Properties

Label 1200.2.v.c.593.2
Level $1200$
Weight $2$
Character 1200.593
Analytic conductor $9.582$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,2,Mod(257,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1200.v (of order \(4\), 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: \(9.58204824255\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 593.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1200.593
Dual form 1200.2.v.c.257.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.292893 - 1.70711i) q^{3} +(3.00000 - 3.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})\) \(q+(-0.292893 - 1.70711i) q^{3} +(3.00000 - 3.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} +1.41421i q^{11} +(-4.24264 - 4.24264i) q^{17} -4.00000i q^{19} +(-6.00000 - 4.24264i) q^{21} +(2.82843 - 2.82843i) q^{23} +(2.53553 + 4.53553i) q^{27} +1.41421 q^{29} +2.00000 q^{31} +(2.41421 - 0.414214i) q^{33} +(2.00000 - 2.00000i) q^{37} +5.65685i q^{41} +(-2.00000 - 2.00000i) q^{43} +(-5.65685 - 5.65685i) q^{47} -11.0000i q^{49} +(-6.00000 + 8.48528i) q^{51} +(-8.48528 + 8.48528i) q^{53} +(-6.82843 + 1.17157i) q^{57} -1.41421 q^{59} -6.00000 q^{61} +(-5.48528 + 11.4853i) q^{63} +(4.00000 - 4.00000i) q^{67} +(-5.65685 - 4.00000i) q^{69} -2.82843i q^{71} +(-3.00000 - 3.00000i) q^{73} +(4.24264 + 4.24264i) q^{77} -10.0000i q^{79} +(7.00000 - 5.65685i) q^{81} +(-2.82843 + 2.82843i) q^{83} +(-0.414214 - 2.41421i) q^{87} -2.82843 q^{89} +(-0.585786 - 3.41421i) q^{93} +(13.0000 - 13.0000i) q^{97} +(-1.41421 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 12 q^{7} - 24 q^{21} - 4 q^{27} + 8 q^{31} + 4 q^{33} + 8 q^{37} - 8 q^{43} - 24 q^{51} - 16 q^{57} - 24 q^{61} + 12 q^{63} + 16 q^{67} - 12 q^{73} + 28 q^{81} + 4 q^{87} - 8 q^{93} + 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) −0.292893 1.70711i −0.169102 0.985599i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000 3.00000i 1.13389 1.13389i 0.144370 0.989524i \(-0.453885\pi\)
0.989524 0.144370i \(-0.0461154\pi\)
\(8\) 0 0
\(9\) −2.82843 + 1.00000i −0.942809 + 0.333333i
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24264 4.24264i −1.02899 1.02899i −0.999567 0.0294245i \(-0.990633\pi\)
−0.0294245 0.999567i \(-0.509367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) −6.00000 4.24264i −1.30931 0.925820i
\(22\) 0 0
\(23\) 2.82843 2.82843i 0.589768 0.589768i −0.347801 0.937568i \(-0.613071\pi\)
0.937568 + 0.347801i \(0.113071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.53553 + 4.53553i 0.487964 + 0.872864i
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) 2.41421 0.414214i 0.420261 0.0721053i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 2.00000i 0.328798 0.328798i −0.523331 0.852129i \(-0.675311\pi\)
0.852129 + 0.523331i \(0.175311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) −2.00000 2.00000i −0.304997 0.304997i 0.537968 0.842965i \(-0.319192\pi\)
−0.842965 + 0.537968i \(0.819192\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 5.65685i −0.825137 0.825137i 0.161703 0.986840i \(-0.448301\pi\)
−0.986840 + 0.161703i \(0.948301\pi\)
\(48\) 0 0
\(49\) 11.0000i 1.57143i
\(50\) 0 0
\(51\) −6.00000 + 8.48528i −0.840168 + 1.18818i
\(52\) 0 0
\(53\) −8.48528 + 8.48528i −1.16554 + 1.16554i −0.182300 + 0.983243i \(0.558354\pi\)
−0.983243 + 0.182300i \(0.941646\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.82843 + 1.17157i −0.904447 + 0.155179i
\(58\) 0 0
\(59\) −1.41421 −0.184115 −0.0920575 0.995754i \(-0.529344\pi\)
−0.0920575 + 0.995754i \(0.529344\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 0 0
\(63\) −5.48528 + 11.4853i −0.691080 + 1.44701i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.00000 4.00000i 0.488678 0.488678i −0.419211 0.907889i \(-0.637693\pi\)
0.907889 + 0.419211i \(0.137693\pi\)
\(68\) 0 0
\(69\) −5.65685 4.00000i −0.681005 0.481543i
\(70\) 0 0
\(71\) 2.82843i 0.335673i −0.985815 0.167836i \(-0.946322\pi\)
0.985815 0.167836i \(-0.0536780\pi\)
\(72\) 0 0
\(73\) −3.00000 3.00000i −0.351123 0.351123i 0.509404 0.860527i \(-0.329866\pi\)
−0.860527 + 0.509404i \(0.829866\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.24264 + 4.24264i 0.483494 + 0.483494i
\(78\) 0 0
\(79\) 10.0000i 1.12509i −0.826767 0.562544i \(-0.809823\pi\)
0.826767 0.562544i \(-0.190177\pi\)
\(80\) 0 0
\(81\) 7.00000 5.65685i 0.777778 0.628539i
\(82\) 0 0
\(83\) −2.82843 + 2.82843i −0.310460 + 0.310460i −0.845088 0.534628i \(-0.820452\pi\)
0.534628 + 0.845088i \(0.320452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.414214 2.41421i −0.0444084 0.258831i
\(88\) 0 0
\(89\) −2.82843 −0.299813 −0.149906 0.988700i \(-0.547897\pi\)
−0.149906 + 0.988700i \(0.547897\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −0.585786 3.41421i −0.0607432 0.354037i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.0000 13.0000i 1.31995 1.31995i 0.406138 0.913812i \(-0.366875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 0 0
\(99\) −1.41421 4.00000i −0.142134 0.402015i
\(100\) 0 0
\(101\) 15.5563i 1.54791i 0.633238 + 0.773957i \(0.281726\pi\)
−0.633238 + 0.773957i \(0.718274\pi\)
\(102\) 0 0
\(103\) −11.0000 11.0000i −1.08386 1.08386i −0.996145 0.0877167i \(-0.972043\pi\)
−0.0877167 0.996145i \(-0.527957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843 + 2.82843i 0.273434 + 0.273434i 0.830481 0.557047i \(-0.188066\pi\)
−0.557047 + 0.830481i \(0.688066\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) −4.00000 2.82843i −0.379663 0.268462i
\(112\) 0 0
\(113\) 4.24264 4.24264i 0.399114 0.399114i −0.478806 0.877920i \(-0.658930\pi\)
0.877920 + 0.478806i \(0.158930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 9.65685 1.65685i 0.870729 0.149394i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.00000 + 3.00000i −0.266207 + 0.266207i −0.827570 0.561363i \(-0.810277\pi\)
0.561363 + 0.827570i \(0.310277\pi\)
\(128\) 0 0
\(129\) −2.82843 + 4.00000i −0.249029 + 0.352180i
\(130\) 0 0
\(131\) 15.5563i 1.35916i −0.733599 0.679582i \(-0.762161\pi\)
0.733599 0.679582i \(-0.237839\pi\)
\(132\) 0 0
\(133\) −12.0000 12.0000i −1.04053 1.04053i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.7279 + 12.7279i 1.08742 + 1.08742i 0.995793 + 0.0916263i \(0.0292065\pi\)
0.0916263 + 0.995793i \(0.470793\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) −8.00000 + 11.3137i −0.673722 + 0.952786i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −18.7782 + 3.22183i −1.54880 + 0.265732i
\(148\) 0 0
\(149\) 4.24264 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 16.2426 + 7.75736i 1.31314 + 0.627145i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −8.00000 + 8.00000i −0.638470 + 0.638470i −0.950178 0.311708i \(-0.899099\pi\)
0.311708 + 0.950178i \(0.399099\pi\)
\(158\) 0 0
\(159\) 16.9706 + 12.0000i 1.34585 + 0.951662i
\(160\) 0 0
\(161\) 16.9706i 1.33747i
\(162\) 0 0
\(163\) 12.0000 + 12.0000i 0.939913 + 0.939913i 0.998294 0.0583818i \(-0.0185941\pi\)
−0.0583818 + 0.998294i \(0.518594\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 13.0000i 1.00000i
\(170\) 0 0
\(171\) 4.00000 + 11.3137i 0.305888 + 0.865181i
\(172\) 0 0
\(173\) 9.89949 9.89949i 0.752645 0.752645i −0.222327 0.974972i \(-0.571365\pi\)
0.974972 + 0.222327i \(0.0713654\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0.414214 + 2.41421i 0.0311342 + 0.181463i
\(178\) 0 0
\(179\) 15.5563 1.16274 0.581368 0.813641i \(-0.302518\pi\)
0.581368 + 0.813641i \(0.302518\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 1.75736 + 10.2426i 0.129908 + 0.757158i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000 6.00000i 0.438763 0.438763i
\(188\) 0 0
\(189\) 21.2132 + 6.00000i 1.54303 + 0.436436i
\(190\) 0 0
\(191\) 2.82843i 0.204658i 0.994751 + 0.102329i \(0.0326294\pi\)
−0.994751 + 0.102329i \(0.967371\pi\)
\(192\) 0 0
\(193\) 7.00000 + 7.00000i 0.503871 + 0.503871i 0.912639 0.408768i \(-0.134041\pi\)
−0.408768 + 0.912639i \(0.634041\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) −8.00000 5.65685i −0.564276 0.399004i
\(202\) 0 0
\(203\) 4.24264 4.24264i 0.297775 0.297775i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.17157 + 10.8284i −0.359449 + 0.752628i
\(208\) 0 0
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −4.82843 + 0.828427i −0.330838 + 0.0567629i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.00000 6.00000i 0.407307 0.407307i
\(218\) 0 0
\(219\) −4.24264 + 6.00000i −0.286691 + 0.405442i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 3.00000 + 3.00000i 0.200895 + 0.200895i 0.800383 0.599489i \(-0.204629\pi\)
−0.599489 + 0.800383i \(0.704629\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.24264 4.24264i −0.281594 0.281594i 0.552151 0.833744i \(-0.313807\pi\)
−0.833744 + 0.552151i \(0.813807\pi\)
\(228\) 0 0
\(229\) 6.00000i 0.396491i 0.980152 + 0.198246i \(0.0635244\pi\)
−0.980152 + 0.198246i \(0.936476\pi\)
\(230\) 0 0
\(231\) 6.00000 8.48528i 0.394771 0.558291i
\(232\) 0 0
\(233\) −7.07107 + 7.07107i −0.463241 + 0.463241i −0.899716 0.436475i \(-0.856227\pi\)
0.436475 + 0.899716i \(0.356227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −17.0711 + 2.92893i −1.10889 + 0.190255i
\(238\) 0 0
\(239\) 25.4558 1.64660 0.823301 0.567605i \(-0.192130\pi\)
0.823301 + 0.567605i \(0.192130\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 0 0
\(243\) −11.7071 10.2929i −0.751011 0.660289i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 5.65685 + 4.00000i 0.358489 + 0.253490i
\(250\) 0 0
\(251\) 12.7279i 0.803379i −0.915776 0.401690i \(-0.868423\pi\)
0.915776 0.401690i \(-0.131577\pi\)
\(252\) 0 0
\(253\) 4.00000 + 4.00000i 0.251478 + 0.251478i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.5563 15.5563i −0.970378 0.970378i 0.0291953 0.999574i \(-0.490706\pi\)
−0.999574 + 0.0291953i \(0.990706\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) −4.00000 + 1.41421i −0.247594 + 0.0875376i
\(262\) 0 0
\(263\) 11.3137 11.3137i 0.697633 0.697633i −0.266266 0.963899i \(-0.585790\pi\)
0.963899 + 0.266266i \(0.0857901\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0.828427 + 4.82843i 0.0506989 + 0.295495i
\(268\) 0 0
\(269\) 9.89949 0.603583 0.301791 0.953374i \(-0.402415\pi\)
0.301791 + 0.953374i \(0.402415\pi\)
\(270\) 0 0
\(271\) 24.0000 1.45790 0.728948 0.684569i \(-0.240010\pi\)
0.728948 + 0.684569i \(0.240010\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.00000 + 2.00000i −0.120168 + 0.120168i −0.764634 0.644465i \(-0.777080\pi\)
0.644465 + 0.764634i \(0.277080\pi\)
\(278\) 0 0
\(279\) −5.65685 + 2.00000i −0.338667 + 0.119737i
\(280\) 0 0
\(281\) 25.4558i 1.51857i −0.650759 0.759284i \(-0.725549\pi\)
0.650759 0.759284i \(-0.274451\pi\)
\(282\) 0 0
\(283\) −4.00000 4.00000i −0.237775 0.237775i 0.578153 0.815928i \(-0.303774\pi\)
−0.815928 + 0.578153i \(0.803774\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 16.9706 + 16.9706i 1.00174 + 1.00174i
\(288\) 0 0
\(289\) 19.0000i 1.11765i
\(290\) 0 0
\(291\) −26.0000 18.3848i −1.52415 1.07773i
\(292\) 0 0
\(293\) −5.65685 + 5.65685i −0.330477 + 0.330477i −0.852768 0.522291i \(-0.825078\pi\)
0.522291 + 0.852768i \(0.325078\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −6.41421 + 3.58579i −0.372190 + 0.208068i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 26.5563 4.55635i 1.52562 0.261755i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.0000 10.0000i 0.570730 0.570730i −0.361602 0.932332i \(-0.617770\pi\)
0.932332 + 0.361602i \(0.117770\pi\)
\(308\) 0 0
\(309\) −15.5563 + 22.0000i −0.884970 + 1.25154i
\(310\) 0 0
\(311\) 31.1127i 1.76424i 0.471025 + 0.882120i \(0.343884\pi\)
−0.471025 + 0.882120i \(0.656116\pi\)
\(312\) 0 0
\(313\) 15.0000 + 15.0000i 0.847850 + 0.847850i 0.989865 0.142014i \(-0.0453579\pi\)
−0.142014 + 0.989865i \(0.545358\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.41421 1.41421i −0.0794301 0.0794301i 0.666276 0.745706i \(-0.267887\pi\)
−0.745706 + 0.666276i \(0.767887\pi\)
\(318\) 0 0
\(319\) 2.00000i 0.111979i
\(320\) 0 0
\(321\) 4.00000 5.65685i 0.223258 0.315735i
\(322\) 0 0
\(323\) −16.9706 + 16.9706i −0.944267 + 0.944267i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 17.0711 2.92893i 0.944032 0.161970i
\(328\) 0 0
\(329\) −33.9411 −1.87123
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) 0 0
\(333\) −3.65685 + 7.65685i −0.200394 + 0.419593i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −15.0000 + 15.0000i −0.817102 + 0.817102i −0.985687 0.168585i \(-0.946080\pi\)
0.168585 + 0.985687i \(0.446080\pi\)
\(338\) 0 0
\(339\) −8.48528 6.00000i −0.460857 0.325875i
\(340\) 0 0
\(341\) 2.82843i 0.153168i
\(342\) 0 0
\(343\) −12.0000 12.0000i −0.647939 0.647939i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.7279 + 12.7279i 0.683271 + 0.683271i 0.960736 0.277465i \(-0.0894943\pi\)
−0.277465 + 0.960736i \(0.589494\pi\)
\(348\) 0 0
\(349\) 2.00000i 0.107058i 0.998566 + 0.0535288i \(0.0170469\pi\)
−0.998566 + 0.0535288i \(0.982953\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.3848 18.3848i 0.978523 0.978523i −0.0212513 0.999774i \(-0.506765\pi\)
0.999774 + 0.0212513i \(0.00676499\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.45584 + 43.4558i 0.394605 + 2.29993i
\(358\) 0 0
\(359\) 22.6274 1.19423 0.597115 0.802156i \(-0.296314\pi\)
0.597115 + 0.802156i \(0.296314\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) −2.63604 15.3640i −0.138356 0.806399i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −7.00000 + 7.00000i −0.365397 + 0.365397i −0.865795 0.500398i \(-0.833187\pi\)
0.500398 + 0.865795i \(0.333187\pi\)
\(368\) 0 0
\(369\) −5.65685 16.0000i −0.294484 0.832927i
\(370\) 0 0
\(371\) 50.9117i 2.64320i
\(372\) 0 0
\(373\) −12.0000 12.0000i −0.621336 0.621336i 0.324537 0.945873i \(-0.394792\pi\)
−0.945873 + 0.324537i \(0.894792\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 24.0000i 1.23280i 0.787434 + 0.616399i \(0.211409\pi\)
−0.787434 + 0.616399i \(0.788591\pi\)
\(380\) 0 0
\(381\) 6.00000 + 4.24264i 0.307389 + 0.217357i
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.65685 + 3.65685i 0.389220 + 0.185888i
\(388\) 0 0
\(389\) 12.7279 0.645331 0.322666 0.946513i \(-0.395421\pi\)
0.322666 + 0.946513i \(0.395421\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 0 0
\(393\) −26.5563 + 4.55635i −1.33959 + 0.229837i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 10.0000 10.0000i 0.501886 0.501886i −0.410138 0.912024i \(-0.634519\pi\)
0.912024 + 0.410138i \(0.134519\pi\)
\(398\) 0 0
\(399\) −16.9706 + 24.0000i −0.849591 + 1.20150i
\(400\) 0 0
\(401\) 14.1421i 0.706225i 0.935581 + 0.353112i \(0.114877\pi\)
−0.935581 + 0.353112i \(0.885123\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.82843 + 2.82843i 0.140200 + 0.140200i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 18.0000 25.4558i 0.887875 1.25564i
\(412\) 0 0
\(413\) −4.24264 + 4.24264i −0.208767 + 0.208767i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 27.3137 4.68629i 1.33756 0.229489i
\(418\) 0 0
\(419\) −24.0416 −1.17451 −0.587255 0.809402i \(-0.699792\pi\)
−0.587255 + 0.809402i \(0.699792\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 0 0
\(423\) 21.6569 + 10.3431i 1.05299 + 0.502901i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18.0000 + 18.0000i −0.871081 + 0.871081i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.6274i 1.08992i −0.838461 0.544962i \(-0.816544\pi\)
0.838461 0.544962i \(-0.183456\pi\)
\(432\) 0 0
\(433\) 1.00000 + 1.00000i 0.0480569 + 0.0480569i 0.730727 0.682670i \(-0.239181\pi\)
−0.682670 + 0.730727i \(0.739181\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.3137 11.3137i −0.541208 0.541208i
\(438\) 0 0
\(439\) 8.00000i 0.381819i −0.981608 0.190910i \(-0.938856\pi\)
0.981608 0.190910i \(-0.0611437\pi\)
\(440\) 0 0
\(441\) 11.0000 + 31.1127i 0.523810 + 1.48156i
\(442\) 0 0
\(443\) −7.07107 + 7.07107i −0.335957 + 0.335957i −0.854843 0.518887i \(-0.826347\pi\)
0.518887 + 0.854843i \(0.326347\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −1.24264 7.24264i −0.0587749 0.342565i
\(448\) 0 0
\(449\) 19.7990 0.934372 0.467186 0.884159i \(-0.345268\pi\)
0.467186 + 0.884159i \(0.345268\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 2.34315 + 13.6569i 0.110091 + 0.641655i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 + 1.00000i −0.0467780 + 0.0467780i −0.730109 0.683331i \(-0.760531\pi\)
0.683331 + 0.730109i \(0.260531\pi\)
\(458\) 0 0
\(459\) 8.48528 30.0000i 0.396059 1.40028i
\(460\) 0 0
\(461\) 32.5269i 1.51493i −0.652876 0.757465i \(-0.726438\pi\)
0.652876 0.757465i \(-0.273562\pi\)
\(462\) 0 0
\(463\) 1.00000 + 1.00000i 0.0464739 + 0.0464739i 0.729962 0.683488i \(-0.239538\pi\)
−0.683488 + 0.729962i \(0.739538\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.4558 25.4558i −1.17796 1.17796i −0.980264 0.197692i \(-0.936655\pi\)
−0.197692 0.980264i \(-0.563345\pi\)
\(468\) 0 0
\(469\) 24.0000i 1.10822i
\(470\) 0 0
\(471\) 16.0000 + 11.3137i 0.737241 + 0.521308i
\(472\) 0 0
\(473\) 2.82843 2.82843i 0.130051 0.130051i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 15.5147 32.4853i 0.710370 1.48740i
\(478\) 0 0
\(479\) −2.82843 −0.129234 −0.0646171 0.997910i \(-0.520583\pi\)
−0.0646171 + 0.997910i \(0.520583\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −28.9706 + 4.97056i −1.31821 + 0.226168i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.00000 + 3.00000i −0.135943 + 0.135943i −0.771804 0.635861i \(-0.780645\pi\)
0.635861 + 0.771804i \(0.280645\pi\)
\(488\) 0 0
\(489\) 16.9706 24.0000i 0.767435 1.08532i
\(490\) 0 0
\(491\) 15.5563i 0.702048i −0.936366 0.351024i \(-0.885834\pi\)
0.936366 0.351024i \(-0.114166\pi\)
\(492\) 0 0
\(493\) −6.00000 6.00000i −0.270226 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.48528 8.48528i −0.380617 0.380617i
\(498\) 0 0
\(499\) 20.0000i 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −25.4558 + 25.4558i −1.13502 + 1.13502i −0.145690 + 0.989330i \(0.546540\pi\)
−0.989330 + 0.145690i \(0.953460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −22.1924 + 3.80761i −0.985599 + 0.169102i
\(508\) 0 0
\(509\) −38.1838 −1.69247 −0.846233 0.532813i \(-0.821135\pi\)
−0.846233 + 0.532813i \(0.821135\pi\)
\(510\) 0 0
\(511\) −18.0000 −0.796273
\(512\) 0 0
\(513\) 18.1421 10.1421i 0.800995 0.447786i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 8.00000i 0.351840 0.351840i
\(518\) 0 0
\(519\) −19.7990 14.0000i −0.869079 0.614532i
\(520\) 0 0
\(521\) 8.48528i 0.371747i 0.982574 + 0.185873i \(0.0595115\pi\)
−0.982574 + 0.185873i \(0.940489\pi\)
\(522\) 0 0
\(523\) 24.0000 + 24.0000i 1.04945 + 1.04945i 0.998712 + 0.0507346i \(0.0161562\pi\)
0.0507346 + 0.998712i \(0.483844\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.48528 8.48528i −0.369625 0.369625i
\(528\) 0 0
\(529\) 7.00000i 0.304348i
\(530\) 0 0
\(531\) 4.00000 1.41421i 0.173585 0.0613716i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.55635 26.5563i −0.196621 1.14599i
\(538\) 0 0
\(539\) 15.5563 0.670059
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) −2.92893 17.0711i −0.125693 0.732590i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 30.0000 30.0000i 1.28271 1.28271i 0.343586 0.939121i \(-0.388358\pi\)
0.939121 0.343586i \(-0.111642\pi\)
\(548\) 0 0
\(549\) 16.9706 6.00000i 0.724286 0.256074i
\(550\) 0 0
\(551\) 5.65685i 0.240990i
\(552\) 0 0
\(553\) −30.0000 30.0000i −1.27573 1.27573i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.9706 + 16.9706i 0.719066 + 0.719066i 0.968414 0.249348i \(-0.0802163\pi\)
−0.249348 + 0.968414i \(0.580216\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −12.0000 8.48528i −0.506640 0.358249i
\(562\) 0 0
\(563\) 24.0416 24.0416i 1.01323 1.01323i 0.0133227 0.999911i \(-0.495759\pi\)
0.999911 0.0133227i \(-0.00424086\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.02944 37.9706i 0.169220 1.59461i
\(568\) 0 0
\(569\) 14.1421 0.592869 0.296435 0.955053i \(-0.404202\pi\)
0.296435 + 0.955053i \(0.404202\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 4.82843 0.828427i 0.201710 0.0346080i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −21.0000 + 21.0000i −0.874241 + 0.874241i −0.992931 0.118690i \(-0.962131\pi\)
0.118690 + 0.992931i \(0.462131\pi\)
\(578\) 0 0
\(579\) 9.89949 14.0000i 0.411409 0.581820i
\(580\) 0 0
\(581\) 16.9706i 0.704058i
\(582\) 0 0
\(583\) −12.0000 12.0000i −0.496989 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.7990 19.7990i −0.817192 0.817192i 0.168508 0.985700i \(-0.446105\pi\)
−0.985700 + 0.168508i \(0.946105\pi\)
\(588\) 0 0
\(589\) 8.00000i 0.329634i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.41421 + 1.41421i −0.0580748 + 0.0580748i −0.735548 0.677473i \(-0.763075\pi\)
0.677473 + 0.735548i \(0.263075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.3137 0.462266 0.231133 0.972922i \(-0.425757\pi\)
0.231133 + 0.972922i \(0.425757\pi\)
\(600\) 0 0
\(601\) 22.0000 0.897399 0.448699 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −7.31371 + 15.3137i −0.297837 + 0.623622i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −25.0000 + 25.0000i −1.01472 + 1.01472i −0.0148286 + 0.999890i \(0.504720\pi\)
−0.999890 + 0.0148286i \(0.995280\pi\)
\(608\) 0 0
\(609\) −8.48528 6.00000i −0.343841 0.243132i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 30.0000 + 30.0000i 1.21169 + 1.21169i 0.970471 + 0.241218i \(0.0775467\pi\)
0.241218 + 0.970471i \(0.422453\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.24264 + 4.24264i 0.170802 + 0.170802i 0.787332 0.616530i \(-0.211462\pi\)
−0.616530 + 0.787332i \(0.711462\pi\)
\(618\) 0 0
\(619\) 48.0000i 1.92928i 0.263566 + 0.964641i \(0.415101\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(620\) 0 0
\(621\) 20.0000 + 5.65685i 0.802572 + 0.227002i
\(622\) 0 0
\(623\) −8.48528 + 8.48528i −0.339956 + 0.339956i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.65685 9.65685i −0.0661684 0.385658i
\(628\) 0 0
\(629\) −16.9706 −0.676661
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) −4.68629 27.3137i −0.186263 1.08562i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 2.82843 + 8.00000i 0.111891 + 0.316475i
\(640\) 0 0
\(641\) 28.2843i 1.11716i 0.829450 + 0.558581i \(0.188654\pi\)
−0.829450 + 0.558581i \(0.811346\pi\)
\(642\) 0 0
\(643\) 8.00000 + 8.00000i 0.315489 + 0.315489i 0.847032 0.531542i \(-0.178387\pi\)
−0.531542 + 0.847032i \(0.678387\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.82843 2.82843i −0.111197 0.111197i 0.649319 0.760516i \(-0.275054\pi\)
−0.760516 + 0.649319i \(0.775054\pi\)
\(648\) 0 0
\(649\) 2.00000i 0.0785069i
\(650\) 0 0
\(651\) −12.0000 8.48528i −0.470317 0.332564i
\(652\) 0 0
\(653\) 4.24264 4.24264i 0.166027 0.166027i −0.619203 0.785231i \(-0.712544\pi\)
0.785231 + 0.619203i \(0.212544\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 11.4853 + 5.48528i 0.448084 + 0.214001i
\(658\) 0 0
\(659\) −46.6690 −1.81797 −0.908984 0.416831i \(-0.863141\pi\)
−0.908984 + 0.416831i \(0.863141\pi\)
\(660\) 0 0
\(661\) −14.0000 −0.544537 −0.272268 0.962221i \(-0.587774\pi\)
−0.272268 + 0.962221i \(0.587774\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.00000 4.00000i 0.154881 0.154881i
\(668\) 0 0
\(669\) 4.24264 6.00000i 0.164030 0.231973i
\(670\) 0 0
\(671\) 8.48528i 0.327571i
\(672\) 0 0
\(673\) 19.0000 + 19.0000i 0.732396 + 0.732396i 0.971094 0.238698i \(-0.0767205\pi\)
−0.238698 + 0.971094i \(0.576721\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 18.3848 + 18.3848i 0.706584 + 0.706584i 0.965815 0.259231i \(-0.0834691\pi\)
−0.259231 + 0.965815i \(0.583469\pi\)
\(678\) 0 0
\(679\) 78.0000i 2.99337i
\(680\) 0 0
\(681\) −6.00000 + 8.48528i −0.229920 + 0.325157i
\(682\) 0 0
\(683\) 25.4558 25.4558i 0.974041 0.974041i −0.0256307 0.999671i \(-0.508159\pi\)
0.999671 + 0.0256307i \(0.00815939\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.2426 1.75736i 0.390781 0.0670474i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 0 0
\(693\) −16.2426 7.75736i −0.617007 0.294678i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 24.0000 24.0000i 0.909065 0.909065i
\(698\) 0 0
\(699\) 14.1421 + 10.0000i 0.534905 + 0.378235i
\(700\) 0 0
\(701\) 1.41421i 0.0534141i 0.999643 + 0.0267071i \(0.00850213\pi\)
−0.999643 + 0.0267071i \(0.991498\pi\)
\(702\) 0 0
\(703\) −8.00000 8.00000i −0.301726 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 46.6690 + 46.6690i 1.75517 + 1.75517i
\(708\) 0 0
\(709\) 6.00000i 0.225335i 0.993633 + 0.112667i \(0.0359394\pi\)
−0.993633 + 0.112667i \(0.964061\pi\)
\(710\) 0 0
\(711\) 10.0000 + 28.2843i 0.375029 + 1.06074i
\(712\) 0 0
\(713\) 5.65685 5.65685i 0.211851 0.211851i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.45584 43.4558i −0.278444 1.62289i
\(718\) 0 0
\(719\) −45.2548 −1.68772 −0.843860 0.536563i \(-0.819722\pi\)
−0.843860 + 0.536563i \(0.819722\pi\)
\(720\) 0 0
\(721\) −66.0000 −2.45797
\(722\) 0 0
\(723\) −5.85786 34.1421i −0.217856 1.26976i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −15.0000 + 15.0000i −0.556319 + 0.556319i −0.928257 0.371938i \(-0.878693\pi\)
0.371938 + 0.928257i \(0.378693\pi\)
\(728\) 0 0
\(729\) −14.1421 + 23.0000i −0.523783 + 0.851852i
\(730\) 0 0
\(731\) 16.9706i 0.627679i
\(732\) 0 0
\(733\) −14.0000 14.0000i −0.517102 0.517102i 0.399592 0.916693i \(-0.369152\pi\)
−0.916693 + 0.399592i \(0.869152\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685 + 5.65685i 0.208373 + 0.208373i
\(738\) 0 0
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.4558 + 25.4558i −0.933884 + 0.933884i −0.997946 0.0640616i \(-0.979595\pi\)
0.0640616 + 0.997946i \(0.479595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.17157 10.8284i 0.189218 0.396191i
\(748\) 0 0
\(749\) 16.9706 0.620091
\(750\) 0 0
\(751\) 30.0000 1.09472 0.547358 0.836899i \(-0.315634\pi\)
0.547358 + 0.836899i \(0.315634\pi\)
\(752\) 0 0
\(753\) −21.7279 + 3.72792i −0.791809 + 0.135853i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.00000 6.00000i 0.218074 0.218074i −0.589613 0.807686i \(-0.700720\pi\)
0.807686 + 0.589613i \(0.200720\pi\)
\(758\) 0 0
\(759\) 5.65685 8.00000i 0.205331 0.290382i
\(760\) 0 0
\(761\) 19.7990i 0.717713i −0.933393 0.358856i \(-0.883167\pi\)
0.933393 0.358856i \(-0.116833\pi\)
\(762\) 0 0
\(763\) 30.0000 + 30.0000i 1.08607 + 1.08607i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 10.0000i 0.360609i −0.983611 0.180305i \(-0.942292\pi\)
0.983611 0.180305i \(-0.0577084\pi\)
\(770\) 0 0
\(771\) −22.0000 + 31.1127i −0.792311 + 1.12050i
\(772\) 0 0
\(773\) 18.3848 18.3848i 0.661254 0.661254i −0.294421 0.955676i \(-0.595127\pi\)
0.955676 + 0.294421i \(0.0951269\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −20.4853 + 3.51472i −0.734905 + 0.126090i
\(778\) 0 0
\(779\) 22.6274 0.810711
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 3.58579 + 6.41421i 0.128146 + 0.229225i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −12.0000 + 12.0000i −0.427754 + 0.427754i −0.887863 0.460109i \(-0.847810\pi\)
0.460109 + 0.887863i \(0.347810\pi\)
\(788\) 0 0
\(789\) −22.6274 16.0000i −0.805557 0.569615i
\(790\) 0 0
\(791\) 25.4558i 0.905106i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.2132 21.2132i −0.751410 0.751410i 0.223332 0.974742i \(-0.428307\pi\)
−0.974742 + 0.223332i \(0.928307\pi\)
\(798\) 0 0
\(799\) 48.0000i 1.69812i
\(800\) 0 0
\(801\) 8.00000 2.82843i 0.282666 0.0999376i
\(802\) 0 0
\(803\) 4.24264 4.24264i 0.149720 0.149720i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.89949 16.8995i −0.102067 0.594890i
\(808\) 0 0
\(809\) −45.2548 −1.59108 −0.795538 0.605904i \(-0.792811\pi\)
−0.795538 + 0.605904i \(0.792811\pi\)
\(810\) 0 0
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) 0 0
\(813\) −7.02944 40.9706i −0.246533 1.43690i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.00000 + 8.00000i −0.279885 + 0.279885i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 26.8701i 0.937771i −0.883259 0.468886i \(-0.844656\pi\)
0.883259 0.468886i \(-0.155344\pi\)
\(822\) 0 0
\(823\) −29.0000 29.0000i −1.01088 1.01088i −0.999940 0.0109363i \(-0.996519\pi\)
−0.0109363 0.999940i \(-0.503481\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.7279 + 12.7279i 0.442593 + 0.442593i 0.892883 0.450289i \(-0.148679\pi\)
−0.450289 + 0.892883i \(0.648679\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i −0.969996 0.243120i \(-0.921829\pi\)
0.969996 0.243120i \(-0.0781709\pi\)
\(830\) 0 0
\(831\) 4.00000 + 2.82843i 0.138758 + 0.0981170i
\(832\) 0 0
\(833\) −46.6690 + 46.6690i −1.61699 + 1.61699i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 5.07107 + 9.07107i 0.175282 + 0.313542i
\(838\) 0 0
\(839\) −39.5980 −1.36707 −0.683537 0.729916i \(-0.739559\pi\)
−0.683537 + 0.729916i \(0.739559\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) −43.4558 + 7.45584i −1.49670 + 0.256793i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 27.0000 27.0000i 0.927731 0.927731i
\(848\) 0 0
\(849\) −5.65685 + 8.00000i −0.194143 + 0.274559i
\(850\) 0 0
\(851\) 11.3137i 0.387829i
\(852\) 0 0
\(853\) 18.0000 + 18.0000i 0.616308 + 0.616308i 0.944582 0.328274i \(-0.106467\pi\)
−0.328274 + 0.944582i \(0.606467\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.7279 + 12.7279i 0.434778 + 0.434778i 0.890250 0.455472i \(-0.150530\pi\)
−0.455472 + 0.890250i \(0.650530\pi\)
\(858\) 0 0
\(859\) 40.0000i 1.36478i −0.730987 0.682391i \(-0.760940\pi\)
0.730987 0.682391i \(-0.239060\pi\)
\(860\) 0 0
\(861\) 24.0000 33.9411i 0.817918 1.15671i
\(862\) 0 0
\(863\) 25.4558 25.4558i 0.866527 0.866527i −0.125559 0.992086i \(-0.540072\pi\)
0.992086 + 0.125559i \(0.0400725\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 32.4350 5.56497i 1.10155 0.188996i
\(868\) 0 0
\(869\) 14.1421 0.479739
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −23.7696 + 49.7696i −0.804477 + 1.68444i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.0000 + 36.0000i −1.21563 + 1.21563i −0.246488 + 0.969146i \(0.579276\pi\)
−0.969146 + 0.246488i \(0.920724\pi\)
\(878\) 0 0
\(879\) 11.3137 + 8.00000i 0.381602 + 0.269833i
\(880\) 0 0
\(881\) 19.7990i 0.667045i 0.942742 + 0.333522i \(0.108237\pi\)
−0.942742 + 0.333522i \(0.891763\pi\)
\(882\) 0 0
\(883\) −12.0000 12.0000i −0.403832 0.403832i 0.475749 0.879581i \(-0.342177\pi\)
−0.879581 + 0.475749i \(0.842177\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.7696 36.7696i −1.23460 1.23460i −0.962178 0.272423i \(-0.912175\pi\)
−0.272423 0.962178i \(-0.587825\pi\)
\(888\) 0 0
\(889\) 18.0000i 0.603701i
\(890\) 0 0
\(891\) 8.00000 + 9.89949i 0.268010 + 0.331646i
\(892\) 0 0
\(893\) −22.6274 + 22.6274i −0.757198 + 0.757198i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.82843 0.0943333
\(900\) 0 0
\(901\) 72.0000 2.39867
\(902\) 0 0
\(903\) 3.51472 + 20.4853i 0.116963 + 0.681707i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −26.0000 + 26.0000i −0.863316 + 0.863316i −0.991722 0.128406i \(-0.959014\pi\)
0.128406 + 0.991722i \(0.459014\pi\)
\(908\) 0 0
\(909\) −15.5563 44.0000i −0.515972 1.45939i
\(910\) 0 0
\(911\) 5.65685i 0.187420i −0.995600 0.0937100i \(-0.970127\pi\)
0.995600 0.0937100i \(-0.0298726\pi\)
\(912\) 0 0
\(913\) −4.00000 4.00000i −0.132381 0.132381i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −46.6690 46.6690i −1.54115 1.54115i
\(918\) 0 0
\(919\) 2.00000i 0.0659739i −0.999456 0.0329870i \(-0.989498\pi\)
0.999456 0.0329870i \(-0.0105020\pi\)
\(920\) 0 0
\(921\) −20.0000 14.1421i −0.659022 0.465999i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 42.1127 + 20.1127i 1.38316 + 0.660588i
\(928\) 0 0
\(929\) 36.7696 1.20637 0.603185 0.797601i \(-0.293898\pi\)
0.603185 + 0.797601i \(0.293898\pi\)
\(930\) 0 0
\(931\) −44.0000 −1.44204
\(932\) 0 0
\(933\) 53.1127 9.11270i 1.73883 0.298336i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −11.0000 + 11.0000i −0.359354 + 0.359354i −0.863575 0.504221i \(-0.831780\pi\)
0.504221 + 0.863575i \(0.331780\pi\)
\(938\) 0 0
\(939\) 21.2132 30.0000i 0.692267 0.979013i
\(940\) 0 0
\(941\) 38.1838i 1.24476i 0.782717 + 0.622378i \(0.213833\pi\)
−0.782717 + 0.622378i \(0.786167\pi\)
\(942\) 0 0
\(943\) 16.0000 + 16.0000i 0.521032 + 0.521032i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.5563 15.5563i −0.505513 0.505513i 0.407633 0.913146i \(-0.366354\pi\)
−0.913146 + 0.407633i \(0.866354\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.00000 + 2.82843i −0.0648544 + 0.0917180i
\(952\) 0 0
\(953\) −21.2132 + 21.2132i −0.687163 + 0.687163i −0.961604 0.274441i \(-0.911507\pi\)
0.274441 + 0.961604i \(0.411507\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.41421 0.585786i 0.110366 0.0189358i
\(958\) 0 0
\(959\) 76.3675 2.46604
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) −10.8284 5.17157i −0.348941 0.166652i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 15.0000 15.0000i 0.482367 0.482367i −0.423520 0.905887i \(-0.639205\pi\)
0.905887 + 0.423520i \(0.139205\pi\)
\(968\) 0 0
\(969\) 33.9411 + 24.0000i 1.09035 + 0.770991i
\(970\) 0 0
\(971\) 41.0122i 1.31614i 0.752955 + 0.658072i \(0.228628\pi\)
−0.752955 + 0.658072i \(0.771372\pi\)
\(972\) 0 0
\(973\) 48.0000 + 48.0000i 1.53881 + 1.53881i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.2132 + 21.2132i 0.678671 + 0.678671i 0.959699 0.281029i \(-0.0906757\pi\)
−0.281029 + 0.959699i \(0.590676\pi\)
\(978\) 0 0
\(979\) 4.00000i 0.127841i
\(980\) 0 0
\(981\) −10.0000 28.2843i −0.319275 0.903047i
\(982\) 0 0
\(983\) −8.48528 + 8.48528i −0.270638 + 0.270638i −0.829357 0.558719i \(-0.811293\pi\)
0.558719 + 0.829357i \(0.311293\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 9.94113 + 57.9411i 0.316430 + 1.84429i
\(988\) 0 0
\(989\) −11.3137 −0.359755
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −4.68629 27.3137i −0.148715 0.866774i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −24.0000 + 24.0000i −0.760088 + 0.760088i −0.976338 0.216250i \(-0.930617\pi\)
0.216250 + 0.976338i \(0.430617\pi\)
\(998\) 0 0
\(999\) 14.1421 + 4.00000i 0.447437 + 0.126554i
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 1200.2.v.c.593.2 4
3.2 odd 2 inner 1200.2.v.c.593.1 4
4.3 odd 2 600.2.r.d.593.1 4
5.2 odd 4 inner 1200.2.v.c.257.1 4
5.3 odd 4 240.2.v.d.17.2 4
5.4 even 2 240.2.v.d.113.1 4
12.11 even 2 600.2.r.d.593.2 4
15.2 even 4 inner 1200.2.v.c.257.2 4
15.8 even 4 240.2.v.d.17.1 4
15.14 odd 2 240.2.v.d.113.2 4
20.3 even 4 120.2.r.a.17.1 4
20.7 even 4 600.2.r.d.257.2 4
20.19 odd 2 120.2.r.a.113.2 yes 4
40.3 even 4 960.2.v.l.257.2 4
40.13 odd 4 960.2.v.b.257.1 4
40.19 odd 2 960.2.v.l.833.1 4
40.29 even 2 960.2.v.b.833.2 4
60.23 odd 4 120.2.r.a.17.2 yes 4
60.47 odd 4 600.2.r.d.257.1 4
60.59 even 2 120.2.r.a.113.1 yes 4
120.29 odd 2 960.2.v.b.833.1 4
120.53 even 4 960.2.v.b.257.2 4
120.59 even 2 960.2.v.l.833.2 4
120.83 odd 4 960.2.v.l.257.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.r.a.17.1 4 20.3 even 4
120.2.r.a.17.2 yes 4 60.23 odd 4
120.2.r.a.113.1 yes 4 60.59 even 2
120.2.r.a.113.2 yes 4 20.19 odd 2
240.2.v.d.17.1 4 15.8 even 4
240.2.v.d.17.2 4 5.3 odd 4
240.2.v.d.113.1 4 5.4 even 2
240.2.v.d.113.2 4 15.14 odd 2
600.2.r.d.257.1 4 60.47 odd 4
600.2.r.d.257.2 4 20.7 even 4
600.2.r.d.593.1 4 4.3 odd 2
600.2.r.d.593.2 4 12.11 even 2
960.2.v.b.257.1 4 40.13 odd 4
960.2.v.b.257.2 4 120.53 even 4
960.2.v.b.833.1 4 120.29 odd 2
960.2.v.b.833.2 4 40.29 even 2
960.2.v.l.257.1 4 120.83 odd 4
960.2.v.l.257.2 4 40.3 even 4
960.2.v.l.833.1 4 40.19 odd 2
960.2.v.l.833.2 4 120.59 even 2
1200.2.v.c.257.1 4 5.2 odd 4 inner
1200.2.v.c.257.2 4 15.2 even 4 inner
1200.2.v.c.593.1 4 3.2 odd 2 inner
1200.2.v.c.593.2 4 1.1 even 1 trivial