Properties

Label 1344.2.u.a.1231.12
Level $1344$
Weight $2$
Character 1344.1231
Analytic conductor $10.732$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(559,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.u (of order \(4\), 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: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1231.12
Character \(\chi\) \(=\) 1344.1231
Dual form 1344.2.u.a.559.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.32544 - 1.32544i) q^{5} +(-2.64192 + 0.142388i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(1.32544 - 1.32544i) q^{5} +(-2.64192 + 0.142388i) q^{7} -1.00000i q^{9} +(1.18514 - 1.18514i) q^{11} +(-2.02086 - 2.02086i) q^{13} +1.87446i q^{15} -0.276387i q^{17} +(-4.58131 + 4.58131i) q^{19} +(1.76743 - 1.96880i) q^{21} +1.81753 q^{23} +1.48640i q^{25} +(0.707107 + 0.707107i) q^{27} +(-2.64137 + 2.64137i) q^{29} -7.11326 q^{31} +1.67603i q^{33} +(-3.31298 + 3.69044i) q^{35} +(-0.786692 - 0.786692i) q^{37} +2.85793 q^{39} -7.33055 q^{41} +(1.69851 - 1.69851i) q^{43} +(-1.32544 - 1.32544i) q^{45} -11.5721 q^{47} +(6.95945 - 0.752353i) q^{49} +(0.195435 + 0.195435i) q^{51} +(3.85262 + 3.85262i) q^{53} -3.14166i q^{55} -6.47895i q^{57} +(4.25247 + 4.25247i) q^{59} +(-1.14204 - 1.14204i) q^{61} +(0.142388 + 2.64192i) q^{63} -5.35708 q^{65} +(3.05161 + 3.05161i) q^{67} +(-1.28519 + 1.28519i) q^{69} -8.36612 q^{71} -16.9603 q^{73} +(-1.05104 - 1.05104i) q^{75} +(-2.96228 + 3.29978i) q^{77} -13.8414i q^{79} -1.00000 q^{81} +(9.90941 - 9.90941i) q^{83} +(-0.366336 - 0.366336i) q^{85} -3.73546i q^{87} -7.94992 q^{89} +(5.62670 + 5.05121i) q^{91} +(5.02983 - 5.02983i) q^{93} +12.1445i q^{95} +13.1657i q^{97} +(-1.18514 - 1.18514i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q+O(q^{10}) \) Copy content Toggle raw display \( 64 q - 8 q^{11} + 16 q^{23} + 16 q^{29} - 24 q^{35} + 16 q^{37} + 8 q^{43} + 16 q^{53} - 56 q^{67} + 128 q^{71} - 64 q^{81} - 8 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 1.32544 1.32544i 0.592756 0.592756i −0.345619 0.938375i \(-0.612331\pi\)
0.938375 + 0.345619i \(0.112331\pi\)
\(6\) 0 0
\(7\) −2.64192 + 0.142388i −0.998551 + 0.0538175i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 1.18514 1.18514i 0.357332 0.357332i −0.505497 0.862829i \(-0.668691\pi\)
0.862829 + 0.505497i \(0.168691\pi\)
\(12\) 0 0
\(13\) −2.02086 2.02086i −0.560487 0.560487i 0.368959 0.929446i \(-0.379714\pi\)
−0.929446 + 0.368959i \(0.879714\pi\)
\(14\) 0 0
\(15\) 1.87446i 0.483983i
\(16\) 0 0
\(17\) 0.276387i 0.0670337i −0.999438 0.0335169i \(-0.989329\pi\)
0.999438 0.0335169i \(-0.0106707\pi\)
\(18\) 0 0
\(19\) −4.58131 + 4.58131i −1.05102 + 1.05102i −0.0523978 + 0.998626i \(0.516686\pi\)
−0.998626 + 0.0523978i \(0.983314\pi\)
\(20\) 0 0
\(21\) 1.76743 1.96880i 0.385686 0.429628i
\(22\) 0 0
\(23\) 1.81753 0.378981 0.189491 0.981883i \(-0.439316\pi\)
0.189491 + 0.981883i \(0.439316\pi\)
\(24\) 0 0
\(25\) 1.48640i 0.297280i
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) −2.64137 + 2.64137i −0.490490 + 0.490490i −0.908461 0.417970i \(-0.862742\pi\)
0.417970 + 0.908461i \(0.362742\pi\)
\(30\) 0 0
\(31\) −7.11326 −1.27758 −0.638790 0.769382i \(-0.720565\pi\)
−0.638790 + 0.769382i \(0.720565\pi\)
\(32\) 0 0
\(33\) 1.67603i 0.291760i
\(34\) 0 0
\(35\) −3.31298 + 3.69044i −0.559997 + 0.623798i
\(36\) 0 0
\(37\) −0.786692 0.786692i −0.129331 0.129331i 0.639478 0.768809i \(-0.279150\pi\)
−0.768809 + 0.639478i \(0.779150\pi\)
\(38\) 0 0
\(39\) 2.85793 0.457636
\(40\) 0 0
\(41\) −7.33055 −1.14484 −0.572420 0.819961i \(-0.693995\pi\)
−0.572420 + 0.819961i \(0.693995\pi\)
\(42\) 0 0
\(43\) 1.69851 1.69851i 0.259020 0.259020i −0.565635 0.824656i \(-0.691369\pi\)
0.824656 + 0.565635i \(0.191369\pi\)
\(44\) 0 0
\(45\) −1.32544 1.32544i −0.197585 0.197585i
\(46\) 0 0
\(47\) −11.5721 −1.68796 −0.843982 0.536372i \(-0.819795\pi\)
−0.843982 + 0.536372i \(0.819795\pi\)
\(48\) 0 0
\(49\) 6.95945 0.752353i 0.994207 0.107479i
\(50\) 0 0
\(51\) 0.195435 + 0.195435i 0.0273664 + 0.0273664i
\(52\) 0 0
\(53\) 3.85262 + 3.85262i 0.529198 + 0.529198i 0.920333 0.391135i \(-0.127917\pi\)
−0.391135 + 0.920333i \(0.627917\pi\)
\(54\) 0 0
\(55\) 3.14166i 0.423621i
\(56\) 0 0
\(57\) 6.47895i 0.858158i
\(58\) 0 0
\(59\) 4.25247 + 4.25247i 0.553625 + 0.553625i 0.927485 0.373860i \(-0.121966\pi\)
−0.373860 + 0.927485i \(0.621966\pi\)
\(60\) 0 0
\(61\) −1.14204 1.14204i −0.146223 0.146223i 0.630206 0.776428i \(-0.282971\pi\)
−0.776428 + 0.630206i \(0.782971\pi\)
\(62\) 0 0
\(63\) 0.142388 + 2.64192i 0.0179392 + 0.332850i
\(64\) 0 0
\(65\) −5.35708 −0.664464
\(66\) 0 0
\(67\) 3.05161 + 3.05161i 0.372813 + 0.372813i 0.868501 0.495688i \(-0.165084\pi\)
−0.495688 + 0.868501i \(0.665084\pi\)
\(68\) 0 0
\(69\) −1.28519 + 1.28519i −0.154718 + 0.154718i
\(70\) 0 0
\(71\) −8.36612 −0.992876 −0.496438 0.868072i \(-0.665359\pi\)
−0.496438 + 0.868072i \(0.665359\pi\)
\(72\) 0 0
\(73\) −16.9603 −1.98505 −0.992527 0.122024i \(-0.961061\pi\)
−0.992527 + 0.122024i \(0.961061\pi\)
\(74\) 0 0
\(75\) −1.05104 1.05104i −0.121364 0.121364i
\(76\) 0 0
\(77\) −2.96228 + 3.29978i −0.337583 + 0.376045i
\(78\) 0 0
\(79\) 13.8414i 1.55728i −0.627470 0.778641i \(-0.715909\pi\)
0.627470 0.778641i \(-0.284091\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 9.90941 9.90941i 1.08770 1.08770i 0.0919340 0.995765i \(-0.470695\pi\)
0.995765 0.0919340i \(-0.0293049\pi\)
\(84\) 0 0
\(85\) −0.366336 0.366336i −0.0397347 0.0397347i
\(86\) 0 0
\(87\) 3.73546i 0.400484i
\(88\) 0 0
\(89\) −7.94992 −0.842690 −0.421345 0.906901i \(-0.638442\pi\)
−0.421345 + 0.906901i \(0.638442\pi\)
\(90\) 0 0
\(91\) 5.62670 + 5.05121i 0.589839 + 0.529511i
\(92\) 0 0
\(93\) 5.02983 5.02983i 0.521569 0.521569i
\(94\) 0 0
\(95\) 12.1445i 1.24600i
\(96\) 0 0
\(97\) 13.1657i 1.33678i 0.743812 + 0.668388i \(0.233016\pi\)
−0.743812 + 0.668388i \(0.766984\pi\)
\(98\) 0 0
\(99\) −1.18514 1.18514i −0.119111 0.119111i
\(100\) 0 0
\(101\) 1.20635 1.20635i 0.120036 0.120036i −0.644537 0.764573i \(-0.722950\pi\)
0.764573 + 0.644537i \(0.222950\pi\)
\(102\) 0 0
\(103\) 11.4220i 1.12544i −0.826647 0.562722i \(-0.809754\pi\)
0.826647 0.562722i \(-0.190246\pi\)
\(104\) 0 0
\(105\) −0.266900 4.95217i −0.0260468 0.483282i
\(106\) 0 0
\(107\) −9.18783 + 9.18783i −0.888221 + 0.888221i −0.994352 0.106131i \(-0.966154\pi\)
0.106131 + 0.994352i \(0.466154\pi\)
\(108\) 0 0
\(109\) 0.541341 0.541341i 0.0518511 0.0518511i −0.680706 0.732557i \(-0.738327\pi\)
0.732557 + 0.680706i \(0.238327\pi\)
\(110\) 0 0
\(111\) 1.11255 0.105599
\(112\) 0 0
\(113\) −11.9427 −1.12347 −0.561735 0.827317i \(-0.689866\pi\)
−0.561735 + 0.827317i \(0.689866\pi\)
\(114\) 0 0
\(115\) 2.40903 2.40903i 0.224644 0.224644i
\(116\) 0 0
\(117\) −2.02086 + 2.02086i −0.186829 + 0.186829i
\(118\) 0 0
\(119\) 0.0393541 + 0.730192i 0.00360759 + 0.0669366i
\(120\) 0 0
\(121\) 8.19091i 0.744628i
\(122\) 0 0
\(123\) 5.18348 5.18348i 0.467379 0.467379i
\(124\) 0 0
\(125\) 8.59736 + 8.59736i 0.768971 + 0.768971i
\(126\) 0 0
\(127\) 18.8618i 1.67371i −0.547422 0.836857i \(-0.684391\pi\)
0.547422 0.836857i \(-0.315609\pi\)
\(128\) 0 0
\(129\) 2.40206i 0.211489i
\(130\) 0 0
\(131\) −8.80983 + 8.80983i −0.769718 + 0.769718i −0.978057 0.208339i \(-0.933194\pi\)
0.208339 + 0.978057i \(0.433194\pi\)
\(132\) 0 0
\(133\) 11.4511 12.7558i 0.992937 1.10606i
\(134\) 0 0
\(135\) 1.87446 0.161328
\(136\) 0 0
\(137\) 4.34380i 0.371116i 0.982633 + 0.185558i \(0.0594092\pi\)
−0.982633 + 0.185558i \(0.940591\pi\)
\(138\) 0 0
\(139\) 1.64886 + 1.64886i 0.139854 + 0.139854i 0.773568 0.633714i \(-0.218470\pi\)
−0.633714 + 0.773568i \(0.718470\pi\)
\(140\) 0 0
\(141\) 8.18271 8.18271i 0.689108 0.689108i
\(142\) 0 0
\(143\) −4.79000 −0.400560
\(144\) 0 0
\(145\) 7.00197i 0.581482i
\(146\) 0 0
\(147\) −4.38908 + 5.45307i −0.362005 + 0.449762i
\(148\) 0 0
\(149\) −10.1553 10.1553i −0.831955 0.831955i 0.155829 0.987784i \(-0.450195\pi\)
−0.987784 + 0.155829i \(0.950195\pi\)
\(150\) 0 0
\(151\) 8.92611 0.726397 0.363198 0.931712i \(-0.381685\pi\)
0.363198 + 0.931712i \(0.381685\pi\)
\(152\) 0 0
\(153\) −0.276387 −0.0223446
\(154\) 0 0
\(155\) −9.42822 + 9.42822i −0.757293 + 0.757293i
\(156\) 0 0
\(157\) 5.83300 + 5.83300i 0.465524 + 0.465524i 0.900461 0.434937i \(-0.143229\pi\)
−0.434937 + 0.900461i \(0.643229\pi\)
\(158\) 0 0
\(159\) −5.44843 −0.432089
\(160\) 0 0
\(161\) −4.80176 + 0.258794i −0.378432 + 0.0203958i
\(162\) 0 0
\(163\) −12.4751 12.4751i −0.977129 0.977129i 0.0226151 0.999744i \(-0.492801\pi\)
−0.999744 + 0.0226151i \(0.992801\pi\)
\(164\) 0 0
\(165\) 2.22149 + 2.22149i 0.172943 + 0.172943i
\(166\) 0 0
\(167\) 21.3378i 1.65117i −0.564280 0.825584i \(-0.690846\pi\)
0.564280 0.825584i \(-0.309154\pi\)
\(168\) 0 0
\(169\) 4.83221i 0.371709i
\(170\) 0 0
\(171\) 4.58131 + 4.58131i 0.350341 + 0.350341i
\(172\) 0 0
\(173\) −10.7377 10.7377i −0.816371 0.816371i 0.169209 0.985580i \(-0.445879\pi\)
−0.985580 + 0.169209i \(0.945879\pi\)
\(174\) 0 0
\(175\) −0.211645 3.92695i −0.0159989 0.296849i
\(176\) 0 0
\(177\) −6.01391 −0.452033
\(178\) 0 0
\(179\) −2.41822 2.41822i −0.180746 0.180746i 0.610935 0.791681i \(-0.290794\pi\)
−0.791681 + 0.610935i \(0.790794\pi\)
\(180\) 0 0
\(181\) 0.318290 0.318290i 0.0236583 0.0236583i −0.695179 0.718837i \(-0.744675\pi\)
0.718837 + 0.695179i \(0.244675\pi\)
\(182\) 0 0
\(183\) 1.61508 0.119390
\(184\) 0 0
\(185\) −2.08543 −0.153324
\(186\) 0 0
\(187\) −0.327556 0.327556i −0.0239533 0.0239533i
\(188\) 0 0
\(189\) −1.96880 1.76743i −0.143209 0.128562i
\(190\) 0 0
\(191\) 24.4401i 1.76842i 0.467088 + 0.884211i \(0.345303\pi\)
−0.467088 + 0.884211i \(0.654697\pi\)
\(192\) 0 0
\(193\) 22.3910 1.61174 0.805871 0.592091i \(-0.201698\pi\)
0.805871 + 0.592091i \(0.201698\pi\)
\(194\) 0 0
\(195\) 3.78803 3.78803i 0.271266 0.271266i
\(196\) 0 0
\(197\) 13.4527 + 13.4527i 0.958463 + 0.958463i 0.999171 0.0407083i \(-0.0129614\pi\)
−0.0407083 + 0.999171i \(0.512961\pi\)
\(198\) 0 0
\(199\) 5.64437i 0.400119i 0.979784 + 0.200059i \(0.0641135\pi\)
−0.979784 + 0.200059i \(0.935886\pi\)
\(200\) 0 0
\(201\) −4.31563 −0.304401
\(202\) 0 0
\(203\) 6.60218 7.35438i 0.463382 0.516176i
\(204\) 0 0
\(205\) −9.71623 + 9.71623i −0.678611 + 0.678611i
\(206\) 0 0
\(207\) 1.81753i 0.126327i
\(208\) 0 0
\(209\) 10.8589i 0.751129i
\(210\) 0 0
\(211\) 13.9095 + 13.9095i 0.957568 + 0.957568i 0.999136 0.0415678i \(-0.0132353\pi\)
−0.0415678 + 0.999136i \(0.513235\pi\)
\(212\) 0 0
\(213\) 5.91574 5.91574i 0.405340 0.405340i
\(214\) 0 0
\(215\) 4.50256i 0.307072i
\(216\) 0 0
\(217\) 18.7926 1.01284i 1.27573 0.0687561i
\(218\) 0 0
\(219\) 11.9928 11.9928i 0.810395 0.810395i
\(220\) 0 0
\(221\) −0.558541 + 0.558541i −0.0375715 + 0.0375715i
\(222\) 0 0
\(223\) 15.4893 1.03724 0.518622 0.855004i \(-0.326445\pi\)
0.518622 + 0.855004i \(0.326445\pi\)
\(224\) 0 0
\(225\) 1.48640 0.0990933
\(226\) 0 0
\(227\) 4.24437 4.24437i 0.281709 0.281709i −0.552081 0.833790i \(-0.686166\pi\)
0.833790 + 0.552081i \(0.186166\pi\)
\(228\) 0 0
\(229\) −2.39096 + 2.39096i −0.157999 + 0.157999i −0.781679 0.623680i \(-0.785637\pi\)
0.623680 + 0.781679i \(0.285637\pi\)
\(230\) 0 0
\(231\) −0.238647 4.42795i −0.0157018 0.291337i
\(232\) 0 0
\(233\) 2.92061i 0.191336i 0.995413 + 0.0956678i \(0.0304987\pi\)
−0.995413 + 0.0956678i \(0.969501\pi\)
\(234\) 0 0
\(235\) −15.3382 + 15.3382i −1.00055 + 1.00055i
\(236\) 0 0
\(237\) 9.78736 + 9.78736i 0.635758 + 0.635758i
\(238\) 0 0
\(239\) 7.83859i 0.507036i 0.967331 + 0.253518i \(0.0815877\pi\)
−0.967331 + 0.253518i \(0.918412\pi\)
\(240\) 0 0
\(241\) 18.3656i 1.18303i −0.806293 0.591516i \(-0.798530\pi\)
0.806293 0.591516i \(-0.201470\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 8.22716 10.2216i 0.525614 0.653031i
\(246\) 0 0
\(247\) 18.5164 1.17817
\(248\) 0 0
\(249\) 14.0140i 0.888103i
\(250\) 0 0
\(251\) 16.2774 + 16.2774i 1.02742 + 1.02742i 0.999613 + 0.0278042i \(0.00885149\pi\)
0.0278042 + 0.999613i \(0.491149\pi\)
\(252\) 0 0
\(253\) 2.15402 2.15402i 0.135422 0.135422i
\(254\) 0 0
\(255\) 0.518077 0.0324432
\(256\) 0 0
\(257\) 13.6309i 0.850274i 0.905129 + 0.425137i \(0.139774\pi\)
−0.905129 + 0.425137i \(0.860226\pi\)
\(258\) 0 0
\(259\) 2.19039 + 1.96636i 0.136104 + 0.122184i
\(260\) 0 0
\(261\) 2.64137 + 2.64137i 0.163497 + 0.163497i
\(262\) 0 0
\(263\) −5.88266 −0.362740 −0.181370 0.983415i \(-0.558053\pi\)
−0.181370 + 0.983415i \(0.558053\pi\)
\(264\) 0 0
\(265\) 10.2129 0.627371
\(266\) 0 0
\(267\) 5.62144 5.62144i 0.344027 0.344027i
\(268\) 0 0
\(269\) 7.28207 + 7.28207i 0.443995 + 0.443995i 0.893352 0.449357i \(-0.148347\pi\)
−0.449357 + 0.893352i \(0.648347\pi\)
\(270\) 0 0
\(271\) 7.65390 0.464941 0.232471 0.972603i \(-0.425319\pi\)
0.232471 + 0.972603i \(0.425319\pi\)
\(272\) 0 0
\(273\) −7.55042 + 0.406935i −0.456972 + 0.0246288i
\(274\) 0 0
\(275\) 1.76159 + 1.76159i 0.106228 + 0.106228i
\(276\) 0 0
\(277\) −20.9351 20.9351i −1.25787 1.25787i −0.952109 0.305758i \(-0.901090\pi\)
−0.305758 0.952109i \(-0.598910\pi\)
\(278\) 0 0
\(279\) 7.11326i 0.425860i
\(280\) 0 0
\(281\) 13.2903i 0.792835i 0.918070 + 0.396417i \(0.129747\pi\)
−0.918070 + 0.396417i \(0.870253\pi\)
\(282\) 0 0
\(283\) 7.97242 + 7.97242i 0.473911 + 0.473911i 0.903178 0.429267i \(-0.141228\pi\)
−0.429267 + 0.903178i \(0.641228\pi\)
\(284\) 0 0
\(285\) −8.58748 8.58748i −0.508678 0.508678i
\(286\) 0 0
\(287\) 19.3667 1.04378i 1.14318 0.0616124i
\(288\) 0 0
\(289\) 16.9236 0.995506
\(290\) 0 0
\(291\) −9.30957 9.30957i −0.545737 0.545737i
\(292\) 0 0
\(293\) 11.7572 11.7572i 0.686863 0.686863i −0.274674 0.961537i \(-0.588570\pi\)
0.961537 + 0.274674i \(0.0885701\pi\)
\(294\) 0 0
\(295\) 11.2728 0.656329
\(296\) 0 0
\(297\) 1.67603 0.0972534
\(298\) 0 0
\(299\) −3.67298 3.67298i −0.212414 0.212414i
\(300\) 0 0
\(301\) −4.24548 + 4.72917i −0.244705 + 0.272585i
\(302\) 0 0
\(303\) 1.70604i 0.0980093i
\(304\) 0 0
\(305\) −3.02741 −0.173349
\(306\) 0 0
\(307\) 20.0939 20.0939i 1.14682 1.14682i 0.159645 0.987174i \(-0.448965\pi\)
0.987174 0.159645i \(-0.0510350\pi\)
\(308\) 0 0
\(309\) 8.07657 + 8.07657i 0.459460 + 0.459460i
\(310\) 0 0
\(311\) 26.0652i 1.47802i 0.673693 + 0.739012i \(0.264707\pi\)
−0.673693 + 0.739012i \(0.735293\pi\)
\(312\) 0 0
\(313\) −13.5651 −0.766743 −0.383372 0.923594i \(-0.625237\pi\)
−0.383372 + 0.923594i \(0.625237\pi\)
\(314\) 0 0
\(315\) 3.69044 + 3.31298i 0.207933 + 0.186666i
\(316\) 0 0
\(317\) 21.5304 21.5304i 1.20927 1.20927i 0.238004 0.971264i \(-0.423507\pi\)
0.971264 0.238004i \(-0.0764931\pi\)
\(318\) 0 0
\(319\) 6.26077i 0.350536i
\(320\) 0 0
\(321\) 12.9935i 0.725229i
\(322\) 0 0
\(323\) 1.26621 + 1.26621i 0.0704541 + 0.0704541i
\(324\) 0 0
\(325\) 3.00381 3.00381i 0.166622 0.166622i
\(326\) 0 0
\(327\) 0.765572i 0.0423362i
\(328\) 0 0
\(329\) 30.5725 1.64772i 1.68552 0.0908420i
\(330\) 0 0
\(331\) −5.85042 + 5.85042i −0.321568 + 0.321568i −0.849369 0.527800i \(-0.823017\pi\)
0.527800 + 0.849369i \(0.323017\pi\)
\(332\) 0 0
\(333\) −0.786692 + 0.786692i −0.0431104 + 0.0431104i
\(334\) 0 0
\(335\) 8.08947 0.441975
\(336\) 0 0
\(337\) 11.1243 0.605980 0.302990 0.952994i \(-0.402015\pi\)
0.302990 + 0.952994i \(0.402015\pi\)
\(338\) 0 0
\(339\) 8.44474 8.44474i 0.458655 0.458655i
\(340\) 0 0
\(341\) −8.43018 + 8.43018i −0.456520 + 0.456520i
\(342\) 0 0
\(343\) −18.2792 + 2.97859i −0.986982 + 0.160829i
\(344\) 0 0
\(345\) 3.40689i 0.183421i
\(346\) 0 0
\(347\) 16.7722 16.7722i 0.900379 0.900379i −0.0950894 0.995469i \(-0.530314\pi\)
0.995469 + 0.0950894i \(0.0303137\pi\)
\(348\) 0 0
\(349\) 11.9769 + 11.9769i 0.641110 + 0.641110i 0.950828 0.309718i \(-0.100235\pi\)
−0.309718 + 0.950828i \(0.600235\pi\)
\(350\) 0 0
\(351\) 2.85793i 0.152545i
\(352\) 0 0
\(353\) 34.7417i 1.84912i −0.381041 0.924558i \(-0.624434\pi\)
0.381041 0.924558i \(-0.375566\pi\)
\(354\) 0 0
\(355\) −11.0888 + 11.0888i −0.588533 + 0.588533i
\(356\) 0 0
\(357\) −0.544151 0.488496i −0.0287995 0.0258540i
\(358\) 0 0
\(359\) −22.2884 −1.17634 −0.588168 0.808739i \(-0.700150\pi\)
−0.588168 + 0.808739i \(0.700150\pi\)
\(360\) 0 0
\(361\) 22.9768i 1.20930i
\(362\) 0 0
\(363\) −5.79185 5.79185i −0.303993 0.303993i
\(364\) 0 0
\(365\) −22.4799 + 22.4799i −1.17665 + 1.17665i
\(366\) 0 0
\(367\) −35.3887 −1.84728 −0.923639 0.383264i \(-0.874800\pi\)
−0.923639 + 0.383264i \(0.874800\pi\)
\(368\) 0 0
\(369\) 7.33055i 0.381613i
\(370\) 0 0
\(371\) −10.7269 9.62974i −0.556912 0.499951i
\(372\) 0 0
\(373\) 6.16397 + 6.16397i 0.319158 + 0.319158i 0.848444 0.529285i \(-0.177540\pi\)
−0.529285 + 0.848444i \(0.677540\pi\)
\(374\) 0 0
\(375\) −12.1585 −0.627862
\(376\) 0 0
\(377\) 10.6757 0.549827
\(378\) 0 0
\(379\) −7.14409 + 7.14409i −0.366967 + 0.366967i −0.866370 0.499403i \(-0.833553\pi\)
0.499403 + 0.866370i \(0.333553\pi\)
\(380\) 0 0
\(381\) 13.3373 + 13.3373i 0.683291 + 0.683291i
\(382\) 0 0
\(383\) 15.8856 0.811718 0.405859 0.913936i \(-0.366972\pi\)
0.405859 + 0.913936i \(0.366972\pi\)
\(384\) 0 0
\(385\) 0.447334 + 8.30001i 0.0227982 + 0.423007i
\(386\) 0 0
\(387\) −1.69851 1.69851i −0.0863401 0.0863401i
\(388\) 0 0
\(389\) 3.30347 + 3.30347i 0.167493 + 0.167493i 0.785876 0.618384i \(-0.212212\pi\)
−0.618384 + 0.785876i \(0.712212\pi\)
\(390\) 0 0
\(391\) 0.502342i 0.0254045i
\(392\) 0 0
\(393\) 12.4590i 0.628472i
\(394\) 0 0
\(395\) −18.3460 18.3460i −0.923088 0.923088i
\(396\) 0 0
\(397\) −11.0837 11.0837i −0.556274 0.556274i 0.371971 0.928244i \(-0.378682\pi\)
−0.928244 + 0.371971i \(0.878682\pi\)
\(398\) 0 0
\(399\) 0.922522 + 17.1168i 0.0461839 + 0.856914i
\(400\) 0 0
\(401\) −12.6471 −0.631566 −0.315783 0.948831i \(-0.602267\pi\)
−0.315783 + 0.948831i \(0.602267\pi\)
\(402\) 0 0
\(403\) 14.3749 + 14.3749i 0.716066 + 0.716066i
\(404\) 0 0
\(405\) −1.32544 + 1.32544i −0.0658618 + 0.0658618i
\(406\) 0 0
\(407\) −1.86467 −0.0924284
\(408\) 0 0
\(409\) 6.39999 0.316459 0.158230 0.987402i \(-0.449421\pi\)
0.158230 + 0.987402i \(0.449421\pi\)
\(410\) 0 0
\(411\) −3.07153 3.07153i −0.151507 0.151507i
\(412\) 0 0
\(413\) −11.8402 10.6292i −0.582617 0.523028i
\(414\) 0 0
\(415\) 26.2687i 1.28948i
\(416\) 0 0
\(417\) −2.33183 −0.114190
\(418\) 0 0
\(419\) −3.92319 + 3.92319i −0.191660 + 0.191660i −0.796413 0.604753i \(-0.793272\pi\)
0.604753 + 0.796413i \(0.293272\pi\)
\(420\) 0 0
\(421\) −19.9120 19.9120i −0.970449 0.970449i 0.0291264 0.999576i \(-0.490727\pi\)
−0.999576 + 0.0291264i \(0.990727\pi\)
\(422\) 0 0
\(423\) 11.5721i 0.562655i
\(424\) 0 0
\(425\) 0.410822 0.0199278
\(426\) 0 0
\(427\) 3.17978 + 2.85455i 0.153880 + 0.138142i
\(428\) 0 0
\(429\) 3.38704 3.38704i 0.163528 0.163528i
\(430\) 0 0
\(431\) 23.6983i 1.14151i −0.821122 0.570753i \(-0.806651\pi\)
0.821122 0.570753i \(-0.193349\pi\)
\(432\) 0 0
\(433\) 25.8198i 1.24082i −0.784278 0.620410i \(-0.786966\pi\)
0.784278 0.620410i \(-0.213034\pi\)
\(434\) 0 0
\(435\) −4.95114 4.95114i −0.237389 0.237389i
\(436\) 0 0
\(437\) −8.32666 + 8.32666i −0.398318 + 0.398318i
\(438\) 0 0
\(439\) 13.7981i 0.658548i 0.944234 + 0.329274i \(0.106804\pi\)
−0.944234 + 0.329274i \(0.893196\pi\)
\(440\) 0 0
\(441\) −0.752353 6.95945i −0.0358263 0.331402i
\(442\) 0 0
\(443\) −9.61155 + 9.61155i −0.456658 + 0.456658i −0.897557 0.440898i \(-0.854660\pi\)
0.440898 + 0.897557i \(0.354660\pi\)
\(444\) 0 0
\(445\) −10.5372 + 10.5372i −0.499510 + 0.499510i
\(446\) 0 0
\(447\) 14.3618 0.679288
\(448\) 0 0
\(449\) −27.2681 −1.28686 −0.643430 0.765505i \(-0.722489\pi\)
−0.643430 + 0.765505i \(0.722489\pi\)
\(450\) 0 0
\(451\) −8.68770 + 8.68770i −0.409088 + 0.409088i
\(452\) 0 0
\(453\) −6.31171 + 6.31171i −0.296550 + 0.296550i
\(454\) 0 0
\(455\) 14.1530 0.762782i 0.663501 0.0357598i
\(456\) 0 0
\(457\) 15.7041i 0.734607i −0.930101 0.367303i \(-0.880281\pi\)
0.930101 0.367303i \(-0.119719\pi\)
\(458\) 0 0
\(459\) 0.195435 0.195435i 0.00912214 0.00912214i
\(460\) 0 0
\(461\) 8.60225 + 8.60225i 0.400647 + 0.400647i 0.878461 0.477814i \(-0.158571\pi\)
−0.477814 + 0.878461i \(0.658571\pi\)
\(462\) 0 0
\(463\) 1.85673i 0.0862897i −0.999069 0.0431449i \(-0.986262\pi\)
0.999069 0.0431449i \(-0.0137377\pi\)
\(464\) 0 0
\(465\) 13.3335i 0.618327i
\(466\) 0 0
\(467\) −12.0605 + 12.0605i −0.558093 + 0.558093i −0.928764 0.370671i \(-0.879128\pi\)
0.370671 + 0.928764i \(0.379128\pi\)
\(468\) 0 0
\(469\) −8.49661 7.62758i −0.392337 0.352209i
\(470\) 0 0
\(471\) −8.24911 −0.380099
\(472\) 0 0
\(473\) 4.02593i 0.185112i
\(474\) 0 0
\(475\) −6.80966 6.80966i −0.312448 0.312448i
\(476\) 0 0
\(477\) 3.85262 3.85262i 0.176399 0.176399i
\(478\) 0 0
\(479\) −14.9496 −0.683063 −0.341531 0.939870i \(-0.610946\pi\)
−0.341531 + 0.939870i \(0.610946\pi\)
\(480\) 0 0
\(481\) 3.17959i 0.144977i
\(482\) 0 0
\(483\) 3.21236 3.57835i 0.146168 0.162821i
\(484\) 0 0
\(485\) 17.4504 + 17.4504i 0.792383 + 0.792383i
\(486\) 0 0
\(487\) −2.67317 −0.121133 −0.0605664 0.998164i \(-0.519291\pi\)
−0.0605664 + 0.998164i \(0.519291\pi\)
\(488\) 0 0
\(489\) 17.6425 0.797823
\(490\) 0 0
\(491\) 7.71425 7.71425i 0.348139 0.348139i −0.511277 0.859416i \(-0.670827\pi\)
0.859416 + 0.511277i \(0.170827\pi\)
\(492\) 0 0
\(493\) 0.730041 + 0.730041i 0.0328794 + 0.0328794i
\(494\) 0 0
\(495\) −3.14166 −0.141207
\(496\) 0 0
\(497\) 22.1026 1.19123i 0.991437 0.0534341i
\(498\) 0 0
\(499\) 10.9644 + 10.9644i 0.490835 + 0.490835i 0.908569 0.417734i \(-0.137176\pi\)
−0.417734 + 0.908569i \(0.637176\pi\)
\(500\) 0 0
\(501\) 15.0881 + 15.0881i 0.674086 + 0.674086i
\(502\) 0 0
\(503\) 16.3842i 0.730533i 0.930903 + 0.365267i \(0.119022\pi\)
−0.930903 + 0.365267i \(0.880978\pi\)
\(504\) 0 0
\(505\) 3.19790i 0.142305i
\(506\) 0 0
\(507\) 3.41689 + 3.41689i 0.151749 + 0.151749i
\(508\) 0 0
\(509\) −18.7512 18.7512i −0.831133 0.831133i 0.156539 0.987672i \(-0.449966\pi\)
−0.987672 + 0.156539i \(0.949966\pi\)
\(510\) 0 0
\(511\) 44.8077 2.41494i 1.98218 0.106831i
\(512\) 0 0
\(513\) −6.47895 −0.286053
\(514\) 0 0
\(515\) −15.1392 15.1392i −0.667113 0.667113i
\(516\) 0 0
\(517\) −13.7145 + 13.7145i −0.603163 + 0.603163i
\(518\) 0 0
\(519\) 15.1854 0.666564
\(520\) 0 0
\(521\) 4.36154 0.191083 0.0955413 0.995425i \(-0.469542\pi\)
0.0955413 + 0.995425i \(0.469542\pi\)
\(522\) 0 0
\(523\) −28.7720 28.7720i −1.25811 1.25811i −0.951994 0.306118i \(-0.900970\pi\)
−0.306118 0.951994i \(-0.599030\pi\)
\(524\) 0 0
\(525\) 2.92643 + 2.62711i 0.127720 + 0.114657i
\(526\) 0 0
\(527\) 1.96601i 0.0856409i
\(528\) 0 0
\(529\) −19.6966 −0.856373
\(530\) 0 0
\(531\) 4.25247 4.25247i 0.184542 0.184542i
\(532\) 0 0
\(533\) 14.8141 + 14.8141i 0.641668 + 0.641668i
\(534\) 0 0
\(535\) 24.3559i 1.05300i
\(536\) 0 0
\(537\) 3.41987 0.147578
\(538\) 0 0
\(539\) 7.35625 9.13953i 0.316856 0.393668i
\(540\) 0 0
\(541\) −11.4795 + 11.4795i −0.493541 + 0.493541i −0.909420 0.415879i \(-0.863474\pi\)
0.415879 + 0.909420i \(0.363474\pi\)
\(542\) 0 0
\(543\) 0.450130i 0.0193169i
\(544\) 0 0
\(545\) 1.43503i 0.0614701i
\(546\) 0 0
\(547\) 27.5701 + 27.5701i 1.17881 + 1.17881i 0.980047 + 0.198764i \(0.0636928\pi\)
0.198764 + 0.980047i \(0.436307\pi\)
\(548\) 0 0
\(549\) −1.14204 + 1.14204i −0.0487409 + 0.0487409i
\(550\) 0 0
\(551\) 24.2019i 1.03103i
\(552\) 0 0
\(553\) 1.97085 + 36.5679i 0.0838090 + 1.55502i
\(554\) 0 0
\(555\) 1.47462 1.47462i 0.0625942 0.0625942i
\(556\) 0 0
\(557\) 7.84102 7.84102i 0.332235 0.332235i −0.521200 0.853435i \(-0.674515\pi\)
0.853435 + 0.521200i \(0.174515\pi\)
\(558\) 0 0
\(559\) −6.86492 −0.290355
\(560\) 0 0
\(561\) 0.463235 0.0195578
\(562\) 0 0
\(563\) 20.5300 20.5300i 0.865238 0.865238i −0.126703 0.991941i \(-0.540439\pi\)
0.991941 + 0.126703i \(0.0404395\pi\)
\(564\) 0 0
\(565\) −15.8293 + 15.8293i −0.665944 + 0.665944i
\(566\) 0 0
\(567\) 2.64192 0.142388i 0.110950 0.00597972i
\(568\) 0 0
\(569\) 8.26625i 0.346539i −0.984874 0.173270i \(-0.944567\pi\)
0.984874 0.173270i \(-0.0554332\pi\)
\(570\) 0 0
\(571\) −11.1033 + 11.1033i −0.464659 + 0.464659i −0.900179 0.435520i \(-0.856565\pi\)
0.435520 + 0.900179i \(0.356565\pi\)
\(572\) 0 0
\(573\) −17.2817 17.2817i −0.721955 0.721955i
\(574\) 0 0
\(575\) 2.70158i 0.112664i
\(576\) 0 0
\(577\) 1.00276i 0.0417454i 0.999782 + 0.0208727i \(0.00664447\pi\)
−0.999782 + 0.0208727i \(0.993356\pi\)
\(578\) 0 0
\(579\) −15.8328 + 15.8328i −0.657991 + 0.657991i
\(580\) 0 0
\(581\) −24.7689 + 27.5908i −1.02759 + 1.14466i
\(582\) 0 0
\(583\) 9.13176 0.378199
\(584\) 0 0
\(585\) 5.35708i 0.221488i
\(586\) 0 0
\(587\) −2.42198 2.42198i −0.0999658 0.0999658i 0.655355 0.755321i \(-0.272519\pi\)
−0.755321 + 0.655355i \(0.772519\pi\)
\(588\) 0 0
\(589\) 32.5880 32.5880i 1.34277 1.34277i
\(590\) 0 0
\(591\) −19.0249 −0.782582
\(592\) 0 0
\(593\) 22.2544i 0.913881i 0.889497 + 0.456940i \(0.151055\pi\)
−0.889497 + 0.456940i \(0.848945\pi\)
\(594\) 0 0
\(595\) 1.01999 + 0.915666i 0.0418155 + 0.0375387i
\(596\) 0 0
\(597\) −3.99117 3.99117i −0.163348 0.163348i
\(598\) 0 0
\(599\) 6.97729 0.285085 0.142542 0.989789i \(-0.454472\pi\)
0.142542 + 0.989789i \(0.454472\pi\)
\(600\) 0 0
\(601\) 9.75706 0.397999 0.198999 0.980000i \(-0.436231\pi\)
0.198999 + 0.980000i \(0.436231\pi\)
\(602\) 0 0
\(603\) 3.05161 3.05161i 0.124271 0.124271i
\(604\) 0 0
\(605\) 10.8566 + 10.8566i 0.441383 + 0.441383i
\(606\) 0 0
\(607\) −0.646351 −0.0262346 −0.0131173 0.999914i \(-0.504175\pi\)
−0.0131173 + 0.999914i \(0.504175\pi\)
\(608\) 0 0
\(609\) 0.531884 + 9.86878i 0.0215530 + 0.399903i
\(610\) 0 0
\(611\) 23.3856 + 23.3856i 0.946082 + 0.946082i
\(612\) 0 0
\(613\) −22.8864 22.8864i −0.924372 0.924372i 0.0729627 0.997335i \(-0.476755\pi\)
−0.997335 + 0.0729627i \(0.976755\pi\)
\(614\) 0 0
\(615\) 13.7408i 0.554084i
\(616\) 0 0
\(617\) 28.2900i 1.13891i 0.822022 + 0.569456i \(0.192846\pi\)
−0.822022 + 0.569456i \(0.807154\pi\)
\(618\) 0 0
\(619\) −27.0007 27.0007i −1.08525 1.08525i −0.996010 0.0892388i \(-0.971557\pi\)
−0.0892388 0.996010i \(-0.528443\pi\)
\(620\) 0 0
\(621\) 1.28519 + 1.28519i 0.0515728 + 0.0515728i
\(622\) 0 0
\(623\) 21.0030 1.13197i 0.841468 0.0453514i
\(624\) 0 0
\(625\) 15.3586 0.614345
\(626\) 0 0
\(627\) −7.67843 7.67843i −0.306647 0.306647i
\(628\) 0 0
\(629\) −0.217431 + 0.217431i −0.00866956 + 0.00866956i
\(630\) 0 0
\(631\) −31.2622 −1.24453 −0.622265 0.782807i \(-0.713787\pi\)
−0.622265 + 0.782807i \(0.713787\pi\)
\(632\) 0 0
\(633\) −19.6710 −0.781851
\(634\) 0 0
\(635\) −25.0002 25.0002i −0.992104 0.992104i
\(636\) 0 0
\(637\) −15.5845 12.5437i −0.617481 0.497000i
\(638\) 0 0
\(639\) 8.36612i 0.330959i
\(640\) 0 0
\(641\) 13.0504 0.515458 0.257729 0.966217i \(-0.417026\pi\)
0.257729 + 0.966217i \(0.417026\pi\)
\(642\) 0 0
\(643\) 0.786696 0.786696i 0.0310242 0.0310242i −0.691424 0.722449i \(-0.743017\pi\)
0.722449 + 0.691424i \(0.243017\pi\)
\(644\) 0 0
\(645\) 3.18379 + 3.18379i 0.125362 + 0.125362i
\(646\) 0 0
\(647\) 22.9721i 0.903125i 0.892239 + 0.451563i \(0.149133\pi\)
−0.892239 + 0.451563i \(0.850867\pi\)
\(648\) 0 0
\(649\) 10.0795 0.395656
\(650\) 0 0
\(651\) −12.5722 + 14.0046i −0.492744 + 0.548883i
\(652\) 0 0
\(653\) −11.7129 + 11.7129i −0.458362 + 0.458362i −0.898118 0.439756i \(-0.855065\pi\)
0.439756 + 0.898118i \(0.355065\pi\)
\(654\) 0 0
\(655\) 23.3539i 0.912511i
\(656\) 0 0
\(657\) 16.9603i 0.661685i
\(658\) 0 0
\(659\) −28.8441 28.8441i −1.12361 1.12361i −0.991195 0.132413i \(-0.957728\pi\)
−0.132413 0.991195i \(-0.542272\pi\)
\(660\) 0 0
\(661\) −20.5775 + 20.5775i −0.800371 + 0.800371i −0.983153 0.182782i \(-0.941490\pi\)
0.182782 + 0.983153i \(0.441490\pi\)
\(662\) 0 0
\(663\) 0.789896i 0.0306770i
\(664\) 0 0
\(665\) −1.72923 32.0848i −0.0670567 1.24420i
\(666\) 0 0
\(667\) −4.80077 + 4.80077i −0.185887 + 0.185887i
\(668\) 0 0
\(669\) −10.9526 + 10.9526i −0.423453 + 0.423453i
\(670\) 0 0
\(671\) −2.70694 −0.104500
\(672\) 0 0
\(673\) 19.3585 0.746215 0.373108 0.927788i \(-0.378292\pi\)
0.373108 + 0.927788i \(0.378292\pi\)
\(674\) 0 0
\(675\) −1.05104 + 1.05104i −0.0404547 + 0.0404547i
\(676\) 0 0
\(677\) −26.0458 + 26.0458i −1.00102 + 1.00102i −0.00102364 + 0.999999i \(0.500326\pi\)
−0.999999 + 0.00102364i \(0.999674\pi\)
\(678\) 0 0
\(679\) −1.87464 34.7828i −0.0719420 1.33484i
\(680\) 0 0
\(681\) 6.00245i 0.230014i
\(682\) 0 0
\(683\) 21.8826 21.8826i 0.837315 0.837315i −0.151189 0.988505i \(-0.548310\pi\)
0.988505 + 0.151189i \(0.0483103\pi\)
\(684\) 0 0
\(685\) 5.75746 + 5.75746i 0.219981 + 0.219981i
\(686\) 0 0
\(687\) 3.38133i 0.129006i
\(688\) 0 0
\(689\) 15.5713i 0.593218i
\(690\) 0 0
\(691\) −12.1294 + 12.1294i −0.461424 + 0.461424i −0.899122 0.437698i \(-0.855794\pi\)
0.437698 + 0.899122i \(0.355794\pi\)
\(692\) 0 0
\(693\) 3.29978 + 2.96228i 0.125348 + 0.112528i
\(694\) 0 0
\(695\) 4.37093 0.165799
\(696\) 0 0
\(697\) 2.02607i 0.0767429i
\(698\) 0 0
\(699\) −2.06518 2.06518i −0.0781125 0.0781125i
\(700\) 0 0
\(701\) −1.76393 + 1.76393i −0.0666225 + 0.0666225i −0.739633 0.673010i \(-0.765001\pi\)
0.673010 + 0.739633i \(0.265001\pi\)
\(702\) 0 0
\(703\) 7.20815 0.271861
\(704\) 0 0
\(705\) 21.6914i 0.816947i
\(706\) 0 0
\(707\) −3.01531 + 3.35885i −0.113402 + 0.126322i
\(708\) 0 0
\(709\) 24.0457 + 24.0457i 0.903055 + 0.903055i 0.995699 0.0926439i \(-0.0295318\pi\)
−0.0926439 + 0.995699i \(0.529532\pi\)
\(710\) 0 0
\(711\) −13.8414 −0.519094
\(712\) 0 0
\(713\) −12.9286 −0.484178
\(714\) 0 0
\(715\) −6.34887 + 6.34887i −0.237434 + 0.237434i
\(716\) 0 0
\(717\) −5.54272 5.54272i −0.206997 0.206997i
\(718\) 0 0
\(719\) −21.8470 −0.814754 −0.407377 0.913260i \(-0.633556\pi\)
−0.407377 + 0.913260i \(0.633556\pi\)
\(720\) 0 0
\(721\) 1.62635 + 30.1760i 0.0605685 + 1.12381i
\(722\) 0 0
\(723\) 12.9864 + 12.9864i 0.482971 + 0.482971i
\(724\) 0 0
\(725\) −3.92613 3.92613i −0.145813 0.145813i
\(726\) 0 0
\(727\) 42.6131i 1.58043i 0.612829 + 0.790216i \(0.290031\pi\)
−0.612829 + 0.790216i \(0.709969\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −0.469446 0.469446i −0.0173631 0.0173631i
\(732\) 0 0
\(733\) −27.0115 27.0115i −0.997693 0.997693i 0.00230393 0.999997i \(-0.499267\pi\)
−0.999997 + 0.00230393i \(0.999267\pi\)
\(734\) 0 0
\(735\) 1.41026 + 13.0452i 0.0520180 + 0.481180i
\(736\) 0 0
\(737\) 7.23314 0.266436
\(738\) 0 0
\(739\) 1.43265 + 1.43265i 0.0527008 + 0.0527008i 0.732966 0.680265i \(-0.238135\pi\)
−0.680265 + 0.732966i \(0.738135\pi\)
\(740\) 0 0
\(741\) −13.0931 + 13.0931i −0.480986 + 0.480986i
\(742\) 0 0
\(743\) 21.5309 0.789892 0.394946 0.918704i \(-0.370763\pi\)
0.394946 + 0.918704i \(0.370763\pi\)
\(744\) 0 0
\(745\) −26.9206 −0.986293
\(746\) 0 0
\(747\) −9.90941 9.90941i −0.362566 0.362566i
\(748\) 0 0
\(749\) 22.9652 25.5817i 0.839132 0.934735i
\(750\) 0 0
\(751\) 4.47866i 0.163429i 0.996656 + 0.0817143i \(0.0260395\pi\)
−0.996656 + 0.0817143i \(0.973960\pi\)
\(752\) 0 0
\(753\) −23.0197 −0.838883
\(754\) 0 0
\(755\) 11.8311 11.8311i 0.430576 0.430576i
\(756\) 0 0
\(757\) −13.9860 13.9860i −0.508329 0.508329i 0.405684 0.914013i \(-0.367033\pi\)
−0.914013 + 0.405684i \(0.867033\pi\)
\(758\) 0 0
\(759\) 3.04624i 0.110572i
\(760\) 0 0
\(761\) −26.8043 −0.971653 −0.485827 0.874055i \(-0.661481\pi\)
−0.485827 + 0.874055i \(0.661481\pi\)
\(762\) 0 0
\(763\) −1.35310 + 1.50726i −0.0489854 + 0.0545664i
\(764\) 0 0
\(765\) −0.366336 + 0.366336i −0.0132449 + 0.0132449i
\(766\) 0 0
\(767\) 17.1873i 0.620599i
\(768\) 0 0
\(769\) 31.2390i 1.12651i −0.826285 0.563253i \(-0.809550\pi\)
0.826285 0.563253i \(-0.190450\pi\)
\(770\) 0 0
\(771\) −9.63852 9.63852i −0.347123 0.347123i
\(772\) 0 0
\(773\) −21.9282 + 21.9282i −0.788703 + 0.788703i −0.981282 0.192578i \(-0.938315\pi\)
0.192578 + 0.981282i \(0.438315\pi\)
\(774\) 0 0
\(775\) 10.5731i 0.379799i
\(776\) 0 0
\(777\) −2.93926 + 0.158413i −0.105446 + 0.00568305i
\(778\) 0 0
\(779\) 33.5835 33.5835i 1.20325 1.20325i
\(780\) 0 0
\(781\) −9.91499 + 9.91499i −0.354786 + 0.354786i
\(782\) 0 0
\(783\) −3.73546 −0.133495
\(784\) 0 0
\(785\) 15.4626 0.551885
\(786\) 0 0
\(787\) 33.9965 33.9965i 1.21184 1.21184i 0.241424 0.970420i \(-0.422385\pi\)
0.970420 0.241424i \(-0.0776145\pi\)
\(788\) 0 0
\(789\) 4.15967 4.15967i 0.148088 0.148088i
\(790\) 0 0
\(791\) 31.5515 1.70049i 1.12184 0.0604624i
\(792\) 0 0
\(793\) 4.61580i 0.163912i
\(794\) 0 0
\(795\) −7.22159 + 7.22159i −0.256123 + 0.256123i
\(796\) 0 0
\(797\) −4.75244 4.75244i −0.168340 0.168340i 0.617909 0.786249i \(-0.287980\pi\)
−0.786249 + 0.617909i \(0.787980\pi\)
\(798\) 0 0
\(799\) 3.19838i 0.113151i
\(800\) 0 0
\(801\) 7.94992i 0.280897i
\(802\) 0 0
\(803\) −20.1003 + 20.1003i −0.709323 + 0.709323i
\(804\) 0 0
\(805\) −6.02145 + 6.70748i −0.212228 + 0.236408i
\(806\) 0 0
\(807\) −10.2984 −0.362521
\(808\) 0 0
\(809\) 32.4123i 1.13956i −0.821798 0.569778i \(-0.807029\pi\)
0.821798 0.569778i \(-0.192971\pi\)
\(810\) 0 0
\(811\) −0.225151 0.225151i −0.00790612 0.00790612i 0.703143 0.711049i \(-0.251780\pi\)
−0.711049 + 0.703143i \(0.751780\pi\)
\(812\) 0 0
\(813\) −5.41212 + 5.41212i −0.189811 + 0.189811i
\(814\) 0 0
\(815\) −33.0702 −1.15840
\(816\) 0 0
\(817\) 15.5628i 0.544473i
\(818\) 0 0
\(819\) 5.05121 5.62670i 0.176504 0.196613i
\(820\) 0 0
\(821\) −37.0047 37.0047i −1.29147 1.29147i −0.933875 0.357599i \(-0.883595\pi\)
−0.357599 0.933875i \(-0.616405\pi\)
\(822\) 0 0
\(823\) 1.21299 0.0422821 0.0211411 0.999777i \(-0.493270\pi\)
0.0211411 + 0.999777i \(0.493270\pi\)
\(824\) 0 0
\(825\) −2.49126 −0.0867345
\(826\) 0 0
\(827\) −22.1226 + 22.1226i −0.769278 + 0.769278i −0.977979 0.208701i \(-0.933076\pi\)
0.208701 + 0.977979i \(0.433076\pi\)
\(828\) 0 0
\(829\) 27.0435 + 27.0435i 0.939258 + 0.939258i 0.998258 0.0589998i \(-0.0187911\pi\)
−0.0589998 + 0.998258i \(0.518791\pi\)
\(830\) 0 0
\(831\) 29.6067 1.02704
\(832\) 0 0
\(833\) −0.207941 1.92350i −0.00720472 0.0666454i
\(834\) 0 0
\(835\) −28.2820 28.2820i −0.978740 0.978740i
\(836\) 0 0
\(837\) −5.02983 5.02983i −0.173856 0.173856i
\(838\) 0 0
\(839\) 18.8066i 0.649277i 0.945838 + 0.324639i \(0.105243\pi\)
−0.945838 + 0.324639i \(0.894757\pi\)
\(840\) 0 0
\(841\) 15.0463i 0.518839i
\(842\) 0 0
\(843\) −9.39768 9.39768i −0.323673 0.323673i
\(844\) 0 0
\(845\) −6.40483 6.40483i −0.220333 0.220333i
\(846\) 0 0
\(847\) −1.16628 21.6397i −0.0400740 0.743549i
\(848\) 0 0
\(849\) −11.2747 −0.386947
\(850\) 0 0
\(851\) −1.42984 1.42984i −0.0490141 0.0490141i
\(852\) 0 0
\(853\) 13.4226 13.4226i 0.459581 0.459581i −0.438937 0.898518i \(-0.644645\pi\)
0.898518 + 0.438937i \(0.144645\pi\)
\(854\) 0 0
\(855\) 12.1445 0.415334
\(856\) 0 0
\(857\) −14.2069 −0.485297 −0.242649 0.970114i \(-0.578016\pi\)
−0.242649 + 0.970114i \(0.578016\pi\)
\(858\) 0 0
\(859\) 12.7166 + 12.7166i 0.433886 + 0.433886i 0.889948 0.456062i \(-0.150741\pi\)
−0.456062 + 0.889948i \(0.650741\pi\)
\(860\) 0 0
\(861\) −12.9563 + 14.4324i −0.441549 + 0.491855i
\(862\) 0 0
\(863\) 0.417666i 0.0142175i −0.999975 0.00710875i \(-0.997737\pi\)
0.999975 0.00710875i \(-0.00226281\pi\)
\(864\) 0 0
\(865\) −28.4644 −0.967818
\(866\) 0 0
\(867\) −11.9668 + 11.9668i −0.406414 + 0.406414i
\(868\) 0 0
\(869\) −16.4040 16.4040i −0.556466 0.556466i
\(870\) 0 0
\(871\) 12.3338i 0.417914i
\(872\) 0 0
\(873\) 13.1657 0.445592
\(874\) 0 0
\(875\) −23.9377 21.4893i −0.809241 0.726472i
\(876\) 0 0
\(877\) −37.3450 + 37.3450i −1.26105 + 1.26105i −0.310468 + 0.950584i \(0.600486\pi\)
−0.950584 + 0.310468i \(0.899514\pi\)
\(878\) 0 0
\(879\) 16.6272i 0.560821i
\(880\) 0 0
\(881\) 40.8076i 1.37484i −0.726259 0.687421i \(-0.758743\pi\)
0.726259 0.687421i \(-0.241257\pi\)
\(882\) 0 0
\(883\) −38.1593 38.1593i −1.28416 1.28416i −0.938277 0.345886i \(-0.887579\pi\)
−0.345886 0.938277i \(-0.612421\pi\)
\(884\) 0 0
\(885\) −7.97109 + 7.97109i −0.267945 + 0.267945i
\(886\) 0 0
\(887\) 34.0533i 1.14340i 0.820464 + 0.571698i \(0.193715\pi\)
−0.820464 + 0.571698i \(0.806285\pi\)
\(888\) 0 0
\(889\) 2.68569 + 49.8313i 0.0900750 + 1.67129i
\(890\) 0 0
\(891\) −1.18514 + 1.18514i −0.0397035 + 0.0397035i
\(892\) 0 0
\(893\) 53.0153 53.0153i 1.77409 1.77409i
\(894\) 0 0
\(895\) −6.41042 −0.214277
\(896\) 0 0
\(897\) 5.19438 0.173435
\(898\) 0 0
\(899\) 18.7888 18.7888i 0.626640 0.626640i
\(900\) 0 0
\(901\) 1.06482 1.06482i 0.0354741 0.0354741i
\(902\) 0 0
\(903\) −0.342023 6.34603i −0.0113818 0.211183i
\(904\) 0 0
\(905\) 0.843750i 0.0280472i
\(906\) 0 0
\(907\) 1.29406 1.29406i 0.0429685 0.0429685i −0.685296 0.728265i \(-0.740327\pi\)
0.728265 + 0.685296i \(0.240327\pi\)
\(908\) 0 0
\(909\) −1.20635 1.20635i −0.0400121 0.0400121i
\(910\) 0 0
\(911\) 21.1193i 0.699715i 0.936803 + 0.349857i \(0.113770\pi\)
−0.936803 + 0.349857i \(0.886230\pi\)
\(912\) 0 0
\(913\) 23.4880i 0.777339i
\(914\) 0 0
\(915\) 2.14070 2.14070i 0.0707694 0.0707694i
\(916\) 0 0
\(917\) 22.0204 24.5292i 0.727178 0.810027i
\(918\) 0 0
\(919\) −0.462843 −0.0152678 −0.00763389 0.999971i \(-0.502430\pi\)
−0.00763389 + 0.999971i \(0.502430\pi\)
\(920\) 0 0
\(921\) 28.4171i 0.936374i
\(922\) 0 0
\(923\) 16.9068 + 16.9068i 0.556494 + 0.556494i
\(924\) 0 0
\(925\) 1.16934 1.16934i 0.0384476 0.0384476i
\(926\) 0 0
\(927\) −11.4220 −0.375148
\(928\) 0 0
\(929\) 7.13095i 0.233959i −0.993134 0.116980i \(-0.962679\pi\)
0.993134 0.116980i \(-0.0373212\pi\)
\(930\) 0 0
\(931\) −28.4366 + 35.3301i −0.931973 + 1.15790i
\(932\) 0 0
\(933\) −18.4309 18.4309i −0.603400 0.603400i
\(934\) 0 0
\(935\) −0.868315 −0.0283969
\(936\) 0 0
\(937\) 5.02900 0.164290 0.0821452 0.996620i \(-0.473823\pi\)
0.0821452 + 0.996620i \(0.473823\pi\)
\(938\) 0 0
\(939\) 9.59195 9.59195i 0.313022 0.313022i
\(940\) 0 0
\(941\) 38.2671 + 38.2671i 1.24747 + 1.24747i 0.956833 + 0.290639i \(0.0938679\pi\)
0.290639 + 0.956833i \(0.406132\pi\)
\(942\) 0 0
\(943\) −13.3235 −0.433873
\(944\) 0 0
\(945\) −4.95217 + 0.266900i −0.161094 + 0.00868226i
\(946\) 0 0
\(947\) 4.42949 + 4.42949i 0.143939 + 0.143939i 0.775404 0.631465i \(-0.217546\pi\)
−0.631465 + 0.775404i \(0.717546\pi\)
\(948\) 0 0
\(949\) 34.2745 + 34.2745i 1.11260 + 1.11260i
\(950\) 0 0
\(951\) 30.4486i 0.987363i
\(952\) 0 0
\(953\) 40.8007i 1.32166i −0.750534 0.660832i \(-0.770204\pi\)
0.750534 0.660832i \(-0.229796\pi\)
\(954\) 0 0
\(955\) 32.3939 + 32.3939i 1.04824 + 1.04824i
\(956\) 0 0
\(957\) −4.42703 4.42703i −0.143106 0.143106i
\(958\) 0 0
\(959\) −0.618504 11.4760i −0.0199725 0.370578i
\(960\) 0 0
\(961\) 19.5985 0.632208
\(962\) 0 0
\(963\) 9.18783 + 9.18783i 0.296074 + 0.296074i
\(964\) 0 0
\(965\) 29.6780 29.6780i 0.955370 0.955370i
\(966\) 0 0
\(967\) 2.09367 0.0673278 0.0336639 0.999433i \(-0.489282\pi\)
0.0336639 + 0.999433i \(0.489282\pi\)
\(968\) 0 0
\(969\) −1.79070 −0.0575255
\(970\) 0 0
\(971\) 12.1592 + 12.1592i 0.390208 + 0.390208i 0.874762 0.484554i \(-0.161018\pi\)
−0.484554 + 0.874762i \(0.661018\pi\)
\(972\) 0 0
\(973\) −4.59092 4.12136i −0.147178 0.132125i
\(974\) 0 0
\(975\) 4.24803i 0.136046i
\(976\) 0 0
\(977\) 45.8415 1.46660 0.733300 0.679906i \(-0.237979\pi\)
0.733300 + 0.679906i \(0.237979\pi\)
\(978\) 0 0
\(979\) −9.42173 + 9.42173i −0.301120 + 0.301120i
\(980\) 0 0
\(981\) −0.541341 0.541341i −0.0172837 0.0172837i
\(982\) 0 0
\(983\) 7.70439i 0.245732i −0.992423 0.122866i \(-0.960791\pi\)
0.992423 0.122866i \(-0.0392085\pi\)
\(984\) 0 0
\(985\) 35.6615 1.13627
\(986\) 0 0
\(987\) −20.4529 + 22.7832i −0.651024 + 0.725196i
\(988\) 0 0
\(989\) 3.08709 3.08709i 0.0981639 0.0981639i
\(990\) 0 0
\(991\) 6.61965i 0.210280i 0.994457 + 0.105140i \(0.0335291\pi\)
−0.994457 + 0.105140i \(0.966471\pi\)
\(992\) 0 0
\(993\) 8.27375i 0.262559i
\(994\) 0 0
\(995\) 7.48130 + 7.48130i 0.237173 + 0.237173i
\(996\) 0 0
\(997\) 20.8831 20.8831i 0.661376 0.661376i −0.294329 0.955704i \(-0.595096\pi\)
0.955704 + 0.294329i \(0.0950960\pi\)
\(998\) 0 0
\(999\) 1.11255i 0.0351995i
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.u.a.1231.12 64
4.3 odd 2 336.2.u.a.139.6 yes 64
7.6 odd 2 inner 1344.2.u.a.1231.21 64
16.3 odd 4 inner 1344.2.u.a.559.21 64
16.13 even 4 336.2.u.a.307.5 yes 64
28.27 even 2 336.2.u.a.139.5 64
112.13 odd 4 336.2.u.a.307.6 yes 64
112.83 even 4 inner 1344.2.u.a.559.12 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.u.a.139.5 64 28.27 even 2
336.2.u.a.139.6 yes 64 4.3 odd 2
336.2.u.a.307.5 yes 64 16.13 even 4
336.2.u.a.307.6 yes 64 112.13 odd 4
1344.2.u.a.559.12 64 112.83 even 4 inner
1344.2.u.a.559.21 64 16.3 odd 4 inner
1344.2.u.a.1231.12 64 1.1 even 1 trivial
1344.2.u.a.1231.21 64 7.6 odd 2 inner