Properties

Label 1620.2.e.a.971.25
Level $1620$
Weight $2$
Character 1620.971
Analytic conductor $12.936$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(971,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.e (of order \(2\), degree \(1\), 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: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 971.25
Character \(\chi\) \(=\) 1620.971
Dual form 1620.2.e.a.971.26

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.192668 - 1.40103i) q^{2} +(-1.92576 - 0.539867i) q^{4} +1.00000i q^{5} -5.01091i q^{7} +(-1.12740 + 2.59403i) q^{8} +O(q^{10})\) \(q+(0.192668 - 1.40103i) q^{2} +(-1.92576 - 0.539867i) q^{4} +1.00000i q^{5} -5.01091i q^{7} +(-1.12740 + 2.59403i) q^{8} +(1.40103 + 0.192668i) q^{10} -4.38659 q^{11} -4.13042 q^{13} +(-7.02043 - 0.965443i) q^{14} +(3.41709 + 2.07930i) q^{16} +1.34325i q^{17} -1.39602i q^{19} +(0.539867 - 1.92576i) q^{20} +(-0.845155 + 6.14573i) q^{22} +7.65734 q^{23} -1.00000 q^{25} +(-0.795799 + 5.78683i) q^{26} +(-2.70522 + 9.64981i) q^{28} +0.167976i q^{29} +3.53478i q^{31} +(3.57153 - 4.38682i) q^{32} +(1.88193 + 0.258802i) q^{34} +5.01091 q^{35} -4.98701 q^{37} +(-1.95586 - 0.268969i) q^{38} +(-2.59403 - 1.12740i) q^{40} +6.52844i q^{41} +11.6725i q^{43} +(8.44751 + 2.36817i) q^{44} +(1.47532 - 10.7281i) q^{46} -0.0180070 q^{47} -18.1093 q^{49} +(-0.192668 + 1.40103i) q^{50} +(7.95419 + 2.22987i) q^{52} +5.66076i q^{53} -4.38659i q^{55} +(12.9984 + 5.64930i) q^{56} +(0.235339 + 0.0323636i) q^{58} -8.57807 q^{59} -3.81677 q^{61} +(4.95232 + 0.681038i) q^{62} +(-5.45794 - 5.84901i) q^{64} -4.13042i q^{65} -3.96182i q^{67} +(0.725176 - 2.58678i) q^{68} +(0.965443 - 7.02043i) q^{70} -2.16175 q^{71} -15.2156 q^{73} +(-0.960838 + 6.98694i) q^{74} +(-0.753665 + 2.68840i) q^{76} +21.9808i q^{77} -7.99947i q^{79} +(-2.07930 + 3.41709i) q^{80} +(9.14653 + 1.25782i) q^{82} +11.7253 q^{83} -1.34325 q^{85} +(16.3536 + 2.24893i) q^{86} +(4.94544 - 11.3789i) q^{88} -13.4268i q^{89} +20.6972i q^{91} +(-14.7462 - 4.13394i) q^{92} +(-0.00346937 + 0.0252283i) q^{94} +1.39602 q^{95} +14.4411 q^{97} +(-3.48908 + 25.3716i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 24 q^{16} - 24 q^{22} - 48 q^{25} + 24 q^{28} - 24 q^{34} - 24 q^{40} + 48 q^{46} - 48 q^{49} + 24 q^{58} + 24 q^{64} + 24 q^{76} + 24 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.192668 1.40103i 0.136237 0.990676i
\(3\) 0 0
\(4\) −1.92576 0.539867i −0.962879 0.269933i
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 5.01091i 1.89395i −0.321311 0.946974i \(-0.604123\pi\)
0.321311 0.946974i \(-0.395877\pi\)
\(8\) −1.12740 + 2.59403i −0.398596 + 0.917127i
\(9\) 0 0
\(10\) 1.40103 + 0.192668i 0.443044 + 0.0609270i
\(11\) −4.38659 −1.32261 −0.661303 0.750119i \(-0.729996\pi\)
−0.661303 + 0.750119i \(0.729996\pi\)
\(12\) 0 0
\(13\) −4.13042 −1.14557 −0.572786 0.819705i \(-0.694137\pi\)
−0.572786 + 0.819705i \(0.694137\pi\)
\(14\) −7.02043 0.965443i −1.87629 0.258025i
\(15\) 0 0
\(16\) 3.41709 + 2.07930i 0.854272 + 0.519826i
\(17\) 1.34325i 0.325786i 0.986644 + 0.162893i \(0.0520826\pi\)
−0.986644 + 0.162893i \(0.947917\pi\)
\(18\) 0 0
\(19\) 1.39602i 0.320269i −0.987095 0.160135i \(-0.948807\pi\)
0.987095 0.160135i \(-0.0511928\pi\)
\(20\) 0.539867 1.92576i 0.120718 0.430613i
\(21\) 0 0
\(22\) −0.845155 + 6.14573i −0.180188 + 1.31027i
\(23\) 7.65734 1.59667 0.798333 0.602216i \(-0.205716\pi\)
0.798333 + 0.602216i \(0.205716\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −0.795799 + 5.78683i −0.156069 + 1.13489i
\(27\) 0 0
\(28\) −2.70522 + 9.64981i −0.511239 + 1.82364i
\(29\) 0.167976i 0.0311923i 0.999878 + 0.0155962i \(0.00496462\pi\)
−0.999878 + 0.0155962i \(0.995035\pi\)
\(30\) 0 0
\(31\) 3.53478i 0.634865i 0.948281 + 0.317432i \(0.102821\pi\)
−0.948281 + 0.317432i \(0.897179\pi\)
\(32\) 3.57153 4.38682i 0.631363 0.775488i
\(33\) 0 0
\(34\) 1.88193 + 0.258802i 0.322749 + 0.0443841i
\(35\) 5.01091 0.846999
\(36\) 0 0
\(37\) −4.98701 −0.819860 −0.409930 0.912117i \(-0.634447\pi\)
−0.409930 + 0.912117i \(0.634447\pi\)
\(38\) −1.95586 0.268969i −0.317283 0.0436325i
\(39\) 0 0
\(40\) −2.59403 1.12740i −0.410151 0.178258i
\(41\) 6.52844i 1.01957i 0.860302 + 0.509786i \(0.170275\pi\)
−0.860302 + 0.509786i \(0.829725\pi\)
\(42\) 0 0
\(43\) 11.6725i 1.78005i 0.455915 + 0.890024i \(0.349312\pi\)
−0.455915 + 0.890024i \(0.650688\pi\)
\(44\) 8.44751 + 2.36817i 1.27351 + 0.357015i
\(45\) 0 0
\(46\) 1.47532 10.7281i 0.217525 1.58178i
\(47\) −0.0180070 −0.00262659 −0.00131330 0.999999i \(-0.500418\pi\)
−0.00131330 + 0.999999i \(0.500418\pi\)
\(48\) 0 0
\(49\) −18.1093 −2.58704
\(50\) −0.192668 + 1.40103i −0.0272474 + 0.198135i
\(51\) 0 0
\(52\) 7.95419 + 2.22987i 1.10305 + 0.309228i
\(53\) 5.66076i 0.777566i 0.921329 + 0.388783i \(0.127104\pi\)
−0.921329 + 0.388783i \(0.872896\pi\)
\(54\) 0 0
\(55\) 4.38659i 0.591487i
\(56\) 12.9984 + 5.64930i 1.73699 + 0.754920i
\(57\) 0 0
\(58\) 0.235339 + 0.0323636i 0.0309015 + 0.00424955i
\(59\) −8.57807 −1.11677 −0.558385 0.829582i \(-0.688579\pi\)
−0.558385 + 0.829582i \(0.688579\pi\)
\(60\) 0 0
\(61\) −3.81677 −0.488687 −0.244343 0.969689i \(-0.578572\pi\)
−0.244343 + 0.969689i \(0.578572\pi\)
\(62\) 4.95232 + 0.681038i 0.628945 + 0.0864919i
\(63\) 0 0
\(64\) −5.45794 5.84901i −0.682242 0.731126i
\(65\) 4.13042i 0.512315i
\(66\) 0 0
\(67\) 3.96182i 0.484014i −0.970275 0.242007i \(-0.922194\pi\)
0.970275 0.242007i \(-0.0778057\pi\)
\(68\) 0.725176 2.58678i 0.0879406 0.313693i
\(69\) 0 0
\(70\) 0.965443 7.02043i 0.115392 0.839102i
\(71\) −2.16175 −0.256553 −0.128276 0.991738i \(-0.540944\pi\)
−0.128276 + 0.991738i \(0.540944\pi\)
\(72\) 0 0
\(73\) −15.2156 −1.78085 −0.890423 0.455133i \(-0.849592\pi\)
−0.890423 + 0.455133i \(0.849592\pi\)
\(74\) −0.960838 + 6.98694i −0.111695 + 0.812216i
\(75\) 0 0
\(76\) −0.753665 + 2.68840i −0.0864513 + 0.308381i
\(77\) 21.9808i 2.50495i
\(78\) 0 0
\(79\) 7.99947i 0.900011i −0.893026 0.450006i \(-0.851422\pi\)
0.893026 0.450006i \(-0.148578\pi\)
\(80\) −2.07930 + 3.41709i −0.232473 + 0.382042i
\(81\) 0 0
\(82\) 9.14653 + 1.25782i 1.01006 + 0.138903i
\(83\) 11.7253 1.28702 0.643512 0.765436i \(-0.277477\pi\)
0.643512 + 0.765436i \(0.277477\pi\)
\(84\) 0 0
\(85\) −1.34325 −0.145696
\(86\) 16.3536 + 2.24893i 1.76345 + 0.242508i
\(87\) 0 0
\(88\) 4.94544 11.3789i 0.527186 1.21300i
\(89\) 13.4268i 1.42324i −0.702563 0.711622i \(-0.747961\pi\)
0.702563 0.711622i \(-0.252039\pi\)
\(90\) 0 0
\(91\) 20.6972i 2.16965i
\(92\) −14.7462 4.13394i −1.53740 0.430993i
\(93\) 0 0
\(94\) −0.00346937 + 0.0252283i −0.000357839 + 0.00260210i
\(95\) 1.39602 0.143229
\(96\) 0 0
\(97\) 14.4411 1.46628 0.733138 0.680080i \(-0.238055\pi\)
0.733138 + 0.680080i \(0.238055\pi\)
\(98\) −3.48908 + 25.3716i −0.352450 + 2.56292i
\(99\) 0 0
\(100\) 1.92576 + 0.539867i 0.192576 + 0.0539867i
\(101\) 11.5704i 1.15130i 0.817696 + 0.575650i \(0.195251\pi\)
−0.817696 + 0.575650i \(0.804749\pi\)
\(102\) 0 0
\(103\) 6.59136i 0.649466i 0.945806 + 0.324733i \(0.105274\pi\)
−0.945806 + 0.324733i \(0.894726\pi\)
\(104\) 4.65663 10.7144i 0.456620 1.05063i
\(105\) 0 0
\(106\) 7.93089 + 1.09065i 0.770316 + 0.105933i
\(107\) −14.2524 −1.37783 −0.688916 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(108\) 0 0
\(109\) 2.72081 0.260607 0.130303 0.991474i \(-0.458405\pi\)
0.130303 + 0.991474i \(0.458405\pi\)
\(110\) −6.14573 0.845155i −0.585972 0.0805824i
\(111\) 0 0
\(112\) 10.4192 17.1227i 0.984523 1.61795i
\(113\) 11.1171i 1.04581i 0.852391 + 0.522904i \(0.175152\pi\)
−0.852391 + 0.522904i \(0.824848\pi\)
\(114\) 0 0
\(115\) 7.65734i 0.714051i
\(116\) 0.0906846 0.323481i 0.00841985 0.0300344i
\(117\) 0 0
\(118\) −1.65272 + 12.0181i −0.152145 + 1.10636i
\(119\) 6.73092 0.617022
\(120\) 0 0
\(121\) 8.24215 0.749286
\(122\) −0.735369 + 5.34740i −0.0665772 + 0.484131i
\(123\) 0 0
\(124\) 1.90831 6.80712i 0.171371 0.611298i
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) 6.21713i 0.551681i 0.961203 + 0.275840i \(0.0889562\pi\)
−0.961203 + 0.275840i \(0.911044\pi\)
\(128\) −9.24619 + 6.51981i −0.817256 + 0.576275i
\(129\) 0 0
\(130\) −5.78683 0.795799i −0.507539 0.0697962i
\(131\) −5.86793 −0.512683 −0.256342 0.966586i \(-0.582517\pi\)
−0.256342 + 0.966586i \(0.582517\pi\)
\(132\) 0 0
\(133\) −6.99534 −0.606573
\(134\) −5.55063 0.763317i −0.479501 0.0659405i
\(135\) 0 0
\(136\) −3.48443 1.51438i −0.298787 0.129857i
\(137\) 6.87606i 0.587461i 0.955888 + 0.293731i \(0.0948969\pi\)
−0.955888 + 0.293731i \(0.905103\pi\)
\(138\) 0 0
\(139\) 9.66986i 0.820187i 0.912043 + 0.410094i \(0.134504\pi\)
−0.912043 + 0.410094i \(0.865496\pi\)
\(140\) −9.64981 2.70522i −0.815558 0.228633i
\(141\) 0 0
\(142\) −0.416500 + 3.02867i −0.0349519 + 0.254161i
\(143\) 18.1184 1.51514
\(144\) 0 0
\(145\) −0.167976 −0.0139496
\(146\) −2.93155 + 21.3174i −0.242617 + 1.76424i
\(147\) 0 0
\(148\) 9.60378 + 2.69232i 0.789426 + 0.221307i
\(149\) 5.02129i 0.411360i 0.978619 + 0.205680i \(0.0659407\pi\)
−0.978619 + 0.205680i \(0.934059\pi\)
\(150\) 0 0
\(151\) 13.5963i 1.10645i −0.833031 0.553227i \(-0.813396\pi\)
0.833031 0.553227i \(-0.186604\pi\)
\(152\) 3.62131 + 1.57387i 0.293727 + 0.127658i
\(153\) 0 0
\(154\) 30.7957 + 4.23500i 2.48159 + 0.341266i
\(155\) −3.53478 −0.283920
\(156\) 0 0
\(157\) −11.4647 −0.914981 −0.457491 0.889214i \(-0.651252\pi\)
−0.457491 + 0.889214i \(0.651252\pi\)
\(158\) −11.2075 1.54124i −0.891620 0.122615i
\(159\) 0 0
\(160\) 4.38682 + 3.57153i 0.346809 + 0.282354i
\(161\) 38.3703i 3.02400i
\(162\) 0 0
\(163\) 21.2335i 1.66313i −0.555424 0.831567i \(-0.687444\pi\)
0.555424 0.831567i \(-0.312556\pi\)
\(164\) 3.52449 12.5722i 0.275216 0.981724i
\(165\) 0 0
\(166\) 2.25910 16.4275i 0.175340 1.27502i
\(167\) 10.1062 0.782037 0.391019 0.920383i \(-0.372123\pi\)
0.391019 + 0.920383i \(0.372123\pi\)
\(168\) 0 0
\(169\) 4.06035 0.312335
\(170\) −0.258802 + 1.88193i −0.0198492 + 0.144338i
\(171\) 0 0
\(172\) 6.30162 22.4785i 0.480494 1.71397i
\(173\) 6.64217i 0.504995i −0.967598 0.252497i \(-0.918748\pi\)
0.967598 0.252497i \(-0.0812519\pi\)
\(174\) 0 0
\(175\) 5.01091i 0.378789i
\(176\) −14.9894 9.12105i −1.12987 0.687525i
\(177\) 0 0
\(178\) −18.8114 2.58692i −1.40997 0.193898i
\(179\) −6.88682 −0.514745 −0.257372 0.966312i \(-0.582857\pi\)
−0.257372 + 0.966312i \(0.582857\pi\)
\(180\) 0 0
\(181\) −6.17574 −0.459039 −0.229520 0.973304i \(-0.573715\pi\)
−0.229520 + 0.973304i \(0.573715\pi\)
\(182\) 28.9973 + 3.98768i 2.14942 + 0.295587i
\(183\) 0 0
\(184\) −8.63288 + 19.8633i −0.636425 + 1.46434i
\(185\) 4.98701i 0.366653i
\(186\) 0 0
\(187\) 5.89229i 0.430887i
\(188\) 0.0346771 + 0.00972138i 0.00252909 + 0.000709004i
\(189\) 0 0
\(190\) 0.268969 1.95586i 0.0195130 0.141893i
\(191\) 1.01197 0.0732234 0.0366117 0.999330i \(-0.488344\pi\)
0.0366117 + 0.999330i \(0.488344\pi\)
\(192\) 0 0
\(193\) −4.26563 −0.307046 −0.153523 0.988145i \(-0.549062\pi\)
−0.153523 + 0.988145i \(0.549062\pi\)
\(194\) 2.78234 20.2324i 0.199761 1.45260i
\(195\) 0 0
\(196\) 34.8741 + 9.77658i 2.49100 + 0.698327i
\(197\) 14.4901i 1.03238i −0.856474 0.516190i \(-0.827350\pi\)
0.856474 0.516190i \(-0.172650\pi\)
\(198\) 0 0
\(199\) 3.54569i 0.251348i −0.992072 0.125674i \(-0.959891\pi\)
0.992072 0.125674i \(-0.0401093\pi\)
\(200\) 1.12740 2.59403i 0.0797192 0.183425i
\(201\) 0 0
\(202\) 16.2105 + 2.22925i 1.14057 + 0.156850i
\(203\) 0.841713 0.0590767
\(204\) 0 0
\(205\) −6.52844 −0.455966
\(206\) 9.23467 + 1.26994i 0.643410 + 0.0884812i
\(207\) 0 0
\(208\) −14.1140 8.58840i −0.978630 0.595498i
\(209\) 6.12377i 0.423590i
\(210\) 0 0
\(211\) 24.7065i 1.70087i −0.526082 0.850434i \(-0.676340\pi\)
0.526082 0.850434i \(-0.323660\pi\)
\(212\) 3.05606 10.9013i 0.209891 0.748702i
\(213\) 0 0
\(214\) −2.74598 + 19.9680i −0.187712 + 1.36499i
\(215\) −11.6725 −0.796061
\(216\) 0 0
\(217\) 17.7125 1.20240
\(218\) 0.524214 3.81193i 0.0355042 0.258177i
\(219\) 0 0
\(220\) −2.36817 + 8.44751i −0.159662 + 0.569531i
\(221\) 5.54819i 0.373212i
\(222\) 0 0
\(223\) 0.482284i 0.0322962i 0.999870 + 0.0161481i \(0.00514032\pi\)
−0.999870 + 0.0161481i \(0.994860\pi\)
\(224\) −21.9820 17.8966i −1.46873 1.19577i
\(225\) 0 0
\(226\) 15.5754 + 2.14191i 1.03606 + 0.142478i
\(227\) −5.09597 −0.338232 −0.169116 0.985596i \(-0.554091\pi\)
−0.169116 + 0.985596i \(0.554091\pi\)
\(228\) 0 0
\(229\) −9.77234 −0.645774 −0.322887 0.946437i \(-0.604653\pi\)
−0.322887 + 0.946437i \(0.604653\pi\)
\(230\) 10.7281 + 1.47532i 0.707393 + 0.0972800i
\(231\) 0 0
\(232\) −0.435734 0.189376i −0.0286073 0.0124331i
\(233\) 12.3013i 0.805882i 0.915226 + 0.402941i \(0.132012\pi\)
−0.915226 + 0.402941i \(0.867988\pi\)
\(234\) 0 0
\(235\) 0.0180070i 0.00117465i
\(236\) 16.5193 + 4.63101i 1.07531 + 0.301453i
\(237\) 0 0
\(238\) 1.29683 9.43020i 0.0840612 0.611269i
\(239\) 18.7092 1.21020 0.605099 0.796150i \(-0.293133\pi\)
0.605099 + 0.796150i \(0.293133\pi\)
\(240\) 0 0
\(241\) −20.9029 −1.34648 −0.673239 0.739425i \(-0.735097\pi\)
−0.673239 + 0.739425i \(0.735097\pi\)
\(242\) 1.58800 11.5475i 0.102080 0.742300i
\(243\) 0 0
\(244\) 7.35017 + 2.06054i 0.470546 + 0.131913i
\(245\) 18.1093i 1.15696i
\(246\) 0 0
\(247\) 5.76615i 0.366891i
\(248\) −9.16930 3.98511i −0.582251 0.253055i
\(249\) 0 0
\(250\) −1.40103 0.192668i −0.0886088 0.0121854i
\(251\) −20.5810 −1.29906 −0.649531 0.760335i \(-0.725035\pi\)
−0.649531 + 0.760335i \(0.725035\pi\)
\(252\) 0 0
\(253\) −33.5896 −2.11176
\(254\) 8.71037 + 1.19784i 0.546537 + 0.0751593i
\(255\) 0 0
\(256\) 7.35299 + 14.2103i 0.459562 + 0.888146i
\(257\) 0.971388i 0.0605935i −0.999541 0.0302967i \(-0.990355\pi\)
0.999541 0.0302967i \(-0.00964523\pi\)
\(258\) 0 0
\(259\) 24.9895i 1.55277i
\(260\) −2.22987 + 7.95419i −0.138291 + 0.493298i
\(261\) 0 0
\(262\) −1.13056 + 8.22113i −0.0698463 + 0.507903i
\(263\) −18.7818 −1.15813 −0.579067 0.815280i \(-0.696583\pi\)
−0.579067 + 0.815280i \(0.696583\pi\)
\(264\) 0 0
\(265\) −5.66076 −0.347738
\(266\) −1.34778 + 9.80067i −0.0826376 + 0.600918i
\(267\) 0 0
\(268\) −2.13886 + 7.62952i −0.130651 + 0.466047i
\(269\) 2.57026i 0.156712i −0.996925 0.0783559i \(-0.975033\pi\)
0.996925 0.0783559i \(-0.0249670\pi\)
\(270\) 0 0
\(271\) 27.5299i 1.67232i −0.548482 0.836162i \(-0.684794\pi\)
0.548482 0.836162i \(-0.315206\pi\)
\(272\) −2.79303 + 4.59001i −0.169352 + 0.278310i
\(273\) 0 0
\(274\) 9.63355 + 1.32480i 0.581984 + 0.0800339i
\(275\) 4.38659 0.264521
\(276\) 0 0
\(277\) −2.45076 −0.147252 −0.0736260 0.997286i \(-0.523457\pi\)
−0.0736260 + 0.997286i \(0.523457\pi\)
\(278\) 13.5477 + 1.86307i 0.812540 + 0.111740i
\(279\) 0 0
\(280\) −5.64930 + 12.9984i −0.337611 + 0.776805i
\(281\) 10.9842i 0.655263i −0.944806 0.327631i \(-0.893750\pi\)
0.944806 0.327631i \(-0.106250\pi\)
\(282\) 0 0
\(283\) 21.3114i 1.26683i −0.773811 0.633416i \(-0.781652\pi\)
0.773811 0.633416i \(-0.218348\pi\)
\(284\) 4.16301 + 1.16706i 0.247029 + 0.0692521i
\(285\) 0 0
\(286\) 3.49084 25.3844i 0.206418 1.50101i
\(287\) 32.7135 1.93101
\(288\) 0 0
\(289\) 15.1957 0.893863
\(290\) −0.0323636 + 0.235339i −0.00190045 + 0.0138196i
\(291\) 0 0
\(292\) 29.3015 + 8.21437i 1.71474 + 0.480710i
\(293\) 22.0638i 1.28898i 0.764611 + 0.644492i \(0.222931\pi\)
−0.764611 + 0.644492i \(0.777069\pi\)
\(294\) 0 0
\(295\) 8.57807i 0.499434i
\(296\) 5.62236 12.9364i 0.326793 0.751915i
\(297\) 0 0
\(298\) 7.03497 + 0.967443i 0.407525 + 0.0560425i
\(299\) −31.6280 −1.82910
\(300\) 0 0
\(301\) 58.4901 3.37132
\(302\) −19.0488 2.61958i −1.09614 0.150740i
\(303\) 0 0
\(304\) 2.90275 4.77033i 0.166484 0.273597i
\(305\) 3.81677i 0.218547i
\(306\) 0 0
\(307\) 9.78159i 0.558265i −0.960253 0.279132i \(-0.909953\pi\)
0.960253 0.279132i \(-0.0900468\pi\)
\(308\) 11.8667 42.3297i 0.676168 2.41196i
\(309\) 0 0
\(310\) −0.681038 + 4.95232i −0.0386804 + 0.281273i
\(311\) −1.58383 −0.0898106 −0.0449053 0.998991i \(-0.514299\pi\)
−0.0449053 + 0.998991i \(0.514299\pi\)
\(312\) 0 0
\(313\) −28.2310 −1.59571 −0.797855 0.602849i \(-0.794032\pi\)
−0.797855 + 0.602849i \(0.794032\pi\)
\(314\) −2.20888 + 16.0623i −0.124654 + 0.906450i
\(315\) 0 0
\(316\) −4.31865 + 15.4051i −0.242943 + 0.866602i
\(317\) 27.2511i 1.53057i 0.643690 + 0.765287i \(0.277403\pi\)
−0.643690 + 0.765287i \(0.722597\pi\)
\(318\) 0 0
\(319\) 0.736841i 0.0412552i
\(320\) 5.84901 5.45794i 0.326970 0.305108i
\(321\) 0 0
\(322\) −53.7578 7.39272i −2.99581 0.411980i
\(323\) 1.87521 0.104339
\(324\) 0 0
\(325\) 4.13042 0.229114
\(326\) −29.7487 4.09101i −1.64763 0.226580i
\(327\) 0 0
\(328\) −16.9349 7.36016i −0.935076 0.406397i
\(329\) 0.0902316i 0.00497463i
\(330\) 0 0
\(331\) 0.737840i 0.0405554i −0.999794 0.0202777i \(-0.993545\pi\)
0.999794 0.0202777i \(-0.00645503\pi\)
\(332\) −22.5802 6.33012i −1.23925 0.347410i
\(333\) 0 0
\(334\) 1.94713 14.1590i 0.106542 0.774746i
\(335\) 3.96182 0.216458
\(336\) 0 0
\(337\) −22.5564 −1.22872 −0.614362 0.789024i \(-0.710586\pi\)
−0.614362 + 0.789024i \(0.710586\pi\)
\(338\) 0.782300 5.68866i 0.0425515 0.309423i
\(339\) 0 0
\(340\) 2.58678 + 0.725176i 0.140288 + 0.0393282i
\(341\) 15.5056i 0.839676i
\(342\) 0 0
\(343\) 55.6675i 3.00576i
\(344\) −30.2789 13.1596i −1.63253 0.709520i
\(345\) 0 0
\(346\) −9.30587 1.27973i −0.500287 0.0687989i
\(347\) −19.7568 −1.06060 −0.530301 0.847809i \(-0.677921\pi\)
−0.530301 + 0.847809i \(0.677921\pi\)
\(348\) 0 0
\(349\) 1.52380 0.0815670 0.0407835 0.999168i \(-0.487015\pi\)
0.0407835 + 0.999168i \(0.487015\pi\)
\(350\) 7.02043 + 0.965443i 0.375258 + 0.0516051i
\(351\) 0 0
\(352\) −15.6668 + 19.2432i −0.835044 + 1.02566i
\(353\) 26.2442i 1.39684i −0.715688 0.698420i \(-0.753887\pi\)
0.715688 0.698420i \(-0.246113\pi\)
\(354\) 0 0
\(355\) 2.16175i 0.114734i
\(356\) −7.24871 + 25.8569i −0.384181 + 1.37041i
\(357\) 0 0
\(358\) −1.32687 + 9.64862i −0.0701272 + 0.509945i
\(359\) −23.6017 −1.24565 −0.622826 0.782360i \(-0.714016\pi\)
−0.622826 + 0.782360i \(0.714016\pi\)
\(360\) 0 0
\(361\) 17.0511 0.897428
\(362\) −1.18987 + 8.65238i −0.0625381 + 0.454759i
\(363\) 0 0
\(364\) 11.1737 39.8577i 0.585661 2.08911i
\(365\) 15.2156i 0.796419i
\(366\) 0 0
\(367\) 20.7055i 1.08082i 0.841402 + 0.540410i \(0.181731\pi\)
−0.841402 + 0.540410i \(0.818269\pi\)
\(368\) 26.1658 + 15.9219i 1.36399 + 0.829989i
\(369\) 0 0
\(370\) −6.98694 0.960838i −0.363234 0.0499516i
\(371\) 28.3656 1.47267
\(372\) 0 0
\(373\) 8.72557 0.451793 0.225896 0.974151i \(-0.427469\pi\)
0.225896 + 0.974151i \(0.427469\pi\)
\(374\) −8.25526 1.13526i −0.426869 0.0587027i
\(375\) 0 0
\(376\) 0.0203011 0.0467106i 0.00104695 0.00240892i
\(377\) 0.693811i 0.0357331i
\(378\) 0 0
\(379\) 18.6877i 0.959924i 0.877289 + 0.479962i \(0.159350\pi\)
−0.877289 + 0.479962i \(0.840650\pi\)
\(380\) −2.68840 0.753665i −0.137912 0.0386622i
\(381\) 0 0
\(382\) 0.194974 1.41779i 0.00997572 0.0725407i
\(383\) −24.5215 −1.25299 −0.626494 0.779426i \(-0.715511\pi\)
−0.626494 + 0.779426i \(0.715511\pi\)
\(384\) 0 0
\(385\) −21.9808 −1.12025
\(386\) −0.821850 + 5.97626i −0.0418310 + 0.304184i
\(387\) 0 0
\(388\) −27.8101 7.79629i −1.41185 0.395796i
\(389\) 10.0847i 0.511314i −0.966768 0.255657i \(-0.917708\pi\)
0.966768 0.255657i \(-0.0822918\pi\)
\(390\) 0 0
\(391\) 10.2857i 0.520172i
\(392\) 20.4164 46.9759i 1.03118 2.37264i
\(393\) 0 0
\(394\) −20.3011 2.79179i −1.02275 0.140648i
\(395\) 7.99947 0.402497
\(396\) 0 0
\(397\) −24.7934 −1.24435 −0.622174 0.782879i \(-0.713750\pi\)
−0.622174 + 0.782879i \(0.713750\pi\)
\(398\) −4.96762 0.683142i −0.249004 0.0342428i
\(399\) 0 0
\(400\) −3.41709 2.07930i −0.170854 0.103965i
\(401\) 10.8737i 0.543008i −0.962437 0.271504i \(-0.912479\pi\)
0.962437 0.271504i \(-0.0875210\pi\)
\(402\) 0 0
\(403\) 14.6001i 0.727283i
\(404\) 6.24649 22.2819i 0.310774 1.10856i
\(405\) 0 0
\(406\) 0.162171 1.17926i 0.00804842 0.0585258i
\(407\) 21.8760 1.08435
\(408\) 0 0
\(409\) 15.6773 0.775192 0.387596 0.921829i \(-0.373306\pi\)
0.387596 + 0.921829i \(0.373306\pi\)
\(410\) −1.25782 + 9.14653i −0.0621194 + 0.451715i
\(411\) 0 0
\(412\) 3.55845 12.6934i 0.175312 0.625357i
\(413\) 42.9840i 2.11510i
\(414\) 0 0
\(415\) 11.7253i 0.575574i
\(416\) −14.7519 + 18.1194i −0.723271 + 0.888377i
\(417\) 0 0
\(418\) 8.57957 + 1.17985i 0.419641 + 0.0577086i
\(419\) 3.90987 0.191010 0.0955048 0.995429i \(-0.469553\pi\)
0.0955048 + 0.995429i \(0.469553\pi\)
\(420\) 0 0
\(421\) −22.0305 −1.07370 −0.536849 0.843678i \(-0.680386\pi\)
−0.536849 + 0.843678i \(0.680386\pi\)
\(422\) −34.6145 4.76016i −1.68501 0.231721i
\(423\) 0 0
\(424\) −14.6842 6.38194i −0.713126 0.309935i
\(425\) 1.34325i 0.0651573i
\(426\) 0 0
\(427\) 19.1255i 0.925547i
\(428\) 27.4467 + 7.69440i 1.32669 + 0.371923i
\(429\) 0 0
\(430\) −2.24893 + 16.3536i −0.108453 + 0.788639i
\(431\) 10.0937 0.486198 0.243099 0.970001i \(-0.421836\pi\)
0.243099 + 0.970001i \(0.421836\pi\)
\(432\) 0 0
\(433\) 25.3622 1.21883 0.609414 0.792852i \(-0.291405\pi\)
0.609414 + 0.792852i \(0.291405\pi\)
\(434\) 3.41262 24.8156i 0.163811 1.19119i
\(435\) 0 0
\(436\) −5.23963 1.46888i −0.250933 0.0703464i
\(437\) 10.6898i 0.511363i
\(438\) 0 0
\(439\) 16.8378i 0.803625i −0.915722 0.401813i \(-0.868380\pi\)
0.915722 0.401813i \(-0.131620\pi\)
\(440\) 11.3789 + 4.94544i 0.542469 + 0.235765i
\(441\) 0 0
\(442\) −7.77317 1.06896i −0.369732 0.0508452i
\(443\) −2.50744 −0.119132 −0.0595659 0.998224i \(-0.518972\pi\)
−0.0595659 + 0.998224i \(0.518972\pi\)
\(444\) 0 0
\(445\) 13.4268 0.636494
\(446\) 0.675694 + 0.0929208i 0.0319950 + 0.00439993i
\(447\) 0 0
\(448\) −29.3089 + 27.3493i −1.38471 + 1.29213i
\(449\) 24.1651i 1.14042i 0.821498 + 0.570211i \(0.193139\pi\)
−0.821498 + 0.570211i \(0.806861\pi\)
\(450\) 0 0
\(451\) 28.6376i 1.34849i
\(452\) 6.00175 21.4088i 0.282299 1.00699i
\(453\) 0 0
\(454\) −0.981831 + 7.13960i −0.0460796 + 0.335078i
\(455\) −20.6972 −0.970298
\(456\) 0 0
\(457\) 29.8994 1.39863 0.699317 0.714812i \(-0.253488\pi\)
0.699317 + 0.714812i \(0.253488\pi\)
\(458\) −1.88282 + 13.6913i −0.0879783 + 0.639753i
\(459\) 0 0
\(460\) 4.13394 14.7462i 0.192746 0.687544i
\(461\) 8.67332i 0.403957i −0.979390 0.201978i \(-0.935263\pi\)
0.979390 0.201978i \(-0.0647371\pi\)
\(462\) 0 0
\(463\) 12.9686i 0.602702i 0.953513 + 0.301351i \(0.0974376\pi\)
−0.953513 + 0.301351i \(0.902562\pi\)
\(464\) −0.349273 + 0.573988i −0.0162146 + 0.0266467i
\(465\) 0 0
\(466\) 17.2344 + 2.37006i 0.798369 + 0.109791i
\(467\) −2.72723 −0.126201 −0.0631007 0.998007i \(-0.520099\pi\)
−0.0631007 + 0.998007i \(0.520099\pi\)
\(468\) 0 0
\(469\) −19.8524 −0.916697
\(470\) −0.0252283 0.00346937i −0.00116370 0.000160030i
\(471\) 0 0
\(472\) 9.67091 22.2517i 0.445140 1.02422i
\(473\) 51.2027i 2.35430i
\(474\) 0 0
\(475\) 1.39602i 0.0640538i
\(476\) −12.9621 3.63380i −0.594118 0.166555i
\(477\) 0 0
\(478\) 3.60467 26.2121i 0.164874 1.19892i
\(479\) 37.5562 1.71599 0.857993 0.513661i \(-0.171711\pi\)
0.857993 + 0.513661i \(0.171711\pi\)
\(480\) 0 0
\(481\) 20.5984 0.939208
\(482\) −4.02733 + 29.2856i −0.183440 + 1.33392i
\(483\) 0 0
\(484\) −15.8724 4.44966i −0.721472 0.202257i
\(485\) 14.4411i 0.655738i
\(486\) 0 0
\(487\) 2.48987i 0.112827i 0.998407 + 0.0564133i \(0.0179665\pi\)
−0.998407 + 0.0564133i \(0.982034\pi\)
\(488\) 4.30302 9.90079i 0.194789 0.448188i
\(489\) 0 0
\(490\) −25.3716 3.48908i −1.14617 0.157620i
\(491\) 22.7785 1.02798 0.513990 0.857796i \(-0.328167\pi\)
0.513990 + 0.857796i \(0.328167\pi\)
\(492\) 0 0
\(493\) −0.225634 −0.0101620
\(494\) 8.07854 + 1.11095i 0.363471 + 0.0499841i
\(495\) 0 0
\(496\) −7.34988 + 12.0786i −0.330019 + 0.542347i
\(497\) 10.8323i 0.485897i
\(498\) 0 0
\(499\) 17.3606i 0.777167i 0.921414 + 0.388583i \(0.127035\pi\)
−0.921414 + 0.388583i \(0.872965\pi\)
\(500\) −0.539867 + 1.92576i −0.0241436 + 0.0861225i
\(501\) 0 0
\(502\) −3.96530 + 28.8346i −0.176980 + 1.28695i
\(503\) −17.0746 −0.761320 −0.380660 0.924715i \(-0.624303\pi\)
−0.380660 + 0.924715i \(0.624303\pi\)
\(504\) 0 0
\(505\) −11.5704 −0.514877
\(506\) −6.47164 + 47.0600i −0.287699 + 2.09207i
\(507\) 0 0
\(508\) 3.35642 11.9727i 0.148917 0.531202i
\(509\) 16.4468i 0.728990i 0.931205 + 0.364495i \(0.118758\pi\)
−0.931205 + 0.364495i \(0.881242\pi\)
\(510\) 0 0
\(511\) 76.2439i 3.37283i
\(512\) 21.3258 7.56386i 0.942474 0.334279i
\(513\) 0 0
\(514\) −1.36094 0.187155i −0.0600285 0.00825507i
\(515\) −6.59136 −0.290450
\(516\) 0 0
\(517\) 0.0789893 0.00347395
\(518\) 35.0110 + 4.81468i 1.53829 + 0.211545i
\(519\) 0 0
\(520\) 10.7144 + 4.65663i 0.469858 + 0.204207i
\(521\) 25.9938i 1.13881i −0.822058 0.569404i \(-0.807174\pi\)
0.822058 0.569404i \(-0.192826\pi\)
\(522\) 0 0
\(523\) 8.16669i 0.357105i 0.983930 + 0.178552i \(0.0571414\pi\)
−0.983930 + 0.178552i \(0.942859\pi\)
\(524\) 11.3002 + 3.16790i 0.493652 + 0.138390i
\(525\) 0 0
\(526\) −3.61865 + 26.3138i −0.157780 + 1.14734i
\(527\) −4.74809 −0.206830
\(528\) 0 0
\(529\) 35.6349 1.54934
\(530\) −1.09065 + 7.93089i −0.0473747 + 0.344496i
\(531\) 0 0
\(532\) 13.4713 + 3.77655i 0.584057 + 0.163734i
\(533\) 26.9652i 1.16799i
\(534\) 0 0
\(535\) 14.2524i 0.616185i
\(536\) 10.2771 + 4.46656i 0.443902 + 0.192926i
\(537\) 0 0
\(538\) −3.60101 0.495208i −0.155251 0.0213499i
\(539\) 79.4379 3.42163
\(540\) 0 0
\(541\) 20.8341 0.895726 0.447863 0.894102i \(-0.352185\pi\)
0.447863 + 0.894102i \(0.352185\pi\)
\(542\) −38.5702 5.30414i −1.65673 0.227832i
\(543\) 0 0
\(544\) 5.89260 + 4.79746i 0.252643 + 0.205689i
\(545\) 2.72081i 0.116547i
\(546\) 0 0
\(547\) 1.79142i 0.0765955i −0.999266 0.0382978i \(-0.987806\pi\)
0.999266 0.0382978i \(-0.0121935\pi\)
\(548\) 3.71215 13.2416i 0.158575 0.565654i
\(549\) 0 0
\(550\) 0.845155 6.14573i 0.0360375 0.262055i
\(551\) 0.234498 0.00998995
\(552\) 0 0
\(553\) −40.0847 −1.70457
\(554\) −0.472184 + 3.43359i −0.0200612 + 0.145879i
\(555\) 0 0
\(556\) 5.22044 18.6218i 0.221396 0.789741i
\(557\) 45.1743i 1.91409i 0.289931 + 0.957047i \(0.406368\pi\)
−0.289931 + 0.957047i \(0.593632\pi\)
\(558\) 0 0
\(559\) 48.2125i 2.03917i
\(560\) 17.1227 + 10.4192i 0.723568 + 0.440292i
\(561\) 0 0
\(562\) −15.3892 2.11630i −0.649153 0.0892709i
\(563\) 17.1060 0.720934 0.360467 0.932772i \(-0.382617\pi\)
0.360467 + 0.932772i \(0.382617\pi\)
\(564\) 0 0
\(565\) −11.1171 −0.467700
\(566\) −29.8579 4.10603i −1.25502 0.172589i
\(567\) 0 0
\(568\) 2.43716 5.60764i 0.102261 0.235291i
\(569\) 16.4106i 0.687969i −0.938975 0.343985i \(-0.888223\pi\)
0.938975 0.343985i \(-0.111777\pi\)
\(570\) 0 0
\(571\) 11.9013i 0.498056i −0.968496 0.249028i \(-0.919889\pi\)
0.968496 0.249028i \(-0.0801111\pi\)
\(572\) −34.8917 9.78154i −1.45890 0.408987i
\(573\) 0 0
\(574\) 6.30284 45.8325i 0.263075 1.91301i
\(575\) −7.65734 −0.319333
\(576\) 0 0
\(577\) 8.29742 0.345426 0.172713 0.984972i \(-0.444747\pi\)
0.172713 + 0.984972i \(0.444747\pi\)
\(578\) 2.92772 21.2896i 0.121777 0.885529i
\(579\) 0 0
\(580\) 0.323481 + 0.0906846i 0.0134318 + 0.00376547i
\(581\) 58.7547i 2.43755i
\(582\) 0 0
\(583\) 24.8314i 1.02841i
\(584\) 17.1540 39.4696i 0.709839 1.63326i
\(585\) 0 0
\(586\) 30.9121 + 4.25100i 1.27697 + 0.175607i
\(587\) −13.0818 −0.539942 −0.269971 0.962868i \(-0.587014\pi\)
−0.269971 + 0.962868i \(0.587014\pi\)
\(588\) 0 0
\(589\) 4.93462 0.203328
\(590\) −12.0181 1.65272i −0.494778 0.0680414i
\(591\) 0 0
\(592\) −17.0411 10.3695i −0.700383 0.426185i
\(593\) 11.5590i 0.474669i 0.971428 + 0.237335i \(0.0762738\pi\)
−0.971428 + 0.237335i \(0.923726\pi\)
\(594\) 0 0
\(595\) 6.73092i 0.275941i
\(596\) 2.71083 9.66980i 0.111040 0.396090i
\(597\) 0 0
\(598\) −6.09371 + 44.3117i −0.249190 + 1.81204i
\(599\) −36.3548 −1.48542 −0.742709 0.669614i \(-0.766460\pi\)
−0.742709 + 0.669614i \(0.766460\pi\)
\(600\) 0 0
\(601\) −19.8417 −0.809361 −0.404680 0.914458i \(-0.632617\pi\)
−0.404680 + 0.914458i \(0.632617\pi\)
\(602\) 11.2692 81.9463i 0.459297 3.33988i
\(603\) 0 0
\(604\) −7.34020 + 26.1832i −0.298669 + 1.06538i
\(605\) 8.24215i 0.335091i
\(606\) 0 0
\(607\) 24.5620i 0.996939i −0.866907 0.498470i \(-0.833896\pi\)
0.866907 0.498470i \(-0.166104\pi\)
\(608\) −6.12409 4.98593i −0.248365 0.202206i
\(609\) 0 0
\(610\) −5.34740 0.735369i −0.216510 0.0297742i
\(611\) 0.0743765 0.00300895
\(612\) 0 0
\(613\) 7.74574 0.312848 0.156424 0.987690i \(-0.450003\pi\)
0.156424 + 0.987690i \(0.450003\pi\)
\(614\) −13.7043 1.88460i −0.553060 0.0760562i
\(615\) 0 0
\(616\) −57.0188 24.7812i −2.29735 0.998462i
\(617\) 4.49553i 0.180983i 0.995897 + 0.0904915i \(0.0288438\pi\)
−0.995897 + 0.0904915i \(0.971156\pi\)
\(618\) 0 0
\(619\) 34.4576i 1.38497i −0.721434 0.692483i \(-0.756517\pi\)
0.721434 0.692483i \(-0.243483\pi\)
\(620\) 6.80712 + 1.90831i 0.273381 + 0.0766395i
\(621\) 0 0
\(622\) −0.305153 + 2.21899i −0.0122355 + 0.0889732i
\(623\) −67.2808 −2.69555
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −5.43921 + 39.5524i −0.217395 + 1.58083i
\(627\) 0 0
\(628\) 22.0782 + 6.18940i 0.881016 + 0.246984i
\(629\) 6.69881i 0.267099i
\(630\) 0 0
\(631\) 7.07657i 0.281714i −0.990030 0.140857i \(-0.955014\pi\)
0.990030 0.140857i \(-0.0449858\pi\)
\(632\) 20.7508 + 9.01861i 0.825424 + 0.358741i
\(633\) 0 0
\(634\) 38.1795 + 5.25041i 1.51630 + 0.208520i
\(635\) −6.21713 −0.246719
\(636\) 0 0
\(637\) 74.7988 2.96364
\(638\) −1.03233 0.141966i −0.0408705 0.00562048i
\(639\) 0 0
\(640\) −6.51981 9.24619i −0.257718 0.365488i
\(641\) 5.47961i 0.216431i −0.994127 0.108216i \(-0.965486\pi\)
0.994127 0.108216i \(-0.0345137\pi\)
\(642\) 0 0
\(643\) 24.7194i 0.974836i 0.873169 + 0.487418i \(0.162061\pi\)
−0.873169 + 0.487418i \(0.837939\pi\)
\(644\) −20.7148 + 73.8919i −0.816278 + 2.91175i
\(645\) 0 0
\(646\) 0.361292 2.62722i 0.0142149 0.103366i
\(647\) −48.5385 −1.90825 −0.954123 0.299416i \(-0.903208\pi\)
−0.954123 + 0.299416i \(0.903208\pi\)
\(648\) 0 0
\(649\) 37.6284 1.47705
\(650\) 0.795799 5.78683i 0.0312138 0.226978i
\(651\) 0 0
\(652\) −11.4632 + 40.8906i −0.448935 + 1.60140i
\(653\) 7.83902i 0.306765i 0.988167 + 0.153382i \(0.0490166\pi\)
−0.988167 + 0.153382i \(0.950983\pi\)
\(654\) 0 0
\(655\) 5.86793i 0.229279i
\(656\) −13.5746 + 22.3083i −0.530000 + 0.870991i
\(657\) 0 0
\(658\) 0.126417 + 0.0173847i 0.00492824 + 0.000677728i
\(659\) −22.1705 −0.863641 −0.431821 0.901959i \(-0.642129\pi\)
−0.431821 + 0.901959i \(0.642129\pi\)
\(660\) 0 0
\(661\) 29.2880 1.13917 0.569585 0.821932i \(-0.307104\pi\)
0.569585 + 0.821932i \(0.307104\pi\)
\(662\) −1.03373 0.142158i −0.0401772 0.00552513i
\(663\) 0 0
\(664\) −13.2191 + 30.4158i −0.513002 + 1.18036i
\(665\) 6.99534i 0.271268i
\(666\) 0 0
\(667\) 1.28625i 0.0498037i
\(668\) −19.4620 5.45597i −0.753008 0.211098i
\(669\) 0 0
\(670\) 0.763317 5.55063i 0.0294895 0.214439i
\(671\) 16.7426 0.646340
\(672\) 0 0
\(673\) 40.8428 1.57437 0.787186 0.616715i \(-0.211537\pi\)
0.787186 + 0.616715i \(0.211537\pi\)
\(674\) −4.34589 + 31.6021i −0.167397 + 1.21727i
\(675\) 0 0
\(676\) −7.81925 2.19205i −0.300741 0.0843095i
\(677\) 41.9168i 1.61099i 0.592600 + 0.805497i \(0.298101\pi\)
−0.592600 + 0.805497i \(0.701899\pi\)
\(678\) 0 0
\(679\) 72.3633i 2.77705i
\(680\) 1.51438 3.48443i 0.0580739 0.133622i
\(681\) 0 0
\(682\) −21.7238 2.98743i −0.831847 0.114395i
\(683\) −46.0471 −1.76194 −0.880972 0.473168i \(-0.843110\pi\)
−0.880972 + 0.473168i \(0.843110\pi\)
\(684\) 0 0
\(685\) −6.87606 −0.262721
\(686\) 77.9918 + 10.7254i 2.97774 + 0.409496i
\(687\) 0 0
\(688\) −24.2708 + 39.8861i −0.925315 + 1.52064i
\(689\) 23.3813i 0.890757i
\(690\) 0 0
\(691\) 41.3646i 1.57358i −0.617219 0.786792i \(-0.711741\pi\)
0.617219 0.786792i \(-0.288259\pi\)
\(692\) −3.58589 + 12.7912i −0.136315 + 0.486249i
\(693\) 0 0
\(694\) −3.80651 + 27.6799i −0.144493 + 1.05071i
\(695\) −9.66986 −0.366799
\(696\) 0 0
\(697\) −8.76934 −0.332162
\(698\) 0.293587 2.13488i 0.0111124 0.0808065i
\(699\) 0 0
\(700\) 2.70522 9.64981i 0.102248 0.364728i
\(701\) 9.16505i 0.346159i −0.984908 0.173080i \(-0.944628\pi\)
0.984908 0.173080i \(-0.0553718\pi\)
\(702\) 0 0
\(703\) 6.96198i 0.262576i
\(704\) 23.9417 + 25.6572i 0.902338 + 0.966992i
\(705\) 0 0
\(706\) −36.7689 5.05643i −1.38382 0.190301i
\(707\) 57.9784 2.18050
\(708\) 0 0
\(709\) 19.0730 0.716302 0.358151 0.933664i \(-0.383407\pi\)
0.358151 + 0.933664i \(0.383407\pi\)
\(710\) −3.02867 0.416500i −0.113664 0.0156310i
\(711\) 0 0
\(712\) 34.8296 + 15.1374i 1.30529 + 0.567299i
\(713\) 27.0670i 1.01367i
\(714\) 0 0
\(715\) 18.1184i 0.677591i
\(716\) 13.2623 + 3.71796i 0.495637 + 0.138947i
\(717\) 0 0
\(718\) −4.54730 + 33.0667i −0.169704 + 1.23404i
\(719\) −36.7611 −1.37096 −0.685478 0.728093i \(-0.740407\pi\)
−0.685478 + 0.728093i \(0.740407\pi\)
\(720\) 0 0
\(721\) 33.0287 1.23005
\(722\) 3.28521 23.8891i 0.122263 0.889060i
\(723\) 0 0
\(724\) 11.8930 + 3.33408i 0.441999 + 0.123910i
\(725\) 0.167976i 0.00623847i
\(726\) 0 0
\(727\) 4.46515i 0.165603i −0.996566 0.0828017i \(-0.973613\pi\)
0.996566 0.0828017i \(-0.0263868\pi\)
\(728\) −53.6890 23.3340i −1.98985 0.864815i
\(729\) 0 0
\(730\) −21.3174 2.93155i −0.788993 0.108502i
\(731\) −15.6792 −0.579915
\(732\) 0 0
\(733\) 9.99033 0.369001 0.184501 0.982832i \(-0.440933\pi\)
0.184501 + 0.982832i \(0.440933\pi\)
\(734\) 29.0090 + 3.98929i 1.07074 + 0.147247i
\(735\) 0 0
\(736\) 27.3484 33.5914i 1.00808 1.23819i
\(737\) 17.3789i 0.640160i
\(738\) 0 0
\(739\) 11.7581i 0.432527i 0.976335 + 0.216264i \(0.0693870\pi\)
−0.976335 + 0.216264i \(0.930613\pi\)
\(740\) −2.69232 + 9.60378i −0.0989717 + 0.353042i
\(741\) 0 0
\(742\) 5.46514 39.7410i 0.200632 1.45894i
\(743\) −21.4878 −0.788309 −0.394155 0.919044i \(-0.628963\pi\)
−0.394155 + 0.919044i \(0.628963\pi\)
\(744\) 0 0
\(745\) −5.02129 −0.183966
\(746\) 1.68114 12.2248i 0.0615509 0.447581i
\(747\) 0 0
\(748\) −3.18105 + 11.3471i −0.116311 + 0.414892i
\(749\) 71.4176i 2.60954i
\(750\) 0 0
\(751\) 13.3689i 0.487838i 0.969796 + 0.243919i \(0.0784331\pi\)
−0.969796 + 0.243919i \(0.921567\pi\)
\(752\) −0.0615315 0.0374420i −0.00224382 0.00136537i
\(753\) 0 0
\(754\) −0.972048 0.133675i −0.0353999 0.00486816i
\(755\) 13.5963 0.494821
\(756\) 0 0
\(757\) 0.899166 0.0326808 0.0163404 0.999866i \(-0.494798\pi\)
0.0163404 + 0.999866i \(0.494798\pi\)
\(758\) 26.1820 + 3.60053i 0.950974 + 0.130777i
\(759\) 0 0
\(760\) −1.57387 + 3.62131i −0.0570904 + 0.131359i
\(761\) 16.7534i 0.607310i −0.952782 0.303655i \(-0.901793\pi\)
0.952782 0.303655i \(-0.0982071\pi\)
\(762\) 0 0
\(763\) 13.6338i 0.493575i
\(764\) −1.94880 0.546327i −0.0705053 0.0197654i
\(765\) 0 0
\(766\) −4.72450 + 34.3553i −0.170703 + 1.24131i
\(767\) 35.4310 1.27934
\(768\) 0 0
\(769\) −44.2917 −1.59720 −0.798600 0.601862i \(-0.794426\pi\)
−0.798600 + 0.601862i \(0.794426\pi\)
\(770\) −4.23500 + 30.7957i −0.152619 + 1.10980i
\(771\) 0 0
\(772\) 8.21456 + 2.30287i 0.295649 + 0.0828820i
\(773\) 7.37167i 0.265141i 0.991174 + 0.132570i \(0.0423230\pi\)
−0.991174 + 0.132570i \(0.957677\pi\)
\(774\) 0 0
\(775\) 3.53478i 0.126973i
\(776\) −16.2809 + 37.4607i −0.584452 + 1.34476i
\(777\) 0 0
\(778\) −14.1289 1.94300i −0.506547 0.0696598i
\(779\) 9.11384 0.326537
\(780\) 0 0
\(781\) 9.48271 0.339318
\(782\) 14.4106 + 1.98173i 0.515322 + 0.0708666i
\(783\) 0 0
\(784\) −61.8809 37.6547i −2.21003 1.34481i
\(785\) 11.4647i 0.409192i
\(786\) 0 0
\(787\) 3.21381i 0.114560i −0.998358 0.0572799i \(-0.981757\pi\)
0.998358 0.0572799i \(-0.0182427\pi\)
\(788\) −7.82274 + 27.9045i −0.278674 + 0.994057i
\(789\) 0 0
\(790\) 1.54124 11.2075i 0.0548350 0.398744i
\(791\) 55.7068 1.98071
\(792\) 0 0
\(793\) 15.7648 0.559826
\(794\) −4.77690 + 34.7363i −0.169526 + 1.23275i
\(795\) 0 0
\(796\) −1.91420 + 6.82815i −0.0678471 + 0.242017i
\(797\) 25.6042i 0.906948i 0.891270 + 0.453474i \(0.149815\pi\)
−0.891270 + 0.453474i \(0.850185\pi\)
\(798\) 0 0
\(799\) 0.0241879i 0.000855708i
\(800\) −3.57153 + 4.38682i −0.126273 + 0.155098i
\(801\) 0 0
\(802\) −15.2344 2.09502i −0.537945 0.0739777i
\(803\) 66.7444 2.35536
\(804\) 0 0
\(805\) 38.3703 1.35237
\(806\) −20.4551 2.81297i −0.720502 0.0990827i
\(807\) 0 0
\(808\) −30.0140 13.0445i −1.05589 0.458904i
\(809\) 39.2342i 1.37940i 0.724095 + 0.689700i \(0.242258\pi\)
−0.724095 + 0.689700i \(0.757742\pi\)
\(810\) 0 0
\(811\) 18.2911i 0.642286i 0.947031 + 0.321143i \(0.104067\pi\)
−0.947031 + 0.321143i \(0.895933\pi\)
\(812\) −1.62094 0.454413i −0.0568837 0.0159468i
\(813\) 0 0
\(814\) 4.21480 30.6488i 0.147729 1.07424i
\(815\) 21.2335 0.743777
\(816\) 0 0
\(817\) 16.2951 0.570094
\(818\) 3.02051 21.9643i 0.105610 0.767964i
\(819\) 0 0
\(820\) 12.5722 + 3.52449i 0.439040 + 0.123080i
\(821\) 21.0781i 0.735632i 0.929899 + 0.367816i \(0.119894\pi\)
−0.929899 + 0.367816i \(0.880106\pi\)
\(822\) 0 0
\(823\) 6.47366i 0.225657i 0.993614 + 0.112829i \(0.0359911\pi\)
−0.993614 + 0.112829i \(0.964009\pi\)
\(824\) −17.0981 7.43109i −0.595642 0.258874i
\(825\) 0 0
\(826\) 60.2217 + 8.28163i 2.09538 + 0.288155i
\(827\) 41.3708 1.43860 0.719301 0.694698i \(-0.244462\pi\)
0.719301 + 0.694698i \(0.244462\pi\)
\(828\) 0 0
\(829\) −43.2056 −1.50059 −0.750297 0.661101i \(-0.770089\pi\)
−0.750297 + 0.661101i \(0.770089\pi\)
\(830\) 16.4275 + 2.25910i 0.570208 + 0.0784144i
\(831\) 0 0
\(832\) 22.5436 + 24.1589i 0.781558 + 0.837557i
\(833\) 24.3253i 0.842821i
\(834\) 0 0
\(835\) 10.1062i 0.349738i
\(836\) 3.30602 11.7929i 0.114341 0.407866i
\(837\) 0 0
\(838\) 0.753306 5.47783i 0.0260225 0.189229i
\(839\) 30.1689 1.04155 0.520774 0.853695i \(-0.325644\pi\)
0.520774 + 0.853695i \(0.325644\pi\)
\(840\) 0 0
\(841\) 28.9718 0.999027
\(842\) −4.24456 + 30.8653i −0.146277 + 1.06369i
\(843\) 0 0
\(844\) −13.3382 + 47.5788i −0.459121 + 1.63773i
\(845\) 4.06035i 0.139680i
\(846\) 0 0
\(847\) 41.3007i 1.41911i
\(848\) −11.7705 + 19.3433i −0.404199 + 0.664253i
\(849\) 0 0
\(850\) −1.88193 0.258802i −0.0645498 0.00887682i
\(851\) −38.1873 −1.30904
\(852\) 0 0
\(853\) 25.7562 0.881875 0.440937 0.897538i \(-0.354646\pi\)
0.440937 + 0.897538i \(0.354646\pi\)
\(854\) 26.7953 + 3.68487i 0.916918 + 0.126094i
\(855\) 0 0
\(856\) 16.0682 36.9711i 0.549199 1.26365i
\(857\) 5.04937i 0.172483i 0.996274 + 0.0862416i \(0.0274857\pi\)
−0.996274 + 0.0862416i \(0.972514\pi\)
\(858\) 0 0
\(859\) 12.0894i 0.412486i 0.978501 + 0.206243i \(0.0661237\pi\)
−0.978501 + 0.206243i \(0.933876\pi\)
\(860\) 22.4785 + 6.30162i 0.766511 + 0.214883i
\(861\) 0 0
\(862\) 1.94474 14.1416i 0.0662382 0.481665i
\(863\) 42.8144 1.45742 0.728709 0.684824i \(-0.240121\pi\)
0.728709 + 0.684824i \(0.240121\pi\)
\(864\) 0 0
\(865\) 6.64217 0.225841
\(866\) 4.88648 35.5331i 0.166049 1.20746i
\(867\) 0 0
\(868\) −34.1099 9.56236i −1.15777 0.324568i
\(869\) 35.0904i 1.19036i
\(870\) 0 0
\(871\) 16.3640i 0.554473i
\(872\) −3.06744 + 7.05786i −0.103877 + 0.239009i
\(873\) 0 0
\(874\) −14.9767 2.05958i −0.506595 0.0696665i
\(875\) −5.01091 −0.169400
\(876\) 0 0
\(877\) −16.2401 −0.548391 −0.274195 0.961674i \(-0.588411\pi\)
−0.274195 + 0.961674i \(0.588411\pi\)
\(878\) −23.5902 3.24411i −0.796132 0.109483i
\(879\) 0 0
\(880\) 9.12105 14.9894i 0.307471 0.505291i
\(881\) 10.2669i 0.345899i −0.984931 0.172949i \(-0.944670\pi\)
0.984931 0.172949i \(-0.0553297\pi\)
\(882\) 0 0
\(883\) 55.9619i 1.88327i −0.336638 0.941634i \(-0.609290\pi\)
0.336638 0.941634i \(-0.390710\pi\)
\(884\) −2.99528 + 10.6845i −0.100742 + 0.359358i
\(885\) 0 0
\(886\) −0.483103 + 3.51299i −0.0162302 + 0.118021i
\(887\) −8.19837 −0.275274 −0.137637 0.990483i \(-0.543951\pi\)
−0.137637 + 0.990483i \(0.543951\pi\)
\(888\) 0 0
\(889\) 31.1535 1.04485
\(890\) 2.58692 18.8114i 0.0867139 0.630559i
\(891\) 0 0
\(892\) 0.260369 0.928763i 0.00871781 0.0310973i
\(893\) 0.0251382i 0.000841216i
\(894\) 0 0
\(895\) 6.88682i 0.230201i
\(896\) 32.6702 + 46.3319i 1.09143 + 1.54784i
\(897\) 0 0
\(898\) 33.8560 + 4.65584i 1.12979 + 0.155368i
\(899\) −0.593757 −0.0198029
\(900\) 0 0
\(901\) −7.60383 −0.253320
\(902\) −40.1220 5.51754i −1.33592 0.183714i
\(903\) 0 0
\(904\) −28.8380 12.5334i −0.959139 0.416855i
\(905\) 6.17574i 0.205289i
\(906\) 0 0
\(907\) 56.1230i 1.86353i −0.363058 0.931766i \(-0.618267\pi\)
0.363058 0.931766i \(-0.381733\pi\)
\(908\) 9.81361 + 2.75115i 0.325676 + 0.0913000i
\(909\) 0 0
\(910\) −3.98768 + 28.9973i −0.132190 + 0.961251i
\(911\) −35.7150 −1.18329 −0.591646 0.806198i \(-0.701522\pi\)
−0.591646 + 0.806198i \(0.701522\pi\)
\(912\) 0 0
\(913\) −51.4342 −1.70222
\(914\) 5.76065 41.8898i 0.190545 1.38559i
\(915\) 0 0
\(916\) 18.8192 + 5.27576i 0.621803 + 0.174316i
\(917\) 29.4037i 0.970995i
\(918\) 0 0
\(919\) 28.1866i 0.929789i 0.885366 + 0.464895i \(0.153908\pi\)
−0.885366 + 0.464895i \(0.846092\pi\)
\(920\) −19.8633 8.63288i −0.654875 0.284618i
\(921\) 0 0
\(922\) −12.1516 1.67107i −0.400191 0.0550338i
\(923\) 8.92893 0.293899
\(924\) 0 0
\(925\) 4.98701 0.163972
\(926\) 18.1694 + 2.49864i 0.597083 + 0.0821103i
\(927\) 0 0
\(928\) 0.736880 + 0.599930i 0.0241893 + 0.0196937i
\(929\) 28.0950i 0.921767i 0.887461 + 0.460883i \(0.152467\pi\)
−0.887461 + 0.460883i \(0.847533\pi\)
\(930\) 0 0
\(931\) 25.2809i 0.828548i
\(932\) 6.64104 23.6893i 0.217534 0.775967i
\(933\) 0 0
\(934\) −0.525451 + 3.82093i −0.0171933 + 0.125025i
\(935\) 5.89229 0.192698
\(936\) 0 0
\(937\) 20.3568 0.665026 0.332513 0.943099i \(-0.392103\pi\)
0.332513 + 0.943099i \(0.392103\pi\)
\(938\) −3.82492 + 27.8137i −0.124888 + 0.908150i
\(939\) 0 0
\(940\) −0.00972138 + 0.0346771i −0.000317076 + 0.00113104i
\(941\) 25.1556i 0.820050i −0.912074 0.410025i \(-0.865520\pi\)
0.912074 0.410025i \(-0.134480\pi\)
\(942\) 0 0
\(943\) 49.9905i 1.62791i
\(944\) −29.3120 17.8364i −0.954025 0.580526i
\(945\) 0 0
\(946\) −71.7364 9.86511i −2.33235 0.320743i
\(947\) 52.6385 1.71052 0.855261 0.518197i \(-0.173397\pi\)
0.855261 + 0.518197i \(0.173397\pi\)
\(948\) 0 0
\(949\) 62.8466 2.04009
\(950\) 1.95586 + 0.268969i 0.0634566 + 0.00872649i
\(951\) 0 0
\(952\) −7.58844 + 17.4602i −0.245943 + 0.565887i
\(953\) 12.7002i 0.411399i −0.978615 0.205700i \(-0.934053\pi\)
0.978615 0.205700i \(-0.0659470\pi\)
\(954\) 0 0
\(955\) 1.01197i 0.0327465i
\(956\) −36.0294 10.1005i −1.16527 0.326673i
\(957\) 0 0
\(958\) 7.23588 52.6173i 0.233781 1.69999i
\(959\) 34.4553 1.11262
\(960\) 0 0
\(961\) 18.5054 0.596947
\(962\) 3.96866 28.8590i 0.127955 0.930452i
\(963\) 0 0
\(964\) 40.2540 + 11.2848i 1.29649 + 0.363459i
\(965\) 4.26563i 0.137315i
\(966\) 0 0
\(967\) 32.7362i 1.05272i 0.850260 + 0.526362i \(0.176444\pi\)
−0.850260 + 0.526362i \(0.823556\pi\)
\(968\) −9.29220 + 21.3803i −0.298663 + 0.687190i
\(969\) 0 0
\(970\) 20.2324 + 2.78234i 0.649624 + 0.0893357i
\(971\) −53.1517 −1.70572 −0.852860 0.522140i \(-0.825134\pi\)
−0.852860 + 0.522140i \(0.825134\pi\)
\(972\) 0 0
\(973\) 48.4549 1.55339
\(974\) 3.48837 + 0.479718i 0.111775 + 0.0153712i
\(975\) 0 0
\(976\) −13.0422 7.93622i −0.417472 0.254032i
\(977\) 1.96623i 0.0629054i −0.999505 0.0314527i \(-0.989987\pi\)
0.999505 0.0314527i \(-0.0100134\pi\)
\(978\) 0 0
\(979\) 58.8980i 1.88239i
\(980\) −9.77658 + 34.8741i −0.312301 + 1.11401i
\(981\) 0 0
\(982\) 4.38869 31.9133i 0.140049 1.01840i
\(983\) 4.44401 0.141742 0.0708709 0.997485i \(-0.477422\pi\)
0.0708709 + 0.997485i \(0.477422\pi\)
\(984\) 0 0
\(985\) 14.4901 0.461694
\(986\) −0.0434724 + 0.316119i −0.00138444 + 0.0100673i
\(987\) 0 0
\(988\) 3.11295 11.1042i 0.0990362 0.353272i
\(989\) 89.3807i 2.84214i
\(990\) 0 0
\(991\) 53.8186i 1.70960i 0.518955 + 0.854802i \(0.326321\pi\)
−0.518955 + 0.854802i \(0.673679\pi\)
\(992\) 15.5064 + 12.6245i 0.492330 + 0.400830i
\(993\) 0 0
\(994\) 15.1764 + 2.08705i 0.481367 + 0.0661971i
\(995\) 3.54569 0.112406
\(996\) 0 0
\(997\) −6.46125 −0.204630 −0.102315 0.994752i \(-0.532625\pi\)
−0.102315 + 0.994752i \(0.532625\pi\)
\(998\) 24.3227 + 3.34483i 0.769920 + 0.105879i
\(999\) 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.e.a.971.25 yes 48
3.2 odd 2 inner 1620.2.e.a.971.24 yes 48
4.3 odd 2 inner 1620.2.e.a.971.23 48
12.11 even 2 inner 1620.2.e.a.971.26 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1620.2.e.a.971.23 48 4.3 odd 2 inner
1620.2.e.a.971.24 yes 48 3.2 odd 2 inner
1620.2.e.a.971.25 yes 48 1.1 even 1 trivial
1620.2.e.a.971.26 yes 48 12.11 even 2 inner