Properties

Label 1456.2.cc.f.673.4
Level $1456$
Weight $2$
Character 1456.673
Analytic conductor $11.626$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(225,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.cc (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 673.4
Root \(-0.268953i\) of defining polynomial
Character \(\chi\) \(=\) 1456.673
Dual form 1456.2.cc.f.225.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.134476 - 0.232920i) q^{3} +1.35585i q^{5} +(-0.866025 - 0.500000i) q^{7} +(1.46383 - 2.53543i) q^{9} +O(q^{10})\) \(q+(-0.134476 - 0.232920i) q^{3} +1.35585i q^{5} +(-0.866025 - 0.500000i) q^{7} +(1.46383 - 2.53543i) q^{9} +(-2.42763 + 1.40159i) q^{11} +(0.303042 + 3.59279i) q^{13} +(0.315805 - 0.182330i) q^{15} +(-2.90934 + 5.03912i) q^{17} +(-2.75793 - 1.59229i) q^{19} +0.268953i q^{21} +(-3.77674 - 6.54150i) q^{23} +3.16167 q^{25} -1.59426 q^{27} +(-3.60830 - 6.24977i) q^{29} +8.04407i q^{31} +(0.652918 + 0.376962i) q^{33} +(0.677925 - 1.17420i) q^{35} +(-6.02532 + 3.47872i) q^{37} +(0.796082 - 0.553731i) q^{39} +(4.38164 - 2.52974i) q^{41} +(-3.13427 + 5.42872i) q^{43} +(3.43767 + 1.98474i) q^{45} +2.64315i q^{47} +(0.500000 + 0.866025i) q^{49} +1.56495 q^{51} -11.0887 q^{53} +(-1.90035 - 3.29150i) q^{55} +0.856503i q^{57} +(5.45213 + 3.14779i) q^{59} +(-6.40404 + 11.0921i) q^{61} +(-2.53543 + 1.46383i) q^{63} +(-4.87129 + 0.410880i) q^{65} +(-4.29803 + 2.48147i) q^{67} +(-1.01576 + 1.75936i) q^{69} +(14.2993 + 8.25572i) q^{71} -15.4351i q^{73} +(-0.425170 - 0.736416i) q^{75} +2.80318 q^{77} -14.7661 q^{79} +(-4.17711 - 7.23496i) q^{81} -9.79310i q^{83} +(-6.83230 - 3.94463i) q^{85} +(-0.970464 + 1.68089i) q^{87} +(-7.24707 + 4.18410i) q^{89} +(1.53395 - 3.26297i) q^{91} +(1.87363 - 1.08174i) q^{93} +(2.15891 - 3.73934i) q^{95} +(-10.5685 - 6.10170i) q^{97} +8.20678i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14 q^{9} - 6 q^{11} + 10 q^{13} - 6 q^{15} + 2 q^{17} - 44 q^{25} + 12 q^{27} - 22 q^{29} + 42 q^{33} + 6 q^{35} + 12 q^{37} - 24 q^{39} + 36 q^{41} - 6 q^{43} - 30 q^{45} + 8 q^{49} + 4 q^{51} + 8 q^{53} - 2 q^{55} + 18 q^{59} + 4 q^{61} + 12 q^{63} - 30 q^{65} - 24 q^{67} - 52 q^{69} - 36 q^{71} + 10 q^{75} - 24 q^{77} - 8 q^{79} + 42 q^{85} - 26 q^{87} - 36 q^{89} + 2 q^{91} - 42 q^{93} + 30 q^{95} - 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.134476 0.232920i −0.0776400 0.134476i 0.824591 0.565729i \(-0.191405\pi\)
−0.902231 + 0.431252i \(0.858072\pi\)
\(4\) 0 0
\(5\) 1.35585i 0.606355i 0.952934 + 0.303177i \(0.0980475\pi\)
−0.952934 + 0.303177i \(0.901953\pi\)
\(6\) 0 0
\(7\) −0.866025 0.500000i −0.327327 0.188982i
\(8\) 0 0
\(9\) 1.46383 2.53543i 0.487944 0.845144i
\(10\) 0 0
\(11\) −2.42763 + 1.40159i −0.731958 + 0.422596i −0.819138 0.573597i \(-0.805548\pi\)
0.0871803 + 0.996193i \(0.472214\pi\)
\(12\) 0 0
\(13\) 0.303042 + 3.59279i 0.0840488 + 0.996462i
\(14\) 0 0
\(15\) 0.315805 0.182330i 0.0815405 0.0470774i
\(16\) 0 0
\(17\) −2.90934 + 5.03912i −0.705618 + 1.22217i 0.260850 + 0.965379i \(0.415997\pi\)
−0.966468 + 0.256787i \(0.917336\pi\)
\(18\) 0 0
\(19\) −2.75793 1.59229i −0.632713 0.365297i 0.149089 0.988824i \(-0.452366\pi\)
−0.781802 + 0.623527i \(0.785699\pi\)
\(20\) 0 0
\(21\) 0.268953i 0.0586903i
\(22\) 0 0
\(23\) −3.77674 6.54150i −0.787504 1.36400i −0.927492 0.373843i \(-0.878040\pi\)
0.139988 0.990153i \(-0.455294\pi\)
\(24\) 0 0
\(25\) 3.16167 0.632334
\(26\) 0 0
\(27\) −1.59426 −0.306816
\(28\) 0 0
\(29\) −3.60830 6.24977i −0.670045 1.16055i −0.977891 0.209116i \(-0.932941\pi\)
0.307845 0.951436i \(-0.400392\pi\)
\(30\) 0 0
\(31\) 8.04407i 1.44476i 0.691498 + 0.722379i \(0.256951\pi\)
−0.691498 + 0.722379i \(0.743049\pi\)
\(32\) 0 0
\(33\) 0.652918 + 0.376962i 0.113658 + 0.0656207i
\(34\) 0 0
\(35\) 0.677925 1.17420i 0.114590 0.198476i
\(36\) 0 0
\(37\) −6.02532 + 3.47872i −0.990557 + 0.571898i −0.905441 0.424473i \(-0.860459\pi\)
−0.0851160 + 0.996371i \(0.527126\pi\)
\(38\) 0 0
\(39\) 0.796082 0.553731i 0.127475 0.0886679i
\(40\) 0 0
\(41\) 4.38164 2.52974i 0.684298 0.395079i −0.117175 0.993111i \(-0.537384\pi\)
0.801472 + 0.598032i \(0.204050\pi\)
\(42\) 0 0
\(43\) −3.13427 + 5.42872i −0.477972 + 0.827872i −0.999681 0.0252518i \(-0.991961\pi\)
0.521709 + 0.853123i \(0.325295\pi\)
\(44\) 0 0
\(45\) 3.43767 + 1.98474i 0.512457 + 0.295867i
\(46\) 0 0
\(47\) 2.64315i 0.385542i 0.981244 + 0.192771i \(0.0617475\pi\)
−0.981244 + 0.192771i \(0.938252\pi\)
\(48\) 0 0
\(49\) 0.500000 + 0.866025i 0.0714286 + 0.123718i
\(50\) 0 0
\(51\) 1.56495 0.219137
\(52\) 0 0
\(53\) −11.0887 −1.52315 −0.761574 0.648079i \(-0.775573\pi\)
−0.761574 + 0.648079i \(0.775573\pi\)
\(54\) 0 0
\(55\) −1.90035 3.29150i −0.256243 0.443826i
\(56\) 0 0
\(57\) 0.856503i 0.113447i
\(58\) 0 0
\(59\) 5.45213 + 3.14779i 0.709807 + 0.409807i 0.810989 0.585061i \(-0.198929\pi\)
−0.101183 + 0.994868i \(0.532263\pi\)
\(60\) 0 0
\(61\) −6.40404 + 11.0921i −0.819953 + 1.42020i 0.0857630 + 0.996316i \(0.472667\pi\)
−0.905716 + 0.423885i \(0.860666\pi\)
\(62\) 0 0
\(63\) −2.53543 + 1.46383i −0.319434 + 0.184426i
\(64\) 0 0
\(65\) −4.87129 + 0.410880i −0.604209 + 0.0509634i
\(66\) 0 0
\(67\) −4.29803 + 2.48147i −0.525088 + 0.303160i −0.739014 0.673690i \(-0.764708\pi\)
0.213926 + 0.976850i \(0.431375\pi\)
\(68\) 0 0
\(69\) −1.01576 + 1.75936i −0.122284 + 0.211801i
\(70\) 0 0
\(71\) 14.2993 + 8.25572i 1.69702 + 0.979773i 0.948564 + 0.316586i \(0.102536\pi\)
0.748453 + 0.663188i \(0.230797\pi\)
\(72\) 0 0
\(73\) 15.4351i 1.80654i −0.429071 0.903271i \(-0.641159\pi\)
0.429071 0.903271i \(-0.358841\pi\)
\(74\) 0 0
\(75\) −0.425170 0.736416i −0.0490944 0.0850340i
\(76\) 0 0
\(77\) 2.80318 0.319452
\(78\) 0 0
\(79\) −14.7661 −1.66131 −0.830656 0.556786i \(-0.812034\pi\)
−0.830656 + 0.556786i \(0.812034\pi\)
\(80\) 0 0
\(81\) −4.17711 7.23496i −0.464123 0.803884i
\(82\) 0 0
\(83\) 9.79310i 1.07493i −0.843285 0.537466i \(-0.819382\pi\)
0.843285 0.537466i \(-0.180618\pi\)
\(84\) 0 0
\(85\) −6.83230 3.94463i −0.741067 0.427855i
\(86\) 0 0
\(87\) −0.970464 + 1.68089i −0.104045 + 0.180211i
\(88\) 0 0
\(89\) −7.24707 + 4.18410i −0.768188 + 0.443513i −0.832228 0.554434i \(-0.812935\pi\)
0.0640399 + 0.997947i \(0.479602\pi\)
\(90\) 0 0
\(91\) 1.53395 3.26297i 0.160802 0.342052i
\(92\) 0 0
\(93\) 1.87363 1.08174i 0.194286 0.112171i
\(94\) 0 0
\(95\) 2.15891 3.73934i 0.221499 0.383648i
\(96\) 0 0
\(97\) −10.5685 6.10170i −1.07306 0.619534i −0.144047 0.989571i \(-0.546012\pi\)
−0.929017 + 0.370037i \(0.879345\pi\)
\(98\) 0 0
\(99\) 8.20678i 0.824813i
\(100\) 0 0
\(101\) −1.21143 2.09825i −0.120541 0.208784i 0.799440 0.600746i \(-0.205130\pi\)
−0.919981 + 0.391962i \(0.871796\pi\)
\(102\) 0 0
\(103\) −14.4909 −1.42783 −0.713917 0.700230i \(-0.753081\pi\)
−0.713917 + 0.700230i \(0.753081\pi\)
\(104\) 0 0
\(105\) −0.364660 −0.0355872
\(106\) 0 0
\(107\) 3.24370 + 5.61826i 0.313581 + 0.543138i 0.979135 0.203212i \(-0.0651381\pi\)
−0.665554 + 0.746350i \(0.731805\pi\)
\(108\) 0 0
\(109\) 2.93644i 0.281260i 0.990062 + 0.140630i \(0.0449129\pi\)
−0.990062 + 0.140630i \(0.955087\pi\)
\(110\) 0 0
\(111\) 1.62053 + 0.935612i 0.153814 + 0.0888044i
\(112\) 0 0
\(113\) 1.66909 2.89095i 0.157015 0.271957i −0.776776 0.629777i \(-0.783146\pi\)
0.933791 + 0.357819i \(0.116480\pi\)
\(114\) 0 0
\(115\) 8.86929 5.12069i 0.827066 0.477507i
\(116\) 0 0
\(117\) 9.55289 + 4.49090i 0.883165 + 0.415184i
\(118\) 0 0
\(119\) 5.03912 2.90934i 0.461936 0.266699i
\(120\) 0 0
\(121\) −1.57108 + 2.72119i −0.142825 + 0.247381i
\(122\) 0 0
\(123\) −1.17846 0.680382i −0.106258 0.0613480i
\(124\) 0 0
\(125\) 11.0660i 0.989773i
\(126\) 0 0
\(127\) 6.27512 + 10.8688i 0.556827 + 0.964452i 0.997759 + 0.0669115i \(0.0213145\pi\)
−0.440932 + 0.897540i \(0.645352\pi\)
\(128\) 0 0
\(129\) 1.68594 0.148439
\(130\) 0 0
\(131\) −1.26001 −0.110087 −0.0550437 0.998484i \(-0.517530\pi\)
−0.0550437 + 0.998484i \(0.517530\pi\)
\(132\) 0 0
\(133\) 1.59229 + 2.75793i 0.138069 + 0.239143i
\(134\) 0 0
\(135\) 2.16158i 0.186039i
\(136\) 0 0
\(137\) −7.20794 4.16151i −0.615816 0.355542i 0.159422 0.987211i \(-0.449037\pi\)
−0.775238 + 0.631669i \(0.782370\pi\)
\(138\) 0 0
\(139\) −2.89438 + 5.01322i −0.245498 + 0.425216i −0.962272 0.272091i \(-0.912285\pi\)
0.716773 + 0.697306i \(0.245618\pi\)
\(140\) 0 0
\(141\) 0.615642 0.355441i 0.0518464 0.0299335i
\(142\) 0 0
\(143\) −5.77131 8.29723i −0.482621 0.693849i
\(144\) 0 0
\(145\) 8.47375 4.89232i 0.703707 0.406285i
\(146\) 0 0
\(147\) 0.134476 0.232920i 0.0110914 0.0192109i
\(148\) 0 0
\(149\) 12.3009 + 7.10193i 1.00773 + 0.581812i 0.910526 0.413452i \(-0.135677\pi\)
0.0972028 + 0.995265i \(0.469010\pi\)
\(150\) 0 0
\(151\) 11.1509i 0.907451i 0.891142 + 0.453725i \(0.149905\pi\)
−0.891142 + 0.453725i \(0.850095\pi\)
\(152\) 0 0
\(153\) 8.51757 + 14.7529i 0.688605 + 1.19270i
\(154\) 0 0
\(155\) −10.9066 −0.876036
\(156\) 0 0
\(157\) 4.04749 0.323025 0.161513 0.986871i \(-0.448363\pi\)
0.161513 + 0.986871i \(0.448363\pi\)
\(158\) 0 0
\(159\) 1.49117 + 2.58278i 0.118257 + 0.204827i
\(160\) 0 0
\(161\) 7.55347i 0.595297i
\(162\) 0 0
\(163\) 4.56675 + 2.63661i 0.357695 + 0.206515i 0.668069 0.744099i \(-0.267121\pi\)
−0.310374 + 0.950615i \(0.600454\pi\)
\(164\) 0 0
\(165\) −0.511105 + 0.885259i −0.0397894 + 0.0689173i
\(166\) 0 0
\(167\) 4.63307 2.67491i 0.358518 0.206990i −0.309912 0.950765i \(-0.600300\pi\)
0.668430 + 0.743775i \(0.266966\pi\)
\(168\) 0 0
\(169\) −12.8163 + 2.17754i −0.985872 + 0.167503i
\(170\) 0 0
\(171\) −8.07429 + 4.66170i −0.617457 + 0.356489i
\(172\) 0 0
\(173\) −4.87155 + 8.43777i −0.370377 + 0.641511i −0.989623 0.143685i \(-0.954105\pi\)
0.619247 + 0.785197i \(0.287438\pi\)
\(174\) 0 0
\(175\) −2.73809 1.58083i −0.206980 0.119500i
\(176\) 0 0
\(177\) 1.69321i 0.127270i
\(178\) 0 0
\(179\) −5.70263 9.87724i −0.426234 0.738259i 0.570301 0.821436i \(-0.306827\pi\)
−0.996535 + 0.0831767i \(0.973493\pi\)
\(180\) 0 0
\(181\) 25.4727 1.89337 0.946685 0.322160i \(-0.104409\pi\)
0.946685 + 0.322160i \(0.104409\pi\)
\(182\) 0 0
\(183\) 3.44477 0.254645
\(184\) 0 0
\(185\) −4.71663 8.16943i −0.346773 0.600629i
\(186\) 0 0
\(187\) 16.3108i 1.19277i
\(188\) 0 0
\(189\) 1.38067 + 0.797131i 0.100429 + 0.0579828i
\(190\) 0 0
\(191\) 8.18905 14.1838i 0.592539 1.02631i −0.401351 0.915924i \(-0.631459\pi\)
0.993889 0.110382i \(-0.0352075\pi\)
\(192\) 0 0
\(193\) 8.28132 4.78122i 0.596103 0.344160i −0.171404 0.985201i \(-0.554830\pi\)
0.767507 + 0.641041i \(0.221497\pi\)
\(194\) 0 0
\(195\) 0.750776 + 1.07937i 0.0537642 + 0.0772951i
\(196\) 0 0
\(197\) 2.60716 1.50525i 0.185753 0.107244i −0.404240 0.914653i \(-0.632464\pi\)
0.589993 + 0.807409i \(0.299131\pi\)
\(198\) 0 0
\(199\) 2.30985 4.00077i 0.163741 0.283607i −0.772467 0.635055i \(-0.780977\pi\)
0.936207 + 0.351448i \(0.114311\pi\)
\(200\) 0 0
\(201\) 1.15597 + 0.667399i 0.0815357 + 0.0470747i
\(202\) 0 0
\(203\) 7.21661i 0.506507i
\(204\) 0 0
\(205\) 3.42995 + 5.94085i 0.239558 + 0.414927i
\(206\) 0 0
\(207\) −22.1140 −1.53703
\(208\) 0 0
\(209\) 8.92697 0.617492
\(210\) 0 0
\(211\) 1.16932 + 2.02533i 0.0804995 + 0.139429i 0.903465 0.428663i \(-0.141015\pi\)
−0.822965 + 0.568092i \(0.807682\pi\)
\(212\) 0 0
\(213\) 4.44080i 0.304279i
\(214\) 0 0
\(215\) −7.36053 4.24960i −0.501984 0.289821i
\(216\) 0 0
\(217\) 4.02203 6.96637i 0.273033 0.472908i
\(218\) 0 0
\(219\) −3.59515 + 2.07566i −0.242937 + 0.140260i
\(220\) 0 0
\(221\) −18.9862 8.92559i −1.27715 0.600400i
\(222\) 0 0
\(223\) 4.21429 2.43312i 0.282210 0.162934i −0.352214 0.935920i \(-0.614571\pi\)
0.634423 + 0.772986i \(0.281238\pi\)
\(224\) 0 0
\(225\) 4.62815 8.01620i 0.308544 0.534413i
\(226\) 0 0
\(227\) 24.3177 + 14.0398i 1.61402 + 0.931855i 0.988426 + 0.151706i \(0.0484766\pi\)
0.625594 + 0.780149i \(0.284857\pi\)
\(228\) 0 0
\(229\) 0.179280i 0.0118471i 0.999982 + 0.00592357i \(0.00188554\pi\)
−0.999982 + 0.00592357i \(0.998114\pi\)
\(230\) 0 0
\(231\) −0.376962 0.652918i −0.0248023 0.0429588i
\(232\) 0 0
\(233\) 15.1384 0.991747 0.495873 0.868395i \(-0.334848\pi\)
0.495873 + 0.868395i \(0.334848\pi\)
\(234\) 0 0
\(235\) −3.58371 −0.233775
\(236\) 0 0
\(237\) 1.98569 + 3.43931i 0.128984 + 0.223407i
\(238\) 0 0
\(239\) 12.8703i 0.832512i −0.909247 0.416256i \(-0.863342\pi\)
0.909247 0.416256i \(-0.136658\pi\)
\(240\) 0 0
\(241\) 8.51452 + 4.91586i 0.548469 + 0.316659i 0.748504 0.663130i \(-0.230772\pi\)
−0.200035 + 0.979789i \(0.564106\pi\)
\(242\) 0 0
\(243\) −3.51484 + 6.08788i −0.225477 + 0.390538i
\(244\) 0 0
\(245\) −1.17420 + 0.677925i −0.0750170 + 0.0433111i
\(246\) 0 0
\(247\) 4.88501 10.3912i 0.310825 0.661177i
\(248\) 0 0
\(249\) −2.28101 + 1.31694i −0.144553 + 0.0834578i
\(250\) 0 0
\(251\) 8.66099 15.0013i 0.546677 0.946872i −0.451823 0.892108i \(-0.649226\pi\)
0.998499 0.0547641i \(-0.0174407\pi\)
\(252\) 0 0
\(253\) 18.3370 + 10.5869i 1.15284 + 0.665592i
\(254\) 0 0
\(255\) 2.12184i 0.132875i
\(256\) 0 0
\(257\) −5.35302 9.27170i −0.333912 0.578353i 0.649363 0.760478i \(-0.275036\pi\)
−0.983275 + 0.182126i \(0.941702\pi\)
\(258\) 0 0
\(259\) 6.95744 0.432314
\(260\) 0 0
\(261\) −21.1278 −1.30778
\(262\) 0 0
\(263\) −7.03484 12.1847i −0.433787 0.751341i 0.563409 0.826178i \(-0.309490\pi\)
−0.997196 + 0.0748372i \(0.976156\pi\)
\(264\) 0 0
\(265\) 15.0346i 0.923567i
\(266\) 0 0
\(267\) 1.94912 + 1.12533i 0.119284 + 0.0688688i
\(268\) 0 0
\(269\) 0.272011 0.471138i 0.0165848 0.0287258i −0.857614 0.514294i \(-0.828054\pi\)
0.874199 + 0.485568i \(0.161387\pi\)
\(270\) 0 0
\(271\) 0.641910 0.370607i 0.0389933 0.0225128i −0.480377 0.877062i \(-0.659500\pi\)
0.519370 + 0.854550i \(0.326167\pi\)
\(272\) 0 0
\(273\) −0.966292 + 0.0815041i −0.0584827 + 0.00493285i
\(274\) 0 0
\(275\) −7.67536 + 4.43137i −0.462842 + 0.267222i
\(276\) 0 0
\(277\) −7.06675 + 12.2400i −0.424600 + 0.735429i −0.996383 0.0849763i \(-0.972919\pi\)
0.571783 + 0.820405i \(0.306252\pi\)
\(278\) 0 0
\(279\) 20.3952 + 11.7752i 1.22103 + 0.704961i
\(280\) 0 0
\(281\) 25.8051i 1.53940i −0.638404 0.769702i \(-0.720405\pi\)
0.638404 0.769702i \(-0.279595\pi\)
\(282\) 0 0
\(283\) −13.6409 23.6267i −0.810865 1.40446i −0.912259 0.409613i \(-0.865664\pi\)
0.101394 0.994846i \(-0.467670\pi\)
\(284\) 0 0
\(285\) −1.16129 −0.0687889
\(286\) 0 0
\(287\) −5.05949 −0.298652
\(288\) 0 0
\(289\) −8.42851 14.5986i −0.495795 0.858742i
\(290\) 0 0
\(291\) 3.28214i 0.192403i
\(292\) 0 0
\(293\) −1.50010 0.866083i −0.0876367 0.0505971i 0.455541 0.890215i \(-0.349446\pi\)
−0.543178 + 0.839618i \(0.682779\pi\)
\(294\) 0 0
\(295\) −4.26793 + 7.39227i −0.248489 + 0.430395i
\(296\) 0 0
\(297\) 3.87028 2.23451i 0.224576 0.129659i
\(298\) 0 0
\(299\) 22.3577 15.5514i 1.29298 0.899360i
\(300\) 0 0
\(301\) 5.42872 3.13427i 0.312906 0.180656i
\(302\) 0 0
\(303\) −0.325817 + 0.564331i −0.0187177 + 0.0324200i
\(304\) 0 0
\(305\) −15.0393 8.68292i −0.861145 0.497182i
\(306\) 0 0
\(307\) 21.4365i 1.22345i 0.791072 + 0.611723i \(0.209523\pi\)
−0.791072 + 0.611723i \(0.790477\pi\)
\(308\) 0 0
\(309\) 1.94869 + 3.37523i 0.110857 + 0.192010i
\(310\) 0 0
\(311\) 27.2724 1.54647 0.773237 0.634117i \(-0.218636\pi\)
0.773237 + 0.634117i \(0.218636\pi\)
\(312\) 0 0
\(313\) 7.28221 0.411615 0.205807 0.978593i \(-0.434018\pi\)
0.205807 + 0.978593i \(0.434018\pi\)
\(314\) 0 0
\(315\) −1.98474 3.43767i −0.111827 0.193691i
\(316\) 0 0
\(317\) 2.54836i 0.143130i 0.997436 + 0.0715651i \(0.0227994\pi\)
−0.997436 + 0.0715651i \(0.977201\pi\)
\(318\) 0 0
\(319\) 17.5192 + 10.1147i 0.980890 + 0.566317i
\(320\) 0 0
\(321\) 0.872404 1.51105i 0.0486928 0.0843384i
\(322\) 0 0
\(323\) 16.0475 9.26503i 0.892907 0.515520i
\(324\) 0 0
\(325\) 0.958120 + 11.3592i 0.0531469 + 0.630096i
\(326\) 0 0
\(327\) 0.683957 0.394883i 0.0378229 0.0218371i
\(328\) 0 0
\(329\) 1.32157 2.28903i 0.0728607 0.126198i
\(330\) 0 0
\(331\) −0.169450 0.0978320i −0.00931381 0.00537733i 0.495336 0.868702i \(-0.335045\pi\)
−0.504650 + 0.863324i \(0.668378\pi\)
\(332\) 0 0
\(333\) 20.3691i 1.11622i
\(334\) 0 0
\(335\) −3.36450 5.82749i −0.183822 0.318390i
\(336\) 0 0
\(337\) 2.00900 0.109437 0.0547185 0.998502i \(-0.482574\pi\)
0.0547185 + 0.998502i \(0.482574\pi\)
\(338\) 0 0
\(339\) −0.897813 −0.0487625
\(340\) 0 0
\(341\) −11.2745 19.5280i −0.610549 1.05750i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −2.38542 1.37722i −0.128427 0.0741473i
\(346\) 0 0
\(347\) 2.27450 3.93956i 0.122102 0.211486i −0.798495 0.602002i \(-0.794370\pi\)
0.920596 + 0.390516i \(0.127703\pi\)
\(348\) 0 0
\(349\) 4.65700 2.68872i 0.249284 0.143924i −0.370153 0.928971i \(-0.620695\pi\)
0.619436 + 0.785047i \(0.287361\pi\)
\(350\) 0 0
\(351\) −0.483129 5.72786i −0.0257875 0.305730i
\(352\) 0 0
\(353\) 7.10223 4.10047i 0.378013 0.218246i −0.298940 0.954272i \(-0.596633\pi\)
0.676954 + 0.736026i \(0.263300\pi\)
\(354\) 0 0
\(355\) −11.1935 + 19.3877i −0.594090 + 1.02899i
\(356\) 0 0
\(357\) −1.35529 0.782475i −0.0717294 0.0414130i
\(358\) 0 0
\(359\) 31.7664i 1.67657i 0.545233 + 0.838284i \(0.316441\pi\)
−0.545233 + 0.838284i \(0.683559\pi\)
\(360\) 0 0
\(361\) −4.42921 7.67162i −0.233117 0.403770i
\(362\) 0 0
\(363\) 0.845093 0.0443559
\(364\) 0 0
\(365\) 20.9277 1.09541
\(366\) 0 0
\(367\) 8.66870 + 15.0146i 0.452502 + 0.783757i 0.998541 0.0540031i \(-0.0171981\pi\)
−0.546038 + 0.837760i \(0.683865\pi\)
\(368\) 0 0
\(369\) 14.8125i 0.771107i
\(370\) 0 0
\(371\) 9.60308 + 5.54434i 0.498567 + 0.287848i
\(372\) 0 0
\(373\) −8.04814 + 13.9398i −0.416717 + 0.721775i −0.995607 0.0936306i \(-0.970153\pi\)
0.578890 + 0.815406i \(0.303486\pi\)
\(374\) 0 0
\(375\) 2.57749 1.48812i 0.133101 0.0768460i
\(376\) 0 0
\(377\) 21.3607 14.8578i 1.10013 0.765218i
\(378\) 0 0
\(379\) 15.2907 8.82811i 0.785432 0.453470i −0.0529197 0.998599i \(-0.516853\pi\)
0.838352 + 0.545129i \(0.183519\pi\)
\(380\) 0 0
\(381\) 1.68771 2.92320i 0.0864641 0.149760i
\(382\) 0 0
\(383\) −7.12262 4.11225i −0.363949 0.210126i 0.306863 0.951754i \(-0.400721\pi\)
−0.670812 + 0.741628i \(0.734054\pi\)
\(384\) 0 0
\(385\) 3.80070i 0.193702i
\(386\) 0 0
\(387\) 9.17609 + 15.8935i 0.466447 + 0.807910i
\(388\) 0 0
\(389\) 15.1688 0.769090 0.384545 0.923106i \(-0.374358\pi\)
0.384545 + 0.923106i \(0.374358\pi\)
\(390\) 0 0
\(391\) 43.9512 2.22271
\(392\) 0 0
\(393\) 0.169441 + 0.293481i 0.00854719 + 0.0148042i
\(394\) 0 0
\(395\) 20.0206i 1.00734i
\(396\) 0 0
\(397\) −24.5139 14.1531i −1.23032 0.710325i −0.263223 0.964735i \(-0.584785\pi\)
−0.967096 + 0.254410i \(0.918119\pi\)
\(398\) 0 0
\(399\) 0.428252 0.741753i 0.0214394 0.0371341i
\(400\) 0 0
\(401\) 14.8257 8.55962i 0.740360 0.427447i −0.0818403 0.996645i \(-0.526080\pi\)
0.822200 + 0.569198i \(0.192746\pi\)
\(402\) 0 0
\(403\) −28.9007 + 2.43769i −1.43965 + 0.121430i
\(404\) 0 0
\(405\) 9.80952 5.66353i 0.487439 0.281423i
\(406\) 0 0
\(407\) 9.75149 16.8901i 0.483364 0.837210i
\(408\) 0 0
\(409\) 9.82072 + 5.66999i 0.485603 + 0.280363i 0.722749 0.691111i \(-0.242878\pi\)
−0.237145 + 0.971474i \(0.576212\pi\)
\(410\) 0 0
\(411\) 2.23850i 0.110417i
\(412\) 0 0
\(413\) −3.14779 5.45213i −0.154893 0.268282i
\(414\) 0 0
\(415\) 13.2780 0.651790
\(416\) 0 0
\(417\) 1.55691 0.0762420
\(418\) 0 0
\(419\) 0.0694276 + 0.120252i 0.00339176 + 0.00587470i 0.867716 0.497060i \(-0.165587\pi\)
−0.864325 + 0.502934i \(0.832254\pi\)
\(420\) 0 0
\(421\) 24.3258i 1.18557i 0.805362 + 0.592783i \(0.201971\pi\)
−0.805362 + 0.592783i \(0.798029\pi\)
\(422\) 0 0
\(423\) 6.70152 + 3.86912i 0.325839 + 0.188123i
\(424\) 0 0
\(425\) −9.19837 + 15.9320i −0.446186 + 0.772818i
\(426\) 0 0
\(427\) 11.0921 6.40404i 0.536785 0.309913i
\(428\) 0 0
\(429\) −1.15649 + 2.46003i −0.0558357 + 0.118772i
\(430\) 0 0
\(431\) −24.9349 + 14.3962i −1.20107 + 0.693440i −0.960794 0.277263i \(-0.910573\pi\)
−0.240280 + 0.970704i \(0.577239\pi\)
\(432\) 0 0
\(433\) −12.6367 + 21.8873i −0.607279 + 1.05184i 0.384408 + 0.923163i \(0.374406\pi\)
−0.991687 + 0.128675i \(0.958928\pi\)
\(434\) 0 0
\(435\) −2.27904 1.31580i −0.109272 0.0630880i
\(436\) 0 0
\(437\) 24.0547i 1.15069i
\(438\) 0 0
\(439\) −3.01512 5.22234i −0.143904 0.249248i 0.785060 0.619420i \(-0.212632\pi\)
−0.928963 + 0.370172i \(0.879299\pi\)
\(440\) 0 0
\(441\) 2.92766 0.139413
\(442\) 0 0
\(443\) −23.4988 −1.11646 −0.558230 0.829686i \(-0.688519\pi\)
−0.558230 + 0.829686i \(0.688519\pi\)
\(444\) 0 0
\(445\) −5.67301 9.82594i −0.268927 0.465794i
\(446\) 0 0
\(447\) 3.82017i 0.180688i
\(448\) 0 0
\(449\) 20.2543 + 11.6938i 0.955859 + 0.551865i 0.894896 0.446275i \(-0.147250\pi\)
0.0609627 + 0.998140i \(0.480583\pi\)
\(450\) 0 0
\(451\) −7.09133 + 12.2826i −0.333918 + 0.578363i
\(452\) 0 0
\(453\) 2.59728 1.49954i 0.122031 0.0704545i
\(454\) 0 0
\(455\) 4.42410 + 2.07981i 0.207405 + 0.0975031i
\(456\) 0 0
\(457\) −22.6616 + 13.0837i −1.06007 + 0.612030i −0.925451 0.378868i \(-0.876313\pi\)
−0.134616 + 0.990898i \(0.542980\pi\)
\(458\) 0 0
\(459\) 4.63825 8.03369i 0.216495 0.374980i
\(460\) 0 0
\(461\) −21.8524 12.6165i −1.01777 0.587609i −0.104312 0.994545i \(-0.533264\pi\)
−0.913457 + 0.406936i \(0.866597\pi\)
\(462\) 0 0
\(463\) 18.1600i 0.843968i −0.906603 0.421984i \(-0.861334\pi\)
0.906603 0.421984i \(-0.138666\pi\)
\(464\) 0 0
\(465\) 1.46668 + 2.54036i 0.0680154 + 0.117806i
\(466\) 0 0
\(467\) 12.6328 0.584577 0.292289 0.956330i \(-0.405583\pi\)
0.292289 + 0.956330i \(0.405583\pi\)
\(468\) 0 0
\(469\) 4.96294 0.229167
\(470\) 0 0
\(471\) −0.544293 0.942743i −0.0250797 0.0434393i
\(472\) 0 0
\(473\) 17.5719i 0.807956i
\(474\) 0 0
\(475\) −8.71966 5.03430i −0.400086 0.230990i
\(476\) 0 0
\(477\) −16.2320 + 28.1146i −0.743211 + 1.28728i
\(478\) 0 0
\(479\) −10.6585 + 6.15368i −0.486999 + 0.281169i −0.723329 0.690504i \(-0.757389\pi\)
0.236330 + 0.971673i \(0.424055\pi\)
\(480\) 0 0
\(481\) −14.3243 20.5935i −0.653130 0.938984i
\(482\) 0 0
\(483\) 1.75936 1.01576i 0.0800534 0.0462189i
\(484\) 0 0
\(485\) 8.27299 14.3292i 0.375657 0.650658i
\(486\) 0 0
\(487\) −13.5393 7.81693i −0.613525 0.354219i 0.160819 0.986984i \(-0.448587\pi\)
−0.774344 + 0.632765i \(0.781920\pi\)
\(488\) 0 0
\(489\) 1.41825i 0.0641355i
\(490\) 0 0
\(491\) −5.10543 8.84286i −0.230405 0.399073i 0.727523 0.686084i \(-0.240672\pi\)
−0.957927 + 0.287011i \(0.907338\pi\)
\(492\) 0 0
\(493\) 41.9911 1.89119
\(494\) 0 0
\(495\) −11.1272 −0.500129
\(496\) 0 0
\(497\) −8.25572 14.2993i −0.370319 0.641412i
\(498\) 0 0
\(499\) 0.681003i 0.0304859i −0.999884 0.0152430i \(-0.995148\pi\)
0.999884 0.0152430i \(-0.00485217\pi\)
\(500\) 0 0
\(501\) −1.24608 0.719424i −0.0556707 0.0321415i
\(502\) 0 0
\(503\) −17.6701 + 30.6056i −0.787873 + 1.36464i 0.139395 + 0.990237i \(0.455484\pi\)
−0.927268 + 0.374399i \(0.877849\pi\)
\(504\) 0 0
\(505\) 2.84492 1.64251i 0.126597 0.0730909i
\(506\) 0 0
\(507\) 2.23069 + 2.69235i 0.0990683 + 0.119572i
\(508\) 0 0
\(509\) 14.5717 8.41300i 0.645881 0.372900i −0.140995 0.990010i \(-0.545030\pi\)
0.786876 + 0.617111i \(0.211697\pi\)
\(510\) 0 0
\(511\) −7.71755 + 13.3672i −0.341404 + 0.591330i
\(512\) 0 0
\(513\) 4.39687 + 2.53853i 0.194126 + 0.112079i
\(514\) 0 0
\(515\) 19.6475i 0.865774i
\(516\) 0 0
\(517\) −3.70461 6.41658i −0.162929 0.282201i
\(518\) 0 0
\(519\) 2.62043 0.115024
\(520\) 0 0
\(521\) 4.76619 0.208811 0.104405 0.994535i \(-0.466706\pi\)
0.104405 + 0.994535i \(0.466706\pi\)
\(522\) 0 0
\(523\) −0.319080 0.552662i −0.0139524 0.0241662i 0.858965 0.512034i \(-0.171108\pi\)
−0.872917 + 0.487868i \(0.837775\pi\)
\(524\) 0 0
\(525\) 0.850340i 0.0371119i
\(526\) 0 0
\(527\) −40.5351 23.4029i −1.76573 1.01945i
\(528\) 0 0
\(529\) −17.0275 + 29.4924i −0.740325 + 1.28228i
\(530\) 0 0
\(531\) 15.9620 9.21567i 0.692692 0.399926i
\(532\) 0 0
\(533\) 10.4167 + 14.9757i 0.451196 + 0.648670i
\(534\) 0 0
\(535\) −7.61752 + 4.39798i −0.329334 + 0.190141i
\(536\) 0 0
\(537\) −1.53374 + 2.65651i −0.0661857 + 0.114637i
\(538\) 0 0
\(539\) −2.42763 1.40159i −0.104565 0.0603708i
\(540\) 0 0
\(541\) 22.2050i 0.954668i −0.878722 0.477334i \(-0.841603\pi\)
0.878722 0.477334i \(-0.158397\pi\)
\(542\) 0 0
\(543\) −3.42548 5.93310i −0.147001 0.254614i
\(544\) 0 0
\(545\) −3.98138 −0.170544
\(546\) 0 0
\(547\) −16.2608 −0.695263 −0.347631 0.937631i \(-0.613014\pi\)
−0.347631 + 0.937631i \(0.613014\pi\)
\(548\) 0 0
\(549\) 18.7489 + 32.4740i 0.800182 + 1.38596i
\(550\) 0 0
\(551\) 22.9819i 0.979062i
\(552\) 0 0
\(553\) 12.7878 + 7.38303i 0.543792 + 0.313958i
\(554\) 0 0
\(555\) −1.26855 + 2.19719i −0.0538470 + 0.0932657i
\(556\) 0 0
\(557\) −18.5915 + 10.7338i −0.787747 + 0.454806i −0.839169 0.543871i \(-0.816958\pi\)
0.0514219 + 0.998677i \(0.483625\pi\)
\(558\) 0 0
\(559\) −20.4541 9.61566i −0.865115 0.406699i
\(560\) 0 0
\(561\) −3.79912 + 2.19342i −0.160399 + 0.0926064i
\(562\) 0 0
\(563\) 5.59103 9.68395i 0.235634 0.408130i −0.723823 0.689986i \(-0.757617\pi\)
0.959457 + 0.281856i \(0.0909500\pi\)
\(564\) 0 0
\(565\) 3.91969 + 2.26304i 0.164903 + 0.0952066i
\(566\) 0 0
\(567\) 8.35421i 0.350844i
\(568\) 0 0
\(569\) −6.58165 11.3998i −0.275917 0.477903i 0.694449 0.719542i \(-0.255648\pi\)
−0.970366 + 0.241639i \(0.922315\pi\)
\(570\) 0 0
\(571\) 17.4199 0.729001 0.364501 0.931203i \(-0.381240\pi\)
0.364501 + 0.931203i \(0.381240\pi\)
\(572\) 0 0
\(573\) −4.40494 −0.184019
\(574\) 0 0
\(575\) −11.9408 20.6821i −0.497965 0.862501i
\(576\) 0 0
\(577\) 8.47023i 0.352620i 0.984335 + 0.176310i \(0.0564161\pi\)
−0.984335 + 0.176310i \(0.943584\pi\)
\(578\) 0 0
\(579\) −2.22729 1.28592i −0.0925629 0.0534412i
\(580\) 0 0
\(581\) −4.89655 + 8.48107i −0.203143 + 0.351854i
\(582\) 0 0
\(583\) 26.9192 15.5418i 1.11488 0.643676i
\(584\) 0 0
\(585\) −6.08899 + 12.9523i −0.251749 + 0.535511i
\(586\) 0 0
\(587\) −18.2459 + 10.5342i −0.753087 + 0.434795i −0.826808 0.562484i \(-0.809846\pi\)
0.0737213 + 0.997279i \(0.476512\pi\)
\(588\) 0 0
\(589\) 12.8085 22.1850i 0.527765 0.914116i
\(590\) 0 0
\(591\) −0.701205 0.404841i −0.0288437 0.0166529i
\(592\) 0 0
\(593\) 13.2761i 0.545185i −0.962130 0.272593i \(-0.912119\pi\)
0.962130 0.272593i \(-0.0878811\pi\)
\(594\) 0 0
\(595\) 3.94463 + 6.83230i 0.161714 + 0.280097i
\(596\) 0 0
\(597\) −1.24248 −0.0508513
\(598\) 0 0
\(599\) −5.39084 −0.220264 −0.110132 0.993917i \(-0.535127\pi\)
−0.110132 + 0.993917i \(0.535127\pi\)
\(600\) 0 0
\(601\) 12.2204 + 21.1663i 0.498480 + 0.863392i 0.999998 0.00175467i \(-0.000558528\pi\)
−0.501519 + 0.865147i \(0.667225\pi\)
\(602\) 0 0
\(603\) 14.5298i 0.591700i
\(604\) 0 0
\(605\) −3.68953 2.13015i −0.150001 0.0866029i
\(606\) 0 0
\(607\) −12.9291 + 22.3939i −0.524776 + 0.908939i 0.474808 + 0.880090i \(0.342518\pi\)
−0.999584 + 0.0288492i \(0.990816\pi\)
\(608\) 0 0
\(609\) 1.68089 0.970464i 0.0681132 0.0393252i
\(610\) 0 0
\(611\) −9.49628 + 0.800985i −0.384178 + 0.0324044i
\(612\) 0 0
\(613\) 12.2706 7.08441i 0.495603 0.286136i −0.231293 0.972884i \(-0.574296\pi\)
0.726896 + 0.686748i \(0.240962\pi\)
\(614\) 0 0
\(615\) 0.922496 1.59781i 0.0371986 0.0644299i
\(616\) 0 0
\(617\) 39.8763 + 23.0226i 1.60536 + 0.926855i 0.990390 + 0.138305i \(0.0441654\pi\)
0.614970 + 0.788550i \(0.289168\pi\)
\(618\) 0 0
\(619\) 12.0284i 0.483464i −0.970343 0.241732i \(-0.922285\pi\)
0.970343 0.241732i \(-0.0777155\pi\)
\(620\) 0 0
\(621\) 6.02111 + 10.4289i 0.241619 + 0.418496i
\(622\) 0 0
\(623\) 8.36820 0.335265
\(624\) 0 0
\(625\) 0.804500 0.0321800
\(626\) 0 0
\(627\) −1.20047 2.07927i −0.0479421 0.0830381i
\(628\) 0 0
\(629\) 40.4831i 1.61417i
\(630\) 0 0
\(631\) −21.0476 12.1518i −0.837890 0.483756i 0.0186564 0.999826i \(-0.494061\pi\)
−0.856546 + 0.516070i \(0.827394\pi\)
\(632\) 0 0
\(633\) 0.314493 0.544718i 0.0125000 0.0216506i
\(634\) 0 0
\(635\) −14.7365 + 8.50812i −0.584800 + 0.337634i
\(636\) 0 0
\(637\) −2.95993 + 2.05884i −0.117277 + 0.0815742i
\(638\) 0 0
\(639\) 41.8636 24.1700i 1.65610 0.956149i
\(640\) 0 0
\(641\) −8.88276 + 15.3854i −0.350848 + 0.607687i −0.986398 0.164373i \(-0.947440\pi\)
0.635550 + 0.772060i \(0.280773\pi\)
\(642\) 0 0
\(643\) 37.3742 + 21.5780i 1.47389 + 0.850953i 0.999568 0.0293964i \(-0.00935852\pi\)
0.474326 + 0.880349i \(0.342692\pi\)
\(644\) 0 0
\(645\) 2.28589i 0.0900067i
\(646\) 0 0
\(647\) 5.39974 + 9.35262i 0.212286 + 0.367689i 0.952429 0.304759i \(-0.0985760\pi\)
−0.740144 + 0.672449i \(0.765243\pi\)
\(648\) 0 0
\(649\) −17.6477 −0.692731
\(650\) 0 0
\(651\) −2.16348 −0.0847933
\(652\) 0 0
\(653\) 1.53280 + 2.65489i 0.0599831 + 0.103894i 0.894458 0.447153i \(-0.147562\pi\)
−0.834475 + 0.551047i \(0.814229\pi\)
\(654\) 0 0
\(655\) 1.70838i 0.0667520i
\(656\) 0 0
\(657\) −39.1346 22.5944i −1.52679 0.881491i
\(658\) 0 0
\(659\) −1.82436 + 3.15988i −0.0710670 + 0.123092i −0.899369 0.437190i \(-0.855974\pi\)
0.828302 + 0.560282i \(0.189307\pi\)
\(660\) 0 0
\(661\) −23.5777 + 13.6126i −0.917068 + 0.529469i −0.882698 0.469940i \(-0.844276\pi\)
−0.0343693 + 0.999409i \(0.510942\pi\)
\(662\) 0 0
\(663\) 0.474246 + 5.62255i 0.0184182 + 0.218362i
\(664\) 0 0
\(665\) −3.73934 + 2.15891i −0.145005 + 0.0837189i
\(666\) 0 0
\(667\) −27.2552 + 47.2074i −1.05533 + 1.82788i
\(668\) 0 0
\(669\) −1.13345 0.654396i −0.0438216 0.0253004i
\(670\) 0 0
\(671\) 35.9034i 1.38604i
\(672\) 0 0
\(673\) 14.1485 + 24.5059i 0.545384 + 0.944633i 0.998583 + 0.0532233i \(0.0169495\pi\)
−0.453199 + 0.891410i \(0.649717\pi\)
\(674\) 0 0
\(675\) −5.04053 −0.194010
\(676\) 0 0
\(677\) −28.8361 −1.10826 −0.554131 0.832429i \(-0.686949\pi\)
−0.554131 + 0.832429i \(0.686949\pi\)
\(678\) 0 0
\(679\) 6.10170 + 10.5685i 0.234162 + 0.405580i
\(680\) 0 0
\(681\) 7.55209i 0.289397i
\(682\) 0 0
\(683\) −35.4243 20.4522i −1.35547 0.782583i −0.366464 0.930432i \(-0.619432\pi\)
−0.989010 + 0.147849i \(0.952765\pi\)
\(684\) 0 0
\(685\) 5.64238 9.77290i 0.215584 0.373403i
\(686\) 0 0
\(687\) 0.0417579 0.0241089i 0.00159316 0.000919813i
\(688\) 0 0
\(689\) −3.36034 39.8393i −0.128019 1.51776i
\(690\) 0 0
\(691\) 36.5934 21.1272i 1.39208 0.803717i 0.398534 0.917154i \(-0.369519\pi\)
0.993545 + 0.113436i \(0.0361858\pi\)
\(692\) 0 0
\(693\) 4.10339 7.10728i 0.155875 0.269983i
\(694\) 0 0
\(695\) −6.79717 3.92435i −0.257831 0.148859i
\(696\) 0 0
\(697\) 29.4395i 1.11510i
\(698\) 0 0
\(699\) −2.03575 3.52603i −0.0769992 0.133367i
\(700\) 0 0
\(701\) −30.3734 −1.14719 −0.573594 0.819140i \(-0.694451\pi\)
−0.573594 + 0.819140i \(0.694451\pi\)
\(702\) 0 0
\(703\) 22.1566 0.835650
\(704\) 0 0
\(705\) 0.481925 + 0.834718i 0.0181503 + 0.0314373i
\(706\) 0 0
\(707\) 2.42285i 0.0911208i
\(708\) 0 0
\(709\) −20.2907 11.7148i −0.762033 0.439960i 0.0679919 0.997686i \(-0.478341\pi\)
−0.830025 + 0.557726i \(0.811674\pi\)
\(710\) 0 0
\(711\) −21.6150 + 37.4383i −0.810627 + 1.40405i
\(712\) 0 0
\(713\) 52.6203 30.3803i 1.97064 1.13775i
\(714\) 0 0
\(715\) 11.2498 7.82503i 0.420719 0.292639i
\(716\) 0 0
\(717\) −2.99776 + 1.73076i −0.111953 + 0.0646363i
\(718\) 0 0
\(719\) 3.51491 6.08800i 0.131084 0.227044i −0.793011 0.609208i \(-0.791488\pi\)
0.924095 + 0.382164i \(0.124821\pi\)
\(720\) 0 0
\(721\) 12.5495 + 7.24547i 0.467369 + 0.269835i
\(722\) 0 0
\(723\) 2.64427i 0.0983415i
\(724\) 0 0
\(725\) −11.4083 19.7597i −0.423692 0.733857i
\(726\) 0 0
\(727\) 20.6810 0.767015 0.383507 0.923538i \(-0.374716\pi\)
0.383507 + 0.923538i \(0.374716\pi\)
\(728\) 0 0
\(729\) −23.1720 −0.858222
\(730\) 0 0
\(731\) −18.2373 31.5880i −0.674531 1.16832i
\(732\) 0 0
\(733\) 39.0035i 1.44063i 0.693648 + 0.720314i \(0.256002\pi\)
−0.693648 + 0.720314i \(0.743998\pi\)
\(734\) 0 0
\(735\) 0.315805 + 0.182330i 0.0116486 + 0.00672534i
\(736\) 0 0
\(737\) 6.95602 12.0482i 0.256228 0.443800i
\(738\) 0 0
\(739\) −24.8910 + 14.3708i −0.915629 + 0.528639i −0.882238 0.470803i \(-0.843964\pi\)
−0.0333913 + 0.999442i \(0.510631\pi\)
\(740\) 0 0
\(741\) −3.07724 + 0.259557i −0.113045 + 0.00953505i
\(742\) 0 0
\(743\) 5.48059 3.16422i 0.201063 0.116084i −0.396088 0.918213i \(-0.629633\pi\)
0.597151 + 0.802129i \(0.296299\pi\)
\(744\) 0 0
\(745\) −9.62915 + 16.6782i −0.352785 + 0.611041i
\(746\) 0 0
\(747\) −24.8297 14.3355i −0.908473 0.524507i
\(748\) 0 0
\(749\) 6.48741i 0.237045i
\(750\) 0 0
\(751\) 15.7203 + 27.2283i 0.573641 + 0.993575i 0.996188 + 0.0872339i \(0.0278028\pi\)
−0.422547 + 0.906341i \(0.638864\pi\)
\(752\) 0 0
\(753\) −4.65880 −0.169776
\(754\) 0 0
\(755\) −15.1190 −0.550237
\(756\) 0 0
\(757\) −12.7212 22.0337i −0.462358 0.800828i 0.536720 0.843760i \(-0.319663\pi\)
−0.999078 + 0.0429328i \(0.986330\pi\)
\(758\) 0 0
\(759\) 5.69475i 0.206706i
\(760\) 0 0
\(761\) −17.8492 10.3053i −0.647034 0.373565i 0.140285 0.990111i \(-0.455198\pi\)
−0.787319 + 0.616546i \(0.788531\pi\)
\(762\) 0 0
\(763\) 1.46822 2.54304i 0.0531532 0.0920641i
\(764\) 0 0
\(765\) −20.0027 + 11.5485i −0.723198 + 0.417539i
\(766\) 0 0
\(767\) −9.65713 + 20.5423i −0.348699 + 0.741739i
\(768\) 0 0
\(769\) 7.95173 4.59093i 0.286747 0.165553i −0.349727 0.936852i \(-0.613726\pi\)
0.636474 + 0.771298i \(0.280392\pi\)
\(770\) 0 0
\(771\) −1.43971 + 2.49365i −0.0518499 + 0.0898066i
\(772\) 0 0
\(773\) −7.35736 4.24777i −0.264626 0.152782i 0.361817 0.932249i \(-0.382157\pi\)
−0.626443 + 0.779467i \(0.715490\pi\)
\(774\) 0 0
\(775\) 25.4327i 0.913569i
\(776\) 0 0
\(777\) −0.935612 1.62053i −0.0335649 0.0581361i
\(778\) 0 0
\(779\) −16.1124 −0.577285
\(780\) 0 0
\(781\) −46.2846 −1.65619
\(782\) 0 0
\(783\) 5.75259 + 9.96377i 0.205581 + 0.356076i
\(784\) 0 0
\(785\) 5.48780i 0.195868i
\(786\) 0 0
\(787\) −1.70464 0.984173i −0.0607638 0.0350820i 0.469310 0.883033i \(-0.344503\pi\)
−0.530074 + 0.847951i \(0.677836\pi\)
\(788\) 0 0
\(789\) −1.89204 + 3.27711i −0.0673585 + 0.116668i
\(790\) 0 0
\(791\) −2.89095 + 1.66909i −0.102790 + 0.0593460i
\(792\) 0 0
\(793\) −41.7924 19.6470i −1.48409 0.697686i
\(794\) 0 0
\(795\) −3.50186 + 2.02180i −0.124198 + 0.0717058i
\(796\) 0 0
\(797\) −19.2449 + 33.3332i −0.681690 + 1.18072i 0.292775 + 0.956181i \(0.405421\pi\)
−0.974465 + 0.224540i \(0.927912\pi\)
\(798\) 0 0
\(799\) −13.3191 7.68981i −0.471197 0.272046i
\(800\) 0 0
\(801\) 24.4993i 0.865639i
\(802\) 0 0
\(803\) 21.6337 + 37.4707i 0.763437 + 1.32231i
\(804\) 0 0
\(805\) −10.2414 −0.360961
\(806\) 0 0
\(807\) −0.146317 −0.00515059
\(808\) 0 0
\(809\) −0.899878 1.55863i −0.0316380 0.0547987i 0.849773 0.527149i \(-0.176739\pi\)
−0.881411 + 0.472350i \(0.843406\pi\)
\(810\) 0 0
\(811\) 49.9235i 1.75305i 0.481357 + 0.876525i \(0.340144\pi\)
−0.481357 + 0.876525i \(0.659856\pi\)
\(812\) 0 0
\(813\) −0.172644 0.0996759i −0.00605488 0.00349579i
\(814\) 0 0
\(815\) −3.57485 + 6.19183i −0.125222 + 0.216890i
\(816\) 0 0
\(817\) 17.2882 9.98135i 0.604838 0.349203i
\(818\) 0 0
\(819\) −6.02759 8.66568i −0.210621 0.302803i
\(820\) 0 0
\(821\) 3.65310 2.10912i 0.127494 0.0736087i −0.434897 0.900480i \(-0.643215\pi\)
0.562391 + 0.826872i \(0.309882\pi\)
\(822\) 0 0
\(823\) −7.24912 + 12.5558i −0.252688 + 0.437669i −0.964265 0.264939i \(-0.914648\pi\)
0.711577 + 0.702608i \(0.247981\pi\)
\(824\) 0 0
\(825\) 2.06431 + 1.19183i 0.0718701 + 0.0414942i
\(826\) 0 0
\(827\) 26.2618i 0.913213i 0.889669 + 0.456607i \(0.150935\pi\)
−0.889669 + 0.456607i \(0.849065\pi\)
\(828\) 0 0
\(829\) −27.5790 47.7682i −0.957858 1.65906i −0.727688 0.685909i \(-0.759405\pi\)
−0.230170 0.973150i \(-0.573928\pi\)
\(830\) 0 0
\(831\) 3.80125 0.131864
\(832\) 0 0
\(833\) −5.81868 −0.201605
\(834\) 0 0
\(835\) 3.62677 + 6.28175i 0.125510 + 0.217389i
\(836\) 0 0
\(837\) 12.8244i 0.443275i
\(838\) 0 0
\(839\) 6.07974 + 3.51014i 0.209896 + 0.121184i 0.601263 0.799051i \(-0.294664\pi\)
−0.391367 + 0.920235i \(0.627998\pi\)
\(840\) 0 0
\(841\) −11.5397 + 19.9874i −0.397922 + 0.689221i
\(842\) 0 0
\(843\) −6.01053 + 3.47018i −0.207014 + 0.119519i
\(844\) 0 0
\(845\) −2.95241 17.3770i −0.101566 0.597788i
\(846\) 0 0
\(847\) 2.72119 1.57108i 0.0935012 0.0539829i
\(848\) 0 0
\(849\) −3.66875 + 6.35446i −0.125911 + 0.218085i
\(850\) 0 0
\(851\) 45.5121 + 26.2764i 1.56013 + 0.900744i
\(852\) 0 0
\(853\) 57.6508i 1.97393i 0.160945 + 0.986963i \(0.448546\pi\)
−0.160945 + 0.986963i \(0.551454\pi\)
\(854\) 0 0
\(855\) −6.32056 10.9475i −0.216159 0.374398i
\(856\) 0 0
\(857\) −2.50654 −0.0856217 −0.0428109 0.999083i \(-0.513631\pi\)
−0.0428109 + 0.999083i \(0.513631\pi\)
\(858\) 0 0
\(859\) −17.1305 −0.584486 −0.292243 0.956344i \(-0.594402\pi\)
−0.292243 + 0.956344i \(0.594402\pi\)
\(860\) 0 0
\(861\) 0.680382 + 1.17846i 0.0231873 + 0.0401617i
\(862\) 0 0
\(863\) 55.6429i 1.89411i −0.321079 0.947053i \(-0.604045\pi\)
0.321079 0.947053i \(-0.395955\pi\)
\(864\) 0 0
\(865\) −11.4403 6.60509i −0.388984 0.224580i
\(866\) 0 0
\(867\) −2.26687 + 3.92634i −0.0769870 + 0.133345i
\(868\) 0 0
\(869\) 35.8465 20.6960i 1.21601 0.702064i
\(870\) 0 0
\(871\) −10.2179 14.6899i −0.346220 0.497750i
\(872\) 0 0
\(873\) −30.9409 + 17.8637i −1.04719 + 0.604596i
\(874\) 0 0
\(875\) 5.53300 9.58344i 0.187050 0.323979i
\(876\) 0 0
\(877\) 34.7751 + 20.0774i 1.17427 + 0.677967i 0.954683 0.297626i \(-0.0961947\pi\)
0.219590 + 0.975592i \(0.429528\pi\)
\(878\) 0 0
\(879\) 0.465871i 0.0157134i
\(880\) 0 0
\(881\) 25.3837 + 43.9658i 0.855197 + 1.48124i 0.876462 + 0.481471i \(0.159897\pi\)
−0.0212650 + 0.999774i \(0.506769\pi\)
\(882\) 0 0
\(883\) 46.3312 1.55917 0.779585 0.626297i \(-0.215430\pi\)
0.779585 + 0.626297i \(0.215430\pi\)
\(884\) 0 0
\(885\) 2.29575 0.0771706
\(886\) 0 0
\(887\) −5.12752 8.88113i −0.172165 0.298199i 0.767011 0.641634i \(-0.221743\pi\)
−0.939177 + 0.343435i \(0.888410\pi\)
\(888\) 0 0
\(889\) 12.5502i 0.420921i
\(890\) 0 0
\(891\) 20.2809 + 11.7092i 0.679436 + 0.392273i
\(892\) 0 0
\(893\) 4.20866 7.28961i 0.140837 0.243938i
\(894\) 0 0
\(895\) 13.3921 7.73191i 0.447647 0.258449i
\(896\) 0 0
\(897\) −6.62882 3.11627i −0.221330 0.104049i
\(898\) 0 0
\(899\) 50.2736 29.0255i 1.67672 0.968053i
\(900\) 0 0
\(901\) 32.2607 55.8772i 1.07476 1.86154i
\(902\) 0 0
\(903\) −1.46007 0.842971i −0.0485881 0.0280523i
\(904\) 0 0
\(905\) 34.5372i 1.14805i
\(906\) 0 0
\(907\) −26.1472 45.2883i −0.868205 1.50377i −0.863830 0.503784i \(-0.831941\pi\)
−0.00437495 0.999990i \(-0.501393\pi\)
\(908\) 0 0
\(909\) −7.09330 −0.235270
\(910\) 0 0
\(911\) −50.9172 −1.68696 −0.843481 0.537159i \(-0.819498\pi\)
−0.843481 + 0.537159i \(0.819498\pi\)
\(912\) 0 0
\(913\) 13.7259 + 23.7740i 0.454262 + 0.786805i
\(914\) 0 0
\(915\) 4.67059i 0.154405i
\(916\) 0 0
\(917\) 1.09120 + 0.630004i 0.0360346 + 0.0208046i
\(918\) 0 0
\(919\) 11.4323 19.8013i 0.377117 0.653186i −0.613524 0.789676i \(-0.710249\pi\)
0.990641 + 0.136490i \(0.0435821\pi\)
\(920\) 0 0
\(921\) 4.99299 2.88270i 0.164525 0.0949884i
\(922\) 0 0
\(923\) −25.3278 + 53.8763i −0.833674 + 1.77336i
\(924\) 0 0
\(925\) −19.0501 + 10.9986i −0.626363 + 0.361631i
\(926\) 0 0
\(927\) −21.2123 + 36.7408i −0.696703 + 1.20673i
\(928\) 0 0
\(929\) 28.8090 + 16.6329i 0.945193 + 0.545708i 0.891584 0.452854i \(-0.149594\pi\)
0.0536089 + 0.998562i \(0.482928\pi\)
\(930\) 0 0
\(931\) 3.18458i 0.104371i
\(932\) 0 0
\(933\) −3.66749 6.35228i −0.120068 0.207964i
\(934\) 0 0
\(935\) 22.1150 0.723239
\(936\) 0 0
\(937\) −51.2179 −1.67322 −0.836608 0.547802i \(-0.815465\pi\)
−0.836608 + 0.547802i \(0.815465\pi\)
\(938\) 0 0
\(939\) −0.979286 1.69617i −0.0319578 0.0553525i
\(940\) 0 0
\(941\) 10.2918i 0.335502i −0.985829 0.167751i \(-0.946350\pi\)
0.985829 0.167751i \(-0.0536504\pi\)
\(942\) 0 0
\(943\) −33.0966 19.1083i −1.07777 0.622253i
\(944\) 0 0
\(945\) −1.08079 + 1.87199i −0.0351581 + 0.0608957i
\(946\) 0 0
\(947\) −30.7341 + 17.7444i −0.998725 + 0.576614i −0.907871 0.419250i \(-0.862293\pi\)
−0.0908539 + 0.995864i \(0.528960\pi\)
\(948\) 0 0
\(949\) 55.4551 4.67749i 1.80015 0.151838i
\(950\) 0 0
\(951\) 0.593565 0.342695i 0.0192477 0.0111126i
\(952\) 0 0
\(953\) 16.6676 28.8691i 0.539916 0.935163i −0.458992 0.888441i \(-0.651789\pi\)
0.998908 0.0467220i \(-0.0148775\pi\)
\(954\) 0 0
\(955\) 19.2312 + 11.1031i 0.622306 + 0.359289i
\(956\) 0 0
\(957\) 5.44078i 0.175875i
\(958\) 0 0
\(959\) 4.16151 + 7.20794i 0.134382 + 0.232757i
\(960\) 0 0
\(961\) −33.7070 −1.08732
\(962\) 0 0
\(963\) 18.9929 0.612039
\(964\) 0 0
\(965\) 6.48263 + 11.2282i 0.208683 + 0.361450i
\(966\) 0 0
\(967\) 29.7131i 0.955509i 0.878494 + 0.477754i \(0.158549\pi\)
−0.878494 + 0.477754i \(0.841451\pi\)
\(968\) 0 0
\(969\) −4.31602 2.49186i −0.138651 0.0800500i
\(970\) 0 0
\(971\) −4.41851 + 7.65308i −0.141797 + 0.245599i −0.928173 0.372148i \(-0.878621\pi\)
0.786377 + 0.617747i \(0.211955\pi\)
\(972\) 0 0
\(973\) 5.01322 2.89438i 0.160716 0.0927896i
\(974\) 0 0
\(975\) 2.51695 1.75071i 0.0806068 0.0560677i
\(976\) 0 0
\(977\) −12.8437 + 7.41533i −0.410907 + 0.237237i −0.691180 0.722683i \(-0.742909\pi\)
0.280272 + 0.959921i \(0.409575\pi\)
\(978\) 0 0
\(979\) 11.7288 20.3149i 0.374854 0.649266i
\(980\) 0 0
\(981\) 7.44516 + 4.29846i 0.237706 + 0.137239i
\(982\) 0 0
\(983\) 20.3390i 0.648714i 0.945935 + 0.324357i \(0.105148\pi\)
−0.945935 + 0.324357i \(0.894852\pi\)
\(984\) 0 0
\(985\) 2.04089 + 3.53493i 0.0650282 + 0.112632i
\(986\) 0 0
\(987\) −0.710882 −0.0226276
\(988\) 0 0
\(989\) 47.3493 1.50562
\(990\) 0 0
\(991\) −25.4453 44.0726i −0.808297 1.40001i −0.914042 0.405619i \(-0.867056\pi\)
0.105745 0.994393i \(-0.466277\pi\)
\(992\) 0 0
\(993\) 0.0526244i 0.00166999i
\(994\) 0 0
\(995\) 5.42445 + 3.13181i 0.171967 + 0.0992849i
\(996\) 0 0
\(997\) 3.69697 6.40334i 0.117084 0.202796i −0.801527 0.597959i \(-0.795979\pi\)
0.918611 + 0.395163i \(0.129312\pi\)
\(998\) 0 0
\(999\) 9.60595 5.54600i 0.303919 0.175468i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1456.2.cc.f.673.4 16
4.3 odd 2 364.2.u.a.309.5 yes 16
12.11 even 2 3276.2.cf.c.1765.4 16
13.4 even 6 inner 1456.2.cc.f.225.4 16
28.3 even 6 2548.2.bb.c.569.4 16
28.11 odd 6 2548.2.bb.d.569.5 16
28.19 even 6 2548.2.bq.c.361.5 16
28.23 odd 6 2548.2.bq.e.361.4 16
28.27 even 2 2548.2.u.c.1765.4 16
52.3 odd 6 4732.2.g.k.337.8 16
52.11 even 12 4732.2.a.s.1.4 8
52.15 even 12 4732.2.a.t.1.4 8
52.23 odd 6 4732.2.g.k.337.7 16
52.43 odd 6 364.2.u.a.225.5 16
156.95 even 6 3276.2.cf.c.2773.5 16
364.95 odd 6 2548.2.bq.e.1941.4 16
364.199 even 6 2548.2.bq.c.1941.5 16
364.251 even 6 2548.2.u.c.589.4 16
364.303 odd 6 2548.2.bb.d.1733.5 16
364.355 even 6 2548.2.bb.c.1733.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
364.2.u.a.225.5 16 52.43 odd 6
364.2.u.a.309.5 yes 16 4.3 odd 2
1456.2.cc.f.225.4 16 13.4 even 6 inner
1456.2.cc.f.673.4 16 1.1 even 1 trivial
2548.2.u.c.589.4 16 364.251 even 6
2548.2.u.c.1765.4 16 28.27 even 2
2548.2.bb.c.569.4 16 28.3 even 6
2548.2.bb.c.1733.4 16 364.355 even 6
2548.2.bb.d.569.5 16 28.11 odd 6
2548.2.bb.d.1733.5 16 364.303 odd 6
2548.2.bq.c.361.5 16 28.19 even 6
2548.2.bq.c.1941.5 16 364.199 even 6
2548.2.bq.e.361.4 16 28.23 odd 6
2548.2.bq.e.1941.4 16 364.95 odd 6
3276.2.cf.c.1765.4 16 12.11 even 2
3276.2.cf.c.2773.5 16 156.95 even 6
4732.2.a.s.1.4 8 52.11 even 12
4732.2.a.t.1.4 8 52.15 even 12
4732.2.g.k.337.7 16 52.23 odd 6
4732.2.g.k.337.8 16 52.3 odd 6