i \(0.712081\pi\)
\(348\) 0 0
\(349\) 20.7945i 1.11310i −0.830813 0.556552i \(-0.812124\pi\)
0.830813 0.556552i \(-0.187876\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.79161 + 16.9596i −0.521155 + 0.902666i 0.478543 + 0.878064i \(0.341165\pi\)
−0.999697 + 0.0246021i \(0.992168\pi\)
\(354\) 0 0
\(355\) −3.97732 + 2.29631i −0.211094 + 0.121875i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −19.1760 + 11.0713i −1.01207 + 0.584319i −0.911797 0.410641i \(-0.865305\pi\)
−0.100273 + 0.994960i \(0.531972\pi\)
\(360\) 0 0
\(361\) −7.83697 + 13.5740i −0.412472 + 0.714423i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.55750i 0.0815231i
\(366\) 0 0
\(367\) 18.1684 + 10.4895i 0.948382 + 0.547549i 0.892578 0.450893i \(-0.148894\pi\)
0.0558039 + 0.998442i \(0.482228\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.68353 + 1.36716i −0.139322 + 0.0709797i
\(372\) 0 0
\(373\) 13.6507 + 23.6437i 0.706805 + 1.22422i 0.966036 + 0.258407i \(0.0831975\pi\)
−0.259231 + 0.965815i \(0.583469\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.44693 −0.332034
\(378\) 0 0
\(379\) 1.96174 0.100768 0.0503840 0.998730i \(-0.483955\pi\)
0.0503840 + 0.998730i \(0.483955\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.10367 + 10.5719i 0.311883 + 0.540197i 0.978770 0.204962i \(-0.0657070\pi\)
−0.666887 + 0.745159i \(0.732374\pi\)
\(384\) 0 0
\(385\) −3.04657 1.97867i −0.155267 0.100842i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −23.8845 13.7897i −1.21099 0.699167i −0.248017 0.968756i \(-0.579779\pi\)
−0.962976 + 0.269589i \(0.913112\pi\)
\(390\) 0 0
\(391\) 7.96175i 0.402643i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.80710 4.86204i 0.141240 0.244636i
\(396\) 0 0
\(397\) −28.7563 + 16.6025i −1.44324 + 0.833255i −0.998064 0.0621887i \(-0.980192\pi\)
−0.445175 + 0.895443i \(0.646859\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −15.1061 + 8.72153i −0.754364 + 0.435532i −0.827269 0.561807i \(-0.810107\pi\)
0.0729044 + 0.997339i \(0.476773\pi\)
\(402\) 0 0
\(403\) 1.94109 3.36206i 0.0966924 0.167476i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.732061i 0.0362869i
\(408\) 0 0
\(409\) −14.3281 8.27231i −0.708477 0.409039i 0.102020 0.994782i \(-0.467469\pi\)
−0.810497 + 0.585743i \(0.800803\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.9079 + 29.2619i 0.733571 + 1.43988i
\(414\) 0 0
\(415\) −2.74724 4.75836i −0.134857 0.233578i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.7781 1.11278 0.556392 0.830920i \(-0.312185\pi\)
0.556392 + 0.830920i \(0.312185\pi\)
\(420\) 0 0
\(421\) 15.6863 0.764504 0.382252 0.924058i \(-0.375149\pi\)
0.382252 + 0.924058i \(0.375149\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.01586 5.22363i −0.146291 0.253383i
\(426\) 0 0
\(427\) −0.0965172 + 1.83996i −0.00467079 + 0.0890421i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.9241 + 12.6579i 1.05604 + 0.609707i 0.924335 0.381581i \(-0.124620\pi\)
0.131709 + 0.991288i \(0.457954\pi\)
\(432\) 0 0
\(433\) 37.9831i 1.82535i 0.408686 + 0.912675i \(0.365987\pi\)
−0.408686 + 0.912675i \(0.634013\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.55491 + 9.62139i −0.265727 + 0.460253i
\(438\) 0 0
\(439\) 7.42813 4.28864i 0.354526 0.204685i −0.312151 0.950032i \(-0.601050\pi\)
0.666677 + 0.745347i \(0.267716\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.94380 + 5.16371i −0.424933 + 0.245335i −0.697186 0.716891i \(-0.745565\pi\)
0.272253 + 0.962226i \(0.412231\pi\)
\(444\) 0 0
\(445\) −2.61165 + 4.52351i −0.123804 + 0.214435i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.5336i 0.969041i −0.874780 0.484520i \(-0.838994\pi\)
0.874780 0.484520i \(-0.161006\pi\)
\(450\) 0 0
\(451\) −22.4594 12.9670i −1.05757 0.610591i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.0828707 1.57981i 0.00388504 0.0740629i
\(456\) 0 0
\(457\) 12.4978 + 21.6469i 0.584623 + 1.01260i 0.994922 + 0.100646i \(0.0320910\pi\)
−0.410299 + 0.911951i \(0.634576\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.6072 1.98442 0.992208 0.124596i \(-0.0397634\pi\)
0.992208 + 0.124596i \(0.0397634\pi\)
\(462\) 0 0
\(463\) 30.1942 1.40325 0.701623 0.712549i \(-0.252459\pi\)
0.701623 + 0.712549i \(0.252459\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.90935 10.2353i −0.273452 0.473633i 0.696291 0.717759i \(-0.254832\pi\)
−0.969743 + 0.244126i \(0.921499\pi\)
\(468\) 0 0
\(469\) −8.67213 17.0220i −0.400442 0.786005i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.19538 + 2.42220i 0.192904 + 0.111373i
\(474\) 0 0
\(475\) 8.41666i 0.386183i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.12346 + 7.14204i −0.188406 + 0.326328i −0.944719 0.327882i \(-0.893665\pi\)
0.756313 + 0.654210i \(0.226999\pi\)
\(480\) 0 0
\(481\) 0.276088 0.159400i 0.0125885 0.00726800i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6.66430 + 3.84763i −0.302610 + 0.174712i
\(486\) 0 0
\(487\) 12.5643 21.7620i 0.569344 0.986132i −0.427287 0.904116i \(-0.640531\pi\)
0.996631 0.0820163i \(-0.0261359\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.80970i 0.352447i −0.984350 0.176223i \(-0.943612\pi\)
0.984350 0.176223i \(-0.0563881\pi\)
\(492\) 0 0
\(493\) −7.57187 4.37162i −0.341020 0.196888i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.4240 + 10.6670i 0.736718 + 0.478481i
\(498\) 0 0
\(499\) −14.6425 25.3616i −0.655489 1.13534i −0.981771 0.190068i \(-0.939129\pi\)
0.326282 0.945273i \(-0.394204\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −5.32335 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(504\) 0 0
\(505\) 2.26439 0.100764
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.7275 25.5087i −0.652784 1.13066i −0.982444 0.186556i \(-0.940268\pi\)
0.329660 0.944100i \(-0.393066\pi\)
\(510\) 0 0
\(511\) 5.91780 3.01491i 0.261788 0.133372i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.93929 1.11965i −0.0854553 0.0493376i
\(516\) 0 0
\(517\) 5.63838i 0.247976i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.43563 16.3430i 0.413382 0.715999i −0.581875 0.813278i \(-0.697681\pi\)
0.995257 + 0.0972791i \(0.0310139\pi\)
\(522\) 0 0
\(523\) 22.1439 12.7848i 0.968285 0.559039i 0.0695719 0.997577i \(-0.477837\pi\)
0.898713 + 0.438537i \(0.144503\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.55958 2.63247i 0.198618 0.114672i
\(528\) 0 0
\(529\) 7.05476 12.2192i 0.306729 0.531270i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 11.2938i 0.489187i
\(534\) 0 0
\(535\) 7.54459 + 4.35587i 0.326181 + 0.188321i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.62071 + 15.4058i −0.0698091 + 0.663575i
\(540\) 0 0
\(541\) 16.3471 + 28.3141i 0.702819 + 1.21732i 0.967473 + 0.252974i \(0.0814088\pi\)
−0.264654 + 0.964343i \(0.585258\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.845227 0.0362055
\(546\) 0 0
\(547\) −41.0972 −1.75719 −0.878594 0.477570i \(-0.841518\pi\)
−0.878594 + 0.477570i \(0.841518\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.10015 + 10.5658i 0.259875 + 0.450117i
\(552\) 0 0
\(553\) −23.9074 1.25409i −1.01665 0.0533292i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.412224 0.237997i −0.0174665 0.0100843i 0.491241 0.871024i \(-0.336543\pi\)
−0.508708 + 0.860939i \(0.669877\pi\)
\(558\) 0 0
\(559\) 2.10965i 0.0892286i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.43275 + 12.8739i −0.313253 + 0.542570i −0.979065 0.203550i \(-0.934752\pi\)
0.665812 + 0.746120i \(0.268085\pi\)
\(564\) 0 0
\(565\) −8.66525 + 5.00288i −0.364550 + 0.210473i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0050 6.93112i 0.503278 0.290567i −0.226788 0.973944i \(-0.572823\pi\)
0.730066 + 0.683377i \(0.239489\pi\)
\(570\) 0 0
\(571\) −1.96776 + 3.40826i −0.0823482 + 0.142631i −0.904258 0.426986i \(-0.859575\pi\)
0.821910 + 0.569617i \(0.192909\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.1137i 1.17242i
\(576\) 0 0
\(577\) 26.6457 + 15.3839i 1.10928 + 0.640440i 0.938642 0.344894i \(-0.112085\pi\)
0.170634 + 0.985335i \(0.445419\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.7617 + 19.6493i −0.529445 + 0.815188i
\(582\) 0 0
\(583\) −1.25954 2.18159i −0.0521649 0.0903523i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 26.7214 1.10291 0.551455 0.834205i \(-0.314073\pi\)
0.551455 + 0.834205i \(0.314073\pi\)
\(588\) 0 0
\(589\) −7.34670 −0.302716
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.21759 + 5.57303i 0.132131 + 0.228857i 0.924498 0.381188i \(-0.124485\pi\)
−0.792367 + 0.610045i \(0.791151\pi\)
\(594\) 0 0
\(595\) 1.16859 1.79929i 0.0479076 0.0737635i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −39.0065 22.5204i −1.59376 0.920159i −0.992653 0.120992i \(-0.961392\pi\)
−0.601109 0.799167i \(-0.705274\pi\)
\(600\) 0 0
\(601\) 6.29317i 0.256704i 0.991729 + 0.128352i \(0.0409687\pi\)
−0.991729 + 0.128352i \(0.959031\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.89322 + 3.27916i −0.0769704 + 0.133317i
\(606\) 0 0
\(607\) −2.21834 + 1.28076i −0.0900396 + 0.0519844i −0.544344 0.838862i \(-0.683221\pi\)
0.454304 + 0.890847i \(0.349888\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.12645 + 1.22771i −0.0860269 + 0.0496677i
\(612\) 0 0
\(613\) 5.95779 10.3192i 0.240633 0.416788i −0.720262 0.693702i \(-0.755978\pi\)
0.960895 + 0.276914i \(0.0893117\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.0171i 0.685083i −0.939503 0.342541i \(-0.888712\pi\)
0.939503 0.342541i \(-0.111288\pi\)
\(618\) 0 0
\(619\) −26.8253 15.4876i −1.07820 0.622498i −0.147789 0.989019i \(-0.547216\pi\)
−0.930410 + 0.366521i \(0.880549\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 22.2428 + 1.16677i 0.891141 + 0.0467457i
\(624\) 0 0
\(625\) −9.68691 16.7782i −0.387476 0.671129i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.432351 0.0172390
\(630\) 0 0
\(631\) −5.21976 −0.207795 −0.103898 0.994588i \(-0.533131\pi\)
−0.103898 + 0.994588i \(0.533131\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.53343 4.38802i −0.100536 0.174133i
\(636\) 0 0
\(637\) −6.16302 + 2.74324i −0.244188 + 0.108691i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −20.0187 11.5578i −0.790693 0.456507i 0.0495138 0.998773i \(-0.484233\pi\)
−0.840206 + 0.542267i \(0.817566\pi\)
\(642\) 0 0
\(643\) 23.2939i 0.918623i −0.888275 0.459312i \(-0.848096\pi\)
0.888275 0.459312i \(-0.151904\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.2570 35.0862i 0.796386 1.37938i −0.125570 0.992085i \(-0.540076\pi\)
0.921956 0.387296i \(-0.126591\pi\)
\(648\) 0 0
\(649\) −23.7886 + 13.7344i −0.933786 + 0.539121i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.3425 + 5.97123i −0.404732 + 0.233672i −0.688524 0.725214i \(-0.741741\pi\)
0.283792 + 0.958886i \(0.408408\pi\)
\(654\) 0 0
\(655\) −1.55355 + 2.69083i −0.0607023 + 0.105139i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 43.4636i 1.69310i −0.532307 0.846552i \(-0.678675\pi\)
0.532307 0.846552i \(-0.321325\pi\)
\(660\) 0 0
\(661\) 43.1033 + 24.8857i 1.67653 + 0.967942i 0.963849 + 0.266450i \(0.0858506\pi\)
0.712677 + 0.701493i \(0.247483\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.66755 + 1.35902i −0.103443 + 0.0527006i
\(666\) 0 0
\(667\) −20.3760 35.2923i −0.788962 1.36652i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.54111 −0.0594939
\(672\) 0 0
\(673\) 32.8822 1.26751 0.633757 0.773532i \(-0.281512\pi\)
0.633757 + 0.773532i \(0.281512\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.7744 + 22.1260i 0.490961 + 0.850370i 0.999946 0.0104056i \(-0.00331226\pi\)
−0.508984 + 0.860776i \(0.669979\pi\)
\(678\) 0 0
\(679\) 27.5197 + 17.8734i 1.05611 + 0.685917i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 37.4260 + 21.6079i 1.43207 + 0.826803i 0.997278 0.0737300i \(-0.0234903\pi\)
0.434787 + 0.900533i \(0.356824\pi\)
\(684\) 0 0
\(685\) 2.92379i 0.111712i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.548508 0.950043i 0.0208965 0.0361938i
\(690\) 0 0
\(691\) 18.2836 10.5561i 0.695542 0.401571i −0.110143 0.993916i \(-0.535131\pi\)
0.805685 + 0.592344i \(0.201797\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.57293 3.21754i 0.211393 0.122048i
\(696\) 0 0
\(697\) 7.65822 13.2644i 0.290076 0.502426i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.7546i 1.27489i −0.770495 0.637446i \(-0.779991\pi\)
0.770495 0.637446i \(-0.220009\pi\)
\(702\) 0 0
\(703\) −0.522475 0.301651i −0.0197055 0.0113770i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.38328 8.60370i −0.164850 0.323576i
\(708\) 0 0
\(709\) 16.5489 + 28.6636i 0.621509 + 1.07648i 0.989205 + 0.146539i \(0.0468132\pi\)
−0.367696 + 0.929946i \(0.619853\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.5398 0.919022
\(714\) 0 0
\(715\) 1.32322 0.0494854
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.6731 34.0748i −0.733684 1.27078i −0.955298 0.295643i \(-0.904466\pi\)
0.221615 0.975134i \(-0.428867\pi\)
\(720\) 0 0
\(721\) −0.500210 + 9.53580i −0.0186288 + 0.355132i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −26.7370 15.4366i −0.992987 0.573301i
\(726\) 0 0
\(727\) 14.7164i 0.545803i 0.962042 + 0.272901i \(0.0879832\pi\)
−0.962042 + 0.272901i \(0.912017\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.43054 + 2.47777i −0.0529104 + 0.0916435i
\(732\) 0 0
\(733\) −39.3009 + 22.6904i −1.45161 + 0.838089i −0.998573 0.0534010i \(-0.982994\pi\)
−0.453040 + 0.891490i \(0.649661\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.8382 7.98947i 0.509735 0.294296i
\(738\) 0 0
\(739\) −4.90885 + 8.50237i −0.180575 + 0.312765i −0.942076 0.335398i \(-0.891129\pi\)
0.761502 + 0.648163i \(0.224462\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 20.2980i 0.744660i 0.928100 + 0.372330i \(0.121441\pi\)
−0.928100 + 0.372330i \(0.878559\pi\)
\(744\) 0 0
\(745\) −4.97465 2.87212i −0.182257 0.105226i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.94601 37.0980i 0.0711057 1.35553i
\(750\) 0 0
\(751\) −8.84315 15.3168i −0.322691 0.558918i 0.658351 0.752711i \(-0.271254\pi\)
−0.981042 + 0.193793i \(0.937921\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.7010 0.389450
\(756\) 0 0
\(757\) −24.2142 −0.880079 −0.440040 0.897978i \(-0.645036\pi\)
−0.440040 + 0.897978i \(0.645036\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.664665 + 1.15123i 0.0240941 + 0.0417322i 0.877821 0.478989i \(-0.158997\pi\)
−0.853727 + 0.520721i \(0.825663\pi\)
\(762\) 0 0
\(763\) −1.63614 3.21149i −0.0592323 0.116264i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.3595 5.98107i −0.374061 0.215964i
\(768\) 0 0
\(769\) 9.99518i 0.360435i −0.983627 0.180218i \(-0.942320\pi\)
0.983627 0.180218i \(-0.0576802\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 20.3473 35.2426i 0.731841 1.26759i −0.224254 0.974531i \(-0.571994\pi\)
0.956095 0.293056i \(-0.0946722\pi\)
\(774\) 0 0
\(775\) 16.1003 9.29552i 0.578340 0.333905i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.5092 + 10.6863i −0.663159 + 0.382875i
\(780\) 0 0
\(781\) −8.19030 + 14.1860i −0.293072 + 0.507615i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.57526i 0.0919148i
\(786\) 0 0
\(787\) −16.0048 9.24038i −0.570510 0.329384i 0.186843 0.982390i \(-0.440174\pi\)
−0.757353 + 0.653006i \(0.773508\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 35.7825 + 23.2399i 1.27228 + 0.826314i
\(792\) 0 0
\(793\) −0.335563 0.581212i −0.0119162 0.0206394i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.8741 1.27073 0.635363 0.772213i \(-0.280850\pi\)
0.635363 + 0.772213i \(0.280850\pi\)
\(798\) 0 0
\(799\) −3.33000 −0.117807
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.77758 + 4.81091i 0.0980187 + 0.169773i
\(804\) 0 0
\(805\) 8.91026 4.53946i 0.314045 0.159995i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.3089 + 20.9630i 1.27655 + 0.737018i 0.976213 0.216814i \(-0.0695665\pi\)
0.300340 + 0.953832i \(0.402900\pi\)
\(810\) 0 0
\(811\) 34.1402i 1.19883i −0.800440 0.599413i \(-0.795401\pi\)
0.800440 0.599413i \(-0.204599\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.95982 + 6.85861i −0.138706 + 0.240247i
\(816\) 0 0
\(817\) 3.45747 1.99617i 0.120962 0.0698372i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27.2818 15.7512i 0.952143 0.549720i 0.0583971 0.998293i \(-0.481401\pi\)
0.893746 + 0.448573i \(0.148068\pi\)
\(822\) 0 0
\(823\) −8.47768 + 14.6838i −0.295513 + 0.511844i −0.975104 0.221747i \(-0.928824\pi\)
0.679591 + 0.733591i \(0.262157\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.7684i 1.52198i −0.648766 0.760988i \(-0.724715\pi\)
0.648766 0.760988i \(-0.275285\pi\)
\(828\) 0 0
\(829\) 4.86284 + 2.80756i 0.168893 + 0.0975107i 0.582064 0.813143i \(-0.302245\pi\)
−0.413171 + 0.910654i \(0.635579\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −9.09859 0.957185i −0.315247 0.0331645i
\(834\) 0 0
\(835\) −6.36228 11.0198i −0.220176 0.381356i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.7401 1.26841 0.634204 0.773166i \(-0.281328\pi\)
0.634204 + 0.773166i \(0.281328\pi\)
\(840\) 0 0
\(841\) −15.7520 −0.543172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.74481 6.48620i −0.128825 0.223132i
\(846\) 0 0
\(847\) 16.1242 + 0.845809i 0.554033 + 0.0290623i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.74519 + 1.00759i 0.0598245 + 0.0345397i
\(852\) 0 0
\(853\) 35.6583i 1.22092i −0.792048 0.610459i \(-0.790985\pi\)
0.792048 0.610459i \(-0.209015\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.444720 + 0.770278i −0.0151914 + 0.0263122i −0.873521 0.486786i \(-0.838169\pi\)
0.858330 + 0.513098i \(0.171502\pi\)
\(858\) 0 0
\(859\) 20.1560 11.6371i 0.687715 0.397052i −0.115040 0.993361i \(-0.536700\pi\)
0.802755 + 0.596308i \(0.203366\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.72738 2.15201i 0.126882 0.0732551i −0.435216 0.900326i \(-0.643328\pi\)
0.562097 + 0.827071i \(0.309995\pi\)
\(864\) 0 0
\(865\) 2.36446 4.09537i 0.0803941 0.139247i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 20.0243i 0.679278i
\(870\) 0 0
\(871\) 6.02627 + 3.47927i 0.204192 + 0.117891i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.59702 13.2369i 0.290632 0.447488i
\(876\) 0 0
\(877\) 29.1826 + 50.5457i 0.985425 + 1.70681i 0.640034 + 0.768347i \(0.278920\pi\)
0.345391 + 0.938459i \(0.387746\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 13.9914 0.471381 0.235691 0.971828i \(-0.424265\pi\)
0.235691 + 0.971828i \(0.424265\pi\)
\(882\) 0 0
\(883\) −1.64531 −0.0553691 −0.0276845 0.999617i \(-0.508813\pi\)
−0.0276845 + 0.999617i \(0.508813\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −0.976428 1.69122i −0.0327852 0.0567857i 0.849167 0.528124i \(-0.177104\pi\)
−0.881952 + 0.471338i \(0.843771\pi\)
\(888\) 0 0
\(889\) −11.7685 + 18.1200i −0.394703 + 0.607725i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.02414 + 2.32334i 0.134663 + 0.0777475i
\(894\) 0 0
\(895\) 4.14642i 0.138599i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 13.4742 23.3381i 0.449391 0.778368i
\(900\) 0 0
\(901\) 1.28844 0.743879i 0.0429240 0.0247822i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.84130 + 1.64042i −0.0944479 + 0.0545295i
\(906\) 0 0
\(907\) −6.98547 + 12.0992i −0.231949 + 0.401747i −0.958382 0.285491i \(-0.907843\pi\)
0.726433 + 0.687237i \(0.241177\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.79122i 0.225003i −0.993652 0.112502i \(-0.964114\pi\)
0.993652 0.112502i \(-0.0358863\pi\)
\(912\) 0 0
\(913\) −16.9717 9.79864i −0.561683 0.324288i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.2312 + 0.694058i 0.436934 + 0.0229198i
\(918\) 0 0
\(919\) 0.292791 + 0.507128i 0.00965828 + 0.0167286i 0.870814 0.491612i \(-0.163592\pi\)
−0.861156 + 0.508341i \(0.830259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.13345 −0.234800
\(924\) 0 0
\(925\) 1.52667 0.0501967
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −18.4832 32.0139i −0.606415 1.05034i −0.991826 0.127596i \(-0.959274\pi\)
0.385411 0.922745i \(-0.374060\pi\)
\(930\) 0 0
\(931\) 10.3274 + 7.50479i 0.338466 + 0.245960i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.55411 + 0.897263i 0.0508247 + 0.0293436i
\(936\) 0 0
\(937\) 33.9658i 1.10961i 0.831979 + 0.554807i \(0.187208\pi\)
−0.831979 + 0.554807i \(0.812792\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −23.0968 + 40.0048i −0.752933 + 1.30412i 0.193462 + 0.981108i \(0.438028\pi\)
−0.946395 + 0.323011i \(0.895305\pi\)
\(942\) 0 0
\(943\) 61.8251 35.6947i 2.01330 1.16238i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.9927 12.1202i 0.682173 0.393853i −0.118500 0.992954i \(-0.537809\pi\)
0.800673 + 0.599101i \(0.204475\pi\)
\(948\) 0 0
\(949\) −1.20959 + 2.09506i −0.0392648 + 0.0680087i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 27.7820i 0.899947i 0.893042 + 0.449974i \(0.148567\pi\)
−0.893042 + 0.449974i \(0.851433\pi\)
\(954\) 0 0
\(955\) 4.00460 + 2.31206i 0.129586 + 0.0748164i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.1091 5.65971i 0.358733 0.182762i
\(960\) 0 0
\(961\) −7.38617 12.7932i −0.238263 0.412684i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −12.0928 −0.389283
\(966\) 0 0
\(967\) 7.92003 0.254691 0.127345 0.991858i \(-0.459354\pi\)
0.127345 + 0.991858i \(0.459354\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −0.806196 1.39637i −0.0258720 0.0448117i 0.852799 0.522238i \(-0.174903\pi\)
−0.878672 + 0.477427i \(0.841570\pi\)
\(972\) 0 0
\(973\) −23.0130 14.9464i −0.737763 0.479159i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11.4103 6.58776i −0.365049 0.210761i 0.306244 0.951953i \(-0.400928\pi\)
−0.671293 + 0.741192i \(0.734261\pi\)
\(978\) 0 0
\(979\) 18.6301i 0.595420i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.52198 + 2.63615i −0.0485437 + 0.0840802i −0.889276 0.457370i \(-0.848791\pi\)
0.840733 + 0.541451i \(0.182125\pi\)
\(984\) 0 0
\(985\) −1.40812 + 0.812981i −0.0448665 + 0.0259037i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.5488 + 6.66770i −0.367230 + 0.212021i
\(990\) 0 0
\(991\) −12.7482 + 22.0806i −0.404961 + 0.701414i −0.994317 0.106460i \(-0.966048\pi\)
0.589356 + 0.807874i \(0.299382\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.08691i 0.161266i
\(996\) 0 0
\(997\) −5.34395 3.08533i −0.169245 0.0977135i 0.412985 0.910738i \(-0.364486\pi\)
−0.582230 + 0.813024i \(0.697820\pi\)
\(998\) 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 1512.2.bl.d.1025.4 yes 16
3.2 odd 2 inner 1512.2.bl.d.1025.5 yes 16
7.5 odd 6 inner 1512.2.bl.d.593.5 yes 16
21.5 even 6 inner 1512.2.bl.d.593.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.bl.d.593.4 16 21.5 even 6 inner
1512.2.bl.d.593.5 yes 16 7.5 odd 6 inner
1512.2.bl.d.1025.4 yes 16 1.1 even 1 trivial
1512.2.bl.d.1025.5 yes 16 3.2 odd 2 inner