Properties

Label 912.2.bb.h.31.1
Level $912$
Weight $2$
Character 912.31
Analytic conductor $7.282$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(31,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 30x^{5} - 5x^{4} + 114x^{3} + 300x^{2} + 116x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(3.14556 + 0.349646i\) of defining polynomial
Character \(\chi\) \(=\) 912.31
Dual form 912.2.bb.h.559.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.73843 - 3.01105i) q^{5} +3.36658i q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(-1.73843 - 3.01105i) q^{5} +3.36658i q^{7} +(-0.500000 + 0.866025i) q^{9} +2.43134i q^{11} +(-5.02115 - 2.89896i) q^{13} +(1.73843 - 3.01105i) q^{15} +(2.67711 + 4.63689i) q^{17} +(-2.84404 - 3.30325i) q^{19} +(-2.91554 + 1.68329i) q^{21} +(1.30585 + 0.753935i) q^{23} +(-3.54429 + 6.13888i) q^{25} -1.00000 q^{27} +(-4.21121 - 2.43134i) q^{29} -10.5654 q^{31} +(-2.10560 + 1.21567i) q^{33} +(10.1369 - 5.85256i) q^{35} +8.46521i q^{37} -5.79792i q^{39} +(-3.41146 + 1.96961i) q^{41} +(-9.02523 + 5.21072i) q^{43} +3.47686 q^{45} +(9.63084 + 5.56037i) q^{47} -4.33385 q^{49} +(-2.67711 + 4.63689i) q^{51} +(-4.41554 - 2.54931i) q^{53} +(7.32090 - 4.22672i) q^{55} +(1.43868 - 4.11463i) q^{57} +(5.58247 + 9.66912i) q^{59} +(-2.80994 + 4.86695i) q^{61} +(-2.91554 - 1.68329i) q^{63} +20.1586i q^{65} +(3.03818 - 5.26229i) q^{67} +1.50787i q^{69} +(-6.15397 - 10.6590i) q^{71} +(-3.97686 - 6.88813i) q^{73} -7.08857 q^{75} -8.18531 q^{77} +(1.19415 + 2.06832i) q^{79} +(-0.500000 - 0.866025i) q^{81} +5.59111i q^{83} +(9.30795 - 16.1218i) q^{85} -4.86269i q^{87} +(-5.04638 - 2.91353i) q^{89} +(9.75958 - 16.9041i) q^{91} +(-5.28272 - 9.14993i) q^{93} +(-5.00209 + 14.3060i) q^{95} +(2.47885 - 1.43117i) q^{97} +(-2.10560 - 1.21567i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 2 q^{5} - 4 q^{9} + 2 q^{15} + 4 q^{17} + 6 q^{21} + 6 q^{23} - 12 q^{25} - 8 q^{27} - 12 q^{29} - 28 q^{31} - 6 q^{33} + 18 q^{35} - 12 q^{41} - 18 q^{43} + 4 q^{45} + 12 q^{47} - 24 q^{49} - 4 q^{51} - 6 q^{53} + 12 q^{55} + 6 q^{57} + 10 q^{59} - 4 q^{61} + 6 q^{63} + 6 q^{67} - 8 q^{71} - 8 q^{73} - 24 q^{75} + 28 q^{77} + 14 q^{79} - 4 q^{81} - 8 q^{85} + 54 q^{89} + 26 q^{91} - 14 q^{93} + 38 q^{95} + 60 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(-1\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −1.73843 3.01105i −0.777450 1.34658i −0.933407 0.358819i \(-0.883180\pi\)
0.155957 0.987764i \(-0.450154\pi\)
\(6\) 0 0
\(7\) 3.36658i 1.27245i 0.771505 + 0.636223i \(0.219504\pi\)
−0.771505 + 0.636223i \(0.780496\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.43134i 0.733078i 0.930403 + 0.366539i \(0.119457\pi\)
−0.930403 + 0.366539i \(0.880543\pi\)
\(12\) 0 0
\(13\) −5.02115 2.89896i −1.39262 0.804027i −0.399011 0.916946i \(-0.630647\pi\)
−0.993604 + 0.112919i \(0.963980\pi\)
\(14\) 0 0
\(15\) 1.73843 3.01105i 0.448861 0.777450i
\(16\) 0 0
\(17\) 2.67711 + 4.63689i 0.649295 + 1.12461i 0.983292 + 0.182037i \(0.0582692\pi\)
−0.333997 + 0.942574i \(0.608397\pi\)
\(18\) 0 0
\(19\) −2.84404 3.30325i −0.652467 0.757817i
\(20\) 0 0
\(21\) −2.91554 + 1.68329i −0.636223 + 0.367324i
\(22\) 0 0
\(23\) 1.30585 + 0.753935i 0.272289 + 0.157206i 0.629928 0.776654i \(-0.283085\pi\)
−0.357638 + 0.933860i \(0.616418\pi\)
\(24\) 0 0
\(25\) −3.54429 + 6.13888i −0.708857 + 1.22778i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.21121 2.43134i −0.782002 0.451489i 0.0551373 0.998479i \(-0.482440\pi\)
−0.837139 + 0.546990i \(0.815774\pi\)
\(30\) 0 0
\(31\) −10.5654 −1.89761 −0.948804 0.315866i \(-0.897705\pi\)
−0.948804 + 0.315866i \(0.897705\pi\)
\(32\) 0 0
\(33\) −2.10560 + 1.21567i −0.366539 + 0.211621i
\(34\) 0 0
\(35\) 10.1369 5.85256i 1.71346 0.989264i
\(36\) 0 0
\(37\) 8.46521i 1.39167i 0.718201 + 0.695836i \(0.244966\pi\)
−0.718201 + 0.695836i \(0.755034\pi\)
\(38\) 0 0
\(39\) 5.79792i 0.928410i
\(40\) 0 0
\(41\) −3.41146 + 1.96961i −0.532780 + 0.307601i −0.742148 0.670236i \(-0.766193\pi\)
0.209368 + 0.977837i \(0.432859\pi\)
\(42\) 0 0
\(43\) −9.02523 + 5.21072i −1.37633 + 0.794627i −0.991716 0.128448i \(-0.959000\pi\)
−0.384619 + 0.923076i \(0.625667\pi\)
\(44\) 0 0
\(45\) 3.47686 0.518300
\(46\) 0 0
\(47\) 9.63084 + 5.56037i 1.40480 + 0.811063i 0.994881 0.101058i \(-0.0322228\pi\)
0.409921 + 0.912121i \(0.365556\pi\)
\(48\) 0 0
\(49\) −4.33385 −0.619121
\(50\) 0 0
\(51\) −2.67711 + 4.63689i −0.374871 + 0.649295i
\(52\) 0 0
\(53\) −4.41554 2.54931i −0.606521 0.350175i 0.165081 0.986280i \(-0.447211\pi\)
−0.771603 + 0.636105i \(0.780545\pi\)
\(54\) 0 0
\(55\) 7.32090 4.22672i 0.987150 0.569931i
\(56\) 0 0
\(57\) 1.43868 4.11463i 0.190558 0.544996i
\(58\) 0 0
\(59\) 5.58247 + 9.66912i 0.726775 + 1.25881i 0.958239 + 0.285968i \(0.0923152\pi\)
−0.231464 + 0.972844i \(0.574351\pi\)
\(60\) 0 0
\(61\) −2.80994 + 4.86695i −0.359776 + 0.623150i −0.987923 0.154945i \(-0.950480\pi\)
0.628148 + 0.778094i \(0.283813\pi\)
\(62\) 0 0
\(63\) −2.91554 1.68329i −0.367324 0.212074i
\(64\) 0 0
\(65\) 20.1586i 2.50036i
\(66\) 0 0
\(67\) 3.03818 5.26229i 0.371173 0.642890i −0.618573 0.785727i \(-0.712289\pi\)
0.989746 + 0.142837i \(0.0456223\pi\)
\(68\) 0 0
\(69\) 1.50787i 0.181526i
\(70\) 0 0
\(71\) −6.15397 10.6590i −0.730342 1.26499i −0.956737 0.290954i \(-0.906027\pi\)
0.226395 0.974036i \(-0.427306\pi\)
\(72\) 0 0
\(73\) −3.97686 6.88813i −0.465457 0.806194i 0.533766 0.845633i \(-0.320777\pi\)
−0.999222 + 0.0394382i \(0.987443\pi\)
\(74\) 0 0
\(75\) −7.08857 −0.818518
\(76\) 0 0
\(77\) −8.18531 −0.932802
\(78\) 0 0
\(79\) 1.19415 + 2.06832i 0.134352 + 0.232704i 0.925350 0.379115i \(-0.123771\pi\)
−0.790998 + 0.611819i \(0.790438\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 5.59111i 0.613704i 0.951757 + 0.306852i \(0.0992757\pi\)
−0.951757 + 0.306852i \(0.900724\pi\)
\(84\) 0 0
\(85\) 9.30795 16.1218i 1.00959 1.74866i
\(86\) 0 0
\(87\) 4.86269i 0.521335i
\(88\) 0 0
\(89\) −5.04638 2.91353i −0.534915 0.308833i 0.208101 0.978107i \(-0.433272\pi\)
−0.743016 + 0.669274i \(0.766605\pi\)
\(90\) 0 0
\(91\) 9.75958 16.9041i 1.02308 1.77203i
\(92\) 0 0
\(93\) −5.28272 9.14993i −0.547792 0.948804i
\(94\) 0 0
\(95\) −5.00209 + 14.3060i −0.513204 + 1.46777i
\(96\) 0 0
\(97\) 2.47885 1.43117i 0.251689 0.145313i −0.368848 0.929490i \(-0.620248\pi\)
0.620538 + 0.784177i \(0.286914\pi\)
\(98\) 0 0
\(99\) −2.10560 1.21567i −0.211621 0.122180i
\(100\) 0 0
\(101\) 9.24254 16.0086i 0.919667 1.59291i 0.119747 0.992804i \(-0.461792\pi\)
0.799920 0.600106i \(-0.204875\pi\)
\(102\) 0 0
\(103\) 3.83108 0.377488 0.188744 0.982026i \(-0.439558\pi\)
0.188744 + 0.982026i \(0.439558\pi\)
\(104\) 0 0
\(105\) 10.1369 + 5.85256i 0.989264 + 0.571152i
\(106\) 0 0
\(107\) 6.17714 0.597167 0.298583 0.954384i \(-0.403486\pi\)
0.298583 + 0.954384i \(0.403486\pi\)
\(108\) 0 0
\(109\) −2.30994 + 1.33364i −0.221252 + 0.127740i −0.606530 0.795061i \(-0.707439\pi\)
0.385278 + 0.922801i \(0.374106\pi\)
\(110\) 0 0
\(111\) −7.33108 + 4.23260i −0.695836 + 0.401741i
\(112\) 0 0
\(113\) 5.29368i 0.497987i −0.968505 0.248994i \(-0.919900\pi\)
0.968505 0.248994i \(-0.0800998\pi\)
\(114\) 0 0
\(115\) 5.24266i 0.488880i
\(116\) 0 0
\(117\) 5.02115 2.89896i 0.464205 0.268009i
\(118\) 0 0
\(119\) −15.6105 + 9.01270i −1.43101 + 0.826193i
\(120\) 0 0
\(121\) 5.08857 0.462597
\(122\) 0 0
\(123\) −3.41146 1.96961i −0.307601 0.177593i
\(124\) 0 0
\(125\) 7.26167 0.649504
\(126\) 0 0
\(127\) −1.00000 + 1.73205i −0.0887357 + 0.153695i −0.906977 0.421180i \(-0.861616\pi\)
0.818241 + 0.574875i \(0.194949\pi\)
\(128\) 0 0
\(129\) −9.02523 5.21072i −0.794627 0.458778i
\(130\) 0 0
\(131\) 8.03133 4.63689i 0.701701 0.405127i −0.106280 0.994336i \(-0.533894\pi\)
0.807981 + 0.589209i \(0.200561\pi\)
\(132\) 0 0
\(133\) 11.1206 9.57467i 0.964283 0.830229i
\(134\) 0 0
\(135\) 1.73843 + 3.01105i 0.149620 + 0.259150i
\(136\) 0 0
\(137\) −3.00000 + 5.19615i −0.256307 + 0.443937i −0.965250 0.261329i \(-0.915839\pi\)
0.708942 + 0.705266i \(0.249173\pi\)
\(138\) 0 0
\(139\) −1.23644 0.713859i −0.104874 0.0605488i 0.446646 0.894711i \(-0.352618\pi\)
−0.551519 + 0.834162i \(0.685952\pi\)
\(140\) 0 0
\(141\) 11.1207i 0.936534i
\(142\) 0 0
\(143\) 7.04837 12.2081i 0.589414 1.02090i
\(144\) 0 0
\(145\) 16.9069i 1.40404i
\(146\) 0 0
\(147\) −2.16692 3.75322i −0.178725 0.309561i
\(148\) 0 0
\(149\) 9.10361 + 15.7679i 0.745797 + 1.29176i 0.949821 + 0.312793i \(0.101265\pi\)
−0.204024 + 0.978966i \(0.565402\pi\)
\(150\) 0 0
\(151\) −1.04628 −0.0851447 −0.0425724 0.999093i \(-0.513555\pi\)
−0.0425724 + 0.999093i \(0.513555\pi\)
\(152\) 0 0
\(153\) −5.35422 −0.432863
\(154\) 0 0
\(155\) 18.3673 + 31.8131i 1.47530 + 2.55529i
\(156\) 0 0
\(157\) −8.85422 15.3360i −0.706644 1.22394i −0.966095 0.258187i \(-0.916875\pi\)
0.259451 0.965756i \(-0.416458\pi\)
\(158\) 0 0
\(159\) 5.09863i 0.404348i
\(160\) 0 0
\(161\) −2.53818 + 4.39626i −0.200037 + 0.346474i
\(162\) 0 0
\(163\) 11.6934i 0.915895i 0.888979 + 0.457948i \(0.151415\pi\)
−0.888979 + 0.457948i \(0.848585\pi\)
\(164\) 0 0
\(165\) 7.32090 + 4.22672i 0.569931 + 0.329050i
\(166\) 0 0
\(167\) 7.69415 13.3267i 0.595391 1.03125i −0.398101 0.917342i \(-0.630331\pi\)
0.993492 0.113906i \(-0.0363361\pi\)
\(168\) 0 0
\(169\) 10.3079 + 17.8539i 0.792919 + 1.37338i
\(170\) 0 0
\(171\) 4.28272 0.811383i 0.327507 0.0620480i
\(172\) 0 0
\(173\) 8.42242 4.86269i 0.640345 0.369703i −0.144403 0.989519i \(-0.546126\pi\)
0.784747 + 0.619816i \(0.212793\pi\)
\(174\) 0 0
\(175\) −20.6670 11.9321i −1.56228 0.901983i
\(176\) 0 0
\(177\) −5.58247 + 9.66912i −0.419604 + 0.726775i
\(178\) 0 0
\(179\) −14.4047 −1.07666 −0.538328 0.842735i \(-0.680944\pi\)
−0.538328 + 0.842735i \(0.680944\pi\)
\(180\) 0 0
\(181\) 8.12065 + 4.68846i 0.603603 + 0.348490i 0.770458 0.637491i \(-0.220028\pi\)
−0.166855 + 0.985982i \(0.553361\pi\)
\(182\) 0 0
\(183\) −5.61987 −0.415433
\(184\) 0 0
\(185\) 25.4892 14.7162i 1.87400 1.08196i
\(186\) 0 0
\(187\) −11.2739 + 6.50898i −0.824428 + 0.475983i
\(188\) 0 0
\(189\) 3.36658i 0.244883i
\(190\) 0 0
\(191\) 21.9024i 1.58480i 0.610001 + 0.792401i \(0.291169\pi\)
−0.610001 + 0.792401i \(0.708831\pi\)
\(192\) 0 0
\(193\) −8.79773 + 5.07937i −0.633275 + 0.365621i −0.782019 0.623254i \(-0.785810\pi\)
0.148745 + 0.988876i \(0.452477\pi\)
\(194\) 0 0
\(195\) −17.4578 + 10.0793i −1.25018 + 0.721793i
\(196\) 0 0
\(197\) −2.52314 −0.179766 −0.0898831 0.995952i \(-0.528649\pi\)
−0.0898831 + 0.995952i \(0.528649\pi\)
\(198\) 0 0
\(199\) 16.0472 + 9.26483i 1.13755 + 0.656766i 0.945823 0.324682i \(-0.105257\pi\)
0.191729 + 0.981448i \(0.438591\pi\)
\(200\) 0 0
\(201\) 6.07636 0.428594
\(202\) 0 0
\(203\) 8.18531 14.1774i 0.574496 0.995056i
\(204\) 0 0
\(205\) 11.8612 + 6.84805i 0.828420 + 0.478289i
\(206\) 0 0
\(207\) −1.30585 + 0.753935i −0.0907631 + 0.0524021i
\(208\) 0 0
\(209\) 8.03133 6.91483i 0.555539 0.478309i
\(210\) 0 0
\(211\) 3.81402 + 6.60608i 0.262568 + 0.454781i 0.966924 0.255066i \(-0.0820973\pi\)
−0.704356 + 0.709847i \(0.748764\pi\)
\(212\) 0 0
\(213\) 6.15397 10.6590i 0.421663 0.730342i
\(214\) 0 0
\(215\) 31.3795 + 18.1170i 2.14006 + 1.23557i
\(216\) 0 0
\(217\) 35.5694i 2.41461i
\(218\) 0 0
\(219\) 3.97686 6.88813i 0.268731 0.465457i
\(220\) 0 0
\(221\) 31.0434i 2.08820i
\(222\) 0 0
\(223\) −1.71728 2.97442i −0.114998 0.199182i 0.802781 0.596274i \(-0.203353\pi\)
−0.917779 + 0.397092i \(0.870019\pi\)
\(224\) 0 0
\(225\) −3.54429 6.13888i −0.236286 0.409259i
\(226\) 0 0
\(227\) 11.4729 0.761482 0.380741 0.924682i \(-0.375669\pi\)
0.380741 + 0.924682i \(0.375669\pi\)
\(228\) 0 0
\(229\) 4.13485 0.273238 0.136619 0.990624i \(-0.456376\pi\)
0.136619 + 0.990624i \(0.456376\pi\)
\(230\) 0 0
\(231\) −4.09265 7.08868i −0.269277 0.466401i
\(232\) 0 0
\(233\) −2.20025 3.81094i −0.144143 0.249663i 0.784910 0.619610i \(-0.212709\pi\)
−0.929053 + 0.369947i \(0.879376\pi\)
\(234\) 0 0
\(235\) 38.6652i 2.52224i
\(236\) 0 0
\(237\) −1.19415 + 2.06832i −0.0775681 + 0.134352i
\(238\) 0 0
\(239\) 7.09898i 0.459195i −0.973286 0.229598i \(-0.926259\pi\)
0.973286 0.229598i \(-0.0737410\pi\)
\(240\) 0 0
\(241\) 7.09134 + 4.09418i 0.456793 + 0.263730i 0.710695 0.703500i \(-0.248381\pi\)
−0.253902 + 0.967230i \(0.581714\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 7.53410 + 13.0494i 0.481336 + 0.833698i
\(246\) 0 0
\(247\) 4.70433 + 24.8309i 0.299330 + 1.57995i
\(248\) 0 0
\(249\) −4.84205 + 2.79556i −0.306852 + 0.177161i
\(250\) 0 0
\(251\) −8.66217 5.00111i −0.546751 0.315667i 0.201059 0.979579i \(-0.435562\pi\)
−0.747811 + 0.663912i \(0.768895\pi\)
\(252\) 0 0
\(253\) −1.83308 + 3.17498i −0.115244 + 0.199609i
\(254\) 0 0
\(255\) 18.6159 1.16577
\(256\) 0 0
\(257\) −0.204332 0.117971i −0.0127459 0.00735885i 0.493614 0.869681i \(-0.335676\pi\)
−0.506359 + 0.862323i \(0.669009\pi\)
\(258\) 0 0
\(259\) −28.4988 −1.77083
\(260\) 0 0
\(261\) 4.21121 2.43134i 0.260667 0.150496i
\(262\) 0 0
\(263\) −5.01096 + 2.89308i −0.308989 + 0.178395i −0.646474 0.762936i \(-0.723757\pi\)
0.337485 + 0.941331i \(0.390424\pi\)
\(264\) 0 0
\(265\) 17.7272i 1.08898i
\(266\) 0 0
\(267\) 5.82706i 0.356610i
\(268\) 0 0
\(269\) 8.79567 5.07818i 0.536281 0.309622i −0.207289 0.978280i \(-0.566464\pi\)
0.743570 + 0.668658i \(0.233131\pi\)
\(270\) 0 0
\(271\) −11.0082 + 6.35557i −0.668698 + 0.386073i −0.795583 0.605844i \(-0.792835\pi\)
0.126885 + 0.991917i \(0.459502\pi\)
\(272\) 0 0
\(273\) 19.5192 1.18135
\(274\) 0 0
\(275\) −14.9257 8.61737i −0.900055 0.519647i
\(276\) 0 0
\(277\) −3.39646 −0.204073 −0.102037 0.994781i \(-0.532536\pi\)
−0.102037 + 0.994781i \(0.532536\pi\)
\(278\) 0 0
\(279\) 5.28272 9.14993i 0.316268 0.547792i
\(280\) 0 0
\(281\) −24.4770 14.1318i −1.46017 0.843031i −0.461154 0.887320i \(-0.652565\pi\)
−0.999019 + 0.0442884i \(0.985898\pi\)
\(282\) 0 0
\(283\) −17.6622 + 10.1973i −1.04991 + 0.606164i −0.922623 0.385702i \(-0.873959\pi\)
−0.127284 + 0.991866i \(0.540626\pi\)
\(284\) 0 0
\(285\) −14.8904 + 2.82107i −0.882032 + 0.167105i
\(286\) 0 0
\(287\) −6.63084 11.4849i −0.391406 0.677935i
\(288\) 0 0
\(289\) −5.83385 + 10.1045i −0.343168 + 0.594384i
\(290\) 0 0
\(291\) 2.47885 + 1.43117i 0.145313 + 0.0838965i
\(292\) 0 0
\(293\) 2.51388i 0.146862i 0.997300 + 0.0734311i \(0.0233949\pi\)
−0.997300 + 0.0734311i \(0.976605\pi\)
\(294\) 0 0
\(295\) 19.4095 33.6182i 1.13006 1.95733i
\(296\) 0 0
\(297\) 2.43134i 0.141081i
\(298\) 0 0
\(299\) −4.37126 7.57124i −0.252796 0.437856i
\(300\) 0 0
\(301\) −17.5423 30.3841i −1.01112 1.75131i
\(302\) 0 0
\(303\) 18.4851 1.06194
\(304\) 0 0
\(305\) 19.5395 1.11883
\(306\) 0 0
\(307\) 4.84404 + 8.39012i 0.276464 + 0.478849i 0.970503 0.241088i \(-0.0775041\pi\)
−0.694040 + 0.719937i \(0.744171\pi\)
\(308\) 0 0
\(309\) 1.91554 + 3.31782i 0.108971 + 0.188744i
\(310\) 0 0
\(311\) 6.56560i 0.372301i −0.982521 0.186151i \(-0.940399\pi\)
0.982521 0.186151i \(-0.0596012\pi\)
\(312\) 0 0
\(313\) 12.5975 21.8195i 0.712053 1.23331i −0.252032 0.967719i \(-0.581099\pi\)
0.964085 0.265594i \(-0.0855680\pi\)
\(314\) 0 0
\(315\) 11.7051i 0.659509i
\(316\) 0 0
\(317\) 8.02321 + 4.63220i 0.450628 + 0.260170i 0.708096 0.706117i \(-0.249555\pi\)
−0.257467 + 0.966287i \(0.582888\pi\)
\(318\) 0 0
\(319\) 5.91143 10.2389i 0.330977 0.573268i
\(320\) 0 0
\(321\) 3.08857 + 5.34956i 0.172387 + 0.298583i
\(322\) 0 0
\(323\) 7.70301 22.0307i 0.428607 1.22582i
\(324\) 0 0
\(325\) 35.5928 20.5495i 1.97433 1.13988i
\(326\) 0 0
\(327\) −2.30994 1.33364i −0.127740 0.0737506i
\(328\) 0 0
\(329\) −18.7194 + 32.4230i −1.03203 + 1.78754i
\(330\) 0 0
\(331\) −16.9879 −0.933737 −0.466868 0.884327i \(-0.654618\pi\)
−0.466868 + 0.884327i \(0.654618\pi\)
\(332\) 0 0
\(333\) −7.33108 4.23260i −0.401741 0.231945i
\(334\) 0 0
\(335\) −21.1267 −1.15427
\(336\) 0 0
\(337\) −3.01298 + 1.73955i −0.164128 + 0.0947591i −0.579814 0.814749i \(-0.696875\pi\)
0.415686 + 0.909508i \(0.363541\pi\)
\(338\) 0 0
\(339\) 4.58446 2.64684i 0.248994 0.143756i
\(340\) 0 0
\(341\) 25.6882i 1.39109i
\(342\) 0 0
\(343\) 8.97580i 0.484648i
\(344\) 0 0
\(345\) 4.54027 2.62133i 0.244440 0.141128i
\(346\) 0 0
\(347\) 14.9790 8.64812i 0.804114 0.464255i −0.0407937 0.999168i \(-0.512989\pi\)
0.844908 + 0.534912i \(0.179655\pi\)
\(348\) 0 0
\(349\) −31.3259 −1.67684 −0.838418 0.545028i \(-0.816519\pi\)
−0.838418 + 0.545028i \(0.816519\pi\)
\(350\) 0 0
\(351\) 5.02115 + 2.89896i 0.268009 + 0.154735i
\(352\) 0 0
\(353\) −0.408665 −0.0217510 −0.0108755 0.999941i \(-0.503462\pi\)
−0.0108755 + 0.999941i \(0.503462\pi\)
\(354\) 0 0
\(355\) −21.3965 + 37.0599i −1.13561 + 1.96693i
\(356\) 0 0
\(357\) −15.6105 9.01270i −0.826193 0.477003i
\(358\) 0 0
\(359\) −14.0545 + 8.11437i −0.741768 + 0.428260i −0.822712 0.568459i \(-0.807540\pi\)
0.0809437 + 0.996719i \(0.474207\pi\)
\(360\) 0 0
\(361\) −2.82292 + 18.7891i −0.148575 + 0.988901i
\(362\) 0 0
\(363\) 2.54429 + 4.40683i 0.133540 + 0.231299i
\(364\) 0 0
\(365\) −13.8270 + 23.9491i −0.723738 + 1.25355i
\(366\) 0 0
\(367\) 5.15185 + 2.97442i 0.268924 + 0.155264i 0.628399 0.777891i \(-0.283711\pi\)
−0.359474 + 0.933155i \(0.617044\pi\)
\(368\) 0 0
\(369\) 3.93921i 0.205067i
\(370\) 0 0
\(371\) 8.58247 14.8653i 0.445579 0.771766i
\(372\) 0 0
\(373\) 21.8930i 1.13358i 0.823863 + 0.566788i \(0.191814\pi\)
−0.823863 + 0.566788i \(0.808186\pi\)
\(374\) 0 0
\(375\) 3.63084 + 6.28879i 0.187496 + 0.324752i
\(376\) 0 0
\(377\) 14.0967 + 24.4163i 0.726019 + 1.25750i
\(378\) 0 0
\(379\) 7.94158 0.407931 0.203966 0.978978i \(-0.434617\pi\)
0.203966 + 0.978978i \(0.434617\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) 15.0143 + 26.0055i 0.767195 + 1.32882i 0.939078 + 0.343703i \(0.111681\pi\)
−0.171883 + 0.985117i \(0.554985\pi\)
\(384\) 0 0
\(385\) 14.2296 + 24.6464i 0.725207 + 1.25610i
\(386\) 0 0
\(387\) 10.4214i 0.529751i
\(388\) 0 0
\(389\) −18.3462 + 31.7765i −0.930187 + 1.61113i −0.147188 + 0.989109i \(0.547022\pi\)
−0.782999 + 0.622023i \(0.786311\pi\)
\(390\) 0 0
\(391\) 8.07347i 0.408293i
\(392\) 0 0
\(393\) 8.03133 + 4.63689i 0.405127 + 0.233900i
\(394\) 0 0
\(395\) 4.15188 7.19127i 0.208904 0.361832i
\(396\) 0 0
\(397\) −15.3291 26.5508i −0.769345 1.33254i −0.937918 0.346856i \(-0.887249\pi\)
0.168573 0.985689i \(-0.446084\pi\)
\(398\) 0 0
\(399\) 13.8522 + 4.84343i 0.693479 + 0.242475i
\(400\) 0 0
\(401\) −13.8066 + 7.97126i −0.689470 + 0.398066i −0.803414 0.595421i \(-0.796985\pi\)
0.113943 + 0.993487i \(0.463652\pi\)
\(402\) 0 0
\(403\) 53.0506 + 30.6288i 2.64264 + 1.52573i
\(404\) 0 0
\(405\) −1.73843 + 3.01105i −0.0863833 + 0.149620i
\(406\) 0 0
\(407\) −20.5818 −1.02020
\(408\) 0 0
\(409\) 6.69006 + 3.86251i 0.330802 + 0.190989i 0.656197 0.754589i \(-0.272164\pi\)
−0.325395 + 0.945578i \(0.605497\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) −32.5518 + 18.7938i −1.60177 + 0.924783i
\(414\) 0 0
\(415\) 16.8351 9.71976i 0.826404 0.477124i
\(416\) 0 0
\(417\) 1.42772i 0.0699157i
\(418\) 0 0
\(419\) 22.5693i 1.10258i 0.834313 + 0.551292i \(0.185865\pi\)
−0.834313 + 0.551292i \(0.814135\pi\)
\(420\) 0 0
\(421\) −35.2174 + 20.3328i −1.71639 + 0.990960i −0.791118 + 0.611663i \(0.790501\pi\)
−0.925275 + 0.379297i \(0.876166\pi\)
\(422\) 0 0
\(423\) −9.63084 + 5.56037i −0.468267 + 0.270354i
\(424\) 0 0
\(425\) −37.9538 −1.84103
\(426\) 0 0
\(427\) −16.3850 9.45987i −0.792925 0.457795i
\(428\) 0 0
\(429\) 14.0967 0.680597
\(430\) 0 0
\(431\) −14.0313 + 24.3030i −0.675866 + 1.17063i 0.300350 + 0.953829i \(0.402897\pi\)
−0.976215 + 0.216804i \(0.930437\pi\)
\(432\) 0 0
\(433\) 15.6711 + 9.04772i 0.753105 + 0.434806i 0.826815 0.562474i \(-0.190150\pi\)
−0.0737094 + 0.997280i \(0.523484\pi\)
\(434\) 0 0
\(435\) −14.6418 + 8.45345i −0.702020 + 0.405312i
\(436\) 0 0
\(437\) −1.22346 6.45778i −0.0585260 0.308918i
\(438\) 0 0
\(439\) −7.65806 13.2641i −0.365499 0.633063i 0.623357 0.781937i \(-0.285768\pi\)
−0.988856 + 0.148874i \(0.952435\pi\)
\(440\) 0 0
\(441\) 2.16692 3.75322i 0.103187 0.178725i
\(442\) 0 0
\(443\) 24.2194 + 13.9831i 1.15070 + 0.664355i 0.949057 0.315104i \(-0.102039\pi\)
0.201640 + 0.979460i \(0.435373\pi\)
\(444\) 0 0
\(445\) 20.2599i 0.960410i
\(446\) 0 0
\(447\) −9.10361 + 15.7679i −0.430586 + 0.745797i
\(448\) 0 0
\(449\) 6.03148i 0.284643i 0.989820 + 0.142322i \(0.0454567\pi\)
−0.989820 + 0.142322i \(0.954543\pi\)
\(450\) 0 0
\(451\) −4.78879 8.29443i −0.225495 0.390569i
\(452\) 0 0
\(453\) −0.523138 0.906101i −0.0245792 0.0425724i
\(454\) 0 0
\(455\) −67.8654 −3.18158
\(456\) 0 0
\(457\) −24.5573 −1.14874 −0.574370 0.818596i \(-0.694753\pi\)
−0.574370 + 0.818596i \(0.694753\pi\)
\(458\) 0 0
\(459\) −2.67711 4.63689i −0.124957 0.216432i
\(460\) 0 0
\(461\) −9.74939 16.8864i −0.454074 0.786480i 0.544560 0.838722i \(-0.316697\pi\)
−0.998634 + 0.0522419i \(0.983363\pi\)
\(462\) 0 0
\(463\) 14.2072i 0.660267i −0.943934 0.330133i \(-0.892906\pi\)
0.943934 0.330133i \(-0.107094\pi\)
\(464\) 0 0
\(465\) −18.3673 + 31.8131i −0.851762 + 1.47530i
\(466\) 0 0
\(467\) 16.9572i 0.784684i −0.919819 0.392342i \(-0.871665\pi\)
0.919819 0.392342i \(-0.128335\pi\)
\(468\) 0 0
\(469\) 17.7159 + 10.2283i 0.818044 + 0.472298i
\(470\) 0 0
\(471\) 8.85422 15.3360i 0.407981 0.706644i
\(472\) 0 0
\(473\) −12.6690 21.9434i −0.582523 1.00896i
\(474\) 0 0
\(475\) 30.3583 5.75154i 1.39294 0.263899i
\(476\) 0 0
\(477\) 4.41554 2.54931i 0.202174 0.116725i
\(478\) 0 0
\(479\) −22.2645 12.8544i −1.01729 0.587332i −0.103972 0.994580i \(-0.533155\pi\)
−0.913318 + 0.407248i \(0.866488\pi\)
\(480\) 0 0
\(481\) 24.5403 42.5051i 1.11894 1.93806i
\(482\) 0 0
\(483\) −5.07636 −0.230983
\(484\) 0 0
\(485\) −8.61863 4.97597i −0.391352 0.225947i
\(486\) 0 0
\(487\) 19.4607 0.881850 0.440925 0.897544i \(-0.354650\pi\)
0.440925 + 0.897544i \(0.354650\pi\)
\(488\) 0 0
\(489\) −10.1268 + 5.84668i −0.457948 + 0.264396i
\(490\) 0 0
\(491\) 35.1119 20.2719i 1.58458 0.914856i 0.590398 0.807112i \(-0.298971\pi\)
0.994179 0.107744i \(-0.0343627\pi\)
\(492\) 0 0
\(493\) 26.0359i 1.17260i
\(494\) 0 0
\(495\) 8.45345i 0.379954i
\(496\) 0 0
\(497\) 35.8843 20.7178i 1.60963 0.929322i
\(498\) 0 0
\(499\) −6.51908 + 3.76379i −0.291834 + 0.168491i −0.638769 0.769399i \(-0.720556\pi\)
0.346934 + 0.937889i \(0.387223\pi\)
\(500\) 0 0
\(501\) 15.3883 0.687498
\(502\) 0 0
\(503\) −1.19084 0.687530i −0.0530968 0.0306554i 0.473217 0.880946i \(-0.343093\pi\)
−0.526313 + 0.850291i \(0.676426\pi\)
\(504\) 0 0
\(505\) −64.2701 −2.85998
\(506\) 0 0
\(507\) −10.3079 + 17.8539i −0.457792 + 0.792919i
\(508\) 0 0
\(509\) 28.0830 + 16.2138i 1.24476 + 0.718662i 0.970059 0.242868i \(-0.0780883\pi\)
0.274700 + 0.961530i \(0.411422\pi\)
\(510\) 0 0
\(511\) 23.1894 13.3884i 1.02584 0.592269i
\(512\) 0 0
\(513\) 2.84404 + 3.30325i 0.125567 + 0.145842i
\(514\) 0 0
\(515\) −6.66008 11.5356i −0.293478 0.508319i
\(516\) 0 0
\(517\) −13.5192 + 23.4159i −0.594572 + 1.02983i
\(518\) 0 0
\(519\) 8.42242 + 4.86269i 0.369703 + 0.213448i
\(520\) 0 0
\(521\) 3.05664i 0.133914i −0.997756 0.0669568i \(-0.978671\pi\)
0.997756 0.0669568i \(-0.0213290\pi\)
\(522\) 0 0
\(523\) −10.2031 + 17.6723i −0.446151 + 0.772756i −0.998132 0.0611006i \(-0.980539\pi\)
0.551980 + 0.833857i \(0.313872\pi\)
\(524\) 0 0
\(525\) 23.8642i 1.04152i
\(526\) 0 0
\(527\) −28.2848 48.9908i −1.23211 2.13407i
\(528\) 0 0
\(529\) −10.3632 17.9495i −0.450572 0.780414i
\(530\) 0 0
\(531\) −11.1649 −0.484517
\(532\) 0 0
\(533\) 22.8393 0.989278
\(534\) 0 0
\(535\) −10.7385 18.5997i −0.464267 0.804135i
\(536\) 0 0
\(537\) −7.20234 12.4748i −0.310804 0.538328i
\(538\) 0 0
\(539\) 10.5371i 0.453864i
\(540\) 0 0
\(541\) −8.20845 + 14.2174i −0.352909 + 0.611256i −0.986758 0.162201i \(-0.948141\pi\)
0.633849 + 0.773457i \(0.281474\pi\)
\(542\) 0 0
\(543\) 9.37692i 0.402402i
\(544\) 0 0
\(545\) 8.03133 + 4.63689i 0.344025 + 0.198623i
\(546\) 0 0
\(547\) −0.882218 + 1.52805i −0.0377209 + 0.0653345i −0.884270 0.466977i \(-0.845343\pi\)
0.846549 + 0.532311i \(0.178676\pi\)
\(548\) 0 0
\(549\) −2.80994 4.86695i −0.119925 0.207717i
\(550\) 0 0
\(551\) 3.94550 + 20.8255i 0.168084 + 0.887196i
\(552\) 0 0
\(553\) −6.96317 + 4.02019i −0.296104 + 0.170956i
\(554\) 0 0
\(555\) 25.4892 + 14.7162i 1.08196 + 0.624667i
\(556\) 0 0
\(557\) 0.388292 0.672541i 0.0164525 0.0284965i −0.857682 0.514181i \(-0.828096\pi\)
0.874134 + 0.485684i \(0.161429\pi\)
\(558\) 0 0
\(559\) 60.4227 2.55561
\(560\) 0 0
\(561\) −11.2739 6.50898i −0.475983 0.274809i
\(562\) 0 0
\(563\) −44.9019 −1.89239 −0.946195 0.323597i \(-0.895108\pi\)
−0.946195 + 0.323597i \(0.895108\pi\)
\(564\) 0 0
\(565\) −15.9395 + 9.20269i −0.670581 + 0.387160i
\(566\) 0 0
\(567\) 2.91554 1.68329i 0.122441 0.0706915i
\(568\) 0 0
\(569\) 33.7523i 1.41497i −0.706729 0.707485i \(-0.749830\pi\)
0.706729 0.707485i \(-0.250170\pi\)
\(570\) 0 0
\(571\) 21.1007i 0.883037i 0.897252 + 0.441518i \(0.145560\pi\)
−0.897252 + 0.441518i \(0.854440\pi\)
\(572\) 0 0
\(573\) −18.9680 + 10.9512i −0.792401 + 0.457493i
\(574\) 0 0
\(575\) −9.25664 + 5.34432i −0.386029 + 0.222874i
\(576\) 0 0
\(577\) 6.84882 0.285120 0.142560 0.989786i \(-0.454467\pi\)
0.142560 + 0.989786i \(0.454467\pi\)
\(578\) 0 0
\(579\) −8.79773 5.07937i −0.365621 0.211092i
\(580\) 0 0
\(581\) −18.8229 −0.780906
\(582\) 0 0
\(583\) 6.19826 10.7357i 0.256706 0.444627i
\(584\) 0 0
\(585\) −17.4578 10.0793i −0.721793 0.416727i
\(586\) 0 0
\(587\) −11.3018 + 6.52511i −0.466476 + 0.269320i −0.714763 0.699366i \(-0.753466\pi\)
0.248287 + 0.968686i \(0.420132\pi\)
\(588\) 0 0
\(589\) 30.0485 + 34.9003i 1.23813 + 1.43804i
\(590\) 0 0
\(591\) −1.26157 2.18510i −0.0518940 0.0898831i
\(592\) 0 0
\(593\) 18.0887 31.3305i 0.742813 1.28659i −0.208397 0.978044i \(-0.566825\pi\)
0.951210 0.308545i \(-0.0998421\pi\)
\(594\) 0 0
\(595\) 54.2754 + 31.3359i 2.22508 + 1.28465i
\(596\) 0 0
\(597\) 18.5297i 0.758368i
\(598\) 0 0
\(599\) −13.4449 + 23.2872i −0.549343 + 0.951490i 0.448977 + 0.893544i \(0.351789\pi\)
−0.998320 + 0.0579467i \(0.981545\pi\)
\(600\) 0 0
\(601\) 3.73240i 0.152248i −0.997098 0.0761240i \(-0.975746\pi\)
0.997098 0.0761240i \(-0.0242545\pi\)
\(602\) 0 0
\(603\) 3.03818 + 5.26229i 0.123724 + 0.214297i
\(604\) 0 0
\(605\) −8.84613 15.3219i −0.359646 0.622926i
\(606\) 0 0
\(607\) −47.3177 −1.92057 −0.960284 0.279026i \(-0.909988\pi\)
−0.960284 + 0.279026i \(0.909988\pi\)
\(608\) 0 0
\(609\) 16.3706 0.663371
\(610\) 0 0
\(611\) −32.2386 55.8388i −1.30423 2.25900i
\(612\) 0 0
\(613\) −8.52115 14.7591i −0.344166 0.596113i 0.641036 0.767511i \(-0.278505\pi\)
−0.985202 + 0.171398i \(0.945172\pi\)
\(614\) 0 0
\(615\) 13.6961i 0.552280i
\(616\) 0 0
\(617\) −5.30107 + 9.18172i −0.213413 + 0.369642i −0.952780 0.303660i \(-0.901791\pi\)
0.739367 + 0.673302i \(0.235125\pi\)
\(618\) 0 0
\(619\) 23.0642i 0.927028i 0.886090 + 0.463514i \(0.153412\pi\)
−0.886090 + 0.463514i \(0.846588\pi\)
\(620\) 0 0
\(621\) −1.30585 0.753935i −0.0524021 0.0302544i
\(622\) 0 0
\(623\) 9.80862 16.9890i 0.392974 0.680651i
\(624\) 0 0
\(625\) 5.09751 + 8.82915i 0.203900 + 0.353166i
\(626\) 0 0
\(627\) 10.0041 + 3.49793i 0.399525 + 0.139694i
\(628\) 0 0
\(629\) −39.2523 + 22.6623i −1.56509 + 0.903605i
\(630\) 0 0
\(631\) 1.71405 + 0.989606i 0.0682352 + 0.0393956i 0.533729 0.845655i \(-0.320790\pi\)
−0.465494 + 0.885051i \(0.654123\pi\)
\(632\) 0 0
\(633\) −3.81402 + 6.60608i −0.151594 + 0.262568i
\(634\) 0 0
\(635\) 6.95372 0.275950
\(636\) 0 0
\(637\) 21.7609 + 12.5637i 0.862198 + 0.497790i
\(638\) 0 0
\(639\) 12.3079 0.486895
\(640\) 0 0
\(641\) 21.2425 12.2644i 0.839030 0.484414i −0.0179044 0.999840i \(-0.505699\pi\)
0.856934 + 0.515425i \(0.172366\pi\)
\(642\) 0 0
\(643\) 29.9660 17.3009i 1.18174 0.682280i 0.225326 0.974283i \(-0.427655\pi\)
0.956417 + 0.292003i \(0.0943218\pi\)
\(644\) 0 0
\(645\) 36.2339i 1.42671i
\(646\) 0 0
\(647\) 40.0103i 1.57297i 0.617610 + 0.786484i \(0.288101\pi\)
−0.617610 + 0.786484i \(0.711899\pi\)
\(648\) 0 0
\(649\) −23.5089 + 13.5729i −0.922807 + 0.532783i
\(650\) 0 0
\(651\) 30.8040 17.7847i 1.20730 0.697036i
\(652\) 0 0
\(653\) 23.7687 0.930140 0.465070 0.885274i \(-0.346029\pi\)
0.465070 + 0.885274i \(0.346029\pi\)
\(654\) 0 0
\(655\) −27.9238 16.1218i −1.09107 0.629932i
\(656\) 0 0
\(657\) 7.95372 0.310304
\(658\) 0 0
\(659\) 10.1016 17.4965i 0.393503 0.681568i −0.599406 0.800446i \(-0.704596\pi\)
0.992909 + 0.118878i \(0.0379296\pi\)
\(660\) 0 0
\(661\) 20.2836 + 11.7107i 0.788940 + 0.455495i 0.839589 0.543222i \(-0.182796\pi\)
−0.0506490 + 0.998717i \(0.516129\pi\)
\(662\) 0 0
\(663\) 26.8843 15.5217i 1.04410 0.602812i
\(664\) 0 0
\(665\) −48.1623 16.8399i −1.86765 0.653025i
\(666\) 0 0
\(667\) −3.66615 6.34996i −0.141954 0.245871i
\(668\) 0 0
\(669\) 1.71728 2.97442i 0.0663940 0.114998i
\(670\) 0 0
\(671\) −11.8332 6.83192i −0.456817 0.263743i
\(672\) 0 0
\(673\) 43.0646i 1.66002i −0.557750 0.830009i \(-0.688335\pi\)
0.557750 0.830009i \(-0.311665\pi\)
\(674\) 0 0
\(675\) 3.54429 6.13888i 0.136420 0.236286i
\(676\) 0 0
\(677\) 19.2557i 0.740057i 0.929020 + 0.370028i \(0.120652\pi\)
−0.929020 + 0.370028i \(0.879348\pi\)
\(678\) 0 0
\(679\) 4.81813 + 8.34525i 0.184903 + 0.320261i
\(680\) 0 0
\(681\) 5.73644 + 9.93581i 0.219821 + 0.380741i
\(682\) 0 0
\(683\) −33.5654 −1.28435 −0.642173 0.766560i \(-0.721967\pi\)
−0.642173 + 0.766560i \(0.721967\pi\)
\(684\) 0 0
\(685\) 20.8612 0.797064
\(686\) 0 0
\(687\) 2.06742 + 3.58088i 0.0788771 + 0.136619i
\(688\) 0 0
\(689\) 14.7807 + 25.6010i 0.563101 + 0.975319i
\(690\) 0 0
\(691\) 15.1717i 0.577159i 0.957456 + 0.288580i \(0.0931830\pi\)
−0.957456 + 0.288580i \(0.906817\pi\)
\(692\) 0 0
\(693\) 4.09265 7.08868i 0.155467 0.269277i
\(694\) 0 0
\(695\) 4.96398i 0.188295i
\(696\) 0 0
\(697\) −18.2657 10.5457i −0.691863 0.399447i
\(698\) 0 0
\(699\) 2.20025 3.81094i 0.0832211 0.144143i
\(700\) 0 0
\(701\) 15.6158 + 27.0473i 0.589800 + 1.02156i 0.994258 + 0.107007i \(0.0341269\pi\)
−0.404458 + 0.914557i \(0.632540\pi\)
\(702\) 0 0
\(703\) 27.9627 24.0754i 1.05463 0.908019i
\(704\) 0 0
\(705\) 33.4851 19.3326i 1.26112 0.728109i
\(706\) 0 0
\(707\) 53.8941 + 31.1157i 2.02689 + 1.17023i
\(708\) 0 0
\(709\) −3.61248 + 6.25700i −0.135670 + 0.234987i −0.925853 0.377884i \(-0.876652\pi\)
0.790183 + 0.612870i \(0.209985\pi\)
\(710\) 0 0
\(711\) −2.38829 −0.0895679
\(712\) 0 0
\(713\) −13.7969 7.96565i −0.516698 0.298316i
\(714\) 0 0
\(715\) −49.0124 −1.83296
\(716\) 0 0
\(717\) 6.14790 3.54949i 0.229598 0.132558i
\(718\) 0 0
\(719\) 12.7051 7.33530i 0.473821 0.273560i −0.244017 0.969771i \(-0.578465\pi\)
0.717838 + 0.696211i \(0.245132\pi\)
\(720\) 0 0
\(721\) 12.8976i 0.480333i
\(722\) 0 0
\(723\) 8.18837i 0.304529i
\(724\) 0 0
\(725\) 29.8515 17.2347i 1.10866 0.640082i
\(726\) 0 0
\(727\) −25.2064 + 14.5529i −0.934852 + 0.539737i −0.888343 0.459181i \(-0.848143\pi\)
−0.0465091 + 0.998918i \(0.514810\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.3231 27.8994i −1.78729 1.03189i
\(732\) 0 0
\(733\) −21.2194 −0.783758 −0.391879 0.920017i \(-0.628175\pi\)
−0.391879 + 0.920017i \(0.628175\pi\)
\(734\) 0 0
\(735\) −7.53410 + 13.0494i −0.277899 + 0.481336i
\(736\) 0 0
\(737\) 12.7944 + 7.38686i 0.471289 + 0.272099i
\(738\) 0 0
\(739\) −2.07225 + 1.19641i −0.0762290 + 0.0440108i −0.537630 0.843181i \(-0.680680\pi\)
0.461401 + 0.887192i \(0.347347\pi\)
\(740\) 0 0
\(741\) −19.1520 + 16.4895i −0.703566 + 0.605757i
\(742\) 0 0
\(743\) −5.71607 9.90052i −0.209702 0.363215i 0.741919 0.670490i \(-0.233916\pi\)
−0.951621 + 0.307275i \(0.900583\pi\)
\(744\) 0 0
\(745\) 31.6520 54.8229i 1.15964 2.00856i
\(746\) 0 0
\(747\) −4.84205 2.79556i −0.177161 0.102284i
\(748\) 0 0
\(749\) 20.7958i 0.759863i
\(750\) 0 0
\(751\) −22.2195 + 38.4853i −0.810801 + 1.40435i 0.101503 + 0.994835i \(0.467635\pi\)
−0.912304 + 0.409514i \(0.865698\pi\)
\(752\) 0 0
\(753\) 10.0022i 0.364501i
\(754\) 0 0
\(755\) 1.81888 + 3.15039i 0.0661958 + 0.114654i
\(756\) 0 0
\(757\) 17.6289 + 30.5341i 0.640732 + 1.10978i 0.985270 + 0.171008i \(0.0547025\pi\)
−0.344537 + 0.938773i \(0.611964\pi\)
\(758\) 0 0
\(759\) −3.66615 −0.133073
\(760\) 0 0
\(761\) 28.0983 1.01856 0.509282 0.860600i \(-0.329911\pi\)
0.509282 + 0.860600i \(0.329911\pi\)
\(762\) 0 0
\(763\) −4.48981 7.77659i −0.162542 0.281531i
\(764\) 0 0
\(765\) 9.30795 + 16.1218i 0.336530 + 0.582886i
\(766\) 0 0
\(767\) 64.7334i 2.33739i
\(768\) 0 0
\(769\) 3.36515 5.82862i 0.121351 0.210185i −0.798950 0.601398i \(-0.794611\pi\)
0.920301 + 0.391212i \(0.127944\pi\)
\(770\) 0 0
\(771\) 0.235943i 0.00849727i
\(772\) 0 0
\(773\) 15.5383 + 8.97107i 0.558875 + 0.322667i 0.752694 0.658371i \(-0.228754\pi\)
−0.193819 + 0.981037i \(0.562087\pi\)
\(774\) 0 0
\(775\) 37.4469 64.8599i 1.34513 2.32984i
\(776\) 0 0
\(777\) −14.2494 24.6807i −0.511194 0.885414i
\(778\) 0 0
\(779\) 16.2084 + 5.66727i 0.580727 + 0.203051i
\(780\) 0 0
\(781\) 25.9157 14.9624i 0.927336 0.535398i
\(782\) 0 0
\(783\) 4.21121 + 2.43134i 0.150496 + 0.0868891i
\(784\) 0 0
\(785\) −30.7849 + 53.3210i −1.09876 + 1.90311i
\(786\) 0 0
\(787\) −16.7441 −0.596861 −0.298431 0.954431i \(-0.596463\pi\)
−0.298431 + 0.954431i \(0.596463\pi\)
\(788\) 0 0
\(789\) −5.01096 2.89308i −0.178395 0.102996i
\(790\) 0 0
\(791\) 17.8216 0.633662
\(792\) 0 0
\(793\) 28.2182 16.2918i 1.00206 0.578539i
\(794\) 0 0
\(795\) −15.3522 + 8.86362i −0.544488 + 0.314360i
\(796\) 0 0
\(797\) 8.30004i 0.294002i −0.989136 0.147001i \(-0.953038\pi\)
0.989136 0.147001i \(-0.0469621\pi\)
\(798\) 0 0
\(799\) 59.5429i 2.10648i
\(800\) 0 0
\(801\) 5.04638 2.91353i 0.178305 0.102944i
\(802\) 0 0
\(803\) 16.7474 9.66912i 0.591003 0.341216i
\(804\) 0 0
\(805\) 17.6498 0.622074
\(806\) 0 0
\(807\) 8.79567 + 5.07818i 0.309622 + 0.178760i
\(808\) 0 0
\(809\) 3.29972 0.116012 0.0580060 0.998316i \(-0.481526\pi\)
0.0580060 + 0.998316i \(0.481526\pi\)
\(810\) 0 0
\(811\) 6.44279 11.1592i 0.226237 0.391854i −0.730453 0.682963i \(-0.760691\pi\)
0.956690 + 0.291109i \(0.0940243\pi\)
\(812\) 0 0
\(813\) −11.0082 6.35557i −0.386073 0.222899i
\(814\) 0 0
\(815\) 35.2093 20.3281i 1.23333 0.712063i
\(816\) 0 0
\(817\) 42.8804 + 14.9931i 1.50019 + 0.524543i
\(818\) 0 0
\(819\) 9.75958 + 16.9041i 0.341027 + 0.590677i
\(820\) 0 0
\(821\) −8.00817 + 13.8706i −0.279487 + 0.484086i −0.971257 0.238032i \(-0.923498\pi\)
0.691770 + 0.722118i \(0.256831\pi\)
\(822\) 0 0
\(823\) 29.6825 + 17.1372i 1.03467 + 0.597366i 0.918319 0.395842i \(-0.129547\pi\)
0.116350 + 0.993208i \(0.462881\pi\)
\(824\) 0 0
\(825\) 17.2347i 0.600037i
\(826\) 0 0
\(827\) −0.865154 + 1.49849i −0.0300844 + 0.0521076i −0.880676 0.473720i \(-0.842911\pi\)
0.850591 + 0.525827i \(0.176244\pi\)
\(828\) 0 0
\(829\) 29.4144i 1.02161i 0.859698 + 0.510803i \(0.170652\pi\)
−0.859698 + 0.510803i \(0.829348\pi\)
\(830\) 0 0
\(831\) −1.69823 2.94142i −0.0589109 0.102037i
\(832\) 0 0
\(833\) −11.6022 20.0956i −0.401992 0.696271i
\(834\) 0 0
\(835\) −53.5030 −1.85155
\(836\) 0 0
\(837\) 10.5654 0.365195
\(838\) 0 0
\(839\) −18.5894 32.1979i −0.641779 1.11159i −0.985035 0.172352i \(-0.944863\pi\)
0.343256 0.939242i \(-0.388470\pi\)
\(840\) 0 0
\(841\) −2.67714 4.63694i −0.0923152 0.159895i
\(842\) 0 0
\(843\) 28.2636i 0.973449i
\(844\) 0 0
\(845\) 35.8393 62.0755i 1.23291 2.13546i
\(846\) 0 0
\(847\) 17.1311i 0.588631i
\(848\) 0 0
\(849\) −17.6622 10.1973i −0.606164 0.349969i
\(850\) 0 0
\(851\) −6.38222 + 11.0543i −0.218780 + 0.378937i
\(852\) 0 0
\(853\) −16.5967 28.7464i −0.568261 0.984258i −0.996738 0.0807048i \(-0.974283\pi\)
0.428477 0.903553i \(-0.359050\pi\)
\(854\) 0 0
\(855\) −9.88832 11.4849i −0.338173 0.392777i
\(856\) 0 0
\(857\) 19.5371 11.2797i 0.667375 0.385309i −0.127707 0.991812i \(-0.540762\pi\)
0.795081 + 0.606503i \(0.207428\pi\)
\(858\) 0 0
\(859\) −12.4006 7.15951i −0.423104 0.244279i 0.273300 0.961929i \(-0.411885\pi\)
−0.696404 + 0.717649i \(0.745218\pi\)
\(860\) 0 0
\(861\) 6.63084 11.4849i 0.225978 0.391406i
\(862\) 0 0
\(863\) −31.0001 −1.05525 −0.527627 0.849476i \(-0.676918\pi\)
−0.527627 + 0.849476i \(0.676918\pi\)
\(864\) 0 0
\(865\) −29.2836 16.9069i −0.995672 0.574852i
\(866\) 0 0
\(867\) −11.6677 −0.396256
\(868\) 0 0
\(869\) −5.02880 + 2.90338i −0.170590 + 0.0984904i
\(870\) 0 0
\(871\) −30.5103 + 17.6151i −1.03380 + 0.596866i
\(872\) 0 0
\(873\) 2.86233i 0.0968753i
\(874\) 0 0
\(875\) 24.4470i 0.826459i
\(876\) 0 0
\(877\) 34.5764 19.9627i 1.16756 0.674093i 0.214458 0.976733i \(-0.431201\pi\)
0.953105 + 0.302640i \(0.0978681\pi\)
\(878\) 0 0
\(879\) −2.17708 + 1.25694i −0.0734311 + 0.0423955i
\(880\) 0 0
\(881\) −17.5857 −0.592479 −0.296239 0.955114i \(-0.595733\pi\)
−0.296239 + 0.955114i \(0.595733\pi\)
\(882\) 0 0
\(883\) −44.6874 25.8003i −1.50385 0.868249i −0.999990 0.00446371i \(-0.998579\pi\)
−0.503861 0.863785i \(-0.668088\pi\)
\(884\) 0 0
\(885\) 38.8189 1.30488
\(886\) 0 0
\(887\) −16.8360 + 29.1608i −0.565297 + 0.979122i 0.431725 + 0.902005i \(0.357905\pi\)
−0.997022 + 0.0771174i \(0.975428\pi\)
\(888\) 0 0
\(889\) −5.83108 3.36658i −0.195568 0.112911i
\(890\) 0 0
\(891\) 2.10560 1.21567i 0.0705404 0.0407265i
\(892\) 0 0
\(893\) −9.02317 47.6269i −0.301949 1.59377i
\(894\) 0 0
\(895\) 25.0415 + 43.3732i 0.837047 + 1.44981i
\(896\) 0 0
\(897\) 4.37126 7.57124i 0.145952 0.252796i
\(898\) 0 0
\(899\) 44.4933 + 25.6882i 1.48393 + 0.856749i
\(900\) 0 0
\(901\) 27.2992i 0.909468i
\(902\) 0 0
\(903\) 17.5423 30.3841i 0.583771 1.01112i
\(904\) 0 0
\(905\) 32.6023i 1.08374i
\(906\) 0 0
\(907\) 0.788790 + 1.36622i 0.0261913 + 0.0453647i 0.878824 0.477146i \(-0.158329\pi\)
−0.852633 + 0.522511i \(0.824995\pi\)
\(908\) 0 0
\(909\) 9.24254 + 16.0086i 0.306556 + 0.530970i
\(910\) 0 0
\(911\) 5.05186 0.167376 0.0836879 0.996492i \(-0.473330\pi\)
0.0836879 + 0.996492i \(0.473330\pi\)
\(912\) 0 0
\(913\) −13.5939 −0.449893
\(914\) 0 0
\(915\) 9.76976 + 16.9217i 0.322978 + 0.559415i
\(916\) 0 0
\(917\) 15.6105 + 27.0381i 0.515503 + 0.892877i
\(918\) 0 0
\(919\) 15.7897i 0.520853i −0.965494 0.260426i \(-0.916137\pi\)
0.965494 0.260426i \(-0.0838631\pi\)
\(920\) 0 0
\(921\) −4.84404 + 8.39012i −0.159616 + 0.276464i
\(922\) 0 0
\(923\) 71.3605i 2.34886i
\(924\) 0 0
\(925\) −51.9669 30.0031i −1.70866 0.986496i
\(926\) 0 0
\(927\) −1.91554 + 3.31782i −0.0629147 + 0.108971i
\(928\) 0 0
\(929\) 17.3039 + 29.9712i 0.567721 + 0.983322i 0.996791 + 0.0800507i \(0.0255082\pi\)
−0.429069 + 0.903272i \(0.641158\pi\)
\(930\) 0 0
\(931\) 12.3256 + 14.3158i 0.403956 + 0.469181i
\(932\) 0 0
\(933\) 5.68598 3.28280i 0.186151 0.107474i
\(934\) 0 0
\(935\) 39.1977 + 22.6308i 1.28190 + 0.740107i
\(936\) 0 0
\(937\) 22.6370 39.2085i 0.739520 1.28089i −0.213192 0.977010i \(-0.568386\pi\)
0.952712 0.303876i \(-0.0982807\pi\)
\(938\) 0 0
\(939\) 25.1950 0.822208
\(940\) 0 0
\(941\) −37.0830 21.4099i −1.20887 0.697943i −0.246360 0.969178i \(-0.579234\pi\)
−0.962513 + 0.271236i \(0.912568\pi\)
\(942\) 0 0
\(943\) −5.93982 −0.193427
\(944\) 0 0
\(945\) −10.1369 + 5.85256i −0.329755 + 0.190384i
\(946\) 0 0
\(947\) 29.4252 16.9887i 0.956191 0.552057i 0.0611923 0.998126i \(-0.480510\pi\)
0.894999 + 0.446069i \(0.147176\pi\)
\(948\) 0 0
\(949\) 46.1151i 1.49696i
\(950\) 0 0
\(951\) 9.26441i 0.300419i
\(952\) 0 0
\(953\) 32.4648 18.7435i 1.05164 0.607163i 0.128530 0.991706i \(-0.458974\pi\)
0.923107 + 0.384543i \(0.125641\pi\)
\(954\) 0 0
\(955\) 65.9492 38.0758i 2.13407 1.23210i
\(956\) 0 0
\(957\) 11.8229 0.382179
\(958\) 0 0
\(959\) −17.4933 10.0997i −0.564887 0.326137i
\(960\) 0 0
\(961\) 80.6284 2.60091
\(962\) 0 0
\(963\) −3.08857 + 5.34956i −0.0995278 + 0.172387i
\(964\) 0 0
\(965\) 30.5885 + 17.6603i 0.984679 + 0.568504i
\(966\) 0 0
\(967\) −36.7557 + 21.2209i −1.18198 + 0.682418i −0.956472 0.291823i \(-0.905738\pi\)
−0.225510 + 0.974241i \(0.572405\pi\)
\(968\) 0 0
\(969\) 22.9306 4.34432i 0.736638 0.139560i
\(970\) 0 0
\(971\) 2.76568 + 4.79030i 0.0887549 + 0.153728i 0.906985 0.421163i \(-0.138378\pi\)
−0.818230 + 0.574891i \(0.805045\pi\)
\(972\) 0 0
\(973\) 2.40326 4.16257i 0.0770451 0.133446i
\(974\) 0 0
\(975\) 35.5928 + 20.5495i 1.13988 + 0.658110i
\(976\) 0 0
\(977\) 15.1102i 0.483419i 0.970349 + 0.241709i \(0.0777081\pi\)
−0.970349 + 0.241709i \(0.922292\pi\)
\(978\) 0 0
\(979\) 7.08379 12.2695i 0.226399 0.392134i
\(980\) 0 0
\(981\) 2.66729i 0.0851599i
\(982\) 0 0
\(983\) 9.01709 + 15.6181i 0.287601 + 0.498139i 0.973237 0.229806i \(-0.0738091\pi\)
−0.685636 + 0.727945i \(0.740476\pi\)
\(984\) 0 0
\(985\) 4.38630 + 7.59730i 0.139759 + 0.242070i
\(986\) 0 0
\(987\) −37.4388 −1.19169
\(988\) 0 0
\(989\) −15.7142 −0.499682
\(990\) 0 0
\(991\) 10.8741 + 18.8344i 0.345426 + 0.598295i 0.985431 0.170076i \(-0.0544012\pi\)
−0.640005 + 0.768370i \(0.721068\pi\)
\(992\) 0 0
\(993\) −8.49393 14.7119i −0.269547 0.466868i
\(994\) 0 0
\(995\) 64.4251i 2.04241i
\(996\) 0 0
\(997\) −11.0995 + 19.2249i −0.351525 + 0.608859i −0.986517 0.163660i \(-0.947670\pi\)
0.634992 + 0.772519i \(0.281003\pi\)
\(998\) 0 0
\(999\) 8.46521i 0.267827i
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 912.2.bb.h.31.1 yes 8
3.2 odd 2 2736.2.bm.r.1855.4 8
4.3 odd 2 912.2.bb.g.31.1 8
12.11 even 2 2736.2.bm.s.1855.4 8
19.8 odd 6 912.2.bb.g.559.1 yes 8
57.8 even 6 2736.2.bm.s.559.4 8
76.27 even 6 inner 912.2.bb.h.559.1 yes 8
228.179 odd 6 2736.2.bm.r.559.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.bb.g.31.1 8 4.3 odd 2
912.2.bb.g.559.1 yes 8 19.8 odd 6
912.2.bb.h.31.1 yes 8 1.1 even 1 trivial
912.2.bb.h.559.1 yes 8 76.27 even 6 inner
2736.2.bm.r.559.4 8 228.179 odd 6
2736.2.bm.r.1855.4 8 3.2 odd 2
2736.2.bm.s.559.4 8 57.8 even 6
2736.2.bm.s.1855.4 8 12.11 even 2