Properties

Label 1620.2.r.h.1189.2
Level $1620$
Weight $2$
Character 1620.1189
Analytic conductor $12.936$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(109,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.r (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: \(12.9357651274\)
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: \( x^{16} + 3x^{14} - 11x^{12} - 90x^{10} - 450x^{8} - 2250x^{6} - 6875x^{4} + 46875x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.2
Root \(1.76348 - 1.37482i\) of defining polynomial
Character \(\chi\) \(=\) 1620.1189
Dual form 1620.2.r.h.109.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.76348 - 1.37482i) q^{5} +(-4.27415 - 2.46768i) q^{7} +O(q^{10})\) \(q+(-1.76348 - 1.37482i) q^{5} +(-4.27415 - 2.46768i) q^{7} +(-1.20635 + 2.08945i) q^{11} +(-2.51942 + 1.45459i) q^{13} -6.86869i q^{17} +4.17891 q^{19} +(2.90917 - 1.67961i) q^{23} +(1.21972 + 4.84895i) q^{25} +(-2.59808 + 4.50000i) q^{29} +(3.08945 + 5.35109i) q^{31} +(4.14474 + 10.2279i) q^{35} +7.84453i q^{37} +(-2.93840 - 5.08945i) q^{41} +(4.27415 + 2.46768i) q^{43} +(-10.3122 - 5.95376i) q^{47} +(8.67891 + 15.0323i) q^{49} +8.54830i q^{53} +(5.00000 - 2.02619i) q^{55} +(0.525704 + 0.910546i) q^{59} +(-4.58945 + 7.94917i) q^{61} +(6.44274 + 0.898624i) q^{65} +(3.50947 - 2.02619i) q^{67} +14.1663 q^{71} -2.02619i q^{73} +(10.3122 - 5.95376i) q^{77} +(3.00000 - 5.19615i) q^{79} +(4.49387 + 2.59454i) q^{83} +(-9.44324 + 12.1128i) q^{85} +3.09334 q^{89} +14.3578 q^{91} +(-7.36942 - 5.74527i) q^{95} +(-0.764681 - 0.441489i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{19} - 6 q^{25} + 4 q^{31} + 48 q^{49} + 80 q^{55} - 28 q^{61} + 48 q^{79} - 22 q^{85} + 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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 0
\(4\) 0 0
\(5\) −1.76348 1.37482i −0.788652 0.614840i
\(6\) 0 0
\(7\) −4.27415 2.46768i −1.61548 0.932696i −0.988070 0.154005i \(-0.950783\pi\)
−0.627407 0.778692i \(-0.715884\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.20635 + 2.08945i −0.363727 + 0.629994i −0.988571 0.150756i \(-0.951829\pi\)
0.624844 + 0.780750i \(0.285163\pi\)
\(12\) 0 0
\(13\) −2.51942 + 1.45459i −0.698760 + 0.403429i −0.806886 0.590708i \(-0.798849\pi\)
0.108125 + 0.994137i \(0.465515\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.86869i 1.66590i −0.553347 0.832951i \(-0.686650\pi\)
0.553347 0.832951i \(-0.313350\pi\)
\(18\) 0 0
\(19\) 4.17891 0.958707 0.479354 0.877622i \(-0.340871\pi\)
0.479354 + 0.877622i \(0.340871\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.90917 1.67961i 0.606604 0.350223i −0.165031 0.986288i \(-0.552772\pi\)
0.771635 + 0.636065i \(0.219439\pi\)
\(24\) 0 0
\(25\) 1.21972 + 4.84895i 0.243943 + 0.969790i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.59808 + 4.50000i −0.482451 + 0.835629i −0.999797 0.0201471i \(-0.993587\pi\)
0.517346 + 0.855776i \(0.326920\pi\)
\(30\) 0 0
\(31\) 3.08945 + 5.35109i 0.554882 + 0.961084i 0.997913 + 0.0645778i \(0.0205701\pi\)
−0.443030 + 0.896507i \(0.646097\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.14474 + 10.2279i 0.700590 + 1.72883i
\(36\) 0 0
\(37\) 7.84453i 1.28963i 0.764337 + 0.644817i \(0.223066\pi\)
−0.764337 + 0.644817i \(0.776934\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.93840 5.08945i −0.458901 0.794839i 0.540003 0.841663i \(-0.318423\pi\)
−0.998903 + 0.0468242i \(0.985090\pi\)
\(42\) 0 0
\(43\) 4.27415 + 2.46768i 0.651802 + 0.376318i 0.789146 0.614205i \(-0.210523\pi\)
−0.137344 + 0.990523i \(0.543857\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.3122 5.95376i −1.50419 0.868445i −0.999988 0.00486027i \(-0.998453\pi\)
−0.504203 0.863585i \(-0.668214\pi\)
\(48\) 0 0
\(49\) 8.67891 + 15.0323i 1.23984 + 2.14747i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.54830i 1.17420i 0.809515 + 0.587100i \(0.199730\pi\)
−0.809515 + 0.587100i \(0.800270\pi\)
\(54\) 0 0
\(55\) 5.00000 2.02619i 0.674200 0.273212i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.525704 + 0.910546i 0.0684408 + 0.118543i 0.898215 0.439556i \(-0.144864\pi\)
−0.829774 + 0.558099i \(0.811531\pi\)
\(60\) 0 0
\(61\) −4.58945 + 7.94917i −0.587619 + 1.01779i 0.406924 + 0.913462i \(0.366601\pi\)
−0.994543 + 0.104325i \(0.966732\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.44274 + 0.898624i 0.799123 + 0.111461i
\(66\) 0 0
\(67\) 3.50947 2.02619i 0.428750 0.247539i −0.270064 0.962842i \(-0.587045\pi\)
0.698814 + 0.715303i \(0.253712\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.1663 1.68123 0.840614 0.541634i \(-0.182194\pi\)
0.840614 + 0.541634i \(0.182194\pi\)
\(72\) 0 0
\(73\) 2.02619i 0.237148i −0.992945 0.118574i \(-0.962168\pi\)
0.992945 0.118574i \(-0.0378323\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3122 5.95376i 1.17519 0.678494i
\(78\) 0 0
\(79\) 3.00000 5.19615i 0.337526 0.584613i −0.646440 0.762964i \(-0.723743\pi\)
0.983967 + 0.178352i \(0.0570765\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.49387 + 2.59454i 0.493267 + 0.284788i 0.725929 0.687770i \(-0.241410\pi\)
−0.232662 + 0.972558i \(0.574744\pi\)
\(84\) 0 0
\(85\) −9.44324 + 12.1128i −1.02426 + 1.31382i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.09334 0.327893 0.163947 0.986469i \(-0.447578\pi\)
0.163947 + 0.986469i \(0.447578\pi\)
\(90\) 0 0
\(91\) 14.3578 1.50511
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.36942 5.74527i −0.756086 0.589452i
\(96\) 0 0
\(97\) −0.764681 0.441489i −0.0776416 0.0448264i 0.460677 0.887568i \(-0.347607\pi\)
−0.538318 + 0.842742i \(0.680940\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.93840 5.08945i 0.292382 0.506420i −0.681991 0.731361i \(-0.738886\pi\)
0.974372 + 0.224941i \(0.0722190\pi\)
\(102\) 0 0
\(103\) −8.54830 + 4.93536i −0.842289 + 0.486296i −0.858042 0.513580i \(-0.828319\pi\)
0.0157525 + 0.999876i \(0.494986\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.7668 + 6.79357i −1.10693 + 0.639085i −0.938032 0.346549i \(-0.887353\pi\)
−0.168896 + 0.985634i \(0.554020\pi\)
\(114\) 0 0
\(115\) −7.43943 1.03764i −0.693731 0.0967606i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −16.9497 + 29.3578i −1.55378 + 2.69123i
\(120\) 0 0
\(121\) 2.58945 + 4.48507i 0.235405 + 0.407733i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.51551 10.2279i 0.403879 0.914812i
\(126\) 0 0
\(127\) 4.93536i 0.437943i 0.975731 + 0.218971i \(0.0702701\pi\)
−0.975731 + 0.218971i \(0.929730\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −7.08314 12.2684i −0.618857 1.07189i −0.989695 0.143194i \(-0.954263\pi\)
0.370838 0.928698i \(-0.379071\pi\)
\(132\) 0 0
\(133\) −17.8613 10.3122i −1.54877 0.894183i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.36376 + 2.51942i 0.372821 + 0.215248i 0.674690 0.738101i \(-0.264277\pi\)
−0.301869 + 0.953349i \(0.597611\pi\)
\(138\) 0 0
\(139\) 2.91055 + 5.04121i 0.246869 + 0.427590i 0.962656 0.270729i \(-0.0872648\pi\)
−0.715786 + 0.698319i \(0.753932\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 7.01894i 0.586953i
\(144\) 0 0
\(145\) 10.7684 4.36376i 0.894264 0.362390i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.58788 + 11.4105i 0.539700 + 0.934788i 0.998920 + 0.0464656i \(0.0147958\pi\)
−0.459220 + 0.888323i \(0.651871\pi\)
\(150\) 0 0
\(151\) 3.08945 5.35109i 0.251416 0.435466i −0.712500 0.701672i \(-0.752437\pi\)
0.963916 + 0.266207i \(0.0857704\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.90862 13.6840i 0.153304 1.09912i
\(156\) 0 0
\(157\) −6.79357 + 3.92227i −0.542186 + 0.313031i −0.745964 0.665986i \(-0.768011\pi\)
0.203779 + 0.979017i \(0.434678\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.5790 −1.30661
\(162\) 0 0
\(163\) 10.7537i 0.842295i 0.906992 + 0.421148i \(0.138373\pi\)
−0.906992 + 0.421148i \(0.861627\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.40305 + 4.27415i −0.572865 + 0.330744i −0.758293 0.651914i \(-0.773966\pi\)
0.185428 + 0.982658i \(0.440633\pi\)
\(168\) 0 0
\(169\) −2.26836 + 3.92892i −0.174489 + 0.302225i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −16.2607 9.38811i −1.23628 0.713765i −0.267945 0.963434i \(-0.586345\pi\)
−0.968331 + 0.249670i \(0.919678\pi\)
\(174\) 0 0
\(175\) 6.75241 23.7350i 0.510434 1.79420i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −7.97961 −0.596424 −0.298212 0.954500i \(-0.596390\pi\)
−0.298212 + 0.954500i \(0.596390\pi\)
\(180\) 0 0
\(181\) 3.82109 0.284020 0.142010 0.989865i \(-0.454644\pi\)
0.142010 + 0.989865i \(0.454644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.7849 13.8337i 0.792919 1.01707i
\(186\) 0 0
\(187\) 14.3518 + 8.28602i 1.04951 + 0.605934i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.30916 5.73164i 0.239443 0.414727i −0.721112 0.692819i \(-0.756369\pi\)
0.960554 + 0.278092i \(0.0897020\pi\)
\(192\) 0 0
\(193\) 18.8513 10.8838i 1.35695 0.783435i 0.367737 0.929930i \(-0.380133\pi\)
0.989211 + 0.146495i \(0.0467992\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.4170i 1.09842i 0.835686 + 0.549208i \(0.185070\pi\)
−0.835686 + 0.549208i \(0.814930\pi\)
\(198\) 0 0
\(199\) −20.3578 −1.44313 −0.721564 0.692348i \(-0.756576\pi\)
−0.721564 + 0.692348i \(0.756576\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 22.2091 12.8225i 1.55878 0.899960i
\(204\) 0 0
\(205\) −1.81530 + 13.0149i −0.126786 + 0.909002i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.04121 + 8.73164i −0.348708 + 0.603980i
\(210\) 0 0
\(211\) −8.08945 14.0113i −0.556901 0.964581i −0.997753 0.0670010i \(-0.978657\pi\)
0.440852 0.897580i \(-0.354676\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.14474 10.2279i −0.282669 0.697538i
\(216\) 0 0
\(217\) 30.4952i 2.07015i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.99110 + 17.3051i 0.672074 + 1.16407i
\(222\) 0 0
\(223\) 22.1354 + 12.7799i 1.48230 + 0.855805i 0.999798 0.0200906i \(-0.00639547\pi\)
0.482500 + 0.875896i \(0.339729\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.3908 + 9.46323i 1.08790 + 0.628097i 0.933015 0.359837i \(-0.117168\pi\)
0.154880 + 0.987933i \(0.450501\pi\)
\(228\) 0 0
\(229\) 3.41055 + 5.90724i 0.225375 + 0.390361i 0.956432 0.291955i \(-0.0943059\pi\)
−0.731057 + 0.682317i \(0.760973\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.150248i 0.00984309i 0.999988 + 0.00492154i \(0.00156658\pi\)
−0.999988 + 0.00492154i \(0.998433\pi\)
\(234\) 0 0
\(235\) 10.0000 + 24.6768i 0.652328 + 1.60974i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.51551 + 7.82109i 0.292084 + 0.505904i 0.974302 0.225244i \(-0.0723180\pi\)
−0.682218 + 0.731149i \(0.738985\pi\)
\(240\) 0 0
\(241\) −9.50000 + 16.4545i −0.611949 + 1.05993i 0.378963 + 0.925412i \(0.376281\pi\)
−0.990912 + 0.134515i \(0.957053\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.36172 38.4411i 0.342547 2.45591i
\(246\) 0 0
\(247\) −10.5284 + 6.07858i −0.669907 + 0.386771i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3205 1.09326 0.546630 0.837374i \(-0.315910\pi\)
0.546630 + 0.837374i \(0.315910\pi\)
\(252\) 0 0
\(253\) 8.10477i 0.509543i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −19.1698 + 11.0677i −1.19578 + 0.690385i −0.959612 0.281327i \(-0.909226\pi\)
−0.236170 + 0.971712i \(0.575892\pi\)
\(258\) 0 0
\(259\) 19.3578 33.5287i 1.20284 2.08337i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.81834 + 3.35922i 0.358774 + 0.207138i 0.668543 0.743674i \(-0.266918\pi\)
−0.309769 + 0.950812i \(0.600252\pi\)
\(264\) 0 0
\(265\) 11.7524 15.0747i 0.721945 0.926034i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −30.0646 −1.83307 −0.916536 0.399952i \(-0.869027\pi\)
−0.916536 + 0.399952i \(0.869027\pi\)
\(270\) 0 0
\(271\) −22.3578 −1.35814 −0.679070 0.734073i \(-0.737617\pi\)
−0.679070 + 0.734073i \(0.737617\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −11.6031 3.30097i −0.699690 0.199056i
\(276\) 0 0
\(277\) −9.31298 5.37685i −0.559563 0.323064i 0.193407 0.981119i \(-0.438046\pi\)
−0.752970 + 0.658055i \(0.771380\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.21712 10.7684i 0.370882 0.642387i −0.618819 0.785533i \(-0.712389\pi\)
0.989701 + 0.143147i \(0.0457221\pi\)
\(282\) 0 0
\(283\) 1.52936 0.882978i 0.0909112 0.0524876i −0.453855 0.891075i \(-0.649952\pi\)
0.544766 + 0.838588i \(0.316618\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 29.0041i 1.71206i
\(288\) 0 0
\(289\) −30.1789 −1.77523
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.8454 + 10.3030i −1.04254 + 0.601910i −0.920551 0.390622i \(-0.872260\pi\)
−0.121987 + 0.992532i \(0.538927\pi\)
\(294\) 0 0
\(295\) 0.324773 2.32848i 0.0189090 0.135569i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.88627 + 8.46327i −0.282581 + 0.489444i
\(300\) 0 0
\(301\) −12.1789 21.0945i −0.701981 1.21587i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 19.0221 7.70850i 1.08920 0.441387i
\(306\) 0 0
\(307\) 8.98775i 0.512958i −0.966550 0.256479i \(-0.917438\pi\)
0.966550 0.256479i \(-0.0825624\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5.35109 9.26836i −0.303433 0.525561i 0.673479 0.739207i \(-0.264799\pi\)
−0.976911 + 0.213646i \(0.931466\pi\)
\(312\) 0 0
\(313\) −6.02889 3.48078i −0.340773 0.196745i 0.319841 0.947471i \(-0.396371\pi\)
−0.660614 + 0.750726i \(0.729704\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 29.4821 + 17.0215i 1.65588 + 0.956021i 0.974587 + 0.224008i \(0.0719143\pi\)
0.681290 + 0.732013i \(0.261419\pi\)
\(318\) 0 0
\(319\) −6.26836 10.8571i −0.350961 0.607882i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 28.7036i 1.59711i
\(324\) 0 0
\(325\) −10.1262 10.4423i −0.561699 0.579237i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 29.3840 + 50.8945i 1.61999 + 2.80591i
\(330\) 0 0
\(331\) 16.2684 28.1776i 0.894190 1.54878i 0.0593863 0.998235i \(-0.481086\pi\)
0.834804 0.550548i \(-0.185581\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −8.97453 1.25176i −0.490331 0.0683907i
\(336\) 0 0
\(337\) −8.54830 + 4.93536i −0.465656 + 0.268846i −0.714419 0.699718i \(-0.753309\pi\)
0.248764 + 0.968564i \(0.419976\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.9078 −0.807303
\(342\) 0 0
\(343\) 51.1196i 2.76020i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.40305 + 4.27415i −0.397416 + 0.229448i −0.685369 0.728196i \(-0.740359\pi\)
0.287952 + 0.957645i \(0.407026\pi\)
\(348\) 0 0
\(349\) 10.0895 17.4754i 0.540076 0.935439i −0.458823 0.888528i \(-0.651729\pi\)
0.998899 0.0469115i \(-0.0149379\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.0397 + 10.9926i 1.01338 + 0.585077i 0.912180 0.409789i \(-0.134398\pi\)
0.101202 + 0.994866i \(0.467731\pi\)
\(354\) 0 0
\(355\) −24.9819 19.4762i −1.32590 1.03369i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.309878 0.0163548 0.00817738 0.999967i \(-0.497397\pi\)
0.00817738 + 0.999967i \(0.497397\pi\)
\(360\) 0 0
\(361\) −1.53673 −0.0808803
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.78566 + 3.57315i −0.145808 + 0.187027i
\(366\) 0 0
\(367\) 8.54830 + 4.93536i 0.446218 + 0.257624i 0.706231 0.707981i \(-0.250394\pi\)
−0.260014 + 0.965605i \(0.583727\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 21.0945 36.5367i 1.09517 1.89689i
\(372\) 0 0
\(373\) −7.78362 + 4.49387i −0.403021 + 0.232684i −0.687786 0.725913i \(-0.741417\pi\)
0.284766 + 0.958597i \(0.408084\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.1165i 0.778539i
\(378\) 0 0
\(379\) 0.357817 0.0183798 0.00918990 0.999958i \(-0.497075\pi\)
0.00918990 + 0.999958i \(0.497075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.8969 6.86869i 0.607904 0.350974i −0.164241 0.986420i \(-0.552517\pi\)
0.772145 + 0.635447i \(0.219184\pi\)
\(384\) 0 0
\(385\) −26.3708 3.67816i −1.34398 0.187456i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −12.8050 + 22.1789i −0.649239 + 1.12452i 0.334066 + 0.942550i \(0.391579\pi\)
−0.983305 + 0.181966i \(0.941754\pi\)
\(390\) 0 0
\(391\) −11.5367 19.9822i −0.583437 1.01054i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −12.4342 + 5.03883i −0.625634 + 0.253531i
\(396\) 0 0
\(397\) 11.0139i 0.552774i 0.961046 + 0.276387i \(0.0891371\pi\)
−0.961046 + 0.276387i \(0.910863\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.3619 17.9473i −0.517447 0.896244i −0.999795 0.0202642i \(-0.993549\pi\)
0.482348 0.875980i \(-0.339784\pi\)
\(402\) 0 0
\(403\) −15.5672 8.98775i −0.775459 0.447712i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.3908 9.46323i −0.812462 0.469075i
\(408\) 0 0
\(409\) 3.41055 + 5.90724i 0.168641 + 0.292094i 0.937942 0.346792i \(-0.112729\pi\)
−0.769302 + 0.638886i \(0.779396\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.18908i 0.255338i
\(414\) 0 0
\(415\) −4.35782 10.7537i −0.213917 0.527879i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.28949 + 14.3578i 0.404968 + 0.701425i 0.994318 0.106453i \(-0.0339493\pi\)
−0.589350 + 0.807878i \(0.700616\pi\)
\(420\) 0 0
\(421\) −14.8578 + 25.7345i −0.724126 + 1.25422i 0.235207 + 0.971945i \(0.424423\pi\)
−0.959333 + 0.282277i \(0.908910\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 33.3059 8.37785i 1.61557 0.406385i
\(426\) 0 0
\(427\) 39.2320 22.6506i 1.89857 1.09614i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.67116 −0.0804972 −0.0402486 0.999190i \(-0.512815\pi\)
−0.0402486 + 0.999190i \(0.512815\pi\)
\(432\) 0 0
\(433\) 36.5737i 1.75762i 0.477170 + 0.878811i \(0.341663\pi\)
−0.477170 + 0.878811i \(0.658337\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.1572 7.01894i 0.581556 0.335761i
\(438\) 0 0
\(439\) 9.26836 16.0533i 0.442355 0.766181i −0.555509 0.831511i \(-0.687477\pi\)
0.997864 + 0.0653296i \(0.0208099\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.1306 9.31298i −0.766386 0.442473i 0.0651979 0.997872i \(-0.479232\pi\)
−0.831584 + 0.555399i \(0.812565\pi\)
\(444\) 0 0
\(445\) −5.45503 4.25280i −0.258593 0.201602i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −25.1783 −1.18824 −0.594120 0.804377i \(-0.702500\pi\)
−0.594120 + 0.804377i \(0.702500\pi\)
\(450\) 0 0
\(451\) 14.1789 0.667659
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −25.3197 19.7395i −1.18701 0.925401i
\(456\) 0 0
\(457\) 23.8902 + 13.7930i 1.11753 + 0.645209i 0.940770 0.339044i \(-0.110104\pi\)
0.176764 + 0.984253i \(0.443437\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.35109 9.26836i 0.249225 0.431671i −0.714086 0.700058i \(-0.753157\pi\)
0.963311 + 0.268387i \(0.0864907\pi\)
\(462\) 0 0
\(463\) 25.6449 14.8061i 1.19182 0.688097i 0.233101 0.972453i \(-0.425113\pi\)
0.958719 + 0.284355i \(0.0917795\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 15.5672i 0.720366i −0.932882 0.360183i \(-0.882714\pi\)
0.932882 0.360183i \(-0.117286\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.3122 + 5.95376i −0.474156 + 0.273754i
\(474\) 0 0
\(475\) 5.09708 + 20.2633i 0.233870 + 0.929744i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −12.6501 + 21.9105i −0.577996 + 1.00112i 0.417713 + 0.908579i \(0.362832\pi\)
−0.995709 + 0.0925394i \(0.970502\pi\)
\(480\) 0 0
\(481\) −11.4105 19.7636i −0.520276 0.901145i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.741529 + 1.82986i 0.0336711 + 0.0830896i
\(486\) 0 0
\(487\) 17.9755i 0.814548i 0.913306 + 0.407274i \(0.133521\pi\)
−0.913306 + 0.407274i \(0.866479\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.7853 + 30.8051i 0.802641 + 1.39021i 0.917872 + 0.396876i \(0.129906\pi\)
−0.115232 + 0.993339i \(0.536761\pi\)
\(492\) 0 0
\(493\) 30.9091 + 17.8454i 1.39208 + 0.803716i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −60.5488 34.9579i −2.71599 1.56808i
\(498\) 0 0
\(499\) 2.08945 + 3.61904i 0.0935368 + 0.162011i 0.908997 0.416803i \(-0.136849\pi\)
−0.815460 + 0.578813i \(0.803516\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 37.2519i 1.66098i 0.557032 + 0.830491i \(0.311940\pi\)
−0.557032 + 0.830491i \(0.688060\pi\)
\(504\) 0 0
\(505\) −12.1789 + 4.93536i −0.541954 + 0.219621i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.77398 6.53673i −0.167279 0.289735i 0.770183 0.637822i \(-0.220165\pi\)
−0.937462 + 0.348087i \(0.886831\pi\)
\(510\) 0 0
\(511\) −5.00000 + 8.66025i −0.221187 + 0.383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.8600 + 3.04900i 0.963267 + 0.134355i
\(516\) 0 0
\(517\) 24.8802 14.3646i 1.09423 0.631755i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.03102 −0.395656 −0.197828 0.980237i \(-0.563389\pi\)
−0.197828 + 0.980237i \(0.563389\pi\)
\(522\) 0 0
\(523\) 7.58430i 0.331638i 0.986156 + 0.165819i \(0.0530268\pi\)
−0.986156 + 0.165819i \(0.946973\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 36.7550 21.2205i 1.60107 0.924380i
\(528\) 0 0
\(529\) −5.85782 + 10.1460i −0.254688 + 0.441132i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14.8061 + 8.54830i 0.641323 + 0.370268i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −41.8791 −1.80386
\(540\) 0 0
\(541\) 3.35782 0.144364 0.0721819 0.997391i \(-0.477004\pi\)
0.0721819 + 0.997391i \(0.477004\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.3444 + 9.62377i 0.528774 + 0.412237i
\(546\) 0 0
\(547\) −17.0966 9.87073i −0.730998 0.422042i 0.0877892 0.996139i \(-0.472020\pi\)
−0.818787 + 0.574097i \(0.805353\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −10.8571 + 18.8051i −0.462529 + 0.801124i
\(552\) 0 0
\(553\) −25.6449 + 14.8061i −1.09053 + 0.629619i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.2320i 1.66231i −0.556037 0.831157i \(-0.687679\pi\)
0.556037 0.831157i \(-0.312321\pi\)
\(558\) 0 0
\(559\) −14.3578 −0.607271
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −10.3122 + 5.95376i −0.434608 + 0.250921i −0.701308 0.712859i \(-0.747400\pi\)
0.266700 + 0.963780i \(0.414067\pi\)
\(564\) 0 0
\(565\) 30.0905 + 4.19698i 1.26592 + 0.176568i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.22731 3.85782i 0.0933738 0.161728i −0.815555 0.578680i \(-0.803568\pi\)
0.908929 + 0.416952i \(0.136902\pi\)
\(570\) 0 0
\(571\) 9.08945 + 15.7434i 0.380382 + 0.658841i 0.991117 0.132994i \(-0.0424592\pi\)
−0.610735 + 0.791835i \(0.709126\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.6927 + 12.0578i 0.487619 + 0.502844i
\(576\) 0 0
\(577\) 7.84453i 0.326572i 0.986579 + 0.163286i \(0.0522094\pi\)
−0.986579 + 0.163286i \(0.947791\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.8050 22.1789i −0.531241 0.920136i
\(582\) 0 0
\(583\) −17.8613 10.3122i −0.739739 0.427088i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.0520 + 5.80351i 0.414890 + 0.239537i 0.692888 0.721045i \(-0.256338\pi\)
−0.277999 + 0.960581i \(0.589671\pi\)
\(588\) 0 0
\(589\) 12.9105 + 22.3617i 0.531970 + 0.921399i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.03883i 0.206920i 0.994634 + 0.103460i \(0.0329914\pi\)
−0.994634 + 0.103460i \(0.967009\pi\)
\(594\) 0 0
\(595\) 70.2524 28.4690i 2.88007 1.16711i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.50531 + 14.7316i 0.347518 + 0.601918i 0.985808 0.167878i \(-0.0536913\pi\)
−0.638290 + 0.769796i \(0.720358\pi\)
\(600\) 0 0
\(601\) 1.67891 2.90795i 0.0684841 0.118618i −0.829750 0.558135i \(-0.811517\pi\)
0.898234 + 0.439517i \(0.144850\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.59973 11.4694i 0.0650383 0.466296i
\(606\) 0 0
\(607\) −4.27415 + 2.46768i −0.173482 + 0.100160i −0.584227 0.811590i \(-0.698602\pi\)
0.410744 + 0.911751i \(0.365269\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 34.6410 1.40143
\(612\) 0 0
\(613\) 37.7170i 1.52337i −0.647945 0.761687i \(-0.724372\pi\)
0.647945 0.761687i \(-0.275628\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.62399 + 2.66966i −0.186155 + 0.107477i −0.590181 0.807271i \(-0.700944\pi\)
0.404026 + 0.914747i \(0.367610\pi\)
\(618\) 0 0
\(619\) 5.82109 10.0824i 0.233969 0.405247i −0.725003 0.688745i \(-0.758162\pi\)
0.958973 + 0.283499i \(0.0914951\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −13.2214 7.63337i −0.529704 0.305825i
\(624\) 0 0
\(625\) −22.0246 + 11.8287i −0.880984 + 0.473147i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 53.8817 2.14840
\(630\) 0 0
\(631\) −22.5367 −0.897173 −0.448586 0.893739i \(-0.648072\pi\)
−0.448586 + 0.893739i \(0.648072\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.78526 8.70341i 0.269265 0.345384i
\(636\) 0 0
\(637\) −43.7316 25.2484i −1.73271 1.00038i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.15551 + 15.8578i −0.361621 + 0.626346i −0.988228 0.152990i \(-0.951110\pi\)
0.626607 + 0.779336i \(0.284443\pi\)
\(642\) 0 0
\(643\) −17.8613 + 10.3122i −0.704380 + 0.406674i −0.808977 0.587841i \(-0.799978\pi\)
0.104597 + 0.994515i \(0.466645\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 44.2709i 1.74047i −0.492639 0.870234i \(-0.663968\pi\)
0.492639 0.870234i \(-0.336032\pi\)
\(648\) 0 0
\(649\) −2.53673 −0.0995752
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.260237 0.150248i 0.0101839 0.00587967i −0.494899 0.868950i \(-0.664795\pi\)
0.505083 + 0.863071i \(0.331462\pi\)
\(654\) 0 0
\(655\) −4.37587 + 31.3731i −0.170979 + 1.22585i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20.7237 + 35.8945i −0.807282 + 1.39825i 0.107458 + 0.994210i \(0.465729\pi\)
−0.914740 + 0.404043i \(0.867605\pi\)
\(660\) 0 0
\(661\) 23.2156 + 40.2107i 0.902983 + 1.56401i 0.823601 + 0.567169i \(0.191961\pi\)
0.0793821 + 0.996844i \(0.474705\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 17.3205 + 42.7415i 0.671660 + 1.65744i
\(666\) 0 0
\(667\) 17.4550i 0.675861i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.0729 19.1789i −0.427466 0.740394i
\(672\) 0 0
\(673\) −12.5971 7.27293i −0.485582 0.280351i 0.237158 0.971471i \(-0.423784\pi\)
−0.722740 + 0.691120i \(0.757117\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.8724 + 17.2469i 1.14809 + 0.662850i 0.948421 0.317013i \(-0.102680\pi\)
0.199669 + 0.979863i \(0.436013\pi\)
\(678\) 0 0
\(679\) 2.17891 + 3.77398i 0.0836188 + 0.144832i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.5714i 1.70548i 0.522339 + 0.852738i \(0.325060\pi\)
−0.522339 + 0.852738i \(0.674940\pi\)
\(684\) 0 0
\(685\) −4.23164 10.4423i −0.161683 0.398981i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.4342 21.5367i −0.473707 0.820484i
\(690\) 0 0
\(691\) 15.3578 26.6005i 0.584239 1.01193i −0.410731 0.911757i \(-0.634726\pi\)
0.994970 0.100175i \(-0.0319402\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.79810 12.8916i 0.0682057 0.489005i
\(696\) 0 0
\(697\) −34.9579 + 20.1829i −1.32412 + 0.764484i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27.3420 −1.03269 −0.516347 0.856379i \(-0.672709\pi\)
−0.516347 + 0.856379i \(0.672709\pi\)
\(702\) 0 0
\(703\) 32.7816i 1.23638i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −25.1183 + 14.5021i −0.944671 + 0.545406i
\(708\) 0 0
\(709\) 7.76836 13.4552i 0.291747 0.505321i −0.682476 0.730908i \(-0.739097\pi\)
0.974223 + 0.225587i \(0.0724301\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 17.9755 + 10.3782i 0.673188 + 0.388665i
\(714\) 0 0
\(715\) −9.64981 + 12.3777i −0.360882 + 0.462902i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 40.3960 1.50652 0.753259 0.657724i \(-0.228481\pi\)
0.753259 + 0.657724i \(0.228481\pi\)
\(720\) 0 0
\(721\) 48.7156 1.81426
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −24.9892 7.10922i −0.928075 0.264030i
\(726\) 0 0
\(727\) −40.7614 23.5336i −1.51176 0.872813i −0.999906 0.0137388i \(-0.995627\pi\)
−0.511851 0.859074i \(-0.671040\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.9497 29.3578i 0.626909 1.08584i
\(732\) 0 0
\(733\) 17.0966 9.87073i 0.631477 0.364584i −0.149847 0.988709i \(-0.547878\pi\)
0.781324 + 0.624126i \(0.214545\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.77717i 0.360147i
\(738\) 0 0
\(739\) 22.8945 0.842189 0.421095 0.907017i \(-0.361646\pi\)
0.421095 + 0.907017i \(0.361646\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −43.0938 + 24.8802i −1.58096 + 0.912767i −0.586239 + 0.810138i \(0.699392\pi\)
−0.994720 + 0.102628i \(0.967275\pi\)
\(744\) 0 0
\(745\) 4.06991 29.1794i 0.149110 1.06905i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 3.17891 + 5.50603i 0.116000 + 0.200918i 0.918179 0.396166i \(-0.129659\pi\)
−0.802179 + 0.597084i \(0.796326\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −12.8050 + 5.18908i −0.466022 + 0.188850i
\(756\) 0 0
\(757\) 1.76596i 0.0641847i 0.999485 + 0.0320924i \(0.0102171\pi\)
−0.999485 + 0.0320924i \(0.989783\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.866025 + 1.50000i 0.0313934 + 0.0543750i 0.881295 0.472566i \(-0.156672\pi\)
−0.849902 + 0.526941i \(0.823339\pi\)
\(762\) 0 0
\(763\) 29.9191 + 17.2738i 1.08314 + 0.625353i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.64893 1.52936i −0.0956474 0.0552221i
\(768\) 0 0
\(769\) 12.8578 + 22.2704i 0.463665 + 0.803091i 0.999140 0.0414599i \(-0.0132009\pi\)
−0.535475 + 0.844551i \(0.679868\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 30.6837i 1.10362i −0.833971 0.551809i \(-0.813938\pi\)
0.833971 0.551809i \(-0.186062\pi\)
\(774\) 0 0
\(775\) −22.1789 + 21.5074i −0.796690 + 0.772569i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.2793 21.2684i −0.439951 0.762018i
\(780\) 0 0
\(781\) −17.0895 + 29.5998i −0.611509 + 1.05916i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.3727 + 2.42313i 0.620060 + 0.0864851i
\(786\) 0 0
\(787\) −41.5261 + 23.9751i −1.48024 + 0.854620i −0.999750 0.0223775i \(-0.992876\pi\)
−0.480495 + 0.876997i \(0.659543\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 67.0574 2.38429
\(792\) 0 0
\(793\) 26.7030i 0.948252i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 10.1821 5.87864i 0.360668 0.208232i −0.308706 0.951158i \(-0.599896\pi\)
0.669374 + 0.742926i \(0.266562\pi\)
\(798\) 0 0
\(799\) −40.8945 + 70.8314i −1.44674 + 2.50584i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.23364 + 2.44429i 0.149402 + 0.0862572i
\(804\) 0 0
\(805\) 29.2367 + 22.7932i 1.03046 + 0.803354i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.19615 0.182687 0.0913435 0.995819i \(-0.470884\pi\)
0.0913435 + 0.995819i \(0.470884\pi\)
\(810\) 0 0
\(811\) 6.89454 0.242100 0.121050 0.992646i \(-0.461374\pi\)
0.121050 + 0.992646i \(0.461374\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 14.7845 18.9639i 0.517877 0.664278i
\(816\) 0 0
\(817\) 17.8613 + 10.3122i 0.624887 + 0.360779i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −26.1053 + 45.2156i −0.911080 + 1.57804i −0.0985382 + 0.995133i \(0.531417\pi\)
−0.812542 + 0.582903i \(0.801917\pi\)
\(822\) 0 0
\(823\) −10.0777 + 5.81834i −0.351285 + 0.202815i −0.665251 0.746620i \(-0.731676\pi\)
0.313966 + 0.949434i \(0.398342\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.7374i 0.477696i 0.971057 + 0.238848i \(0.0767697\pi\)
−0.971057 + 0.238848i \(0.923230\pi\)
\(828\) 0 0
\(829\) 4.17891 0.145139 0.0725697 0.997363i \(-0.476880\pi\)
0.0725697 + 0.997363i \(0.476880\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 103.252 59.6127i 3.57748 2.06546i
\(834\) 0 0
\(835\) 18.9313 + 2.64051i 0.655145 + 0.0913788i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −11.9085 + 20.6262i −0.411128 + 0.712095i −0.995013 0.0997414i \(-0.968198\pi\)
0.583885 + 0.811836i \(0.301532\pi\)
\(840\) 0 0
\(841\) 1.00000 + 1.73205i 0.0344828 + 0.0597259i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.40178 3.80997i 0.323431 0.131067i
\(846\) 0 0
\(847\) 25.5598i 0.878245i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.1758 + 22.8211i 0.451659 + 0.782297i
\(852\) 0 0
\(853\) 10.8423 + 6.25983i 0.371235 + 0.214333i 0.673998 0.738733i \(-0.264576\pi\)
−0.302763 + 0.953066i \(0.597909\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.7820 28.1643i −1.66636 0.962075i −0.969574 0.244798i \(-0.921278\pi\)
−0.696788 0.717277i \(-0.745388\pi\)
\(858\) 0 0
\(859\) 19.2684 + 33.3738i 0.657428 + 1.13870i 0.981279 + 0.192591i \(0.0616890\pi\)
−0.323851 + 0.946108i \(0.604978\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 8.54830i 0.290988i −0.989359 0.145494i \(-0.953523\pi\)
0.989359 0.145494i \(-0.0464771\pi\)
\(864\) 0 0
\(865\) 15.7684 + 38.9113i 0.536140 + 1.32302i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.23808 + 12.5367i 0.245535 + 0.425279i
\(870\) 0 0
\(871\) −5.89454 + 10.2096i −0.199729 + 0.345941i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −44.5392 + 32.5728i −1.50570 + 1.10116i
\(876\) 0 0
\(877\) 41.7515 24.1052i 1.40985 0.813975i 0.414473 0.910061i \(-0.363966\pi\)
0.995373 + 0.0960863i \(0.0306325\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −23.9388 −0.806520 −0.403260 0.915085i \(-0.632123\pi\)
−0.403260 + 0.915085i \(0.632123\pi\)
\(882\) 0 0
\(883\) 50.2366i 1.69060i 0.534295 + 0.845298i \(0.320577\pi\)
−0.534295 + 0.845298i \(0.679423\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 22.2091 12.8225i 0.745710 0.430536i −0.0784318 0.996919i \(-0.524991\pi\)
0.824142 + 0.566384i \(0.191658\pi\)
\(888\) 0 0
\(889\) 12.1789 21.0945i 0.408467 0.707486i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −43.0938 24.8802i −1.44208 0.832585i
\(894\) 0 0
\(895\) 14.0719 + 10.9706i 0.470371 + 0.366706i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −32.1065 −1.07081
\(900\) 0 0
\(901\) 58.7156 1.95610
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.73841 5.25333i −0.223992 0.174627i
\(906\) 0 0
\(907\) 51.2898 + 29.6122i 1.70305 + 0.983256i 0.942639 + 0.333815i \(0.108336\pi\)
0.760411 + 0.649442i \(0.224997\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 17.7853 30.8051i 0.589254 1.02062i −0.405076 0.914283i \(-0.632755\pi\)
0.994330 0.106335i \(-0.0339116\pi\)
\(912\) 0 0
\(913\) −10.8423 + 6.25983i −0.358829 + 0.207170i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 69.9158i 2.30882i
\(918\) 0 0
\(919\) 19.8211 0.653837 0.326919 0.945052i \(-0.393990\pi\)
0.326919 + 0.945052i \(0.393990\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.6908 + 20.6061i −1.17478 + 0.678257i
\(924\) 0 0
\(925\) −38.0377 + 9.56810i −1.25067 + 0.314597i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.37206 + 11.0367i −0.209060 + 0.362103i −0.951419 0.307900i \(-0.900374\pi\)
0.742358 + 0.670003i \(0.233707\pi\)
\(930\) 0 0
\(931\) 36.2684 + 62.8186i 1.18865 + 2.05880i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.9173 34.3435i −0.455144 1.12315i
\(936\) 0 0
\(937\) 12.7799i 0.417501i 0.977969 + 0.208751i \(0.0669397\pi\)
−0.977969 + 0.208751i \(0.933060\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −26.9408 46.6629i −0.878246 1.52117i −0.853264 0.521479i \(-0.825380\pi\)
−0.0249824 0.999688i \(-0.507953\pi\)
\(942\) 0 0
\(943\) −17.0966 9.87073i −0.556742 0.321435i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.07858 3.50947i −0.197527 0.114042i 0.397974 0.917397i \(-0.369713\pi\)
−0.595502 + 0.803354i \(0.703047\pi\)
\(948\) 0 0
\(949\) 2.94727 + 5.10482i 0.0956725 + 0.165710i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.7573i 0.380855i 0.981701 + 0.190428i \(0.0609875\pi\)
−0.981701 + 0.190428i \(0.939013\pi\)
\(954\) 0 0
\(955\) −13.7156 + 5.55810i −0.443827 + 0.179856i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.4342 21.5367i −0.401522 0.695457i
\(960\) 0 0
\(961\) −3.58945 + 6.21712i −0.115789 + 0.200552i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −48.2073 6.72388i −1.55185 0.216449i
\(966\) 0 0
\(967\) 11.2931 6.52007i 0.363161 0.209671i −0.307305 0.951611i \(-0.599427\pi\)
0.670467 + 0.741940i \(0.266094\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.7142 −0.696843 −0.348422 0.937338i \(-0.613282\pi\)
−0.348422 + 0.937338i \(0.613282\pi\)
\(972\) 0 0
\(973\) 28.7292i 0.921016i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −23.5336 + 13.5871i −0.752907 + 0.434691i −0.826743 0.562579i \(-0.809809\pi\)
0.0738365 + 0.997270i \(0.476476\pi\)
\(978\) 0 0
\(979\) −3.73164 + 6.46339i −0.119264 + 0.206571i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 22.4694 + 12.9727i 0.716662 + 0.413765i 0.813523 0.581533i \(-0.197547\pi\)
−0.0968610 + 0.995298i \(0.530880\pi\)
\(984\) 0 0
\(985\) 21.1957 27.1875i 0.675350 0.866267i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.5790 0.527181
\(990\) 0 0
\(991\) −32.5367 −1.03356 −0.516782 0.856117i \(-0.672870\pi\)
−0.516782 + 0.856117i \(0.672870\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 35.9006 + 27.9884i 1.13812 + 0.887293i
\(996\) 0 0
\(997\) 25.8703 + 14.9362i 0.819320 + 0.473035i 0.850182 0.526489i \(-0.176492\pi\)
−0.0308620 + 0.999524i \(0.509825\pi\)
\(998\) 0 0
\(999\) 0 0
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 1620.2.r.h.1189.2 16
3.2 odd 2 inner 1620.2.r.h.1189.7 16
5.4 even 2 inner 1620.2.r.h.1189.4 16
9.2 odd 6 1620.2.d.e.649.2 yes 8
9.4 even 3 inner 1620.2.r.h.109.4 16
9.5 odd 6 inner 1620.2.r.h.109.5 16
9.7 even 3 1620.2.d.e.649.7 yes 8
15.14 odd 2 inner 1620.2.r.h.1189.5 16
45.2 even 12 8100.2.a.be.1.1 8
45.4 even 6 inner 1620.2.r.h.109.2 16
45.7 odd 12 8100.2.a.be.1.2 8
45.14 odd 6 inner 1620.2.r.h.109.7 16
45.29 odd 6 1620.2.d.e.649.1 8
45.34 even 6 1620.2.d.e.649.8 yes 8
45.38 even 12 8100.2.a.be.1.7 8
45.43 odd 12 8100.2.a.be.1.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.d.e.649.1 8 45.29 odd 6
1620.2.d.e.649.2 yes 8 9.2 odd 6
1620.2.d.e.649.7 yes 8 9.7 even 3
1620.2.d.e.649.8 yes 8 45.34 even 6
1620.2.r.h.109.2 16 45.4 even 6 inner
1620.2.r.h.109.4 16 9.4 even 3 inner
1620.2.r.h.109.5 16 9.5 odd 6 inner
1620.2.r.h.109.7 16 45.14 odd 6 inner
1620.2.r.h.1189.2 16 1.1 even 1 trivial
1620.2.r.h.1189.4 16 5.4 even 2 inner
1620.2.r.h.1189.5 16 15.14 odd 2 inner
1620.2.r.h.1189.7 16 3.2 odd 2 inner
8100.2.a.be.1.1 8 45.2 even 12
8100.2.a.be.1.2 8 45.7 odd 12
8100.2.a.be.1.7 8 45.38 even 12
8100.2.a.be.1.8 8 45.43 odd 12