Properties

Label 2548.2.y.e.961.4
Level $2548$
Weight $2$
Character 2548.961
Analytic conductor $20.346$
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] = [2548,2,Mod(753,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.753");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2548.y (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.3458824350\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.11348687176217973595570176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 16x^{14} + 176x^{12} - 1016x^{10} + 4224x^{8} - 8512x^{6} + 12304x^{4} - 8448x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 961.4
Root \(1.12031 + 0.646813i\) of defining polynomial
Character \(\chi\) \(=\) 2548.961
Dual form 2548.2.y.e.753.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.646813 - 1.12031i) q^{3} +(2.42353 + 1.39923i) q^{5} +(0.663266 - 1.14881i) q^{9} +O(q^{10})\) \(q+(-0.646813 - 1.12031i) q^{3} +(2.42353 + 1.39923i) q^{5} +(0.663266 - 1.14881i) q^{9} +(-0.894522 + 0.516453i) q^{11} +(1.79846 - 3.12499i) q^{13} -3.62016i q^{15} +(-3.47818 - 6.02438i) q^{17} +(2.19774 + 1.26887i) q^{19} +(1.91568 - 3.31806i) q^{23} +(1.41568 + 2.45203i) q^{25} -5.59691 q^{27} -5.10175 q^{29} +(-3.28119 + 1.89440i) q^{31} +(1.15718 + 0.668097i) q^{33} +(-5.37577 - 3.10370i) q^{37} +(-4.66423 + 0.00644825i) q^{39} +0.990338i q^{41} +0.560979 q^{43} +(3.21489 - 1.85612i) q^{45} +(6.22741 + 3.59540i) q^{47} +(-4.49946 + 7.79329i) q^{51} +(-0.589150 - 1.02044i) q^{53} -2.89054 q^{55} -3.28288i q^{57} +(10.3260 - 5.96172i) q^{59} +(6.92345 - 11.9918i) q^{61} +(8.73120 - 5.05707i) q^{65} +(-11.9348 + 6.89054i) q^{67} -4.95635 q^{69} -3.80432i q^{71} +(2.76331 - 1.59540i) q^{73} +(1.83136 - 3.17201i) q^{75} +(-3.66620 + 6.35004i) q^{79} +(1.63036 + 2.82387i) q^{81} -16.6260i q^{83} -19.4670i q^{85} +(3.29988 + 5.71555i) q^{87} +(10.3026 + 5.94820i) q^{89} +(4.24464 + 2.45064i) q^{93} +(3.55087 + 6.15029i) q^{95} -8.60355i q^{97} +1.37018i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} - 4 q^{13} + 4 q^{17} + 6 q^{23} - 2 q^{25} - 24 q^{27} - 4 q^{29} - 12 q^{43} + 8 q^{51} - 22 q^{53} + 40 q^{55} + 8 q^{61} + 6 q^{65} + 40 q^{69} - 20 q^{75} + 26 q^{79} + 24 q^{81} - 32 q^{87} + 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) −0.646813 1.12031i −0.373438 0.646813i 0.616654 0.787234i \(-0.288488\pi\)
−0.990092 + 0.140421i \(0.955154\pi\)
\(4\) 0 0
\(5\) 2.42353 + 1.39923i 1.08384 + 0.625754i 0.931929 0.362641i \(-0.118125\pi\)
0.151909 + 0.988395i \(0.451458\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.663266 1.14881i 0.221089 0.382937i
\(10\) 0 0
\(11\) −0.894522 + 0.516453i −0.269709 + 0.155716i −0.628755 0.777603i \(-0.716435\pi\)
0.359047 + 0.933320i \(0.383102\pi\)
\(12\) 0 0
\(13\) 1.79846 3.12499i 0.498802 0.866716i
\(14\) 0 0
\(15\) 3.62016i 0.934721i
\(16\) 0 0
\(17\) −3.47818 6.02438i −0.843581 1.46113i −0.886847 0.462062i \(-0.847110\pi\)
0.0432661 0.999064i \(-0.486224\pi\)
\(18\) 0 0
\(19\) 2.19774 + 1.26887i 0.504197 + 0.291098i 0.730445 0.682971i \(-0.239313\pi\)
−0.226248 + 0.974070i \(0.572646\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.91568 3.31806i 0.399447 0.691863i −0.594211 0.804309i \(-0.702535\pi\)
0.993658 + 0.112447i \(0.0358688\pi\)
\(24\) 0 0
\(25\) 1.41568 + 2.45203i 0.283136 + 0.490406i
\(26\) 0 0
\(27\) −5.59691 −1.07713
\(28\) 0 0
\(29\) −5.10175 −0.947370 −0.473685 0.880694i \(-0.657077\pi\)
−0.473685 + 0.880694i \(0.657077\pi\)
\(30\) 0 0
\(31\) −3.28119 + 1.89440i −0.589320 + 0.340244i −0.764828 0.644234i \(-0.777176\pi\)
0.175509 + 0.984478i \(0.443843\pi\)
\(32\) 0 0
\(33\) 1.15718 + 0.668097i 0.201439 + 0.116301i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.37577 3.10370i −0.883772 0.510246i −0.0118717 0.999930i \(-0.503779\pi\)
−0.871900 + 0.489684i \(0.837112\pi\)
\(38\) 0 0
\(39\) −4.66423 + 0.00644825i −0.746875 + 0.00103255i
\(40\) 0 0
\(41\) 0.990338i 0.154665i 0.997005 + 0.0773324i \(0.0246403\pi\)
−0.997005 + 0.0773324i \(0.975360\pi\)
\(42\) 0 0
\(43\) 0.560979 0.0855485 0.0427743 0.999085i \(-0.486380\pi\)
0.0427743 + 0.999085i \(0.486380\pi\)
\(44\) 0 0
\(45\) 3.21489 1.85612i 0.479248 0.276694i
\(46\) 0 0
\(47\) 6.22741 + 3.59540i 0.908362 + 0.524443i 0.879904 0.475152i \(-0.157607\pi\)
0.0284580 + 0.999595i \(0.490940\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.49946 + 7.79329i −0.630050 + 1.09128i
\(52\) 0 0
\(53\) −0.589150 1.02044i −0.0809260 0.140168i 0.822722 0.568444i \(-0.192454\pi\)
−0.903648 + 0.428276i \(0.859121\pi\)
\(54\) 0 0
\(55\) −2.89054 −0.389760
\(56\) 0 0
\(57\) 3.28288i 0.434828i
\(58\) 0 0
\(59\) 10.3260 5.96172i 1.34433 0.776150i 0.356892 0.934146i \(-0.383837\pi\)
0.987440 + 0.157996i \(0.0505032\pi\)
\(60\) 0 0
\(61\) 6.92345 11.9918i 0.886456 1.53539i 0.0424210 0.999100i \(-0.486493\pi\)
0.844035 0.536288i \(-0.180174\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.73120 5.05707i 1.08297 0.627252i
\(66\) 0 0
\(67\) −11.9348 + 6.89054i −1.45806 + 0.841813i −0.998916 0.0465465i \(-0.985178\pi\)
−0.459148 + 0.888360i \(0.651845\pi\)
\(68\) 0 0
\(69\) −4.95635 −0.596674
\(70\) 0 0
\(71\) 3.80432i 0.451490i −0.974186 0.225745i \(-0.927518\pi\)
0.974186 0.225745i \(-0.0724816\pi\)
\(72\) 0 0
\(73\) 2.76331 1.59540i 0.323421 0.186727i −0.329495 0.944157i \(-0.606879\pi\)
0.652917 + 0.757430i \(0.273545\pi\)
\(74\) 0 0
\(75\) 1.83136 3.17201i 0.211467 0.366272i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −3.66620 + 6.35004i −0.412480 + 0.714436i −0.995160 0.0982656i \(-0.968671\pi\)
0.582681 + 0.812701i \(0.302004\pi\)
\(80\) 0 0
\(81\) 1.63036 + 2.82387i 0.181151 + 0.313763i
\(82\) 0 0
\(83\) 16.6260i 1.82494i −0.409140 0.912472i \(-0.634171\pi\)
0.409140 0.912472i \(-0.365829\pi\)
\(84\) 0 0
\(85\) 19.4670i 2.11150i
\(86\) 0 0
\(87\) 3.29988 + 5.71555i 0.353784 + 0.612772i
\(88\) 0 0
\(89\) 10.3026 + 5.94820i 1.09207 + 0.630508i 0.934127 0.356940i \(-0.116180\pi\)
0.157945 + 0.987448i \(0.449513\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.24464 + 2.45064i 0.440148 + 0.254120i
\(94\) 0 0
\(95\) 3.55087 + 6.15029i 0.364312 + 0.631007i
\(96\) 0 0
\(97\) 8.60355i 0.873558i −0.899569 0.436779i \(-0.856119\pi\)
0.899569 0.436779i \(-0.143881\pi\)
\(98\) 0 0
\(99\) 1.37018i 0.137708i
\(100\) 0 0
\(101\) 4.18455 + 7.24785i 0.416378 + 0.721188i 0.995572 0.0940017i \(-0.0299659\pi\)
−0.579194 + 0.815190i \(0.696633\pi\)
\(102\) 0 0
\(103\) 0.260721 0.451582i 0.0256896 0.0444957i −0.852895 0.522083i \(-0.825155\pi\)
0.878584 + 0.477587i \(0.158489\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.71761 + 8.17115i −0.456069 + 0.789934i −0.998749 0.0500050i \(-0.984076\pi\)
0.542680 + 0.839939i \(0.317410\pi\)
\(108\) 0 0
\(109\) 11.8524 6.84298i 1.13525 0.655439i 0.190003 0.981784i \(-0.439150\pi\)
0.945251 + 0.326345i \(0.105817\pi\)
\(110\) 0 0
\(111\) 8.03007i 0.762180i
\(112\) 0 0
\(113\) −12.2655 −1.15384 −0.576921 0.816800i \(-0.695746\pi\)
−0.576921 + 0.816800i \(0.695746\pi\)
\(114\) 0 0
\(115\) 9.28544 5.36095i 0.865872 0.499911i
\(116\) 0 0
\(117\) −2.39716 4.13878i −0.221618 0.382631i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −4.96655 + 8.60232i −0.451505 + 0.782029i
\(122\) 0 0
\(123\) 1.10949 0.640564i 0.100039 0.0577577i
\(124\) 0 0
\(125\) 6.06884i 0.542814i
\(126\) 0 0
\(127\) −10.9331 −0.970156 −0.485078 0.874471i \(-0.661209\pi\)
−0.485078 + 0.874471i \(0.661209\pi\)
\(128\) 0 0
\(129\) −0.362849 0.628472i −0.0319470 0.0553339i
\(130\) 0 0
\(131\) −3.29363 + 5.70473i −0.287765 + 0.498424i −0.973276 0.229638i \(-0.926246\pi\)
0.685511 + 0.728063i \(0.259579\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −13.5643 7.83136i −1.16743 0.674016i
\(136\) 0 0
\(137\) 11.1140 6.41665i 0.949531 0.548212i 0.0565956 0.998397i \(-0.481975\pi\)
0.892935 + 0.450185i \(0.148642\pi\)
\(138\) 0 0
\(139\) 18.5998 1.57761 0.788805 0.614643i \(-0.210700\pi\)
0.788805 + 0.614643i \(0.210700\pi\)
\(140\) 0 0
\(141\) 9.30221i 0.783387i
\(142\) 0 0
\(143\) 0.00514865 + 3.72419i 0.000430552 + 0.311432i
\(144\) 0 0
\(145\) −12.3643 7.13851i −1.02680 0.592821i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.64012 4.98838i −0.707827 0.408664i 0.102429 0.994740i \(-0.467338\pi\)
−0.810256 + 0.586076i \(0.800672\pi\)
\(150\) 0 0
\(151\) −12.1714 + 7.02715i −0.990493 + 0.571861i −0.905422 0.424513i \(-0.860445\pi\)
−0.0850714 + 0.996375i \(0.527112\pi\)
\(152\) 0 0
\(153\) −9.22782 −0.746025
\(154\) 0 0
\(155\) −10.6028 −0.851636
\(156\) 0 0
\(157\) −3.88429 6.72779i −0.310000 0.536936i 0.668362 0.743836i \(-0.266996\pi\)
−0.978362 + 0.206900i \(0.933662\pi\)
\(158\) 0 0
\(159\) −0.762140 + 1.32006i −0.0604416 + 0.104688i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.40289 + 3.69671i 0.501513 + 0.289549i 0.729338 0.684153i \(-0.239828\pi\)
−0.227825 + 0.973702i \(0.573161\pi\)
\(164\) 0 0
\(165\) 1.86964 + 3.23831i 0.145551 + 0.252102i
\(166\) 0 0
\(167\) 15.7122i 1.21585i 0.793995 + 0.607925i \(0.207998\pi\)
−0.793995 + 0.607925i \(0.792002\pi\)
\(168\) 0 0
\(169\) −6.53110 11.2403i −0.502393 0.864640i
\(170\) 0 0
\(171\) 2.91538 1.68319i 0.222944 0.128717i
\(172\) 0 0
\(173\) −0.207792 + 0.359907i −0.0157982 + 0.0273632i −0.873816 0.486256i \(-0.838362\pi\)
0.858018 + 0.513619i \(0.171696\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −13.3580 7.71224i −1.00405 0.579688i
\(178\) 0 0
\(179\) −1.95396 3.38436i −0.146046 0.252959i 0.783717 0.621118i \(-0.213321\pi\)
−0.929763 + 0.368160i \(0.879988\pi\)
\(180\) 0 0
\(181\) −4.79151 −0.356150 −0.178075 0.984017i \(-0.556987\pi\)
−0.178075 + 0.984017i \(0.556987\pi\)
\(182\) 0 0
\(183\) −17.9127 −1.32414
\(184\) 0 0
\(185\) −8.68559 15.0439i −0.638577 1.10605i
\(186\) 0 0
\(187\) 6.22261 + 3.59262i 0.455042 + 0.262719i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.31453 7.47298i 0.312188 0.540726i −0.666648 0.745373i \(-0.732271\pi\)
0.978836 + 0.204647i \(0.0656047\pi\)
\(192\) 0 0
\(193\) 21.1773 12.2267i 1.52438 0.880100i 0.524794 0.851229i \(-0.324142\pi\)
0.999583 0.0288702i \(-0.00919095\pi\)
\(194\) 0 0
\(195\) −11.3129 6.51070i −0.810137 0.466241i
\(196\) 0 0
\(197\) 7.14128i 0.508795i −0.967100 0.254398i \(-0.918123\pi\)
0.967100 0.254398i \(-0.0818772\pi\)
\(198\) 0 0
\(199\) −0.501418 0.868482i −0.0355446 0.0615651i 0.847706 0.530467i \(-0.177983\pi\)
−0.883250 + 0.468902i \(0.844650\pi\)
\(200\) 0 0
\(201\) 15.4391 + 8.91378i 1.08899 + 0.628730i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.38571 + 2.40012i −0.0967822 + 0.167632i
\(206\) 0 0
\(207\) −2.54121 4.40151i −0.176626 0.305926i
\(208\) 0 0
\(209\) −2.62124 −0.181315
\(210\) 0 0
\(211\) 26.2655 1.80819 0.904096 0.427329i \(-0.140546\pi\)
0.904096 + 0.427329i \(0.140546\pi\)
\(212\) 0 0
\(213\) −4.26203 + 2.46069i −0.292030 + 0.168603i
\(214\) 0 0
\(215\) 1.35955 + 0.784938i 0.0927207 + 0.0535323i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3.57469 2.06385i −0.241555 0.139462i
\(220\) 0 0
\(221\) −25.0815 + 0.0346748i −1.68716 + 0.00233248i
\(222\) 0 0
\(223\) 17.7540i 1.18890i −0.804133 0.594449i \(-0.797370\pi\)
0.804133 0.594449i \(-0.202630\pi\)
\(224\) 0 0
\(225\) 3.75589 0.250393
\(226\) 0 0
\(227\) −15.4391 + 8.91378i −1.02473 + 0.591629i −0.915471 0.402385i \(-0.868181\pi\)
−0.109260 + 0.994013i \(0.534848\pi\)
\(228\) 0 0
\(229\) −3.64963 2.10712i −0.241175 0.139242i 0.374542 0.927210i \(-0.377800\pi\)
−0.615716 + 0.787968i \(0.711133\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 14.1440 24.4981i 0.926604 1.60492i 0.137642 0.990482i \(-0.456048\pi\)
0.788961 0.614443i \(-0.210619\pi\)
\(234\) 0 0
\(235\) 10.0616 + 17.4272i 0.656344 + 1.13682i
\(236\) 0 0
\(237\) 9.48538 0.616142
\(238\) 0 0
\(239\) 8.94385i 0.578530i 0.957249 + 0.289265i \(0.0934108\pi\)
−0.957249 + 0.289265i \(0.906589\pi\)
\(240\) 0 0
\(241\) 14.7241 8.50097i 0.948464 0.547596i 0.0558604 0.998439i \(-0.482210\pi\)
0.892603 + 0.450843i \(0.148876\pi\)
\(242\) 0 0
\(243\) −6.28629 + 10.8882i −0.403266 + 0.698477i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.91775 4.58592i 0.503794 0.291795i
\(248\) 0 0
\(249\) −18.6264 + 10.7539i −1.18040 + 0.681503i
\(250\) 0 0
\(251\) 6.70637 0.423303 0.211651 0.977345i \(-0.432116\pi\)
0.211651 + 0.977345i \(0.432116\pi\)
\(252\) 0 0
\(253\) 3.95743i 0.248802i
\(254\) 0 0
\(255\) −21.8092 + 12.5915i −1.36574 + 0.788513i
\(256\) 0 0
\(257\) −7.33278 + 12.7007i −0.457406 + 0.792251i −0.998823 0.0485035i \(-0.984555\pi\)
0.541417 + 0.840754i \(0.317888\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.38381 + 5.86093i −0.209453 + 0.362783i
\(262\) 0 0
\(263\) 3.01314 + 5.21891i 0.185798 + 0.321812i 0.943845 0.330388i \(-0.107180\pi\)
−0.758047 + 0.652200i \(0.773846\pi\)
\(264\) 0 0
\(265\) 3.29742i 0.202559i
\(266\) 0 0
\(267\) 15.3895i 0.941822i
\(268\) 0 0
\(269\) −0.412748 0.714900i −0.0251657 0.0435882i 0.853168 0.521636i \(-0.174678\pi\)
−0.878334 + 0.478048i \(0.841345\pi\)
\(270\) 0 0
\(271\) −8.51092 4.91378i −0.517002 0.298491i 0.218705 0.975791i \(-0.429817\pi\)
−0.735707 + 0.677300i \(0.763150\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.53272 1.46226i −0.152729 0.0881778i
\(276\) 0 0
\(277\) −13.9258 24.1203i −0.836722 1.44925i −0.892620 0.450809i \(-0.851135\pi\)
0.0558978 0.998436i \(-0.482198\pi\)
\(278\) 0 0
\(279\) 5.02596i 0.300896i
\(280\) 0 0
\(281\) 4.04257i 0.241159i 0.992704 + 0.120580i \(0.0384753\pi\)
−0.992704 + 0.120580i \(0.961525\pi\)
\(282\) 0 0
\(283\) −14.8798 25.7726i −0.884512 1.53202i −0.846272 0.532751i \(-0.821158\pi\)
−0.0382402 0.999269i \(-0.512175\pi\)
\(284\) 0 0
\(285\) 4.59350 7.95618i 0.272096 0.471283i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.6954 + 27.1852i −0.923259 + 1.59913i
\(290\) 0 0
\(291\) −9.63867 + 5.56489i −0.565029 + 0.326219i
\(292\) 0 0
\(293\) 29.8091i 1.74147i 0.491755 + 0.870733i \(0.336355\pi\)
−0.491755 + 0.870733i \(0.663645\pi\)
\(294\) 0 0
\(295\) 33.3673 1.94272
\(296\) 0 0
\(297\) 5.00656 2.89054i 0.290510 0.167726i
\(298\) 0 0
\(299\) −6.92362 11.9539i −0.400403 0.691310i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 5.41324 9.37601i 0.310983 0.538638i
\(304\) 0 0
\(305\) 33.5584 19.3750i 1.92155 1.10941i
\(306\) 0 0
\(307\) 18.5928i 1.06115i −0.847639 0.530574i \(-0.821977\pi\)
0.847639 0.530574i \(-0.178023\pi\)
\(308\) 0 0
\(309\) −0.674552 −0.0383739
\(310\) 0 0
\(311\) 7.95597 + 13.7801i 0.451142 + 0.781400i 0.998457 0.0555262i \(-0.0176836\pi\)
−0.547316 + 0.836926i \(0.684350\pi\)
\(312\) 0 0
\(313\) −0.897873 + 1.55516i −0.0507508 + 0.0879029i −0.890285 0.455404i \(-0.849495\pi\)
0.839534 + 0.543307i \(0.182828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.8833 + 8.01553i 0.779764 + 0.450197i 0.836347 0.548201i \(-0.184687\pi\)
−0.0565827 + 0.998398i \(0.518020\pi\)
\(318\) 0 0
\(319\) 4.56362 2.63481i 0.255514 0.147521i
\(320\) 0 0
\(321\) 12.2057 0.681253
\(322\) 0 0
\(323\) 17.6534i 0.982260i
\(324\) 0 0
\(325\) 10.2086 0.0141133i 0.566272 0.000782865i
\(326\) 0 0
\(327\) −15.3326 8.85226i −0.847893 0.489531i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −5.90448 3.40895i −0.324540 0.187373i 0.328875 0.944374i \(-0.393331\pi\)
−0.653414 + 0.757001i \(0.726664\pi\)
\(332\) 0 0
\(333\) −7.13113 + 4.11716i −0.390784 + 0.225619i
\(334\) 0 0
\(335\) −38.5658 −2.10707
\(336\) 0 0
\(337\) 0.418615 0.0228034 0.0114017 0.999935i \(-0.496371\pi\)
0.0114017 + 0.999935i \(0.496371\pi\)
\(338\) 0 0
\(339\) 7.93349 + 13.7412i 0.430888 + 0.746320i
\(340\) 0 0
\(341\) 1.95673 3.38916i 0.105963 0.183533i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −12.0119 6.93507i −0.646698 0.373371i
\(346\) 0 0
\(347\) 5.03291 + 8.71725i 0.270180 + 0.467966i 0.968908 0.247422i \(-0.0795833\pi\)
−0.698727 + 0.715388i \(0.746250\pi\)
\(348\) 0 0
\(349\) 8.27233i 0.442808i 0.975182 + 0.221404i \(0.0710639\pi\)
−0.975182 + 0.221404i \(0.928936\pi\)
\(350\) 0 0
\(351\) −10.0658 + 17.4903i −0.537273 + 0.933563i
\(352\) 0 0
\(353\) −14.1949 + 8.19541i −0.755516 + 0.436197i −0.827684 0.561195i \(-0.810342\pi\)
0.0721674 + 0.997393i \(0.477008\pi\)
\(354\) 0 0
\(355\) 5.32312 9.21991i 0.282522 0.489342i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 27.8437 + 16.0756i 1.46954 + 0.848437i 0.999416 0.0341658i \(-0.0108774\pi\)
0.470120 + 0.882603i \(0.344211\pi\)
\(360\) 0 0
\(361\) −6.27995 10.8772i −0.330524 0.572484i
\(362\) 0 0
\(363\) 12.8497 0.674436
\(364\) 0 0
\(365\) 8.92931 0.467382
\(366\) 0 0
\(367\) −3.89787 6.75131i −0.203467 0.352416i 0.746176 0.665749i \(-0.231888\pi\)
−0.949643 + 0.313333i \(0.898554\pi\)
\(368\) 0 0
\(369\) 1.13771 + 0.656857i 0.0592268 + 0.0341946i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −15.8686 + 27.4851i −0.821643 + 1.42313i 0.0828156 + 0.996565i \(0.473609\pi\)
−0.904458 + 0.426562i \(0.859725\pi\)
\(374\) 0 0
\(375\) −6.79900 + 3.92541i −0.351099 + 0.202707i
\(376\) 0 0
\(377\) −9.17527 + 15.9429i −0.472550 + 0.821101i
\(378\) 0 0
\(379\) 30.3236i 1.55762i 0.627261 + 0.778809i \(0.284176\pi\)
−0.627261 + 0.778809i \(0.715824\pi\)
\(380\) 0 0
\(381\) 7.07168 + 12.2485i 0.362293 + 0.627510i
\(382\) 0 0
\(383\) 12.1411 + 7.00966i 0.620381 + 0.358177i 0.777017 0.629479i \(-0.216732\pi\)
−0.156636 + 0.987656i \(0.550065\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.372078 0.644458i 0.0189138 0.0327597i
\(388\) 0 0
\(389\) −9.94435 17.2241i −0.504198 0.873297i −0.999988 0.00485467i \(-0.998455\pi\)
0.495790 0.868443i \(-0.334879\pi\)
\(390\) 0 0
\(391\) −26.6523 −1.34786
\(392\) 0 0
\(393\) 8.52144 0.429850
\(394\) 0 0
\(395\) −17.7703 + 10.2597i −0.894122 + 0.516222i
\(396\) 0 0
\(397\) 12.7495 + 7.36095i 0.639881 + 0.369436i 0.784569 0.620042i \(-0.212884\pi\)
−0.144688 + 0.989477i \(0.546218\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.0098 + 9.82062i 0.849429 + 0.490418i 0.860458 0.509521i \(-0.170177\pi\)
−0.0110289 + 0.999939i \(0.503511\pi\)
\(402\) 0 0
\(403\) 0.0188858 + 13.6607i 0.000940766 + 0.680487i
\(404\) 0 0
\(405\) 9.12499i 0.453424i
\(406\) 0 0
\(407\) 6.41166 0.317814
\(408\) 0 0
\(409\) 20.8826 12.0566i 1.03258 0.596160i 0.114856 0.993382i \(-0.463359\pi\)
0.917722 + 0.397223i \(0.130026\pi\)
\(410\) 0 0
\(411\) −14.3773 8.30075i −0.709181 0.409446i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 23.2636 40.2938i 1.14197 1.97794i
\(416\) 0 0
\(417\) −12.0306 20.8375i −0.589139 1.02042i
\(418\) 0 0
\(419\) 19.7818 0.966406 0.483203 0.875508i \(-0.339473\pi\)
0.483203 + 0.875508i \(0.339473\pi\)
\(420\) 0 0
\(421\) 19.1237i 0.932033i 0.884776 + 0.466016i \(0.154311\pi\)
−0.884776 + 0.466016i \(0.845689\pi\)
\(422\) 0 0
\(423\) 8.26086 4.76941i 0.401657 0.231897i
\(424\) 0 0
\(425\) 9.84797 17.0572i 0.477697 0.827395i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.16893 2.41462i 0.201278 0.116579i
\(430\) 0 0
\(431\) −7.68673 + 4.43794i −0.370257 + 0.213768i −0.673571 0.739123i \(-0.735240\pi\)
0.303314 + 0.952891i \(0.401907\pi\)
\(432\) 0 0
\(433\) 36.9341 1.77494 0.887470 0.460866i \(-0.152461\pi\)
0.887470 + 0.460866i \(0.152461\pi\)
\(434\) 0 0
\(435\) 18.4691i 0.885527i
\(436\) 0 0
\(437\) 8.42035 4.86149i 0.402800 0.232557i
\(438\) 0 0
\(439\) −10.3296 + 17.8913i −0.493003 + 0.853906i −0.999968 0.00806061i \(-0.997434\pi\)
0.506964 + 0.861967i \(0.330768\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.42746 + 2.47244i −0.0678207 + 0.117469i −0.897942 0.440114i \(-0.854938\pi\)
0.830121 + 0.557583i \(0.188271\pi\)
\(444\) 0 0
\(445\) 16.6458 + 28.8314i 0.789086 + 1.36674i
\(446\) 0 0
\(447\) 12.9062i 0.610442i
\(448\) 0 0
\(449\) 30.0443i 1.41788i 0.705269 + 0.708940i \(0.250826\pi\)
−0.705269 + 0.708940i \(0.749174\pi\)
\(450\) 0 0
\(451\) −0.511463 0.885879i −0.0240838 0.0417144i
\(452\) 0 0
\(453\) 15.7452 + 9.09051i 0.739775 + 0.427109i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 24.8109 + 14.3246i 1.16060 + 0.670075i 0.951449 0.307807i \(-0.0995952\pi\)
0.209156 + 0.977882i \(0.432929\pi\)
\(458\) 0 0
\(459\) 19.4670 + 33.7179i 0.908644 + 1.57382i
\(460\) 0 0
\(461\) 0.922363i 0.0429587i 0.999769 + 0.0214794i \(0.00683762\pi\)
−0.999769 + 0.0214794i \(0.993162\pi\)
\(462\) 0 0
\(463\) 26.0929i 1.21264i 0.795220 + 0.606321i \(0.207355\pi\)
−0.795220 + 0.606321i \(0.792645\pi\)
\(464\) 0 0
\(465\) 6.85802 + 11.8784i 0.318033 + 0.550849i
\(466\) 0 0
\(467\) 14.7777 25.5957i 0.683829 1.18443i −0.289974 0.957034i \(-0.593647\pi\)
0.973803 0.227392i \(-0.0730199\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −5.02482 + 8.70324i −0.231532 + 0.401024i
\(472\) 0 0
\(473\) −0.501808 + 0.289719i −0.0230732 + 0.0133213i
\(474\) 0 0
\(475\) 7.18525i 0.329682i
\(476\) 0 0
\(477\) −1.56305 −0.0715672
\(478\) 0 0
\(479\) 13.7658 7.94771i 0.628977 0.363140i −0.151379 0.988476i \(-0.548371\pi\)
0.780356 + 0.625336i \(0.215038\pi\)
\(480\) 0 0
\(481\) −19.3671 + 11.2174i −0.883066 + 0.511467i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.0383 20.8510i 0.546632 0.946795i
\(486\) 0 0
\(487\) 0.0737279 0.0425668i 0.00334093 0.00192889i −0.498329 0.866988i \(-0.666053\pi\)
0.501670 + 0.865059i \(0.332719\pi\)
\(488\) 0 0
\(489\) 9.56433i 0.432514i
\(490\) 0 0
\(491\) 2.56324 0.115678 0.0578388 0.998326i \(-0.481579\pi\)
0.0578388 + 0.998326i \(0.481579\pi\)
\(492\) 0 0
\(493\) 17.7448 + 30.7348i 0.799184 + 1.38423i
\(494\) 0 0
\(495\) −1.91720 + 3.32068i −0.0861716 + 0.149254i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −24.8498 14.3470i −1.11243 0.642262i −0.172973 0.984927i \(-0.555337\pi\)
−0.939458 + 0.342664i \(0.888671\pi\)
\(500\) 0 0
\(501\) 17.6026 10.1629i 0.786428 0.454044i
\(502\) 0 0
\(503\) 32.7056 1.45827 0.729136 0.684369i \(-0.239922\pi\)
0.729136 + 0.684369i \(0.239922\pi\)
\(504\) 0 0
\(505\) 23.4206i 1.04220i
\(506\) 0 0
\(507\) −8.36827 + 14.5873i −0.371648 + 0.647843i
\(508\) 0 0
\(509\) 22.5979 + 13.0469i 1.00164 + 0.578295i 0.908732 0.417381i \(-0.137052\pi\)
0.0929037 + 0.995675i \(0.470385\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −12.3006 7.10175i −0.543084 0.313550i
\(514\) 0 0
\(515\) 1.26373 0.729617i 0.0556868 0.0321508i
\(516\) 0 0
\(517\) −7.42741 −0.326657
\(518\) 0 0
\(519\) 0.537611 0.0235985
\(520\) 0 0
\(521\) −0.877849 1.52048i −0.0384592 0.0666134i 0.846155 0.532937i \(-0.178912\pi\)
−0.884614 + 0.466323i \(0.845578\pi\)
\(522\) 0 0
\(523\) −16.1890 + 28.0401i −0.707893 + 1.22611i 0.257744 + 0.966213i \(0.417021\pi\)
−0.965637 + 0.259894i \(0.916312\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.8251 + 13.1781i 0.994278 + 0.574047i
\(528\) 0 0
\(529\) 4.16033 + 7.20591i 0.180884 + 0.313300i
\(530\) 0 0
\(531\) 15.8168i 0.686392i
\(532\) 0 0
\(533\) 3.09480 + 1.78108i 0.134050 + 0.0771472i
\(534\) 0 0
\(535\) −22.8666 + 13.2020i −0.988609 + 0.570774i
\(536\) 0 0
\(537\) −2.52769 + 4.37809i −0.109078 + 0.188929i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.38318 + 4.26268i 0.317428 + 0.183267i 0.650245 0.759724i \(-0.274666\pi\)
−0.332818 + 0.942991i \(0.607999\pi\)
\(542\) 0 0
\(543\) 3.09921 + 5.36799i 0.133000 + 0.230362i
\(544\) 0 0
\(545\) 38.2996 1.64057
\(546\) 0 0
\(547\) 1.14355 0.0488945 0.0244473 0.999701i \(-0.492217\pi\)
0.0244473 + 0.999701i \(0.492217\pi\)
\(548\) 0 0
\(549\) −9.18417 15.9074i −0.391971 0.678913i
\(550\) 0 0
\(551\) −11.2123 6.47344i −0.477661 0.275778i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −11.2359 + 19.4612i −0.476937 + 0.826080i
\(556\) 0 0
\(557\) 37.0897 21.4137i 1.57154 0.907329i 0.575560 0.817760i \(-0.304784\pi\)
0.995981 0.0895693i \(-0.0285491\pi\)
\(558\) 0 0
\(559\) 1.00890 1.75305i 0.0426718 0.0741463i
\(560\) 0 0
\(561\) 9.29503i 0.392436i
\(562\) 0 0
\(563\) 6.78771 + 11.7567i 0.286068 + 0.495484i 0.972868 0.231363i \(-0.0743183\pi\)
−0.686800 + 0.726847i \(0.740985\pi\)
\(564\) 0 0
\(565\) −29.7259 17.1622i −1.25058 0.722021i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.43555 7.68259i 0.185948 0.322071i −0.757948 0.652315i \(-0.773798\pi\)
0.943895 + 0.330244i \(0.107131\pi\)
\(570\) 0 0
\(571\) −3.14047 5.43945i −0.131424 0.227634i 0.792801 0.609480i \(-0.208622\pi\)
−0.924226 + 0.381846i \(0.875288\pi\)
\(572\) 0 0
\(573\) −11.1628 −0.466331
\(574\) 0 0
\(575\) 10.8480 0.452392
\(576\) 0 0
\(577\) 21.7376 12.5502i 0.904950 0.522473i 0.0261471 0.999658i \(-0.491676\pi\)
0.878803 + 0.477185i \(0.158343\pi\)
\(578\) 0 0
\(579\) −27.3955 15.8168i −1.13852 0.657325i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.05401 + 0.608536i 0.0436528 + 0.0252030i
\(584\) 0 0
\(585\) −0.0185042 13.3847i −0.000765053 0.553388i
\(586\) 0 0
\(587\) 28.2963i 1.16791i −0.811784 0.583957i \(-0.801503\pi\)
0.811784 0.583957i \(-0.198497\pi\)
\(588\) 0 0
\(589\) −9.61496 −0.396178
\(590\) 0 0
\(591\) −8.00047 + 4.61907i −0.329095 + 0.190003i
\(592\) 0 0
\(593\) −11.1408 6.43213i −0.457497 0.264136i 0.253494 0.967337i \(-0.418420\pi\)
−0.710991 + 0.703201i \(0.751753\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.648648 + 1.12349i −0.0265474 + 0.0459814i
\(598\) 0 0
\(599\) −12.7842 22.1429i −0.522350 0.904736i −0.999662 0.0260025i \(-0.991722\pi\)
0.477312 0.878734i \(-0.341611\pi\)
\(600\) 0 0
\(601\) −6.63763 −0.270755 −0.135377 0.990794i \(-0.543225\pi\)
−0.135377 + 0.990794i \(0.543225\pi\)
\(602\) 0 0
\(603\) 18.2810i 0.744461i
\(604\) 0 0
\(605\) −24.0732 + 13.8987i −0.978716 + 0.565062i
\(606\) 0 0
\(607\) 11.1613 19.3319i 0.453023 0.784660i −0.545549 0.838079i \(-0.683679\pi\)
0.998572 + 0.0534195i \(0.0170121\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.4353 12.9944i 0.907636 0.525698i
\(612\) 0 0
\(613\) 3.66910 2.11836i 0.148194 0.0855596i −0.424070 0.905630i \(-0.639399\pi\)
0.572263 + 0.820070i \(0.306066\pi\)
\(614\) 0 0
\(615\) 3.58518 0.144568
\(616\) 0 0
\(617\) 14.0193i 0.564397i 0.959356 + 0.282198i \(0.0910636\pi\)
−0.959356 + 0.282198i \(0.908936\pi\)
\(618\) 0 0
\(619\) −20.6520 + 11.9234i −0.830074 + 0.479244i −0.853878 0.520473i \(-0.825756\pi\)
0.0238037 + 0.999717i \(0.492422\pi\)
\(620\) 0 0
\(621\) −10.7219 + 18.5709i −0.430255 + 0.745224i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15.5701 26.9682i 0.622804 1.07873i
\(626\) 0 0
\(627\) 1.69545 + 2.93661i 0.0677099 + 0.117277i
\(628\) 0 0
\(629\) 43.1809i 1.72174i
\(630\) 0 0
\(631\) 7.18624i 0.286080i 0.989717 + 0.143040i \(0.0456877\pi\)
−0.989717 + 0.143040i \(0.954312\pi\)
\(632\) 0 0
\(633\) −16.9889 29.4256i −0.675247 1.16956i
\(634\) 0 0
\(635\) −26.4968 15.2979i −1.05149 0.607079i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.37044 2.52328i −0.172892 0.0998193i
\(640\) 0 0
\(641\) 4.02943 + 6.97918i 0.159153 + 0.275661i 0.934563 0.355796i \(-0.115790\pi\)
−0.775411 + 0.631457i \(0.782457\pi\)
\(642\) 0 0
\(643\) 28.4148i 1.12057i 0.828300 + 0.560286i \(0.189309\pi\)
−0.828300 + 0.560286i \(0.810691\pi\)
\(644\) 0 0
\(645\) 2.03083i 0.0799640i
\(646\) 0 0
\(647\) 18.8396 + 32.6311i 0.740659 + 1.28286i 0.952195 + 0.305490i \(0.0988201\pi\)
−0.211536 + 0.977370i \(0.567847\pi\)
\(648\) 0 0
\(649\) −6.15789 + 10.6658i −0.241718 + 0.418669i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −18.5243 + 32.0850i −0.724911 + 1.25558i 0.234100 + 0.972213i \(0.424786\pi\)
−0.959011 + 0.283370i \(0.908547\pi\)
\(654\) 0 0
\(655\) −15.9644 + 9.21707i −0.623782 + 0.360141i
\(656\) 0 0
\(657\) 4.23269i 0.165133i
\(658\) 0 0
\(659\) 44.4553 1.73173 0.865867 0.500274i \(-0.166767\pi\)
0.865867 + 0.500274i \(0.166767\pi\)
\(660\) 0 0
\(661\) −34.9366 + 20.1706i −1.35888 + 0.784547i −0.989472 0.144722i \(-0.953771\pi\)
−0.369403 + 0.929269i \(0.620438\pi\)
\(662\) 0 0
\(663\) 16.2619 + 28.0767i 0.631558 + 1.09041i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.77332 + 16.9279i −0.378424 + 0.655450i
\(668\) 0 0
\(669\) −19.8901 + 11.4835i −0.768995 + 0.443980i
\(670\) 0 0
\(671\) 14.3025i 0.552143i
\(672\) 0 0
\(673\) −16.8024 −0.647684 −0.323842 0.946111i \(-0.604975\pi\)
−0.323842 + 0.946111i \(0.604975\pi\)
\(674\) 0 0
\(675\) −7.92345 13.7238i −0.304974 0.528230i
\(676\) 0 0
\(677\) −23.2825 + 40.3265i −0.894819 + 1.54987i −0.0607909 + 0.998151i \(0.519362\pi\)
−0.834028 + 0.551722i \(0.813971\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 19.9725 + 11.5311i 0.765346 + 0.441873i
\(682\) 0 0
\(683\) −5.01860 + 2.89749i −0.192031 + 0.110869i −0.592933 0.805252i \(-0.702030\pi\)
0.400902 + 0.916121i \(0.368697\pi\)
\(684\) 0 0
\(685\) 35.9135 1.37218
\(686\) 0 0
\(687\) 5.45164i 0.207993i
\(688\) 0 0
\(689\) −4.24842 + 0.00587339i −0.161852 + 0.000223758i
\(690\) 0 0
\(691\) −23.4877 13.5606i −0.893515 0.515871i −0.0184246 0.999830i \(-0.505865\pi\)
−0.875091 + 0.483959i \(0.839198\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 45.0771 + 26.0253i 1.70987 + 0.987196i
\(696\) 0 0
\(697\) 5.96617 3.44457i 0.225985 0.130472i
\(698\) 0 0
\(699\) −36.5941 −1.38412
\(700\) 0 0
\(701\) 11.0283 0.416535 0.208267 0.978072i \(-0.433218\pi\)
0.208267 + 0.978072i \(0.433218\pi\)
\(702\) 0 0
\(703\) −7.87638 13.6423i −0.297063 0.514529i
\(704\) 0 0
\(705\) 13.0159 22.5442i 0.490208 0.849064i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −43.0844 24.8748i −1.61807 0.934193i −0.987419 0.158126i \(-0.949455\pi\)
−0.630651 0.776067i \(-0.717212\pi\)
\(710\) 0 0
\(711\) 4.86333 + 8.42353i 0.182389 + 0.315907i
\(712\) 0 0
\(713\) 14.5162i 0.543638i
\(714\) 0 0
\(715\) −5.19851 + 9.03291i −0.194413 + 0.337811i
\(716\) 0 0
\(717\) 10.0199 5.78500i 0.374201 0.216045i
\(718\) 0 0
\(719\) 14.2825 24.7380i 0.532647 0.922572i −0.466626 0.884455i \(-0.654531\pi\)
0.999273 0.0381175i \(-0.0121361\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19.0475 10.9971i −0.708384 0.408986i
\(724\) 0 0
\(725\) −7.22244 12.5096i −0.268235 0.464596i
\(726\) 0 0
\(727\) −36.5250 −1.35464 −0.677318 0.735691i \(-0.736858\pi\)
−0.677318 + 0.735691i \(0.736858\pi\)
\(728\) 0 0
\(729\) 26.0464 0.964681
\(730\) 0 0
\(731\) −1.95118 3.37955i −0.0721671 0.124997i
\(732\) 0 0
\(733\) 16.2331 + 9.37219i 0.599584 + 0.346170i 0.768878 0.639396i \(-0.220815\pi\)
−0.169294 + 0.985566i \(0.554149\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.11727 12.3275i 0.262168 0.454089i
\(738\) 0 0
\(739\) −9.68550 + 5.59192i −0.356287 + 0.205702i −0.667451 0.744654i \(-0.732615\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(740\) 0 0
\(741\) −10.2590 5.90412i −0.376873 0.216893i
\(742\) 0 0
\(743\) 32.7986i 1.20326i 0.798774 + 0.601632i \(0.205482\pi\)
−0.798774 + 0.601632i \(0.794518\pi\)
\(744\) 0 0
\(745\) −13.9598 24.1790i −0.511446 0.885851i
\(746\) 0 0
\(747\) −19.1001 11.0275i −0.698838 0.403474i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 14.3726 24.8940i 0.524463 0.908396i −0.475132 0.879915i \(-0.657600\pi\)
0.999594 0.0284812i \(-0.00906708\pi\)
\(752\) 0 0
\(753\) −4.33777 7.51324i −0.158077 0.273798i
\(754\) 0 0
\(755\) −39.3304 −1.43138
\(756\) 0 0
\(757\) 12.4952 0.454145 0.227072 0.973878i \(-0.427085\pi\)
0.227072 + 0.973878i \(0.427085\pi\)
\(758\) 0 0
\(759\) 4.43356 2.55972i 0.160928 0.0929119i
\(760\) 0 0
\(761\) 25.6888 + 14.8314i 0.931218 + 0.537639i 0.887196 0.461392i \(-0.152650\pi\)
0.0440211 + 0.999031i \(0.485983\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −22.3639 12.9118i −0.808570 0.466828i
\(766\) 0 0
\(767\) −0.0594340 42.9906i −0.00214604 1.55230i
\(768\) 0 0
\(769\) 40.8488i 1.47305i 0.676412 + 0.736523i \(0.263534\pi\)
−0.676412 + 0.736523i \(0.736466\pi\)
\(770\) 0 0
\(771\) 18.9718 0.683251
\(772\) 0 0
\(773\) −16.8048 + 9.70226i −0.604427 + 0.348966i −0.770781 0.637100i \(-0.780134\pi\)
0.166354 + 0.986066i \(0.446800\pi\)
\(774\) 0 0
\(775\) −9.29025 5.36373i −0.333715 0.192671i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.25661 + 2.17651i −0.0450227 + 0.0779816i
\(780\) 0 0
\(781\) 1.96475 + 3.40305i 0.0703044 + 0.121771i
\(782\) 0 0
\(783\) 28.5540 1.02044
\(784\) 0 0
\(785\) 21.7400i 0.775935i
\(786\) 0 0
\(787\) 8.73934 5.04566i 0.311524 0.179858i −0.336084 0.941832i \(-0.609103\pi\)
0.647608 + 0.761974i \(0.275769\pi\)
\(788\) 0 0
\(789\) 3.89787 6.75131i 0.138768 0.240353i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −25.0226 43.2024i −0.888578 1.53416i
\(794\) 0 0
\(795\) −3.69414 + 2.13282i −0.131018 + 0.0756432i
\(796\) 0 0
\(797\) 11.2735 0.399329 0.199665 0.979864i \(-0.436015\pi\)
0.199665 + 0.979864i \(0.436015\pi\)
\(798\) 0 0
\(799\) 50.0217i 1.76964i
\(800\) 0 0
\(801\) 13.6667 7.89048i 0.482889 0.278796i
\(802\) 0 0
\(803\) −1.64790 + 2.85424i −0.0581530 + 0.100724i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.533941 + 0.924813i −0.0187956 + 0.0325550i
\(808\) 0 0
\(809\) −8.71180 15.0893i −0.306290 0.530511i 0.671257 0.741224i \(-0.265755\pi\)
−0.977548 + 0.210714i \(0.932421\pi\)
\(810\) 0 0
\(811\) 35.9555i 1.26257i −0.775552 0.631284i \(-0.782528\pi\)
0.775552 0.631284i \(-0.217472\pi\)
\(812\) 0 0
\(813\) 12.7132i 0.445871i
\(814\) 0 0
\(815\) 10.3451 + 17.9182i 0.362373 + 0.627648i
\(816\) 0 0
\(817\) 1.23289 + 0.711808i 0.0431333 + 0.0249030i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.0928 + 7.55913i 0.456942 + 0.263816i 0.710758 0.703437i \(-0.248352\pi\)
−0.253816 + 0.967253i \(0.581686\pi\)
\(822\) 0 0
\(823\) −21.6167 37.4413i −0.753512 1.30512i −0.946111 0.323843i \(-0.895025\pi\)
0.192599 0.981278i \(-0.438308\pi\)
\(824\) 0 0
\(825\) 3.78325i 0.131716i
\(826\) 0 0
\(827\) 50.4513i 1.75436i −0.480158 0.877182i \(-0.659421\pi\)
0.480158 0.877182i \(-0.340579\pi\)
\(828\) 0 0
\(829\) 0.898637 + 1.55649i 0.0312110 + 0.0540590i 0.881209 0.472727i \(-0.156730\pi\)
−0.849998 + 0.526786i \(0.823397\pi\)
\(830\) 0 0
\(831\) −18.0148 + 31.2026i −0.624927 + 1.08241i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −21.9850 + 38.0792i −0.760823 + 1.31778i
\(836\) 0 0
\(837\) 18.3646 10.6028i 0.634772 0.366486i
\(838\) 0 0
\(839\) 4.72104i 0.162988i −0.996674 0.0814942i \(-0.974031\pi\)
0.996674 0.0814942i \(-0.0259692\pi\)
\(840\) 0 0
\(841\) −2.97220 −0.102490
\(842\) 0 0
\(843\) 4.52894 2.61479i 0.155985 0.0900580i
\(844\) 0 0
\(845\) −0.100590 36.3798i −0.00346039 1.25150i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −19.2489 + 33.3401i −0.660620 + 1.14423i
\(850\) 0 0
\(851\) −20.5965 + 11.8914i −0.706040 + 0.407633i
\(852\) 0 0
\(853\) 48.8428i 1.67234i 0.548467 + 0.836172i \(0.315212\pi\)
−0.548467 + 0.836172i \(0.684788\pi\)
\(854\) 0 0
\(855\) 9.42069 0.322181
\(856\) 0 0
\(857\) 7.49820 + 12.9873i 0.256134 + 0.443636i 0.965203 0.261502i \(-0.0842180\pi\)
−0.709069 + 0.705139i \(0.750885\pi\)
\(858\) 0 0
\(859\) −16.5232 + 28.6190i −0.563764 + 0.976468i 0.433399 + 0.901202i \(0.357314\pi\)
−0.997163 + 0.0752662i \(0.976019\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.8004 + 23.5561i 1.38886 + 0.801859i 0.993187 0.116532i \(-0.0371778\pi\)
0.395674 + 0.918391i \(0.370511\pi\)
\(864\) 0 0
\(865\) −1.00718 + 0.581498i −0.0342453 + 0.0197715i
\(866\) 0 0
\(867\) 40.6080 1.37912
\(868\) 0 0
\(869\) 7.57367i 0.256919i
\(870\) 0 0
\(871\) 0.0686936 + 49.6883i 0.00232759 + 1.68363i
\(872\) 0 0
\(873\) −9.88384 5.70644i −0.334517 0.193134i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.84222 3.37301i −0.197278 0.113898i 0.398107 0.917339i \(-0.369667\pi\)
−0.595385 + 0.803441i \(0.703000\pi\)
\(878\) 0 0
\(879\) 33.3955 19.2809i 1.12640 0.650329i
\(880\) 0 0
\(881\) −22.5679 −0.760333 −0.380166 0.924918i \(-0.624133\pi\)
−0.380166 + 0.924918i \(0.624133\pi\)
\(882\) 0 0
\(883\) −19.2628 −0.648245 −0.324122 0.946015i \(-0.605069\pi\)
−0.324122 + 0.946015i \(0.605069\pi\)
\(884\) 0 0
\(885\) −21.5824 37.3818i −0.725484 1.25657i
\(886\) 0 0
\(887\) −0.426329 + 0.738424i −0.0143147 + 0.0247938i −0.873094 0.487552i \(-0.837890\pi\)
0.858779 + 0.512346i \(0.171223\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.91679 1.68401i −0.0977160 0.0564164i
\(892\) 0 0
\(893\) 9.12417 + 15.8035i 0.305329 + 0.528845i
\(894\) 0 0
\(895\) 10.9361i 0.365555i
\(896\) 0 0
\(897\) −8.91378 + 15.4885i −0.297623 + 0.517147i
\(898\) 0 0
\(899\) 16.7398 9.66473i 0.558304 0.322337i
\(900\) 0 0
\(901\) −4.09833 + 7.09852i −0.136535 + 0.236486i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −11.6124 6.70441i −0.386009 0.222862i
\(906\) 0 0
\(907\) −23.3797 40.4948i −0.776310 1.34461i −0.934055 0.357129i \(-0.883756\pi\)
0.157745 0.987480i \(-0.449578\pi\)
\(908\) 0 0
\(909\) 11.1019 0.368226
\(910\) 0 0
\(911\) −33.7415 −1.11791 −0.558954 0.829199i \(-0.688797\pi\)
−0.558954 + 0.829199i \(0.688797\pi\)
\(912\) 0 0
\(913\) 8.58655 + 14.8723i 0.284173 + 0.492203i
\(914\) 0 0
\(915\) −43.4121 25.0640i −1.43516 0.828589i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11.7170 20.2945i 0.386509 0.669453i −0.605468 0.795869i \(-0.707014\pi\)
0.991977 + 0.126416i \(0.0403475\pi\)
\(920\) 0 0
\(921\) −20.8298 + 12.0261i −0.686364 + 0.396272i
\(922\) 0 0
\(923\) −11.8885 6.84191i −0.391314 0.225204i
\(924\) 0 0
\(925\) 17.5754i 0.577876i
\(926\) 0 0
\(927\) −0.345855 0.599038i −0.0113594 0.0196750i
\(928\) 0 0
\(929\) −9.51123 5.49131i −0.312053 0.180164i 0.335792 0.941936i \(-0.390996\pi\)
−0.647845 + 0.761772i \(0.724330\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 10.2920 17.8264i 0.336947 0.583609i
\(934\) 0 0
\(935\) 10.0538 + 17.4137i 0.328795 + 0.569489i
\(936\) 0 0
\(937\) −47.5028 −1.55185 −0.775924 0.630826i \(-0.782716\pi\)
−0.775924 + 0.630826i \(0.782716\pi\)
\(938\) 0 0
\(939\) 2.32302 0.0758090
\(940\) 0 0
\(941\) −7.00541 + 4.04458i −0.228370 + 0.131849i −0.609820 0.792540i \(-0.708758\pi\)
0.381450 + 0.924390i \(0.375425\pi\)
\(942\) 0 0
\(943\) 3.28600 + 1.89717i 0.107007 + 0.0617804i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.2176 + 9.36323i 0.527001 + 0.304264i 0.739794 0.672833i \(-0.234923\pi\)
−0.212793 + 0.977097i \(0.568256\pi\)
\(948\) 0 0
\(949\) −0.0159050 11.5046i −0.000516297 0.373454i
\(950\) 0 0
\(951\) 20.7382i 0.672482i
\(952\) 0 0
\(953\) 47.2451 1.53042 0.765209 0.643781i \(-0.222635\pi\)
0.765209 + 0.643781i \(0.222635\pi\)
\(954\) 0 0
\(955\) 20.9128 12.0740i 0.676723 0.390706i
\(956\) 0 0
\(957\) −5.90362 3.40846i −0.190837 0.110180i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8.32252 + 14.4150i −0.268468 + 0.465001i
\(962\) 0 0
\(963\) 6.25806 + 10.8393i 0.201663 + 0.349291i
\(964\) 0 0
\(965\) 68.4320 2.20290
\(966\) 0 0
\(967\) 39.7889i 1.27952i −0.768573 0.639762i \(-0.779033\pi\)
0.768573 0.639762i \(-0.220967\pi\)
\(968\) 0 0
\(969\) −19.7773 + 11.4184i −0.635339 + 0.366813i
\(970\) 0 0
\(971\) 5.00663 8.67174i 0.160670 0.278289i −0.774439 0.632649i \(-0.781968\pi\)
0.935109 + 0.354359i \(0.115301\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −6.61888 11.4277i −0.211974 0.365980i
\(976\) 0 0
\(977\) 51.0164 29.4543i 1.63216 0.942328i 0.648734 0.761015i \(-0.275299\pi\)
0.983425 0.181313i \(-0.0580347\pi\)
\(978\) 0 0
\(979\) −12.2879 −0.392722
\(980\) 0 0
\(981\) 18.1549i 0.579640i
\(982\) 0 0
\(983\) −4.46527 + 2.57802i −0.142420 + 0.0822262i −0.569517 0.821980i \(-0.692870\pi\)
0.427097 + 0.904206i \(0.359536\pi\)
\(984\) 0 0
\(985\) 9.99229 17.3071i 0.318381 0.551452i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.07466 1.86136i 0.0341721 0.0591878i
\(990\) 0 0
\(991\) 31.3453 + 54.2916i 0.995715 + 1.72463i 0.577941 + 0.816078i \(0.303856\pi\)
0.417774 + 0.908551i \(0.362810\pi\)
\(992\) 0 0
\(993\) 8.81982i 0.279889i
\(994\) 0 0
\(995\) 2.80640i 0.0889687i
\(996\) 0 0
\(997\) −7.06884 12.2436i −0.223872 0.387758i 0.732108 0.681188i \(-0.238537\pi\)
−0.955981 + 0.293430i \(0.905203\pi\)
\(998\) 0 0
\(999\) 30.0877 + 17.3712i 0.951934 + 0.549600i
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 2548.2.y.e.961.4 16
7.2 even 3 2548.2.g.g.2157.5 8
7.3 odd 6 2548.2.y.f.753.6 16
7.4 even 3 inner 2548.2.y.e.753.3 16
7.5 odd 6 364.2.g.a.337.4 yes 8
7.6 odd 2 2548.2.y.f.961.5 16
13.12 even 2 inner 2548.2.y.e.961.3 16
21.5 even 6 3276.2.e.f.2521.1 8
28.19 even 6 1456.2.k.d.337.6 8
91.5 even 12 4732.2.a.p.1.2 4
91.12 odd 6 364.2.g.a.337.3 8
91.25 even 6 inner 2548.2.y.e.753.4 16
91.38 odd 6 2548.2.y.f.753.5 16
91.47 even 12 4732.2.a.o.1.2 4
91.51 even 6 2548.2.g.g.2157.6 8
91.90 odd 2 2548.2.y.f.961.6 16
273.194 even 6 3276.2.e.f.2521.8 8
364.103 even 6 1456.2.k.d.337.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
364.2.g.a.337.3 8 91.12 odd 6
364.2.g.a.337.4 yes 8 7.5 odd 6
1456.2.k.d.337.5 8 364.103 even 6
1456.2.k.d.337.6 8 28.19 even 6
2548.2.g.g.2157.5 8 7.2 even 3
2548.2.g.g.2157.6 8 91.51 even 6
2548.2.y.e.753.3 16 7.4 even 3 inner
2548.2.y.e.753.4 16 91.25 even 6 inner
2548.2.y.e.961.3 16 13.12 even 2 inner
2548.2.y.e.961.4 16 1.1 even 1 trivial
2548.2.y.f.753.5 16 91.38 odd 6
2548.2.y.f.753.6 16 7.3 odd 6
2548.2.y.f.961.5 16 7.6 odd 2
2548.2.y.f.961.6 16 91.90 odd 2
3276.2.e.f.2521.1 8 21.5 even 6
3276.2.e.f.2521.8 8 273.194 even 6
4732.2.a.o.1.2 4 91.47 even 12
4732.2.a.p.1.2 4 91.5 even 12