Properties

Label 896.2.u.c.113.4
Level $896$
Weight $2$
Character 896.113
Analytic conductor $7.155$
Analytic rank $0$
Dimension $52$
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] = [896,2,Mod(113,896)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(896, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("896.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 896 = 2^{7} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 896.u (of order \(8\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.15459602111\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(13\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 113.4
Character \(\chi\) \(=\) 896.113
Dual form 896.2.u.c.785.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.694865 - 1.67755i) q^{3} +(-0.286541 - 0.118689i) q^{5} +(0.707107 + 0.707107i) q^{7} +(-0.210023 + 0.210023i) q^{9} +O(q^{10})\) \(q+(-0.694865 - 1.67755i) q^{3} +(-0.286541 - 0.118689i) q^{5} +(0.707107 + 0.707107i) q^{7} +(-0.210023 + 0.210023i) q^{9} +(1.06259 - 2.56533i) q^{11} +(-0.401442 + 0.166283i) q^{13} +0.563161i q^{15} +1.58480i q^{17} +(7.27546 - 3.01360i) q^{19} +(0.694865 - 1.67755i) q^{21} +(-1.26201 + 1.26201i) q^{23} +(-3.46752 - 3.46752i) q^{25} +(-4.53439 - 1.87821i) q^{27} +(-3.29808 - 7.96227i) q^{29} +0.0589875 q^{31} -5.04183 q^{33} +(-0.118689 - 0.286541i) q^{35} +(1.50897 + 0.625037i) q^{37} +(0.557896 + 0.557896i) q^{39} +(2.22267 - 2.22267i) q^{41} +(1.16639 - 2.81591i) q^{43} +(0.0851079 - 0.0352528i) q^{45} +5.51458i q^{47} +1.00000i q^{49} +(2.65858 - 1.10122i) q^{51} +(-1.25750 + 3.03587i) q^{53} +(-0.608955 + 0.608955i) q^{55} +(-10.1109 - 10.1109i) q^{57} +(-10.1068 - 4.18638i) q^{59} +(0.240133 + 0.579733i) q^{61} -0.297018 q^{63} +0.134766 q^{65} +(-6.03571 - 14.5715i) q^{67} +(2.99401 + 1.24016i) q^{69} +(-5.06754 - 5.06754i) q^{71} +(-6.46348 + 6.46348i) q^{73} +(-3.40748 + 8.22639i) q^{75} +(2.56533 - 1.06259i) q^{77} -7.34488i q^{79} +9.80283i q^{81} +(7.41929 - 3.07317i) q^{83} +(0.188098 - 0.454110i) q^{85} +(-11.0654 + 11.0654i) q^{87} +(-5.60338 - 5.60338i) q^{89} +(-0.401442 - 0.166283i) q^{91} +(-0.0409884 - 0.0989547i) q^{93} -2.44240 q^{95} +10.6589 q^{97} +(0.315609 + 0.761949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52 q+O(q^{10}) \) Copy content Toggle raw display \( 52 q + 20 q^{23} + 24 q^{27} - 48 q^{33} + 24 q^{39} + 44 q^{43} + 40 q^{45} - 16 q^{51} - 36 q^{53} - 32 q^{55} - 32 q^{61} - 68 q^{63} + 80 q^{65} - 28 q^{67} - 32 q^{69} - 32 q^{75} - 12 q^{77} + 64 q^{85} + 56 q^{87} + 64 q^{95} - 72 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(645\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{8}\right)\)

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
\(3\) −0.694865 1.67755i −0.401180 0.968535i −0.987380 0.158369i \(-0.949377\pi\)
0.586200 0.810167i \(-0.300623\pi\)
\(4\) 0 0
\(5\) −0.286541 0.118689i −0.128145 0.0530795i 0.317689 0.948195i \(-0.397093\pi\)
−0.445835 + 0.895115i \(0.647093\pi\)
\(6\) 0 0
\(7\) 0.707107 + 0.707107i 0.267261 + 0.267261i
\(8\) 0 0
\(9\) −0.210023 + 0.210023i −0.0700078 + 0.0700078i
\(10\) 0 0
\(11\) 1.06259 2.56533i 0.320384 0.773476i −0.678847 0.734280i \(-0.737520\pi\)
0.999232 0.0391965i \(-0.0124798\pi\)
\(12\) 0 0
\(13\) −0.401442 + 0.166283i −0.111340 + 0.0461185i −0.437658 0.899142i \(-0.644192\pi\)
0.326318 + 0.945260i \(0.394192\pi\)
\(14\) 0 0
\(15\) 0.563161i 0.145408i
\(16\) 0 0
\(17\) 1.58480i 0.384370i 0.981359 + 0.192185i \(0.0615573\pi\)
−0.981359 + 0.192185i \(0.938443\pi\)
\(18\) 0 0
\(19\) 7.27546 3.01360i 1.66911 0.691366i 0.670389 0.742010i \(-0.266127\pi\)
0.998717 + 0.0506436i \(0.0161272\pi\)
\(20\) 0 0
\(21\) 0.694865 1.67755i 0.151632 0.366072i
\(22\) 0 0
\(23\) −1.26201 + 1.26201i −0.263147 + 0.263147i −0.826331 0.563184i \(-0.809576\pi\)
0.563184 + 0.826331i \(0.309576\pi\)
\(24\) 0 0
\(25\) −3.46752 3.46752i −0.693503 0.693503i
\(26\) 0 0
\(27\) −4.53439 1.87821i −0.872644 0.361461i
\(28\) 0 0
\(29\) −3.29808 7.96227i −0.612438 1.47856i −0.860314 0.509765i \(-0.829732\pi\)
0.247876 0.968792i \(-0.420268\pi\)
\(30\) 0 0
\(31\) 0.0589875 0.0105945 0.00529724 0.999986i \(-0.498314\pi\)
0.00529724 + 0.999986i \(0.498314\pi\)
\(32\) 0 0
\(33\) −5.04183 −0.877671
\(34\) 0 0
\(35\) −0.118689 0.286541i −0.0200622 0.0484343i
\(36\) 0 0
\(37\) 1.50897 + 0.625037i 0.248074 + 0.102755i 0.503256 0.864138i \(-0.332135\pi\)
−0.255182 + 0.966893i \(0.582135\pi\)
\(38\) 0 0
\(39\) 0.557896 + 0.557896i 0.0893348 + 0.0893348i
\(40\) 0 0
\(41\) 2.22267 2.22267i 0.347122 0.347122i −0.511914 0.859037i \(-0.671063\pi\)
0.859037 + 0.511914i \(0.171063\pi\)
\(42\) 0 0
\(43\) 1.16639 2.81591i 0.177873 0.429423i −0.809647 0.586917i \(-0.800342\pi\)
0.987520 + 0.157494i \(0.0503415\pi\)
\(44\) 0 0
\(45\) 0.0851079 0.0352528i 0.0126871 0.00525518i
\(46\) 0 0
\(47\) 5.51458i 0.804385i 0.915555 + 0.402192i \(0.131752\pi\)
−0.915555 + 0.402192i \(0.868248\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) 2.65858 1.10122i 0.372275 0.154202i
\(52\) 0 0
\(53\) −1.25750 + 3.03587i −0.172731 + 0.417009i −0.986409 0.164306i \(-0.947462\pi\)
0.813679 + 0.581315i \(0.197462\pi\)
\(54\) 0 0
\(55\) −0.608955 + 0.608955i −0.0821114 + 0.0821114i
\(56\) 0 0
\(57\) −10.1109 10.1109i −1.33922 1.33922i
\(58\) 0 0
\(59\) −10.1068 4.18638i −1.31579 0.545020i −0.389224 0.921143i \(-0.627257\pi\)
−0.926570 + 0.376123i \(0.877257\pi\)
\(60\) 0 0
\(61\) 0.240133 + 0.579733i 0.0307459 + 0.0742272i 0.938507 0.345261i \(-0.112210\pi\)
−0.907761 + 0.419488i \(0.862210\pi\)
\(62\) 0 0
\(63\) −0.297018 −0.0374207
\(64\) 0 0
\(65\) 0.134766 0.0167156
\(66\) 0 0
\(67\) −6.03571 14.5715i −0.737379 1.78019i −0.616246 0.787554i \(-0.711347\pi\)
−0.121133 0.992636i \(-0.538653\pi\)
\(68\) 0 0
\(69\) 2.99401 + 1.24016i 0.360436 + 0.149298i
\(70\) 0 0
\(71\) −5.06754 5.06754i −0.601406 0.601406i 0.339280 0.940686i \(-0.389817\pi\)
−0.940686 + 0.339280i \(0.889817\pi\)
\(72\) 0 0
\(73\) −6.46348 + 6.46348i −0.756493 + 0.756493i −0.975682 0.219190i \(-0.929659\pi\)
0.219190 + 0.975682i \(0.429659\pi\)
\(74\) 0 0
\(75\) −3.40748 + 8.22639i −0.393462 + 0.949902i
\(76\) 0 0
\(77\) 2.56533 1.06259i 0.292346 0.121094i
\(78\) 0 0
\(79\) 7.34488i 0.826363i −0.910649 0.413182i \(-0.864417\pi\)
0.910649 0.413182i \(-0.135583\pi\)
\(80\) 0 0
\(81\) 9.80283i 1.08920i
\(82\) 0 0
\(83\) 7.41929 3.07317i 0.814373 0.337324i 0.0636756 0.997971i \(-0.479718\pi\)
0.750697 + 0.660646i \(0.229718\pi\)
\(84\) 0 0
\(85\) 0.188098 0.454110i 0.0204021 0.0492551i
\(86\) 0 0
\(87\) −11.0654 + 11.0654i −1.18634 + 1.18634i
\(88\) 0 0
\(89\) −5.60338 5.60338i −0.593958 0.593958i 0.344741 0.938698i \(-0.387967\pi\)
−0.938698 + 0.344741i \(0.887967\pi\)
\(90\) 0 0
\(91\) −0.401442 0.166283i −0.0420825 0.0174312i
\(92\) 0 0
\(93\) −0.0409884 0.0989547i −0.00425030 0.0102611i
\(94\) 0 0
\(95\) −2.44240 −0.250585
\(96\) 0 0
\(97\) 10.6589 1.08225 0.541125 0.840942i \(-0.317999\pi\)
0.541125 + 0.840942i \(0.317999\pi\)
\(98\) 0 0
\(99\) 0.315609 + 0.761949i 0.0317199 + 0.0765787i
\(100\) 0 0
\(101\) 11.6268 + 4.81597i 1.15691 + 0.479207i 0.876844 0.480775i \(-0.159645\pi\)
0.280063 + 0.959982i \(0.409645\pi\)
\(102\) 0 0
\(103\) 10.2382 + 10.2382i 1.00880 + 1.00880i 0.999961 + 0.00883982i \(0.00281384\pi\)
0.00883982 + 0.999961i \(0.497186\pi\)
\(104\) 0 0
\(105\) −0.398215 + 0.398215i −0.0388618 + 0.0388618i
\(106\) 0 0
\(107\) −4.42716 + 10.6881i −0.427989 + 1.03326i 0.551935 + 0.833887i \(0.313890\pi\)
−0.979924 + 0.199370i \(0.936110\pi\)
\(108\) 0 0
\(109\) 13.9220 5.76668i 1.33349 0.552348i 0.401838 0.915711i \(-0.368371\pi\)
0.931648 + 0.363363i \(0.118371\pi\)
\(110\) 0 0
\(111\) 2.96570i 0.281491i
\(112\) 0 0
\(113\) 8.04001i 0.756340i −0.925736 0.378170i \(-0.876553\pi\)
0.925736 0.378170i \(-0.123447\pi\)
\(114\) 0 0
\(115\) 0.511404 0.211831i 0.0476887 0.0197533i
\(116\) 0 0
\(117\) 0.0493889 0.119235i 0.00456601 0.0110233i
\(118\) 0 0
\(119\) −1.12062 + 1.12062i −0.102727 + 0.102727i
\(120\) 0 0
\(121\) 2.32636 + 2.32636i 0.211488 + 0.211488i
\(122\) 0 0
\(123\) −5.27309 2.18419i −0.475459 0.196941i
\(124\) 0 0
\(125\) 1.17548 + 2.83785i 0.105138 + 0.253825i
\(126\) 0 0
\(127\) 0.950863 0.0843754 0.0421877 0.999110i \(-0.486567\pi\)
0.0421877 + 0.999110i \(0.486567\pi\)
\(128\) 0 0
\(129\) −5.53433 −0.487270
\(130\) 0 0
\(131\) 1.54070 + 3.71958i 0.134611 + 0.324981i 0.976784 0.214228i \(-0.0687235\pi\)
−0.842172 + 0.539209i \(0.818723\pi\)
\(132\) 0 0
\(133\) 7.27546 + 3.01360i 0.630863 + 0.261312i
\(134\) 0 0
\(135\) 1.07637 + 1.07637i 0.0926390 + 0.0926390i
\(136\) 0 0
\(137\) 8.92167 8.92167i 0.762230 0.762230i −0.214495 0.976725i \(-0.568811\pi\)
0.976725 + 0.214495i \(0.0688106\pi\)
\(138\) 0 0
\(139\) 1.64625 3.97441i 0.139633 0.337105i −0.838557 0.544813i \(-0.816600\pi\)
0.978191 + 0.207708i \(0.0666005\pi\)
\(140\) 0 0
\(141\) 9.25100 3.83189i 0.779075 0.322703i
\(142\) 0 0
\(143\) 1.20652i 0.100894i
\(144\) 0 0
\(145\) 2.67297i 0.221978i
\(146\) 0 0
\(147\) 1.67755 0.694865i 0.138362 0.0573115i
\(148\) 0 0
\(149\) −4.15685 + 10.0355i −0.340543 + 0.822142i 0.657119 + 0.753787i \(0.271775\pi\)
−0.997661 + 0.0683551i \(0.978225\pi\)
\(150\) 0 0
\(151\) 6.51727 6.51727i 0.530368 0.530368i −0.390314 0.920682i \(-0.627634\pi\)
0.920682 + 0.390314i \(0.127634\pi\)
\(152\) 0 0
\(153\) −0.332844 0.332844i −0.0269089 0.0269089i
\(154\) 0 0
\(155\) −0.0169024 0.00700119i −0.00135763 0.000562349i
\(156\) 0 0
\(157\) 1.42079 + 3.43009i 0.113391 + 0.273751i 0.970380 0.241585i \(-0.0776672\pi\)
−0.856988 + 0.515336i \(0.827667\pi\)
\(158\) 0 0
\(159\) 5.96663 0.473184
\(160\) 0 0
\(161\) −1.78475 −0.140658
\(162\) 0 0
\(163\) 6.70768 + 16.1938i 0.525386 + 1.26839i 0.934517 + 0.355919i \(0.115832\pi\)
−0.409131 + 0.912476i \(0.634168\pi\)
\(164\) 0 0
\(165\) 1.44469 + 0.598412i 0.112469 + 0.0465863i
\(166\) 0 0
\(167\) 8.90516 + 8.90516i 0.689102 + 0.689102i 0.962033 0.272932i \(-0.0879933\pi\)
−0.272932 + 0.962033i \(0.587993\pi\)
\(168\) 0 0
\(169\) −9.05888 + 9.05888i −0.696837 + 0.696837i
\(170\) 0 0
\(171\) −0.895092 + 2.16094i −0.0684494 + 0.165251i
\(172\) 0 0
\(173\) 20.4231 8.45954i 1.55274 0.643167i 0.568933 0.822384i \(-0.307356\pi\)
0.983810 + 0.179217i \(0.0573563\pi\)
\(174\) 0 0
\(175\) 4.90381i 0.370693i
\(176\) 0 0
\(177\) 19.8637i 1.49304i
\(178\) 0 0
\(179\) −6.69861 + 2.77466i −0.500678 + 0.207388i −0.618706 0.785623i \(-0.712343\pi\)
0.118028 + 0.993010i \(0.462343\pi\)
\(180\) 0 0
\(181\) −6.85410 + 16.5473i −0.509461 + 1.22995i 0.434733 + 0.900559i \(0.356843\pi\)
−0.944194 + 0.329389i \(0.893157\pi\)
\(182\) 0 0
\(183\) 0.805672 0.805672i 0.0595570 0.0595570i
\(184\) 0 0
\(185\) −0.358198 0.358198i −0.0263352 0.0263352i
\(186\) 0 0
\(187\) 4.06553 + 1.68400i 0.297301 + 0.123146i
\(188\) 0 0
\(189\) −1.87821 4.53439i −0.136619 0.329829i
\(190\) 0 0
\(191\) −23.2866 −1.68496 −0.842478 0.538730i \(-0.818904\pi\)
−0.842478 + 0.538730i \(0.818904\pi\)
\(192\) 0 0
\(193\) 10.5900 0.762288 0.381144 0.924516i \(-0.375530\pi\)
0.381144 + 0.924516i \(0.375530\pi\)
\(194\) 0 0
\(195\) −0.0936439 0.226076i −0.00670598 0.0161897i
\(196\) 0 0
\(197\) −13.6629 5.65934i −0.973438 0.403211i −0.161447 0.986881i \(-0.551616\pi\)
−0.811991 + 0.583670i \(0.801616\pi\)
\(198\) 0 0
\(199\) 18.9071 + 18.9071i 1.34029 + 1.34029i 0.895769 + 0.444519i \(0.146626\pi\)
0.444519 + 0.895769i \(0.353374\pi\)
\(200\) 0 0
\(201\) −20.2504 + 20.2504i −1.42835 + 1.42835i
\(202\) 0 0
\(203\) 3.29808 7.96227i 0.231480 0.558842i
\(204\) 0 0
\(205\) −0.900693 + 0.373079i −0.0629071 + 0.0260570i
\(206\) 0 0
\(207\) 0.530102i 0.0368446i
\(208\) 0 0
\(209\) 21.8662i 1.51252i
\(210\) 0 0
\(211\) 22.4979 9.31893i 1.54882 0.641542i 0.565716 0.824600i \(-0.308600\pi\)
0.983102 + 0.183058i \(0.0585997\pi\)
\(212\) 0 0
\(213\) −4.97980 + 12.0223i −0.341211 + 0.823755i
\(214\) 0 0
\(215\) −0.668438 + 0.668438i −0.0455871 + 0.0455871i
\(216\) 0 0
\(217\) 0.0417105 + 0.0417105i 0.00283149 + 0.00283149i
\(218\) 0 0
\(219\) 15.3341 + 6.35158i 1.03618 + 0.429200i
\(220\) 0 0
\(221\) −0.263524 0.636204i −0.0177266 0.0427957i
\(222\) 0 0
\(223\) 19.1991 1.28567 0.642834 0.766005i \(-0.277758\pi\)
0.642834 + 0.766005i \(0.277758\pi\)
\(224\) 0 0
\(225\) 1.45652 0.0971012
\(226\) 0 0
\(227\) 5.57737 + 13.4650i 0.370183 + 0.893701i 0.993719 + 0.111906i \(0.0356956\pi\)
−0.623536 + 0.781795i \(0.714304\pi\)
\(228\) 0 0
\(229\) −18.5925 7.70125i −1.22862 0.508913i −0.328482 0.944510i \(-0.606537\pi\)
−0.900142 + 0.435597i \(0.856537\pi\)
\(230\) 0 0
\(231\) −3.56511 3.56511i −0.234567 0.234567i
\(232\) 0 0
\(233\) 9.44778 9.44778i 0.618945 0.618945i −0.326316 0.945261i \(-0.605807\pi\)
0.945261 + 0.326316i \(0.105807\pi\)
\(234\) 0 0
\(235\) 0.654522 1.58016i 0.0426963 0.103078i
\(236\) 0 0
\(237\) −12.3214 + 5.10370i −0.800362 + 0.331521i
\(238\) 0 0
\(239\) 29.8950i 1.93375i −0.255254 0.966874i \(-0.582159\pi\)
0.255254 0.966874i \(-0.417841\pi\)
\(240\) 0 0
\(241\) 10.5234i 0.677874i 0.940809 + 0.338937i \(0.110067\pi\)
−0.940809 + 0.338937i \(0.889933\pi\)
\(242\) 0 0
\(243\) 2.84158 1.17702i 0.182288 0.0755060i
\(244\) 0 0
\(245\) 0.118689 0.286541i 0.00758278 0.0183065i
\(246\) 0 0
\(247\) −2.41957 + 2.41957i −0.153953 + 0.153953i
\(248\) 0 0
\(249\) −10.3108 10.3108i −0.653421 0.653421i
\(250\) 0 0
\(251\) −11.7000 4.84628i −0.738495 0.305895i −0.0184573 0.999830i \(-0.505875\pi\)
−0.720038 + 0.693935i \(0.755875\pi\)
\(252\) 0 0
\(253\) 1.89646 + 4.57847i 0.119230 + 0.287846i
\(254\) 0 0
\(255\) −0.892496 −0.0558902
\(256\) 0 0
\(257\) 7.39861 0.461513 0.230756 0.973012i \(-0.425880\pi\)
0.230756 + 0.973012i \(0.425880\pi\)
\(258\) 0 0
\(259\) 0.625037 + 1.50897i 0.0388379 + 0.0937630i
\(260\) 0 0
\(261\) 2.36494 + 0.979589i 0.146386 + 0.0606350i
\(262\) 0 0
\(263\) 14.1801 + 14.1801i 0.874385 + 0.874385i 0.992947 0.118562i \(-0.0378284\pi\)
−0.118562 + 0.992947i \(0.537828\pi\)
\(264\) 0 0
\(265\) 0.720651 0.720651i 0.0442693 0.0442693i
\(266\) 0 0
\(267\) −5.50637 + 13.2936i −0.336985 + 0.813553i
\(268\) 0 0
\(269\) −23.9345 + 9.91400i −1.45931 + 0.604468i −0.964394 0.264471i \(-0.914803\pi\)
−0.494920 + 0.868939i \(0.664803\pi\)
\(270\) 0 0
\(271\) 20.7120i 1.25816i 0.777339 + 0.629082i \(0.216569\pi\)
−0.777339 + 0.629082i \(0.783431\pi\)
\(272\) 0 0
\(273\) 0.788984i 0.0477515i
\(274\) 0 0
\(275\) −12.5799 + 5.21076i −0.758595 + 0.314221i
\(276\) 0 0
\(277\) −3.88572 + 9.38096i −0.233470 + 0.563647i −0.996581 0.0826204i \(-0.973671\pi\)
0.763111 + 0.646268i \(0.223671\pi\)
\(278\) 0 0
\(279\) −0.0123888 + 0.0123888i −0.000741696 + 0.000741696i
\(280\) 0 0
\(281\) 19.1224 + 19.1224i 1.14075 + 1.14075i 0.988314 + 0.152432i \(0.0487104\pi\)
0.152432 + 0.988314i \(0.451290\pi\)
\(282\) 0 0
\(283\) −5.71215 2.36605i −0.339552 0.140647i 0.206391 0.978470i \(-0.433828\pi\)
−0.545942 + 0.837823i \(0.683828\pi\)
\(284\) 0 0
\(285\) 1.69714 + 4.09726i 0.100530 + 0.242701i
\(286\) 0 0
\(287\) 3.14333 0.185545
\(288\) 0 0
\(289\) 14.4884 0.852260
\(290\) 0 0
\(291\) −7.40651 17.8809i −0.434177 1.04820i
\(292\) 0 0
\(293\) −22.7513 9.42391i −1.32915 0.550551i −0.398735 0.917066i \(-0.630551\pi\)
−0.930412 + 0.366516i \(0.880551\pi\)
\(294\) 0 0
\(295\) 2.39914 + 2.39914i 0.139683 + 0.139683i
\(296\) 0 0
\(297\) −9.63644 + 9.63644i −0.559163 + 0.559163i
\(298\) 0 0
\(299\) 0.296773 0.716473i 0.0171628 0.0414347i
\(300\) 0 0
\(301\) 2.81591 1.16639i 0.162307 0.0672296i
\(302\) 0 0
\(303\) 22.8510i 1.31275i
\(304\) 0 0
\(305\) 0.194619i 0.0111438i
\(306\) 0 0
\(307\) −4.81779 + 1.99559i −0.274966 + 0.113895i −0.515905 0.856646i \(-0.672544\pi\)
0.240940 + 0.970540i \(0.422544\pi\)
\(308\) 0 0
\(309\) 10.0610 24.2893i 0.572348 1.38177i
\(310\) 0 0
\(311\) −12.8134 + 12.8134i −0.726582 + 0.726582i −0.969937 0.243356i \(-0.921752\pi\)
0.243356 + 0.969937i \(0.421752\pi\)
\(312\) 0 0
\(313\) −1.20989 1.20989i −0.0683871 0.0683871i 0.672086 0.740473i \(-0.265399\pi\)
−0.740473 + 0.672086i \(0.765399\pi\)
\(314\) 0 0
\(315\) 0.0851079 + 0.0352528i 0.00479529 + 0.00198627i
\(316\) 0 0
\(317\) 0.324788 + 0.784107i 0.0182419 + 0.0440398i 0.932738 0.360555i \(-0.117413\pi\)
−0.914496 + 0.404595i \(0.867413\pi\)
\(318\) 0 0
\(319\) −23.9304 −1.33984
\(320\) 0 0
\(321\) 21.0061 1.17245
\(322\) 0 0
\(323\) 4.77594 + 11.5301i 0.265740 + 0.641553i
\(324\) 0 0
\(325\) 1.96859 + 0.815418i 0.109198 + 0.0452313i
\(326\) 0 0
\(327\) −19.3478 19.3478i −1.06994 1.06994i
\(328\) 0 0
\(329\) −3.89940 + 3.89940i −0.214981 + 0.214981i
\(330\) 0 0
\(331\) −0.967319 + 2.33531i −0.0531686 + 0.128360i −0.948232 0.317579i \(-0.897130\pi\)
0.895063 + 0.445939i \(0.147130\pi\)
\(332\) 0 0
\(333\) −0.448192 + 0.185647i −0.0245608 + 0.0101734i
\(334\) 0 0
\(335\) 4.89171i 0.267263i
\(336\) 0 0
\(337\) 5.25318i 0.286159i −0.989711 0.143079i \(-0.954300\pi\)
0.989711 0.143079i \(-0.0457004\pi\)
\(338\) 0 0
\(339\) −13.4875 + 5.58672i −0.732542 + 0.303429i
\(340\) 0 0
\(341\) 0.0626798 0.151323i 0.00339430 0.00819457i
\(342\) 0 0
\(343\) −0.707107 + 0.707107i −0.0381802 + 0.0381802i
\(344\) 0 0
\(345\) −0.710714 0.710714i −0.0382635 0.0382635i
\(346\) 0 0
\(347\) 22.2993 + 9.23669i 1.19709 + 0.495851i 0.890058 0.455847i \(-0.150664\pi\)
0.307033 + 0.951699i \(0.400664\pi\)
\(348\) 0 0
\(349\) −11.7749 28.4272i −0.630298 1.52167i −0.839249 0.543748i \(-0.817005\pi\)
0.208951 0.977926i \(-0.432995\pi\)
\(350\) 0 0
\(351\) 2.13261 0.113830
\(352\) 0 0
\(353\) 7.78602 0.414408 0.207204 0.978298i \(-0.433564\pi\)
0.207204 + 0.978298i \(0.433564\pi\)
\(354\) 0 0
\(355\) 0.850597 + 2.05352i 0.0451450 + 0.108990i
\(356\) 0 0
\(357\) 2.65858 + 1.10122i 0.140707 + 0.0582827i
\(358\) 0 0
\(359\) −16.7745 16.7745i −0.885325 0.885325i 0.108745 0.994070i \(-0.465317\pi\)
−0.994070 + 0.108745i \(0.965317\pi\)
\(360\) 0 0
\(361\) 30.4156 30.4156i 1.60082 1.60082i
\(362\) 0 0
\(363\) 2.28609 5.51911i 0.119989 0.289678i
\(364\) 0 0
\(365\) 2.61920 1.08491i 0.137095 0.0567867i
\(366\) 0 0
\(367\) 4.18075i 0.218234i −0.994029 0.109117i \(-0.965198\pi\)
0.994029 0.109117i \(-0.0348022\pi\)
\(368\) 0 0
\(369\) 0.933624i 0.0486025i
\(370\) 0 0
\(371\) −3.03587 + 1.25750i −0.157615 + 0.0652861i
\(372\) 0 0
\(373\) −4.80301 + 11.5955i −0.248691 + 0.600392i −0.998093 0.0617222i \(-0.980341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(374\) 0 0
\(375\) 3.94385 3.94385i 0.203659 0.203659i
\(376\) 0 0
\(377\) 2.64798 + 2.64798i 0.136378 + 0.136378i
\(378\) 0 0
\(379\) −5.48718 2.27286i −0.281857 0.116749i 0.237276 0.971442i \(-0.423745\pi\)
−0.519134 + 0.854693i \(0.673745\pi\)
\(380\) 0 0
\(381\) −0.660721 1.59512i −0.0338498 0.0817206i
\(382\) 0 0
\(383\) −20.7157 −1.05852 −0.529262 0.848459i \(-0.677531\pi\)
−0.529262 + 0.848459i \(0.677531\pi\)
\(384\) 0 0
\(385\) −0.861192 −0.0438904
\(386\) 0 0
\(387\) 0.346439 + 0.836377i 0.0176105 + 0.0425154i
\(388\) 0 0
\(389\) −25.5609 10.5877i −1.29599 0.536815i −0.375223 0.926935i \(-0.622434\pi\)
−0.920764 + 0.390119i \(0.872434\pi\)
\(390\) 0 0
\(391\) −2.00002 2.00002i −0.101146 0.101146i
\(392\) 0 0
\(393\) 5.16920 5.16920i 0.260752 0.260752i
\(394\) 0 0
\(395\) −0.871759 + 2.10461i −0.0438629 + 0.105895i
\(396\) 0 0
\(397\) 11.9912 4.96691i 0.601820 0.249282i −0.0609066 0.998143i \(-0.519399\pi\)
0.662727 + 0.748861i \(0.269399\pi\)
\(398\) 0 0
\(399\) 14.2990i 0.715846i
\(400\) 0 0
\(401\) 10.0744i 0.503094i 0.967845 + 0.251547i \(0.0809393\pi\)
−0.967845 + 0.251547i \(0.919061\pi\)
\(402\) 0 0
\(403\) −0.0236801 + 0.00980861i −0.00117959 + 0.000488601i
\(404\) 0 0
\(405\) 1.16349 2.80892i 0.0578144 0.139576i
\(406\) 0 0
\(407\) 3.20685 3.20685i 0.158958 0.158958i
\(408\) 0 0
\(409\) 20.7261 + 20.7261i 1.02484 + 1.02484i 0.999684 + 0.0251572i \(0.00800864\pi\)
0.0251572 + 0.999684i \(0.491991\pi\)
\(410\) 0 0
\(411\) −21.1659 8.76722i −1.04404 0.432455i
\(412\) 0 0
\(413\) −4.18638 10.1068i −0.205998 0.497323i
\(414\) 0 0
\(415\) −2.49069 −0.122263
\(416\) 0 0
\(417\) −7.81120 −0.382516
\(418\) 0 0
\(419\) 1.64415 + 3.96932i 0.0803218 + 0.193914i 0.958939 0.283613i \(-0.0915332\pi\)
−0.878617 + 0.477527i \(0.841533\pi\)
\(420\) 0 0
\(421\) 26.7236 + 11.0693i 1.30243 + 0.539484i 0.922666 0.385599i \(-0.126005\pi\)
0.379764 + 0.925084i \(0.376005\pi\)
\(422\) 0 0
\(423\) −1.15819 1.15819i −0.0563132 0.0563132i
\(424\) 0 0
\(425\) 5.49531 5.49531i 0.266561 0.266561i
\(426\) 0 0
\(427\) −0.240133 + 0.579733i −0.0116209 + 0.0280552i
\(428\) 0 0
\(429\) 2.02400 0.838369i 0.0977198 0.0404769i
\(430\) 0 0
\(431\) 8.46979i 0.407975i −0.978973 0.203988i \(-0.934610\pi\)
0.978973 0.203988i \(-0.0653902\pi\)
\(432\) 0 0
\(433\) 41.2749i 1.98354i −0.128017 0.991772i \(-0.540861\pi\)
0.128017 0.991772i \(-0.459139\pi\)
\(434\) 0 0
\(435\) 4.48404 1.85735i 0.214993 0.0890532i
\(436\) 0 0
\(437\) −5.37851 + 12.9849i −0.257289 + 0.621151i
\(438\) 0 0
\(439\) 28.1447 28.1447i 1.34327 1.34327i 0.450497 0.892778i \(-0.351247\pi\)
0.892778 0.450497i \(-0.148753\pi\)
\(440\) 0 0
\(441\) −0.210023 0.210023i −0.0100011 0.0100011i
\(442\) 0 0
\(443\) 18.7826 + 7.78002i 0.892390 + 0.369640i 0.781289 0.624169i \(-0.214562\pi\)
0.111101 + 0.993809i \(0.464562\pi\)
\(444\) 0 0
\(445\) 0.940540 + 2.27066i 0.0445859 + 0.107640i
\(446\) 0 0
\(447\) 19.7236 0.932893
\(448\) 0 0
\(449\) 6.56124 0.309644 0.154822 0.987942i \(-0.450520\pi\)
0.154822 + 0.987942i \(0.450520\pi\)
\(450\) 0 0
\(451\) −3.34008 8.06367i −0.157278 0.379703i
\(452\) 0 0
\(453\) −15.4617 6.40444i −0.726453 0.300907i
\(454\) 0 0
\(455\) 0.0952937 + 0.0952937i 0.00446744 + 0.00446744i
\(456\) 0 0
\(457\) −22.5764 + 22.5764i −1.05608 + 1.05608i −0.0577483 + 0.998331i \(0.518392\pi\)
−0.998331 + 0.0577483i \(0.981608\pi\)
\(458\) 0 0
\(459\) 2.97658 7.18609i 0.138935 0.335418i
\(460\) 0 0
\(461\) −31.6098 + 13.0932i −1.47222 + 0.609812i −0.967364 0.253393i \(-0.918454\pi\)
−0.504854 + 0.863205i \(0.668454\pi\)
\(462\) 0 0
\(463\) 4.26935i 0.198413i 0.995067 + 0.0992067i \(0.0316305\pi\)
−0.995067 + 0.0992067i \(0.968370\pi\)
\(464\) 0 0
\(465\) 0.0332195i 0.00154052i
\(466\) 0 0
\(467\) 14.1292 5.85251i 0.653822 0.270822i −0.0310145 0.999519i \(-0.509874\pi\)
0.684836 + 0.728697i \(0.259874\pi\)
\(468\) 0 0
\(469\) 6.03571 14.5715i 0.278703 0.672849i
\(470\) 0 0
\(471\) 4.76690 4.76690i 0.219647 0.219647i
\(472\) 0 0
\(473\) −5.98435 5.98435i −0.275161 0.275161i
\(474\) 0 0
\(475\) −35.6775 14.7781i −1.63699 0.678065i
\(476\) 0 0
\(477\) −0.373500 0.901708i −0.0171014 0.0412864i
\(478\) 0 0
\(479\) 13.7054 0.626217 0.313108 0.949717i \(-0.398630\pi\)
0.313108 + 0.949717i \(0.398630\pi\)
\(480\) 0 0
\(481\) −0.709698 −0.0323594
\(482\) 0 0
\(483\) 1.24016 + 2.99401i 0.0564292 + 0.136232i
\(484\) 0 0
\(485\) −3.05422 1.26510i −0.138685 0.0574453i
\(486\) 0 0
\(487\) 15.3828 + 15.3828i 0.697063 + 0.697063i 0.963776 0.266713i \(-0.0859375\pi\)
−0.266713 + 0.963776i \(0.585938\pi\)
\(488\) 0 0
\(489\) 22.5050 22.5050i 1.01771 1.01771i
\(490\) 0 0
\(491\) 2.87621 6.94379i 0.129802 0.313369i −0.845595 0.533824i \(-0.820754\pi\)
0.975397 + 0.220456i \(0.0707544\pi\)
\(492\) 0 0
\(493\) 12.6186 5.22679i 0.568312 0.235403i
\(494\) 0 0
\(495\) 0.255789i 0.0114969i
\(496\) 0 0
\(497\) 7.16658i 0.321465i
\(498\) 0 0
\(499\) −16.9429 + 7.01799i −0.758470 + 0.314168i −0.728192 0.685373i \(-0.759639\pi\)
−0.0302776 + 0.999542i \(0.509639\pi\)
\(500\) 0 0
\(501\) 8.75098 21.1267i 0.390965 0.943873i
\(502\) 0 0
\(503\) −4.67325 + 4.67325i −0.208370 + 0.208370i −0.803574 0.595204i \(-0.797071\pi\)
0.595204 + 0.803574i \(0.297071\pi\)
\(504\) 0 0
\(505\) −2.75995 2.75995i −0.122816 0.122816i
\(506\) 0 0
\(507\) 21.4914 + 8.90205i 0.954469 + 0.395354i
\(508\) 0 0
\(509\) 8.28515 + 20.0021i 0.367233 + 0.886578i 0.994201 + 0.107534i \(0.0342954\pi\)
−0.626969 + 0.779044i \(0.715705\pi\)
\(510\) 0 0
\(511\) −9.14074 −0.404362
\(512\) 0 0
\(513\) −38.6500 −1.70644
\(514\) 0 0
\(515\) −1.71850 4.14884i −0.0757264 0.182820i
\(516\) 0 0
\(517\) 14.1467 + 5.85977i 0.622172 + 0.257712i
\(518\) 0 0
\(519\) −28.3826 28.3826i −1.24586 1.24586i
\(520\) 0 0
\(521\) 19.8777 19.8777i 0.870857 0.870857i −0.121709 0.992566i \(-0.538838\pi\)
0.992566 + 0.121709i \(0.0388376\pi\)
\(522\) 0 0
\(523\) −15.8527 + 38.2718i −0.693190 + 1.67351i 0.0450633 + 0.998984i \(0.485651\pi\)
−0.738253 + 0.674524i \(0.764349\pi\)
\(524\) 0 0
\(525\) −8.22639 + 3.40748i −0.359029 + 0.148715i
\(526\) 0 0
\(527\) 0.0934833i 0.00407219i
\(528\) 0 0
\(529\) 19.8147i 0.861508i
\(530\) 0 0
\(531\) 3.00190 1.24343i 0.130271 0.0539602i
\(532\) 0 0
\(533\) −0.522681 + 1.26186i −0.0226398 + 0.0546573i
\(534\) 0 0
\(535\) 2.53713 2.53713i 0.109690 0.109690i
\(536\) 0 0
\(537\) 9.30926 + 9.30926i 0.401724 + 0.401724i
\(538\) 0 0
\(539\) 2.56533 + 1.06259i 0.110497 + 0.0457692i
\(540\) 0 0
\(541\) −12.4194 29.9831i −0.533952 1.28907i −0.928886 0.370365i \(-0.879233\pi\)
0.394934 0.918709i \(-0.370767\pi\)
\(542\) 0 0
\(543\) 32.5216 1.39563
\(544\) 0 0
\(545\) −4.67367 −0.200198
\(546\) 0 0
\(547\) 9.86098 + 23.8065i 0.421625 + 1.01789i 0.981868 + 0.189564i \(0.0607076\pi\)
−0.560243 + 0.828328i \(0.689292\pi\)
\(548\) 0 0
\(549\) −0.172191 0.0713238i −0.00734893 0.00304403i
\(550\) 0 0
\(551\) −47.9901 47.9901i −2.04445 2.04445i
\(552\) 0 0
\(553\) 5.19361 5.19361i 0.220855 0.220855i
\(554\) 0 0
\(555\) −0.351996 + 0.849795i −0.0149414 + 0.0360718i
\(556\) 0 0
\(557\) −27.4760 + 11.3809i −1.16420 + 0.482226i −0.879270 0.476323i \(-0.841969\pi\)
−0.284927 + 0.958549i \(0.591969\pi\)
\(558\) 0 0
\(559\) 1.32438i 0.0560151i
\(560\) 0 0
\(561\) 7.99028i 0.337350i
\(562\) 0 0
\(563\) −22.7852 + 9.43795i −0.960282 + 0.397762i −0.807086 0.590434i \(-0.798957\pi\)
−0.153196 + 0.988196i \(0.548957\pi\)
\(564\) 0 0
\(565\) −0.954263 + 2.30379i −0.0401461 + 0.0969214i
\(566\) 0 0
\(567\) −6.93165 + 6.93165i −0.291102 + 0.291102i
\(568\) 0 0
\(569\) −14.7197 14.7197i −0.617080 0.617080i 0.327701 0.944781i \(-0.393726\pi\)
−0.944781 + 0.327701i \(0.893726\pi\)
\(570\) 0 0
\(571\) 18.4362 + 7.63654i 0.771533 + 0.319579i 0.733493 0.679697i \(-0.237889\pi\)
0.0380396 + 0.999276i \(0.487889\pi\)
\(572\) 0 0
\(573\) 16.1810 + 39.0644i 0.675972 + 1.63194i
\(574\) 0 0
\(575\) 8.75206 0.364986
\(576\) 0 0
\(577\) −3.73589 −0.155527 −0.0777635 0.996972i \(-0.524778\pi\)
−0.0777635 + 0.996972i \(0.524778\pi\)
\(578\) 0 0
\(579\) −7.35865 17.7653i −0.305815 0.738302i
\(580\) 0 0
\(581\) 7.41929 + 3.07317i 0.307804 + 0.127497i
\(582\) 0 0
\(583\) 6.45180 + 6.45180i 0.267206 + 0.267206i
\(584\) 0 0
\(585\) −0.0283039 + 0.0283039i −0.00117022 + 0.00117022i
\(586\) 0 0
\(587\) 3.25930 7.86864i 0.134526 0.324773i −0.842234 0.539112i \(-0.818760\pi\)
0.976759 + 0.214339i \(0.0687598\pi\)
\(588\) 0 0
\(589\) 0.429162 0.177765i 0.0176833 0.00732466i
\(590\) 0 0
\(591\) 26.8526i 1.10457i
\(592\) 0 0
\(593\) 8.91561i 0.366120i −0.983102 0.183060i \(-0.941400\pi\)
0.983102 0.183060i \(-0.0586003\pi\)
\(594\) 0 0
\(595\) 0.454110 0.188098i 0.0186167 0.00771128i
\(596\) 0 0
\(597\) 18.5798 44.8555i 0.760419 1.83581i
\(598\) 0 0
\(599\) 20.4876 20.4876i 0.837100 0.837100i −0.151376 0.988476i \(-0.548371\pi\)
0.988476 + 0.151376i \(0.0483706\pi\)
\(600\) 0 0
\(601\) 4.48561 + 4.48561i 0.182972 + 0.182972i 0.792650 0.609678i \(-0.208701\pi\)
−0.609678 + 0.792650i \(0.708701\pi\)
\(602\) 0 0
\(603\) 4.32799 + 1.79271i 0.176249 + 0.0730049i
\(604\) 0 0
\(605\) −0.390485 0.942714i −0.0158755 0.0383268i
\(606\) 0 0
\(607\) −11.9396 −0.484615 −0.242307 0.970200i \(-0.577904\pi\)
−0.242307 + 0.970200i \(0.577904\pi\)
\(608\) 0 0
\(609\) −15.6488 −0.634123
\(610\) 0 0
\(611\) −0.916980 2.21378i −0.0370970 0.0895602i
\(612\) 0 0
\(613\) 19.9328 + 8.25643i 0.805078 + 0.333474i 0.746988 0.664837i \(-0.231499\pi\)
0.0580894 + 0.998311i \(0.481499\pi\)
\(614\) 0 0
\(615\) 1.25172 + 1.25172i 0.0504742 + 0.0504742i
\(616\) 0 0
\(617\) −22.7234 + 22.7234i −0.914808 + 0.914808i −0.996646 0.0818380i \(-0.973921\pi\)
0.0818380 + 0.996646i \(0.473921\pi\)
\(618\) 0 0
\(619\) −2.70370 + 6.52730i −0.108671 + 0.262354i −0.968855 0.247629i \(-0.920349\pi\)
0.860184 + 0.509984i \(0.170349\pi\)
\(620\) 0 0
\(621\) 8.09275 3.35213i 0.324751 0.134516i
\(622\) 0 0
\(623\) 7.92438i 0.317484i
\(624\) 0 0
\(625\) 23.5664i 0.942654i
\(626\) 0 0
\(627\) −36.6817 + 15.1940i −1.46492 + 0.606792i
\(628\) 0 0
\(629\) −0.990556 + 2.39141i −0.0394961 + 0.0953519i
\(630\) 0 0
\(631\) 3.43447 3.43447i 0.136724 0.136724i −0.635432 0.772157i \(-0.719178\pi\)
0.772157 + 0.635432i \(0.219178\pi\)
\(632\) 0 0
\(633\) −31.2660 31.2660i −1.24271 1.24271i
\(634\) 0 0
\(635\) −0.272462 0.112857i −0.0108123 0.00447860i
\(636\) 0 0
\(637\) −0.166283 0.401442i −0.00658836 0.0159057i
\(638\) 0 0
\(639\) 2.12860 0.0842062
\(640\) 0 0
\(641\) −17.3679 −0.685992 −0.342996 0.939337i \(-0.611442\pi\)
−0.342996 + 0.939337i \(0.611442\pi\)
\(642\) 0 0
\(643\) −16.1938 39.0954i −0.638622 1.54177i −0.828516 0.559966i \(-0.810814\pi\)
0.189893 0.981805i \(-0.439186\pi\)
\(644\) 0 0
\(645\) 1.58581 + 0.656865i 0.0624413 + 0.0258641i
\(646\) 0 0
\(647\) −7.10802 7.10802i −0.279445 0.279445i 0.553442 0.832888i \(-0.313314\pi\)
−0.832888 + 0.553442i \(0.813314\pi\)
\(648\) 0 0
\(649\) −21.4789 + 21.4789i −0.843119 + 0.843119i
\(650\) 0 0
\(651\) 0.0409884 0.0989547i 0.00160646 0.00387834i
\(652\) 0 0
\(653\) −6.59052 + 2.72988i −0.257907 + 0.106829i −0.507891 0.861421i \(-0.669575\pi\)
0.249984 + 0.968250i \(0.419575\pi\)
\(654\) 0 0
\(655\) 1.24868i 0.0487898i
\(656\) 0 0
\(657\) 2.71496i 0.105921i
\(658\) 0 0
\(659\) 8.47271 3.50951i 0.330050 0.136711i −0.211504 0.977377i \(-0.567836\pi\)
0.541554 + 0.840666i \(0.317836\pi\)
\(660\) 0 0
\(661\) −0.0390952 + 0.0943841i −0.00152063 + 0.00367111i −0.924638 0.380847i \(-0.875632\pi\)
0.923117 + 0.384518i \(0.125632\pi\)
\(662\) 0 0
\(663\) −0.884151 + 0.884151i −0.0343376 + 0.0343376i
\(664\) 0 0
\(665\) −1.72704 1.72704i −0.0669717 0.0669717i
\(666\) 0 0
\(667\) 14.2106 + 5.88624i 0.550239 + 0.227916i
\(668\) 0 0
\(669\) −13.3408 32.2075i −0.515785 1.24522i
\(670\) 0 0
\(671\) 1.74237 0.0672634
\(672\) 0 0
\(673\) 3.92984 0.151484 0.0757421 0.997127i \(-0.475867\pi\)
0.0757421 + 0.997127i \(0.475867\pi\)
\(674\) 0 0
\(675\) 9.21037 + 22.2358i 0.354507 + 0.855856i
\(676\) 0 0
\(677\) −18.1313 7.51024i −0.696844 0.288642i 0.00600459 0.999982i \(-0.498089\pi\)
−0.702848 + 0.711340i \(0.748089\pi\)
\(678\) 0 0
\(679\) 7.53700 + 7.53700i 0.289243 + 0.289243i
\(680\) 0 0
\(681\) 18.7127 18.7127i 0.717071 0.717071i
\(682\) 0 0
\(683\) 2.18867 5.28391i 0.0837471 0.202183i −0.876459 0.481477i \(-0.840100\pi\)
0.960206 + 0.279294i \(0.0901004\pi\)
\(684\) 0 0
\(685\) −3.61534 + 1.49752i −0.138135 + 0.0572174i
\(686\) 0 0
\(687\) 36.5411i 1.39413i
\(688\) 0 0
\(689\) 1.42783i 0.0543959i
\(690\) 0 0
\(691\) 23.9531 9.92169i 0.911218 0.377439i 0.122695 0.992444i \(-0.460846\pi\)
0.788523 + 0.615006i \(0.210846\pi\)
\(692\) 0 0
\(693\) −0.315609 + 0.761949i −0.0119890 + 0.0289440i
\(694\) 0 0
\(695\) −0.943440 + 0.943440i −0.0357867 + 0.0357867i
\(696\) 0 0
\(697\) 3.52247 + 3.52247i 0.133423 + 0.133423i
\(698\) 0 0
\(699\) −22.4141 9.28422i −0.847778 0.351161i
\(700\) 0 0
\(701\) 7.87457 + 19.0109i 0.297418 + 0.718031i 0.999979 + 0.00640805i \(0.00203976\pi\)
−0.702561 + 0.711623i \(0.747960\pi\)
\(702\) 0 0
\(703\) 12.8621 0.485103
\(704\) 0 0
\(705\) −3.10560 −0.116964
\(706\) 0 0
\(707\) 4.81597 + 11.6268i 0.181123 + 0.437270i
\(708\) 0 0
\(709\) 28.4087 + 11.7673i 1.06691 + 0.441929i 0.845900 0.533342i \(-0.179064\pi\)
0.221012 + 0.975271i \(0.429064\pi\)
\(710\) 0 0
\(711\) 1.54260 + 1.54260i 0.0578519 + 0.0578519i
\(712\) 0 0
\(713\) −0.0744427 + 0.0744427i −0.00278790 + 0.00278790i
\(714\) 0 0
\(715\) 0.143201 0.345718i 0.00535542 0.0129291i
\(716\) 0 0
\(717\) −50.1505 + 20.7730i −1.87290 + 0.775782i
\(718\) 0 0
\(719\) 22.7491i 0.848398i 0.905569 + 0.424199i \(0.139444\pi\)
−0.905569 + 0.424199i \(0.860556\pi\)
\(720\) 0 0
\(721\) 14.4790i 0.539227i
\(722\) 0 0
\(723\) 17.6536 7.31237i 0.656545 0.271950i
\(724\) 0 0
\(725\) −16.1732 + 39.0454i −0.600656 + 1.45011i
\(726\) 0 0
\(727\) −4.54966 + 4.54966i −0.168737 + 0.168737i −0.786424 0.617687i \(-0.788070\pi\)
0.617687 + 0.786424i \(0.288070\pi\)
\(728\) 0 0
\(729\) 16.8459 + 16.8459i 0.623923 + 0.623923i
\(730\) 0 0
\(731\) 4.46265 + 1.84849i 0.165057 + 0.0683689i
\(732\) 0 0
\(733\) 14.9141 + 36.0057i 0.550864 + 1.32990i 0.916831 + 0.399275i \(0.130738\pi\)
−0.365968 + 0.930628i \(0.619262\pi\)
\(734\) 0 0
\(735\) −0.563161 −0.0207725
\(736\) 0 0
\(737\) −43.7942 −1.61318
\(738\) 0 0
\(739\) −4.49399 10.8494i −0.165314 0.399103i 0.819414 0.573202i \(-0.194299\pi\)
−0.984728 + 0.174099i \(0.944299\pi\)
\(740\) 0 0
\(741\) 5.74022 + 2.37768i 0.210872 + 0.0873462i
\(742\) 0 0
\(743\) −27.2587 27.2587i −1.00003 1.00003i −1.00000 2.62449e-5i \(-0.999992\pi\)
−2.62449e−5 1.00000i \(-0.500008\pi\)
\(744\) 0 0
\(745\) 2.38222 2.38222i 0.0872778 0.0872778i
\(746\) 0 0
\(747\) −0.912786 + 2.20366i −0.0333971 + 0.0806277i
\(748\) 0 0
\(749\) −10.6881 + 4.42716i −0.390535 + 0.161765i
\(750\) 0 0
\(751\) 30.0673i 1.09717i 0.836095 + 0.548585i \(0.184833\pi\)
−0.836095 + 0.548585i \(0.815167\pi\)
\(752\) 0 0
\(753\) 22.9948i 0.837977i
\(754\) 0 0
\(755\) −2.64100 + 1.09394i −0.0961158 + 0.0398125i
\(756\) 0 0
\(757\) 6.77290 16.3512i 0.246165 0.594296i −0.751707 0.659497i \(-0.770769\pi\)
0.997872 + 0.0652018i \(0.0207691\pi\)
\(758\) 0 0
\(759\) 6.36283 6.36283i 0.230956 0.230956i
\(760\) 0 0
\(761\) 25.7438 + 25.7438i 0.933212 + 0.933212i 0.997905 0.0646936i \(-0.0206070\pi\)
−0.0646936 + 0.997905i \(0.520607\pi\)
\(762\) 0 0
\(763\) 13.9220 + 5.76668i 0.504010 + 0.208768i
\(764\) 0 0
\(765\) 0.0558686 + 0.134879i 0.00201993 + 0.00487655i
\(766\) 0 0
\(767\) 4.75342 0.171636
\(768\) 0 0
\(769\) 2.19445 0.0791339 0.0395669 0.999217i \(-0.487402\pi\)
0.0395669 + 0.999217i \(0.487402\pi\)
\(770\) 0 0
\(771\) −5.14103 12.4116i −0.185150 0.446991i
\(772\) 0 0
\(773\) 26.7318 + 11.0727i 0.961476 + 0.398256i 0.807532 0.589823i \(-0.200803\pi\)
0.153944 + 0.988080i \(0.450803\pi\)
\(774\) 0 0
\(775\) −0.204540 0.204540i −0.00734730 0.00734730i
\(776\) 0 0
\(777\) 2.09706 2.09706i 0.0752317 0.0752317i
\(778\) 0 0
\(779\) 9.47271 22.8692i 0.339395 0.819372i
\(780\) 0 0
\(781\) −18.3846 + 7.61517i −0.657854 + 0.272492i
\(782\) 0 0
\(783\) 42.2985i 1.51163i
\(784\) 0 0
\(785\) 1.15150i 0.0410986i
\(786\) 0 0
\(787\) 4.67661 1.93712i 0.166703 0.0690507i −0.297771 0.954637i \(-0.596243\pi\)
0.464474 + 0.885587i \(0.346243\pi\)
\(788\) 0 0
\(789\) 13.9346 33.6412i 0.496086 1.19766i
\(790\) 0 0
\(791\) 5.68514 5.68514i 0.202140 0.202140i
\(792\) 0 0
\(793\) −0.192799 0.192799i −0.00684649 0.00684649i
\(794\) 0 0
\(795\) −1.70969 0.708175i −0.0606363 0.0251164i
\(796\) 0 0
\(797\) 14.4293 + 34.8354i 0.511112 + 1.23393i 0.943237 + 0.332120i \(0.107764\pi\)
−0.432125 + 0.901814i \(0.642236\pi\)
\(798\) 0 0
\(799\) −8.73949 −0.309181
\(800\) 0 0
\(801\) 2.35368 0.0831633
\(802\) 0 0
\(803\) 9.71289 + 23.4490i 0.342761 + 0.827497i
\(804\) 0 0
\(805\) 0.511404 + 0.211831i 0.0180246 + 0.00746605i
\(806\) 0 0
\(807\) 33.2625 + 33.2625i 1.17090 + 1.17090i
\(808\) 0 0
\(809\) 34.5198 34.5198i 1.21365 1.21365i 0.243836 0.969816i \(-0.421594\pi\)
0.969816 0.243836i \(-0.0784059\pi\)
\(810\) 0 0
\(811\) 7.88782 19.0429i 0.276979 0.668686i −0.722770 0.691088i \(-0.757132\pi\)
0.999749 + 0.0224024i \(0.00713151\pi\)
\(812\) 0 0
\(813\) 34.7455 14.3920i 1.21858 0.504751i
\(814\) 0 0
\(815\) 5.43632i 0.190426i
\(816\) 0 0
\(817\) 24.0021i 0.839728i
\(818\) 0 0
\(819\) 0.119235 0.0493889i 0.00416642 0.00172579i
\(820\) 0 0
\(821\) 11.0386 26.6495i 0.385250 0.930076i −0.605682 0.795707i \(-0.707099\pi\)
0.990931 0.134368i \(-0.0429005\pi\)
\(822\) 0 0
\(823\) −17.5633 + 17.5633i −0.612216 + 0.612216i −0.943523 0.331307i \(-0.892511\pi\)
0.331307 + 0.943523i \(0.392511\pi\)
\(824\) 0 0
\(825\) 17.4826 + 17.4826i 0.608667 + 0.608667i
\(826\) 0 0
\(827\) 1.69624 + 0.702605i 0.0589840 + 0.0244320i 0.411980 0.911193i \(-0.364837\pi\)
−0.352996 + 0.935625i \(0.614837\pi\)
\(828\) 0 0
\(829\) −10.6418 25.6915i −0.369603 0.892302i −0.993815 0.111047i \(-0.964580\pi\)
0.624212 0.781255i \(-0.285420\pi\)
\(830\) 0 0
\(831\) 18.4371 0.639576
\(832\) 0 0
\(833\) −1.58480 −0.0549099
\(834\) 0 0
\(835\) −1.49475 3.60864i −0.0517279 0.124882i
\(836\) 0 0
\(837\) −0.267473 0.110791i −0.00924521 0.00382949i
\(838\) 0 0
\(839\) 13.5414 + 13.5414i 0.467500 + 0.467500i 0.901104 0.433604i \(-0.142758\pi\)
−0.433604 + 0.901104i \(0.642758\pi\)
\(840\) 0 0
\(841\) −32.0143 + 32.0143i −1.10394 + 1.10394i
\(842\) 0 0
\(843\) 18.7913 45.3663i 0.647207 1.56250i
\(844\) 0 0
\(845\) 3.67094 1.52055i 0.126284 0.0523086i
\(846\) 0 0
\(847\) 3.28998i 0.113045i
\(848\) 0 0
\(849\) 11.2265i 0.385293i
\(850\) 0 0
\(851\) −2.69314 + 1.11553i −0.0923195 + 0.0382400i
\(852\) 0 0
\(853\) 4.32133 10.4326i 0.147960 0.357206i −0.832472 0.554067i \(-0.813075\pi\)
0.980431 + 0.196862i \(0.0630750\pi\)
\(854\) 0 0
\(855\) 0.512962 0.512962i 0.0175429 0.0175429i
\(856\) 0 0
\(857\) 22.0539 + 22.0539i 0.753349 + 0.753349i 0.975103 0.221754i \(-0.0711782\pi\)
−0.221754 + 0.975103i \(0.571178\pi\)
\(858\) 0 0
\(859\) 23.1538 + 9.59062i 0.789997 + 0.327228i 0.740943 0.671568i \(-0.234379\pi\)
0.0490549 + 0.998796i \(0.484379\pi\)
\(860\) 0 0
\(861\) −2.18419 5.27309i −0.0744369 0.179707i
\(862\) 0 0
\(863\) 4.96957 0.169166 0.0845830 0.996416i \(-0.473044\pi\)
0.0845830 + 0.996416i \(0.473044\pi\)
\(864\) 0 0
\(865\) −6.85613 −0.233116
\(866\) 0 0
\(867\) −10.0675 24.3051i −0.341910 0.825444i
\(868\) 0 0
\(869\) −18.8420 7.80463i −0.639172 0.264754i
\(870\) 0 0
\(871\) 4.84597 + 4.84597i 0.164199 + 0.164199i
\(872\) 0 0
\(873\) −2.23862 + 2.23862i −0.0757659 + 0.0757659i
\(874\) 0 0
\(875\) −1.17548 + 2.83785i −0.0397383 + 0.0959369i
\(876\) 0 0
\(877\) 30.6215 12.6839i 1.03402 0.428304i 0.199856 0.979825i \(-0.435953\pi\)
0.834160 + 0.551522i \(0.185953\pi\)
\(878\) 0 0
\(879\) 44.7149i 1.50820i
\(880\) 0 0
\(881\) 19.9074i 0.670697i 0.942094 + 0.335349i \(0.108854\pi\)
−0.942094 + 0.335349i \(0.891146\pi\)
\(882\) 0 0
\(883\) −8.54376 + 3.53894i −0.287521 + 0.119095i −0.521782 0.853079i \(-0.674733\pi\)
0.234262 + 0.972174i \(0.424733\pi\)
\(884\) 0 0
\(885\) 2.35760 5.69176i 0.0792500 0.191326i
\(886\) 0 0
\(887\) −8.05704 + 8.05704i −0.270529 + 0.270529i −0.829313 0.558784i \(-0.811268\pi\)
0.558784 + 0.829313i \(0.311268\pi\)
\(888\) 0 0
\(889\) 0.672361 + 0.672361i 0.0225503 + 0.0225503i
\(890\) 0 0
\(891\) 25.1475 + 10.4164i 0.842473 + 0.348964i
\(892\) 0 0
\(893\) 16.6187 + 40.1212i 0.556125 + 1.34260i
\(894\) 0 0
\(895\) 2.24875 0.0751675
\(896\) 0 0
\(897\) −1.40814 −0.0470163
\(898\) 0 0
\(899\) −0.194546 0.469675i −0.00648846 0.0156645i
\(900\) 0 0
\(901\) −4.81124 1.99288i −0.160286 0.0663925i
\(902\) 0 0
\(903\) −3.91336 3.91336i −0.130228 0.130228i
\(904\) 0 0
\(905\) 3.92797 3.92797i 0.130570 0.130570i
\(906\) 0 0
\(907\) 5.39566 13.0263i 0.179160 0.432531i −0.808631 0.588316i \(-0.799791\pi\)
0.987791 + 0.155786i \(0.0497910\pi\)
\(908\) 0 0
\(909\) −3.45336 + 1.43043i −0.114541 + 0.0474443i
\(910\) 0 0
\(911\) 28.0139i 0.928141i 0.885798 + 0.464071i \(0.153612\pi\)
−0.885798 + 0.464071i \(0.846388\pi\)
\(912\) 0 0
\(913\) 22.2985i 0.737971i
\(914\) 0 0
\(915\) −0.326483 + 0.135234i −0.0107932 + 0.00447069i
\(916\) 0 0
\(917\) −1.54070 + 3.71958i −0.0508784 + 0.122831i
\(918\) 0 0
\(919\) −18.5061 + 18.5061i −0.610461 + 0.610461i −0.943066 0.332605i \(-0.892072\pi\)
0.332605 + 0.943066i \(0.392072\pi\)
\(920\) 0 0
\(921\) 6.69543 + 6.69543i 0.220622 + 0.220622i
\(922\) 0 0
\(923\) 2.87696 + 1.19168i 0.0946964 + 0.0392246i
\(924\) 0 0
\(925\) −3.06506 7.39971i −0.100779 0.243301i
\(926\) 0 0
\(927\) −4.30053 −0.141248
\(928\) 0 0
\(929\) 49.4795 1.62337 0.811685 0.584095i \(-0.198550\pi\)
0.811685 + 0.584095i \(0.198550\pi\)
\(930\) 0 0
\(931\) 3.01360 + 7.27546i 0.0987666 + 0.238444i
\(932\) 0 0
\(933\) 30.3987 + 12.5916i 0.995210 + 0.412230i
\(934\) 0 0
\(935\) −0.965069 0.965069i −0.0315611 0.0315611i
\(936\) 0 0
\(937\) −23.8520 + 23.8520i −0.779212 + 0.779212i −0.979697 0.200485i \(-0.935748\pi\)
0.200485 + 0.979697i \(0.435748\pi\)
\(938\) 0 0
\(939\) −1.18894 + 2.87037i −0.0387997 + 0.0936709i
\(940\) 0 0
\(941\) 5.35340 2.21745i 0.174516 0.0722868i −0.293715 0.955893i \(-0.594892\pi\)
0.468231 + 0.883606i \(0.344892\pi\)
\(942\) 0 0
\(943\) 5.61004i 0.182688i
\(944\) 0 0
\(945\) 1.52221i 0.0495176i
\(946\) 0 0
\(947\) −27.8691 + 11.5438i −0.905624 + 0.375122i −0.786380 0.617743i \(-0.788047\pi\)
−0.119244 + 0.992865i \(0.538047\pi\)
\(948\) 0 0
\(949\) 1.51995 3.66947i 0.0493395 0.119116i
\(950\) 0 0
\(951\) 1.08970 1.08970i 0.0353358 0.0353358i
\(952\) 0 0
\(953\) −21.9681 21.9681i −0.711616 0.711616i 0.255257 0.966873i \(-0.417840\pi\)
−0.966873 + 0.255257i \(0.917840\pi\)
\(954\) 0 0
\(955\) 6.67257 + 2.76387i 0.215919 + 0.0894366i
\(956\) 0 0
\(957\) 16.6284 + 40.1444i 0.537519 + 1.29769i
\(958\) 0 0
\(959\) 12.6172 0.407429
\(960\) 0 0
\(961\) −30.9965 −0.999888
\(962\) 0 0
\(963\) −1.31494 3.17456i −0.0423735 0.102299i
\(964\) 0 0
\(965\) −3.03448 1.25692i −0.0976835 0.0404618i
\(966\) 0 0
\(967\) 23.9430 + 23.9430i 0.769956 + 0.769956i 0.978098 0.208143i \(-0.0667418\pi\)
−0.208143 + 0.978098i \(0.566742\pi\)
\(968\) 0 0
\(969\) 16.0238 16.0238i 0.514757 0.514757i
\(970\) 0 0
\(971\) 0.212732 0.513581i 0.00682690 0.0164816i −0.920429 0.390909i \(-0.872161\pi\)
0.927256 + 0.374427i \(0.122161\pi\)
\(972\) 0 0
\(973\) 3.97441 1.64625i 0.127414 0.0527765i
\(974\) 0 0
\(975\) 3.86902i 0.123908i
\(976\) 0 0
\(977\) 27.3205i 0.874061i −0.899447 0.437031i \(-0.856030\pi\)
0.899447 0.437031i \(-0.143970\pi\)
\(978\) 0 0
\(979\) −20.3287 + 8.42040i −0.649707 + 0.269117i
\(980\) 0 0
\(981\) −1.71281 + 4.13508i −0.0546857 + 0.132023i
\(982\) 0 0
\(983\) 4.92072 4.92072i 0.156947 0.156947i −0.624266 0.781212i \(-0.714602\pi\)
0.781212 + 0.624266i \(0.214602\pi\)
\(984\) 0 0
\(985\) 3.24327 + 3.24327i 0.103339 + 0.103339i
\(986\) 0 0
\(987\) 9.25100 + 3.83189i 0.294463 + 0.121970i
\(988\) 0 0
\(989\) 2.08171 + 5.02570i 0.0661946 + 0.159808i
\(990\) 0 0
\(991\) −43.4505 −1.38025 −0.690126 0.723689i \(-0.742445\pi\)
−0.690126 + 0.723689i \(0.742445\pi\)
\(992\) 0 0
\(993\) 4.58977 0.145652
\(994\) 0 0
\(995\) −3.17360 7.66174i −0.100610 0.242893i
\(996\) 0 0
\(997\) −15.2304 6.30863i −0.482351 0.199796i 0.128239 0.991743i \(-0.459068\pi\)
−0.610590 + 0.791947i \(0.709068\pi\)
\(998\) 0 0
\(999\) −5.66833 5.66833i −0.179338 0.179338i
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 896.2.u.c.113.4 52
4.3 odd 2 224.2.u.c.197.12 yes 52
32.13 even 8 inner 896.2.u.c.785.4 52
32.19 odd 8 224.2.u.c.141.12 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.u.c.141.12 52 32.19 odd 8
224.2.u.c.197.12 yes 52 4.3 odd 2
896.2.u.c.113.4 52 1.1 even 1 trivial
896.2.u.c.785.4 52 32.13 even 8 inner