Properties

Label 3240.2.q.f.2161.1
Level $3240$
Weight $2$
Character 3240.2161
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,2,Mod(1081,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1081");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (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: \(25.8715302549\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2161.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.2161
Dual form 3240.2.q.f.1081.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{11} -7.00000 q^{19} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-3.50000 + 6.06218i) q^{29} +(-0.500000 - 0.866025i) q^{31} -2.00000 q^{37} +(4.50000 + 7.79423i) q^{41} +(3.00000 - 5.19615i) q^{43} +(-1.00000 + 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} -1.00000 q^{55} +(1.50000 + 2.59808i) q^{59} +(5.00000 - 8.66025i) q^{61} +(1.00000 + 1.73205i) q^{67} -1.00000 q^{71} +(-2.00000 + 3.46410i) q^{79} +(-3.00000 + 5.19615i) q^{83} +7.00000 q^{89} +(3.50000 + 6.06218i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + q^{11} - 14 q^{19} + 6 q^{23} - q^{25} - 7 q^{29} - q^{31} - 4 q^{37} + 9 q^{41} + 6 q^{43} - 2 q^{47} + 7 q^{49} - 2 q^{55} + 3 q^{59} + 10 q^{61} + 2 q^{67} - 2 q^{71} - 4 q^{79} - 6 q^{83} + 14 q^{89} + 7 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.500000 0.866025i 0.150756 0.261116i −0.780750 0.624844i \(-0.785163\pi\)
0.931505 + 0.363727i \(0.118496\pi\)
\(12\) 0 0
\(13\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 + 5.19615i 0.625543 + 1.08347i 0.988436 + 0.151642i \(0.0484560\pi\)
−0.362892 + 0.931831i \(0.618211\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.50000 + 6.06218i −0.649934 + 1.12572i 0.333205 + 0.942855i \(0.391870\pi\)
−0.983138 + 0.182864i \(0.941463\pi\)
\(30\) 0 0
\(31\) −0.500000 0.866025i −0.0898027 0.155543i 0.817625 0.575751i \(-0.195290\pi\)
−0.907428 + 0.420208i \(0.861957\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.50000 + 7.79423i 0.702782 + 1.21725i 0.967486 + 0.252924i \(0.0813924\pi\)
−0.264704 + 0.964330i \(0.585274\pi\)
\(42\) 0 0
\(43\) 3.00000 5.19615i 0.457496 0.792406i −0.541332 0.840809i \(-0.682080\pi\)
0.998828 + 0.0484030i \(0.0154132\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.00000 + 1.73205i −0.145865 + 0.252646i −0.929695 0.368329i \(-0.879930\pi\)
0.783830 + 0.620975i \(0.213263\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.50000 + 2.59808i 0.195283 + 0.338241i 0.946993 0.321253i \(-0.104104\pi\)
−0.751710 + 0.659494i \(0.770771\pi\)
\(60\) 0 0
\(61\) 5.00000 8.66025i 0.640184 1.10883i −0.345207 0.938527i \(-0.612191\pi\)
0.985391 0.170305i \(-0.0544754\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 + 1.73205i 0.122169 + 0.211604i 0.920623 0.390453i \(-0.127682\pi\)
−0.798454 + 0.602056i \(0.794348\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −2.00000 + 3.46410i −0.225018 + 0.389742i −0.956325 0.292306i \(-0.905577\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 + 5.19615i −0.329293 + 0.570352i −0.982372 0.186938i \(-0.940144\pi\)
0.653079 + 0.757290i \(0.273477\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.00000 0.741999 0.370999 0.928633i \(-0.379015\pi\)
0.370999 + 0.928633i \(0.379015\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.50000 + 6.06218i 0.359092 + 0.621966i
\(96\) 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
\(99\) 0 0
\(100\) 0 0
\(101\) −4.50000 + 7.79423i −0.447767 + 0.775555i −0.998240 0.0592978i \(-0.981114\pi\)
0.550474 + 0.834853i \(0.314447\pi\)
\(102\) 0 0
\(103\) 3.00000 + 5.19615i 0.295599 + 0.511992i 0.975124 0.221660i \(-0.0711475\pi\)
−0.679525 + 0.733652i \(0.737814\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(114\) 0 0
\(115\) 3.00000 5.19615i 0.279751 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 + 8.66025i 0.454545 + 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.50000 + 9.52628i 0.480537 + 0.832315i 0.999751 0.0223297i \(-0.00710836\pi\)
−0.519213 + 0.854645i \(0.673775\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 + 17.3205i −0.854358 + 1.47979i 0.0228820 + 0.999738i \(0.492716\pi\)
−0.877240 + 0.480053i \(0.840618\pi\)
\(138\) 0 0
\(139\) −3.50000 6.06218i −0.296866 0.514187i 0.678551 0.734553i \(-0.262608\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 7.00000 0.581318
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 + 8.66025i 0.409616 + 0.709476i 0.994847 0.101391i \(-0.0323294\pi\)
−0.585231 + 0.810867i \(0.698996\pi\)
\(150\) 0 0
\(151\) 5.50000 9.52628i 0.447584 0.775238i −0.550645 0.834740i \(-0.685618\pi\)
0.998228 + 0.0595022i \(0.0189513\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.500000 + 0.866025i −0.0401610 + 0.0695608i
\(156\) 0 0
\(157\) 9.00000 + 15.5885i 0.718278 + 1.24409i 0.961681 + 0.274169i \(0.0884028\pi\)
−0.243403 + 0.969925i \(0.578264\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.0000 + 17.3205i 0.773823 + 1.34030i 0.935454 + 0.353450i \(0.114991\pi\)
−0.161630 + 0.986851i \(0.551675\pi\)
\(168\) 0 0
\(169\) 6.50000 11.2583i 0.500000 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −7.00000 + 12.1244i −0.532200 + 0.921798i 0.467093 + 0.884208i \(0.345301\pi\)
−0.999293 + 0.0375896i \(0.988032\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00000 + 1.73205i 0.0735215 + 0.127343i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.50000 7.79423i 0.325609 0.563971i −0.656027 0.754738i \(-0.727764\pi\)
0.981635 + 0.190767i \(0.0610975\pi\)
\(192\) 0 0
\(193\) 7.00000 + 12.1244i 0.503871 + 0.872730i 0.999990 + 0.00447566i \(0.00142465\pi\)
−0.496119 + 0.868255i \(0.665242\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.0000 1.42494 0.712470 0.701702i \(-0.247576\pi\)
0.712470 + 0.701702i \(0.247576\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.50000 7.79423i 0.314294 0.544373i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.50000 + 6.06218i −0.242100 + 0.419330i
\(210\) 0 0
\(211\) −4.50000 7.79423i −0.309793 0.536577i 0.668524 0.743690i \(-0.266926\pi\)
−0.978317 + 0.207114i \(0.933593\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −10.0000 + 17.3205i −0.669650 + 1.15987i 0.308353 + 0.951272i \(0.400222\pi\)
−0.978002 + 0.208595i \(0.933111\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −10.0000 + 17.3205i −0.663723 + 1.14960i 0.315906 + 0.948790i \(0.397691\pi\)
−0.979630 + 0.200812i \(0.935642\pi\)
\(228\) 0 0
\(229\) 5.00000 + 8.66025i 0.330409 + 0.572286i 0.982592 0.185776i \(-0.0594799\pi\)
−0.652183 + 0.758062i \(0.726147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 2.00000 0.130466
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) 5.50000 9.52628i 0.354286 0.613642i −0.632709 0.774389i \(-0.718057\pi\)
0.986996 + 0.160748i \(0.0513906\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.50000 6.06218i 0.223607 0.387298i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.0000 19.0526i −0.686161 1.18847i −0.973070 0.230508i \(-0.925961\pi\)
0.286909 0.957958i \(-0.407372\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.00000 5.19615i 0.184988 0.320408i −0.758585 0.651575i \(-0.774109\pi\)
0.943572 + 0.331166i \(0.107442\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.500000 + 0.866025i 0.0301511 + 0.0522233i
\(276\) 0 0
\(277\) −11.0000 + 19.0526i −0.660926 + 1.14476i 0.319447 + 0.947604i \(0.396503\pi\)
−0.980373 + 0.197153i \(0.936830\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.00000 8.66025i 0.298275 0.516627i −0.677466 0.735554i \(-0.736922\pi\)
0.975741 + 0.218926i \(0.0702554\pi\)
\(282\) 0 0
\(283\) −13.0000 22.5167i −0.772770 1.33848i −0.936039 0.351895i \(-0.885537\pi\)
0.163270 0.986581i \(-0.447796\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.00000 + 12.1244i 0.408944 + 0.708312i 0.994772 0.102123i \(-0.0325637\pi\)
−0.585827 + 0.810436i \(0.699230\pi\)
\(294\) 0 0
\(295\) 1.50000 2.59808i 0.0873334 0.151266i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.5000 + 26.8468i 0.878924 + 1.52234i 0.852523 + 0.522690i \(0.175072\pi\)
0.0264017 + 0.999651i \(0.491595\pi\)
\(312\) 0 0
\(313\) 10.0000 17.3205i 0.565233 0.979013i −0.431795 0.901972i \(-0.642119\pi\)
0.997028 0.0770410i \(-0.0245472\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.0000 19.0526i 0.617822 1.07010i −0.372061 0.928208i \(-0.621349\pi\)
0.989882 0.141890i \(-0.0453179\pi\)
\(318\) 0 0
\(319\) 3.50000 + 6.06218i 0.195962 + 0.339417i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 15.5000 26.8468i 0.851957 1.47563i −0.0274825 0.999622i \(-0.508749\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 1.73205i 0.0546358 0.0946320i
\(336\) 0 0
\(337\) −10.0000 17.3205i −0.544735 0.943508i −0.998624 0.0524499i \(-0.983297\pi\)
0.453889 0.891058i \(-0.350036\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.00000 −0.0541530
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(348\) 0 0
\(349\) −8.50000 + 14.7224i −0.454995 + 0.788074i −0.998688 0.0512103i \(-0.983692\pi\)
0.543693 + 0.839284i \(0.317025\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.00000 + 12.1244i −0.372572 + 0.645314i −0.989960 0.141344i \(-0.954858\pi\)
0.617388 + 0.786659i \(0.288191\pi\)
\(354\) 0 0
\(355\) 0.500000 + 0.866025i 0.0265372 + 0.0459639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −11.0000 + 19.0526i −0.574195 + 0.994535i 0.421933 + 0.906627i \(0.361352\pi\)
−0.996129 + 0.0879086i \(0.971982\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 + 17.3205i 0.517780 + 0.896822i 0.999787 + 0.0206542i \(0.00657489\pi\)
−0.482006 + 0.876168i \(0.660092\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.0000 + 25.9808i −0.760530 + 1.31728i 0.182047 + 0.983290i \(0.441728\pi\)
−0.942578 + 0.333987i \(0.891606\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.00000 + 1.73205i 0.0499376 + 0.0864945i 0.889914 0.456129i \(-0.150764\pi\)
−0.839976 + 0.542623i \(0.817431\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.00000 + 1.73205i −0.0495682 + 0.0858546i
\(408\) 0 0
\(409\) 5.00000 + 8.66025i 0.247234 + 0.428222i 0.962757 0.270367i \(-0.0871450\pi\)
−0.715523 + 0.698589i \(0.753812\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 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
\(421\) 15.5000 26.8468i 0.755424 1.30843i −0.189740 0.981834i \(-0.560764\pi\)
0.945163 0.326598i \(-0.105902\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.0000 −0.915198 −0.457599 0.889159i \(-0.651290\pi\)
−0.457599 + 0.889159i \(0.651290\pi\)
\(432\) 0 0
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −21.0000 36.3731i −1.00457 1.73996i
\(438\) 0 0
\(439\) −18.5000 + 32.0429i −0.882957 + 1.52933i −0.0349192 + 0.999390i \(0.511117\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.0000 29.4449i 0.807694 1.39897i −0.106763 0.994285i \(-0.534048\pi\)
0.914457 0.404683i \(-0.132618\pi\)
\(444\) 0 0
\(445\) −3.50000 6.06218i −0.165916 0.287375i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.0000 1.55737 0.778683 0.627417i \(-0.215888\pi\)
0.778683 + 0.627417i \(0.215888\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.0000 + 32.9090i −0.888783 + 1.53942i −0.0474665 + 0.998873i \(0.515115\pi\)
−0.841316 + 0.540544i \(0.818219\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.50000 16.4545i 0.442459 0.766362i −0.555412 0.831575i \(-0.687440\pi\)
0.997871 + 0.0652135i \(0.0207728\pi\)
\(462\) 0 0
\(463\) −7.00000 12.1244i −0.325318 0.563467i 0.656259 0.754536i \(-0.272138\pi\)
−0.981577 + 0.191069i \(0.938805\pi\)
\(464\) 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
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 5.19615i −0.137940 0.238919i
\(474\) 0 0
\(475\) 3.50000 6.06218i 0.160591 0.278152i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.50000 14.7224i 0.388375 0.672685i −0.603856 0.797093i \(-0.706370\pi\)
0.992231 + 0.124408i \(0.0397032\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4.50000 7.79423i −0.203082 0.351749i 0.746438 0.665455i \(-0.231763\pi\)
−0.949520 + 0.313707i \(0.898429\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −18.5000 32.0429i −0.828174 1.43444i −0.899469 0.436984i \(-0.856047\pi\)
0.0712957 0.997455i \(-0.477287\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −40.0000 −1.78351 −0.891756 0.452517i \(-0.850526\pi\)
−0.891756 + 0.452517i \(0.850526\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.00000 8.66025i −0.221621 0.383859i 0.733679 0.679496i \(-0.237801\pi\)
−0.955300 + 0.295637i \(0.904468\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.00000 5.19615i 0.132196 0.228970i
\(516\) 0 0
\(517\) 1.00000 + 1.73205i 0.0439799 + 0.0761755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 + 11.2583i −0.282609 + 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.00000 1.73205i −0.0432338 0.0748831i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) 39.0000 1.67674 0.838370 0.545101i \(-0.183509\pi\)
0.838370 + 0.545101i \(0.183509\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.50000 2.59808i −0.0642529 0.111289i
\(546\) 0 0
\(547\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 24.5000 42.4352i 1.04374 1.80780i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −17.0000 29.4449i −0.716465 1.24095i −0.962392 0.271665i \(-0.912426\pi\)
0.245927 0.969288i \(-0.420908\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.50000 2.59808i 0.0628833 0.108917i −0.832870 0.553469i \(-0.813304\pi\)
0.895753 + 0.444552i \(0.146637\pi\)
\(570\) 0 0
\(571\) −19.5000 33.7750i −0.816050 1.41344i −0.908572 0.417729i \(-0.862826\pi\)
0.0925222 0.995711i \(-0.470507\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 20.0000 0.832611 0.416305 0.909225i \(-0.363325\pi\)
0.416305 + 0.909225i \(0.363325\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.0000 + 32.9090i −0.784214 + 1.35830i 0.145254 + 0.989394i \(0.453600\pi\)
−0.929468 + 0.368904i \(0.879733\pi\)
\(588\) 0 0
\(589\) 3.50000 + 6.06218i 0.144215 + 0.249788i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.50000 14.7224i −0.347301 0.601542i 0.638468 0.769648i \(-0.279568\pi\)
−0.985769 + 0.168106i \(0.946235\pi\)
\(600\) 0 0
\(601\) 9.50000 16.4545i 0.387513 0.671192i −0.604601 0.796528i \(-0.706668\pi\)
0.992114 + 0.125336i \(0.0400009\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 8.66025i 0.203279 0.352089i
\(606\) 0 0
\(607\) 19.0000 + 32.9090i 0.771186 + 1.33573i 0.936913 + 0.349562i \(0.113670\pi\)
−0.165727 + 0.986172i \(0.552997\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −10.0000 17.3205i −0.402585 0.697297i 0.591452 0.806340i \(-0.298555\pi\)
−0.994037 + 0.109043i \(0.965221\pi\)
\(618\) 0 0
\(619\) −10.0000 + 17.3205i −0.401934 + 0.696170i −0.993959 0.109749i \(-0.964995\pi\)
0.592025 + 0.805919i \(0.298329\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 41.0000 1.63218 0.816092 0.577922i \(-0.196136\pi\)
0.816092 + 0.577922i \(0.196136\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.0000 + 17.3205i 0.396838 + 0.687343i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.5000 18.1865i 0.414725 0.718325i −0.580674 0.814136i \(-0.697211\pi\)
0.995400 + 0.0958109i \(0.0305444\pi\)
\(642\) 0 0
\(643\) 7.00000 + 12.1244i 0.276053 + 0.478138i 0.970400 0.241502i \(-0.0776401\pi\)
−0.694347 + 0.719640i \(0.744307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.0000 17.3205i −0.391330 0.677804i 0.601295 0.799027i \(-0.294652\pi\)
−0.992625 + 0.121223i \(0.961318\pi\)
\(654\) 0 0
\(655\) 5.50000 9.52628i 0.214903 0.372223i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.0000 17.3205i 0.389545 0.674711i −0.602844 0.797859i \(-0.705966\pi\)
0.992388 + 0.123148i \(0.0392990\pi\)
\(660\) 0 0
\(661\) −10.5000 18.1865i −0.408403 0.707374i 0.586308 0.810088i \(-0.300581\pi\)
−0.994711 + 0.102714i \(0.967247\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −42.0000 −1.62625
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.00000 8.66025i −0.193023 0.334325i
\(672\) 0 0
\(673\) −23.0000 + 39.8372i −0.886585 + 1.53561i −0.0426985 + 0.999088i \(0.513595\pi\)
−0.843886 + 0.536522i \(0.819738\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.00000 + 1.73205i −0.0384331 + 0.0665681i −0.884602 0.466347i \(-0.845570\pi\)
0.846169 + 0.532915i \(0.178903\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.0000 1.53056 0.765279 0.643699i \(-0.222601\pi\)
0.765279 + 0.643699i \(0.222601\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 17.3205i 0.380418 0.658903i −0.610704 0.791859i \(-0.709113\pi\)
0.991122 + 0.132956i \(0.0424468\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.50000 + 6.06218i −0.132763 + 0.229952i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.00000 −0.339925 −0.169963 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.00000 5.19615i 0.112351 0.194597i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −27.0000 −1.00693 −0.503465 0.864016i \(-0.667942\pi\)
−0.503465 + 0.864016i \(0.667942\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.50000 6.06218i −0.129987 0.225144i
\(726\) 0 0
\(727\) −20.0000 + 34.6410i −0.741759 + 1.28476i 0.209935 + 0.977715i \(0.432675\pi\)
−0.951694 + 0.307049i \(0.900659\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −23.0000 39.8372i −0.849524 1.47142i −0.881633 0.471935i \(-0.843556\pi\)
0.0321090 0.999484i \(-0.489778\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00000 0.0736709
\(738\) 0 0
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.0000 17.3205i −0.366864 0.635428i 0.622209 0.782851i \(-0.286235\pi\)
−0.989073 + 0.147423i \(0.952902\pi\)
\(744\) 0 0
\(745\) 5.00000 8.66025i 0.183186 0.317287i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 20.0000 + 34.6410i 0.729810 + 1.26407i 0.956963 + 0.290209i \(0.0937250\pi\)
−0.227153 + 0.973859i \(0.572942\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.0000 −0.400331
\(756\) 0 0
\(757\) −20.0000 −0.726912 −0.363456 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 9.50000 + 16.4545i 0.344375 + 0.596475i 0.985240 0.171179i \(-0.0547576\pi\)
−0.640865 + 0.767653i \(0.721424\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.50000 + 2.59808i 0.0540914 + 0.0936890i 0.891803 0.452423i \(-0.149440\pi\)
−0.837712 + 0.546113i \(0.816107\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −31.5000 54.5596i −1.12860 1.95480i
\(780\) 0 0
\(781\) −0.500000 + 0.866025i −0.0178914 + 0.0309888i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.00000 15.5885i 0.321224 0.556376i
\(786\) 0 0
\(787\) −20.0000 34.6410i −0.712923 1.23482i −0.963755 0.266788i \(-0.914038\pi\)
0.250832 0.968031i \(-0.419296\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −20.0000 34.6410i −0.708436 1.22705i −0.965437 0.260637i \(-0.916068\pi\)
0.257001 0.966411i \(-0.417266\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\)