Properties

Label 1344.2.q.z.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.z.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 0.958605 0.284738i \(-0.0919067\pi\)
−0.725893 + 0.687807i \(0.758573\pi\)
\(12\) 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 \(0.410142\pi\)
−0.971028 + 0.238967i \(0.923191\pi\)
\(20\) 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 \(-0.547934\pi\)
0.931235 0.364419i \(-0.118732\pi\)
\(24\) 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 0.524518 0.851399i \(-0.324246\pi\)
−0.999592 + 0.0285462i \(0.990912\pi\)
\(32\) 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 \(-0.584892\pi\)
−0.263545 + 0.964647i \(0.584892\pi\)
\(44\) 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 \(0.410095\pi\)
−0.971063 + 0.238823i \(0.923239\pi\)
\(48\) 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 0.988103 0.153796i \(-0.0491499\pi\)
−0.627243 + 0.778824i \(0.715817\pi\)
\(60\) 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 0.715991 0.698110i \(-0.245975\pi\)
−0.962576 + 0.271012i \(0.912642\pi\)
\(68\) 0 0
\(69\) 7.49314 0.902068
\(70\) 0 0
\(71\) −5.49314 −0.651915 −0.325958 0.945384i \(-0.605687\pi\)
−0.325958 + 0.945384i \(0.605687\pi\)
\(72\) 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 −0.558964 0.829192i \(-0.688801\pi\)
0.997583 + 0.0694809i \(0.0221343\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 6.52991 0.716751 0.358375 0.933578i \(-0.383331\pi\)
0.358375 + 0.933578i \(0.383331\pi\)
\(84\) 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 −0.980979 0.194114i \(-0.937817\pi\)
0.658597 + 0.752496i \(0.271150\pi\)
\(104\) 0 0
\(105\) −7.26496 + 0.160897i −0.708987 + 0.0157020i
\(106\) 0 0
\(107\) −2.26496 + 3.92302i −0.218961 + 0.379252i −0.954491 0.298241i \(-0.903600\pi\)
0.735529 + 0.677493i \(0.236934\pi\)
\(108\) 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 \(-0.446118\pi\)
0.168468 + 0.985707i \(0.446118\pi\)
\(128\) 0 0
\(129\) −1.72818 2.99330i −0.152158 0.263545i
\(130\) 0 0
\(131\) 0.771819 1.33683i 0.0674341 0.116799i −0.830337 0.557262i \(-0.811852\pi\)
0.897771 + 0.440462i \(0.145185\pi\)
\(132\) 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 \(-0.460079\pi\)
0.125087 + 0.992146i \(0.460079\pi\)
\(140\) 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 0.597208 0.802087i \(-0.296277\pi\)
−0.993231 + 0.116154i \(0.962944\pi\)
\(152\) 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 \(-0.555434\pi\)
0.939562 0.342380i \(-0.111233\pi\)
\(164\) 0 0
\(165\) 2.11985 + 3.67169i 0.165030 + 0.285841i
\(166\) 0 0
\(167\) −11.8990 −0.920772 −0.460386 0.887719i \(-0.652289\pi\)
−0.460386 + 0.887719i \(0.652289\pi\)
\(168\) 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 0.804814 0.593527i \(-0.202265\pi\)
−0.916416 + 0.400226i \(0.868932\pi\)
\(180\) 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 \(0.379882\pi\)
−0.989327 + 0.145716i \(0.953452\pi\)
\(192\) 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 \(0.136446\pi\)
−0.0947970 + 0.995497i \(0.530220\pi\)
\(200\) 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 \(-0.533890\pi\)
−0.106268 + 0.994337i \(0.533890\pi\)
\(212\) 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 \(-0.775591\pi\)
−0.761611 + 0.648034i \(0.775591\pi\)
\(224\) 0 0
\(225\) 2.54364 0.169576
\(226\) 0 0
\(227\) 11.3017 + 19.5752i 0.750122 + 1.29925i 0.947763 + 0.318975i \(0.103339\pi\)
−0.197641 + 0.980274i \(0.563328\pi\)
\(228\) 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 \(-0.577917\pi\)
−0.242345 + 0.970190i \(0.577917\pi\)
\(240\) 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 \(-0.781826\pi\)
−0.774157 + 0.632994i \(0.781826\pi\)
\(252\) 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 0.995534 0.0944035i \(-0.0300944\pi\)
−0.579523 + 0.814956i \(0.696761\pi\)
\(264\) 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 \(-0.622595\pi\)
0.990431 0.138012i \(-0.0440713\pi\)
\(272\) 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 \(-0.957597\pi\)
0.380544 0.924763i \(-0.375737\pi\)
\(284\) 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 \(-0.639277\pi\)
−0.423724 + 0.905791i \(0.639277\pi\)
\(308\) 0 0
\(309\) −6.54364 −0.372255
\(310\) 0 0
\(311\) 0.949499 + 1.64458i 0.0538412 + 0.0932556i 0.891690 0.452647i \(-0.149520\pi\)
−0.837849 + 0.545903i \(0.816187\pi\)
\(312\) 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 −0.931954 0.362577i \(-0.881897\pi\)
0.779978 + 0.625807i \(0.215230\pi\)
\(332\) 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 \(0.0569627\pi\)
−0.337863 + 0.941195i \(0.609704\pi\)
\(348\) 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 \(-0.509333\pi\)
0.880312 0.474396i \(-0.157333\pi\)
\(360\) 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 0.811084 0.584930i \(-0.198878\pi\)
−0.912106 + 0.409954i \(0.865545\pi\)
\(368\) 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 \(-0.612326\pi\)
−0.345603 + 0.938381i \(0.612326\pi\)
\(380\) 0 0
\(381\) 1.89853 + 3.28836i 0.0972649 + 0.168468i
\(382\) 0 0
\(383\) −2.08727 + 3.61527i −0.106655 + 0.184731i −0.914413 0.404782i \(-0.867347\pi\)
0.807758 + 0.589514i \(0.200681\pi\)
\(384\) 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 \(-0.796911\pi\)
−0.803275 + 0.595608i \(0.796911\pi\)
\(420\) 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 0.827822 0.560991i \(-0.189580\pi\)
−0.899744 + 0.436419i \(0.856247\pi\)
\(432\) 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 −0.872770 0.488132i \(-0.837679\pi\)
0.859119 + 0.511775i \(0.171012\pi\)
\(440\) 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 \(-0.498870\pi\)
0.864246 0.503070i \(-0.167796\pi\)
\(444\) 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 \(-0.557295\pi\)
−0.179028 + 0.983844i \(0.557295\pi\)
\(464\) 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 \(-0.518022\pi\)
0.892931 0.450193i \(-0.148645\pi\)
\(468\) 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 \(-0.958941\pi\)
0.384447 0.923147i \(-0.374392\pi\)
\(480\) 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 \(0.201842\pi\)
0.110281 + 0.993900i \(0.464825\pi\)
\(488\) 0 0
\(489\) 19.5667 0.884836
\(490\) 0 0
\(491\) −14.6309 −0.660284 −0.330142 0.943931i \(-0.607097\pi\)
−0.330142 + 0.943931i \(0.607097\pi\)
\(492\) 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 \(0.466257\pi\)
−0.914068 + 0.405561i \(0.867076\pi\)
\(500\) 0 0
\(501\) −5.94950 10.3048i −0.265804 0.460386i
\(502\) 0 0
\(503\) 19.4931 0.869156 0.434578 0.900634i \(-0.356898\pi\)
0.434578 + 0.900634i \(0.356898\pi\)
\(504\) 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 \(0.383006\pi\)
−0.987849 + 0.155418i \(0.950328\pi\)
\(524\) 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 \(-0.538159\pi\)
−0.119594 + 0.992823i \(0.538159\pi\)
\(548\) 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 0.576669 0.816978i \(-0.304352\pi\)
−0.995858 + 0.0909207i \(0.971019\pi\)
\(564\) 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 \(-0.970062\pi\)
0.416457 0.909155i \(-0.363272\pi\)
\(572\) 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 \(-0.603944\pi\)
−0.320777 + 0.947155i \(0.603944\pi\)
\(588\) 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 0.996626 0.0820735i \(-0.0261542\pi\)
−0.569391 + 0.822067i \(0.692821\pi\)
\(600\) 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 −0.892212 0.451618i \(-0.850847\pi\)
0.837218 + 0.546869i \(0.184181\pi\)
\(608\) 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 \(-0.965233\pi\)
0.402617 0.915369i \(-0.368101\pi\)
\(620\) 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 \(-0.751624\pi\)
−0.710705 + 0.703491i \(0.751624\pi\)
\(632\) 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 \(-0.564532\pi\)
−0.201348 + 0.979520i \(0.564532\pi\)
\(644\) 0 0
\(645\) −9.49314 −0.373792
\(646\) 0 0
\(647\) 10.2765 + 17.7994i 0.404010 + 0.699766i 0.994206 0.107494i \(-0.0342827\pi\)
−0.590196 + 0.807260i \(0.700949\pi\)
\(648\) 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 \(-0.635724\pi\)
−0.413587 + 0.910465i \(0.635724\pi\)
\(660\) 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 0.868709 0.495323i \(-0.164950\pi\)
−0.863317 + 0.504662i \(0.831617\pi\)
\(684\) 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 \(-0.606167\pi\)
0.981992 0.188922i \(-0.0604995\pi\)
\(692\) 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 −0.856683 0.515842i \(-0.827479\pi\)
0.875074 + 0.483988i \(0.160812\pi\)
\(720\) 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 \(-0.724289\pi\)
−0.647749 + 0.761854i \(0.724289\pi\)
\(728\) 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 0.667388 0.744710i \(-0.267412\pi\)
−0.978632 + 0.205620i \(0.934079\pi\)
\(740\) 0 0
\(741\) 36.4426 1.33875
\(742\) 0 0
\(743\) 6.81172 0.249898 0.124949 0.992163i \(-0.460123\pi\)
0.124949 + 0.992163i \(0.460123\pi\)
\(744\) 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 −0.522091 0.852890i \(-0.674848\pi\)
0.999670 + 0.0256996i \(0.00818133\pi\)
\(752\) 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 0.675025 0.737795i \(-0.264133\pi\)
−0.976462 + 0.215691i \(0.930800\pi\)
\(788\) 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 \(-0.172140\pi\)
0.857300 + 0.514817i \(0.172140\pi\)
\(812\) 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 0.679545 0.733634i \(-0.262177\pi\)
−0.975118 + 0.221686i \(0.928844\pi\)
\(824\) 0 0
\(825\) 3.92645 0.136702
\(826\) 0 0
\(827\) −42.4289 −1.47540 −0.737699 0.675130i \(-0.764088\pi\)
−0.737699 + 0.675130i \(0.764088\pi\)
\(828\) 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 \(-0.173470\pi\)
0.855142 + 0.518394i \(0.173470\pi\)
\(840\) 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 −0.891158 0.453694i \(-0.850106\pi\)
0.838489 + 0.544918i \(0.183439\pi\)
\(860\) 0 0
\(861\) 6.98627 + 11.5045i 0.238092 + 0.392074i
\(862\) 0 0
\(863\) 9.07355 15.7158i 0.308867 0.534974i −0.669248 0.743039i \(-0.733383\pi\)
0.978115 + 0.208066i \(0.0667168\pi\)
\(864\) 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 \(-0.339466\pi\)
0.483222 + 0.875498i \(0.339466\pi\)
\(884\) 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 \(-0.585623\pi\)
0.967763 0.251863i \(-0.0810433\pi\)
\(888\) 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 \(0.231389\pi\)
0.201935 + 0.979399i \(0.435277\pi\)
\(908\) 0 0
\(909\) 2.58041 0.0855868
\(910\) 0 0
\(911\) 24.4059 0.808602 0.404301 0.914626i \(-0.367515\pi\)
0.404301 + 0.914626i \(0.367515\pi\)
\(912\) 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 \(-0.545063\pi\)
0.927910 0.372803i \(-0.121603\pi\)
\(920\) 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 −0.668612 0.743611i \(-0.733111\pi\)
0.978292 + 0.207229i \(0.0664446\pi\)
\(948\) 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 \(-0.641715\pi\)
−0.430647 + 0.902520i \(0.641715\pi\)
\(968\) 0 0
\(969\) 22.6172 + 39.1741i 0.726569 + 1.25845i
\(970\) 0 0
\(971\) −5.75809 + 9.97331i −0.184786 + 0.320059i −0.943504 0.331360i \(-0.892493\pi\)
0.758718 + 0.651419i \(0.225826\pi\)
\(972\) 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 \(-0.869540\pi\)
0.113501 0.993538i \(-0.463793\pi\)
\(984\) 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 0.948991 0.315304i \(-0.102107\pi\)
−0.747557 + 0.664198i \(0.768773\pi\)
\(992\) 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.z.961.3 6
4.3 odd 2 1344.2.q.y.961.3 6
7.2 even 3 9408.2.a.eh.1.1 3
7.4 even 3 inner 1344.2.q.z.193.3 6
7.5 odd 6 9408.2.a.ei.1.3 3
8.3 odd 2 672.2.q.l.289.1 yes 6
8.5 even 2 672.2.q.k.289.1 yes 6
24.5 odd 2 2016.2.s.u.289.3 6
24.11 even 2 2016.2.s.v.289.3 6
28.11 odd 6 1344.2.q.y.193.3 6
28.19 even 6 9408.2.a.eg.1.3 3
28.23 odd 6 9408.2.a.ej.1.1 3
56.5 odd 6 4704.2.a.bt.1.1 3
56.11 odd 6 672.2.q.l.193.1 yes 6
56.19 even 6 4704.2.a.bv.1.1 3
56.37 even 6 4704.2.a.bu.1.3 3
56.51 odd 6 4704.2.a.bs.1.3 3
56.53 even 6 672.2.q.k.193.1 6
168.11 even 6 2016.2.s.v.865.3 6
168.53 odd 6 2016.2.s.u.865.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.q.k.193.1 6 56.53 even 6
672.2.q.k.289.1 yes 6 8.5 even 2
672.2.q.l.193.1 yes 6 56.11 odd 6
672.2.q.l.289.1 yes 6 8.3 odd 2
1344.2.q.y.193.3 6 28.11 odd 6
1344.2.q.y.961.3 6 4.3 odd 2
1344.2.q.z.193.3 6 7.4 even 3 inner
1344.2.q.z.961.3 6 1.1 even 1 trivial
2016.2.s.u.289.3 6 24.5 odd 2
2016.2.s.u.865.3 6 168.53 odd 6
2016.2.s.v.289.3 6 24.11 even 2
2016.2.s.v.865.3 6 168.11 even 6
4704.2.a.bs.1.3 3 56.51 odd 6
4704.2.a.bt.1.1 3 56.5 odd 6
4704.2.a.bu.1.3 3 56.37 even 6
4704.2.a.bv.1.1 3 56.19 even 6
9408.2.a.eg.1.3 3 28.19 even 6
9408.2.a.eh.1.1 3 7.2 even 3
9408.2.a.ei.1.3 3 7.5 odd 6
9408.2.a.ej.1.1 3 28.23 odd 6