Properties

Label 1456.2.cc.d.225.4
Level $1456$
Weight $2$
Character 1456.225
Analytic conductor $11.626$
Analytic rank $0$
Dimension $12$
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] = [1456,2,Mod(225,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.cc (of order \(6\), 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: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 39 x^{10} - 140 x^{9} + 460 x^{8} - 1066 x^{7} + 2127 x^{6} - 3172 x^{5} + 3842 x^{4} - 3394 x^{3} + 2141 x^{2} - 832 x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 225.4
Root \(0.500000 + 0.613147i\) of defining polynomial
Character \(\chi\) \(=\) 1456.225
Dual form 1456.2.cc.d.673.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.126439 - 0.218999i) q^{3} +1.14776i q^{5} +(0.866025 - 0.500000i) q^{7} +(1.46803 + 2.54270i) q^{9} +O(q^{10})\) \(q+(0.126439 - 0.218999i) q^{3} +1.14776i q^{5} +(0.866025 - 0.500000i) q^{7} +(1.46803 + 2.54270i) q^{9} +(-3.84935 - 2.22243i) q^{11} +(-3.54343 + 0.666437i) q^{13} +(0.251357 + 0.145121i) q^{15} +(1.35488 + 2.34672i) q^{17} +(-5.68371 + 3.28149i) q^{19} -0.252878i q^{21} +(1.04000 - 1.80133i) q^{23} +3.68266 q^{25} +1.50110 q^{27} +(-3.59960 + 6.23469i) q^{29} +7.90895i q^{31} +(-0.973417 + 0.562002i) q^{33} +(0.573878 + 0.993985i) q^{35} +(-8.35199 - 4.82202i) q^{37} +(-0.302078 + 0.860269i) q^{39} +(-8.22266 - 4.74735i) q^{41} +(1.70160 + 2.94725i) q^{43} +(-2.91839 + 1.68494i) q^{45} -1.67435i q^{47} +(0.500000 - 0.866025i) q^{49} +0.685238 q^{51} +13.2815 q^{53} +(2.55080 - 4.41812i) q^{55} +1.65963i q^{57} +(-0.0586805 + 0.0338792i) q^{59} +(-4.05023 - 7.01521i) q^{61} +(2.54270 + 1.46803i) q^{63} +(-0.764907 - 4.06699i) q^{65} +(0.444700 + 0.256747i) q^{67} +(-0.262993 - 0.455517i) q^{69} +(-9.34208 + 5.39365i) q^{71} +8.02452i q^{73} +(0.465632 - 0.806497i) q^{75} -4.44485 q^{77} -10.7404 q^{79} +(-4.21428 + 7.29935i) q^{81} +15.3479i q^{83} +(-2.69346 + 1.55507i) q^{85} +(0.910259 + 1.57661i) q^{87} +(9.40465 + 5.42978i) q^{89} +(-2.73548 + 2.34886i) q^{91} +(1.73205 + 1.00000i) q^{93} +(-3.76635 - 6.52351i) q^{95} +(1.84198 - 1.06347i) q^{97} -13.0503i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 6 q^{9} + 18 q^{11} - 8 q^{13} + 6 q^{15} + 4 q^{17} - 12 q^{19} + 6 q^{23} - 24 q^{25} - 40 q^{27} - 10 q^{29} + 12 q^{33} - 2 q^{35} - 6 q^{37} + 54 q^{39} - 24 q^{41} - 26 q^{43} + 72 q^{45} + 6 q^{49} + 36 q^{51} + 36 q^{53} + 6 q^{55} - 6 q^{59} - 28 q^{61} - 34 q^{65} + 42 q^{67} + 32 q^{69} - 48 q^{71} + 48 q^{75} - 4 q^{77} - 44 q^{79} - 34 q^{81} + 54 q^{85} - 2 q^{87} + 12 q^{89} + 16 q^{91} - 32 q^{95} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.126439 0.218999i 0.0729996 0.126439i −0.827215 0.561885i \(-0.810076\pi\)
0.900215 + 0.435446i \(0.143409\pi\)
\(4\) 0 0
\(5\) 1.14776i 0.513292i 0.966505 + 0.256646i \(0.0826174\pi\)
−0.966505 + 0.256646i \(0.917383\pi\)
\(6\) 0 0
\(7\) 0.866025 0.500000i 0.327327 0.188982i
\(8\) 0 0
\(9\) 1.46803 + 2.54270i 0.489342 + 0.847565i
\(10\) 0 0
\(11\) −3.84935 2.22243i −1.16062 0.670086i −0.209170 0.977879i \(-0.567076\pi\)
−0.951453 + 0.307793i \(0.900410\pi\)
\(12\) 0 0
\(13\) −3.54343 + 0.666437i −0.982769 + 0.184837i
\(14\) 0 0
\(15\) 0.251357 + 0.145121i 0.0649001 + 0.0374701i
\(16\) 0 0
\(17\) 1.35488 + 2.34672i 0.328606 + 0.569163i 0.982236 0.187652i \(-0.0600877\pi\)
−0.653629 + 0.756815i \(0.726754\pi\)
\(18\) 0 0
\(19\) −5.68371 + 3.28149i −1.30393 + 0.752826i −0.981076 0.193623i \(-0.937976\pi\)
−0.322856 + 0.946448i \(0.604643\pi\)
\(20\) 0 0
\(21\) 0.252878i 0.0551825i
\(22\) 0 0
\(23\) 1.04000 1.80133i 0.216855 0.375603i −0.736990 0.675904i \(-0.763754\pi\)
0.953845 + 0.300300i \(0.0970869\pi\)
\(24\) 0 0
\(25\) 3.68266 0.736531
\(26\) 0 0
\(27\) 1.50110 0.288886
\(28\) 0 0
\(29\) −3.59960 + 6.23469i −0.668428 + 1.15775i 0.309915 + 0.950764i \(0.399699\pi\)
−0.978344 + 0.206988i \(0.933634\pi\)
\(30\) 0 0
\(31\) 7.90895i 1.42049i 0.703955 + 0.710245i \(0.251416\pi\)
−0.703955 + 0.710245i \(0.748584\pi\)
\(32\) 0 0
\(33\) −0.973417 + 0.562002i −0.169450 + 0.0978321i
\(34\) 0 0
\(35\) 0.573878 + 0.993985i 0.0970030 + 0.168014i
\(36\) 0 0
\(37\) −8.35199 4.82202i −1.37306 0.792736i −0.381746 0.924267i \(-0.624677\pi\)
−0.991312 + 0.131532i \(0.958011\pi\)
\(38\) 0 0
\(39\) −0.302078 + 0.860269i −0.0483712 + 0.137753i
\(40\) 0 0
\(41\) −8.22266 4.74735i −1.28416 0.741412i −0.306556 0.951852i \(-0.599177\pi\)
−0.977607 + 0.210441i \(0.932510\pi\)
\(42\) 0 0
\(43\) 1.70160 + 2.94725i 0.259491 + 0.449452i 0.966106 0.258147i \(-0.0831118\pi\)
−0.706614 + 0.707599i \(0.749778\pi\)
\(44\) 0 0
\(45\) −2.91839 + 1.68494i −0.435048 + 0.251175i
\(46\) 0 0
\(47\) 1.67435i 0.244229i −0.992516 0.122114i \(-0.961033\pi\)
0.992516 0.122114i \(-0.0389674\pi\)
\(48\) 0 0
\(49\) 0.500000 0.866025i 0.0714286 0.123718i
\(50\) 0 0
\(51\) 0.685238 0.0959525
\(52\) 0 0
\(53\) 13.2815 1.82436 0.912180 0.409789i \(-0.134398\pi\)
0.912180 + 0.409789i \(0.134398\pi\)
\(54\) 0 0
\(55\) 2.55080 4.41812i 0.343950 0.595739i
\(56\) 0 0
\(57\) 1.65963i 0.219824i
\(58\) 0 0
\(59\) −0.0586805 + 0.0338792i −0.00763956 + 0.00441070i −0.503815 0.863812i \(-0.668071\pi\)
0.496175 + 0.868222i \(0.334737\pi\)
\(60\) 0 0
\(61\) −4.05023 7.01521i −0.518579 0.898205i −0.999767 0.0215878i \(-0.993128\pi\)
0.481188 0.876617i \(-0.340205\pi\)
\(62\) 0 0
\(63\) 2.54270 + 1.46803i 0.320350 + 0.184954i
\(64\) 0 0
\(65\) −0.764907 4.06699i −0.0948751 0.504447i
\(66\) 0 0
\(67\) 0.444700 + 0.256747i 0.0543287 + 0.0313667i 0.526918 0.849916i \(-0.323347\pi\)
−0.472590 + 0.881283i \(0.656681\pi\)
\(68\) 0 0
\(69\) −0.262993 0.455517i −0.0316606 0.0548378i
\(70\) 0 0
\(71\) −9.34208 + 5.39365i −1.10870 + 0.640109i −0.938493 0.345300i \(-0.887777\pi\)
−0.170208 + 0.985408i \(0.554444\pi\)
\(72\) 0 0
\(73\) 8.02452i 0.939199i 0.882880 + 0.469599i \(0.155602\pi\)
−0.882880 + 0.469599i \(0.844398\pi\)
\(74\) 0 0
\(75\) 0.465632 0.806497i 0.0537665 0.0931263i
\(76\) 0 0
\(77\) −4.44485 −0.506538
\(78\) 0 0
\(79\) −10.7404 −1.20839 −0.604193 0.796838i \(-0.706504\pi\)
−0.604193 + 0.796838i \(0.706504\pi\)
\(80\) 0 0
\(81\) −4.21428 + 7.29935i −0.468254 + 0.811039i
\(82\) 0 0
\(83\) 15.3479i 1.68465i 0.538967 + 0.842327i \(0.318815\pi\)
−0.538967 + 0.842327i \(0.681185\pi\)
\(84\) 0 0
\(85\) −2.69346 + 1.55507i −0.292147 + 0.168671i
\(86\) 0 0
\(87\) 0.910259 + 1.57661i 0.0975900 + 0.169031i
\(88\) 0 0
\(89\) 9.40465 + 5.42978i 0.996891 + 0.575555i 0.907327 0.420426i \(-0.138119\pi\)
0.0895643 + 0.995981i \(0.471453\pi\)
\(90\) 0 0
\(91\) −2.73548 + 2.34886i −0.286756 + 0.246228i
\(92\) 0 0
\(93\) 1.73205 + 1.00000i 0.179605 + 0.103695i
\(94\) 0 0
\(95\) −3.76635 6.52351i −0.386419 0.669298i
\(96\) 0 0
\(97\) 1.84198 1.06347i 0.187025 0.107979i −0.403564 0.914951i \(-0.632229\pi\)
0.590589 + 0.806973i \(0.298895\pi\)
\(98\) 0 0
\(99\) 13.0503i 1.31161i
\(100\) 0 0
\(101\) −3.00944 + 5.21251i −0.299451 + 0.518664i −0.976010 0.217724i \(-0.930137\pi\)
0.676560 + 0.736388i \(0.263470\pi\)
\(102\) 0 0
\(103\) −5.75670 −0.567224 −0.283612 0.958939i \(-0.591533\pi\)
−0.283612 + 0.958939i \(0.591533\pi\)
\(104\) 0 0
\(105\) 0.290242 0.0283247
\(106\) 0 0
\(107\) −2.77468 + 4.80589i −0.268239 + 0.464603i −0.968407 0.249375i \(-0.919775\pi\)
0.700168 + 0.713978i \(0.253108\pi\)
\(108\) 0 0
\(109\) 7.96986i 0.763374i −0.924292 0.381687i \(-0.875343\pi\)
0.924292 0.381687i \(-0.124657\pi\)
\(110\) 0 0
\(111\) −2.11203 + 1.21938i −0.200465 + 0.115739i
\(112\) 0 0
\(113\) 2.18535 + 3.78514i 0.205580 + 0.356076i 0.950318 0.311282i \(-0.100758\pi\)
−0.744737 + 0.667358i \(0.767425\pi\)
\(114\) 0 0
\(115\) 2.06749 + 1.19366i 0.192794 + 0.111310i
\(116\) 0 0
\(117\) −6.89639 8.03151i −0.637571 0.742513i
\(118\) 0 0
\(119\) 2.34672 + 1.35488i 0.215123 + 0.124202i
\(120\) 0 0
\(121\) 4.37835 + 7.58352i 0.398032 + 0.689411i
\(122\) 0 0
\(123\) −2.07933 + 1.20050i −0.187487 + 0.108246i
\(124\) 0 0
\(125\) 9.96557i 0.891347i
\(126\) 0 0
\(127\) −3.43247 + 5.94522i −0.304583 + 0.527553i −0.977168 0.212467i \(-0.931850\pi\)
0.672586 + 0.740019i \(0.265184\pi\)
\(128\) 0 0
\(129\) 0.860593 0.0757710
\(130\) 0 0
\(131\) 16.1996 1.41537 0.707683 0.706530i \(-0.249741\pi\)
0.707683 + 0.706530i \(0.249741\pi\)
\(132\) 0 0
\(133\) −3.28149 + 5.68371i −0.284541 + 0.492840i
\(134\) 0 0
\(135\) 1.72289i 0.148283i
\(136\) 0 0
\(137\) 4.50527 2.60112i 0.384911 0.222229i −0.295042 0.955484i \(-0.595334\pi\)
0.679953 + 0.733256i \(0.262000\pi\)
\(138\) 0 0
\(139\) 2.87013 + 4.97122i 0.243442 + 0.421653i 0.961692 0.274131i \(-0.0883902\pi\)
−0.718251 + 0.695784i \(0.755057\pi\)
\(140\) 0 0
\(141\) −0.366680 0.211703i −0.0308800 0.0178286i
\(142\) 0 0
\(143\) 15.1210 + 5.30964i 1.26448 + 0.444015i
\(144\) 0 0
\(145\) −7.15589 4.13146i −0.594265 0.343099i
\(146\) 0 0
\(147\) −0.126439 0.218999i −0.0104285 0.0180627i
\(148\) 0 0
\(149\) 2.62042 1.51290i 0.214673 0.123941i −0.388808 0.921319i \(-0.627113\pi\)
0.603481 + 0.797377i \(0.293780\pi\)
\(150\) 0 0
\(151\) 1.27030i 0.103376i −0.998663 0.0516879i \(-0.983540\pi\)
0.998663 0.0516879i \(-0.0164601\pi\)
\(152\) 0 0
\(153\) −3.97800 + 6.89009i −0.321602 + 0.557031i
\(154\) 0 0
\(155\) −9.07754 −0.729126
\(156\) 0 0
\(157\) −4.11859 −0.328699 −0.164350 0.986402i \(-0.552553\pi\)
−0.164350 + 0.986402i \(0.552553\pi\)
\(158\) 0 0
\(159\) 1.67931 2.90864i 0.133178 0.230670i
\(160\) 0 0
\(161\) 2.08000i 0.163927i
\(162\) 0 0
\(163\) 9.71606 5.60957i 0.761021 0.439376i −0.0686413 0.997641i \(-0.521866\pi\)
0.829662 + 0.558266i \(0.188533\pi\)
\(164\) 0 0
\(165\) −0.645041 1.11724i −0.0502164 0.0869774i
\(166\) 0 0
\(167\) 3.46184 + 1.99869i 0.267885 + 0.154664i 0.627926 0.778273i \(-0.283904\pi\)
−0.360041 + 0.932936i \(0.617237\pi\)
\(168\) 0 0
\(169\) 12.1117 4.72294i 0.931671 0.363303i
\(170\) 0 0
\(171\) −16.6877 9.63463i −1.27614 0.736778i
\(172\) 0 0
\(173\) −6.08396 10.5377i −0.462555 0.801168i 0.536533 0.843879i \(-0.319734\pi\)
−0.999087 + 0.0427113i \(0.986400\pi\)
\(174\) 0 0
\(175\) 3.18927 1.84133i 0.241087 0.139191i
\(176\) 0 0
\(177\) 0.0171346i 0.00128792i
\(178\) 0 0
\(179\) 5.93554 10.2806i 0.443643 0.768412i −0.554314 0.832308i \(-0.687019\pi\)
0.997957 + 0.0638960i \(0.0203526\pi\)
\(180\) 0 0
\(181\) −4.79134 −0.356137 −0.178069 0.984018i \(-0.556985\pi\)
−0.178069 + 0.984018i \(0.556985\pi\)
\(182\) 0 0
\(183\) −2.04843 −0.151424
\(184\) 0 0
\(185\) 5.53450 9.58604i 0.406905 0.704780i
\(186\) 0 0
\(187\) 12.0445i 0.880779i
\(188\) 0 0
\(189\) 1.29999 0.750549i 0.0945602 0.0545944i
\(190\) 0 0
\(191\) 7.79263 + 13.4972i 0.563855 + 0.976625i 0.997155 + 0.0753756i \(0.0240156\pi\)
−0.433300 + 0.901250i \(0.642651\pi\)
\(192\) 0 0
\(193\) −15.2295 8.79275i −1.09624 0.632916i −0.161011 0.986953i \(-0.551475\pi\)
−0.935231 + 0.354037i \(0.884809\pi\)
\(194\) 0 0
\(195\) −0.987379 0.346712i −0.0707077 0.0248285i
\(196\) 0 0
\(197\) −0.458833 0.264907i −0.0326905 0.0188739i 0.483566 0.875308i \(-0.339341\pi\)
−0.516256 + 0.856434i \(0.672675\pi\)
\(198\) 0 0
\(199\) −12.5732 21.7775i −0.891294 1.54377i −0.838326 0.545170i \(-0.816465\pi\)
−0.0529680 0.998596i \(-0.516868\pi\)
\(200\) 0 0
\(201\) 0.112455 0.0649258i 0.00793195 0.00457951i
\(202\) 0 0
\(203\) 7.19919i 0.505284i
\(204\) 0 0
\(205\) 5.44880 9.43760i 0.380561 0.659150i
\(206\) 0 0
\(207\) 6.10698 0.424465
\(208\) 0 0
\(209\) 29.1715 2.01783
\(210\) 0 0
\(211\) 2.72085 4.71265i 0.187311 0.324432i −0.757042 0.653366i \(-0.773356\pi\)
0.944353 + 0.328934i \(0.106689\pi\)
\(212\) 0 0
\(213\) 2.72787i 0.186911i
\(214\) 0 0
\(215\) −3.38273 + 1.95302i −0.230700 + 0.133195i
\(216\) 0 0
\(217\) 3.95448 + 6.84935i 0.268447 + 0.464964i
\(218\) 0 0
\(219\) 1.75736 + 1.01461i 0.118751 + 0.0685611i
\(220\) 0 0
\(221\) −6.36485 7.41248i −0.428146 0.498617i
\(222\) 0 0
\(223\) −8.00684 4.62275i −0.536177 0.309562i 0.207351 0.978267i \(-0.433516\pi\)
−0.743528 + 0.668704i \(0.766849\pi\)
\(224\) 0 0
\(225\) 5.40624 + 9.36388i 0.360416 + 0.624259i
\(226\) 0 0
\(227\) 1.30318 0.752389i 0.0864948 0.0499378i −0.456129 0.889914i \(-0.650764\pi\)
0.542624 + 0.839976i \(0.317431\pi\)
\(228\) 0 0
\(229\) 25.4380i 1.68099i 0.541818 + 0.840496i \(0.317736\pi\)
−0.541818 + 0.840496i \(0.682264\pi\)
\(230\) 0 0
\(231\) −0.562002 + 0.973417i −0.0369770 + 0.0640461i
\(232\) 0 0
\(233\) 12.1004 0.792724 0.396362 0.918094i \(-0.370273\pi\)
0.396362 + 0.918094i \(0.370273\pi\)
\(234\) 0 0
\(235\) 1.92174 0.125361
\(236\) 0 0
\(237\) −1.35800 + 2.35213i −0.0882116 + 0.152787i
\(238\) 0 0
\(239\) 5.57964i 0.360917i −0.983583 0.180458i \(-0.942242\pi\)
0.983583 0.180458i \(-0.0577581\pi\)
\(240\) 0 0
\(241\) 20.1291 11.6215i 1.29663 0.748608i 0.316807 0.948490i \(-0.397389\pi\)
0.979820 + 0.199882i \(0.0640559\pi\)
\(242\) 0 0
\(243\) 3.31734 + 5.74581i 0.212808 + 0.368594i
\(244\) 0 0
\(245\) 0.993985 + 0.573878i 0.0635034 + 0.0366637i
\(246\) 0 0
\(247\) 17.9529 15.4156i 1.14231 0.980868i
\(248\) 0 0
\(249\) 3.36118 + 1.94058i 0.213006 + 0.122979i
\(250\) 0 0
\(251\) 11.8707 + 20.5607i 0.749273 + 1.29778i 0.948171 + 0.317760i \(0.102930\pi\)
−0.198898 + 0.980020i \(0.563736\pi\)
\(252\) 0 0
\(253\) −8.00664 + 4.62264i −0.503373 + 0.290623i
\(254\) 0 0
\(255\) 0.786486i 0.0492517i
\(256\) 0 0
\(257\) 3.50520 6.07118i 0.218648 0.378710i −0.735747 0.677257i \(-0.763169\pi\)
0.954395 + 0.298547i \(0.0965020\pi\)
\(258\) 0 0
\(259\) −9.64405 −0.599252
\(260\) 0 0
\(261\) −21.1372 −1.30836
\(262\) 0 0
\(263\) 15.4226 26.7127i 0.950998 1.64718i 0.207728 0.978187i \(-0.433393\pi\)
0.743270 0.668991i \(-0.233274\pi\)
\(264\) 0 0
\(265\) 15.2440i 0.936429i
\(266\) 0 0
\(267\) 2.37823 1.37307i 0.145545 0.0840306i
\(268\) 0 0
\(269\) 2.81595 + 4.87737i 0.171692 + 0.297379i 0.939011 0.343886i \(-0.111743\pi\)
−0.767320 + 0.641265i \(0.778410\pi\)
\(270\) 0 0
\(271\) 2.60224 + 1.50240i 0.158075 + 0.0912645i 0.576951 0.816779i \(-0.304242\pi\)
−0.418876 + 0.908043i \(0.637576\pi\)
\(272\) 0 0
\(273\) 0.168527 + 0.896054i 0.0101997 + 0.0542317i
\(274\) 0 0
\(275\) −14.1759 8.18443i −0.854836 0.493540i
\(276\) 0 0
\(277\) −12.0866 20.9346i −0.726214 1.25784i −0.958473 0.285185i \(-0.907945\pi\)
0.232259 0.972654i \(-0.425388\pi\)
\(278\) 0 0
\(279\) −20.1101 + 11.6106i −1.20396 + 0.695105i
\(280\) 0 0
\(281\) 1.74575i 0.104143i −0.998643 0.0520713i \(-0.983418\pi\)
0.998643 0.0520713i \(-0.0165823\pi\)
\(282\) 0 0
\(283\) −13.0572 + 22.6156i −0.776167 + 1.34436i 0.157969 + 0.987444i \(0.449505\pi\)
−0.934136 + 0.356917i \(0.883828\pi\)
\(284\) 0 0
\(285\) −1.90485 −0.112834
\(286\) 0 0
\(287\) −9.49471 −0.560455
\(288\) 0 0
\(289\) 4.82861 8.36339i 0.284036 0.491964i
\(290\) 0 0
\(291\) 0.537855i 0.0315296i
\(292\) 0 0
\(293\) −4.89767 + 2.82767i −0.286125 + 0.165194i −0.636193 0.771530i \(-0.719492\pi\)
0.350068 + 0.936724i \(0.386159\pi\)
\(294\) 0 0
\(295\) −0.0388851 0.0673509i −0.00226398 0.00392132i
\(296\) 0 0
\(297\) −5.77825 3.33608i −0.335288 0.193579i
\(298\) 0 0
\(299\) −2.48468 + 7.07597i −0.143693 + 0.409214i
\(300\) 0 0
\(301\) 2.94725 + 1.70160i 0.169877 + 0.0980785i
\(302\) 0 0
\(303\) 0.761021 + 1.31813i 0.0437195 + 0.0757245i
\(304\) 0 0
\(305\) 8.05174 4.64868i 0.461041 0.266182i
\(306\) 0 0
\(307\) 20.4767i 1.16867i −0.811514 0.584333i \(-0.801356\pi\)
0.811514 0.584333i \(-0.198644\pi\)
\(308\) 0 0
\(309\) −0.727871 + 1.26071i −0.0414071 + 0.0717193i
\(310\) 0 0
\(311\) −1.23712 −0.0701506 −0.0350753 0.999385i \(-0.511167\pi\)
−0.0350753 + 0.999385i \(0.511167\pi\)
\(312\) 0 0
\(313\) −9.01151 −0.509361 −0.254680 0.967025i \(-0.581970\pi\)
−0.254680 + 0.967025i \(0.581970\pi\)
\(314\) 0 0
\(315\) −1.68494 + 2.91839i −0.0949353 + 0.164433i
\(316\) 0 0
\(317\) 0.580644i 0.0326122i 0.999867 + 0.0163061i \(0.00519062\pi\)
−0.999867 + 0.0163061i \(0.994809\pi\)
\(318\) 0 0
\(319\) 27.7122 15.9997i 1.55159 0.895810i
\(320\) 0 0
\(321\) 0.701656 + 1.21530i 0.0391626 + 0.0678317i
\(322\) 0 0
\(323\) −15.4015 8.89204i −0.856961 0.494767i
\(324\) 0 0
\(325\) −13.0492 + 2.45426i −0.723841 + 0.136138i
\(326\) 0 0
\(327\) −1.74539 1.00770i −0.0965202 0.0557260i
\(328\) 0 0
\(329\) −0.837173 1.45003i −0.0461549 0.0799426i
\(330\) 0 0
\(331\) 27.1632 15.6827i 1.49303 0.861999i 0.493058 0.869996i \(-0.335879\pi\)
0.999968 + 0.00799735i \(0.00254566\pi\)
\(332\) 0 0
\(333\) 28.3154i 1.55168i
\(334\) 0 0
\(335\) −0.294683 + 0.510406i −0.0161003 + 0.0278865i
\(336\) 0 0
\(337\) 9.43033 0.513703 0.256851 0.966451i \(-0.417315\pi\)
0.256851 + 0.966451i \(0.417315\pi\)
\(338\) 0 0
\(339\) 1.10525 0.0600292
\(340\) 0 0
\(341\) 17.5771 30.4444i 0.951851 1.64865i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0.522822 0.301851i 0.0281478 0.0162511i
\(346\) 0 0
\(347\) 3.23650 + 5.60578i 0.173744 + 0.300934i 0.939726 0.341928i \(-0.111080\pi\)
−0.765982 + 0.642862i \(0.777747\pi\)
\(348\) 0 0
\(349\) 8.57811 + 4.95258i 0.459176 + 0.265105i 0.711698 0.702486i \(-0.247927\pi\)
−0.252522 + 0.967591i \(0.581260\pi\)
\(350\) 0 0
\(351\) −5.31903 + 1.00039i −0.283909 + 0.0533967i
\(352\) 0 0
\(353\) 24.9206 + 14.3879i 1.32639 + 0.765790i 0.984739 0.174037i \(-0.0556814\pi\)
0.341649 + 0.939828i \(0.389015\pi\)
\(354\) 0 0
\(355\) −6.19059 10.7224i −0.328562 0.569087i
\(356\) 0 0
\(357\) 0.593434 0.342619i 0.0314078 0.0181333i
\(358\) 0 0
\(359\) 12.1382i 0.640631i 0.947311 + 0.320315i \(0.103789\pi\)
−0.947311 + 0.320315i \(0.896211\pi\)
\(360\) 0 0
\(361\) 12.0364 20.8476i 0.633493 1.09724i
\(362\) 0 0
\(363\) 2.21438 0.116225
\(364\) 0 0
\(365\) −9.21019 −0.482083
\(366\) 0 0
\(367\) −13.5388 + 23.4498i −0.706717 + 1.22407i 0.259351 + 0.965783i \(0.416491\pi\)
−0.966068 + 0.258287i \(0.916842\pi\)
\(368\) 0 0
\(369\) 27.8770i 1.45122i
\(370\) 0 0
\(371\) 11.5022 6.64077i 0.597162 0.344772i
\(372\) 0 0
\(373\) 7.81661 + 13.5388i 0.404729 + 0.701010i 0.994290 0.106713i \(-0.0340327\pi\)
−0.589561 + 0.807724i \(0.700699\pi\)
\(374\) 0 0
\(375\) 2.18245 + 1.26004i 0.112701 + 0.0650680i
\(376\) 0 0
\(377\) 8.59987 24.4910i 0.442916 1.26135i
\(378\) 0 0
\(379\) 32.3295 + 18.6654i 1.66065 + 0.958778i 0.972404 + 0.233304i \(0.0749536\pi\)
0.688249 + 0.725475i \(0.258380\pi\)
\(380\) 0 0
\(381\) 0.867997 + 1.50341i 0.0444688 + 0.0770223i
\(382\) 0 0
\(383\) 32.8193 18.9482i 1.67699 0.968209i 0.713421 0.700736i \(-0.247145\pi\)
0.963565 0.267473i \(-0.0861886\pi\)
\(384\) 0 0
\(385\) 5.10160i 0.260002i
\(386\) 0 0
\(387\) −4.99598 + 8.65329i −0.253960 + 0.439872i
\(388\) 0 0
\(389\) −10.1053 −0.512361 −0.256180 0.966629i \(-0.582464\pi\)
−0.256180 + 0.966629i \(0.582464\pi\)
\(390\) 0 0
\(391\) 5.63629 0.285039
\(392\) 0 0
\(393\) 2.04826 3.54769i 0.103321 0.178957i
\(394\) 0 0
\(395\) 12.3273i 0.620254i
\(396\) 0 0
\(397\) 4.77041 2.75419i 0.239420 0.138229i −0.375490 0.926826i \(-0.622526\pi\)
0.614910 + 0.788597i \(0.289192\pi\)
\(398\) 0 0
\(399\) 0.829817 + 1.43728i 0.0415428 + 0.0719542i
\(400\) 0 0
\(401\) −23.1657 13.3747i −1.15684 0.667903i −0.206296 0.978490i \(-0.566141\pi\)
−0.950545 + 0.310587i \(0.899474\pi\)
\(402\) 0 0
\(403\) −5.27082 28.0248i −0.262558 1.39601i
\(404\) 0 0
\(405\) −8.37787 4.83696i −0.416300 0.240351i
\(406\) 0 0
\(407\) 21.4332 + 37.1233i 1.06240 + 1.84014i
\(408\) 0 0
\(409\) −27.7124 + 15.9998i −1.37029 + 0.791137i −0.990964 0.134127i \(-0.957177\pi\)
−0.379325 + 0.925264i \(0.623844\pi\)
\(410\) 0 0
\(411\) 1.31553i 0.0648904i
\(412\) 0 0
\(413\) −0.0338792 + 0.0586805i −0.00166709 + 0.00288748i
\(414\) 0 0
\(415\) −17.6157 −0.864719
\(416\) 0 0
\(417\) 1.45159 0.0710846
\(418\) 0 0
\(419\) 5.84782 10.1287i 0.285685 0.494821i −0.687090 0.726572i \(-0.741112\pi\)
0.972775 + 0.231751i \(0.0744456\pi\)
\(420\) 0 0
\(421\) 25.8565i 1.26017i −0.776527 0.630084i \(-0.783021\pi\)
0.776527 0.630084i \(-0.216979\pi\)
\(422\) 0 0
\(423\) 4.25736 2.45799i 0.207000 0.119511i
\(424\) 0 0
\(425\) 4.98956 + 8.64216i 0.242029 + 0.419206i
\(426\) 0 0
\(427\) −7.01521 4.05023i −0.339490 0.196004i
\(428\) 0 0
\(429\) 3.07469 2.64014i 0.148447 0.127467i
\(430\) 0 0
\(431\) −9.27186 5.35311i −0.446610 0.257850i 0.259788 0.965666i \(-0.416347\pi\)
−0.706397 + 0.707816i \(0.749681\pi\)
\(432\) 0 0
\(433\) 11.2150 + 19.4249i 0.538956 + 0.933500i 0.998961 + 0.0455830i \(0.0145146\pi\)
−0.460004 + 0.887917i \(0.652152\pi\)
\(434\) 0 0
\(435\) −1.80957 + 1.04475i −0.0867621 + 0.0500922i
\(436\) 0 0
\(437\) 13.6510i 0.653015i
\(438\) 0 0
\(439\) 6.35913 11.0143i 0.303505 0.525686i −0.673423 0.739258i \(-0.735177\pi\)
0.976927 + 0.213572i \(0.0685099\pi\)
\(440\) 0 0
\(441\) 2.93605 0.139812
\(442\) 0 0
\(443\) 7.86448 0.373653 0.186826 0.982393i \(-0.440180\pi\)
0.186826 + 0.982393i \(0.440180\pi\)
\(444\) 0 0
\(445\) −6.23206 + 10.7942i −0.295428 + 0.511696i
\(446\) 0 0
\(447\) 0.765157i 0.0361907i
\(448\) 0 0
\(449\) −14.5815 + 8.41864i −0.688144 + 0.397300i −0.802916 0.596092i \(-0.796720\pi\)
0.114772 + 0.993392i \(0.463386\pi\)
\(450\) 0 0
\(451\) 21.1013 + 36.5485i 0.993620 + 1.72100i
\(452\) 0 0
\(453\) −0.278195 0.160616i −0.0130707 0.00754639i
\(454\) 0 0
\(455\) −2.69592 3.13966i −0.126387 0.147189i
\(456\) 0 0
\(457\) 7.96539 + 4.59882i 0.372605 + 0.215124i 0.674596 0.738187i \(-0.264318\pi\)
−0.301991 + 0.953311i \(0.597651\pi\)
\(458\) 0 0
\(459\) 2.03380 + 3.52265i 0.0949299 + 0.164423i
\(460\) 0 0
\(461\) −18.8812 + 10.9010i −0.879383 + 0.507712i −0.870455 0.492248i \(-0.836175\pi\)
−0.00892828 + 0.999960i \(0.502842\pi\)
\(462\) 0 0
\(463\) 26.0636i 1.21128i 0.795740 + 0.605638i \(0.207082\pi\)
−0.795740 + 0.605638i \(0.792918\pi\)
\(464\) 0 0
\(465\) −1.14776 + 1.98797i −0.0532259 + 0.0921899i
\(466\) 0 0
\(467\) −28.6668 −1.32654 −0.663271 0.748379i \(-0.730832\pi\)
−0.663271 + 0.748379i \(0.730832\pi\)
\(468\) 0 0
\(469\) 0.513495 0.0237110
\(470\) 0 0
\(471\) −0.520751 + 0.901966i −0.0239949 + 0.0415604i
\(472\) 0 0
\(473\) 15.1267i 0.695526i
\(474\) 0 0
\(475\) −20.9312 + 12.0846i −0.960387 + 0.554480i
\(476\) 0 0
\(477\) 19.4977 + 33.7709i 0.892736 + 1.54626i
\(478\) 0 0
\(479\) 15.5679 + 8.98812i 0.711315 + 0.410678i 0.811548 0.584286i \(-0.198625\pi\)
−0.100233 + 0.994964i \(0.531959\pi\)
\(480\) 0 0
\(481\) 32.8082 + 11.5204i 1.49593 + 0.525285i
\(482\) 0 0
\(483\) −0.455517 0.262993i −0.0207267 0.0119666i
\(484\) 0 0
\(485\) 1.22060 + 2.11414i 0.0554247 + 0.0959983i
\(486\) 0 0
\(487\) −18.9847 + 10.9608i −0.860278 + 0.496681i −0.864105 0.503311i \(-0.832115\pi\)
0.00382768 + 0.999993i \(0.498782\pi\)
\(488\) 0 0
\(489\) 2.83707i 0.128297i
\(490\) 0 0
\(491\) 10.8003 18.7067i 0.487411 0.844221i −0.512484 0.858697i \(-0.671275\pi\)
0.999895 + 0.0144759i \(0.00460798\pi\)
\(492\) 0 0
\(493\) −19.5081 −0.878599
\(494\) 0 0
\(495\) 14.9786 0.673237
\(496\) 0 0
\(497\) −5.39365 + 9.34208i −0.241938 + 0.419049i
\(498\) 0 0
\(499\) 38.6105i 1.72844i −0.503112 0.864221i \(-0.667812\pi\)
0.503112 0.864221i \(-0.332188\pi\)
\(500\) 0 0
\(501\) 0.875423 0.505426i 0.0391110 0.0225807i
\(502\) 0 0
\(503\) −4.73503 8.20132i −0.211125 0.365679i 0.740942 0.671569i \(-0.234379\pi\)
−0.952067 + 0.305890i \(0.901046\pi\)
\(504\) 0 0
\(505\) −5.98268 3.45410i −0.266226 0.153706i
\(506\) 0 0
\(507\) 0.497075 3.24962i 0.0220759 0.144321i
\(508\) 0 0
\(509\) 11.3043 + 6.52651i 0.501052 + 0.289283i 0.729148 0.684356i \(-0.239917\pi\)
−0.228096 + 0.973639i \(0.573250\pi\)
\(510\) 0 0
\(511\) 4.01226 + 6.94944i 0.177492 + 0.307425i
\(512\) 0 0
\(513\) −8.53180 + 4.92584i −0.376688 + 0.217481i
\(514\) 0 0
\(515\) 6.60728i 0.291152i
\(516\) 0 0
\(517\) −3.72111 + 6.44515i −0.163654 + 0.283457i
\(518\) 0 0
\(519\) −3.07700 −0.135065
\(520\) 0 0
\(521\) 10.3444 0.453196 0.226598 0.973988i \(-0.427240\pi\)
0.226598 + 0.973988i \(0.427240\pi\)
\(522\) 0 0
\(523\) −1.06684 + 1.84782i −0.0466497 + 0.0807996i −0.888407 0.459056i \(-0.848188\pi\)
0.841758 + 0.539855i \(0.181521\pi\)
\(524\) 0 0
\(525\) 0.931263i 0.0406437i
\(526\) 0 0
\(527\) −18.5601 + 10.7157i −0.808490 + 0.466782i
\(528\) 0 0
\(529\) 9.33681 + 16.1718i 0.405948 + 0.703123i
\(530\) 0 0
\(531\) −0.172289 0.0994712i −0.00747671 0.00431668i
\(532\) 0 0
\(533\) 32.3002 + 11.3420i 1.39908 + 0.491277i
\(534\) 0 0
\(535\) −5.51599 3.18466i −0.238477 0.137685i
\(536\) 0 0
\(537\) −1.50097 2.59975i −0.0647715 0.112187i
\(538\) 0 0
\(539\) −3.84935 + 2.22243i −0.165803 + 0.0957266i
\(540\) 0 0
\(541\) 8.41225i 0.361671i −0.983513 0.180836i \(-0.942120\pi\)
0.983513 0.180836i \(-0.0578802\pi\)
\(542\) 0 0
\(543\) −0.605812 + 1.04930i −0.0259979 + 0.0450297i
\(544\) 0 0
\(545\) 9.14745 0.391834
\(546\) 0 0
\(547\) 1.00730 0.0430692 0.0215346 0.999768i \(-0.493145\pi\)
0.0215346 + 0.999768i \(0.493145\pi\)
\(548\) 0 0
\(549\) 11.8917 20.5970i 0.507525 0.879059i
\(550\) 0 0
\(551\) 47.2482i 2.01284i
\(552\) 0 0
\(553\) −9.30143 + 5.37018i −0.395537 + 0.228363i
\(554\) 0 0
\(555\) −1.39955 2.42410i −0.0594078 0.102897i
\(556\) 0 0
\(557\) 37.9237 + 21.8953i 1.60688 + 0.927732i 0.990063 + 0.140624i \(0.0449110\pi\)
0.616816 + 0.787108i \(0.288422\pi\)
\(558\) 0 0
\(559\) −7.99365 9.30937i −0.338095 0.393744i
\(560\) 0 0
\(561\) −2.63772 1.52289i −0.111365 0.0642965i
\(562\) 0 0
\(563\) −2.11334 3.66041i −0.0890665 0.154268i 0.818050 0.575147i \(-0.195055\pi\)
−0.907117 + 0.420879i \(0.861722\pi\)
\(564\) 0 0
\(565\) −4.34441 + 2.50825i −0.182771 + 0.105523i
\(566\) 0 0
\(567\) 8.42856i 0.353966i
\(568\) 0 0
\(569\) 4.71320 8.16349i 0.197587 0.342231i −0.750158 0.661258i \(-0.770023\pi\)
0.947746 + 0.319027i \(0.103356\pi\)
\(570\) 0 0
\(571\) −25.5304 −1.06842 −0.534208 0.845353i \(-0.679390\pi\)
−0.534208 + 0.845353i \(0.679390\pi\)
\(572\) 0 0
\(573\) 3.94117 0.164645
\(574\) 0 0
\(575\) 3.82996 6.63368i 0.159720 0.276644i
\(576\) 0 0
\(577\) 31.7623i 1.32228i 0.750262 + 0.661141i \(0.229927\pi\)
−0.750262 + 0.661141i \(0.770073\pi\)
\(578\) 0 0
\(579\) −3.85120 + 2.22349i −0.160050 + 0.0924052i
\(580\) 0 0
\(581\) 7.67396 + 13.2917i 0.318370 + 0.551432i
\(582\) 0 0
\(583\) −51.1254 29.5172i −2.11740 1.22248i
\(584\) 0 0
\(585\) 9.21820 7.91537i 0.381126 0.327260i
\(586\) 0 0
\(587\) 29.7429 + 17.1721i 1.22762 + 0.708768i 0.966532 0.256546i \(-0.0825846\pi\)
0.261090 + 0.965314i \(0.415918\pi\)
\(588\) 0 0
\(589\) −25.9532 44.9522i −1.06938 1.85222i
\(590\) 0 0
\(591\) −0.116029 + 0.0669892i −0.00477278 + 0.00275557i
\(592\) 0 0
\(593\) 15.1751i 0.623168i 0.950219 + 0.311584i \(0.100860\pi\)
−0.950219 + 0.311584i \(0.899140\pi\)
\(594\) 0 0
\(595\) −1.55507 + 2.69346i −0.0637516 + 0.110421i
\(596\) 0 0
\(597\) −6.35899 −0.260256
\(598\) 0 0
\(599\) −7.48641 −0.305886 −0.152943 0.988235i \(-0.548875\pi\)
−0.152943 + 0.988235i \(0.548875\pi\)
\(600\) 0 0
\(601\) −18.2071 + 31.5356i −0.742683 + 1.28637i 0.208586 + 0.978004i \(0.433114\pi\)
−0.951269 + 0.308361i \(0.900219\pi\)
\(602\) 0 0
\(603\) 1.50765i 0.0613962i
\(604\) 0 0
\(605\) −8.70403 + 5.02527i −0.353869 + 0.204306i
\(606\) 0 0
\(607\) 2.26168 + 3.91735i 0.0917989 + 0.159000i 0.908268 0.418389i \(-0.137405\pi\)
−0.816469 + 0.577389i \(0.804072\pi\)
\(608\) 0 0
\(609\) 1.57661 + 0.910259i 0.0638877 + 0.0368856i
\(610\) 0 0
\(611\) 1.11585 + 5.93292i 0.0451424 + 0.240020i
\(612\) 0 0
\(613\) −34.8434 20.1168i −1.40731 0.812512i −0.412183 0.911101i \(-0.635234\pi\)
−0.995128 + 0.0985892i \(0.968567\pi\)
\(614\) 0 0
\(615\) −1.37788 2.38656i −0.0555615 0.0962354i
\(616\) 0 0
\(617\) −34.2107 + 19.7516i −1.37727 + 0.795168i −0.991830 0.127565i \(-0.959284\pi\)
−0.385440 + 0.922733i \(0.625950\pi\)
\(618\) 0 0
\(619\) 22.9229i 0.921348i 0.887569 + 0.460674i \(0.152392\pi\)
−0.887569 + 0.460674i \(0.847608\pi\)
\(620\) 0 0
\(621\) 1.56114 2.70397i 0.0626463 0.108507i
\(622\) 0 0
\(623\) 10.8596 0.435079
\(624\) 0 0
\(625\) 6.97525 0.279010
\(626\) 0 0
\(627\) 3.68841 6.38852i 0.147301 0.255133i
\(628\) 0 0
\(629\) 26.1330i 1.04199i
\(630\) 0 0
\(631\) −3.11897 + 1.80074i −0.124164 + 0.0716862i −0.560796 0.827954i \(-0.689505\pi\)
0.436632 + 0.899640i \(0.356171\pi\)
\(632\) 0 0
\(633\) −0.688043 1.19173i −0.0273472 0.0473668i
\(634\) 0 0
\(635\) −6.82365 3.93964i −0.270788 0.156340i
\(636\) 0 0
\(637\) −1.19456 + 3.40191i −0.0473302 + 0.134789i
\(638\) 0 0
\(639\) −27.4288 15.8360i −1.08507 0.626464i
\(640\) 0 0
\(641\) 2.60928 + 4.51940i 0.103060 + 0.178506i 0.912944 0.408085i \(-0.133803\pi\)
−0.809884 + 0.586590i \(0.800470\pi\)
\(642\) 0 0
\(643\) −18.1006 + 10.4504i −0.713820 + 0.412124i −0.812474 0.582998i \(-0.801880\pi\)
0.0986540 + 0.995122i \(0.468546\pi\)
\(644\) 0 0
\(645\) 0.987751i 0.0388927i
\(646\) 0 0
\(647\) 3.81055 6.60007i 0.149808 0.259476i −0.781348 0.624095i \(-0.785468\pi\)
0.931156 + 0.364620i \(0.118801\pi\)
\(648\) 0 0
\(649\) 0.301176 0.0118222
\(650\) 0 0
\(651\) 2.00000 0.0783862
\(652\) 0 0
\(653\) 12.8910 22.3278i 0.504462 0.873755i −0.495524 0.868594i \(-0.665024\pi\)
0.999987 0.00516051i \(-0.00164265\pi\)
\(654\) 0 0
\(655\) 18.5932i 0.726495i
\(656\) 0 0
\(657\) −20.4039 + 11.7802i −0.796033 + 0.459590i
\(658\) 0 0
\(659\) 7.82964 + 13.5613i 0.305000 + 0.528275i 0.977261 0.212039i \(-0.0680103\pi\)
−0.672262 + 0.740314i \(0.734677\pi\)
\(660\) 0 0
\(661\) −27.4434 15.8444i −1.06742 0.616277i −0.139947 0.990159i \(-0.544693\pi\)
−0.927476 + 0.373881i \(0.878027\pi\)
\(662\) 0 0
\(663\) −2.42809 + 0.456668i −0.0942992 + 0.0177355i
\(664\) 0 0
\(665\) −6.52351 3.76635i −0.252971 0.146053i
\(666\) 0 0
\(667\) 7.48715 + 12.9681i 0.289904 + 0.502128i
\(668\) 0 0
\(669\) −2.02475 + 1.16899i −0.0782815 + 0.0451958i
\(670\) 0 0
\(671\) 36.0054i 1.38997i
\(672\) 0 0
\(673\) 8.38642 14.5257i 0.323273 0.559925i −0.657889 0.753115i \(-0.728550\pi\)
0.981161 + 0.193190i \(0.0618835\pi\)
\(674\) 0 0
\(675\) 5.52803 0.212774
\(676\) 0 0
\(677\) −8.34791 −0.320836 −0.160418 0.987049i \(-0.551284\pi\)
−0.160418 + 0.987049i \(0.551284\pi\)
\(678\) 0 0
\(679\) 1.06347 1.84198i 0.0408122 0.0706887i
\(680\) 0 0
\(681\) 0.380525i 0.0145818i
\(682\) 0 0
\(683\) 23.5376 13.5895i 0.900642 0.519986i 0.0232337 0.999730i \(-0.492604\pi\)
0.877408 + 0.479744i \(0.159270\pi\)
\(684\) 0 0
\(685\) 2.98545 + 5.17095i 0.114068 + 0.197572i
\(686\) 0 0
\(687\) 5.57089 + 3.21636i 0.212543 + 0.122712i
\(688\) 0 0
\(689\) −47.0622 + 8.85132i −1.79293 + 0.337208i
\(690\) 0 0
\(691\) −22.7403 13.1291i −0.865080 0.499454i 0.000629844 1.00000i \(-0.499800\pi\)
−0.865710 + 0.500545i \(0.833133\pi\)
\(692\) 0 0
\(693\) −6.52516 11.3019i −0.247870 0.429324i
\(694\) 0 0
\(695\) −5.70574 + 3.29421i −0.216431 + 0.124957i
\(696\) 0 0
\(697\) 25.7284i 0.974531i
\(698\) 0 0
\(699\) 1.52996 2.64997i 0.0578685 0.100231i
\(700\) 0 0
\(701\) −26.4443 −0.998786 −0.499393 0.866376i \(-0.666444\pi\)
−0.499393 + 0.866376i \(0.666444\pi\)
\(702\) 0 0
\(703\) 63.2937 2.38717
\(704\) 0 0
\(705\) 0.242983 0.420859i 0.00915127 0.0158505i
\(706\) 0 0
\(707\) 6.01888i 0.226363i
\(708\) 0 0
\(709\) 10.0688 5.81322i 0.378141 0.218320i −0.298868 0.954294i \(-0.596609\pi\)
0.677009 + 0.735975i \(0.263276\pi\)
\(710\) 0 0
\(711\) −15.7671 27.3095i −0.591314 1.02419i
\(712\) 0 0
\(713\) 14.2466 + 8.22530i 0.533541 + 0.308040i
\(714\) 0 0
\(715\) −6.09417 + 17.3552i −0.227909 + 0.649048i
\(716\) 0 0
\(717\) −1.22193 0.705484i −0.0456339 0.0263468i
\(718\) 0 0
\(719\) 22.4379 + 38.8636i 0.836793 + 1.44937i 0.892562 + 0.450925i \(0.148906\pi\)
−0.0557687 + 0.998444i \(0.517761\pi\)
\(720\) 0 0
\(721\) −4.98545 + 2.87835i −0.185668 + 0.107195i
\(722\) 0 0
\(723\) 5.87765i 0.218592i
\(724\) 0 0
\(725\) −13.2561 + 22.9602i −0.492319 + 0.852721i
\(726\) 0 0
\(727\) −19.5156 −0.723793 −0.361896 0.932218i \(-0.617871\pi\)
−0.361896 + 0.932218i \(0.617871\pi\)
\(728\) 0 0
\(729\) −23.6079 −0.874368
\(730\) 0 0
\(731\) −4.61092 + 7.98635i −0.170541 + 0.295386i
\(732\) 0 0
\(733\) 10.7037i 0.395349i 0.980268 + 0.197675i \(0.0633390\pi\)
−0.980268 + 0.197675i \(0.936661\pi\)
\(734\) 0 0
\(735\) 0.251357 0.145121i 0.00927144 0.00535287i
\(736\) 0 0
\(737\) −1.14120 1.97662i −0.0420368 0.0728099i
\(738\) 0 0
\(739\) −0.413619 0.238803i −0.0152152 0.00878452i 0.492373 0.870384i \(-0.336130\pi\)
−0.507588 + 0.861600i \(0.669463\pi\)
\(740\) 0 0
\(741\) −1.10604 5.88079i −0.0406315 0.216036i
\(742\) 0 0
\(743\) −8.43019 4.86717i −0.309274 0.178559i 0.337328 0.941387i \(-0.390477\pi\)
−0.646601 + 0.762828i \(0.723810\pi\)
\(744\) 0 0
\(745\) 1.73644 + 3.00760i 0.0636181 + 0.110190i
\(746\) 0 0
\(747\) −39.0251 + 22.5312i −1.42785 + 0.824372i
\(748\) 0 0
\(749\) 5.54937i 0.202769i
\(750\) 0 0
\(751\) −18.1084 + 31.3646i −0.660784 + 1.14451i 0.319626 + 0.947544i \(0.396443\pi\)
−0.980410 + 0.196967i \(0.936891\pi\)
\(752\) 0 0
\(753\) 6.00369 0.218787
\(754\) 0 0
\(755\) 1.45800 0.0530619
\(756\) 0 0
\(757\) −23.2347 + 40.2436i −0.844479 + 1.46268i 0.0415945 + 0.999135i \(0.486756\pi\)
−0.886073 + 0.463545i \(0.846577\pi\)
\(758\) 0 0
\(759\) 2.33793i 0.0848614i
\(760\) 0 0
\(761\) 18.5037 10.6831i 0.670760 0.387263i −0.125605 0.992080i \(-0.540087\pi\)
0.796365 + 0.604817i \(0.206754\pi\)
\(762\) 0 0
\(763\) −3.98493 6.90210i −0.144264 0.249873i
\(764\) 0 0
\(765\) −7.90814 4.56577i −0.285919 0.165076i
\(766\) 0 0
\(767\) 0.185352 0.159155i 0.00669266 0.00574677i
\(768\) 0 0
\(769\) −39.8297 22.9957i −1.43630 0.829246i −0.438706 0.898631i \(-0.644563\pi\)
−0.997590 + 0.0693848i \(0.977896\pi\)
\(770\) 0 0
\(771\) −0.886387 1.53527i −0.0319225 0.0552913i
\(772\) 0 0
\(773\) 17.2754 9.97393i 0.621351 0.358737i −0.156044 0.987750i \(-0.549874\pi\)
0.777395 + 0.629013i \(0.216541\pi\)
\(774\) 0 0
\(775\) 29.1260i 1.04624i
\(776\) 0 0
\(777\) −1.21938 + 2.11203i −0.0437451 + 0.0757688i
\(778\) 0 0
\(779\) 62.3136 2.23262
\(780\) 0 0
\(781\) 47.9479 1.71571
\(782\) 0 0
\(783\) −5.40334 + 9.35887i −0.193100 + 0.334459i
\(784\) 0 0
\(785\) 4.72714i 0.168719i
\(786\) 0 0
\(787\) −38.0726 + 21.9812i −1.35714 + 0.783546i −0.989238 0.146318i \(-0.953258\pi\)
−0.367904 + 0.929864i \(0.619925\pi\)
\(788\) 0 0
\(789\) −3.90004 6.75506i −0.138845 0.240487i
\(790\) 0 0
\(791\) 3.78514 + 2.18535i 0.134584 + 0.0777021i
\(792\) 0 0
\(793\) 19.0269 + 22.1586i 0.675665 + 0.786876i
\(794\) 0 0
\(795\) 3.33841 + 1.92743i 0.118401 + 0.0683590i
\(796\) 0 0
\(797\) 15.9862 + 27.6889i 0.566260 + 0.980792i 0.996931 + 0.0782827i \(0.0249437\pi\)
−0.430671 + 0.902509i \(0.641723\pi\)
\(798\) 0 0
\(799\) 3.92922 2.26854i 0.139006 0.0802551i
\(800\) 0 0
\(801\) 31.8842i 1.12657i
\(802\) 0 0
\(803\) 17.8339 30.8892i 0.629345 1.09006i
\(804\) 0 0
\(805\) 2.38733 0.0841423
\(806\) 0 0
\(807\) 1.42419 0.0501337
\(808\) 0 0
\(809\) 14.0866 24.3986i 0.495257 0.857811i −0.504728 0.863279i \(-0.668407\pi\)
0.999985 + 0.00546789i \(0.00174049\pi\)
\(810\) 0 0
\(811\) 39.0534i 1.37135i 0.727908 + 0.685675i \(0.240493\pi\)
−0.727908 + 0.685675i \(0.759507\pi\)
\(812\) 0 0
\(813\) 0.658049 0.379925i 0.0230788 0.0133245i
\(814\) 0 0
\(815\) 6.43842 + 11.1517i 0.225528 + 0.390626i
\(816\) 0 0
\(817\) −19.3428 11.1676i −0.676718 0.390703i
\(818\) 0 0
\(819\) −9.98820 3.50729i −0.349016 0.122555i
\(820\) 0 0
\(821\) 20.4243 + 11.7920i 0.712812 + 0.411542i 0.812101 0.583516i \(-0.198324\pi\)
−0.0992893 + 0.995059i \(0.531657\pi\)
\(822\) 0 0
\(823\) −12.3566 21.4023i −0.430725 0.746038i 0.566211 0.824260i \(-0.308409\pi\)
−0.996936 + 0.0782228i \(0.975075\pi\)
\(824\) 0 0
\(825\) −3.58476 + 2.06966i −0.124805 + 0.0720564i
\(826\) 0 0
\(827\) 45.2456i 1.57334i 0.617371 + 0.786672i \(0.288198\pi\)
−0.617371 + 0.786672i \(0.711802\pi\)
\(828\) 0 0
\(829\) 22.4064 38.8091i 0.778208 1.34790i −0.154765 0.987951i \(-0.549462\pi\)
0.932974 0.359945i \(-0.117204\pi\)
\(830\) 0 0
\(831\) −6.11287 −0.212053
\(832\) 0 0
\(833\) 2.70976 0.0938875
\(834\) 0 0
\(835\) −2.29401 + 3.97334i −0.0793875 + 0.137503i
\(836\) 0 0
\(837\) 11.8721i 0.410360i
\(838\) 0 0
\(839\) 26.7511 15.4447i 0.923550 0.533212i 0.0387839 0.999248i \(-0.487652\pi\)
0.884766 + 0.466036i \(0.154318\pi\)
\(840\) 0 0
\(841\) −11.4142 19.7700i −0.393593 0.681723i
\(842\) 0 0
\(843\) −0.382316 0.220730i −0.0131677 0.00760236i
\(844\) 0 0
\(845\) 5.42078 + 13.9013i 0.186481 + 0.478219i
\(846\) 0 0
\(847\) 7.58352 + 4.37835i 0.260573 + 0.150442i
\(848\) 0 0
\(849\) 3.30187 + 5.71900i 0.113320 + 0.196276i
\(850\) 0 0
\(851\) −17.3721 + 10.0298i −0.595508 + 0.343817i
\(852\) 0 0
\(853\) 27.9201i 0.955965i 0.878369 + 0.477982i \(0.158632\pi\)
−0.878369 + 0.477982i \(0.841368\pi\)
\(854\) 0 0
\(855\) 11.0582 19.1534i 0.378182 0.655031i
\(856\) 0 0
\(857\) −17.5921 −0.600935 −0.300468 0.953792i \(-0.597143\pi\)
−0.300468 + 0.953792i \(0.597143\pi\)
\(858\) 0 0
\(859\) −22.1337 −0.755193 −0.377596 0.925970i \(-0.623249\pi\)
−0.377596 + 0.925970i \(0.623249\pi\)
\(860\) 0 0
\(861\) −1.20050 + 2.07933i −0.0409130 + 0.0708633i
\(862\) 0 0
\(863\) 4.58867i 0.156200i −0.996946 0.0781000i \(-0.975115\pi\)
0.996946 0.0781000i \(-0.0248854\pi\)
\(864\) 0 0
\(865\) 12.0947 6.98289i 0.411233 0.237426i
\(866\) 0 0
\(867\) −1.22105 2.11492i −0.0414690 0.0718264i
\(868\) 0 0
\(869\) 41.3434 + 23.8697i 1.40248 + 0.809722i
\(870\) 0 0
\(871\) −1.74687 0.613401i −0.0591903 0.0207843i
\(872\) 0 0
\(873\) 5.40815 + 3.12240i 0.183038 + 0.105677i
\(874\) 0 0
\(875\) 4.98278 + 8.63043i 0.168449 + 0.291762i
\(876\) 0 0
\(877\) −34.6248 + 19.9906i −1.16920 + 0.675035i −0.953490 0.301423i \(-0.902538\pi\)
−0.215705 + 0.976459i \(0.569205\pi\)
\(878\) 0 0
\(879\) 1.43011i 0.0482365i
\(880\) 0 0
\(881\) −23.9548 + 41.4910i −0.807058 + 1.39787i 0.107834 + 0.994169i \(0.465608\pi\)
−0.914893 + 0.403697i \(0.867725\pi\)
\(882\) 0 0
\(883\) −34.0091 −1.14450 −0.572248 0.820080i \(-0.693929\pi\)
−0.572248 + 0.820080i \(0.693929\pi\)
\(884\) 0 0
\(885\) −0.0196664 −0.000661077
\(886\) 0 0
\(887\) 1.88672 3.26789i 0.0633497 0.109725i −0.832611 0.553858i \(-0.813155\pi\)
0.895961 + 0.444133i \(0.146488\pi\)
\(888\) 0 0
\(889\) 6.86494i 0.230243i
\(890\) 0 0
\(891\) 32.4445 18.7319i 1.08693 0.627541i
\(892\) 0 0
\(893\) 5.49435 + 9.51650i 0.183862 + 0.318457i
\(894\) 0 0
\(895\) 11.7997 + 6.81254i 0.394419 + 0.227718i
\(896\) 0 0
\(897\) 1.23547 + 1.43882i 0.0412511 + 0.0480408i
\(898\) 0 0
\(899\) −49.3098 28.4690i −1.64457 0.949496i
\(900\) 0 0
\(901\) 17.9949 + 31.1681i 0.599497 + 1.03836i
\(902\) 0 0
\(903\) 0.745296 0.430297i 0.0248019 0.0143194i
\(904\) 0 0
\(905\) 5.49929i 0.182802i
\(906\) 0 0
\(907\) −7.13753 + 12.3626i −0.236998 + 0.410492i −0.959851 0.280509i \(-0.909497\pi\)
0.722854 + 0.691001i \(0.242830\pi\)
\(908\) 0 0
\(909\) −17.6718 −0.586135
\(910\) 0 0
\(911\) −17.4161 −0.577020 −0.288510 0.957477i \(-0.593160\pi\)
−0.288510 + 0.957477i \(0.593160\pi\)
\(912\) 0 0
\(913\) 34.1096 59.0796i 1.12886 1.95525i
\(914\) 0 0
\(915\) 2.35110i 0.0777248i
\(916\) 0 0
\(917\) 14.0293 8.09980i 0.463287 0.267479i
\(918\) 0 0
\(919\) 18.1179 + 31.3811i 0.597654 + 1.03517i 0.993166 + 0.116707i \(0.0372338\pi\)
−0.395512 + 0.918461i \(0.629433\pi\)
\(920\) 0 0
\(921\) −4.48437 2.58905i −0.147765 0.0853121i
\(922\) 0 0
\(923\) 29.5084 25.3379i 0.971281 0.834007i
\(924\) 0 0
\(925\) −30.7575 17.7579i −1.01130 0.583875i
\(926\) 0 0
\(927\) −8.45098 14.6375i −0.277567 0.480760i
\(928\) 0 0
\(929\) −12.4656 + 7.19701i −0.408983 + 0.236126i −0.690353 0.723473i \(-0.742545\pi\)
0.281370 + 0.959599i \(0.409211\pi\)
\(930\) 0 0
\(931\) 6.56298i 0.215093i
\(932\) 0 0
\(933\) −0.156420 + 0.270927i −0.00512096 + 0.00886977i
\(934\) 0 0
\(935\) 13.8241 0.452097
\(936\) 0 0
\(937\) −6.25633 −0.204385 −0.102193 0.994765i \(-0.532586\pi\)
−0.102193 + 0.994765i \(0.532586\pi\)
\(938\) 0 0
\(939\) −1.13941 + 1.97351i −0.0371831 + 0.0644031i
\(940\) 0 0
\(941\) 29.1506i 0.950284i 0.879909 + 0.475142i \(0.157603\pi\)
−0.879909 + 0.475142i \(0.842397\pi\)
\(942\) 0 0
\(943\) −17.1031 + 9.87448i −0.556954 + 0.321557i
\(944\) 0 0
\(945\) 0.861446 + 1.49207i 0.0280228 + 0.0485370i
\(946\) 0 0
\(947\) 9.43915 + 5.44970i 0.306731 + 0.177091i 0.645463 0.763792i \(-0.276665\pi\)
−0.338732 + 0.940883i \(0.609998\pi\)
\(948\) 0 0
\(949\) −5.34784 28.4343i −0.173598 0.923016i
\(950\) 0 0
\(951\) 0.127160 + 0.0734160i 0.00412345 + 0.00238068i
\(952\) 0 0
\(953\) 13.7652 + 23.8421i 0.445900 + 0.772321i 0.998114 0.0613812i \(-0.0195505\pi\)
−0.552215 + 0.833702i \(0.686217\pi\)
\(954\) 0 0
\(955\) −15.4915 + 8.94403i −0.501294 + 0.289422i
\(956\) 0 0
\(957\) 8.09193i 0.261575i
\(958\) 0 0
\(959\) 2.60112 4.50527i 0.0839945 0.145483i
\(960\) 0 0
\(961\) −31.5515 −1.01779
\(962\) 0 0
\(963\) −16.2932 −0.525042
\(964\) 0 0
\(965\) 10.0919 17.4797i 0.324870 0.562692i
\(966\) 0 0
\(967\) 21.0297i 0.676271i −0.941097 0.338135i \(-0.890204\pi\)
0.941097 0.338135i \(-0.109796\pi\)
\(968\) 0 0
\(969\) −3.89469 + 2.24860i −0.125116 + 0.0722355i
\(970\) 0 0
\(971\) 2.18410 + 3.78297i 0.0700911 + 0.121401i 0.898941 0.438070i \(-0.144338\pi\)
−0.828850 + 0.559471i \(0.811004\pi\)
\(972\) 0 0
\(973\) 4.97122 + 2.87013i 0.159370 + 0.0920123i
\(974\) 0 0
\(975\) −1.11245 + 3.16808i −0.0356269 + 0.101460i
\(976\) 0 0
\(977\) −46.8829 27.0679i −1.49992 0.865978i −0.499918 0.866073i \(-0.666637\pi\)
−1.00000 9.46999e-5i \(0.999970\pi\)
\(978\) 0 0
\(979\) −24.1346 41.8023i −0.771344 1.33601i
\(980\) 0 0
\(981\) 20.2649 11.7000i 0.647009 0.373551i
\(982\) 0 0
\(983\) 44.3574i 1.41478i −0.706823 0.707391i \(-0.749872\pi\)
0.706823 0.707391i \(-0.250128\pi\)
\(984\) 0 0
\(985\) 0.304049 0.526628i 0.00968780 0.0167798i
\(986\) 0 0
\(987\) −0.423405 −0.0134771
\(988\) 0 0
\(989\) 7.07864 0.225088
\(990\) 0 0
\(991\) 13.6936 23.7181i 0.434992 0.753429i −0.562303 0.826932i \(-0.690084\pi\)
0.997295 + 0.0735026i \(0.0234177\pi\)
\(992\) 0 0
\(993\) 7.93162i 0.251702i
\(994\) 0 0
\(995\) 24.9952 14.4310i 0.792402 0.457494i
\(996\) 0 0
\(997\) 22.9017 + 39.6669i 0.725303 + 1.25626i 0.958849 + 0.283916i \(0.0916338\pi\)
−0.233546 + 0.972346i \(0.575033\pi\)
\(998\) 0 0
\(999\) −12.5371 7.23832i −0.396658 0.229010i
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 1456.2.cc.d.225.4 12
4.3 odd 2 182.2.m.b.43.2 12
12.11 even 2 1638.2.bj.g.1135.5 12
13.10 even 6 inner 1456.2.cc.d.673.4 12
28.3 even 6 1274.2.v.d.667.5 12
28.11 odd 6 1274.2.v.e.667.5 12
28.19 even 6 1274.2.o.e.459.5 12
28.23 odd 6 1274.2.o.d.459.5 12
28.27 even 2 1274.2.m.c.589.2 12
52.7 even 12 2366.2.a.bf.1.4 6
52.19 even 12 2366.2.a.bh.1.4 6
52.23 odd 6 182.2.m.b.127.2 yes 12
52.35 odd 6 2366.2.d.r.337.10 12
52.43 odd 6 2366.2.d.r.337.4 12
156.23 even 6 1638.2.bj.g.127.5 12
364.23 odd 6 1274.2.v.e.361.5 12
364.75 even 6 1274.2.v.d.361.5 12
364.179 odd 6 1274.2.o.d.569.2 12
364.283 even 6 1274.2.o.e.569.2 12
364.335 even 6 1274.2.m.c.491.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
182.2.m.b.43.2 12 4.3 odd 2
182.2.m.b.127.2 yes 12 52.23 odd 6
1274.2.m.c.491.2 12 364.335 even 6
1274.2.m.c.589.2 12 28.27 even 2
1274.2.o.d.459.5 12 28.23 odd 6
1274.2.o.d.569.2 12 364.179 odd 6
1274.2.o.e.459.5 12 28.19 even 6
1274.2.o.e.569.2 12 364.283 even 6
1274.2.v.d.361.5 12 364.75 even 6
1274.2.v.d.667.5 12 28.3 even 6
1274.2.v.e.361.5 12 364.23 odd 6
1274.2.v.e.667.5 12 28.11 odd 6
1456.2.cc.d.225.4 12 1.1 even 1 trivial
1456.2.cc.d.673.4 12 13.10 even 6 inner
1638.2.bj.g.127.5 12 156.23 even 6
1638.2.bj.g.1135.5 12 12.11 even 2
2366.2.a.bf.1.4 6 52.7 even 12
2366.2.a.bh.1.4 6 52.19 even 12
2366.2.d.r.337.4 12 52.43 odd 6
2366.2.d.r.337.10 12 52.35 odd 6