Properties

Label 1344.2.q.y.961.3
Level $1344$
Weight $2$
Character 1344.961
Analytic conductor $10.732$
Analytic rank $0$
Dimension $6$
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] = [1344,2,Mod(193,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.3
Root \(0.500000 + 0.0585812i\) of defining polynomial
Character \(\chi\) \(=\) 1344.961
Dual form 1344.2.q.y.193.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(1.37328 - 2.37860i) q^{5} +(2.64510 - 0.0585812i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{3} +(1.37328 - 2.37860i) q^{5} +(2.64510 - 0.0585812i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(-0.771819 - 1.33683i) q^{11} -6.03677 q^{13} -2.74657 q^{15} +(-3.74657 - 6.48925i) q^{17} +(3.01839 - 5.22800i) q^{19} +(-1.37328 - 2.26144i) q^{21} +(-3.74657 + 6.48925i) q^{23} +(-1.27182 - 2.20285i) q^{25} +1.00000 q^{27} -1.25343 q^{29} +(2.64510 + 4.58145i) q^{31} +(-0.771819 + 1.33683i) q^{33} +(3.49314 - 6.37208i) q^{35} +(2.47475 - 4.28639i) q^{37} +(3.01839 + 5.22800i) q^{39} -5.08727 q^{41} +3.45636 q^{43} +(1.37328 + 2.37860i) q^{45} +(4.74657 - 8.22130i) q^{47} +(6.99314 - 0.309906i) q^{49} +(-3.74657 + 6.48925i) q^{51} +(-1.91692 - 3.32021i) q^{53} -4.23970 q^{55} -6.03677 q^{57} +(-2.77182 - 4.80093i) q^{59} +(-7.29021 + 12.6270i) q^{61} +(-1.27182 + 2.32002i) q^{63} +(-8.29021 + 14.3591i) q^{65} +(2.01839 + 3.49595i) q^{67} +7.49314 q^{69} +5.49314 q^{71} +(-6.27182 - 10.8631i) q^{73} +(-1.27182 + 2.20285i) q^{75} +(-2.11985 - 3.49084i) q^{77} +(-3.89853 + 6.75246i) q^{79} +(-0.500000 - 0.866025i) q^{81} -6.52991 q^{83} -20.5804 q^{85} +(0.626716 + 1.08550i) q^{87} +(4.74657 - 8.22130i) q^{89} +(-15.9679 + 0.353641i) q^{91} +(2.64510 - 4.58145i) q^{93} +(-8.29021 - 14.3591i) q^{95} +1.54364 q^{97} +1.54364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 3 q^{7} - 3 q^{9} + 6 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{23} - 3 q^{25} + 6 q^{27} - 24 q^{29} + 3 q^{31} - 12 q^{35} + 3 q^{37} - 3 q^{39} - 12 q^{41} + 30 q^{43} + 12 q^{47} + 9 q^{49} - 6 q^{51} + 6 q^{53} + 24 q^{55} + 6 q^{57} - 12 q^{59} - 18 q^{61} - 3 q^{63} - 24 q^{65} - 9 q^{67} + 12 q^{69} - 33 q^{73} - 3 q^{75} + 12 q^{77} - 27 q^{79} - 3 q^{81} + 36 q^{83} - 72 q^{85} + 12 q^{87} + 12 q^{89} - 51 q^{91} + 3 q^{93} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 0.866025i −0.288675 0.500000i
\(4\) 0 0
\(5\) 1.37328 2.37860i 0.614151 1.06374i −0.376381 0.926465i \(-0.622832\pi\)
0.990533 0.137277i \(-0.0438349\pi\)
\(6\) 0 0
\(7\) 2.64510 0.0585812i 0.999755 0.0221416i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −0.771819 1.33683i −0.232712 0.403069i 0.725893 0.687807i \(-0.241427\pi\)
−0.958605 + 0.284738i \(0.908093\pi\)
\(12\) 0 0
\(13\) −6.03677 −1.67430 −0.837150 0.546974i \(-0.815780\pi\)
−0.837150 + 0.546974i \(0.815780\pi\)
\(14\) 0 0
\(15\) −2.74657 −0.709161
\(16\) 0 0
\(17\) −3.74657 6.48925i −0.908676 1.57387i −0.815905 0.578186i \(-0.803761\pi\)
−0.0927713 0.995687i \(-0.529573\pi\)
\(18\) 0 0
\(19\) 3.01839 5.22800i 0.692465 1.19939i −0.278562 0.960418i \(-0.589858\pi\)
0.971028 0.238967i \(-0.0768088\pi\)
\(20\) 0 0
\(21\) −1.37328 2.26144i −0.299675 0.493486i
\(22\) 0 0
\(23\) −3.74657 + 6.48925i −0.781213 + 1.35310i 0.150022 + 0.988683i \(0.452066\pi\)
−0.931235 + 0.364419i \(0.881268\pi\)
\(24\) 0 0
\(25\) −1.27182 2.20285i −0.254364 0.440571i
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.25343 −0.232756 −0.116378 0.993205i \(-0.537128\pi\)
−0.116378 + 0.993205i \(0.537128\pi\)
\(30\) 0 0
\(31\) 2.64510 + 4.58145i 0.475074 + 0.822853i 0.999592 0.0285462i \(-0.00908778\pi\)
−0.524518 + 0.851399i \(0.675754\pi\)
\(32\) 0 0
\(33\) −0.771819 + 1.33683i −0.134356 + 0.232712i
\(34\) 0 0
\(35\) 3.49314 6.37208i 0.590448 1.07708i
\(36\) 0 0
\(37\) 2.47475 4.28639i 0.406846 0.704679i −0.587688 0.809088i \(-0.699962\pi\)
0.994534 + 0.104409i \(0.0332951\pi\)
\(38\) 0 0
\(39\) 3.01839 + 5.22800i 0.483329 + 0.837150i
\(40\) 0 0
\(41\) −5.08727 −0.794499 −0.397249 0.917711i \(-0.630035\pi\)
−0.397249 + 0.917711i \(0.630035\pi\)
\(42\) 0 0
\(43\) 3.45636 0.527090 0.263545 0.964647i \(-0.415108\pi\)
0.263545 + 0.964647i \(0.415108\pi\)
\(44\) 0 0
\(45\) 1.37328 + 2.37860i 0.204717 + 0.354580i
\(46\) 0 0
\(47\) 4.74657 8.22130i 0.692358 1.19920i −0.278705 0.960377i \(-0.589905\pi\)
0.971063 0.238823i \(-0.0767615\pi\)
\(48\) 0 0
\(49\) 6.99314 0.309906i 0.999019 0.0442723i
\(50\) 0 0
\(51\) −3.74657 + 6.48925i −0.524624 + 0.908676i
\(52\) 0 0
\(53\) −1.91692 3.32021i −0.263309 0.456065i 0.703810 0.710388i \(-0.251481\pi\)
−0.967119 + 0.254323i \(0.918147\pi\)
\(54\) 0 0
\(55\) −4.23970 −0.571682
\(56\) 0 0
\(57\) −6.03677 −0.799590
\(58\) 0 0
\(59\) −2.77182 4.80093i −0.360860 0.625028i 0.627243 0.778824i \(-0.284183\pi\)
−0.988103 + 0.153796i \(0.950850\pi\)
\(60\) 0 0
\(61\) −7.29021 + 12.6270i −0.933415 + 1.61672i −0.155979 + 0.987760i \(0.549853\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(62\) 0 0
\(63\) −1.27182 + 2.32002i −0.160234 + 0.292295i
\(64\) 0 0
\(65\) −8.29021 + 14.3591i −1.02827 + 1.78102i
\(66\) 0 0
\(67\) 2.01839 + 3.49595i 0.246585 + 0.427098i 0.962576 0.271012i \(-0.0873581\pi\)
−0.715991 + 0.698110i \(0.754025\pi\)
\(68\) 0 0
\(69\) 7.49314 0.902068
\(70\) 0 0
\(71\) 5.49314 0.651915 0.325958 0.945384i \(-0.394313\pi\)
0.325958 + 0.945384i \(0.394313\pi\)
\(72\) 0 0
\(73\) −6.27182 10.8631i −0.734061 1.27143i −0.955134 0.296173i \(-0.904289\pi\)
0.221073 0.975257i \(-0.429044\pi\)
\(74\) 0 0
\(75\) −1.27182 + 2.20285i −0.146857 + 0.254364i
\(76\) 0 0
\(77\) −2.11985 3.49084i −0.241580 0.397818i
\(78\) 0 0
\(79\) −3.89853 + 6.75246i −0.438619 + 0.759711i −0.997583 0.0694809i \(-0.977866\pi\)
0.558964 + 0.829192i \(0.311199\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −6.52991 −0.716751 −0.358375 0.933578i \(-0.616669\pi\)
−0.358375 + 0.933578i \(0.616669\pi\)
\(84\) 0 0
\(85\) −20.5804 −2.23226
\(86\) 0 0
\(87\) 0.626716 + 1.08550i 0.0671910 + 0.116378i
\(88\) 0 0
\(89\) 4.74657 8.22130i 0.503135 0.871456i −0.496858 0.867832i \(-0.665513\pi\)
0.999993 0.00362404i \(-0.00115357\pi\)
\(90\) 0 0
\(91\) −15.9679 + 0.353641i −1.67389 + 0.0370717i
\(92\) 0 0
\(93\) 2.64510 4.58145i 0.274284 0.475074i
\(94\) 0 0
\(95\) −8.29021 14.3591i −0.850557 1.47321i
\(96\) 0 0
\(97\) 1.54364 0.156733 0.0783663 0.996925i \(-0.475030\pi\)
0.0783663 + 0.996925i \(0.475030\pi\)
\(98\) 0 0
\(99\) 1.54364 0.155141
\(100\) 0 0
\(101\) −1.29021 2.23470i −0.128380 0.222361i 0.794669 0.607043i \(-0.207644\pi\)
−0.923049 + 0.384682i \(0.874311\pi\)
\(102\) 0 0
\(103\) 3.27182 5.66696i 0.322382 0.558382i −0.658597 0.752496i \(-0.728850\pi\)
0.980979 + 0.194114i \(0.0621832\pi\)
\(104\) 0 0
\(105\) −7.26496 + 0.160897i −0.708987 + 0.0157020i
\(106\) 0 0
\(107\) 2.26496 3.92302i 0.218961 0.379252i −0.735529 0.677493i \(-0.763066\pi\)
0.954491 + 0.298241i \(0.0963998\pi\)
\(108\) 0 0
\(109\) −3.22132 5.57949i −0.308546 0.534418i 0.669498 0.742814i \(-0.266509\pi\)
−0.978045 + 0.208396i \(0.933176\pi\)
\(110\) 0 0
\(111\) −4.94950 −0.469786
\(112\) 0 0
\(113\) 8.00000 0.752577 0.376288 0.926503i \(-0.377200\pi\)
0.376288 + 0.926503i \(0.377200\pi\)
\(114\) 0 0
\(115\) 10.2902 + 17.8232i 0.959567 + 1.66202i
\(116\) 0 0
\(117\) 3.01839 5.22800i 0.279050 0.483329i
\(118\) 0 0
\(119\) −10.2902 16.9452i −0.943302 1.55337i
\(120\) 0 0
\(121\) 4.30859 7.46270i 0.391690 0.678427i
\(122\) 0 0
\(123\) 2.54364 + 4.40571i 0.229352 + 0.397249i
\(124\) 0 0
\(125\) 6.74657 0.603431
\(126\) 0 0
\(127\) −3.79707 −0.336935 −0.168468 0.985707i \(-0.553882\pi\)
−0.168468 + 0.985707i \(0.553882\pi\)
\(128\) 0 0
\(129\) −1.72818 2.99330i −0.152158 0.263545i
\(130\) 0 0
\(131\) −0.771819 + 1.33683i −0.0674341 + 0.116799i −0.897771 0.440462i \(-0.854815\pi\)
0.830337 + 0.557262i \(0.188148\pi\)
\(132\) 0 0
\(133\) 7.67768 14.0054i 0.665739 1.21442i
\(134\) 0 0
\(135\) 1.37328 2.37860i 0.118193 0.204717i
\(136\) 0 0
\(137\) −0.746568 1.29309i −0.0637836 0.110476i 0.832370 0.554220i \(-0.186983\pi\)
−0.896154 + 0.443744i \(0.853650\pi\)
\(138\) 0 0
\(139\) −2.94950 −0.250173 −0.125087 0.992146i \(-0.539921\pi\)
−0.125087 + 0.992146i \(0.539921\pi\)
\(140\) 0 0
\(141\) −9.49314 −0.799466
\(142\) 0 0
\(143\) 4.65929 + 8.07013i 0.389630 + 0.674858i
\(144\) 0 0
\(145\) −1.72132 + 2.98141i −0.142948 + 0.247593i
\(146\) 0 0
\(147\) −3.76496 5.90128i −0.310528 0.486729i
\(148\) 0 0
\(149\) −0.746568 + 1.29309i −0.0611613 + 0.105934i −0.894985 0.446097i \(-0.852814\pi\)
0.833823 + 0.552031i \(0.186147\pi\)
\(150\) 0 0
\(151\) 4.86642 + 8.42889i 0.396024 + 0.685933i 0.993231 0.116154i \(-0.0370565\pi\)
−0.597208 + 0.802087i \(0.703723\pi\)
\(152\) 0 0
\(153\) 7.49314 0.605784
\(154\) 0 0
\(155\) 14.5299 1.16707
\(156\) 0 0
\(157\) 4.54364 + 7.86981i 0.362622 + 0.628079i 0.988391 0.151929i \(-0.0485484\pi\)
−0.625770 + 0.780008i \(0.715215\pi\)
\(158\) 0 0
\(159\) −1.91692 + 3.32021i −0.152022 + 0.263309i
\(160\) 0 0
\(161\) −9.52991 + 17.3842i −0.751062 + 1.37007i
\(162\) 0 0
\(163\) −9.78334 + 16.9452i −0.766290 + 1.32725i 0.173271 + 0.984874i \(0.444566\pi\)
−0.939562 + 0.342380i \(0.888767\pi\)
\(164\) 0 0
\(165\) 2.11985 + 3.67169i 0.165030 + 0.285841i
\(166\) 0 0
\(167\) 11.8990 0.920772 0.460386 0.887719i \(-0.347711\pi\)
0.460386 + 0.887719i \(0.347711\pi\)
\(168\) 0 0
\(169\) 23.4426 1.80328
\(170\) 0 0
\(171\) 3.01839 + 5.22800i 0.230822 + 0.399795i
\(172\) 0 0
\(173\) 1.29021 2.23470i 0.0980925 0.169901i −0.812803 0.582539i \(-0.802059\pi\)
0.910895 + 0.412638i \(0.135393\pi\)
\(174\) 0 0
\(175\) −3.49314 5.75227i −0.264056 0.434831i
\(176\) 0 0
\(177\) −2.77182 + 4.80093i −0.208343 + 0.360860i
\(178\) 0 0
\(179\) 1.49314 + 2.58619i 0.111602 + 0.193301i 0.916416 0.400226i \(-0.131068\pi\)
−0.804814 + 0.593527i \(0.797735\pi\)
\(180\) 0 0
\(181\) 10.5436 0.783702 0.391851 0.920029i \(-0.371835\pi\)
0.391851 + 0.920029i \(0.371835\pi\)
\(182\) 0 0
\(183\) 14.5804 1.07781
\(184\) 0 0
\(185\) −6.79707 11.7729i −0.499730 0.865559i
\(186\) 0 0
\(187\) −5.78334 + 10.0170i −0.422920 + 0.732519i
\(188\) 0 0
\(189\) 2.64510 0.0585812i 0.192403 0.00426115i
\(190\) 0 0
\(191\) 8.58041 14.8617i 0.620857 1.07536i −0.368470 0.929640i \(-0.620118\pi\)
0.989327 0.145716i \(-0.0465485\pi\)
\(192\) 0 0
\(193\) −5.44950 9.43881i −0.392264 0.679420i 0.600484 0.799637i \(-0.294975\pi\)
−0.992748 + 0.120216i \(0.961641\pi\)
\(194\) 0 0
\(195\) 16.5804 1.18735
\(196\) 0 0
\(197\) −5.41959 −0.386130 −0.193065 0.981186i \(-0.561843\pi\)
−0.193065 + 0.981186i \(0.561843\pi\)
\(198\) 0 0
\(199\) −11.4931 19.9067i −0.814727 1.41115i −0.909524 0.415652i \(-0.863554\pi\)
0.0947970 0.995497i \(-0.469780\pi\)
\(200\) 0 0
\(201\) 2.01839 3.49595i 0.142366 0.246585i
\(202\) 0 0
\(203\) −3.31546 + 0.0734275i −0.232699 + 0.00515360i
\(204\) 0 0
\(205\) −6.98627 + 12.1006i −0.487942 + 0.845141i
\(206\) 0 0
\(207\) −3.74657 6.48925i −0.260404 0.451034i
\(208\) 0 0
\(209\) −9.31859 −0.644580
\(210\) 0 0
\(211\) 3.08727 0.212537 0.106268 0.994337i \(-0.466110\pi\)
0.106268 + 0.994337i \(0.466110\pi\)
\(212\) 0 0
\(213\) −2.74657 4.75720i −0.188192 0.325958i
\(214\) 0 0
\(215\) 4.74657 8.22130i 0.323713 0.560688i
\(216\) 0 0
\(217\) 7.26496 + 11.9635i 0.493177 + 0.812132i
\(218\) 0 0
\(219\) −6.27182 + 10.8631i −0.423810 + 0.734061i
\(220\) 0 0
\(221\) 22.6172 + 39.1741i 1.52140 + 2.63514i
\(222\) 0 0
\(223\) 22.7466 1.52322 0.761611 0.648034i \(-0.224409\pi\)
0.761611 + 0.648034i \(0.224409\pi\)
\(224\) 0 0
\(225\) 2.54364 0.169576
\(226\) 0 0
\(227\) −11.3017 19.5752i −0.750122 1.29925i −0.947763 0.318975i \(-0.896661\pi\)
0.197641 0.980274i \(-0.436672\pi\)
\(228\) 0 0
\(229\) −5.67768 + 9.83403i −0.375192 + 0.649851i −0.990356 0.138548i \(-0.955756\pi\)
0.615164 + 0.788399i \(0.289090\pi\)
\(230\) 0 0
\(231\) −1.96323 + 3.58126i −0.129171 + 0.235630i
\(232\) 0 0
\(233\) −1.70979 + 2.96145i −0.112012 + 0.194011i −0.916582 0.399848i \(-0.869063\pi\)
0.804569 + 0.593859i \(0.202396\pi\)
\(234\) 0 0
\(235\) −13.0368 22.5804i −0.850425 1.47298i
\(236\) 0 0
\(237\) 7.79707 0.506474
\(238\) 0 0
\(239\) 7.49314 0.484691 0.242345 0.970190i \(-0.422083\pi\)
0.242345 + 0.970190i \(0.422083\pi\)
\(240\) 0 0
\(241\) 9.30173 + 16.1111i 0.599177 + 1.03781i 0.992943 + 0.118594i \(0.0378388\pi\)
−0.393766 + 0.919211i \(0.628828\pi\)
\(242\) 0 0
\(243\) −0.500000 + 0.866025i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 8.86642 17.0594i 0.566455 1.08989i
\(246\) 0 0
\(247\) −18.2213 + 31.5602i −1.15939 + 2.00813i
\(248\) 0 0
\(249\) 3.26496 + 5.65507i 0.206908 + 0.358375i
\(250\) 0 0
\(251\) 24.5299 1.54831 0.774157 0.632994i \(-0.218174\pi\)
0.774157 + 0.632994i \(0.218174\pi\)
\(252\) 0 0
\(253\) 11.5667 0.727191
\(254\) 0 0
\(255\) 10.2902 + 17.8232i 0.644398 + 1.11613i
\(256\) 0 0
\(257\) 8.74657 15.1495i 0.545596 0.945000i −0.452973 0.891524i \(-0.649637\pi\)
0.998569 0.0534758i \(-0.0170300\pi\)
\(258\) 0 0
\(259\) 6.29487 11.4829i 0.391144 0.713514i
\(260\) 0 0
\(261\) 0.626716 1.08550i 0.0387927 0.0671910i
\(262\) 0 0
\(263\) −6.74657 11.6854i −0.416011 0.720553i 0.579523 0.814956i \(-0.303239\pi\)
−0.995534 + 0.0944035i \(0.969906\pi\)
\(264\) 0 0
\(265\) −10.5299 −0.646847
\(266\) 0 0
\(267\) −9.49314 −0.580971
\(268\) 0 0
\(269\) 3.17035 + 5.49121i 0.193300 + 0.334805i 0.946342 0.323167i \(-0.104748\pi\)
−0.753042 + 0.657972i \(0.771414\pi\)
\(270\) 0 0
\(271\) −10.1199 + 17.5281i −0.614737 + 1.06476i 0.375693 + 0.926744i \(0.377405\pi\)
−0.990431 + 0.138012i \(0.955929\pi\)
\(272\) 0 0
\(273\) 8.29021 + 13.6518i 0.501746 + 0.826243i
\(274\) 0 0
\(275\) −1.96323 + 3.40041i −0.118387 + 0.205052i
\(276\) 0 0
\(277\) −1.27182 2.20285i −0.0764162 0.132357i 0.825285 0.564716i \(-0.191014\pi\)
−0.901701 + 0.432360i \(0.857681\pi\)
\(278\) 0 0
\(279\) −5.29021 −0.316716
\(280\) 0 0
\(281\) 26.5804 1.58565 0.792827 0.609447i \(-0.208608\pi\)
0.792827 + 0.609447i \(0.208608\pi\)
\(282\) 0 0
\(283\) 10.2718 + 17.7913i 0.610596 + 1.05758i 0.991140 + 0.132821i \(0.0424035\pi\)
−0.380544 + 0.924763i \(0.624263\pi\)
\(284\) 0 0
\(285\) −8.29021 + 14.3591i −0.491069 + 0.850557i
\(286\) 0 0
\(287\) −13.4564 + 0.298018i −0.794304 + 0.0175915i
\(288\) 0 0
\(289\) −19.5735 + 33.9024i −1.15139 + 1.99426i
\(290\) 0 0
\(291\) −0.771819 1.33683i −0.0452448 0.0783663i
\(292\) 0 0
\(293\) −5.25343 −0.306909 −0.153454 0.988156i \(-0.549040\pi\)
−0.153454 + 0.988156i \(0.549040\pi\)
\(294\) 0 0
\(295\) −15.2260 −0.886491
\(296\) 0 0
\(297\) −0.771819 1.33683i −0.0447855 0.0775707i
\(298\) 0 0
\(299\) 22.6172 39.1741i 1.30799 2.26550i
\(300\) 0 0
\(301\) 9.14243 0.202478i 0.526961 0.0116706i
\(302\) 0 0
\(303\) −1.29021 + 2.23470i −0.0741204 + 0.128380i
\(304\) 0 0
\(305\) 20.0230 + 34.6809i 1.14652 + 1.98582i
\(306\) 0 0
\(307\) 14.8485 0.847449 0.423724 0.905791i \(-0.360723\pi\)
0.423724 + 0.905791i \(0.360723\pi\)
\(308\) 0 0
\(309\) −6.54364 −0.372255
\(310\) 0 0
\(311\) −0.949499 1.64458i −0.0538412 0.0932556i 0.837849 0.545903i \(-0.183813\pi\)
−0.891690 + 0.452647i \(0.850480\pi\)
\(312\) 0 0
\(313\) 0.0436371 0.0755817i 0.00246652 0.00427213i −0.864790 0.502135i \(-0.832548\pi\)
0.867256 + 0.497862i \(0.165882\pi\)
\(314\) 0 0
\(315\) 3.77182 + 6.21119i 0.212518 + 0.349961i
\(316\) 0 0
\(317\) 4.11985 7.13579i 0.231394 0.400786i −0.726825 0.686823i \(-0.759005\pi\)
0.958219 + 0.286037i \(0.0923380\pi\)
\(318\) 0 0
\(319\) 0.967422 + 1.67562i 0.0541652 + 0.0938169i
\(320\) 0 0
\(321\) −4.52991 −0.252835
\(322\) 0 0
\(323\) −45.2344 −2.51691
\(324\) 0 0
\(325\) 7.67768 + 13.2981i 0.425881 + 0.737648i
\(326\) 0 0
\(327\) −3.22132 + 5.57949i −0.178139 + 0.308546i
\(328\) 0 0
\(329\) 12.0735 22.0242i 0.665636 1.21424i
\(330\) 0 0
\(331\) 2.76496 4.78904i 0.151976 0.263230i −0.779978 0.625807i \(-0.784770\pi\)
0.931954 + 0.362577i \(0.118103\pi\)
\(332\) 0 0
\(333\) 2.47475 + 4.28639i 0.135615 + 0.234893i
\(334\) 0 0
\(335\) 11.0873 0.605763
\(336\) 0 0
\(337\) 7.17455 0.390823 0.195411 0.980721i \(-0.437396\pi\)
0.195411 + 0.980721i \(0.437396\pi\)
\(338\) 0 0
\(339\) −4.00000 6.92820i −0.217250 0.376288i
\(340\) 0 0
\(341\) 4.08308 7.07210i 0.221111 0.382976i
\(342\) 0 0
\(343\) 18.4794 1.22940i 0.997794 0.0663814i
\(344\) 0 0
\(345\) 10.2902 17.8232i 0.554006 0.959567i
\(346\) 0 0
\(347\) −12.0368 20.8483i −0.646168 1.11920i −0.984030 0.178000i \(-0.943037\pi\)
0.337863 0.941195i \(-0.390296\pi\)
\(348\) 0 0
\(349\) 31.1608 1.66800 0.834000 0.551764i \(-0.186045\pi\)
0.834000 + 0.551764i \(0.186045\pi\)
\(350\) 0 0
\(351\) −6.03677 −0.322219
\(352\) 0 0
\(353\) 5.08727 + 8.81142i 0.270768 + 0.468984i 0.969059 0.246830i \(-0.0793889\pi\)
−0.698290 + 0.715814i \(0.746056\pi\)
\(354\) 0 0
\(355\) 7.54364 13.0660i 0.400375 0.693469i
\(356\) 0 0
\(357\) −9.52991 + 17.3842i −0.504376 + 0.920070i
\(358\) 0 0
\(359\) −16.1240 + 27.9277i −0.850995 + 1.47397i 0.0293169 + 0.999570i \(0.490667\pi\)
−0.880312 + 0.474396i \(0.842667\pi\)
\(360\) 0 0
\(361\) −8.72132 15.1058i −0.459017 0.795040i
\(362\) 0 0
\(363\) −8.61718 −0.452285
\(364\) 0 0
\(365\) −34.4520 −1.80330
\(366\) 0 0
\(367\) 1.93531 + 3.35205i 0.101022 + 0.174976i 0.912106 0.409954i \(-0.134455\pi\)
−0.811084 + 0.584930i \(0.801122\pi\)
\(368\) 0 0
\(369\) 2.54364 4.40571i 0.132416 0.229352i
\(370\) 0 0
\(371\) −5.26496 8.66999i −0.273343 0.450123i
\(372\) 0 0
\(373\) 17.3454 30.0431i 0.898109 1.55557i 0.0682000 0.997672i \(-0.478274\pi\)
0.829909 0.557899i \(-0.188392\pi\)
\(374\) 0 0
\(375\) −3.37328 5.84270i −0.174196 0.301716i
\(376\) 0 0
\(377\) 7.56668 0.389704
\(378\) 0 0
\(379\) 13.4564 0.691207 0.345603 0.938381i \(-0.387674\pi\)
0.345603 + 0.938381i \(0.387674\pi\)
\(380\) 0 0
\(381\) 1.89853 + 3.28836i 0.0972649 + 0.168468i
\(382\) 0 0
\(383\) 2.08727 3.61527i 0.106655 0.184731i −0.807758 0.589514i \(-0.799319\pi\)
0.914413 + 0.404782i \(0.132653\pi\)
\(384\) 0 0
\(385\) −11.2145 + 0.248367i −0.571542 + 0.0126579i
\(386\) 0 0
\(387\) −1.72818 + 2.99330i −0.0878484 + 0.152158i
\(388\) 0 0
\(389\) −9.32698 16.1548i −0.472897 0.819081i 0.526622 0.850099i \(-0.323458\pi\)
−0.999519 + 0.0310185i \(0.990125\pi\)
\(390\) 0 0
\(391\) 56.1471 2.83948
\(392\) 0 0
\(393\) 1.54364 0.0778662
\(394\) 0 0
\(395\) 10.7076 + 18.5461i 0.538757 + 0.933155i
\(396\) 0 0
\(397\) −0.728181 + 1.26125i −0.0365464 + 0.0633002i −0.883720 0.468016i \(-0.844969\pi\)
0.847174 + 0.531316i \(0.178302\pi\)
\(398\) 0 0
\(399\) −15.9679 + 0.353641i −0.799394 + 0.0177042i
\(400\) 0 0
\(401\) 14.6172 25.3177i 0.729947 1.26431i −0.226958 0.973905i \(-0.572878\pi\)
0.956905 0.290401i \(-0.0937888\pi\)
\(402\) 0 0
\(403\) −15.9679 27.6572i −0.795417 1.37770i
\(404\) 0 0
\(405\) −2.74657 −0.136478
\(406\) 0 0
\(407\) −7.64023 −0.378712
\(408\) 0 0
\(409\) −13.0299 22.5685i −0.644288 1.11594i −0.984466 0.175578i \(-0.943821\pi\)
0.340178 0.940361i \(-0.389513\pi\)
\(410\) 0 0
\(411\) −0.746568 + 1.29309i −0.0368255 + 0.0637836i
\(412\) 0 0
\(413\) −7.61299 12.5366i −0.374611 0.616885i
\(414\) 0 0
\(415\) −8.96742 + 15.5320i −0.440193 + 0.762437i
\(416\) 0 0
\(417\) 1.47475 + 2.55434i 0.0722188 + 0.125087i
\(418\) 0 0
\(419\) 32.8853 1.60655 0.803275 0.595608i \(-0.203089\pi\)
0.803275 + 0.595608i \(0.203089\pi\)
\(420\) 0 0
\(421\) 17.1976 0.838160 0.419080 0.907949i \(-0.362353\pi\)
0.419080 + 0.907949i \(0.362353\pi\)
\(422\) 0 0
\(423\) 4.74657 + 8.22130i 0.230786 + 0.399733i
\(424\) 0 0
\(425\) −9.52991 + 16.5063i −0.462269 + 0.800673i
\(426\) 0 0
\(427\) −18.5436 + 33.8268i −0.897389 + 1.63699i
\(428\) 0 0
\(429\) 4.65929 8.07013i 0.224953 0.389630i
\(430\) 0 0
\(431\) 1.49314 + 2.58619i 0.0719219 + 0.124572i 0.899744 0.436419i \(-0.143753\pi\)
−0.827822 + 0.560991i \(0.810420\pi\)
\(432\) 0 0
\(433\) 18.4426 0.886297 0.443148 0.896448i \(-0.353862\pi\)
0.443148 + 0.896448i \(0.353862\pi\)
\(434\) 0 0
\(435\) 3.44264 0.165062
\(436\) 0 0
\(437\) 22.6172 + 39.1741i 1.08193 + 1.87395i
\(438\) 0 0
\(439\) 0.286010 0.495384i 0.0136505 0.0236434i −0.859119 0.511775i \(-0.828988\pi\)
0.872770 + 0.488132i \(0.162321\pi\)
\(440\) 0 0
\(441\) −3.22818 + 6.21119i −0.153723 + 0.295771i
\(442\) 0 0
\(443\) −18.2650 + 31.6358i −0.867794 + 1.50306i −0.00354850 + 0.999994i \(0.501130\pi\)
−0.864246 + 0.503070i \(0.832204\pi\)
\(444\) 0 0
\(445\) −13.0368 22.5804i −0.618002 1.07041i
\(446\) 0 0
\(447\) 1.49314 0.0706229
\(448\) 0 0
\(449\) −22.5530 −1.06434 −0.532170 0.846638i \(-0.678623\pi\)
−0.532170 + 0.846638i \(0.678623\pi\)
\(450\) 0 0
\(451\) 3.92645 + 6.80082i 0.184889 + 0.320238i
\(452\) 0 0
\(453\) 4.86642 8.42889i 0.228644 0.396024i
\(454\) 0 0
\(455\) −21.0873 + 38.4668i −0.988587 + 1.80335i
\(456\) 0 0
\(457\) −10.1172 + 17.5235i −0.473262 + 0.819714i −0.999532 0.0306040i \(-0.990257\pi\)
0.526270 + 0.850318i \(0.323590\pi\)
\(458\) 0 0
\(459\) −3.74657 6.48925i −0.174875 0.302892i
\(460\) 0 0
\(461\) −25.4931 −1.18733 −0.593667 0.804711i \(-0.702320\pi\)
−0.593667 + 0.804711i \(0.702320\pi\)
\(462\) 0 0
\(463\) 7.70446 0.358057 0.179028 0.983844i \(-0.442705\pi\)
0.179028 + 0.983844i \(0.442705\pi\)
\(464\) 0 0
\(465\) −7.26496 12.5833i −0.336904 0.583535i
\(466\) 0 0
\(467\) −18.0735 + 31.3043i −0.836344 + 1.44859i 0.0565874 + 0.998398i \(0.481978\pi\)
−0.892931 + 0.450193i \(0.851355\pi\)
\(468\) 0 0
\(469\) 5.54364 + 9.12890i 0.255981 + 0.421534i
\(470\) 0 0
\(471\) 4.54364 7.86981i 0.209360 0.362622i
\(472\) 0 0
\(473\) −2.66769 4.62057i −0.122660 0.212454i
\(474\) 0 0
\(475\) −15.3554 −0.704552
\(476\) 0 0
\(477\) 3.83384 0.175540
\(478\) 0 0
\(479\) 13.2902 + 23.0193i 0.607245 + 1.05178i 0.991692 + 0.128632i \(0.0410587\pi\)
−0.384447 + 0.923147i \(0.625608\pi\)
\(480\) 0 0
\(481\) −14.9395 + 25.8760i −0.681183 + 1.17984i
\(482\) 0 0
\(483\) 19.8201 0.438957i 0.901846 0.0199732i
\(484\) 0 0
\(485\) 2.11985 3.67169i 0.0962575 0.166723i
\(486\) 0 0
\(487\) −20.2118 35.0078i −0.915883 1.58636i −0.805603 0.592456i \(-0.798158\pi\)
−0.110281 0.993900i \(-0.535175\pi\)
\(488\) 0 0
\(489\) 19.5667 0.884836
\(490\) 0 0
\(491\) 14.6309 0.660284 0.330142 0.943931i \(-0.392903\pi\)
0.330142 + 0.943931i \(0.392903\pi\)
\(492\) 0 0
\(493\) 4.69607 + 8.13383i 0.211500 + 0.366329i
\(494\) 0 0
\(495\) 2.11985 3.67169i 0.0952803 0.165030i
\(496\) 0 0
\(497\) 14.5299 0.321794i 0.651756 0.0144344i
\(498\) 0 0
\(499\) 18.0552 31.2725i 0.808260 1.39995i −0.105808 0.994387i \(-0.533743\pi\)
0.914068 0.405561i \(-0.132924\pi\)
\(500\) 0 0
\(501\) −5.94950 10.3048i −0.265804 0.460386i
\(502\) 0 0
\(503\) −19.4931 −0.869156 −0.434578 0.900634i \(-0.643102\pi\)
−0.434578 + 0.900634i \(0.643102\pi\)
\(504\) 0 0
\(505\) −7.08727 −0.315380
\(506\) 0 0
\(507\) −11.7213 20.3019i −0.520562 0.901640i
\(508\) 0 0
\(509\) −11.9537 + 20.7044i −0.529838 + 0.917707i 0.469556 + 0.882903i \(0.344414\pi\)
−0.999394 + 0.0348040i \(0.988919\pi\)
\(510\) 0 0
\(511\) −17.2260 28.3666i −0.762032 1.25487i
\(512\) 0 0
\(513\) 3.01839 5.22800i 0.133265 0.230822i
\(514\) 0 0
\(515\) −8.98627 15.5647i −0.395982 0.685862i
\(516\) 0 0
\(517\) −14.6540 −0.644480
\(518\) 0 0
\(519\) −2.58041 −0.113267
\(520\) 0 0
\(521\) 3.54364 + 6.13776i 0.155250 + 0.268900i 0.933150 0.359488i \(-0.117048\pi\)
−0.777900 + 0.628388i \(0.783715\pi\)
\(522\) 0 0
\(523\) 14.3737 24.8961i 0.628520 1.08863i −0.359329 0.933211i \(-0.616994\pi\)
0.987849 0.155418i \(-0.0496724\pi\)
\(524\) 0 0
\(525\) −3.23504 + 5.90128i −0.141189 + 0.257553i
\(526\) 0 0
\(527\) 19.8201 34.3294i 0.863378 1.49541i
\(528\) 0 0
\(529\) −16.5735 28.7062i −0.720589 1.24810i
\(530\) 0 0
\(531\) 5.54364 0.240573
\(532\) 0 0
\(533\) 30.7107 1.33023
\(534\) 0 0
\(535\) −6.22085 10.7748i −0.268951 0.465837i
\(536\) 0 0
\(537\) 1.49314 2.58619i 0.0644336 0.111602i
\(538\) 0 0
\(539\) −5.81172 9.10944i −0.250329 0.392371i
\(540\) 0 0
\(541\) −3.51152 + 6.08214i −0.150972 + 0.261491i −0.931585 0.363523i \(-0.881574\pi\)
0.780613 + 0.625015i \(0.214907\pi\)
\(542\) 0 0
\(543\) −5.27182 9.13106i −0.226235 0.391851i
\(544\) 0 0
\(545\) −17.6951 −0.757976
\(546\) 0 0
\(547\) 5.59414 0.239188 0.119594 0.992823i \(-0.461841\pi\)
0.119594 + 0.992823i \(0.461841\pi\)
\(548\) 0 0
\(549\) −7.29021 12.6270i −0.311138 0.538907i
\(550\) 0 0
\(551\) −3.78334 + 6.55294i −0.161176 + 0.279165i
\(552\) 0 0
\(553\) −9.91646 + 18.0893i −0.421691 + 0.769237i
\(554\) 0 0
\(555\) −6.79707 + 11.7729i −0.288520 + 0.499730i
\(556\) 0 0
\(557\) 14.3228 + 24.8078i 0.606876 + 1.05114i 0.991752 + 0.128172i \(0.0409108\pi\)
−0.384876 + 0.922968i \(0.625756\pi\)
\(558\) 0 0
\(559\) −20.8653 −0.882507
\(560\) 0 0
\(561\) 11.5667 0.488346
\(562\) 0 0
\(563\) 9.94637 + 17.2276i 0.419189 + 0.726057i 0.995858 0.0909207i \(-0.0289810\pi\)
−0.576669 + 0.816978i \(0.695648\pi\)
\(564\) 0 0
\(565\) 10.9863 19.0288i 0.462196 0.800547i
\(566\) 0 0
\(567\) −1.37328 2.26144i −0.0576725 0.0949714i
\(568\) 0 0
\(569\) −17.8201 + 30.8653i −0.747058 + 1.29394i 0.202169 + 0.979351i \(0.435201\pi\)
−0.949227 + 0.314592i \(0.898132\pi\)
\(570\) 0 0
\(571\) 13.8385 + 23.9690i 0.579123 + 1.00307i 0.995580 + 0.0939155i \(0.0299384\pi\)
−0.416457 + 0.909155i \(0.636728\pi\)
\(572\) 0 0
\(573\) −17.1608 −0.716904
\(574\) 0 0
\(575\) 19.0598 0.794849
\(576\) 0 0
\(577\) 3.00686 + 5.20804i 0.125177 + 0.216814i 0.921802 0.387660i \(-0.126717\pi\)
−0.796625 + 0.604474i \(0.793383\pi\)
\(578\) 0 0
\(579\) −5.44950 + 9.43881i −0.226473 + 0.392264i
\(580\) 0 0
\(581\) −17.2723 + 0.382530i −0.716575 + 0.0158700i
\(582\) 0 0
\(583\) −2.95903 + 5.12519i −0.122551 + 0.212264i
\(584\) 0 0
\(585\) −8.29021 14.3591i −0.342758 0.593674i
\(586\) 0 0
\(587\) 15.5436 0.641555 0.320777 0.947155i \(-0.396056\pi\)
0.320777 + 0.947155i \(0.396056\pi\)
\(588\) 0 0
\(589\) 31.9358 1.31589
\(590\) 0 0
\(591\) 2.70979 + 4.69350i 0.111466 + 0.193065i
\(592\) 0 0
\(593\) 24.3270 42.1356i 0.998989 1.73030i 0.460568 0.887624i \(-0.347646\pi\)
0.538421 0.842676i \(-0.319021\pi\)
\(594\) 0 0
\(595\) −54.4373 + 1.20562i −2.23171 + 0.0494258i
\(596\) 0 0
\(597\) −11.4931 + 19.9067i −0.470383 + 0.814727i
\(598\) 0 0
\(599\) −10.4564 18.1110i −0.427235 0.739993i 0.569391 0.822067i \(-0.307179\pi\)
−0.996626 + 0.0820735i \(0.973846\pi\)
\(600\) 0 0
\(601\) −7.98627 −0.325767 −0.162883 0.986645i \(-0.552079\pi\)
−0.162883 + 0.986645i \(0.552079\pi\)
\(602\) 0 0
\(603\) −4.03677 −0.164390
\(604\) 0 0
\(605\) −11.8338 20.4968i −0.481114 0.833314i
\(606\) 0 0
\(607\) 1.35490 2.34675i 0.0549936 0.0952517i −0.837218 0.546869i \(-0.815819\pi\)
0.892212 + 0.451618i \(0.149153\pi\)
\(608\) 0 0
\(609\) 1.72132 + 2.83456i 0.0697513 + 0.114862i
\(610\) 0 0
\(611\) −28.6540 + 49.6301i −1.15922 + 2.00782i
\(612\) 0 0
\(613\) −8.25343 14.2954i −0.333353 0.577384i 0.649814 0.760093i \(-0.274847\pi\)
−0.983167 + 0.182709i \(0.941513\pi\)
\(614\) 0 0
\(615\) 13.9725 0.563427
\(616\) 0 0
\(617\) −23.1608 −0.932420 −0.466210 0.884674i \(-0.654381\pi\)
−0.466210 + 0.884674i \(0.654381\pi\)
\(618\) 0 0
\(619\) 14.7145 + 25.4862i 0.591424 + 1.02438i 0.994041 + 0.109008i \(0.0347673\pi\)
−0.402617 + 0.915369i \(0.631899\pi\)
\(620\) 0 0
\(621\) −3.74657 + 6.48925i −0.150345 + 0.260404i
\(622\) 0 0
\(623\) 12.0735 22.0242i 0.483716 0.882382i
\(624\) 0 0
\(625\) 15.6240 27.0616i 0.624962 1.08247i
\(626\) 0 0
\(627\) 4.65929 + 8.07013i 0.186074 + 0.322290i
\(628\) 0 0
\(629\) −37.0873 −1.47877
\(630\) 0 0
\(631\) 35.7054 1.42141 0.710705 0.703491i \(-0.248376\pi\)
0.710705 + 0.703491i \(0.248376\pi\)
\(632\) 0 0
\(633\) −1.54364 2.67366i −0.0613541 0.106268i
\(634\) 0 0
\(635\) −5.21445 + 9.03170i −0.206929 + 0.358412i
\(636\) 0 0
\(637\) −42.2160 + 1.87083i −1.67266 + 0.0741252i
\(638\) 0 0
\(639\) −2.74657 + 4.75720i −0.108653 + 0.188192i
\(640\) 0 0
\(641\) 1.68141 + 2.91229i 0.0664118 + 0.115029i 0.897319 0.441382i \(-0.145512\pi\)
−0.830908 + 0.556410i \(0.812178\pi\)
\(642\) 0 0
\(643\) 10.2113 0.402695 0.201348 0.979520i \(-0.435468\pi\)
0.201348 + 0.979520i \(0.435468\pi\)
\(644\) 0 0
\(645\) −9.49314 −0.373792
\(646\) 0 0
\(647\) −10.2765 17.7994i −0.404010 0.699766i 0.590196 0.807260i \(-0.299051\pi\)
−0.994206 + 0.107494i \(0.965717\pi\)
\(648\) 0 0
\(649\) −4.27868 + 7.41089i −0.167953 + 0.290903i
\(650\) 0 0
\(651\) 6.72818 12.2734i 0.263698 0.481031i
\(652\) 0 0
\(653\) −5.67722 + 9.83323i −0.222167 + 0.384804i −0.955466 0.295102i \(-0.904646\pi\)
0.733299 + 0.679906i \(0.237980\pi\)
\(654\) 0 0
\(655\) 2.11985 + 3.67169i 0.0828295 + 0.143465i
\(656\) 0 0
\(657\) 12.5436 0.489374
\(658\) 0 0
\(659\) 21.2344 0.827174 0.413587 0.910465i \(-0.364276\pi\)
0.413587 + 0.910465i \(0.364276\pi\)
\(660\) 0 0
\(661\) −14.4610 25.0472i −0.562469 0.974224i −0.997280 0.0737028i \(-0.976518\pi\)
0.434812 0.900521i \(-0.356815\pi\)
\(662\) 0 0
\(663\) 22.6172 39.1741i 0.878379 1.52140i
\(664\) 0 0
\(665\) −22.7696 37.4955i −0.882968 1.45401i
\(666\) 0 0
\(667\) 4.69607 8.13383i 0.181832 0.314943i
\(668\) 0 0
\(669\) −11.3733 19.6991i −0.439717 0.761611i
\(670\) 0 0
\(671\) 22.5069 0.868868
\(672\) 0 0
\(673\) −34.2618 −1.32070 −0.660348 0.750960i \(-0.729591\pi\)
−0.660348 + 0.750960i \(0.729591\pi\)
\(674\) 0 0
\(675\) −1.27182 2.20285i −0.0489523 0.0847879i
\(676\) 0 0
\(677\) 17.3733 30.0914i 0.667710 1.15651i −0.310833 0.950464i \(-0.600608\pi\)
0.978543 0.206042i \(-0.0660585\pi\)
\(678\) 0 0
\(679\) 4.08308 0.0904281i 0.156694 0.00347031i
\(680\) 0 0
\(681\) −11.3017 + 19.5752i −0.433083 + 0.750122i
\(682\) 0 0
\(683\) −0.140907 0.244058i −0.00539166 0.00933863i 0.863317 0.504662i \(-0.168383\pi\)
−0.868709 + 0.495323i \(0.835050\pi\)
\(684\) 0 0
\(685\) −4.10100 −0.156691
\(686\) 0 0
\(687\) 11.3554 0.433234
\(688\) 0 0
\(689\) 11.5720 + 20.0433i 0.440859 + 0.763590i
\(690\) 0 0
\(691\) −17.2076 + 29.8044i −0.654608 + 1.13381i 0.327384 + 0.944891i \(0.393833\pi\)
−0.981992 + 0.188922i \(0.939501\pi\)
\(692\) 0 0
\(693\) 4.08308 0.0904281i 0.155103 0.00343508i
\(694\) 0 0
\(695\) −4.05050 + 7.01567i −0.153644 + 0.266120i
\(696\) 0 0
\(697\) 19.0598 + 33.0126i 0.721942 + 1.25044i
\(698\) 0 0
\(699\) 3.41959 0.129341
\(700\) 0 0
\(701\) 0.498472 0.0188270 0.00941352 0.999956i \(-0.497004\pi\)
0.00941352 + 0.999956i \(0.497004\pi\)
\(702\) 0 0
\(703\) −14.9395 25.8760i −0.563454 0.975931i
\(704\) 0 0
\(705\) −13.0368 + 22.5804i −0.490993 + 0.850425i
\(706\) 0 0
\(707\) −3.54364 5.83543i −0.133272 0.219464i
\(708\) 0 0
\(709\) −2.08727 + 3.61527i −0.0783892 + 0.135774i −0.902555 0.430574i \(-0.858311\pi\)
0.824166 + 0.566348i \(0.191644\pi\)
\(710\) 0 0
\(711\) −3.89853 6.75246i −0.146206 0.253237i
\(712\) 0 0
\(713\) −39.6402 −1.48454
\(714\) 0 0
\(715\) 25.5941 0.957166
\(716\) 0 0
\(717\) −3.74657 6.48925i −0.139918 0.242345i
\(718\) 0 0
\(719\) −0.493136 + 0.854137i −0.0183909 + 0.0318540i −0.875074 0.483988i \(-0.839188\pi\)
0.856683 + 0.515842i \(0.172521\pi\)
\(720\) 0 0
\(721\) 8.32232 15.1813i 0.309939 0.565383i
\(722\) 0 0
\(723\) 9.30173 16.1111i 0.345935 0.599177i
\(724\) 0 0
\(725\) 1.59414 + 2.76113i 0.0592048 + 0.102546i
\(726\) 0 0
\(727\) 34.9304 1.29550 0.647749 0.761854i \(-0.275711\pi\)
0.647749 + 0.761854i \(0.275711\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.9495 22.4292i −0.478955 0.829574i
\(732\) 0 0
\(733\) 2.56202 4.43756i 0.0946305 0.163905i −0.814824 0.579709i \(-0.803166\pi\)
0.909454 + 0.415804i \(0.136500\pi\)
\(734\) 0 0
\(735\) −19.2071 + 0.851179i −0.708466 + 0.0313962i
\(736\) 0 0
\(737\) 3.11566 5.39648i 0.114767 0.198782i
\(738\) 0 0
\(739\) 8.46102 + 14.6549i 0.311244 + 0.539090i 0.978632 0.205620i \(-0.0659211\pi\)
−0.667388 + 0.744710i \(0.732588\pi\)
\(740\) 0 0
\(741\) 36.4426 1.33875
\(742\) 0 0
\(743\) −6.81172 −0.249898 −0.124949 0.992163i \(-0.539877\pi\)
−0.124949 + 0.992163i \(0.539877\pi\)
\(744\) 0 0
\(745\) 2.05050 + 3.55157i 0.0751245 + 0.130120i
\(746\) 0 0
\(747\) 3.26496 5.65507i 0.119458 0.206908i
\(748\) 0 0
\(749\) 5.76122 10.5095i 0.210511 0.384008i
\(750\) 0 0
\(751\) −13.0877 + 22.6686i −0.477578 + 0.827190i −0.999670 0.0256996i \(-0.991819\pi\)
0.522091 + 0.852890i \(0.325152\pi\)
\(752\) 0 0
\(753\) −12.2650 21.2435i −0.446960 0.774157i
\(754\) 0 0
\(755\) 26.7319 0.972874
\(756\) 0 0
\(757\) −49.7138 −1.80688 −0.903439 0.428717i \(-0.858966\pi\)
−0.903439 + 0.428717i \(0.858966\pi\)
\(758\) 0 0
\(759\) −5.78334 10.0170i −0.209922 0.363596i
\(760\) 0 0
\(761\) 6.32698 10.9586i 0.229353 0.397251i −0.728264 0.685297i \(-0.759672\pi\)
0.957616 + 0.288046i \(0.0930057\pi\)
\(762\) 0 0
\(763\) −8.84757 14.5696i −0.320304 0.527455i
\(764\) 0 0
\(765\) 10.2902 17.8232i 0.372043 0.644398i
\(766\) 0 0
\(767\) 16.7328 + 28.9821i 0.604188 + 1.04648i
\(768\) 0 0
\(769\) −20.2344 −0.729670 −0.364835 0.931072i \(-0.618875\pi\)
−0.364835 + 0.931072i \(0.618875\pi\)
\(770\) 0 0
\(771\) −17.4931 −0.630000
\(772\) 0 0
\(773\) −9.87062 17.0964i −0.355021 0.614915i 0.632100 0.774887i \(-0.282193\pi\)
−0.987122 + 0.159972i \(0.948860\pi\)
\(774\) 0 0
\(775\) 6.72818 11.6536i 0.241683 0.418608i
\(776\) 0 0
\(777\) −13.0919 + 0.289947i −0.469671 + 0.0104018i
\(778\) 0 0
\(779\) −15.3554 + 26.5963i −0.550163 + 0.952910i
\(780\) 0 0
\(781\) −4.23970 7.34338i −0.151709 0.262767i
\(782\) 0 0
\(783\) −1.25343 −0.0447940
\(784\) 0 0
\(785\) 24.9588 0.890818
\(786\) 0 0
\(787\) 8.45636 + 14.6469i 0.301437 + 0.522104i 0.976462 0.215691i \(-0.0692004\pi\)
−0.675025 + 0.737795i \(0.735867\pi\)
\(788\) 0 0
\(789\) −6.74657 + 11.6854i −0.240184 + 0.416011i
\(790\) 0 0
\(791\) 21.1608 0.468649i 0.752392 0.0166633i
\(792\) 0 0
\(793\) 44.0093 76.2264i 1.56282 2.70688i
\(794\) 0 0
\(795\) 5.26496 + 9.11917i 0.186729 + 0.323424i
\(796\) 0 0
\(797\) −3.48475 −0.123436 −0.0617180 0.998094i \(-0.519658\pi\)
−0.0617180 + 0.998094i \(0.519658\pi\)
\(798\) 0 0
\(799\) −71.1334 −2.51652
\(800\) 0 0
\(801\) 4.74657 + 8.22130i 0.167712 + 0.290485i
\(802\) 0 0
\(803\) −9.68141 + 16.7687i −0.341650 + 0.591754i
\(804\) 0 0
\(805\) 28.2628 + 46.5413i 0.996131 + 1.64036i
\(806\) 0 0
\(807\) 3.17035 5.49121i 0.111602 0.193300i
\(808\) 0 0
\(809\) 9.59414 + 16.6175i 0.337312 + 0.584241i 0.983926 0.178576i \(-0.0571490\pi\)
−0.646614 + 0.762817i \(0.723816\pi\)
\(810\) 0 0
\(811\) −48.8285 −1.71460 −0.857300 0.514817i \(-0.827860\pi\)
−0.857300 + 0.514817i \(0.827860\pi\)
\(812\) 0 0
\(813\) 20.2397 0.709837
\(814\) 0 0
\(815\) 26.8706 + 46.5413i 0.941237 + 1.63027i
\(816\) 0 0
\(817\) 10.4326 18.0699i 0.364992 0.632184i
\(818\) 0 0
\(819\) 7.67768 14.0054i 0.268280 0.489389i
\(820\) 0 0
\(821\) −13.2723 + 22.9883i −0.463206 + 0.802296i −0.999119 0.0419773i \(-0.986634\pi\)
0.535913 + 0.844273i \(0.319968\pi\)
\(822\) 0 0
\(823\) 8.47941 + 14.6868i 0.295574 + 0.511949i 0.975118 0.221686i \(-0.0711559\pi\)
−0.679545 + 0.733634i \(0.737823\pi\)
\(824\) 0 0
\(825\) 3.92645 0.136702
\(826\) 0 0
\(827\) 42.4289 1.47540 0.737699 0.675130i \(-0.235912\pi\)
0.737699 + 0.675130i \(0.235912\pi\)
\(828\) 0 0
\(829\) −15.9679 27.6572i −0.554588 0.960574i −0.997935 0.0642243i \(-0.979543\pi\)
0.443348 0.896350i \(-0.353791\pi\)
\(830\) 0 0
\(831\) −1.27182 + 2.20285i −0.0441189 + 0.0764162i
\(832\) 0 0
\(833\) −28.2113 44.2191i −0.977464 1.53210i
\(834\) 0 0
\(835\) 16.3407 28.3029i 0.565493 0.979463i
\(836\) 0 0
\(837\) 2.64510 + 4.58145i 0.0914281 + 0.158358i
\(838\) 0 0
\(839\) −49.5392 −1.71028 −0.855142 0.518394i \(-0.826530\pi\)
−0.855142 + 0.518394i \(0.826530\pi\)
\(840\) 0 0
\(841\) −27.4289 −0.945824
\(842\) 0 0
\(843\) −13.2902 23.0193i −0.457739 0.792827i
\(844\) 0 0
\(845\) 32.1934 55.7606i 1.10749 1.91822i
\(846\) 0 0
\(847\) 10.9595 19.9920i 0.376573 0.686934i
\(848\) 0 0
\(849\) 10.2718 17.7913i 0.352528 0.610596i
\(850\) 0 0
\(851\) 18.5436 + 32.1185i 0.635668 + 1.10101i
\(852\) 0 0
\(853\) −11.4564 −0.392258 −0.196129 0.980578i \(-0.562837\pi\)
−0.196129 + 0.980578i \(0.562837\pi\)
\(854\) 0 0
\(855\) 16.5804 0.567038
\(856\) 0 0
\(857\) 16.1240 + 27.9277i 0.550787 + 0.953991i 0.998218 + 0.0596726i \(0.0190057\pi\)
−0.447431 + 0.894318i \(0.647661\pi\)
\(858\) 0 0
\(859\) 1.54364 2.67366i 0.0526682 0.0912240i −0.838489 0.544918i \(-0.816561\pi\)
0.891158 + 0.453694i \(0.149894\pi\)
\(860\) 0 0
\(861\) 6.98627 + 11.5045i 0.238092 + 0.392074i
\(862\) 0 0
\(863\) −9.07355 + 15.7158i −0.308867 + 0.534974i −0.978115 0.208066i \(-0.933283\pi\)
0.669248 + 0.743039i \(0.266617\pi\)
\(864\) 0 0
\(865\) −3.54364 6.13776i −0.120487 0.208690i
\(866\) 0 0
\(867\) 39.1471 1.32951
\(868\) 0 0
\(869\) 12.0358 0.408288
\(870\) 0 0
\(871\) −12.1845 21.1042i −0.412858 0.715090i
\(872\) 0 0
\(873\) −0.771819 + 1.33683i −0.0261221 + 0.0452448i
\(874\) 0 0
\(875\) 17.8454 0.395222i 0.603283 0.0133609i
\(876\) 0 0
\(877\) 4.44264 7.69487i 0.150017 0.259837i −0.781216 0.624260i \(-0.785400\pi\)
0.931234 + 0.364423i \(0.118734\pi\)
\(878\) 0 0
\(879\) 2.62672 + 4.54961i 0.0885969 + 0.153454i
\(880\) 0 0
\(881\) −41.5667 −1.40042 −0.700209 0.713938i \(-0.746910\pi\)
−0.700209 + 0.713938i \(0.746910\pi\)
\(882\) 0 0
\(883\) −28.7182 −0.966444 −0.483222 0.875498i \(-0.660534\pi\)
−0.483222 + 0.875498i \(0.660534\pi\)
\(884\) 0 0
\(885\) 7.61299 + 13.1861i 0.255908 + 0.443245i
\(886\) 0 0
\(887\) −20.9074 + 36.2127i −0.702001 + 1.21590i 0.265761 + 0.964039i \(0.414377\pi\)
−0.967763 + 0.251863i \(0.918957\pi\)
\(888\) 0 0
\(889\) −10.0436 + 0.222437i −0.336853 + 0.00746029i
\(890\) 0 0
\(891\) −0.771819 + 1.33683i −0.0258569 + 0.0447855i
\(892\) 0 0
\(893\) −28.6540 49.6301i −0.958868 1.66081i
\(894\) 0 0
\(895\) 8.20200 0.274163
\(896\) 0 0
\(897\) −45.2344 −1.51033
\(898\) 0 0
\(899\) −3.31546 5.74254i −0.110577 0.191524i
\(900\) 0 0
\(901\) −14.3638 + 24.8787i −0.478526 + 0.828831i
\(902\) 0 0
\(903\) −4.74657 7.81634i −0.157956 0.260112i
\(904\) 0 0
\(905\) 14.4794 25.0791i 0.481312 0.833657i
\(906\) 0 0
\(907\) −28.5851 49.5108i −0.949152 1.64398i −0.747217 0.664580i \(-0.768611\pi\)
−0.201935 0.979399i \(-0.564723\pi\)
\(908\) 0 0
\(909\) 2.58041 0.0855868
\(910\) 0 0
\(911\) −24.4059 −0.808602 −0.404301 0.914626i \(-0.632485\pi\)
−0.404301 + 0.914626i \(0.632485\pi\)
\(912\) 0 0
\(913\) 5.03991 + 8.72937i 0.166797 + 0.288900i
\(914\) 0 0
\(915\) 20.0230 34.6809i 0.661942 1.14652i
\(916\) 0 0
\(917\) −1.96323 + 3.58126i −0.0648314 + 0.118264i
\(918\) 0 0
\(919\) −23.8522 + 41.3133i −0.786812 + 1.36280i 0.141098 + 0.989996i \(0.454937\pi\)
−0.927910 + 0.372803i \(0.878397\pi\)
\(920\) 0 0
\(921\) −7.42425 12.8592i −0.244637 0.423724i
\(922\) 0 0
\(923\) −33.1608 −1.09150
\(924\) 0 0
\(925\) −12.5897 −0.413948
\(926\) 0 0
\(927\) 3.27182 + 5.66696i 0.107461 + 0.186127i
\(928\) 0 0
\(929\) −19.6172 + 33.9780i −0.643619 + 1.11478i 0.341000 + 0.940063i \(0.389234\pi\)
−0.984619 + 0.174717i \(0.944099\pi\)
\(930\) 0 0
\(931\) 19.4878 37.4955i 0.638687 1.22887i
\(932\) 0 0
\(933\) −0.949499 + 1.64458i −0.0310852 + 0.0538412i
\(934\) 0 0
\(935\) 15.8843 + 27.5125i 0.519474 + 0.899755i
\(936\) 0 0
\(937\) 15.3491 0.501433 0.250717 0.968061i \(-0.419334\pi\)
0.250717 + 0.968061i \(0.419334\pi\)
\(938\) 0 0
\(939\) −0.0872743 −0.00284809
\(940\) 0 0
\(941\) −6.08308 10.5362i −0.198303 0.343470i 0.749676 0.661806i \(-0.230210\pi\)
−0.947978 + 0.318335i \(0.896876\pi\)
\(942\) 0 0
\(943\) 19.0598 33.0126i 0.620673 1.07504i
\(944\) 0 0
\(945\) 3.49314 6.37208i 0.113632 0.207284i
\(946\) 0 0
\(947\) −9.52991 + 16.5063i −0.309680 + 0.536382i −0.978292 0.207229i \(-0.933555\pi\)
0.668612 + 0.743611i \(0.266889\pi\)
\(948\) 0 0
\(949\) 37.8615 + 65.5781i 1.22904 + 2.12876i
\(950\) 0 0
\(951\) −8.23970 −0.267191
\(952\) 0 0
\(953\) 17.1334 0.555004 0.277502 0.960725i \(-0.410493\pi\)
0.277502 + 0.960725i \(0.410493\pi\)
\(954\) 0 0
\(955\) −23.5667 40.8187i −0.762600 1.32086i
\(956\) 0 0
\(957\) 0.967422 1.67562i 0.0312723 0.0541652i
\(958\) 0 0
\(959\) −2.05050 3.37663i −0.0662141 0.109037i
\(960\) 0 0
\(961\) 1.50686 2.60996i 0.0486085 0.0841924i
\(962\) 0 0
\(963\) 2.26496 + 3.92302i 0.0729872 + 0.126417i
\(964\) 0 0
\(965\) −29.9348 −0.963637
\(966\) 0 0
\(967\) 26.7833 0.861294 0.430647 0.902520i \(-0.358285\pi\)
0.430647 + 0.902520i \(0.358285\pi\)
\(968\) 0 0
\(969\) 22.6172 + 39.1741i 0.726569 + 1.25845i
\(970\) 0 0
\(971\) 5.75809 9.97331i 0.184786 0.320059i −0.758718 0.651419i \(-0.774174\pi\)
0.943504 + 0.331360i \(0.107507\pi\)
\(972\) 0 0
\(973\) −7.80173 + 0.172785i −0.250112 + 0.00553924i
\(974\) 0 0
\(975\) 7.67768 13.2981i 0.245883 0.425881i
\(976\) 0 0
\(977\) −10.6677 18.4770i −0.341289 0.591131i 0.643383 0.765544i \(-0.277530\pi\)
−0.984672 + 0.174414i \(0.944197\pi\)
\(978\) 0 0
\(979\) −14.6540 −0.468343
\(980\) 0 0
\(981\) 6.44264 0.205698
\(982\) 0 0
\(983\) 25.1976 + 43.6435i 0.803678 + 1.39201i 0.917180 + 0.398474i \(0.130460\pi\)
−0.113501 + 0.993538i \(0.536207\pi\)
\(984\) 0 0
\(985\) −7.44264 + 12.8910i −0.237142 + 0.410742i
\(986\) 0 0
\(987\) −25.1103 + 0.556119i −0.799270 + 0.0177015i
\(988\) 0 0
\(989\) −12.9495 + 22.4292i −0.411770 + 0.713207i
\(990\) 0 0
\(991\) −6.34117 10.9832i −0.201434 0.348894i 0.747557 0.664198i \(-0.231227\pi\)
−0.948991 + 0.315304i \(0.897893\pi\)
\(992\) 0 0
\(993\) −5.52991 −0.175486
\(994\) 0 0
\(995\) −63.1334 −2.00146
\(996\) 0 0
\(997\) −20.1708 34.9369i −0.638816 1.10646i −0.985693 0.168551i \(-0.946091\pi\)
0.346877 0.937911i \(-0.387242\pi\)
\(998\) 0 0
\(999\) 2.47475 4.28639i 0.0782976 0.135615i
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 1344.2.q.y.961.3 6
4.3 odd 2 1344.2.q.z.961.3 6
7.2 even 3 9408.2.a.ej.1.1 3
7.4 even 3 inner 1344.2.q.y.193.3 6
7.5 odd 6 9408.2.a.eg.1.3 3
8.3 odd 2 672.2.q.k.289.1 yes 6
8.5 even 2 672.2.q.l.289.1 yes 6
24.5 odd 2 2016.2.s.v.289.3 6
24.11 even 2 2016.2.s.u.289.3 6
28.11 odd 6 1344.2.q.z.193.3 6
28.19 even 6 9408.2.a.ei.1.3 3
28.23 odd 6 9408.2.a.eh.1.1 3
56.5 odd 6 4704.2.a.bv.1.1 3
56.11 odd 6 672.2.q.k.193.1 6
56.19 even 6 4704.2.a.bt.1.1 3
56.37 even 6 4704.2.a.bs.1.3 3
56.51 odd 6 4704.2.a.bu.1.3 3
56.53 even 6 672.2.q.l.193.1 yes 6
168.11 even 6 2016.2.s.u.865.3 6
168.53 odd 6 2016.2.s.v.865.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.1 6 56.11 odd 6
672.2.q.k.289.1 yes 6 8.3 odd 2
672.2.q.l.193.1 yes 6 56.53 even 6
672.2.q.l.289.1 yes 6 8.5 even 2
1344.2.q.y.193.3 6 7.4 even 3 inner
1344.2.q.y.961.3 6 1.1 even 1 trivial
1344.2.q.z.193.3 6 28.11 odd 6
1344.2.q.z.961.3 6 4.3 odd 2
2016.2.s.u.289.3 6 24.11 even 2
2016.2.s.u.865.3 6 168.11 even 6
2016.2.s.v.289.3 6 24.5 odd 2
2016.2.s.v.865.3 6 168.53 odd 6
4704.2.a.bs.1.3 3 56.37 even 6
4704.2.a.bt.1.1 3 56.19 even 6
4704.2.a.bu.1.3 3 56.51 odd 6
4704.2.a.bv.1.1 3 56.5 odd 6
9408.2.a.eg.1.3 3 7.5 odd 6
9408.2.a.eh.1.1 3 28.23 odd 6
9408.2.a.ei.1.3 3 28.19 even 6
9408.2.a.ej.1.1 3 7.2 even 3