Properties

Label 484.2.e.a.81.1
Level $484$
Weight $2$
Character 484.81
Analytic conductor $3.865$
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] = [484,2,Mod(9,484)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(484, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("484.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.e (of order \(5\), degree \(4\), 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: \(3.86475945783\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.1
Root \(0.809017 - 0.587785i\) of defining polynomial
Character \(\chi\) \(=\) 484.81
Dual form 484.2.e.a.245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.309017 - 0.951057i) q^{3} +(2.42705 - 1.76336i) q^{5} +(0.618034 + 1.90211i) q^{7} +(1.61803 + 1.17557i) q^{9} +O(q^{10})\) \(q+(0.309017 - 0.951057i) q^{3} +(2.42705 - 1.76336i) q^{5} +(0.618034 + 1.90211i) q^{7} +(1.61803 + 1.17557i) q^{9} +(3.23607 + 2.35114i) q^{13} +(-0.927051 - 2.85317i) q^{15} +(-4.85410 + 3.52671i) q^{17} +(2.47214 - 7.60845i) q^{19} +2.00000 q^{21} -3.00000 q^{23} +(1.23607 - 3.80423i) q^{25} +(4.04508 - 2.93893i) q^{27} +(-4.04508 - 2.93893i) q^{31} +(4.85410 + 3.52671i) q^{35} +(-0.309017 - 0.951057i) q^{37} +(3.23607 - 2.35114i) q^{39} -10.0000 q^{43} +6.00000 q^{45} +(2.42705 - 1.76336i) q^{49} +(1.85410 + 5.70634i) q^{51} +(4.85410 + 3.52671i) q^{53} +(-6.47214 - 4.70228i) q^{57} +(0.927051 + 2.85317i) q^{59} +(3.23607 - 2.35114i) q^{61} +(-1.23607 + 3.80423i) q^{63} +12.0000 q^{65} -1.00000 q^{67} +(-0.927051 + 2.85317i) q^{69} +(-12.1353 + 8.81678i) q^{71} +(-1.23607 - 3.80423i) q^{73} +(-3.23607 - 2.35114i) q^{75} +(-1.61803 - 1.17557i) q^{79} +(0.309017 + 0.951057i) q^{81} +(-4.85410 + 3.52671i) q^{83} +(-5.56231 + 17.1190i) q^{85} -9.00000 q^{89} +(-2.47214 + 7.60845i) q^{91} +(-4.04508 + 2.93893i) q^{93} +(-7.41641 - 22.8254i) q^{95} +(5.66312 + 4.11450i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} + 3 q^{15} - 6 q^{17} - 8 q^{19} + 8 q^{21} - 12 q^{23} - 4 q^{25} + 5 q^{27} - 5 q^{31} + 6 q^{35} + q^{37} + 4 q^{39} - 40 q^{43} + 24 q^{45} + 3 q^{49} - 6 q^{51} + 6 q^{53} - 8 q^{57} - 3 q^{59} + 4 q^{61} + 4 q^{63} + 48 q^{65} - 4 q^{67} + 3 q^{69} - 15 q^{71} + 4 q^{73} - 4 q^{75} - 2 q^{79} - q^{81} - 6 q^{83} + 18 q^{85} - 36 q^{89} + 8 q^{91} - 5 q^{93} + 24 q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.309017 0.951057i 0.178411 0.549093i −0.821362 0.570408i \(-0.806785\pi\)
0.999773 + 0.0213149i \(0.00678525\pi\)
\(4\) 0 0
\(5\) 2.42705 1.76336i 1.08541 0.788597i 0.106792 0.994281i \(-0.465942\pi\)
0.978618 + 0.205685i \(0.0659421\pi\)
\(6\) 0 0
\(7\) 0.618034 + 1.90211i 0.233595 + 0.718931i 0.997305 + 0.0733714i \(0.0233759\pi\)
−0.763710 + 0.645560i \(0.776624\pi\)
\(8\) 0 0
\(9\) 1.61803 + 1.17557i 0.539345 + 0.391857i
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 3.23607 + 2.35114i 0.897524 + 0.652089i 0.937829 0.347098i \(-0.112833\pi\)
−0.0403050 + 0.999187i \(0.512833\pi\)
\(14\) 0 0
\(15\) −0.927051 2.85317i −0.239364 0.736685i
\(16\) 0 0
\(17\) −4.85410 + 3.52671i −1.17729 + 0.855353i −0.991864 0.127304i \(-0.959367\pi\)
−0.185429 + 0.982658i \(0.559367\pi\)
\(18\) 0 0
\(19\) 2.47214 7.60845i 0.567147 1.74550i −0.0943381 0.995540i \(-0.530073\pi\)
0.661485 0.749958i \(-0.269927\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.23607 3.80423i 0.247214 0.760845i
\(26\) 0 0
\(27\) 4.04508 2.93893i 0.778477 0.565597i
\(28\) 0 0
\(29\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(30\) 0 0
\(31\) −4.04508 2.93893i −0.726519 0.527847i 0.161942 0.986800i \(-0.448224\pi\)
−0.888460 + 0.458954i \(0.848224\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.85410 + 3.52671i 0.820493 + 0.596123i
\(36\) 0 0
\(37\) −0.309017 0.951057i −0.0508021 0.156353i 0.922437 0.386148i \(-0.126194\pi\)
−0.973239 + 0.229795i \(0.926194\pi\)
\(38\) 0 0
\(39\) 3.23607 2.35114i 0.518186 0.376484i
\(40\) 0 0
\(41\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(48\) 0 0
\(49\) 2.42705 1.76336i 0.346722 0.251908i
\(50\) 0 0
\(51\) 1.85410 + 5.70634i 0.259626 + 0.799047i
\(52\) 0 0
\(53\) 4.85410 + 3.52671i 0.666762 + 0.484431i 0.868940 0.494918i \(-0.164802\pi\)
−0.202178 + 0.979349i \(0.564802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.47214 4.70228i −0.857255 0.622832i
\(58\) 0 0
\(59\) 0.927051 + 2.85317i 0.120692 + 0.371451i 0.993092 0.117342i \(-0.0374373\pi\)
−0.872400 + 0.488793i \(0.837437\pi\)
\(60\) 0 0
\(61\) 3.23607 2.35114i 0.414336 0.301033i −0.361019 0.932559i \(-0.617571\pi\)
0.775355 + 0.631526i \(0.217571\pi\)
\(62\) 0 0
\(63\) −1.23607 + 3.80423i −0.155730 + 0.479287i
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −1.00000 −0.122169 −0.0610847 0.998133i \(-0.519456\pi\)
−0.0610847 + 0.998133i \(0.519456\pi\)
\(68\) 0 0
\(69\) −0.927051 + 2.85317i −0.111604 + 0.343481i
\(70\) 0 0
\(71\) −12.1353 + 8.81678i −1.44019 + 1.04636i −0.452187 + 0.891923i \(0.649356\pi\)
−0.988003 + 0.154436i \(0.950644\pi\)
\(72\) 0 0
\(73\) −1.23607 3.80423i −0.144671 0.445251i 0.852298 0.523057i \(-0.175209\pi\)
−0.996969 + 0.0778060i \(0.975209\pi\)
\(74\) 0 0
\(75\) −3.23607 2.35114i −0.373669 0.271486i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.61803 1.17557i −0.182043 0.132262i 0.493032 0.870011i \(-0.335889\pi\)
−0.675075 + 0.737749i \(0.735889\pi\)
\(80\) 0 0
\(81\) 0.309017 + 0.951057i 0.0343352 + 0.105673i
\(82\) 0 0
\(83\) −4.85410 + 3.52671i −0.532807 + 0.387107i −0.821407 0.570343i \(-0.806810\pi\)
0.288600 + 0.957450i \(0.406810\pi\)
\(84\) 0 0
\(85\) −5.56231 + 17.1190i −0.603317 + 1.85682i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −2.47214 + 7.60845i −0.259150 + 0.797582i
\(92\) 0 0
\(93\) −4.04508 + 2.93893i −0.419456 + 0.304752i
\(94\) 0 0
\(95\) −7.41641 22.8254i −0.760907 2.34183i
\(96\) 0 0
\(97\) 5.66312 + 4.11450i 0.575003 + 0.417764i 0.836919 0.547327i \(-0.184354\pi\)
−0.261916 + 0.965091i \(0.584354\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.5623 10.5801i −1.44900 1.05276i −0.986063 0.166374i \(-0.946794\pi\)
−0.462941 0.886389i \(-0.653206\pi\)
\(102\) 0 0
\(103\) 2.47214 + 7.60845i 0.243587 + 0.749683i 0.995866 + 0.0908382i \(0.0289546\pi\)
−0.752279 + 0.658845i \(0.771045\pi\)
\(104\) 0 0
\(105\) 4.85410 3.52671i 0.473712 0.344172i
\(106\) 0 0
\(107\) 1.85410 5.70634i 0.179243 0.551653i −0.820559 0.571562i \(-0.806338\pi\)
0.999802 + 0.0199092i \(0.00633772\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) −4.63525 + 14.2658i −0.436048 + 1.34202i 0.455961 + 0.890000i \(0.349296\pi\)
−0.892009 + 0.452018i \(0.850704\pi\)
\(114\) 0 0
\(115\) −7.28115 + 5.29007i −0.678971 + 0.493301i
\(116\) 0 0
\(117\) 2.47214 + 7.60845i 0.228549 + 0.703402i
\(118\) 0 0
\(119\) −9.70820 7.05342i −0.889950 0.646586i
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.927051 + 2.85317i 0.0829180 + 0.255195i
\(126\) 0 0
\(127\) 12.9443 9.40456i 1.14862 0.834520i 0.160322 0.987065i \(-0.448747\pi\)
0.988297 + 0.152545i \(0.0487468\pi\)
\(128\) 0 0
\(129\) −3.09017 + 9.51057i −0.272074 + 0.837359i
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 0 0
\(135\) 4.63525 14.2658i 0.398939 1.22781i
\(136\) 0 0
\(137\) −7.28115 + 5.29007i −0.622071 + 0.451961i −0.853644 0.520856i \(-0.825613\pi\)
0.231573 + 0.972817i \(0.425613\pi\)
\(138\) 0 0
\(139\) 4.32624 + 13.3148i 0.366947 + 1.12935i 0.948753 + 0.316018i \(0.102346\pi\)
−0.581806 + 0.813327i \(0.697654\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.927051 2.85317i −0.0764619 0.235325i
\(148\) 0 0
\(149\) −4.85410 + 3.52671i −0.397664 + 0.288919i −0.768589 0.639743i \(-0.779041\pi\)
0.370925 + 0.928663i \(0.379041\pi\)
\(150\) 0 0
\(151\) −3.09017 + 9.51057i −0.251474 + 0.773959i 0.743029 + 0.669259i \(0.233388\pi\)
−0.994504 + 0.104700i \(0.966612\pi\)
\(152\) 0 0
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) −15.0000 −1.20483
\(156\) 0 0
\(157\) 1.54508 4.75528i 0.123311 0.379513i −0.870278 0.492560i \(-0.836061\pi\)
0.993590 + 0.113047i \(0.0360612\pi\)
\(158\) 0 0
\(159\) 4.85410 3.52671i 0.384955 0.279686i
\(160\) 0 0
\(161\) −1.85410 5.70634i −0.146124 0.449723i
\(162\) 0 0
\(163\) 3.23607 + 2.35114i 0.253468 + 0.184156i 0.707263 0.706951i \(-0.249930\pi\)
−0.453794 + 0.891107i \(0.649930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.70820 + 7.05342i 0.751243 + 0.545810i 0.896212 0.443626i \(-0.146308\pi\)
−0.144969 + 0.989436i \(0.546308\pi\)
\(168\) 0 0
\(169\) 0.927051 + 2.85317i 0.0713116 + 0.219475i
\(170\) 0 0
\(171\) 12.9443 9.40456i 0.989873 0.719185i
\(172\) 0 0
\(173\) 5.56231 17.1190i 0.422894 1.30153i −0.482102 0.876115i \(-0.660127\pi\)
0.904996 0.425420i \(-0.139873\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) −2.78115 + 8.55951i −0.207873 + 0.639768i 0.791710 + 0.610897i \(0.209191\pi\)
−0.999583 + 0.0288706i \(0.990809\pi\)
\(180\) 0 0
\(181\) 10.5172 7.64121i 0.781739 0.567967i −0.123762 0.992312i \(-0.539496\pi\)
0.905500 + 0.424345i \(0.139496\pi\)
\(182\) 0 0
\(183\) −1.23607 3.80423i −0.0913728 0.281216i
\(184\) 0 0
\(185\) −2.42705 1.76336i −0.178440 0.129644i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 8.09017 + 5.87785i 0.588473 + 0.427551i
\(190\) 0 0
\(191\) −6.48936 19.9722i −0.469553 1.44514i −0.853164 0.521642i \(-0.825320\pi\)
0.383611 0.923495i \(-0.374680\pi\)
\(192\) 0 0
\(193\) −16.1803 + 11.7557i −1.16469 + 0.846194i −0.990363 0.138494i \(-0.955774\pi\)
−0.174323 + 0.984689i \(0.555774\pi\)
\(194\) 0 0
\(195\) 3.70820 11.4127i 0.265550 0.817279i
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −0.309017 + 0.951057i −0.0217964 + 0.0670824i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.85410 3.52671i −0.337383 0.245123i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −16.1803 11.7557i −1.11390 0.809296i −0.130627 0.991432i \(-0.541699\pi\)
−0.983273 + 0.182135i \(0.941699\pi\)
\(212\) 0 0
\(213\) 4.63525 + 14.2658i 0.317602 + 0.977480i
\(214\) 0 0
\(215\) −24.2705 + 17.6336i −1.65524 + 1.20260i
\(216\) 0 0
\(217\) 3.09017 9.51057i 0.209774 0.645619i
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) 5.25329 16.1680i 0.351786 1.08269i −0.606063 0.795416i \(-0.707252\pi\)
0.957850 0.287270i \(-0.0927478\pi\)
\(224\) 0 0
\(225\) 6.47214 4.70228i 0.431476 0.313485i
\(226\) 0 0
\(227\) 1.85410 + 5.70634i 0.123061 + 0.378743i 0.993543 0.113458i \(-0.0361926\pi\)
−0.870482 + 0.492201i \(0.836193\pi\)
\(228\) 0 0
\(229\) 10.5172 + 7.64121i 0.694998 + 0.504945i 0.878299 0.478111i \(-0.158679\pi\)
−0.183302 + 0.983057i \(0.558679\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.4164 + 14.1068i 1.27201 + 0.924170i 0.999281 0.0379203i \(-0.0120733\pi\)
0.272730 + 0.962090i \(0.412073\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.61803 + 1.17557i −0.105103 + 0.0763615i
\(238\) 0 0
\(239\) 1.85410 5.70634i 0.119932 0.369112i −0.873012 0.487699i \(-0.837836\pi\)
0.992944 + 0.118587i \(0.0378363\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 0 0
\(245\) 2.78115 8.55951i 0.177681 0.546847i
\(246\) 0 0
\(247\) 25.8885 18.8091i 1.64725 1.19680i
\(248\) 0 0
\(249\) 1.85410 + 5.70634i 0.117499 + 0.361625i
\(250\) 0 0
\(251\) 7.28115 + 5.29007i 0.459582 + 0.333906i 0.793367 0.608743i \(-0.208326\pi\)
−0.333785 + 0.942649i \(0.608326\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 14.5623 + 10.5801i 0.911927 + 0.662554i
\(256\) 0 0
\(257\) −5.56231 17.1190i −0.346967 1.06785i −0.960522 0.278203i \(-0.910261\pi\)
0.613555 0.789652i \(-0.289739\pi\)
\(258\) 0 0
\(259\) 1.61803 1.17557i 0.100540 0.0730464i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) −2.78115 + 8.55951i −0.170204 + 0.523833i
\(268\) 0 0
\(269\) 4.85410 3.52671i 0.295960 0.215027i −0.429889 0.902882i \(-0.641447\pi\)
0.725849 + 0.687854i \(0.241447\pi\)
\(270\) 0 0
\(271\) 6.18034 + 19.0211i 0.375429 + 1.15545i 0.943189 + 0.332257i \(0.107810\pi\)
−0.567760 + 0.823194i \(0.692190\pi\)
\(272\) 0 0
\(273\) 6.47214 + 4.70228i 0.391711 + 0.284595i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.09017 + 5.87785i 0.486091 + 0.353166i 0.803679 0.595063i \(-0.202873\pi\)
−0.317588 + 0.948229i \(0.602873\pi\)
\(278\) 0 0
\(279\) −3.09017 9.51057i −0.185004 0.569383i
\(280\) 0 0
\(281\) 14.5623 10.5801i 0.868714 0.631158i −0.0615273 0.998105i \(-0.519597\pi\)
0.930242 + 0.366947i \(0.119597\pi\)
\(282\) 0 0
\(283\) −1.23607 + 3.80423i −0.0734766 + 0.226138i −0.981050 0.193756i \(-0.937933\pi\)
0.907573 + 0.419894i \(0.137933\pi\)
\(284\) 0 0
\(285\) −24.0000 −1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.87132 18.0701i 0.345372 1.06295i
\(290\) 0 0
\(291\) 5.66312 4.11450i 0.331978 0.241196i
\(292\) 0 0
\(293\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(294\) 0 0
\(295\) 7.28115 + 5.29007i 0.423925 + 0.308000i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.70820 7.05342i −0.561440 0.407910i
\(300\) 0 0
\(301\) −6.18034 19.0211i −0.356229 1.09636i
\(302\) 0 0
\(303\) −14.5623 + 10.5801i −0.836583 + 0.607813i
\(304\) 0 0
\(305\) 3.70820 11.4127i 0.212331 0.653488i
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 3.70820 11.4127i 0.210273 0.647154i −0.789183 0.614159i \(-0.789495\pi\)
0.999456 0.0329949i \(-0.0105045\pi\)
\(312\) 0 0
\(313\) 0.809017 0.587785i 0.0457283 0.0332236i −0.564686 0.825306i \(-0.691003\pi\)
0.610415 + 0.792082i \(0.291003\pi\)
\(314\) 0 0
\(315\) 3.70820 + 11.4127i 0.208934 + 0.643032i
\(316\) 0 0
\(317\) −26.6976 19.3969i −1.49948 1.08944i −0.970586 0.240754i \(-0.922605\pi\)
−0.528898 0.848685i \(-0.677395\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4.85410 3.52671i −0.270930 0.196842i
\(322\) 0 0
\(323\) 14.8328 + 45.6507i 0.825320 + 2.54007i
\(324\) 0 0
\(325\) 12.9443 9.40456i 0.718019 0.521671i
\(326\) 0 0
\(327\) 0.618034 1.90211i 0.0341774 0.105187i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 0.618034 1.90211i 0.0338681 0.104235i
\(334\) 0 0
\(335\) −2.42705 + 1.76336i −0.132604 + 0.0963424i
\(336\) 0 0
\(337\) 0.618034 + 1.90211i 0.0336665 + 0.103615i 0.966478 0.256751i \(-0.0826520\pi\)
−0.932811 + 0.360366i \(0.882652\pi\)
\(338\) 0 0
\(339\) 12.1353 + 8.81678i 0.659097 + 0.478862i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.1803 + 11.7557i 0.873656 + 0.634748i
\(344\) 0 0
\(345\) 2.78115 + 8.55951i 0.149732 + 0.460828i
\(346\) 0 0
\(347\) −9.70820 + 7.05342i −0.521164 + 0.378648i −0.817042 0.576578i \(-0.804388\pi\)
0.295878 + 0.955226i \(0.404388\pi\)
\(348\) 0 0
\(349\) 4.32624 13.3148i 0.231578 0.712724i −0.765979 0.642866i \(-0.777745\pi\)
0.997557 0.0698585i \(-0.0222548\pi\)
\(350\) 0 0
\(351\) 20.0000 1.06752
\(352\) 0 0
\(353\) −21.0000 −1.11772 −0.558859 0.829263i \(-0.688761\pi\)
−0.558859 + 0.829263i \(0.688761\pi\)
\(354\) 0 0
\(355\) −13.9058 + 42.7975i −0.738041 + 2.27146i
\(356\) 0 0
\(357\) −9.70820 + 7.05342i −0.513813 + 0.373307i
\(358\) 0 0
\(359\) −11.1246 34.2380i −0.587135 1.80701i −0.590524 0.807020i \(-0.701079\pi\)
0.00338942 0.999994i \(-0.498921\pi\)
\(360\) 0 0
\(361\) −36.4058 26.4503i −1.91609 1.39212i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.70820 7.05342i −0.508151 0.369193i
\(366\) 0 0
\(367\) −5.87132 18.0701i −0.306481 0.943250i −0.979121 0.203280i \(-0.934840\pi\)
0.672640 0.739970i \(-0.265160\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.70820 + 11.4127i −0.192520 + 0.592517i
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −23.4615 + 17.0458i −1.20514 + 0.875583i −0.994780 0.102042i \(-0.967462\pi\)
−0.210356 + 0.977625i \(0.567462\pi\)
\(380\) 0 0
\(381\) −4.94427 15.2169i −0.253303 0.779586i
\(382\) 0 0
\(383\) 21.8435 + 15.8702i 1.11615 + 0.810929i 0.983621 0.180250i \(-0.0576906\pi\)
0.132528 + 0.991179i \(0.457691\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −16.1803 11.7557i −0.822493 0.597576i
\(388\) 0 0
\(389\) −8.34346 25.6785i −0.423030 1.30195i −0.904868 0.425692i \(-0.860031\pi\)
0.481838 0.876260i \(-0.339969\pi\)
\(390\) 0 0
\(391\) 14.5623 10.5801i 0.736447 0.535060i
\(392\) 0 0
\(393\) −1.85410 + 5.70634i −0.0935271 + 0.287847i
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) −34.0000 −1.70641 −0.853206 0.521575i \(-0.825345\pi\)
−0.853206 + 0.521575i \(0.825345\pi\)
\(398\) 0 0
\(399\) 4.94427 15.2169i 0.247523 0.761798i
\(400\) 0 0
\(401\) −14.5623 + 10.5801i −0.727207 + 0.528347i −0.888679 0.458531i \(-0.848376\pi\)
0.161472 + 0.986877i \(0.448376\pi\)
\(402\) 0 0
\(403\) −6.18034 19.0211i −0.307865 0.947510i
\(404\) 0 0
\(405\) 2.42705 + 1.76336i 0.120601 + 0.0876219i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.61803 1.17557i −0.0800066 0.0581282i 0.547063 0.837091i \(-0.315746\pi\)
−0.627070 + 0.778963i \(0.715746\pi\)
\(410\) 0 0
\(411\) 2.78115 + 8.55951i 0.137184 + 0.422209i
\(412\) 0 0
\(413\) −4.85410 + 3.52671i −0.238855 + 0.173538i
\(414\) 0 0
\(415\) −5.56231 + 17.1190i −0.273043 + 0.840340i
\(416\) 0 0
\(417\) 14.0000 0.685583
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −3.09017 + 9.51057i −0.150606 + 0.463517i −0.997689 0.0679432i \(-0.978356\pi\)
0.847084 + 0.531460i \(0.178356\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.41641 + 22.8254i 0.359749 + 1.10719i
\(426\) 0 0
\(427\) 6.47214 + 4.70228i 0.313209 + 0.227559i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.5623 10.5801i −0.701442 0.509627i 0.178960 0.983856i \(-0.442727\pi\)
−0.880401 + 0.474229i \(0.842727\pi\)
\(432\) 0 0
\(433\) 8.96149 + 27.5806i 0.430662 + 1.32544i 0.897467 + 0.441081i \(0.145405\pi\)
−0.466805 + 0.884360i \(0.654595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.41641 + 22.8254i −0.354775 + 1.09188i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −6.48936 + 19.9722i −0.308319 + 0.948907i 0.670099 + 0.742271i \(0.266252\pi\)
−0.978418 + 0.206636i \(0.933748\pi\)
\(444\) 0 0
\(445\) −21.8435 + 15.8702i −1.03548 + 0.752320i
\(446\) 0 0
\(447\) 1.85410 + 5.70634i 0.0876960 + 0.269901i
\(448\) 0 0
\(449\) −2.42705 1.76336i −0.114540 0.0832179i 0.529041 0.848596i \(-0.322552\pi\)
−0.643580 + 0.765378i \(0.722552\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.09017 + 5.87785i 0.380109 + 0.276166i
\(454\) 0 0
\(455\) 7.41641 + 22.8254i 0.347687 + 1.07007i
\(456\) 0 0
\(457\) 22.6525 16.4580i 1.05964 0.769872i 0.0856167 0.996328i \(-0.472714\pi\)
0.974021 + 0.226456i \(0.0727139\pi\)
\(458\) 0 0
\(459\) −9.27051 + 28.5317i −0.432710 + 1.33175i
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) 23.0000 1.06890 0.534450 0.845200i \(-0.320519\pi\)
0.534450 + 0.845200i \(0.320519\pi\)
\(464\) 0 0
\(465\) −4.63525 + 14.2658i −0.214955 + 0.661563i
\(466\) 0 0
\(467\) −2.42705 + 1.76336i −0.112311 + 0.0815984i −0.642523 0.766267i \(-0.722112\pi\)
0.530212 + 0.847865i \(0.322112\pi\)
\(468\) 0 0
\(469\) −0.618034 1.90211i −0.0285382 0.0878314i
\(470\) 0 0
\(471\) −4.04508 2.93893i −0.186388 0.135419i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −25.8885 18.8091i −1.18785 0.863022i
\(476\) 0 0
\(477\) 3.70820 + 11.4127i 0.169787 + 0.522551i
\(478\) 0 0
\(479\) 9.70820 7.05342i 0.443579 0.322279i −0.343476 0.939161i \(-0.611605\pi\)
0.787055 + 0.616882i \(0.211605\pi\)
\(480\) 0 0
\(481\) 1.23607 3.80423i 0.0563598 0.173458i
\(482\) 0 0
\(483\) −6.00000 −0.273009
\(484\) 0 0
\(485\) 21.0000 0.953561
\(486\) 0 0
\(487\) 8.96149 27.5806i 0.406084 1.24980i −0.513903 0.857848i \(-0.671801\pi\)
0.919987 0.391950i \(-0.128199\pi\)
\(488\) 0 0
\(489\) 3.23607 2.35114i 0.146340 0.106322i
\(490\) 0 0
\(491\) 7.41641 + 22.8254i 0.334698 + 1.03009i 0.966871 + 0.255267i \(0.0821632\pi\)
−0.632173 + 0.774827i \(0.717837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24.2705 17.6336i −1.08868 0.790973i
\(498\) 0 0
\(499\) −1.23607 3.80423i −0.0553340 0.170301i 0.919570 0.392926i \(-0.128537\pi\)
−0.974904 + 0.222626i \(0.928537\pi\)
\(500\) 0 0
\(501\) 9.70820 7.05342i 0.433731 0.315124i
\(502\) 0 0
\(503\) −9.27051 + 28.5317i −0.413352 + 1.27217i 0.500365 + 0.865814i \(0.333199\pi\)
−0.913717 + 0.406351i \(0.866801\pi\)
\(504\) 0 0
\(505\) −54.0000 −2.40297
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −6.48936 + 19.9722i −0.287636 + 0.885252i 0.697961 + 0.716136i \(0.254091\pi\)
−0.985596 + 0.169115i \(0.945909\pi\)
\(510\) 0 0
\(511\) 6.47214 4.70228i 0.286310 0.208017i
\(512\) 0 0
\(513\) −12.3607 38.0423i −0.545737 1.67961i
\(514\) 0 0
\(515\) 19.4164 + 14.1068i 0.855589 + 0.621622i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −14.5623 10.5801i −0.639214 0.464416i
\(520\) 0 0
\(521\) −8.34346 25.6785i −0.365534 1.12500i −0.949646 0.313324i \(-0.898557\pi\)
0.584113 0.811673i \(-0.301443\pi\)
\(522\) 0 0
\(523\) −6.47214 + 4.70228i −0.283007 + 0.205616i −0.720228 0.693738i \(-0.755963\pi\)
0.437221 + 0.899354i \(0.355963\pi\)
\(524\) 0 0
\(525\) 2.47214 7.60845i 0.107893 0.332060i
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −1.85410 + 5.70634i −0.0804612 + 0.247634i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −5.56231 17.1190i −0.240479 0.740120i
\(536\) 0 0
\(537\) 7.28115 + 5.29007i 0.314205 + 0.228283i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 12.9443 + 9.40456i 0.556518 + 0.404334i 0.830183 0.557491i \(-0.188236\pi\)
−0.273665 + 0.961825i \(0.588236\pi\)
\(542\) 0 0
\(543\) −4.01722 12.3637i −0.172395 0.530579i
\(544\) 0 0
\(545\) 4.85410 3.52671i 0.207927 0.151068i
\(546\) 0 0
\(547\) 2.47214 7.60845i 0.105701 0.325314i −0.884193 0.467121i \(-0.845291\pi\)
0.989894 + 0.141807i \(0.0452913\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.23607 3.80423i 0.0525630 0.161772i
\(554\) 0 0
\(555\) −2.42705 + 1.76336i −0.103023 + 0.0748503i
\(556\) 0 0
\(557\) −5.56231 17.1190i −0.235682 0.725356i −0.997030 0.0770122i \(-0.975462\pi\)
0.761348 0.648344i \(-0.224538\pi\)
\(558\) 0 0
\(559\) −32.3607 23.5114i −1.36871 0.994427i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 29.1246 + 21.1603i 1.22746 + 0.891799i 0.996697 0.0812119i \(-0.0258790\pi\)
0.230759 + 0.973011i \(0.425879\pi\)
\(564\) 0 0
\(565\) 13.9058 + 42.7975i 0.585020 + 1.80051i
\(566\) 0 0
\(567\) −1.61803 + 1.17557i −0.0679510 + 0.0493693i
\(568\) 0 0
\(569\) 0 0 −0.951057 0.309017i \(-0.900000\pi\)
0.951057 + 0.309017i \(0.100000\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 0 0
\(573\) −21.0000 −0.877288
\(574\) 0 0
\(575\) −3.70820 + 11.4127i −0.154643 + 0.475942i
\(576\) 0 0
\(577\) −13.7533 + 9.99235i −0.572557 + 0.415987i −0.836033 0.548679i \(-0.815131\pi\)
0.263476 + 0.964666i \(0.415131\pi\)
\(578\) 0 0
\(579\) 6.18034 + 19.0211i 0.256846 + 0.790491i
\(580\) 0 0
\(581\) −9.70820 7.05342i −0.402764 0.292625i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 19.4164 + 14.1068i 0.802770 + 0.583246i
\(586\) 0 0
\(587\) −3.70820 11.4127i −0.153054 0.471052i 0.844905 0.534917i \(-0.179657\pi\)
−0.997959 + 0.0638654i \(0.979657\pi\)
\(588\) 0 0
\(589\) −32.3607 + 23.5114i −1.33340 + 0.968771i
\(590\) 0 0
\(591\) 1.85410 5.70634i 0.0762676 0.234727i
\(592\) 0 0
\(593\) −36.0000 −1.47834 −0.739171 0.673517i \(-0.764783\pi\)
−0.739171 + 0.673517i \(0.764783\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 0 0
\(597\) 2.47214 7.60845i 0.101178 0.311393i
\(598\) 0 0
\(599\) 0 0 −0.587785 0.809017i \(-0.700000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(600\) 0 0
\(601\) 8.03444 + 24.7275i 0.327732 + 1.00865i 0.970192 + 0.242336i \(0.0779136\pi\)
−0.642461 + 0.766319i \(0.722086\pi\)
\(602\) 0 0
\(603\) −1.61803 1.17557i −0.0658914 0.0478729i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −11.3262 8.22899i −0.459718 0.334005i 0.333703 0.942678i \(-0.391702\pi\)
−0.793421 + 0.608674i \(0.791702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 9.88854 30.4338i 0.399395 1.22921i −0.526091 0.850428i \(-0.676343\pi\)
0.925486 0.378782i \(-0.123657\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 5.25329 16.1680i 0.211148 0.649845i −0.788257 0.615346i \(-0.789016\pi\)
0.999405 0.0344993i \(-0.0109837\pi\)
\(620\) 0 0
\(621\) −12.1353 + 8.81678i −0.486971 + 0.353805i
\(622\) 0 0
\(623\) −5.56231 17.1190i −0.222849 0.685859i
\(624\) 0 0
\(625\) 23.4615 + 17.0458i 0.938460 + 0.681831i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.85410 + 3.52671i 0.193546 + 0.140619i
\(630\) 0 0
\(631\) −13.2877 40.8954i −0.528976 1.62802i −0.756316 0.654206i \(-0.773003\pi\)
0.227340 0.973815i \(-0.426997\pi\)
\(632\) 0 0
\(633\) −16.1803 + 11.7557i −0.643111 + 0.467247i
\(634\) 0 0
\(635\) 14.8328 45.6507i 0.588622 1.81159i
\(636\) 0 0
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) −30.0000 −1.18678
\(640\) 0 0
\(641\) 12.0517 37.0912i 0.476012 1.46501i −0.368576 0.929598i \(-0.620155\pi\)
0.844588 0.535417i \(-0.179845\pi\)
\(642\) 0 0
\(643\) 10.5172 7.64121i 0.414759 0.301340i −0.360767 0.932656i \(-0.617485\pi\)
0.775526 + 0.631316i \(0.217485\pi\)
\(644\) 0 0
\(645\) 9.27051 + 28.5317i 0.365026 + 1.12343i
\(646\) 0 0
\(647\) −2.42705 1.76336i −0.0954172 0.0693247i 0.539054 0.842271i \(-0.318782\pi\)
−0.634471 + 0.772947i \(0.718782\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −8.09017 5.87785i −0.317079 0.230371i
\(652\) 0 0
\(653\) 0.927051 + 2.85317i 0.0362783 + 0.111653i 0.967556 0.252658i \(-0.0813046\pi\)
−0.931277 + 0.364311i \(0.881305\pi\)
\(654\) 0 0
\(655\) −14.5623 + 10.5801i −0.568996 + 0.413400i
\(656\) 0 0
\(657\) 2.47214 7.60845i 0.0964472 0.296834i
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 0 0
\(663\) −7.41641 + 22.8254i −0.288029 + 0.886463i
\(664\) 0 0
\(665\) 38.8328 28.2137i 1.50587 1.09408i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −13.7533 9.99235i −0.531733 0.386327i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 27.5066 + 19.9847i 1.06030 + 0.770354i 0.974144 0.225928i \(-0.0725414\pi\)
0.0861567 + 0.996282i \(0.472541\pi\)
\(674\) 0 0
\(675\) −6.18034 19.0211i −0.237881 0.732124i
\(676\) 0 0
\(677\) 33.9787 24.6870i 1.30591 0.948798i 0.305913 0.952059i \(-0.401038\pi\)
0.999995 + 0.00326161i \(0.00103820\pi\)
\(678\) 0 0
\(679\) −4.32624 + 13.3148i −0.166026 + 0.510975i
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) −8.34346 + 25.6785i −0.318787 + 0.981126i
\(686\) 0 0
\(687\) 10.5172 7.64121i 0.401257 0.291530i
\(688\) 0 0
\(689\) 7.41641 + 22.8254i 0.282543 + 0.869577i
\(690\) 0 0
\(691\) 0.809017 + 0.587785i 0.0307765 + 0.0223604i 0.603067 0.797690i \(-0.293945\pi\)
−0.572291 + 0.820051i \(0.693945\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 33.9787 + 24.6870i 1.28889 + 0.936431i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 19.4164 14.1068i 0.734396 0.533570i
\(700\) 0 0
\(701\) 5.56231 17.1190i 0.210085 0.646576i −0.789381 0.613904i \(-0.789598\pi\)
0.999466 0.0326724i \(-0.0104018\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.1246 34.2380i 0.418384 1.28765i
\(708\) 0 0
\(709\) 29.9336 21.7481i 1.12418 0.816765i 0.139343 0.990244i \(-0.455501\pi\)
0.984838 + 0.173479i \(0.0555008\pi\)
\(710\) 0 0
\(711\) −1.23607 3.80423i −0.0463562 0.142670i
\(712\) 0 0
\(713\) 12.1353 + 8.81678i 0.454469 + 0.330191i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −4.85410 3.52671i −0.181280 0.131707i
\(718\) 0 0
\(719\) 13.9058 + 42.7975i 0.518598 + 1.59608i 0.776639 + 0.629945i \(0.216923\pi\)
−0.258042 + 0.966134i \(0.583077\pi\)
\(720\) 0 0
\(721\) −12.9443 + 9.40456i −0.482070 + 0.350244i
\(722\) 0 0
\(723\) 2.47214 7.60845i 0.0919397 0.282961i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 4.01722 12.3637i 0.148786 0.457916i
\(730\) 0 0
\(731\) 48.5410 35.2671i 1.79535 1.30440i
\(732\) 0 0
\(733\) −1.23607 3.80423i −0.0456552 0.140512i 0.925630 0.378429i \(-0.123536\pi\)
−0.971286 + 0.237917i \(0.923536\pi\)
\(734\) 0 0
\(735\) −7.28115 5.29007i −0.268569 0.195127i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 27.5066 + 19.9847i 1.01185 + 0.735149i 0.964595 0.263734i \(-0.0849541\pi\)
0.0472504 + 0.998883i \(0.484954\pi\)
\(740\) 0 0
\(741\) −9.88854 30.4338i −0.363265 1.11801i
\(742\) 0 0
\(743\) −9.70820 + 7.05342i −0.356159 + 0.258765i −0.751448 0.659792i \(-0.770644\pi\)
0.395289 + 0.918557i \(0.370644\pi\)
\(744\) 0 0
\(745\) −5.56231 + 17.1190i −0.203787 + 0.627192i
\(746\) 0 0
\(747\) −12.0000 −0.439057
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 10.8156 33.2870i 0.394667 1.21466i −0.534554 0.845134i \(-0.679520\pi\)
0.929221 0.369525i \(-0.120480\pi\)
\(752\) 0 0
\(753\) 7.28115 5.29007i 0.265340 0.192781i
\(754\) 0 0
\(755\) 9.27051 + 28.5317i 0.337388 + 1.03837i
\(756\) 0 0
\(757\) 17.7984 + 12.9313i 0.646893 + 0.469995i 0.862211 0.506549i \(-0.169079\pi\)
−0.215318 + 0.976544i \(0.569079\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −29.1246 21.1603i −1.05577 0.767059i −0.0824659 0.996594i \(-0.526280\pi\)
−0.973300 + 0.229535i \(0.926280\pi\)
\(762\) 0 0
\(763\) 1.23607 + 3.80423i 0.0447487 + 0.137722i
\(764\) 0 0
\(765\) −29.1246 + 21.1603i −1.05300 + 0.765051i
\(766\) 0 0
\(767\) −3.70820 + 11.4127i −0.133895 + 0.412088i
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 12.9787 39.9444i 0.466812 1.43670i −0.389878 0.920867i \(-0.627483\pi\)
0.856689 0.515833i \(-0.172517\pi\)
\(774\) 0 0
\(775\) −16.1803 + 11.7557i −0.581215 + 0.422277i
\(776\) 0 0
\(777\) −0.618034 1.90211i −0.0221718 0.0682379i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.63525 14.2658i −0.165439 0.509170i
\(786\) 0 0
\(787\) −25.8885 + 18.8091i −0.922827 + 0.670473i −0.944226 0.329298i \(-0.893188\pi\)
0.0213991 + 0.999771i \(0.493188\pi\)
\(788\) 0 0
\(789\) 5.56231 17.1190i 0.198023 0.609453i
\(790\) 0 0
\(791\) −30.0000 −1.06668
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 5.56231 17.1190i 0.197275 0.607149i
\(796\) 0 0
\(797\) −7.28115 + 5.29007i −0.257912 + 0.187384i −0.709226 0.704981i \(-0.750955\pi\)
0.451314 + 0.892365i \(0.350955\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −14.5623 10.5801i −0.514534 0.373831i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −14.5623 10.5801i −0.513254 0.372901i
\(806\) 0 0
\(807\) −1.85410 5.70634i −0.0652675 0.200873i
\(808\) 0 0
\(809\) 19.4164 14.1068i 0.682645 0.495970i −0.191589 0.981475i \(-0.561364\pi\)
0.874234 + 0.485505i \(0.161364\pi\)
\(810\) 0 0
\(811\) 11.7426 36.1401i 0.412340 1.26905i −0.502268 0.864712i \(-0.667501\pi\)
0.914609 0.404340i \(-0.132499\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −24.7214 + 76.0845i −0.864891 + 2.66186i
\(818\) 0 0
\(819\) −12.9443 + 9.40456i −0.452309 + 0.328622i
\(820\) 0 0
\(821\) 9.27051 + 28.5317i 0.323543 + 0.995763i 0.972094 + 0.234592i \(0.0753753\pi\)
−0.648551 + 0.761171i \(0.724625\pi\)
\(822\) 0 0
\(823\) 34.7877 + 25.2748i 1.21262 + 0.881023i 0.995466 0.0951136i \(-0.0303214\pi\)
0.217158 + 0.976137i \(0.430321\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.1246 + 21.1603i 1.01276 + 0.735815i 0.964787 0.263034i \(-0.0847231\pi\)
0.0479754 + 0.998849i \(0.484723\pi\)
\(828\) 0 0
\(829\) −5.87132 18.0701i −0.203919 0.627600i −0.999756 0.0220881i \(-0.992969\pi\)
0.795837 0.605512i \(-0.207031\pi\)
\(830\) 0 0
\(831\) 8.09017 5.87785i 0.280645 0.203900i
\(832\) 0 0
\(833\) −5.56231 + 17.1190i −0.192722 + 0.593139i
\(834\) 0 0
\(835\) 36.0000 1.24583
\(836\) 0 0
\(837\) −25.0000 −0.864126
\(838\) 0 0
\(839\) −12.0517 + 37.0912i −0.416070 + 1.28053i 0.495221 + 0.868767i \(0.335087\pi\)
−0.911291 + 0.411764i \(0.864913\pi\)
\(840\) 0 0
\(841\) 23.4615 17.0458i 0.809017 0.587785i
\(842\) 0 0
\(843\) −5.56231 17.1190i −0.191576 0.589610i
\(844\) 0 0
\(845\) 7.28115 + 5.29007i 0.250479 + 0.181984i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.23607 + 2.35114i 0.111062 + 0.0806910i
\(850\) 0 0
\(851\) 0.927051 + 2.85317i 0.0317789 + 0.0978054i
\(852\) 0 0
\(853\) −30.7426 + 22.3358i −1.05261 + 0.764765i −0.972707 0.232038i \(-0.925461\pi\)
−0.0799015 + 0.996803i \(0.525461\pi\)
\(854\) 0 0
\(855\) 14.8328 45.6507i 0.507272 1.56122i
\(856\) 0 0
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −38.8328 + 28.2137i −1.32188 + 0.960405i −0.321978 + 0.946747i \(0.604348\pi\)
−0.999907 + 0.0136580i \(0.995652\pi\)
\(864\) 0 0
\(865\) −16.6869 51.3571i −0.567372 1.74619i
\(866\) 0 0
\(867\) −15.3713 11.1679i −0.522037 0.379282i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −3.23607 2.35114i −0.109650 0.0796654i
\(872\) 0 0
\(873\) 4.32624 + 13.3148i 0.146421 + 0.450637i
\(874\) 0 0
\(875\) −4.85410 + 3.52671i −0.164099 + 0.119225i
\(876\) 0 0
\(877\) −16.0689 + 49.4549i −0.542608 + 1.66997i 0.184003 + 0.982926i \(0.441094\pi\)
−0.726611 + 0.687049i \(0.758906\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −27.0000 −0.909653 −0.454827 0.890580i \(-0.650299\pi\)
−0.454827 + 0.890580i \(0.650299\pi\)
\(882\) 0 0
\(883\) −1.23607 + 3.80423i −0.0415970 + 0.128022i −0.969698 0.244305i \(-0.921440\pi\)
0.928101 + 0.372327i \(0.121440\pi\)
\(884\) 0 0
\(885\) 7.28115 5.29007i 0.244753 0.177824i
\(886\) 0 0
\(887\) 9.27051 + 28.5317i 0.311273 + 0.958001i 0.977261 + 0.212038i \(0.0680101\pi\)
−0.665988 + 0.745962i \(0.731990\pi\)
\(888\) 0 0
\(889\) 25.8885 + 18.8091i 0.868274 + 0.630838i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 8.34346 + 25.6785i 0.278891 + 0.858338i
\(896\) 0 0
\(897\) −9.70820 + 7.05342i −0.324147 + 0.235507i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −20.0000 −0.665558
\(904\) 0 0
\(905\) 12.0517 37.0912i 0.400611 1.23295i
\(906\) 0 0
\(907\) 3.23607 2.35114i 0.107452 0.0780684i −0.532762 0.846265i \(-0.678846\pi\)
0.640214 + 0.768197i \(0.278846\pi\)
\(908\) 0 0
\(909\) −11.1246 34.2380i −0.368980 1.13560i
\(910\) 0 0
\(911\) −9.70820 7.05342i −0.321647 0.233690i 0.415231 0.909716i \(-0.363701\pi\)
−0.736878 + 0.676026i \(0.763701\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −9.70820 7.05342i −0.320943 0.233179i
\(916\) 0 0
\(917\) −3.70820 11.4127i −0.122456 0.376880i
\(918\) 0 0
\(919\) 27.5066 19.9847i 0.907358 0.659234i −0.0329871 0.999456i \(-0.510502\pi\)
0.940345 + 0.340221i \(0.110502\pi\)
\(920\) 0 0
\(921\) −4.94427 + 15.2169i −0.162919 + 0.501414i
\(922\) 0 0
\(923\) −60.0000 −1.97492
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −4.94427 + 15.2169i −0.162391 + 0.499789i
\(928\) 0 0
\(929\) −14.5623 + 10.5801i −0.477774 + 0.347123i −0.800463 0.599382i \(-0.795413\pi\)
0.322689 + 0.946505i \(0.395413\pi\)
\(930\) 0 0
\(931\) −7.41641 22.8254i −0.243063 0.748071i
\(932\) 0 0
\(933\) −9.70820 7.05342i −0.317832 0.230919i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −25.8885 18.8091i −0.845742 0.614467i 0.0782268 0.996936i \(-0.475074\pi\)
−0.923969 + 0.382468i \(0.875074\pi\)
\(938\) 0 0
\(939\) −0.309017 0.951057i −0.0100844 0.0310366i
\(940\) 0 0
\(941\) 24.2705 17.6336i 0.791196 0.574838i −0.117122 0.993118i \(-0.537367\pi\)
0.908318 + 0.418280i \(0.137367\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 30.0000 0.975900
\(946\) 0 0
\(947\) 27.0000 0.877382 0.438691 0.898638i \(-0.355442\pi\)
0.438691 + 0.898638i \(0.355442\pi\)
\(948\) 0 0
\(949\) 4.94427 15.2169i 0.160498 0.493962i
\(950\) 0 0
\(951\) −26.6976 + 19.3969i −0.865728 + 0.628988i
\(952\) 0 0
\(953\) 12.9787 + 39.9444i 0.420422 + 1.29393i 0.907311 + 0.420461i \(0.138132\pi\)
−0.486889 + 0.873464i \(0.661868\pi\)
\(954\) 0 0
\(955\) −50.9681 37.0305i −1.64929 1.19828i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.5623 10.5801i −0.470241 0.341650i
\(960\) 0 0
\(961\) −1.85410 5.70634i −0.0598097 0.184075i
\(962\) 0 0
\(963\) 9.70820 7.05342i 0.312842 0.227293i
\(964\) 0 0
\(965\) −18.5410 + 57.0634i −0.596857 + 1.83694i
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 48.0000 1.54198
\(970\) 0 0
\(971\) −4.63525 + 14.2658i −0.148752 + 0.457813i −0.997474 0.0710264i \(-0.977373\pi\)
0.848722 + 0.528839i \(0.177373\pi\)
\(972\) 0 0
\(973\) −22.6525 + 16.4580i −0.726205 + 0.527619i
\(974\) 0 0
\(975\) −4.94427 15.2169i −0.158343 0.487331i
\(976\) 0 0
\(977\) −36.4058 26.4503i −1.16472 0.846221i −0.174356 0.984683i \(-0.555784\pi\)
−0.990368 + 0.138461i \(0.955784\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.23607 + 2.35114i 0.103320 + 0.0750662i
\(982\) 0 0
\(983\) 13.9058 + 42.7975i 0.443525 + 1.36503i 0.884093 + 0.467311i \(0.154777\pi\)
−0.440568 + 0.897719i \(0.645223\pi\)
\(984\) 0 0
\(985\) 14.5623 10.5801i 0.463994 0.337111i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −2.16312 + 6.65740i −0.0686445 + 0.211266i
\(994\) 0 0
\(995\) 19.4164 14.1068i 0.615542 0.447217i
\(996\) 0 0
\(997\) −3.09017 9.51057i −0.0978667 0.301203i 0.890124 0.455719i \(-0.150618\pi\)
−0.987990 + 0.154517i \(0.950618\pi\)
\(998\) 0 0
\(999\) −4.04508 2.93893i −0.127981 0.0929835i
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 484.2.e.a.81.1 4
11.2 odd 10 484.2.e.b.269.1 4
11.3 even 5 inner 484.2.e.a.245.1 4
11.4 even 5 inner 484.2.e.a.9.1 4
11.5 even 5 44.2.a.a.1.1 1
11.6 odd 10 484.2.a.a.1.1 1
11.7 odd 10 484.2.e.b.9.1 4
11.8 odd 10 484.2.e.b.245.1 4
11.9 even 5 inner 484.2.e.a.269.1 4
11.10 odd 2 484.2.e.b.81.1 4
33.5 odd 10 396.2.a.c.1.1 1
33.17 even 10 4356.2.a.j.1.1 1
44.27 odd 10 176.2.a.a.1.1 1
44.39 even 10 1936.2.a.c.1.1 1
55.27 odd 20 1100.2.b.c.749.1 2
55.38 odd 20 1100.2.b.c.749.2 2
55.49 even 10 1100.2.a.b.1.1 1
77.5 odd 30 2156.2.i.c.1145.1 2
77.16 even 15 2156.2.i.b.1145.1 2
77.27 odd 10 2156.2.a.a.1.1 1
77.38 odd 30 2156.2.i.c.177.1 2
77.60 even 15 2156.2.i.b.177.1 2
88.5 even 10 704.2.a.f.1.1 1
88.27 odd 10 704.2.a.i.1.1 1
88.61 odd 10 7744.2.a.m.1.1 1
88.83 even 10 7744.2.a.bc.1.1 1
99.5 odd 30 3564.2.i.a.1189.1 2
99.16 even 15 3564.2.i.j.2377.1 2
99.38 odd 30 3564.2.i.a.2377.1 2
99.49 even 15 3564.2.i.j.1189.1 2
132.71 even 10 1584.2.a.p.1.1 1
143.38 even 10 7436.2.a.d.1.1 1
165.38 even 20 9900.2.c.g.5149.1 2
165.104 odd 10 9900.2.a.h.1.1 1
165.137 even 20 9900.2.c.g.5149.2 2
176.5 even 20 2816.2.c.e.1409.1 2
176.27 odd 20 2816.2.c.k.1409.2 2
176.93 even 20 2816.2.c.e.1409.2 2
176.115 odd 20 2816.2.c.k.1409.1 2
220.27 even 20 4400.2.b.k.4049.2 2
220.159 odd 10 4400.2.a.v.1.1 1
220.203 even 20 4400.2.b.k.4049.1 2
264.5 odd 10 6336.2.a.j.1.1 1
264.203 even 10 6336.2.a.i.1.1 1
308.27 even 10 8624.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.2.a.a.1.1 1 11.5 even 5
176.2.a.a.1.1 1 44.27 odd 10
396.2.a.c.1.1 1 33.5 odd 10
484.2.a.a.1.1 1 11.6 odd 10
484.2.e.a.9.1 4 11.4 even 5 inner
484.2.e.a.81.1 4 1.1 even 1 trivial
484.2.e.a.245.1 4 11.3 even 5 inner
484.2.e.a.269.1 4 11.9 even 5 inner
484.2.e.b.9.1 4 11.7 odd 10
484.2.e.b.81.1 4 11.10 odd 2
484.2.e.b.245.1 4 11.8 odd 10
484.2.e.b.269.1 4 11.2 odd 10
704.2.a.f.1.1 1 88.5 even 10
704.2.a.i.1.1 1 88.27 odd 10
1100.2.a.b.1.1 1 55.49 even 10
1100.2.b.c.749.1 2 55.27 odd 20
1100.2.b.c.749.2 2 55.38 odd 20
1584.2.a.p.1.1 1 132.71 even 10
1936.2.a.c.1.1 1 44.39 even 10
2156.2.a.a.1.1 1 77.27 odd 10
2156.2.i.b.177.1 2 77.60 even 15
2156.2.i.b.1145.1 2 77.16 even 15
2156.2.i.c.177.1 2 77.38 odd 30
2156.2.i.c.1145.1 2 77.5 odd 30
2816.2.c.e.1409.1 2 176.5 even 20
2816.2.c.e.1409.2 2 176.93 even 20
2816.2.c.k.1409.1 2 176.115 odd 20
2816.2.c.k.1409.2 2 176.27 odd 20
3564.2.i.a.1189.1 2 99.5 odd 30
3564.2.i.a.2377.1 2 99.38 odd 30
3564.2.i.j.1189.1 2 99.49 even 15
3564.2.i.j.2377.1 2 99.16 even 15
4356.2.a.j.1.1 1 33.17 even 10
4400.2.a.v.1.1 1 220.159 odd 10
4400.2.b.k.4049.1 2 220.203 even 20
4400.2.b.k.4049.2 2 220.27 even 20
6336.2.a.i.1.1 1 264.203 even 10
6336.2.a.j.1.1 1 264.5 odd 10
7436.2.a.d.1.1 1 143.38 even 10
7744.2.a.m.1.1 1 88.61 odd 10
7744.2.a.bc.1.1 1 88.83 even 10
8624.2.a.w.1.1 1 308.27 even 10
9900.2.a.h.1.1 1 165.104 odd 10
9900.2.c.g.5149.1 2 165.38 even 20
9900.2.c.g.5149.2 2 165.137 even 20