Properties

Label 336.3.z.f.47.10
Level $336$
Weight $3$
Character 336.47
Analytic conductor $9.155$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,3,Mod(47,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.z (of order \(6\), degree \(2\), 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: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 83 x^{18} + 2702 x^{16} + 44346 x^{14} + 396449 x^{12} + 1961403 x^{10} + 5268164 x^{8} + \cdots + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 47.10
Root \(0.0416279i\) of defining polynomial
Character \(\chi\) \(=\) 336.47
Dual form 336.3.z.f.143.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.99894 + 0.0796850i) q^{3} +(4.89508 - 8.47853i) q^{5} +(1.55722 + 6.82459i) q^{7} +(8.98730 + 0.477941i) q^{9} +O(q^{10})\) \(q+(2.99894 + 0.0796850i) q^{3} +(4.89508 - 8.47853i) q^{5} +(1.55722 + 6.82459i) q^{7} +(8.98730 + 0.477941i) q^{9} +(2.66502 + 4.61595i) q^{11} +15.7574i q^{13} +(15.3557 - 25.0366i) q^{15} +(-7.79558 - 13.5023i) q^{17} +(10.9740 - 19.0076i) q^{19} +(4.12619 + 20.5906i) q^{21} +(-11.4600 + 19.8494i) q^{23} +(-35.4237 - 61.3556i) q^{25} +(26.9143 + 2.14947i) q^{27} -24.5758i q^{29} +(-10.8812 - 18.8467i) q^{31} +(7.62442 + 14.0553i) q^{33} +(65.4853 + 20.2040i) q^{35} +(-15.4747 + 26.8030i) q^{37} +(-1.25563 + 47.2557i) q^{39} +14.7950 q^{41} +51.1900i q^{43} +(48.0458 - 73.8596i) q^{45} +(10.3662 + 5.98495i) q^{47} +(-44.1501 + 21.2548i) q^{49} +(-22.3025 - 41.1139i) q^{51} +(-47.4641 + 27.4034i) q^{53} +52.1820 q^{55} +(34.4251 - 56.1281i) q^{57} +(40.2875 - 23.2600i) q^{59} +(-27.2055 - 15.7071i) q^{61} +(10.7334 + 62.0789i) q^{63} +(133.600 + 77.1340i) q^{65} +(-14.4100 + 8.31959i) q^{67} +(-35.9497 + 58.6140i) q^{69} -73.0574 q^{71} +(34.4201 - 19.8724i) q^{73} +(-101.344 - 186.825i) q^{75} +(-27.3520 + 25.3757i) q^{77} +(-4.29574 - 2.48015i) q^{79} +(80.5431 + 8.59080i) q^{81} -56.0926i q^{83} -152.640 q^{85} +(1.95832 - 73.7014i) q^{87} +(-4.12044 + 7.13681i) q^{89} +(-107.538 + 24.5378i) q^{91} +(-31.1301 - 57.3872i) q^{93} +(-107.438 - 186.087i) q^{95} +63.5054i q^{97} +(21.7452 + 42.7587i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{3} + 16 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 3 q^{3} + 16 q^{7} + 11 q^{9} + 44 q^{19} - 5 q^{21} - 98 q^{25} + 144 q^{27} + 70 q^{31} + 15 q^{33} - 64 q^{37} - 6 q^{39} + 21 q^{45} - 184 q^{49} + 33 q^{51} - 120 q^{55} + 218 q^{57} - 102 q^{61} - 27 q^{63} + 144 q^{67} + 372 q^{73} + 246 q^{75} - 258 q^{79} + 287 q^{81} - 108 q^{85} - 318 q^{87} - 510 q^{91} - 173 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 2.99894 + 0.0796850i 0.999647 + 0.0265617i
\(4\) 0 0
\(5\) 4.89508 8.47853i 0.979017 1.69571i 0.313030 0.949743i \(-0.398656\pi\)
0.665987 0.745964i \(-0.268011\pi\)
\(6\) 0 0
\(7\) 1.55722 + 6.82459i 0.222460 + 0.974942i
\(8\) 0 0
\(9\) 8.98730 + 0.477941i 0.998589 + 0.0531046i
\(10\) 0 0
\(11\) 2.66502 + 4.61595i 0.242275 + 0.419632i 0.961362 0.275288i \(-0.0887731\pi\)
−0.719087 + 0.694920i \(0.755440\pi\)
\(12\) 0 0
\(13\) 15.7574i 1.21211i 0.795422 + 0.606056i \(0.207249\pi\)
−0.795422 + 0.606056i \(0.792751\pi\)
\(14\) 0 0
\(15\) 15.3557 25.0366i 1.02371 1.66910i
\(16\) 0 0
\(17\) −7.79558 13.5023i −0.458563 0.794255i 0.540322 0.841458i \(-0.318302\pi\)
−0.998885 + 0.0472034i \(0.984969\pi\)
\(18\) 0 0
\(19\) 10.9740 19.0076i 0.577580 1.00040i −0.418176 0.908366i \(-0.637330\pi\)
0.995756 0.0920323i \(-0.0293363\pi\)
\(20\) 0 0
\(21\) 4.12619 + 20.5906i 0.196485 + 0.980507i
\(22\) 0 0
\(23\) −11.4600 + 19.8494i −0.498263 + 0.863017i −0.999998 0.00200458i \(-0.999362\pi\)
0.501735 + 0.865021i \(0.332695\pi\)
\(24\) 0 0
\(25\) −35.4237 61.3556i −1.41695 2.45423i
\(26\) 0 0
\(27\) 26.9143 + 2.14947i 0.996826 + 0.0796100i
\(28\) 0 0
\(29\) 24.5758i 0.847442i −0.905793 0.423721i \(-0.860724\pi\)
0.905793 0.423721i \(-0.139276\pi\)
\(30\) 0 0
\(31\) −10.8812 18.8467i −0.351005 0.607958i 0.635421 0.772166i \(-0.280827\pi\)
−0.986426 + 0.164208i \(0.947493\pi\)
\(32\) 0 0
\(33\) 7.62442 + 14.0553i 0.231043 + 0.425919i
\(34\) 0 0
\(35\) 65.4853 + 20.2040i 1.87101 + 0.577258i
\(36\) 0 0
\(37\) −15.4747 + 26.8030i −0.418235 + 0.724404i −0.995762 0.0919674i \(-0.970684\pi\)
0.577527 + 0.816371i \(0.304018\pi\)
\(38\) 0 0
\(39\) −1.25563 + 47.2557i −0.0321957 + 1.21168i
\(40\) 0 0
\(41\) 14.7950 0.360854 0.180427 0.983588i \(-0.442252\pi\)
0.180427 + 0.983588i \(0.442252\pi\)
\(42\) 0 0
\(43\) 51.1900i 1.19047i 0.803553 + 0.595233i \(0.202940\pi\)
−0.803553 + 0.595233i \(0.797060\pi\)
\(44\) 0 0
\(45\) 48.0458 73.8596i 1.06769 1.64132i
\(46\) 0 0
\(47\) 10.3662 + 5.98495i 0.220558 + 0.127339i 0.606209 0.795306i \(-0.292690\pi\)
−0.385650 + 0.922645i \(0.626023\pi\)
\(48\) 0 0
\(49\) −44.1501 + 21.2548i −0.901023 + 0.433771i
\(50\) 0 0
\(51\) −22.3025 41.1139i −0.437305 0.806155i
\(52\) 0 0
\(53\) −47.4641 + 27.4034i −0.895549 + 0.517045i −0.875753 0.482759i \(-0.839635\pi\)
−0.0197954 + 0.999804i \(0.506301\pi\)
\(54\) 0 0
\(55\) 52.1820 0.948764
\(56\) 0 0
\(57\) 34.4251 56.1281i 0.603949 0.984704i
\(58\) 0 0
\(59\) 40.2875 23.2600i 0.682840 0.394238i −0.118084 0.993004i \(-0.537675\pi\)
0.800924 + 0.598766i \(0.204342\pi\)
\(60\) 0 0
\(61\) −27.2055 15.7071i −0.445993 0.257494i 0.260144 0.965570i \(-0.416230\pi\)
−0.706136 + 0.708076i \(0.749563\pi\)
\(62\) 0 0
\(63\) 10.7334 + 62.0789i 0.170372 + 0.985380i
\(64\) 0 0
\(65\) 133.600 + 77.1340i 2.05539 + 1.18668i
\(66\) 0 0
\(67\) −14.4100 + 8.31959i −0.215074 + 0.124173i −0.603667 0.797236i \(-0.706294\pi\)
0.388593 + 0.921409i \(0.372961\pi\)
\(68\) 0 0
\(69\) −35.9497 + 58.6140i −0.521010 + 0.849478i
\(70\) 0 0
\(71\) −73.0574 −1.02898 −0.514489 0.857497i \(-0.672018\pi\)
−0.514489 + 0.857497i \(0.672018\pi\)
\(72\) 0 0
\(73\) 34.4201 19.8724i 0.471508 0.272225i −0.245363 0.969431i \(-0.578907\pi\)
0.716871 + 0.697206i \(0.245574\pi\)
\(74\) 0 0
\(75\) −101.344 186.825i −1.35126 2.49100i
\(76\) 0 0
\(77\) −27.3520 + 25.3757i −0.355221 + 0.329555i
\(78\) 0 0
\(79\) −4.29574 2.48015i −0.0543765 0.0313943i 0.472565 0.881296i \(-0.343328\pi\)
−0.526942 + 0.849901i \(0.676661\pi\)
\(80\) 0 0
\(81\) 80.5431 + 8.59080i 0.994360 + 0.106059i
\(82\) 0 0
\(83\) 56.0926i 0.675815i −0.941179 0.337907i \(-0.890281\pi\)
0.941179 0.337907i \(-0.109719\pi\)
\(84\) 0 0
\(85\) −152.640 −1.79576
\(86\) 0 0
\(87\) 1.95832 73.7014i 0.0225095 0.847143i
\(88\) 0 0
\(89\) −4.12044 + 7.13681i −0.0462971 + 0.0801889i −0.888245 0.459369i \(-0.848075\pi\)
0.841948 + 0.539558i \(0.181409\pi\)
\(90\) 0 0
\(91\) −107.538 + 24.5378i −1.18174 + 0.269646i
\(92\) 0 0
\(93\) −31.1301 57.3872i −0.334733 0.617067i
\(94\) 0 0
\(95\) −107.438 186.087i −1.13092 1.95881i
\(96\) 0 0
\(97\) 63.5054i 0.654695i 0.944904 + 0.327347i \(0.106155\pi\)
−0.944904 + 0.327347i \(0.893845\pi\)
\(98\) 0 0
\(99\) 21.7452 + 42.7587i 0.219648 + 0.431906i
\(100\) 0 0
\(101\) 16.0741 + 27.8412i 0.159150 + 0.275656i 0.934562 0.355799i \(-0.115791\pi\)
−0.775412 + 0.631455i \(0.782458\pi\)
\(102\) 0 0
\(103\) −85.1203 + 147.433i −0.826411 + 1.43139i 0.0744256 + 0.997227i \(0.476288\pi\)
−0.900836 + 0.434159i \(0.857046\pi\)
\(104\) 0 0
\(105\) 194.777 + 65.8089i 1.85501 + 0.626751i
\(106\) 0 0
\(107\) −101.021 + 174.974i −0.944124 + 1.63527i −0.186629 + 0.982431i \(0.559756\pi\)
−0.757495 + 0.652840i \(0.773577\pi\)
\(108\) 0 0
\(109\) 17.6522 + 30.5745i 0.161947 + 0.280500i 0.935567 0.353150i \(-0.114889\pi\)
−0.773620 + 0.633650i \(0.781556\pi\)
\(110\) 0 0
\(111\) −48.5435 + 79.1474i −0.437329 + 0.713039i
\(112\) 0 0
\(113\) 92.2111i 0.816027i −0.912976 0.408014i \(-0.866222\pi\)
0.912976 0.408014i \(-0.133778\pi\)
\(114\) 0 0
\(115\) 112.196 + 194.329i 0.975616 + 1.68982i
\(116\) 0 0
\(117\) −7.53113 + 141.617i −0.0643686 + 1.21040i
\(118\) 0 0
\(119\) 80.0085 74.2277i 0.672340 0.623762i
\(120\) 0 0
\(121\) 46.2953 80.1858i 0.382606 0.662693i
\(122\) 0 0
\(123\) 44.3694 + 1.17894i 0.360727 + 0.00958489i
\(124\) 0 0
\(125\) −448.854 −3.59083
\(126\) 0 0
\(127\) 75.3464i 0.593279i 0.954990 + 0.296640i \(0.0958660\pi\)
−0.954990 + 0.296640i \(0.904134\pi\)
\(128\) 0 0
\(129\) −4.07907 + 153.516i −0.0316207 + 1.19005i
\(130\) 0 0
\(131\) 36.9356 + 21.3248i 0.281951 + 0.162785i 0.634307 0.773082i \(-0.281286\pi\)
−0.352355 + 0.935866i \(0.614619\pi\)
\(132\) 0 0
\(133\) 146.808 + 45.2943i 1.10382 + 0.340559i
\(134\) 0 0
\(135\) 149.972 217.672i 1.11090 1.61239i
\(136\) 0 0
\(137\) 88.8407 51.2922i 0.648472 0.374396i −0.139398 0.990236i \(-0.544517\pi\)
0.787871 + 0.615841i \(0.211184\pi\)
\(138\) 0 0
\(139\) −90.8399 −0.653524 −0.326762 0.945107i \(-0.605958\pi\)
−0.326762 + 0.945107i \(0.605958\pi\)
\(140\) 0 0
\(141\) 30.6108 + 18.7746i 0.217098 + 0.133153i
\(142\) 0 0
\(143\) −72.7356 + 41.9939i −0.508641 + 0.293664i
\(144\) 0 0
\(145\) −208.367 120.301i −1.43701 0.829660i
\(146\) 0 0
\(147\) −134.097 + 60.2237i −0.912227 + 0.409685i
\(148\) 0 0
\(149\) 147.329 + 85.0603i 0.988784 + 0.570874i 0.904910 0.425602i \(-0.139938\pi\)
0.0838731 + 0.996476i \(0.473271\pi\)
\(150\) 0 0
\(151\) 65.0630 37.5641i 0.430881 0.248769i −0.268841 0.963185i \(-0.586641\pi\)
0.699722 + 0.714415i \(0.253307\pi\)
\(152\) 0 0
\(153\) −63.6079 125.075i −0.415738 0.817486i
\(154\) 0 0
\(155\) −213.057 −1.37456
\(156\) 0 0
\(157\) −180.575 + 104.255i −1.15016 + 0.664045i −0.948926 0.315498i \(-0.897828\pi\)
−0.201234 + 0.979543i \(0.564495\pi\)
\(158\) 0 0
\(159\) −144.526 + 78.3990i −0.908966 + 0.493076i
\(160\) 0 0
\(161\) −153.310 47.3003i −0.952235 0.293791i
\(162\) 0 0
\(163\) −256.887 148.314i −1.57599 0.909900i −0.995410 0.0956983i \(-0.969492\pi\)
−0.580582 0.814202i \(-0.697175\pi\)
\(164\) 0 0
\(165\) 156.491 + 4.15812i 0.948429 + 0.0252007i
\(166\) 0 0
\(167\) 184.743i 1.10625i 0.833100 + 0.553123i \(0.186564\pi\)
−0.833100 + 0.553123i \(0.813436\pi\)
\(168\) 0 0
\(169\) −79.2971 −0.469213
\(170\) 0 0
\(171\) 107.711 165.582i 0.629891 0.968315i
\(172\) 0 0
\(173\) 127.316 220.518i 0.735931 1.27467i −0.218383 0.975863i \(-0.570078\pi\)
0.954314 0.298807i \(-0.0965885\pi\)
\(174\) 0 0
\(175\) 363.565 337.296i 2.07751 1.92741i
\(176\) 0 0
\(177\) 122.673 66.5451i 0.693070 0.375961i
\(178\) 0 0
\(179\) −58.7396 101.740i −0.328154 0.568380i 0.653991 0.756502i \(-0.273093\pi\)
−0.982146 + 0.188122i \(0.939760\pi\)
\(180\) 0 0
\(181\) 20.4562i 0.113017i 0.998402 + 0.0565087i \(0.0179969\pi\)
−0.998402 + 0.0565087i \(0.982003\pi\)
\(182\) 0 0
\(183\) −80.3362 49.2726i −0.438996 0.269249i
\(184\) 0 0
\(185\) 151.500 + 262.405i 0.818918 + 1.41841i
\(186\) 0 0
\(187\) 41.5508 71.9680i 0.222197 0.384856i
\(188\) 0 0
\(189\) 27.2422 + 187.026i 0.144139 + 0.989558i
\(190\) 0 0
\(191\) 92.0960 159.515i 0.482178 0.835157i −0.517613 0.855615i \(-0.673179\pi\)
0.999791 + 0.0204581i \(0.00651248\pi\)
\(192\) 0 0
\(193\) 2.02801 + 3.51262i 0.0105078 + 0.0182001i 0.871232 0.490872i \(-0.163322\pi\)
−0.860724 + 0.509072i \(0.829989\pi\)
\(194\) 0 0
\(195\) 394.512 + 241.966i 2.02314 + 1.24085i
\(196\) 0 0
\(197\) 176.763i 0.897272i −0.893714 0.448636i \(-0.851910\pi\)
0.893714 0.448636i \(-0.148090\pi\)
\(198\) 0 0
\(199\) 21.5178 + 37.2699i 0.108130 + 0.187286i 0.915013 0.403425i \(-0.132181\pi\)
−0.806883 + 0.590711i \(0.798847\pi\)
\(200\) 0 0
\(201\) −43.8776 + 23.8017i −0.218296 + 0.118416i
\(202\) 0 0
\(203\) 167.720 38.2699i 0.826206 0.188522i
\(204\) 0 0
\(205\) 72.4229 125.440i 0.353283 0.611903i
\(206\) 0 0
\(207\) −112.482 + 172.915i −0.543390 + 0.835339i
\(208\) 0 0
\(209\) 116.984 0.559732
\(210\) 0 0
\(211\) 151.863i 0.719731i 0.933004 + 0.359866i \(0.117177\pi\)
−0.933004 + 0.359866i \(0.882823\pi\)
\(212\) 0 0
\(213\) −219.095 5.82158i −1.02861 0.0273313i
\(214\) 0 0
\(215\) 434.016 + 250.579i 2.01868 + 1.16549i
\(216\) 0 0
\(217\) 111.677 103.608i 0.514639 0.477456i
\(218\) 0 0
\(219\) 104.807 56.8535i 0.478572 0.259605i
\(220\) 0 0
\(221\) 212.762 122.838i 0.962725 0.555830i
\(222\) 0 0
\(223\) −159.992 −0.717453 −0.358726 0.933443i \(-0.616789\pi\)
−0.358726 + 0.933443i \(0.616789\pi\)
\(224\) 0 0
\(225\) −289.039 568.352i −1.28462 2.52601i
\(226\) 0 0
\(227\) 179.764 103.787i 0.791912 0.457210i −0.0487235 0.998812i \(-0.515515\pi\)
0.840635 + 0.541602i \(0.182182\pi\)
\(228\) 0 0
\(229\) 318.721 + 184.014i 1.39179 + 0.803552i 0.993514 0.113711i \(-0.0362739\pi\)
0.398280 + 0.917264i \(0.369607\pi\)
\(230\) 0 0
\(231\) −84.0491 + 73.9208i −0.363849 + 0.320004i
\(232\) 0 0
\(233\) 13.9743 + 8.06808i 0.0599757 + 0.0346270i 0.529688 0.848193i \(-0.322309\pi\)
−0.469712 + 0.882820i \(0.655642\pi\)
\(234\) 0 0
\(235\) 101.487 58.5937i 0.431861 0.249335i
\(236\) 0 0
\(237\) −12.6850 7.78012i −0.0535234 0.0328275i
\(238\) 0 0
\(239\) 86.2929 0.361058 0.180529 0.983570i \(-0.442219\pi\)
0.180529 + 0.983570i \(0.442219\pi\)
\(240\) 0 0
\(241\) 259.957 150.086i 1.07866 0.622765i 0.148126 0.988968i \(-0.452676\pi\)
0.930535 + 0.366203i \(0.119342\pi\)
\(242\) 0 0
\(243\) 240.860 + 32.1814i 0.991192 + 0.132434i
\(244\) 0 0
\(245\) −35.9094 + 478.372i −0.146569 + 1.95254i
\(246\) 0 0
\(247\) 299.511 + 172.923i 1.21259 + 0.700092i
\(248\) 0 0
\(249\) 4.46974 168.218i 0.0179508 0.675576i
\(250\) 0 0
\(251\) 298.636i 1.18979i 0.803805 + 0.594893i \(0.202805\pi\)
−0.803805 + 0.594893i \(0.797195\pi\)
\(252\) 0 0
\(253\) −122.165 −0.482866
\(254\) 0 0
\(255\) −457.758 12.1631i −1.79513 0.0476985i
\(256\) 0 0
\(257\) 72.5522 125.664i 0.282304 0.488965i −0.689648 0.724145i \(-0.742235\pi\)
0.971952 + 0.235180i \(0.0755679\pi\)
\(258\) 0 0
\(259\) −207.017 63.8704i −0.799292 0.246604i
\(260\) 0 0
\(261\) 11.7458 220.870i 0.0450030 0.846246i
\(262\) 0 0
\(263\) 111.748 + 193.553i 0.424898 + 0.735945i 0.996411 0.0846482i \(-0.0269767\pi\)
−0.571513 + 0.820593i \(0.693643\pi\)
\(264\) 0 0
\(265\) 536.568i 2.02478i
\(266\) 0 0
\(267\) −12.9257 + 21.0745i −0.0484107 + 0.0789309i
\(268\) 0 0
\(269\) −6.39468 11.0759i −0.0237720 0.0411744i 0.853895 0.520446i \(-0.174234\pi\)
−0.877667 + 0.479272i \(0.840901\pi\)
\(270\) 0 0
\(271\) 54.9222 95.1281i 0.202665 0.351026i −0.746721 0.665137i \(-0.768373\pi\)
0.949386 + 0.314111i \(0.101706\pi\)
\(272\) 0 0
\(273\) −324.456 + 65.0182i −1.18848 + 0.238162i
\(274\) 0 0
\(275\) 188.810 327.028i 0.686581 1.18919i
\(276\) 0 0
\(277\) −245.860 425.843i −0.887583 1.53734i −0.842725 0.538345i \(-0.819050\pi\)
−0.0448582 0.998993i \(-0.514284\pi\)
\(278\) 0 0
\(279\) −88.7846 174.582i −0.318224 0.625740i
\(280\) 0 0
\(281\) 80.2824i 0.285703i −0.989744 0.142851i \(-0.954373\pi\)
0.989744 0.142851i \(-0.0456271\pi\)
\(282\) 0 0
\(283\) −141.957 245.877i −0.501615 0.868823i −0.999998 0.00186632i \(-0.999406\pi\)
0.498383 0.866957i \(-0.333927\pi\)
\(284\) 0 0
\(285\) −307.371 566.626i −1.07849 1.98816i
\(286\) 0 0
\(287\) 23.0391 + 100.970i 0.0802756 + 0.351812i
\(288\) 0 0
\(289\) 22.9580 39.7644i 0.0794394 0.137593i
\(290\) 0 0
\(291\) −5.06042 + 190.449i −0.0173898 + 0.654464i
\(292\) 0 0
\(293\) −327.509 −1.11778 −0.558890 0.829242i \(-0.688773\pi\)
−0.558890 + 0.829242i \(0.688773\pi\)
\(294\) 0 0
\(295\) 455.439i 1.54386i
\(296\) 0 0
\(297\) 61.8054 + 129.964i 0.208099 + 0.437588i
\(298\) 0 0
\(299\) −312.776 180.581i −1.04607 0.603950i
\(300\) 0 0
\(301\) −349.351 + 79.7140i −1.16063 + 0.264831i
\(302\) 0 0
\(303\) 45.9869 + 84.7751i 0.151772 + 0.279786i
\(304\) 0 0
\(305\) −266.347 + 153.775i −0.873268 + 0.504182i
\(306\) 0 0
\(307\) 204.668 0.666671 0.333335 0.942808i \(-0.391826\pi\)
0.333335 + 0.942808i \(0.391826\pi\)
\(308\) 0 0
\(309\) −267.019 + 435.359i −0.864139 + 1.40893i
\(310\) 0 0
\(311\) −391.185 + 225.851i −1.25783 + 0.726209i −0.972652 0.232266i \(-0.925386\pi\)
−0.285178 + 0.958475i \(0.592053\pi\)
\(312\) 0 0
\(313\) 278.509 + 160.797i 0.889806 + 0.513730i 0.873879 0.486143i \(-0.161597\pi\)
0.0159272 + 0.999873i \(0.494930\pi\)
\(314\) 0 0
\(315\) 578.879 + 212.878i 1.83771 + 0.675802i
\(316\) 0 0
\(317\) 203.355 + 117.407i 0.641498 + 0.370369i 0.785191 0.619253i \(-0.212565\pi\)
−0.143693 + 0.989622i \(0.545898\pi\)
\(318\) 0 0
\(319\) 113.441 65.4951i 0.355614 0.205314i
\(320\) 0 0
\(321\) −316.900 + 516.687i −0.987227 + 1.60962i
\(322\) 0 0
\(323\) −342.195 −1.05943
\(324\) 0 0
\(325\) 966.808 558.187i 2.97479 1.71750i
\(326\) 0 0
\(327\) 50.5015 + 93.0977i 0.154439 + 0.284702i
\(328\) 0 0
\(329\) −24.7024 + 80.0652i −0.0750831 + 0.243359i
\(330\) 0 0
\(331\) −136.729 78.9403i −0.413077 0.238490i 0.279034 0.960281i \(-0.409986\pi\)
−0.692111 + 0.721791i \(0.743319\pi\)
\(332\) 0 0
\(333\) −151.886 + 233.490i −0.456114 + 0.701172i
\(334\) 0 0
\(335\) 162.900i 0.486270i
\(336\) 0 0
\(337\) 308.611 0.915759 0.457880 0.889014i \(-0.348609\pi\)
0.457880 + 0.889014i \(0.348609\pi\)
\(338\) 0 0
\(339\) 7.34784 276.536i 0.0216750 0.815739i
\(340\) 0 0
\(341\) 57.9970 100.454i 0.170079 0.294586i
\(342\) 0 0
\(343\) −213.807 268.208i −0.623343 0.781949i
\(344\) 0 0
\(345\) 320.984 + 591.721i 0.930387 + 1.71513i
\(346\) 0 0
\(347\) 97.4526 + 168.793i 0.280843 + 0.486435i 0.971593 0.236659i \(-0.0760526\pi\)
−0.690749 + 0.723094i \(0.742719\pi\)
\(348\) 0 0
\(349\) 353.944i 1.01417i 0.861897 + 0.507084i \(0.169276\pi\)
−0.861897 + 0.507084i \(0.830724\pi\)
\(350\) 0 0
\(351\) −33.8702 + 424.101i −0.0964962 + 1.20826i
\(352\) 0 0
\(353\) −322.725 558.976i −0.914234 1.58350i −0.808018 0.589157i \(-0.799460\pi\)
−0.106216 0.994343i \(-0.533873\pi\)
\(354\) 0 0
\(355\) −357.622 + 619.420i −1.00739 + 1.74484i
\(356\) 0 0
\(357\) 245.856 216.229i 0.688671 0.605684i
\(358\) 0 0
\(359\) 265.046 459.074i 0.738291 1.27876i −0.214973 0.976620i \(-0.568966\pi\)
0.953264 0.302138i \(-0.0977002\pi\)
\(360\) 0 0
\(361\) −60.3585 104.544i −0.167198 0.289596i
\(362\) 0 0
\(363\) 145.227 236.784i 0.400073 0.652296i
\(364\) 0 0
\(365\) 389.109i 1.06605i
\(366\) 0 0
\(367\) 83.5711 + 144.749i 0.227714 + 0.394412i 0.957130 0.289658i \(-0.0935415\pi\)
−0.729416 + 0.684070i \(0.760208\pi\)
\(368\) 0 0
\(369\) 132.967 + 7.07115i 0.360345 + 0.0191630i
\(370\) 0 0
\(371\) −260.929 281.250i −0.703313 0.758086i
\(372\) 0 0
\(373\) −56.0705 + 97.1169i −0.150323 + 0.260367i −0.931346 0.364135i \(-0.881365\pi\)
0.781023 + 0.624502i \(0.214698\pi\)
\(374\) 0 0
\(375\) −1346.09 35.7669i −3.58956 0.0953783i
\(376\) 0 0
\(377\) 387.252 1.02719
\(378\) 0 0
\(379\) 427.746i 1.12862i 0.825564 + 0.564309i \(0.190857\pi\)
−0.825564 + 0.564309i \(0.809143\pi\)
\(380\) 0 0
\(381\) −6.00398 + 225.960i −0.0157585 + 0.593070i
\(382\) 0 0
\(383\) −537.538 310.348i −1.40349 0.810308i −0.408745 0.912649i \(-0.634033\pi\)
−0.994749 + 0.102341i \(0.967367\pi\)
\(384\) 0 0
\(385\) 81.2588 + 356.121i 0.211062 + 0.924990i
\(386\) 0 0
\(387\) −24.4658 + 460.060i −0.0632191 + 1.18879i
\(388\) 0 0
\(389\) 259.669 149.920i 0.667530 0.385398i −0.127610 0.991824i \(-0.540731\pi\)
0.795140 + 0.606426i \(0.207397\pi\)
\(390\) 0 0
\(391\) 357.351 0.913940
\(392\) 0 0
\(393\) 109.069 + 66.8951i 0.277528 + 0.170216i
\(394\) 0 0
\(395\) −42.0560 + 24.2811i −0.106471 + 0.0614710i
\(396\) 0 0
\(397\) 398.346 + 229.985i 1.00339 + 0.579307i 0.909249 0.416252i \(-0.136657\pi\)
0.0941399 + 0.995559i \(0.469990\pi\)
\(398\) 0 0
\(399\) 436.659 + 147.533i 1.09438 + 0.369758i
\(400\) 0 0
\(401\) −592.401 342.023i −1.47731 0.852925i −0.477637 0.878557i \(-0.658507\pi\)
−0.999671 + 0.0256326i \(0.991840\pi\)
\(402\) 0 0
\(403\) 296.976 171.459i 0.736913 0.425457i
\(404\) 0 0
\(405\) 467.103 640.835i 1.15334 1.58231i
\(406\) 0 0
\(407\) −164.962 −0.405311
\(408\) 0 0
\(409\) −325.542 + 187.952i −0.795945 + 0.459539i −0.842051 0.539397i \(-0.818652\pi\)
0.0461061 + 0.998937i \(0.485319\pi\)
\(410\) 0 0
\(411\) 270.515 146.743i 0.658188 0.357039i
\(412\) 0 0
\(413\) 221.477 + 238.725i 0.536263 + 0.578027i
\(414\) 0 0
\(415\) −475.583 274.578i −1.14598 0.661634i
\(416\) 0 0
\(417\) −272.423 7.23857i −0.653294 0.0173587i
\(418\) 0 0
\(419\) 285.762i 0.682010i −0.940061 0.341005i \(-0.889233\pi\)
0.940061 0.341005i \(-0.110767\pi\)
\(420\) 0 0
\(421\) 350.727 0.833081 0.416540 0.909117i \(-0.363242\pi\)
0.416540 + 0.909117i \(0.363242\pi\)
\(422\) 0 0
\(423\) 90.3041 + 58.7430i 0.213485 + 0.138872i
\(424\) 0 0
\(425\) −552.296 + 956.605i −1.29952 + 2.25084i
\(426\) 0 0
\(427\) 64.8298 210.126i 0.151826 0.492099i
\(428\) 0 0
\(429\) −221.476 + 120.141i −0.516261 + 0.280050i
\(430\) 0 0
\(431\) −233.727 404.826i −0.542289 0.939272i −0.998772 0.0495402i \(-0.984224\pi\)
0.456483 0.889732i \(-0.349109\pi\)
\(432\) 0 0
\(433\) 217.745i 0.502874i −0.967874 0.251437i \(-0.919097\pi\)
0.967874 0.251437i \(-0.0809032\pi\)
\(434\) 0 0
\(435\) −615.294 377.378i −1.41447 0.867536i
\(436\) 0 0
\(437\) 251.526 + 435.655i 0.575574 + 0.996923i
\(438\) 0 0
\(439\) 230.686 399.560i 0.525481 0.910160i −0.474078 0.880483i \(-0.657219\pi\)
0.999560 0.0296775i \(-0.00944802\pi\)
\(440\) 0 0
\(441\) −406.949 + 169.922i −0.922787 + 0.385310i
\(442\) 0 0
\(443\) −78.2121 + 135.467i −0.176551 + 0.305795i −0.940697 0.339248i \(-0.889827\pi\)
0.764146 + 0.645043i \(0.223161\pi\)
\(444\) 0 0
\(445\) 40.3398 + 69.8706i 0.0906513 + 0.157013i
\(446\) 0 0
\(447\) 435.052 + 266.831i 0.973271 + 0.596937i
\(448\) 0 0
\(449\) 49.2939i 0.109786i 0.998492 + 0.0548930i \(0.0174818\pi\)
−0.998492 + 0.0548930i \(0.982518\pi\)
\(450\) 0 0
\(451\) 39.4291 + 68.2932i 0.0874259 + 0.151426i
\(452\) 0 0
\(453\) 198.113 107.468i 0.437336 0.237236i
\(454\) 0 0
\(455\) −318.364 + 1031.88i −0.699701 + 2.26787i
\(456\) 0 0
\(457\) 130.664 226.317i 0.285917 0.495222i −0.686914 0.726738i \(-0.741035\pi\)
0.972831 + 0.231516i \(0.0743686\pi\)
\(458\) 0 0
\(459\) −180.790 380.162i −0.393877 0.828240i
\(460\) 0 0
\(461\) −261.457 −0.567151 −0.283576 0.958950i \(-0.591521\pi\)
−0.283576 + 0.958950i \(0.591521\pi\)
\(462\) 0 0
\(463\) 410.629i 0.886887i 0.896302 + 0.443444i \(0.146243\pi\)
−0.896302 + 0.443444i \(0.853757\pi\)
\(464\) 0 0
\(465\) −638.944 16.9774i −1.37407 0.0365105i
\(466\) 0 0
\(467\) −377.906 218.184i −0.809220 0.467203i 0.0374651 0.999298i \(-0.488072\pi\)
−0.846685 + 0.532095i \(0.821405\pi\)
\(468\) 0 0
\(469\) −79.2173 85.3866i −0.168907 0.182061i
\(470\) 0 0
\(471\) −549.842 + 298.266i −1.16739 + 0.633261i
\(472\) 0 0
\(473\) −236.291 + 136.422i −0.499557 + 0.288420i
\(474\) 0 0
\(475\) −1554.96 −3.27360
\(476\) 0 0
\(477\) −439.671 + 223.598i −0.921743 + 0.468758i
\(478\) 0 0
\(479\) 712.054 411.104i 1.48654 0.858256i 0.486660 0.873592i \(-0.338215\pi\)
0.999882 + 0.0153361i \(0.00488182\pi\)
\(480\) 0 0
\(481\) −422.346 243.842i −0.878058 0.506947i
\(482\) 0 0
\(483\) −455.998 154.067i −0.944095 0.318980i
\(484\) 0 0
\(485\) 538.432 + 310.864i 1.11017 + 0.640957i
\(486\) 0 0
\(487\) 617.211 356.347i 1.26737 0.731719i 0.292884 0.956148i \(-0.405385\pi\)
0.974490 + 0.224429i \(0.0720518\pi\)
\(488\) 0 0
\(489\) −758.570 465.254i −1.55127 0.951440i
\(490\) 0 0
\(491\) 806.962 1.64351 0.821754 0.569842i \(-0.192996\pi\)
0.821754 + 0.569842i \(0.192996\pi\)
\(492\) 0 0
\(493\) −331.831 + 191.583i −0.673085 + 0.388606i
\(494\) 0 0
\(495\) 468.976 + 24.9399i 0.947425 + 0.0503837i
\(496\) 0 0
\(497\) −113.766 498.587i −0.228906 1.00319i
\(498\) 0 0
\(499\) 45.1352 + 26.0588i 0.0904513 + 0.0522221i 0.544543 0.838733i \(-0.316703\pi\)
−0.454092 + 0.890955i \(0.650036\pi\)
\(500\) 0 0
\(501\) −14.7212 + 554.033i −0.0293837 + 1.10586i
\(502\) 0 0
\(503\) 716.425i 1.42430i −0.702025 0.712152i \(-0.747721\pi\)
0.702025 0.712152i \(-0.252279\pi\)
\(504\) 0 0
\(505\) 314.737 0.623242
\(506\) 0 0
\(507\) −237.807 6.31878i −0.469048 0.0124631i
\(508\) 0 0
\(509\) −122.636 + 212.412i −0.240935 + 0.417312i −0.960981 0.276615i \(-0.910787\pi\)
0.720046 + 0.693926i \(0.244121\pi\)
\(510\) 0 0
\(511\) 189.221 + 203.957i 0.370295 + 0.399134i
\(512\) 0 0
\(513\) 336.214 487.987i 0.655389 0.951242i
\(514\) 0 0
\(515\) 833.342 + 1443.39i 1.61814 + 2.80270i
\(516\) 0 0
\(517\) 63.8001i 0.123404i
\(518\) 0 0
\(519\) 399.385 651.175i 0.769529 1.25467i
\(520\) 0 0
\(521\) 278.097 + 481.679i 0.533776 + 0.924527i 0.999222 + 0.0394506i \(0.0125608\pi\)
−0.465446 + 0.885077i \(0.654106\pi\)
\(522\) 0 0
\(523\) −196.110 + 339.673i −0.374972 + 0.649470i −0.990323 0.138783i \(-0.955681\pi\)
0.615351 + 0.788253i \(0.289014\pi\)
\(524\) 0 0
\(525\) 1117.19 982.562i 2.12798 1.87155i
\(526\) 0 0
\(527\) −169.650 + 293.842i −0.321916 + 0.557575i
\(528\) 0 0
\(529\) 1.83458 + 3.17759i 0.00346802 + 0.00600678i
\(530\) 0 0
\(531\) 373.193 189.790i 0.702812 0.357419i
\(532\) 0 0
\(533\) 233.132i 0.437396i
\(534\) 0 0
\(535\) 989.015 + 1713.02i 1.84863 + 3.20192i
\(536\) 0 0
\(537\) −168.050 309.793i −0.312941 0.576896i
\(538\) 0 0
\(539\) −215.772 147.151i −0.400319 0.273007i
\(540\) 0 0
\(541\) 166.601 288.561i 0.307950 0.533384i −0.669964 0.742394i \(-0.733691\pi\)
0.977914 + 0.209009i \(0.0670239\pi\)
\(542\) 0 0
\(543\) −1.63005 + 61.3468i −0.00300193 + 0.112978i
\(544\) 0 0
\(545\) 345.636 0.634194
\(546\) 0 0
\(547\) 173.083i 0.316422i −0.987405 0.158211i \(-0.949427\pi\)
0.987405 0.158211i \(-0.0505726\pi\)
\(548\) 0 0
\(549\) −236.997 154.167i −0.431689 0.280815i
\(550\) 0 0
\(551\) −467.126 269.696i −0.847779 0.489466i
\(552\) 0 0
\(553\) 10.2366 33.1788i 0.0185110 0.0599979i
\(554\) 0 0
\(555\) 433.429 + 799.011i 0.780954 + 1.43966i
\(556\) 0 0
\(557\) 918.753 530.442i 1.64947 0.952320i 0.672182 0.740386i \(-0.265357\pi\)
0.977284 0.211934i \(-0.0679763\pi\)
\(558\) 0 0
\(559\) −806.624 −1.44298
\(560\) 0 0
\(561\) 130.343 212.517i 0.232341 0.378818i
\(562\) 0 0
\(563\) 550.654 317.920i 0.978071 0.564689i 0.0763835 0.997079i \(-0.475663\pi\)
0.901687 + 0.432389i \(0.142329\pi\)
\(564\) 0 0
\(565\) −781.815 451.381i −1.38374 0.798904i
\(566\) 0 0
\(567\) 66.7946 + 563.052i 0.117803 + 0.993037i
\(568\) 0 0
\(569\) −704.449 406.714i −1.23805 0.714787i −0.269352 0.963042i \(-0.586810\pi\)
−0.968695 + 0.248255i \(0.920143\pi\)
\(570\) 0 0
\(571\) 637.778 368.221i 1.11695 0.644871i 0.176329 0.984331i \(-0.443578\pi\)
0.940620 + 0.339461i \(0.110245\pi\)
\(572\) 0 0
\(573\) 288.902 471.037i 0.504191 0.822055i
\(574\) 0 0
\(575\) 1623.83 2.82405
\(576\) 0 0
\(577\) 106.581 61.5346i 0.184716 0.106646i −0.404791 0.914409i \(-0.632656\pi\)
0.589507 + 0.807764i \(0.299322\pi\)
\(578\) 0 0
\(579\) 5.80198 + 10.6957i 0.0100207 + 0.0184728i
\(580\) 0 0
\(581\) 382.809 87.3485i 0.658880 0.150342i
\(582\) 0 0
\(583\) −252.986 146.061i −0.433938 0.250534i
\(584\) 0 0
\(585\) 1163.84 + 757.079i 1.98947 + 1.29415i
\(586\) 0 0
\(587\) 939.308i 1.60018i 0.599878 + 0.800092i \(0.295216\pi\)
−0.599878 + 0.800092i \(0.704784\pi\)
\(588\) 0 0
\(589\) −477.640 −0.810934
\(590\) 0 0
\(591\) 14.0853 530.101i 0.0238330 0.896956i
\(592\) 0 0
\(593\) 64.5058 111.727i 0.108779 0.188410i −0.806497 0.591238i \(-0.798639\pi\)
0.915276 + 0.402828i \(0.131973\pi\)
\(594\) 0 0
\(595\) −237.694 1041.71i −0.399485 1.75077i
\(596\) 0 0
\(597\) 61.5607 + 113.485i 0.103117 + 0.190092i
\(598\) 0 0
\(599\) −444.320 769.585i −0.741770 1.28478i −0.951689 0.307064i \(-0.900653\pi\)
0.209919 0.977719i \(-0.432680\pi\)
\(600\) 0 0
\(601\) 634.763i 1.05618i −0.849189 0.528089i \(-0.822909\pi\)
0.849189 0.528089i \(-0.177091\pi\)
\(602\) 0 0
\(603\) −133.483 + 67.8836i −0.221365 + 0.112576i
\(604\) 0 0
\(605\) −453.239 785.033i −0.749155 1.29757i
\(606\) 0 0
\(607\) 472.304 818.054i 0.778095 1.34770i −0.154943 0.987923i \(-0.549520\pi\)
0.933038 0.359777i \(-0.117147\pi\)
\(608\) 0 0
\(609\) 506.032 101.404i 0.830922 0.166510i
\(610\) 0 0
\(611\) −94.3075 + 163.345i −0.154349 + 0.267341i
\(612\) 0 0
\(613\) 50.2939 + 87.1115i 0.0820455 + 0.142107i 0.904128 0.427261i \(-0.140521\pi\)
−0.822083 + 0.569368i \(0.807188\pi\)
\(614\) 0 0
\(615\) 227.188 370.417i 0.369411 0.602304i
\(616\) 0 0
\(617\) 544.570i 0.882610i 0.897357 + 0.441305i \(0.145484\pi\)
−0.897357 + 0.441305i \(0.854516\pi\)
\(618\) 0 0
\(619\) 33.3779 + 57.8123i 0.0539224 + 0.0933963i 0.891727 0.452574i \(-0.149494\pi\)
−0.837804 + 0.545971i \(0.816161\pi\)
\(620\) 0 0
\(621\) −351.105 + 509.599i −0.565386 + 0.820611i
\(622\) 0 0
\(623\) −55.1223 17.0068i −0.0884788 0.0272982i
\(624\) 0 0
\(625\) −1311.58 + 2271.73i −2.09853 + 3.63477i
\(626\) 0 0
\(627\) 350.828 + 9.32187i 0.559535 + 0.0148674i
\(628\) 0 0
\(629\) 482.537 0.767149
\(630\) 0 0
\(631\) 738.431i 1.17026i −0.810941 0.585128i \(-0.801044\pi\)
0.810941 0.585128i \(-0.198956\pi\)
\(632\) 0 0
\(633\) −12.1012 + 455.429i −0.0191173 + 0.719477i
\(634\) 0 0
\(635\) 638.827 + 368.827i 1.00603 + 0.580830i
\(636\) 0 0
\(637\) −334.921 695.693i −0.525778 1.09214i
\(638\) 0 0
\(639\) −656.589 34.9171i −1.02753 0.0546434i
\(640\) 0 0
\(641\) 148.471 85.7195i 0.231623 0.133728i −0.379697 0.925111i \(-0.623972\pi\)
0.611321 + 0.791383i \(0.290639\pi\)
\(642\) 0 0
\(643\) 749.426 1.16551 0.582757 0.812646i \(-0.301974\pi\)
0.582757 + 0.812646i \(0.301974\pi\)
\(644\) 0 0
\(645\) 1281.62 + 786.057i 1.98701 + 1.21869i
\(646\) 0 0
\(647\) −589.924 + 340.593i −0.911784 + 0.526419i −0.881005 0.473108i \(-0.843132\pi\)
−0.0307791 + 0.999526i \(0.509799\pi\)
\(648\) 0 0
\(649\) 214.734 + 123.977i 0.330870 + 0.191028i
\(650\) 0 0
\(651\) 343.168 301.815i 0.527140 0.463617i
\(652\) 0 0
\(653\) −727.697 420.136i −1.11439 0.643394i −0.174428 0.984670i \(-0.555808\pi\)
−0.939963 + 0.341276i \(0.889141\pi\)
\(654\) 0 0
\(655\) 361.606 208.773i 0.552070 0.318738i
\(656\) 0 0
\(657\) 318.842 162.149i 0.485299 0.246802i
\(658\) 0 0
\(659\) −1035.77 −1.57172 −0.785862 0.618401i \(-0.787781\pi\)
−0.785862 + 0.618401i \(0.787781\pi\)
\(660\) 0 0
\(661\) −624.364 + 360.477i −0.944575 + 0.545351i −0.891392 0.453234i \(-0.850270\pi\)
−0.0531835 + 0.998585i \(0.516937\pi\)
\(662\) 0 0
\(663\) 647.850 351.431i 0.977149 0.530062i
\(664\) 0 0
\(665\) 1102.67 1023.00i 1.65814 1.53834i
\(666\) 0 0
\(667\) 487.815 + 281.640i 0.731356 + 0.422249i
\(668\) 0 0
\(669\) −479.807 12.7490i −0.717200 0.0190567i
\(670\) 0 0
\(671\) 167.439i 0.249537i
\(672\) 0 0
\(673\) −705.661 −1.04853 −0.524266 0.851555i \(-0.675660\pi\)
−0.524266 + 0.851555i \(0.675660\pi\)
\(674\) 0 0
\(675\) −821.522 1727.49i −1.21707 2.55924i
\(676\) 0 0
\(677\) −5.03682 + 8.72402i −0.00743990 + 0.0128863i −0.869721 0.493543i \(-0.835702\pi\)
0.862281 + 0.506429i \(0.169035\pi\)
\(678\) 0 0
\(679\) −433.398 + 98.8918i −0.638289 + 0.145643i
\(680\) 0 0
\(681\) 547.372 296.926i 0.803776 0.436015i
\(682\) 0 0
\(683\) −73.9512 128.087i −0.108274 0.187536i 0.806797 0.590829i \(-0.201199\pi\)
−0.915071 + 0.403292i \(0.867866\pi\)
\(684\) 0 0
\(685\) 1004.32i 1.46616i
\(686\) 0 0
\(687\) 941.162 + 577.243i 1.36996 + 0.840237i
\(688\) 0 0
\(689\) −431.808 747.913i −0.626716 1.08550i
\(690\) 0 0
\(691\) −605.873 + 1049.40i −0.876805 + 1.51867i −0.0219783 + 0.999758i \(0.506996\pi\)
−0.854827 + 0.518913i \(0.826337\pi\)
\(692\) 0 0
\(693\) −257.949 + 214.987i −0.372220 + 0.310226i
\(694\) 0 0
\(695\) −444.669 + 770.189i −0.639811 + 1.10819i
\(696\) 0 0
\(697\) −115.336 199.767i −0.165475 0.286610i
\(698\) 0 0
\(699\) 41.2653 + 25.3093i 0.0590348 + 0.0362078i
\(700\) 0 0
\(701\) 34.2631i 0.0488775i −0.999701 0.0244387i \(-0.992220\pi\)
0.999701 0.0244387i \(-0.00777986\pi\)
\(702\) 0 0
\(703\) 339.639 + 588.273i 0.483129 + 0.836803i
\(704\) 0 0
\(705\) 309.023 167.632i 0.438331 0.237776i
\(706\) 0 0
\(707\) −164.974 + 153.054i −0.233344 + 0.216484i
\(708\) 0 0
\(709\) 207.128 358.756i 0.292141 0.506003i −0.682175 0.731189i \(-0.738966\pi\)
0.974316 + 0.225186i \(0.0722990\pi\)
\(710\) 0 0
\(711\) −37.4218 24.3429i −0.0526326 0.0342376i
\(712\) 0 0
\(713\) 498.794 0.699571
\(714\) 0 0
\(715\) 822.255i 1.15001i
\(716\) 0 0
\(717\) 258.787 + 6.87625i 0.360931 + 0.00959031i
\(718\) 0 0
\(719\) −68.9380 39.8014i −0.0958804 0.0553566i 0.451293 0.892376i \(-0.350963\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(720\) 0 0
\(721\) −1138.72 351.326i −1.57936 0.487277i
\(722\) 0 0
\(723\) 791.556 429.386i 1.09482 0.593895i
\(724\) 0 0
\(725\) −1507.86 + 870.566i −2.07981 + 1.20078i
\(726\) 0 0
\(727\) −9.43364 −0.0129761 −0.00648806 0.999979i \(-0.502065\pi\)
−0.00648806 + 0.999979i \(0.502065\pi\)
\(728\) 0 0
\(729\) 719.760 + 115.703i 0.987324 + 0.158715i
\(730\) 0 0
\(731\) 691.185 399.056i 0.945533 0.545904i
\(732\) 0 0
\(733\) 193.869 + 111.930i 0.264487 + 0.152701i 0.626379 0.779518i \(-0.284536\pi\)
−0.361893 + 0.932220i \(0.617869\pi\)
\(734\) 0 0
\(735\) −145.809 + 1431.75i −0.198380 + 1.94796i
\(736\) 0 0
\(737\) −76.8057 44.3438i −0.104214 0.0601680i
\(738\) 0 0
\(739\) −541.551 + 312.665i −0.732816 + 0.423092i −0.819452 0.573148i \(-0.805722\pi\)
0.0866354 + 0.996240i \(0.472388\pi\)
\(740\) 0 0
\(741\) 884.436 + 542.451i 1.19357 + 0.732053i
\(742\) 0 0
\(743\) 209.834 0.282415 0.141208 0.989980i \(-0.454902\pi\)
0.141208 + 0.989980i \(0.454902\pi\)
\(744\) 0 0
\(745\) 1442.37 832.755i 1.93607 1.11779i
\(746\) 0 0
\(747\) 26.8090 504.121i 0.0358888 0.674861i
\(748\) 0 0
\(749\) −1351.44 416.956i −1.80432 0.556684i
\(750\) 0 0
\(751\) −836.749 483.097i −1.11418 0.643272i −0.174271 0.984698i \(-0.555757\pi\)
−0.939909 + 0.341426i \(0.889090\pi\)
\(752\) 0 0
\(753\) −23.7968 + 895.592i −0.0316027 + 1.18937i
\(754\) 0 0
\(755\) 735.518i 0.974196i
\(756\) 0 0
\(757\) 318.215 0.420363 0.210181 0.977662i \(-0.432595\pi\)
0.210181 + 0.977662i \(0.432595\pi\)
\(758\) 0 0
\(759\) −366.366 9.73472i −0.482696 0.0128257i
\(760\) 0 0
\(761\) 333.550 577.725i 0.438304 0.759165i −0.559255 0.828996i \(-0.688913\pi\)
0.997559 + 0.0698308i \(0.0222459\pi\)
\(762\) 0 0
\(763\) −181.170 + 168.080i −0.237444 + 0.220288i
\(764\) 0 0
\(765\) −1371.82 72.9529i −1.79323 0.0953633i
\(766\) 0 0
\(767\) 366.518 + 634.829i 0.477860 + 0.827678i
\(768\) 0 0
\(769\) 774.930i 1.00771i 0.863788 + 0.503856i \(0.168086\pi\)
−0.863788 + 0.503856i \(0.831914\pi\)
\(770\) 0 0
\(771\) 227.593 371.078i 0.295192 0.481295i
\(772\) 0 0
\(773\) 61.0922 + 105.815i 0.0790326 + 0.136889i 0.902833 0.429992i \(-0.141484\pi\)
−0.823800 + 0.566880i \(0.808150\pi\)
\(774\) 0 0
\(775\) −770.901 + 1335.24i −0.994711 + 1.72289i
\(776\) 0 0
\(777\) −615.742 208.040i −0.792460 0.267747i
\(778\) 0 0
\(779\) 162.361 281.218i 0.208422 0.360998i
\(780\) 0 0
\(781\) −194.700 337.230i −0.249295 0.431792i
\(782\) 0 0
\(783\) 52.8250 661.441i 0.0674648 0.844752i
\(784\) 0 0
\(785\) 2041.35i 2.60045i
\(786\) 0 0
\(787\) 351.265 + 608.408i 0.446334 + 0.773073i 0.998144 0.0608970i \(-0.0193961\pi\)
−0.551810 + 0.833970i \(0.686063\pi\)
\(788\) 0 0
\(789\) 319.703 + 589.360i 0.405200 + 0.746971i
\(790\) 0 0
\(791\) 629.303 143.593i 0.795579 0.181533i
\(792\) 0 0
\(793\) 247.504 428.690i 0.312111 0.540593i
\(794\) 0 0
\(795\) −42.7564 + 1609.14i −0.0537816 + 2.02407i
\(796\) 0 0
\(797\) 615.706 0.772530 0.386265 0.922388i \(-0.373765\pi\)
0.386265 + 0.922388i \(0.373765\pi\)
\(798\) 0 0
\(799\) 186.625i 0.233573i
\(800\) 0 0
\(801\) −40.4426 + 62.1714i −0.0504902 + 0.0776172i
\(802\) 0 0
\(803\) 183.461 + 105.921i 0.228469 + 0.131907i
\(804\) 0 0
\(805\) −1151.50 + 1068.30i −1.43044 + 1.32708i
\(806\) 0 0
\(807\) −18.2947 33.7255i −0.0226700 0.0417913i
\(808\) 0 0
\(809\) −1077.98 + 622.375i −1.33249 + 0.769313i −0.985681 0.168622i \(-0.946068\pi\)
−0.346809 + 0.937936i \(0.612735\pi\)
\(810\) 0 0
\(811\) −1339.58 −1.65176 −0.825881 0.563845i \(-0.809322\pi\)
−0.825881 + 0.563845i \(0.809322\pi\)
\(812\) 0 0
\(813\) 172.289 280.907i 0.211917 0.345519i
\(814\) 0 0
\(815\) −2514.97 + 1452.02i −3.08585 + 1.78161i
\(816\) 0 0
\(817\) 972.998 + 561.760i 1.19094 + 0.687589i
\(818\) 0 0
\(819\) −978.205 + 169.132i −1.19439 + 0.206510i
\(820\) 0 0
\(821\) 263.793 + 152.301i 0.321307 + 0.185507i 0.651975 0.758241i \(-0.273941\pi\)
−0.330668 + 0.943747i \(0.607274\pi\)
\(822\) 0 0
\(823\) −297.149 + 171.559i −0.361056 + 0.208456i −0.669544 0.742773i \(-0.733510\pi\)
0.308488 + 0.951228i \(0.400177\pi\)
\(824\) 0 0
\(825\) 592.289 965.693i 0.717926 1.17054i
\(826\) 0 0
\(827\) −652.212 −0.788648 −0.394324 0.918972i \(-0.629021\pi\)
−0.394324 + 0.918972i \(0.629021\pi\)
\(828\) 0 0
\(829\) −314.990 + 181.859i −0.379963 + 0.219372i −0.677802 0.735244i \(-0.737067\pi\)
0.297839 + 0.954616i \(0.403734\pi\)
\(830\) 0 0
\(831\) −703.388 1296.67i −0.846435 1.56037i
\(832\) 0 0
\(833\) 631.165 + 430.437i 0.757701 + 0.516731i
\(834\) 0 0
\(835\) 1566.35 + 904.332i 1.87587 + 1.08303i
\(836\) 0 0
\(837\) −252.348 530.635i −0.301491 0.633972i
\(838\) 0 0
\(839\) 1070.44i 1.27585i −0.770099 0.637925i \(-0.779793\pi\)
0.770099 0.637925i \(-0.220207\pi\)
\(840\) 0 0
\(841\) 237.030 0.281843
\(842\) 0 0
\(843\) 6.39730 240.762i 0.00758873 0.285602i
\(844\) 0 0
\(845\) −388.166 + 672.323i −0.459368 + 0.795648i
\(846\) 0 0
\(847\) 619.328 + 191.080i 0.731201 + 0.225596i
\(848\) 0 0
\(849\) −406.129 748.683i −0.478361 0.881841i
\(850\) 0 0
\(851\) −354.681 614.326i −0.416782 0.721887i
\(852\) 0 0
\(853\) 1557.63i 1.82606i 0.407891 + 0.913031i \(0.366264\pi\)
−0.407891 + 0.913031i \(0.633736\pi\)
\(854\) 0 0
\(855\) −876.635 1723.77i −1.02530 2.01611i
\(856\) 0 0
\(857\) 117.943 + 204.284i 0.137624 + 0.238371i 0.926597 0.376057i \(-0.122720\pi\)
−0.788973 + 0.614428i \(0.789387\pi\)
\(858\) 0 0
\(859\) 557.231 965.153i 0.648698 1.12358i −0.334737 0.942312i \(-0.608647\pi\)
0.983434 0.181265i \(-0.0580193\pi\)
\(860\) 0 0
\(861\) 61.0471 + 304.639i 0.0709026 + 0.353820i
\(862\) 0 0
\(863\) 375.824 650.946i 0.435486 0.754283i −0.561849 0.827239i \(-0.689910\pi\)
0.997335 + 0.0729562i \(0.0232433\pi\)
\(864\) 0 0
\(865\) −1246.45 2158.91i −1.44098 2.49585i
\(866\) 0 0
\(867\) 72.0183 117.422i 0.0830661 0.135434i
\(868\) 0 0
\(869\) 26.4386i 0.0304242i
\(870\) 0 0
\(871\) −131.095 227.064i −0.150511 0.260694i
\(872\) 0 0
\(873\) −30.3518 + 570.742i −0.0347673 + 0.653771i
\(874\) 0 0
\(875\) −698.963 3063.24i −0.798815 3.50085i
\(876\) 0 0
\(877\) −613.699 + 1062.96i −0.699770 + 1.21204i 0.268776 + 0.963203i \(0.413381\pi\)
−0.968546 + 0.248835i \(0.919952\pi\)
\(878\) 0 0
\(879\) −982.182 26.0976i −1.11739 0.0296901i
\(880\) 0 0
\(881\) 1209.71 1.37311 0.686557 0.727076i \(-0.259121\pi\)
0.686557 + 0.727076i \(0.259121\pi\)
\(882\) 0 0
\(883\) 990.879i 1.12217i −0.827757 0.561087i \(-0.810383\pi\)
0.827757 0.561087i \(-0.189617\pi\)
\(884\) 0 0
\(885\) 36.2916 1365.83i 0.0410075 1.54332i
\(886\) 0 0
\(887\) 1395.29 + 805.571i 1.57304 + 0.908198i 0.995793 + 0.0916309i \(0.0292080\pi\)
0.577251 + 0.816567i \(0.304125\pi\)
\(888\) 0 0
\(889\) −514.209 + 117.331i −0.578413 + 0.131981i
\(890\) 0 0
\(891\) 174.995 + 394.678i 0.196402 + 0.442961i
\(892\) 0 0
\(893\) 227.519 131.358i 0.254780 0.147097i
\(894\) 0 0
\(895\) −1150.14 −1.28507
\(896\) 0 0
\(897\) −923.606 566.476i −1.02966 0.631522i
\(898\) 0 0
\(899\) −463.173 + 267.413i −0.515209 + 0.297456i
\(900\) 0 0
\(901\) 740.020 + 427.251i 0.821332 + 0.474196i
\(902\) 0 0
\(903\) −1054.04 + 211.220i −1.16726 + 0.233909i
\(904\) 0 0
\(905\) 173.438 + 100.135i 0.191644 + 0.110646i
\(906\) 0 0
\(907\) 967.470 558.569i 1.06667 0.615842i 0.139400 0.990236i \(-0.455483\pi\)
0.927270 + 0.374394i \(0.122149\pi\)
\(908\) 0 0
\(909\) 131.157 + 257.900i 0.144287 + 0.283718i
\(910\) 0 0
\(911\) −223.900 −0.245774 −0.122887 0.992421i \(-0.539215\pi\)
−0.122887 + 0.992421i \(0.539215\pi\)
\(912\) 0 0
\(913\) 258.921 149.488i 0.283594 0.163733i
\(914\) 0 0
\(915\) −811.012 + 439.940i −0.886352 + 0.480808i
\(916\) 0 0
\(917\) −88.0162 + 285.278i −0.0959828 + 0.311099i
\(918\) 0 0
\(919\) 412.009 + 237.874i 0.448324 + 0.258840i 0.707122 0.707092i \(-0.249993\pi\)
−0.258798 + 0.965931i \(0.583327\pi\)
\(920\) 0 0
\(921\) 613.787 + 16.3090i 0.666436 + 0.0177079i
\(922\) 0 0
\(923\) 1151.20i 1.24724i
\(924\) 0 0
\(925\) 2192.68 2.37047
\(926\) 0 0
\(927\) −835.466 + 1284.34i −0.901258 + 1.38548i
\(928\) 0 0
\(929\) 415.674 719.968i 0.447442 0.774992i −0.550777 0.834653i \(-0.685668\pi\)
0.998219 + 0.0596603i \(0.0190018\pi\)
\(930\) 0 0
\(931\) −80.5033 + 1072.44i −0.0864697 + 1.15192i
\(932\) 0 0
\(933\) −1191.14 + 646.142i −1.27668 + 0.692542i
\(934\) 0 0
\(935\) −406.789 704.579i −0.435068 0.753561i
\(936\) 0 0
\(937\) 358.471i 0.382573i −0.981534 0.191286i \(-0.938734\pi\)
0.981534 0.191286i \(-0.0612660\pi\)
\(938\) 0 0
\(939\) 822.420 + 504.415i 0.875847 + 0.537183i
\(940\) 0 0
\(941\) 485.287 + 840.541i 0.515714 + 0.893243i 0.999834 + 0.0182409i \(0.00580657\pi\)
−0.484120 + 0.875002i \(0.660860\pi\)
\(942\) 0 0
\(943\) −169.552 + 293.672i −0.179800 + 0.311423i
\(944\) 0 0
\(945\) 1719.06 + 684.536i 1.81911 + 0.724377i
\(946\) 0 0
\(947\) −884.688 + 1532.32i −0.934200 + 1.61808i −0.158147 + 0.987416i \(0.550552\pi\)
−0.776054 + 0.630667i \(0.782781\pi\)
\(948\) 0 0
\(949\) 313.139 + 542.373i 0.329967 + 0.571520i
\(950\) 0 0
\(951\) 600.494 + 368.301i 0.631434 + 0.387278i
\(952\) 0 0
\(953\) 1028.42i 1.07914i 0.841941 + 0.539570i \(0.181413\pi\)
−0.841941 + 0.539570i \(0.818587\pi\)
\(954\) 0 0
\(955\) −901.635 1561.68i −0.944121 1.63527i
\(956\) 0 0
\(957\) 345.421 187.376i 0.360942 0.195796i
\(958\) 0 0
\(959\) 488.393 + 526.429i 0.509273 + 0.548935i
\(960\) 0 0
\(961\) 243.701 422.103i 0.253591 0.439233i
\(962\) 0 0
\(963\) −991.536 + 1524.26i −1.02963 + 1.58283i
\(964\) 0 0
\(965\) 39.7091 0.0411494
\(966\) 0 0
\(967\) 1136.76i 1.17555i 0.809024 + 0.587776i \(0.199996\pi\)
−0.809024 + 0.587776i \(0.800004\pi\)
\(968\) 0 0
\(969\) −1026.22 27.2678i −1.05905 0.0281402i
\(970\) 0 0
\(971\) 418.792 + 241.790i 0.431300 + 0.249011i 0.699900 0.714241i \(-0.253228\pi\)
−0.268600 + 0.963252i \(0.586561\pi\)
\(972\) 0 0
\(973\) −141.458 619.945i −0.145383 0.637148i
\(974\) 0 0
\(975\) 2943.88 1596.93i 3.01936 1.63788i
\(976\) 0 0
\(977\) −105.815 + 61.0925i −0.108306 + 0.0625307i −0.553175 0.833065i \(-0.686584\pi\)
0.444868 + 0.895596i \(0.353250\pi\)
\(978\) 0 0
\(979\) −43.9243 −0.0448665
\(980\) 0 0
\(981\) 144.033 + 283.219i 0.146822 + 0.288704i
\(982\) 0 0
\(983\) 1120.41 646.870i 1.13979 0.658057i 0.193410 0.981118i \(-0.438045\pi\)
0.946378 + 0.323061i \(0.104712\pi\)
\(984\) 0 0
\(985\) −1498.69 865.268i −1.52151 0.878445i
\(986\) 0 0
\(987\) −80.4609 + 238.143i −0.0815207 + 0.241279i
\(988\) 0 0
\(989\) −1016.09 586.640i −1.02739 0.593165i
\(990\) 0 0
\(991\) −411.728 + 237.711i −0.415468 + 0.239870i −0.693136 0.720807i \(-0.743772\pi\)
0.277669 + 0.960677i \(0.410438\pi\)
\(992\) 0 0
\(993\) −403.751 247.633i −0.406597 0.249378i
\(994\) 0 0
\(995\) 421.325 0.423443
\(996\) 0 0
\(997\) −291.989 + 168.580i −0.292868 + 0.169087i −0.639235 0.769012i \(-0.720749\pi\)
0.346367 + 0.938099i \(0.387415\pi\)
\(998\) 0 0
\(999\) −474.103 + 688.120i −0.474577 + 0.688809i
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 336.3.z.f.47.10 yes 20
3.2 odd 2 inner 336.3.z.f.47.4 20
4.3 odd 2 336.3.z.g.47.1 yes 20
7.3 odd 6 336.3.z.g.143.7 yes 20
12.11 even 2 336.3.z.g.47.7 yes 20
21.17 even 6 336.3.z.g.143.1 yes 20
28.3 even 6 inner 336.3.z.f.143.4 yes 20
84.59 odd 6 inner 336.3.z.f.143.10 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.z.f.47.4 20 3.2 odd 2 inner
336.3.z.f.47.10 yes 20 1.1 even 1 trivial
336.3.z.f.143.4 yes 20 28.3 even 6 inner
336.3.z.f.143.10 yes 20 84.59 odd 6 inner
336.3.z.g.47.1 yes 20 4.3 odd 2
336.3.z.g.47.7 yes 20 12.11 even 2
336.3.z.g.143.1 yes 20 21.17 even 6
336.3.z.g.143.7 yes 20 7.3 odd 6