Properties

Label 980.2.i.l.961.2
Level $980$
Weight $2$
Character 980.961
Analytic conductor $7.825$
Analytic rank $0$
Dimension $4$
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] = [980,2,Mod(361,980)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("980.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 980.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.82533939809\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 980.961
Dual form 980.2.i.l.361.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.20711 + 2.09077i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-1.41421 + 2.44949i) q^{9} +O(q^{10})\) \(q+(1.20711 + 2.09077i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(-1.41421 + 2.44949i) q^{9} +(-0.914214 - 1.58346i) q^{11} -6.41421 q^{13} -2.41421 q^{15} +(1.79289 + 3.10538i) q^{17} +(-3.82843 + 6.63103i) q^{19} +(-1.70711 + 2.95680i) q^{23} +(-0.500000 - 0.866025i) q^{25} +0.414214 q^{27} -4.65685 q^{29} +(3.70711 + 6.42090i) q^{31} +(2.20711 - 3.82282i) q^{33} +(0.292893 - 0.507306i) q^{37} +(-7.74264 - 13.4106i) q^{39} -3.41421 q^{41} +0.343146 q^{43} +(-1.41421 - 2.44949i) q^{45} +(5.44975 - 9.43924i) q^{47} +(-4.32843 + 7.49706i) q^{51} +(6.12132 + 10.6024i) q^{53} +1.82843 q^{55} -18.4853 q^{57} +(0.292893 + 0.507306i) q^{59} +(5.41421 - 9.37769i) q^{61} +(3.20711 - 5.55487i) q^{65} +(-1.53553 - 2.65962i) q^{67} -8.24264 q^{69} -10.4853 q^{71} +(5.41421 + 9.37769i) q^{73} +(1.20711 - 2.09077i) q^{75} +(7.57107 - 13.1135i) q^{79} +(4.74264 + 8.21449i) q^{81} +8.00000 q^{83} -3.58579 q^{85} +(-5.62132 - 9.73641i) q^{87} +(-8.48528 + 14.6969i) q^{89} +(-8.94975 + 15.5014i) q^{93} +(-3.82843 - 6.63103i) q^{95} +9.72792 q^{97} +5.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{5} + 2 q^{11} - 20 q^{13} - 4 q^{15} + 10 q^{17} - 4 q^{19} - 4 q^{23} - 2 q^{25} - 4 q^{27} + 4 q^{29} + 12 q^{31} + 6 q^{33} + 4 q^{37} - 14 q^{39} - 8 q^{41} + 24 q^{43} + 2 q^{47} - 6 q^{51} + 16 q^{53} - 4 q^{55} - 40 q^{57} + 4 q^{59} + 16 q^{61} + 10 q^{65} + 8 q^{67} - 16 q^{69} - 8 q^{71} + 16 q^{73} + 2 q^{75} + 2 q^{79} + 2 q^{81} + 32 q^{83} - 20 q^{85} - 14 q^{87} - 16 q^{93} - 4 q^{95} - 12 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\) \(491\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 1.20711 + 2.09077i 0.696923 + 1.20711i 0.969528 + 0.244981i \(0.0787816\pi\)
−0.272605 + 0.962126i \(0.587885\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.41421 + 2.44949i −0.471405 + 0.816497i
\(10\) 0 0
\(11\) −0.914214 1.58346i −0.275646 0.477432i 0.694652 0.719346i \(-0.255558\pi\)
−0.970298 + 0.241913i \(0.922225\pi\)
\(12\) 0 0
\(13\) −6.41421 −1.77898 −0.889491 0.456952i \(-0.848941\pi\)
−0.889491 + 0.456952i \(0.848941\pi\)
\(14\) 0 0
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) 1.79289 + 3.10538i 0.434840 + 0.753166i 0.997283 0.0736709i \(-0.0234714\pi\)
−0.562442 + 0.826837i \(0.690138\pi\)
\(18\) 0 0
\(19\) −3.82843 + 6.63103i −0.878301 + 1.52126i −0.0250976 + 0.999685i \(0.507990\pi\)
−0.853204 + 0.521578i \(0.825344\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.70711 + 2.95680i −0.355956 + 0.616535i −0.987281 0.158984i \(-0.949178\pi\)
0.631325 + 0.775519i \(0.282511\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −4.65685 −0.864756 −0.432378 0.901692i \(-0.642325\pi\)
−0.432378 + 0.901692i \(0.642325\pi\)
\(30\) 0 0
\(31\) 3.70711 + 6.42090i 0.665816 + 1.15323i 0.979063 + 0.203556i \(0.0652498\pi\)
−0.313247 + 0.949672i \(0.601417\pi\)
\(32\) 0 0
\(33\) 2.20711 3.82282i 0.384208 0.665468i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.292893 0.507306i 0.0481513 0.0834006i −0.840945 0.541120i \(-0.818000\pi\)
0.889097 + 0.457720i \(0.151334\pi\)
\(38\) 0 0
\(39\) −7.74264 13.4106i −1.23981 2.14742i
\(40\) 0 0
\(41\) −3.41421 −0.533211 −0.266605 0.963806i \(-0.585902\pi\)
−0.266605 + 0.963806i \(0.585902\pi\)
\(42\) 0 0
\(43\) 0.343146 0.0523292 0.0261646 0.999658i \(-0.491671\pi\)
0.0261646 + 0.999658i \(0.491671\pi\)
\(44\) 0 0
\(45\) −1.41421 2.44949i −0.210819 0.365148i
\(46\) 0 0
\(47\) 5.44975 9.43924i 0.794927 1.37685i −0.127958 0.991780i \(-0.540842\pi\)
0.922885 0.385075i \(-0.125824\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.32843 + 7.49706i −0.606101 + 1.04980i
\(52\) 0 0
\(53\) 6.12132 + 10.6024i 0.840828 + 1.45636i 0.889196 + 0.457527i \(0.151265\pi\)
−0.0483676 + 0.998830i \(0.515402\pi\)
\(54\) 0 0
\(55\) 1.82843 0.246545
\(56\) 0 0
\(57\) −18.4853 −2.44844
\(58\) 0 0
\(59\) 0.292893 + 0.507306i 0.0381314 + 0.0660456i 0.884461 0.466614i \(-0.154526\pi\)
−0.846330 + 0.532659i \(0.821193\pi\)
\(60\) 0 0
\(61\) 5.41421 9.37769i 0.693219 1.20069i −0.277558 0.960709i \(-0.589525\pi\)
0.970777 0.239982i \(-0.0771415\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.20711 5.55487i 0.397793 0.688997i
\(66\) 0 0
\(67\) −1.53553 2.65962i −0.187595 0.324925i 0.756853 0.653586i \(-0.226736\pi\)
−0.944448 + 0.328661i \(0.893403\pi\)
\(68\) 0 0
\(69\) −8.24264 −0.992297
\(70\) 0 0
\(71\) −10.4853 −1.24437 −0.622187 0.782869i \(-0.713756\pi\)
−0.622187 + 0.782869i \(0.713756\pi\)
\(72\) 0 0
\(73\) 5.41421 + 9.37769i 0.633686 + 1.09758i 0.986792 + 0.161993i \(0.0517921\pi\)
−0.353106 + 0.935583i \(0.614875\pi\)
\(74\) 0 0
\(75\) 1.20711 2.09077i 0.139385 0.241421i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.57107 13.1135i 0.851812 1.47538i −0.0277599 0.999615i \(-0.508837\pi\)
0.879572 0.475766i \(-0.157829\pi\)
\(80\) 0 0
\(81\) 4.74264 + 8.21449i 0.526960 + 0.912722i
\(82\) 0 0
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) −3.58579 −0.388933
\(86\) 0 0
\(87\) −5.62132 9.73641i −0.602669 1.04385i
\(88\) 0 0
\(89\) −8.48528 + 14.6969i −0.899438 + 1.55787i −0.0712241 + 0.997460i \(0.522691\pi\)
−0.828214 + 0.560412i \(0.810643\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −8.94975 + 15.5014i −0.928046 + 1.60742i
\(94\) 0 0
\(95\) −3.82843 6.63103i −0.392788 0.680329i
\(96\) 0 0
\(97\) 9.72792 0.987721 0.493860 0.869541i \(-0.335585\pi\)
0.493860 + 0.869541i \(0.335585\pi\)
\(98\) 0 0
\(99\) 5.17157 0.519763
\(100\) 0 0
\(101\) 0.414214 + 0.717439i 0.0412158 + 0.0713878i 0.885897 0.463881i \(-0.153544\pi\)
−0.844682 + 0.535269i \(0.820210\pi\)
\(102\) 0 0
\(103\) 2.20711 3.82282i 0.217473 0.376674i −0.736562 0.676370i \(-0.763552\pi\)
0.954035 + 0.299696i \(0.0968854\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.58579 9.67487i 0.539998 0.935305i −0.458905 0.888485i \(-0.651758\pi\)
0.998903 0.0468193i \(-0.0149085\pi\)
\(108\) 0 0
\(109\) −4.50000 7.79423i −0.431022 0.746552i 0.565940 0.824447i \(-0.308513\pi\)
−0.996962 + 0.0778949i \(0.975180\pi\)
\(110\) 0 0
\(111\) 1.41421 0.134231
\(112\) 0 0
\(113\) 9.07107 0.853334 0.426667 0.904409i \(-0.359688\pi\)
0.426667 + 0.904409i \(0.359688\pi\)
\(114\) 0 0
\(115\) −1.70711 2.95680i −0.159189 0.275723i
\(116\) 0 0
\(117\) 9.07107 15.7116i 0.838621 1.45253i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.82843 6.63103i 0.348039 0.602821i
\(122\) 0 0
\(123\) −4.12132 7.13834i −0.371607 0.643642i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −2.10051 −0.186390 −0.0931948 0.995648i \(-0.529708\pi\)
−0.0931948 + 0.995648i \(0.529708\pi\)
\(128\) 0 0
\(129\) 0.414214 + 0.717439i 0.0364695 + 0.0631670i
\(130\) 0 0
\(131\) −10.1213 + 17.5306i −0.884304 + 1.53166i −0.0377944 + 0.999286i \(0.512033\pi\)
−0.846509 + 0.532374i \(0.821300\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.207107 + 0.358719i −0.0178249 + 0.0308737i
\(136\) 0 0
\(137\) 10.4853 + 18.1610i 0.895818 + 1.55160i 0.832789 + 0.553591i \(0.186743\pi\)
0.0630291 + 0.998012i \(0.479924\pi\)
\(138\) 0 0
\(139\) −5.89949 −0.500389 −0.250194 0.968196i \(-0.580495\pi\)
−0.250194 + 0.968196i \(0.580495\pi\)
\(140\) 0 0
\(141\) 26.3137 2.21601
\(142\) 0 0
\(143\) 5.86396 + 10.1567i 0.490369 + 0.849344i
\(144\) 0 0
\(145\) 2.32843 4.03295i 0.193365 0.334919i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.2426 + 17.7408i −0.839110 + 1.45338i 0.0515300 + 0.998671i \(0.483590\pi\)
−0.890640 + 0.454709i \(0.849743\pi\)
\(150\) 0 0
\(151\) 3.08579 + 5.34474i 0.251118 + 0.434949i 0.963834 0.266504i \(-0.0858685\pi\)
−0.712716 + 0.701453i \(0.752535\pi\)
\(152\) 0 0
\(153\) −10.1421 −0.819943
\(154\) 0 0
\(155\) −7.41421 −0.595524
\(156\) 0 0
\(157\) −3.24264 5.61642i −0.258791 0.448239i 0.707127 0.707086i \(-0.249991\pi\)
−0.965918 + 0.258847i \(0.916657\pi\)
\(158\) 0 0
\(159\) −14.7782 + 25.5965i −1.17199 + 2.02994i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.77817 8.27604i 0.374256 0.648230i −0.615960 0.787778i \(-0.711232\pi\)
0.990215 + 0.139548i \(0.0445650\pi\)
\(164\) 0 0
\(165\) 2.20711 + 3.82282i 0.171823 + 0.297606i
\(166\) 0 0
\(167\) 0.414214 0.0320528 0.0160264 0.999872i \(-0.494898\pi\)
0.0160264 + 0.999872i \(0.494898\pi\)
\(168\) 0 0
\(169\) 28.1421 2.16478
\(170\) 0 0
\(171\) −10.8284 18.7554i −0.828071 1.43426i
\(172\) 0 0
\(173\) −6.27817 + 10.8741i −0.477321 + 0.826744i −0.999662 0.0259926i \(-0.991725\pi\)
0.522341 + 0.852736i \(0.325059\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −0.707107 + 1.22474i −0.0531494 + 0.0920575i
\(178\) 0 0
\(179\) 5.24264 + 9.08052i 0.391853 + 0.678710i 0.992694 0.120659i \(-0.0385007\pi\)
−0.600841 + 0.799369i \(0.705167\pi\)
\(180\) 0 0
\(181\) −10.2426 −0.761329 −0.380665 0.924713i \(-0.624305\pi\)
−0.380665 + 0.924713i \(0.624305\pi\)
\(182\) 0 0
\(183\) 26.1421 1.93248
\(184\) 0 0
\(185\) 0.292893 + 0.507306i 0.0215339 + 0.0372979i
\(186\) 0 0
\(187\) 3.27817 5.67796i 0.239724 0.415214i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.67157 + 4.62730i −0.193308 + 0.334820i −0.946345 0.323159i \(-0.895255\pi\)
0.753036 + 0.657979i \(0.228588\pi\)
\(192\) 0 0
\(193\) 2.82843 + 4.89898i 0.203595 + 0.352636i 0.949684 0.313210i \(-0.101404\pi\)
−0.746089 + 0.665846i \(0.768071\pi\)
\(194\) 0 0
\(195\) 15.4853 1.10892
\(196\) 0 0
\(197\) 3.55635 0.253379 0.126690 0.991942i \(-0.459565\pi\)
0.126690 + 0.991942i \(0.459565\pi\)
\(198\) 0 0
\(199\) −0.636039 1.10165i −0.0450876 0.0780940i 0.842601 0.538539i \(-0.181023\pi\)
−0.887688 + 0.460445i \(0.847690\pi\)
\(200\) 0 0
\(201\) 3.70711 6.42090i 0.261479 0.452895i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.70711 2.95680i 0.119230 0.206512i
\(206\) 0 0
\(207\) −4.82843 8.36308i −0.335599 0.581274i
\(208\) 0 0
\(209\) 14.0000 0.968400
\(210\) 0 0
\(211\) −1.68629 −0.116089 −0.0580445 0.998314i \(-0.518487\pi\)
−0.0580445 + 0.998314i \(0.518487\pi\)
\(212\) 0 0
\(213\) −12.6569 21.9223i −0.867233 1.50209i
\(214\) 0 0
\(215\) −0.171573 + 0.297173i −0.0117012 + 0.0202670i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −13.0711 + 22.6398i −0.883261 + 1.52985i
\(220\) 0 0
\(221\) −11.5000 19.9186i −0.773574 1.33987i
\(222\) 0 0
\(223\) −9.92893 −0.664890 −0.332445 0.943123i \(-0.607874\pi\)
−0.332445 + 0.943123i \(0.607874\pi\)
\(224\) 0 0
\(225\) 2.82843 0.188562
\(226\) 0 0
\(227\) −8.86396 15.3528i −0.588322 1.01900i −0.994452 0.105188i \(-0.966456\pi\)
0.406131 0.913815i \(-0.366878\pi\)
\(228\) 0 0
\(229\) 2.05025 3.55114i 0.135485 0.234666i −0.790298 0.612723i \(-0.790074\pi\)
0.925782 + 0.378057i \(0.123408\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.41421 12.8418i 0.485721 0.841294i −0.514144 0.857704i \(-0.671890\pi\)
0.999865 + 0.0164099i \(0.00522367\pi\)
\(234\) 0 0
\(235\) 5.44975 + 9.43924i 0.355502 + 0.615748i
\(236\) 0 0
\(237\) 36.5563 2.37459
\(238\) 0 0
\(239\) 19.8284 1.28259 0.641297 0.767293i \(-0.278397\pi\)
0.641297 + 0.767293i \(0.278397\pi\)
\(240\) 0 0
\(241\) 13.7782 + 23.8645i 0.887530 + 1.53725i 0.842786 + 0.538249i \(0.180914\pi\)
0.0447444 + 0.998998i \(0.485753\pi\)
\(242\) 0 0
\(243\) −10.8284 + 18.7554i −0.694644 + 1.20316i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 24.5563 42.5328i 1.56248 2.70630i
\(248\) 0 0
\(249\) 9.65685 + 16.7262i 0.611978 + 1.05998i
\(250\) 0 0
\(251\) 12.9289 0.816067 0.408033 0.912967i \(-0.366215\pi\)
0.408033 + 0.912967i \(0.366215\pi\)
\(252\) 0 0
\(253\) 6.24264 0.392471
\(254\) 0 0
\(255\) −4.32843 7.49706i −0.271057 0.469484i
\(256\) 0 0
\(257\) 6.89949 11.9503i 0.430379 0.745438i −0.566527 0.824043i \(-0.691713\pi\)
0.996906 + 0.0786054i \(0.0250467\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 6.58579 11.4069i 0.407650 0.706070i
\(262\) 0 0
\(263\) −5.00000 8.66025i −0.308313 0.534014i 0.669680 0.742650i \(-0.266431\pi\)
−0.977993 + 0.208635i \(0.933098\pi\)
\(264\) 0 0
\(265\) −12.2426 −0.752059
\(266\) 0 0
\(267\) −40.9706 −2.50736
\(268\) 0 0
\(269\) 4.36396 + 7.55860i 0.266075 + 0.460856i 0.967845 0.251548i \(-0.0809396\pi\)
−0.701769 + 0.712404i \(0.747606\pi\)
\(270\) 0 0
\(271\) −2.48528 + 4.30463i −0.150970 + 0.261488i −0.931584 0.363525i \(-0.881573\pi\)
0.780614 + 0.625013i \(0.214906\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.914214 + 1.58346i −0.0551292 + 0.0954865i
\(276\) 0 0
\(277\) 2.29289 + 3.97141i 0.137767 + 0.238619i 0.926651 0.375923i \(-0.122674\pi\)
−0.788884 + 0.614542i \(0.789341\pi\)
\(278\) 0 0
\(279\) −20.9706 −1.25547
\(280\) 0 0
\(281\) −1.00000 −0.0596550 −0.0298275 0.999555i \(-0.509496\pi\)
−0.0298275 + 0.999555i \(0.509496\pi\)
\(282\) 0 0
\(283\) −13.6213 23.5928i −0.809703 1.40245i −0.913069 0.407804i \(-0.866295\pi\)
0.103366 0.994643i \(-0.467039\pi\)
\(284\) 0 0
\(285\) 9.24264 16.0087i 0.547487 0.948275i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.07107 3.58719i 0.121828 0.211011i
\(290\) 0 0
\(291\) 11.7426 + 20.3389i 0.688366 + 1.19228i
\(292\) 0 0
\(293\) 11.5858 0.676849 0.338424 0.940994i \(-0.390106\pi\)
0.338424 + 0.940994i \(0.390106\pi\)
\(294\) 0 0
\(295\) −0.585786 −0.0341058
\(296\) 0 0
\(297\) −0.378680 0.655892i −0.0219732 0.0380587i
\(298\) 0 0
\(299\) 10.9497 18.9655i 0.633240 1.09680i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.00000 + 1.73205i −0.0574485 + 0.0995037i
\(304\) 0 0
\(305\) 5.41421 + 9.37769i 0.310017 + 0.536965i
\(306\) 0 0
\(307\) −6.41421 −0.366079 −0.183039 0.983106i \(-0.558594\pi\)
−0.183039 + 0.983106i \(0.558594\pi\)
\(308\) 0 0
\(309\) 10.6569 0.606247
\(310\) 0 0
\(311\) −13.4853 23.3572i −0.764680 1.32446i −0.940416 0.340027i \(-0.889564\pi\)
0.175735 0.984437i \(-0.443770\pi\)
\(312\) 0 0
\(313\) −7.27817 + 12.6062i −0.411387 + 0.712543i −0.995042 0.0994591i \(-0.968289\pi\)
0.583655 + 0.812002i \(0.301622\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.48528 + 16.4290i −0.532746 + 0.922744i 0.466522 + 0.884509i \(0.345507\pi\)
−0.999269 + 0.0382345i \(0.987827\pi\)
\(318\) 0 0
\(319\) 4.25736 + 7.37396i 0.238366 + 0.412863i
\(320\) 0 0
\(321\) 26.9706 1.50535
\(322\) 0 0
\(323\) −27.4558 −1.52768
\(324\) 0 0
\(325\) 3.20711 + 5.55487i 0.177898 + 0.308129i
\(326\) 0 0
\(327\) 10.8640 18.8169i 0.600778 1.04058i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.414214 0.717439i 0.0227672 0.0394340i −0.854417 0.519587i \(-0.826086\pi\)
0.877185 + 0.480153i \(0.159419\pi\)
\(332\) 0 0
\(333\) 0.828427 + 1.43488i 0.0453975 + 0.0786308i
\(334\) 0 0
\(335\) 3.07107 0.167790
\(336\) 0 0
\(337\) 6.72792 0.366493 0.183247 0.983067i \(-0.441339\pi\)
0.183247 + 0.983067i \(0.441339\pi\)
\(338\) 0 0
\(339\) 10.9497 + 18.9655i 0.594709 + 1.03007i
\(340\) 0 0
\(341\) 6.77817 11.7401i 0.367059 0.635764i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.12132 7.13834i 0.221884 0.384315i
\(346\) 0 0
\(347\) 2.29289 + 3.97141i 0.123089 + 0.213196i 0.920984 0.389600i \(-0.127387\pi\)
−0.797895 + 0.602796i \(0.794053\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) −2.65685 −0.141812
\(352\) 0 0
\(353\) 6.03553 + 10.4539i 0.321239 + 0.556402i 0.980744 0.195298i \(-0.0625674\pi\)
−0.659505 + 0.751700i \(0.729234\pi\)
\(354\) 0 0
\(355\) 5.24264 9.08052i 0.278250 0.481944i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.65685 9.79796i 0.298557 0.517116i −0.677249 0.735754i \(-0.736828\pi\)
0.975806 + 0.218638i \(0.0701613\pi\)
\(360\) 0 0
\(361\) −19.8137 34.3183i −1.04283 1.80623i
\(362\) 0 0
\(363\) 18.4853 0.970226
\(364\) 0 0
\(365\) −10.8284 −0.566786
\(366\) 0 0
\(367\) −2.86396 4.96053i −0.149498 0.258937i 0.781544 0.623850i \(-0.214432\pi\)
−0.931042 + 0.364912i \(0.881099\pi\)
\(368\) 0 0
\(369\) 4.82843 8.36308i 0.251358 0.435365i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.242641 0.420266i 0.0125635 0.0217605i −0.859675 0.510841i \(-0.829334\pi\)
0.872239 + 0.489080i \(0.162667\pi\)
\(374\) 0 0
\(375\) 1.20711 + 2.09077i 0.0623347 + 0.107967i
\(376\) 0 0
\(377\) 29.8701 1.53839
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −2.53553 4.39167i −0.129899 0.224992i
\(382\) 0 0
\(383\) −10.2426 + 17.7408i −0.523374 + 0.906511i 0.476255 + 0.879307i \(0.341994\pi\)
−0.999630 + 0.0272042i \(0.991340\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.485281 + 0.840532i −0.0246682 + 0.0427266i
\(388\) 0 0
\(389\) 7.57107 + 13.1135i 0.383868 + 0.664880i 0.991611 0.129254i \(-0.0412583\pi\)
−0.607743 + 0.794134i \(0.707925\pi\)
\(390\) 0 0
\(391\) −12.2426 −0.619137
\(392\) 0 0
\(393\) −48.8701 −2.46517
\(394\) 0 0
\(395\) 7.57107 + 13.1135i 0.380942 + 0.659810i
\(396\) 0 0
\(397\) 5.79289 10.0336i 0.290737 0.503571i −0.683247 0.730187i \(-0.739433\pi\)
0.973984 + 0.226616i \(0.0727662\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.74264 + 15.1427i −0.436587 + 0.756190i −0.997424 0.0717360i \(-0.977146\pi\)
0.560837 + 0.827926i \(0.310479\pi\)
\(402\) 0 0
\(403\) −23.7782 41.1850i −1.18448 2.05157i
\(404\) 0 0
\(405\) −9.48528 −0.471327
\(406\) 0 0
\(407\) −1.07107 −0.0530909
\(408\) 0 0
\(409\) 13.5858 + 23.5313i 0.671774 + 1.16355i 0.977401 + 0.211395i \(0.0678007\pi\)
−0.305627 + 0.952151i \(0.598866\pi\)
\(410\) 0 0
\(411\) −25.3137 + 43.8446i −1.24863 + 2.16270i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 + 6.92820i −0.196352 + 0.340092i
\(416\) 0 0
\(417\) −7.12132 12.3345i −0.348733 0.604023i
\(418\) 0 0
\(419\) −21.5563 −1.05310 −0.526548 0.850145i \(-0.676514\pi\)
−0.526548 + 0.850145i \(0.676514\pi\)
\(420\) 0 0
\(421\) −2.31371 −0.112763 −0.0563816 0.998409i \(-0.517956\pi\)
−0.0563816 + 0.998409i \(0.517956\pi\)
\(422\) 0 0
\(423\) 15.4142 + 26.6982i 0.749465 + 1.29811i
\(424\) 0 0
\(425\) 1.79289 3.10538i 0.0869681 0.150633i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −14.1569 + 24.5204i −0.683499 + 1.18386i
\(430\) 0 0
\(431\) 2.08579 + 3.61269i 0.100469 + 0.174017i 0.911878 0.410462i \(-0.134632\pi\)
−0.811409 + 0.584479i \(0.801299\pi\)
\(432\) 0 0
\(433\) −34.2843 −1.64760 −0.823798 0.566883i \(-0.808149\pi\)
−0.823798 + 0.566883i \(0.808149\pi\)
\(434\) 0 0
\(435\) 11.2426 0.539043
\(436\) 0 0
\(437\) −13.0711 22.6398i −0.625274 1.08301i
\(438\) 0 0
\(439\) 7.87868 13.6463i 0.376029 0.651301i −0.614452 0.788955i \(-0.710623\pi\)
0.990480 + 0.137653i \(0.0439560\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.72792 + 13.3852i −0.367165 + 0.635948i −0.989121 0.147104i \(-0.953005\pi\)
0.621956 + 0.783052i \(0.286338\pi\)
\(444\) 0 0
\(445\) −8.48528 14.6969i −0.402241 0.696702i
\(446\) 0 0
\(447\) −49.4558 −2.33918
\(448\) 0 0
\(449\) 18.1716 0.857570 0.428785 0.903407i \(-0.358942\pi\)
0.428785 + 0.903407i \(0.358942\pi\)
\(450\) 0 0
\(451\) 3.12132 + 5.40629i 0.146977 + 0.254572i
\(452\) 0 0
\(453\) −7.44975 + 12.9033i −0.350020 + 0.606252i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.464466 0.804479i 0.0217268 0.0376319i −0.854958 0.518698i \(-0.826417\pi\)
0.876684 + 0.481066i \(0.159750\pi\)
\(458\) 0 0
\(459\) 0.742641 + 1.28629i 0.0346635 + 0.0600389i
\(460\) 0 0
\(461\) −33.6569 −1.56756 −0.783778 0.621041i \(-0.786710\pi\)
−0.783778 + 0.621041i \(0.786710\pi\)
\(462\) 0 0
\(463\) 37.4558 1.74072 0.870360 0.492415i \(-0.163886\pi\)
0.870360 + 0.492415i \(0.163886\pi\)
\(464\) 0 0
\(465\) −8.94975 15.5014i −0.415035 0.718861i
\(466\) 0 0
\(467\) −9.44975 + 16.3674i −0.437282 + 0.757395i −0.997479 0.0709647i \(-0.977392\pi\)
0.560197 + 0.828360i \(0.310726\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 7.82843 13.5592i 0.360715 0.624777i
\(472\) 0 0
\(473\) −0.313708 0.543359i −0.0144243 0.0249837i
\(474\) 0 0
\(475\) 7.65685 0.351321
\(476\) 0 0
\(477\) −34.6274 −1.58548
\(478\) 0 0
\(479\) 3.29289 + 5.70346i 0.150456 + 0.260598i 0.931395 0.364010i \(-0.118592\pi\)
−0.780939 + 0.624607i \(0.785259\pi\)
\(480\) 0 0
\(481\) −1.87868 + 3.25397i −0.0856604 + 0.148368i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.86396 + 8.42463i −0.220861 + 0.382543i
\(486\) 0 0
\(487\) 2.29289 + 3.97141i 0.103901 + 0.179962i 0.913289 0.407313i \(-0.133534\pi\)
−0.809388 + 0.587275i \(0.800201\pi\)
\(488\) 0 0
\(489\) 23.0711 1.04331
\(490\) 0 0
\(491\) −39.2843 −1.77287 −0.886437 0.462849i \(-0.846827\pi\)
−0.886437 + 0.462849i \(0.846827\pi\)
\(492\) 0 0
\(493\) −8.34924 14.4613i −0.376031 0.651305i
\(494\) 0 0
\(495\) −2.58579 + 4.47871i −0.116222 + 0.201303i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −15.8137 + 27.3901i −0.707919 + 1.22615i 0.257709 + 0.966223i \(0.417032\pi\)
−0.965628 + 0.259929i \(0.916301\pi\)
\(500\) 0 0
\(501\) 0.500000 + 0.866025i 0.0223384 + 0.0386912i
\(502\) 0 0
\(503\) 9.04163 0.403146 0.201573 0.979473i \(-0.435395\pi\)
0.201573 + 0.979473i \(0.435395\pi\)
\(504\) 0 0
\(505\) −0.828427 −0.0368645
\(506\) 0 0
\(507\) 33.9706 + 58.8387i 1.50869 + 2.61312i
\(508\) 0 0
\(509\) −10.4645 + 18.1250i −0.463829 + 0.803376i −0.999148 0.0412743i \(-0.986858\pi\)
0.535319 + 0.844650i \(0.320192\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −1.58579 + 2.74666i −0.0700142 + 0.121268i
\(514\) 0 0
\(515\) 2.20711 + 3.82282i 0.0972567 + 0.168454i
\(516\) 0 0
\(517\) −19.9289 −0.876473
\(518\) 0 0
\(519\) −30.3137 −1.33062
\(520\) 0 0
\(521\) 14.6569 + 25.3864i 0.642128 + 1.11220i 0.984957 + 0.172801i \(0.0552818\pi\)
−0.342828 + 0.939398i \(0.611385\pi\)
\(522\) 0 0
\(523\) 15.0711 26.1039i 0.659012 1.14144i −0.321860 0.946787i \(-0.604308\pi\)
0.980872 0.194655i \(-0.0623586\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.2929 + 23.0240i −0.579048 + 1.00294i
\(528\) 0 0
\(529\) 5.67157 + 9.82345i 0.246590 + 0.427107i
\(530\) 0 0
\(531\) −1.65685 −0.0719014
\(532\) 0 0
\(533\) 21.8995 0.948572
\(534\) 0 0
\(535\) 5.58579 + 9.67487i 0.241495 + 0.418281i
\(536\) 0 0
\(537\) −12.6569 + 21.9223i −0.546184 + 0.946018i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.98528 15.5630i 0.386307 0.669104i −0.605642 0.795737i \(-0.707084\pi\)
0.991950 + 0.126633i \(0.0404171\pi\)
\(542\) 0 0
\(543\) −12.3640 21.4150i −0.530588 0.919006i
\(544\) 0 0
\(545\) 9.00000 0.385518
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) 0 0
\(549\) 15.3137 + 26.5241i 0.653573 + 1.13202i
\(550\) 0 0
\(551\) 17.8284 30.8797i 0.759517 1.31552i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −0.707107 + 1.22474i −0.0300150 + 0.0519875i
\(556\) 0 0
\(557\) −15.8995 27.5387i −0.673683 1.16685i −0.976852 0.213917i \(-0.931378\pi\)
0.303169 0.952937i \(-0.401955\pi\)
\(558\) 0 0
\(559\) −2.20101 −0.0930928
\(560\) 0 0
\(561\) 15.8284 0.668277
\(562\) 0 0
\(563\) 12.3137 + 21.3280i 0.518961 + 0.898867i 0.999757 + 0.0220344i \(0.00701433\pi\)
−0.480796 + 0.876832i \(0.659652\pi\)
\(564\) 0 0
\(565\) −4.53553 + 7.85578i −0.190811 + 0.330495i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.89949 + 6.75412i −0.163475 + 0.283148i −0.936113 0.351700i \(-0.885604\pi\)
0.772637 + 0.634848i \(0.218937\pi\)
\(570\) 0 0
\(571\) 0.414214 + 0.717439i 0.0173343 + 0.0300239i 0.874562 0.484913i \(-0.161149\pi\)
−0.857228 + 0.514937i \(0.827815\pi\)
\(572\) 0 0
\(573\) −12.8995 −0.538884
\(574\) 0 0
\(575\) 3.41421 0.142383
\(576\) 0 0
\(577\) 0.0355339 + 0.0615465i 0.00147930 + 0.00256222i 0.866764 0.498718i \(-0.166196\pi\)
−0.865285 + 0.501281i \(0.832862\pi\)
\(578\) 0 0
\(579\) −6.82843 + 11.8272i −0.283780 + 0.491521i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11.1924 19.3858i 0.463541 0.802877i
\(584\) 0 0
\(585\) 9.07107 + 15.7116i 0.375042 + 0.649593i
\(586\) 0 0
\(587\) 25.1716 1.03894 0.519471 0.854488i \(-0.326129\pi\)
0.519471 + 0.854488i \(0.326129\pi\)
\(588\) 0 0
\(589\) −56.7696 −2.33915
\(590\) 0 0
\(591\) 4.29289 + 7.43551i 0.176586 + 0.305856i
\(592\) 0 0
\(593\) −3.86396 + 6.69258i −0.158674 + 0.274831i −0.934391 0.356250i \(-0.884055\pi\)
0.775717 + 0.631081i \(0.217389\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.53553 2.65962i 0.0628452 0.108851i
\(598\) 0 0
\(599\) −13.3995 23.2086i −0.547489 0.948278i −0.998446 0.0557324i \(-0.982251\pi\)
0.450957 0.892546i \(-0.351083\pi\)
\(600\) 0 0
\(601\) −28.3431 −1.15614 −0.578071 0.815987i \(-0.696194\pi\)
−0.578071 + 0.815987i \(0.696194\pi\)
\(602\) 0 0
\(603\) 8.68629 0.353733
\(604\) 0 0
\(605\) 3.82843 + 6.63103i 0.155648 + 0.269590i
\(606\) 0 0
\(607\) 14.0355 24.3103i 0.569685 0.986723i −0.426912 0.904293i \(-0.640399\pi\)
0.996597 0.0824298i \(-0.0262680\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34.9558 + 60.5453i −1.41416 + 2.44940i
\(612\) 0 0
\(613\) 9.34315 + 16.1828i 0.377366 + 0.653617i 0.990678 0.136223i \(-0.0434965\pi\)
−0.613312 + 0.789841i \(0.710163\pi\)
\(614\) 0 0
\(615\) 8.24264 0.332375
\(616\) 0 0
\(617\) −12.8701 −0.518129 −0.259065 0.965860i \(-0.583414\pi\)
−0.259065 + 0.965860i \(0.583414\pi\)
\(618\) 0 0
\(619\) −21.4350 37.1266i −0.861547 1.49224i −0.870436 0.492282i \(-0.836163\pi\)
0.00888927 0.999960i \(-0.497170\pi\)
\(620\) 0 0
\(621\) −0.707107 + 1.22474i −0.0283752 + 0.0491473i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 16.8995 + 29.2708i 0.674901 + 1.16896i
\(628\) 0 0
\(629\) 2.10051 0.0837526
\(630\) 0 0
\(631\) −22.5147 −0.896297 −0.448148 0.893959i \(-0.647916\pi\)
−0.448148 + 0.893959i \(0.647916\pi\)
\(632\) 0 0
\(633\) −2.03553 3.52565i −0.0809052 0.140132i
\(634\) 0 0
\(635\) 1.05025 1.81909i 0.0416780 0.0721884i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 14.8284 25.6836i 0.586604 1.01603i
\(640\) 0 0
\(641\) −8.34315 14.4508i −0.329534 0.570770i 0.652885 0.757457i \(-0.273558\pi\)
−0.982420 + 0.186687i \(0.940225\pi\)
\(642\) 0 0
\(643\) 38.2132 1.50698 0.753491 0.657458i \(-0.228368\pi\)
0.753491 + 0.657458i \(0.228368\pi\)
\(644\) 0 0
\(645\) −0.828427 −0.0326193
\(646\) 0 0
\(647\) −13.5563 23.4803i −0.532955 0.923105i −0.999259 0.0384809i \(-0.987748\pi\)
0.466304 0.884624i \(-0.345585\pi\)
\(648\) 0 0
\(649\) 0.535534 0.927572i 0.0210215 0.0364104i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.6569 42.7069i 0.964897 1.67125i 0.255006 0.966939i \(-0.417922\pi\)
0.709891 0.704312i \(-0.248744\pi\)
\(654\) 0 0
\(655\) −10.1213 17.5306i −0.395473 0.684979i
\(656\) 0 0
\(657\) −30.6274 −1.19489
\(658\) 0 0
\(659\) −9.34315 −0.363957 −0.181979 0.983302i \(-0.558250\pi\)
−0.181979 + 0.983302i \(0.558250\pi\)
\(660\) 0 0
\(661\) −2.92893 5.07306i −0.113922 0.197319i 0.803426 0.595404i \(-0.203008\pi\)
−0.917348 + 0.398085i \(0.869675\pi\)
\(662\) 0 0
\(663\) 27.7635 48.0877i 1.07824 1.86757i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.94975 13.7694i 0.307815 0.533152i
\(668\) 0 0
\(669\) −11.9853 20.7591i −0.463378 0.802594i
\(670\) 0 0
\(671\) −19.7990 −0.764332
\(672\) 0 0
\(673\) −44.4853 −1.71478 −0.857391 0.514666i \(-0.827916\pi\)
−0.857391 + 0.514666i \(0.827916\pi\)
\(674\) 0 0
\(675\) −0.207107 0.358719i −0.00797154 0.0138071i
\(676\) 0 0
\(677\) −17.8640 + 30.9413i −0.686568 + 1.18917i 0.286374 + 0.958118i \(0.407550\pi\)
−0.972941 + 0.231052i \(0.925783\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 21.3995 37.0650i 0.820030 1.42033i
\(682\) 0 0
\(683\) 13.5563 + 23.4803i 0.518719 + 0.898448i 0.999763 + 0.0217518i \(0.00692435\pi\)
−0.481044 + 0.876696i \(0.659742\pi\)
\(684\) 0 0
\(685\) −20.9706 −0.801244
\(686\) 0 0
\(687\) 9.89949 0.377689
\(688\) 0 0
\(689\) −39.2635 68.0063i −1.49582 2.59083i
\(690\) 0 0
\(691\) 10.2426 17.7408i 0.389648 0.674891i −0.602754 0.797927i \(-0.705930\pi\)
0.992402 + 0.123036i \(0.0392632\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.94975 5.10911i 0.111890 0.193800i
\(696\) 0 0
\(697\) −6.12132 10.6024i −0.231862 0.401596i
\(698\) 0 0
\(699\) 35.7990 1.35404
\(700\) 0 0
\(701\) 17.4853 0.660410 0.330205 0.943909i \(-0.392882\pi\)
0.330205 + 0.943909i \(0.392882\pi\)
\(702\) 0 0
\(703\) 2.24264 + 3.88437i 0.0845828 + 0.146502i
\(704\) 0 0
\(705\) −13.1569 + 22.7883i −0.495516 + 0.858259i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 20.8137 36.0504i 0.781675 1.35390i −0.149290 0.988793i \(-0.547699\pi\)
0.930965 0.365108i \(-0.118968\pi\)
\(710\) 0 0
\(711\) 21.4142 + 37.0905i 0.803096 + 1.39100i
\(712\) 0 0
\(713\) −25.3137 −0.948006
\(714\) 0 0
\(715\) −11.7279 −0.438599
\(716\) 0 0
\(717\) 23.9350 + 41.4567i 0.893870 + 1.54823i
\(718\) 0 0
\(719\) 12.1213 20.9947i 0.452049 0.782972i −0.546464 0.837482i \(-0.684027\pi\)
0.998513 + 0.0545107i \(0.0173599\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −33.2635 + 57.6140i −1.23708 + 2.14269i
\(724\) 0 0
\(725\) 2.32843 + 4.03295i 0.0864756 + 0.149780i
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 0.615224 + 1.06560i 0.0227549 + 0.0394126i
\(732\) 0 0
\(733\) 14.3492 24.8536i 0.530001 0.917989i −0.469386 0.882993i \(-0.655525\pi\)
0.999387 0.0349962i \(-0.0111419\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.80761 + 4.86293i −0.103420 + 0.179128i
\(738\) 0 0
\(739\) −4.67157 8.09140i −0.171847 0.297647i 0.767219 0.641385i \(-0.221640\pi\)
−0.939065 + 0.343738i \(0.888307\pi\)
\(740\) 0 0
\(741\) 118.569 4.35572
\(742\) 0 0
\(743\) −23.0711 −0.846395 −0.423198 0.906037i \(-0.639092\pi\)
−0.423198 + 0.906037i \(0.639092\pi\)
\(744\) 0 0
\(745\) −10.2426 17.7408i −0.375261 0.649972i
\(746\) 0 0
\(747\) −11.3137 + 19.5959i −0.413947 + 0.716977i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.5711 23.5058i 0.495215 0.857738i −0.504769 0.863254i \(-0.668423\pi\)
0.999985 + 0.00551611i \(0.00175584\pi\)
\(752\) 0 0
\(753\) 15.6066 + 27.0314i 0.568736 + 0.985080i
\(754\) 0 0
\(755\) −6.17157 −0.224607
\(756\) 0 0
\(757\) 42.8284 1.55663 0.778313 0.627877i \(-0.216076\pi\)
0.778313 + 0.627877i \(0.216076\pi\)
\(758\) 0 0
\(759\) 7.53553 + 13.0519i 0.273523 + 0.473755i
\(760\) 0 0
\(761\) 17.4350 30.1984i 0.632019 1.09469i −0.355119 0.934821i \(-0.615560\pi\)
0.987138 0.159868i \(-0.0511070\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5.07107 8.78335i 0.183345 0.317563i
\(766\) 0 0
\(767\) −1.87868 3.25397i −0.0678352 0.117494i
\(768\) 0 0
\(769\) 0.142136 0.00512554 0.00256277 0.999997i \(-0.499184\pi\)
0.00256277 + 0.999997i \(0.499184\pi\)
\(770\) 0 0
\(771\) 33.3137 1.19976
\(772\) 0 0
\(773\) −9.13604 15.8241i −0.328600 0.569153i 0.653634 0.756811i \(-0.273244\pi\)
−0.982234 + 0.187658i \(0.939910\pi\)
\(774\) 0 0
\(775\) 3.70711 6.42090i 0.133163 0.230645i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 13.0711 22.6398i 0.468320 0.811153i
\(780\) 0 0
\(781\) 9.58579 + 16.6031i 0.343006 + 0.594105i
\(782\) 0 0
\(783\) −1.92893 −0.0689344
\(784\) 0 0
\(785\) 6.48528 0.231470
\(786\) 0 0
\(787\) −9.13604 15.8241i −0.325665 0.564068i 0.655982 0.754777i \(-0.272255\pi\)
−0.981647 + 0.190709i \(0.938921\pi\)
\(788\) 0 0
\(789\) 12.0711 20.9077i 0.429741 0.744334i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −34.7279 + 60.1505i −1.23322 + 2.13601i
\(794\) 0 0
\(795\) −14.7782 25.5965i −0.524128 0.907816i
\(796\) 0 0
\(797\) −1.38478 −0.0490513 −0.0245256 0.999699i \(-0.507808\pi\)
−0.0245256 + 0.999699i \(0.507808\pi\)
\(798\) 0 0
\(799\) 39.0833 1.38267
\(800\) 0 0
\(801\) −24.0000 41.5692i −0.847998 1.46878i
\(802\) 0 0
\(803\) 9.89949 17.1464i 0.349346 0.605084i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.5355 + 18.2481i −0.370868 + 0.642363i
\(808\) 0 0
\(809\) −19.9853 34.6155i −0.702645 1.21702i −0.967535 0.252738i \(-0.918669\pi\)
0.264890 0.964279i \(-0.414664\pi\)
\(810\) 0 0
\(811\) −11.5563 −0.405798 −0.202899 0.979200i \(-0.565036\pi\)
−0.202899 + 0.979200i \(0.565036\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) 0 0
\(815\) 4.77817 + 8.27604i 0.167372 + 0.289897i
\(816\) 0 0
\(817\) −1.31371 + 2.27541i −0.0459608 + 0.0796065i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −21.0858 + 36.5217i −0.735899 + 1.27461i 0.218429 + 0.975853i \(0.429907\pi\)
−0.954328 + 0.298762i \(0.903426\pi\)
\(822\) 0 0
\(823\) 14.2635 + 24.7050i 0.497193 + 0.861163i 0.999995 0.00323843i \(-0.00103082\pi\)
−0.502802 + 0.864402i \(0.667697\pi\)
\(824\) 0 0
\(825\) −4.41421 −0.153683
\(826\) 0 0
\(827\) 38.0416 1.32284 0.661419 0.750017i \(-0.269955\pi\)
0.661419 + 0.750017i \(0.269955\pi\)
\(828\) 0 0
\(829\) −1.36396 2.36245i −0.0473723 0.0820513i 0.841367 0.540464i \(-0.181751\pi\)
−0.888739 + 0.458413i \(0.848418\pi\)
\(830\) 0 0
\(831\) −5.53553 + 9.58783i −0.192026 + 0.332598i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −0.207107 + 0.358719i −0.00716723 + 0.0124140i
\(836\) 0 0
\(837\) 1.53553 + 2.65962i 0.0530758 + 0.0919300i
\(838\) 0 0
\(839\) 14.3848 0.496618 0.248309 0.968681i \(-0.420125\pi\)
0.248309 + 0.968681i \(0.420125\pi\)
\(840\) 0 0
\(841\) −7.31371 −0.252197
\(842\) 0 0
\(843\) −1.20711 2.09077i −0.0415750 0.0720100i
\(844\) 0 0
\(845\) −14.0711 + 24.3718i −0.484059 + 0.838416i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 32.8848 56.9581i 1.12860 1.95480i
\(850\) 0 0
\(851\) 1.00000 + 1.73205i 0.0342796 + 0.0593739i
\(852\) 0 0
\(853\) −26.2843 −0.899956 −0.449978 0.893040i \(-0.648568\pi\)
−0.449978 + 0.893040i \(0.648568\pi\)
\(854\) 0 0
\(855\) 21.6569 0.740649
\(856\) 0 0
\(857\) −7.58579 13.1390i −0.259126 0.448819i 0.706882 0.707331i \(-0.250101\pi\)
−0.966008 + 0.258513i \(0.916768\pi\)
\(858\) 0 0
\(859\) 2.72792 4.72490i 0.0930755 0.161211i −0.815728 0.578435i \(-0.803664\pi\)
0.908804 + 0.417224i \(0.136997\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.1213 24.4588i 0.480695 0.832589i −0.519059 0.854738i \(-0.673718\pi\)
0.999755 + 0.0221495i \(0.00705097\pi\)
\(864\) 0 0
\(865\) −6.27817 10.8741i −0.213464 0.369731i
\(866\) 0 0
\(867\) 10.0000 0.339618
\(868\) 0 0
\(869\) −27.6863 −0.939193
\(870\) 0 0
\(871\) 9.84924 + 17.0594i 0.333729 + 0.578035i
\(872\) 0 0
\(873\) −13.7574 + 23.8284i −0.465616 + 0.806471i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.6569 + 28.8505i −0.562462 + 0.974213i 0.434819 + 0.900518i \(0.356812\pi\)
−0.997281 + 0.0736948i \(0.976521\pi\)
\(878\) 0 0
\(879\) 13.9853 + 24.2232i 0.471712 + 0.817029i
\(880\) 0 0
\(881\) −0.284271 −0.00957734 −0.00478867 0.999989i \(-0.501524\pi\)
−0.00478867 + 0.999989i \(0.501524\pi\)
\(882\) 0 0
\(883\) 55.1127 1.85469 0.927345 0.374208i \(-0.122085\pi\)
0.927345 + 0.374208i \(0.122085\pi\)
\(884\) 0 0
\(885\) −0.707107 1.22474i −0.0237691 0.0411693i
\(886\) 0 0
\(887\) −10.7990 + 18.7044i −0.362595 + 0.628032i −0.988387 0.151957i \(-0.951442\pi\)
0.625792 + 0.779990i \(0.284776\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 8.67157 15.0196i 0.290509 0.503176i
\(892\) 0 0
\(893\) 41.7279 + 72.2749i 1.39637 + 2.41859i
\(894\) 0 0
\(895\) −10.4853 −0.350484
\(896\) 0 0
\(897\) 52.8701 1.76528
\(898\) 0 0
\(899\) −17.2635 29.9012i −0.575768 0.997260i
\(900\) 0 0
\(901\) −21.9497 + 38.0181i −0.731252 + 1.26657i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.12132 8.87039i 0.170238 0.294862i
\(906\) 0 0
\(907\) 21.2635 + 36.8294i 0.706041 + 1.22290i 0.966314 + 0.257365i \(0.0828541\pi\)
−0.260273 + 0.965535i \(0.583813\pi\)
\(908\) 0 0
\(909\) −2.34315 −0.0777172
\(910\) 0 0
\(911\) 23.5980 0.781836 0.390918 0.920426i \(-0.372158\pi\)
0.390918 + 0.920426i \(0.372158\pi\)
\(912\) 0 0
\(913\) −7.31371 12.6677i −0.242048 0.419240i
\(914\) 0 0
\(915\) −13.0711 + 22.6398i −0.432116 + 0.748447i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17.7426 30.7312i 0.585276 1.01373i −0.409565 0.912281i \(-0.634320\pi\)
0.994841 0.101446i \(-0.0323471\pi\)
\(920\) 0 0
\(921\) −7.74264 13.4106i −0.255129 0.441896i
\(922\) 0 0
\(923\) 67.2548 2.21372
\(924\) 0 0
\(925\) −0.585786 −0.0192605
\(926\) 0 0
\(927\) 6.24264 + 10.8126i 0.205035 + 0.355131i
\(928\) 0 0
\(929\) −23.3640 + 40.4676i −0.766547 + 1.32770i 0.172878 + 0.984943i \(0.444693\pi\)
−0.939425 + 0.342755i \(0.888640\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 32.5563 56.3893i 1.06585 1.84610i
\(934\) 0 0
\(935\) 3.27817 + 5.67796i 0.107208 + 0.185689i
\(936\) 0 0
\(937\) −17.7279 −0.579146 −0.289573 0.957156i \(-0.593513\pi\)
−0.289573 + 0.957156i \(0.593513\pi\)
\(938\) 0 0
\(939\) −35.1421 −1.14682
\(940\) 0 0
\(941\) −2.00000 3.46410i −0.0651981 0.112926i 0.831584 0.555399i \(-0.187435\pi\)
−0.896782 + 0.442473i \(0.854101\pi\)
\(942\) 0 0
\(943\) 5.82843 10.0951i 0.189800 0.328743i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.02082 + 13.8925i −0.260641 + 0.451444i −0.966413 0.256996i \(-0.917267\pi\)
0.705771 + 0.708440i \(0.250601\pi\)
\(948\) 0 0
\(949\) −34.7279 60.1505i −1.12732 1.95257i
\(950\) 0 0
\(951\) −45.7990 −1.48513
\(952\) 0 0
\(953\) 14.8701 0.481688 0.240844 0.970564i \(-0.422576\pi\)
0.240844 + 0.970564i \(0.422576\pi\)
\(954\) 0 0
\(955\) −2.67157 4.62730i −0.0864501 0.149736i
\(956\) 0 0
\(957\) −10.2782 + 17.8023i −0.332246 + 0.575467i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11.9853 + 20.7591i −0.386622 + 0.669649i
\(962\) 0 0
\(963\) 15.7990 + 27.3647i 0.509115 + 0.881814i
\(964\) 0 0
\(965\) −5.65685 −0.182101
\(966\) 0 0
\(967\) 43.4558 1.39745 0.698723 0.715392i \(-0.253752\pi\)
0.698723 + 0.715392i \(0.253752\pi\)
\(968\) 0 0
\(969\) −33.1421 57.4039i −1.06468 1.84408i
\(970\) 0 0
\(971\) 21.2635 36.8294i 0.682377 1.18191i −0.291877 0.956456i \(-0.594280\pi\)
0.974254 0.225455i \(-0.0723870\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −7.74264 + 13.4106i −0.247963 + 0.429484i
\(976\) 0 0
\(977\) −10.0711 17.4436i −0.322202 0.558070i 0.658740 0.752371i \(-0.271090\pi\)
−0.980942 + 0.194300i \(0.937756\pi\)
\(978\) 0 0
\(979\) 31.0294 0.991705
\(980\) 0 0
\(981\) 25.4558 0.812743
\(982\) 0 0
\(983\) 20.0061 + 34.6516i 0.638095 + 1.10521i 0.985850 + 0.167628i \(0.0536108\pi\)
−0.347755 + 0.937585i \(0.613056\pi\)
\(984\) 0 0
\(985\) −1.77817 + 3.07989i −0.0566574 + 0.0981334i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −0.585786 + 1.01461i −0.0186269 + 0.0322628i
\(990\) 0 0
\(991\) −1.34315 2.32640i −0.0426664 0.0739004i 0.843904 0.536495i \(-0.180252\pi\)
−0.886570 + 0.462594i \(0.846919\pi\)
\(992\) 0 0
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) 1.27208 0.0403276
\(996\) 0 0
\(997\) 18.4203 + 31.9049i 0.583377 + 1.01044i 0.995076 + 0.0991186i \(0.0316023\pi\)
−0.411699 + 0.911320i \(0.635064\pi\)
\(998\) 0 0
\(999\) 0.121320 0.210133i 0.00383841 0.00664831i
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 980.2.i.l.961.2 4
7.2 even 3 980.2.a.j.1.1 2
7.3 odd 6 980.2.i.k.361.1 4
7.4 even 3 inner 980.2.i.l.361.2 4
7.5 odd 6 980.2.a.k.1.2 yes 2
7.6 odd 2 980.2.i.k.961.1 4
21.2 odd 6 8820.2.a.bg.1.1 2
21.5 even 6 8820.2.a.bl.1.1 2
28.19 even 6 3920.2.a.bo.1.1 2
28.23 odd 6 3920.2.a.bx.1.2 2
35.2 odd 12 4900.2.e.q.2549.4 4
35.9 even 6 4900.2.a.z.1.2 2
35.12 even 12 4900.2.e.r.2549.1 4
35.19 odd 6 4900.2.a.x.1.1 2
35.23 odd 12 4900.2.e.q.2549.1 4
35.33 even 12 4900.2.e.r.2549.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.j.1.1 2 7.2 even 3
980.2.a.k.1.2 yes 2 7.5 odd 6
980.2.i.k.361.1 4 7.3 odd 6
980.2.i.k.961.1 4 7.6 odd 2
980.2.i.l.361.2 4 7.4 even 3 inner
980.2.i.l.961.2 4 1.1 even 1 trivial
3920.2.a.bo.1.1 2 28.19 even 6
3920.2.a.bx.1.2 2 28.23 odd 6
4900.2.a.x.1.1 2 35.19 odd 6
4900.2.a.z.1.2 2 35.9 even 6
4900.2.e.q.2549.1 4 35.23 odd 12
4900.2.e.q.2549.4 4 35.2 odd 12
4900.2.e.r.2549.1 4 35.12 even 12
4900.2.e.r.2549.4 4 35.33 even 12
8820.2.a.bg.1.1 2 21.2 odd 6
8820.2.a.bl.1.1 2 21.5 even 6